Perry's Chemical Engineers' Handbook, Eighth Edition

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Section 1

Conversion Factors and Mathematical Symbols*

James O. Maloney, Ph.D., P.E. Emeritus Professor of Chemical Engineering, University of Kansas; Fellow, American Institute of Chemical Engineering; Fellow, American Association for the Advancement of Science; Member, American Chemical Society; Member, American Society for Engineering Education

Table 1-1 Table 1-2a Table 1-2b Table 1-3 Table 1-4 Table 1-5 Table 1-6 Table 1-7 Table 1-8

CONVERSION FACTORS SI Base and Supplementary Quantities and Units. . . . . . . Derived Units of SI that Have Special Names. . . . . . . . . . Additional Common Derived Units of SI . . . . . . . . . . . . . SI Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion Factors: U.S. Customary and Commonly Used Units to SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric Conversion Factors as Exact Numerical Multiples of SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alphabetical Listing of Common Conversions . . . . . . . . . Common Units and Conversion Factors . . . . . . . . . . . . . . Kinematic-Viscosity Conversion Formulas . . . . . . . . . . . .

1-2 1-2 1-2 1-2 1-3 1-12 1-14 1-17 1-17

Table 1-9 Values of the Gas-Law Constant. . . . . . . . . . . . . . . . . . . . . Table 1-10 United States Customary System of Weights and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-11 Temperature Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-12 Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-13 Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot . . . Table 1-14 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . .

1-17 1-18 1-18 1-18 1-19 1-20

CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS

*Much of the material was taken from Sec. 1. of the fifth edition. The contribution of Cecil H. Chilton in developing that material is acknowledged. 1-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

TABLE 1-1

SI Base and Supplementary Quantities and Units

Quantity or “dimension” Base quantity or “dimension” length mass time electric current thermodynamic temperature amount of substance luminous intensity Supplementary quantity or “dimension” plane angle solid angle

SI unit

SI unit symbol (“abbreviation”); Use roman (upright) type

meter kilogram second ampere kelvin mole* candela

m kg s A K mol cd

radian steradian

rad sr

*When the mole is used, the elementary entities must be specified; they may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

TABLE 1-2a

Derived Units of SI that Have Special Names Quantity

Unit

Symbol

Formula

frequency (of a periodic phenomenon) force pressure, stress energy, work, quantity of heat power, radiant flux quantity of electricity, electric charge electric potential, potential difference, electromotive force capacitance electric resistance conductance magnetic flux magnetic-flux density inductance luminous flux illuminance activity (of radionuclides) absorbed dose

hertz newton pascal joule watt coulomb volt

Hz N Pa J W C V

l/s (kg⋅m)/s2 N/m2 N⋅m J/s A⋅s W/A

farad ohm siemens weber tesla henry lumen lux becquerel gray

F Ω S Wb T H lm lx Bq Gy

C/V V/A A/V V⋅s Wb/m2 Wb/A cd⋅sr lm/m2 l/s J/kg

TABLE 1-2b

Additional Common Derived Units of SI

Quantity

Unit

acceleration angular acceleration angular velocity area concentration (of amount of substance) current density density, mass electric-charge density electric-field strength electric-flux density energy density entropy heat capacity heat-flux density, irradiance luminance magnetic-field strength molar energy molar entropy molar-heat capacity moment of force permeability permittivity radiance

meter per second squared radian per second squared radian per second square meter mole per cubic meter

m/s2 rad/s2 rad/s m2 mol/m3

ampere per square meter kilogram per cubic meter coulomb per cubic meter volt per meter coulomb per square meter joule per cubic meter joule per kelvin joule per kelvin watt per square meter

A/m2 kg/m3 C/m3 V/m C/m2 J/m3 J/K J/K W/m2

candela per square meter ampere per meter joule per mole joule per mole-kelvin joule per mole-kelvin newton-meter henry per meter farad per meter watt per square-metersteradian watt per steradian joule per kilogram-kelvin joule per kilogram joule per kilogram-kelvin cubic meter per kilogram newton per meter watt per meter-kelvin meter per second pascal-second square meter per second cubic meter 1 per meter

cd/m2 A/m J/mol J/(mol⋅K) J/(mol⋅K) N⋅m H/m F/m W/(m2⋅sr)

radiant intensity specific-heat capacity specific energy specific entropy specific volume surface tension thermal conductivity velocity viscosity, dynamic viscosity, kinematic volume wave number

TABLE 1-3

W/sr J/(kg⋅K) J/kg J/(kg⋅K) m3/kg N/m W/(m⋅K) m/s Pa⋅s m2/s m3 1/m

SI Prefixes

Multiplication factor

Prefix

Symbol

000 = 1018 000 = 1015 000 = 1012 000 = 109 000 = 106 000 = 103 100 = 102 10 = 101 0.1 = 10−1 0.01 = 10−2 0.001 = 10−3 0.000 001 = 10−6 0.000 000 001 = 10−9 0.000 000 000 001 = 10−12 0.000 000 000 000 001 = 10−15 0.000 000 000 000 000 001 = 10−18

exa peta tera giga mega kilo hecto* deka* deci* centi milli micro nano pico femto atto

E P T G M k h da d c m µ n p f a

1 000 000 000 1 000 000 1 000 1

000 000 000 000 1

000 000 000 000 000 1

*Generally to be avoided.

1-2

Symbol

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Space,† time Length

naut mi mi chain link fathom yd ft

1.852* E + 00 1.609 344* E + 00 2.011 68* E + 01 2.011 68* E − 01 1.828 8* E + 00 9.144* E − 01 3.048* E − 01 3.048* E + 01 2.54* E + 01 2.54 E + 00 2.54* E + 01

in in mil

km km m m m m m cm mm cm µm

Length/length

ft/mi

m/km

1.893 939 E − 01

Length/volume

ft/U.S. gal ft/ft3 ft/bbl

m/m3 m/m3 m/m3

8.051 964 E + 01 1.076 391 E + 01 1.917 134 E + 00

Area

mi2 section acre ha yd2 ft2 in2

km2 ha ha m2 m2 m2 mm2 cm2

2.589 988 E + 00 2.589 988 E + 02 4.046 856 E − 01 1.000 000* E + 04 8.361 274 E − 01 9.290 304* E − 02 6.451 6* E + 02 6.451 6* E + 00

Area/volume

ft2/in3 ft2/ft3

m2/cm3 m2/m3

5.699 291 E − 03 3.280 840 E + 00

Volume

cubem acre⋅ft

km3 m3 ha⋅m m3 m3 m3 dm3 m3 dm3 m3 dm3 dm3 dm3 dm3 cm3 cm3 cm3

4.168 182 1.233 482 1.233 482 7.645 549 1.589 873 2.831 685 2.831 685 4.546 092 4.546 092 3.785 412 3.785 412 1.136 523 9.463 529 4.731 765 2.841 307 2.957 353 1.638 706

yd3 bbl (42 U.S. gal) ft3 U.K. gal U.S. gal U.K. qt U.S. qt U.S. pt U.K. fl oz U.S. fl oz in3

L L L L L L

E + 00 E + 03 E − 01 E − 01 E − 01 E − 02 E + 01 E − 03 E + 00 E − 03 E + 00 E + 00 E − 01 E − 01 E + 01 E + 01 E + 01

Volume/length (linear displacement)

bbl/in bbl/ft ft3/ft U.S. gal/ft

m3m m3/m m3/m m3/m L/m

6.259 342 E + 00 5.216 119 E − 01 9.290 304* E − 02 1.241 933 E − 02 1.241 933 E + 01

Plane angle

rad deg (°) min (′) sec (″)

rad rad rad rad

1 1.745 329 E − 02 2.908 882 E − 04 4.848 137 E − 06

Solid angle

sr

sr

1

Time

year week h

a d s min s h ns

1 7.0* 3.6* 6.0* 6.0* 1.666 667 1

E + 00 E + 03 E + 01 E + 01 E − 02

1.016 047 9.071 847 5.080 234 4.535 924 4.535 924 3.110 348 2.834 952 6.479 891

E + 00 E − 01 E + 01 E + 01 E − 01 E + 01 E + 01 E + 01

min mµs Mass, amount of substance Mass

U.K. ton U.S. ton U.K. cwt U.S. cwt lbm oz (troy) oz (av) gr

Mg Mg kg kg kg g g mg

t t

1-3

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity Amount of substance

lbm⋅mol std m3(0°C, 1 atm) std ft3 (60°F, 1 atm)

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit 4.535 924 E − 01 4.461 58 E − 02 1.195 30 E − 03

kmol kmol kmol

Enthalpy, calorific value, heat, entropy, heat capacity Calorific value, enthalpy (mass basis)

Btu/lbm cal/g cal/lbm

MJ/kg kJ/kg kWh/kg kJ/kg J/kg

Caloric value, enthalpy (mole basis)

kcal/(g⋅mol) Btu/(lb⋅mol)

kJ/kmol kJ/kmol

Calorific value (volume basis—solids and liquids)

Btu/U.S. gal

MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3

kJ/dm3

Btu/U.K. gal Btu/ft3

cal/mL (ft⋅lbf)/U.S. gal

J/g J/g

2.326 000 2.326 000 6.461 112 4.184* 9.224 141

E − 03 E + 00 E − 04 E + 00 E + 00

4.184* E + 03 2.326 000 E + 00

kJ/dm3 kJ/dm3

2.787 163 2.787 163 7.742 119 2.320 800 2.320 800 6.446 667 3.725 895 3.725 895 1.034 971 4.184* 3.581 692

E − 01 E + 02 E − 02 E − 01 E + 02 E − 02 E − 02 E + 01 E − 02 E + 00 E − 01

Calorific value (volume basis—gases)

cal/mL kcal/m3 Btu/ft3

kJ/m3 kJ/m3 kJ/m3 kWh/m3

J/dm3 J/dm3 J/dm3

4.184* 4.184* 3.725 895 1.034 971

E + 03 E + 00 E + 01 E − 02

Specific entropy

Btu/(lbm⋅°R) cal/(g⋅K) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

4.186 8* 4.184* 4.184*

E + 00 E + 00 E + 00

Specific-heat capacity (mass basis)

kWh/(kg⋅°C) Btu/(lbm⋅°F) kcal/(kg⋅°C)

kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)

J/(g⋅K) J/(g⋅K) J/(g⋅K)

3.6* 4.186 8* 4.184*

E + 03 E + 00 E + 00

Specific-heat capacity (mole basis)

Btu/(lb⋅mol⋅°F) cal/(g⋅mol⋅°C)

kJ/(kmol⋅K) kJ/(kmol⋅K)

4.186 8* 4.184*

E + 00 E + 00

Temperature (absolute)

°R K

K K

5/9 1

Temperature (traditional)

°F

°C

5/9(°F − 32)

Temperature (difference)

°F

K, °C

5/9

Pressure

atm (760 mmHg at 0°C or 14,696 psi)

µmHg (0°C) µ bar mmHg = torr (0°C) cmH2O (4°C) lbf/ft2 (psf) mHg (0°C) bar dyn/cm2

MPa kPa bar MPa kPa MPa kPa bar kPa kPa kPa kPa kPa Pa Pa Pa

1.013 250* E − 01 1.013 250* E + 02 1.013 250* E + 00 1.0* E − 01 1.0* E + 02 6.894 757 E − 03 6.894 757 E + 00 6.894 757 E − 02 3.376 85 E + 00 2.488 4 E − 01 1.333 224 E − 01 9.806 38 E − 02 4.788 026 E − 02 1.333 224 E − 01 1.0* E + 05 1.0* E − 01

Vacuum, draft

inHg (60°F) inH2O (39.2°F) inH2O (60°F) mmHg (0°C) = torr cmH2O (4°C)

kPa kPa kPa kPa kPa

3.376 85 2.490 82 2.488 4 1.333 224 9.806 38

E + 00 E − 01 E − 01 E − 01 E − 02

Liquid head

ft in

m mm cm

3.048* 2.54* 2.54*

E − 01 E + 01 E + 00

Pressure drop/length

psi/ft

kPa/m

2.262 059 E + 01

Temperature, pressure, vacuum

bar mmHg (0°C) = torr

1-4

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Density, specific volume, concentration, dosage kg/m3 g/m3 kg/m3 g/cm3 kg/m3 kg/m3 g/cm3 kg/m3 kg/m3

1.601 846 1.601 846 1.198 264 1.198 264 9.977 633 1.601 846 1.601 846 1.0* 1.601 846

E + 01 E + 04 E + 02 E − 01 E + 01 E + 01 E − 02 E + 03 E + 01

ft /lbm U.K. gal/lbm U.S. gal/lbm

m3/kg m3/g dm3/kg dm3/kg dm3/kg

6.242 796 6.242 796 6.242 796 1.002 242 8.345 404

E − 02 E − 05 E + 01 E + 01 E + 00

Specific volume (mole basis)

L/(g⋅mol) ft3/(lb⋅mol)

m3/kmol m3/kmol

1 6.242 796 E − 02

Specific volume

bbl/U.S. ton bbl/U.K. ton

m3/t m3/t

1.752 535 E − 01 1.564 763 E − 01

Yield

bbl/U.S. ton bbl/U.K. ton U.S. gal/U.S. ton U.S. gal/U.K. ton

dm3/t dm3/t dm3/t dm3/t

Concentration (mass/mass)

wt % wt ppm

kg/kg g/kg mg/kg

lbm/bbl g/U.S. gal g/U.K. gal lbm/1000 U.S. gal lbm/1000 U.K. gal gr/U.S. gal gr/ft3 lbm/1000 bbl mg/U.S. gal gr/100 ft3

kg/m3 kg/m3 kg/m3 g/m3 g/m3 g/m3 mg/m3 g/m3 g/m3 mg/m3

ft3/ft3 bbl/(acre⋅ft) vol% U.K. gal/ft3 U.S. gal/ft3 mL/U.S. gal mL/U.K. gal vol ppm U.K. gal/1000 bbl U.S. gal/1000 bbl U.K. pt/1000 bbl

m3/m3 m3/m3 m3/m3 dm3/m3 dm3/m3 dm3/m3 dm3/m3 cm3/m3 dm3/m3 cm3/m3 cm3/m3 cm3/m3

Concentration (mole/volume)

(lb⋅mol)/U.S. gal (lb⋅mol)/U.K. gal (lb⋅mol)/ft3 std ft3 (60°F, 1 atm)/bbl

kmol/m3 kmol/m3 kmol/m3 kmol/m3

Concentration (volume/mole)

U.S. gal/1000 std ft3 (60°F/60°F) bbl/million std ft3 (60°F/60°F)

dm3/kmol dm3/kmol

Throughput (mass basis)

U.K. ton/year U.S. ton/year U.K. ton/day

Density

lbm/ft3 lbm/U.S. gal lbm/U.K. gal lbm/ft3 g/cm3 lbm/ft3

Specific volume

ft3/lbm 3

Concentration (mass/volume)

Concentration (volume/volume)

cm3/g cm3/g

L/t L/t L/t L/t

g/dm3 g/L mg/dm3 mg/dm3 mg/dm3 mg/dm3 mg/dm3

L/m3 L/m3 L/m3 L/m3 L/m3

L/kmol L/kmol

1.752 535 1.564 763 4.172 702 3.725 627

E + 02 E + 02 E + 00 E + 00

1.0* 1.0* 1

E − 02 E + 01

2.853 010 2.641 720 2.199 692 1.198 264 9.977 633 1.711 806 2.288 351 2.853 010 2.641 720 2.288 351

E + 00 E − 01 E − 01 E + 02 E + 01 E + 01 E + 03 E + 00 E − 01 E + 01

1 1.288 931 1.0* 1.605 437 1.336 806 2.641 720 2.199 692 1 1.0* 2.859 403 2.380 952 3.574 253

E − 04 E − 02 E + 02 E + 02 E − 01 E − 01 E − 03 E + 01 E + 01 E + 00

1.198 264 9.977 644 1.601 846 7.518 21

E + 02 E + 01 E + 01 E − 03

3.166 91 1.330 10

E + 00 E − 01

1.016 047 9.071 847 1.016 047 4.233 529 9.071 847 3.779 936 1.016 047 9.071 847 4.535 924

E + 00 E − 01 E + 00 E − 02 E − 01 E − 02 E + 00 E − 01 E − 01

Facility throughput, capacity

U.S. ton/day U.K. ton/h U.S. ton/h lbm/h

t/a t/a t/d t/h t/d t/h t/h t/h kg/h

1-5

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity Throughput (volume basis)

bbl/day 3

ft /day bbl/h ft3/h U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit E + 01 E − 01 E − 03 E − 01 E − 02 E − 03 E − 03 E − 03 E − 03 E − 01 E − 02 E − 01 E − 02

t/a m3/d m3/h m3/h m3/h m3/h L/s m3/h L/s m3/h L/s m3/h L/s

5.803 036 1.589 873 1.179 869 1.589 873 2.831 685 4.546 092 1.262 803 3.785 412 1.051 503 2.727 655 7.576 819 2.271 247 6.309 020

kmol/h kmol/s

4.535 924 E − 01 1.259 979 E − 04

Throughput (mole basis)

(lbm⋅mol)/h

Flow rate (mass basis)

U.K. ton/min U.S. ton/min U.K. ton/h U.S. ton/h U.K. ton/day U.S. ton/day million lbm/year U.K. ton/year U.S. ton/year lbm/s lbm/min lbm/h

kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s

1.693 412 1.511 974 2.822 353 2.519 958 1.175 980 1.049 982 5.249 912 3.221 864 2.876 664 4.535 924 7.559 873 1.259 979

E + 01 E + 01 E − 01 E − 01 E − 02 E − 02 E + 00 E − 05 E − 05 E − 01 E − 03 E − 04

Flow rate (volume basis)

bbl/day

U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min ft3/min ft3/s

m3/d L/s m3/d L/s m3/s L/s m3/s L/s dm3/s dm3/s dm3/s dm3/s dm3/s dm3/s

1.589 873 1.840 131 2.831 685 3.277 413 4.416 314 4.416 314 7.865 791 7.865 791 1.262 803 1.051 503 7.576 820 6.309 020 4.719 474 2.831 685

E − 01 E − 03 E − 02 E − 04 E − 05 E − 02 E − 06 E − 03 E − 03 E − 03 E − 02 E − 02 E − 01 E + 01

Flow rate (mole basis)

(lb⋅mol)/s (lb⋅mol)/h million scf/D

kmol/s kmol/s kmol/s

4.535 924 E − 01 1.259 979 E − 04 1.383 45 E − 02

Flow rate/length (mass basis)

lbm/(s⋅ft) lbm/(h⋅ft)

kg/(s⋅m) kg/(s⋅m)

1.488 164 E + 00 4.133 789 E − 04

Flow rate/length (volume basis)

U.K. gal/(min⋅ft) U.S. gal/(min⋅ft) U.K. gal/(h⋅in) U.S. gal/(h⋅in) U.K. gal/(h⋅ft) U.S. gal/(h⋅ft)

m2/s m2/s m2/s m2/s m2/s m2/s

Flow rate/area (mass basis)

lbm/(s⋅ft2) lbm/(h⋅ft2)

kg/(s⋅m2) kg/(s⋅m2)

Flow rate/area (volume basis)

ft3/(s⋅ft2) ft3/(min⋅ft2) U.K. gal/(h⋅in2) U.S. gal/(h⋅in2) U.K. gal/(min⋅ft2) U.S. gal/(min⋅ft2) U.K. gal/(h⋅ft2) U.S. gal/(h⋅ft2)

m/s m/s m/s m/s m/s m/s m/s m/s

Flow rate

3

ft /day bbl/h ft3/h

1-6

L/s L/s L/s L/s L/s L/s

m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m)

2.485 833 2.069 888 4.971 667 4.139 776 4.143 055 3.449 814

E − 04 E − 04 E − 05 E − 05 E − 06 E − 06

4.882 428 E + 00 1.356 230 E − 03 m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2)

3.048* 5.08* 1.957 349 1.629 833 8.155 621 6.790 972 1.359 270 1.131 829

E − 01 E − 03 E − 03 E − 03 E − 04 E − 04 E − 05 E − 05

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Energy, work, power Energy, work

therm

E + 02 E + 05 E + 01 E + 01 E + 00 E + 03 E − 01 E + 00 E + 03 E − 01 E + 00 E + 03 E + 00 E − 04 E + 00 E − 04 E + 00 E − 03 E − 03 E − 03 E − 03 E − 05 E − 07

kcal cal ft⋅lbf lbf⋅ft J (lbf⋅ft2)/s2 erg

MJ kJ kWh MJ MJ kJ kWh MJ kJ kWh MJ kJ kJ kWh kJ kWh kJ kJ kJ kJ kJ kJ J

1.055 056 1.055 056 2.930 711 1.431 744 2.684 520 2.684 520 7.456 999 2.647 780 2.647 780 7.354 999 3.6* 3.6* 1.899 101 5.275 280 1.055 056 2.930 711 4.184* 4.184* 1.355 818 1.355 818 1.0* 4.214 011 1.0*

Impact energy

kgf⋅m lbf⋅ft

J J

9.806 650* E + 00 1.355 818 E + 00

Surface energy

erg/cm2

mJ/m2

1.0*

Specific-impact energy

(kgf⋅m)/cm2 (lbf⋅ft)/in2

J/cm2 J/cm2

9.806 650* E − 02 2.101 522 E − 03

Power

million Btu/h ton of refrigeration Btu/s kW hydraulic horsepower—hhp hp (electric) hp [(550 ft⋅lbf)/s] ch or CV Btu/min (ft⋅lbf)/s kcal/h Btu/h (ft⋅lbf)/min

MW kW kW kW kW kW kW kW kW kW W W W

2.930 711 3.516 853 1.055 056 1 7.460 43 7.46* 7.456 999 7.354 999 1.758 427 1.355 818 1.162 222 2.930 711 2.259 697

Power/area

Btu/(s⋅ft2) cal/(h⋅cm2) Btu/(h⋅ft2)

kW/m2 kW/m2 kW/m2

1.135 653 E + 01 1.162 222 E − 02 3.154 591 E − 03

Heat-release rate, mixing power

hp/ft3 cal/(h⋅cm3) Btu/(s⋅ft3) Btu/(h⋅ft3)

kW/m3 kW/m3 kW/m3 kW/m3

2.633 414 1.162 222 3.725 895 1.034 971

Cooling duty (machinery)

Btu/(bhp⋅h)

W/kW

Specific fuel consumption (mass basis)

lbm/(hp⋅h)

mg/J kg/kWh

kg/MJ

1.689 659 E − 01 6.082 774 E − 01

Specific fuel consumption (volume basis)

m3/kWh U.S. gal/(hp⋅h) U.K. pt/(hp⋅h)

dm3/MJ dm3/MJ dm3/MJ

mm3/J mm3/J mm3/J

2.777 778 E + 02 1.410 089 E + 00 2.116 806 E − 01

Fuel consumption

U.K. gal/mi U.S. gal/mi mi/U.S. gal mi/U.K. gal

dm3/100 km dm3/100 km km/dm3 km/dm3

L/100 km L/100 km km/L km/L

2.824 807 2.352 146 4.251 437 3.540 064

U.S. tonf⋅mi hp⋅h ch⋅h or CV⋅h kWh Chu Btu

E + 00

E − 01 E + 00 E + 00 E − 01 E − 01 E − 01 E − 01 E − 02 E − 03 E + 00 E − 01 E − 02

E + 01 E + 00 E + 01 E − 02

3.930 148 E − 01

E + 02 E + 02 E − 01 E − 01

1-7

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

in/s in/min

km/h km/h m/s cm/s m/s mm/s mm/s m/d mm/s mm/s

1.852* E + 00 1.609 344* E + 00 3.048* E − 01 3.048* E + 01 5.08* E − 03 8.466 667 E − 02 3.527 778 E − 03 3.048* E − 01 2.54* E + 01 4.233 333 E − 01

Corrosion rate

in/year (ipy) mil/year

mm/a mm/a

2.54* 2.54*

Rotational frequency

r/min

r/s rad/s

1.666 667 E − 02 1.047 198 E − 01

Acceleration (linear)

ft/s2

m/s2 cm/s2

3.048* 3.048*

Acceleration (rotational)

rpm/s

rad/s2

1.047 198 E − 01

Momentum

(lbm⋅ft)/s

(kg⋅m)/s

1.382 550 E − 01

Force

U.K. tonf U.S. tonf kgf (kp) lbf dyn

kN kN N N mN

9.964 016 E + 00 8.896 443 E + 00 9.806 650* E + 00 4.448 222 E + 00 1.0 E − 02

Bending moment, torque

U.S. tonf⋅ft kgf⋅m lbf⋅ft lbf⋅in

kN⋅m N⋅m N⋅m N⋅m

2.711 636 E + 00 9.806 650* E + 00 1.355 818 E + 00 1.129 848 E − 01

Bending moment/length

(lbf⋅ft)/in (lbf⋅in)/in

(N⋅m)/m (N⋅m)/m

5.337 866 E + 01 4.448 222 E + 00

Moment of inertia

lbm⋅ft2

Velocity (linear), speed

knot mi/h ft/s ft/min ft/h ft/day

MPa MPa MPa MPa kPa Pa

E − 01 E + 01

4.214 011 E − 02

kg⋅m2 2

E + 01 E − 02

N/mm2 N/mm2 N/mm2 N/mm2

1.378 951 E + 01 9.806 650* E + 00 9.576 052 E − 02 6.894 757 E − 03 4.788 026 E − 02 1.0* E − 01

Stress

U.S. tonf/in kgf/mm2 U.S. tonf/ft2 lbf/in2 (psi) lbf/ft2 (psf) dyn/cm2

Mass/length

lbm/ft

kg/m

1.488 164 E + 00

Mass/area structural loading, bearing capacity (mass basis)

U.S. ton/ft2 lbm/ft2

Mg/m2 kg/m2

9.764 855 E + 00 4.882 428 E + 00

Diffusivity

ft2/s m2/s ft2/h

m2/s mm2/s m2/s

9.290 304* E − 02 1.0* E + 06 2.580 64* E − 05

Thermal resistance

(°C⋅m2⋅h)/kcal (°F⋅ft2⋅h)/Btu

(K⋅m2)/kW (K⋅m2)/kW

8.604 208 E + 02 1.761 102 E + 02

Heat flux

Btu/(h⋅ft2)

kW/m2

3.154 591 E − 03

Thermal conductivity

(cal⋅cm)/(s⋅cm2⋅°C) (Btu⋅ft)/(h⋅ft2⋅°F)

W/(m⋅K) W/(m⋅K) (kJ⋅m)/(h⋅m2⋅K) W/(m⋅K) W/(m⋅K) W/(m⋅K)

4.184* 1.730 735 6.230 646 1.162 222 1.442 279 1.162 222

E + 02 E + 00 E + 00 E + 00 E − 01 E − 01

kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kJ/(h⋅m2⋅K) kW/(m2⋅K) kW/(m2⋅K)

4.184* 2.044 175 1.162 222 5.678 263 2.044 175 5.678 263 1.162 222

E + 01 E + 01 E − 02 E − 03 E + 01 E − 03 E − 03

Miscellaneous transport properties

(kcal⋅m)/(h⋅m2⋅°C) (Btu⋅in)/(h⋅ft2⋅°F) (cal⋅cm)/(h⋅cm2⋅°C) Heat-transfer coefficient

cal/(s⋅cm2⋅°C) Btu/(s⋅ft2⋅°F) cal/(h⋅cm2⋅°C) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°R) kcal/(h⋅m2⋅°C)

1-8

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

kW/(m3⋅K) kW/(m3⋅K)

Conversion factor; multiply customary unit by factor to obtain SI unit

Volumetric heat-transfer coefficient

Btu/(s⋅ft3⋅°F) Btu/(h⋅ft3⋅°F)

6.706 611 E + 01 1.862 947 E − 02

Surface tension

dyn/cm

mN/m

Viscosity (dynamic)

(lbf⋅s)/in2 (lbf⋅s)/ft2 (kgf⋅s)/m2 lbm/(ft⋅s) (dyn⋅s)/cm2 cP lbm/(ft⋅h)

Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s

Viscosity (kinematic)

ft2/s in2/s m2/h ft2/h cSt

m2/s mm2/s mm2/s m2/s mm2/s

9.290 304* E − 02 6.451 6* E + 02 2.777 778 E + 02 2.580 64* E − 05 1

Permeability

darcy millidarcy

µm2 µm2

9.869 233 E − 01 9.869 233 E − 04

Thermal flux

Btu/(h⋅ft2) Btu/(s⋅ft2) cal/(s⋅cm2)

W/m2 W/m2 W/m2

3.152 1.135 4.184

E + 00 E + 04 E + 04

Mass-transfer coefficient

(lb⋅mol)/[h⋅ft2(lb⋅mol/ft3)] (g⋅mol)/[s⋅m2(g⋅mol/L)]

m/s m/s

8.467 1.0

E − 05 E + 01

Admittance

S

S

1

Capacitance

µF

µF

1

Charge density

C/mm3

C/mm3

1

Conductance

S


S S

1 1

S/m
/m
m /m

S/m S/m mS/m

1 1 1

Current density

A/mm2

A/mm2

1

Displacement

C/cm2

C/cm2

1

Electric charge

C

C

1

Electric current

A

A

1

Electric-dipole moment

C⋅m

C⋅m

1

1 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2

6.894 757 E + 03 4.788 026 E + 01 9.806 650* E + 00 1.488 164 E + 00 1.0* E − 01 1.0* E − 03 4.133 789 E − 04

Electricity, magnetism

Conductivity

(mho)

Electric-field strength

V/m

V/m

1

Electric flux

C

C

1

Electric polarization

C/cm2

C/cm2

1

Electric potential

V mV

V mV

1 1

Electromagnetic moment

A⋅m2

A⋅m2

1

Electromotive force

V

V

1

Flux of displacement

C

C

1

Frequency

cycles/s

Hz

1

Impedance

Ω

Ω

1

Linear-current density

A/mm

A/mm

1

Magnetic-dipole moment

Wb⋅m

Wb⋅m

1

Magnetic-field strength

A/mm Oe gamma

A/mm A/m A/m

1 7.957 747 E + 01 7.957 747 E − 04

Magnetic flux

mWb

mWb

1

1-9

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued) Customary or commonly used unit

Quantity Magnetic-flux density

mT G gamma

SI unit mT T nT

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit 1 1.0* 1

Magnetic induction

mT

mT

1

Magnetic moment

A⋅m2

A⋅m2

1

Magnetic polarization

mT

mT

1

Magnetic potential difference

A

A

1

Magnetic-vector potential

Wb/mm

Wb/mm

1

Magnetization

A/mm

A/mm

1

Modulus of admittance

S

S

1

Modulus of impedance

Ω

Ω

1

Mutual inductance

H

H

1

Permeability

µH/m

µH/m

1

E − 04

Permeance

H

H

1

Permittivity

µF/m

µF/m

1

Potential difference

V

V

1

Quantity of electricity

C

C

1

Reactance

Ω

Ω

1

Reluctance

H−1

H−1

1

Resistance

Ω

Ω

1

Resistivity

Ω⋅cm Ω⋅m

Ω⋅cm Ω⋅m

1 1

Self-inductance

mH

mH

1

Surface density of change

mC/m2

mC/m2

1

Susceptance

S

S

1

Volume density of charge

C/mm3

C/mm3

1

Absorbed dose

rad

Gy

1.0*

Acoustical energy

J

J

1

Acoustical intensity

W/cm2

W/m2

1.0*

Acoustical power

W

W

1

Sound pressure

N/m2

N/m2

1.0*

Illuminance

fc

lx

1.076 391 E + 01

Illumination

fc

lx

1.076 391 E + 01

Irradiance

W/m2

W/m2

1

Light exposure

fc⋅s

lx⋅s

1.076 391 E + 01

Luminance

cd/m2

cd/m2

1

Acoustics, light, radiation

Luminous efficacy

lm/W

lm/W

1

Luminous exitance

lm/m2

lm/m2

1

Luminous flux

lm

lm

1

Luminous intensity

cd

cd

1

Radiance

W/m2⋅sr

W/m2⋅sr

1

Radiant energy

J

J

1

Radiant flux

W

W

1

Radiant intensity

W/sr

W/sr

1

Radiant power

W

W

1

1-10

E − 02 E + 04

TABLE 1-4

Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Concluded) Customary or commonly used unit

Quantity

SI unit

Alternate SI unit

Conversion factor; multiply customary unit by factor to obtain SI unit

Wavelength

Å

nm

1.0*

E − 01

Capture unit

10−3 cm−1

m−1

E + 01

m−1

m−1

1.0* 1 1

Ci

Bq

3.7*

E + 10

Radioactivity

10−3 cm−1

*An asterisk indicates that the conversion factor is exact. † Conversion factors for length, area, and volume are based on the international foot. The international foot is longer by 2 parts in 1 million than the U.S. Survey foot (land-measurement use). NOTE: The following unit symbols are used in the table: Unit symbol

Name

Unit symbol

A a Bq C cd Ci d °C ° dyn F fc G g gr Gy H h ha Hz J K L, ᐉ, l

ampere annum (year) becquerel coulomb candela curie day degree Celsius degree dyne farad footcandle gauss gram grain gray henry hour hectare hertz joule kelvin liter

lm lx m min ′ N naut mi Oe Ω Pa rad r S s ″ sr St T t V W Wb

NOTE:

Name lumen lux meter minute minute newton U.S. nautical mile oersted ohm pascal radian revolution siemens second second steradian stokes tesla tonne volt watt weber

Copyright SPE-AIME, The SI Metric System of Units and SPE’s Tentative Metric Standard, Society of Petroleum Engineers, Dallas, 1977.

1-11

TABLE 1-5

Metric Conversion Factors as Exact Numerical Multiples of SI Units

The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2 m. To convert from abampere abcoulomb abfarad abhenry abmho abohm abvolt acre ampere (international of 1948) angstrom are astronomical unit atmosphere bar barn barrel (petroleum 42 gal) barye British thermal unit (ISO/ TC 12) British thermal unit (International Steam Table) British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (39°F) British thermal unit (60°F) bushel (U.S.) cable caliber calorie (International Steam Table) calorie (mean) calorie (thermochemical) calorie (15°C) calorie (20°C) calorie (kilogram, International Steam Table) calorie (kilogram, mean) calorie (kilogram, thermochemical) carat (metric) Celsius (temperature) centimeter of mercury (0°C) centimeter of water (4°C) chain (engineer’s) chain (surveyor’s or Gunter’s) circular mil cord coulomb (international of 1948) cubit cup curie day (mean solar) day (sidereal) degree (angle) denier (international) dram (avoirdupois) dram (troy or apothecary) dram (U.S. fluid) dyne electron volt erg Fahrenheit (temperature)

Multiply by

To convert from

To

Multiply by

ampere coulomb farad henry mho ohm volt meter2 ampere

To

+01 1.00 +01 1.00 +09 1.00 −09 1.00 +09 1.00 −09 1.00 −08 1.00 +03 4.046 856 −01 9.998 35

meter meter2 meter newton/meter2 newton/meter2 meter2 meter3 newton/meter2 joule

−10 1.00 +02 1.00 +11 1.495 978 +05 1.013 25 +05 1.00 −28 1.00 −01 1.589 873 −01 1.00 +03 1.055 06

joule

+03 1.055 04

joule joule

+03 1.055 87 +03 1.054 350

joule joule meter3 meter meter joule

+03 1.059 67 +03 1.054 68 −02 3.523 907 +02 2.194 56 −04 2.54 +00 4.1868

joule joule joule joule joule

+00 4.190 02 +00 4.184 +00 4.185 80 +00 4.181 90 +03 4.186 8

joule joule

+03 4.190 02 +03 4.184

kilogram kelvin newton/meter2 newton/meter2 meter meter

−04 2.00 tK = tc + 273.15 +03 1.333 22 +01 9.806 38 +01 3.048 +01 2.011 68

meter3 meter meter newton/meter2 lumen/meter2 candela/meter2 meter meter/second2 meter3 meter3 meter3 tesla tesla ampere turn meter3 meter3 degree (angular) radian kilogram kilogram meter meter2 henry meter3 watt watt watt watt watt watt second (mean solar) second (mean solar) kilogram kilogram meter newton/meter2 newton/meter2 newton/meter2 newton/meter2 joule 1/meter joule

−05 2.957 352 −01 3.048 −01 3.048 006 +03 2.988 98 +01 1.076 391 +00 3.426 259 +02 2.011 68 −02 1.00 −03 4.546 087 −03 4.404 883 −03 3.785 411 −09 1.00 −04 1.00 −01 7.957 747 −04 1.420 652 −04 1.182 941 −01 9.00 −02 1.570 796 −05 6.479 891 −03 1.00 −01 1.016 +04 1.00 +00 1.000 495 −01 2.384 809 +02 7.456 998 +03 9.809 50 +02 7.46 +02 7.354 99 +02 7.457 +02 7.460 43 +03 3.60 +03 3.590 170 +01 5.080 234 +01 4.535 923 −02 2.54 +03 3.386 389 +03 3.376 85 +02 2.490 82 +02 2.4884 +00 1.000 165 +02 1.00 +03 4.186 74

meter2 meter3 coulomb

−10 5.067 074 +00 3.624 556 −01 9.998 35

meter meter3 disintegration/second second (mean solar) second (mean solar) radian kilogram/meter kilogram kilogram meter3 newton joule joule kelvin

fluid ounce (U.S.) foot foot (U.S. survey) foot of water (39.2°F) footcandle footlambert furlong gal (galileo) gallon (U.K. liquid) gallon (U.S. dry) gallon (U.S. liquid) gamma gauss gilbert gill (U.K.) gill (U.S.) grad grad grain gram hand hectare henry (international of 1948) hogshead (U.S.) horsepower (550 ft lbf/s) horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (U.K.) horsepower (water) hour (mean solar) hour (sidereal) hundredweight (long) hundredweight (short) inch inch of mercury (32°F) inch of mercury (60°F) inch of water (39.2°F) inch of water (60°F) joule (international of 1948) kayser kilocalorie (International Steam Table) kilocalorie (mean) kilocalorie (thermochemical) kilogram mass kilogram-force (kgf) kilopond-force kip knot (international) lambert lambert langley lbf (pound-force, avoirdupois) lbm (pound-mass, avoirdupois) league (British nautical) league (international nautical) league (statute) light-year link (engineer’s) link (surveyor’s or Gunter’s) liter lux maxwell meter micrometer mil mile (U.S. statute) mile (U.K. nautical) mile (international nautical) mile (U.S. nautical) millibar millimeter of mercury (0°C)

joule joule kilogram newton newton newton meter/second candela/meter2 candela/meter2 joule/meter2 newton

+03 4.190 02 +03 4.184 +00 1.00 +00 9.806 65 +00 9.806 65 +03 4.448 221 −01 5.144 444 +04 1/π +03 3.183 098 +04 4.184 +00 4.448 221

kilogram

−01 4.535 923

meter meter

+03 5.559 552 +03 5.556

meter meter meter meter meter3 lumen/meter2 weber wavelengths Kr 86 meter meter meter meter meter meter newton/meter2 newton/meter2

+03 4.828 032 +15 9.460 55 −01 3.048 −01 2.011 68 −03 1.00 +00 1.00 −08 1.00 +06 1.650 763 −06 1.00 −05 2.54 +03 1.609 344 +03 1.853 184 +03 1.852 +03 1.852 +02 1.00 +02 1.333 224

Fahrenheit (temperature)

Celsius

farad (international of 1948) faraday (based on carbon 12) faraday (chemical) faraday (physical) fathom fermi (femtometer)

farad coulomb

−01 4.572 −04 2.365 882 +10 3.70 +04 8.64 +04 8.616 409 −02 1.745 329 −07 1.111 111 −03 1.771 845 −03 3.887 934 −06 3.696 691 −05 1.00 −19 1.602 10 −07 1.00 tK = (5/9)(tF + 459.67) tc = (5/9)(tF − 32) −01 9.995 05 +04 9.648 70

coulomb coulomb meter meter

+04 9.649 57 +04 9.652 19 +00 1.828 8 −15 1.00

1-12

TABLE 1-5

Metric Conversion Factors as Exact Numerical Multiples of SI Units (Concluded)

The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2. To convert from minute (angle) minute (mean solar) minute (sidereal) month (mean calendar) nautical mile (international) nautical mile (U.S.) nautical mile (U.K.) oersted ohm (international of 1948) ounce-force (avoirdupois) ounce-mass (avoirdupois) ounce-mass (troy or apothecary) ounce (U.S. fluid) pace parsec pascal peck (U.S.) pennyweight perch phot pica (printer’s) pint (U.S. dry) pint (U.S. liquid) point (printer’s) poise pole pound-force (lbf avoirdupois) pound-mass (lbm avoirdupois) pound-mass (troy or apothecary) poundal quart (U.S. dry) quart (U.S. liquid) rad (radiation dose absorbed) Rankine (temperature) rayleigh (rate of photon emission) rhe rod roentgen rutherford second (angle)

To

Multiply by

radian second (mean solar) second (mean solar) second (mean solar) meter meter meter ampere/meter ohm newton kilogram kilogram meter3 meter meter newton/meter2 meter3 kilogram meter lumen/meter2 meter meter3 meter3 meter (newton-second)/meter2 meter newton

−04 2.908 882 +01 6.00 +01 5.983 617 +06 2.628 +03 1.852 +03 1.852 +03 1.853 184 +01 7.957 747 +00 1.000 495 −01 2.780 138 −02 2.834 952 −02 3.110 347 −05 2.957 352 −01 7.62 +16 3.083 74 +00 1.00 −03 8.809 767 −03 1.555 173 +00 5.0292 +04 1.00 −03 4.217 517 −04 5.506 104 −04 4.731 764 −04 3.514 598 −01 1.00 +00 5.0292 +00 4.448 221

kilogram

−01 4.535 923

kilogram

−01 3.732 417

newton meter3 meter3 joule/kilogram

−01 1.382 549 −03 1.101 220 −04 9.463 529 −02 1.00

kelvin 1/second-meter2

tK = (5/9)tR +10 1.00

meter2/(newtonsecond) meter coulomb/kilogram disintegration/second radian

+01 1.00 +00 5.0292 −04 2.579 76 +06 1.00 −06 4.848 136

To

Multiply by

second (ephemeris) second (mean solar)

To convert from

second second (ephemeris)

second (sidereal) section scruple (apothecary) shake skein slug span statampere statcoulomb statfarad stathenry statmho statohm statute mile (U.S.) statvolt stere stilb stoke tablespoon teaspoon ton (assay) ton (long) ton (metric) ton (nuclear equivalent of TNT) ton (register) ton (short, 2000 lb) tonne torr (0°C) township unit pole volt (international of 1948) watt (international of 1948) yard year (calendar) year (sidereal) year (tropical) year 1900, tropical, Jan., day 0, hour 12 year 1900, tropical, Jan., day 0, hour 12

second (mean solar) meter2 kilogram second meter kilogram meter ampere coulomb farad henry mho ohm meter volt meter3 candela/meter2 meter2/second meter3 meter3 kilogram kilogram kilogram joule meter3 kilogram kilogram newton/meter2 meter2 weber volt watt meter second (mean solar) second (mean solar) second (mean solar) second (ephemeris)

+00 1.000 000 Consult American Ephemeris and Nautical Almanac −01 9.972 695 +06 2.589 988 −03 1.295 978 −08 1.00 +02 1.097 28 +01 1.459 390 −01 2.286 −10 3.335 640 −10 3.335 640 −12 1.112 650 +11 8.987 554 −12 1.112 650 +11 8.987 554 +03 1.609 344 +02 2.997 925 +00 1.00 +04 1.00 −04 1.00 −05 1.478 676 −06 4.928 921 −02 2.916 666 +03 1.016 046 +03 1.00 +09 4.20 +00 2.831 684 +02 9.071 847 +03 1.00 +02 1.333 22 +07 9.323 957 −07 1.256 637 +00 1.000 330 +00 1.000 165 −01 9.144 +07 3.1536 +07 3.155 815 +07 3.155 692 +07 3.155 692

second

+07 3.155 692

1-13

1-14 TABLE 1-6

Alphabetical Listing of Common Conversions

To convert from Acres Acres Acres Acre-feet Ampere-hours (absolute) Angstrom units Angstrom units Angstrom units Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Bags (cement) Barrels (cement) Barrels (oil) Barrels (oil) Barrels (U.S. liquid) Barrels (U.S. liquid) Barrels per day Bars Bars Bars Board feet Boiler horsepower Boiler horsepower B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. per cubic foot B.t.u. per hour B.t.u. per minute B.t.u. per pound B.t.u. per pound per degree Fahrenheit B.t.u. per pound per degree Fahrenheit B.t.u. per second B.t.u. per square foot per hour B.t.u. per square foot per minute B.t.u. per square foot per second for a temperature gradient of 1°F. per inch

To

Multiply by

Square feet Square meters Square miles Cubic meters Coulombs (absolute) Inches Meters Microns Millimeters of mercury at 32°F Dynes per square centimeter Newtons per square meter Feet of water at 39.1°F Grams per square centimeter Inches of mercury at 32°F Pounds per square foot Pounds per square inch Pounds (cement) Pounds (cement) Cubic meters Gallons Cubic meters Gallons Gallons per minute Atmospheres Newtons per square meter Pounds per square inch Cubic feet B.t.u. per hour Kilowatts Calories (gram) Centigrade heat units (c.h.u. or p.c.u.) Foot-pounds Horsepower-hours Joules Liter-atmospheres Pounds carbon to CO2 Pounds water evaporated from and at 212°F Cubic foot-atmospheres Kilowatt-hours Joules per cubic meter Watts Horsepower Joules per kilogram Calories per gram per degree centigrade Joules per kilogram per degree Kelvin Watts Joules per square meter per second Kilowatts per square foot Calories, gram (15°C.), per square centimeter per second for a temperature gradient of 1°C. per centimeter

43,560 4074 0.001563 1233 3600 3.937 × 10−9 1 × 10−10 1 × 10−4 760 1.0133 × 106 101,325 33.90 1033.3 29.921 2116.3 14.696 94 376 0.15899 42 0.11924 31.5 0.02917 0.9869 1 × 105 14.504 1⁄12 33,480 9.803 252 0.55556 777.9 3.929 × 10−4 1055.1 10.41 6.88 × 10−5 0.001036 0.3676 2.930 × 10−4 37,260 0.29307 0.02357 2326 1 4186.8 1054.4 3.1546 0.1758 1.2405

To convert from Drams (avoirdupois) Dynes Ergs Faradays Fathoms Feet Feet per minute Feet per minute Feet per (second)2 Feet of water at 39.2°F. Foot-poundals Foot-poundals Foot-poundals Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds force Foot-pounds per second Foot-pounds per second Furlongs Gallons (U.S. liquid) Gallons Gallons Gallons Gallons Gallons Gallons per minute Gallons per minute Grains Grains Grains per cubic foot Grains per gallon Grams Grams Grams Grams Grams Grams Grams per cubic centimeter Grams per cubic centimeter Grams per liter Grams per liter Grams per square centimeter Grams per square centimeter Hectares Hectares Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British)

To Grams Newtons Joules Coulombs (abs.) Feet Meters Centimeters per second Miles per hour Meters per (second)2 Newtons per square meter B.t.u. Joules Liter-atmospheres B.t.u. Calories, gram Foot-poundals Horsepower-hours Kilowatt-hours Liter-atmospheres Joules Horsepower Kilowatts Miles Barrels (U.S. liquid) Cubic meters Cubic feet Gallons (Imperial) Liters Ounces (U.S. fluid) Cubic feet per hour Cubic feet per second Grams Pounds Grams per cubic meter Parts per million Drams (avoirdupois) Drams (troy) Grains Kilograms Pounds (avoirdupois) Pounds (troy) Pounds per cubic foot Pounds per gallon Grains per gallon Pounds per cubic foot Pounds per square foot Pounds per square inch Acres Square meters B.t.u. per minute B.t.u. per hour Foot-pounds per minute Foot-pounds per second Watts

Multiply by 1.7719 1 × 10−5 1 × 10−7 96,500 6 0.3048 0.5080 0.011364 0.3048 2989 3.995 × 10−5 0.04214 4.159 × 10−4 0.0012856 0.3239 32.174 5.051 × 10−7 3.766 × 10−7 0.013381 1.3558 0.0018182 0.0013558 0.125 0.03175 0.003785 0.13368 0.8327 3.785 128 8.021 0.002228 0.06480 1⁄ 7000 2.2884 17.118 0.5644 0.2572 15.432 0.001 0.0022046 0.002679 62.43 8.345 58.42 0.0624 2.0482 0.014223 2.471 10,000 42.42 2545 33,000 550 745.7

B.t.u. (60°F.) per degree Fahrenheit Bushels (U.S. dry) Bushels (U.S. dry) Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram, per gram per degree C. Calories, kilogram Calories, kilogram per second Candle power (spherical) Carats (metric) Centigrade heat units Centimeters Centimeters Centimeters Centimeters Centimeters Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters of mercury at 0°C Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters per second Centimeters of water at 4°C. Centistokes Circular mils Circular mils Circular mils Cords Cubic centimeters Cubic centimeters Cubic centimeters Cubic centimeters Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic foot-atmospheres Cubic foot-atmospheres Cubic feet of water (60°F.) Cubic feet per minute Cubic feet per minute Cubic feet per second Cubic feet per second Cubic inches Cubic yards Curies Curies Degrees Drams (apothecaries’ or troy)

Calories per degree centigrade Cubic feet Cubic meters B.t.u. Foot-pounds Joules Liter-atmospheres Horsepower-hours Joules per kilogram per degree Kelvin Kilowatt-hours Kilowatts Lumens Grams B.t.u. Angstrom units Feet Inches Meters Microns Atmospheres Feet of water at 39.1°F. Newtons per square meter Pounds per square foot Pounds per square inch Feet per minute Newtons per square meter Square meters per second Square centimeters Square inches Square mils Cubic feet Cubic feet Gallons Ounces (U.S. fluid) Quarts (U.S. fluid) Bushels (U.S.) Cubic centimeters Cubic meters Cubic yards Gallons Liters Foot-pounds Liter-atmospheres Pounds Cubic centimeters per second Gallons per second Gallons per minute Million gallons per day Cubic meters Cubic meters Disintegrations per minute Coulombs per minute Radians Grams

453.6 1.2444 0.03524 3.968 × 10−3 3.087 4.1868 4.130 × 10−2 1.5591 × 10−6 4186.8 0.0011626 4.185 12.556 0.2 1.8 1 × 108 0.03281 0.3937 0.01 10,000 0.013158 0.4460 1333.2 27.845 0.19337 1.9685 98.064 1 × 10−6 5.067 × 10−6 7.854 × 10−7 0.7854 128 3.532 × 10−5 2.6417 × 10−4 0.03381 0.0010567 0.8036 28,317 0.028317 0.03704 7.481 28.316 2116.3 28.316 62.37 472.0 0.1247 448.8 0.64632 1.6387 × 10−5 0.76456 2.2 × 1012 1.1 × 1012 0.017453 3.888

Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (metric) Horsepower (metric) Hours (mean solar) Inches Inches of mercury at 60°F Inches of water at 60°F Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Kilocalories Kilograms Kilograms force Kilograms per square centimeter Kilometers Kilowatt-hours Kilowatt-hours Kilowatts Knots (international) Knots (nautical miles per hour) Lamberts Liter-atmospheres Liter-atmospheres Liters Liters Liters Lumens Micromicrons Microns Microns Miles (nautical) Miles (nautical) Miles Miles Miles per hour Miles per hour Milliliters Millimeters Millimeters of mercury at 0°C. Millimicrons Mils Mils Minims (U.S.) Minutes (angle) Minutes (mean solar) Newtons Ounces (avoirdupois) Ounces (avoirdupois) Ounces (U.S. fluid) Ounces (troy)

Horsepower (metric) Pounds carbon to CO2 per hour Pounds water evaporated per hour at 212°F Foot-pounds per second Kilogram-meters per second Seconds Meters Newtons per square meter Newtons per square meter B.t.u. (mean) Calories, gram (mean) Cubic foot-atmospheres Foot-pounds Kilowatt-hours Liter-atmospheres Joules Pounds (avoirdupois) Newtons Pounds per square inch Miles B.t.u. Foot-pounds Horsepower Meters per second Miles per hour Candles per square inch Cubic foot-atmospheres Foot-pounds Cubic feet Cubic meters Gallons Watts Microns Angstrom units Meters Feet Miles (U.S. statute) Feet Meters Feet per second Meters per second Cubic centimeters Meters Newtons per square meter Microns Inches Meters Cubic centimeters Radians Seconds Kilograms Kilograms Ounces (troy) Cubic meters Ounces (apothecaries’)

1.0139 0.175 2.64 542.47 75.0 3600 0.0254 3376.9 248.84 9.480 × 10−4 0.2389 0.3485 0.7376 2.7778 × 10−7 0.009869 4186.8 2.2046 9.807 14.223 0.6214 3414 2.6552 × 106 1.3410 0.5144 1.1516 2.054 0.03532 74.74 0.03532 0.001 0.26418 0.001496 −6 1 × 10 1 × 104 1 × 10−6 6080 1.1516 5280 1609.3 1.4667 0.4470 1 0.001 133.32 0.001 0.001 2.54 × 10−5 0.06161 2.909 × 10−4 60 0.10197 0.02835 0.9115 2.957 × 10−5 1.000

1-15

1-16

TABLE 1-6

Alphabetical Listing of Common Conversions (Concluded)

To convert from Pints (U.S. liquid) Poundals Pounds (avoirdupois) Pounds (avoirdupois) Pounds (avoirdupois) Pounds per cubic foot Pounds per cubic foot Pounds per square foot Pounds per square foot Pounds per square inch Pounds per square inch Pounds per square inch Pounds force Pounds force per square foot Pounds water evaporated from and at 212°F. Pound-centigrade units (p.c.u.) Quarts (U.S. liquid) Radians Revolutions per minute Seconds (angle) Slugs Slugs Slugs

To Cubic meters Newtons Grains Kilograms Pounds (troy) Grams per cubic centimeter Kilograms per cubic meter Atmospheres Kilograms per square meter Atmospheres Kilograms per square centimeter Newtons per square meter Newtons Newtons per square meter Horsepower-hours B.t.u. Cubic meters Degrees Radians per second Radians Gee pounds Kilograms Pounds

Multiply by 4.732 × 10−4 0.13826 7000 0.45359 1.2153 0.016018 16.018 4.725 × 10−4 4.882 0.06805 0.07031 6894.8 4.4482 47.88 0.379 1.8 9.464 × 10−4 57.30 0.10472 4.848 × 10−6 1 14.594 32.17

To convert from Square centimeters Square feet Square feet per hour Square inches Square inches Square yards Stokes Tons (long) Tons (long) Tons (metric) Tons (metric) Tons (metric) Tons (short) Tons (short) Tons (refrigeration) Tons (British shipping) Tons (U.S. shipping) Torr (mm. mercury, 0°C.) Watts Watts Watts Watt-hours Yards

To Square feet Square meters Square meters per second Square centimeters Square meters Square meters Square meters per second Kilograms Pounds Kilograms Pounds Tons (short) Kilograms Pounds B.t.u. per hour Cubic feet Cubic feet Newtons per square meter B.t.u. per hour Joules per second Kilogram-meters per second Joules Meters

Multiply by 0.0010764 0.0929 2.581 × 10−5 6.452 6.452 × 10−4 0.8361 1 × 10−4 1016 2240 1000 2204.6 1.1023 907.18 2000 12,000 42.00 40.00 133.32 3.413 1 0.10197 3600 0.9144

TABLE 1-7 Mass (M)

Length (L)

Common Units and Conversion Factors* 1 pound mass = 453.5924 grams = 0.45359 kilograms = 7000 grains 1 slug = 32.174 pounds mass 1 ton (short) = 2000 pounds mass 1 ton (long) = 2240 pounds mass 1 ton (metric) = 1000 kilograms = 2204.62 pounds mass 1 pound mole = 453.59 gram moles = 30.480 centimeters = 0.3048 meters 1 inch = 2.54 centimeters = 0.0254 meters 1 mile (U.S.) = 1.60935 kilometers 1 yard = 0.9144 meters 1 foot

Area (L2)

Volume (L3)

1 square foot = 929.0304 square centimeters = 0.09290304 square meters 1 square inch = 6.4516 square centimeters 1 square yard = 0.836127 square meters 1 cubic foot

1 gallon Time (θ)

= 28,316.85 cubic centimeters = 0.02831685 cubic meters = 28.31685 liters = 7.481 gallons (U.S.) = 3.7853 liters = 231 cubic inches

1 hour = 60 minutes = 3600 seconds Temperature (T) 1 centigrade or Celsius degree = 1.8 Fahrenheit degree Temperature, Kelvin = T°C + 273.15 Temperature, Rankine = T°F + 459.7 Temperature, Fahrenheit = 9/5 T°C + 32 Temperature, centigrade or Celsius = 5/9 (T°F − 32) Temperature, Rankine = 1.8 T K Force (F) 1 pound force = 444,822.2 dynes = 4.448222 Newtons = 32.174 poundals 2 Pressure (F/L ) Normal atmospheric pressure

1 atm = 760 millimeters of mercury at 0°C (density 13.5951 g/cm3) = 29.921 inches of mercury at 32°F = 14.696 pounds force/square inch = 33.899 feet of water at 39.1°F = 1.01325 × 106 dynes/square centimeter = 1.01325 × 105 Newtons/square meter Density (M/L3) 1 pound mass/cubic foot = 0.01601846 grams/cubic centimeter = 16.01846 kilogram/cubic meter Energy (H or FL) 1 British thermal unit = 251.98 calories = 1054.4 joules = 777.97 foot-pounds force = 10.409 liter-atmospheres = 0.2930 watt-hour Diffusivity (L2/θ) 1 square foot/hour = 0.258 cm2/s = 2.58 × 10−5 m2/s Viscosity (M/Lθ) 1 pound mass/foot hour = 0.00413 g/cm s 0.000413 kg/m s 1 centipoise = 0.01 poise = 0.01 g/cm s = 0.001 kg/m s = 0.000672 lbm/ft s = 0.0000209 lbfs/ft2 2 Thermal conductivity [H/θL (T/L)] 2 1 Btu/hr ft (°F/ft) = 0.00413 cal/s cm2 (°C/cm) = 1.728 J/s m2 (°C/m) Heat transfer coefficient 1 Btu/hr ft2 °F = 5.678 J/s m2 °C Heat capacity (H/MT) 1 Btu/lbm °F = 1 cal/g °C = 4184 J/kg °C Gas constant 1.987 Btu/lbm mole °R = 1.987 cal/mol K = 82.057 atm cm3/mol K = 0.7302 atm ft3/lb mole °F = 10.73 (lbf /in.2) (ft3)/lb mole °R = 1545 (lbf /ft2) (ft3)/lb mole °R = 8.314 (N/m2) (m3)/mol K Gravitational acceleration g = 9.8066 m/s2 = 32.174 ft/s2

NOTE:

U.S. customary units, or British units, on left and SI units on right. *Adapted from Faust et al., Principles of Unit Operations, John Wiley and Sons, 1980.

TABLE 1-8

Kinematic-Viscosity Conversion Formulas

Viscosity scale Saybolt Universal Saybolt Furol Redwood No. 1 Redwood Admiralty Engler

Range of t, sec 32 < t < 100 t > 100 25 < t < 40 t > 40 34 < t < 100 t > 100

Kinematic viscosity, stokes 0.00226t − 1.95/t 0.00220t − 1.35/t 0.0224t − 1.84/t 0.0216t − 0.60/t 0.00260t − 1.79/t 0.00247t − 0.50/t 0.027t − 20/t 0.00147t − 3.74/t

TABLE 1-9 Temp. scale

Values of the Gas-Law Constant Press. units

Vol. units

Kelvin atm. atm. mm. Hg bar kg/cm2 atm mm Hg

cm3 liters liters liters liters ft3 ft3

atm in Hg mm Hg lb/in2abs lb/ft2abs

ft3 ft3 ft3 ft3 ft3

Rankine

Wt. units

Energy units*

R

g-moles g-moles g-moles g-moles g-moles g-moles g-moles g-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles

calories joules (abs) joules (int) atm cm3 atm liters mm Hg-liters bar-liters kg/(cm2)(liters) atm-ft3 mm Hg-ft3 chu or pcu Btu hp-hr kw-hr atm-ft3 in Hg-ft3 mm Hg-ft3 (lb)(ft3)/in2 ft-lb

1.9872 8.3144 8.3130 82.057 0.08205 62.361 0.08314 0.08478 1.314 998.9 1.9872 1.9872 0.0007805 0.0005819 0.7302 21.85 555.0 10.73 1545.0

*Energy units are the product of pressure units and volume units. 1-17

TABLE 1-10 United States Customary System of Weights and Measures Linear Measure 12 inches (in) or (″) = 1 foot (ft) or (′) 3 feet = 1 yard (yd) 16.5 feet = 1 rod (rd) 5.5 yards





5280 feet = 1 mile (mi) 320 rods 1 mil = 0.001 inch Nautical:

Square Measure 144 sq. inches (sq. in) or (in ) or (ⵧ″) = 1 sq. foot (ft2) or (ⵧ′) 9 sq. feet (ft2) (ⵧ′) = 1 sq. yard (yd2) 30.25 sq. yards = 1 sq. rod, pole, or perch 10 sq. chains 160 sq. rods = = 1 acre 43,560 sq. ft 640 acres = 1 sq. mile = 1 section 1 circular inch (area of circle of 1 inch diameter) = 0.7854 sq. inch 1 sq. inch = 1.2732 circular inch 1 circular mil = area of circle of 0.001 inch diameter 1,000,000 circular mils = 1 circular inch 2




Circular Measure 60 seconds (″) (sec) = 1 minute (min) or (′) 60 minutes (′) = 1 degree (°) 90 degrees (°) = 1 quadrant 360 degrees (°) = 1 circumference = 1 radian (rad.) 57.29578 degrees = 57° 17′ 44.81″



Volume Measure Solid:

1728 cubic in (cu. in) (in3) = 1 cubic foot (cu. ft)(ft3) 27 cu. ft = 1 cubic yard (cu. yd) Dry Measure: 2 pints = 1 quart 8 quarts = 1 peck 4 pecks = 1 bushel 1 United States Winchester bushel = 2150.42 cubic inches Liquid: 4 gills = 1 pint (pt) 2 pints = 1 quart (qt) 4 quarts = 1 gallon (gal) 7.4805 gallons = 1 cubic foot Apothecaries’ Liquid: 60 minims (min. or ) = 1 fluid dram or drachm 8 drams ( ) = 1 fluid ounce 16 ounces (oz. ) = 1 pint Avoirdupois Weight 16 drams = 437.5 grains = 1 ounce (oz) 16 ounces = 7000 grains = 1 pound (lb) 100 pounds = 1 hundredweight (cwt) 2000 pounds = 1 short ton: 2240 pounds = 1 long ton Troy Weight 24 grains = 1 pennyweight (dwt) 20 pennyweights = 1 ounce (oz) 12 ounces = 1 pound (lb) Apothecaries’ Weight 20 grains (gr) = 1 scruple ( ) 3 scruples = 1 dram ( ) 8 drams = 1 ounce ( ) 12 ounces = 1 pound (lb)

1-18

Temperature Conversion Formulas

°F = (°C × 5/9) + 32 °C = (°F − 32) × 5/9 °R = °F + 459.69 °K = °C + 273.15 °K = °R × 5/9 Temperature difference, T °F = °C × 9/5 NOTE: An extensive table of temperature conversions may be found in the sixth edition of the Handbook (Table 1-12).

6080.2 feet = 1 nautical mile 6 feet = 1 fathom 120 fathoms = 1 cable length 1 knot = 1 nautical mile per hour 60 nautical miles = 1° of latitude



TABLE 1-11

TABLE 1-12 Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu

Greek Alphabet

= Α, α = A, a = Β, β = B, b = Γ, γ = G, g = ∆, δ = D, d = Ε, ε = E, e = Ζ, ζ = Z, z = Η, η = E, e = Θ, θ = Th, th = Ι,  = I, i = Κ, κ = K, k = Λ, λ = L, l = Μ, µ = M, m

Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega

= Ν, ν = N, n = Ξ, ξ = X, x = Ο, ο = O, o = Π, π = P, p = Ρ, ρ = R, r = Σ, σ = S, s = Τ, τ = T, t = , υ = U, u = Φ, φ = Ph, ph = Χ, χ = Ch, ch = Ψ, ψ = Ps, ps = Ω, ω = O, o

TABLE 1-13

Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot* 145 140 sp gr 60°/60°F − 1 141.5 °Bé = 145 −  (heavier than H2O); °Bé =  − 130 (lighter than H2O); °Tw =  °API =  − 131.5 sp gr sp gr 0.005 sp gr Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

104.33 102.38 100.47 98.58 96.73

4.9929 5.0346 5.0763 5.1180 5.1597

37.350 37.662 37.973 38.285 38.597

0.700 .705 .710 .715 .720

70.00 68.58 67.18 65.80 64.44

94.00 92.22 90.47 88.75 87.05

94.90 93.10 91.33 89.59 87.88

5.2014 5.2431 5.2848 5.3265 5.3682

39.910 39.222 39.534 39.845 40.157

.725 .730 .735 .740 .745

.650 .655 .660 .665 .670

85.38 83.74 82.12 80.53 78.96

86.19 84.53 82.89 81.28 79.69

5.4098 5.4515 5.4932 5.5349 5.5766

40.468 40.780 41.092 41.404 41.716

.675 .680 .685 .690 .695

77.41 75.88 74.38 72.90 71.44

78.13 76.59 75.07 73.57 72.10

5.6183 5.6600 5.7017 5.7434 5.7851

42.028 42.340 42.652 42.963 43.275

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

Sp gr 60°/ 60°

°Bé

°API

0.600 .605 .610 .615 .620

103.33 101.40 99.51 97.64 95.81

.625 .630 .635 .640 .645

°API

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

70.64 69.21 67.80 66.40 65.03

5.8268 5.8685 5.9101 5.9518 5.9935

43.587 43.899 44.211 44.523 44.834

0.800 .805 .810 .815 .820

45.00 43.91 42.84 41.78 40.73

45.38 44.28 43.19 42.12 41.06

63.10 61.78 60.48 59.19 57.92

63.67 62.34 61.02 59.72 58.43

6.0352 6.0769 6.1186 6.1603 6.2020

45.146 45.458 45.770 46.082 46.394

.825 .830 .835 .840 .845

39.70 38.67 37.66 36.67 35.68

.750 .755 .760 .765 .770

56.67 55.43 54.21 53.01 51.82

57.17 55.92 54.68 53.47 52.27

6.2437 6.2854 6.3271 6.3688 6.4104

46.706 47.018 47.330 47.642 47.953

.850 .855 .860 .865 .870

.775 .780 .785 .790 .795

50.65 49.49 48.34 47.22 46.10

51.08 49.91 48.75 47.61 46.49

6.4521 6.4938 6.5355 6.5772 6.6189

47.265 48.577 48.889 49.201 49.513

.875 .880 .885 .890 .895

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per gal at 60°F wt in air

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

°Bé

°API

Lb per gal at 60°F wt in air

6.6606 6.7023 6.7440 6.7857 6.8274

49.825 50.137 50.448 50.760 51.072

0.900 .905 .910 .915 .920

25.56 24.70 23.85 23.01 22.17

25.72 24.85 23.99 23.14 22.30

7.4944 7.5361 7.5777 7.6194 7.6612

56.062 56.374 56.685 56.997 57.310

40.02 38.98 37.96 36.95 35.96

6.8691 6.9108 6.9525 6.9941 7.0358

51.384 51.696 52.008 52.320 52.632

.925 .930 .935 .940 .945

21.35 20.54 19.73 18.94 18.15

21.47 20.65 19.84 19.03 18.24

7.7029 7.7446 7.7863 7.8280 7.8697

57.622 57.934 58.246 58.557 58.869

34.71 33.74 32.79 31.85 30.92

34.97 34.00 33.03 32.08 31.14

7.0775 7.1192 7.1609 7.2026 7.2443

52.943 53.255 53.567 53.879 54.191

.950 .955 .960 .965 .970

17.37 16.60 15.83 15.08 14.33

17.45 16.67 15.90 15.13 14.38

7.9114 7.9531 7.9947 8.0364 8.0780

59.181 59.493 59.805 60.117 60.428

30.00 29.09 28.19 27.30 26.42

30.21 29.30 28.39 27.49 26.60

7.2860 7.3277 7.3694 7.4111 7.4528

54.503 54.815 55.127 55.438 55.750

.975 .980 .985 .990 .995 1.000

13.59 12.86 12.13 11.41 10.70 10.00

13.63 12.89 12.15 11.43 10.71 10.00

8.1197 8.1615 8.2032 8.2449 8.2866 8.3283

60.740 61.052 61.364 61.676 61.988 62.300 Lb per ft3 at 60°F. wt. in air

°Tw

Lb per gal at 60°F wt in air

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

°Bé

°Tw

Lb per gal at 60°F wt in air

1.005 1.010 1.015 1.020 1.025

0.72 1.44 2.14 2.84 3.54

1 2 3 4 5

8.3700 8.4117 8.4534 8.4950 8.5367

62.612 62.924 63.236 63.547 63.859

1.255 1.260 1.265 1.270 1.275

29.46 29.92 30.38 30.83 31.27

51 52 53 54 55

10.4546 10.4963 10.5380 10.5797 10.6214

78.206 78.518 78.830 79.141 79.453

1.505 1.510 1.515 1.520 1.525

48.65 48.97 49.29 49.61 49.92

101 102 103 104 105

12.5392 12.5809 12.6226 12.6643 12.7060

93.800 94.112 94.424 94.735 95.047

1.755 1.760 1.765 1.770 1.775

62.38 62.61 62.85 63.08 63.31

151 152 153 154 155

14.6238 14.6655 14.7072 14.7489 14.7906

109.394 109.705 110.017 110.329 110.641

1.030 1.035 1.040 1.045 1.050

4.22 4.90 5.58 6.24 6.91

6 7 8 9 10

8.5784 8.6201 8.6618 8.7035 8.7452

64.171 64.483 64.795 65.107 65.419

1.280 1.285 1.290 1.295 1.300

31.72 32.16 32.60 33.03 33.46

56 57 58 59 60

10.6630 10.7047 10.7464 10.7881 10.8298

79.765 80.077 80.389 80.701 81.013

1.530 1.535 1.540 1.545 1.550

50.23 50.54 50.84 51.15 51.45

106 107 108 109 110

12.7477 12.7894 12.8310 12.8727 12.9144

95.359 95.671 95.983 96.295 96.606

1.780 1.785 1.790 1.795 1.800

63.54 63.77 63.99 64.22 64.44

156 157 158 159 160

14.8323 14.8740 14.9157 14.9574 14.9990

110.953 111.265 111.577 111.889 112.200

1.055 1.060 1.065 1.070 1.075

7.56 8.21 8.85 9.49 10.12

11 12 13 14 15

8.7869 8.8286 8.8703 8.9120 8.9537

65.731 66.042 66.354 66.666 66.978

1.305 1.310 1.315 1.320 1.325

33.89 34.31 34.73 35.15 35.57

61 62 63 64 65

10.8715 10.9132 10.9549 10.9966 11.0383

81.325 81.636 81.948 82.260 82.572

1.555 1.560 1.565 1.570 1.575

51.75 52.05 52.35 52.64 52.94

111 112 113 114 115

12.9561 12.9978 13.0395 13.0812 13.1229

96.918 97.230 97.542 97.854 98.166

1.805 1.810 1.815 1.820 1.825

64.67 64.89 65.11 65.33 65.55

161 162 163 164 165

15.0407 15.0824 15.1241 15.1658 15.2075

112.512 112.824 113.136 113.448 113.760

1.080 1.085 1.090 1.095 1.100

10.74 11.36 11.97 12.58 13.18

16 17 18 19 20

8.9954 9.0371 9.0787 9.1204 9.1621

67.290 67.602 67.914 68.226 68.537

1.330 1.335 1.340 1.345 1.350

35.98 36.39 36.79 37.19 37.59

66 67 68 69 70

11.0800 11.1217 11.1634 11.2051 11.2467

82.884 83.196 83.508 83.820 84.131

1.580 1.585 1.590 1.595 1.600

53.23 53.52 53.81 54.09 54.38

116 117 118 119 120

13.1646 13.2063 13.2480 13.2897 13.3313

98.478 98.790 99.102 99.414 99.725

1.830 1.835 1.840 1.845 1.850

65.77 65.98 66.20 66.41 66.62

166 167 168 169 170

15.2492 15.2909 15.3326 15.3743 15.4160

114.072 114.384 114.696 115.007 115.318

1.105 1.110 1.115 1.120 1.125

13.78 14.37 14.96 15.54 16.11

21 22 23 24 25

9.2038 9.2455 9.2872 9.3289 9.3706

68.849 69.161 69.473 69.785 70.097

1.355 1.360 1.365 1.370 1.375

37.99 38.38 38.77 39.16 39.55

71 72 73 74 75

11.2884 11.3301 11.3718 11.4135 11.4552

84.443 84.755 85.067 85.379 85.691

1.605 1.610 1.615 1.620 1.625

54.66 54.94 55.22 55.49 55.77

121 122 123 124 125

13.3730 13.4147 13.4564 13.4981 13.5398

100.037 100.349 100.661 100.973 101.285

1.855 1.860 1.865 1.870 1.875

66.83 67.04 67.25 67.46 67.67

171 172 173 174 175

15.4577 15.4993 15.5410 15.5827 15.6244

115.630 115.943 116.255 116.567 116.879

1.130 1.135 1.140 1.145 1.150

16.68 17.25 17.81 18.36 18.91

26 27 28 29 30

9.4123 9.4540 9.4957 9.5374 9.5790

70.409 70.721 71.032 71.344 71.656

1.380 1.385 1.390 1.395 1.400

39.93 40.31 40.68 41.06 41.43

76 77 78 79 80

11.4969 11.5386 11.5803 11.6220 11.6637

86.003 86.315 86.626 86.938 87.250

1.630 1.635 1.640 1.645 1.650

56.04 56.32 56.59 56.85 57.12

126 127 128 129 130

13.5815 13.6232 13.6649 13.7066 13.7483

101.597 101.909 102.220 102.532 102.844

1.880 1.885 1.890 1.895 1.900

67.87 68.08 68.28 68.48 68.68

176 177 178 179 180

15.6661 15.7078 15.7495 15.7912 15.8329

117.191 117.503 117.814 118.126 118.438

1.155 1.160 1.165 1.170 1.175

19.46 20.00 20.54 21.07 21.60

31 32 33 34 35

9.6207 9.6624 9.7041 9.7458 9.7875

71.968 72.280 72.592 72.904 73.216

1.405 1.410 1.415 1.420 1.425

41.80 42.16 42.53 42.89 43.25

81 82 83 84 85

11.7054 11.7471 11.7888 11.8304 11.8721

87.562 87.874 88.186 88.498 88.810

1.655 1.660 1.665 1.670 1.675

57.39 57.65 57.91 58.17 58.43

131 132 133 134 135

13.7900 13.8317 13.8734 13.9150 13.9567

103.156 103.468 103.780 104.092 104.404

1.905 1.910 1.915 1.920 1.925

68.88 69.08 69.28 69.48 69.68

181 182 183 184 185

15.8746 15.9163 15.9580 15.9996 16.0413

118.740 119.062 119.374 119.686 119.998

1.180 1.185 1.190 1.195 1.200

22.12 22.64 23.15 23.66 24.17

36 37 38 39 40

9.8292 9.8709 9.9126 9.9543 9.9960

73.528 73.840 74.151 74.463 74.775

1.430 1.435 1.440 1.445 1.450

43.60 43.95 44.31 44.65 45.00

86 87 88 89 90

11.9138 11.9555 11.9972 12.0389 12.0806

89.121 89.433 89.745 90.057 90.369

1.680 1.685 1.690 1.695 1.700

58.69 58.95 59.20 59.45 59.71

136 137 138 139 140

13.9984 14.0401 14.0818 14.1235 14.1652

104.715 105.027 105.339 105.651 105.963

1.930 1.935 1.940 1.945 1.950

69.87 70.06 70.26 70.45 70.64

186 187 188 189 190

16.0830 16.1247 16.1664 16.2081 16.2498

120.309 120.621 120.933 121.245 121.557

1.205 1.210 1.215 1.220 1.225

24.67 25.17 25.66 26.15 26.63

41 42 43 44 45

10.0377 10.0793 10.1210 10.1627 10.2044

75.087 75.399 75.711 76.022 76.334

1.455 1.460 1.465 1.470 1.475

45.34 45.68 46.02 46.36 46.69

91 92 93 94 95

12.1223 12.1640 12.2057 12.2473 12.2890

90.681 90.993 91.305 91.616 91.928

1.705 1.710 1.715 1.720 1.725

59.96 60.20 60.45 60.70 60.94

141 142 143 144 145

14.2069 14.2486 14.2903 14.3320 14.3737

106.275 106.587 106.899 107.210 107.522

1.955 1.960 1.965 1.970 1.975

70.83 71.02 71.21 71.40 71.58

191 192 193 194 195

16.2915 16.3332 16.3749 16.4166 16.4583

121.869 122.181 122.493 122.804 123.116

1.230 1.235 1.240 1.245 1.250

27.11 27.59 28.06 28.53 29.00

46 47 48 49 50

10.2461 10.2878 10.3295 10.3712 10.4129

76.646 76.958 77.270 77.582 77.894

1.480 1.485 1.490 1.495 1.500

47.03 47.36 47.68 48.01 48.33

96 97 98 99 100

12.3307 12.3724 12.4141 12.4558 12.4975

92.240 92.552 92.864 93.176 93.488

1.730 1.735 1.740 1.745 1.750

61.18 61.34 61.67 61.91 62.14

146 147 148 149 150

14.4153 14.4570 14.4987 14.5404 14.5821

107.834 108.146 108.458 108.770 109.082

1.980 1.985 1.990 1.995 2.000

71.77 71.95 72.14 72.32 72.50

196 197 198 199 200

16.5000 16.5417 16.5833 16.6250 16.6667

123.428 123.740 124.052 124.364 124.676

Sp gr 60°/ 60°

°Bé

Lb per ft3 at 60°F wt in air

Sp gr 60°/ 60°

Lb per ft3 at 60°F. wt. in air

*Prepared by Lewis V. Judson, Ph.D., Chief of Length Section of National Bureau of Standards with the advice and assistance of E. L. Peffer, B.S., A.M., late Chief of Capacity and Density Section, National Bureau of Standards.

1-19

TABLE 1-14

Fundamental Physical Constants

1 sec = 1.00273791 sidereal seconds g0 = 9.80665 m/sec2 1 liter = 0.001 cu. m 1 atm = 101,325 newtons/sq m 1 mm Hg (pressure) = (1⁄ 760) atm = 133.3224 newtons/sq m 1 int ohm = 1.000495  0.000015 abs ohm 1 int amp = 0.999835  0.000025 abs amp 1 int coul = 0.999835  0.000025 abs coul 1 int volt = 1.000330  0.000029 abs volt 1 int watt = 1.000165  0.000052 abs watt 1 int joule = 1.000165  0.000052 abs joule T0°C = 273.150  0.010°K (PV)0°CP=0 = (RT)0°C = 2271.16  0.04 abs joule/mole = 22,414.6  0.4 cu. cm atm/mole = 22.4146  0.0004 liter atm/mole R = 8.31439  0.00034 abs joule/deg mole = 1.98719  0.00013 cal/deg mole = 82.0567  0.0034 cu. cm atm/deg mole = 0.0820567  0.0000034 liter atm/deg mole ln 10 = 2.302585 R ln 10 = 19.14460  0.00078 abs joule/deg mole = 4.57567  0.00030 cal/deg mole N = (6.02283  0.0022) × 1023/mole h = (6.6242  0.0044) × 10−34 joule sec c = (2.99776  0.00008) × 108 m/sec (h2/8π2k) = (4.0258  0.0037) × 10−39 g sq cm deg (h/8π2c) = (2.7986  0.0018) × 10−39 g cm Z = Nhc = 11.9600  0.0036 abs joule cm/mole = 2.85851  0.0009 cal cm/mole (Z/R) = (hc/k) = c2 = 1.43847  0.00045 cm deg Ᏺ = 96,501.2  10.0 int coul/g-equiv or int joule/int volt g-equiv = 96,485.3  10.0 abs coul/g-equiv or abs joule/abs volt g-equiv = 23,068.1  2.4 cal/int volt g-equiv = 23,060.5  2.4 cal/abs volt g-equiv e = (1.60199  0.00060) × 10−19 abs coul = (1.60199  0.00060) × 10−20 abs emu = (4.80239  0.00180) × 10−10 abs esu 1 int electron-volt/molecule = 96,501.2  10 int joule/mole = 23,068.1  2.4 cal/mole 1 abs electron-volt/molecule = 96,485.3  10. abs joule/mole = 23,060.5  2.4 cal/mole 1 int electron-volt = (1.60252  0.00060) × 10−12 erg 1 abs electron-volt = (1.60199  0.00060) × 10−12 erg hc = (1.23916  0.00032) × 10−4 int electron-volt cm = (1.23957  0.00032) × 10−4 abs electron-volt cm k = (8.61442  0.00100) × 10−5 int electron-volt/deg = (8.61727  0.00100) × 10−5 abs electron-volt/deg = (R/N) = (1.38048  0.00050) × 10−23 joule/deg 1 IT cal = (1⁄ 860) = 0.00116279 int watt-hr = 4.18605 int joule = 4.18674 abs joule = 1.000654 cal 1 cal = 4.1840 abs joule = 4.1833 int joule = 41.2929  0.0020 cu. cm atm = 0.0412929  0.0000020 liter atm 1 IT cal/g = 1.8 Btu/lb 1 Btu = 251.996 IT cal = 0.293018 int watt-hr = 1054.866 int joule = 1055.040 abs joule = 252.161 cal 1 horsepower = 550 ft-lb (wt)/sec = 745.578 int watt = 745.70 abs watt 1 in = (1/0.3937) = 2.54 cm 1 ft = 0.304800610 m 1 lb = 453.5924277 g 1 gal = 231 cu. in = 0.133680555 cu. ft = 3.785412 × 10−3 cu. m = 3.785412 liter

1-20

sec = mean solar second Definition: g0 = standard gravity Definition: atm = standard atmosphere mm Hg (pressure) = standard millimeter mercury int = international; abs = absolute amp = ampere coul = coulomb

Absolute temperature of the ice point, 0°C PV product for ideal gas at 0°C R = gas constant per mole

ln = natural logarithm (base e) N = Avogadro number h = Planck constant c = velocity of light Constant in rotational partition function of gases Constant relating wave number and moment of inertia Z = constant relating wave number and energy per mole c2 = second radiation constant Ᏺ = Faraday constant

e = electronic charge

Constant relating wave number and energy per molecule k = Boltzmann constant Definition of IT cal: IT = International steam tables cal = thermochemical calorie Definition: cal = thermochemical calorie

Definition of Btu: Btu = IT British Thermal Unit

cal = thermochemical calorie Definition of horsepower (mechanical): lb (wt) = weight of 1 lb at standard gravity Definition of in: in = U.S. inch ft = U.S. foot (1 ft = 12 in) Definition; lb = avoirdupois pound Definition; gal = U.S. gallon

CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. Table 1-7 provides a number of useful conversion factors. To make a conversion of an element in U.S. customary units to SI units, one multiplies the value of the U.S. customary unit, found on the left side in the table, by the equivalent value on the right side. For example, to convert 10 British thermal units to joules, one multiplies 10 by 1054.4 to obtain 10544 joules. In each example, the initial values of the factors are expressed in U.S. customary units, and the dimensionless value is calculated. Then the factors are converted to SI units, and the dimensionless value is recalculated. The two dimensionless values will be approximately the same. (Small variations occur due to the number of significant figures carried in the solution.)

Example 1. Calculation of a Reynolds Number DVρ NRe =  µ

U.S. customary units D = 3 in. = 3⁄12 ft V = 6 ft/s ρ = 0.08 lbm/ft3 µ = 0.015 cp = (0.015)(0.000672) lbm/ft⋅s

(3/12)(6)(0.08) NRe =  = 11,904 (0.015)(0.000672) SI units D = (3)(0.0254) m V = (6)(0.3048) m/s ρ = (0.08)(16.018) kg/m3 µ = (0.015)(0.001) kg/m⋅s

(Difference due to rounding)

Example 4. Calculation of a Grashof Number NGr = L3ρ2gβ(∆T)/µ2 U.S. Customary units L = 3 ft ρ = 0.0725 lbm/ft3 g = 32.174 ft/s2 β = 0.00168/°R ∆T = 99 °R µ = 0.019 centipoise = 0.019 × 0.000672 lbm/ft⋅s = 1.277 × 10−5 lbm/ft⋅s (33) (0.0725)2(32.174) (0.00168) (99) = 4.66 × 109 NGr =  (1.277 × 10−5)2 SI units L = (3)(0.3048) = 0.9144 m ρ = (0.0725)(16.018) = 1.1613 kg/m3 g = 9.807 m/s2 β = (0.00168)/(1.8) = 0.000933/°K ∆T = (99)(1.8) = 178.2 °K µ = (0.019)(0.001) = 1.9 × 10−5 kg/m⋅s (0.9144)3(1.1613)2(9.807)(0.000933)(178.2) = 4.66 × 109 NGr =  (1.9 × 10−5)2

Example 5. Calculation of a Schmidt Number µ NSc =  ρD

(3 × 0.0254) (6 × 0.3048) (0.08 × 16.018) NRe =  = 11,904 (0.015) (0.001)

Example 2. Calculation of a Prandtl Number Cp µ NPr =  k U.S. customary units γp = 0.47 Btu/lbm °F µ = 15 centipoise = (15) (0.000672) (3600) lbm/ft⋅hr k = 0.065 Btu/hr⋅ft2 (°F/ft) (0.47) (15 × 0.000672 × 3600) NPr =  = 262.4 0.065 SI units γ = (0.47)(4184) J/kg °C µ = (15)(0.001) kg/m⋅s k = (0.065)(1.728) J/s⋅m2 (°C/m) (0.47) (4184) (15) (0.001) NPr =  = 262.6 (0.065) (1.728) (Difference due to rounding)

Example 3. Calculation of a Nusselt Number hD NNu =  k U.S. customary units h = 200 Btu/hr⋅ft2⋅°F D = 1.5 in. = 1.5/12 ft k = 0.07 Btu/hr⋅ft2 (°F/ft) (200)(1.5/12) NNu =  = 357.1 0.07 SI units h = (200)(5.678) J/(s⋅m2⋅°C) D = (1.5)(0.0254) m k = (0.07)(1.728) J/s⋅m2 (°C/m)

(200) (5.678) (1.5) (0.0254) NNu =  = 357.7 (0.07) (1.728)

U.S. customary units µ = 0.02 centipoise = (0.02)(2.42) lbm/ft⋅hr ρ = 0.08 lbm/ft3 D = 1.0 ft2/hr (diffusivity) (0.02) (2.42) NSc =  = 0.605 (0.08)(1.0) SI units µ = (0.02)(0.001) kg/m⋅s ρ = (0.08)(16.02) kg/m2 D = (1.0)(2.58 × 10−5) m2/s (0.02) (0.001) = 0.605 NSc =  (0.08)(16.02)(1.0) (2.58 × 10−5)

Example 6. Calculation of an Archimedes Number d 3ρf(ρp − ρf)g NAr =  µ2 U.S. customary units d = 2 mm = 2/[(1000)(0.3048)] = 0.00656 ft ρf = 0.0175 lbm/ft3 ρp = 168.5 lbm/ft3 g = 32.174 ft/s2 µ = 0.04 centipoise = 0.04 × 0.000672 = 2.688−5 lbm/ft⋅s (0.00656)3 (0.0175) (168.5 − 0.017) (32.174) = 37,064 NAr =  (2.688 × 10−5)2 SI units d = 2/1000 m ρp = 168.5 × 16.02 = 2699.37 kg/m3 ρf = 0.0175 × 16.02 = 0.2804 g/m3 g = 9.807 m/s2 µ = 0.04 × 0.001 = 4 × 10−5 kg/m⋅s (2/1000)3 (0.2804) (2699.37 − 0.28) (9.807) = 37,118 NAr =  (4 × 10−5)2 (Difference due to rounding)

1-21

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Section 2

Physical and Chemical Data*

Bruce E. Poling Department of Chemical Engineering, University of Toledo (Physical and Chemical Data) George H. Thomson AIChE Design Institute for Physical Properties (Physical and Chemical Data) Daniel G. Friend National Institute of Standards and Technology (Physical and Chemical Data) Richard L. Rowley Department of Chemical Engineering, Brigham Young University (Prediction and Correlation of Physical Properties) W. Vincent Wilding Department of Chemical Engineering, Brigham Young University (Prediction and Correlation of Physical Properties)

2-10

GENERAL REFERENCES

Vapor Pressures of Organic Compounds, up to 1 atm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-65

PHYSICAL PROPERTIES OF PURE SUBSTANCES Tables 2-1 2-2

Physical Properties of the Elements and Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Properties of Organic Compounds . . . . . . . . . . . . . .

VAPOR PRESSURES OF PURE SUBSTANCES Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-3 Vapor Pressure of Water Ice from 0 to −40 °C . . . . . . . . . . . . 2-4 Vapor Pressure of Supercooled Liquid Water from 0 to −40 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Vapor Pressure (MPa) of Liquid Water from 0 to 100 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 Sorted by Chemical Family . . . . . . . . . . . . 2-7 Formula Index of Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8 Vapor Pressure of Inorganic and Organic Liquids, ln P = C1 + C2/T + C3 ln T + C4 T C5, P in Pa . . . . . . . . . . . . . 2-9 Vapor Pressures of Inorganic Compounds, up to 1 atm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-7 2-28

2-48 2-48 2-48 2-48 2-48 2-49 2-52 2-55 2-61

VAPOR PRESSURES OF SOLUTIONS Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-11 Partial Pressures of Water over Aqueous Solutions of HCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12 Partial Pressures of HCl over Aqueous Solutions of HCl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of H3PO4 Aqueous: Partial Pressure of H2O Vapor (Fig. 2-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of H3PO4 Aqueous: Weight of H2O in Saturated Air (Fig. 2-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13 Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . 2-14 Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15 Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16 Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-17 Total Pressure, bar, of Aqueous Sulfuric Acid Solutions . . . . . . 2-18 Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19 Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55 °C . . . . . . . . . . . . . . . . . . . . . .

2-80 2-80 2-80 2-81 2-81 2-81 2-82 2-84 2-86 2-87 2-88 2-89

*Contribution in part of the National Institute of Standards and Technology; not subject to copyright in the United States. 2-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

2-2 2-20 2-21 2-22

2-23 2-24 2-25 2-26 2-27 2-28 2-29

PHYSICAL AND CHEMICAL DATA Partial Pressures of HI over Aqueous Solutions of HI at 25 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Vapor Pressures of Aqueous Solutions of CH3COOH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressure of Aqueous Diethylene Glycol Solutions (Fig. 2-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressure of H2O over Aqueous Solutions of NH3 (psia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mole Percentages of H2O over Aqueous Solutions of NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of NH3 over Aqueous Solutions of NH3 (psia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Vapor Pressures of Aqueous Solutions of NH3 (psia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-89 2-89 2-89 2-89 2-90 2-91 2-92 2-93 2-94 2-94 2-94

WATER-VAPOR CONTENT OF GASES Chart for Gases at High Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Content of Air (Fig. 2-4) . . . . . . . . . . . . . . . . . . . . . . .

2-95 2-95

DENSITIES OF PURE SUBSTANCES Tables 2-30 2-31 2-32

Density (kg/m3) of Saturated Liquid Water from the Triple Point to the Critical Point . . . . . . . . . . . . . . . . . . Density (kg/m3) of Mercury from 0 to 350°C . . . . . . . . . . . . Densities of Inorganic and Organic Liquids (mol/dm3) . . . . .

2-96 2-97 2-98

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-104 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-104 Tables 2-33 Aluminum Sulfate [Al2(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . 2-104 2-34 Ammonia (NH3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-104 2-35 Ammonium Acetate (CH3COONH4) . . . . . . . . . . . . . . . . . . 2-104 2-36 Ammonium Bichromate [(NH4)2Cr2O7] . . . . . . . . . . . . . . . . 2-104 2-37 Ammonium Chloride (NH4Cl) . . . . . . . . . . . . . . . . . . . . . . . 2-104 2-38 Ammonium Chromate [(NH4)2CrO4] . . . . . . . . . . . . . . . . . . 2-104 2-39 Ammonium Nitrate (NH4NO3) . . . . . . . . . . . . . . . . . . . . . . . 2-104 2-40 Ammonium Sulfate [(NH4)2SO4] . . . . . . . . . . . . . . . . . . . . . . 2-104 2-41 Arsenic Acid (H3AsO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-104 2-42 Barium Chloride (BaCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-43 Cadmium Nitrate [Cd(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-44 Calcium Chloride (CaCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-45 Calcium Hydroxide [Ca(OH)2] . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-46 Calcium Hypochlorite (CaOCl2) . . . . . . . . . . . . . . . . . . . . . . 2-105 2-47 Calcium Nitrate [Ca(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-48 Chromic Acid (CrO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-49 Chromium Chloride (CrCl3) . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-50 Copper Nitrate [Cu(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-51 Copper Sulfate (CuSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-52 Cuprous Chloride (CuCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-53 Ferric Chloride (FeCl3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-105 2-54 Ferric Sulfate [Fe2(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-55 Ferric Nitrate [Fe(NO3)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-56 Ferrous Sulfate (FeSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-57 Hydrogen Bromide (HBr) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-58 Hydrogen Cyanide (HCN) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-59 Hydrogen Chloride (HCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-60 Hydrogen Fluoride (HF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-61 Hydrogen Peroxide (H2O2) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-62 Hydrofluosilic Acid (H2SiF6) . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-63 Magnesium Chloride (MgCl2) . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-64 Magnesium Sulfate (MgSO4) . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-65 Nickel Chloride (NiCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-66 Nickel Nitrate [Ni(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-67 Nickel Sulfate (NiSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-106 2-68 Nitric Acid (HNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-107 2-69 Perchloric Acid (HClO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108 2-70 Phosphoric Acid (H3PO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108 2-71 Potassium Bicarbonate (KHCO3) . . . . . . . . . . . . . . . . . . . . . 2-108 2-72 Potassium Bromide (KBr) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108

2-73 2-74 2-75 2-76 2-77 2-78 2-79 2-80 2-81 2-82 2-83 2-84 2-85 2-86 2-87 2-88 2-89 2-90 2-91 2-92 2-93 2-94 2-95 2-96 2-97 2-98 2-99 2-100 2-101 2-102 2-103 2-104 2-105 2-106 2-107

Potassium Carbonate (K2CO3) . . . . . . . . . . . . . . . . . . . . . . . Potassium Chromate (K2CrO4) . . . . . . . . . . . . . . . . . . . . . . Potassium Chlorate (KClO3) . . . . . . . . . . . . . . . . . . . . . . . . Potassium Chloride (KCl) . . . . . . . . . . . . . . . . . . . . . . . . . . Potassium Chrome Alum [K2Cr2(SO4)4] . . . . . . . . . . . . . . . Potassium Hydroxide (KOH) . . . . . . . . . . . . . . . . . . . . . . . . Potassium Nitrate (KNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . Potassium Dichromate (K2Cr2O7) . . . . . . . . . . . . . . . . . . . . Potassium Sulfate (K2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . Potassium Sulfite (K2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Acetate (NaC2H3O2) . . . . . . . . . . . . . . . . . . . . . . . . Sodium Arsenate (Na3AsO4) . . . . . . . . . . . . . . . . . . . . . . . . Sodium Bichromate (Na2Cr2O7) . . . . . . . . . . . . . . . . . . . . . Sodium Bromide (NaBr) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Formate (HCOONa) . . . . . . . . . . . . . . . . . . . . . . . Sodium Carbonate (Na2CO3) . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chlorate (NaClO3) . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chloride (NaCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chromate (Na2CrO4) . . . . . . . . . . . . . . . . . . . . . . . Sodium Hydroxide (NaOH) . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrate (NaNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrite (NaNO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Silicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfate (Na2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfide (Na2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfite (Na2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate (Na2S2O3) . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate Pentahydrate (Na2S2O3 ⋅5H2O) . . . . . . Stannic Chloride (SnCl4) . . . . . . . . . . . . . . . . . . . . . . . . . . . Stannous Chloride (SnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid (H2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Bromide (ZnBr2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Chloride (ZnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Nitrate [Zn(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Sulfate (ZnSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-108 Formic Acid (HCOOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-109 Acetic Acid (CH3COOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-110 Oxalic Acid (H2C2O4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-111 Methyl Alcohol (CH3OH) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-112 Ethyl Alcohol (C2H5OH) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-113 Densities of Mixtures of C2H5OH and H2O at 20°C . . . . . 2-114 Specific Gravity {60°/60°F [(15.56°/15.56°C)]} of Mixtures by Volume of C2H5OH and H2O . . . . . . . . . . . . 2-115 n-Propyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . 2-116 Isopropyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . 2-117 Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-118 Hydrazine (N2H4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-119 Densities of Aqueous Solutions of Miscellaneous Organic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-109 2-110 2-110 2-110 2-110 2-110 2-110 2-110 2-110 2-111 2-111 2-111 2-111 2-111 2-111 2-111 2-112 2-114 2-114 2-114 2-114

2-114 2-114 2-115 2-116 2-116 2-117 2-118 2-119 2-120 2-120 2-121 2-121 2-122

DENSITIES OF MISCELLANEOUS MATERIALS Tables 2-120 2-121

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids . . . . . . . . . . . . . . . . . . . Density (kg/m3) of Selected Elements as a Function of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . .

SOLUBILITIES Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-122 Solubilities of Inorganic Compounds in Water at Various Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-123 Solubility as a Function of Temperature and Henry’s Constant at 25°C for Gases in Water . . . . . . . . . . 2-124 Henry’s Constant H for Various Compounds in Water at 25°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-125 Henry’s Constant H for Various Compounds in Water at 25°C from Infinite Dilution Activity Coefficients . . . . . . 2-126 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-127 Ammonia-Water at 10 and 20°C . . . . . . . . . . . . . . . . . . . . . 2-128 Carbon Dioxide (CO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-129 Carbonyl Sulfide (COS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-130 Chlorine (Cl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-131 Chlorine Dioxide (ClO2) . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-124 2-125

2-125 2-126 2-130 2-130 2-131 2-131 2-131 2-131 2-131 2-132 2-132

PHYSICAL AND CHEMICAL DATA 2-132 2-133 2-134

Hydrogen Chloride (HCl) . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Sulfide (H2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Vapor Pressure of Sulfur Dioxide over Water, mmHg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

THERMAL EXPANSION Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Expansion of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-135 Linear Expansion of the Solid Elements . . . . . . . . . . . . . . . 2-136 Linear Expansion of Miscellaneous Substances . . . . . . . . . 2-137 Volume Expansion of Liquids . . . . . . . . . . . . . . . . . . . . . . . 2-138 Volume Expansion of Solids . . . . . . . . . . . . . . . . . . . . . . . . . JOULE-THOMSON EFFECT Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-139 Additional References Available for the Joule-Thomson Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 2-140 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc) . . . . . . . . . . . . . . . . . . . CRITICAL CONSTANTS Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-141 Critical Constants and Acentric Factors of Inorganic and Organic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . COMPRESSIBILITIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-142 Composition of Selected Refrigerant Mixtures . . . . . . . . . . 2-143 Compressibility Factors for R 407A (Klea 60) . . . . . . . . . . 2-144 Compressibility Factors for R 407B (Klea 61) . . . . . . . . . . 2-145 Compressibilities of Liquids . . . . . . . . . . . . . . . . . . . . . . . . 2-146 Compressibilities of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . LATENT HEATS Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-147 Heats of Fusion and Vaporization of the Elements and Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . 2-148 Heats of Fusion of Miscellaneous Materials . . . . . . . . . . . . 2-149 Heats of Fusion of Organic Compounds . . . . . . . . . . . . . . . 2-150 Heats of Vaporization of Inorganic and Organic Liquids (J/kmol) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SPECIFIC HEATS OF PURE COMPOUNDS Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-151 Heat Capacities of the Elements and Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-152 Specific Heat [kJ/(kg⋅K)] of Selected Elements . . . . . . . . . 2-153 Heat Capacities of Inorganic and Organic Liquids [J/(kmol⋅K)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-154 Specific Heats of Organic Solids . . . . . . . . . . . . . . . . . . . . . 2-155 Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to a Polynomial Cp [J/(kmol⋅K)] . . . . . . . . . . . . . . . . . . . . . . . . 2-156 Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions Cp [J/(kmol⋅K)] . . . . . . . . . . . . . 2-157 Cp/Cv: Ratios of Specific Heats of Gases at 1 atm Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SPECIFIC HEATS OF AQUEOUS SOLUTIONS Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-158 Acetic Acid (at 38°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-159 Ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-160 Aniline (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-161 Copper Sulfate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-132 2-132 2-133

2-133 2-133 2-133 2-134 2-135 2-136 2-136

2-162 2-163 2-164 2-165 2-166 2-167 2-168 2-169 2-170 2-171 2-172 2-173 2-174 2-175

Ethyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrochloric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phosphoric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potassium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potassium Hydroxide (at 19°C) . . . . . . . . . . . . . . . . . . . . . . Normal Propyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Carbonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Hydroxide (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Sulfate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-3 2-183 2-183 2-183 2-183 2-183 2-183 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184

SPECIFIC HEATS OF MISCELLANEOUS MATERIALS 2-137 2-137

Tables 2-176 2-177

Specific Heats of Miscellaneous Liquids and Solids . . . . . . Oils (Animal, Vegetable, Mineral Oils) . . . . . . . . . . . . . . . .

2-143 2-143

PROPERTIES OF FORMATION AND COMBUSTION REACTIONS Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-178 Heats and Free Energies of Formation of Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-179 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K . . . . . . . . . . . . . . . . . . . 2-180 Ideal Gas Sensible Enthalpies, hT − h298 (kJ/kmol), of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-181 Ideal Gas Entropies s°, kJ/(kmol⋅K), of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-143 2-143 2-143 2-144 2-144

Tables 2-182 2-183

2-137

2-138 2-138

2-144 2-145 2-147 2-148 2-150

2-156 2-156 2-156 2-164 2-165 2-171 2-174 2-176 2-182

2-183 2-183 2-183 2-183 2-183 2-183

2-185 2-185

2-185 2-186 2-195 2-201 2-202

HEATS OF SOLUTION Heats of Solution of Inorganic Compounds in Water . . . . . Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

THERMODYNAMIC PROPERTIES Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-184 List of Substances for Which Thermodynamic Property Tables Were Generated from NIST Standard Reference Database 23 . . . . . . . . . . . . . . . . . . . . 2-185 Thermodynamic Properties of Acetone . . . . . . . . . . . . . . . . 2-186 Saturated Acetylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-187 Thermodynamic Properties of Air . . . . . . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Dry Air (Fig. 2-5) . . . . . . . 2-188 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air, Moist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-189 Thermodynamic Properties of Ammonia . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Ammonia (Fig. 2-6) . . . . . Enthalpy-Concentration Diagram for Aqueous Ammonia (Fig. 2-7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-190 Thermodynamic Properties of Argon . . . . . . . . . . . . . . . . . 2-191 Liquid-Vapor Equilibrium Data for the ArgonNitrogen-Oxygen System . . . . . . . . . . . . . . . . . . . . . . . . . . 2-192 Thermodynamic Properties of the International Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-193 Thermodynamic Properties of Benzene . . . . . . . . . . . . . . . 2-194 Saturated Bromine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-195 Thermodynamic Properties of Butane . . . . . . . . . . . . . . . . 2-196 Thermodynamic Properties of 1-Butene . . . . . . . . . . . . . . . 2-197 Thermodynamic Properties of cis-2-Butene . . . . . . . . . . . . 2-198 Thermodynamic Properties of trans-2-Butene . . . . . . . . . . 2-199 Thermodynamic Properties of Carbon Dioxide . . . . . . . . . 2-200 Thermodynamic Properties of Carbon Monoxide . . . . . . . Temperature-Entropy Diagram for Carbon Monoxide (Fig. 2-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-201 Thermophysical Properties of Saturated Carbon Tetrachloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-203 2-206

2-207 2-207 2-207 2-207

2-208 2-209 2-210 2-211 2-215 2-216 2-216 2-217 2-219 2-220 2-221 2-224 2-228 2-229 2-231 2-232 2-234 2-236 2-238 2-240 2-242 2-244 2-245

2-4 Tables 2-202 2-203 2-204 2-205

PHYSICAL AND CHEMICAL DATA

Saturated Carbon Tetrafluoride (R14) . . . . . . . . . . . . . . . . Thermodynamic Properties of Carbonyl Sulfide . . . . . . . . Saturated Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Chlorine . . . . . . . Enthalpy–Log-Pressure Diagram for Chlorine (Fig. 2-9) . . 2-206 Saturated Chloroform (R20) . . . . . . . . . . . . . . . . . . . . . . . . 2-207 Thermodynamic Properties of Cyclohexane . . . . . . . . . . . . 2-208 Thermodynamic Properties of Decane . . . . . . . . . . . . . . . . 2-209 Thermodynamic Properties of Deuterium Oxide (Heavy Water) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-210 Thermodynamic Properties of 2,2-Dimethylpropane (Neopentane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-211 Saturated Diphenyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-212 Thermodynamic Properties of Dodecane . . . . . . . . . . . . . . 2-213 Thermodynamic Properties of Ethane . . . . . . . . . . . . . . . . 2-214 Thermodynamic Properties of Ethanol . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Ethyl Alcohol (Fig. 2-10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-215 Thermodynamic Properties of Ethylene . . . . . . . . . . . . . . . 2-216 Thermodynamic Properties of Fluorine . . . . . . . . . . . . . . . 2-217 Flutec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-218 Halon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-219 Thermodynamic Properties of Helium . . . . . . . . . . . . . . . . 2-220 Thermodynamic Properties of Heptane . . . . . . . . . . . . . . . 2-221 Thermodynamic Properties of Hexane . . . . . . . . . . . . . . . . 2-222 Saturated Hydrazine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-223 Thermodynamic Properties of Normal Hydrogen . . . . . . . 2-224 Thermodynamic Properties of para-Hydrogen . . . . . . . . . . 2-225 Saturated Hydrogen Peroxide . . . . . . . . . . . . . . . . . . . . . . . 2-226 Thermodynamic Properties of Hydrogen Sulfide . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Hydrogen Chloride at 1 atm (Fig. 2-11) . . . . . . . . . . . . . . . . . . . . . . . 2-227 Thermodynamic Properties of Isobutane . . . . . . . . . . . . . . 2-228 Thermodynamic Properties of Isobutene (2-Methyl 1-Propene) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-229 Thermodynamic Properties of Krypton . . . . . . . . . . . . . . . . 2-230 Saturated Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-231 Lithium Bromide—Water Solutions. . . . . . . . . . . . . . . . . . . 2-232 Saturated Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Mercury (Fig. 2-12) . 2-233 Thermodynamic Properties of Methane . . . . . . . . . . . . . . . 2-234 Thermodynamic Properties of Methanol . . . . . . . . . . . . . . 2-235 Thermodynamic Properties of 2-Methyl Butane (Isopentane) 2-236 Thermodynamic Properties of 2-Methyl Pentane (Isohexane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-237 Saturated Methyl Chloride . . . . . . . . . . . . . . . . . . . . . . . . . 2-238 Thermodynamic Properties of Neon . . . . . . . . . . . . . . . . . . 2-239 Thermodynamic Properties of Nitrogen . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Nitrogen (Fig. 2-13) . . . . . 2-240 Saturated Nitrogen Tetroxide . . . . . . . . . . . . . . . . . . . . . . . 2-241 Thermodynamic Properties of Nitrogen Trifluoride . . . . . 2-242 Thermodynamic Properties of Nitrous Oxide . . . . . . . . . . . Mollier Diagram for Nitrous Oxide (Fig. 2-14). . . . . . . . . . 2-243 Thermodynamic Properties of Nonane . . . . . . . . . . . . . . . . 2-244 Thermodynamic Properties of Octane . . . . . . . . . . . . . . . . 2-245 Thermodynamic Properties of Oxygen . . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Oxygen (Fig. 2-15) . . . . . . Enthalpy-Concentration Diagram for Oxygen-Nitrogen Mixture at 1 atm (Fig. 2-16). . . . . . . . . . . . . . . . . . . . . . . . 2-246 Thermodynamic Properties of Pentane . . . . . . . . . . . . . . . . 2-247 Saturated Potassium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Potassium (Fig. 2-17) . . . . . . . . . . . . . 2-248 Thermodynamic Properties of Propane . . . . . . . . . . . . . . . 2-249 Thermodynamic Properties of Propylene . . . . . . . . . . . . . . 2-250 Thermodynamic Properties of R-11, Trichlorofluoromethane Pressure-Enthalpy Diagram for Refrigerant 11 (Fig. 2-18) 2-251 Thermodynamic Properties of R-12, Dichlorodifluoromethane . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Refrigerant 12 (Fig. 2-19) 2-252 Thermodynamic Properties of R-13, Chlorotrifluoromethane Refrigerant 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-253 Saturated Refrigerant 13B1, Bromotrifluoromethane . . . . . 2-254 Saturated Refrigerant 21, Dichlorofluoromethane . . . . . . . . 2-255 Thermodynamic Properties of R-22, Chlorodifluoromethane Pressure-Enthalpy Diagram for Refrigerant 22 (Fig. 2-20) . 2-256 Thermodynamic Properties of R-23, Trifluoromethane . . . . 2-257 Thermodynamic Properties of R-32, Difluoromethane . . . . Pressure-Enthalpy Diagram for Refrigerant 32 (Fig. 2-21) .

2-245 2-246 2-248 2-249 2-250 2-251 2-252 2-254 2-256 2-258 2-260 2-261 2-263 2-265 2-267 2-268 2-270 2-271 2-271 2-272 2-274 2-276 2-278 2-279 2-281 2-282 2-283 2-285 2-286 2-288 2-290 2-292 2-292 2-293 2-295 2-296 2-298 2-300 2-302 2-304 2-305 2-307 2-309 2-310 2-311 2-313 2-315 2-316 2-318 2-320 2-322 2-323 2-324 2-326 2-326 2-327 2-329 2-331 2-333 2-334 2-336 2-337 2-339 2-339 2-339 2-339 2-340 2-342 2-343 2-345 2-347

2-258 2-259 2-260 2-261 2-262

2-263 2-264 2-265 2-266 2-267 2-268 2-269 2-270 2-271 2-272 2-273 2-274 2-275

2-276 2-277 2-278 2-279 2-280 2-281 2-282 2-283 2-284 2-285 2-286 2-287 2-288 2-289 2-290 2-291 2-292 2-293 2-294 2-295 2-296 2-297 2-298 2-299

2-300 2-301 2-302 2-303 2-304 2-305 2-306 2-307 2-308 2-309

Thermodynamic Properties of R-41, Fluoromethane . . . . . Saturated R-401A (SUVA MP 39) . . . . . . . . . . . . . . . . . . . . . R-401A (SUVA MP 39) at Atmospheric Pressure . . . . . . . . . Thermodynamic Properties of Saturated R-407A (Klea 60) Thermodynamic Properties of Saturated R-407B (Klea 61) Enthalpy–Log-Pressure Diagram for R-407A (Klea 60) (Fig. 2-22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for R-407B (Klea 61) (Fig. 2-23). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated R-404A (SUVA HP 62) . . . . . . . . . . . . . . . . . . . . . R-404A (SUVA HP 62) at Atmospheric Pressure . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 123 Saturated R-401B (SUVA MP 66) . . . . . . . . . . . . . . . . . . . . . R-401B (SUVA MP 66) at Atmospheric Pressure . . . . . . . . . Saturated R-402A (SUVA HP 80) . . . . . . . . . . . . . . . . . . . . . R-402A (SUVA HP 80) at Atmospheric Pressure . . . . . . . . . Saturated R-402B (SUVA HP 81) . . . . . . . . . . . . . . . . . . . . . R-402B (SUVA HP 81) at Atmospheric Pressure . . . . . . . . . Thermodynamic Properties of R-113, 1,1, 2-Trichlorotrifluoroethane . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-114, 1, 2-Dichlorotetrafluoroethane . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 115, Chloropentafluoroethane . . . . . Thermodynamic Properties of R-116, Hexafluoroethane . . Thermodynamic Properties of R-123, 2,2-Dichloro-1,1,1-Trifluoroethane . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 123 (Fig. 2-24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-124, 2-Chloro-1,1,1,2-Tetrafluoroethane . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-125, Pentafluoroethane . . Enthalpy–Log-Pressure Diagram for Refrigerant 125 (Fig. 2-25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-134a, 1,1,1,2Tetrafluoroethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Refrigerant 134a (Fig. 2-26) Thermodynamic Properties of R-141b, 1,1-Dichloro-1Fluoroethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-142b, 1-Chloro-1,1Difluoroethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-143a, 1,1,1-Trifluoroethane Thermodynamic Properties of R-152a, 1,1-Difluoroethane . Saturated Refrigerant 216a, 1,3-Dichloro-1,1,2,2,3,3Hexafluoropropane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-218, Octafluoropropane . . Thermodynamic Properties of R-227ea, 1,1,1,2,3,3,3Heptafluoropropane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 245cb 1,1,1,2,2-Pentafluoropropane . Refrigerant RC 318, Octafluorocyclobutane . . . . . . . . . . . . . Thermodynamic Properties of R-404A . . . . . . . . . . . . . . . . . Thermodynamic Properties of R-407C . . . . . . . . . . . . . . . . . Pressure-Enthalpy Diagram for Refrigerant 407C (Fig. 2-27) Thermodynamic Properties of R-410A . . . . . . . . . . . . . . . . . Saturated Refrigerant 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 502 . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Refrigerant 507 . . . . . . . . . . Thermodynamic Properties of R-507A . . . . . . . . . . . . . . . . . Saturated Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Seawater . . . . . . . . Saturated Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Sodium (Fig. 2-28) . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sodium Hydroxide at 1 atm (Fig. 2-29) . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Sulfur Dioxide . . . . . . . . . . . Thermodynamic Properties of Sulfur Hexafluoride . . . . . . . Pressure-Enthalpy Diagram for Sulfur Hexafluoride (SF6) (Fig. 2-30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated SUVA AC 9000 . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sulfuric Acid at 1 atm (Fig. 2-31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Toluene . . . . . . . . . . . . . . . . . Saturated Solid/Vapor Water . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Water . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Water Substance along the Melting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Xenon . . . . . . . . . . . . . . . . . . Surface Tension (N/m) of Saturated Liquid Refrigerants . . Surface Tension σ (dyn/cm) of Various Liquids . . . . . . . . . .

2-348 2-350 2-350 2-351 2-351 2-352 2-353 2-354 2-354 2-355 2-355 2-355 2-356 2-356 2-356 2-357 2-359 2-361 2-362 2-365 2-366 2-367 2-369 2-371 2-372 2-374 2-375 2-377 2-379 2-381 2-383 2-384 2-386 2-388 2-388 2-389 2-391 2-393 2-394 2-396 2-396 2-397 2-397 2-397 2-398 2-400 2-400 2-401 2-402 2-403 2-404 2-406 2-408 2-409 2-409 2-410 2-412 2-413 2-416 2-417 2-419 2-419

PHYSICAL AND CHEMICAL DATA TRANSPORT PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-310 Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-311 Velocity of Sound (m/s) in Saturated Liquid Refrigerants . . 2-312 Vapor Viscosity of Inorganic and Organic Substances (Pas) 2-313 Viscosity of Inorganic and Organic Liquids (Pas) . . . . . . . . 2-314 Vapor Thermal Conductivity of Inorganic and Organic Substances [W/(mK)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-315 Thermal Conductivity of Inorganic and Organic Liquids [W/(mK)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-316 Transport Properties of Selected Gases at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-317 Lower and Upper Flammability Limits, Flash Point, and Autoignition Temperature for Selected Hydrocarbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-318 Viscosities of Liquids: Coordinates for Use with Fig. 2-32 . . Nomograph for Viscosities of Liquids at 1 atm (Fig. 2-32) . . Tables 2-319 Viscosity of Sucrose Solutions . . . . . . . . . . . . . . . . . . . . . . . . Nomograph for Thermal Conductivity of Organic Liquids (Fig. 2-33) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-320 Thermal Conductivity Nomograph Coordinates . . . . . . . . . . 2-321 Prandtl Number of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-322 Prandtl Number of Liquid Refrigerants . . . . . . . . . . . . . . . . 2-323 Thermophysical Properties of Miscellaneous Saturated Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-324 Diffusivities of Pairs of Gases and Vapors (1 atm) . . . . . . . . 2-325 Diffusivities in Liquids (25°C) . . . . . . . . . . . . . . . . . . . . . . . . 2-326 Thermal Conductivities of Some Building and Insulating Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-327 Thermal-Conductivity-Temperature for Metals . . . . . . . . . . 2-328 Thermal Conductivity of Chromium Alloys . . . . . . . . . . . . . 2-329 Thermal Conductivity of Some Alloys at High Temperature 2-330 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation . . . . . . . . . . . . . . . . . 2-331 Thermal Conductivities of Insulating Materials at High Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-332 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt) . . . . . . . . . . . . . . . . . . . 2-333 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber) . . . . . . . . . . . . . . . . . . . . . . 2-334 Thermal Diffusivity (m2/s) of Selected Elements . . . . . . . . . 2-335 Thermophysical Properties of Selected Nonmetallic Solid Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-420 2-420 2-420 2-420 2-420 2-421 2-427 2-433 2-439 2-445 2-446 2-448 2-449 2-450 2-450 2-450 2-451 2-451 2-452 2-454 2-456 2-459 2-460 2-461 2-461 2-461 2-461 2-462 2-462 2-462 2-463

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-463 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-464 Classification of Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-467 Theory and Empirical Extension of Theory . . . . . . . . . . . . . . . . . . 2-467 Corresponding States (CS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-467 Group Contributions (GC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-467 Computational Chemistry (CC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-468 Empirical QSPR Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-468 Molecular Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-468 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-468 Critical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-468 Tables 2-336 Ambrose Group Contributions for Critical Constants . . . . . 2-469 2-337 Joback Group Contributions for Critical Constants . . . . . . . 2-470 Normal Melting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-471 Normal Boiling Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-471 2-338 Fedors Method Atomic and Structural Contributions . . . . . 2-471 2-339 First-Order Groups and Their Contributions for Melting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-472 2-340 Second-Order Groups and Their Contributions for Melting Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-472 Characterizing and Correlating Constants . . . . . . . . . . . . . . . . . . . . . 2-473

2-341 Group Contributions for the Nannoolal Method for Normal Boiling Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-342 Intermolecular Interaction Corrections for the Nannoolal et al. Method for Normal Boiling Point . . . . . . . Vapor Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-343 Domalski-Hearing Group Contribution Values for Standard State Thermal Properties . . . . . . . . . . . . . . . . . . Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs’ Energy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Latent Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Sublimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-344 Cs (CH) Group Values for Chickos Estimation of ∆Hfus . 2-345 Ct (Functional) Group Values for Chickos Estimation of ∆Hfus Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-346 Group Contributions and Corrections for ∆Hsub . . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-347 Benson and CHETAH Group Contributions for Ideal Gas Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-348 Liquid Heat Capacity Group Parameters for Ruzicka-Domalski Method . . . . . . . . . . . . . . . . . . . . . . . . . . Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-349 Group Values and Nonlinear Correction Terms for Estimation of Solid Heat Capacity with the Goodman et al. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-350 Element Contributions to Solid Heat Capacity for the Modified Kopp’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-351 Simple Fluid Compressibility Factors Z(0) . . . . . . . . . . . . . . 2-352 Acentric Deviations Z(1) from the Simple Fluid Compressibility Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-353 Constants for the Two Reference Fluids Used in Lee-Kesler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-354 Relationships for Eq. (2-66) for Common Cubic EoS . . . . . Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-355 Reichenberg Group Contribution Values . . . . . . . . . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-356 Group Contributions for the Hsu et al. Method . . . . . . . . . . 2-357 UNIFAC-VISCO Group Interaction Parameters αmn . . . . . . Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-358 Correlation Parameters for Baroncini et al. Method for Estimation of Thermal Conductivity . . . . . . . . . . . . . . . . . . 2-359 Sastri-Rao Group Contributions for Liquid Thermal Conductivity at the Normal Boiling Point . . . . . . . . . . . . . . Liquid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-360 Knotts Group Contributions for the Parachor in Estimating Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flammability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-361 Group Contributions for Pintar Flammability Limits Method for Organic Compounds . . . . . . . . . . . . . . . . . . . . . 2-362 Group Contributions for Pintar Flammability Limits Method for Inorganic Compounds . . . . . . . . . . . . . . . . . . . 2-363 Group Contributions for Pintar Autoignition Temperature Method for Organic Compounds . . . . . . . . . . . . . . . . . . . . . 2-364 Group Contributions for Pintar Autoignition Temperature Method for Inorganic Compounds. . . . . . . . . . . . . . . . . . . .

2-5

2-474 2-476 2-477 2-477 2-478 2-478 2-478 2-479 2-485 2-486 2-486 2-486 2-487 2-488 2-488 2-488 2-489 2-489 2-489 2-490 2-491 2-495 2-496 2-497 2-497 2-497 2-498 2-498 2-500 2-501 2-502 2-502 2-503 2-503 2-504 2-504 2-505 2-506 2-506 2-507 2-509 2-509 2-510 2-510 2-511 2-511 2-512 2-513 2-513 2-514 2-514 2-515 2-516 2-516 2-517 2-517

GENERAL REFERENCES Considerations of reader interest, space availability, the system or systems of units employed, copyright considerations, etc., have all influenced the revision of material in previous editions for the present edition. Reference is made at numerous places to various specialized works and, when appropriate, to more general works. A listing of general works may be useful to readers in need of further information.

ASHRAE Handbook—Fundamentals, SI edition, ASHRAE, Atlanta, 2005; Benedek, P., and F. Olti, Computer-Aided Chemical Thermodynamics of Gases and Liquids, Wiley, New York, 1985; Brule, M. R., L. L. Lee, and K. E. Starling, Chem. Eng., 86, 25, Nov. 19, 1979, pp. 155–164; Cox, J. D., and G. Pilcher, Thermochemistry of Organic and Organometallic Compounds, Academic Press, New York, 1970; Cox, J. D., D. D. Wagman, and V. A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., New York, 1989; Daubert, T. E., R. P. Danner, H. M. Sibel, and C. C. Stebbins, Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, Taylor & Francis, Washington, 1997; Domalski, E. S., and E. D. Hearing, Heat capacities and entropies of organic compounds in the condensed phase, vol. 3, J. Phys. Chem. Ref. Data 25(1):1–525, Jan-Feb 1996; Dykyj, J., and M. Repas, Saturated vapor pressures of organic compounds, Veda, Bratislava, 1979 (Slovak); Dykyj, J., M. Repas, and J. Svoboda, Saturated vapor pressures of organic compounds, Veda, Bratislava, 1984 (Slovak); Glushko, V. P., Ed., Thermal Constants of Compounds, Issues I–X., Moscow, 1965–1982 (Russian only); Gmehling, J., Azeotropic Data, 2 vols., VCH Weinheim, Germany, 1994; Gmehling, J., and U. Onken, Vapor-Liquid Equilibrium Data Collection, Dechema Chemistry Data Series, Frankfurt, 1977–1978; International Data Series, Selected Data on Mixtures, Series A: Thermodynamics Research Center, National Institute of Standards and Technology, Boulder, Colo.; Kaye, S. M., Encyclopedia of Explosives and Related Items, U.S. Army R&D command, Dover, N.J., 1980; King, M. B., Phase Equilibrium in Mixtures, Pergamon, Oxford, 1969; Landolt-Boernstein, Numerical Data and Functional Relationships in Science and Technology (New Series), http://www.springeronline.com/sgw/cda/frontpage/0,11855,4-10113-295859-0,00.html; Lide, D. R., CRC Handbook of Chemistry and Physics, 86th ed., CRC Press, Boca Raton, Fla., 2005; Lyman, W. J., W. F. Reehl, and D. H. Rosenblatt, Handbook of Chemical Property Estimation Methods, McGraw-Hill, New York, 1990; Majer, V., and V. Svoboda, Enthalpies of Vaporization of Organic Compounds: A Critical Review and Data Compilation, Blackwell Science, 1985; Majer V., V. Svoboda, and J. Pick, Heats of Vaporization of Fluids, Elsevier, Amsterdam, 1989 (general discussion); Marsh, K. N., Recommended Reference Materials for the Realization of Physicochemical Properties, Blackwell Science, 1987; NIST-IUPAC Solubility Data Series, Pergamon Press, http://www.iupac.org/publications/ci/1999/march/solubility.html; Ohse, R. W., and H. von Tippelskirch, High Temp.—High Press., 9:367–385, 1977; Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Science Pubs., Oxford, England, 1985; Pedley, J. B., R. D. Naylor, and S. P. Kirby, Thermochemical Data of Organic Compounds, Chapman and Hall, New York, 1986; Physical Property Data for the Design Engineer, Hemisphere, New York, 1989; Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of

2-6

Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001; Rothman, D, et al., Max Planck Inst. f. Stromungsforschung, Ber 6, 1978; Smith, B. D., and R. Srivastava, Thermodynamic Data for Pure Compounds, Part A: Hydrocarbons and Ketones, Elsevier, Amsterdam, 1986, Physical sciences data 25, http://www.elsevier.com/wps/find/bookseriesdescription.librarians/BS_PSD/description; Sterbacek, Z., B. Biskup, and P. Tausk, Calculation of Properties Using Corresponding States Methods, Elsevier, Amsterdam, 1979; Stull, D. R., E. F. Westrum, and G. C. Sink, The Chemical Thermodynamics of Organic Compounds, Wiley, New York, 1969; TRC Thermodynamic Tables—Hydrocarbons, Thermodynamics Research Center, National Institute of Standards and Technology, Boulder, Colo.; TRC Thermodynamic Tables—NonHydrocarbons, Thermodynamics Research Center, National Institute of Standards and Technology, Boulder, Colo.; Young, D. A., “Phase Diagrams of the Elements,” UCRL Rep. 51902, 1975 republished in expanded form by the University of California Press, 1991; Zabransky, M., V. Ruzicka, Jr., V. Majer, and E. S. Domalski, Heat Capacity of Liquids: Critical Review and Recommended Values, J. Phy. Chem. Ref. Data, Monograph No. 6, 1996. CRITICAL DATA ARE COMPILED IN:

Ambrose, D., “Vapor-Liquid Critical Properties,” N. P. L. Teddington, Middlesex, Rep. 107, 1980; Kudchaker, A. P., G. H. Alani, and B. J. Zwolinski, Chem. Revs. 68:659–735, 1968; Matthews, J. F., Chem. Revs. 72:71–100, 1972; Simmrock, K., R. Janowsky, and A. Ohnsorge, Critical Data of Pure Substances, Parts 1 and 2, Dechema Chemistry Data Series, 1986; Other recent references for critical data can be found in Lide, D. R., CRC Handbook of Chemistry and Physics, 86th ed., CRC Press, Boca Raton, Fla., 2005. PUBLICATIONS ON THERMOCHEMISTRY

Pedley, J. B., Thermochemical Data and Structures of Organic Compounds, 1, Thermodynamic Research Center, Texas A&M Univ., 1994 (976 pp., 3000 cpds.); Frenkel, M., et al., Thermodynamics of Organic Compounds in the Gas State, 2 vols., Thermodynamic Research Center, Texas A&M Univ., 1994 (1825 pp., 2000 cpds.); Barin, I., Thermochemical Data of Pure Substances, 2 vols., 2d ed., VCH Weinheim, Germany, 1993 (1834 pp., 2400 substances); Gurvich, L.V., et al., Thermodynamic Properties of Individual Substances, 3 vols., 4th ed., Hemisphere, New York, 1989, 1990, and 1993 (2520 pp.); Lide, D. R., and G. W. A. Milne, Handbook of Data on Organic Compounds, 7 vols., 3d ed., Chemical Rubber, Miami, 1993 (7000 pp.); Daubert, T. E., et al., Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, extant 1995, Taylor & Francis, Bristol, Pa., 1995; Database 11, NIST, Gaithersburg, Md. U.S. Bureau of Mines publications include Bulletins 584, 1960 (232 pp.); 592, 1961 (149 pp.); 595, 1961 (68 pp.); 654, 1970 (26 pp.); Chase, M. W., et al., JANAF Thermochemical Tables, 3d ed., J. Phys. Chem. Ref. Data 14 suppl 1., 1986 (1896 pp.); Journal of Physical and Chemical Reference Data is available online at http://listserv. nd.edu/cgi-bin/wa?A2=ind0501&L=pamnet&F=&S=&P=8490 and at http://www.nist.gov/srd/reprints.htm

PHYSICAL PROPERTIES OF PURE SUBSTANCES TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds* Abbreviations Used in the Table

a., acid A., specific gravity with reference to air = 1 abs., absolute ac., acetic acid act., acetone al., 95 percent ethyl alcohol alk, alkali (i.e., aq. NaOH or KOH) am., amyl (C5H11) amor., amorphous anh., anhydrous aq., aqueous or water aq. reg., aqua regia

atm., atmosphere or 760 mm. of mercury pressure bk., black brn., brown bz., benzene c., cold cb., cubic cc, cubic centimeter chl., chloroform col., colorless or white conc., concentrated cr., crystals or crystalline d., decomposes D., specific gravity with reference to hydrogen = 1

d. 50, decomposes at 50°C; 50 d., melts at 50°C with decomposition delq., deliquescent dil., dilute dk., dark eff., effloresces or efflorescent et., ethyl ether expl., explodes gel., gelatinous gly., glycerol (glycerin) gn., green h., hot hex., hexagonal

hyg., hygroscopic i., insoluble ign., ignites lq., liquid lt., light m. al., methyl alcohol mn., monoclinic nd., needles NH3, liquid ammonia NH4OH, ammonium hydroxide solution oct., octahedral or., orange pd., powder

Formula weights are based upon the International Atomic Weights in “Atomic Weights of the Elements 2001,” Pure Appl. Chem., 75, 1107, 2003, and are computed to the nearest hundredth. Refractive index, where given for a uniaxial crystal, is for the ordinary (ω) ray; where given for a biaxial crystal, the index given is for the median (β) value. Unless otherwise specified, the index is given for the sodium D-line (λ = 589.3 mµ). Specific gravity values are given at room temperatures (15 to 20 °C) unless otherwise indicated by the small figures which follow the value: thus, “5.6 18° 4 ” indicates a specific gravity of 5.6 for the substance at 18 °C referred to water at 4°C. In this table the values for the specific gravity of gases are given with reference to air (A) = 1, or hydrogen (D) = 1. Melting point is recorded in a certain case as “82 d.” and in some other case as “d. 82,” the distinction being made in this manner to indicate that the former is a melting point with decomposition at 82°C, while in the latter decomposition only occurs at 82 °C. Where a value such as “−2H2O, 82” is given it indicates loss of 2 moles of water per formula weight of the compound at a temperature of 82 °C. Boiling point is given at atmospheric pressure (760 mm. of mercury) unless otherwise indicated; thus, “8215 mm.” indicates the boiling point is 82°C when the pressure is 15 mm.

Name Aluminum acetate, normal acetate, basic bromide bromide carbide chloride chloride fluoride (fluellite) fluoride hydroxide nitrate nitride oxide oxide (corundum) phosphate

Formula

Formula weight

Color, crystalline form and refractive index

Al Al(C2H3O2)3 Al(OH)(C2H3O2)2 AlBr3 AlBr36H2O Al4C3 AlCl3

26.98 204.11 162.08 266.69 374.78 143.96 133.34

silv., cb. wh. pd. wh., amor. trig. col., delq. cr. yel., hex., 2.70 wh., delq., hex.

AlCl3·6H2O AlF3H2O Al2F67H2O Al(OH)3 Al(NO3)39H2O Al2N2 Al2O3 Al2O3 AlPO4

241.43 101.99 294.06 78.00 375.13 81.98 101.96 101.96 121.95

col., delq., trig., 1.560 col., rhb., 1.490 wh., cr. pd. wh., mn. rhb., delq. yel., hex. col., hex., 1.67–8 wh., trig., 1.768 col., hex.

pl., plates pr., prisms or prismatic pyr., pyridine rhb., rhombic (orthorhombic) s., soluble satd., saturated sl., slightly soln., solution subl., sublimes sulf., sulfides tart. a., tartaric acid tet., tetragonal tr., transition tri., triclinic

trig., trigonal v., very vac., in vacuo vl., violet volt., volatile or volatilizes wh., white yel., yellow ∞, soluble in all proportions <, less than >, greater than 42, about or near 42 −3H2O, 100, loses 3 moles of water per formula weight at 100°C

Solubility is given in parts by weight (of the formula shown at the extreme left) per 100 parts by weight of the solvent; the small superscript indicates the temperature. In the case of gases the solubility is often expressed in some manner as “510° cc” which indicates that at 10 °C, 5 cc. of the gas are soluble in 100 g of the solvent. The symbols of the common mineral acids: H2SO4, HNO3, HCl, etc., represent dilute aqueous solutions of these acids. See also special tables on Solubility. REFERENCES: The information given in this table has been collected mainly from the following sources: Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry, Longmans, New York, 1922. Abegg, Handbuch der anorganischen Chemie, S. Hirzel, Leipzig, 1905. Gmelin-Kraut, Handbuch der anorganischen Chemie, 7th ed., Carl Winter, Heidelberg; 8th ed., Verlag Chemie, Berlin, 1924. Friend, Textbook of Inorganic Chemistry, Griffin, London, 1914. Winchell, Microscopic Character of Artificial Inorganic Solid Substances or Artificial Minerals, Wiley, New York, 1931. International Critical Tables, McGraw-Hill, New York, 1926. Tables annuelles internationales de constants et donnes numeriques, McGraw-Hill, New York. Annual Tables of Physical Constants and Numerical Data, National Research Council, Princeton, N.J., 1943. Comey and Hahn, A Dictionary of Chemical Solubilities, Macmillan, New York, 1921. Seidell, Solubilities of Inorganic and Metal Organic Compounds, Van Nostrand, New York, 1940.

Specific gravity 2.7020° 3.01 25° 4 2.95 2.44 25° 4 2.17 2.42 3.05 3.99 4.00 2.59

25° 4

Melting point, °C 660 d. 200 d. 97.5 d. 100 d. >2200 1945.2atm. d. −4H2O, 120 −2H2O, 300 73 21504atm. 1999 to 2032 1999 to 2032

Boiling point, °C 2056 268 752mm

182.7 ; subl. 178 −6H2O, 250 d. 134 d. >1400 2210

Solubility in 100 parts Cold water i. s. i. s. s. d. to CH4 69.8715° 400 sl. s. i. 0.00010418° v. s. d. slowly i. i. i.

Hot water i. d.

Other reagents s. HCl, H2SO4, alk.

s. d.

s.a.; i. NH4 salts s.al., act., CS2 s. al., CS2 s. a.; i. act. s. et., chl., CCl4; i. bz.

v. s.

50 al.; s. et.

s.

sl. s. i. v. s. d. i. i. i.

2-7

*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Inorganic Compounds in Lange’s Handbook of Chemistry.

s. a., alk.; i. a. s. al., CS2 s. alk. d. v. sl. s. a., alk. v. sl. s. a., alk. s. a., alk.; i. ac.

2-8

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued)

Name Aluminum (Cont.) potassium silicate (muscovite)

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

Boiling point, °C

Solubility in 100 parts Cold water

3Al2O3K2O6SiO2· 2H2O Al2O3K2O6SiO2 AlK(C4H4O6)2 AlF33NaF Al2O3Na2O6SiO2 Al2(SO4)3 Al2(SO4)3(NH4)2SO4 24H2O Cr2(SO4)3(NH4)2SO4 24H2O Fe2(SO4)3(NH4)2SO4 24H2O Al2(SO4)3K2SO4 24H2O Cr2(SO4)3K2SO4 24H2O Al2(SO4)3Na2SO4 24H2O NH3

796.61

mn., 1.590

2.9

d.

i.

556.66 362.22 209.94 524.44 342.15 906.66

col., mn., 1.524 col. wh., mn., 1.3389 col., tri., 1.529 wh. cr. col., oct., 1.4594

2.56

1450 (1150)

2.90 2.61 2.71 1.64 20° 4

1000 1100 d. 770 93.5

i. s. sl. s. i. 31.30° 3.90°

17.03

col. gas, 1.325 (lq.)

77.08 337.09 79.06 97.94 114.10 157.13

wh., hyg. cr. pl. mn. or rhb., 1.5358 col., cb., 1.7108 col. pl. wh. cr.

272.21

wh.

chloride (salammoniac) chloroplatinate chloroplatinite chlorostannate chromate cyanide dichromate ferrocyanide fluoride fluoride, acid formate

NH4C2H3O2 NH4CNAu(CN)3H2O NH4HCO3 NH4Br (NH4)2CO3H2O NH4HCO3 NH2CO2NH4‡ (NH4)2CO3 2NH4HCO3H2O NH4Cl (NH4)2PtCl6 (NH4)2PtCl4 (NH4)2SnCl6 (NH4)2CrO4 NH4CN (NH4)2Cr2O7 (NH4)4Fe(CN)66H2O NH4F NH4FHF HCO2NH4

53.49 443.87 372.97 367.50 152.07 44.06 252.06 392.19 37.04 57.04 63.06

wh., cb., 1.639, 1.6426 yel., cb. tet. pink., cb. yel., mn. col., cb. or., mn. mn. wh., hex. wh., rhb., 1.390 col., mn., delq.

hydrosulfide hydroxide molybdate molybdate, heptanitrate (α), stable −16° to 32° nitrate (β), stable 32° to 84°

NH4HS NH4OH (NH4)2MoO4 (NH4)6Mo7O244H2O‡ NH4NO3 NH4NO3

51.11 35.05 196.01 1235.86 80.04 80.04

col., rhb. in soln. only mn. col., mn. col., tet., 1.611 col., rhb. or mn.

nitrite osmochloride oxalate oxalate, acid perchlorate persulfate phosphate, monobasic

NH4NO2 (NH4)2OsCl6 (NH4)2C2O4H2O NH4HC2O4H2O NH4ClO4 (NH4)2S2O8 NH4H2PO4

64.04 439.02 142.11 125.08 117.49 228.20 115.03

wh. nd. cb. col., rhb. col., trimetric col., rhb., 1.4833 wh., mn., 1.5016 col., tet., 1.5246

potassium silicate (orthoclase) Aluminum potassium tartrate sodium fluoride (cryolite) sodium silicate sulfate Alum, ammonium (tschermigite) ammonium chrome ammonium iron potassium (kalinite) potassium chrome sodium Ammonia† Ammonium acetate auricyanide bicarbonate bromide carbonate carbonate, carbamate carbonate, sesqui-

956.69 964.38

gn. or vl., oct., 1.4842

1.72

100 d.

vl., oct., 1.485

1.71

948.78

col., mn., 1.4564

1.76 26° 4

92

998.81

red or gn., cb., 1.4814

1.83

89

916.56

col., oct., 1.4388

−20H2O, 120; −24H2O, 200

1.675

−79°

0.817 0.5971 (A) 1.073 1.573 2.327 15° 4

2.4 1.91712° 0.79100° (A) 2.15 12° 2.21 12 1.266

25°

21.2

5.7 °

i. al. s. al. i. al.

93°

50 0°

61

i. HCl d. a.

∞

0

20

i. al. 45°

106.4

121.7

0°

96°

i. al.

−77.7

−33.4

89.9

7.4

14.820° al.; s. et.

114 d. 200 d. 35–60 subl. 542 d. 58 subl.

d.

1484° s. 11.90° 6810° 10015° 2515°

v. s. 2730° 145.6100°

s. al.; sl. s. act. i. al. i. al. s. al., et., act. i. al., CS2, NH3

2015°

5049°

29.40° 0.715° s. 33.315° 40.530° s. 47.230° s. v. s. v. s. 1020°

77.3100° 1.25100° v. s.

s. NH3; sl. s. al., m. al. 0.005 al.

d. v. s. v. s. d.

sl. s. act., NH3; i. al. s. al. s. al.; i. act. i. al. s. al.; i. NH3

53180°

s. al.

d. 1.5317° 3.065

i. 89100° ∞ 100°

124 −18H2O, 64.5

Other reagents

s.

25°

40

20° 4

Hot water

d. 350 d. d.

subl. 520

d. 180 36 d. 185 d. 114–116 d.

2.27

d.

1.66 25° 4 1.725 25° 4

169.6

1.69 2.93 20° 4 1.501 1.556 1.95 1.98 1.803 19° 4

expl.

d. 180; subl. in vac. subl. 120

d. 210 d. 210

v. s. s. d. 4425° 118.30° 365.835°

241.830° 58080°

s.

d. 0°

d. d. d. 120

6765°

2.5 s. 10.90° 58.20° 22.70°

s. al. d.

11.850° 100°

46.9 d. 173.2100°

i. al., NH3 i. al. 3.820° al., 17.120° m. al.; v. s. NH3 s. al. sl. s. al.; i. NH3 220° al.; s. act.; i. et. i. ac.

phosphate, dibasic phosphate, metaAmmonium phosphomolybdate silicofluoride sulfamate sulfate (mascagnite) sulfate, acid sulfide sulfide, pentasulfite sulfite, acid tartrate thiocyanate vanadate, metaAntimony

(NH4)2HPO4 (NH4)4P4O12 (NH4)3PO412MoO3 3H2O (?) (NH4)2SiF6 NH4SO3NH2 (NH4)2SO4 NH4HSO4 (NH4)2S (NH4)2S5 (NH4)2SO3H2O NH4HSO3 (NH4)2C4H4O6 NH4CNS NH4VO3 Sb

132.06 388.04 1930.39 178.15 114.12 132.14 115.11 68.14 196.40 134.16 99.11 184.15 76.12 116.98 121.76

col., mn., 1.53 col., mn. yel.

13115° s. 0.0315°

1.619 2.21 d. 2.01

i.

s. alk.; i. al., HNO3

55.5 35750° 103.3100°

s. al.; i. act.

1380

18.5 1340° 70.60° 100 v. s. s. 10012° s. 450° 1200° 0.4418° i.

∞72°

1.41 2.03 12° 4 1.60 1.305 2.326 6.68425°

d. d. d. 149.6 d. 630.5 73.4

220.2

601.60°

1570

v. sl. s.

1.78

132 235 d. 146.9 d.

subl. d. 160

17.5°

cb., 1.3696 col. pl. col., rhb., 1.5230 col., rhb., 1.480 yel.-wh. or.-red pr. col., mn. rhb. col., mn. col., mn., 1.685 col. cr. tin wh., trig.

1.769 20° 4

i. act.

490

d. 170

i. al., act., CS2 v. sl. s. al.; i. act. 12025° NH3 i. al., act.

60°

87 17020° 3.0570° i.

sl. s. al. s. al., act., NH3, SO2 i. al., NH4Cl s. aq. reg., h. conc. H2SO4 s. al., HCl, HBr, H2C4H4O6 s. HCl, KOH, H2C4H4O6

chloride, tri- (butter of antimony)* oxide, tri- (valentinite) oxide, tri- (senarmontite) sulfide, tri- (stibnite)

SbCl3

228.12

col., rhb., delq.

3.14 20° 4

Sb2O3 Sb2O3 Sb2S3

291.52 291.52 339.72

rhb., 2.35 cb., 2.087 bk., rhb., 4.046

5.67 5.2 4.64

656 652 550

0.00017

d.

sulfide, pentatelluride, triAntimonyl potassium tartrate (tartar emetic) sulfate, normal sulfate, basic Argon

Sb2S5 Sb2Te3

403.85 626.32

golden gray

4.1200°

−2S, 135 629

i.

i.

(SbO)KC4H4O6aH2O (SbO)2SO4 (SbO)2SO4Sb2(OH)4 Ar

333.94 371.58 683.20 39.95

wh., rhb. wh. pd. wh. pd. col. gas

2.60 4.89

−aH2O, 100 −189.2

−185.7

35.7100° d. d. 2.2350° cc

s. gly.; i. al.

Arsenic (crystalline) (α) Arsenic (black) (β)

As4 As4

299.69 299.69

met., hex. bk., amor.

1.65−288°; 1.402−185.7°; 1.38 (A) 5.72714° 4.720°

5.268.7° d. i. 5.60° cc

81436atm.

subl. 615

i. i.

i. i.

Arsenic (yellow)(γ) acid, orthoacid, metaacid, pyropentoxide sulfide, di- (realgar)

As4 H3AsO4aH2O HAsO3 H4As2O7 As2O5 As2S2

299.69 150.95 123.93 265.87 229.84 213.97

yel., cb. col., hyg. wh., hyg. col. wh., amor. red, mn., 2.68

2.020° 2.0–2.5

s. HNO3 s. HNO3, aq. reg., aq. Cl2, h. alk.

d. 358 35.5 d. d. 206

−H2O, 160

50 H3AsO4 H3AsO4 76.7100° d.

s. alk.

d. 565

16.7 d. to form d. to form 59.50° i.

s. alk., al. s. K2S, NaHCO3

sulfide, pentaArsenious chloride (butter of arsenic) hydride (arsine) oxide (arsenolite) oxide (claudetite) oxide

As2S5

310.17

yel.

d. 500

0.0001360°

i.

s. HNO3, alk.

AsCl3 AsH3 As2O3 As2O3 As2O3

181.28 77.95 197.84 197.84 197.84

oily lq. col. gas col., cb., fibrous, 1.755 col., mn., 1.92 amor. or vitreous

130 −55; d. 230

d. 20 cc sl. s. sl. s. 1.210°

d. sl. s. sl. s. sl. s. 2.9340°

Auric chloride

AuCl32H2O

339.36

or. cr.

v. s.

v. s.

cyanide Aurous chloride cyanide Cf. also under Gold Barium acetate acetate bromide

Au(CN)36H2O AuCl AuCN

383.11 232.42 222.98

yel. cr. yel. cr.

7.4

v. s. d. i.

v. s. d. i.

s. HCl, HBr, PCl3 sl. s. alk. i. al., et. i. al., et. s. HCl, alk., Na2CO3; i. al., et. s. HCl, al., et.; sl. s. NH3 s. al. s. HCl, HBr; d. al. s. KCN; i. al., et.

Ba Ba(C2H3O2)2 Ba(C2H3O2)2H2O BaBr2

137.33 255.42 273.43 297.14

silv. met. col. wh., tri. pr., 1.517 col.

3.5 2.468 2.19 4.781 24° 4

d. 58.80° 7530°(anh.) 980°

d. 75.0100° 7940°(anh.) 149100°

2-9

*Usually the solution. †See special tables. ‡Usual commercial form.

4.086 (α)3.50619°; (β)3.25419° lq. 2.163 2.695 (A) 3.865 25° 4 3.85 3.738

(α)tr. 267; (β)307 −18 −113.5 subl. subl. 315

sl. s. 18°

d. d. 50 AuCl3, 170 d.

d. 290

850

1140

−H2O, 41 847

d.

s. HCl; alk., NH4HS, K2S; i. ac. s. HCl, alk., NH4HS

5.1515° gly. 2425° cc al.

s. a.; d. al. i. al. v. s. m. al.; v. sl. s. act.

2-10

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued)

Name Barium (Cont.) bromide carbonate (witherite) carbonate (α) carbonate (β) Barium chlorate chlorate chloride chloride chloride hydroxide hydroxide nitrate (nitrobarite) oxalate oxide peroxide peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, pyrosilicofluoride sulfate (barite, barytes) sulfide, monosulfide, trisulfide, tetraBeryllium (glucinum) Bismuth carbonate, subchloride, dichloride, trinitrate nitrate, suboxide, trioxide, trioxide, trioxychloride

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

BaBr22H2O BaCO3 BaCO3 BaCO3 Ba(ClO3)2 Ba(ClO3)2H2O* BaCl2 BaCl2 BaCl22H2O† Ba(OH)2 Ba(OH)28H2O Ba(NO3)2 BaC2O4 BaO

333.17 197.34 197.34 197.34 304.23 322.24 208.23 208.23 244.26 171.34 315.46 261.34 225.35 153.33

col., mn., 1.7266 wh., rhb., 1.676 wh., hex. wh. col. col., mn., 1.577 col., mn., 1.7361 col., cb. col., mn., 1.646 col., mn. col., mn., 1.5017 col., cb., 1.572 wh. cr. col., cb., 1.98

BaO2* BaO28H2O BaH4(PO4)2 BaHPO4 Ba3(PO4)2 Ba2P2O7 BaSiF6 BaSO4

169.33 313.45 331.30 233.31 601.92 448.60 279.40 233.39

gray or wh. pd. pearly sc. tri. wh., rhb. nd., 1.635 wh., cb. wh., rhb. pr. col., rhb., 1.636

BaS BaS3 BaS42H2O Be(Gl) Bi

169.39 233.52 301.62 9.01 208.98

Bi2O3CO2H2O BiCl2(?) BiCl3* Bi(NO3)35H2O BiONO3H2O Bi2O3 Bi2O3 Bi2O3 BiOCl

527.98 279.89 315.34 485.07 305.00 465.96 465.96 465.96 260.43

col., cb., 2.155 yel.-gn. red, rhb. gray, met., hex. silv. wh. or reddish, hex. wh. pd. bk. nd. wh. cr. col., tri. hex. pl. yel., rhb. yel., tet. yel., cb. wh., amor.

6.86 4.86 4.75 2.82 4.92815° 8.9 8.55 8.20 7.7215°

d. 163 230 d. 30 d. 260 820 860 tr. 704

3.69 4.29

3.179 3.856 24° 4 3.097 24° 4 4.495 16° 2.188 3.24428° 2.658 5.72 4.958 4°

2.9 4.16515° 4.116° 3.920° 4.27915° 4.49915°

Boiling point, °C

−2H2O, 100 tr. 811 to α tr. 982 to β 174090atm 414 d. 120 tr. 925 962 −2H2O, 100

d. d. 1450

77.9 592

−8H2O, 550 d.

1923

2.988 1.816 9.8020°

d. 400 d. 200 1284 271

Other reagents

v. s. 0.0065100°

s. al. s. a.; i. al.

0.002218° 20.350° s. 310°

0.0065100° 84.880° s. 59100°

s. a.; i. al.

76.8100° 101.480°

sl. s. HCl, HNO3; i. al.

2000

39.30° 1.670° 5.615° 5.00° 0.00168° 1.50°

d. d. d.

tr. to mn. 1149

v. sl. s. 0.168 d. 0.015 i. 0.01 0.02617° 0.0001150°

0.09100° 0.00028530°

2767 1450

d. s. 4115° i. i.

d. s. v. s. sl. s. d. i.

i. d. d. d. i. i. i. i. sl. s.

i.

i. i. i. i. sl. s.

2.660°

40.2100°

1560 1560

4.2515° 20°

Hot water

v. s. 0.002218°

−O, 800 −8H2O, 100

1580 d.

Solubility in 100 parts Cold water

d. 300 447 −5H2O, 80 1900

34.2100° 0.002424° 90.880°

sl. s. al., act. sl. s. HCl, HNO3; i. al.

v. sl. s. al.; i. et. sl. s. a.; i. al. s. a., NH4Cl; i. al. s. HCl, HNO3, abs. al.; i. NH3, act. s. dil. a.; i. act. s. dil. a.; i. al., et., act. s. a. s. a., NH4 salts s. a. s. a., NH4 salts sl. s. HCl, NH4Cl; i. al. s. conc. H2SO4; 0.006, 3% HCl d. HCl; i. al. i. al., CS2 s. dil. a., alk. s. aq. reg., conc. H2SO4, HNO3 s. a. s. al. 4219° act.; s. a.; i. al. s. a. s. a. s. a. s. a. s. a.; i. act., NH3, H2C4H4O6 22.220° gly., 0.2425° et.; s. al. s. HNO3; i. al.

Boric acid

H3BO3

61.83

wh., tri.

1.43515°

185 d.

Boron

B

10.81

2.32

2300

2550

i.

i.

carbide oxide oxide (sassolite) Bromic acid Bromine

B4C B2O3 B2O33H2O HBrO3 Br2

55.25 69.62 123.67 128.91 159.81

gray or bk., amor. or mn. bk. cr. col. glass, 1.459 tri., 1.456 col.; in soln. only rhb., or red lq.

2.54 1.85 1.49

2450 577 d. d. 100 −7.2

>3500 >1500

i. 1.10° sl. s. v. s. 4.220°

i. 15.7100° s. d. 3.1330°

i. a. s. a., al., gly.

hydrate Cadmium acetate acetate carbonate

Br210H2O Cd Cd(C2H3O2)2 Cd(C2H3O2)22H2O* CdCO3

339.96 112.41 230.50 266.53 172.42

red, oct. silv. met., hex. col. col., mn. wh., trig.

8.6520° 2.341 2.01 4.2584°

d. 6.8 320.9 256 −H2O, 130 d. <500

i.

wh., cb.

4.047 25° 4

568

s. a., NH4NO3 s. m. al. s. al. s. a., KCN, NH4 salts; i. NH3 1.5215° al.; i. et., act.

chloride

CdCl2

183.32

3.11920°; 5.87 (A)

58.78 767 d.

960

s. i. v. s. v. s. i. 900°

i. 147100°

s. al., et., alk., CS2

chloride cyanide hydroxide nitrate nitrate oxide oxide oxide, subCadmium sulfate sulfate sulfate sulfate sulfate sulfide (greenockite) Calcium acetate aluminate aluminum silicate (anorthite) arsenate bromide carbonate (aragonite) carbonate (calcite) chloride (hydrophilite) chloride chloride citrate cyanamide ferrocyanide fluoride (fluorite) formate hydride hydroxide hypochlorite hypophosphate lactate magnesium carbonate (dolomite) magnesium silicate (diopside) nitrate (nitrocalcite) nitrate nitride nitrite oxalate oxalate oxide

CdCl2 2aH2O Cd(CN)2 Cd(OH)2 Cd(NO3)2 Cd(NO3)24H2O* CdO CdO Cd2O CdSO4 CdSO4H2O 3CdSO48H2O* CdSO44H2O CdSO47H2O CdS Ca Ca(C2H2O2)2H2O Ca(AlO2)2 CaOAl2O32SiO2 Ca3(AsO4)2 CaBr2 CaCO3 CaCO3 CaCl2* CaCl2H2O CaCl26H2O Ca3(C6H5O7)24H2O CaCN2 Ca2Fe(CN)612H2O CaF2 Ca(HCO2)2 CaH2 Ca(OH)2 Ca(ClO)24H2O Ca2P2O62H2O Ca(C3H5O3)25H2O

228.36 164.45 146.43 236.42 308.48 128.41 128.41 240.82 208.47 226.49 769.54 280.53 334.58 144.48 40.08 176.18 158.04 278.21 398.07 199.89 100.09 100.09 110.98 129.00 219.08 570.49 80.10 508.29 78.07 130.11 42.09 74.09 215.04 274.13 308.29

col., mn., 1.6513

3.327

wh., trig. col. col. nd. brn., cb. brn., amor, 2.49 gn., amor. rhb. mn. col., mn., 1.565 col. mn. yel.-or., hex., 2.506 silv. met., cb. wh. nd. col., rhb. or mn. tri., 1.5832 wh. pd. delq. nd. col., rhb., 1.6809 col., hex., 1.550 wh., delq., cb, 1.52 col., delq. col., trig., 1.417 col. nd. col., rhombohedral yel., tri., 1.5818 wh., cb., 1.4339 col., rhb. wh. cr. or pd. col., hex., 1.574 wh., feathery cr. granular col., eff.

4.79 15° 4

CaOMgO2CO2 CaOMgO2SiO2 Ca(NO3)2 Ca(NO3)24H2O* Ca3N2 Ca(NO2)2H2O CaC2O4 CaC2O4H2O CaO

184.40 216.55 164.09 236.15 148.25 150.10 128.10 146.11 56.08

trig., 1.68174 wh., mn. col., cb. col., mn., 1.498 brn. cr. delq., hex. col., cb. col. col., cb., 1.837

peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyro- (brushite) phosphide silicate (α) (pseudowollastonite)

CaO28H2O CaH4(PO4)2H2O CaHPO42H2O Ca3(PO4)2 Ca(PO3)2 Ca2P2O7 Ca2P2O75H2O Ca3P2 CaSiO3

216.20 252.07 172.09 310.18 198.02 254.10 344.18 182.18 116.16

silicate (β) (wollastonite) sulfate (anhydrite)

CaSiO3 CaSO4

116.16 136.14

pearly, tet. wh., tri. wh., mn. pl. wh., amor. wh., tet., 1.588 col., biaxial, 1.60 wh., mn. red cr. col., pseudo hex., 1.6150 or mn.(?) col., mn., 1.610 col., rhb., 1.576, or mn, 1.50

2-11

*Usual commercial form. †The solubility of CaCO3 in H2O is greatly increased by increasing the amount of CO2 in the H2O.

2.455 17° 4 8.15 6.95 8.192 18° 4 4.691 24° 4 3.78620° 3.09 3.05 2.48 20° 4 4.58 20° 1.55

tr. 34 d. >200 d. 300 350 59.4

132

d. 900–1000 d. 1000 tr. 108 tr. 41.5

3.6720° 2.765

tr. 4 1750100atm 810 d. 1600 1551

3.353 25° 4 2.93 25° 2.711 4 2.152 15° 4

760 d. 825 1339103atm. 772

1810

1.6817°

29.92 −2H2O, 130

−6H2O, 200 −4H2O, 185

1.7 3.18020° 2.015 1.7 2.2

2.872 3.3 2.36 1.82 2.6317° 2.2334° 2.24° 2.2 3.32 2.220 16° 4 2.306 16° 4 3.14 2.82 3.09 2.25 2.5115° 2.905 2.915 2.96

subl. in N2, 980 1200  30

>1600

1330 d. d. 675 −H2O, 580 d. −2H2O, 200 −3H2O, 100

−8H2O, 100 −H2O, 100 d. 1670 975 1230

2850 expl. 275 d. 200

>1600 1540 tr. 1190 to α 1450(mn.)

180100°

76.50° s. 114.20° s. 350−5° 0.000001 d. 520° d.

60.8100° s. 127.660° s.

0.01325° 1250° 0.001220°† 0.001425° 59.50° s. v. s. 0.08518° s. d. s. 0.001618° 16.10° d. 0.1850° delq.; d. i. 10.5

32659.5° i. i.

Colloidal d. 45.580° i. 312105° 0.002100° 0.002100° 347260° s. v. s. 0.09626° d. 15090° 0.001726° 18.4100 0.077100° d. ∞

2.0515° m. al. s. a.; NH4OH, KCN s. a., NH4 salts; i. alk. v. s. a. s. al., NH3; i. HNO3 s. a., NH4 salts; i. alk. s. a., NH4 salts; i. alk. d. a., alk. i.act., NH3 i. al. i. al. i. al. s. a.; v. s. NH4OH s, a.; sl. s. al. sl. s. al. s. HCl s. dil. a. s. al., act.; sl. s. NH3 s. a., NH4Cl s. a., NH4Cl s. al. s. al. s. al. 0.006518° al. i. al. sl. s. a. i. al., et. d. a.; i. bz. s. NH4Cl d. a. s. HCl, H4P2O6 ∞h. al.; i. et.

18°

d. 730–760 1391 561 42.7 900 d. −H2O, 200 2570

16820° 0.024718° 0.0002625° 109.70° 2150° i. i.

tr. 1193 to rhb.

0.032 i. 1020° 2660° d. 770° 0.0006713° i. Forms Ca(OH)2 sl. s. 0.0224.5° 0.0025 i. i. sl. s. d. 0.009517° 0.29820°

i. 376151° v. s. d. 41790° 0.001495° i. d. d. 0.075100° d. i.

0.1619100°

1415° al.; s. amyl al., NH3 s. dil. a.; i. abs. al. s. 90% al. s. a.; i. ac. s. a.; i. ac s. a.; i. al. s. a. d.; i. al., et. s. a.; i. al., ac. i. a. s. a. s. a.; i. NH4Cl s. dil. a.; i. al., et. s. HCl s. a., Na2S2O3, NH4 salts

Physical Properties of the Elements and Inorganic Compounds (Continued)

TABLE 2-1

2-12

Name Calcium (Cont.) sulfate (gypsum) sulfhydrate sulfide (oldhamite) sulfite tartrate thiocyanate thiosulfate tungstate (scheelite) Carbon, cf. table of organic compounds Carbon, amorphous Carbon, diamond Carbon, graphite dioxide disulfide

Formula

172.17

col., mn., 1.5226

Ca(SH)26H2O CaS CaSO32H2O CaC4H4O64H2O Ca(CNS)23H2O CaS2O36H2O CaWO4

214.32 72.14 156.17 260.21 210.29 260.30 287.92

col. pr. col., cb. wh., cr., 1.595 col., rhb. wh., delq. cr. col., tri., 1.56 wh., tet., 1.9200

C C C CO2

12.01 12.01 12.01 44.01

bk., amor. col., cb., 2.4195 bk., hex. col. gas

CS2

76.14

col. lq.

CO

oxychloride (phosgene) oxysulfide

COCl2 COS C3O2 CSCl2 2CeO23H2O Ce(OH)(NO3)33H2O CeO2 Ce(SO4)24H2O Ce

28.01

98.92 60.08 68.03 114.98 398.28 397.18 172.11 404.30 140.12

Cerous sulfate sulfate Cesium Chloric acid Chlorine

Ce2(SO4)3 Ce2(SO4)38H2O Cs HClO37H2O Cl2

568.42 712.54 132.91 210.57 70.91

hydrate Chloroplatinic acid Chlorostannic acid Chlorosulfonic acid Chromic acetate chloride chloride fluoride hydroxide

Cl28H2O H2PtCl66H2O H2SnCl66H2O HOSO2Cl Cr2(C2H3O2)62H2O CrCl3 CrCl36H2O* CrF3 Cr(OH)3

215.03 517.90 441.54 116.52 494.29 158.36 266.45 108.99 103.02

Cr(OH)32H2O Cr(NO3)39H2O* Cr(NO3)37aH2O Cr2O3 Cr2(SO4)3 Cr2(SO4)35H2O Cr2(SO4)315H2O Cr2(SO4)318H2O Cr2S3

139.05 400.15 373.13 151.99 392.18 482.26 662.41 716.46 200.19

hydroxide nitrate nitrate oxide sulfate sulfate sulfate sulfate sulfide

Color, crystalline form and refractive index

CaSO42H2O

monoxide

suboxide thionyl chloride Ceric hydroxide hydroxynitrate oxide sulfate Cerium

Formula weight

col., poisonous, odorless gas poisonous gas gas gas yel.-red lq. yel., gelatinous red, mn. wh. or pa. yel., cb. yel., rhb. steel gray, cb. or hex. wh., mn. or rhb. tri. silv. met., hex. lq. rhb., or gn.-yel. gas rhb. red-brn., delq. delq. col. lq. gn. pink, trig. vl. or gn., hex. pl. gn., rhb. gn. or blue, gelatinous gn. purple pr. purple, mn. dark gn., hex. rose pd. gn. vl. vl., cb., 1.564 brn.-bk. pd.

Specific gravity 2.32

Melting point, °C

Boiling point, °C

−1aH2O, 128

−2H2O, 163

d. 15

2.815°

−2H2O, 100 d.

d. 650

Solubility in 100 parts Cold water

Hot water

0.2230°

0.25750°

v. s. d. 0.004318° 0.0370° s. 71.29° 0.2

v. s. d. 0.002790° 0.2285° v. s. d.

Other reagents s. a., gly., Na2S2O3, NH4 salts s. al. s. a. s. H2SO3 sl. s. al. v. s. al. i. al. s. NH4Cl; i. a.

1.87316° 6.06

d.

1.8–2.1 3.5120° 2.2620° lq. 1.101−87°; 1.53 (A); solid 1.56−79° 22° lq. 1.261 20 ; 2.63 (A) lq. 0.814−195° 4 ; 0.968 (A) 1.392 19° 4 lq. 1.24−87°; 2.10 (A) lq. 1.1140° 1.50915°

>3500 >3500 >3500 −56.65.2atm.

4200 4200 4200 subl. −78.5

i. i. i. 179.70° cc

i. i. i. 90.120° cc

i. a., alk. i. a., alk. i. a., alk. s. a., alk.

−108.6

46.3

0.20°

0.01450°

s. al.; et.

−207

−192

0.00440°; 3.50° cc

0.001850° 2.3220° cc

s. al., Cu2Cl2

−104 −138.2

8.2756mm −50.2760mm

v. s. sl. d. 1330° cc

d. 40.330° cc

s. ac., CCl4, bs.; d.a. v. s. alk., al.

−107

7761mm 73.5

d.

s. et. s. a.; sl. s. alk. carb.; i. alk

7.3 3.91 6.920° cb.; 6.7 hex. 3.91 2.88617° 1.9020° 1.28214.2° lq. 1.56−33.6°; 2.490° (A) 1.23 2.431 1.97128° 1.78725°

1950 645

5.21 3.012 1.86717° 1.722° 3.7719°

i.

0°

−8H2O, 630 28.5 <−20 −101.6 d. 9.6 60 19.2 −80

15°

2.757 1.835 25° 4 3.8

1400

d. i. s. d. i.

670 d. 40 −34.6

151.5765mm 1200–1500 d.

subl. 83 >1000

d.

−2H2O, 100 36.5 100 1900

d. 100 d.

100

−10H2O, 100 −12H2O, 100

−S, 1350

18.98 250° d. v. s. 1.460°; 31010° cc s. v. s. s. d. s. i.§ v. s. d. i. i. i. s. s. i. i.† s. s. 12020° i.

Slowly oxidized 0.4100° 7.640°

s. H2SO4, HCl s. dil. H2SO4 s. dil. a.; i. al.

s. a., al., NH3 0.57 ; 17730° cc 30°

v. s.

sl. s.

i. s. s. i. d. 67° d. d.

s. alk. s. alk. s. al., et. d. al.; i. CS2 4.7615° m. al. i. a., act., CS2 s. al.; i. et. sl. s. a.; i. al., NH3 s. a., alk.; sl. s. NH3 s. a., alk. s. a., alk., al., act. sl. s. a. i. a. s. al., H2SO4 sl. s. al. s. al. s. h. HNO3

Chromium trioxide (chromic acid) Chromous chloride hydroxide oxide sulfate sulfide (daubrelite) Chromyl chloride Cobalt carbonyl sulfide, diCobaltic chloride chloride, dichro chloride, luteo chloride, praseo Cobaltic chloride, purpureo chloride, roseo hydroxide oxide sulfate sulfide Cobalto-cobaltic oxide Cobaltous acetate chloride chloride nitrate oxide sulfate sulfate

Cr

52.00

gray, met., cb.

CrO3 CrCl2 Cr(OH)2 CrO CrSO47H2O CrS CrO2Cl2 Co Co(CO)4 CoS2 CoCl3 Co(NH3)3Cl3H2O Co(NH3)6Cl3 Co(NH3)4Cl3H2O Co(NH3)5Cl3 Co(NH3)5Cl3H2O Co(OH)3 Co2O3 Co2(SO4)3 Co2S3 Co3O4 Co(C2H3O2)24H2O CoCl2 CoCl26H2O* Co(NO3)26H2O

99.99 122.90 86.01 68.00 274.17 84.06 154.90 58.93 170.97 123.06 165.29 234.40 267.48 251.43 250.44 268.46 109.96 165.86 406.05 214.06 240.80 249.08 129.84 237.93 291.03

red, rhb. wh., delq. yel.-brn. bk. pd. blue bk. pd. dark red lq. silv. met., cb. or. cr. bk., cb. red cr.

CoO CoSO4 CoSO4H2O

74.93 155.00 173.01

brn., cb. red pd. red pd., mn.(?), 1.639 red, mn., 1.483 brn. nd. yel.-red met., cb.

or., mn. gn., rhb. rhb. brick red bk. bk. blue cr. bk. cr. bk., cb. red-vl., mn., 1.542 blue cr. red, mn. red, mn., 1.4

7.1 2.70 2.75

1615

d. 3.97 1.92 8.920° 1.7318° 4.269 2.94 1.701620° 1.847 25° 1.819 25

1550 −96.5 1480 51

5.18

d. 100 −1aH2O, 100 d. 900

4.8 6.07 1.705318.7° 3.356 25° 1.924 25 25° 1.883 25

−4H2O, 140 subl. 86 <100

5.68 3.71025° 3.13

d. 1800 d. 880 d.

25° 1.948 25 5.4518° 8.9220° 1.930 20° 4 1.882

96.8 >1100 1083

−7H2O, 420

115

240 d.

1.98

d. 110

1.81 3.88

d. 150 d. 220

ammonium sulfate carbonate, basic (azurite)

CuSO44NH3H2O 2CuCO3Cu(OH)2

245.75 344.67

blue, tet., 1.670, 1.744 blue, rhb. blue, mn., 1.758

carbonate, basic (malachite) chloride (eriochalcite)

CuCO3Cu(OH)2 CuCl2

221.12 134.45

dark gn., mn., 1.875 brn.-yel. pd.

3.9 3.054

d. 498

chloride chromate, basic cyanide dichromate ferricyanide ferrocyanide formate hydroxide lactate nitrate nitrate

CuCl22H2O CuCrO42CuO2H2O Cu(CN)2 CuCr2O72H2O Cu3[Fe(CN)6]2 Cu2Fe(CN)67H2O Cu(HCO2)2 Cu(OH)2 Cu(C3H5O3)22H2O Cu(NO3)23H2O* Cu(NO3)26H2O

170.48 374.66 115.58 315.56 614.54 465.15 153.58 97.56 277.72 241.60 295.65

gn., rhb., 1.684 yel.-brn. yel.-gn. bk., tri. yel.-gn. red-brn. blue, mn. blue, gelatinous dark blue, mn. blue, delq. blue, rhb.

2.3922.4° 2.28618°

−2H2O, 110 −2H2O, 260 d. −2H2O, 100

1.831 3.368

−H2O

2.0473.9° 2.074

114.5 −3H2O, 26.4

*Usual commercial form. †Also a soluble modification.

277.47

dark gn., mn. gn.

117.6 2900 d. 52

subl.

ammonium chloride

281.10 91.00 63.55 181.63 199.65 1013.79

i.

i. 0°

197 d.

CoSO47H2O* CoS Cu Cu(C2H3O2)2 Cu(C2H3O2)2H2O (CuOAs2O3)3 Cu(C2H3O2)2* CuCl22NH4Cl2H2O

sulfate (biebeorite) sulfide (syeporite) Copper Cupric acetate acetate aceto-arsenite (Paris green)

2200

1049 −6H2O, 110 d.

164.9 v. s. d. i. 12.350 i. d. i. i. i. s. s. 4.260° v. s. 0.2320° 16.120° i. i. d. i. i. s. 457° 116.50° 84.030°(anh.) i. 25.60° s.

2300

−HNO3, 170

206.7 v. s. i.

i. d.

s. HCl, dil. H2SO4; i. HNO3 s. H2SO4, al., et. sl. s. al.; i. et. s. conc. a. i. dil. HNO3 sl. s. al. v. s. a. s. et. s. a. s. al., et., CS2 s. HNO3, aq. reg.

s. 12.7446.5° 1.03146.5° 24.8716° i. i. i. s. 10596° 17780° 334.990° (anh.) i. 83100° s.

s. a.; al. i. al., NH4OH s. a.; i. al. i. al. sl. s. HCl s. a.; i. al. s. a. s. H2SO4 d. a. s. H2SO4; i. HCl, HNO3 s. a., al. 31 al.; 8.6 act. v. s. et., act. 10012.5° al.; s. act.; sl. s. NH3 s. a., NH4OH; i. al. 1.0418° m. al.; i. NH8

3380° 0.0003818° i. s. 7.2 i.

s.

20

7 al.; s. et.; gly. s. a., NH4OH

33.80°

99.380°

s. a.

18.05 i.

d. d.

i. 70.70°

d. 107.9100°

i. al. s. NH4OH, h. aq. NaHCO3 s. KCN; 0.03 aq. CO 5315° al.; 6815° m. al.

110.40° i. i. sl. s. i. i. 12.5 i. 16.7 38140° 243.70°

192.4100°

21.5°

Forms Cu2Cl2 993 d.

100°

i.

d. i. d. d. 45100° 66680° ∞

2.58° al. s. a., aq. reg. s. HNO3, h. H2SO4

s. al.; et., NH4Cl s. HNO3, NH4OH s. KCN, C5H5N s. a.; NH4OH s. NH4OH; i. HCl s. NH4OH; i. a., NH8 0.25 al. s. a., NH4OH, KCN, al. sl. s. al. 10012.5° al. s. al.

2-13

2-14

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued) Formula weight

Color, crystalline form and refractive index

CuO CuO CuCl22CuO4H2O Cu3P2 CuSO4

79.55 79.55 365.60 252.59 159.61

CuSO45H2O* CuS

249.69 95.61

tartate Cuprous ammonium iodide carbonate chloride (nantokite) cyanide

CuC4H4O63H2O CuINH4IH2O Cu2CO3 Cu2Cl2 Cu2(CN)2

265.66 353.41 187.10 198.00 179.13

bk., cb. bk., tri., 2.63 blue-gn. bk. gn.-wh., rhb., 1.733 blue, tri., 1.5368 blue, hex. or mn., 1.45 1 gn. pd. rhb. pl. yel. wh., cb., 1.973 wh., mn.

ferricyanide ferrocyanide fluoride hydroxide oxide (cuprite) Cuprous phosphide sulfide (chalcocite) sulfide Cyanogen

Cu3Fe(CN)6 Cu4Fe(CN)6 Cu2F2 CuOH Cu2O Cu6P2 Cu2S Cu2S C2N2

402.59 466.13 165.09 80.55 143.09 443.22 159.16 159.16 52.03

brn.-red brn.-red red cr. yel. red, cb., 2.705 gray-bk. bk., rhb. bk., cb. poisonous gas

Fe(OH)(C2H3O2)2

190.94

brn., amor.

Name Cupric acetate (Cont.) oxide (paramelaconite) oxide (tenorite) oxychloride phosphide sulfate (hydrocyanite) sulfate (blue vitriol or chalcanthite) sulfide (covellite)

Cyanogen compounds, cf. table of organic compounds Ferric acetate, basic ammonium sulfate, cf. Alum chloride (molysite) chloride ferrocyanide (Prussian blue) hydroxide lactate nitrate oxide (hematite) sulfate sulfate (coquimbite) Ferroso-ferric chloride ferricyanide (Prussian green) oxide (magnetite; magnetic iron oxide) oxide, hydrated Ferrous ammonium sulfate

Formula

Specific gravity 6.40 6.45

Melting point, °C

6.35 3.60615°

d. 1026 d. 1026 −3H2O, 140 d. d. >600

2.286 15.6° 4 4.6

−4H2O, 110 tr. 103

4.4 3.53 2.9

d. 422 474.5

3.4 6.0 6.4 to 6.8 5.6 5.80 lq. 0.866−17.2°; 1.806 (A)

−O, 1800

1100 1130 −34.4

−20.5

2.804

282 37 d.

Fe(OH)3 Fe(C3H5O3)3 Fe(NO3)36H2O Fe2O3

106.87 323.06 349.95 159.69

red-brn. brn., amor., delq. rhb., delq. red or bk., trig., 3.042 rhb., 1.814 yel., trig. yel., delq. gn. bk., cb., 2.42

3.4 to 3.9

−1aH2O, 500

303.59 392.14

chloride (lawrencite) chloroplatinate ferricyanide (Turnbull’s blue) ferrocyanide formate hydroxide nitrate oxide

FePtCl66H2O Fe3[Fe(CN)6]2 Fe2Fe(CN)6 Fe(HCO2)22H2O Fe(OH)2 Fe(NO3)26H2O FeO

571.73 591.43 323.64 181.91 89.86 287.95 71.84

126.75

bk. blue-gn., mn., 1.4915 gn.-yel., hex., 1.567 yel., hex. dark blue blue-wh., amor.

i. i. i. i. 14.30°

Hot water i. i. 75.4100°

s. a.; KCN, NH4Cl s. a., KCN, NH4Cl s. a. s. HNO3; i. HCl i. al.

24.30° 0.00003318°

205100°

1.18° al. s. HNO3, KCN

0.0215° d. i. 1.5225° i.

0.1485°

s. a., KOH s. NH4I s. a., NH4OH s. HCl, NH4OH, al. s. KCN, HCl, NH4OH; sl. s. NH3 s. NH4OH; i. HCl s. NH4OH; i. NH4Cl s. HF, HCl, HNO3; i. al. s. a., NH4OH s. HCl, NH4Cl, NH4OH s. HNO3; i. HCl s. HNO3, NH4OH; i. act. s. HNO3, NH4OH; i. act. 230020° cc al.; 50018° cc et.

i. i. i. i. i. i. 0.000518° 0.000518° 45020° cc

i. i.

i. i.

s. a.; al. 0°

100°

v. s. al.; et. +HCl s. al., act., gly. s. HCl, conc. H2SO4; i. al., et. s. a.; i. al., et. i. et. s. al., act. s. HCl

535.8 ∞ d.

i. v. s. 1500° i.

i. v. s. ∞ d. d. s.

5.2

d. 50 d. 180 1538 d.

sl. s. 440 s. i. i.

i.

s. d. h. HCl i. al.

1.864

d. d.

i. 180°

i. 10075°

s. a. i. al.

64.410°

105.7100°

100 al.; s. act.; i. et.

v. s. i. i. sl. s. 0.00067 2000° i.

v. s.

20°

1.684 5.12

35 1560 d.

3.09718° 2.1

d. 480

2.7

d.

delq.

2.714 d. 3.4 5.7

315 280

Other reagents

74.4 2460° i.

d. lt. gn. cr. bk.

subl. 1100

Solubility in 100 parts Cold water

i. 11°

bk.-brn., hex. delq. red-yel., delq. dark blue

Fe3O44H2O FeSO4(NH4)2SO4 6H2O FeCl2

1366 d.

908 −aH2O, 360 1235

162.20 270.30 859.23

399.88 562.02 775.43 1662.61 231.53

Forms CuO, 650 −5H2O, 250 d. 220

d.

FeCl3 FeCl36H2O* Fe4[Fe(CN)6]3

Fe2(SO4)3 Fe2(SO4)39H2O FeCl22FeCl318H2O Fe′′′ 4 Fe′′ 3 [Fe(CN)6]6 Fe3O4

Boiling point, °C

60.5 1420

i. H2SO4, NH3 s. abs. al.

i. dil. a., al.

30025° i.

s. a., NH4Cl s. a.; i. alk.

phosphate (vivianite) silicate sulfate (siderotilate) sulfate (copperas) sulfide cf. also under iron Fluoboric acid Fluorine

Fe3(PO4)28H2O

501.60

FeSiO3 FeSO45H2O FeSO47H2O* FeS

131.93 241.98 278.01 87.91

blue, mn., 1.592, 1.603 mn. gn., tri., 1.536 blue-gn., mn. bk., hex.

1550

s. 14950°

i. al. i. al. s. a.; i. NH3

−223

130 d. −187

∞ d.

∞

s. al.

s.

s.

s.

s.

i. s.

i.

s. aq. reg., KCN; i. a. s. aq. reg., KCN; i. a.

0.970° cc ∞ s. ∞ v. s. s. 174.910° v. s. v. s. 3.05522° ∞ 4250010° cc ∞ ∞ ∞ ∞ 2210°

1.0850° cc ∞

Absorbed by Pt s. al.

∞ v. s. v. s. v. s.

∞ al.; i. et. sl. s. al. s. al.

144.09 157.25 309.44

yel. met., cb. blue to vl.

19.3

1063

2600

12.1 0.1368 (A) 1.011 15° 4

>1700 <−272.2 1.4 128 −40

>3200(?) −268.9 113.5

Au Au

196.97 196.97

Hf He N2H4 N2H42HCO2H N2H4H2O N2H4HCl N2H42HCl N2H4HNO3 N2H42HNO3 N2H4aH2SO4 N2H4H2SO4 HN3 HI HIH2O HI2H2O HI3H2O HI4H2O HBr HBrH2O

178.49 4.00 32.05 124.10 50.06 68.51 104.97 95.06 158.07 81.08 130.12 43.03 127.91 145.93 163.94 181.96 199.97 80.91 98.93

hex. col. gas col. lq. cb. col. yel. lq. wh., cb. cr. nd. delq. pl. rhb. col. lq. col. gas col. lq. col. lq. col. lq. col. lq. col. gas; 1.325 (lq.) col. lq.

Hydrobromic acid Hydrobromic acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrocyanic acid (prussic acid)

HBr (47.8% in H2O) HBr2H2O HCl† HCl (45.2% in H2O) HCl2H2O HCl3H2O HCN

80.91 118.96 36.46 36.46 72.49 90.51 27.03

Hydrofluoric acid Hydrofluoric acid Hydrogen

HF HF (35.35% in H2O) H2

20.01 20.01 2.02

col. lq. wh. cr. col. gas; 1.256 (lq.) col. lq. col. lq. col. lq. poisonous gas or col. lq., 1.254 gas or col. lq. col. lq. col. gas or cb.

H2O2‡ H2Se H2S NH2OH NH2OHHCl NH2OHHNO3 NH2OHaH2SO4

34.01 80.98 34.08 33.03 69.49 96.04 82.07

2-15

*Usual commercial form. †Usual commercial form about 31 percent. ‡Usual commercial forms 3 or 30 percent.

delq. cr.

col. lq., 1.333 col. gas col. gas rhb., delq. col., mn. col. cr. col., mn.

20°

1.0321° 1.42

1.378 4.40° (A) 1.715°

2.710° (A) 1.78 1.486 2.11−15° 1.2680° (A) 1.48 1.46 −18.3° 4 18°

0.697

0.98813.6° 1.15 lq. 0.0709−252.7° 0.06948 (A) 1.438 20° 4 2.12−42° 1.1895 (A) 1.3518° 1.6717°

s. a.; i. ac.

s. 32.80° 0.00061618°

H2SiF6 Gd GaBr3

lq. 1.51−187°; 1.3115° (A)

i.

−5H2O, 300 −7H2O, 300 d.

87.81 38.00

col. lq. gn.-yel. gas

i. 64 1193

HBF4 F2

Fluosilicic acid Gadolinium Gallium bromide Glucinum cf. Beryllium Gold Gold, colloidal Gold salts cf. under Auric and Aurous Hafnium Helium Hydrazine formate hydrate hydrochloride hydrochloride, dinitrate nitrate, disulfate sulfate Hydrazoic acid (azoimide) Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydrobromic acid Hydrobromic acid

peroxide selenide sulfide Hydroxylamine hydrochloride nitrate sulfate

2.58 3.5 2.2 1.89914.8° 4.84

198 70.7 104 85 254 −80 −50.8 −43 −48 −36.5 −86

−11 −111 −15.35 0 −24.4 −14 −83 −35 −259.1 −0.89 −64 −82.9 34 151 48 170 d.

118.5739.5mm subl. 140 d. 37 −35.5 127774mm

−67

d. d. 26

∞ s. 82.30° ∞ ∞ ∞ ∞

19.4 120 −252.7

∞ 0° to 19.4° v. s. 2.10° cc

126 −85

760mm

151.4 −42 −59.6 56.522mm d. d. <100

∞ 3774° cc 4370° cc s. 83.317° v. s. 32.90°

27.6560° ∞ v. s.

130100°

s. 56.160°

i. al. v. sl. s. abs. al. ∞ al. s. al. ∞ al. s. al. s. al. s. al. s. al. Stable at −15.5° and 1 atm., and at −11.3° and 2.5 atm. s. al. s. al., et. s. al. s. al. s. al. ∞ al., et.

v. s. 0.8580° cc

sl. s. Fe, Pd, Pt

27022.5° cc 18640° cc d. v. s. d. 68.590°

s. a., et.; i. petr. et s. CS2, COCl2 9.5415° cc al.; s. CS2 s. a., al. s. al.; i. et. v. s. abs. al. v. sl. s. al.; i. et., abs. al.

2-16

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued)

Name

Formula

Hypobromous acid Indium Iodic acid

HBrO In HIO3

Iodine oxide, pentaIodoplatinic acid Iridium Iron, cast† pure steel white pig wrought carbide (cementite) carbonyl nitride

I2 I2O5 H2PtI69H2O Ir Fe Fe Fe Fe Fe Fe3C Fe(CO)5 Fe2N

silicide sulfide, di- (marcasite) sulfide, di- (pyrite) sulfide (pyrrhotite) Cf. also under ferric and ferrous Krypton Lanthanum Lead

FeSi FeS2 FeS2 Fe7S8

acetate acetate (sugar of lead) acetate acetate, basic acetate, basic acetate, basic arsenate, monobasic arsenate, dibasic (schultenite) arsenate, metaarsenate, pyroLead azide bromide

Formula weight 96.91 114.82 175.91

Color, crystalline form and refractive index yel. soft, tet. met. col., rhb.

Specific gravity

Melting point, °C

7.320° 4.6290°

155 110 d.

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

4050mm 1450

s. i. 2860°

d. i. 576101°

0.01620° 187.412° s. d. i. i. i. i. i. i. i. i. d.

0.0956660°

blue-bk., rhb. wh., trimetric brn., delq. mn. wh. met., cb. gray silv. met., cb. silv. gray gray gray pseudo hex. pa. yel. lq. gray

4.9320° 4.799 25° 4

113.5 d. 300

184.35

22.420° 7.03 7.8620° 7.6 to 7.8 7.6 to 7.8 7.86 7.4 1.45721° 6.35

2350 1275 1535 1375 1075 1505 1837 −21 d. >560

>4800

83.93 119.98 119.98 647.44

yel.-gray, oct. yel., rhb. yel., cb. hex.

6.1 20° 4 4.87 5.0 4.6 20° 4

d. d.

i. 0.00049 0.0005 i.

i.

tr. 450 1171 d. >700

Kr La Pb

83.80 138.91 207.20

col. gas lead gray silv. met., cb.

2.818 (A) 6.1520° 20° 11.337 20

−169 826 327.5

−151.8 1800 1620

11.050° cc d. i.

3.5760° cc

Pb(C2H3O2)2 Pb(C2H3O2)23H2O† Pb(C2H3O2)210H2O Pb2(C2H3O2)3OH Pb(C2H3O2)2 Pb(OH)2H2O Pb(C2H3O2)2 2Pb(OH)2 PbH4(AsO4)2 PbHAsO4 Pb(AsO3)2 Pb2As2O7 PbN6 PbBr2

325.29 379.33 505.44 608.54 584.52

wh. cr. wh., mn. wh., rhb. wh. wh. nd.

3.251 20° 4 2.55 1.689

280 −3H2O, 75 22

807.72

wh. nd.

489.07 347.13 453.04 676.24 291.24 367.01

tri., 1.82 wh., mn., 1.9097 hex. rhb., 2.03 col. nd. col., rhb.

4.46 5.94 6.4215° 15° 6.85 15

253.81 333.81 1120.66 192.22 55.85 55.85 55.85 55.85 55.85 179.55 195.90 125.70

15°

3000

102.5760mm

d. 140 d. >200

−H2O, 280

6.66

802 expl. 350 373

918

22150° 200100° s.

5.55

18.2

d. 0.05100° 4.75100°

0.0001120°

d. i. 3.34100° i. i. 18100° d.

i.

267.21

wh., rhb., 2.0763

6.6

d. 315

2PbCO3Pb(OH)2† PbCl2 PbCrO4 PbCrO4PbO Pb(HCO2)2 3PbOH2O Pb(NO3)2

775.63 278.11 323.19 546.39 297.23 687.61 331.21

6.14 5.80 6.12

d. 400 501 844

4.56 7.592 4.53

d. 190 −H2O, 130 d. 470

i. 0.6730° 0.00000720° i. 1.616° 0.014 38.80°

oxide, suboxide, mono- (litharge)

Pb2O PbO

430.40 223.20

wh., hex. wh., rhb., 2.2172 yel., mn., 2.42 or.-yel. nd. wh., rhb. cb. col., cb. or mn., 1.7815 bk., amor. yel., tet.

8.34 9.53

d. red heat 888

i. 0.006818°

oxide, mono (massicotite)

PbO

223.20

yel., rhb., 2.61

s. a. v. s. 87% al.; i. abs. al. et., chl. s. al., KI, et. i. abs. al., et., chl. sl. s. aq. reg., aq. Cl2 s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a. s. al., H2SO4, alk. s. HCl, H2SO4 i. aq. reg. i. dil. a. i. dil. a.

sl. s. al., bz. s. a. s. HNO3; i. c. HCl, H2SO4 s. gly.; v. sl. s. al. s. gly.; sl. s. al. sl. s. al. s. al.

d. i. d. i. i. 0.45540°

PbCO3

8.0

i.

19.70° 45.6415° s. v. s. v. s.

carbonate (cerussite) carbonate, basic (hydrocerussite; white lead) chloride (cotunnite) chromate (crocoite) chromate, basic formate hydroxide nitrate

954760mm d.

i. i. i. i. i. i. i.

Other reagents

sl. s.

138.8100°

s. al. s. HNO3 s. HNO3, NaOH s. HNO3 s. HCl, HNO3; i. sc. v. s. ac.; i. NH4OH s. a., KBr.; sl. s. NH3; i. al. s. a., alk.; i. NH3, al. s. ac.; sl. s. aq. CO2 sl. s. dil. HCl, NH3, i. al. s. a., alk.; i. NH3, ac. s. a., alk. i. al. s. a., alk. 8.822° al. s. a., alk. s. alk., PbAc, NH4Cl, CaCl2

oxide, mono-

PbO

223.20

amor.

9.2 to 9.5

oxide, red (minium) oxide, sesquioxide, di- (plattnerite) silicate sulfate (anglesite)

Pb3O4 Pb2O3 PbO2 PbSiO3 PbSO4

685.60 462.40 239.20 283.28 303.26

9.1

Pb(HSO4)2 H2O PbSO4PbO PbS Pb(CNS)2 Li LiC7H5O2 LiBr

419.36 526.46 239.27 323.36 6.94 128.05 86.85

LiBr2H2O Li2CO3 LiCl

122.88 73.89 42.39

citrate fluoride formate hydride hydroxide hydroxide nitrate nitrate oxide phosphate, monobasic phosphate, tribasic phosphate, tribasic salicylate sulfate sulfate sulfate, acid Lutecium Magnesium acetate acetate aluminate (spinel)

Li3C6H5O74H2O LiF LiHCO2H2O LiH LiOH LiOHH2O LiNO3 LiNO33H2O Li2O LiH2PO4 Li3PO4 Li3PO412H2O LiC7H5O3 Li2SO4 Li2SO4H2O† LiHSO4 Lu Mg Mg(C2H3O2)2 Mg(C2H3O2)24H2O† MgO·Al2O3

281.98 25.94 69.97 7.95 23.95 41.96 68.95 122.99 29.88 103.93 115.79 331.98 144.05 109.94 127.96 104.01 174.97 24.31 142.39 214.45 142.26

red, amor. red-yel., amor. brn., tet., 2.229 col., mn., 1.961 wh., mn. or rhb., 1.8823 cr. col., mn. lead gray, cb., 3.912 col., mn. silv. met. cb. wh. leaflets wh., delq., cb., 1.784 wh. pr. col., mn., 1.567 wh., delq., cb., 1.662 wh. cr. wh., cb., 1.3915 col., rhb. wh., cb. wh. cr. col., mn. col., trig., 1.735 col. col., 1.644 col. wh., rhb. wh., trig. col. col., mn., 1.465 col., mn., 1.477 pr.

ammonium chloride ammonium phosphate (struvite) ammonium sulfate (boussingaultite) benzoate carbonate (magnesite) carbonate (nesquehonite) carbonate, basic (hydromagnesite) Magnesium chloride (chloromagnesite) chloride (bischofite) hydroxide (brucite) nitride oxide (magnesia; periclase) perchlorate

MgCl2NH4Cl6H2O MgNH4PO46H2O

256.79 245.41

MgSO4(NH4)2SO4 6H2O Mg(C7H5O2)23H2O MgCO3 MgCO33H2O 3MgCO3Mg(OH)23H2O

360.60

col., mn.

320.58 84.31 138.36 365.31

wh. pd. wh., trig. 1.700 col., rhb., 1.501 wh., rhb., 1.530

95.21

sulfate, acid sulfate, basic (lanarkite) sulfide (galena) thiocyanate Lithium benzoate bromide bromide carbonate chloride

2-17

*See also a table of alloys. †Usual commercial form.

MgCl2 MgCl26H2O† Mg(OH)2 Mg3N2 MgO Mg(ClO4)2†

203.30 58.32 100.93 40.30 223.21

i.

i.

d. 500 d. 360 d. 290 766

i. i. i. i.

i. i. i.

1336  5

3.464 25° 4

547

1265

0.00280° 0.000118° 0.004418° 0.0000918° 0.0520° d. 3325° 1430° (2H2O)

0.005640°

6.92 7.5 3.82 0.5320°

1170 d. 977 1120 d. 190 186

2.110° 2.068 25° 4

44 618 614

d. 1360

24620° 1.540° 670°

9.375 6.49 6.2

2.29521.5° 1.46 0.820 2.54 1.83 2.38 2.013 25° 4

d. 870 −H2O, 94 680 445 261 29.88

1670 925 d. subl. <1000

61.215° 0.2718° 49.20° d. 12.70° 22.310° 53.40° v. s. forms LiOH

s. alk., PbAc, NH4Cl, CaCl2 s. ac., h. HCl s. a., alk. s. ac., h. alk.; i. al. s. a.

i. s. d. 40100° 266100° (1H2O) 0.72100° 127.5100° 66.7100° 0.13535° 346.6104° 17.5100° 26.880° 19470° ∞

s. conc. a., NH4 salts; i. al. sl. s. H2SO4 sl. s. H2SO4 s. a.; i. alk. s. KCNS, HNO3 s. a., NH3 7.725°, 1078° al. s. al., act. s. al. s. dil. a.; i. al., act., NH3 2.4815° al.; s. et. sl. s. al., et. s. HF; i. act. sl. s. al., et. i. et. sl. s. al. sl. s. al. s. al., NH3

2.22 2.06 2.12313°

>100 837 100 d. 860 −H2O, 130 170.5

silv. met., hex. wh. wh., mn. pr., 1.491 col. cb., 1.718–23

1.7420° 1.42 1.454 3.6

651 323 80 2135

wh., rhb., delq. col., rhb., 1.496

1.456 1.715

−4H2O, 195 d. 100

1.72

>120

16.86

130

4.525° (anh.) 0.0106 0.151819° 0.04

s.

3.037 1.852 2.16

−3H2O, 110 d. 350 −H2O, 100 d.

d. 0.011

s. act. s. a., aq. CO2; i. act., NH3 s. a., aq. CO2 s. a., NH4 salts; i. al.

col., hex., 1.675

2.32525°

712

1412

52.80°

73100°

50 al.

wh., delq., mn., 1.507 wh., trig., 1.5617 gn.-yel., amor. col., cb., 1.7364 wh., delq.

1.56 2.4

118 d. d. d. 2800 d.

d.

2810° 0.000918° i. 0.00062 99.625°

918100°

50 al. s. NH4 salts, dil. a. s. a.; i. al. s. a., NH4 salts; i. al. 2425 al., 51.825° m. al.; 0.29 et.

2.461 2.53717.5° 1.645

3.65 2.6025°

1110

0.03418° v. sl. s. 12826° 35.340° 43.60° d.

v. sl. s. v. sl. s.

i. v. s. v. s. i.

sl. s. d. v. s. v. s.

16.7 0.02310°

s. 0.019580°

0°

3600

29.9100° 35100°

s. a., NH4Cl; i. act. v. s. al. i. act., 80% al. i. 80% al. s. a., NH4 salts 5.2515° m. al. v. s. al. v. sl. s. dil. HCl; i. dil. HNO3 s. a.; i. al.

100°

d. v. s.

2-18

Physical Properties of the Elements and Inorganic Compounds (Continued)

TABLE 2-1

Name

Formula

Formula weight

Magnesium chloride (Cont.) peroxide phosphate, pyrophosphate, pyropotassium chloride (carnallite) potassium sulfate (picromerite) silicofluoride sodium chloride sulfate sulfate (epsom salt; epsomite) Manganese acetate acetate carbonate (rhodocrosite)

MgO2 Mg2P2O7 Mg2P2O73H2O MgCl2KCl6H2O MgSO4K2SO46H2O MgSiF66H2O MgCl2NaClH2O MgSO4 MgSO47H2O* Mn Mn(C2H3O2)2 Mn(C2H3O2)24H2O* MnCO3

56.30 222.55 276.60 277.85 402.72 274.47 171.67 120.37 246.47 54.94 173.03 245.09 114.95

wh. pd. col., mn., 1.604 wh., amor. delq., rhb., 1.475 mn., 1.4629 col., trig., 1.3439 col. col. col., rhb., 1.4554 gray-pink met.

chloride (scacchite) chloride

MnCl2 MnCl24H2O*

125.84 197.91

chloride, perhydroxide (ous) (pyrochroite) hydroxide (ic) (manganite) nitrate oxide (ous) (manganosite) oxide (ic) oxide, di- (pyrolusite; polianite) sulfate (ous) sulfate (ous) (szmikite)

MnCl4 Mn(OH)2 Mn2O3H2O Mn(NO3)26H2O MnO Mn2O3 MnO2*

196.75 88.95 175.89 287.04 70.94 157.87 86.94

rose, delq., cb. rose red, delq., mn. 1.575 gn. wh., trig. brn., rhb., 2.24 rose red, mn. gray-gn., cb., 2.16 brn.-bk., cb. bk., rhb.

MnSO4 MnSO4H2O

151.00 169.02

red-wh. pa. pink, mn., 1.595

sulfate (ous)

MnSO42H2O

187.03

2.52615°

sulfate (ous)

MnSO43H2O

205.05

2.35615°

sulfate (ous)

MnSO44H2O*

223.06

sulfate (ous)

MnSO45H2O

sulfate (ous)

MnSO46H2O

259.09

sulfate (ous)

MnSO47H2O

277.11

pink, mn. or rhb.

2.092

sulfate (ic)

Mn2(SO4)3

398.06

gn., delq. cr.

3.24

Mercuric acetate bromide carbonate, basic chloride (corrosive sublimate) fulminate hydroxide oxide (montroydite) oxychloride (kleinite) silicofluoride, basic sulfate sulfate, basic (turpeth) Mercurous acetate bromide carbonate

Hg(C2H3O2)2 HgBr2 HgCO32HgO HgCl2 Hg(CNO)2 Hg(OH)2 HgO HgCl23HgO HgSiF6HgO3H2O HgSO4 HgSO42HgO HgC2H3O2 HgBr Hg2CO3

241.08

318.68 360.40 693.78 271.50 284.62 234.60 216.59 921.26 613.30 296.65 729.83 259.63 280.49 461.19

Color, crystalline form and refractive index

pa. pink, mn. rose, trig., 1.817

pink, rhb. or mn., 1.518 pink, tri., 1.508

wh. pl. wh., rhb. brn.-red wh., rhb., 1.859 cb. yel. or red, rhb., 2.5 yel., hex. yel. nd. wh., rhb. yel., tet. wh. sc. wh., tet. yel. pd.

Specific gravity

Melting point, °C

2.59822° 2.56 1.60 19.4° 4 2.15 17.5° 1.788 4

expl. 275 1383 −3H2O, 100 265 d. 72 d.

2.66 1.68 7.220° 1.74 20° 4 1.589 3.125

1185 70 d. 1260

2.977 25° 4 2.01

650 58.0

3.25818° 3.258 1.8221° 5.18 4.81 5.026

d. d. 25.8 1650 −0, 1080 −0, >230

3.235 2.87

700 Stable 57 to 117 Stable 40 to 57 Stable 30 to 40 Stable 18 to 30 Stable 8 to 18 Stable −5 to +8 Stable −10 to −5; 19 d. d. 160

2.107 15°

2.103

d. 237

5.44 4.42

277 expl. −H2O, 175 d. 100 d. 260

6.47 6.44 7.307

d. d. subl. 345 d. 130

Solubility in 100 parts Cold water

Hot water

Other reagents

i. i. i. 64.519° d. 19.260° 64.817.5° s. 26.90° 72.40° d. s. s. 0.006525°

i. i. sl. s. d. 81.775° s. s. 68.3100° 17840°

63.40° 1518°

123.8100° ∞

s. al., m. al. s. aq. CO2, dil. a.; l. NH3, al. s. al.; i. et., NH3 s. al.; i. et.

129.5

s. 0.00220° i. 4260° i. i. i.

s. i. i. ∞ i. i. i.

s. al., et. s. a., NH4 salts; i. alk. s. h. H2SO4 v. s. al. s. a., NH4Cl s. a.; i. act. s. HCl; i. HNO3, act.

d. 850

530° 98.4748°

7350° 79.77100°

s. al.; i. et.

85.2735°

106.855°

74.225°

99.3157°

13616°

16950°

1900

d.

3.270 6.053

11.14 7.93

Boiling point, °C

1190 −H2O, 106; −4H2O, 200

−4H2O, 450

5°

200

0°

2479°

142 204 −7H2O, 280

25114°

v. s.

d.

304

25 0.520° i. 3.60° sl. s. i. 0.005225° i. d. d. 0.005 0.7513° 7 × 10−9 i.

d. HF s. al. s. al. s. dil. a.

i. al.

35°

1760°

10°

322

s. 64.550°

s. a. s. a.; i. alk. s. a.; i. al. d. al.

100°

100 25100°

61.3100° i. 0.041100° d. 0.167100° d. i. d.

s. HCl, dil. H2SO4; l. conc. H2SO4, HNO3 s. al. sl. d. 25.20° al.; v. sl. s. et. s. aq. CO2, NH4Cl 3325° 99% al.; 33 et. s. NH4OH, al. s. a. s. a.; i. al. s. HCl s. a. s. a.; i. al., act., NH8 s. a.; i. al. s. H2SO4, HNO3; i. al. s. a.; i. al., act. s. NH4Cl

HgCl

236.04

wh., tet., 1.9733

7.150

302

383.7

0.00140°

0.000743°

iodide nitrate Mercurous oxide

HgI HgNO3H2O Hg2O

327.49 280.61 417.18

yel., tet. wh. mn. bk.

7.70 4.7853.9° 9.8

290 d. 70 d. 100

subl. 140; 310d. expl.

2 × 10−8 v. s. i.

v. sl. s. d. 0.0007

sulfate Mercury† Molybdenum

Hg2SO4 Hg Mo

497.24 200.59 95.94

wh., mn. silv. lq. or hex.(?) gray, cb.

7.56 13.54620° 10.2

d. −38.87 2620  10

0.05516.5° i. i.

0.092100° i. i.

MoCl2

166.85

yel., amor.

3.714 25° 4

d.

i.

i.

3.578 25° 4

d.

chloride (calomel)

chloride, dichloride, tri-

MoCl3

202.30

dark red pd.

chloride, tetra-

MoCl4

237.75

brn., delq.

chloride, pentaoxide, tri- (molybdite) sulfide, di- (molybdenite) sulfide, trisulfide, tetraMolybdic acid Molybdic acid Neodymium Neon Neptunium Nickel

MoCl5 MoO3 MoS2 MoS3 MoS4 H2MoO4 H2MoO4H2O Nd Ne 239

273.21 143.94 160.07 192.14 224.20 161.95 161.95 144.24 20.18

Np Ni

239.05 58.69 176.78 291.18 394.99

carbonyl chloride chloride

Ni(C2H3O2)2 NiCl2NH4Cl6H2O NiSO4(NH4)2SO4 6H2O Ni(BrO3)26H2O NiBr2 NiBr23H2O NiBr26NH3 NiPtBr66H2O NiCO3 2NiCO33Ni(OH)2 4H2O Ni(CO)4 NiCl2 NiCl26H2O*

chloride, ammonia cyanide dimethylglyoxime

NiCl26NH3 Ni(CN)24H2O NiC8H14O4N4

231.78 182.79 288.91

formate hydroxide (ic) hydroxide (ous) nitrate nitrate, ammonia oxide, mono- (bunsenite) potassium cyanide sulfate

Ni(HCO2)22H2O Ni(OH)3 Ni(OH)2dH2O Ni(NO3)26H2O Ni(NO3)24NH32H2O NiO Ni(CN)22KCNH2O NiSO4

184.76 109.72 97.21 290.79 286.86 74.69 258.97 154.76

acetate ammonium chloride ammonium sulfate bromate bromide bromide bromide, ammonia bromoplatinate carbonate carbonate, basic

2-19

*Usual commercial form. †See also Tables 2-28 and 2-280.

422.59 218.50 272.55 320.68 841.29 118.70 587.59 170.73 129.60 237.69

bk. cr. col., rhb. bk., hex., 4.7 red-brn. brn. pd. yel-wh., hex. yel., mn. yellowish col. gas

2.928 25° 4 19.5°

4.50 4.80114°

3.12415° 6.920° lq. 1.204−245.9° 0.674 (A)

1.145° cc

s. lq. O2, al., act., bz.

i.

s. dil. HNO3; sl. s. H2SO4, HCl; i. NH3 i. al.

d.

d.

s.

d.

194

268

s.

795 1185 d. d. d. 115 −H2O, 70 840 −248.67

subl.

−2H2O, 200 −245.9

d. 18°

0.107 i. sl. s. i. v. sl. s. 0.13318° d. 2.60° cc 238

Produced by Neutron bombardment of U 1452 2900 i.

8.9020

gn. pr. gn., delq., mn. blue-gn., mn., 1.5007 gn., cb. yel., delq. gn., delq. vl. pd. trig. lt. gn., rhb. lt. gn.

1.798 1.645 1.923

d.

2.575 4.64 28° 4

lq. yel., delq. gn., delq., mn., 1.57

1.3117° 3.544

gn. pl. scarlet red cr.

2.10679° i. s. i. sl. s. 2.1370°

i.

volt.

silv. met., cb.

1.837 3.715

356.9 3700

s. aq. reg., Hg(NO3)2; sl. s. HNO3, HCl; i. al., etc. s. KI; i. al. s. HNO3; i. al., et. s. h. ac.; i. alk., dil. HCl, NH3 s. H2SO4, HNO3 s. HNO3; i. HCl s. h. conc. H2SO4; i. HCl, HF, NH3, dil. H2SO4, Hg s. HCl, H2SO4, NH4OH, al., et. s. HNO3, H2SO4; v. sl. s. al., et. s. HNO3, H2SO4; sl. s. al., et. s. HNO3, H2SO4; i. abs. al., et. s. a., NH4OH s. H2SO4, aq. reg. s. alk. sulfides s. alk. sulfides; i. NH3 s. NH4OH, H2SO4; i. NH s. a., NH4OH, NH4, salts

16.6 15025° 2.53.5°

v. s. 39.288°

v. sl. s. (NH4)2SO4

d. d. −3H2O, 200

28 112.80° 1999° v. s.

156100° 316100° d.

s. NH4OH s. al., et., NH4OH s. al., et., NH4OH i. c. NH4OH

d. d.

0.009325° i.

i. d.

s. a. s. a., NH4 salts

0.0189.8° 53.80° 180

i. 87.6100° v. s.

s. aq. reg., HNO3, al., et. s. NH4OH, al.; i. NH3 v. s. al.

s. i. i.

d. i. i.

s. NH4OH; i. al. s. KCN; i. dil. KCl s. abs. al., a.; i. ac., NH4OH

i. v. sl. s. ∞56.7°

s. a., NH4OH, NH4Cl s. a., NH4OH; i. alk. s. NH4OH; i. abs. al. i. al. s. a., NH4OH d. a. i. al., et., act.

−25 subl.

43751mm 973

−4H2O, 200 subl. 250

d.

gn. cr. bk. lt. gn. gn., mn.

2.154 4.36 2.05

d. d. d. 56.7

gn.-bk., cb., 2.37 red yel., mn. yel., cb.

7.45 1.87511° 3.68

Forms Ni2O3 at 400 −H2O, 100 −SO3, 840

136.7

s. i. v. sl. s. 243.00° v. s. i. s. 27.20°

i. 76.7100°

2-20

Physical Properties of the Elements and Inorganic Compounds (Continued)

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index gn. mn. or blue, tet., 1.5109 gn., rhb., 1.4893 col. lq. col. lq. col. lq. col., rhb. col. gas or cb. cr.

Nickel (Cont.) sulfate

NiSO46H2O*

262.85

sulfate (morenosite) Nitric acid Nitric acid Nitric acid Nitro acid sulfite Nitrogen

NiSO47H2O HNO3 HNO3H2O HNO33H2O NO2HSO3 N2

280.86 63.01 81.03 117.06 127.08 28.01

Nitrogen oxide, mono- (ous)

N2O

44.01

oxide, di- (ic)

NO or (NO)2

oxide, tri-

N2O3

30.01 60.01 76.01

oxide, tetra- (per- or di-)

NO2 or (NO2)2

col. gas col. gas

N2O5

46.01 92.01 108.01

red-brn. gas or blue lq. or solid yel. lq., col. solid, red-brn. gas wh., rhb.

oxybromide oxychloride

NOBr NOCl

109.91 65.46

brn. lq. red-yel. lq. or gas

Nitroxyl chloride Osmium chloride, dichloride, trichloride, tetraOxygen

NO2Cl Os OsCl2 OsCl3 OsCl4 O2

81.46 190.23 261.14 296.59 332.04 32.00

yel.-brn. gas blue, hex. gn., delq. brn., cb. red-yel. nd. col. gas or hex. solid

Ozone

O3

48.00

Palladium

Pd

106.42

silv. met., cb.

PdBr2 PdCl2 PdCl22H2O Pd(CN)2

266.23 177.33 213.36 158.45

brn. brn., cb. brn. pr. yel.

213.85 211.39 100.46 118.47 136.49

met. red or yel., tet. unstable, col. lq fairly stable nd. stable lq., col.

Periodic acid Periodic acid Permanganic acid Permolybdic acid Persulfuric acid Phosphamic acid Phosphatomolybdic acid Phosphine

Pd2H Pd(NH3)2Cl2 HClO4 HClO4H2O HClO42H2O* 73.6% anh. HIO4 HIO42H2O HMnO4 HMoO42H2O H2S2O8 PONH2(OH)2 H7P(Mo2O7)628H2O PH3

Phosphonium chloride

PH4Cl

oxide, penta-

bromide (ous) chloride chloride cyanide hydride Palladous dichlorodiammine Perchloric acid Perchloric acid Perchloric acid

191.91 227.94 119.94 196.98 194.14 97.01 2365.71 34.00 70.46

col. gas

wh. cr. delq., mn. exists only in solution wh. cr. hyg. cr. cb. yel. cb. col. gas wh., cb.

Specific gravity 2.07 1.948 1.502

Melting point, °C

Boiling point, °C

Solubility in 100 parts Cold water

tr. 53.3

−6H2O, 280

13150°

98–100 −42 −38 −18.5 73 d. −209.86

−6H2O, 103 86

0°

1.026−252.5° 0.808−195.8° 12.50° (D) lq. 1.226−89° −102.3 1.530 (A) −150.2° lq. 1.269 −161 1.0367 (A) 1.4472° −102

−195.8 −90.7

130.520° cc

60.8224° cc 0.0100°

7.34 cc s.

−9.3

21.3

d.

1.63

30

47

s.

>1.0 1.417−12° 2.31 (A) lq. 1.3214° 22.4820°

−55.5 −64.5

−2 −5.5

d. d.

<−30 2700

5 >5300

1.14 1.426−252.5° 1.1053 (A) 1.71−183° 3.03−80° 1.658 (A) 12.020° 111550°

s. al. expl. with al. d. al. d. al. s. H2SO4 sl. s. al. s. H2SO4, al. 26.6 cc al.; 3.5 cc H2SO4; s. aq. FeSO4 s. a., et. s. HNO3, H2SO4, chl., CS2

Forms HNO3 s. fuming H2SO4

−183

−251

−112

0.4940° cc

060° cc

s. oil turp., oil cinn.

1555

2200

i.

i.

i. s. s. i.

i. s. s. i.

s. aq. reg., h. H2SO4; i. NH3 s. HBr s. HCl, act., al. s. HCl, act., al. s. HCN, KCN, NH4OH; i. dil. a.

d.

lq. 0.746−90° 1.146 (A)

v. s. NH4OH, al.

−218.4

500 d.

11.06 2.5 1.768 22° 4 1.88 25° 1.71 4

0°

1.5520° cc

Other reagents

d i. s. d. sl. s. s. d. 4.890° cc

d. 560–600 −188°

30°

117.8 ∞ ∞ ∞

3.5

18°

280100°

63.5 ∞ ∞ 263−20° d. 2.350° cc

−151

1.44820°

Hot water

i.

2.630° cc 1.7100° cc

sl. s. aq. reg., HNO3; i. NH3 s. NaCl, al., et. s. a., alk., al.; sl. s. et. s. HCl, al. sl. s. al., s. fused Ag

d. −112 50 −17.8

1618mm d. 200

d. 138 d. 110

subl. 110

s. s. s. v. s.

<60 d. 78 −132.5

−25H2O, 140 −85

s. v. s. v. s. v. s. v. s. v. s. s. 2617° cc

2846atm

subl.

d.

s. a., NH4OH s. al. v. s. d. v. s. d. v. s. d. i.100°

sl. s. al., et. d. al. i. al. s. HNO8 s. Cu2Cl2, al., et.

Phosphoric acid, hypoPhosphoric acid, meta-

H4P2O6 HPO3

161.98 79.98

cr. vitreous, delq.

2.2–2.5

55 subl.

d. 70

s. s.

Phosphoric acid, orthoPhosphoric acid, pyro-

H3PO4† H4P2O7

98.00 177.98

col., rhb. wh. nd.

1.834

42.35 61

−aH2O, 213

2340 80028°

Phosphorous acid, hypoPhosphorous acid, orthoPhosphorous acid, pyroPhosphorus, black Phosphorus, red Phosphorus, yellow

H3PO2 H3PO3 H4P2O5 P4 P4 P4

66.00 82.00 145.98 123.90 123.90 123.90

syrupy col. nd. rhombohedral red, cb. yel., hex., 2.1168

1.49318.8° 1.65121.2°

26.5 74 38

d. d. 200 d. 130 ign. in air, 400 ign. in air, 725 280

∞ 307.30° d. i. i. 0.0003

PCl3 PCl5

137.33 208.24

col., fuming lq. delq., tet.

−111.8 148 under pressure subl. 250 2

75.95760mm subl. 160

d. d.

1755

4300

chloride, trichloride, pentaoxide, pentaoxychloride Phosphotungstic acid Platinum chloride (ic) chloride (ous) chloride (ic) cyanide (ous) Plutonium Plutonium Potassium acetate acetate, acid aluminate amide arsenate (monobasic) auricyanide aurocyanide bicarbonate bisulfate bromate bromide carbonate carbonate carbonate chlorate chloride (sylvite) chloroplatinate chromate (tarapacaite) cyanate cyanide dichromate ferricyanide ferrocyanide formate hydride hydrosulfide hydroxide iodate iodide

P2O5 POCl3 H3PO412WO3xH2O Pt PtCl4

141.94 153.33 2880.05 195.08 336.89

wh., delq., amor. col., fuming lq. yel.-gn. cr. silv. met., cb.

18.2°

2.69 2.2020° 1.8220°; lq. 1.74544.5° 1.574 20.8° 4 solid 1.6; 295° 3.60 (A) 2.387 1.675 21.4520° lq. 191755°

brn. 11°

Forms H3PO4 d. s. i.

v. s.

s. H2SO4; i. NH3, act. d. al. s. al., et. s. aq. reg., fused alk.

d. 370

14025°

v. s.

i.

i.

v. s. i.

v. s. i.

brn.

5.87

d. 581

PtCl48H2O Pt(CN)2 Pu Pu K

481.01 247.11 238.05 239.05 39.10

red, mn. yel.-brn.

2.43

−4H2O, 100

98.14 158.19 250.20 55.12 180.03 367.16 288.10 100.12 136.17 167.00 119.00 138.21 174.24 330.46 122.55 74.55 485.99 194.19 81.12 65.12 294.18 329.24 422.39 84.12 40.11 72.17 56.11 214.00 166.00

2-21

*One commercial form 70 to 72 per cent. †Common commercial form 85 per cent H3PO4 in aqueous solution.

wh. pd. delq. nd. or pl. cr. yel.-grn. col., tet., 1.5674 pl. rhb. mn., 1.482 rhb., or mn., 1.480 trig. col., cb., 1.5594 wh., delq. pd., 1.531 rhb. mn. col., mn., 1.5167 col., cb., 1.4904 yel., cb., 1.825 yel., rhb., 1.7261 wh., tet. wh., cb., delq., 1.410 red, tri. red, mn. pr., 1.5689 yel., mn., 1.5772 col., rhb. cb., 1.453 wh., delq., rhb. wh., delq., rhb. col., mn. wh., cb., 1.6670

107.2760mm

238

Produced by deuteron bombardment on U Produced by neutron bombardment on U238 0.8620° 62.3 760 lq. 0.8342° 1.8 292 148 d. 200 2.867 2.17 2.35 3.2717.5° 2.7525° 2.29 2.043 2.13 2.32 1.988 3.499 2.73218° 2.048 1.5216° 2.69 1.84 1.85317° 1.91 0.80 2.0 2.044 3.89 3.13

s. al. v. s. al., et.

i. CS2 s. alk.; i. CS2, NH3, et. 0.4 al.; 100010° CS2; 1.50°, 1081° bs.; s. NH3 s. et., chl., CS2 s. CS2, C6H5COCl

265.98

KC2H3O2 KH(C2H3O2)2 K2(AlO2)23H2O KNH2 KH2AsO4 KAu(CN)415H2O KAu(CN)2 KHCO3 KHSO4 KBrO3 KBr K2CO3 K2CO32H2O 2K2CO33H3O KClO3 KCl K2PtCl6 K2CrO4 KCNO KCN K2Cr2O7 K3Fe(CN)6 K4Fe(CN)63H2O KHCO2 KH KHS KOH KIO3 KI

i. lq. CO2

i. i. sl. s.

59043atm 44.1; ign. 34

PtCl2

silv. met., cb.

26°

45062° Forms H3PO4 v. s. Forms H3PO4 ∞ 73040°

338 288 d. 200 d. 100–200 210 370 d. 730 891 368 790 d. 250 975 634.5 398 d. −3HO2, 70 167.5 d. 455 380 560 723

subl. 400

d. 1380 d. d. 400 1500

d. d. 1320 1330

d. 0°

217 d. s. d. 18.876° s. 14.3 22.40° 36.30° 3.110° 53.50° 105.50° 1830° 129.40° 3.30° 27.60° 0.740° 58.00° s. s. 4.90° 334.4° 27.812.2° 33118° d. s. 970° 4.730° 127.50°

i.

Forms KOH 39690° d. v. s. v. s. 200100° 6060° 121.6100° 49.75100° 104100° 156100° 331100° 268100° 57100° 56.7100° 5.2100° 75.6100° d. 122.2108.8° 80100° 77.5100° 90.696.8° 65790° s. d. 178100° 32.2100° 208100°

s. al., act.; sl. s. NH2; i. et. s. HCl, NH4OH; sl. s. NH3; i. al., et. s. al., et. i. alk. s. a., al., Hg 33 al.; i. et. s. ac. s. alk.; i al. d. al.; 3.625° NH3 i. al. s. al. sl. s. al.; i. et. i. satd. K2CO3, al. d. al. sl. s. al.; i. act. sl. s. al., et. i. al. 0.83 al.; s. alk. s. al., alk. i. al., et. i. al. v. sl. s. al. s. gly.; 0.919.5° al.; 1.3 h. al. i. al. s. act.; sl. s. al.; i. NH3 s. act.; i. NH3, al., et. sl. s. al.; i. et. i. et., bz., CS2 s. al. v. s. al., et.; i. NH3 s. KI; i. al., NH3 420° al.; s. NH3; sl. s. et.

2-22

TABLE 2-1

Physical Properties of the Elements and Inorganic Compounds (Continued)

Formula

Formula weight

Color, crystalline form and refractive index

Potassium (Cont.) iodide, triiodoplatinate manganate metabisulfite nitrate (saltpeter) nitrite oxalate oxalate, acid oxalate, acid oxide perchlorate permanganate persulfate phosphate, monobasic

KI3 K2PtI6 K2MnO4 K2S2O5 KNO3 KNO2 K2C2O4H2O KHC2O4* KHC2O4aH2O K2O KClO4 KMnO4 K2S2O3 KH2PO4

419.81 1034.70 197.13 222.32 101.10 85.10 184.23 128.13 137.13 94.20 138.55 158.03 190.32 136.09

phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, metaphosphate, pyrophthalate, acid platinocyanide silicate silicate, tetrasulfate (arcanite) Potassium sulfate, pyrosulfide, monosulfite sulfite, acid tartrate tartrate, acid thiocyanate

K2HPO4 K3PO4 KPO3 K4P4O122H2O K4P2O73H2O KHC8H4O4 K2Pt(CN)43H2O K2SiO3 K2Si4O9H2O K2SO4 K2S2O7 K2S5H2O K2SO32H2O KHSO3 K2C4H4O6aH2O KHC4H4O6* KCNS

174.18 212.27 118.07 508.31 384.38 204.22 431.39 154.28 352.55 174.26 254.32 200.34 194.29 120.17 235.28 188.18 97.18

thiosulfate thiosulfate Praseodymium Radium bromide Radon (Niton)

K2S2O3 3K2S2O3H2O Pr Ra RaBr2 Rn

190.32 588.99 140.91 226.03 385.83 222.02

dark blue, delq., mn. cb. gn., rhb. mn., pl. col., rhb., 1.5038 pr. wh., mn. mn., 1.545 trimetric wh., cb. col., rhb., 1.4737 purple, rhb. wh., tri., 1.4669 col., delq., tet., 1.5095 wh., delq. wh., rhb. wh. pd. amor. delq. wh. cr. yel., rhb., 1.62 hyg. 1.521 rhb., 1.530 col., rhb., 1.4947 col. rhb., delq. wh., rhb. wh., mn. col., mn., 1.526 col., rhb. col., delq., mn., 1.660 col., cb. delq., mn. yel. wh., met. wh., mn. gas

Rhenium

Re

186.21

hex.

Rhodium chloride chloride Rubidium

Rh RhCl3 RhCl34H2O Rb

102.91 209.26 281.33 85.47

gray-wh., cb. red dark red silv. wh.

Ruthenium Ruthenium Samarium Scandium Selenic acid Selenic acid Selenium Selenium

Ru Ru Sm (also Sa) Sc H2SeO4 H2SeO4H2O Se8 Se8

101.07 101.07 150.36 44.96 144.97 162.99 631.68 631.68

bk., porous gray, hex.

Name

hex. pr. nd. red pd., amor., 2.92 gray, trig., 3.00; red, hex.

Specific gravity 3.498 5.18 10.6°

2.11 1.915 2.13 2.0

2.32 20° 4 2.524 11° 4 2.703 2.338 2.56417° 2.25814.5° 2.26414.5° 2.33 1.63 2.4516° 2.417 2.662 2.277

1.98 1.956 1.886 2.23 6.520° 5? 5.79 lq. 5.5; 111 (D)

Melting point, °C 45 d. 190 d. 150 tr. 129; 333 297 d. d. d.

Boiling point, °C d. 225

d. 400 d. 350

d. 400 d. <240 d. <100 256 d. 1340 tr. 450; 798 −2H2O, 100 −2H2O, 180 d. 976 d. 400 tr. 588 300 60 d. d. 190

1320 d. −3H2O, 300

−3H2O, 150 d.

172.3 d. 400 −H2O, 180 940 960 728 −71

d. 500

Solubility in 100 parts Cold water v. s. s. d. 250° 13.30° 2810° 28.70° 14.350° 2.20° Forms KOH 0.750° 2.830° 1.770° 14.80°

Hot water

s. KI, al. 12094° 246100° 413100° 83.2100° 48.1100° 51.5100° v. s. 21.8100° 32.3575° 1040° 83.590°

3325° 193.125° s. s. s. 10.225° sl. s. s. s. 7.350° s. s. 100 45.515° 12.517.5° 0.370° 1770°

>100 91.575° 278100° 6.1100° 21720°

96.10°

311.290°

v. s. v. s. s. 83 v. s. 36 v. s. s. s. 24.1100° d.

d. 1140 subl. 900 −62

lq. 1.47588.5; 1.5320° 8.6 12.220° 7.7 2.5? 2.950 15° 4 2.627 15° 4 4.2625° 4.80; 4.50

1955 d. 450

>2500 subl. 800

38.5 >1950 2450 >1300 1200 58 26 50 220

s. KOH sl. s. al.; i. et. 0.130° al.; i. et. v. s. NH3; sl. s. al.

s. al., et. 0.10520° m. al.; i. et. s. H2SO4; d. al. i. al. i. al. sl. s. al. i. al. s. a. i. al. s. al., et. i. al. i. al. i. al., act., CS2 s. al., gly.; i. et. sl. s. al.; i. NH3 i. abs. al. sl. s. al. s. a., alk.; i. al., ac. 20.822° act.; s. al. i. al.

d. d. +H2 7020° 510° cc

s. 8.560° cc

3440 12.5

Other reagents

d. a. s. al. i. HF, HCl; s. H2SO4; HNO3 sl. s. aq. reg., a. v. sl. s. alk.; i. aq. reg., a. s. HCl, al.; i. et. s. a., al.

i. i.

700

i. i. v. s. d.

>2700

i. i.

i. i.

sl. s. aq. reg., a.

2400 260 205 688 688

130030° v. s. i. i.

∞60°

s. H2SO4; d. al.; i. NH3

i. i.

s. CS2, H2SO4, CH2I2 s. CS2, H2SO4

Selenium Selenous acid Silicic acid, metaSilicic acid, orthoSilicon, crystalline

Se8 H2SeO3 H2SiO3 H4SiO4 Si

631.68 128.97 78.10 96.11 28.09

steel gray hex. amor., 1.41 amor. gray, cb., 3.736

4.825° 3.004 15° 4 2.1–2.3 17° 1.576 2.420°

cr. brn., amor. blue-bk., trig., 2.654 lf. or lq. col., fuming lq., 1.412 gas col. gas iridescent, amor. col., cb. or tet., 1.487

217 d.

688

2600

i. 900° i. sl. s. i.

i. 40090° i. sl. s. i.

1420

2.0–2.5

2600

i.

i.

2 3.17 1.580° 1.50

>2700 −1 −70

2600 subl. 2200 144760mm 57.6

i. i. d. d.

i. i.

3.57 (A) lq. 0.68−185° 2.2 2.32

−95.7 −185 1600–1750 1710

−651810mm −112760mm subl. 1750 2230

v. s. d. i. i. i.

Silicon, graphitic

Si

Silicon, amorphous carbide chloride, trichloride, tetra-

Si SiC Si2Cl6 SiCl4

28.09 40.10 268.89 169.90

SiF4 SiH4 SiO2xH2O SiO2

104.08 32.12

SiO2 SiO2 SiO2 Ag AgBr

60.08 60.08 60.08 107.87 187.77

hex., 1.5442 trig., rhb., 1.469 silv. met., cb. pa. yel., cb., 2.252

2.20 2.65020° 2.26 10.520° 6.473 25° 4

tr. <1425 tr. 1670 960.5 434

2230 2230 2230 1950 d. 700

i. i. i. i. 0.0000220°

i. 0.00037100°

carbonate chloride (cerargyrite)

Ag2CO3 AgCl

275.75 143.32

yel. pd. wh., cb., 2.071

6.077 5.56

218 d. 455

1550

0.00320° 0.00008910°

0.05100° 0.00217100°

cyanide nitrate (lunar caustic) Sodium

AgCN AgNO3 Na

133.89 169.87 22.99

wh., 1.685 col., rhb., 1.744 silv. met, cb.

3.95 4.352 19° 4 0.9720°

−(CN)2, 320 212 97.5

444 d. 880

acetate acetate aluminate amide Sodium ammonium phosphate antimonate, metaarsenate arsenate, acid (monobasic) arsenate, acid (dibasic) arsenate, acid (dibasic) arsenite, acid benzoate bicarbonate bifluoride bisulfate bisulfite borate, tetraborate, tetra

NaC2H3O2 NaC2H3O23H2O NaAlO2 NaNH2 NaNH4HPO44H2O 2NaSbO37H2O Na3AsO412H2O NaH2AsO4H2O Na2HAsO47H2O* Na2HAsO412H2O Na2HAsO3 NaC7H5O2 NaHCO3 NaHF2 NaHSO4 NaHSO3 Na2B4O7 Na2B4O75H2O

82.03 136.08 81.97 39.01 209.07 511.60 424.07 181.94 312.01 402.09 169.91 144.10 84.01 61.99 120.06 104.06 201.22 291.30

wh., mn., 1.464 wh., mn. amor. olive gn. col., mn. cb. hex., 1.4589 rhb., 1.5535 col., mn., 1.4658 mn., 1.4496 col. col. cr. wh., mn., 1.500 col. cr. col., tri. col., mn., 1.526

1.528 1.45

324 58 1650 210 79 d.

col., rhb., 1.461

2.742 1.48 2.367 1.815

Na2B4O710H2O*

381.37

wh., mn., 1.4694

1.73

fluoride hydride (silane) oxide, di- (opal) oxide, di- (cristobalite) oxide, di- (lechatelierite) oxide, di- (quartz) oxide, di- (tridymite) Silver bromide (bromyrite)

borate, tetra- (borax)

28.09

60.08

1.574

400

1.759 2.535 1.871 1.72 1.87

86.3 d. 100 125 28

2.20

−CO2, 270 d. >315 d. 741

d., −H2O

75

−10H2O, 200

17.5°

2-23

bromate bromide bromide

NaBrO3 NaBr NaBr2H2O

150.89 102.89 138.92

col., cb. col., cb., 1.6412 col., mn.

3.339 3.20517.5° 2.176

381 755 50.7

carbonate (soda ash) carbonate

Na2CO3 Na2CO3H2O

105.99 124.00

2.533 1.55

851 −H2O, 100

carbonate carbonate (sal soda)

Na2CO37H2O Na2CO310H2O

232.10 286.14

wh. pd., 1.535 wh., rhb., 1.506– 1.509 rhb. or trig. wh., mn., 1.425

1.51 1.46

d. 35.1

*Usual commercial form.

−3H2O, 120

−7H2O, 100 −12H2O, 100

0.00002220° 1220° d., forms NaOH 46.520° v. s. s. d. 16.7 0.03112.8° 26.717° s. 6115° 5.590.1° v. s. 62.525° 6.90° 3.720° 500° sl. s. 1.30° 2262° (anh.)

i. i. i. i.

952100° 170100° v. s. v. s. 100

s. HF; i. alk. s. HF; i. alk. s. HF; i. alk. s. HNO3, h. H2SO4; i. alk. 0.5118° NH4OH; s. KCN, Na2S2O3 s. NH4OH, Na2S2O3; i. al. s. NH4OH, KCN; sl. s. HCl s. NH4OH, KCN, HNO3 s. gly.; v. sl. s. al. i. bz.; d. al. 2.118° al. 7.825° abs. al. i. al. d. al. i. al. sl. s. al., NH4 salts; i. ac. 1.67 al., 5015° gly. sl. s. al. sl. s. al. 2.325°, 8.378° al. i. al.

1390

27.5 9020° 79.50° (anh.)

d.

7.10° s. s. 21.50°

s. 23830°

0°

s. HNO3, al., et. i. al., et.; d. KOH s. HF, h. alk., fused CaCl2 s. HF; i. alk.

v. s. 140.730° 76.9100° 16.460° s. 100100° s. 8.7940° 52.3100° (anh.) 20.380° (anh.) 90.9100° 121100° 118.380° (anh.) 48.5104° s.

1.30.5 (anh).

i. CS2; s. H2SO4 v. s. al.; i. NH3 s. alk.; i. NH4Cl s. alk.; i. NH4Cl s. HNO3 + HF, Ag; sl. s. Pb, Zn; i. HF s. HNO3 + HF, fused alk.; i. HF. s. HF, KOH s. fused alk.; i. a. d. alk. d. conc. H2SO4, al.

d. al.; i. NH3 i. al., act. i. al. s. gly.; i. abs. al. i. al. sl. s. al. sl. s. al. i. al., et. s. gly.; i. al., et. i. al.

2-24

Physical Properties of the Elements and Inorganic Compounds (Continued)

TABLE 2-1

Name Sodium ammonium phosphate (Cont.) carbonate, sesqui- (trona) chlorate

Formula

Na3H(CO3)22H2O NaClO3

Formula weight

226.03 106.44

Color, crystalline form and refractive index

Specific gravity

chloride chromate chromate citrate cyanide dichromate

NaCl Na2CrO4 Na2CrO410H2O 2Na3C6H5O711H2O NaCN Na2Cr2O72H2O

58.44 161.97 342.13 714.31 49.01 298.00

ferricyanide ferrocyanide

Na3Fe(CN)6H2O Na4Fe(CN)610H2O

298.93 484.06

red, delq. yel., mn.

1.458

fluoride (villiaumite) formate hydride

NaF NaHCO2 NaH

tet., 1.3258 wh., mn. silv. nd., 1.470

2.79 1.919 0.92

992 253 d. 800

hydrosulfide hydrosulfide hydrosulfite hydroxide hydroxide hypochlorite iodide iodide lactate nitrate (soda niter) nitrite

NaSH2H2O NaSH3H2O Na2S2O42H2O NaOH NaOH3aH2O NaOCl NaI* NaI2H2O NaC3H5O3 NaNO3 NaNO2

2.130

d. 22 d. 318.4 15.5 d. 651

1300

2.257 2.1680°

d. 308 271

d. 380 d. 320

oxide

Na2O

2.27

subl.

perborate perchlorate perchlorate peroxide peroxide phosphate, monobasic phosphate, monobasic phosphate, dibasic phosphate, dibasic phosphate, tribasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyrophosphate (pyrodisodium) phosphate (pyrodisodium) potassium tartrate silicate, metaSodium silicate, metasilicate, orthosilicofluoride stannate sulfate (thenardite) sulfate

NaBO3H2O NaClO4 NaClO4H2O Na2O2* Na2O28H2O NaH2PO4H2O* NaH2PO42H2O Na2HPO47H2O Na2HPO412H2O Na3PO4 Na3PO412H2O* Na4P4O12 Na4P2O7* Na4P2O710H2O Na2H2P2O7 Na2H2P2O76H2O NaKC4H4O64H2O Na2SiO3 Na2SiO39H2O Na4SiO4 Na2SiF6 Na2SnO33H2O Na2SO4 Na2SO4

92.09 110.11 210.14 40.00 103.05 74.44 149.89 185.92 112.06 84.99 69.00 61.98 99.81 122.44 140.46 77.98 222.10 137.99 156.01 268.07 358.14 163.94 380.12 407.85 265.90 446.06 221.94 330.03 282.22 122.06 284.20 184.04 188.06 266.73 142.04 142.04

col., delq., nd. rhb. col. cr. wh., delq. mn. pa. yel., in soln. only col., cb., 1.7745 col., mn. col., amor. col., trig., 1.5874 pa. yel., rhb. wh., delq. wh. pd. rhb., 1.4617 hex. yel.-wh. pd. wh., hex. col., rhb., 1.4852 col., rhb., 1.4629 col., mn., 1.4424 col., mn., 1.4361 wh. wh., trig., 1.4458 col. wh. mn., 1.4525 col., mn., 1.510 col., mn., 1.4645 rhb., 1.493 col., rhb., 1.520 rhb. col., hex., 1.530 wh., hex., 1.312 hex. tablets col., rhb., 1.477 col., mn.

2.163 2.723 1.483 1.857 23.5° 4 18°

2.52

3.6670° 2.448

2.02 2.805 2.040 1.91 1.679 1.52 2.53717.5° 1.62 2.476 2.45 1.82 1.862 1.848 1.790

2.679 2.698

d. 248

Boiling point, °C

wh., mn., 1.5073 wh., cb., or trig., 1.5151 col., cb., 1.5443 yel., rhb. yel., delq., mn. wh., rhb. wh., cb., 1.452 red, mn., 1.6994

41.99 68.01 24.00

2.112 2.49015°

Melting point, °C

800.4 392 19.9 −11H2O, 150 563.7 −2H2O, 84.6; 356 (anh.)

d. 40 482 d. d. 130 d. d. 30 −H2O, 100 60 d. 34.6 1340 73.4 616 d. 988 d. d. 220 70 to 80 1088 47 1018 d. d. 140 tr. 100 to mn. tr. 500 to hex.

d. 1413 d. 1496 d. 400

Solubility in 100 parts Cold water

130° 790°

42100° 230100° 0°

1390

d. 200 −12H2O, 180 −11H2O, 100

−4H2O, 215 −6H2O, 100

100°

35.7 320° v. s. 9125° 4810° 2380°

39.8 126100° ∞ 250100° 8235° 50880°

18.90° 17.920° (anh.)

67100° 6398.5° (anh.) 5100° 160100°

40° 440° d. d.

Hot water

s. s. 2220° 420° s. 260° 158.70° v. s. v. s. 730° 72.10°

s. s. d. 347100° v. s. 15856° 302100° v. s. v. s. 180100° 163.2100°

Forms NaOH sl. s. 1700° 20915° s. d. s. d. 710° 91.10° 18540° 4.30° 4.50° 28.315° s. 2.260° 5.40° 4.50° 6.90° 260° s. v. s. s. 0.440° 500° 50° 48.840°

d. 320100° 28450° d. d. 39083° 30840° 2000100° 76.730° 77100° ∞ s. 4596° 93100° 2140° 3640° 6626° s. d. v. s. s. 2.45100° 6750° 42100° 42.5100°

Other reagents

s. al. sl. s. al.; i. conc. HCl sl. s. al. i. al. s. NH3; sl. s. al. i. al. i. al. v. sl. s. al. sl. s. al.; i. et. i. bz., CS2, CCl4, NH3; s. molten metal s. al.; d. a. s. al.; d. a. i. al. v. s. al., et., gly.; i. act. v. s. al., act. v. s. NH3 s. al.; i. et. s. NH3; sl. s. gly., al. 0.320° et.; 0.3 abs. al.; 4.420° m. al.; v. s. NH3 d. al. s. gly., alk. s. al.; 51 m. al.; 52 act.; i. et. s. al. s. dil. a. i. al. i. al. i. CS2 s. a., alk. d. a. i. al., NH3 sl. s. al. i. Na or K salts, al. 2918°, aN NaOH i. al. i. al., act. i. al. d. HI; s. H2SO4

sulfate sulfate sulfate (Glauber’s salt) sulfide, monosulfide, tetrasulfide, pentasulfite sulfite tartrate thiocyanate thiosulfate thiosulfate (hypo) tungstate tungstate tungstate, parauranate vanadate vanadate, pyroStannic chloride

Na2SO4 Na2SO47H2O Na2SO410H2O Na2S Na2S4 Na2S5 Na2SO3 Na2SO37H2O Na2C4H4O62H2O NaCNS Na2S2O3 Na2S2O35H2O* Na2WO4 Na2WO42H2O* Na6W7O2416H2O Na2UO4 Na3VO416H2O Na4V2O7 SnCl4

142.04 268.15 322.19 78.04 174.24 206.30 126.04 252.15 230.08 81.07 158.11 248.18 293.82 329.85 2097.05 348.01 472.15 305.84 260.52

col., hex. tet. col., mn., 1.396 pink or wh., amor. yel., cb. yel. hex. pr., 1.565 mn. rhb. delq., rhb., 1.625 mn. mn. pr., 1.5079 wh., rhb. wh., rhb. wh., tri. yel. col. nd. hex. col., fuming lq.

884 1.464 1.856 2.633 15° 4 1.561 1.818

2.226

866 (anh.) 654 −30.2

7.0

1127

215.5 246.8 37.7 −SO2, 360 800

sulfate

Sn(SO4)22H2O

346.87

col., delq., hex.

SnBr2 SnCl2 SnCl22H2O* SnSO4 Sr

278.52 189.62 225.65 214.77 87.62

yel., rhb. wh., rhb. wh., tri. wh. cr. silv. met.

5.1217°

acetate carbonate (strontianite) chloride chloride hydroxide hydroxide

Sr(C2H3O2)2 SrCO3 SrCl2 SrCl26H2O* Sr(OH)2 Sr(OH)28H2O*

205.71 147.63 158.53 266.62 121.63 265.76

wh. cr. wh., rhb., 1.664 wh., cb., 1.6499 wh., rhb., 1.5364 wh., delq. col., tet., 1.499

2.099 3.70 3.052 1.93317° 3.625 1.90

nitrate nitrate oxide (strontia)

Sr(NO3)2* Sr(NO3)24H2O SrO

211.63 283.69 103.62

col., cb., 1.5878 wh., mn. col., cb., 1.870

2.986 2.2 4.7

SrO2 SrO28H2O SrSO4 Sr(HSO4)2 NH2SO3H S S8 S8 S2Br2 S2Cl2 SCl2 SCl4 SO2

119.62 263.74 183.68 281.76 97.09 32.07 256.52 256.52 223.94 135.04 102.97 173.88 64.06

wh. pd. wh. cr. col., rhb., 1.6237 col., granular wh., rhb. pa. yel. pd., 2.0–2.9 pa. yel., mn. pa. yel., rhb. red, fuming lq. red-yel. lq. dark red fuming lq. yel.-brn. lq. col. gas

oxide, tri-(α)

2-25

oxide, tri-(β) Sulfuric acid Sulfuric acid *Usual commercial form.

SO3 (SO3)2 H2SO4* H2SO4H2O

80.06 160.13 98.08 116.09

col. pr. col., silky, nd. col., viscous lq. pr. or lq.

d.

d. 48.0 692 −2H2O, 100 −16H2O, 300

150.71

peroxide peroxide sulfate (celestite) sulfate, acid Sulfamic acid Sulfur, amorphous Sulfur, monoclinic Sulfur, rhombic Sulfur bromide, monochloride, monochloride, dichloride, tetraoxide, di-

275 251.8 d. −7H2O, 150

1.667 1.685 4.179 3.245 3.98714°

SnO2

Stannous bromide chloride chloride (tin salt) sulfate Strontium

−10H2O, 100

287

oxide (cassiterite)

wh., tet., 1.9968

32.4

2.7115.5° 2.6

3.96 2.03 12° 4 2.046 1.96 2.07 2.635 1.687 1.621 15° 15 lq., 1.4340°; 2.264 (A) lq., 1.923; 2.75 (A) 1.9720° 1.834 18° 4 1.842 15° 4

149760atm 873 −4H2O, 61 375 −7H2O in dry air 570

114.1

620 623 d. 1150 d. −CO2, 1350 −6H2O, 100

19.420° 44.90° 3615° 15.410° s. s. 13.90° 34.72° 296° 11010° 500° 74.70° 57.580° 880° 8 i. v. s. s. s.

45.360° 202.626° 41234° 57.390° s. s. 28.384° 67.818° 6643° 225100° 23180° 301.860° 97100° 123.5100° d. i. d.

i.

i.

v. s.

d.

s. 83.90° 118.70° 1919° d.

d. 269.815° ∞ 18100° Forms Sr(OH)2 36.497° 0.065100° 100.8100° 19840° 21.83100° 47.7100°

36.90° 0.001118° 43.50° 1040° 0.410° 0.900°

d. −8H2O, 100 1580 d. d. 205 d. 120 119.0 112.8 −46 −80 −78 −30 −75.5

444.6 444.6 444.6 540.18mm 138 59 d. > −20 −10.0

400° 62.20° Forms Sr(OH)2 0.00820° 0.01820° 0.01130° d. 200° i. i. i. d. d. d. d. 22.80°

16.83

44.6

d.

50 10.49 8.62

d. 340 290

Forms H2SO4 ∞ ∞

2430 d.

d.

i. al. sl. s. al.; i. et. s. al. s. al. i. al., NH i. al. i. al. v. s. al. s. NH3; v. sl. s. al. sl. s. NH3; i. a., al. s. alk. carb., dil. a. i. al. i. al. s. abs. al., act., NH3; s. ∞ CS2 s. conc. H2SO4; i. alk.; NH4OH, NH3 s. dil. H2SO4, HCl; d. abs. al. s. C6H5N s. alk., abs. al., et. s. tart. a., alk., al. s. H2SO4 s. al., a. 0.2615° m. al. s. a., NH4 salts, aq. CO2 v. sl. s. act., abs. al.; i. NH3 s. NH4Cl s. NH4Cl; i. act.

10089° 12420°

s. NH3; 0.012 abs. al. i. HNO3 sl. s. al.; i. et.

d. d. 0.011432°

s. al., NH4Cl; i. act. s. al.; i. NH4OH sl. s. a.; i. dil. H2SO4, al. 1470° H2SO4 sl. s. al., act.; i. et. sl. s. CS2 s. CS2, al. 240°, 18155° CS2

4070° i. i. i.

s. CS2, et., bz. d. al. 4.550°

s. H2SO4; al., ac. s. H2SO4

∞ ∞

s. H2SO4 d. al. d. al.

2-26

Physical Properties of the Elements and Inorganic Compounds (Concluded)

TABLE 2-1

Name

Formula

Formula weight

Color, crystalline form and refractive index

Specific gravity

Melting point, °C

Boiling point, °C

Solubility in 100 parts Cold water

Hot water

H2SO42H2O H2S2O7 SO2Cl2 SOBr2 SOCl2 Ta

134.11 178.14 134.97 207.87 118.97 180.95

col. lq. cr. col. lq. or.-yel. lq. col. lq. bk.-gray, cb.

1.650 40° 1.920° 1.667 20° 4 2.6818° 1.638 16.6

−38.9 35 −54.1 −50 −104.5 2850

167 d. 69.1760mm 6840mm 78.8 >4100

∞ d. d. d. d. i.

i.

Tellurium

Te

127.60

met., hex.

(α) 6.24; (β) 6.00

452

1390

i.

i.

Terbium Thallium acetate chloride, monochloride, sesquichloride, trichloride, trisulfate (ic) sulfate (ous) sulfate, acid Thio, cf. sulfo or sulfur Thorium

Tb Tl TlC2H3O2 TlCl Tl2Cl3 TlCl3 TlCl34H2O Tl2(SO4)37H2O Tl2SO4 TlHSO4

158.93 204.38 263.43 239.84 515.13 310.74 382.80 823.06 504.83 301.45

blue-wh., tet. silky nd. wh., cb. yel., hex. hex. pl. nd. lf. col., rhb., 1.8671 trimorphous

11.85 3.68 7.00 5.9

303.5 110 430 400–500 25 37 −6H2O, 200 632 115 d.

1650

i. v. s. 0.210° 0.2615° v. s. 86.217° d. 2.700°

i.

Th

232.04

cb.

11.2

1845

>3000

i.

i.

oxide, di- (thorianite) sulfate sulfate Thulium Tin

ThO2 Th(SO4)2 Th(SO4)29H2O Tm Sn

264.04 424.16 586.30 168.93 118.71

wh., cb.

>2800

4400

mn. pr.

9.69 4.22517° 2.77

silv. met., tet.

7.31

231.85

2260

i. 0.740° sl. s. i. i.

5.2250° sl. s. i. i.

Tin

Sn

118.71

gray, cb.

5.750

Stable −163 to +18

2260

i.

i.

Tin salts, cf. stannic and stannous Titanic acid

H2TiO3

97.88

wh. pd.

i.

i.

i. d.

d.

s. s. i.

s. d. i.

i.

i.

i.

i.

s. H2SO4, alk. s. h. conc. KOH; sl. s. NH3, HNO3, aq. reg. s. F2; i. a. s. h. HNO3; sl. s. HCl, H2SO4 s. alk.; i. a. s. HF, alk., NH3

Ti TiCl2

47.87 118.77

dark gray, cb. bk., delq.

chloride, trichloride, tetraoxide, di- (anatase)

TiCl3 TiCl4* TiO2

154.23 189.68 79.87

oxide, di- (brookite)

TiO2

79.87

oxide, di- (rutile)

TiO2

79.87

Titanium chloride, di-

6.77

17.5°

4.50

806 d. d. −4H2O, 100 d. d.

d. al. d. al. s. ac.; d. al. s. bz., CS2, CCl4; d. act. s. bz., chl. s. fused alk., HF; i. HCl, HNO3, H2SO4 s. H2SO4, HNO3, KCN, KOH, aq. reg.; i. CS2

100°

1.8 1.9100° d. d. 18.45100°

>3000

W

183.84

vl., delq. col. lq. brn. or bk., tet., 2.534–2.564 brn. or bk., rhb., 2.586 col. if pure, tet., 2.615 gray-bk., cb.

19.3

3370

5900

i.

i.

WC W2C

195.85 379.69

gray pd., cb. iron gray

15.718° 16.0618°

2777 2877

6000 6000

i. i.

i. i.

oxide, triTungstic acid (tungstite)

WO3 H2WO4

231.84 249.85

yel., rhb. yel., rhb. 2.24

7.16 5.5

i. i.

i. sl. s.

Uranic acid

H2UO4

304.04

yel. pd.

5.92615°

Uranium carbide oxide, di- (uraninite)

U U2C3 UO2

238.03 512.09 270.03

wh. cr. cr. bk., rhb.

18.485 13° 4 11.28 10.9

>2130 −aH2O, 100; 1473 −H2O, 250 to 300 1133 2400 2176

Tungsten carbide carbide

lq., 1.726 3.84

136.4

4.17 4.26

1640 d.

s. HNO3, H2SO4; i. NH3 v. s. al. sl. s. HCl; i. al., NH4OH s. al., et. s. al., et. s. dil. H2SO4 v. sl. s. dil. H2SO4

−9H2O, 400

1800 Unstable in air d. 440 −30

∞

Other reagents

Sulfuric acid Sulfuric acid, pyroSulfuric oxychloride Sulfurous oxybromide oxychloride Tantalum

<3000

3500

s. HCl, H2SO; sl. s. HNO3; i. HF, alk. s. h. H2SO4; i. alk.

s. HCl, H2SO4, dil. HNO3 h. aq KOH s. a., h. alk. solns. s. alk.; v. sl. s. dil. a.; i. al. s. a. i. CS2, et., chl. s. dil. HCl sl. s. alk.

i.

i.

s. a., alk. carb.; i. alk.

i. d. i.

i.

s. a.; i. alk. d. a. s. HNO3, conc. H2SO4

i.

oxide (pitchblende) sulfate (ous) Uranyl acetate carbonate (rutherfordine) nitrate sulfate Vanadic acid, metaVanadic acid, pyroVanadium chloride, dichloride, trichloride, tetraoxide, dioxide, trioxide, tetraoxide, pentaoxychloride, monoVanadyl chloride chloride, dichloride, triWater†

U3O8 U(SO4)24H2O UO2(C2H3O2)22H2O UO2CO3 UO2(NO3)26H2O UO2SO43H2O HVO3 H4V2O7 V VCl2 VCl3 VCl4 V2O2 V2O3 V2O4 V2O5 VOCl (VO)2Cl VOCl2 VOCl3 H2O

842.08 502.22 424.15 330.04 502.13 420.14 99.95 217.91 50.94 121.85 157.30 192.75 133.88 149.88 165.88 181.88 86.39 169.33 137.85 173.30 18.02

Water, heavy Xenon

D2O Xe

20.029 131.29

olive gn. gn., rhb. yel., rhb. tet. yel., rhb., 1.4967 yel. cr. yel. scales pa. yel., amor. lt. gray, cb. gn., hex., delq. pink, tabular, delq. red lq. lt. gray cr. bk. cr. blue cr. red-yel., rhb. brn. pd. yel. cr. gn., delq. yel. lq. col. lq., 1.3330020°; hex. solid, 1.309 col. lq., 1.3284420° col. gas

Ytterbium Yttrium Zinc acetate acetate bromide carbonate

Yb Y Zn Zn(C2H3O2)2 Zn(C2H3O2)22H2O* ZnBr2 ZnCO3

173.04 88.91 65.41 183.50 219.53 225.22 125.42

dark gray, hex. silv. met., hex. mn. wh., mn., 1.494 rhb. wh., trig., 1.818

ZnCl2

136.32

wh., delq., 1.687, uniaxial col., rhb. col., rhb. cb.

chloride

117.44 99.42 319.22

2-27

cyanide hydroxide iodide

Zn(CN)2 Zn(OH)2 ZnI2

nitrate oxide (zincite) oxide peroxide phosphide silicate

Zn(NO3)26H2O ZnO ZnO ZnO2 Zn3P2 ZnSiO3

297.51 81.41 81.41 97.41 258.17 141.49

sulfate (zincosite)

ZnSO4

sulfate sulfate sulfate (goslarite) sulfide (α) (wurzite) sulfide (β) (sphalerite)

ZnSO4H2O ZnSO46H2O ZnSO47H2O* ZnS ZnS

161.47 179.49 269.56 287.58 97.47 97.47

sulfide (blende) sulfite Zirconium oxide, di- (baddeleyite) oxide, di- (free from Hf)

ZnS ZnSO32aH2O Zr ZrO2 ZrO2

97.47 190.51 91.22 123.22 123.22

col., tet. wh., hex., 2.004 wh., amor. yel. steel gray, cb. hex. or rhb.; glass, 1.650 wh., rhb., 1.669 col. mn. rhb., 1.4801 wh., hex., 2.356 wh., cb.; glass (?) 2.18–2.25 wh., granular mn. cb., pd. ign. easily yel. or brn., mn., 2.19 wh., mn.

*Usual commercial form. †Cf. special tables on water and steam, Tables 2-3, 2-4, 2-5, 2-185, 2-186 and 2-351 through 2-357. NOTE: °F = 9⁄ 5 °C + 32.

7.31 15°

2.89 5.6 2.807 3.2816.5° 5.96 3.2318° 3.0018° 1.81630° 3.64 4.87 18° 4 4.399 3.357 18° 4 2.824 3.64 2.8813° 1.829 1.004° (lq.); 0.9150° (ice) 1.10720° lq., 3.06−109.1 2.7−140° 4.53 (A)

d. −4H2O, 300 −2H2O, 110 60.2 d. 100

118

1710

3000

d. −109 ign. 1970 1967 800

148.5755mm

d. 1750

d. in air

i. 2311° 9.217°

i. 963° d.

s. HNO3, H2SO4 s. dil. a. s. al., act.

170.30° 18.913.2° i. i. i. s. s. s. d. i. sl. s. i. 0.820° i. i. d. s. d.

∞60° 23025°

v. s. ac., al., et.; i. dil., alk. 4 al.; s. a. s. a., alk.; i. NH3 s. a., alk., NH4OH s. HNO3, H2SO4; i. aq., alk. s. al., et. s. abs. al., et. s. abs. al., et., chl., ac. s. a. s. HNO3, HF, alk. s. a., alk. s. a., alk.; i. abs. al. v. s. HNO3 s. HNO3 s. abs. al., dil. HNO3 s. al., et., ∞Br2 ∞ al.; sl. s. et.

i. d. d. i. s. i.

<−15 0

127.19 100

3.82 −140

101.42 −109.1

∞ 24.20° cc

∞ 7.350° cc

∞ al.; sl. s. et.

5.51 7.140 1.840 1.735 4.2194° 4.42

1490 419.4 242 237 394 −CO2, 300

2500 907 subl. in vac. −2H2O, 100 650

sl. d. i. 3025° 4025° 3900° 0.00115°

d. i. 44.6100° 66.6100° 670100°

2.91 25° 4

283

732

43225°

615100°

v. s. dil. a., h. KOH s. a., ac., alk. 2.825°, 16679° al. v. s. al. v. s. NH4OH, al., et. s. a., alk., NH4 salts; i. act., NH3 10012.5° al.; v. s. et.; i. NH3 s. KCN, NH3, alk.; i. al. s. a., alk., NH4OH s. a., al., NH3, aq. (NH4)2CO3 v. s. al. s. a., alk., NH4Cl; i. NH3

3.053 4.666 14.2° 4 2.065 14° 4

d. 80 d. 125 446

5.606 5.47 1.571 4.55 13° 4 3.52

36.4 >1800 >1800 expl. 212 >420 1437

3.74 15° 4 3.28 15° 4 2.072 15° 4 1.96616.5° 4.087 4.102 25° 4

d. 740 d. 238 −5H2O, 70 tr. 39 1850150atm tr. 1020

4.04 6.4 5.49 5.73

−2aH2O, 100 1700 2700

18°

624 −6H2O, 105

1100

−7H2O, 280 subl. 1185

d. 200 >2900 4300

0.0005 0.0005218° 4300°

sl. s.

324.5 0.0004218° 0.0004218° 0.0022 i. i.

∞36.4°

420° s. s. 115.20° 0.0006918° i.

61100° 89.5100° s. 653.6100° i. i.

sl. s. al.; s. gly.

i. 0.16 i. i. i.

i. d. i. i. i.

v. s. a.; i. ac. s. H2SO3, NH4OH; i. al. s. HF, aq. reg.; sl. s. a. s. H2SO4, HF s. H2SO4, HF

510100°

i. NH4OH; d. a. s. dil. a.

sl. s. al.; i. act.; NH3 sl. s. al.; i. act.; NH3 v. s. a.; i. ac. s. a.

2-28

TABLE 2-2

Physical Properties of Organic Compounds* Abbreviations Used in the Table

(A), density referred to air al., ethyl alcohol amor., amorphous aq., aqua, water brn., brown bz., benzene c., cubic cc., cubic centimeter chl., chloroform col., colorless

cr., crystalline d., decomposes d-, dextrorotatory dl-, dextro-laevorotatory et., ethyl ether expl., explodes gn., green h., hot hex., hexagonal

i-, iso-, containing the group (CH3)2CHi., insoluble ign., ignites l-, laevorotatory lf., leaflets lq., liquid m-, meta mn., monoclinic n-, normal

are usually referred to water at 4°C, e.g., 1.02895/4 a density of 1.028 at 95° C referred to water at 4° C, the 4 being omitted when it is not clear whether the reference is to water at 4°C or at the temperature indicated by the upper figure. The melting and boiling points given have been selected from available data as probably the most accurate. The solubility is given in grams of the substance in 100 of the solvent. In the case of gases, the solubility is often expressed in some manner as “510 cc.” which indicates that, at 10°C, 5 cc. of the gas are soluble in 100 of the solvent.

This table of the physical properties includes the organic compounds of most general interest. For the properties of other organic compounds, reference must be made to larger tables in Lange’s Handbook of Chemistry (Handbook Publishers), Handbook of Chemistry and Physics (Chemical Rubber Publishing Co.), Van Nostrand’s Chemical Annual, International Critical Tables (McGraw-Hill), and similar works. The molecular weights are based on the atomic weight values in “Atomic weights of the Elements 2001,” PURE Appl. Chem., 75, 1107, 2003. The densities are given for the temperature indicated and Name Abietic acid Acenaphthene Acetal Acet-aldehyde -aldehyde, par-aldehyde ammonia -amide -anilide -phenetidide (o-) (m-) -toluidide (o-) (p-) Acetic acid anhydride nitrile Acetone Acetonyl urea Acetophenone benzoyl hydride Acetyl-chloride -phenylenediamine (-p) Acetylene dichloride (cis) (trans) Aconitic acid Acridine Acrolein ethylene aldehyde Acrylic acid nitrile Adipic acid amide nitrile Adrenaline (1-) (3,4,1) Alanine (α) (dl-) Aldol acetaldol Alizarin Allyl alcohol bromide chloride thiocyanate (i) thiourea Aluminum ethoxide Amino-anthraquinone (α) (β) -azobenzene -benzoic acid (m-) (p-)

Synonym sylvic acid, abietinic acid naphthylene ethylene acetaldehyde diethylacetal ethanal paraldehyde ethanamide antifebrin o-ethoxyacetanilide acetyl-m-phenetidine N-tolylacetamide N-tolylacetamide ethanoic acid, vinegar acid acetyl oxide, acetic oxide methyl cyanide propanone, dimethyl ketone dimethyl hydantoin methyl-phenyl ketone ethanoyl chloride amino-acetanilide (p) ethyne; ethine 1,2-dichloroethene dioform equisetic acid; citridic acid acrylic aldehyde, propenal propenoic acid vinyl cyanide hexandioc acid, adipinic acid tetramethylene 1-suprarenine 2-hydroxybutyraldehyde Anthraquinoic acid propen-1-ol-3,propenyl alcohol 3-bromo-propene-1 3-chloro-propene-1 mustard oil thiosinamide

aminodracylic acid

v. s., very soluble v. sl. s., very slightly soluble wh., white yel., yellow (+), right rotation >, greater than <, less than ∞, infinitely

s-, sec-, secondary silv., silvery sl., slightly subl., sublimes sym., symmetrical t-, tertiary tet., tetragonal tri., triclinic uns., unsymmetrical v., very

nd., needles o-, ortho or., orange p-, para pd., powder pet., petroleum ether pl., plates pr., prisms rhb., rhombic s., soluble

Formula

Formula weight

Form and color

C20H30O2 C10H6(CH2)2 CH3CH(OC2H5)2 CH3CHO (C2H4O)3 CH3CHOHNH2 CH3CONH2 C6H5NHCOCH3 CH3CONHC6H4OC2H5 CH3CONHC6H4OC2H5 CH3C6H4NHCOCH3 CH3C6H4NHCOCH3 CH3CO2H (CH3CO)2O CH3CN CH3COCH3 (CH3)2 CH3COC6H5 CH3COCl C2H3ONHC6H4NH2 HC⯗CH CHCl:CHCl CHCl:CHCl C3H3(CO2H)3 C6H4 < (CH)(N) > C6H4 CH2:CHCHO CH2:CHCO2H CH2:CHCN (CH2CH2CO2H)2 (CH2CH2CONH2)2 (CH2CH2CN)2 C6H3(OH)2(CHOHCH2NHCH3) CH3CH(NH2)CO2H CH3CH(OH)CH2COH C6H4(CO)2C6H2(OH)2 CH2:CHCH2OH CH2:CHCH2Br CH2:CHCH2Cl CH2:CHCH2NCS CH2:CHCH2NHCSNH2 Al(OCH2CH3)3 C6H4(CO)2C6H3NH2 C6H4(CO)2C6H3NH2 C6H5N:NC6H4NH2 H2NC6H4CO2H H2NC6H4CO2H

302.45 154.21 118.17 44.05 132.16 61.08 59.07 135.16 179.22 179.22 149.19 149.19 60.05 102.09 41.05 58.08 128.13 120.15 78.50 150.18 26.04 96.94 96.94 174.11 179.22 56.06 72.06 53.06 146.14 144.17 108.14 183.20 89.09 88.11 240.21 58.08 120.98 76.52 99.15 116.18 162.16 223.23 223.23 197.24 137.14 137.14

lf. rhb./al. lq. col. lq. col. cr. col. cr. col. cr. rhb./al. lf./al. lf./al. rhb. rhb. or mn. col. lq. col. lq. col. lq. col. lq. tri./al. lf. col. lq. nd./aq. col. gas col. lq. col. lq. cr./aq. rhb./aq. al. col. lq. col. lq. col. lq. mn. pr. cr. pd. col. oil col. pd. nd./aq. col. lq. red rhb. col. lq. lq. col. lq. col. oil col. pr. pd. red nd. red nd. yel. mn. nd./aq. mn. pr.

Specific gravity 1.06995/95 0.82122/4 0.78318/4 0.99420/4 1.159 1.214 1.16815 1.21215 1.04920/4 1.08220/4 0.78320/4 0.79220/4 1.03315/15 1.10520/4 (A) 0.906 1.29115/4 1.26515/4 0.84120/4 1.06216/4 0.81120 1.36025/4 19/19

0.951

1.10320/4 0.85420/4 1.39820/4 0.93820/4 1.01320/4 1.21920/20 1.14220/0

1.5114°

Melting point, °C 182 95 −123.5 10.5–12 97 81(69.4) 113–4 79 96–7 110 153 16.7 −73 −41 −94.6 175 20.5 −112.0 162 −81.5891 −80.5 −50 192 d. 110–1 −87.7 12–13 −82 151–3 226–7 1 d. 207–11 295 d. 289–90 −129 −119.4 −136.4 −80 77–8 150–60 256 302 126–7 173–4 187–8

Boiling point, °C 278–9 102.2 20.2 124.4752 100–10 d. 222 305 >250 296 306–7 118.1 139.6 81.6–2.0 56.5 subl. 202.3749 51–2 −84760 60.3 48.4 346 52.5 141–2 78–9 26510 295 subl. >200 8320 430 96.6 70–1753 44.6 152 200–510 subl. subl. 225120

Solubility in 100 parts Water

Alcohol

Ether

i. i. 625 ∞ 1213 v. s. s. 0.56 i. sl. s. 0.8619 0.0922 ∞ 12 c. ∞ ∞ s. i. d. s. h. 100 cc.18 0.3520 0.6320 3315 sl. s. h. 40 ∞ s. 1.415 0.412 v. sl. s. 0.0320 2217 ∞ 0.03100 ∞ i. <0.1 0.2 30 d. i. i. sl. s. h. v. sl. s. 0.313

v. s. s. h. ∞ ∞ ∞ v. s. s. 2120 s. s. s. 1025 ∞ ∞ ∞ ∞ s. s. d. v. s. 600 cc.18 ∞ ∞ sl. s. s. s. ∞

v. s. s. chl. ∞ ∞ ∞ sl. s. v. sl. s. 7 25

v. s.

0.615

s. v. sl. s. v. sl. s. ∞ v. s. ∞ ∞ ∞ ∞ s. i. s. s. s. h. 210 1110

v. sl. s. i. i. s. v. s. ∞ ∞ ∞ ∞ v. sl. s. v. sl. s. s. i. s. 1.86 8.26

s. s. ∞ ∞ ∞ ∞ s. s. ∞ v. s. ∞ ∞ v. sl. s. s. s.

Amino-diphenylamine (p-) -G-acid (2-)(6-,8-), Na2 salt -mono-potassium salt -sodium salt -J-acid (2-)(5-,7-) -mono-potassium salt -naphthol sulfonic (1-,2-,4-)(α-) (1-,8-,4-) -phenol (o-) (m-) (p-) -toluene sulfonic acid (1-,2-,3-) (1-,4-,2-) (1-,4-,3-) (1-,2-,5-) Amyl acetate (n-) (i-) (s-) (s-) (t-) alcohol (n-) fusel oil, (s-,n-) methyl-propyl carbinol, (prim.-,i-) isobutyl carbinol, (s-,i-) (t-) (d-) -amine (n-) (s-,n-) (i-) (t-)

aniline (i-) benzoate (i-) bromide (n-) (i-) (t-) n-butyrate (n-) (i-) (t-) i-butyrate (i-) chloride (n-) (s-) (s-) (i-) (s-,i-) (t-) i-cyanide (i-) formate (n-) (i-) iodide (n-) (i-) (s-,n-) (t-)

2-29

mercaptan (n-) (n-) (i-) phenol (t-)(p-) propionate (n-) (i-) (act.) salicylate (n-) Amyl i-valerate (i) (t-)

2-aminophenol 3-aminophenol p-hydroxyaniline

common amyl acetate α-Me-Bu-acetate di Et-carbinol acetate pentanol-1 pentanol-2 2-methyl-butanol-4 2-methyl-butanol-3 2-methyl-butanol-2 active amyl alcohol

1-NH2-2-Me-butane 3-amino pentane 3-NH2-2-Me-butane 1-bromopentane 4-Br-2-Me-butane 2-Br-2-Me-butane

1-chloropentane 2-chloropentane 3-chloropentane 4-Cl-2-Me-butane 3-Cl-2-Me-butane 2-Cl-2-Me-butane 1-Cl-2-Me-butane iso-caproic iso-nitrile 1-iodopentane 4-I-2-Me-butane 2-iodopentane 2-I-2-Me-butane pentanthiol-1 pentanthiol-3 2-Me-butanthiol-4 pentaphen

H2NC6H4NHC6H5 C10H5(NH2)(SO3Na)2 C10H5(NH2)S2O6HK C10H5(NH2)S2O6HNa C10H5(NH2)(SO3H)2 C10H5(NH2)S2O6HK C10H5OHNH2SO3HaH2O NH2(OH)C10H5SO3H H2NC6H4OH H2NC6H4OH H2NC6H4OH C6H3(CH3)(NH2)SO3H C6H3(CH3)(NH2)SO3HH2O C6H3(CH3)(NH2)SO3HaH2O C6H3(CH3)(NH2)SO3HH2O CH3CO2CH2(CH2)3CH3 CH3CO2CH2CH2CH(CH3)2 CH3CO2CH2CH(CH3)C2H5 CH3CO2CH(CH3)CH2C2H5 CH3CO2CH(C2H5)2 CH3CO2C(CH3)2C2H5 CH3(CH2)3CH2OH C2H5CH2CH(OH)CH3 (CH3)2CHCH2CH2OH (C2H5)2CHOH (CH3)2CHCH(OH)CH3 (CH3)2C(OH)C2H5 (CH3)3CCH2OH C2H5CH(CH3)CH2OH CH3(CH2)4NH2 (C3H7)(CH3)CHNH2 (CH3)2CH(CH2)2NH2 (C2H5)(CH3)2CNH2 C2H5CH(CH3)CH2NH2 (C2H5)2CHNH2 (CH3)2CHCH(CH3)NH2 C6H5NHC5H11 C6H5CO2C5H11 CH3(CH2)3CH2Br (CH3)2CH(CH2)2Br (CH3)2C(Br)C2H5 C2H5CH2CO2(CH2)4CH3 C2H5CH2CO2C5H11 C3H7CO2C(CH3)2C2H5 (CH3)2CHCO2C5H11 CH3(CH2)3CH2Cl C2H5CH2CHClCH3 (C2H5)2CHCl (CH3)2CH(CH2)2Cl (CH3)CHCHClCH3 (CH3)2CClC2H5 (CH3)(C2H5)CHCH2Cl (CH3)2CH(CH2)2NC HCO2CH2(CH2)3CH3 HCO2CH2CH2CH(CH3)2 CH3(CH2)3CH2I (CH3)2CHCH2CH2I C2H5CH2CHICH3 (CH3)2CIC2H5 C2H5CH(CH3)CH2I CH3(CH2)3CH2SH (C2H5)2CHSH (CH3)2CH(CH2)2SH C5H11C6H4OH C2H5CO2(CH2)4CH3 C2H5CO2(CH2)2CH(CH3)2 C2H5CO2C5H11 HOC6H4CO2C5H11 C4H9CO2C5H11 C4H9CO2C5H11

184.24 347.28 341.40 325.29 303.31 341.40 248.26 239.25 109.13 109.13 109.13 187.22 205.23 196.22 205.23 130.18 130.18 130.18 130.18 130.18 130.18 88.15 88.15 88.15 88.15 88.15 88.15 88.15 88.15 87.16 87.16 87.16 87.16 87.16 87.16 87.16 163.26 192.25 151.04 151.04 151.04 158.24 158.24 158.24 158.24 106.59 106.59 106.59 106.59 106.59 106.59 106.59 97.16 116.16 116.16 198.05 198.05 198.05 198.05 198.05 104.21 104.21 104.21 164.24 144.21 144.21 144.21 208.25 172.26 172.26

nd./aq. al.

67

354

col. nd. pr. lf. nd. mn. nd. tri./aq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. col. lq. lq. cr. lq. col. lq. col. lq. lq. col. lq. col. lq.

173 122–3 184–6 d.

subl.

subl.

d. 0.87920/20 0.87615/4 0.88013 0.9220 0.87120/4 0.87419 0.817 20/20 0.81020/20 0.81315/4 0.815 25/4 0.81919 0.809 20/4 20/4

0.816 0.76619 0.749 20/4 0.75118/4 0.73125/4 0.75518 0.749 20/4 0.75718 0.92815/4 0.99214/14 1.21820/4 1.22017/15 1.21619/0 0.87115/4 0.86619/15 0.86515/0 0.8760/4 0.878 20/4 0.870 20/4 0.895 21 0.89320/4 0.8830 0.87120/4 0.88117.5 0.9020 0.882 20/4 1.51020/4 1.515 18/4 1.507 17/4 1.47119/15 1.524 20/4 0.857 20

−H2O, 120 −70.8

−78.5 −117.2 −11.9 52–3 −55 −105

−95 −73.2

−99

−72.9 −73.5 −93.5 −86

0.83520/4 0.87615/4 0.870 20/4 0.866 20/4 1.06515 0.85820/15 0.86114/0

93 −73.1

148.4737 142757 141–2 133.5 133 124.5 749 137.9 119.5 132.0 115.6 113–4 102 113–4 128 103–4 91–2 95 77–8 95–6 90–1 83–4 254.5 261746 129.7 120745 108765 186.4 178.6 164 168.8 108.4 96.7 97.3 99.7 758 91753 85.7 98–9 137–9 132 123.5 157.0 147 765 144–5 127 765 148 126767 105 120 265–7 168.7 160.2 5816 265 194 173–4

*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Organic Compounds in Lange’s Handbook of Chemistry.

sl. s. v. sl. s. 12.820 2.718 10.0 20 3.418 v. s. v. sl. s. 1.7 0 2.6 0 1.10 0.97 11 0.5 20 0.47 311 v. sl. s. 0.315 v. sl. s. sl. s. sl. s. v. sl. s. 2.7 22 420 214 5.5 30 2.830 sl. s. sl. s. 3.630 s. ∞ ∞ ∞ ∞ ∞ ∞ i. i. i. 0.0216 i. 0.05 50 i. sl. s. i. i. i. i. i. i. i. i. i. v. sl. s. 0.322 i. i. i. i. i. i. i. i. sl. s. i. 0.125 v. sl. s. i. v. sl. s. sl. s.

s.

s.

4.30 s. 40

v. s. sl. s. i. bz.

i. i. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ s. s. s. ∞ s. s. s. s. s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. ∞ ∞ ∞ s. s. s. ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.

2-30

TABLE 2-2

Physical Properties of Organic Compounds (Continued)

Name Amylene (n-)(α-) (i-) (α-) (-n)(β-) (i-)(β-) Anethole (p-) Anhydroformald-aniline Aniline hydrochloride nitrate sulfate Anisal-acetone (p-) Anisic acid (p-) aldehyde (p-) Anisidine (o-) (m-) (p-) Anisole Anthracene Anthramine (α) (β) Anthranil Anthranilic acid (o-) Anthrapurpurin (1-,2-,7-) Anthraquinone disulfonate Na2 (1-,5-) (1-,8-) (2-,6-) (2-,7-) sulfonate Na (1-) (2-) Anthrarufin (1-,5-) Antipyrene Apiole Arabinose (α)(d- or l-) (dl-) Arachidic acid Arsanilic acid (p-) Asparagine (l-) Aspirin (o-) Atropic acid Auramine Aurine, coralline (4-,4′-) Azo-anisole (2-,2′-) benzene Azoxybenzene Barbituric acid Benzal acetone Benzaldehyde Benzamide Benzanilide Benzene sulfinic acid sulfonic acid sulfonic amide sulfonic chloride Benzidine (4-,4′-) disulfonic acid (2-,2′-) (3-,3′-) Benzil Benzoic acid anhydride nitrile

Synonym pentene-1 2-methyl-butene-3 2-methyl-butene-1 pentene-2 2-methyl-butene-2 p-propenyl anisole methylene aniline amino benzene, phenyl amine, cyanol aniline salt, aniline chloride MeO-benzalacetone 2-amino-anisole MeO-aniline(m) 4-amino anisole methyl phenyl ether paranaphthalene, anthracin green oil α-amino-anthracene β-amino-anthracene

diphenyleneketone, dihydrodiketoanthracene ρ-anthraquinone disulfonate x-anthraquinone disulfonate

1-ph-2,3-diMepyrazolone-5 1-allyl-2, 5-diMeO-3,4 methylenedioxybenzene eicosanoic acid

α-phenyl acrylic acid 4,4′-dimethylaminobenzophenomide diMeO-azobenzene diphenyldiimide malonyl urea Me-cinnamyl ketone artificial almond oil benzol, phenyl hydride, cyclohexatriene benzene sulfonamide benzene sulfonyl chloride

dibenzoyl phenyl cyanide

Formula

Formula weight

Form and color

Specific gravity

Melting point, °C

C2H5CH2CH:CH2 (CH3)2CHCH:CH2 (C2H5)(CH3)C:CH2 C2H5CH:CHCH3 (CH3)2C:CHCH3 CH3CH:CHC6H4OCH3 (CH2NC6H5)3 C6H5NH2

70.13 70.13 70.13 70.13 70.13 148.20 315.41 93.13

lq. col. lq. col. lq. col. lq. col. lq. lf./al. pr./al. col. oil

0.644 20 0.63215 0.667 0/0 0.650 20/4 0.66319/4 0.99120/20

C6H5NH2HCl C6H5NH2HNO3 (C6H5NH2)2H2SO4 CH3OC6H4CH:CHCOCH3 CH3OC6H4CO2H CH3OC6H4CHO CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H5 C6H4:(CH)2:C6H4

129.59 156.14 284.33 176.21 152.15 136.15 123.15 123.15 123.15 108.14 178.23

cr. rhb. lf./al. lf./et. mn./aq. col. oil col. lq. oil pl./aq. col. lq. col. mn.

1.222 4 1.356 4 1.377 4

C6H4:(CH)2:C6H3NH2 C6H4:(CH)2:C6H3NH2 C6H4:(NH)CO H2NC6H4CO2H C14H5O2(OH)3 C6H4:(CO)2:C6H4

193.24 193.24 119.12 137.14 256.21 208.21

yel./al. yel./al. col. oil col. rhb. or. nd./al. yel. rhb.

C14H6O2(SO3Na)25H2O C14H6O2(SO3Na)24H2O C14H6O2(SO3Na)27H2O C14H6O2(SO3Na)24H2O C14H7O2SO3Na C14H7O2SO3Na C14H6O2(OH)2 C11H12ON2 C12H14O4

502.38 484.36 538.41 484.36 310.26 310.26 240.21 188.23 222.24

yel. lf. yel. pr. col. cr. cr. yel. lf. silv. lf. yel. lf. mn./aq. col. nd.

1.088113/4 1.0220/4

CH2OH(CHOH)3CHO CH2OH(CHOH)3CHO CH3(CH2)18CO2H H2NC6H4.AsO3H2 HO2CC2H3(NH2)CONH2 CH3CO2C6H4CO2H C6H5C(:CH2)CO2H [(CH3)2NC6H4]2C:NH

150.13 150.13 312.53 217.05 132.12 180.16 148.16 267.37

rhb. pr.

1.585 20/4

col. lf. nd./aq. rhb. nd./aq. nd./aq. col./al.

(HOC6H4)2C:C6H4:O (CH3OC6H4N:)2 C6H5N:NC6H5 (C6H5)2N2O CO:(NHCO)2:CH22H2O C6H5CH:CHCOCH3 C6H5CHO C6H5CONH2 C6H5CONHC6H5 C6H6

290.31 242.27 182.22 198.22 164.12 146.19 106.12 121.14 197.23 78.11

red or. pr. or. mn. yel. rhb. col./aq. pl. col. lq. col. pr. lf./al. col. lq.

C6H5SO2H C6H5SO3H C6H5SO2NH2 C6H5SO2Cl NH2C6H4C6H4NH2 (C6H3(NH2)SO3H)23H2O (C6H3(NH2)SO3H)2 C6H5COCOC6H5 C6H5CO2H (C6H5CO)2O C6H5CN

142.18 158.18 157.19 176.62 184.24 398.41 344.36 210.23 122.12 226.23 103.12

pr./aq. col. nd. mn./aq. cr. cr./aq. pr./aq.

1.38415/15

83–4 65–6 156 14.5 128–9 d. >175

pr. mn. pr. rhb./et. col. lq.

1.2315 1.26615/4 1.19915/4 1.00125/6

95 121.7 42 −12.9

1.02220/4

1.385 4 1.123 20/4 1.09815/15 1.096 20/4 1.089 55/55 0.990 22/4 1.25 27/4

1.18715/4 1.438 20/4

1.54315/4

1.20320/4 1.248 20/20 1.035 20/20 1.046 20/4 1.341 1.314 0.879 20/4

−135 −139 −124 22.5 143 −6.2 198 d. 190 d. 73–4 184.2 2.5 5.2 <−12 57.2 −37.3 217–8 130 238 <−18 144–5 369 286

280 113(109) 30 159.5 164.5 77 232 227–35 135–6 106–7 136 310 d. 153 68 36 d. 245 41–2 −26 130 163 5.5

Boiling point, °C

Solubility in 100 parts Alcohol

Ether

30–1 20.5 771 31–2758 36.4 37–8 235.3 185 184.4

i. i. i. v. sl. s. i. v. sl. s. i. 3.618

∞ ∞ ∞ ∞ s. s. sl. s. ∞

∞ ∞ ∞ ∞ ∞ ∞ s. ∞

245

275–80 247–8 225 251 243 154–5 340–2

1815 s. 514 i. 0.0319 v. sl. s. v. sl. s. v. sl. s. s. h. i. i.

s. s. sl. s. v. s. v. s. ∞ ∞ s. s. s. 1.520

i. sl. s. i. v. s. v. s. ∞ ∞ s. s. s.

subl. d. >215 subl. 462 379–81

i. i. sl. s. h. 0.35 14 sl. s. h. i.

s. sl. s. s. 1110 v. s. h. 0.0518

sl. s. s. 167 sl. s. v. sl. s.

i.

i.

subl. 319174 294

v. s. sl. s. 3.920 30.5 20 0.5320 0.8425 i. 100 25 i

v. sl. s. i. i. sl. s. 100 s.

i. i. i. s. sl. s. s.

460 16.910 i. v. s. h. 3.128 137 0.1 c. i.

0.59°

i.

s. h. v. s. h. i. c. s. s. 720

v. s. i.

i. i. i. i. s. h. i. 0.3 1.35 25 i. 0.07 22

s. s. 4.220 11.415 sl. s. s. ∞ 1725 430 s.

s. s. ∞ sl. s. sl. s. ∞

v. s. h. v. s. 0.4316 i. 1 h. 0.09 25 v. sl. s. i. 0.217 i. 1100

v. s. v. s. v. s. v. s. 1 h. i.

v. s. i. v. s. s. 2 i.

v. s. 4615 s. ∞

v. s. 6615 s. ∞

328 d. 235 267 d.

297 d. 260–2 179 290 117–910 80.1 d. > 100 d. 251.5 400740 348 d. 249.2 360 190.7

Water

520 s. 2.320 s. s.

2-31

Benzoin (dl-) Benzophenone Benzotrichloride Benzoyl-benzoic acid (o-) -chloride -peroxide Benzyl acetate alcohol amine aniline benzoate butyrate chloride ether formate propionate Berberonic acid (2-,4-,5-) Biuret Borneol (dl-) (d- or l-) (iso-) Bornyl acetate (d-) Bromo-aniline (p-) -benzene -camphor (3-)(d-) -diphenyl (p-) -naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -styrene (ω)(1) (2) -toluene (o-) (m-) (p-) Bromoform Butadiene (1-,2-) (1-,3-) Butadienyl acetylene Butane (i-) Butyl acetate (n-) (s-) (i-) (tert-) alcohol (n-) (s-) (i-) (tert-) amine (n-) (s-) (i-) (t-) p-aminophenol (N)(n) (N)(i-) aniline (n-) (i-) arsonic acid (n-) benzoate (n-) (i-) bromide (n-) (s-) (i-) (t-) butyrate (n-)(n-) (n-)(i-) (i-)(i-) caproate carbamate (i-) cellosolve (n-)

diphenyl ketone phenyl chloroform

phenyl carbinol ω-amino toluene phenyl-benzylamine ω-chlorotoluene dibenzyl ether

allophanamide

phenyl bromide α-bromocamphor α-naphthyl bromide β-naphthyl bromide

o-tolyl bromide tribromo-methane methyl-allene erythrene diethyl trimethyl-methane

butanol-1 butanol-2 2-methyl-propanol-1 2-methyl-propanol-2

1-bromo-butane 2-bromo-butane 1-Br-2-Me-propane 2-Br-2-Me-propane

2-BuO-ethanol-1

C6H5COCHOHC6H5 C6H5COC6H5 C6H5CCl3 C6H5COC6H4CO2HH2O C6H5COCl (C6H5CO)2O2 CH3CO2CH2C6H5 C6H5CH2OH C6H5CH2NH2 C6H5CH2NHC6H5 C6H5CO2CH2C6H5 C2H5CH2CO2CH2C6H5 C6H5CH2Cl (C6H5CH2)2O HCO2CH2C6H5 C2H5CO2CH2C6H5 C5H2N(CO2H)32H2O NH(CONH2)2 C10H17OH C10H17OH C10H17OH CH3CO2C10H17 BrC6H4NH2 C6H5Br BrC10H15O BrC6H4C6H5 C10H7Br C10H7Br BrC6H4OH BrC6H4OH BrC6H4OH C6H5CH:CHBr C6H5CH:CHBr CH3C6H4Br CH3C6H4Br CH3C6H4Br CHBr3 CH3CH:C:CH2 CH2:CHCH:CH2 CH2:(CH)2:CHC⯗CH CH3CH2CH2CH3 (CH3)2CHCH3 CH3CO2(CH2)2C2H5 CH3CO2CH(CH3)C2H5 CH3CO2CH2CH(CH3)2 CH3CO2C(CH3)3 C2H5CH2CH2OH C2H5CH(OH)CH3 (CH3)2CHCH2OH (CH3)3COH C2H5CH2CH2NH2 C2H5CH(NH2)CH3 (CH3)2CHCH2NH2 (CH3)3CNH2 C4H9NHC6H4OH C4H9NHC6H4OH C4H9NHC6H5 C4H9NHC6H5 C4H9AsO(OH)2 C6H5CO2C4H9 C6H5CO2C4H9 C2H5CH2CH2Br C2H5CH(Br)CH3 (CH3)2CHCH2Br (CH3)3CBr C2H5CH2CO2CH2CH2C2H5 C2H5CH2CO2CH2CH(CH3)2 (CH3)2CHCO2CH2CH(CH3)2 CH3(CH2)4CO2C4H9 NH2CO2CH2CH(CH3)2 C4H9OCH2CH2OH

212.24 182.22 195.47 244.24 140.57 242.23 150.17 108.14 107.15 183.25 212.24 178.23 126.58 198.26 136.15 164.20 247.16 103.08 154.25 154.25 154.25 196.29 172.02 157.01 231.13 233.10 207.07 207.07 173.01 173.01 173.01 183.05 183.05 171.03 171.03 171.03 252.73 54.09 54.09 78.11 58.12 58.12 116.16 116.16 116.16 116.16 74.12 74.12 74.12 74.12 73.14 73.14 73.14 73.14 165.23 165.23 149.23 149.23 182.05 178.23 178.23 137.02 137.02 137.02 137.02 144.21 144.21 144.21 172.26 117.15 118.17

mn. col. rhb. col. lq. tri./aq. col. lq. rhb./et. col. lq. col. lq. lq. mn. pr. nd. col. lq. col. lq. lq. col. lq. lq. tri. nd./al. col. cr. col. cr. col. cr. rhb./pet. rhb. col. lq. cr. cr./al. col. oil lf./al. col. lq. cr. tet. cr. lq. lq. col. lq. col. lq. cr./al. col. lq. lq. col. gas col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. lq. oil col. lf. col. oil col. oil lq. lq. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lf. col. lq.

1.08354 1.38014 1.21220/4 1.05717 1.04320/4 0.98220/4 1.065 25/25 1.1220/4 1.01616/18 1.100 20/20 1.03616 1.08123 1.03616/17 20/4

1.011 1.01120/4 15

0.991 1.820 1.495 20/4 1.449 20/4 1.48220/4 1.605 0 1.55380 1.588 80 1.42220/4 1.427 20/4 1.42220/4 1.410 20/4 1.390 20/4 2.890 20/4 0.62120/4 0.773 20/4 0.600 0.600 0.882 20 0.865 25/4 0.87120/4 0.866 20/4 0.810 20/4 0.808 20/4 0.80517.5 0.779 26 0.739 25/4 0.724 20/4 0.73220/20 0.69818/4

133–7 48.5 −4.75 93(128) −0.5 108 d. −51.5 −15.3 37–8 21 238–40 −39 3.6 243 192–3 d. 210.5 208–9 212 29 63–4 −30.6 77–8 90–1 5–6 59 5.6 32–3 63.5 7 −7.5 −28 −39.8 28.5 8–9 −108.9 −135 −145 −76.3 −98.9 −79.9 −114.7 −108 25.5 −50 −104 −85 −67.5 71 79

0.940 20/4 1.005 25/25 0.997 25/25 1.277 20/4 1.25125/4 1.258 25/4 1.21120/4 0.87220/20 0.86318/4 0.875 0/4 0.8820/0 0.95676/4 0.90320/4

158–9 −22 −112.4 −112 −118.5 −16.2 −80.7 65

344768 305.4 220.7 197.2 expl. 213.5 204.7 184.5 306750 323–4 i. 179.4 295–8 202–3747 220–2 subl. 212–3 226–7 156.2 274 310 281.1 281–2 194–5 236–7 238 221 10826 181.8 183.7 184–5 150.5 18–9 −4.41 83–6 −0.6 −10 125 740 112744 118 95–6760 117 99.5 107–8 82.9 77.8 66772 68–9 45.2 235720 231–2 249–50 241.5 101.6 91.3 91.5 73.3 165.7736 156.9 148–9 204.3 206–7 171.2

v. sl. s. i. i. sl. s. d. i. i. 417 ∞ i. i. v. s. i. i. i. i. v. sl. s. 1.30 v. sl. s. v. sl. s. i. i. i. c. i. i. i. i. i. s. 1.415 i. i. i. i. i. 0.1 c. i. i. i. i. i. 0.7 i. 0.625 i. 915 12.520 1015 ∞ ∞ ∞ ∞ i. i. i. 0.0115 s. i. i. 0.0616 i. 0.0618 i. i. i. i. i. i. ∞

s. h. 6.515 s.

sl. s. 1513 s.

d. h. s. h. ∞ ∞ ∞

∞ s. ∞ ∞ ∞ s. ∞

∞ v. s. ∞ s. h. s.

∞ s. ∞

sl. s. h. s.

i.

v. s.

v. s.

s. v. s. s. 2026 s. s. 620 s. s. v. s. ∞ ∞ s. s. s. ∞ ∞ ∞

s. v. s. ∞ v. s. 34 25 ∞ v. s. ∞ s. v. s. ∞ ∞ ∞25 s. ∞25 ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

v. s. v. s. s. s. ∞ ∞

v. s. v. s. i. s. ∞ ∞

∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞

s. ∞

s. ∞

2-32

TABLE 2-2

Physical Properties of Organic Compounds (Continued)

Name chloride (n-) (s-) (i-) (t-) dimethylbenzene (t-)(1-,3-,5-) formate (n-) (s-) (i-) furoate (n-) iodide (n-) (s-) (i-) (t-) lactate (n-) mercaptan (n-) (i-) (t-) methacrylate (n-) (i-) phenol (p-)(t-) propionate (n-) (s-) (i-) stearate (n-) (i-) iso-thiocyanate (n-) (i-) (s-)(d-) (t-) valerate (n-)(n-) (i-)(n-) (i-)(s-) (i-)(i-) Butylene (α-) (β-) Butyraldehyde (n-) (i-) Butyric acid (n-) (i-) amide (n-) (i-) anhydride (n-) (i-) anilide (n-) Caffeic acid (3-,4-) Caffeine Camphene (dl-) (d- or l-) Camphor (d-) Camphoric acid (d-) Cantharidine Capric acid Caproic acid (n-) (i-) Caprylic acid (n-) Carbazole Carbitol Carbon disulfide monoxide suboxide tetrabromide tetrachloride tetrafluoride Carbonyl sulfide Carminic acid Carvacrol (1-,2-,4-)

Synonym 1-chloro-butane 2-chloro-butane 1-Cl2-2-Me-propane 2-Cl2-2-Me-propane

1-iodo-butane 2-iodo-butane 1-iodo-2-Me-propane 2-iodo-2-Me-propane butanthiol-1 2-Me-propanthiol-1

butyl mustard oil iso-Bu mustard oil

butene-1 butene-2 2-Me-propanol butanoic acid 2-Me-propanoic acid n-butyramide iso-butyramide n-butyranilide

decanoic acid hexanoic acid 2-Me-pentanoic-5 acid octanoic acid diphenylenelimine, dibenzopyrrole diethylene glycol mono-Et ether

tetrabromomethane tetrachloromethane tetrafluoromethane

Formula C2H5CH2CH2Cl C2H5CHClCH3 (CH3)2CHCH2Cl (CH3)3CCl (CH3)3CC6H3:(CH3)2 HCO2CH2CH2C2H5 HCO2CH(CH3)C2H5 HCO2CH2CH(CH3)2 OC4H3CO2C4H9 C2H5CH2CH2I C2H5CHICH3 (CH3)2CHCH2I (CH3)3CI CH3CH(OH)CO2C4H9 C2H5CH2CH2SH (CH3)2CHCH2SH (CH3)3CSH CH2:C(CH3)CO2C4H9 CH2:C(CH3)CO2C4H9 (CH3)3CC6H4OH C2H5CO2C4H9 C2H5CO2C4H9 C2H5CO2C4H9 CH3(CH2)16CO2C4H9 CH3(CH2)16CO2C4H9 C2H5CH2CH2N:CS (CH3)2CHCH2N:CS C4H9N:CS (CH3)3CN:CS CH3(CH2)3CO2(CH2)3CH3 (CH3)2CHCH2CO2(CH2)3CH3 (CH3)2CHCH2CO2C4H9 C4H9CO2C4H9 C2H5CH:CH2 CH3CH:CHCH3 CH3CH2CH2CHO (CH3)2CHCHO C2H5CH2CO2H (CH3)2CHCO2H C2H5CH2CONH2 (CH3)2CHCONH2 (C2H5CH2CO)2O [(CH3)2CHCO]2O C3H7CONHC6H5 (HO)2C6H3C2H2CO2H C8H10O2N4H2O C10H16 C10H16 C10H16O C8H14(CO2H)2 C10H12O4 CH3(CH2)8CO2H CH3(CH2)4CO2H (CH3)2CH(CH2)2CO2H CH3(CH2)6CO2H (C6H4)2NH C2H5O(CH2)2O(CH2)2OH CS2 CO OC:C:CO CBr4 CCl4 CF4 COS C22H20O13 CH3C6H3(OH)CH(CH3)2

Formula weight 92.57 92.57 92.57 92.57 162.27 102.13 102.13 102.13 168.19 184.02 184.02 184.02 184.02 146.18 90.19 90.19 90.19 142.20 142.20 150.22 130.18 130.18 130.18 340.58 340.58 115.20 115.20 115.20 115.20 158.24 158.24 158.24 158.24 56.11 56.11 72.11 72.11 88.11 88.11 87.12 87.12 158.19 158.19 163.22 180.16 212.21 136.23 136.23 152.23 200.23 196.20 172.26 116.16 116.16 144.21 167.21 134.17 76.14 28.01 68.03 331.63 153.82 88.00 60.08 492.39 150.22

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. lq. col. lq. lq. lq. lq. lq. col. lq. col. lq. lq. lq. lq. lq. nd./aq. col. lq. col. lq. col. lq. col. lq. wax lq. lq. lq. lq. lq. lq. col. lq. col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. rhb. mn. pl. col. lq. col. lq. mn. pr. yel./aq. nd./al. cr. cr. trig. mn. cr. col. nd. oily lq. col. oil col. lf. lf. col. lq. col. lq. col. gas gas col. mn. col. lq. gas col. gas red pd. col. lq.

0.887 20 0.87120/4 0.88415 0.84715

−123.1 −131 −131.2 −26.5

77.9763 67.8767 68.9 51–2 200–2147 106.9 97 98.2 118–2025 129.9 118–9 120 99 75–66 97–8 88 65–7 155 155 236–8 146 132.5 136.8 220–525

0.9110 0.88220/4 0.885 20/4 1.056 20/4 1.617 20/4 1.595 20 1.606 20/4 1.370 19/15 0.968 0.837 25/4 0.836 20/4 0.889 15.6 0.889 15.6 0.908 112/4 0.88315 0.866 20/4 0.888 0/4 0.855 25/25 0.95611 0.96414/4 0.943 20/4 0.91910 0.87015/4 0.862 25/4 0.848 20/4 0.8740/4 0.69 20/4

0.817 0.79420/4 0.96420/4 0.949 20/4 1.032 1.013 0.968 20/20 0.950 25/4 1.134 1.2319 0.82278 0.845 50/4 0.999 9/9 1.186 0.889 87 0.922 20/4 0.925 20/4 0.910 20/4 0.990 20/20 1.263 20/4 0.81−195/4 1.1140 3.42 1.595 20/4 1.24−87 0.977 20/4

−95.3 −103.5 −104 −90.7 −34 −116 <−79

99 −89.55 −71.4 27.5 25

10.5 −93

−130 −127 −99 −65.9 −4.7 −47 115–6 129–30 −75 −53.5 92 195–213 237 50 42.7 178–9 187 212 31.5 −1.5 −35 16 244.8 −108.6 −207 −107 90.1(48) −22.6 −138.2 d. 136 0.5

165724 162 159–63 140770 186 168.8 163–4752 168.7 −5758 3746 75.7 64757 163.5757 154.5 216 216–20 199.5 181.5734 18915 d. subl. 160 159.6 209.1759 268–70 202761 207.7 237.5 354.8 201.9 46.3 −192 7761 189.5 76.8 −128 −50.2760 238

Solubility in 100 parts Water 0.0718 i. i. i. i. v. sl. s. sl. s. 1.122 i. i. i. i. i. sl. s. sl. s. v. sl. s. i. i. sl. s. i. i. i. 0.3 25 i. i. i. i. i. v. sl. s. i. i. i. i. 4 1120 ∞ 20 20 16.315 v. s. d. d. i. s. h. 2 i. i. 0.1 0.612 0.003 0.003 1.120 v. sl. s. 0.0715 i. ∞ 0.20 3.50 cc. d. 0.0230 0.0820 sl. s. 8014 cc. s. v. sl. s.

Alcohol

Ether

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.

s. ∞ ∞ ∞ s.

s. ∞ ∞ ∞ s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

s. s. s. s. ∞ ∞ ∞ ∞ v. s.

∞ ∞ ∞ ∞ s. s. d d s. s. 2 s. s. 12012 s.

∞ ∞ ∞ ∞ sl. s. sl. s. ∞ ∞ s. sl. s. 0.3 s. s. v. s.

s. s. s. s. 0.9214 v. s. ∞ s.

s. s. s. s. sl. s. s. ∞

s. ∞

s. s. ∞

s. s. ∞

s. v. sl. s. ∞

2-33

Carvacrylamine (2-,1-,4-) Carvone (d-) Cellosolve acetate Cellulose Cetyl acetate alcohol Chloral hydrate Chloranil Chloretone Chloro-acetanilide (p-) -acetic acid -acetone -acetophenone (ω-) -acetyl chloride -aniline (o-) (m-) (p-) -anthraquinone (1-) (2-) -benzaldehyde (o-) (m-) (p-) -benzene -benzoic acid (o-) (m-) (p-) -buta-1,3-diene (2-) (1-) -buta-1,2-diene (4-) -dimethylhydantoin -dinitrobenzene (α)(1-,2-)(4-) (α)(1-,3-)(4-) -diphenyl (o-) (m-) (p-) -hydroquinone -naphthalene (α-) (β-) -nitrobenzene (o-) (m-) (p-) -nitrotoluene (2-,4-) (2-,6-) -phenol (o-) (m-) (p-) -propionic acid (α)(dl-) -toluene (o-) (m-) (p-) Chloroform Chlorophyll (α-) Chloropicrin Cholesterol Chrysene Chrysoidine (2-,4-) Chrysophanic acid Cinchomeronic acid (3-,4-) Cineole, eucalyptole Cinnamic acid (cis-) (trans-) aldehyde Cinnamyl alcohol cinnamate Citraconic acid (cis-) Citral (α) Citric acid Citronellal (d-) Citronellol (d-) Coniine (d-)(2-)

H2NC6H3(CH3)C3H7 C10H14O C2H5O(CH2)2OH CH3CO2CH2CH2OC2H5 (C6H10O5)x CH3CO2(CH2)15CH3 CH3(CH2)14CH2OH CCl3CHO CCl3CH(OH)2 OC:(CClCCl)2:CO Cl3CC(OH)(CH3)2 CH3CONHC6H4CI ClCH2CO2H CH3COCH2Cl C6H5COCH2Cl ClCH2COCl ClC6H4NH2 ClC6H4NH2 ClC6H4NH2 C6H4(CO)2C6H3Cl C6H4(CO)2C6H3Cl ClC6H4CHO ClC6H4CHO ClC6H4CHO C6H5Cl ClC6H4CO2H ClC6H4CO2H ClC6H4CO2H CH2:CClCH:CH2 CH2:CHCH:CHCl CH2:C:CHCH2Cl  C(CH3)2N(Cl)CON(Cl)CO ClC6H3(NO2)2 ClC6H3(NO2)2 C6H5C6H4Cl C6H5C6H4Cl C6H5C6H4Cl ClC6H3(OH)2 C10H7Cl C10H7Cl ClC6H4NO2 ClC6H4NO2 ClC6H4NO2 CH3C6H3(NO2)(Cl) CH3C6H3(NO2)(Cl) ClC6H4OH ClC6H4OH ClC6H4OH CH3CHClCO2H CH3C6H4Cl CH3C6H4Cl CH3C6H4Cl CHCl3 C55H72O5N4Mg Cl3CNO2 C27H45OHH2O C18H12 C6H5N:NC6H3(NH2)2 C14H5(OH)2(CH3)O2 C5H3N(CO2H)2 C10H18O C6H5CH:CHCO2H C6H5CH:CHCO2H C6H5CH:CHCHO C6H5CH:CHCH2OH C8H7CO2C9H9 CH3C(CO2H):CHCO2H C9H15CHO C3H4(OH)(CO2H)3 C9H17CHO C10H20O C3H7C5H10N

149.23 150.22 90.12 132.16 162.14 284.48 242.44 147.39 165.40 245.88 177.46 169.61 94.50 92.52 154.59 112.94 127.57 127.57 127.57 242.66 242.66 140.57 140.57 140.57 112.56 156.57 156.57 156.57 88.54 88.54 88.54 197.02 202.55 202.55 188.65 188.65 188.65 144.56 162.62 162.62 157.55 157.55 157.55 171.58 171.58 128.56 128.56 128.56 108.52 126.58 126.58 126.58 119.38 893.49 164.38 404.67 228.29 212.25 254.24 167.12 154.25 148.16 148.16 132.16 134.18 264.32 130.10 152.23 192.12 154.25 156.27 127.23

oil col. lq. col. lq. col. lq. amor. nd. lf. col. lq. mn. pr. yel./bz. col. cr. rhb. col. cr. col. lq. rhb. col. lq. lq. lq. rhb. yel. nd. nd./al. nd. pr. pr. col. lq. mn./aq. pr. tri. col. lq. col. lq. col. lq. cr./et. rhb./et. cr. cr. lf. mn. col. lq. lf./al. mn. nd. yel./al. mn. pr. cr. cr. col. lq. nd. nd. col. lq. col. lq. col. lq. col. lq. col. lq. lq. rhb./al. col. rhb. yel. cr. yel./al. cr./HCl col. oil mn. pr. mn. pr. lq. nd. nd. or pr. nd. col. oil cr. col. oil col. oil col. lq.

0.99420 0.96120/4 0.93120/4 0.975 20/4 1.3–1.4 0.858 20 0.818 50/4 1.505 25/4 1.619 50/4 1.385 22 1.58 20/20 1.16216 1.32415 1.498 20/20 1.21320/4 1.216 20/4 1.42719 1.298 1.25015 1.196 61 1.107 20/4 1.544 25/4 1.496 25/4 1.54124 0.958 20/20 0.965 20/20 0.99120/20 1.5 20/20 1.697 22

1.194 20/4 1.26616 1.305 80/4 1.34350/4 1.298 91 1.256 80 1.24118/15 1.268 25 1.306 20/4 1.306 9 1.082 20/4 1.07220/4 1.070 20/4 1.489 20 1.65123/4 1.067

20

0.927 1.2844 1.245 1.110 20/20 1.040 35/35 1.08516.5 1.617 0.89017/4 1.54220/4 0.85517.5 0.84820/4 0.84717

−16 −70 22–3 49–50 −57 51.7 290 97 175–6 61.2 −44.5 58–9 0 −10.4 70–1 162 208–9 11 17–8 47.8 −45.2 141–2 158 242–3

130 39(36) 53(43) 34 89 77.5 106 −20 56–7 32.5 44.4(24) 83–4 38.2 37.5 7(0) 32–3 41–3 <−20 −34 −47.8 7.5 −63.5 d. −64 149–51 253–4 117.5 195 258–9 d. 1.5 68 133 −7.5 33 44 92–3 153 −2

241 230766 135.1 156.3 i. 20015 189.515 97.6768 d. 98 subl. 167 189.5 121 245–7 105 210.5 230767 230–1 subl. 208748 213–4 213748 132.1 subl. 59.4 69 88 315 d. 315 d. 267–8 284–5 282 263 sl. d. 259.3 264751 245.5753 235.6 242761 240718 238 175–6 214 217 186 159.5 161.6 162.2 61.2 i. 112.3766 subl. 448 subl. subl. d. 176–7 12519 300 252 sl. d. 257.5 229 d. 204–8 224–5 166–7

v. sl. s. i. ∞ 22 i. i. i. v. s. 47417 i. 0.8 c. sl. s. v. s. ∞ 0.11 d. i. i. s. h. i. i. v. sl. s. v. sl. s. s. h. 0.049 20 0.20825 0.04125 0.00825 v. sl. s. v. sl. s. d. 0.2125 i. i. i. i. i. v. s. i. i. i. i. i. i. i. 2.8520 2.6020 2.7120 ∞ i. i. i. 0.8220 s. 0.1718 0.2620 i. sl. s. h. i. c. v. sl. s. 1.915

s. ∞ ∞ ∞ i. v. sl. s. c. s. ∞ v. s. i. c. 111 s. s. ∞ v. s. d.

0.0418 v. sl. s. sl. s. i. 36025 i. 207.725 v. sl. s. v. sl. s. 1.1

s. sl. s. h.

s. ∞ ∞ ∞ s. ∞ s. i. c. s. v. s. s. ∞ v. s. s. s. s.

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. v. s. v. s. ∞ s. s. s. ∞ ∞

v. s. h. s. h.

v. s. s.

v. s. s. v. s. s. h. v. s. h. v. s. h.

v. s. ∞ v. s. s. v. s. v. s.

s. s. v. s. ∞ s. s. s. ∞

s. s. v. s. ∞ ∞ ∞ ∞ ∞

s. 1.117 0.116 s. s. h. sl. s. ∞

s. 18 v. sl. s. s. sl. s. i. ∞

2420 s. v. s. 4 c. s. ∞ 7615 ∞ ∞ v. s.

v. s. ∞ v. s. 33 s. ∞ 215 ∞ ∞ v. s.

2-34

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

Coumaric acid (o-) (p-) Coumarin Coumarone Creatine Creatinine Creosol (3-,1-,4-) Cresidine (1-,2-,4-) Cresol (o-) (m-) (p-) Cresyl benzoate (o-) (m-) (p-) Crotonic acid (α-) acid (β-)(cis-) aldehyde (α) Cumene Cumic acid (p-) Cumidine (p-) Cyanamide Cyanic acid Cyanoacetic acid Cyanogen bromide chloride Cyanuric acid Cyclo-butane -heptane -hexane -hexanol -hexanone -hexene -hexyl acetate amine bromide chloride -pentadiene (1-,3-) -pentane -pentanone -propane Cymene (o-) (m-) (p-) Cystine (l-) Dambose Decahydronaphthalene (cis-) (trans-) Decane (n-) Decyl alcohol Dextrin Diacetone alcohol Diamino-benzophenone (4-,4′-) -diphenylamine (4-,4′-) -diphenylmethane (4-,4′-) -diphenylurea (4-,4′-) Diamyl-amine (i-) ether (n-) (i-) Diamyl ketone (i-) phthalate (n-) (i-) tartrate (i-) Dianisidine (o-)(4-,3-)2 Diazo-aminobenzene -aminotoluene (2-,2′-) -methane

Formula HOC6H4CH:CHCO2H HOC6H4CH:CHCO2H C9H6O2 C8H6O C4H9N3O2H2O C4H7N3O CH3OC6H3(CH3)OH CH3(NH2)C6H3OCH3 CH3C6H4OH CH3C6H4OH CH3C6H4OH C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 CH3CH:CHCO2H CH3CH:CHCO2H CH3CH:CHCHO C6H5CH(CH3)2 (CH3)2CHC6H4CO2H (CH3)2CHC6H4NH2 H2NCN HOCN or HNCO CH2(CN)CO2H (CN)2 BrCN ClCN C3H3O3N32H2O CH2 < (CH2)2 > CH2 CH2 < (CH2CH2CH2)2 > CH2 < (CH2CH2)2 > CH2 CH2 < (CH2CH2)2 > CHOH CH2 < (CH2CH2)2 > CO (CH2CH2CH:)2 CH3CO2C6H11 CH2 < (CH2CH2)2 > CHNH2 CH2 < (CH2CH2)2 > CHBr CH2 < (CH2CH2)2 > CHCl CH2 < (CH:CH)2 > CH2 < (CH2CH2)2 > < (CH2CH2)2 > CO < CH2CH2CH2 > CH3C6H4CH(CH3)2 CH3C6H4CH(CH3)2 CH3C6H4CH(CH3)2 [SCH2CH(NH2)CO2H]2 C6H6(OH)6 C10H18 C10H18 CH3(CH2)8CH3 CH3(CH2)8CH2OH (C6H10O5)x (CH3)2C(OH)CH2COCH3 H2NC6H4COC6H4NH2 H2NC6H4NHC6H4NH2 H2NC6H4CH2C6H4NH2 (H2NC6H4NH)2CO [(CH3)2CHCH2CH2]2NH (C2H5CH2CH2CH2)2O [(CH3)2CH(CH2)2]2O [(CH3)2CHCH2CH2]2CO C6H4(CO2C5H11)2 C6H4(CO2C5H11)2 (HOCHCO2C5H11)2 [NH2(OCH3)C6H3]2 C6H5N:NNHC6H5 C7H7N:NNHC7H7 CH2:N2

Formula weight 164.16 164.16 146.14 118.13 149.15 113.12 138.16 137.18 108.14 108.14 108.14 212.24 212.24 212.24 86.09 86.09 70.09 120.19 164.20 135.21 42.04 43.02 85.06 52.03 105.92 61.47 165.10 56.11 98.19 84.16 100.16 98.14 82.14 142.20 99.17 163.06 118.60 66.10 70.13 84.12 42.08 134.22 134.22 134.22 240.30 180.16 138.25 138.25 142.28 158.28 162.14 116.16 212.25 199.25 198.26 242.28 157.30 158.28 158.28 170.29 306.40 306.40 290.35 244.29 197.24 225.29 42.04

Form and color nd./aq. cr./aq. rhb./et. oil mn./aq. mn. pr. nd./pet. cr. lq. pr. lq. cr. cr. col. mn. nd. col. lq. col. lq. tri. lq. col. nd. gas col. lq. col. gas nd. gas mn./aq. col. gas oil col. lq. col. nd. col. oil lq. oil col. lq. col. lq. col. lq. col. lq. col. oil col. oil col. gas col. lq. col. lq. col. lq. pl. mn./aq. lq. lq. col. lq. col. oil amor. lq. yel. nd. lf./aq. nd./aq. cr. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. lq. col. lf. yel. lf. or. cr. gas

Specific gravity

0.93520/4 1.07815/15 1.09220/20 20/4

1.048 1.03420/4 1.03520/4

0.96479.7 1.03115/4 0.85320/20 0.86220/4 1.1624 0.953 1.07348/4 1.1400 0.86617 2.01520/4 1.2220 1.7680/4 0.7030/4 0.81020/4 0.77920/4 0.96220/4 0.94719/4 0.81020/4 0.9850/4 0.86520/0 1.32420/20 0.97718/4 0.80519/4 0.74520/4 0.94820 0.720−79 0.87520/4 0.86220 0.85720/4 1.752 0.89518/4 0.87220/4 0.7302 0.83020/4 1.038 0.93125

0.76721/4 0.77420/4 0.77720/4 0.82125/4

Melting point, °C

Boiling point, °C

207–8 206–7 d. 70 <−18 295 260 d. 5.5 93–4 30.8 10.9 35–6

subl.

55 71.5 72 15.5 −69 −96.9 116–7 <−20 44–5 −80 65–6 −34.4 52 −6.5 >360 −50 −12 6.5 23.9 −45 −103.7

−43.9 −85 −93.3 −58.2 −126.6 <−25 −73.5 d. 258–61 253 −51 −32 −29.7 7 −47 237–9 158 93–4 subl. 310 −44 −69 14.6

1.03 1.06315/4 131.5 96–8 51 −145

290–1 173–4 221–2765 235 190.8 202.8 202 308 314 316 189 170–1 d. 102.2 152.5 subl. 225761 14019 −640 1080.2 −21 61.3750 12.5–13 d. 11–12726 118–20 80–1 160–1 155–6 83.3 174750 134 165714 142 41–2 49–50 129–30 −34749 177 175–6 176–7 31915 193.3 185.3 174.0 232.9 167.9 d. 249–5315 188–90 190 173.4 228 204–611 22540 19516 expl. −23

Solubility in 100 parts Water sl. s. c. s. h. 0.3 c. i. 1.418 8.716 v. sl. s. v. sl. s. 2.5 0.5 1.8 i. i. i. 8.315 ∞25 18 i. 0.0225 i. v. s. sl. s. s. 45020 cc. s. 250020 cc. 0.2717 i. i. i. 3.620 s. v. sl. s. i. i. i. i. i. i. v. sl. s. i. i. i. i. 0.0119 212 i. i. i. i. s. ∞ sl. s. h. sl. s. sl. s. c. v. sl. s. sl. s. i. i. i. i. i. i. i. 0.05 d.

Alcohol

Ether

s. v. s. h. v. s.

v. sl. s. v. s. s. s. i.

0.0117 116 ∞ s. ∞30 ∞ ∞36

∞ s. ∞30 ∞ ∞36

s. ∞ ∞ s.

∞ ∞ s.

v. s. s. 230020 cc. s. v. s. 0.122 v. s. ∞ s. s. v. s. ∞ s. s. ∞

v. s. s. s. 50020 cc. s. 500020 cc.

∞ s. s. v. s. ∞ s. ∞ ∞

s. s. s. s. i. i. s. s. ∞ s. i. ∞ s. s. s.

s. s. s. s. i. s. s. ∞

s. ∞ ∞ s.

∞ ∞ ∞ s.

s.

s.

s. s. h.

s. v. s.

i. ∞ s. s. s.

s.

2-35

Dibenzothiazyl-disulfide (2-,2′-) Dibensoyl methane Dibensyl-amine -aniline ketone phthalate (o-) succinate Dibromo-benzene (o-) (m-) (p-) -diphenyl (4-,4′-) Dibutyl-adipate (n-) (i-) -amine (n-) (i-) -p-aminophenol (s-) -aniline (n-) carbonate (n-) (i-) (s-) ether (n-) (i-) (s-) ketone (n-) (i-) malate (l-)(n-) oxalate (n-) phthalate (n-) tartrate (d-)(n-) (d-)(i-) Dichloro-acetic acid -acetone (αα-) -aniline (2-,5-) -anthraquinone (1-,3-) (1-,4-) (1-,5-) (1-,6-) (1-,8-) (2-,3-) (2-,6-) (2-,7-) -benzene (o-) (m-) (p-) -butane (n-)(1-,4-) -diphenyl (4-,4′-) -ethane (1-,2-) -naphthalene (β-)(1-,4-) (γ-)(1-,5-) -nitrobenzene (2-,5-) -pentane (1-,5-) -phenol (2-,4-) Dichloramine T (p-) Dicyandiamide Diethanolamine Diethyl adipate -amine -aminophenol (m-) -aniline sulfonic acid (m-) carbonate diethyl malonate Diethyl dimethyl malonate glutarate ketone malonate -malonic acid -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide

(C6H4NSC)2S2 (C6H5CO)2CH2 (C6H5CH2)2NH C6H5N(CH2C6H5)2 (C6H5CH2)2CO C6H4(CO2CH2C6H5)2 (CH2CO2CH2C6H5)2 C6H4Br2 C6H4Br2 C6H4Br2 BrC6H4C6H4Br (CH2CH2CO2C4H9)2 (CH2CH2CO2C4H9)2 (C2H5CH2CH2)2NH [(CH3)2CHCH2]2NH (C4H9)2NC6H4OH C6H5N(C4H9)2 CO(OC4H9)2 CO(OC4H9)2 CO(OC4H9)2 (C2H5CH2CH2)2O [(CH3)2CHCH2]2O [C2H5(CH3)CH]2O (C2H5CH2CH2)2CO [(CH3)2CHCH2]2CO C2H4O(CO2C4H9)2 (CO2C4H9)2 C6H4(CO2C4H9)2 (CHOHCO2C4H9)2 (CHOHCO2C4H9)2 Cl2CHCO2H Cl2CHCOCH3 Cl2C6H3NH2 C6H4:(CO)2:C6H2Cl2 C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4Cl2 C6H4Cl2 C6H4Cl2 ClCH2(CH2)2CH2Cl ClC6H4C6H4Cl ClCH2CH2Cl C10H6Cl2 C10H6Cl2 Cl2C6H3NO2 ClCH2(CH2)3CH2Cl Cl2C6H3OH CH3C6H4SO2NCl2 H2NC(:NH)NHCN HN(CH2CH2OH)2 (CH2CH2CO2C2H5)2 (C2H5)2NH (C2H5)2NC6H4OH (C2H5)2NC6H5 (C2H5)2NC6H4SO3H OC(OC2H5)2 (C2H5)2C(CO2C2H5)2 (CH3)2C(CO2C2H5)2 CH2(CH2CO2C2H5)2 (C2H5)2CO CH2(CO2C2H5)2 (C2H5)2C(CO2H)2 C10H7N(C2H5)2 C10H7N(C2H5)2 (CO2C2H5)2 C6H4(CO2C2H5)2 O2S(OC2H5)2 (C2H5)2S

332.49 224.25 197.28 273.37 210.27 346.38 298.33 235.90 235.90 235.90 312.00 258.35 258.35 129.24 129.24 221.34 205.34 174.24 174.24 174.24 130.23 130.23 130.23 142.24 142.24 246.30 202.25 278.34 262.30 262.30 128.94 126.97 162.02 277.10 277.10 277.10 277.10 277.10 277.10 277.10 277.10 147.00 147.00 147.00 127.01 223.10 98.96 197.06 197.06 192.00 141.04 163.00 240.11 84.08 105.14 202.25 73.14 165.23 149.23 229.30 118.13 216.27 188.22 188.22 86.13 160.17 160.17 199.29 199.29 146.14 222.24 154.18 90.19

cr. rhb./al. col. oil pr./al. cr. pr./al. lf./al. col. lq. col. lq. pl./al. mn. pr. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. col. lq. lq. lq. lq. lq. oil lq. col. lq. col. lq. pr. cr. lq. lq. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. col. lq. col. lq. col. mn. lq. pr. col. lq. nd./al. lf./al. tri./al. col. lq. nd. cr. mn. pl. pr. col. lq. col. lq. rhb. oil cr. col. lq. col. lq. col. lq. syrup col. lq. col. lq. pr./aq. col. oil col. oil col. lq. col. lq. col. lq. col. lq.

1.50 1.028

25/25

20/4

1.956 1.952 20/4 2.26118 1.897 0.965 20/4 0.950 25 0.768 20/20 0.74125/4

180 78 −26 70–1 34–5 42–3 45–6 1.8 −6.9 87–8 164–5 −38 −20 −70

0.924 20/4 0.91915 0.769 20/20 0.76215 0.756 21 0.82718/4 0.805 21/4 1.038 20/4 0.986 20/4 1.045 21 1.09815 1.03175/4 1.560 25/25 1.23415

1.305 20/4 1.288 20/4 1.458 21 1.442 0/4 1.256 20/20 1.300 76/4 1.669 22 1.094 25/4 1.383 60/25 1.4014 1.09720/4 1.00920/4 0.712 15/15 0.934 20/4 0.975 20/4 0.985 20/4 0.994 25/25 1.025 21 0.816 19/4 1.055 20/4 1.005 1.026 1.079 20/4 1.121 25/25 1.172 25/4 0.837 20/4

−98 −5.9 −29.6 22–2.5 73–4 9.7(−4) 50 208–9 187.5 251 203–4 202–3 268–70 282 210–11 −17.6 −24.8 53 −38.7 148 −35.3 67–8 107 54.6 45 83 207–8 28 −21 −38.9 78 −34.4 270 d. −43 −24 −42 −49.8 125 −40.6 −25 −99.5

d. 219–2118 268–71250 >300 330.6 27412 23814 221–2 219755 218.6758 355–60 18314 278–80 159761 139–40 17010 262.8 207740 190 178–80 142.4 122.5 121 187.7 168.1 170–118 245.5 340 200–318 323–5 194.4 120 251

179 172766 174764 161–3 315–9 83.7 286–7 740 subl. 266 180–1 209–10 d. 270 748 239–41761 55.5759 276–80 216 126759 230 196.7 237 101.7 198.9 d. 170–80 285–90 318 186 298–9 210 92–3 754

i. i. i. i. i. v. sl. s. i. i. i. i. i. i. i. ∞ v. sl. s. i. i. i. i.

4.420 s. v. s. h.

s. s. s.

s. s. s. s. 1.6 v. sl. s. h. ∞

s. s. s. s. 7125 ∞

∞ s.

∞ s.

∞ s.

∞

<0.05 i. i. i. <0.06 v. sl. s. i. 0.04 25 i. v. sl. s. ∞ v. sl. s. v. sl. s. i. i. i. i. i. i. i. i. i. i. i.

∞ ∞ ∞ s. ∞

∞ ∞ ∞ v. s. ∞

s. ∞

s. ∞

∞ s. s. i. v. sl. s. sl. s.

∞ s. s.

∞ s. v. s.

∞ s. v. s.

i. 0.90 i. i. i. i. 0.45 20 sl. s. 2.318 ∞ 0.4380 v. s. s. 1.412 s. i. i. i. 0.88 20 4.7 20 2.08 20 65 16 i. i. v. sl. s. i. i. 0.3120

v. sl. s. ∞ v. sl. s. s. v. s. h. s. v. s.

425 ∞

1.318 ∞ s. ∞

0.0118 v. sl. s. s. ∞

v. sl. s.

sl. s. sl. s.

s. s. v. s.

s.

s.

∞ ∞ ∞ v. s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ ∞ ∞

2-36

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

tartrate (d-) -toluidine (o-) (m-) (p-) Diethyleneglycol dinitrate Difluorodichloromethane Diglycerol Dihydroxy-dinaphthyl (α-) (-2,-2′,-1,-1′) -diphenyl (4-,4′-) -ethyl formal (β-) -naphthalene (1-,5-) (1-,8-) Dimethoxy-benzene (p-) -diphenylamine (4-,4′-) -ethyl adipate Dimethyl adipate -amine -aminoasobenzene (p-) -aminoethanol -aminophenol (m-) -aniline sulfonic acid (m-) (p-) carbonate ether -formamide fumarate glutarate glyoxime -naphthalene (1-,4-) (2-,3-) -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide tartrate (d-) -vinyl-ethenyl carbinol Dinaphthyl (αα-) -methane (αα′-) (β,β′-) Dinitro-anisole (1-)(2-,4-) -benzene (o-) (m-) (p-) sulfonic acid (2-,4-)(1-) -benzoic acid (2-,4-) (3-,5-) -benzophenone (4-,4′-) -diphenyl (4-,4′-) (2-,4′-) -naphthalene (1-,5-) (1-,8-) Dinitro-phenol (2-,3-) (2-,4-) (2-,6-) -salicylic acid (3-,5-) -stilbene (4-,4′-) -toluene (2-,4-) (3-,4-) (3-,5-) Dioxane Dipentene

Formula (CHOHCO2C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 O(CH2CH2ONO2)2 F2CCl2 [(HO)2C3H5]2O (HOC10H6)2 (HOC10H6)2 (HOC6H4)2 CH2(OCH2CH2OH)2 C10H6(OH)2 C10H6(OH)2 (CH3O)2C6H4 HN(C6H4OCH3)2 (CH2)4(CO2C2H4OCH3)2 [(CH2)2CO2CH3]2 (CH3)2NH C6H5N:NC6H4N(CH3)2 (CH3)2NCH2CH2OH (CH3)2NC6H4OH (CH3)2NC6H5 (CH3)2NC6H4SO3H (CH3)2NC6H4SO3HH2O OC(OCH3)2 CH3OCH3 HCON(CH3)2 (:CHCO2CH3)2 (CH2)3(CO2CH3)2 (CH3C:NOH)2 C10H6(CH3)2 C10H6(CH3)2 C10H7N(CH3)2 C10H7N(CH3)2 (CO2CH3)2 C6H4(CO2CH3)2 (CH3O)2SO2 (CH3)2S (CHOHCO2CH3)2 (CH3)2COHC⯗CCH:CH2 C10H7C10H7 (C10H7)2CH2 (C10H7)2CH2 CH3OC6H3(NO2)2 C6H4(NO2)2 C6H4(NO2)2 C6H4(NO2)2 (NO2)2C6H3SO3H3H2O (NO2)2C6H3CO2H (NO2)2C6H3CO2H (NO2C6H4)2CO (NO2C6H4)2 (NO2C6H4)2 C10H6(NO2)2 C10H6(NO2)2 (NO2)2C6H3OH (NO2)2C6H3OH (NO2)2C6H3OH (NO2)2C6H2(OH)CO2HH2O (NO2C6H4CH:)2 (NO2)2C6H3CH3 (NO2)2C6H3CH3 (NO2)2C6H3CH3 O < (CH2CH2)2 > O C10H16

Formula weight 206.19 163.26 163.26 163.26 196.12 120.91 166.17 286.32 286.32 186.21 136.15 160.17 160.17 138.16 229.27 262.30 174.19 45.08 225.29 89.14 137.18 121.18 201.24 219.26 90.08 46.07 73.09 144.13 160.17 116.12 156.22 156.22 171.24 171.24 118.09 194.18 126.13 62.13 178.14 110.15 254.33 268.35 268.35 198.13 168.11 168.11 168.11 302.22 212.12 212.12 272.21 244.20 244.20 218.17 218.17 184.11 184.11 184.11 246.13 270.24 182.13 182.13 182.13 88.11 136.23

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

lq. lq. lq. lq. lq. gas lq. pl./al. nd./al. rhb./al. lq. pr./aq. nd. lf. cr. lq. col. lq. col. lq. yel./al. col. lq. nd. yel. lq. cr. pr. col. lq. gas lq. col. tri. lq. col. cr. lq. lf./al. col. oil col. cr. col. mn. col. lq. col. oil oil cr. lq. lf./al. pr./al. nd./al. col. mn. col. mn. col. rhb. col. mn. pr. cr./aq. mn. pr. col. nd. nd./al. mn. nd. rhb. yel. mn. yel. rhb. yel. rhb. pl./aq. yel. lf. nd. nd. mn. pr. col. lq. col. lq.

1.204 20/4

17

0.924 15.5 1.377 25/4 1.486 −30

−11.3 −155

280 208–9755 231–2 228–9

1.25 1.154 25 1.053 55/55 1.075 15.6 1.063 20/4 0.680 0/4 0.887

20/4

0.956 20/4 1.070 20/4 0.94525 1.08915.6 1.016 20/4 20

1.042 1.03970/70 1.14854 1.18925/25 1.3520/4 0.84621/4 1.32820/4 0.887 20/4

1.341 20 1.5918 1.575 20/4 1.625 18

1.445 1.474 1.681 20 1.683 24

1.321 71 1.259 111 1.277 111 1.033 20/4 0.865 18

300 218 270–2 −5.3 258–60 140 56 103 10–1 −96 116–7 85 2.5 d. 266 257 0.5 −138.5 −58.3 102 −37 240–6 <−18 104 46 54 −26.8 −83.2 61.5 160 109 92 94–5 117–8 89.8 173–4 106–8 179–80 204–5 189 233 93.5 216 170–2 144–5 114–5 63–4 173 d. 210–6 70 60–1 92–3 9.5–10.5

−29.2 220–3010 subl. subl. 264 d. 212.6 145–502 11518 7.4 d. 135756 265–8 193 89–90 −23.7 152.8 192 13050 264–6 265767 274.5711 304–5 163.3 280734 188.3 37.3 280 150 240–412 >360 319774 300–2 299777 subl.

subl. d. subl.

300 subl. 101.1 178

Solubility in 100 parts Water

Alcohol

Ether

sl. s. i. i. i. i. 5.7 cc.26 s. h. i. i. sl. s. ∞ sl. s. sl. s. h. v. sl. s. i. 5 i. v. s. i. ∞ sl. s. h. i. s. s. h. i. 3700 cc.18 ∞ i.

∞ s. s.

∞ s. s.

s.

s. i. v. s. v. s. v. s.

sl. s.

sl. s.

0.0620 i. i. i. i. 6 0.43 v. sl. s. i. s. 620 i. i. i. sl. s. h. 0.01 c. 0.399 0.18100 s. 1.8525 s h. i. i. i. i. i. sl. s. 0.5 c. s. h. s. c. i. 0.0322 i. sl. s. ∞ i.

v. s.

v. s.

sl. s. s. s. s.

s. s. s.

s. s. v. s.

v. s.

s.

v. s. v. s. v. s.

s. s.

s. s.

s. s.

s. s.

v. sl. s. ∞ s.

v. sl. s. ∞ s.

∞ s. 20015

∞ s.

s. h. 0.8 c. s. 1.520 1.921 320 0.1821 s. s. v. s.

s. v. s.

v. sl. s. sl. s.

20

1.5 v. s. h. 0.2 c. v. s. h. 420 s. h. v. s. v. sl. s. 1.215

v. s. v. s. h. s. v. s. v. sl. s. 916

s. h. s.

s. s.

2-37

Diphenyl -amine carbonate -chloroarsine -ethane ether guanidine -methane phenylenediamine (p-) succinate sulfide sulfone urea (uns.) Diphenylene oxide Dipropyl adipate (n-) -amine (n-) (i-) aniline (n-) carbonate (n-) ether (n-) (i-) ketone (n-) (i-) oxalate (n-) (i-) Disalicylal ethylenediamine Ditolyl guanidine (o-) Divinyl acetylene Docosane (n-) Dodecane (n-) Dulcitol Durene (1-,2-,4-,5-) Elaidic acid Eosine Ephedrine (l-) Epichlorhydrin (α-) Epidichlorohydrin (α-) Erythritol (dl-) tetranitrate Ethane Ethanol-amine formamide Ether Ethyl abietate acetate acetoacetate alcohol -amine hydrochloride aniline sulfonic acid (m-) anisate (p-) anthranilate (o-) benzene benzoate -benzyl-aniline bromide butyrate (n-) (i-) caprate (n-) Ethyl caproate (n-) caprylate (n-) chloride chloroacetate chlorocarbonate cinnamate (trans-) cyanoacetate formate furoate (α) heptoate hypochlorite iodide lactate

C6H5C6H5 C6H5NHC6H5 CO(OC6H5)2 (C6H5)2AsCl (C6H5CH2)2 C6H5OC6H5 (C6H5NH)2C:NH (C6H5)2CH2 (C6H5NH)2C6H4 (CH2CO2C6H5)2 (C6H5)2S (C6H5)2SO2 (C6H5)2NCONH2 < (C6H4)2O (CH2CH2CO2C3H7)2 (C2H5CH2)2NH [(CH3)2CH]2NH C6H5N(C3H7)2 CO(OCH2C2H5)2 (C2H5CH2)2O [(CH3)2CH]2O (C2H5CH2)2CO [(CH3)2CH]2CO (CO2CH2C2H5)2 [CO2CH(CH3)2]2 [HOC6H4CH:NCH2]2 (C7H7NH)2C:NH (H2C:CHC⯗)2 CH3(CH2)20CH3 CH3(CH2)10CH3 CH2OH(CHOH)4CH2OH (CH3)4C6H2 C8H17CH:CH(CH2)7CO2H C20H8O5Br4 C6H5CHOHCH(CH3)NHCH3 C2H3OCH2Cl CH2:CClCH2Cl CH2OH(CHOH)2CH2OH C4H6(ONO2)4 CH3CH3 HOCH2CH2NH2 HCONHCH2CH2OH (CH3CH2)2O C19H29CO2C2H5 CH3CO2C2H5 CH3COCH2CO2C2H5 CH3CH2OH C2H5NH2 C2H5NH2HCl C6H5NHC2H5 C2H5NHC6H4SO3H CH3OC6H4CO2C2H5 NH2C6H4CO2C2H5 C6H5C2H5 C6H5CO2C2H5 C6H5N(C2H5)CH2C6H5 C2H5Br C2H5CH2CO2C2H5 (CH3)2CHCO2C2H5 CH3(CH2)8CO2C2H5 CH3(CH2)4CO2C2H5 CH3(CH2)6CO2C2H5 CH3CH2Cl ClCH2CO2C2H5 ClCO2CH2CH3 C6H5CH:CHCO2C2H5 CH2(CN)CO2C2H5 HCO2CH2CH3 OC4H3CO2C2H5 CH3(CH2)5CO2C2H5 ClOCH2CH3 CH3CH2I CH3CH(OH)CO2C2H5

154.21 169.22 214.22 264.58 182.26 170.21 211.26 168.23 260.33 270.28 186.27 218.27 212.25 168.19 230.30 101.19 101.19 177.29 146.18 102.17 102.17 114.19 114.19 174.19 174.19 268.31 239.32 78.11 310.60 170.33 182.17 134.22 282.46 647.89 165.23 92.52 110.97 122.12 302.11 30.07 61.08 89.09 74.12 330.50 88.11 130.14 46.07 45.08 81.54 121.18 201.24 180.20 165.19 106.17 150.17 211.30 108.97 116.16 116.16 200.32 144.21 172.26 64.51 122.55 108.52 176.21 113.11 74.08 140.14 158.24 80.51 155.97 118.13

col. mn. col. mn. nd./al. rhb. col. pr. col. rhb. mn./al. col. pr. cr. lf./al. col. lq. nd./aq. rhb. lf./al. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. cr. lq. cr. lq. mn. mn. lf./al. col. cr. cr./et. lq. col. lq. tet. pr. lf./al. col. gas col. oil lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. mn. lq. nd./aq. lq. cr. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lf. col. lq. yel. lq. col. lq. oil

0.992 73/4 1.160 20/20 1.272 14 1.583 40 0.978 50/50 1.073 20 1.001 26/4 1.119 15/15 1.248 25/4 1.276 0.979 20/4 0.739 20/4 0.722 22 0.910 20 0.968 22 0.744 21/0 0.725 21/0 0.822 20/4 0.806 20/4 1.038 0/0 1.34 1.1020/4 0.776 20/4 0.778 44/4 0.751 20/4 1.466 15 0.838 81/4 0.851 79/4 1.183 25/25 1.204 25 1.451 20/4 0.546 −88 1.022 20 1.169 25 0.708 25/4 1.020 20/20 0.901 20/4 1.025 20/4 0.789 20/4 0.689 15/15 1.216 0.963 20/4 1.103 25/25 1.117 20/4 0.867 20/4 1.052 15/15 1.034 18.5 1.431 20/4 0.879 20/4 0.871 20/4 0.859 28 0.873 20/20 0.878 17 0.917 6/6 1.159 20/4 1.138 20/4 1.049 20/4 1.062 20/4 0.923 20/4 1.117 21/4 0.872 20/20 1.013 −6/4 1.933 20/4 1.030 25/4

69–70 52.9 80 43–4 52–3 27 147–8 26–7 152 122–3 <−40 128–9 189 86–7 −20.3 −39.6 −61 −122 −60 −32.6 −51.7

254.9 302 302–6 d. 327 284 259 d. > 170 265 330 296–7 379 287–8 143–510 110–1 83.5743 245.4 168.2 91 69 144.2 123.7 213.5 190

125–6 178–9 44.5 −9.6 189 79–80 51–2 40 −25.6 126 61 −172 10.5 <−40 −116.3 −82.4 −45 −112 −80.6 108–9 −63.5 d. 294 7–8 13 −94.4 −34.6 −117.8 −93.3 −88.2 −20 −67.5 −45 −139 −26 −80.6 12 −22.5 −79 34 −66.1 expl. −105

85 224.515 214.5 290–53 193–5 288100 255 117756 94 329–31 expl. −88.6 171757 d. 34.6 2004 77.1 180755 78.4 16.6 204 269–70 266–8 136.2 211–2 28510 38.4 120–1 110–1 244.6758 165–6736 207–8753 13 144 94–5 271 208753 54760 195766 187–8 36752 72.4 155

i. 0.0325 i. 0.2 d. i. v. sl. s. v. sl. s. i. i. i. i. sl. s. h. v. sl. s. i. i. s. s. i. v. sl. s. sl. s. 0.2 0.43 v. sl. s. d. h. 0.0328 v. sl. s. i. i. i. 3.215 i. i. i. 5 <5 i. 60 i. c. 4.7 cc.20 ∞ ∞ 7.520 i. 8.515 1317 ∞ ∞ 24017 i. 2.1515 i. v. sl. s. 0.0115 0.0820 i. 1.060 0.6825 sl. s. 0.00220 i. i. 0.450 i. d. i. 225 1118 i. 0.02920 0.420 ∞

1020 5619.5 v. s. 20 s. s. 920 v. s.

6.620 s. s. s. v. s. ∞ sl. s. v. s. s. ∞

s. h. s. h. s. s. h. s. ∞ s. s.

s. v. s. s. ∞

∞ ∞ ∞ ∞

∞ ∞ ∞ ∞

s.

s. h.

s.

4 h. v. s. v. sl. s. s. v. s. s. 500 ∞ ∞ sl. s. c. s. 150 cc. ∞

v. s. v. s. i. s. v. s. s. ∞ ∞ i. s. 1

∞ ∞ ∞ ∞ v. s. ∞ s. s. ∞ ∞ 18 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ i. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

2-38

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

laurate mercaptan methacrylate naphthylamine (α-) naphthyl ether (α-) nitrate nitrite oleate palmitate pelargonate propionate salicylate (o-) stearate toluate (o-) (m-) toluene sulfonate (p-) toluidine (o-) (p-) urea valerate (n-) (i-) Ethylal Ethylene bromide bromohydrin chlorobromide chlorohydrin diamine oxide Ethylidene diacetate Eugenol (1-,4-,3-) (i-)(1-,3-,4-) Fenchyl alcohol (dl-) (d-)(α-) (i-)(l-) Ferric dimethyl-dithiocarbamate Fluorene Fluorescein Fluoro-dichloromethane -trichloromethane Formaldehyde (m-) (p-) Formamide Formanilide Formic acid Fructose Fuchsin Fulminic acid Fumaric acid (trans-) Furfural Furfuran Furfuryl acetate alcohol butyrate propionate Furoic acid G-acid, K salt (2-)(6-,8-) Na salt (2-)(6-,8-) Galactose (d-)(α-) Gallic acid (3-,4-,5-) Gamma acid (2-,8-,6-) Geraniol Glucose (d-)(α-) (d-)(β-) Glucuronic acid Glutam(in)ic acid (dl-)

Formula CH3(CH2)10CO2C2H5 CH3CH2SH CH2:C(CH3)CO2C2H5 C10H7NHC2H5 C10H7OC2H5 C2H5ONO2 C2H5ONO C17H33CO2C2H5 CH3(CH2)14CO2C2H5 CH3(CH2)7CO2C2H5 CH3CH2CO2C2H5 HOC6H4CO2C2H5 CH3(CH2)16CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4SO3C2H5 CH3C6H4NHC2H5 CH3C6H4NHC2H5 C2H5NHCONH2 CH3(CH2)3CO2C2H5 (CH3)2CH(CH2)CO2C2H5 CH2(OC2H5)2 H2C:CH2 BrCH2CH2Br BrCH2CH2OH ClCH2CH2Br ClCH2CH2OH H2NCH2CH2NH2 < (CH2)2 > O CH3CH(O2CCH3)2 C3H5C6H3(OH)OCH3 C3H5C6H3(OCH3)OH C10H17OH C10H17OH C10H17OH Fe[SSCN(CH3)2]3 (C6H4)2 > CH2 C20H12O5 FCHCl2 Cl3CF HCHO (CH2O)3 (CH2O)xxH2O HCONH2 HCONHC6H5 HCO2H CH2OH(CHOH)3COCH2OH C20H19N3HCl C:NOH HO2CCH:CHCO2H C4H3OCHO C4H4O CH3CO2CH2C4H3O C4H3OCH2OH C3H7CO2CH2C4H3O C2H5CO2CH2C4H3O C4H3OCO2H HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C5H11O5CHO (HO)3C6H2CO2HH2O C10H5(NH2)(OH)SO3H C9H15CH2OH C5H11O5CHO C6H12O6H2O CHO(CHOH)4CO2H [CHNH2(CH2)2](CO2H)2

Formula weight 228.37 62.13 114.14 171.24 172.22 91.07 75.07 310.51 284.48 186.29 102.13 166.17 312.53 164.20 164.20 200.25 135.21 135.21 88.11 130.18 130.18 104.15 28.05 187.86 124.96 143.41 80.51 60.10 44.05 146.14 164.20 164.20 154.25 154.25 154.25 416.49 166.22 332.31 102.92 137.37 30.03 90.08 (30.03) 45.04 121.14 46.03 180.16 337.85 43.02 116.07 96.08 68.07 140.14 98.10 168.19 154.16 112.08 380.48 348.26 180.16 188.13 239.25 154.25 180.16 198.17 194.14 147.13

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

oil lq. col. lq. oil cr. col. lq. lq. oil col. nd. col. lq. col. lq. col. lq. col. cr. lq. lq. pr./al. lq. lq. nd. col. lq. col. lq. lq. col. gas col. lq. col. lq. lq. col. lq. col. lq. lq. col. lq. oil oil col. cr. col. pr. col. cr. cr. cr./al. yel. red gas col. lq. gas wh. amor. lq. mn. col. lq. nd./aq. red

0.868 13/4 0.839 20/4 0.913 15.6 1.060 20/4 1.061 20/20 1.100 25/4 0.900 15.5 0.867 25 0.858 25/4 0.866 17.5 0.891 20/4 1.136 15/4 0.848 36.3 1.032 25/25 1.030 20/20 1.166 48/4 0.948 25/4 0.942 25/4 1.213 18 0.877 20 0.867 20/4 0.824 25/4 0.57−102/4 2.180 20/4 1.772 20/4 1.689 19 1.213 20/4 0.900 20/20 0.887 7/4 1.061 12 1.070 15/15 1.091 15/15 0.935 40 0.964 20/4 0.961

−10.7 −121

269 36–7 118 303723 276.4 87–8 17 216–815 19110 227–8757 99.1 233–4 20110 227 231750 221.3 215–6 217

col. pr. lq. col. lq. col. oil oil col. lq. col. lq. mn. pr. cr. cr. pr. mn./aq. cr. col. lq. rhb. cr. cr. cr./aq.

1.203 0/4 1.4260 1.494 17.2 0.815 −20 1.1765 1.139 20/4 1.147 15/15 1.220 20/4 1.669 17.5 1.22 1.635 20/4 1.159 20/4 0.937 20/4 1.118 20/4 1.129 25/4 1.053 20/4 1.109 20/4

5.5 −102 <−15 24–5 −44.5 −72.6 1.3 33.4(31) <−10 33–4 <−15 92 −91.2 −99.3 −66.5 −169 10 −16.6 −69 8.5 −111.3 18.85 10.3 −10 35 45–7 61–2 d. 100–30 115–6 d. > 290 −127 −92 64 150–60 2 47 8.6 95–105 d. >200 286–7 −38.7

133–4

1.694 4/4 0.883 15 1.544 25 1.562 18/4 1.460

145.5 135 89 −103.9 131.5 150.3 106.7 128.8 117.2 13.5747 168740 253.5 267.5 201 201–2 201–2 ign. >150 293–5 14.5 24.9 −21 114.5759 subl. 193 216120 100.8

290 161.7760 31–2756 175–7 169.5752 212–3 195–6 230–2

165.5 d. 220 <−15 146 150 154 199 d.

230 d.

Solubility in 100 parts Water

Alcohol

Ether

i. 1.5 i. i. i. 1.355 v. sl. s. i. i. i. 2.420 i. i. i. i. i. i. i. v. s. 0.2425 0.1720 918 26 cc.0 0.4380 sl. s. 0.6980 ∞ ∞ ∞ sl. s. v. sl. s. v. sl. s. sl. s. sl. s. i. v. sl. s. i. v. sl. s. h. i. i. v. s. 2125 20–3018 ∞ sl. s. ∞ v. s. 0.3

s. s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

∞ s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.

80 ∞ ∞ ∞ 360 cc. ∞ s.

i. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ ∞ ∞ ∞

∞ 0.3 v. s. ∞ ∞

s.

s.

0.717 9.113 i. i. ∞ v. sl. s. v. sl. s. 3.615 825 3420 10.30 113

5.830 ∞ s. s. s. s. s. s.

0.725 ∞ s. s. s. ∞ ∞ s.

0.640 2815

2.515

i. 8217.5 15415 v. s. 1.5 20

∞ sl. s.

∞ i.

v. sl. s.

v. sl. s.

s. h. s. h. s. ∞ v. s. s. i. ∞ v. s. ∞ 818 s.

s. s. ∞ v. s. s. i. v. sl. s. s. i.

Glutaric acid Glycerol acetate (mono-) (di-) nitrate (mono-) (α-) (β-) dinitrate (1-,3-) Glyceryl triacetate tribenzoate tributyrate tricaprate tricaproate tricaprylate trilaurate trimyristate trinitrate trinitrite trioleate tripalmitate tristearate Glycide Glycine, Glycocoll Glycol diacetate dibenzoate dibutyrate dicaprylate diformate dilaurate dinitrate dinitrite dipalmitate dipropionate ether formal formate (mono-) Glycolic acid Guaiacol (o-) Guanidine H-acid, Na salt (1-,8-,3-,6-) Heptacosane (n-) Heptane (n-) (i-)

Heptoic acid aldehyde Heptyl acetate (n-) alcohol (n-) mercaptan Hexachloro-benzene -ethane Hexacosane (n-) Hexadecane (n-) Hexaethylbenzene Hexamethylbenzene Hexamethylene-diamine -diisocyanate -glycol tetramine Hexane (n-) (i-) (neo-)

2-39

CH2(CH2CO2H)2 CH2OHCHOHCH2OH C5H10O4 (CH3CO2)2C3H5OH CH2OHCHOHCH2NO3 CH2OHCHNO3CH2OH CHOH(CH2ONO2)2 (CH3CO2)3C3H5 (C6H5CO2)3C3H5 (C2H5CH2CO2)3C3H5 [CH3(CH2)8CO2]3C3H5 [CH3(CH2)4CO2]3C3H5 [CH3(CH2)6CO2]3C3H5 [CH3(CH2)10CO2]3C3H5 [CH3(CH2)12CO2]3C3H5 CH2NO3CHNO3CH2NO3 CH2NO2CHNO2CH2NO2 (C17H33CO2)3C3H5 [CH3(CH2)14CO2]3C3H5 [CH3(CH2)16CO2]3C3H5 C2H3OCH2OH NH2CH2CO2H CH2OHCH2OH (CH3CO2CH2)2 (C6H5CO2CH2)2 (C3H7CO2CH2)2 (C7H15CO2CH2)2 (HCO2CH2)2 (C11H23CO2CH2)2 (O2NOCH2)2 (ONOCH2)2 (C15H31CO2CH2)2 (C2H5CO2CH2)2 (HOCH2CH2)2O < OCH2CH2OCH2 > HCO2CH2CH2OH HOCH2CO2H CH3OC6H4OH NH:C(NH2)2 C10H8O7NS2Na1aH2O CH3(CH2)25CH3 CH3(CH2)5CH3 (CH3)2CH(CH2)3CH3 C3H7CH(CH3)C2H5 (CH3)3CCH2C2H5 [(CH3)2CH]2CH2 (CH3)2C(C2H5)2 (C2H5)3CH (CH3)3CCH(CH3)2 CH3(CH2)5CO2H CH3(CH2)5CHO CH3CO2CH2(CH2)5CH3 CH3(CH2)5CH2OH [(CH3)2CH]2CHOH (C2H5CH2)2CHOH CH3CH(SH)C5H11 C6Cl6 CCl3CCl3 CH3(CH2)24CH3 CH3(CH2)14CH3 C6(C2H5)6 C6(CH3)6 NH2(CH2)6NH2 OCN(CH2)6NCO HO(CH2)6OH (CH2)6N4 CH3(CH2)4CH3 (CH3)2CH(CH2)2CH3 (CH3)3CC2H5 (CH3)2CHCH(CH3)2 (C2H5)2CHCH3

132.11 92.09 134.13 176.17 137.09 137.09 182.09 218.20 404.41 302.36 554.84 386.52 470.68 639.00 723.16 227.09 179.09 885.43 807.32 891.48 74.08 75.07 62.07 146.14 270.28 202.25 314.46 118.09 426.67 152.06 120.06 538.89 174.19 106.12 74.08 90.08 76.05 124.14 59.07 368.32 380.73 100.20 100.20 100.20 100.20 100.20 100.20 100.20 100.20 130.18 114.19 158.24 116.20 116.20 116.20 132.27 284.78 236.74 366.71 226.44 246.43 162.27 116.20 168.19 118.17 140.19 86.18 86.18 86.18 86.18 86.18

col. cr. col. lq. col. oil col. lq. col. pr. lf. oil col. lq. nd. col. lq. col. cr. col. lq. col. lq. col. nd. lf. yel. oil yel. lq. col. oil col. nd. col. pr. col. lq. mn. col. lq. col. lq. rhb./et. col. lq. lq. lq. amor. yel. lq. lq. nd. lq. lq. lq. lq. nd./aq. pr. col. cr. cr. col. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. mn. rhb. cr. lf. pr./al. pl./al. lf. lq. nd./aq. col. rhb. col. lq. lq. lq. lq. lq.

1.429 15 1.260 50/4 1.20 20/4 1.178 15/15 1.4015 1.4015 1.4715 1.16117/4 1.228 12 1.032 20/4 0.921 40/4 0.987 20/4 0.954 20/4 0.894 60/4 0.885 60/6 1.601 15 1.291 10/16 0.915 15 0.866 80/4 0.862 80/4 1.114 16/16 1.161 1.113 19/4 1.109 14/4 1.024 0

97.5 17.9 40 58–9 54 <−30 −78 75–6 <−75 31(25) −25 8.3(−21) 45–6 56.5 13.3(2) −4 65.1 70.8(55)

20020 290 158165 175–640 155–60 155–60 146–815 258–9 d. 305–9

16015 150 sl. d. 24018 310–200.1 166 sl. d.

232–6 d. −15.6 −31 73–4

197.4 190.5 >360 240

22 1.482 21/2 1.216 0 1.045 25 1.118 20/20 1.060 20/4 1.199 15/4 1.140 15/15 0.780 60/4 0.684 20/4 0.679 20/4 0.687 20/4 0.674 20/4 0.675 20/4 0.693 20/4 0.698 20/4 0.690 20/4 0.918 20 0.850 20/ᐉ 0.874 16/16 0.824 20/4 0.829 20/4 0.820 20/4 0.835 20 2.044 24 2.091 20/4 0.779 57/4 0.774 20/4 0.831 130/4 1.0428 20/4

0.659 0.654 20/4 0.649 20/20 0.662 20/4 0.664 20/4

52–4 −20 <−15 71–2 −10.5 79(63) 28.3 50 59.5 −90.6 −118.2 −119.4 −125 −119.4 −135.0 −118.7 −25 −10 −42 −34 −37 228–31 186–7 56.6 18.5 130 166 42 42 subl. −94 −153.7 −98.2 −129.8 −118

174 18820 expl. 114 96–8 2600.1 211–2 244.8 75–6 180 d. 205 27015 98.4760 90.0 91.8 79.1 80.8 86.0 93.5 80.8 221–2 155 191.5 759 175 756 140 156 174–5 765 309742 186777 26215 287.5 298.3 265 204–5 143–420 250 69 60.2 49.7 58.0760 63.2

63.920 ∞ v. s. s. 7015 7.1715 i. i. i. i. i. i. i. 0.1820 d. i. i. i. ∞ 23 c. ∞ 14.322 i. i. i. v. sl. s. i. 0.9225 i. i. sl. s. ∞ ∞ ∞ ∞ 1.715 v. s. 0.1720 i. 0.00515 i. i. i. i. i. i. i. 0.2515 0.0220 i. 0.18 25 v. sl. s. i. i. i. 0.00522 i. i. i. i. v. s. d. s. 8112 0.01415 i. i. i. i.

v. s. ∞ v. s. s. v. s. v. s. v. s. ∞ s. h. s. s. h. s. s. sl. s. c. s. 5020 d. sl. s. 0.00421 s. h. ∞ 0.1 c. ∞ ∞

v. s. i. sl. s. sl. s. v. sl. s. sl. s. v. s. ∞ s. s. v. s. s. s. v. s.

v. s.

∞ s. v. s. v. s. s. h. ∞ i. 1.0 ∞ s. v. s.

v. s. s. s. d. s. ∞ ∞

v. s. ∞ s. s. ∞ i.

9025 v. s. s.

v. s. v. s.

sl. s. s. s. s. s. s. s. s. s. ∞ s. ∞ ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ s. ∞ ∞ s.

v. sl. s. h. v. s. v. sl. s. ∞ 0.7525 0.20 s. d. s. 3 5033

s. h. v. s. ∞ 825 v. s. sl. s. h. v. sl. s. ∞ s. s. s. s.

2-40

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

Hexyl acetate (n-) alcohol (n-) formate (n-) resorcinol (2-,4-) Hippuric acid Histidine (l-) Homophthalic acid (o-) Hydracrylic acid Hydro-cyanic acid -quinone (p-) Hydroxy-benzaldehyde (p-) -benzanilide (o-) -quinoline (2-)(α-) (8-)(o-) Indigo White Indole Indoxyl Iodo-benzene -phenol (p-) Iodoform Ionone (α-) (β-) Irone (β-) Isatin Isoprene Ketene Koch acid (1-)(3-,6-,8-) Lactic acid (dl-) anhydride Lactide (dl-) Lactose Lauric acid Laurone Lauryl alcohol Lead tetraethyl tetramethyl Lepidine (py-4) Leucine (l-) Levulinic acid Limonene (d- or l-) Linalool (d- or l-) Linalyl acetate Linoleic acid Maleic acid anhydride Malic acid (dl-) (d- or l-) Malonic acid Maltose Mandelic acid (dl-) Mannitol (d-) Mannose (d-) Margaric acid Mellitic acid Menthol (l-)(α-) Mercapto-benzothiazole (2-) -thiazoline (2-) Mercuric cyanide fulminate Mesityl oxide Mesitylene (1-,3-,5-) Metanilic acid (m-) Methane

Formula CH3CO2(CH2)5CH3 CH3(CH2)4CH2OH (CH3)2CHC(CH3)2OH (CH3)2COHCH2C2H5 HCO2CH2(CH2)4CH3 CH3(CH2)5C6H3(OH)2 C6H5CONHCH2CO2H C6H9O2N3 HO2CC6H4CH2CO2H HOCH2CH2CO2H HCN C6H4(OH)2 HOC6H4CHO HOC6H4CONHC6H5 C9H6NOH C9H6NOH [C6H4(CO)(NH)C:]2 C16H12O2N2 C8H7N C8H6NOH C6H5I IC6H4OH HCI3 C10H16:CHCOCH3 C10H16:CHCOCH3 C14H22O C6H4 < (CO)(N) > COH CH2:CHC(CH3):CH2 H2C:CO C10H4(NH2)S3O9HNa2 CH3CH(OH)CO2H C6H10O5 C6H8O4 C12H22O11H2O CH3(CH2)10CO2H [CH3(CH2)10]2CO CH3(CH2)10CH2OH Pb(CH2CH3)4 Pb(CH3)4 C9H6NCH3 (CH3)2CHCH2CH(NH2)CO2H CH3CO(CH2)2CO2H C10H16 C10H17OH CH3CO2C10H17 C17H31CO2H HO2CCH:CHCO2H < (CHCO)2 > O HO2CCH2CH(OH)CO2H HO2CCH2CH(OH)CO2H H2C(CO2H)2 C12H22O11H2O C6H5CH(OH)CO2H CH2OH(CHOH)4CH2OH CH2OH(CHOH)4CHO CH3(CH2)15CO2H C6(CO2H)6 C10H19OH < C6H4N:C(SH)S > < CH2N:C(SH)SCH2 > Hg(CN)2 Hg(ONC)2aH2O (CH3)2C:CHCOCH3 C6H3(CH3)3 H2NC6H4SO3H CH4

Formula weight 144.21 102.17 102.17 102.17 130.18 194.27 179.17 155.15 180.16 90.08 27.03 110.11 122.12 213.23 145.16 145.16 262.26 264.28 117.15 133.15 204.01 220.01 393.73 192.30 192.30 206.32 147.13 68.12 42.04 427.34 90.08 162.14 144.13 360.31 200.32 338.61 186.33 323.44 267.34 143.19 131.17 116.12 136.23 154.25 196.29 280.45 116.07 98.06 134.09 134.09 104.06 360.31 152.15 182.17 180.16 270.45 342.17 156.27 167.25 119.21 252.62 293.63 98.14 120.19 173.19 16.04

Form and color

Specific gravity

col. lq. col. lq. lq. lq. lq. col. nd. rhb. lf./aq. cr./aq. syrup lq. cr. nd./aq. pr./al. pr./al. pr. cr. gray lf./aq. yel. pr. col. lq. nd./aq. yel. hex. col. oil col. oil col. oil yel. red col. lq. col. gas cr. hyg. yel. oil tri./al. col. rhb. col. nd. pl. lf. col. lq. col. lq. lq. cr. lf. lq. col. oil col. lq. yel. oil mn. cr. col. cr. col. cr. col. tri. col. nd. rhb./aq. col. rhb. rhb. col. pl. nd./al. col. cr. nd. cr. cr. cr./aq. lq. col. lq. col. nd. gas

0.890 0/0 0.820 20/20 0.821 20/0 0.809 20/4 0.898 0 1.371 20/4

0.697 18 1.332 15 1.129 130

1.35

25/4

1.824 1.857 112 4.008 17 0.930 20 0.944 20 0.939 20

Melting point, °C −51.6 −14 −107 68–70 187–8 d. 287 175–80 −12 170.3 116–7 135 199–200 75–6 390–2 52 85 −28.5 93–4 119

0.681 20/4

200–1 −120 −151

1.249 15/4

16.8

0.862 10/4 1.525 20 0.869 50/4 0.809 69/4 0.831 24/4 1.659 18/4 1.995 20/4 1.086 20 1.29318 1.140 20/20 0.842 20/4 0.868 20 0.895 20 0.903 18/4 1.609 1.5 1.601 20/4 1.595 20/4 1.631 15 1.540 17 1.300 20/4 1.489 20/4 1.539 20/4 0.853 60

124.5 202 48(44) 69–70 24 −136 −27.5 9–10 295 33.5 −96.9

15/15

0.890 1.4220/4 1.50 4.003 22 4.4 0.858 20/4 0.865 20/4

0.415 −164

−9.5 130.5 57–60 128–9 99–100 130–5 d. d. 118.1 166 132 60–1 286–8 42–3 179 106 d. 320 expl. −59 −45(−52) d. −182.6

Boiling point, °C 169.2 157.2 120–1 123762 153.6 1797 d. d. 25–6 285730 subl. d. subl. 266.6752 subl. 253–4 110 188.6 d. subl. 136.117 14018 14416 subl. 34 −56 12214 d. 250 255757 d. 225100 255–9 152291 110760 261–3 subl. 245–6 177 198–200 220762 d. 229–3016 135 d. 202 150 d. 140 d. d. 290–53 227100 d. 212 d.

130750 164.8 −161.4

Solubility in 100 parts Water i. 0.620 v. sl. s. v. sl. s. 0.05 0.420 s. s. h. ∞ 615 1.3831 v. sl. s. h. s. h. v. sl. s. c. i. i. s. h. s. 0.03420 sl. s. 0.0125 sl. s. sl. s. v. sl. s. s. h. i. d. 7.220 ∞ v. sl. s. v. sl. s. 1710 i. i. i. i. i. sl. s. 2.218 v. s. i. v. sl. s. v. sl. s. i. 7925 16.380 14426 v. s. 13816 10825 1620 1314 24817 i. v. s. 0.04 c. i. 1.660 12.515 0.0712 320 i. 215 0.420 cc.

Alcohol

Ether

v. s. ∞ ∞ ∞ ∞ v. s. s. h. v. sl. s. v. s.

v. s. ∞ ∞ ∞ ∞ s. 0.2518 i. sl. s.

∞ v. s.

∞ v. s.

s. v. s. s. i. s. s. h. s. s. v. s. 1.517 ∞ ∞ v. s. v. s. h. ∞ d.

s. v. s. sl. s. i. s. s. s. ∞ v. s. 13.625 ∞ ∞ v. s. sl. s. ∞ s.

∞ s. v. sl. s. c. i. s. i. c. s. sl. s. ∞ ∞

∞ s.

s. ∞ ∞ ∞

v. s. ∞ s. ∞ ∞ 7030

v. s. ∞ ∞ ∞ ∞ 825

v. s. v. s. 4225 v. sl. s. c. s. 0.0114 v. sl. s. 3228 v. s. v. s. s.

v. s. 8.415 815 i. s. i. i. v. s.

s. ∞ s. v. sl. s. 4720 cc.

i. s.

v. s. sl. s.

∞ ∞ v. sl. s. 10410 cc.

2-41

Methoxy-methoxyethanol Methyl acetate acrylic acid (α-) alcohol -amine -amine hydrochloride aniline anthracene (α-) (β-) anthranilate (o-) anthraquinone (2-) benzoate benzylaniline bromide butyrate (n-) (i-) caprate caproate (n-) caprylate cellosolve chloride chloroacetate chloroformate cinnamate cyclohexane ethyl carbonate ethyl ketone ethyl oxalate formate furoate glucamine glycolate heptoate hypochlorite iodide lactate laurate mercaptan methacrylate myristate naphthalene (α-) (β-) nitrate nitrite nonyl ketone (n-) oleate orange palmitate phosphine propionate propyl ketone (n-) salicylate (o-) stearate toluate (o-) (m-) (p-) Methyl toluidine (o-) (m-) (p-) valerate (n-) (i-) vinyl ketone Methylal Methylene-bis-(phenyl-4-isocyanate) bromide chloride dianiline iodide Michler’s hydrol (p-,p′-) ketone Morphine Mucic acid

CH3(OCH2)2CH2OH CH3CO2CH3 CH2:C(CH3)CO2H CH3OH CH3NH2 CH3NH2HCl C6H5NHCH3 C6H4:(CH)2:C6H3CH3 C6H4:(CH)2:C6H3CH3 NH2C6H4CO2CH3 C6H4:(CO)2:C6H3CH3 C6H5CO2CH3 C6H5N(CH3)CH2C6H5 CH3Br CH3(CH2)2CO2CH3 (CH3)2CHCO2CH3 CH3(CH2)8CO2CH3 CH3(CH2)4CO2CH3 CH3(CH2)6CO2CH3 CH3OCH2CH2OH CH3Cl ClCH2CO2CH3 ClCO2CH3 C6H5CH:CHCO2CH3 CH2 < (CH2CH2)2 > CHCH3 CH3OCOOC2H5 CH3.COC2H5 CH3OCOCO2C2H5 HCO2CH3 C4H3OCO2CH3 CH2OH(CHOH)4CH2NHCH3 HOCH2CO2CH3 CH3(CH2)5CO2CH3 ClOCH3 CH3I CH3CH(OH)CO2CH3 CH3(CH2)10CO2CH3 CH3SH CH2:C(CH3)CO2CH3 CH3(CH2)12CO2CH3 C10H7CH3 C10H7CH3 CH3ONO2 CH3ONO CH3(CH2)8COCH3 C17H33CO2CH3 (CH3)2NC6H4N2C6H4SO3Na CH3(CH2)14CO2CH3 CH3PH2 CH3CH2CO2CH3 CH3COCH2CH2CH3 HOC6H4CO2CH3 CH3(CH2)16CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3(CH2)3CO2CH3 (CH3)2CHCH2CO2CH3 CH3COCH:CH2 HCH(OCH3)2 (OCNC6H4)2CH2 CH2Br2 CH2Cl2 (C6H5NH)2CH2 CH2I2 [(CH3)2NC6H4]2CHOH [(CH3)2NC6H4]2CO C17H19O3NH2O (CHOHCHOHCO2H)2

106.12 74.08 86.09 32.04 31.06 67.52 107.15 192.26 192.26 151.16 222.24 136.15 197.28 94.94 102.13 102.13 186.29 130.18 158.24 76.09 50.49 108.52 94.50 162.19 98.19 104.10 72.11 132.11 60.05 126.11 195.21 90.08 144.21 66.49 141.94 104.10 214.34 48.11 100.12 242.40 142.20 142.20 77.04 61.04 170.29 296.49 327.33 270.45 48.02 88.11 86.13 152.15 298.50 150.17 150.17 150.17 121.18 121.18 121.18 116.16 116.16 70.09 76.09 250.25 173.83 84.93 198.26 267.84 270.37 268.35 303.35 210.14

lq. col. lq. pr. col. lq. col. gas pl./al. lq. lf./al. col. lf. col. lq. col. nd. col. lq. lq. gas col. lq. col. lq. lq. col. lq. col. lq. col. lq. gas col. lq. col. lq. cr. col. lq. lq. col. lq. lq. lq. col. lq. lq. lq. gas col. lq. lq. lq. gas lq. cr./al. oil mn. lq. gas col. oil oil red pd. col. cr. gas col. lq. col. lq. col. lq. col. cr. col. lq. col. lq. cr. lq. lq. lq. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. cr. col. lq. gn. lf./al. pr./al. pd.

1.038 25 0.924 20/4 1.015 20/4 0.792 20/4 0.699 −11 1.23 0.989 20/4 1.047 99.4 1.181 0/4 1.168 19/4 1.087 25/25 1.732 0/0 0.898 20/4 0.891 20/4 0.904 0/0 0.887 18 0.965 20/4 0.952 0 1.236 20/4 1.236 15 1.042 36/0 0.769 20/4 1.002 27 0.805 20/4 1.156 0/0 0.974 20/4 1.179 21/4

<−70 −98.7 15–16 −97–8 −92.5 226–8 −57 86 207 24 176–7 −12.5 9.2 −93 <−95 −84.7 −18 −40 −97.7 −32.7 33.4 −126.3 −14.5 −85.9 −99.8

18

1.168 0.881 15/4 20/4

2.279 1.090 19

0.896 0 0.950 15.6 1.025 14/4 0.994 40/4 1.203 25 0.991 15 0.828 20/20 0.879 18

−64.4 5 −121 −48 18–9 −19 35–6 expl. 13.5 30–1

20/4

0.915 0.812 15/15 1.182 25/25 1.073 15 1.066 15 0.973 15 0.935 55/4 0.895 15/4 0.881 20/4 0.836 20/4 0.866 15/4 1.222 30 2.495 20/4 1.336 20/4 3.325 20/4 1.317

−87.5 −77.8 −8.3 38–9 <−50 33–4

−91 −104.8 −52.8 −96.7 65 5.7 96–7 174 254 d. 206–14

167.5 57.1 161–3 64.7 −6.7 758 23015 195.5 135.5 15 subl. 198–9 305–6 4.5 758 102.3 92.6 223–4 149.5 192–4 124–5 −24 130740 71–2 263 101 109.2 79.6 173.7 32 181.3 151.2 172–3 12726 42.4 144.8 14818 5.8752 100.3 295715 244.6 241–2 65 −12 228 190–110 19615 −14759 79.7 102 222.2 21515 213 215 217 206–7 206–7 211761 127.3 116.7764 81 42–3 210–213 98.5756 40–1 208–9 d. 180 d. >360 d.

∞ 3322 s. h. ∞ v. s. v. s. 0.0125 i. i. sl. s. i. 0.0230 i. v. sl. s. 1.7 v. sl. s. i. i. i. ∞ 28016 cc. v. sl. s. d. i. i. i. 3510 i. 3020 i.

∞ ∞ ∞ v. s. 23 h. s.

∞ ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s. ∞ ∞

v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s.

∞ s.

∞ s.

v. s.

v. s.

v. s. v. s. s. s. s. ∞

v. s. v. s. s. s. s. ∞

s. sl. s. ∞ ∞ ∞ s. ∞

∞ ∞ ∞ s. ∞

v. s. ∞ ∞ ∞ ∞ ∞

v. s. ∞ ∞ ∞ ∞ ∞

∞ d. ∞ ∞ s. ∞ s. h. sl. s. sl. s. i.

∞

i. ∞

∞

i. 1.815 ∞ i. s. i. i. i. i. sl. s. i. i. 0.2 c. i. i. 0.520 v. sl. s. 0.0730 i. i. i. i. i. i. i. v. sl. s. v. sl. s. >85 33 d. 1.170 220 i. 1.420 i. i. 0.0220 0.3314

s.

∞ ∞ s. ∞ s. v. sl. s. s. i.

2-42

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

Mustard gas Myricyl alcohol Myristic acid Myristyl alcohol Naphthalene disulfonic acid (1-,5-) (1-,6-) sulfonic acid (α-) (β-) Naphthasultam (1-,8-) disulfonate Na (1-,8-) (2-,4-) Naphthoic acid (α-) (β-) Naphthol (α-) (β-) sulfonic acid (α-)(1-,2-) (β-)(2-,6-) Naphthyl acetate (α-) (β-) amine (α-) (β-) amine hydrochloride (α-) (β-) amine sulfonic acid (1-,4-) (1-,5-) (1-,7-) (1-,8-) (2-,5-) (2-,6-) (2-,7-) isocyanate (α-) Nicotine Nicotinic acid (3-) (i-)(4-) Nitro-acetanilide (p-) -acetophenone (m-) -aminoanisole (4-,1-,2-) (5-,1-,2-) (3-,1-,4-) -aminophenol (4-,2-,1-) -aniline (o-) (m-) (p-) -anisole (o-) (p-) -anthraquinone (α-) -anthraquinone sulfonic acid (1-,5-) -benzal chloride (m-) -benzaldehyde (m-) Nitro-benzene -benzidine (2-) -benzoic acid (o-) (m-) (p-) -benzyl alcohol (m-) -benzyl bromide (p-) -chlorotoluene (1-,2-,6-) -cresol (1-,3-,4-) -cymene (1-,2-,4-) -dimethylaniline (o-) (m-) (p-) -diphenyl (o-) (p-) -diphenylamine (o-) -guanidine

Formula (ClCH2CH2)2S C31H63OH(?) CH3(CH2)12CO2H CH3(CH2)12CH2OH C10H8 C10H6(SO3H)2 C10H6(SO3H)2 C10H7SO3H2H2O C10H7SO3HH2O C10H7O2NS C10H5O8NS3Na22H2O C10H4O8NS3Na38aH2O C10H7CO2H C10H7CO2H C10H7OH C10H7OH HOC10H6SO3H HOC10H6SO3H CH3CO2C10H7 CH3CO2C10H7 C10H7NH2 C10H7NH2 C10H7NH2HCl C10H7NH2HCl NH2C10H6SO3H NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3H NH2C10H6.SO3HH2O NH2C10H6SO3HH2O C10H7N:CO C10H14N2 C5H4NCO2H C5H4NCO2H CH3CONHC6H4NO2 CH3COC6H4NO2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(NH2)OH NO2C6H4NH2 NO2C6H4NH2 NO2C6H4NH2 CH3OC6H4NO2 CH3OC6H4NO2 C6H4:(CO)2:C6H3NO2 NO2C14H6O2SO3H NO2C6H4CHCl2 NO2C6H4CHO C6H5NO2 NH2C6H4C6H3(NH2)NO2 NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CH2OH NO2C6H4CH2Br CH3C6H3(NO2)Cl CH3C6H3(NO2)OH CH3C6H3(NO2)CH(CH3)2 NO2C6N4N(CH3)2 NO2C6H4N(CH3)2 NO2C6H4N(CH3)2 C6H5C6H4NO2 C6H5C6H4NO2 C6H5NHC6H4NO2 H2NC(NH)NHNO2

Formula weight 159.08 452.84 228.37 214.39 128.17 288.30 288.30 244.26 226.25 205.23 445.35 584.43 172.18 172.18 144.17 144.17 224.23 224.23 186.21 186.21 143.19 143.19 179.65 179.65 223.25 241.26 241.26 241.26 223.25 241.26 241.26 169.18 162.23 123.11 123.11 180.16 165.15 168.15 168.15 168.15 154.12 138.12 138.12 138.12 153.14 153.14 253.21 333.27 206.03 151.12 123.11 229.23 167.12 167.12 167.12 153.14 216.03 171.58 153.14 179.22 166.18 166.18 166.18 199.21 199.21 214.22 104.07

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

oil cr. col. lf. cr. pl./al. lf. cr. cr. cr. nd. cr. lf. nd. mn. mn. mn. pl./aq. lf. nd./al. nd./al. rhb. lf./aq. nd. lf. nd. cr. cr. cr. cr. cr. cr. col. lq. oil nd./al. nd./aq. rhb. nd. red nd. yel. nd. red or. pr. yel. rhb. yel. rhb. yel. mn. col. cr. pr./al. nd. yel. cr. mn. nd./aq. yel. lq. red nd. tri./aq. mn. yel. mn. cr. nd./al. cr. yel. oil yel. oil red mn. yel. nd. rhb. nd./al. or. cr. nd./aq.

1.275 20/4 0.777 95 0.853 70/4 0.824 38/4 1.145 20/4

13–4 88 57–8 38 80.2 d. d. 125 90 125 177–8

217

160–1 184 96 122–3 >250 125 46–9 69–70 50 111–2

300 >300 278–80 285–6

1.077 100/4 1.224 4 1.217 4

1.123 25/25 1.061 98/4

250.5100 16715 217.9

300.8 306.1 subl.

d.

1.18 1.009 20/4

1.207 156 1.211 156 1.442 15 1.43 1.437 14 1.254 20/4 1.233 20

1.205 18/4 1.575 4/4 1.494 4/4 1.550 22/4

89/4

1.240 1.067 20/4 1.179 20/4 1.313 17 1.44

<−80 235.2 317 215–6 80–1 118 139–40 123 142–3 71.5 114 146–7 9.4 54 230 65 58 5.7 143 147.5 140–1 240–2 27 99–100 37.5 32 60–1 163–4 37 113–4 75–6 246–7

269–70 246730 subl. d. 202

284.1 306.4 331.7 272–3 274 2707 16423 210.9

subl. 175–803 238 12522 15215 151–380 280–5 320 340

Solubility in 100 parts Water

Alcohol

Ether

0.0725 i. i. <0.02 0.00325 10220 16420 v. s. 7730 s. h. v. s. v. s. v. sl. s. h. 0.00725 sl. s. h. 0.07425 v. s. h. v. s. sl. s. h. i. 0.17 c. v. s. h. 3.820 v. s. 0.2100 sl. s. 0.4625 0.42100 0.08 0.38100 0.28100 d. s. s. h. s. h. s. h. i. i.

s. v. sl. s. v. s. sl. s. 9.520 s. s. v. s.

s. v. s. v. s. s. v. s. i. i. sl. s.

sl. s.

s.

s. ∞ s. h. sl. s. h. s. s. s.

s. ∞ v. sl. s. v. sl. s. s.

sl. s. sl. s. c. s. h. 0.1120 0.0819 0.1730 0.0630 i. s. i. 1.95112 0.1920 sl. s. h. 0.6520 0.24165 0.0215

s. v. s. v. s. 7.120 5.820 ∞ v. s. sl. s. i. v. s. h. v. s. h. v. s.

s. v. s. v. s. 7.920 6.120 ∞ v. s. v. sl. s. i. v. s. v. s. ∞

2811 3112 0.910

2211 2510 2.218

i. i. v. sl. s. i. v. sl. s. i. i. i. i.

219

v. s.

v. s.

v. s.

v. s. s. s. h. s. sl. s. c.

v. s. s.

9100

sl. s.

v. sl. s.

sl. s. s. h. s. v. s. v. s. v. s. s. s. v. s. s. s. v. s. i.

s. s. v. s. v. s. i. s. s. v. s. s. s. i.

v. s. v. s.

2-43

-naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -phenol sulfonic acid (1-,4-,2-) (1-,2-,4-) -phthalic acid (3-) (4-) -toluene (o-) (m-) (p-) -toluene sulfonic acid (1-,4-,2-) -toluidine (4-,1-,2-) (3-,1-,4-) Nitron Nitroso-dimethylaniline (p-) -naphthol (β-)(1-) Nonadecane (n-) Nonane (n-) Octadecane (n-) Octane (n-) (iso-) Octyl acetate (n-) (sec-) alcohol (n-) (sec-) Octylene (n-) Oleic acid Orcinol (1-,3-,5-) Oxalic acid Palmitic acid Pelargonic acid Penta-chloroethane -decane (n-) -erythritol Pentandiol Pentane (n-) (i-) (neo-) Phenacetin Phenanthrene Phenetidine (o-) (p-) Phenetole Phenol -phthalein -sulfonic acid (o-) Phenyl acetaldehyde acetic acid -acetylene aniline (o-) (p-) Phenyl-ethyl alcohol -glycine -hydrazine -hydrazine sulfonic acid (p-) isocyanate -methylpyrazolone (3-)(N-) -mustard oil naphthalene (α-) (β-) naphthylamine (α-) (β-) phenol (o-) (p-) propyl alcohol (γ-) quinoline (2-)(α-) (8-)(0-) salicylate, salol stearate urethane

C10H7NO2 C10H7NO2 NO2C6H4OH NO2C6H4OH NO2C6H4OH HOC6H3(NO2)SO3H3H2O HOC6H3(NO2)SO3H3H2O NO2C6H3(CO2H)2 NO2C6H3(CO2H)2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H3(NO2)SO3H2H2O NO2C6H3(CH3)NH2 NO2C6H3(CH3)NH2 C20H16N4 ONC6H4N(CH3)2 ONC10H6OH CH3(CH2)17CH3 CH3(CH2)7CH3 CH3(CH2)16CH3 CH3(CH2)6CH3 (CH3)3CCH2CH(CH3)2 CH3CO2CH2(CH2)6CH3 CH3CO2CH(CH3)C6H13 CH3(CH2)6CH2OH CH3(CH2)5CH(OH)CH3 CH3(CH2)5CH:CH2 C8H17CH:CH(CH2)7CO2H (HO)2C6H3CH3 HO2CCO2H2H2O CH3(CH2)14CO2H CH3(CH2)7CO2H CHCl2CCl3 CH3(CH2)13CH3 C(CH2OH)4 HOCH2(CH2)3CH2OH CH3(CH2)3CH3 (CH3)2CHCH2CH3 (CH3)2C(CH3)2 C2H5OC6H4NHCOCH3 < (C6H4CH)2 > C2H5OC6H4NH2 C2H5OC6H4NH2 C2H5OC6H5 C6H5OH C20H14O4 HOC6H4SO3HeH2O C6H5CH2CHO C6H5CH2CO2H C6H5C:CH C6H5C6H4NH2 C6H5C6H4NH2 C6H5CH2CH2OH C6H5NHCH2CO2H C6H5NHNH2 H2NNHC6H4SO3H C6H5N:CO C4H5ON2C6H5 C6H5N:CS C10H7C6H5 C10H7C6H5 C10H7NHC6H5 C10H7NHC6H5 C6H5C6H4OH C6H5C6H4OH C6H5(CH2)3OH C6H5C9H6N C6H5C9H6N HOC6H4CO2C6H5 CH3(CH2)16CO2C6H5 C6H5NHCO2C2H5

173.17 173.17 139.11 139.11 139.11 273.22 273.22 211.13 211.13 137.14 137.14 137.14 253.23 152.15 152.15 312.37 150.18 173.17 268.52 128.26 254.49 114.23 114.23 172.26 172.26 130.23 130.23 112.21 282.46 124.14 126.07 256.42 158.24 202.29 212.41 136.15 104.15 72.15 72.15 72.15 179.22 178.23 137.18 137.18 122.16 94.11 318.32 187.69 120.15 136.15 102.13 169.22 169.22 122.16 151.16 108.14 188.20 119.12 174.20 135.19 204.27 204.27 219.28 219.28 170.21 170.21 136.19 205.25 205.25 214.22 360.57 165.19

yel./al. col./al. yel. mn. col. mn. yel. pr. nd. nd./aq. yel./aq. yel. cr. yel. lq. lq. rhb. pl./aq. yel. mn. red mn. yel. lf. gn. tri. brn. pr. cr. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. nd. pr./bz. col. mn. col. pl. col. oil col. lq. col. lq. cr. lq. col. lq. col. lq. col. lq. col. mn. pl./al. oil lq. col. lq. col. nd. col. rhb. cr. lq. lf. col. lq. cr. lf. col. oil cr. yel. oil cr./al. lq. pr./aq. col. lq. waxy lf./al. pr./al. rhb. nd. nd. oil nd. lq. rhb./al. cr. pl./al.

1.223 62 1.295 45 1.485 20 1.479 20

1.163 20/4 1.160 18/4 1.139 55/55 1.365 15 1.312 17

0.777 32/4 0.718 20/4 0.775 28/4 0.703 20/4 0.692 20/4 0.885 0/4 0.863 14/4 0.827 20/4 0.822 20/4 0.721 18/4 0.854 78/4 1.290 4 1.653 19/4 0.849 70/4 0.906 20/4 1.671 25/4 0.770 20/4 0.994 20/4 0.630 18/4 0.621 19 0.613 20/4 1.179 25 1.061 15 0.967 20/4 1.071 25/4 1.299 25/4 1.025 20 1.081 80/4 0.930 20/4 1.023 18/4 1.097 23/4 1.096 20/4 1.138 15/15 1.17 1.18 1.008 20/4 1.250 20/4 1.106

30/4

59–60 79 44–5 96–7 113–4 d. 110 51.5 222 164–5 −4.1 15–16 51.9 130 105–7 116–7 189–90 d. 86–7 109.5 32 −53.7 28 −56.5 −107.4 −38.5 −16 −38.6 14 107–8 101.5 63–4 12.5 −22 10 262 −129.7 −160.0 −20 134–5 99–100 <−21 3–4 −30.2 42–3 261–2 50 d.

304 16515 214.5 19470 subl.

222.3 230–1 237.7

330 150.5759 317 125.7 99.3760 210 195 194–5 179–80 126 285–6100 287–90 subl. 271.5100 253–4 162 270.5 27630 239.4 36.3 27.95 9.5 d. 340 228–9 254–5 172 181.4

76–7 −43 45–6 50–2

193–4 265.5 142–3 299760 302 219–21750

127 19.6 286

243.5

128 −21 45 102.5 62 107–8 56–7 164–5 <−18 86 42–3 52 52–3

166769 19117 219–20 336–7 345–6 335258 399.5 275 305–8 235–7 363 283187 172–312 26715 237–8

i. i. 1.08100 1.3520 1.625 v. s. v. s. 2.0525 v. s. 0.0780 0.0580 0.0480 47.728 v. sl. s. sl. s. h. i. i. 0.120 i. i. i. 0.00216 i. i. i. 0.05425 0.09625 i. i. v. s. s. i. v. sl. s. 0.0520 i. 5.615 ∞ 0.03616 i. i. 0.720 i. i. i. i. 8.215 0.220 v. s. v. sl. s. 1.6620 i. v. sl. s. s. h. 1.620 s. sl. s. h. 0.612 d. 120 i. i. i. 0.0860 0.460 i. i. sl. s. sl. s. sl. s. 0.01525 i. i. c.

s. v. s. v. s. v. s. v. s. v. s. v. s. v. s. h. v. s. ∞ ∞ 8.615 v. s. s. s. s. h. s. 2.418 sl. s. sl. s. sl. s. sl. s. sl. s. s. s. ∞ ∞ ∞ ∞ v. s. s. 920 s. ∞ v. s. v. sl. s.

s. v. s. v. s. s. v. s. sl. s. sl. s. s. ∞ ∞ 80.815 v. s. s. v. sl. s. s. s. s. s. s. s. s. s. ∞ ∞ ∞ ∞ v. s. 1.3 s. s. ∞ v. s. i.

∞ ∞ s. 40 h. 10 h. s. s. ∞ ∞ 1025 v. s. ∞ v. s. ∞ s. s. s. s. ∞ sl. s. d. v. s. h. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. h. s. v. s.

∞ ∞ s. 1.625 v. s. s. s. ∞ ∞ 5.9 c.

s.

s.

∞ v. s. ∞ s. s. ∞ sl. s. ∞ v. s. v. sl. s. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. s. s.

2-44

TABLE 2-2

Physical Properties of Organic Compounds (Continued) Name

Phenylene-diamine (o-) (m-) (p-) Phloroglucinol (1-,3-,5-) Phorone Phosgene Phthalic acid (o-) (m-)(iso-) anhydride (o-) nitrile (o-) Phthalide Phthalimide (o-) Picoline (α-) (β-) (γ-) Picramic acid (1-,2-,4-,6-) Picric acid (2-,4-,6-) Picryl chloride (2-,4-,6-) Pinacol Pinacoline Pinene (α-)(dl-) hydrochloride Pinol (dl-) Piperidine carboxylic acid (α-)(dl-) Piperidinium pentamethylene dithiocarbamate Propane Propionic acid aldehyde anhydride Propyl acetate (n-) (i-) alcohol (n-) (i-) amine (n-) (i-) aniline (n-) benzoate (n-) (i-) bromide (n-) (i-) n-butyrate (n-) i-butyrate (n-) n-butyrate (i-) i-butyrate (i-) chloride (n-) (i-) Propyl formate (n-) (i-) furoate (n-) lactate (n-) (i-) mercaptan (n-) (i-) propionate (n-) (i-) thiocyanate (i-) n-valerate (n-) i-valerate (n-) i-valerate (i-) Propylene bromide chlorohydrin chloride glycol oxide Protocatechuic acid (3-,4-)

Formula C6H4(NH2)2 C6H4(NH2)2 C6H4(NH2)2 C6H3(OH)32H2O [(CH3)2C:CH]2CO OCCl2 C6H4(CO2H)2 C6H4(CO2H)2 C6H4 < (CO)2 > O C6H4(CN)2 C6H4(CH2)(CO) > O C6H4 < (CO)2 > NH C5H4NCH3 C5H4NCH3 C5H4NCH3 HOC6H2(NH2)(NO2)2 HOC6H2(NO2)3 ClC6H2(NO2)3 [(CH3)2COH]2 CH3COC(CH3)3 C10H16 C10H17Cl C10H16O CH2 < (CH2CH2)2 > NH HO2CCH < (CH2CH2)2 > NH (CH2)5CS2HHN(CH2)5 CH3CH2CH3 CH3CH2CO2H CH3CH2CHO (CH3CH2CO)2O CH3CO2CH2CH2CH3 CH3CO2CH(CH3)2 CH3CH2CH2OH (CH3)2CHOH CH3CH2CH2NH2 (CH3)2CHNH2 C6H5NHCH2CH2CH3 C6H5CO2CH2CH2CH3 C6H5CO2CH(CH3)2 CH3CH2CH2Br (CH3)2CHBr C2H5CH2CO2CH2C2H5 (CH3)2CHCO2CH2C2H5 C2H5CH2CO2CH(CH3)2 (CH3)2CHCO2CH(CH3)2 CH3CH2CH2Cl (CH3)2CHCl HCO2CH2CH2CH3 HCO2CH(CH3)2 C4H3OCO2C3H7 CH3CH(OH)CO2CH2C2H5 CH3CH(OH)CO2CH(CH3)2 CH3CH2CH2SH (CH3)2CHSH C2H5CO2CH2C2H5 C2H5CO2CH(CH3)2 (CH3)2CHCNS CH3(CH2)3CO2CH2C2H5 (CH3)2CHCH2CO2C3H7 (CH3)2CHCH2CO2C3H7 CH3CH:CH2 CH3CHBrCH2Br CH3CHClCH2OH CH3CHClCH2Cl CH3CH(OH)CH2OH CH3(CHCH2)O (HO)2C6H3CO2HH2O

Formula weight

Form and color

108.14 108.14 108.14 162.14 138.21 98.92 166.13 166.13 148.12 128.13 134.13 147.13 93.13 93.13 93.13 199.12 229.10 247.55 118.17 100.16 136.23 172.69 152.23 85.15 129.16 232.43 44.10 74.08 58.08 130.14 102.13 102.13 60.10 60.10 59.11 59.11 135.21 164.20 164.20 122.99 122.99 130.18 130.18 130.18 130.18 78.54 78.54 88.11 88.11 154.16 132.16 132.16 76.16 76.16 116.16 116.16 101.17 144.21 144.21 144.21 42.08 201.89 94.54 112.99 76.09 58.08 172.14

lf./aq. rhb. mn. rhb. yel. pr. gas mn./aq. nd./aq. rhb. cr. nd./aq. cr./et. col. lq. col. lq. lq. red nd. yel. rhb. yel. mn. col. nd. col. lq. col. lq. lf. lq. lq. cr. cr. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. gas col. lq. col. lq. col. lq. col. oil col. lq. nd./aq.

Specific gravity 1.139 15/15 0.885 20/4 1.392 19/4 1.593 20/4 1.527

4

1.164 99/4 0.950 15/4 0.961 15/4 0.957 15/4 1.763 20/4 1.797 20 0.967 15 0.800 16 0.878 20/4 0.953 20/20 0.860 20/4 1.13 0.585 −45/4 0.992 20/4 0.807 20/4 1.012 20/4 0.886 20/4 0.874 20/20 0.804 20/4 0.78920/4 0.718 20/20 0.694 15/4 0.949 18 1.021 25/25 1.010 25/25 1.353 20/4 1.310 20/4 0.879 15 0.884 0/4 0.865 18 0.869 0/4 0.890 20/4 0.859 20 0.901 20/4 0.873 20/4 1.075 26/4 0.836 25/4 0.809 25/4 0.883 20/4 0.893 0 0.963 20 0.874 15 0.863 20/4 0.854 17 0.609 −47/4 1.933 20/4 1.103 20 1.159 20/20 1.040 19.4 0.831 20/20 1.542 4/4

Melting point, °C

Boiling point, °C

103–4 62.8 140 117 28 −104 208 330 130.8 141 73(65) 238 −70

256–8 284–7 267 subl. 197.2743 8.2756 d. subl. 284.5

169 121.8 83 43(38) −52.5 −55 131–2 −9 264 175 −187.1 −22 −81 −45 −92.5 −73.4 −127 −85.8 −83 −101 −51.6 −109.9 −89 −95.2

−122.8 −117 −92.9

−112 −130.7 −76 −70.7 −185 −55.5 <−70 199 d.

290 subl. 128.8 143.5 143.1 expl. d. 171–2789 106.2 154–6 207–8 183–4 106 −42.2 141.1 49.5740 168.8780 101.6 88.4 97.8 82.5 49–50761 33–4 222 231 218.5 70.8 60 142.7 134–5 128 120.8 46.4 36.5 81.3 68-71751 211 122-3150 167.5 67-8 58-60 122-3 109-11750 152-3754 67.5 155.9 142756 −48749 141.6 133-4 96.8 188-9 35

Solubility in 100 parts Water 73381 35.125 669107 1.1325 0.150 v. sl. s. 0.7025 0.2100 v. sl. s. sl. s. c. v. sl. s. 0.0425 v. s. ∞ ∞ 0.1422 1.2320 0.01815 sl. s. c. 2.515 v. sl. s. i. ∞ s. 628 6.518 cc. ∞ 2020 d. 1.616 320 ∞ ∞ ∞ ∞ i. i. i. 0.2520 0.3220 0.1717 v. sl. s. v. sl. s. v. sl. s. 0.2720 0.3120 12.222 2.122 v. sl. s. s. s. v. sl. s. v. sl. s. 0.5625 0.625 i. i. i. 44.6 cc. 0.2520 s. 0.2720 ∞ 3320 1.8214

Alcohol

Ether

v. s. v. s. s. v. s. s.

v. s. s. s. v. s. s.

1218 s. s.

0.6815

s. 5 ∞ ∞ ∞ s. 620 4.817 v. s. s. s. 33 s. ∞

sl. s. s. h. ∞ ∞ ∞ sl. s. 113 717 v. s. s. ∞ s. s.

s. ∞ ∞ d. ∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

v. s. ∞ ∞

∞ ∞ ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ ∞ ∞ ∞

1200 cc. s. s. v. s. ∞ ∞ v. s.

v. s. s. v. s. 8 ∞ s.

∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞

2-45

Pulegol (iso-)(d-) Pulegone Pyrazole Pyrazoline Pyrazolone Pyrene Pyridazine Pyridine Pyrocatechol (o-) Pyrogallol (1-,2-,3-) Pyrone Pyrrole Pyrrolidine Pyrroline Pyruvic acid Quercitrin Quinaldine (py-2) Quinoline (iso-) -diol (1-,3-) Quinone (p-) R-acid Ca salt (2-)(3-,6-) K salt Na salt Raffinose Resorcinol (m-) Retene Rhamnose (β-) Ricinoleic acid Rosaniline Rosolic acid Saccharin Safrole (1-,3-,4-) (iso-)(1-,3-,4-) Salicylic acid (o-) aldehyde (o-) Saligenin Schaeffer’s salt, Ca K Na Semicarbazide hydrochloride Skatole (3-) Sodium methylate Sorbitol Sorbose (d- or l-) Starch Stearic acid amide Styrene Suberic acid Succinic acid Sucrose Sulfanilic acid (p-) Sylvestrene (d-) Tartaric acid (meso-) (racemic) (d- or l-) Tartronic acid Terephthalic acid (p-) Terpin hydrate (cis-) Terpineol (α-)(d- or l-) (dl-) Terpinyl acetate (α-)(dl-) Tetrabromo-ethane (sym) (uns) Tetrachloro-ethane (sym) (uns) -ethylene Tetracosane (n-) Tetradecane (n-) Tetraethyl-thiuram disulfide

C10H17OH C10H16O NHN:CHCH:CH NHN:CHCH2CH2 NHCOCH2CH:N C16H10 N2 < (CHCH)2 > CH < (CHCH)2 > N C6H4(OH)2 C6H3(OH)3 CO < (CHCH)2 > O < (CH:CH)2 > NH < (CH2CH2)2 > NH < (CHCH2)2 > NH CH3COCO2H C21H20O112H2O CH3C9H6N C9H7N C9H7N C6H4CH:C(OH)N:C(OH) CO < (CHCH)2 > CO HOC10H5(SO3)2Ca HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C18H32O165H2O C6H4(OH)2 C18H18 CH3(CHOH)4CHOH2O C17H32(OH)CO2H C20H21ON3 C20H16O3 C6H4(CO)(SO2) > NH CH2:CHCH2C6H3:O2CH2 CH3CH:CHC6H3:O2CH2 HOC6H4CO2H HOC6H4CHO HOC6H4CH2OH (HOC10H6SO3)2Ca5H2O HOC10H6SO3K HOC10H6SO3Na NH2CONHNH2 NH2CONHNH3Cl CH3C8H6N CH3ONa [CH2OH(CHOH)2]2 C6H12O6 (C6H10O5)x CH3(CH2)16CO2H CH3(CH2)16CONH2 C6H5CH:CH2 HO2C(CH2)6CO2H HO2C(CH2)2CO2H C12H22O11 H2NC6H4SO3H C10H16 (CHOHCO2H)2 (CHOHCO2H)2H2O (CHOHCO2H)2 CH(OH)(CO2H)2aH2O C6H4(CO2H)2 C10H20O2H2O C10H18O C10H18O CH3CO2C10H17 Br2CHCHBr2 Br3CCH2Br Cl2CHCHCl2 Cl3CCH2Cl Cl2C:CCl2 CH3(CH2)22CH3 CH3(CH2)12CH3 [(C2H5)2NCS]2S2

154.25 152.23 68.08 70.09 84.08 202.25 80.09 79.10 110.11 126.11 96.08 67.09 71.12 69.11 88.06 484.41 143.19 129.16 129.16 161.16 108.09 342.36 380.48 348.26 594.51 110.11 234.34 182.17 298.46 319.40 304.34 183.18 162.19 162.19 138.12 122.12 124.14 576.60 262.32 246.21 75.07 111.53 131.17 54.02 182.17 180.16 162.14 284.48 283.49 104.15 174.19 118.09 342.30 173.19 136.23 150.09 168.10 150.09 129.07 166.13 190.28 154.25 154.25 196.29 345.65 345.65 167.85 167.85 165.83 338.65 198.39 296.54

col. lq. col. lq. nd./et. lq. nd. yel. pr. lq. col. lq. nd./aq. nd. cr. lq. lq. lq. col. lq. yel. nd. lq. lq. pl. cr. yel. mn. cr. cr. cr. cr./aq. col. rhb. lf./al. col. mn. lq. col. nd. red lf. mn. col. mn. col. lq. mn. col. oil rhb./aq. cr. cr. cr. pr./al. pr. lf. pd. cr. rhb. amor. mn. col. cr. col. lq. nd./aq. col. mn. col. mn. col. cr. lq. cr. tri. mn. pr./aq. cr. rhb. col. cr. col. cr. lq. col. lq. col. lq. col. lq. lq. col. lq. cr. col. lq. cr.

0.911 20/4 0.932 20/20 70 1.277 0/4 1.107 20/4 0.982 20/4 1.344 4 1.453 4 1.190 40.3 0.948 20/4 0.852 22.5 0.910 20/4 1.267 20/4 1.059 20/4 1.095 20 1.099 21/4 1.318 20/4

1.465 0 1.272 15 1.1316 1.47120/4 0.954 16

1.100 20/4 1.122 20/4 1.443 20/4 1.153 25/4 1.161 25

1.654 15 1.5021 0.847 69.3 0.903 20/4 1.266 25/4 1.572 25/4 1.588 15 0.863 20/4 1.737 1.697 20/4 1.760 20/4 1.510 0.935 15 0.935 20/20 0.966 20/4 2.964 20/4 2.875 20/4 1.600 20/4 1.588 20/4 1.624 15/4 0.779 51/4 0.765 20/4 1.17

165 149-50 −8 −42 104-5 133-4 32.5

13.6 182-5 −1 −15 24.6 237 115.7

119 110.7 98-9 126 4-5 186 d. 308-10 d. 225-8 11.2 6-7 159 -7 86-7

96 173 d. 95 d. 300 110-2 165 d. 70-1 108-9 −31 140-4 189-90 170-86 d. d. > 280 159-60 205-6 168-70 d. 155-8 subl. 117 38-40 35 < −50 −1.0 0 −36 −19 51.1 5.5 70

86-9 10 224754 186-8 144 subl. d. >360 208 115-6 240-5 309 215-7 131 87-8 90-1 165 244-5750 237.1747 240.5763 subl.

d. 130 276.5 390-4 226-810 subl. 233-4 252-3 21120 196.5 subl.

265-6755

291110 25112 145-6 279100 235 d. 176-7 d. subl. d. 219-21 218-9752 220 d. 15154 10413 146.3 129-30 120.8 324 252.5

v. sl. s. i. s. ∞ s. i. ∞ ∞ 45.120 40 13 v. sl. s. i. ∞ v. s. ∞ 0.04 20 v. sl. s. 6 sl. s. v. sl. s. sl. s. h. 30.6 25 29.5 25 25.2 25 14.3 20 14712 i. 60.8 21 i. v. sl. s. 0.1225 0.4 25 i. i. 0.223 1.7 86 6.615 4.7620 3.4625 6.2925 v. s. v. s. 0.05 c. d. v. s. 5517 i. 0.0325 i. v. sl. s. 0.1416 6.820 1790 0.810 12015 20.620 13920 v. s. 0.001 c. 0.415 i. i. i. i. 0.2920 i. 0.0220 i. i.

∞ s. ∞ v. s. 3 h. s. ∞ v. s. s. s. s. ∞ ∞ ∞ s. ∞ s.

0.120 v. s. 69 h.

∞ s. sl. s. v. sl. s. v. s. s. s. v. s. s. v. s. s. ∞ ∞ ∞ sl. s. s. ∞ s. s.

∞ sl. s. v. s. h. 3.1 c. s. ∞ 4915 ∞ v. s.

v. s. v. s. h. i. ∞ i. sl. s. 1.05 c. ∞ ∞ 5115 ∞ v. s.

v. s. sl. s. s.

i. i. s.

v. s. h. sl. s. i. 220 s. h. ∞ s. 9.915 0.9 v. sl. s.

i. 6g s. h: ∞ 0.815 1.215 i. v. sl. s.

20 2515 v. s. sl. s. h. 1015 v. s. v. s. 20 ∞ s. ∞ ∞ ∞ v. s.

0.09 0.415 i. i. 115 v. s. v. s. ∞ ∞ ∞ ∞ s. v. s.

2-46

TABLE 2-2

Physical Properties of Organic Compounds (Concluded) Name

Tetrafluoro-ethylene Tetrahydro-furan -furfuryl alcohol -pyran Tetralin Tetramethyl-thiuram disulfide Tetryl (2-,4-,6-) Theobromine Thio-acetic acid -aniline (4-, 4′-) -carbanilide -naphthol (β-) -phenol -salicylic acid (o-) -urea Thiophene Thymol (5-,2-,1-) Tolidine (0-)(3-,3′-,4-,4′-) Toluene sulfonic acid (o-) (p-) sulfonic amide (p-) sulfonic chloride (p-) Toluic acid (o-) (m-) (p-) Toluidine (o-) (m-) (p-) hydrochloride (o-) sulfonic acid (1-,2-,3-) Toluylenediamine (1-,2-,4-) Tolylene diisocyanate (1-,2-,4-) Trehalose Triamylamine (n-) (i-) Tributyl-amine (n-) phosphite Trichloro-acetic acid -benzene (s-)(1-,3-,5-) -ethane (1-,1-,1-) -ethylene -phenol Tricosane (n-) Tricresyl phosphate (o-) Tridecane (n-) Triethanol amine Triethyl-amine -benzene (1-,3-,5-) (1-,2-,4-) borate citrate Triethylene glycol Trifluoro-chloromethane -chloroethylene -trichloroethane Trimethoxybutane (1-,3-,3-) Trimethylamine Trimethylene bromide chloride glycol Trinitro-benzene (1-,3-,5-) -benzoic acid (2-,4-,6-) -tert-butylxylene -naphthalene (α-)(1-,3-,5-) (β-)(1-,3-,8-) (γ-)(1-,4-,5-)

Formula F2C:CF2 CH2(CH2)2CH2O C4H7OCH2OH CH2(CH2)3CH2O C6H4CH2(CH2)2CH2 [(CH3)2NCS]2S2 (NO2)3C6H2N(CH3)NO2 C7H8O2N4 CH3COSH (NH2C6H4)2S (C6H5NH)2CS C10H7SH C6H5SH HSC6H4CO2H NH2CSNH2 < (CH:CH)2 > S (CH3)(C3H7)C6H3OH [CH3(NH2)C6H3]2 C6H5CH3 CH3C6H4SO3H2H2O CH3C6H4SO3HH2O CH3C6H4SO2NH2 CH3C6H4SO2Cl CH3C6H4CO2H CH3C6H4CO2H CH3C6H4CO2H CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH3Cl CH3(NH2)C6H3SO3H CH3C6H3(NH2)2 CH3C6H3(NCO)2 C12H22O112H2O [CH3(CH2)3CH2]3N [(CH3)2CH(CH2)2]3N [CH3(CH2)2CH2]3N [CH3(CH2)3O]3P Cl3CCO2H C6H3Cl3 Cl3CCH3 Cl2C:CHCl Cl3C6H2OH CH3(CH2)21CH3 OP(OC6H4CH3)3 CH3(CH2)11CH3 (HOCH2CH2)3N (CH3CH2)3N (C2H5)3C6H3 (C2H5)3C6H3 B(OCH2CH3)3 HOC3H4(CO2C2H5)3 (CH2OCH2CH2OH)2 CF3Cl F2C:CFCl Cl2CFCClF2 CH2(OCH3)CH2C(OCH3)2CH3 (CH3)3N BrCH2CH2CH2Br ClCH2CH2CH2Cl HOCH2CH2CH2OH C6H3(NO2)3 (NO2)3C6H2CO2H (NO2)3C6(CH3)2C4H9 C10H5(NO2)3 C10H5(NO2)3 C10H5(NO2)3

Formula weight

Form and color

Specific gravity

Melting point, °C

Boiling point, °C

100.02 72.11 102.13 86.13 132.20 240.43 287.14 180.16 76.12 216.30 228.31 160.24 110.18 154.19 76.12 84.14 150.22 212.29 92.14 208.23 190.22 171.22 190.65 136.15 136.15 136.15 107.15 107.15 107.15 143.61 187.22 122.17 174.16 378.33 227.43 227.43 185.35 250.31 163.39 181.45 133.40 131.39 197.45 324.63 368.36 184.36 149.19 101.19 162.27 162.27 145.99 276.28 150.17 104.46 116.47 187.38 148.20 59.11 201.89 112.99 76.09 213.10 257.11 297.26 263.16 263.16 263.16

gas col. lq. col. lq. lq. col. lq. cr. yel. mn. rhb. yel. lq. nd./aq. rhb./al. cr./al. col. lq. yel. nd. rhb./al. col. lq. cr. lf. col. lq. cr. mn. mn. tri. cr./aq. pr./aq. cr./aq. col. lq. col. lq. cr. mn. pr. cr. rhb. lq. rhb./al. lq. col. lq. col. lq. lq. cr. nd. lq. col. lq. nd. lf. lq. col. lq. col. lq. col. oil lq. lq. lq. oil col. lq. gas gas lq. lq. gas lq. lq. oil col. rhb. rhb./aq. nd./al. rhb. cr./al. yel. cr.

1.58−78 0.88821/4 1.05020/4 0.88120/4 0.97318/4 1.29 1.5719

−142.5 −65

−76.3 65-6 177-8743 88 206764

10

1.074 1.324

1.07423/4 1.40520/4 1.07015/4 0.97225/25 0.86620/4

1.062115/4 1.054112/4 0.99920/4 0.98920/4 1.04620/4

1.2328

−31 155-6 130.5 330 < −17 108 154 81 164 180-2 −30 51.5 128-9 −95 d. 104-5 137 69 104-5 110-1 179-80 −16.3 −31.5 44-5 218-20 99

expl. 93 d. 286-8 168-9 subl. d. 84 232752 110.8 128.80 146-70 134.510 259751 263 274-5 199.7 203.3 200.3 242 283-5 134.520

97 20/4

0.786 0.77820/20 0.92520/4 1.61746/15 26/4

1.325 1.46620/20 1.49075/4 0.77948/4 0.75720/4 1.12620/20 0.72920/20 0.86120/4 0.88217/4 0.86420/20 1.13720/4 1.12520/20 1.726−130 1.57620/4 0.932 0.662 −5 1.987 15/4 1.201 15 1.060 20/4 1.688 20/4

58 63.5 −73 68-9 47.7 −6.2 20-1 −114.8

−5 −182 −157.5 −35 −124 −34.4 121 210-20 d. 110 122-3 218-9 148-9

240-5 235 216.5761 122-312 195.5754 208.5764 74.1 87.2 246 23415 234 277-9150 89.4 215 217-8755 120 294 290 −80 −27.9 47.6 63-525 3.5 167.5 123-5 214 d.

Solubility in 100 parts Water 0.0130 s. ∞ s. i. i. i. 0.0615 s. sl. s. h. i. v. sl. s. v. sl. s. sl. s. h. 9.213 i. 0.0919 v. sl. s. 0.0516 v. s. v. s. 0.29 i. 2.17100 1.6100 1.3100 1.525 sl. s. 0.7421 s. 0.9711 s. h. d. s. h. i. i. i. i. 12025 i. i. 0.125 0.0925 i. i. i. ∞ ∞ > 190 i. i. d. i. ∞ d. i. d. 4119 0.1730 0.2725 ∞ 0.0315 2.0524 i. i. 0.02100 i.

Alcohol

Ether

s. ∞

s. ∞

s.

s.

s. h. 0.06 c. ∞ s. v. s. v. s. v. s. s. s. s. v. s. s. s. s. s. 7.45 s. v. s. v. s. v. s. ∞ ∞ v. s. sl. s.

s. 0.03 h. ∞ s. v. s. v. s. ∞ sl. s. v. s. s. ∞

s. v. s. v. s. ∞ ∞ v. s.

s. d. sl. s. h.

s. i.

s.

∞

s. sl. s. ∞ ∞ v. s.

s. ∞ ∞ v. s.

v. s. ∞ ∞ s. s.

v. s. sl. s. ∞ s. s.

∞ ∞

∞ v. sl. s.

∞

∞

s. s. s. ∞ 1.918

s. s. s. 1.518

sl. s. s. 0.0523 0.1119

0.1315 0.419

s.

-phenol (2-,3-,6-) -toluene (β-)(2-,3-,4-) (γ-)(2-,4-,5-) (α-)(2-,4-,6-) Trional Triphenyl-arsine carbinol guanidine (α-) methane methyl phosphate Tripropylamine (n-) Undecane (n-) Urea nitrate Uric acid Valeric acid (n-) (i-) aldehyde (n-) (i-) amide (n-) (i-) Vanillic acid (3-,4-,1-) alcohol (3-,4-,1-) hyl-thiuram disulfide Vanillin (3-,4-,1-) Veratrole (o-) Vinyl acetate (poly-) acetic acid acetylene alcohol (poly-) chloride propionate Xylene (o-) (m-) (p-) sulfonic acid (1-,4-,2-) Xylidine (1:2)(3-) (1:2)(4-) (1:3)(2-) (1:3)(4-) (1:3)(5-) (1:4)(2-) Xylose (l-)(+) Xylylene dichloride (p-) Zinc diethyl dimethyl dimethyl-dithiocarbamate NOTE:

°F = 9⁄5 °C + 32.

(NO2)3C6H2OH CH3C6H2(NO2)3 CH3C6H2(NO2)3 CH3C6H3(NO2)3 (C2H5SO2C2H4)2 (C6H5)3As (C6H5)3COH C6H5N:C(NHC6H5)2 (C6H5)3CH (C6H5)3C . . . OP(OC6H5)3 (CH3CH2CH2)3N CH3(CH2)9CH3 H2NCONH2 CO(NH2)2HNO3 C5H4O3N4 C2H5CH2CH2CO2H (CH3)2CHCH2CO2H C2H5CH2CH2CHO (CH3)2CHCH2CHO C2H5CH2CH2CONH2 (CH3)2CHCH2CONH2 CH3O(OH)C6H3CO2H CH3O(OH)C6H3CH2OH [(C2H5)2NCS]2S2 CH3O(OH)C6H3CHO C6H4(OCH3)2 CH3CO2CH:CH2 (CH3CO2CH:CH2)x CH2:CHCH2CO2H CH2:CHC:CH CH2:CHOH (CH2:CHOH)x CH2:CHCl C2H5CO2CH:CH2 C6H4(CH3)2 C6H4(CH3)2 C6H4(CH3)2 (CH3)2C6H3SO3H2H2O (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 CH2OH(CHOH)3CHO C6H4(CH2Cl)2 Zn(CH2CH3)2 Zn(CH3)2 Zn[S2CN(CH3)2]2

229.10 227.13 227.13 227.13 242.36 306.23 260.33 287.36 244.33 243.32 326.28 143.27 156.31 60.06 123.07 168.11 102.13 102.13 86.13 86.13 101.15 101.15 168.15 154.16 296.54 152.15 138.16 86.09 (86.09) 86.09 52.07 44.05 (44.05) 62.50 100.12 106.17 106.17 106.17 222.26 121.18 121.18 121.18 121.18 121.18 121.18 150.13 175.06 123.53 95.48 305.84

nd. cr. yel. pl. cr./al. pl./al. pl. cr. rhb./al. cr. col. cr. pr./al. col. lq. col. lq. col. pr. col. mn. cr. col. lq. col. lq. lq. col. lq. mn. pl. mn. nd./aq. mn./aq. cr. mn. cr. col. lq. col. lq. gas gas lq. col. lq. col. lq. col. lq. col. lf. lq. pr. lq. lq. oil oil nd. mn. col. lq. col. lq.

1.620 20/4 1.620 20/4 1.654 1.199 85/4 1.306 1.188 20/4 1.13 1.014 99/4 1.206 58/4 0.757 20/4 0.741 20/4 1.335 20/4 1.893 20 0.939 20/4 0.931 20/20 0.819 11 0.803 17 1.023 0.965 20/4 1.17 1.056 1.091 15/15 0.932 20/4 1.1920 1.013 15/15 0.705 1.5

117-8 112 104 80.8 76 59-60 162.5 144-5 93.4 145-7 49-50 −93.5 −25.6 132.7 152 d. d. −34.5 −37.6 −92 −51 106 135-7 207 115 70 81-2 22.5 < −60 100-25 −39

1.320 0.908 25/25

d. >200 −160

0.881 20/4 0.867 17/4 0.861 20/4

−25 −47.4 13.2 86 < −15 49-50

0.991 15 1.076 17.5 0.980 15 0.978 20/4 0.972 20/4 0.979 21/4 1.535 0 1.417 0 1.182 18 1.386 11 2.0040/4

15.5 153-4 100.5 −28 −40 248-50

expl. expl. expl. d. >360 >360 d. 359754 d. 24511 156.5 194.5 d. 187 176 103.4 92.5 232 subl. d. 285 207.1 72-3 163 5.5 −12 93-5 144 139.3 138.5 1490.1 223 224-6 216-7 213-4 221-2 215789 240-5 d. 118 46

s. h. i. i. 0.0120 0.315 i. i. i. i. i. i. v. sl. s. i. 10017 v. s. h. 0.06 h. 3.316 4.220 v. sl. s. sl. s. v. s. s. 0.1214 v. s. h. i. 114 v. sl. s. 220 i. s. 0.670.6 s. sl. s. v. sl. s. i. i. i. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. 11720 i. d. d. i.

v. s. sl. s. c. s. h. 1.522 50 s. v. s. 40 v. s. h. sl. s. h. 15525 ∞ ∞ 2020 s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.

v. s. s. v. s. 533 6.615 v. s. v. s.

v. s. ∞ ∞ sl. s.

v. s. s. ∞

v. s. s. ∞

∞

∞

v. s.

i. ∞ ∞ s. s. v. s. s. v. s. v. s.

s.

v. s.

s. s. s.

∞ ∞ v. s.

s.

s.

v. sl. s. s. d. d.

i. v. sl. s.

2-47

2-48

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-3

Vapor Pressure of Water Ice from 0 to −40 ⬚C

Vapor pressure t, ºC mmHg 0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5 −4.0 −4.5 −5.0 −5.5 −6.0 −6.5 −7.0 −7.5 −8.0 −8.5 −9.0 −9.5 −10.0 −10.5 −11.0 −11.5 −12.0 −12.5 −13.0

4.584 4.399 4.220 4.049 3.883 3.724 3.571 3.423 3.281 3.145 3.013 2.887 2.766 2.649 2.537 2.429 2.325 2.225 2.130 2.038 1.949 1.865 1.783 1.705 1.630 1.558 1.489

Vapor pressure

UNITS CONVERSIONS

Vapor pressure

kPa

t, ºC

mmHg

kPa

t, ºC

mmHg

kPa

0.6112 0.5865 0.5627 0.5398 0.5177 0.4965 0.4761 0.4564 0.4375 0.4193 0.4018 0.3849 0.3687 0.3532 0.3382 0.3238 0.3100 0.2967 0.2839 0.2717 0.2599 0.2486 0.2377 0.2273 0.2173 0.2077 0.1985

−13.5 −14.0 −14.5 −15.0 −15.5 −16.0 −16.5 −17.0 −17.5 −18.0 −18.5 −19.0 −19.5 −20.0 −20.5 −21.0 −21.5 −22.0 −22.5 −23.0 −23.5 −24.0 −24.5 −25.0 −25.5 −26.0 −26.5

1.423 1.359 1.298 1.240 1.184 1.130 1.079 1.029 0.9822 0.9370 0.8937 0.8522 0.8125 0.7745 0.7381 0.7034 0.6701 0.6383 0.6078 0.5787 0.5509 0.5243 0.4989 0.4747 0.4515 0.4294 0.4083

0.1897 0.1812 0.1731 0.1653 0.1578 0.1507 0.1438 0.1372 0.1310 0.1249 0.1191 0.1136 0.1083 0.1033 0.09841 0.09377 0.08934 0.08510 0.08104 0.07716 0.07345 0.06991 0.06652 0.06329 0.06020 0.05725 0.05443

−27.0 −27.5 −28.0 −28.5 −29.0 −29.5 −30.0 −30.5 −31.0 −31.5 −32.0 −32.5 −33.0 −33.5 −34.0 −34.5 −35.0 −35.5 −36.0 −36.5 −37.0 −37.5 −38.0 −38.5 −39.0 −39.5 −40.0

0.3881 0.3688 0.3505 0.3330 0.3162 0.3003 0.2851 0.2706 0.2568 0.2437 0.2311 0.2192 0.2078 0.1970 0.1867 0.1769 0.1676 0.1587 0.1503 0.1423 0.1347 0.1274 0.1206 0.1140 0.1078 0.1019 0.0963

0.05174 0.04918 0.04673 0.04439 0.04216 0.04004 0.03801 0.03608 0.03424 0.03249 0.03082 0.02923 0.02771 0.02627 0.02490 0.02359 0.02235 0.02116 0.02004 0.01897 0.01796 0.01699 0.01607 0.01520 0.01437 0.01359 0.01284

SOURCE: Formulation of Wagner, Saul, and Pruss, J. Phys. Chem. Ref. Data, 23, 515 (1994), implemented in Harvey, Peskin, and Klein, NIST/ASME Steam Properties, NIST Standard Reference Database 10, Version 2.2, National Institute of Standards and Technology, Gaithersburg, Md., 2000. This source provides data down to 190 K (−83.15 ºC). A formula extending to 110 K may be found in Murphy and Koop, Q. J. R. Meteorol. Soc., 131, 1539 (2005).

TABLE 2-4 Vapor Pressure of Supercooled Liquid Water from 0 to −40 ⬚C* Vapor pressure t, ºC mmHg 0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5 −4.0 −4.5 −5.0 −5.5 −6.0 −6.5 −7.0 −7.5 −8.0 −8.5 −9.0 −9.5 −10.0 −10.5 −11.0 −11.5 −12.0 −12.5 −13.0

4.584 4.421 4.262 4.108 3.959 3.816 3.676 3.542 3.411 3.285 3.163 3.046 2.932 2.822 2.715 2.612 2.513 2.417 2.324 2.235 2.149 2.065 1.985 1.907 1.832 1.760 1.690

Vapor pressure

Vapor pressure

kPa

t, ºC

mmHg

kPa

t, ºC

mmHg

kPa

0.6112 0.5894 0.5682 0.5477 0.5279 0.5087 0.4901 0.4722 0.4548 0.4380 0.4218 0.4061 0.3909 0.3762 0.3620 0.3483 0.3351 0.3223 0.3099 0.2980 0.2865 0.2753 0.2646 0.2542 0.2442 0.2346 0.2253

−13.5 −14.0 −14.5 −15.0 −15.5 −16.0 −16.5 −17.0 −17.5 −18.0 −18.5 −19.0 −19.5 −20.0 −20.5 −21.0 −21.5 −22.0 −22.5 −23.0 −23.5 −24.0 −24.5 −25.0 −25.5 −26.0 −26.5

1.623 1.558 1.495 1.435 1.377 1.321 1.267 1.215 1.165 1.117 1.070 1.026 0.9827 0.9414 0.9016 0.8633 0.8265 0.7911 0.7571 0.7244 0.6930 0.6628 0.6337 0.6059 0.5791 0.5534 0.5288

0.2163 0.2077 0.1993 0.1913 0.1836 0.1761 0.1689 0.1620 0.1553 0.1489 0.1427 0.1367 0.1310 0.1255 0.1202 0.1151 0.1102 0.1055 0.1009 0.0965 0.0923 0.08836 0.08449 0.08078 0.07721 0.07379 0.07050

−27.0 −27.5 −28.0 −28.5 −29.0 −29.5 −30.0 −30.5 −31.0 −31.5 −32.0 −32.5 −33.0 −33.5 −34.0 −34.5 −35.0 −35.5 −36.0 −36.5 −37.0 −37.5 −38.0 −38.5 −39.0 −39.5 −40.0

0.5051 0.4824 0.4606 0.4397 0.4197 0.4005 0.3820 0.3644 0.3475 0.3313 0.3158 0.3009 0.2867 0.2731 0.2600 0.2476 0.2356 0.2242 0.2133 0.2029 0.1929 0.1834 0.1743 0.1656 0.1573 0.1494 0.1419

0.06734 0.06431 0.06141 0.05862 0.05595 0.05339 0.05094 0.04858 0.04633 0.04417 0.04210 0.04012 0.03822 0.03640 0.03467 0.03300 0.03141 0.02989 0.02844 0.02705 0.02572 0.02445 0.02324 0.02208 0.02098 0.01992 0.01891

*SOURCE: Murphy and Koop, Q. J. R. Meteorol. Soc., 131, 1539 (2005). The formula in the reference extends down to 123 K (-150.15 ºC), although in practice pure liquid water cannot be supercooled below about 235 K.

For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. ADDITIONAL REFERENCES Additional vapor-pressure data may be found in major thermodynamic property databases, such as those produced by the AIChE’s DIPPER program (www.aiche.org/dippr/), NIST’s Thermodynamics Research Center (trc.nist.gov), the Dortmund Databank (www.ddbst.de), and the Physical Property Data Service (www.ppds.co.uk). Additional sources include the NIST Chemistry Webbook (webbook.nist.gov/chemistry/); Boublik, Fried, and Hala, The Vapor Pressures of Pure Substances, 2d ed., Elsevier, Amsterdam, 1984; Poling, Prausnitz, and O’Connell, The Properties of Gases and Liquids, 5th ed., McGrawHill, New York, 2001; Vapor Pressure of Chemicals (Subvolumes A, B, and C), vol. IV/20 in Landolt-Bornstein: Numerical Data nad Functional Relationships in Science and Technology—New Series, Springer-Verlag, Berlin, 1999–2001. The most recent work on water may be found at The International Association for the Properties of Water and Steam website http://iapws.org.

TABLE 2-5 Vapor Pressure (MPa) of Liquid Water from 0 to 100°C t, °C 0.01 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Pvp, MPa

t, °C

Pvp, MPa

t, °C

Pvp, MPa

0.00061165 0.00065709 0.00070599 0.00075808 0.00081355 0.00087258 0.00093536 0.0010021 0.0010730 0.0011483 0.0012282 0.0013130 0.0014028 0.0014981 0.0015990 0.0017058 0.0018188 0.0019384 0.0020647 0.0021983 0.0023393 0.0024882 0.0026453 0.0028111 0.0029858 0.0031699 0.0033639 0.0035681 0.0037831 0.0040092 0.0042470 0.0044969 0.0047596 0.0050354

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

0.0053251 0.0056290 0.0059479 0.0062823 0.0066328 0.0070002 0.0073849 0.0077878 0.0082096 0.0086508 0.0091124 0.0095950 0.010099 0.010627 0.011177 0.011752 0.012352 0.012978 0.013631 0.014312 0.015022 0.015762 0.016533 0.017336 0.018171 0.019041 0.019946 0.020888 0.021867 0.022885 0.023943 0.025042 0.026183 0.027368

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.028599 0.029876 0.031201 0.032575 0.034000 0.035478 0.037009 0.038595 0.040239 0.041941 0.043703 0.045527 0.047414 0.049367 0.051387 0.053476 0.055635 0.057867 0.060173 0.062556 0.065017 0.067558 0.070182 0.072890 0.075684 0.078568 0.081541 0.084608 0.087771 0.091030 0.094390 0.097852 0.10142

From E. W. Lemmon, M. O. McLinden, and D. G. Friend, “Thermophysical Properties of Fluid Systems” in NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P. J. Linstrom and W. G. Mallard, June 2005, National Institute of Standards and Technology, Gaithersburg, Md. (http://webbook.nist.gov) and Wagner, W., and A., Pruss, “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data 31(2):387–535, 2002. The website mentioned above allows users to generate their own tables of thermodynamic properties. The user can select the units as well as the temperatures and/or pressures for which properties are to be generated. The results can be then be copied into spreadsheets or other files.

VAPOR PRESSURES OF PURE SUBSTANCES

2-49

TABLE 2-6 Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 Sorted by Chemical Family Name

Cmpd. No.

Formula

Paraffins Methane Ethane Propane Butane Pentane Hexane Heptane Octane Nonane Decane Undecane Dodecane Tridecane Tetradecane Pentadecane Hexadecane Heptadecane Octadecane Nonadecane Eicosane 2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,2,3,3-Tetramethylbutane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane Cyclopropane Cyclobutane Cyclopentane Cyclohexane Methylcyclopentane Ethylcyclopentane Methylcyclohexane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Ethylcyclohexane

193 125 295 31 279 171 160 265 256 74 336 123 327 319 277 169 158 263 254 124 236 202 107 234 114 323 332 333 71 64 69 65 217 134 213 108 109 110 133

CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22 C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42 C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18 C8H18 C3H6 C4H8 C5H10 C6H12 C6H12 C7H14 C7H14 C8H16 C8H16 C8H16 C8H16

1-Octyne 1-Nonyne 1-Decyne Methyl acetylene Vinyl acetylene Dimethyl acetylene 2-Methyl -1-butene-3-yne 3-Methyl-1-butyne

135 305 36 37 38 70 285 68 177 166 271 260 77 238 205 206 218 219 298 294 29 30 201

C2H4 C3H6 C4H8 C4H8 C4H8 C5H8 C5H10 C6H10 C6H12 C7H14 C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C6H10 C6H10 C9H14 C3H4 C4H6 C4H6 C5H8

Ketones

7 43 288 289 178 180 181 168

C2H2 C4H6 C5H8 C5H8 C6H10 C6H10 C6H10 C7H12

Olefins Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene Cyclopentene 1-Pentene Cyclohexene 1-Hexene 1-Heptene 1-Octene 1-Nonene 1-Decene 2-Methyl propene 2-Methyl-1-butene 2-Methyl-2-butene 1-Methylcyclopentene 3-Methylcyclopentene Propenylcyclohexene Propadiene 1,2-Butadiene 1,3-Butadiene 3-Methyl-1,2-butadiene Acetylenes Acetylene 1-Butyne 1-Pentyne 2-Pentyne 3-Hexyne 1-Hexyne 2-Hexyne 1-Heptyne

Name

Cmpd. No.

Formula

Acetylenes 273 262 79 197 339 105 207 210

C8H14 C9H16 C10H18 C3H4 C4H4 C4H6 C5H6 C5H8

16 325 312 129 343 344 345 243 62 304 330 331 246 321 40 24 290 318

C6H6 C7H8 C8H8 C8H10 C8H10 C8H10 C8H10 C9H10 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H14 C12H10 C14H10 C18H14

153 1 299 44 278 170 159 264 255 73

CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O

8 5 222 229 283 284 310 67 144 175 176 226 102 164 165 269 270 20

C3H4O C3H6O C4H8O C5H10O C5H10O C5H10O C6H4O2 C6H10O C6H12O C6H12O C6H12O C6H12O C7H14O C7H14O C7H14O C8H16O C8H16O C13H10O

156 324 320 322

C4H4O C4H4S C4H8O C4H8S

14 25 52 80 149

Ar Br2 Cl2 D2 F2

Aromatics Benzene Toluene Styrene Ethylbenzene m-Xylene o-Xylene p-Xylene alpha-Methyl styrene Cumene Propylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene Naphthalene 1,2,3,4-Tetrahydronaphthalene Butylbenzene Biphenyl Phenanthrene o-Terphenyl Aldehydes Formaldehyde Acetaldehyde Propionaldehyde Butyraldehyde Pentanal Hexanal Heptanal Octanal Nonanal Decanal

Acrolein Acetone Methylethyl ketone Methylisopropyl ketone 2-Pentanone 3-Pentanone Quinone Cyclohexanone Ethylisopropyl ketone 2-Hexanone 3-Hexanone Methylisobutyl ketone Diisopropyl ketone 3-Heptanone 2-Heptanone 2-Octanone 3-Octanone Benzophenone Heterocyclics Furan Thiophene Tetrahydrofuran Tetrahydrothiophene Elements Argon Bromine Chlorine Deuterium Fluorine

2-50

PHYSICAL AND CHEMICAL DATA

TABLE 2-6 Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 Sorted by Chemical Family (Continued) Name

Cmpd. No.

Formula

Elements Hydrogen Helium-4 Nitrogen Neon Oxygen

183 157 249 247 275

H2 He N2 Ne O2

194 126 296 297 34 35 281 282 66 173 174 162 163 267 268 258 259 76 337 237 204 21 214 215 216 137 309 32 33

CH4O C2H6O C3H8O C3H8O C4H10O C4H10O C5H12O C5H12O C6H12O C6H14O C6H14O C7H16O C7H16O C8H18O C8H18O C9H20O C9H20O C10H22O C11H24O C4H10O C5H12O C7H8O C7H14O C7H14O C7H14O C2H6O2 C3H8O2 C4H10O2 C4H10O2

291 59 60 61

C6H6O C7H8O C7H8O C7H8O

112 245 221 120 95 240 228 103 208 225 244 147 143 104 101 235 13 84 142 22 121

C2H6O C3H6O C3H8O C4H8O2 C4H10O C4H10O C4H10O C4H10O2 C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O2 C6H14O C6H14O C7H8O C8H18O C8H18O C9H12O C12H10O

155 274 3 9 191 300

CH2O2 C2H2O4 C2H4O2 C3H4O2 C3H4O4 C3H6O2

Alcohols Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol 2-Butanol 1-Pentanol 2-Pentanol Cyclohexanol 1-Hexanol 2-Hexanol 1-Heptanol 2-Heptanol 1-Octanol 2-Octanol 1-Nonanol 2-Nonanol 1-Decanol 1-Undecanol 2-Methyl-2-propanol 3-Methyl-1-butanol Benzyl alcohol 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Ethylene glycol 1,2-Propylene glycol 1,2-Butanediol 1,3-Butanediol Phenols Phenol m-Cresol o-Cresol p-Cresol Ethers Dimethyl ether Methyl vinyl ether Methylethyl ether 1,4-Dioxane Diethyl ether Methylpropyl ether Methylisopropyl ether 1,1-Dimethoxyethane Methylbutyl ether Methylisobutyl ether Methyl tert-butyl ether Ethylpropyl ether Ethylisopropyl ether 1,2-Dimethoxypropane Di-isopropyl ether Methyl pentyl ether Anisole Dibutyl ether Ethylhexyl ether Benzyl ethyl ether Diphenyl ether Acids Formic acid Oxalic acid Acetic acid Acrylic acid Malonic acid Propionic acid

Name

Cmpd. No.

Formula

Methacrylic acid Acetic anhydride Succinic acid Butyric acid Isobutyric acid 2-Methylbutanoic acid Pentanoic acid 2-Ethyl butanoic acid Hexanoic acid Benzoic acid Heptanoic acid Phthalic anhydride Terephthalic acid 2-Ethyl hexanoic acid Octanoic acid 2-Methyloctanoic acid Nonanoic acid Decanoic acid

192 4 313 45 189 203 280 131 172 18 161 293 317 141 266 233 257 75

C4H6O2 C4H6O3 C4H6O4 C4H8O2 C4H8O2 C5H10O2 C5H10O2 C6H12O2 C6H12O2 C7H6O2 C7H14O2 C8H4O3 C8H6O4 C8H16O2 C8H16O2 C9H18O2 C9H18O2 C10H20O2

224 140 196 198 338 127 239 306 232 146 211 302 39 132 200 130 115 119

C2H4O2 C3H6O2 C3H6O2 C4H6O2 C4H6O2 C4H8O2 C4H8O2 C4H8O2 C5H8O2 C5H10O2 C5H10O2 C5H10O2 C6H12O2 C6H12O2 C8H8O2 C9H10O2 C10H10O4 C10H10O4

199 138 106 128 136 190 303 329 94 93 100 122 328

CH5N C2H5N C2H7N C2H7N C2H8N2 C3H9N C3H9N C3H9N C4H11N C4H11NO2 C6H15N C6H15N C6H15N

154 2 113 195 15

CH3NO C2H5NO C3H7NO C3H7NO C7H7NO

6 63 10 301 46 19

C2H3N C2N2 C3H3N C3H5N C4H7N C7H5N

251 248

CH3NO2 C2H5NO2

Acids

Esters Methyl formate Ethyl formate Methyl acetate Methyl acrylate Vinyl acetate Ethyl acetate Methyl propionate Propyl formate Methyl methacrylate Ethyl propionate Methyl butyrate Propyl acetate Butyl acetate Ethyl butyrate Methyl benzoate Ethyl benzoate Dimethyl phthalate Dimethyl terephthalate Amines Methyl amine Ethyleneimine Dimethyl amine Ethyl amine Ethylenediamine Isopropyl amine Propyl amine Trimethyl amine Diethyl amine Diethanol amine Di-isopropyl amine Dipropyl amine Triethyl amine Amides Formamide Acetamide N,N-Dimethyl formamide N-Methyl acetamide Benzamide Nitriles Acetonitrile Cyanogen Acrylonitrile Propionitrile Butyronitrile Benzonitrile Nitro Compounds Nitromethane Nitroethane

VAPOR PRESSURES OF PURE SUBSTANCES

2-51

TABLE 2-6 Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 Sorted by Chemical Family (Concluded) Name

Cmpd. No.

Formula

1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene

334 335

C6H3N3O6 C7H5N3O6

227 292

C2H3NO C7H5NO

231 145 308 307 41 42 287 286 17 72 179 23 167 272 261 78

CH4S C2H6S C3H8S C3H8S C4H10S C4H10S C5H12S C5H12S C6H6S C6H12S C6H14S C7H8S C7H16S C8H18S C9H20S C10H22S

117 111 223 96 230 241 209

C2H6S C2H6S2 C3H8S C4H10S C4H10S C4H10S C5H12S

50 51 55 83 90 99 28 56 152 340 326 81 82 88

CCl4 CF4 CHCl3 CH2Br2 CH2Cl2 CH2F2 CH3Br CH3Cl CH3F C2H3Cl C2H3Cl3 C2H4Br2 C2H4Br2 C2H4Cl2

Isocyanates Methyl isocyanate Phenyl isocyanate Mercaptans Methyl mercaptan Ethyl mercaptan Propyl mercaptan 2-Propyl mercaptan Butyl mercaptan sec-Butyl mercaptan Pentyl mercaptan 2-Pentyl mercaptan Benzenethiol Cyclohexyl mercaptan Hexyl mercaptan Benzyl mercaptan Heptyl mercaptan Octyl mercaptan Nonyl mercaptan Decyl mercaptan Sulfides Dimethyl sulfide Dimethyl disulfide Methylethyl sulfide Diethyl sulfide Methylisopropyl sulfide Methylpropyl sulfide Methylbutyl sulfide

Cmpd. No.

Formula

1,2-Dichloroethane 1,1-Difluoroethane 1,2-Difluoroethane Bromoethane Chloroethane Fluoroethane 1,1-Dichloropropane 1,2-Dichloropropane 1-Chloropropane 2-Chloropropane m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene Bromobenzene Chlorobenzene Fluorobenzene

89 97 98 27 54 151 91 92 57 58 85 86 87 26 53 150

C2H4Cl2 C2H4F2 C2H4F2 C2H5Br C2H5Cl C2H5F C3H6Cl2 C3H6Cl2 C3H7Cl C3H7Cl C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H5Br C6H5Cl C6H5F

242 212 220 341 148 116 311

CH6Si CH5ClSi CH4Cl2Si C2H3Cl3Si C2H5Cl3Si C2H8Si F4Si

186 49 47 48 250 315 184 185 187 188 12 182 253 252 314 276 316

CHN CO CO2 CS2 F3N F6S HBr HCl HF H2S H3N H4N2 NO N2O O2S O3 O3S

11 139 118 342

Mixture C2H4O C2H6OS H2O

Silanes Methylsilane Methylchlorosilane Methyldichlorosilane Vinyl trichlorosilane Ethyltrichlorosilane Dimethylsilane Silicon tetrafluoride Light Gases

Halogenated Hydrocarbons Carbon tetrachloride Carbon tetrafluoride Chloroform Dibromomethane Dichloromethane Difluoromethane Bromomethane Chloromethane Fluoromethane Vinyl chloride 1,1,2-Trichloroethane 1,1-Dibromoethane 1,2-Dibromoethane 1,1-Dichloroethane

Name Halogenated Hydrocarbons

Nitro Compounds

Hydrogen cyanide Carbon monoxide Carbon dioxide Carbon disulfide Nitrogen trifluoride Sulfur hexafluoride Hydrogen bromide Hydrogen chloride Hydrogen fluoride Hydrogen sulfide Ammonia Hydrazine Nitric oxide Nitrous oxide Sulfur dioxide Ozone Sulfur trioxide Others Air Ethylene oxide Dimethyl sulfoxide Water

2-52

PHYSICAL AND CHEMICAL DATA

TABLE 2-7 Formula Index of Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 Formula

No.

Name

Formula

No.

Name

Ar Br2 CCl4 CF4 CHCl3 CHN CH2Br2 CH2Cl2 CH2F2 CH2O CH2O2 CH3Br CH3Cl CH3F CH3NO CH3NO2 CH4 CH4Cl2Si CH4O CH4S CH5ClSi CH5N CH6Si CO CO2 CS2 C2H2 C2H2O4 C2H3Cl C2H3Cl3 C2H3Cl3Si C2H3N C2H3NO C2H4 C2H4Br2 C2H4Br2 C2H4Cl2 C2H4Cl2 C2H4F2 C2H4F2 C2H4O C2H4O C2H4O2 C2H4O2 C2H5Br C2H5Cl C2H5Cl3Si C2H5F C2H5N C2H5NO C2H5NO2 C2H6 C2H6O C2H6O C2H6O2 C2H6OS C2H6S C2H6S C2H6S2 C2H7N C2H7N C2H8N2 C2H8Si C2N2 C3H3N C3H4 C3H4 C3H4O C3H4O2 C3H4O4 C3H5N C3H6 C3H6 C3H6Cl2

11 14 25 50 51 55 186 83 90 99 153 155 28 56 152 154 251 193 220 194 231 212 199 242 49 47 48 7 274 340 326 341 6 227 135 81 82 88 89 97 98 1 139 3 224 27 54 148 151 138 2 248 125 112 126 137 118 117 145 111 106 128 136 116 63 10 197 294 8 9 191 301 71 305 91

Air Argon Bromine Carbon tetrachloride Carbon tetrafluoride Chloroform Hydrogen cyanide Dibromomethane Dichloromethane Difluoromethane Formaldehyde Formic acid Bromomethane Chloromethane Fluoromethane Formamide Nitromethane Methane Methyldichlorosilane Methanol Methyl mercaptan Methylchlorosilane Methyl amine Methylsilane Carbon monoxide Carbon dioxide Carbon disulfide Acetylene Oxalic acid Vinyl chloride 1,1,2-Trichloroethane Vinyl trichlorosilane Acetonitrile Methyl Isocyanate Ethylene 1,1-Dibromoethane 1,2-Dibromoethane 1,1-Dichloroethane 1,2-Dichloroethane 1,1-Difluoroethane 1,2-Difluoroethane Acetaldehyde Ethylene oxide Acetic acid Methyl formate Bromoethane Chloroethane Ethyltrichlorosilane Fluoroethane Ethyleneimine Acetamide Nitroethane Ethane Dimethyl ether Ethanol Ethylene glycol Dimethyl sulfoxide Dimethyl sulfide Ethyl mercaptan Dimethyl disulfide Dimethyl amine Ethyl amine Ethylenediamine Dimethylsilane Cyanogen Acrylonitrile Methyl acetylene Propadiene Acrolein Acrylic acid Malonic acid Propionitrile Cyclopropane Propylene 1,1-Dichloropropane

C3H6Cl2 C3H6O C3H6O C3H6O C3H6O2 C3H6O2 C3H6O2 C3H7Cl C3H7Cl C3H7NO C3H7NO C3H8 C3H8O C3H8O C3H8O C3H8O2 C3H8S C3H8S C3H8S C3H9N C3H9N C3H9N C4H4 C4H4O C4H4S C4H6 C4H6 C4H6 C4H6 C4H6O2 C4H6O2 C4H6O2 C4H6O3 C4H6O4 C4H7N C4H8 C4H8 C4H8 C4H8 C4H8 C4H8O C4H8O C4H8O C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8O2 C4H8S C4H10 C4H10 C4H10O C4H10O C4H10O C4H10O C4H10O C4H10O C4H10O2 C4H10O2 C4H10O2 C4H10S C4H10S C4H10S C4H10S C4H10S C4H11N C4H11NO2 C5H6 C5H8 C5H8 C5H8 C5H8 C5H8 C5H8O2

92 5 245 299 140 196 300 57 58 113 195 295 221 296 297 309 223 308 307 190 303 329 339 156 324 29 30 43 105 192 198 338 4 313 46 36 37 38 64 238 44 222 320 45 120 127 189 239 306 322 31 236 34 35 95 237 240 228 32 33 103 41 42 96 230 241 94 93 207 70 201 210 288 289 232

1,2-Dichloropropane Acetone Methyl vinyl ether Propionaldehyde Ethyl formate Methyl acetate Propionic acid 1-Chloropropane 2-Chloropropane N,N-Dimethyl formamide N-Methyl acetamide Propane Methylethyl ether 1-Propanol 2-Propanol 1,2-Propylene glycol Methylethyl sulfide Propyl mercaptan 2-Propyl mercaptan Isopropyl amine Propyl amine Trimethyl amine Vinyl acetylene Furan Thiophene 1,2-Butadiene 1,3-Butadiene 1-Butyne Dimethyl acetylene Methacrylic acid Methyl acrylate Vinyl acetate Acetic anhydride Succinic acid Butyronitrile 1-Butene cis-2-Butene trans-2-Butene Cyclobutane 2-Methyl propene Butyraldehyde Methylethyl ketone Tetrahydrofuran Butyric acid 1,4-Dioxane Ethyl acetate Isobutyric acid Methyl propionate Propyl formate Tetrahydrothiophene Butane 2-Methylpropane 1-Butanol 2-Butanol Diethyl ether 2-Methyl-2-propanol Methylpropyl ether Methylisopropyl ether 1,2-Butanediol 1,3-Butanediol 1,1-Dimethoxyethane Butyl mercaptan sec-Butyl mercaptan Diethyl sulfide Methylisopropyl sulfide Methylpropyl sulfide Diethyl amine Diethanol amine 2-Methyl-1-butene-3-yne Cyclopentene 3-Methyl-1,2-butadiene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne Methyl methacrylate

VAPOR PRESSURES OF PURE SUBSTANCES

2-53

TABLE 2-7 Formula Index of Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 (Continued) Formula

No.

Name

Formula

No.

Name

C5H10 C5H10 C5H10 C5H10 C5H10O C5H10O C5H10O C5H10O C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H10O2 C5H12 C5H12 C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O C5H12O2 C5H12S C5H12S C5H12S C6H3N3O6 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H4O2 C6H5Br C6H5Cl C6H5F C6H6 C6H6O C6H6S C6H10 C6H10 C6H10 C6H10 C6H10 C6H10 C6H10O C6H12 C6H12 C6H12 C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O C6H12O2 C6H12O2 C6H12O2 C6H12O2 C6H12S C6H14 C6H14 C6H14 C6H14O C6H14O C6H14O C6H14O C6H14S C6H15N C6H15N C6H15N C7H5N C7H5N3O6 C7H5NO C7H6O2 C7H7NO

69 205 206 285 229 278 283 284 146 203 211 280 302 202 279 143 147 204 208 225 244 281 282 104 209 286 287 334 85 86 87 310 26 53 150 16 291 17 218 68 178 180 181 219 67 65 177 217 66 144 170 175 176 226 39 131 132 172 72 107 171 234 101 173 174 235 179 100 122 328 19 335 292 18 15

Cyclopentane 2-Methyl-1-butene 2-Methyl-2-butene 1-Pentene Methylisopropyl ketone Pentanal 2-Pentanone 3-Pentanone Ethyl propionate 2-Methylbutanoic acid Methyl butyrate Pentanoic acid Propyl acetate 2-Methylbutane Pentane Ethylisopropyl ether Ethylpropyl ether 3-Methyl-1-butanol Methylbutyl ether Methylisobutyl ether Methyl tert-butyl ether 1-Pentanol 2-Pentanol 1,2-Dimethoxypropane Methylbutyl sulfide 2-Pentyl mercaptan Pentyl mercaptan 1,3,5-Trinitrobenzene m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene Quinone Bromobenzene Chlorobenzene Fluorobenzene Benzene Phenol Benzenethiol 1-Methylcyclopentene Cyclohexene 3-Hexyne 1-Hexyne 2-Hexyne 3-Methylcyclopentene Cyclohexanone Cyclohexane 1-Hexene Methylcyclopentane Cyclohexanol Ethylisopropyl ketone Hexanal 2-Hexanone 3-Hexanone Methylisobutyl ketone Butyl acetate 2-Ethyl butanoic acid Ethyl butyrate Hexanoic acid Cyclohexyl mercaptan 2,3-Dimethylbutane Hexane 2-Methylpentane Di-isopropyl ether 1-Hexanol 2-Hexanol Methyl pentyl ether Hexyl mercaptan Di-isopropyl amine Dipropyl amine Triethyl amine Benzonitrile 2,4,6-Trinitrotoluene Phenyl isocyanate Benzoic acid Benzamide

C7H8 C7H8O C7H8O C7H8O C7H8O C7H8O C7H8S C7H12 C7H14 C7H14 C7H14 C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O C7H14O2 C7H16 C7H16 C7H16O C7H16O C7H16S C8H4O3 C8H6O4 C8H8 C8H8O2 C8H10 C8H10 C8H10 C8H10 C8H14 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16O C8H16O C8H16O C8H16O2 C8H16O2 C8H18 C8H18 C8H18 C8H18 C8H18O C8H18O C8H18O C8H18O C8H18S C9H10 C9H10O2 C9H12 C9H12 C9H12 C9H12 C9H12O C9H14 C9H16 C9H18 C9H18O C9H18O2 C9H18O2 C9H20 C9H20O C9H20O C9H20S C10H8 C10H10O4 C10H10O4 C10H12 C10H14 C10H18

325 13 21 59 60 61 23 168 134 166 213 102 159 164 165 214 215 216 161 114 160 162 163 167 293 317 312 200 129 343 344 345 273 108 109 110 133 271 264 269 270 141 266 265 323 332 333 84 142 267 268 272 243 130 62 304 330 331 22 298 262 260 255 233 257 256 258 259 261 246 115 119 321 40 79

Toluene Anisole Benzyl alcohol m-Cresol o-Cresol p-Cresol Benzyl mercaptan 1-Heptyne Ethylcyclopentane 1-Heptene Methylcyclohexane Di-isopropyl ketone Heptanal 3-Heptanone 2-Heptanone 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Heptanoic acid 2,3-Dimethylpentane Heptane 1-Heptanol 2-Heptanol Heptyl mercaptan Phthalic anhydride Terephthalic acid Styrene Methyl benzoate Ethylbenzene m-Xylene o-Xylene p-Xylene 1-Octyne 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Ethylcyclohexane 1-Octene Octanal 2-Octanone 3-Octanone 2-Ethyl hexanoic acid Octanoic acid Octane 2,2,3,3-Tetramethylbutane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane Dibutyl ether Ethylhexyl ether 1-Octanol 2-Octanol Octyl mercaptan alpha-Methyl styrene Ethyl benzoate Cumene Propylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene Benzyl ethyl ether Propenylcyclohexene 1-Nonyne 1-Nonene Nonanal 2-Methyloctanoic acid Nonanoic acid Nonane 1-Nonanol 2-Nonanol Nonyl mercaptan Naphthalene Dimethyl phthalate Dimethyl terephthalate 1,2,3,4-Tetrahydronaphthalene Butylbenzene 1-Decyne

2-54

PHYSICAL AND CHEMICAL DATA

TABLE 2-7 Formula Index of Substances in Tables 2-8, 2-32, 2-141, 2-150, 2-153, 2-155, 2-156, 2-179, 2-312, 2-313, 2-314, and 2-315 (Concluded) Formula

No.

Name

Formula

No.

Name

C10H20 C10H20O C10H20O2 C10H22 C10H22O C10H22S C11H24 C11H24O C12H10 C12H10O C12H26 C13H10O C13H28 C14H10 C14H30 C15H32 C16H34 C17H36 C18H14 C18H38 C19H40 C20H42 Cl2

77 73 75 74 76 78 336 337 24 121 123 20 327 290 319 277 169 158 318 263 254 124 52

1-Decene Decanal Decanoic acid Decane 1-Decanol Decyl mercaptan Undecane 1-Undecanol Biphenyl Diphenyl ether Dodecane Benzophenone Tridecane Phenanthrene Tetradecane Pentadecane Hexadecane Heptadecane o-Terphenyl Octadecane Nonadecane Eicosane Chlorine

D2 F2 F3N F4Si F6S HBr HCl HF H2 H2O H2S H3N H4N2 He NO N2 N2O Ne O2 O2S O3 O3S

80 149 250 311 315 184 185 187 183 342 188 12 182 157 253 249 252 247 275 314 276 316

Deuterium Fluorine Nitrogen trifluoride Silicon tetrafluoride Sulfur hexafluoride Hydrogen bromide Hydrogen chloride Hydrogen fluoride Hydrogen Water Hydrogen sulfide Ammonia Hydrazine Helium-4 Nitric oxide Nitrogen Nitrous oxide Neon Oxygen Sulfur dioxide Ozone Sulfur trioxide

TABLE 2-8

Vapor Pressure of Inorganic and Organic Liquids, ln P = C1 + C2/T + C3 ln T + C4 T C5, P in Pa

2-55

No.

Name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol o-Cresol

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O

CAS no. 75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4 95-48-7

C1 193.69 125.81 53.27 100.95 69.006 58.302 39.63 138.4 46.745 87.604 21.662 90.483 128.06 42.127 85.474 83.107 77.765 88.513 138.5 88.404 100.68 68.541 118.02 77.314 108.26 63.749 62.217 72.586 39.714 75.572 66.343 103.28 123.22 106.295 114.68 51.836 72.541 71.704 122.82 101.22 65.382 60.649 77.004 99.33 93.815 66.32 140.54 67.114 45.698 78.441 61.89 71.334 54.144 65.988 146.43 64.697 79.24 46.854 95.403 210.88

C2 −8,036.7 −12,376 −6,304.5 −8,873.2 −5,599.6 −5,385.6 −2,552.2 −7,122.7 −6,587.1 −6,392.7 −692.39 −4,669.7 −9,307.7 −1,093.1 −11,932 −6,486.2 −8,455.1 −11,829 −11,195 −11,769 −11,059 −7,886.2 −10,527 −9,910.4 −6,592 −7,130.2 −5,113.3 −4,698.6 −3,769.9 −4,621.9 −4,363.2 −11,548 −12,620 −9,866.4 −9,850.2 −4,019.2 −4,691.2 −4,563.1 −9,253.2 −9,255.4 −6,262.4 −5,785.9 −5,054.5 −7,083.6 −9,942.2 −6,714.9 −4,735 −4,820.4 −1,076.6 −6,128.1 −2,296.3 −3,855 −6,244.4 −4,661.3 −7,792.3 −4,048.1 −5,718.8 −4,445.5 −10,581 −13,928

C3 −29.502 −14.589 −4.2985 −11.451 −7.0985 −5.4954 −2.78 −19.638 −3.2208 −10.101 −0.392 −11.607 −16.693 −4.1425 −8.3348 −9.2194 −7.7404 −8.6826 −17.085 −8.9014 −10.709 −6.5804 −13.91 −7.5079 −14.16 −5.879 −5.9761 −7.9966 −2.6407 −8.5323 −7.046 −10.925 −13.986 −11.655 −12.963 −4.5229 −7.9776 −7.9053 −14.99 −11.538 −6.2585 −5.6113 −8.5665 −11.733 −9.8019 −6.3087 −21.268 −7.5303 −4.8814 −8.5766 −7.086 −8.5171 −4.5343 −6.8586 −20.614 −6.8066 −8.789 −3.6533 −10.004 −29.483

C4

C5

Tmin, K

P at Tmin

Tmax, K

P at Tmax

4.3678E-02 5.0824E-06 8.8865E-18 6.1316E-06 6.2237E-06 5.3634E-06 2.3930E-16 2.6447E-02 5.2253E-07 1.0891E-05 4.7574E-03 1.7194E-02 1.4919E-02 5.7254E-05 1.2850E-18 6.9844E-06 4.3089E-18 2.3248E-19 9.5641E-06 1.9334E-18 3.0582E-18 2.4285E-06 6.4794E-06 2.2385E-18 1.6043E-02 5.2136E-18 4.7174E-17 1.1553E-05 6.9379E-18 1.2269E-05 9.4509E-06 4.2560E-18 3.9260E-06 1.0832E-17 1.8738E-17 4.8833E-17 1.0368E-05 1.1319E-05 1.0470E-05 5.9208E-06 1.4943E-17 1.5877E-17 1.0161E-05 1.0027E-05 9.3124E-18 1.3516E-17 4.0909E-02 9.1695E-03 7.5673E-05 6.8465E-06 3.4687E-05 1.2378E-02 4.7030E-18 7.9404E-06 2.4578E-02 1.0371E-05 8.4486E-06 1.3260E-17 4.3032E-18 2.5182E-02

1 2 6 2 2 2 6 1 2 2 1 1 1 2 6 2 6 6 2 6 6 2 2 6 1 6 6 2 6 2 2 6 2 6 6 6 2 2 2 2 6 6 2 2 6 6 1 1 2 2 2 1 6 2 1 2 2 6 6 1

150.15 353.33 289.81 200.15 178.45 229.32 192.4 185.45 286.15 189.63 59.15 195.41 235.65 83.78 403 278.68 258.27 395.45 260.4 321.35 257.85 275.65 243.95 342.2 265.85 242.43 154.55 179.47 136.95 164.25 134.86 220 196.15 183.85 158.45 87.8 134.26 167.62 199.65 185.3 157.46 133.02 147.43 176.75 267.95 161.25 216.58 161.11 68.15 250.33 89.56 172.12 227.95 134.8 207.15 175.43 150.35 155.97 285.39 304.19

3.23E−01 3.36E+02 1.28E+03 2.20E−02 2.79E+00 1.87E+02 1.27E+05 1.03E+01 2.57E+02 3.68E+00 5.64E+03 6.11E+03 2.45E+00 6.87E+04 3.55E+02 4.76E+03 7.68E+00 7.96E+02 3.08E+00 1.49E+00 1.88E−01 2.31E+01 2.98E−01 9.42E+01 5.85E+03 7.84E+00 3.72E−01 1.95E+02 4.47E−01 6.92E+01 6.74E−01 2.93E−04 3.74E−07 2.90E−04 1.95E−06 6.94E−07 2.72E−01 7.45E+01 8.17E−02 1.54E−04 2.35E−03 3.40E−05 1.18E+00 3.17E−01 6.78E+00 6.18E−04 5.19E+05 1.49E+00 1.54E+04 1.12E+03 1.08E+02 1.37E+03 8.45E+00 1.25E−01 5.25E+01 8.71E+02 6.96E−02 9.08E−01 5.86E+00 6.53E+01

466 761 591.95 606 508.2 545.5 308.3 506 615 535 132.45 405.65 645.6 150.86 824 562.05 689 751 699.35 830 720.15 662 718 773 584.15 670.15 503.8 467 452 425 425.12 680 676 563.1 535.9 419.5 435.5 428.6 575.4 660.5 570.1 554 440 537.2 615.7 582.25 304.21 552 132.92 556.35 227.51 417.15 632.35 460.35 536.4 416.25 503.15 489 705.85 697.55

5.565E+06 6.569E+06 5.739E+06 3.970E+06 4.709E+06 4.852E+06 6.106E+06 5.020E+06 5.661E+06 4.480E+06 3.793E+06 1.130E+07 4.273E+06 4.896E+06 5.047E+06 4.875E+06 4.728E+06 4.469E+06 4.243E+06 3.357E+06 4.372E+06 3.113E+06 4.074E+06 3.407E+06 1.028E+07 4.520E+06 6.290E+06 7.997E+06 4.361E+06 4.303E+06 3.770E+06 5.202E+06 4.033E+06 4.401E+06 4.182E+06 4.021E+06 4.238E+06 4.100E+06 3.087E+06 2.882E+06 3.973E+06 4.060E+06 4.599E+06 4.323E+06 4.071E+06 3.787E+06 7.390E+06 8.041E+06 3.494E+06 4.544E+06 3.742E+06 7.793E+06 4.529E+06 5.327E+06 5.554E+06 6.691E+06 4.581E+06 4.510E+06 4.522E+06 5.058E+06

2-56

TABLE 2-8

Vapor Pressure of Inorganic and Organic Liquids, ln P = C1 + C2/T + C3 ln T + C4 T C5, P in Pa (Continued)

No.

Name

Formula

CAS no.

C1

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane

C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2

106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1

118.53 102.81 81.565 85.899 51.087 189.19 85.424 88.184 66.341 67.952 40.608 85.146 201.64 112.73 123.36 156.239 68.401 91.91 142.94 18.947 62.711 43.751 86.295 72.227 53.187 77.105 88.31 66.611 92.355 101.6 83.495 65.955 106.38 49.314 136.9 46.705 73.491 84.625 69.132 462.84 41.631 50.868 53.637 62.097 66.592 71.738 77.161 81.184 78.952 78.429 81.045 44.704 82.762 78.335 72.517 63.08 84.39 56.273 43.541 44.494

C2 −11,957 −8,674.6 −4,808.9 −4,884.4 −5,226.4 −14,337 −7,944.4 −6,624.9 −5,198.5 −5,187.5 −3,179.6 −7,843.7 −15,133 −9,749.6 −14,680 −15,212 −7,776.9 −10,565 −11,119 −154.47 −6,503.5 −5,587.7 −7,010.3 −7,537.6 −6,827.5 −8,111.1 −8,463.4 −5,493.1 −6,920.4 −6,541.6 −6,661.4 −6,015.6 −13,714 −4,949 −6,954.3 −5,177.4 −4,385.9 −5,217.4 −3,847.7 −18,227 −4,668.7 −6,036.5 −5,251.2 −6,174.9 −4,999.8 −5,302 −5,691.1 −6,927 −7,075.4 −6,882.1 −6,941.3 −3,525.6 −7,955.5 −6,348.7 −10,415 −4,062.3 −5,740.6 −7,620.6 −8,204.8 −5,406.7

C3

C4

C5

Tmin, K

P at Tmin

Tmax, K

P at Tmax

−13.293 −11.922 −9.3748 −10.883 −4.2278 −24.148 −9.2862 −10.059 −6.8103 −7.0785 −2.8937 −9.2982 −26.264 −13.245 −13.474 −18.424 −6.4637 −9.5957 −17.818 −0.5723 −5.7669 −3.0891 −9.5972 −7.0596 −4.3233 −7.8886 −9.6308 −6.7301 −10.651 −12.247 −9.2386 −6.5509 −11.06 −3.9256 −19.254 −3.5985 −8.1851 −9.871 −7.5868 −73.734 −2.8551 −4.066 −4.5649 −5.715 −6.8387 −7.3324 −8.501 −8.8498 −8.4344 −8.4129 −8.777 −3.4444 −8.8038 −8.5105 −6.755 −6.425 −9.6454 −4.6279 −2.7519 −3.1287

8.6988E-18 7.0048E-06 1.3901E-05 1.4934E-02 9.7554E-18 1.0740E-05 4.9957E-06 8.2566E-06 6.1930E-06 6.8165E-06 5.6131E-17 5.1788E-06 1.4625E-05 7.1266E-06 1.9491E-18 8.5006E-18 6.3750E-18 5.7028E-18 1.1020E-05 3.8899E-02 1.0427E-06 8.2664E-07 6.7794E-06 9.1442E-18 2.3112E-18 2.7267E-06 4.5833E-06 5.3579E-06 9.1426E-06 1.2311E-05 6.7652E-06 4.3172E-06 3.2645E-18 9.1978E-18 2.4508E-02 1.7147E-06 1.2978E-05 1.3050E-05 1.5065E-05 9.2794E-02 6.3693E-04 1.1326E-06 1.6754E-17 1.2323E-17 6.6793E-06 6.4200E-17 8.0325E-06 5.4580E-06 4.5035E-06 4.9831E-06 5.5501E-06 5.4574E-17 4.2431E-06 6.4311E-06 1.3269E-06 1.5115E-16 1.0073E-05 4.3819E-07 1.0466E-18 2.8913E-18

6 2 2 1 6 2 2 2 2 2 6 2 2 2 6 6 6 6 2 1 2 2 2 6 6 2 2 2 2 2 2 2 6 6 1 2 2 2 2 1 1 2 6 6 2 6 2 2 2 2 2 6 2 2 2 6 2 2 6 6

307.93 177.14 245.25 182.48 279.69 296.6 242 169.67 179.28 138.13 145.59 189.64 267.15 243.51 304.55 280.05 206.89 247.56 229.15 18.73 210.15 282.85 220.6 175.3 248.39 256.15 326.14 176.19 237.49 178.01 200 172.71 301.15 223.35 156.85 169.20 154.56 215 136.95 176.85 187.65 204.81 159.95 226.1 240.91 180.96 145.19 239.66 223.16 184.99 188.44 131.65 212.72 160 274.18 122.93 174.88 291.67 413.8 284.95

3.45E+01 4.71E−04 7.39E+04 1.80E+02 5.36E+03 7.65E+01 6.80E+00 1.04E−01 9.07E+00 1.28E−02 7.80E+01 8.24E−03 4.86E−01 1.39E+00 1.50E−01 1.51E−01 2.59E−02 2.59E−02 1.60E−01 1.72E+04 2.64E+00 7.53E+02 2.13E+01 7.14E−04 6.41E+00 6.49E+00 1.23E+03 2.21E+00 2.37E+02 5.93E+00 4.52E+00 8.25E−02 1.02E−01 3.74E+02 3.95E−01 9.93E−02 6.45E+01 2.83E+03 5.43E+01 4.47E−03 6.86E+00 8.21E−01 9.45E−02 4.50E+01 6.12E+03 7.56E+01 1.52E−02 6.06E+01 6.41E+00 8.04E−02 2.07E−01 3.05E+00 1.95E−01 1.26E−02 3.72E−02 4.15E−01 7.86E+00 5.02E+01 1.26E+03 2.53E+03

704.65 631 400.15 459.93 553.8 650.1 653 560.4 511.7 507 398 664 674.2 617.7 722.1 688 616.6 696 619.85 38.35 628 650.15 611 584.1 683.95 705 684.75 523 561.6 510 560 572 736.6 496.6 466.7 557.15 386.44 445 351.255 523.1 500.05 576 507.8 543 473.2 437.2 500 591.15 606.15 596.15 615 400.1 649.6 537.3 766 402 503.04 729 772 587

5.151E+06 3.226E+06 5.961E+06 4.991E+06 4.094E+06 4.265E+06 3.989E+06 4.392E+06 4.513E+06 4.799E+06 5.494E+06 3.970E+06 2.599E+06 2.091E+06 2.233E+06 2.309E+06 2.223E+06 2.130E+06 2.363E+06 1.663E+06 6.034E+06 5.375E+06 7.170E+06 2.459E+06 4.070E+06 4.074E+06 4.070E+06 5.106E+06 5.318E+06 6.093E+06 4.239E+06 4.232E+06 4.260E+06 3.674E+06 3.641E+06 3.961E+06 4.507E+06 4.372E+06 5.760E+06 3.199E+06 2.869E+06 3.017E+06 3.773E+06 3.447E+06 4.870E+06 5.258E+06 3.130E+06 2.939E+06 2.939E+06 2.938E+06 5.363E+06 5.274E+06 4.365E+06 2.882E+06 2.780E+06 3.561E+06 5.533E+06 5.648E+06 2.778E+06 5.158E+06

2-57

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183

Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne Hydrazine Hydrogen

C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10 H4N2 H2

101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2 302-01-2 1333-74-0

59.969 54 137.47 203.66 51.857 73.304 66.824 81.56 89.063 52.923 90.464 57.661 80.208 88.671 53.963 73.51 84.09 66.51 91.944 73.833 117.52 77.523 57.723 57.459 65.551 105.64 86.898 62.614 42.393 51.915 56.639 59.123 101.51 100.3 50.323 74.738 11.533 156.95 92.252 87.829 120.47 147.41 124.23 78.463 75.494 65.922 79.858 59.083 156.06 81.507 104.65 114.05 135.421 109.42 107.44 73.155 51.024 47.091 68.467 133.2 123.71 76.858 12.69

−8,585.5 −6,018.5 −11,976 −19,441 −2,598.7 −7,122.3 −6,227.6 −5,596.9 −7,733.7 −7,531.7 −10,243 −6,346.5 −7,203.2 −7,012.7 −2,443 −7,572.7 −10,411 −6,019.2 −5,293.4 −5,817 −12,991 −7,978.8 −5,236.9 −6,356.8 −5,027.4 −8,007 −6,646.4 −6,148.2 −1,103.3 −5,439 −3,576.5 −3,043.7 −4,917.2 −10,763 −5,378.2 −5,417 −8.99 −15,557 −8,349 −6,996.4 −13,106 −13,466 −11,637 −8,077.2 −7,896.5 −6,189 −8,501.8 −6,031.8 −15,015 −7,776.8 −6,995.5 −12,332 −12,288 −10,449 −8,528.6 −7,242.9 −4,986.4 −5,104 −7,390.5 −7,492.9 −7,639 −7,245.2 −94.896

−5.1538 −4.4981 −16.698 −25.525 −5.1283 −7.1424 −6.41 −9.0779 −9.917 −4.2347 −9.2836 −5.032 −8.6023 −10.045 −5.5643 −7.1435 −8.1976 −6.3332 −11.682 −7.809 −12.895 −7.7757 −5.2136 −4.9545 −6.6853 −12.477 −9.5758 −5.84 −4.1203 −4.2896 −5.5801 −6.1845 −13.765 −10.946 −4.203 −8.0636 0.6724 −18.966 −10.274 −9.8802 −13.31 −17.353 −14.148 −7.9062 −7.5047 −6.3629 −8.1043 −5.3072 −18.941 −8.4516 −12.702 −12.45 −15.732 −12.051 −12.679 −7.2569 −4.2463 −3.6371 −6.5456 −18.405 −16.451 −8.22 1.1125

1.9983E-18 9.9684E-18 8.0906E-06 8.8382E-06 1.4913E-05 2.8853E-06 1.7914E-17 8.7920E-06 5.9860E-06 1.1835E-06 5.2573E-18 8.2534E-18 4.5901E-06 7.4578E-06 1.9079E-05 1.2124E-17 1.6536E-18 1.0394E-17 1.4902E-02 6.3200E-06 6.1306E-18 1.0076E-17 2.2998E-17 5.2015E-18 6.3208E-06 9.0000E-06 5.9615E-17 1.0900E-17 5.7815E-05 8.7527E-18 9.8969E-06 1.6637E-05 2.2031E-02 3.8503E-06 3.4697E-06 7.4700E-06 2.7430E-01 6.4559E-06 5.9252E-06 7.2099E-06 5.8384E-18 1.1284E-17 6.9486E-17 8.0521E-18 8.9130E-18 2.0091E-17 8.1501E-18 1.4357E-17 6.8172E-06 1.5143E-17 1.2381E-05 5.6253E-18 1.2701E-17 2.6122E-46 8.4606E-06 1.2741E-17 1.6768E-17 5.1621E-04 7.7611E-18 2.2062E-02 1.6495E-02 6.1557E-03 3.2915E-04

6 6 2 2 2 2 6 2 2 2 6 6 2 2 2 6 6 6 1 2 6 6 6 6 2 2 6 6 2 6 2 2 1 2 2 2 1 2 2 2 6 6 5.7 6 6 6 6 6 2 6 2 6 6 16 2 6 6 1 6 1 1 1 2

300.03 210.15 263.57 309.58 90.35 159.05 189.6 192.15 178.2 238.45 258.15 175.15 161.84 134.71 104 284.29 260.15 195.2 160.65 193.55 235 180 140 204.15 125.26 199.25 145.65 167.55 53.48 230.94 129.95 131.35 181.15 275.6 281.45 187.55 1.76 295.13 229.8 182.57 265.83 239.15 230 234.15 238.15 154.12 229.92 192.22 291.31 217.15 177.83 269.25 228.55 223 217.35 217.5 133.39 170.05 192.62 141.25 183.65 274.69 13.95

7.09E+00 3.69E+00 6.15E−01 9.26E−03 1.13E+00 4.96E−04 1.43E+00 1.52E+02 3.91E−03 1.69E−01 4.63E−01 1.04E−02 3.57E−04 3.71E−06 1.26E+02 6.78E+02 2.19E−01 9.71E+00 7.79E+00 1.81E+01 2.86E−04 7.60E−04 4.31E−03 9.70E−01 1.14E−03 7.80E−01 1.61E−03 1.85E−02 2.53E+02 1.51E+02 8.37E+00 4.33E+02 8.87E+02 1.04E+00 2.40E+03 5.00E+01 1.46E+03 4.65E−02 1.45E+00 1.83E−01 4.34E−02 1.95E−02 3.68E−02 2.30E+00 3.54E+00 1.86E−03 3.05E−01 8.15E−01 9.23E−02 1.25E+00 9.02E−01 2.43E−01 2.25E−02 7.44E−02 1.45E+00 2.22E+00 7.96E−04 2.20E−01 1.31E−02 3.92E−04 5.40E−01 4.08E+02 7.21E+03

766.8 550 658 768 305.32 514 523.3 456.15 617.15 698 655 571 609.15 569.5 282.34 593 720 537 469.15 508.4 674.6 583 489 567 499.15 546 500.23 559.95 144.12 560.09 375.31 317.42 408 771 588 490.15 5.2 736 616.8 540.2 677.3 632.3 608.3 606.6 611.4 537.4 645 547 723 591 507.6 660.2 611.3 585.3 587.61 582.82 504 544 623 516.2 549 653.15 33.19

3.097E+06 3.111E+06 1.822E+06 1.175E+06 4.852E+06 6.109E+06 3.850E+06 5.594E+06 3.590E+06 3.203E+06 3.403E+06 2.935E+06 3.041E+06 3.412E+06 5.032E+06 6.290E+06 8.257E+06 6.850E+06 7.255E+06 4.708E+06 2.788E+06 2.460E+06 3.415E+06 3.293E+06 5.492E+06 3.337E+06 3.372E+06 3.320E+06 5.167E+06 4.544E+06 5.006E+06 5.875E+06 6.594E+06 7.751E+06 5.807E+06 5.550E+06 2.285E+05 1.344E+06 3.155E+06 2.719E+06 3.039E+06 3.013E+06 2.995E+06 2.919E+06 2.946E+06 2.921E+06 2.772E+06 3.209E+06 1.411E+06 3.461E+06 3.045E+06 3.284E+06 3.441E+06 3.298E+06 3.286E+06 3.322E+06 3.212E+06 3.540E+06 3.079E+06 3.635E+06 3.530E+06 1.473E+07 1.315E+06

2-58

TABLE 2-8

Vapor Pressure of Inorganic and Organic Liquids, ln P = C1 + C2/T + C3 ln T + C4 T C5, P in Pa (Continued)

No.

Name

Formula

CAS no.

C1

184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242

Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane

HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si

10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9

29.315 104.27 36.75 59.544 85.584 110.38 136.66 122.92 109.53 39.205 82.718 79.128 61.267 50.242 107.69 75.206 84.828 66.575 71.308 85.383 121.85 93.131 83.927 95.453 60.164 96.344 69.459 71.87 95.984 92.684 134.63 125.1 54.179 55.368 52.732 52.601 79.788 78.586 72.698 79.07 77.184 57.984 80.503 57.612 53.867 45.242 52.82 54.15 107.36 105.7 53.579 61.907 108.43 172.27 78.01 70.717 67.942 83.711 37.205

C2 −2,424.5 −3,731.2 −3,927.1 −4,143.8 −3,839.9 −10,540 −7,201.5 −16,258 −10,410 −1,324.4 −6,904.5 −9,523.9 −5,618.6 −3,811.9 −7,027.2 −5,082.8 −9,334.7 −5,213.4 −4,976 −9,575.4 −10,976 −5,525.4 −5,640.5 −5,448.8 −5,621.7 −7,856.3 −5,250 −6,885.7 −5,401.7 −7,080.8 −10,682 −10,288 −7,477.2 −5,149.8 −5,286.9 −5,120.3 −5,420 −5,176.3 −6,143.6 −6,114.1 −5,606.1 −5,339.6 −7,421.8 −5,197.9 −4,701 −5,324.4 −5,437.7 −4,337.7 −8,085.3 −12,458 −5,041.2 −6,188.9 −5,039.9 −11,589 −4,634.1 −6,439.7 −5,419.1 −6,786.9 −2,590.3

C3

C4

C5

Tmin, K

P at Tmin

Tmax, K

P at Tmax

−1.1354 −15.047 −2.1245 −6.1764 −11.199 −12.262 −18.934 −13.113 −12.289 −3.4366 −8.8622 −7.7355 −5.6473 −4.2526 −13.916 −8.0919 −8.7063 −6.7693 −7.7169 −8.6164 −13.869 −11.852 −9.6453 −12.384 −5.53 −11.058 −7.1125 −7.0944 −11.829 −10.695 −16.511 −15.157 −4.22 −5.0136 −4.4509 −4.4554 −9.0702 −8.7501 −7.5779 −8.631 −8.392 −5.2362 −8.379 −5.1269 −4.7052 −3.2551 −4.442 −4.8127 −12.72 −11.234 −4.6404 −5.706 −15.012 −22.113 −8.9575 −6.9845 −6.8067 −9.2526 −2.5993

2.3806E-18 3.1340E-02 3.8948E-17 1.4161E-05 1.8848E-02 1.4310E-17 2.2255E-02 2.0609E-18 3.1990E-06 3.1019E-05 7.4664E-06 3.1616E-18 2.1080E-17 6.5326E-17 1.5185E-02 8.1130E-06 6.1723E-18 4.8106E-06 8.7271E-06 5.6124E-18 1.4283E-17 1.4205E-02 1.1121E-05 1.5643E-02 1.8629E-17 7.3080E-06 7.9289E-17 1.4903E-17 1.8092E-05 8.1366E-06 8.4427E-06 1.0918E-05 3.5225E-18 3.2220E-06 1.0883E-17 1.3288E-17 1.1489E-05 9.1727E-06 5.6476E-06 6.5333E-06 7.8468E-06 2.0767E-17 1.8114E-17 2.1702E-17 2.8791E-17 3.0363E-18 9.5103E-18 4.5000E-17 8.3307E-06 4.4629E-18 1.9443E-17 1.1767E-17 2.2725E-02 1.3703E-05 1.3413E-05 2.0129E-17 4.7778E-17 6.6666E-06 6.0508E-06

6 1 6 2 1 6 1 6 2 2 2 6 6 6 1 2 6 2 2 6 6 1 2 1 6 2 6 6 2 2 2 2 6 2 6 6 2 2 2 2 2 6 6 6 6 6 6 6 2 6 6 6 1 2 2 6 6 2 2

185.15 158.97 259.83 189.79 187.68 227.15 177.95 407.95 288.15 90.69 175.47 301.15 175.15 170.45 196.32 179.69 260.75 159.53 113.25 193 155.95 135.58 139.39 160.15 157.48 175.3 183.45 187.35 139.05 146.58 299.15 280.15 269.15 130.73 146.62 115 182.55 160 186.48 167.23 174.15 150 189.15 256.15 127.93 180.15 171.64 150.18 224.95 240 119.55 176 113.54 298.97 132.81 185.65 133.97 160.17 116.34

2.95E+04 1.35E+04 1.87E+04 3.37E+02 2.29E+04 7.82E−02 7.73E+00 7.03E+01 5.86E+01 1.17E+04 1.11E−01 2.86E+01 1.02E+00 4.15E+02 4.07E+00 1.77E+02 1.81E+00 7.28E−01 1.21E−04 6.94E−05 8.67E−09 2.05E−02 1.94E−02 2.92E+00 2.99E−02 4.61E−03 4.36E+01 1.34E−01 4.12E−01 1.52E−04 2.57E+02 4.56E+01 1.62E+01 2.25E−04 3.98E−03 2.12E−06 2.58E+01 7.85E+00 1.39E+00 2.25E−01 6.88E+00 2.13E−02 6.99E−02 7.28E+03 3.32E−03 2.95E−01 1.80E−01 3.15E+00 1.91E+01 4.19E−04 2.07E−05 6.33E−02 1.21E−02 5.88E+03 6.45E−01 6.34E−01 2.90E−03 4.26E−03 1.43E+01

363.15 324.65 456.65 461.15 373.53 605 471.85 805 662 190.56 512.5 718 506.55 402.4 536 430.05 693 490 460.4 643 577.2 465 470 492 512.74 593 463.2 554.5 442 572.1 686 614 617 532.7 542 526 483 437.8 535.5 533 487.2 497 574.6 488 464.48 553.4 553.1 469.95 566 694 497.7 546.49 407.8 506.2 417.9 530.6 476.25 565 352.5

8.463E+06 8.356E+06 5.353E+06 6.487E+06 8.999E+06 3.683E+06 4.540E+06 5.652E+06 4.812E+06 4.590E+06 8.146E+06 4.997E+06 4.695E+06 5.619E+06 4.277E+06 7.414E+06 3.590E+06 3.831E+06 3.366E+06 3.887E+06 3.916E+06 3.465E+06 3.394E+06 4.469E+06 3.377E+06 3.464E+06 4.199E+06 3.480E+06 4.170E+06 3.486E+06 3.994E+06 3.808E+06 3.767E+06 3.759E+06 4.130E+06 4.129E+06 3.964E+06 4.433E+06 4.120E+06 4.261E+06 5.983E+06 3.416E+06 3.272E+06 5.480E+06 3.764E+06 3.792E+06 4.022E+06 7.231E+06 3.674E+06 2.545E+06 3.044E+06 3.041E+06 3.630E+06 3.957E+06 4.004E+06 4.028E+06 3.802E+06 3.972E+06 4.702E+06

2-59

243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305

-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene

C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N C9H12 C3H6

98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8 103-65-1 115-07-1

56.485 57.1511 51.085 62.964 29.755 75.632 58.282 68.149 57.278 96.512 72.974 182.54 337.71 109.35 137.6 162.854 146.46 63.313 106.2 114.77 157.68 83.601 96.084 140.16 144.111 133.41 63.775 72.382 74.936 78.368 64.612 122.04 51.245 40.067 135.57 149.58 78.741 101.7 114.748 122.26 84.635 44.286 46.994 58.985 67.309 82.805 137.29 72.958 95.444 86.779 126.5 57.069 59.078 84.6642 96.094 64.268 80.581 54.552 82.699 115.16 58.398 91.379 43.905

−6,954.2 −5,201.7 −4,271 −8,137.5 −271.06 −7,202.3 −1,084.1 −2,257.9 −6,089 −4,045 −2,650 −17,897 −18,506 −9,030.4 −14,948 −15,205 −13,813 −7,040.4 −10,982 −9,430.8 −16,093 −8,865.8 −7,900.2 −14,813 −13,667 −12,630 −7,711.3 −8,054.8 −7,155.9 −8,855.4 −6,802.5 −16,050 −1,200.2 −2,204.8 −13,478 −8,890 −5,420.3 −10,955 −10,643 −10,774 −7,078.4 −5,415.1 −4,289.5 −6,193.1 −6,880.8 −5,683.8 −7,447.1 −10,943 −10,113 −8,101.8 −12,551 −3,682.7 −3,492.6 −8,307.2 −8,575.4 −7,298.9 −5,896.1 −7,149.4 −6,703.5 −8,433.9 −5,312.7 −8,276.8 −3,097.8

−4.7889 −5.1429 −4.307 −5.6317 −2.6081 −7.6464 −8.3144 −8.9118 −4.9821 −12.277 −8.261 −22.498 −50.224 −12.882 −15.618 −19.424 −17.158 −5.8055 −11.696 −13.631 −18.954 −8.5711 −11.003 −16.004 −16.826 −15.369 −5.7359 −7.0002 −7.5843 −7.8202 −6.0261 −12.986 −6.4361 −2.9351 −16.022 −20.697 −8.8253 −10.829 −12.858 −13.943 −9.3 −3.0913 −3.7345 −5.2746 −6.4449 −9.4301 −19.01 −6.7902 −10.09 −9.5303 −15.002 −5.5662 −6.0669 −8.5767 −10.292 −5.9109 −8.9301 −4.2769 −9.1506 −13.934 −5.2876 −10.176 −3.4425

2.7753E-18 1.6529E-17 3.0530E-17 2.2675E-18 5.2700E-04 1.8250E-17 4.4127E-02 2.3233E-02 1.2154E-17 2.8860E-05 9.7000E-15 7.4008E-06 4.7345E-02 7.8544E-06 5.5660E-18 1.0722E-17 8.6279E-40 7.5753E-18 8.8955E-18 8.1918E-06 5.9272E-06 7.9446E-18 7.1802E-06 6.4239E-18 9.3666E-18 2.9939E-41 3.0902E-18 5.8276E-18 1.7106E-17 5.6629E-18 1.1013E-17 2.0871E-18 2.8405E-02 7.7520E-16 5.6136E-06 2.2101E-02 9.6171E-06 7.1880E-18 1.2491E-17 1.0700E-42 6.2702E-06 1.8580E-18 2.5424E-17 7.3986E-18 1.0148E-17 1.0767E-05 2.1415E-02 1.0850E-18 6.7603E-18 6.1367E-06 7.7521E-06 6.5133E-06 1.0919E-05 7.5091E-18 1.6665E-17 4.8482E-18 8.2236E-06 1.1843E-18 7.5424E-06 1.0346E-05 1.9913E-06 5.6240E-06 9.9989E-17

6 6 6 6 2 6 1 1 6 2 6 2 1 2 6 6 14 6 6 2 2 6 2 6 6 14 6 6 6 6 6 6 1 6 2 1 2 6 6 15 2 6 6 6 6 2 1 6 6 2 2 2 2 6 6 6 2 6 2 2 2 2 6

249.95 164.55 151.15 353.43 24.56 183.63 63.15 66.46 244.6 182.3 109.5 305.04 255.15 219.66 285.55 268.15 238.15 191.91 253.05 223.15 301.31 246 216.38 289.65 257.65 241.55 252.85 255.55 171.45 223.95 193.55 462.65 54.36 80.15 283.07 182 143.42 239.15 195.56 200 196.29 234.18 108.02 160.75 197.45 167.45 163.83 372.38 314.06 243.15 404.15 136.87 85.47 146.95 185.26 199 170 252.45 180.26 178.15 188.36 173.55 87.89

9.23E+00 4.93E−01 3.37E+00 9.91E+02 4.38E+04 3.18E-02 1.25E+04 1.86E−01 1.47E+02 8.69E+04 2.20E+04 1.59E−02 3.42E−01 4.31E−01 4.71E−02 8.55E−02 4.32E−03 2.04E−02 1.47E−01 4.50E−01 3.39E−02 1.46E+00 2.11E+00 1.83E−01 9.60E−02 4.04E−02 4.68E+00 7.84E+00 2.98E−03 3.05E−02 1.04E−01 2.15E+03 1.48E+02 7.35E−01 1.29E−01 5.23E−02 6.86E−02 3.28E−02 5.48E−04 4.15E−03 7.52E−01 7.34E+01 3.70E−05 1.77E−03 2.01E−01 2.40E+00 2.05E−01 2.93E+01 1.88E+02 4.33E+00 7.90E+02 1.83E+01 1.68E−04 4.28E−07 1.95E−02 2.48E−02 1.31E+00 1.31E+01 1.69E−01 1.71E−02 1.30E+01 1.81E−04 1.17E−03

654 497.1 437 748.4 44.4 593 126.2 234 588.15 309.57 180.15 758 658 594.6 710.7 670.9 649.5 593.1 681 598.05 747 638.9 568.7 694.26 652.3 629.8 632.7 627.7 566.9 667.3 574 804 154.58 261 708 566.1 469.7 639.16 588.1 561 561.08 560.95 464.8 584.3 598 481.2 519 869 694.25 653 791 394 369.83 536.8 508.3 636 504.4 600.81 564.4 549.73 496.95 638.35 364.85

3.341E+06 3.285E+06 4.583E+06 4.069E+06 2.665E+06 5.159E+06 3.391E+06 4.500E+06 6.309E+06 7.278E+06 6.516E+06 1.208E+06 2.743E+06 2.305E+06 2.502E+06 2.522E+06 2.551E+06 2.427E+06 2.330E+06 2.620E+06 1.256E+06 2.951E+06 2.467E+06 2.761E+06 2.782E+06 2.754E+06 2.647E+06 2.705E+06 2.663E+06 2.523E+06 2.880E+06 7.060E+06 5.021E+06 5.566E+06 1.474E+06 3.969E+06 3.364E+06 3.589E+06 3.896E+06 3.709E+06 3.706E+06 3.699E+06 3.562E+06 3.537E+06 3.474E+06 4.170E+06 4.020E+06 2.902E+06 6.059E+06 4.063E+06 4.734E+06 5.218E+06 4.214E+06 5.170E+06 4.783E+06 3.130E+06 4.919E+06 4.608E+06 4.191E+06 3.366E+06 4.738E+06 3.202E+06 4.599E+06

2-60 TABLE 2-8

Vapor Pressure of Inorganic and Organic Liquids, ln P = C1 + C2/T + C3 ln T + C4 T C5, P in Pa (Concluded)

No.

Name

Formula

CAS no.

306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

C1 104.08 60.43 62.165 212.8 48.651 272.85 105.93 128.65 47.365 29.16 180.99 248.72 110.52 140.47 54.898 137.23 75.881 57.963 93.193 76.945 54.153 137.45 56.55 134.68 78.341 85.301 84.912 83.105 506.33 302 131 182.571 57.406 55.682 91.432 54.571 73.649 85.099 90.405 88.72

C2

C3

C4

C5

Tmin, K

P at Tmin

Tmax, K

P at Tmax

−7,535.9 −5,276.9 −5,624 −15,420 −7,289.5 −9,548.9 −8,685.9 −16,958 −4,084.5 −2,383.6 −12,060 −32,238 −14,045 −13,231 −5,305.4 −10,620 −6,910.6 −5,901.5 −7,001.5 −6,729.8 −6,041.8 −12,549 −5,681.9 −6,055.8 −8,019.8 −8,215.9 −6,722.2 −6,903.7 −37,483 −24,324 −11,143 −17,112 −5,702.8 −4,439.3 −5,141.7 −5,561.5 −7,258.2 −7,615.9 −7,955.2 −7,741.2

−12.348 −5.6572 −5.8595 −28.109 −3.4453 −40.089 −12.42 −13.872 −3.6469 −1.1342 −22.839 −30.009 −11.861 −16.859 −4.7627 −17.908 −7.9499 −5.2048 −10.738 −8.179 −4.5383 −16.543 −4.9815 −19.415 −8.1458 −9.2166 −9.5157 −9.1858 −69.22 −40.13 −15.855 −22.125 −5.0307 −5.0136 −10.981 −4.712 −7.3037 −9.3072 −10.086 −9.8693

9.6020E−06 2.6039E−17 2.0597E−17 2.1564E−05 1.0068E−18 6.3699E−15 7.5583E−06 2.1559E−18 1.7990E−17

2 6 6 2 6 6 2 6 6

7.2350E−17 4.7950E−06 2.2121E−18 6.5877E−06 1.4291E−17 1.4506E−02 4.4315E−06 9.1301E−18 8.2308E−06 5.3017E−06 4.9833E−18 7.1275E−06 1.2363E−17 2.8619E−02 3.8971E−06 4.7979E−06 7.2244E−06 6.4703E−06 2.7381E−05 1.7403E−05 8.1871E−06 1.12835E−17 1.1042E−17 1.9650E−17 1.4318E−05 1.0702E−17 4.1653E−06 5.5643E−06 5.9594E−06 6.0770E−06

6 2 6 2 6 1 2 6 2 2 6 2 6 1 2 2 2 2 2 2 2 6 6 6 2 6 2 2 2 2

180.25 142.61 159.95 213.15 388.85 186.35 242.54 460.65 197.67 223.15 289.95 700.15 329.35 279.01 164.65 237.38 176.99 373.96 234.94 178.18 236.5 267.76 158.45 156.08 247.79 229.33 165.78 172.22 398.4 354 247.57 288.45 180.35 173.15 119.36 178.35 273.16 225.3 247.98 286.41

2.11E−01 9.73E−03 6.51E−02 9.29E−05 1.17E+04 2.21E+05 1.06E+01 8.85E+02 1.67E+03 2.30E+05 2.09E+04 4.57E+03 4.14E−01 2.53E−01 1.96E−01 1.33E−01 1.54E−02 8.69E+04 1.86E+02 4.75E−02 4.47E+01 2.51E−01 1.06E−02 9.92E+00 3.71E+00 6.93E−01 1.71E−02 1.68E−02 8.50E+00 9.36E−01 4.08E−01 1.26E−01 7.06E−01 6.69E+01 1.92E−02 3.54E−01 6.11E+02 3.18E+00 2.18E+01 5.76E+02

538 517 536.6 626 683 259 636 806 430.75 318.69 490.85 1113 857 693 540.15 720 631.95 568 579.35 591.75 602 675 535.15 433.25 664.5 649.1 543.8 573.5 846 828 639 703.9 519.13 454 432 543.15 647.096 617 630.3 616.2

4.031E+06 4.752E+06 4.627E+06 6.041E+06 5.925E+06 3.748E+06 3.823E+06 4.727E+06 7.860E+06 3.771E+06 8.192E+06 3.943E+06 2.974E+06 1.569E+06 5.203E+06 3.624E+06 5.117E+06 2.871E+06 5.702E+06 4.080E+06 4.447E+06 1.679E+06 3.037E+06 4.102E+06 3.447E+06 3.212E+06 2.550E+06 2.812E+06 3.410E+06 3.019E+06 1.949E+06 2.120E+06 3.930E+06 4.887E+06 5.750E+06 3.058E+06 2.193E+07 3.528E+06 3.741E+06 3.501E+06

Vapor pressure Ps is calculated by

Ps = exp(C1 + C2/T + C3 ln T + C4T C5) where Ps is in Pa and T is in K. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AIChE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-9

2-61

Vapor Pressures of Inorganic Compounds, up to 1 atm* Compound

Pressure, mmHg 1

Name Aluminum borohydride bromide chloride fluoride iodide oxide Ammonia heavy Ammonium bromide carbamate chloride cyanide hydrogen sulfide iodide Antimony tribromide trichloride pentachloride triiodide trioxide Argon Arsenic Arsenic tribromide trichloride trifluoride pentafluoride trioxide Arsine Barium Beryllium borohydride bromide chloride iodide Bismuth tribromide trichloride Diborane hydrobromide Borine carbonyl triamine Boron hydrides dihydrodecaborane dihydrodiborane dihydropentaborane tetrahydropentaborane tetrahydrotetraborane Boron tribromide trichloride trifluoride Bromine pentafluoride Cadmium chloride fluoride iodide oxide Calcium Carbon (graphite) dioxide disulfide monoxide oxyselenide oxysulfide selenosulfide subsulfide tetrabromide tetrachloride tetrafluoride Cesium bromide chloride fluoride iodide

5

10

20

Al Al(BH4)3 AlBr3 Al2Cl6 AlF3 AlI3 Al2O3 NH3 ND3 NH4Br N2H6CO2 NH4Cl NH4CN NH4HS NH4I Sb SbBr3 SbCl3 SbCl5 SbI3 Sb4O6 A As AsBr3 AsCl3 AsF3 AsF5 As2O3 AsH3 Ba Be(BH4)2 BeBr2 BeCl2 BeI2 Bi BiBr3 BiCl3 B2H5Br BH3CO B3N3H6 B10H14 B2H6 B5H9 B5H11 B4H10 BBr3 BCl3 BF3 Br2 BrF5 Cd CdCl2 CdF2 CdI2 CdO Ca C CO2 CS2 CO COSe COS CSeS C3S2 CBr4 CCl4 CF4 Cs CsBr CsCl CsF CsI

40

60

100

200

400

760

Melting point, °C

1749 −3.9 176.1 152.0 1422 294.5 2665 −68.4 −67.4 320.0 26.7 271.5 −0.5 0.0 331.8 1223 203.5 143.3 114.1 303.5 957 −200.5 518 145.2 70.9 13.2 −84.3 332.5 −98.0 1301 58.6 405 411 411 1271 360 343 −29.0 −95.3 +4.0

1844 +11.2 199.8 161.8 1457 322.0 2766 −57.0 −57.0 345.3 37.2 293.2 +9.6 +10.5 355.8 1288 225.7 165.9

1947 28.1 227.0 171.6 1496 354.0 2874 −45.4 −45.4 370.8 48.0 316.5 20.5 21.8 381.0 1364 250.2 192.2

2056 45.9 256.3 180.2 1537 385.5 2977 −33.6 −33.4 396.0 58.3 337.8 31.7 33.3 404.9 1440 275.0 219.0

660 −64. 97. 192.4 1040

333.8 1085 −195.6 548 167.7 89.2 26.7 −75.5 370.0 −87.2 1403 69.0 427 435 435 1319 392 372 −15.4 −85.5 18.5

368.5 1242 −190.6 579 193.6 109.7 41.4 −64.0 412.2 −75.2 1518 79.7 451 461 461 1370 425 405 0.0 −74.8 34.3

401.0 1425 −185.6 610 220.0 130.4 56.3 −52.8 457.2 −62.1 1638 90.0 474 487 487 1420 461 441 +16.3 −64.0 50.6

142.3 −120.9 +9.6 20.1 −28.1 33.5 −32.4 −123.0 +9.3 −4.5 611 797 1486 640 1341 1207 4373 −100.2 −5.1 −205.7 −61.7 −85.9 28.3 109.9 119.7 23.0 −150.7 509 1072 1069 1025 1055

163.8 −111.2 24.6 34.8 −14.0 50.3 −18.9 −115.9 24.3 +9.9 658 847 1561 688 1409 1288 4516 −93.0 +10.4 −201.3 −49.8 −75.0 45.7 130.8 139.7 38.3 −143.6 561 1140 1139 1092 1124

−99.6 40.8 51.2 +0.8 70.0 −3.6 −108.3 41.0 25.7 711 908 1651 742 1484 1388 4660 −85.7 28.0 −196.3 −35.6 −62.7 65.2

−86.5 58.1 67.0 16.1 91.7 +12.7 −100.7 58.2 40.4 765 967 1751 796 1559 1487 4827 −78.2 46.5 −191.3 −21.9 −49.9 85.6

163.5 57.8 −135.5 624 1221 1217 1170 1200

189.5 76.7 −127.7 690 1300 1300 1251 1280

Temperature, °C

Formula 1284 81.3 100.0 1238 178.0 2148 −109.1

1421 −52.2 103.8 116.4 1298 207.7 2306 −97.5

1487 −42.9 118.0 123.8 1324 225.8 2385 −91.9

1555 −32.5 134.0 131.8 1350 244.2 2465 −85.8

1635 −20.9 150.6 139.9 1378 265.0 2549 −79.2

198.3 −26.1 160.4 −50.6 −51.1 210.9 886 93.9 49.2 22.7 163.6 574 −218.2 372 41.8 −11.4

234.5 −10.4 193.8 −35.7 −36.0 247.0 984 126.0 71.4 48.6 203.8 626 −213.9 416 70.6 +11.7

252.0 −2.9 209.8 −28.6 −28.7 263.5 1033 142.7 85.2 61.8 223.5 666 −210.9 437 85.2 +23.5

270.6 +5.3 226.1 −20.9 −20.8 282.8 1084 158.3 100.6 75.8 244.8 729 −207.9 459 101.3 36.0

−117.9 212.5 −142.6

−108.0 242.6 −130.8 984 19.8 325 328 322 1099 261 242 −75.3 −127.3 −45.0

−103.1 259.7 −124.7 1049 28.1 342 346 341 1136 282 264 −66.3 −121.1 −35.3

−98.0 279.2 −117.7 1120 36.8 361 365 361 1177 305 287 −56.4 −114.1 −25.0

290.0 14.0 245.0 −12.6 −12.3 302.8 1141 177.4 117.8 91.0 267.8 812 −204.9 483 118.7 50.0 −2.5 −92.4 299.2 −110.2 1195 46.2 379 384 382 1217 327 311 −45.4 −106.6 −13.2

1684 −13.4 161.7 145.4 1398 277.8 2599 −74.3 −74.0 303.8 19.6 256.2 −7.4 −7.0 316.0 1176 188.1 128.3 101.0 282.5 873 −202.9 498 130.0 58.7 +4.2 −88.5 310.3 −104.8 1240 51.7 390 395 394 1240 340 324 −38.2 −101.9 −5.8

3586 −134.3 −73.8 −222.0 −117.1 −132.4 −47.3 14.0

80.8 −149.5 −40.4 −29.9 −73.1 −20.4 −75.2 −145.4 −32.8 −51.0 455 618 1231 481 1100 926 3828 −124.4 −54.3 −217.2 −102.3 −119.8 −26.5 41.2

90.2 −144.3 −30.7 −19.9 −64.3 −10.1 −66.9 −141.3 −25.0 −41.9 484 656 1286 512 1149 983 3946 −119.5 −44.7 −215.0 −95.0 −113.3 −16.0 54.9

100.0 −138.5 −20.0 −9.2 −54.8 +1.5 −57.9 −136.4 −16.8 −32.0 516 695 1344 546 1200 1046 4069 −114.4 −34.3 −212.8 −86.3 −106.0 −4.4 69.3

−50.0 −184.6 279 748 744 712 738

−30.0 −174.1 341 838 837 798 828

−19.6 −169.3 375 887 884 844 873

−8.2 −164.3 409 938 934 893 923

117.4 −131.6 −8.0 +2.7 −44.3 14.0 −47.8 −131.0 −8.0 −21.0 553 736 1400 584 1257 1111 4196 −108.6 −22.5 −210.0 −76.4 −98.3 +8.6 85.6 96.3 +4.3 −158.8 449 993 989 947 976

127.8 −127.2 −0.4 10.2 −37.4 22.1 −41.2 −127.6 −0.6 −14.0 578 762 1436 608 1295 1152 4273 −104.8 −15.3 −208.1 −70.2 −93.0 17.0 96.0 106.3 12.3 −155.4 474 1026 1023 980 1009

+1.0 289 291 283 1021 −93.3 −139.2 −63.0 60.0 −159.7 −50.2 −90.9 −41.4 −91.5 −154.6 −48.7 −69.3 394 1112 416 1000

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947).

2050 −77.7 −74.0 520 36 630.5 96.6 73.4 2.8 167 656 −189.2 814 −18 −5.9 −79.8 312.8 −116.3 850 123 490 405 488 271 218 230 −104.2 −137.0 −58.2 99.6 −169 −47.0 −119.9 −45 −107 −126.8 −7.3 −61.4 320.9 568 520 385 851 −57.5 −110.8 −205.0 −138.8 −75.2 +0.4 90.1 −22.6 −183.7 28.5 636 646 683 621

2-62

PHYSICAL AND CHEMICAL DATA

TABLE 2-9

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued) Compound

Pressure, mmHg 1

Name Chlorine fluoride trifluoride monoxide dioxide heptoxide Chlorosulfonic acid Chromium carbonyl oxychloride Cobalt chloride nitrosyl tricarbonyl Columbium fluoride Copper Cuprous bromide chloride iodide Cyanogen bromide chloride fluoride Deuterium cyanide Fluorine oxide Germanium bromide chloride hydride Trichlorogermane Tetramethylgermane Digermane Trigermane Gold Helium para-Hydrogen Hydrogen bromide chloride cyanide fluoride iodide oxide (water) sulfide disulfide selenide telluride Iodine heptafluoride Iron pentacarbonyl Ferric chloride Ferrous chloride Krypton Lead bromide chloride fluoride iodide oxide sulfide Lithium bromide chloride fluoride iodide Magnesium chloride Manganese chloride Mercury Mercuric bromide chloride iodide Molybdenum hexafluoride oxide

5

10

20

Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 HSO3Cl Cr Cr(CO)6 CrO2Cl2 CoCl2 Co(CO)3NO CbF5 Cu Cu2Br2 Cu2Cl2 Cu2I2 C2N2 CNBr CNCl CNF DCN F2 F2O GeBr4 GeCl4 GeH4 GeHCl3 Ge(CH3)4 Ge2H6 Ge3H8 Au He H2 HBr HCl HCN H2F2 HI H2O H2S HSSH H2Se H2Te I2 IF7 Fe Fe(CO)5 Fe2Cl6 FeCl2 Kr Pb PbBr2 PbCl2 PbF2 PbI2 PbO PbS Li LiBr LiCl LiF LiI Mg MgCl2 Mn MnCl2 Hg HgBr2 HgCl2 HgI2 Mo MoF6 MoO3

40

60

100

200

400

760

Melting point, °C

−71.7 −120.8 −34.7 −39.4 −29.4 29.1 105.3 2139 108.0 58.0 843 29.0 148.5 2207 951 960 907 −51.8 22.6 −24.9 −97.0 −17.5 −202.7 −165.8 113.2 27.5 −120.3 26.5 −6.3 −20.3 47.9 2521 −270.3 −257.9 −97.7 −114.0 −17.8 −28.2 −72.1 51.6 −91.6 22.0 −74.2 −45.7 116.5 −31.9 2360 50.3 272.5 842 −171.8 1421 745 784 1080 701 1265 1108 1097 1076 1129 1425 993 909 1142 1792 960 261.7 237.8 237.0 261.8 4109 −8.0 955

−60.2 −114.4 −20.7 −26.5 −17.8 44.6 120.0 2243 121.8 75.2 904 44.4 172.2 2325 1052 1077 1018 −42.6 33.8 −14.1 −89.2 −5.4 −198.3 −159.0 135.4 44.4 −111.2 41.6 +8.8 −4.7 67.0 2657 −269.8 −256.3 −88.1 −105.2 −5.3 −13.2 −60.3 66.5 −82.3 35.3 −65.2 −32.4 137.3 −20.7 2475 68.0 285.0 897 −165.9 1519 796 833 1144 750 1330 1160 1178 1147 1203 1503 1049 967 1223 1900 1028 290.7 262.7 256.5 291.0 4322 +4.1 1014

−47.3 −107.0 −4.9 −12.5 −4.0 62.2 136.1 2361 137.2 95.2 974 62.0 198.0 2465 1189 1249 1158 −33.0 46.0 −2.3 −80.5 +10.0 −193.2 −151.9 161.6 63.8 −100.2 58.3 26.0 +13.3 88.6 2807 −269.3 −254.5 −78.0 −95.3 +10.2 +2.5 −48.3 83.0 −71.8 49.6 −53.6 −17.2 159.8 −8.3 2605 86.1 298.0 961 −159.0 1630 856 893 1219 807 1402 1221 1273 1226 1290 1591 1110 1034 1316 2029 1108 323.0 290.0 275.5 324.2 4553 17.2 1082

−33.8 −100.5 +11.5 +2.2 +11.1 78.8 151.0 2482 151.0 117.1 1050 80.0 225.0 2595 1355 1490 1336 −21.0 61.5 +13.1 −72.6 26.2 −187.9 −144.6 189.0 84.0 −88.9 75.0 44.0 31.5 110.8 2966 −268.6 −252.5 −66.5 −84.8 25.9 19.7 −35.1 100.0 −60.4 64.0 −41.1 −2.0 183.0 +4.0 2735 105.0 319.0 1026 −152.0 1744 914 954 1293 872 1472 1281 1372 1310 1382 1681 1171 1107 1418 2151 1190 357.0 319.0 304.0 354.0 4804 36.0 1151

−100.7 −145 −83 −116 −59 −91 −80 1615

Temperature, °C

Formula −118.0 −98.5

−106.7 −143.4 −80.4 −81.6

−45.3 32.0 1616 36.0 −18.4

−23.8 53.5 1768 58.0 +3.2

1628 572 546 −95.8 −35.7 −76.7 −134.4 −68.9 −223.0 −196.1 −45.0 −163.0 −41.3 −73.2 −88.7 −36.9 1869 −271.7 −263.3 −138.8 −150.8 −71.0 −123.3 −17.3 −134.3 −43.2 −115.3 −96.4 38.7 −87.0 1787 194.0 −199.3 973 513 547 479 943 852 723 748 783 1047 723 621 778 1292 126.2 136.5 136.2 157.5 3102 −65.5 734

1795 666 645 610 −83.2 −18.3 −61.4 −123.8 −54.0 −216.9 −186.6 43.3 −24.9 −151.0 −22.3 −54.6 −69.8 −12.8 2059 −271.5 −261.9 −127.4 −140.7 −55.3 −74.7 −109.6 +1.2 −122.4 −24.4 −103.4 −82.4 62.2 −70.7 1957 −6.5 221.8 −191.3 1099 578 615 861 540 1039 928 838 840 880 1156 802 702 877 1434 736 164.8 165.3 166.0 189.2 3393 −49.0 785

−101.6 −139.0 −71.8 −73.1 −59.0 −13.2 64.0 1845 68.3 13.8

−93.3 −134.3 −62.3 −64.3 −51.2 −2.1 75.3 1928 79.5 25.7

86.3 1879 718 702 656 −76.8 −10.0 −53.8 −118.5 −46.7 −214.1 −182.3 56.8 −15.0 −145.3 −13.0 −45.2 −60.1 −0.9 2154 −271.3 −261.3 −121.8 −135.6 −47.7 −65.8 −102.3 11.2 −116.3 −15.2 −97.9 −75.4 73.2 −63.0 2039 +4.6 235.5 700 −187.2 1162 610 648 904 571 1085 975 881 888 932 1211 841 743 930 1505 778 184.0 179.8 180.2 204.5 3535 −40.8 814

−1.3 103.0 1970 777 766 716 −70.1 −1.0 −46.1 −112.8 −38.8 −211.0 −177.8 71.8 −4.1 −139.2 −3.0 −35.0 −49.9 +11.8 2256 −271.1 −260.4 −115.4 −130.0 −39.7 −56.0 −94.5 22.1 −109.7 −5.1 −91.8 −67.8 84.7 −54.5 2128 16.7 246.0 737 −182.9 1234 646 684 950 605 1134 1005 940 939 987 1270 883 789 988 1583 825 204.6 194.3 195.8 220.0 3690 −32.0 851

−84.5 −128.8 −51.3 −54.3 −42.8 +10.3 87.6 2013 91.2 38.5 770 +11.0 121.5 2067 844 838 786 −62.7 +8.6 −37.5 −106.4 −30.1 −207.7 −173.0 88.1 +8.0 −131.6 +8.8 −23.4 −38.2 26.3 2363 −270.7 −259.6 −108.3 −123.8 −30.9 −45.0 −85.6 34.0 −102.3 +6.0 −84.7 −59.1 97.5 −45.3 2224 30.3 256.8 779 −178.4 1309 686 725 1003 644 1189 1048 1003 994 1045 1333 927 838 1050 1666 879 228.8 211.5 212.5 238.2 3859 −22.1 892

−79.0 −125.3 −44.1 −48.0 −37.2 +18.2 95.2 2067 98.3 46.7 801 18.5 133.2 2127 887 886 836 −57.9 14.7 −32.1 −102.3 −24.7 −205.6 −170.0 98.8 16.2 −126.7 16.2 −16.2 −30.7 35.5 2431 −270.6 −258.9 −103.8 −119.6 −25.1 −37.9 −79.8 41.5 −97.9 12.8 −80.2 −53.7 105.4 −39.4 2283 39.1 263.7 805 −175.7 1358 711 750 1036 668 1222 1074 1042 1028 1081 1372 955 868 1088 1720 913 242.0 221.0 222.2 249.0 3964 −16.2 917

735 −11 75.5 1083 504 422 605 −34.4 58 −6.5 −12 −223 −223.9 26.1 −49.5 −165 −71.1 −88 −109 −105.6 1063 −259.1 −87.0 −114.3 −13.2 −83.7 −50.9 0.0 −85.5 −89.7 −64 −49.0 112.9 5.5 1535 −21 304 −156.7 327.5 373 501 855 402 890 1114 186 547 614 870 446 651 712 1260 650 −38.9 237 277 259 2622 17 795

VAPOR PRESSURES OF PURE SUBSTANCES

TABLE 2-9

2-63

Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued) Compound

Pressure, mmHg 1

Name Neon Nickel carbonyl chloride Nitrogen Nitric oxide Nitrogen dioxide Nitrogen pentoxide Nitrous oxide Nitrosyl chloride fluoride Osmium tetroxide (yellow) (white) Oxygen Ozone Phosgene Phosphorus (yellow) (violet) tribromide trichloride pentachloride Phosphine Phosphonium bromide chloride iodide Phosphorus trioxide pentoxide oxychloride thiobromide thiochloride Platinum Potassium bromide chloride fluoride hydroxide iodide Radon Rhenium heptoxide Rubidium bromide chloride fluoride iodide Selenium dioxide hexafluoride oxychloride tetrachloride Silicon dioxide tetrachloride tetrafluoride Trichlorofluorosilane Iodosilane Diiodosilane Disiloxan Trisilane Trisilazane Tetrasilane Octachlorotrisilane Hexachlorodisiloxane Hexachlorodisilane Tribromosilane Trichlorosilane Trifluorosilane Dibromosilane Difluorosilane Monobromosilane Monochlorosilane Monofluorosilane Tribromofluorosilane Dichlorodifluorosilane Trifluorobromosilane

5

10

20

Ne Ni Ni(CO)4 NiCl2 N2 NO NO2 N2O5 N2O NOCl NOF OsO4 OsO4 O2 O3 COCl2 P P PBr3 PCl3 PCl5 PH3 PH4Br PH4Cl PH4I P4O6 P4O10 POCl3 PSBr3 PSCl3 Pt K KBr KCl KF KOH KI Rn Re2O7 Rb RbBr RbCl RbF RbI Se SeO2 SeF6 SeOCl2 SeCl4 Si SiO2 SiCl4 SiF4 SiFCl3 SiH3I SiH2I2 (SiH3)2O Si3H8 (SiH3)3N Si4H10 Si3Cl3 (SiCl3)2O Si2Cl6 SiHBr3 SiHCl3 SiHF3 SiH2Br2 SiH2F2 SiH3Br SiH3Cl SiH3F SiFBr3 SiF2Cl2 SiF3Br

40

60

100

200

400

760

Melting point, °C

−251.0 2364 −6.0 866 −209.7 −166.0 −14.7 7.4 −110.3 −46.3 −88.8 71.5 71.5 −198.8 −141.0 −35.6 197.3 349 103.6 21.0 117.0 −118.8 7.4 −52.0 29.3 108.3 510 47.4 126.3 63.8 3714 586 1137 1164 1245 1064 1080 −99.0 289.0 514 1114 1133 1168 1072 554 258.0 −73.9 118.0 147.5 2083 1969 +5.4 −113.3 −33.2 −4.4 79.4 −55.9 +1.6 −1.1 47.4 146.0 75.4 85.4 51.6 −16.4 −118.7 14.1 −107.3 −42.3 −68.5 −122.4 28.6 −70.3

−249.7 2473 +8.8 904 −205.6 −162.3 −5.0 15.6 −103.6 −34.0 −79.2 89.5 89.5 −194.0 −132.6 −22.3 222.7 370 125.2 37.6 131.3 −109.4 17.6 −44.0 39.9 129.0 532 65.0 141.8 82.0 3923 643 1212 1239 1323 1142 1152 −87.7 307.0 563 1186 1207 1239 1141 594 277.0 −64.8 134.6 161.0 2151 2053 21.0 −170.2 −19.3 +10.7 101.8 −43.5 17.8 +14.0 63.6 166.2 92.5 102.2 70.2 −1.8 −111.3 31.6 −98.3 −28.6 −57.0 −115.2 45.7 −58.0 −69.8

−248.1 2603 25.8 945 −200.9 −156.8 +8.0 24.4 −96.2 −20.3 −68.2 109.3 109.3 −188.8 −122.5 −7.6 251.0 391 149.7 56.9 147.2 −98.3 28.0 −35.4 51.6 150.3 556 84.3 157.8 102.3 4169 708 1297 1322 1411 1233 1238 −75.0 336.0 620 1267 1294 1322 1223 637 297.7 −55.2 151.7 176.4 2220 2141 38.4 −100.7 −4.0 27.9 125.5 −29.3 35.5 31.0 81.7 189.5 113.6 120.6 90.2 +14.5 −102.8 50.7 −87.6 −13.3 −44.5 −106.8 64.6 −45.0 −55.9

−246.0 2732 42.5 987 −195.8 −151.7 21.0 32.4 −85.5 −6.4 −56.0 130.0 130.0 −183.1 −111.1 +8.3 280.0 417 175.3 74.2 162.0 −87.5 38.3 −27.0 62.3 173.1 591 105.1 175.0 124.0 4407 774 1383 1407 1502 1327 1324 −61.8 362.4 679 1352 1381 1408 1304 680 317.0 −45.8 168.0 191.5 2287 2227 56.8 −94.8 +12.2 45.4 149.5 −15.4 53.1 48.7 100.0 211.4 135.6 139.0 111.8 31.8 −95.0 70.5 −77.8 +2.4 −30.4 −98.0 83.8 −31.8 −41.7

−248.7 1452 −25 1001 −210.0 −161 −9.3 30 −90.9 −64.5 −134 56 42 −218.7 −251 −104 44.1 590 −40 −111.8

Temperature, °C

Formula −257.3 1810

−255.5 1979

−254.6 2057

−253.7 2143

671 −226.1 −184.5 −55.6 −36.8 −143.4

731 −221.3 −180.6 −42.7 −23.0 −133.4

759 −219.1 −178.2 −36.7 −16.7 −128.7

789 −216.8 −175.3 −30.4 −10.0 −124.0

−132.0 3.2 −5.6 −219.1 −180.4 −92.9 76.6 237 7.8 −51.6 55.5

−120.3 22.0 +15.6 −213.4 −168.6 −77.0 111.2 271 34.4 −31.5 74.0

−114.3 31.3 26.0 −210.6 −163.2 −69.3 128.0 287 47.8 −21.3 83.2

−107.8 41.0 37.4 −207.5 −157.2 −60.3 146.2 306 62.4 −10.2 92.5

−43.7 −91.0 −25.2 384

−28.5 −79.6 −9.0 39.7 424

50.0 −18.3 2730 341 795 821 885 719 745 −144.2 212.5 297 781 792 921 748 356 157.0 −118.6 34.8 74.0 1724

72.4 +4.6 3007 408 892 919 988 814 840 −132.4 237.5 358 876 887 982 839 413 187.7 −105.2 59.8 96.3 1835

−63.4 −144.0 −92.6

−44.1 −134.8 −76.4 −53.0 3.8 −95.8 −49.7 −49.9 −6.2 74.7 17.8 27.4 −8.0 −62.6 −142.7 −40.0 −136.0 −85.7 −104.3 −145.5 −25.4 −110.5

−21.2 −74.0 −1.1 53.0 442 2.0 83.6 16.1 3146 443 940 968 1039 863 887 −126.3 248.0 389 923 937 1016 884 442 202.5 −98.9 71.9 107.4 1888 1732 −34.4 −130.4 −68.3 −47.7 18.0 −88.2 −40.0 −40.4 +4.3 89.3 29.4 38.8 +3.4 −53.4 −138.2 −29.4 −130.4 −77.3 −97.7 −141.2 −15.1 −102.9

−13.3 −68.0 +7.3 67.8 462 13.6 95.5 29.0 3302 483 994 1020 1096 918 938 −119.2 261.0 422 975 990 1052 935 473 217.5 −92.3 84.2 118.1 1942 1798 −24.0 −125.9 −59.0 −33.4 34.1 −79.8 −29.0 −30.0 15.8 104.2 41.5 51.5 16.0 −43.8 −132.9 −18.0 −124.3 −68.3 −90.1 −136.3 −3.7 −94.5

−112.5 −68.9 −68.7 −27.7 46.3 −5.0 +4.0 −30.5 −80.7 −152.0 −60.9 −146.7 −117.8 −153.0 −46.1 −124.7

−252.6 2234 −23.0 821 −214.0 −171.7 −23.9 −2.9 −118.3 −60.2 −100.3 51.7 50.5 −204.1 −150.7 −50.3 166.7 323 79.0 +2.3 102.5 −129.4 −5.0 −61.5 16.1 84.0 481 27.3 108.0 42.7 3469 524 1050 1078 1156 976 995 −111.3 272.0 459 1031 1047 1096 991 506 234.1 −84.7 98.0 130.1 2000 1867 −12.1 −120.8 −48.8 −21.8 52.6 −70.4 −16.9 −18.5 28.4 121.5 55.2 65.3 30.0 −32.9 −127.3 −5.2 −117.6 −57.8 −81.8 −130.8 +9.2 −85.0

−251.9 2289 −15.9 840 −212.3 −168.9 −19.9 +1.8 −114.9 −54.2 −95.7 59.4 59.4 −201.9 −146.7 −44.0 179.8 334 89.8 10.2 108.3 −125.0 +0.3 −57.3 21.9 94.2 493 35.8 116.0 51.8 3574 550 1087 1115 1193 1013 1030 −106.2 280.0 482 1066 1084 1123 1026 527 244.6 −80.0 106.5 137.8 2036 1911 −4.8 −117.5 −42.2 −14.3 64.0 −64.2 −9.0 −11.0 36.6 132.0 63.8 73.9 39.2 −25.8 −123.7 +3.2 −113.3 −51.1 −76.0 −127.2 17.4 −78.6

−132.5 −28.5 22.5 569 2 38 −36.2 1755 62.3 730 790 880 380 723 −71 296 38.5 682 715 760 642 217 340 −34.7 8.5 1420 1710 −68.8 −90 −120.8 −57.0 −1.0 −144.2 −117.2 −105.7 −93.6 −33.2 −1.2 −73.5 −126.6 −131.4 −70.2 −93.9 −82.5 −139.7 −70.5

2-64

PHYSICAL AND CHEMICAL DATA

TABLE 2-9

Vapor Pressures of Inorganic Compounds, up to 1 atm (Concluded) Compound

Pressure, mmHg 1

Name Trifluorochlorosilane Hexafluorodisilane Dichlorofluorobromosilane Dibromochlorofluorosilane Silane Disilane Silver chloride iodide Sodium bromide chloride cyanide fluoride hydroxide iodide Strontium Strontium oxide Sulfur monochloride hexafluoride Sulfuryl chloride Sulfur dioxide trioxide (α) trioxide (β) trioxide (γ) Tellurium chloride fluoride Thallium Thallous bromide chloride iodide Thionyl bromide Thionyl chloride Tin Stannic bromide Stannous chloride Stannic chloride iodide hydride Tin tetramethyl trimethyl-ethyl trimethyl-propyl Titanium chloride Tungsten Tungsten hexafluoride Uranium hexafluoride Vanadyl trichloride Xenon Zinc chloride fluoride diethyl Zirconium bromide chloride iodide

5

10

20

SiF3Cl Si2F6 SiFCl2Br SiFClBr2 SiH4 Si2H6 Ag AgCl AgI Na NaBr NaCl NaCN NaF NaOH NaI Sr SrO S S2Cl2 SF6 SO2Cl2 SO2 SO3 SO3 SO3 Te TeCl4 TeF6 Tl TlBr TlCl TlI SOBr2 SOCl2 Sn SnBr4 SnCl2 SnCl4 SnI4 SnH4 Sn(CH3)4 Sn(CH3)3·C2H5 Sn(CH3)3·C3H7 TiCl4 W WF6 UF6 VOCl3 Xe Zn ZnCl2 ZnF2 Zn(C2H5)2 ZrBr4 ZrCl4 ZrI4

40

100

200

400

760

Melting point, °C

−108.2 −46.7 −29.0 −4.7 −146.3 −66.4 1795 1242 1152 662 1099 1169 1156 1403 1057 1039 1057

−101.7 −41.7 −19.5 +6.3 −140.5 −57.5 1865 1297 1210 701 1148 1220 1214 1455 1111 1083 1111

−91.7 −34.2 −3.2 23.0 −131.6 −44.6 1971 1379 1297 758 1220 1296 1302 1531 1192 1150 1192

−81.0 −26.4 +15.4 43.0 −122.0 −29.0 2090 1467 1400 823 1304 1379 1401 1617 1286 1225 1285

−70.0 −18.9 35.4 59.5 −111.5 −14.3 2212 1564 1506 892 1392 1465 1497 1704 1378 1304 1384

305.5 63.2 −96.8 +7.2 −54.6 +4.0 8.0 21.4 789 287 −73.8 1143 621 612 631 68.3 10.4 1903 116.2 467 43.5 234.2 −96.6 11.7 38.4 57.5 58.0 5007 −27.5 10.4 49.8 −137.7 700 584 1207 47.2 289 268 355

327.2 75.3 −90.9 17.8 −46.9 10.5 14.3 28.0 838 304 −67.9 1196 653 645 663 80.6 21.4 1968 131.0 493 54.7 254.2 −89.2 22.8 50.0 69.8 71.0 5168 −20.3 18.2 62.5 −132.8 736 610 1254 59.1 301 279 369

359.7 93.5 −82.3 33.7 −35.4 20.5 23.7 35.8 910 330 −57.3 1274 703 694 712 99.0 37.9 2063 152.8 533 72.0 283.5 −78.0 39.8 67.3 88.0 90.5 5403 −10.0 30.0 82.0 −125.4 788 648 1329 77.0 318 295 389

399.6 115.4 −72.6 51.3 −23.0 32.6 32.6 44.0 997 360 −48.2 1364 759 748 763 119.2 56.5 2169 177.7 577 92.1 315.5 −65.2 58.5 87.6 109.6 112.7 5666 +1.2 42.7 103.5 −117.1 844 689 1417 97.3 337 312 409

444.6 138.0 −63.5 69.2 −10.0 44.8 44.8 51.6 1087 392 −38.6 1457 819 807 823 139.5 75.4 2270 204.7 623 113.0 348.0 −52.3 78.0 108.8 131.7 136.0 5927 17.3 55.7 127.2 −108.0 907 732 1497 118.0 357 331 431

−142 −18.6 −112.3 −99.3 −185 −132.6 960.5 455 552 97.5 755 800 564 992 318 651 800 2430 112.8 −80 −50.2 −54.1 −73.2 16.8 32.3 62.1 452 224 −37.8 3035 460 430 440 −52.2 −104.5 231.9 31.0 246.8 −30.2 144.5 −149.9

60

Temperature, °C

Formula −144.0 −81.0 −86.5 −65.2 −179.3 −114.8 1357 912 820 439 806 865 817 1077 739 767 2068 183.8 −7.4 −132.7 −95.5 −39.0 −34.0 −15.3 520 −111.3 825 440 −6.7 −52.9 1492 316 −22.7 −140.0 −51.3 −30.0 −12.0 −13.9 3990 −71.4 −38.8 −23.2 −168.5 487 428 970 −22.4 207 190 264

−133.0 −68.8 −68.4 −45.5 −168.6 −99.3 1500 1019 927 511 903 967 928 1186 843 857 847 2198 223.0 +15.7 −120.6 −35.1 −83.0 −23.7 −19.2 −2.0 605 −98.8 931 490 487 502 +18.4 −32.4 1634 58.3 366 −1.0 156.0 −125.8 −31.0 −7.6 +10.7 +9.4 4337 −56.5 −22.0 +0.2 −158.2 558 481 1055 0.0 237 217 297

−127.0 −63.1 −59.0 −35.6 −163.0 −91.4 1575 1074 983 549 952 1017 983 1240 897 903 898 2262 243.8 27.5 −114.7 −24.8 −76.8 −16.5 −12.3 +4.3 650 233 −92.4 983 522 517 531 31.0 −21.9 1703 72.7 391 +10.0 175.8 −118.5 −20.6 +3.8 21.8 21.3 4507 −49.2 −13.8 12.2 −152.8 593 508 1086 +11.7 250 230 311

−120.5 −57.0 −48.8 −24.5 −156.9 −82.7 1658 1134 1045 589 1005 1072 1046 1300 953 952 953 2333 264.7 40.0 −108.4 −13.4 −69.7 −9.1 −4.9 11.1 697 253 −86.0 1040 559 550 567 44.1 −10.5 1777 88.1 420 22.0 196.2 −111.2 −9.3 16.1 34.0 34.2 4690 −41.5 −5.2 26.6 −147.1 632 536 1129 24.2 266 243 329

−112.8 −50.6 −37.0 −12.0 −150.3 −72.8 1743 1200 1111 633 1063 1131 1115 1363 1017 1005 1018 2410 288.3 54.1 −101.5 −1.0 −60.5 −1.0 +3.2 17.9 753 273 −78.4 1103 598 589 607 58.8 +2.2 1855 105.5 450 35.2 218.8 −102.3 +3.5 30.0 48.5 48.4 4886 −33.0 +4.4 40.0 −141.2 673 566 1175 38.0 281 259 344

−30 3370 −0.5 69.2 −111.6 419.4 365 872 −28 450 437 499

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-65

Vapor Pressures of Organic Compounds, up to 1 atm* Pressure, mmHg Compound

1

Name

Formula

Acenaphthalene Acetal Acetaldehyde Acetamide Acetanilide Acetic acid anhydride Acetone Acetonitrile Acetophenone Acetyl chloride Acetylene Acridine Acrolein (2-propenal) Acrylic acid Adipic acid Allene (propadiene) Allyl alcohol (propen-1-ol-3) chloride (3-chloropropene) isopropyl ether isothiocyanate n-propyl ether 4-Allylveratrole iso-Amyl acetate n-Amyl alcohol iso-Amyl alcohol sec-Amyl alcohol (2-pentanol) tert-Amyl alcohol sec-Amylbenzene iso-Amyl benzoate bromide (1-bromo-3-methylbutane) n-butyrate formate iodide (1-iodo-3-methylbutane) isobutyrate Amyl isopropionate iso-Amyl isovalerate n-Amyl levulinate iso-Amyl levulinate nitrate 4-tert-Amylphenol Anethole Angelonitrile Aniline 2-Anilinoethanol Anisaldehyde o-Anisidine (2-methoxyaniline) Anthracene Anthraquinone Azelaic acid Azelaldehyde Azobenzene Benzal chloride (α,α-Dichlorotoluene) Benzaldehyde Benzanthrone Benzene Benzenesulfonylchloride Benzil Benzoic acid anhydride Benzoin Benzonitrile Benzophenone Benzotrichloride (α,α,α-Trichlorotoluene) Benzotrifluoride (α,α,α-Trifluorotoluene) Benzoyl bromide chloride nitrile Benzyl acetate alcohol

C12H10 C6H14O2 C2H4O C2H5NO C8H9NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H3OCl C2H2 C13H9N C3H4O C3H4O2 C6H10O4 C3H4 C3H6O C3H5Cl C6H12O C4H5NS C6H12O C11H14O2 C7H14O2 C5H12O C5H12O C5H12O C5H12O C11H16 C12H16O2 C5H11Br C9H18O2 C6H12O2 C5H11I C9H18O2 C8H16O2 C10H20O2 C10H18O3 C10H18O3 C5H11NO3 C11H16O C10H12O C5H7N C6H7N C8H11NO C8H8O2 C7H9NO C14H10 C14H8O2 C9H16O4 C9H18O C12H10N2 C7H6Cl2 C7H6O C17H10O C6H6 C6H5ClO2S C14H10O2 C7H6O2 C14H10O3 C14H12O2 C7H5N C13H10O C7H5Cl3 C7H5F3 C7H5BrO C7H5ClO C8H5NO C9H10O2 C7H8O

5

10

20

114.8 −2.3 −65.1 92.0 146.6 +6.3 24.8 −40.5 −26.6 64.0 −35.0 −133.0 165.8 −46.0 27.3 191.0 −108.0 +0.2 −52.0 −23.1 +25.3 −18.2 113.9 +23.7 34.7 30.9 22.1 +7.2 55.8 104.5 +2.1 47.1 +5.4 +21.9 40.1 33.7 54.4 110.0 104.0 28.8 109.8 91.6 +15.0 57.9 134.3 102.6 88.0 173.5 219.4 210.4 58.4 135.7 64.0 50.1 274.5 −19.6 96.5 165.2 119.5 180.0 170.2 55.3 141.7 73.7 −10.3 75.4 59.1 71.7 73.4 80.8

131.2 +8.0 −56.8 105.0 162.0 17.5 36.0 −31.1 −16.3 78.0 −27.6 −128.2 184.0 −36.7 39.0 205.5 −101.0 10.5 −42.9 −12.9 38.3 −7.9 127.0 35.2 44.9 40.8 32.2 17.2 69.2 121.6 13.6 59.9 17.1 34.1 52.8 46.3 68.6 124.0 118.8 40.3 125.5 106.0 28.0 69.4 149.6 117.8 101.7 187.2 234.2 225.5 71.6 151.5 78.7 62.0 297.2 −11.5 112.0 183.0 132.1 198.0 188.1 69.2 157.6 87.6 −0.4 89.8 73.0 85.5 87.6 92.6

148.7 19.6 −47.8 120.0 180.0 29.9 48.3 −20.8 −5.0 92.4 −19.6 −122.8 203.5 −26.3 52.0 222.0 −93.4 21.7 −32.8 −1.8 52.1 +3.7 142.8 47.8 55.8 51.7 42.6 27.9 83.8 139.7 26.1 74.0 30.0 47.6 66.6 60.0 83.8 139.7 134.4 53.5 142.3 121.8 41.0 82.0 165.7 133.5 116.1 201.9 248.3 242.4 85.0 168.3 94.3 75.0 322.5 −2.6 129.0 202.8 146.7 218.0 207.0 83.4 175.8 102.7 12.2 105.4 87.6 100.2 102.3 105.8

40

60

100

200

400

760

197.5 50.1 −22.6 158.0 227.2 63.0 82.2 +7.7 27.0 133.6 +3.2 −107.9 256.0 +2.5 86.1 265.0 −72.5 50.0 −4.5 29.0 89.5 35.8 183.7 83.2 85.8 80.7 70.7 55.3 124.1 186.8 60.4 113.1 65.4 84.4 104.4 97.6 125.1 180.5 177.0 88.6 189.0 164.2 77.5 119.9 209.5 176.7 155.2 250.0 285.0 286.5 123.0 216.0 138.3 112.5 390.0 26.1 174.5 255.8 186.2 270.4 258.0 123.5 224.4 144.3 45.3 147.7 128.0 141.0 144.0 141.7

222.1 66.3 −10.0 178.3 250.5 80.0 100.0 22.7 43.7 154.2 16.1 −100.3 284.0 17.5 103.3 287.8 −61.3 64.5 10.4 44.3 108.0 52.6 204.0 101.3 102.0 95.8 85.7 69.7 145.2 210.2 78.7 133.2 83.2 103.8 124.2 117.3 146.1 203.1 198.1 106.7 213.0 186.1 96.3 140.1 230.6 199.0 175.3 279.0 314.6 309.6 142.1 240.0 160.7 131.7 426.5 42.2 198.0 283.5 205.8 299.1 284.4 144.1 249.8 165.6 62.5 169.2 149.5 161.3 165.5 160.0

250.0 84.0 +4.9 200.0 277.0 99.0 119.8 39.5 62.5 178.0 32.0 −92.0 314.3 34.5 122.0 312.5 −48.5 80.2 27.5 61.7 129.8 71.4 226.2 121.5 119.8 113.7 102.3 85.7 168.0 235.8 99.4 155.3 102.7 125.8 146.0 138.4 169.5 227.4 222.7 126.5 239.5 210.5 117.7 161.9 254.5 223.0 197.3 310.2 346.2 332.8 163.4 266.1 187.0 154.1

277.5 102.2 20.2 222.0 303.8 118.1 139.6 56.5 81.8 202.4 50.8 −84.0 346.0 52.5 141.0 337.5 −35.0 96.6 44.6 79.5 150.7 90.5 248.0 142.0 137.8 130.6 119.7 101.7 193.0 262.0 120.4 178.6 123.3 148.2 168.8 160.2 194.0 253.2 247.9 147.5 266.0 235.3 140.0 184.4 279.6 248.0 218.5 342.0 379.9 356.5 185.0 293.0 214.0 179.0

60.6 224.0 314.3 227.0 328.8 313.5 166.7 276.8 189.2 82.0 193.7 172.8 185.0 189.0 183.0

80.1 251.5 347.0 249.2 360.0 343.0 190.6 305.4 213.5 102.2 218.5 197.2 208.0 213.5 204.7

Temperature, °C −23.0 −81.5 65.0 114.0 −17.2 1.7 −59.4 −47.0 37.1 −50.0 −142.9 129.4 −64.5 +3.5 159.5 −120.6 −20.0 −70.0 −43.7 −2.0 −39.0 85.0 0.0 +13.6 +10.0 +1.5 −12.9 29.0 72.0 −20.4 21.2 −17.5 −2.5 14.8 +8.5 27.0 81.3 75.6 +5.2 62.6 −8.0 34.8 104.0 73.2 61.0 145.0 190.0 178.3 33.3 103.5 35.4 26.2 225.0 −36.7 65.9 128.4 96.0 143.8 135.6 28.2 108.2 45.8 −32.0 47.0 32.1 44.5 45.0 58.0

168.2 31.9 −37.8 135.8 199.6 43.0 62.1 −9.4 +7.7 109.4 −10.4 −116.7 224.2 −15.0 66.2 240.5 −85.2 33.4 −21.2 +10.9 67.4 16.4 158.3 62.1 68.0 63.4 54.1 38.8 100.0 158.3 39.8 90.0 44.0 62.3 81.8 75.5 100.6 155.8 151.7 67.6 160.3 139.3 55.8 96.7 183.7 150.5 132.0 217.5 264.3 260.0 100.2 187.9 112.1 90.1 350.0 +7.6 147.7 224.5 162.6 239.8 227.6 99.6 195.7 119.8 25.7 122.6 103.8 116.6 119.6 119.8

181.2 39.8 −31.4 145.8 211.8 51.7 70.8 −2.0 15.9 119.8 −4.5 −112.8 238.7 −7.5 75.0 251.0 −78.8 40.3 −14.1 18.7 76.2 25.0 169.6 71.0 75.5 71.0 61.5 46.0 110.4 171.4 48.7 99.8 53.3 71.9 91.7 85.2 110.3 165.2 162.6 76.3 172.6 149.8 65.2 106.0 194.0 161.7 142.1 231.8 273.3 271.8 110.0 199.8 123.4 99.6 368.8 15.4 158.2 238.2 172.8 252.7 241.7 109.8 208.2 130.0 34.0 133.4 114.7 127.0 129.8 129.3

Melting point, °C 95 −123.5 81 113.5 16.7 −73 −94.6 −41 20.5 −112.0 −81.5 110.5 −87.7 14 152 −136 −129 −136.4 −80

−117.2 −11.9

93 22.5 −6.2 2.5 5.2 217.5 286 106.5 68 −16.1 −26 174 +5.5 14.5 95 121.7 42 132 −12.9 48.5 −21.2 −29.3 0 −0.5 33.5 −51.5 −15.3

*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947). For information on fuels see Hibbard, N.A.C.A. Research Mem. E56I21, 1956. For methane see Johnson (ed.), WADD-TR-60-56, 1960.

2-66

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Benzylamine Benzyl bromide (α-bromotoluene) chloride (α-chlorotoluene) cinnamate Benzyldichlorosilane Benzyl ethyl ether phenyl ether isothiocyanate Biphenyl 1-Biphenyloxy-2,3-epoxypropane d-Bornyl acetate Bornyl n-butyrate formate isobutyrate propionate Brassidic acid Bromoacetic acid 4-Bromoanisole Bromobenzene 4-Bromobiphenyl 1-Bromo-2-butanol 1-Bromo-2-butanone cis-1-Bromo-1-butene trans-1-Bromo-1-butene 2-Bromo-1-butene cis-2-Bromo-2-butene trans-2-Bromo-2-butene 1,4-Bromochlorobenzene 1-Bromo-1-chloroethane 1-Bromo-2-chloroethane 2-Bromo-4,6-dichlorophenol 1-Bromo-4-ethyl benzene (2-Bromoethyl)-benzene 2-Bromoethyl 2-chloroethyl ether (2-Bromoethyl)-cyclohexane 1-Bromoethylene Bromoform (tribromomethane) 1-Bromonaphthalene 2-Bromo-4-phenylphenol 3-Bromopyridine 2-Bromotoluene 3-Bromotoluene 4-Bromotoluene 3-Bromo-2,4,6-trichlorophenol 2-Bromo-1,4-xylene 1,2-Butadiene (methyl allene) 1,3-Butadiene n-Butane iso-Butane (2-methylpropane) 1,3-Butanediol 1,2,3-Butanetriol 1-Butene cis-2-Butene trans-2-Butene 3-Butenenitrile iso-Butyl acetate n-Butyl acrylate alcohol iso-Butyl alcohol sec-Butyl alcohol tert-Butyl alcohol iso-Butyl amine n-Butylbenzene iso-Butylbenzene sec-Butylbenzene tert-Butylbenzene iso-Butyl benzoate n-Butyl bromide (1-bromobutane) iso-Butyl n-butyrate carbamate Butyl carbitol (diethylene glycol butyl ether) n-Butyl chloride (1-chlorobutane) iso-Butyl chloride

1

5

10

20

29.0 32.2 22.0 173.8 45.3 26.0 95.4 79.5 70.6 135.3 46.9 74.0 47.0 70.0 64.6 209.6 54.7 48.8 +2.9 98.0 23.7 +6.2 −44.0 −38.4 −47.3 −39.0 −45.0 32.0 −36.0 −28.8 84.0 30.4 48.0 36.5 38.7 −95.4

70.0

54.8 59.6 47.8 206.3 70.2 52.0 127.7 107.8 101.8 169.9 75.7 103.4 74.8 99.8 93.7 241.7 81.6 77.8 27.8 133.7 45.4 30.0 −23.2 −17.0 −27.0 −17.9 −24.1 59.5 −18.0 −7.0 115.6 42.5 76.2 63.2 66.6 −77.8 22.0 117.5 135.4 42.0 49.7 50.8 47.5 146.2 65.0 −72.7 −87.6 −85.7 −94.1 67.5 132.0 −89.4 −81.1 −84.0 +2.9 +1.4 +23.5 +20.0 +11.6 +7.2 −3.0 −31.0 48.8 40.5 44.2 39.0 93.6 −11.2 30.0 83.7 95.7

67.7 73.4 60.8 221.5 83.2 65.0 144.0 121.8 117.0 187.2 90.2 118.0 89.3 114.0 108.0 256.0 94.1 91.9 40.0 150.6 55.8 41.8 −12.8 −6.4 −16.8 −7.2 −13.8 72.7 −9.4 +4.1 130.8 74.0 90.5 76.3 80.5 −68.8 34.0 133.6 152.3 55.2 62.3 64.0 61.1 163.2 78.8 −64.2 −79.7 −77.8 −86.4 85.3 146.0 −81.6 −73.4 −76.3 14.1 12.8 35.5 30.2 21.7 16.9 +5.5 −21.0 62.0 53.7 57.0 51.7 108.6 −0.3 42.2 96.4 107.8

81.8 88.3 75.0 239.3 96.7 79.6 160.7 137.0 134.2 205.8 106.0 133.8 104.0 130.0 123.7 272.9 108.2 107.8 53.8 169.8 67.2 54.2 −1.4 +5.4 −5.3 +4.6 −2.4 87.8 0.0 16.0 147.7 90.2 105.8 90.8 95.8 −58.8 48.0 150.2 171.8 69.1 76.0 78.1 75.2 181.8 94.0 −54.9 −71.0 −68.9 −77.9 100.0 161.0 −73.0 −64.6 −67.5 26.6 25.5 48.6 41.5 32.4 27.3 14.3 −10.3 76.3 67.8 70.6 65.6 124.2 +11.6 56.1 110.1 120.5

97.3 104.8 90.7 255.8 111.8 95.4 180.1 153.0 152.5 226.3 123.7 150.7 121.2 147.2 140.4 290.0 124.0 125.0 68.6 190.8 79.5 68.2 +11.5 18.4 +7.2 17.7 +10.5 103.8 +10.4 29.7 165.8 108.5 123.2 106.6 113.0 −48.1 63.6 170.2 193.8 84.1 91.0 93.9 91.8 200.5 110.6 −44.3 −61.3 −59.1 −68.4 117.4 178.0 −63.4 −54.7 −57.6 40.0 39.2 63.4 53.4 44.1 38.1 24.5 +1.3 92.4 83.3 86.2 80.8 141.8 24.8 71.7 125.3 135.5

C4H9Cl C4H9Cl

−49.0 −53.8

−28.9 −34.3

−18.6 −24.5

−7.4 −13.8

+5.0 −1.9

60

100

200

400

760

107.3 115.6 100.5 267.0 121.3 105.5 192.6 163.8 165.2 239.7 135.7 161.8 131.7 157.6 151.2 301.5 133.8 136.0 78.1 204.5 87.0 77.3 19.8 27.2 15.4 26.2 18.7 114.8 17.0 38.0 177.6 121.0 133.8 116.4 123.7 −41.2 73.4 183.5 207.0 94.1 100.0 104.1 102.3 213.0 121.6 −37.5 −55.1 −52.8 −62.4 127.5 188.0 −57.2 −48.4 −51.3 48.8 48.0 72.6 60.3 51.7 45.2 31.0 8.8 102.6 93.3 96.0 90.6 152.0 33.4 81.3 134.6 146.0

120.0 129.8 114.2 281.5 133.5 118.9 209.2 177.7 180.7 255.0 149.8 176.4 145.8 172.2 165.7 316.2 146.3 150.1 90.8 221.8 97.6 89.2 30.8 38.1 26.3 37.5 29.9 128.0 28.0 49.5 193.2 135.5 148.2 129.8 138.0 −31.9 85.9 198.8 224.5 107.8 112.0 117.8 116.4 229.3 135.7 −28.3 −46.8 −44.2 −54.1 141.2 202.5 −48.9 −39.8 −42.7 60.2 59.7 85.1 70.1 61.5 54.1 39.8 18.8 116.2 107.0 109.5 103.8 166.4 44.7 94.0 147.2 159.8

140.0 150.8 134.0 303.8 152.0 139.6 233.2 198.0 204.2 280.4 172.0 198.0 166.4 194.2 187.5 336.8 165.8 172.7 110.1 248.2 112.1 107.0 47.8 55.7 42.8 54.5 46.5 149.5 44.7 66.8 216.5 156.5 169.8 150.0 160.0 −17.2 106.1 224.2 251.0 127.7 133.6 138.0 137.4 253.0 156.4 −14.2 −33.9 −31.2 −41.5 161.0 222.0 −36.2 −26.8 −29.7 78.0 77.6 104.0 84.3 75.9 67.9 52.7 32.0 136.9 127.2 128.8 123.7 188.2 62.0 113.9 165.7 181.2

161.3 175.2 155.8 326.7 173.0 161.5 259.8 220.4 229.4 309.8 197.5 222.2 190.2 218.2 211.2 359.6 186.7 197.5 132.3 277.7 128.3 126.3 66.8 75.0 61.9 74.0 66.0 172.6 63.4 86.0 242.0 182.0 194.0 172.3 186.2 −1.1 127.9 252.0 280.2 150.0 157.3 160.0 160.2 278.0 181.0 +1.8 −19.3 −16.3 −27.1 183.8 243.5 −21.7 −12.0 −14.8 98.0 97.5 125.2 100.8 91.4 83.9 68.0 50.7 159.2 149.6 150.3 145.8 212.8 81.7 135.7 186.0 205.0

184.5 198.5 179.4 350.0 194.3 185.0 287.0 243.0 254.9 340.0 223.0 247.0 214.0 243.0 235.0 382.5 208.0 223.0 156.2 310.0 145.0 147.0 86.2 94.7 81.0 93.9 85.5 196.9 82.7 106.7 268.0 206.0 219.0 195.8 213.0 +15.8 150.5 281.1 311.0 173.4 181.8 183.7 184.5 305.8 206.7 18.5 −4.5 −0.5 −11.7 206.5 264.0 −6.3 +3.7 +0.9 119.0 118.0 147.4 117.5 108.0 99.5 82.9 68.6 183.1 172.8 173.5 168.5 237.0 101.6 156.9 206.5 231.2

13.0 +5.9

24.0 16.0

40.0 32.0

58.8 50.0

77.8 68.9

Temperature, °C

Formula C7H9N C7H7Br C7H7Cl C16H14O2 C7H8Cl2Si C9H12O C13H12O C8H7NS C12H10 C15H14O2 C12H20O2 C14H24O2 C11H18O2 C14H24O2 C13H22O2 C22H42O2 C2H3BrO2 C7H7BrO C6H5Br C12H9Br C4H9BrO C4H7BrO C4H7Br C4H7Br C4H7Br C4H7Br C4H7Br C6H4BrCl C2H4BrCl C2H4BrCl C6H3BrCl2O C8H9Br C8H9Br C4H8BrClO C8H15Br C2H3Br CHBr3 C10H7Br C12H9BrO C5H4BrN C7H7Br C7H7Br C7H7Br C6H2BrCl3O C8H9Br C4H6 C4H6 C4H10 C4H10 C4H10O2 C4H10O3 C4H8 C4H8 C4H8 C4H5N C6H12O2 C7H12O2 C4H10O C4H10O C4H10O C4H10O C4H11N C10H14 C10H14 C10H14 C10H14 C11H14O2 C4H9Br C8H16O2 C5H11NO2 C8H18O3

40

84.2 100.0 16.8 24.4 14.8 10.3 112.4 37.5 −89.0 −102.8 −101.5 −109.2 22.2 102.0 −104.8 −96.4 −99.4 −19.6 −21.2 −0.5 −1.2 −9.0 −12.2 −20.4 −50.0 22.7 14.1 18.6 13.0 64.0 −33.0 +4.6

Melting point, °C −4 −39 39

69.5 29

61.5 49.5 12.5 −30.7 90.5

−100.3 −133.4 −111.2 −114.6 16.6 −16.6 68 −45.0

−138 8.5 5.5 95 −28 39.8 28.5 +9.5 −108.9 −135 −145 77 −130 −138.9 −105.4 −98.9 −64.6 −79.9 −108 −114.7 25.3 −85.0 −88.0 −51.5 −75.5 −58 −112.4 65 −123.1 −131.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-67

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Formula

sec-Butyl chloride (2-Chlorobutane) tert-Butyl chloride sec-Butyl chloroacetate 2-tert-Butyl-4-cresol 4-tert-Butyl-2-cresol iso-Butyl dichloroacetate 2,3-Butylene glycol (2,3-butanediol) 2-Butyl-2-ethylbutane-1,3-diol 2-tert-Butyl-4-ethylphenol n-Butyl formate iso-Butyl formate sec-Butyl formate sec-Butyl glycolate iso-Butyl iodide (1-iodo-2-methylpropane) isobutyrate isovalerate levulinate naphthylketone (1-isovaleronaphthone) 2-sec-Butylphenol 2-tert-Butylphenol 4-iso-Butylphenol 4-sec-Butylphenol 4-tert-Butylphenol 2-(4-tert-Butylphenoxy)ethyl acetate 4-tert-Butylphenyl dichlorophosphate

C4H9Cl C4H9Cl C6H11ClO2 C11H16O C11H16O C6H10Cl2O2 C4H10O2 C10H22O2 C12H15O C5H10O2 C5H10O2 C5H10O2 C6H12O3 C4H9I C8H16O2 C9H18O2 C9H16O3 C15H16O C10H14O C10H14O C10H14O C10H14O C10H14O C14H20O3 C10H13Cl2 O2P C11H14O C7H14O2 C12H18O C12H18O C12H18O C12H18O C4H8O2 C4H8O2 C4H7N C11H14O C10H16 C10H16O2 C10H16O C10H19N C10H20O C10H20O2 C6H12O2 C6H12O2 C6H10O2 C6H11N C8H18O C8H16O C8H16O2 C8H15N C12H9N CO2 CS2 CO COSe COS CBr4 CCl4 CF4 C10H14O C10H14O C10H12O2 C2HCl3O C2H3Cl3O2

tert-Butyl phenyl ketone (pivalophenone) iso-Butyl propionate 4-tert-Butyl-2,5-xylenol 4-tert-Butyl-2,6-xylenol 6-tert-Butyl-2,4-xylenol 6-tert-Butyl-3,4-xylenol Butyric acid iso-Butyric acid Butyronitrile iso-Valerophenone Camphene Campholenic acid d-Camphor Camphylamine Capraldehyde Capric acid n-Caproic acid iso-Caproic acid iso-Caprolactone Capronitrile Capryl alcohol (2-octanol) Caprylaldehyde Caprylic acid (octanoic acid) Caprylonitrile Carbazole Carbon dioxide disulfide monoxide oxyselenide (carbonyl selenide) oxysulfide (carbonyl sulfide) tetrabromide tetrachloride tetrafluoride Carvacrol Carvone Chavibetol Chloral (trichloroacetaldehyde) hydrate (trichloroacetaldehyde hydrate) Chloranil Chloroacetic acid anhydride 2-Chloroaniline 3-Chloroaniline 4-Chloroaniline Chlorobenzene 2-Chlorobenzotrichloride (2-α,α,α-tetrachlorotoluene)

C6Cl4O2 C2H3ClO2 C4H4Cl2O3 C6H6ClN C6H6ClN C6H6ClN C6H5Cl C7H4Cl4

1

5

10

20

−60.2

−39.8

−29.2

−17.7

17.0 70.0 74.3 28.6 44.0 94.1 76.3 −26.4 −32.7 −34.4 28.3 −17.0 +4.1 16.0 65.0 136.0 57.4 56.6 72.1 71.4 70.0 118.0 96.0

41.8 98.0 103.7 54.3 68.4 122.6 106.2 −4.7 −11.4 −13.3 53.6 +5.8 28.0 41.2 92.1 167.9 86.0 84.2 100.9 100.5 99.2 150.0 129.6

54.6 112.0 118.0 67.5 80.3 136.8 121.0 +6.1 −0.8 −3.1 66.0 17.0 39.9 53.8 105.9 184.0 100.8 98.1 115.5 114.8 114.0 165.8 146.0

57.8 −2.3 88.2 74.0 70.3 83.9 25.5 14.7 −20.0 58.3

85.7 +20.9 119.8 103.9 100.2 113.6 49.8 39.3 +2.1 87.0

97.6 41.5 45.3 51.9 125.0 71.4 66.2 38.3 9.2 32.8 73.4 92.3 43.0

40

60

100

200

400

760

31.5 +14.6 124.1 187.8 197.8 139.2 145.6 212.0 200.3 67.9 60.0 56.8 135.5 81.0 106.3 124.8 181.8 269.7 179.7 173.8 192.1 194.3 191.5 250.3 240.0

50.0 32.6 146.0 210.0 221.8 160.0 164.0 233.5 223.8 86.2 79.0 75.2 155.6 100.3 126.3 146.4 205.5 294.0 203.8 196.3 214.7 217.6 214.0 277.6 268.2

68.0 51.0 167.8 232.6 247.0 183.0 182.0 255.0 247.8 106.0 98.2 93.6 177.5 120.4 147.5 168.7 229.9 320.0 228.0 219.5 237.0 242.1 238.0 304.4 299.0

Temperature, °C

Melting point, °C

68.2 127.2 134.0 81.4 93.4 151.2 137.0 18.0 +11.0 +8.4 79.8 29.8 52.4 67.7 120.2 201.6 116.1 113.0 130.3 130.3 129.5 183.3 164.0

−5.0 −19.0 83.6 143.9 150.8 96.7 107.8 167.8 154.0 31.6 24.1 21.3 94.2 42.8 67.2 82.7 136.2 219.7 133.4 129.2 147.2 147.8 146.0 201.5 184.3

+3.4 −11.4 93.0 153.7 161.7 106.6 116.3 178.0 165.4 39.8 32.4 29.6 104.0 51.8 75.9 92.4 147.0 231.5 143.9 140.0 157.0 157.9 156.0 212.8 197.2

14.2 −1.0 105.5 167.0 176.2 119.8 127.8 191.9 179.0 51.0 43.4 40.2 116.4 63.5 88.0 105.2 160.2 246.7 157.3 153.5 171.2 172.4 170.2 228.0 214.3

125.7 68.6 74.0 78.8 142.0 89.5 83.0 66.4 34.6 57.6 92.0 114.1 67.6

99.0 32.3 135.0 119.0 115.0 127.0 61.5 51.2 13.4 101.4 47.2 139.8 82.3 83.7 92.0 152.2 99.5 94.0 80.3 47.5 70.0 101.2 124.0 80.4

114.3 44.8 151.0 135.0 131.0 143.0 74.0 64.0 25.7 116.8 60.4 153.9 97.5 97.6 106.3 165.0 111.8 107.0 95.7 61.7 83.3 110.2 136.4 94.6

130.4 58.5 169.8 152.2 148.5 159.7 88.0 77.8 38.4 133.8 75.7 170.0 114.0 112.5 122.2 179.9 125.0 120.4 112.3 76.9 98.0 120.0 150.6 110.6

−134.3 −73.8 −222.0 −117.1 −132.4

−124.4 −54.3 −217.2 −102.3 −119.8

−119.5 −44.7 −215.0 −95.0 −113.3

−114.4 −34.3 −212.8 −86.3 −106.0

−50.0 −184.6 70.0 57.4 83.6 −37.8 −9.8

−30.0 −174.1 98.4 86.1 113.3 −16.0 +10.0

−19.6 −169.3 113.2 100.4 127.0 −5.0 19.5

−8.2 −164.3 127.9 116.1 143.2 +7.2 29.2

−108.6 −22.5 −210.0 −76.4 −98.3 96.3 +4.3 −158.8 145.2 133.0 159.8 20.2 39.7

140.8 67.6 180.3 163.6 158.2 170.0 96.5 86.3 47.3 144.6 85.0 180.0 124.0 122.0 132.0 189.8 133.3 129.6 123.2 86.8 107.4 126.0 160.0 121.2 248.2 −104.8 −15.3 −208.1 −70.2 −93.0 106.3 12.3 −155.4 155.3 143.8 170.7 29.1 46.2

154.0 79.5 195.0 176.0 172.0 184.0 108.0 98.0 59.0 158.0 97.9 193.7 138.0 134.6 145.3 200.0 144.0 141.4 137.2 99.8 119.8 133.9 172.2 134.8 265.0 −100.2 −5.1 −205.7 −61.7 −85.9 119.7 23.0 −150.7 169.7 157.3 185.5 40.2 55.0

70.7 43.0 67.2 46.3 63.5 59.3 −13.0

89.3 68.3 94.1 72.3 89.8 87.9 +10.6

97.8 81.0 108.0 84.8 102.0 102.1 22.2

106.4 94.2 122.4 99.2 116.7 117.8 35.3

116.1 109.2 138.2 115.6 133.6 135.0 49.7

122.0 118.3 148.0 125.7 144.1 145.8 58.3

129.5 130.7 159.8 139.5 158.0 159.9 70.7

140.3 149.0 177.8 160.0 179.5 182.3 89.4

151.3 169.0 197.0 183.7 203.5 206.6 110.0

162.6 189.5 217.0 208.8 228.5 230.5 132.2

290 61.2 46 0 −10.4 70.5 −45.2

69.0

101.8

117.9

135.8

155.0

167.8

185.0

208.0

233.0

262.1

28.7

175.0 197.7 220.0 97.0 116.4 136.8 217.5 241.3 265.3 196.0 217.8 239.8 192.3 214.2 236.5 204.5 226.7 249.5 125.5 144.5 163.5 115.8 134.5 154.5 76.7 96.8 117.5 180.1 204.2 228.0 117.5 138.7 160.5 212.7 234.0 256.0 157.9 182.0 209.2 153.0 173.8 195.0 164.8 186.3 208.5 217.1 240.3 268.4 160.8 181.0 202.0 158.3 181.0 207.7 157.8 182.1 207.0 119.7 141.0 163.7 138.0 157.5 178.5 145.4 156.5 168.5 190.3 213.9 237.5 155.2 179.5 204.5 292.5 323.0 354.8 −93.0 −85.7 −78.2 +10.4 28.0 46.5 −201.3 −196.3 −191.3 −49.8 −35.6 −21.9 −75.0 −62.7 −49.9 139.7 163.5 189.5 38.3 57.8 76.7 −143.6 −135.5 −127.7 191.2 213.8 237.0 179.6 203.5 227.5 206.8 229.8 254.0 57.8 77.5 97.7 68.0 82.1 96.2

−131.3 −26.5

22.5

−95.3 −90.7 −80.7

99

−71

−74 −47 50 178.5 31.5 −1.5 −35 −38.6 16 244.8 −57.5 −110.8 −205.0 −138.8 90.1 −22.6 −183.7 +0.5 −57 51.7

2-68

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

2-Chlorobenzotrifluoride (2-chloro-α,α,α-trifluorotoluene) 2-Chlorobiphenyl 4-Chlorobiphenyl α-Chlorocrotonic acid Chlorodifluoromethane Chlorodimethylphenylsilane 1-Chloro-2-ethoxybenzene 2-(2-Chloroethoxy) ethanol bis-2-Chloroethyl acetacetal 1-Chloro-2-ethylbenzene 1-Chloro-3-ethylbenzene 1-Chloro-4-ethylbenzene 2-Chloroethyl chloroacetate 2-Chloroethyl 2-chloroisopropyl ether 2-Chloroethyl 2-chloropropyl ether 2-Chloroethyl α-methylbenzyl ether Chloroform (trichloromethane) 1-Chloronaphthalene 4-Chlorophenethyl alcohol 2-Chlorophenol 3-Chlorophenol 4-Chlorophenol 2-Chloro-3-phenylphenol 2-Chloro-6-phenylphenol Chloropicrin (trichloronitromethane) 1-Chloropropene 2-Chloropyridine 3-Chlorostyrene 4-Chlorostyrene 1-Chlorotetradecane 2-Chlorotoluene 3-Chlorotoluene 4-Chlorotoluene Chlorotriethylsilane 1-Chloro-1,2,2-trifluoroethylene Chlorotrifluoromethane Chlorotrimethylsilane trans-Cinnamic acid Cinnamyl alcohol Cinnamylaldehyde Citraconic anhydride cis-α-Citral d-Citronellal Citronellic acid Citronellol Citronellyl acetate Coumarin o-Cresol (2-cresol; 2-methylphenol) m-Cresol (3-cresol; 3-methylphenol) p-Cresol (4-cresol; 4-methylphenol) cis-Crotonic acid trans-Crotonic acid cis-Crotononitrile trans-Crotononitrile Cumene 4-Cumidene Cuminal Cuminyl alcohol 2-Cyano-2-n-butyl acetate Cyanogen bromide chloride iodide Cyclobutane Cyclobutene Cyclohexane Cyclohexaneethanol Cyclohexanol Cyclohexanone 2-Cyclohexyl-4,6-dinitrophenol Cyclopentane Cyclopropane Cymene

1

5

10

20

60

100

200

400

760

88.3 197.0 212.5 155.9 −76.4 124.7 141.8 139.5 150.7 110.0 113.6 116.0 140.0 115.8 125.6 164.8 10.4 180.4 188.1 106.0 143.0 150.0 237.0 237.1 53.8 −15.1 104.6 121.2 122.0 215.5 94.7 96.3 96.6 82.3 −66.7 −111.7 +6.0 232.4 177.8 177.7 145.4 160.0 140.1 195.4 159.8 161.0 216.5 127.4 138.0 140.0 116.3 128.0 50.1 62.8 88.1 158.0 160.0 176.2 133.8 −51.8 22.6 −24.9 97.6 −32.8 −41.2 25.5 142.7 103.7 90.4 229.0 −1.3 −70.0 110.8

108.3 219.6 237.8 173.8 −65.8 145.5 162.0 157.2 169.8 130.2 133.8 137.0 159.8 135.7 146.3 186.3 25.9 204.2 210.0 126.4 164.8 172.0 261.3 261.6 71.8 +1.3 125.0 142.2 143.5 240.3 115.0 116.6 117.1 101.6 −55.0 −102.5 21.9 253.3 199.8 199.3 165.8 181.8 160.0 214.5 179.8 178.8 240.0 146.7 157.3 157.3 133.9 146.0 68.0 81.1 107.3 180.0 182.8 197.9 152.2 −42.6 33.8 −14.1 111.5 −18.9 −27.8 42.0 161.7 121.7 110.3 248.7 +13.8 −59.1 131.4

130.0 243.8 264.5 193.2 −53.6 168.6 185.5 176.5 190.5 152.2 156.7 159.8 182.2 156.5 169.8 210.8 42.7 230.8 234.5 149.8 188.7 196.0 289.4 289.5 91.8 18.0 147.7 165.7 166.0 267.5 137.1 139.7 139.8 123.6 −41.7 −92.7 39.4 276.7 224.6 222.4 189.8 205.0 183.8 236.6 201.0 197.8 264.7 168.4 179.0 179.4 152.2 165.5 88.0 101.5 129.2 203.2 206.7 221.7 173.4 −33.0 46.0 −2.3 126.1 −3.4 −12.2 60.8 183.5 141.4 132.5 269.8 31.0 −46.9 153.5

152.2 267.5 292.9 212.0 −40.8 193.5 208.0 196.0 212.6 177.6 181.1 184.3 205.0 180.0 194.1 235.0 61.3 259.3 259.3 174.5 214.0 220.0 317.5 317.0 111.9 37.0 170.2 190.0 191.0 296.0 159.3 162.3 162.3 146.3 −27.9 −81.2 57.9 300.0 250.0 246.0 213.5 228.0 206.5 257.0 221.5 217.0 291.0 190.8 202.8 201.8 171.9 185.0 108.0 122.8 152.4 227.0 232.0 246.6 195.2 −21.0 61.5 +13.1 141.1 +12.9 +2.4 80.7 205.4 161.0 155.6 291.5 49.3 −33.5 177.2

Temperature, °C

Formula C7H4ClF3 C12H9Cl C12H9Cl C4H5ClO2 CHClF2 C8H11ClSi C8H9ClO C4H9ClO2 C6H12Cl2O2 C8H9Cl C8H9Cl C8H9Cl C4H6Cl2O2 C5H10Cl2O C5H10Cl2O C10H13ClO CHCl3 C10H7Cl C8H9ClO C6H5ClO C6H5ClO C6H5ClO C12H9ClO C12H9ClO CCl3NO2 C3H5Cl C5H4ClN C8H7Cl C8H7Cl C14H29Cl C7H7Cl C7H7Cl C7H7Cl C6H15ClSi C2ClF3 CClF3 C3H9ClSi C9H8O2 C9H10O C9H8O C5H4O3 C10H16O C10H18O C10H18O2 C10H20O C12H22O2 C9H6O2 C7H8O C7H8O C7H8O C4H6O2 C4H6O2 C4H5N C4H5N C9H12 C9H13N C10H12O C10H14O C7H11NO2 C2N2 CBrN CClN CIN C4H8 C4H6 C6H12 C8H16O C6H12O C6H10O C12H14N2O5 C5H10 C3H6 C10H14

40

0.0 89.3 96.4 70.0 −122.8 29.8 45.8 53.0 56.2 17.2 18.6 19.2 46.0 24.7 29.8 62.3 −58.0 80.6 84.0 12.1 44.2 49.8 118.0 119.8 −25.5 −81.3 13.3 25.3 28.0 98.5 +5.4 +4.8 +5.5 −4.9 −116.0 −149.5 −62.8 127.5 72.6 76.1 47.1 61.7 44.0 99.5 66.4 74.7 106.0 38.2 52.0 53.0 33.5

24.7 109.8 129.8 95.6 −110.2 56.7 72.8 78.3 83.7 43.0 45.2 46.4 72.1 50.1 56.5 91.4 −39.1 104.8 114.3 38.2 72.0 78.2 152.2 153.7 −3.3 −63.4 38.8 51.3 54.5 131.8 30.6 30.3 31.0 +19.8 −102.5 −139.2 −43.6 157.8 102.5 105.8 74.8 90.0 71.4 127.3 93.6 100.2 137.8 64.0 76.0 76.5 57.4

−29.0 −19.5 +2.9 60.0 58.0 74.2 42.0 −95.8 −35.7 −76.7 25.2 −92.0 −99.1 −45.3 50.4 21.0 +1.4 132.8 −68.0 −116.8 17.3

−7.1 +3.5 26.8 88.2 87.3 103.7 68.7 −83.2 −18.3 −61.4 47.2 −76.0 −83.4 −25.4 77.2 44.0 26.4 161.8 −49.6 −104.2 43.9

37.1 134.7 146.0 108.0 −103.7 70.0 86.5 90.7 97.6 56.1 58.1 60.0 86.0 63.0 70.0 106.0 −29.7 118.6 129.0 51.2 86.1 92.2 169.7 170.7 +7.8 −54.1 51.7 65.2 67.5 148.2 43.2 43.2 43.8 32.0 −95.9 −134.1 −34.0 173.0 117.8 120.0 88.9 103.9 84.8 141.4 107.0 113.0 153.4 76.7 87.8 88.6 69.0 80.0 +4.0 15.0 38.3 102.2 102.0 118.0 82.0 −76.8 −10.0 −53.8 57.7 −67.9 −75.4 −15.9 90.0 56.0 38.7 175.9 −40.4 −97.5 57.0

50.6 151.2 164.0 121.2 −96.5 84.7 101.5 104.1 112.2 70.3 73.0 75.5 100.0 77.2 84.8 121.8 −19.0 134.4 145.0 65.9 101.7 108.1 186.7 189.8 20.0 −44.0 65.8 80.0 82.0 166.2 56.9 57.4 57.8 45.5 −88.2 −128.5 −23.2 189.5 133.7 135.7 103.8 119.4 99.8 155.6 121.5 126.0 170.0 90.5 101.4 102.3 82.0 93.0 16.4 27.8 51.5 117.8 117.9 133.8 96.2 −70.1 −1.0 −46.1 68.6 −58.7 −66.6 −5.0 104.0 68.8 52.5 191.2 −30.1 −90.3 71.1

65.9 169.9 183.8 135.6 −88.6 101.2 117.8 118.4 127.8 86.2 89.2 91.8 116.0 92.4 101.5 139.6 −7.1 153.2 162.0 82.0 118.0 125.0 207.4 208.2 33.8 −32.7 81.7 96.5 98.0 187.0 72.0 73.0 73.5 60.2 −79.7 −121.9 −11.4 207.1 151.0 152.2 120.3 135.9 116.1 171.9 137.2 140.5 189.0 105.8 116.0 117.7 96.0 107.8 30.0 41.8 66.1 134.2 135.2 150.3 111.8 −62.7 +8.6 −37.5 80.3 −48.4 −56.4 +6.7 119.8 83.0 67.8 206.7 −18.6 −82.3 87.0

75.4 182.1 196.0 144.4 −83.4 111.5 127.8 127.5 138.0 96.4 99.6 102.0 126.2 102.2 111.8 150.0 +0.5 165.6 173.5 92.0 129.4 136.1 219.6 220.0 42.3 −25.1 91.6 107.2 108.5 199.8 81.8 83.2 83.3 69.5 −74.1 −117.3 −4.0 217.8 162.0 163.7 131.3 146.3 126.2 182.1 147.2 149.7 200.5 115.5 125.8 127.0 104.5 116.7 38.5 50.9 75.4 145.0 146.0 161.7 121.5 −57.9 14.7 −32.1 88.0 −41.8 −50.0 14.7 129.8 91.8 77.5 216.0 −11.3 −77.0 97.2

Melting point, °C −6.0 34 75.5 −160

−80.2 −53.3 −62.6

−63.5 −20 7 32.5 42 +6 −64 −99.0 −15.0 +0.9 +7.3 −157.5 133 33 −7.5

70 30.8 10.9 35.5 15.5 72 −96.0

−34.4 58 −6.5 −50 +6.6 23.9 −45.0 −93.7 −126.6 −68.2

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-69

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

cis-Decalin trans-Decalin Decane Decan-2-one 1-Decene Decyl alcohol Decyltrimethylsilane Dehydroacetic acid Desoxybenzoin Diacetamide Diacetylene (1,3-butadiyne) Diallyldichlorosilane Diallyl sulfide Diisoamyl ether oxalate sulfide Dibenzylamine Dibenzyl ketone (1,3-diphenyl2-propanone) 1,4-Dibromobenzene 1,2-Dibromobutane dl-2,3-Dibromobutane meso-2,3-Dibromobutane 1,2-Dibromodecane Di(2-bromoethyl) ether α,β-Dibromomaleic anhydride 1,2-Dibromo-2-methylpropane 1,3-Dibromo-2-methylpropane 1,2-Dibromopentane 1,2-Dibromopropane 1,3-Dibromopropane 2,3-Dibromopropene 2,3-Dibromo-1-propanol Diisobutylamine 2,6-Ditert-butyl-4-cresol 4,6-Ditert-butyl-2-cresol 4,6-Ditert-butyl-3-cresol 2,6-Ditert-butyl-4-ethylphenol 4,6-Ditert-butyl-3-ethylphenol Diisobutyl oxalate 2,4-Ditert-butylphenol Dibutyl phthalate sulfide Diisobutyl d-tartrate Dicarvacryl-mono-(6-chloro-2-xenyl) phosphate Dicarvacryl-2-tolyl phosphate Dichloroacetic acid 1,2-Dichlorobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene 1,2-Dichlorobutane 2,3-Dichlorobutane 1,2-Dichloro-1,2-difluoroethylene Dichlorodifluoromethane Dichlorodiphenyl silane Dichlorodiisopropyl ether Di(2-chloroethoxy) methane Dichloroethoxymethylsilane 1,2-Dichloro-3-ethylbenzene 1,2-Dichloro-4-ethylbenzene 1,4-Dichloro-2-ethylbenzene cis-1,2-Dichloroethylene trans-1,2-Dichloro ethylene Di(2-chloroethyl) ether Dichlorofluoromethane 1,5-Dichlorohexamethyltrisiloxane Dichloromethylphenylsilane 1,1-Dichloro-2-methylpropane 1,2-Dichloro-2-methylpropane 1,3-Dichloro-2-methylpropane 2,4-Dichlorophenol 2,6-Dichlorophenol

1

5

10

20

22.5 −0.8 16.5 44.2 14.7 69.5 67.4 91.7 123.3 70.0 −82.5 +9.5 −9.5 18.6 85.4 43.0 118.3 125.5

50.1 +30.6 42.3 71.9 40.3 97.3 96.4 122.0 156.2 95.0 −68.0 34.8 +14.4 44.3 116.0 73.0 149.8 159.8

64.2 47.2 55.7 85.8 53.7 111.3 111.0 137.3 173.5 108.0 −61.2 47.4 26.6 57.0 131.4 87.6 165.6 177.6

79.8 65.3 69.8 100.7 67.8 125.8 126.5 153.0 192.0 122.6 −53.8 61.3 39.7 70.7 147.7 102.7 182.2 195.7

97.2 85.7 85.5 117.1 83.3 142.1 144.0 171.0 212.0 138.2 −45.9 76.4 54.2 86.3 165.7 120.0 200.2 216.6

C6H4Br2 61.0 C4H8Br2 7.5 C4H8Br2 +5.0 C4H8Br2 +1.5 C10H20Br2 95.7 C4H8Br2O 47.7 C4H2Br2O3 50.0 C4H8Br2 −28.8 C4H8Br2 14.0 C5H10Br2 19.8 C3H6Br2 −7.0 C3H6Br2 +9.7 C3H4Br2 −6.0 C3H6Br2O 57.0 C8H19N −5.1 C15H24O 85.8 C15H24O 86.2 C15H24O 103.7 C16H26O 89.1 C16H26O 111.5 C10H18O4 63.2 C14H22O 84.5 C16H22O4 148.2 C8H18S +21.7 C12H22O6 117.8 C32H34ClO4P 204.2

79.3 33.2 30.0 26.6 123.6 75.3 78.0 −3.0 40.0 45.4 +17.3 35.4 +17.9 84.5 +18.4 116.2 117.3 135.2 121.4 142.6 91.2 115.4 182.1 51.8 151.8 234.5

87.7 46.1 41.6 39.3 137.3 88.5 92.0 +10.5 53.0 58.0 29.4 48.0 30.0 98.2 30.6 131.0 132.4 150.0 137.0 157.4 105.3 130.0 198.2 66.4 169.0 249.3

103.6 60.0 56.4 53.2 151.0 103.6 106.7 25.7 67.5 72.0 42.3 62.1 43.2 113.5 43.7 147.0 149.0 167.0 154.0 174.0 120.3 146.0 216.2 80.5 188.0 264.5

C27H33O4P 180.2 C2H2Cl2O2 44.0 C6H4Cl2 20.0 C6H4Cl2 12.1 C6H4Cl2 C4H8Cl2 −23.6 C4H8Cl2 −25.2 C2Cl2F2 −82.0 CCl2F2 −118.5 C12H10Cl2Si 109.6 C6H12Cl2O 29.6 C5H10Cl2O2 53.0 C8H8Cl2OSi −33.8 C8H8Cl2 46.0 C8H8Cl2 47.0 C8H8Cl2 38.5 C2H2Cl2 −58.4 C2H2Cl2 −65.4 C4H8Cl2O 23.5 CHCl2F −91.3 C6H18Cl2 26.0 O2Si3 C7H8Cl2Si 35.7 C4H8Cl2 −31.0 C4H8Cl2 −25.8 C4H8Cl2 −3.0 53.0 C6H4Cl2O C6H4Cl2O 59.5

209.3 69.8 46.0 39.0 −0.3 −3.0 −65.6 −104.6 142.4 55.2 80.4 −12.1 75.0 77.2 68.0 −39.2 −47.2 49.3 −75.5 52.0

221.8 82.6 59.1 52.0 54.8 +11.5 +8.5 −57.3 −97.8 158.0 68.2 94.0 −1.3 90.0 92.3 83.2 −29.9 −38.0 62.0 −67.5 65.1

63.5 −8.4 −4.2 +20.6 80.0 87.6

77.4 +2.6 +6.7 32.0 92.8 101.0

60

Melting point, °C

100

200

400

760

108.0 98.4 95.5 127.8 93.5 152.0 154.3 181.5 224.5 148.0 −41.0 86.3 63.7 96.0 177.0 130.6 212.2 229.4

123.2 114.6 108.6 142.0 106.5 165.8 169.5 197.5 241.3 160.6 −34.0 99.7 75.8 109.6 192.2 145.3 227.3 246.6

145.4 136.2 128.4 163.2 126.7 186.2 191.0 219.5 265.2 180.8 −20.9 119.4 94.8 129.0 215.0 166.4 249.8 272.3

169.9 160.1 150.6 186.7 149.2 208.8 215.5 244.5 293.0 202.0 −6.1 142.0 116.1 150.3 240.0 191.0 274.3 301.7

194.6 186.7 174.1 211.0 172.0 231.0 240.0 269.0 321.0 223.0 +9.7 165.3 138.6 173.4 265.0 216.0 300.0 330.5

−43.3 −30.7 −29.7 +3.5

120.8 76.0 72.0 68.0 167.4 119.8 123.5 42.3 83.5 87.4 57.2 77.8 57.8 129.8 57.8 164.1 167.4 185.3 172.1 192.3 137.5 164.3 235.8 96.0 208.5 280.5

131.6 86.0 82.0 78.0 177.5 130.0 133.8 53.7 93.7 97.4 66.4 87.8 67.0 140.0 67.0 175.2 179.0 196.1 183.9 204.4 147.8 175.8 247.8 105.8 221.6 290.7

146.5 99.8 95.3 91.7 190.2 144.0 147.7 68.8 107.4 110.1 78.7 101.3 79.5 153.0 79.2 190.0 194.0 211.0 198.0 218.0 161.8 190.0 263.7 118.6 239.5 304.9

168.5 120.2 115.7 111.8 209.6 165.0 168.0 92.1 117.8 130.2 97.8 121.7 98.0 173.8 97.6 212.8 217.5 233.0 220.0 241.7 183.5 212.5 287.0 138.0 264.7 323.8

192.5 143.5 138.0 134.2 229.8 188.0 192.0 119.8 150.6 151.8 118.5 144.1 119.5 196.0 118.0 237.6 243.4 257.1 244.0 264.6 205.8 237.0 313.5 159.0 294.0 342.0

218.6 166.3 160.5 157.3 250.4 212.5 215.0 149.0 174.6 175.0 141.6 167.5 141.2 219.0 139.5 262.5 269.3 282.0 268.6 290.0 229.5 260.8 340.0 182.0 324.0 361.0

87.5 −64.5

237.0 96.3 73.4 66.2 69.2 24.5 21.2 −48.3 −90.1 176.0 82.2 109.5 +11.3 105.9 109.6 99.8 −19.4 −28.0 76.0 −58.6 79.0

251.5 111.8 89.4 82.0 84.8 37.7 35.0 −38.2 −81.6 195.5 97.3 125.5 24.4 123.8 127.5 118.0 −7.9 −17.0 91.5 −48.8 94.8

260.3 121.5 99.5 92.2 95.2 47.8 43.9 −31.8 −76.1 207.5 106.9 135.8 32.6 135.0 139.0 129.0 −0.5 −10.0 101.5 −42.6 105.0

272.5 134.0 112.9 105.0 108.4 60.2 56.0 −23.0 −68.6 223.8 119.7 149.6 44.1 149.8 153.3 144.0 +9.5 −0.2 114.5 −33.9 118.2

290.0 152.3 133.4 125.9 128.3 79.7 74.0 −10.0 −57.0 248.0 139.0 170.0 61.0 172.0 176.0 166.2 24.6 +14.3 134.0 −20.9 138.3

309.8 173.7 155.8 149.0 150.2 100.8 94.2 +5.0 −43.9 275.5 159.8 192.0 80.3 197.0 201.7 191.5 41.0 30.8 155.4 −6.2 160.2

330.0 194.4 179.0 173.0 173.9 123.5 116.0 20.9 −29.8 304.0 182.7 215.0 100.6 222.1 226.6 216.3 59.0 47.8 178.5 +8.9 184.0

92.4 14.6 18.7 44.8 107.7 115.5

109.5 28.2 32.0 58.6 123.4 131.6

120.0 37.0 40.2 67.5 133.5 141.8

134.2 48.2 51.7 78.8 146.0 154.6

155.5 65.8 68.9 96.1 165.2 175.5

180.2 85.4 87.8 115.4 187.5 197.7

205.5 106.0 108.0 135.0 210.0 220.0

Temperature, °C

Formula C10H18 C10H18 C10H22 C10H20O C10H20 C10H22O C13H30Si C8H8O4 C14H12O C4H7NO2 C4H2 C6H10Cl2Si C6H10S C10H22O C12H22O4 C10H22S C14H15N C15H14O

40

+7 60 78.5 −34.9 −83

−26 34.5

−34.5

−70.3 −55.5 −34.4 −70

−79.7 73.5

9.7 −17.6 −24.2 53.0 −80.4 −112

−40.8 −76.4 −61.2 −80.5 −50.0 −135 −53.0

45.0

2-70

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

α,α-Dichlorophenylacetonitrile Dichlorophenylarsine 1,2-Dichloropropane 2,3-Dichlorostyrene 2,4-Dichlorostyrene 2,5-Dichlorostyrene 2,6-Dichlorostyrene 3,4-Dichlorostyrene 3,5-Dichlorostyrene 1,2-Dichlorotetraethylbenzene 1,4-Dichlorotetraethylbenzene 1,2-Dichloro-1,1,2,2-tetrafluoroethane Dichloro-4-tolylsilane 3,4-Dichloro-α,α,α-trifluorotoluene Dicyclopentadiene Diethoxydimethylsilane Diethoxydiphenylsilane Diethyl adipate Diethylamine N-Diethylaniline Diethyl arsanilate 1,2-Diethylbenzene 1,3-Diethylbenzene 1,4-Diethylbenzene Diethyl carbonate cis-Diethyl citraconate Diethyl dioxosuccinate Diethylene glycol Diethyleneglycol-bis-chloroacetate Diethylene glycol dimethyl ether Di(2-methoxyethyl) ether glycol ethyl ether Diethyl ether ethylmalonate fumarate glutarate Diethylhexadecylamine Diethyl itaconate ketone (3-pentanone) malate maleate malonate mesaconate oxalate phthalate sebacate 2,5-Diethylstyrene Diethyl succinate isosuccinate sulfate sulfide sulfite d-Diethyl tartrate dl-Diethyl tartrate 3,5-Diethyltoluene Diethylzinc 1-Dihydrocarvone Dihydrocitronellol 1,4-Dihydroxyanthraquinone Dimethylacetylene (2-butyne) Dimethylamine N,N-Dimethylaniline Dimethyl arsanilate Di(α-methylbenzyl) ether 2,2-Dimethylbutane 2,3-Dimethylbutane Dimethyl citraconate 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane trans-1,3-Dimethylcyclohexane cis-1,3-Dimethylcyclohexane cis-1,4-Dimethylcyclohexane trans-1,4-Dimethylcyclohexane

1

5

10

20

56.0 61.8 −38.5 61.0 53.5 55.5 47.8 57.2 53.5 105.6 91.7 −95.4 46.2 11.0 −19.1 111.5 74.0

84.0 100.0 −17.0 90.1 82.2 83.9 75.7 86.0 82.2 138.7 126.1 −80.0 71.7 38.3 34.1 +2.4 142.8 106.6

49.7

78.0

98.1 116.0 −6.1 104.6 97.4 98.2 90.0 100.4 97.4 155.0 143.8 −72.3 84.2 52.2 47.6 13.3 157.6 123.0 −33.0 91.9

113.8 133.1 +6.0 120.5 111.8 114.0 105.5 116.2 111.8 172.5 162.0 −63.5 97.8 67.3 62.0 25.3 174.3 138.3 −22.6 107.2

130.0 151.0 19.4 137.8 129.2 131.0 122.4 133.7 129.2 192.2 183.2 −53.7 113.2 84.0 77.9 38.0 193.2 154.6 −11.3 123.6

38.0 22.3 20.7 20.7 −10.1 59.8 70.0 91.8 148.3

62.6 48.7 46.8 47.1 +12.3 88.3 98.0 120.0 180.0

74.8 62.0 59.9 60.3 23.8 103.0 112.0 133.8 195.8

88.0 76.4 74.5 74.7 36.0 118.2 126.8 148.0 212.0

C6H14O3 C6H14O3 C4H10O C9H16O4 C8H12O4 C9H16O4 C20H43N C9H14O4 C5H10O C8H14O5 C8H12O4 C7H12O4 C9H14O4 C6H10O4 C12H14O4 C14H26O4 C12H16 C8H14O4 C8H14O4 C4H10O4S C4H10S C4H10O3S C8H14O6 C8H14O6 C11H16 C4H10Zn C10H16O C10H22O C14H8O4 C4H6 C2H7N C8H11N C8H12AsNO3 C16H18O C6H14 C6H14 C7H10O4 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16

13.0 45.3 −74.3 50.8 53.2 65.6 139.8 51.3 −12.7 80.7 57.3 40.0 62.8 47.4 108.8 125.3 49.7 54.6 39.8 47.0 −39.6 10.0 102.0 100.0 34.0 −22.4 46.6 68.0 196.7 −73.0 −87.7 29.5 15.0 96.7 −69.3 −63.6 50.8 −24.4 −15.9 −21.1 −19.4 −22.7 −20.0 −24.3

37.6 72.0 −56.9 77.8 81.2 94.7 175.8 80.2 +7.5 110.4 85.6 67.5 91.0 71.8 140.7 156.2 78.4 83.0 66.7 74.0 −18.6 34.2 133.0 131.7 61.5 0.0 75.5 91.7 239.8 −57.9 −72.2 56.3 39.6 128.3 −50.7 −44.5 78.2 −1.4 +7.3 +1.7 +3.4 0.0 +3.2 −1.7

50.0 85.8 −48.1 91.6 95.3 109.7 194.0 95.2 17.2 125.3 100.0 81.3 105.3 83.8 156.0 172.1 92.6 96.6 80.0 87.7 −8.0 46.4 148.0 147.2 75.3 +11.7 90.0 103.0 259.8 −50.5 −64.6 70.0 51.8 144.0 −41.5 −34.9 91.8 +10.3 18.4 13.0 14.9 +11.2 14.5 +10.1

63.0 100.3 −38.5 106.0 110.2 125.4 213.5 111.0 27.9 141.2 115.3 95.9 120.3 96.8 173.6 189.8 108.5 111.7 94.7 102.1 +3.5 59.7 164.2 163.8 90.2 24.2 106.0 115.0 282.0 −42.5 −56.0 84.8 65.0 160.3 −31.1 −24.1 106.5 23.0 31.1 25.6 27.4 23.6 27.1 22.6

60

100

200

400

760

141.0 163.2 28.0 149.0 140.0 142.0 133.3 144.6 140.0 204.8 195.8 −47.5 122.6 95.0 88.0 46.3 205.0 165.8 −4.0 133.8

154.5 178.9 39.4 163.5 153.8 155.8 147.6 158.2 153.8 220.7 212.0 −39.1 135.5 109.2 101.7 57.6 220.0 179.0 +6.0 147.3

176.2 202.8 57.0 185.7 176.0 178.0 169.0 181.5 176.0 245.6 238.5 −26.3 153.5 129.0 121.8 74.2 243.8 198.2 21.0 168.2

199.5 228.8 76.0 210.0 200.0 202.5 193.5 205.7 200.0 272.8 265.8 −12.0 175.2 150.5 144.2 93.2 259.7 219.1 38.0 192.4

223.5 256.5 96.8 235.0 225.0 227.0 217.0 230.0 225.0 302.0 296.5 +3.5 196.3 172.8 166.6 113.5 296.0 240.0 55.5 215.5

102.6 92.5 90.4 91.1 49.5 135.7 143.8 164.3 229.0

111.8 102.6 100.7 101.3 57.9 146.2 153.7 174.0 239.5

123.8 116.2 114.4 115.3 69.7 160.0 167.7 187.5 252.0

141.9 136.7 134.8 136.1 86.5 182.3 188.0 207.0 271.5

161.0 159.0 156.9 159.0 105.8 206.5 210.8 226.5 291.8

181.0 183.5 181.1 183.8 125.8 230.3 233.5 244.8 313.0

77.5 116.7 27.7 122.4 126.7 142.8 235.0 128.2 39.4 157.8 131.8 113.3 137.3 110.6 192.1 207.5 125.8 127.8 111.0 118.0 16.1 74.2 182.3 181.7 107.0 38.0 123.7 127.6 307.4 −33.9 −46.7 101.6 79.7 179.6 −19.5 −12.4 122.6 37.3 45.3 39.7 41.4 37.5 41.1 36.5

86.8 126.8 −21.8 132.4 137.7 153.2 248.5 139.9 46.7 169.0 142.4 123.0 147.9 119.7 204.1 218.4 136.8 138.2 121.4 128.6 24.2 83.8 194.0 193.2 117.7 47.2 134.7 136.7 323.3 −27.8 −40.7 111.9 88.6 191.5 −12.1 −4.9 132.7 45.7 54.4 48.7 50.4 46.4 50.1 45.4

99.5 140.3 −11.5 146.0 151.1 167.8 265.5 154.3 56.2 183.9 156.0 136.2 161.6 130.8 219.5 234.4 151.0 151.1 134.8 142.5 35.0 96.3 208.5 208.0 131.7 59.1 149.7 145.9 344.5 −18.8 −32.6 125.8 101.0 206.8 −2.0 +5.4 145.8 57.9 66.8 61.0 62.5 58.5 62.3 57.6

118.0 159.0 +2.2 166.0 172.2 189.5 292.8 177.5 70.6 205.3 177.8 155.5 183.2 147.9 243.0 255.8 173.2 171.7 155.1 162.5 51.3 115.8 230.4 230.0 152.4 77.0 171.8 160.2 377.8 −5.0 −20.4 146.5 119.8 229.7 +13.4 21.1 165.8 76.2 85.6 79.6 81.0 76.9 80.8 76.0

138.5 180.3 17.9 188.7 195.8 212.8 324.6 203.1 86.3 229.5 201.7 176.8 205.8 166.2 267.5 280.3 198.0 193.8 177.7 185.5 69.7 137.0 254.8 254.3 176.5 97.3 197.0 176.8 413.0 +10.6 −7.1 169.2 140.3 254.8 31.0 39.0 188.0 97.2 107.0 100.9 102.1 97.8 101.9 97.0

159.8 201.9 34.6 211.5 218.5 237.0 355.0 227.9 102.7 253.4 225.0 198.9 229.0 185.7 294.0 305.5 223.0 216.5 201.3 209.5 88.0 159.0 280.0 280.0 200.7 118.0 223.0 193.5 450.0 27.2 +7.4 193.1 160.5 281.0 49.7 58.0 210.5 119.5 129.7 123.4 124.4 120.1 124.3 119.3

Temperature, °C

Formula C8H5Cl2N C6H5AsCl2 C3H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C14H20Cl2 C14H20Cl2 C2Cl2F4 C7H8Cl2Si C7H3Cl2F3 C10H8 C6H16O2Si C16H20O2Si C10H18O4 C4H11N C10H15N C10H16As NO3 C10H14 C10H14 C10H14 C5H10O3 C9H14O4 C8H10O6 C4H10O3 C8H12Cl2O5

40

Melting point, °C

−94 −12.1 32.9 −21 −38.9 −34.4 −31.4 −83.9 −43.2 −43

−116.3 +0.6

−42 −49.8 −40.6 1.3 −20.8 −25.0 −99.5 17 −28 194 −32.5 −96 +2.5 −99.8 −128.2 −34 −50.0 −88.0 −92.0 −76.2 −87.4 −36.9

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-71

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Dimethyl ether 2,2-Dimethylhexane 2,3-Dimethylhexane 2,4-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 3,4-Dimethylhexane Dimethyl itaconate 1-Dimethyl malate Dimethyl maleate malonate trans-Dimethyl mesaconate 2,7-Dimethyloctane Dimethyl oxalate 2,2-Dimethylpentane 2,3-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,3-Dimethylphenol (2,3-xylenol) 2,4-Dimethylphenol (2,4-xylenol) 2,5-Dimethylphenol (2,5-xylenol) 3,4-Dimethylphenol (3,4-xylenol) 3,5-Dimethylphenol (3,5-xylenol) Dimethylphenylsilane Dimethyl phthalate 3,5-Dimethyl-1,2-pyrone 4,6-Dimethylresorcinol Dimethyl sebacate 2,4-Dimethylstyrene 2,5-Dimethylstyrene α,α-Dimethylsuccinic anhydride Dimethyl sulfide d-Dimethyl tartrate dl-Dimethyl tartrate N,N-Dimethyl-2-toluidine N,N-Dimethyl-4-toluidine Di(nitrosomethyl) amine Diosphenol 1,4-Dioxane Dipentene Diphenylamine Diphenyl carbinol (benzhydrol) chlorophosphate disulfide 1,2-Diphenylethane (dibenzyl) Diphenyl ether 1,1-Diphenylethylene trans-Diphenylethylene 1,1-Diphenylhydrazine Diphenylmethane Diphenyl sulfide Diphenyl-2-tolyl thiophosphate 1,2-Dipropoxyethane 1,2-Diisopropylbenzene 1,3-Diisopropylbenzene Dipropylene glycol Dipropyleneglycol monobutyl ether isopropyl ether Di-n-propyl ether Diisopropyl ether Di-n-propyl ketone (4-heptanone) Di-n-propyl oxalate Diisopropyl oxalate Di-n-propyl succinate Di-n-propyl d-tartrate Diisopropyl d-tartrate Divinyl acetylene (1,5-hexadiene-3-yne) 1,3-Divinylbenzene Docosane n-Dodecane 1-Dodecene n-Dodecyl alcohol Dodecylamine Dodecyltrimethylsilane Elaidic acid

1

5

10

20

−115.7 −29.7 −23.0 −26.9 −26.7 −25.8 −22.1 69.3 75.4 45.7 35.0 46.8 +6.3 20.0 −49.0 −42.0 −48.0 −45.9 56.0 51.8 51.8 66.2 62.0 +5.3 100.3 78.6 49.0 104.0 34.2 29.0 61.4 −75.6 102.1 100.4 28.8 50.1 +3.2 66.7 −35.8 14.0 108.3 110.0 121.5 131.6 86.8 66.1 87.4 113.2 126.0 76.0 96.1 159.7 −38.8 40.0 34.7 73.8 64.7 46.0 −43.3 −57.0 23.0 53.4 43.2 77.5 115.6 103.7 −45.1 32.7 157.8 47.8 47.2 91.0 82.8 91.2 171.3

−101.1 −7.9 −1.1 −5.3 −5.5 −4.4 +0.2 94.0 104.0 73.0 59.8 74.0 30.5 44.0 −28.7 −20.8 −27.4 −25.0 83.8 78.0 78.0 93.8 89.2 30.3 131.8 107.6 76.8 139.8 61.9 55.9 88.1 −58.0 133.2 131.8 54.1 74.3 27.8 95.4 −12.8 40.4 141.7 145.0 160.5 164.0 119.8 97.8 119.6 145.8 159.3 107.4 129.0 179.8 −10.3 67.8 62.3 102.1 92.0 72.8 −22.3 −37.4 44.4 80.2 69.0 107.6 147.7 133.7 −24.4 60.0 195.4 75.8 74.0 120.2 111.8 122.1 206.7

−93.3 +3.1 +9.9 +5.2 +5.3 +6.1 11.3 106.6 118.3 86.4 72.0 87.8 42.3 56.0 −18.7 −10.3 −17.1 −14.4 97.6 91.3 91.3 107.7 102.4 42.6 147.6 122.0 90.7 156.2 75.8 69.0 102.0 −49.2 148.2 147.5 66.2 86.7 40.0 109.0 −1.2 53.8 157.0 162.0 182.0 180.0 136.0 114.0 135.0 161.0 176.1 122.8 145.0 201.6 +5.0 81.8 76.0 116.2 106.0 86.2 −11.8 −27.4 55.0 93.9 81.9 122.2 163.5 148.2 −14.0 73.8 213.0 90.0 87.8 134.7 127.8 137.7 223.5

−85.2 15.0 22.1 17.2 17.2 18.2 23.5 119.7 133.8 101.3 85.0 102.1 55.8 69.4 −7.5 +1.1 −5.9 −2.9 112.0 105.0 105.0 122.0 117.0 56.2 164.0 136.4 105.8 175.8 90.8 84.0 116.3 −39.4 164.3 164.0 80.2 100.0 53.7 124.0 +12.0 68.2 175.2 180.9 203.8 197.0 153.7 130.8 151.8 179.8 194.0 139.8 162.0 215.5 22.3 96.8 91.2 131.3 120.4 100.8 0.0 −16.7 66.2 108.6 95.6 138.0 180.4 164.0 −2.8 88.7 233.5 104.6 102.4 150.0 141.6 153.8 242.3

60

100

200

400

760

−62.7 48.2 56.0 50.6 50.5 52.5 57.7 153.7 175.1 140.4 121.9 141.5 93.9 104.8 23.9 33.3 25.4 29.3 152.2 143.0 143.0 161.0 156.0 94.2 210.0 177.5 147.3 222.6 132.3 124.7 155.3 −12.0 208.8 209.5 118.1 140.3 90.3 165.6 45.1 108.3 222.8 227.5 265.0 241.3 202.8 178.8 198.6 227.4 242.5 186.3 211.8 252.5 74.2 138.7 132.3 169.9 159.8 140.3 33.0 13.7 96.0 148.1 132.6 180.3 227.0 207.3 29.5 130.0 286.0 146.2 142.3 192.0 182.1 199.5 288.0

−50.9 65.7 73.8 68.1 68.0 70.0 75.6 171.0 196.3 160.0 140.0 161.0 114.0 123.3 40.3 50.1 41.8 46.2 173.0 161.5 161.5 181.5 176.2 114.2 232.7 198.0 167.8 245.0 153.2 145.6 175.8 +2.6 230.5 232.3 138.3 161.6 110.0 186.2 62.3 128.2 247.5 250.0 299.5 262.6 227.8 203.3 222.8 251.7 267.2 210.7 236.8 270.3 103.8 159.8 153.7 189.9 180.0 160.0 50.3 30.0 111.2 168.0 151.2 202.5 250.1 228.2 46.0 151.4 314.2 167.2 162.2 213.0 203.0 222.0 312.4

−37.8 85.6 94.1 88.2 87.9 90.4 96.0 189.8 219.5 182.2 159.8 183.5 136.0 143.3 59.2 69.4 60.6 65.5 196.0 184.2 184.2 203.6 197.8 136.4 257.8 221.0 192.0 269.6 177.5 168.7 197.5 18.7 255.0 257.4 161.5 185.4 131.3 209.5 81.8 150.5 274.1 275.6 337.2 285.8 255.0 230.7 249.8 278.3 294.0 237.5 263.9 290.0 140.0 184.3 177.6 210.5 203.8 183.1 69.5 48.2 127.3 190.3 171.8 226.5 275.6 251.8 64.4 175.2 343.5 191.0 185.5 235.7 225.0 248.0 337.0

−23.7 106.8 115.6 109.4 109.1 112.0 117.7 208.0 242.6 205.0 180.7 206.0 159.7 163.3 79.2 89.8 80.5 86.1 218.0 211.5 211.5 225.2 219.5 159.3 283.7 245.0 215.0 293.5 202.0 193.0 219.5 36.0 280.0 282.0 184.8 209.5 153.0 232.0 101.1 174.6 302.0 301.0 378.0 310.0 284.0 258.5 277.0 306.5 322.2 264.5 292.5 310.0 180.0 209.0 202.0 231.8 227.0 205.6 89.5 67.5 143.7 213.5 193.5 250.8 303.0 275.0 84.0 199.5 376.0 216.2 208.0 259.0 248.0 273.0 362.0

Temperature, °C

Formula C2H6O C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C7H10O4 C6H10O5 C6H8O4 C5H8O4 C7H10O4 C10H22 C4H6O4 C7H16 C7H16 C7H16 C7H16 C8H10O C8H10O C8H10O C8H10O C8H10O C8H12Si C10H10O4 C7H8O2 C8H10O2 C12H22O4 C10H12 C10H12 C6H8O3 C2H6S C6H10O6 C6H10O6 C9H13N C9H13N C2H5N3O2 C10H16O2 C4H8O2 C10H16 C12H11N C13H12O C12H10ClPO3 C12H10S2 C14H14 C12H10O C14H12 C14H12 C12H12N2 C13H12 C12H10S C18H17O3PS C8H18O2 C12H18 C12H18 C6H14O3 C10H22O3 C9H20O3 C6H14O C6H14O C7H14O C8H14O4 C8H14O4 C10H18O4 C10H18O6 C10H18O6 C6H6 C10H10 C22H46 C12H26 C12H24 C12H26O C12H27N C15H34Si C18H34O2

40 −76.2 28.2 35.6 30.5 30.4 31.7 37.1 133.7 150.1 117.2 100.0 118.0 71.2 83.6 +5.0 13.9 +6.5 +9.9 129.2 121.5 121.5 138.0 133.3 71.4 182.8 152.7 122.5 196.0 107.7 100.2 132.3 −28.4 182.4 182.4 95.0 116.3 68.2 141.2 25.2 84.3 194.3 200.0 227.9 214.8 173.7 150.0 170.8 199.0 213.5 157.8 182.8 230.6 42.3 114.0 107.9 147.4 136.3 117.0 +13.2 −4.5 78.1 124.6 110.5 154.8 199.7 181.8 +10.0 105.5 254.5 121.7 118.6 167.2 157.4 172.1 260.8

−70.4 36.7 44.2 39.0 38.9 40.4 45.8 142.6 160.4 127.1 109.7 127.8 80.8 92.8 13.0 22.1 14.5 18.1 139.5 131.0 131.0 148.0 143.5 81.3 194.0 163.8 133.2 208.0 118.0 110.7 142.4 −21.4 193.8 193.8 105.2 126.4 77.7 151.3 33.8 94.6 206.9 212.0 244.2 226.2 186.0 162.0 183.4 211.5 225.9 170.2 194.8 240.4 55.8 124.3 118.2 156.5 146.3 126.8 21.6 +3.4 85.8 134.8 120.0 166.0 211.7 192.6 18.1 116.0 268.3 132.1 128.5 177.8 168.0 184.2 273.0

Melting point, °C −138.5

−90.7 38 −62 −52.8 −123.7 −135 −119.5 −135.0 75 25.5 74.5 62.5 68 51.5 38

−83.2 61.5 89 −61

10 52.9 68.5 61 51.5 27 124 44 26.5

−105

−122 −60 −32.6

−66.9 44.5 −9.6 −31.5 24 51.5

2-72

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound

1

5

10

20

−16.5 −69.0 206.7 52.6 −159.5 36.3 −50.9 167.0 −43.4 28.5 −92.5 −29.5 47.0 −29.0 −31.3 −82.3 52.0 38.5 29.7 33.7 33.5 −9.8 44.0 107.6 −74.3 10.6 −18.4 −24.3 118.2 11.0 107.8 133.2 −89.8 +1.0 −5.1 +6.6 87.6 28.3 31.5 67.8 −14.5 −32.2 9.6 76.0 98.3 −168.3 112.0 −4.0 −11.0 −27.0 −44.5 53.0 −33.5

+5.6 −50.0 239.7 80.0 −148.5 63.1 −31.0 198.2 −23.5 54.0 −76.7 −8.7 70.7 −6.4 −12.0 −66.4 80.0 66.4 55.9 60.3 60.2 +13.9 72.0 136.4 −56.4 35.8 +4.0 −2.4 149.8 35.8 65.8 131.8 168.2 −73.9 25.4 +18.0 30.2 108.5 55.5 58.4 93.5 +9.2 −10.8 34.0 106.3 130.2 −158.3 142.4 +19.0 +10.5 +4.7 −24.0 79.7 −10.2

16.6 −40.3 254.5 93.7 −142.9 76.2 −20.7 213.5 −13.5 67.3 −68.7 +2.0 82.0 +5.0 −2.3 −58.3 93.8 80.6 69.0 73.9 73.9 25.9 86.0 150.3 −47.5 48.0 15.3 +8.4 165.0 48.0 77.8 143.7 186.0 −65.8 37.5 29.9 41.9 134.0 68.8 72.0 106.0 20.6 −0.1 46.3 121.7 146.0 −153.2 158.0 30.3 21.5 18.6 −13.6 92.1 +1.6

29.0 −29.5 270.6 108.4 −136.7 91.0 −9.8 230.0 −3.0 81.1 −59.9 13.0 94.4 17.7 +8.0 −48.6 109.0 96.0 83.1 88.5 88.5 38.6 101.4 166.8 −37.8 61.8 27.8 20.6 181.8 61.7 91.0 155.5 205.5 −56.8 50.4 42.0 54.3 150.3 83.6 86.7 119.8 33.4 +11.7 59.5 137.7 162.8 −147.6 173.5 42.5 33.0 32.7 −2.4 105.8 14.7

42.0 −17.3 289.1 124.6 −129.8 107.2 +3.7 247.0 +9.1 96.2 −50.0 26.0 108.1 31.8 19.0 −39.8 125.7 113.2 98.8 104.8 104.7 52.8 118.2 181.8 −26.7 77.0 41.5 33.8 199.8 76.3 105.6 168.8 226.5 −47.0 65.2 56.0 68.2 169.2 99.9 103.3 133.8 47.6 25.0 74.0 154.4 182.0 −141.3 191.0 56.0 45.8 48.0 +10.0 120.0 29.7

C4H10O2

−48.0

−26.2

−15.3

−3.0

+10.7

19.7

31.8

50.0

70.8

93.0

C3H8O2

−13.5

+10.2

22.0

34.3

47.8

56.4

68.0

85.3

104.3

124.4

−89.7 40.5 −117.0 −60.5 37.6 14.3 −20.0 50.0 −60.7 −112.5 −54.4 27.8 47.3 −76.7 26.5 −91.0

−73.8 67.3 −103.8 −42.2 63.8 38.8 +2.1 77.7 −41.9 −98.4 −34.3 57.3 74.0 −59.1 51.0 −75.6

−65.7 80.2 −97.7 −33.0 77.1 50.5 12.8 91.8 −32.3 −91.7 −24.3 72.1 87.3 −50.2 63.2 −67.8

−56.6 94.6 −90.0 −22.7 91.5 63.9 25.0 106.3 −21.9 −84.1 −13.1 88.0 101.8 −40.7 76.1 −59.1

−46.9 110.3 −81.8 −11.5 107.5 78.1 38.5 123.7 −10.2 −75.8 −0.9 106.0 117.7 −29.8 91.0 −49.4

−40.7 120.6 −76.4 −4.3 117.5 87.6 47.1 134.0 −2.9 −70.4 +7.2 117.8 127.6 −22.4 100.0 −43.3

−32.1 133.8 −69.3 −5.4 130.4 99.8 58.9 147.9 +7.2 −63.2 18.0 131.8 141.3 −13.0 112.0 −34.8

−19.5 153.2 −58.0 20.0 150.1 117.8 76.7 168.2 22.4 −52.0 34.1 149.8 160.2 +1.5 130.0 −22.0

−4.9 175.6 −45.5 37.1 172.5 138.0 97.0 192.2 39.8 −39.5 52.3 167.3 183.0 17.7 149.8 −7.8

+10.7 198.0 −32.0 54.3 195.0 158.2 118.5 216.0 57.4 −26.5 72.4 184.0 206.2 35.0 170.0 +7.5

Name

Formula

Epichlorohydrin 1,2-Epoxy-2-methylpropane Erucic acid Estragole (p-methoxy allyl benzene) Ethane Ethoxydimethylphenylsilane Ethoxytrimethylsilane Ethoxytriphenylsilane Ethyl acetate acetoacetate Ethylacetylene (1-butyne) Ethyl acrylate α-Ethylacrylic acid α-Ethylacrylonitrile Ethyl alcohol (ethanol) Ethylamine 4-Ethylaniline N-Ethylaniline 2-Ethylanisole 3-Ethylanisole 4-Ethylanisole Ethylbenzene Ethyl benzoate benzoylacetate bromide α-bromoisobutyrate n-butyrate isobutyrate Ethylcamphoronic anhydride Ethyl isocaproate carbamate carbanilate Ethylcetylamine Ethyl chloride chloroacetate chloroglyoxylate α-chloropropionate trans-cinnamate 3-Ethylcumene 4-Ethylcumene Ethyl cyanoacetate Ethylcyclohexane Ethylcyclopentane Ethyl dichloroacetate N,N-diethyloxamate N-Ethyldiphenylamine Ethylene Ethylene-bis-(chloroacetate) Ethylene chlorohydrin (2-chloroethanol) diamine (1,2-ethanediamine) dibromide (1,2-dibromethane) dichloride (1,2-dichloroethane) glycol (1,2-ethanediol) glycol diethyl ether (1,2-diethoxyethane) glycol dimethyl ether (1,2-dimethoxyethane) glycol monomethyl ether (2-methoxyethanol) oxide Ethyl α-ethylacetoacetate fluoride formate 2-furoate glycolate 3-Ethylhexane 2-Ethylhexyl acrylate Ethylidene chloride (1,1-dichloroethane) fluoride (1,1-difluoroethane) Ethyl iodide Ethyl l-leucinate Ethyl levulinate Ethyl mercaptan (ethanethiol) Ethyl methylcarbamate Ethyl methyl ether

C3H5ClO C4H8O C22H42O2 C10H12O C2H6 C10H16OSi C5H14OSi C20H20OSi C4H8O2 C6H10O3 C4H6 C5H8O2 C5H8O2 C5H7N C2H6O C2H7N C8H11N C8H11N C9H12O C9H12O C9H12O C8H10 C9H10O2 C11H12O3 C2H5Br C6H11BrO2 C6H12O2 C6H12O2 C11H16O5 C8H16O2 C3H7NO2 C9H11NO2 C18H39N C2H5Cl C4H7ClO2 C4H5ClO3 C5H9ClO2 C11H12O2 C11H16 C11H16 C5H7NO2 C8H16 C7H14 C4H6Cl2O2 C8H15NO3 C14H15N C2H4 C6H8Cl2O4 C2H5ClO C2H8N2 C2H4Br2 C2H4Cl2 C2H6O2 C6H14O2

C2H4O C8H14O3 C2H5F C3H6O2 C7H8O3 C4H8O3 C8H18 C11H20O2 C2H4Cl2 C2H4F2 C2H5I C8H17NO2 C7H12O3 C2H6S C4H9NO2 C3H8O

40

60

100

200

400

760

Temperature, °C 50.6 −9.7 300.2 135.2 −125.4 127.5 11.5 258.3 16.6 106.0 −43.4 33.5 116.7 40.6 26.0 −33.4 136.0 123.6 109.0 115.5 115.4 61.8 129.0 191.9 −19.5 86.7 50.1 42.3 211.5 85.8 114.8 177.3 239.8 −40.6 74.0 65.2 77.3 181.2 110.2 113.8 142.1 56.7 33.4 83.6 166.0 193.7 −137.3 201.8 64.1 53.8 57.9 18.1 129.5 39.0

62.0 +1.2 314.4 148.5 −119.3 131.4 22.1 273.5 27.0 118.5 −34.9 44.5 127.5 53.0 34.9 −25.1 149.8 137.3 122.3 129.2 128.4 74.1 143.2 205.0 −10.0 99.8 62.0 53.5 226.6 98.4 126.2 187.9 256.8 −32.0 86.0 76.6 89.3 196.0 124.3 127.2 152.8 69.0 45.0 96.1 180.3 209.8 −131.8 215.0 75.0 62.5 70.4 29.4 141.8 51.8

79.3 98.0 117.9 17.5 36.0 55.5 336.5 358.8 381.5 168.7 192.0 215.0 −110.2 −99.7 −88.6 151.5 175.0 199.5 38.1 56.3 75.7 295.0 319.5 344.0 42.0 59.3 77.1 138.0 158.2 180.8 −21.6 −6.9 +8.7 61.5 80.0 99.5 144.0 160.7 179.2 71.6 92.2 114.0 48.4 63.5 78.4 −12.3 +2.0 16.6 170.6 194.2 217.4 156.9 180.8 204.0 142.1 164.2 187.1 149.7 172.8 196.5 149.2 172.3 196.5 92.7 113.8 136.2 164.8 188.4 213.4 223.8 244.7 265.0 +4.5 21.0 38.4 119.7 141.2 163.6 79.8 100.0 121.0 71.0 90.0 110.0 248.5 272.8 298.0 117.8 139.2 160.4 144.2 164.0 184.0 203.8 220.0 237.0 283.3 313.0 342.0 −18.6 −3.9 +12.3 103.8 123.8 144.2 94.5 114.7 135.0 107.2 126.2 146.5 219.3 245.0 271.0 145.4 168.2 193.0 148.3 171.8 195.8 169.8 187.8 206.0 87.8 109.1 131.8 62.4 82.3 103.4 115.2 135.9 156.5 202.8 226.5 252.0 233.0 258.8 286.0 −123.4 −113.9 −103.7 237.3 259.5 283.5 91.8 110.0 128.8 81.0 99.0 117.2 89.8 110.1 131.5 45.7 64.0 82.4 158.5 178.5 197.3 71.8 94.1 119.5

Melting point, °C −25.6 33.5 −183.2

−82.4 −45 −130 −71.2 −112 −80.6 −4 −63.5

−94.9 −34.6 −117.8 −93.3 −88.2 49 52.5 −139 −26 12

−111.3 −138.6

−169 −69 8.5 10 −35.3 −15.6

−111.3 −79 34

−96.7 −117 −105 −121

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-73

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

1-Ethylnaphthalene Ethyl α-naphthyl ketone (1-propionaphthone) Ethyl 3-nitrobenzoate 3-Ethylpentane 4-Ethylphenetole 2-Ethylphenol 3-Ethylphenol 4-Ethylphenol Ethyl phenyl ether (phenetole) Ethyl propionate Ethyl propyl ether Ethyl salicylate 3-Ethylstyrene 4-Ethylstyrene Ethylisothiocyanate 2-Ethyltoluene 3-Ethyltoluene 4-Ethyltoluene Ethyl trichloroacetate Ethyltrimethylsilane Ethyltrimethyltin Ethyl isovalerate 2-Ethyl-1,4-xylene 4-Ethyl-1,3-xylene 5-Ethyl-1,3-xylene Eugenol iso-Eugenol Eugenyl acetate Fencholic acid d-Fenchone dl-Fenchyl alcohol Fluorene Fluorobenzene 2-Fluorotoluene 3-Fluorotoluene 4-Fluorotoluene Formaldehyde Formamide Formic acid trans-Fumaryl chloride Furfural (2-furaldehyde) Furfuryl alcohol Geraniol Geranyl acetate Geranyl n-butyrate Geranyl isobutyrate Geranyl formate Glutaric acid Glutaric anhydride Glutaronitrile Glutaryl chloride Glycerol Glycerol dichlorohydrin (1,3-dichloro-2-propanol) Glycol diacetate Glycolide (1,4-dioxane-2,6-dione) Guaicol (2-methoxyphenol) Heneicosane Heptacosane Heptadecane Heptaldehyde (enanthaldehyde) n-Heptane Heptanoic acid (enanthic acid) 1-Heptanol Heptanoyl chloride (enanthyl chloride) 2-Heptene Heptylbenzene Heptyl cyanide (enanthonitrile) Hexachlorobenzene Hexachloroethane Hexacosane Hexadecane 1-Hexadecene n-Hexadecyl alcohol (cetyl alcohol)

1

100

200

400

760

Melting point, °C

164.1

180.0

204.6

230.8

258.1

−27

206.9 192.6 17.5 119.8 117.9 130.0 131.3 86.6 27.2 −12.0 136.7 99.2 97.3 50.8 76.4 73.3 73.6 85.5 −9.0 30.0 55.2 96.0 97.2 92.6 155.8 167.0 183.0 171.8 99.5 110.8 185.2 +11.5 34.7 37.0 37.8 −70.6 137.5 24.0 79.5 82.1 95.7 141.8 150.0 170.1 164.0 136.2 226.3 185.5 176.4 128.3 198.0 93.0

218.2 205.0 25.7 129.8 127.9 139.8 141.7 95.4 35.1 −4.0 147.6 109.6 107.6 59.8 86.0 82.9 83.2 94.4 −1.2 38.4 64.0 106.2 107.4 103.0 167.3 178.2 194.0 181.5 109.8 120.2 197.8 19.6 43.7 45.8 46.5 −65.0 147.0 32.4 89.0 91.5 104.0 151.5 160.3 180.2 174.0 147.2 235.5 196.2 189.5 139.1 208.0 102.0

233.5 220.3 36.9 143.5 141.8 152.0 154.2 108.4 45.2 +6.8 161.5 123.2 121.5 71.9 99.0 95.9 96.3 107.4 +9.2 50.0 75.9 120.0 121.2 116.5 182.2 194.0 209.7 194.0 123.6 132.3 214.7 30.4 55.3 57.5 58.1 −57.3 157.5 43.8 101.0 103.4 115.9 165.3 175.2 193.8 187.7 160.7 247.0 212.5 205.5 151.8 220.1 114.8

255.5 244.6 53.8 163.2 161.6 171.8 175.0 127.9 61.7 23.3 183.7 144.0 142.0 90.0 119.0 115.5 116.1 125.8 25.0 67.3 93.8 140.2 141.8 137.4 204.7 217.2 232.5 215.0 144.0 150.0 240.3 47.2 73.0 75.4 76.0 −46.0 175.5 61.4 120.0 121.8 133.1 185.6 196.3 214.0 207.6 182.6 265.0 236.5 230.0 172.4 240.0 133.3

280.2 270.6 73.0 185.7 184.5 193.3 197.4 149.8 79.8 41.6 207.0 167.2 165.0 110.1 141.4 137.8 136.4 146.0 42.8 87.6 114.0 163.1 164.4 159.6 228.3 242.3 257.4 237.8 166.8 173.2 268.6 65.7 92.8 95.4 96.1 −33.0 193.5 80.3 140.0 141.8 151.8 207.8 219.8 235.0 228.5 205.8 283.5 261.0 257.3 195.3 263.0 153.5

306.0 298.0 93.5 208.0 207.5 214.0 219.0 172.0 99.1 61.7 231.5 191.5 189.0 131.0 165.1 161.3 162.0 167.0 62.0 108.8 134.3 186.9 188.4 183.7 253.5 267.5 282.0 264.1 191.0 201.0 295.0 84.7 114.0 116.0 117.0 −19.5 210.5 100.6 160.0 161.8 170.0 230.0 243.3 257.4 251.0 230.0 303.0 287.0 286.2 217.0 290.0 174.3

106.1 148.6 121.6 243.4 305.7 195.8 66.3 22.3 139.5 99.8 86.4 21.5 144.0 92.6 206.0 102.3 295.2 181.3 178.8 219.8

115.8 158.2 131.0 255.3 318.3 207.3 74.0 30.6 148.5 108.0 93.5 30.0 154.8 103.0 219.0 112.0 307.8 193.2 190.8 234.3

128.0 173.2 144.0 272.0 333.5 223.0 84.0 41.8 160.0 119.5 102.7 41.3 170.2 116.8 235.5 124.2 323.2 208.5 205.3 251.7

147.8 194.0 162.7 296.5 359.4 247.8 102.0 58.7 179.5 136.6 116.3 58.6 193.3 137.7 258.5 143.1 348.4 231.7 226.8 280.2

168.3 217.0 184.1 323.8 385.0 274.5 125.5 78.0 199.6 155.6 130.7 78.1 217.8 160.0 283.5 163.8 374.6 258.3 250.0 312.7

190.5 240.0 205.0 350.5 410.6 303.0 155.0 98.4 221.5 175.8 145.0 98.5 244.0 184.6 309.4 185.6 399.8 287.5 274.0 344.0

5

10

20

70.0

101.4

116.8

133.8

152.0

C13H12O C9H9NO4 C7H16 C10H14O C8H10O C8H10O C8H10O C8H10O C5H10O2 C5H12O C9H10O3 C10H12 C10H12 C3H5NS C9H12 C9H12 C9H12 C4H5Cl3O2 C5H14Si C5H14Sn C7H14O2 C10H14 C10H14 C10H14 C10H12O2 C10H12O2 C12H14O3 C10H16O2 C10H16O C10H18O C13H10 C6H5F C7H7F C7H7F C7H7F CH2O CH3NO CH2O2 C4H2Cl2O2 C5H4O2 C5H6O2 C10H18O C12H20O2 C14H24O2 C14H24O2 C11H18O2 C5H8O4 C5H6O3 C5H6N2 C5H6Cl2O2 C3H8O3 C3H6Cl2O

124.0 108.1 −37.8 48.5 46.2 60.0 59.3 18.1 −28.0 −64.3 61.2 28.3 26.0 −13.2 9.4 7.2 7.6 20.7 −60.6 −30.0 −6.1 25.7 26.3 22.1 78.4 86.3 101.6 101.7 28.0 45.8 −43.4 −24.2 −22.4 −21.8

155.5 140.2 −17.0 75.7 73.4 86.8 86.5 43.7 −7.2 −45.0 90.0 55.0 52.7 +10.6 34.8 32.3 32.7 45.5 −41.4 −7.6 +17.0 52.0 53.0 48.8 108.1 117.0 132.3 128.7 54.7 70.3 129.3 −22.8 −2.2 −0.3 +0.3

70.5 −20.0 +15.0 18.5 31.8 69.2 73.5 96.8 90.9 61.8 155.5 100.8 91.3 56.1 125.5 28.0

96.3 −5.0 38.5 42.6 56.0 96.8 102.7 125.2 119.6 90.3 183.8 133.3 123.7 84.0 153.8 52.2

171.0 155.0 −6.8 89.5 87.0 100.2 100.2 56.4 +3.4 −35.0 104.2 68.3 66.3 22.8 47.6 44.7 44.9 57.7 −31.8 +3.8 28.7 65.6 66.4 62.1 123.0 132.4 148.0 142.3 68.3 82.1 146.0 −12.4 +8.9 +11.0 11.8 −88.0 109.5 +2.1 51.8 54.8 68.0 110.0 117.9 139.0 133.0 104.3 196.0 149.5 140.0 97.8 167.2 64.7

188.1 173.6 +4.7 103.8 101.5 114.5 115.0 70.3 14.3 −24.0 119.3 82.8 80.8 36.1 61.2 58.2 58.5 70.6 −21.0 16.1 41.3 79.8 80.6 76.5 138.7 149.0 164.2 155.8 83.0 95.6 164.2 −1.2 21.4 23.4 24.0 −79.6 122.5 10.3 65.0 67.8 81.0 125.6 133.0 153.8 147.9 119.8 210.5 166.0 156.5 112.3 182.2 78.0

C6H10O4 C4H4O4 C7H8O2 C21H44 C27H56 C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H13ClO C7H14 C13H20 C7H13N C6Cl6 C2Cl6 C26H54 C16H34 C16H32 C16H34O

38.3

64.1 103.0 79.1 188.0 248.6 145.2 32.7 −12.7 101.3 64.3 54.6 −14.1 94.6 47.8 149.3 49.8 240.0 135.2 131.7 158.3

77.1 116.6 92.0 205.4 266.8 160.0 43.0 −2.1 113.2 74.7 64.6 −3.5 110.0 61.6 166.4 73.5 257.4 149.8 146.2 177.8

90.8 132.0 106.0 223.2 284.6 177.7 54.0 +9.5 125.6 85.8 75.0 +8.3 126.0 76.3 185.7 87.6 275.8 164.7 162.0 197.8

60

Temperature, °C

Formula C12H12

40

52.4 152.6 211.7 115.0 12.0 −34.0 78.0 42.4 34.2 −35.8 64.0 21.0 114.4 32.7 204.0 105.3 101.6 122.7

47 −118.6 −45 −4 46.5 −30.2 −72.6 1.3 −5.9 −95.5

−99.3

−10 295 19 5 35 113 −42.1 −80 −110.8 −92 8.2

97.5

17.9 −31 97 28.3 40.4 59.5 22.5 −42 −90.6 −10 34.6

230 186.6 56.6 18.5 4 49.3

2-74

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Formula

n-Hexadecylamine (cetylamine) Hexaethylbenzene n-Hexane 1-Hexanol 2-Hexanol 3-Hexanol 1-Hexene n-Hexyl levulinate n-Hexyl phenyl ketone (enanthophenone) Hydrocinnamic acid Hydrogen cyanide (hydrocyanic acid) Hydroquinone 4-Hydroxybenzaldehyde α-Hydroxyisobutyric acid α-Hydroxybutyronitrile 4-Hydroxy-3-methyl-2-butanone 4-Hydroxy-4-methyl-2-pentanone 3-Hydroxypropionitrile Indene Iodobenzene Iodononane 2-Iodotoluene α-Ionone Isoprene Lauraldehyde Lauric acid Levulinaldehyde Levulinic acid d-Limonene Linalyl acetate Maleic anhydride Menthane 1-Menthol Menthyl acetate benzoate formate Mesityl oxide Methacrylic acid Methacrylonitrile Methane Methanethiol Methoxyacetic acid N-Methylacetanilide Methyl acetate acetylene (propyne) acrylate alcohol (methanol) Methylamine N-Methylaniline Methyl anthranilate benzoate 2-Methylbenzothiazole α-Methylbenzyl alcohol Methyl bromide 2-Methyl-1-butene 2-Methyl-2-butene Methyl isobutyl carbinol (2-methyl4-pentanol) n-butyl ketone (2-hexanone) isobutyl ketone (4-methyl-2-pentanone) n-butyrate isobutyrate caprate caproate caprylate chloride chloroacetate cinnamate α-Methylcinnamic acid Methylcyclohexane Methylcyclopentane Methylcyclopropane Methyl n-decyl ketone (n-dodecan-2-one) dichloroacetate N-Methyldiphenylamine

C16H35N C18H30 C6H14 C6H14O C6H14O C6H14O C6H12 C11H20O3 C13H18O C9H10O2 CHN C6H6O2 C7H6O2 C4H8O3 C5H9NO C5H10O2 C6H12O2 C3H5NO C9H8 C6H5I C9H19I C7H7I C13H20O C5H8 C12H24O C12H24O2 C5H8O2 C5H8O3 C10H16 C12H20O2 C4H2O3 C10H20 C10H20O C12H22O2 C17H24O2 C11H20O2 C6H10O C4H6O2 C4H5N CH4 CH4S C3H6O3 C9H11NO C3H6O2 C3H4 C4H6O2 CH4O CH5N C7H9N C8H9NO2 C8H8O2 C8H7NS C8H10O CH3Br C5H10 C5H10 C6H14O C6H12O C6H12O C5H10O2 C5H10O2 C11H22O2 C7H14O2 C9H18O2 CH3Cl C3H5ClO2 C10H10O2 C10H10O2 C7H14 C6H12 C4H8 C12H24O C3H4Cl2O2 C13H13N

1

5

10

20

40

60

100

123.6

157.8 134.3 −34.5 47.2 34.8 25.7 −38.0 120.0 130.3 133.5 −55.3 153.3 153.2 98.5 65.8 69.3 46.7 87.8 44.3 50.6 96.2 65.9 108.8 −62.3 108.4 150.6 54.9 128.1 40.4 82.5 63.4 35.7 83.2 85.8 154.2 75.8 +14.1 48.5 −23.3 −199.0 −75.3 79.3 103.8 −38.6 −97.5 −23.6 −25.3 −81.3 62.8 109.0 64.4 97.5 75.2 −80.6 −72.8 −57.0

176.0 150.3 −25.0 58.2 45.0 36.7 −28.1 134.7 145.5 148.7 −47.7 163.5 169.7 110.5 77.8 81.0 58.8 102.0 58.5 64.0 109.0 79.8 123.0 −53.3 123.7 166.0 68.0 141.8 53.8 96.0 78.7 48.3 96.0 100.0 170.0 90.0 26.0 60.0 −12.5 −195.5 −67.5 92.0 118.6 −29.3 −90.5 −13.5 −16.2 −73.8 76.2 124.2 77.3 111.2 88.0 −72.8 −64.3 −47.9

195.7 168.0 −14.1 70.3 55.9 49.0 −17.2 150.2 161.0 165.0 −39.7 174.6 186.8 123.8 90.7 94.0 72.0 117.9 73.9 78.3 123.0 95.6 139.0 −43.5 140.2 183.6 82.7 154.1 68.2 111.4 95.0 62.7 110.3 115.4 186.3 105.8 37.9 72.7 −0.6 −191.8 −58.8 106.5 135.1 −19.1 −82.9 −2.7 −6.0 −65.9 90.5 141.5 91.8 125.5 102.1 −64.0 −54.8 −37.9

215.7 187.7 −2.3 83.7 67.9 62.2 −5.0 167.8 178.9 183.3 −30.9 192.0 206.0 138.0 104.8 108.2 86.7 134.1 90.7 94.4 138.1 112.4 155.6 −32.6 157.8 201.4 98.3 169.5 84.3 127.7 111.8 78.3 126.1 132.1 204.3 123.0 51.7 86.4 +12.8 −187.7 −49.2 122.0 152.2 −7.9 −74.3 +9.2 +5.0 −56.9 106.0 159.7 107.8 141.2 117.8 −54.2 −44.1 −26.7

228.8 199.7 +5.4 92.0 76.0 70.7 +2.8 179.0 189.8 194.0 −25.1 203.0 217.5 146.4 113.9 117.4 96.0 144.7 100.8 105.0 147.7 123.8 166.3 −25.4 168.7 212.7 108.4 178.0 94.6 138.1 122.0 88.6 136.1 143.2 215.8 133.8 60.4 95.3 21.5 −185.1 −43.1 131.8 164.2 −0.5 −68.8 17.3 12.1 −51.3 115.8 172.0 117.4 150.4 127.4 −48.0 −37.3 −19.4

245.8 216.0 15.8 102.8 87.3 81.8 13.0 193.6 204.2 209.0 −17.8 216.5 233.5 157.7 125.0 129.0 108.2 157.7 114.7 118.3 159.8 138.1 181.2 −16.0 184.5 227.5 121.8 190.2 108.3 151.8 135.8 102.1 149.4 156.7 230.4 148.0 72.1 106.6 32.8 −181.4 −34.8 144.5 179.8 +9.4 −61.3 28.0 21.2 −43.7 129.8 187.8 130.8 163.9 140.3 −39.4 −28.0 −9.9

+22.1 28.8 +19.7 −5.5 −13.0 93.5 30.0 61.7 −99.5 19.0 108.1 155.0 −14.0 −33.8 −80.6 106.0 26.7 134.0

33.3 38.8 30.0 +5.0 −2.9 108.0 42.0 74.9 −92.4 30.0 123.0 169.8 −3.2 −23.7 −72.8 120.4 38.1 149.7

45.4 50.0 40.8 16.7 +8.4 123.0 55.4 89.0 −84.8 41.5 140.0 185.2 +8.7 −12.8 −64.0 136.0 50.7 165.8

58.2 62.0 52.8 29.6 21.0 139.0 70.0 105.3 −76.0 54.5 157.9 201.8 22.0 −0.6 −54.2 152.4 64.7 184.0

67.0 69.8 60.4 37.4 28.9 148.6 79.7 115.3 −70.4 63.0 170.0 212.0 30.5 +7.2 −48.0 163.8 73.6 195.4

78.0 79.8 70.4 48.0 39.6 161.5 91.4 128.0 −63.0 73.5 185.8 224.8 42.1 17.9 −39.3 177.5 85.4 210.1

200

400

760

Temperature, °C

−53.9 24.4 14.6 +2.5 −57.5 90.0 100.0 102.2 −71.0 132.4 121.2 73.5 41.0 44.6 22.0 58.7 16.4 24.1 70.0 37.2 79.5 −79.8 77.7 121.0 28.1 102.0 14.0 55.4 44.0 +9.7 56.0 57.4 123.2 47.3 −8.7 25.5 −44.5 −205.9 −90.7 52.5 −57.2 −111.0 −43.7 −44.0 −95.8 36.0 77.6 39.0 70.0 49.0 −96.3 −89.1 −75.4 −0.3 +7.7 −1.4 −26.8 −34.1 63.7 +5.0 34.2 −2.9 77.4 125.7 −35.9 −53.7 −96.0 77.1 3.2 103.5

272.2 300.4 330.0 241.7 268.5 298.3 31.6 49.6 68.7 119.6 138.0 157.0 103.7 121.8 139.9 98.3 117.0 135.5 29.0 46.8 66.0 215.7 241.0 266.8 225.0 248.3 271.3 230.8 255.0 279.8 −5.3 +10.2 25.9 238.0 262.5 286.2 256.8 282.6 310.0 175.2 193.8 212.0 142.0 159.8 178.8 146.5 165.5 185.0 126.8 147.5 167.9 178.0 200.0 221.0 135.6 157.8 181.6 139.8 163.9 188.6 179.0 199.3 219.5 160.0 185.7 211.0 202.5 225.2 250.0 −1.2 +15.4 32.6 207.8 231.8 257.0 249.8 273.8 299.2 142.0 164.0 187.0 208.3 227.4 245.8 128.5 151.4 175.0 173.3 196.2 220.0 155.9 179.5 202.0 122.7 146.0 169.5 168.3 190.2 212.0 178.8 202.8 227.0 253.2 277.1 301.0 169.8 194.2 219.0 90.0 109.8 130.0 123.9 142.5 161.0 50.0 70.3 90.3 −175.5 −168.8 −161.5 −22.1 −7.9 +6.8 163.5 184.2 204.0 202.3 227.4 253.0 24.0 40.0 57.8 −49.8 −37.2 −23.3 43.9 61.8 80.2 34.8 49.9 64.7 −32.4 −19.7 −6.3 149.3 172.0 195.5 212.4 238.5 266.5 151.4 174.7 199.5 183.2 204.5 225.5 159.0 180.7 204.0 −26.5 −11.9 +3.6 −13.8 +2.5 20.2 +4.9 21.6 38.5 94.9 94.3 85.6 64.3 55.7 181.6 109.8 148.1 −51.2 90.5 209.6 245.0 59.6 34.0 −26.0 199.0 103.2 232.8

113.5 111.0 102.0 83.1 73.6 202.9 129.8 170.0 −38.0 109.5 235.0 266.8 79.6 52.3 −11.3 222.5 122.6 257.0

131.7 127.5 119.0 102.3 92.6 224.0 150 193.0 −24.0 130.3 263.0 288.0 100.9 71.8 +4.5 246.5 143.0 282.0

Melting point, °C 130 −95.3 −51.6 −98.5 48.5 −13.2 170.3 115.5 79 −47 −2 −28.5

−146.7 44.5 48 33.5 −96.9 58 42.5 54.5 −59 15 −182.5 −121 102 −98.7 −102.7 −97.8 −93.5 −57 24 −12.5 15.4 −93 −135 −133 −56.9 −84.7 −84.7 −18 −40 −97.7 −31.9 33.4 −126.4 −142.4

−7.6

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-75

Vapor Pressures of Organic Compounds, up to 1 atm* (Continued) Pressure, mmHg Compound Name

Formula

Methyl n-dodecyl ketone (2-tetradecanone) Methylene bromide (dibromomethane) chloride (dichloromethane) Methyl ethyl ketone (2-butanone) 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane Methyl fluoride formate α-Methylglutaric anhydride Methyl glycolate 2-Methylheptadecane 2-Methylheptane 3-Methylheptane 4-Methylheptane 2-Methyl-2-heptene 6-Methyl-3-hepten-2-ol 6-Methyl-5-hepten-2-ol 2-Methylhexane 3-Methylhexane Methyl iodide laurate levulinate methacrylate myristate α-naphthyl ketone (1-acetonaphthone) β-naphthyl ketone (2-acetonaphthone) n-nonyl ketone (undecan-2-one) palmitate n-pentadecyl ketone (2-heptdecanone) 2-Methylpentane 3-Methylpentane 2-Methyl-1-pentanol 2-Methyl-2-pentanol Methyl n-pentyl ketone (2-heptanone) phenyl ether (anisole) 2-Methylpropene Methyl propionate 4-Methylpropiophenone 2-Methylpropionyl bromide Methyl propyl ether n-propyl ketone (2-pentanone) isopropyl ketone (3-Methyl-2-butanone) 2-Methylquinoline Methyl salicylate α-Methyl styrene 4-Methyl styrene Methyl n-tetradecyl ketone (2-hexadecanone) thiocyanate isothiocyanate undecyl ketone (2-tridecanone) isovalerate Monovinylacetylene (butenyne) Myrcene Myristaldehyde Myristic acid (tetradecanoic acid) Naphthalene 1-Naphthoic acid 2-Naphthoic acid 1-Naphthol 2-Naphthol 1-Naphthylamine 2-Naphthylamine Nicotine 2-Nitroaniline 3-Nitroaniline 4-Nitroaniline 2-Nitrobenzaldehyde 3-Nitrobenzaldehyde Nitrobenzene Nitroethane Nitroglycerin Nitromethane 2-Nitrophenol 2-Nitrophenyl acetate

C14H28O CH2Br2 CH2Cl2 C4H8O C8H18 C8H18 CH3F C2H4O2 C6H8O3 C3H6O3 C18H38 C8H18 C8H18 C8H18 C8H16 C8H16O C8H16O C7H16 C7H16 CH3I C13H26O2 C6H10O3 C5H8O2 C15H30O2 C12H10O C12H10O C11H22O C17H34O2 C17H34O C6H14 C6H14 C6H14O C6H14O C7H14O C7H8O C4H8 C4H8O2 C10H12O C4H7BrO C4H10O C5H10O C5H10O C10H9N C8H8O3 C9H10 C9H10 C16H32O C2H3NS C2H3NS C13H26O C6H12O2 C4H4 C10H16 C14H28O C14H28O2 C10H8 C11H8O2 C11H8O2 C10H8O C10H8O C10H9N C10H9N C10H14N2 C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO3 C7H5NO3 C6H5NO2 C2H5NO2 C3H5N3O9 CH3NO2 C6H5NO3 C8H7NO4

1

5

10

20

40

99.3 −35.1 −70.0 −48.3 −24.0 −23.9 −147.3 −74.2 93.8 +9.6 119.8 −21.0 −19.8 −20.4 −16.1 41.6 41.9 −40.4 −39.0

130.0 −13.2 −52.1 −28.0 −1.8 −1.4 −137.0 −57.0 125.4 33.7 152.0 +1.3 +2.6 +1.5 +6.7 65.0 66.0 −19.5 −18.1 −55.0 117.9 66.4 −10.0 145.7 146.3 152.3 95.5 166.8 161.6 −41.7 −39.8 38.0 +16.8 43.6 30.0 −96.5 −21.5 89.3 38.4 −54.3 +8.0 −1.0 104.0 81.6 34.0 42.0

145.5 −2.4 −43.3 −17.7 +9.5 +9.9 −131.6 −48.6 141.8 45.3 168.7 12.3 13.3 12.4 17.8 76.7 77.8 −9.1 −7.8 −45.8 133.2 79.7 +1.0 160.8 161.5 168.5 108.9 184.3 178.0 −32.1 −30.1 49.6 27.6 55.5 42.2 −81.9 −11.8 103.8 50.6 −45.4 17.9 +8.3 119.0 95.3 47.1 55.1

161.3 +9.7 −33.4 −6.5 21.7 22.3 −125.9 −39.2 157.7 58.1 186.0 24.4 25.4 24.5 30.4 89.3 90.4 +2.3 +3.6 −35.6 149.0 93.7 11.0 177.8 178.4 185.7 123.1 202.0 196.4 −21.4 −19.4 61.6 38.8 67.7 55.8 −73.4 −1.0 120.2 64.1 −35.4 28.5 18.3 134.0 110.0 61.8 69.2

179.8 23.3 −22.3 +6.0 35.2 36.2 −119.1 −28.7 177.5 72.3 204.8 37.9 38.9 38.0 44.0 102.7 104.0 14.9 16.4 −24.2 166.0 109.5 25.5 195.8 196.8 203.8 139.0

151.5 +9.8 −8.3 117.0 +2.9 −77.7 40.0 132.0 174.1 74.2 184.0 189.7 125.5 128.6 137.7 141.6 91.8 135.7 151.5 177.6 117.7 127.4 71.6 +1.5 167 −7.9 76.8 128.0

167.3 21.6 +5.4 131.8 14.0 −70.0 53.2 148.3 190.8 85.8 196.8 202.8 142.0 145.5 153.8 157.6 107.2 150.4 167.8 194.4 133.4 142.8 84.9 12.5 188 +2.8 90.4 142.0

184.6 34.5 20.4 147.8 26.4 −61.3 67.0 166.2 207.6 101.7 211.2 216.9 158.0 161.8 171.6 175.8 123.7 167.7 185.5 213.2 150.0 159.0 99.3 24.8 210 14.1 105.8 155.8

60

100

200

400

760

191.4 31.6 −15.7 14.0 43.9 45.0 −115.0 −21.9 189.9 81.8 216.3 46.6 47.6 46.6 52.8 111.5 112.8 23.0 24.5 −16.9 176.8 119.3 34.5 207.5 208.6 214.7 148.6

206.0 42.3 −6.3 25.0 55.7 57.1 −109.0 −12.9 205.0 93.7 231.5 58.3 59.4 58.3 64.6 122.6 123.8 34.1 35.6 −7.0 190.8 133.0 47.0 222.6 223.8 229.8 161.0

228.2 58.5 +8.0 41.6 73.6 75.3 −99.9 +0.8 229.1 111.8 254.5 76.0 77.1 76.1 82.3 139.5 140.0 50.8 52.4 +8.0

253.3 79.0 24.1 60.0 94.0 96.2 −89.5 16.0 255.5 131.7 279.8 96.2 97.4 96.3 102.2 156.6 156.6 69.8 71.6 25.3

278.0 98.6 40.7 79.6 115.6 118.3 −78.2 32.0 282.5 151.5 306.5 117.6 118.9 117.7 122.5 175.5 174.3 90.0 91.9 42.4

153.4 63.0 245.3 246.7 251.6 181.2

175.8 82.0 269.8 270.5 275.8 202.3

197.7 101.0 295.8 295.5 301.0 224.0

214.3 −9.7 −7.3 74.7 51.3 81.2 70.7 −63.8 +11.0 138.0 79.4 −24.3 39.8 29.6 150.8 126.2 77.8 85.0

226.7 −1.9 +0.1 83.4 58.8 89.8 80.1 −57.7 18.7 149.3 88.8 −17.4 47.3 36.2 161.7 136.7 88.3 95.0

242.0 +8.1 10.5 94.2 69.2 100.0 93.0 −49.3 29.0 164.2 101.6 −8.1 56.8 45.5 176.2 150.0 102.2 108.6

265.8 24.1 26.5 111.3 85.0 116.1 112.3 −36.7 44.2 187.4 120.5 +6.0 71.0 59.0 197.8 172.6 121.8 128.7

291.7 41.6 44.2 129.8 102.6 133.2 133.8 −22.2 61.8 212.7 141.7 22.5 86.8 73.8 211.7 197.5 143.0 151.2

319.5 60.3 63.3 147.9 121.2 150.2 155.5 −6.9 79.8 238.5 163.0 39.1 103.3 88.9 246.5 223.2 165.4 175.0

203.7 49.0 38.2 165.7 39.8 −51.7 82.6 186.0 223.5 119.3 225.0 231.5 177.8 181.7 191.5 195.7 142.1 186.0 204.2 234.2 168.8 177.7 115.4 38.0 235 27.5 122.1 172.8

215.0 58.1 47.5 176.6 48.2 −45.3 92.6 198.3 237.2 130.2 234.5 241.3 190.0 193.7 203.8 208.1 154.7 197.8 216.5 245.9 180.7 189.5 125.8 46.5 251 35.5 132.6 181.7

230.5 70.4 59.3 191.5 59.8 −37.1 106.0 214.5 250.5 145.5 245.8 252.7 206.0 209.8 220.0 224.3 169.5 213.0 232.1 261.8 196.2 204.3 139.9 57.8

254.4 89.8 77.5 214.0 77.3 −24.1 126.0 240.4 272.3 167.7 263.5 270.3 229.6 234.0 244.9 249.7 193.8 236.3 255.3 284.5 220.0 227.4 161.2 74.8

279.8 110.8 97.8 238.3 96.7 −10.1 148.3 267.9 294.6 193.2 281.4 289.5 255.8 260.6 272.2 277.4 219.8 260.0 280.2 310.2 246.8 252.1 185.8 94.0

307.0 132.9 119.0 262.5 116.7 +5.3 171.5 297.8 318.0 217.9 300.0 308.5 282.5 288.0 300.8 306.1 247.3 284.5 305.7 336.0 273.5 278.3 210.6 114.0

46.6 146.4 194.1

63.5 167.6 213.0

82.0 191.0 233.5

101.2 214.5 253.0

Temperature, °C

87.8 39.8 −30.5 115.0 115.6 120.2 68.2 134.3 129.6 −60.9 −59.0 15.4 −4.5 19.3 +5.4 −105.1 −42.0 59.6 13.5 −72.2 −12.0 −19.9 75.3 54.0 7.4 16.0 109.8 −14.0 −34.7 86.8 −19.2 −93.2 14.5 99.0 142.0 52.6 156.0 160.8 94.0 104.3 108.0 61.8 104.0 119.3 142.4 85.8 96.2 44.4 −21.0 127 −29.0 49.3 100.0

Melting point, °C −52.8 −96.7 −85.9 −114.5 −90 −99.8

−109.5 −120.8 −121.1

−118.2 −64.4 5 18.5 55.5 15 30 −154 −118 −103 −37.3 −140.3 −87.5

−77.8 −92 −1 −8.3 −23.2

−51 35.5 28.5

23.5 57.5 80.2 160.5 184 96 122.5 50 111.5 71.5 114 146.5 40.9 58 +5.7 −90 11 −29 45

2-76

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

1-Nitropropane 2-Nitropropane 2-Nitrotoluene 3-Nitrotoluene 4-Nitrotoluene 4-Nitro-1,3-xylene (4-nitro-m-xylene) Nonacosane Nonadecane n-Nonane 1-Nonanol 2-Nonanone Octacosane Octadecane n-Octane n-Octanol (1-octanol) 2-Octanone n-Octyl acrylate iodide (1-Iodooctane) Oleic acid Palmitaldehyde Palmitic acid Palmitonitrile Pelargonic acid Pentachlorobenzene Pentachloroethane Pentachloroethylbenzene Pentachlorophenol Pentacosane Pentadecane 1,3-Pentadiene 1,4-Pentadiene Pentaethylbenzene Pentaethylchlorobenzene n-Pentane iso-Pentane (2-methylbutane) neo-Pentane (2,2-dimethylpropane) 2,3,4-Pentanetriol 1-Pentene α-Phellandrene Phenanthrene Phenethyl alcohol (phenyl cellosolve) 2-Phenetidine Phenol 2-Phenoxyethanol 2-Phenoxyethyl acetate Phenyl acetate Phenylacetic acid Phenylacetonitrile Phenylacetyl chloride Phenyl benzoate 4-Phenyl-3-buten-2-one Phenyl isocyanate isocyanide Phenylcyclohexane Phenyl dichlorophosphate m-Phenylene diamine (1,3-phenylenediamine) Phenylglyoxal Phenylhydrazine N-Phenyliminodiethanol 1-Phenyl-1,3-pentanedione 2-Phenylphenol 4-Phenylphenol 3-Phenyl-1-propanol Phenyl isothiocyanate Phorone iso-Phorone Phosgene (carbonyl chloride) Phthalic anhydride Phthalide Phthaloyl chloride 2-Picoline Pimelic acid α-Pinene β-Pinene

1

5

10

20

40

C3H7NO2 −9.6 C3H7NO2 −18.8 C7H7NO2 50.0 C7H7NO2 50.2 C7H7NO2 53.7 C8H9NO2 65.6 C29H60 234.2 C19H40 133.2 C9H20 +1.4 C9H20O 59.5 C9H18O 32.1 C28H58 226.5 C18H38 119.6 C8H18 −14.0 C8H18O 54.0 C8H16O 23.6 C11H20O2 58.5 C8H17I 45.8 C18H34O2 176.5 C16H32O 121.6 C16H32O2 153.6 C16H31N 134.3 C9H18O2 108.2 C6HCl5 98.6 C2HCl5 +1.0 C8H5Cl5 96.2 C6HCl5O C25H52 194.2 C15H32 91.6 C5H8 −71.8 C5H8 −83.5 C16H26 86.0 C16H25Cl 90.0 C5H12 −76.6 C5H12 −82.9 C5H12 −102.0 C5H12O3 155.0 C5H10 −80.4 C10H16 20.0 C14H10 118.2 C8H10O2 58.2 C8H11NO 67.0 C6H6O 40.1 C8H10O2 78.0 C10H12O3 82.6 C8H8O2 38.2 C8H8O2 97.0 C8H7N 60.0 C8H7ClO 48.0 C13H10O2 106.8 C10H10O 81.7 C7H5NO 10.6 C7H5N 12.0 C12H16 67.5 C6H5Cl2O2P 66.7

+13.5 +4.1 79.1 81.0 85.0 95.0 269.8 166.3 25.8 86.1 59.0 260.3 152.1 +8.3 76.5 48.4 87.7 74.8 208.5 154.6 188.1 168.3 126.0 129.7 27.2 130.0

25.3 15.8 93.8 96.0 100.5 109.8 286.4 183.5 38.0 99.7 72.3 277.4 169.6 19.2 88.3 60.9 102.0 90.0 223.0 171.8 205.8 185.8 137.4 144.3 39.8 148.0

230.0 121.0 −53.8 −66.2 120.0 123.8 −62.5 −65.8 −85.4 189.3 −63.3 45.7 154.3 85.9 94.7 62.5 106.6 113.5 64.8 127.0 89.0 75.3 141.5 112.2 36.0 37.0 96.5 95.9

248.2 135.4 −45.0 −57.1 135.8 140.7 −50.1 −57.0 −76.7 204.5 −54.5 58.0 173.0 100.0 108.6 73.8 121.2 128.0 78.0 141.3 103.5 89.0 157.8 127.4 48.5 49.7 111.3 110.0

37.9 28.2 109.6 112.8 117.7 125.8 303.6 200.8 51.2 113.8 87.2 295.4 187.5 31.5 101.0 74.3 117.8 105.9 240.0 190.0 223.8 204.2 149.8 160.0 53.9 166.0 192.2 266.1 150.2 −34.8 −47.7 152.4 158.1 −40.2 −47.3 −67.2 220.5 −46.0 72.1 193.7 114.8 123.7 86.0 136.0 144.5 92.3 156.0 119.4 103.6 177.0 143.8 62.5 63.4 126.4 125.9

51.8 41.8 126.3 130.7 136.0 143.3 323.2 220.0 66.0 129.0 103.4 314.2 207.4 45.1 115.2 89.8 135.6 123.8 257.2 210.0 244.4 223.8 163.7 178.5 69.9 186.2 211.2 285.6 167.7 −23.4 −37.0 171.9 178.2 −29.2 −36.5 −56.1 239.6 −34.1 87.8 215.8 130.5 139.9 100.1 152.2 162.3 108.1 173.6 136.3 119.8 197.6 161.3 77.7 78.3 144.0 143.4

C6H8N2 C8H6O2 C6H8N2 C10H15NO2 C11H12O2 C12H10O C12H10O C9H12O C7H5NS C9H14O C9H14O CCl2O C8H4O3 C8H6O2 C8H4Cl2O2 C6H7N C7H12O4 C10H16 C10H16

71.8 145.0 98.0 100.0

131.2 75.0 101.6 179.2 128.5 131.6

74.7 47.2 42.0 38.0 −92.9 96.5 95.5 86.3 −11.1 163.4 −1.0 +4.2

102.4 75.6 68.3 66.7 −77.0 121.3 127.7 118.3 +12.6 196.2 +24.6 30.0

147.0 87.8 115.8 195.8 144.0 146.2 176.2 116.0 89.8 81.5 81.2 −69.3 134.0 144.0 134.2 24.4 212.0 37.3 42.3

163.8 100.7 131.5 213.4 159.9 163.3 193.8 131.2 115.5 95.6 96.8 −60.3 151.7 161.3 151.0 37.4 229.3 51.4 58.1

182.5 115.5 148.2 233.0 178.0 180.3 213.0 147.4 122.5 111.3 114.5 −50.3 172.0 181.0 170.0 51.2 247.0 66.8 71.5

60

200

400

760

60.5 50.3 137.6 142.5 147.9 153.8 334.8 232.8 75.5 139.0 113.8 326.8 219.7 53.8 123.8 99.0 145.6 135.4 269.8 222.6 256.0 236.6 172.3 190.1 80.0 199.0 223.4 298.4 178.4 −16.5 −30.0 184.2 191.0 −22.2 −29.6 −49.0 249.8 −27.1 97.6 229.9 141.2 149.8 108.4 163.2 174.0 118.1 184.5 147.7 129.8 210.8 172.6 87.7 88.0 154.2 153.6

72.3 62.0 151.5 156.9 163.0 168.5 350.0 248.0 88.1 151.3 127.4 341.8 236.0 65.7 135.2 111.7 159.1 150.0 286.0 239.5 271.5 251.5 184.4 205.5 93.5 216.0 239.6 314.0 194.0 −6.7 −20.6 200.0 208.0 −12.6 −20.2 −39.1 263.5 −17.7 110.6 249.0 154.0 163.5 121.4 176.5 189.2 131.6 198.2 161.8 143.5 227.8 187.8 100.6 101.0 169.3 168.0

90.2 80.0 173.7 180.3 186.7 191.7 373.2 271.8 107.5 170.5 148.2 364.8 260.6 83.6 152.0 130.4 180.2 173.3 309.8 264.1 298.7 277.1 203.1 227.0 114.0 241.8 261.8 339.0 216.1 +8.0 −6.7 224.1 230.3 +1.9 −5.9 −23.7 284.5 −3.4 130.6 277.1 175.0 184.0 139.0 197.6 211.3 151.2 219.5 184.2 163.8 254.0 211.0 120.8 120.8 191.3 189.8

110.6 99.8 197.7 206.8 212.5 217.5 397.2 299.8 128.2 192.1 171.2 388.9 288.0 104.0 173.8 151.0 204.0 199.3 334.7 292.3 326.0 304.5 227.5 251.6 137.2 269.3 285.0 365.4 242.8 24.7 +8.3 250.2 257.2 18.5 +10.5 −7.1 307.0 +12.8 152.0 308.0 197.5 207.0 160.0 221.0 235.0 173.5 243.0 208.5 186.0 283.5 235.4 142.7 142.3 214.6 213.0

131.6 120.3 222.3 231.9 238.3 244.0 421.8 330.0 150.8 213.5 195.0 412.5 317.0 125.6 195.2 172.9 227.0 225.5 360.0 321.0 353.8 332.0 253.5 276.0 160.5 299.0 309.3 390.3 270.5 42.1 26.1 277.0 285.0 36.1 27.8 +9.5 327.2 30.1 175.0 340.2 219.5 228.0 181.9 245.3 259.7 195.9 265.5 233.5 210.0 314.0 261.0 165.6 165.0 240.0 239.5

−108 −93 −4.1 15.5 51.9 +2 63.8 32 −53.7 −5 −19 61.6 28 −56.8 −15.4 −16

194.0 124.2 158.7 245.3 189.8 192.2 225.3 156.8 133.3 121.4 125.6 −44.0 185.3 193.5 182.2 59.9 258.2 76.8 81.2

209.9 136.2 173.5 260.6 204.5 205.9 240.9 170.3 147.7 134.0 140.6 −35.6 202.3 210.0 197.8 71.4 272.0 90.1 94.0

233.0 153.8 195.4 284.5 226.7 227.9 263.2 191.2 169.6 153.5 163.3 −22.3 228.0 234.5 222.0 89.0 294.5 110.2 114.1

259.0 173.5 218.2 311.3 251.2 251.8 285.5 212.8 194.0 175.3 188.7 −7.6 256.8 261.8 248.3 108.4 318.5 132.3 136.1

285.5 193.5 243.5 337.8 276.5 275.0 308.0 235.0 218.5 197.2 215.2 +8.3 284.5 290.0 275.8 128.8 342.1 155.0 158.3

62.8 73 19.5

Temperature, °C

Formula

99.8

Melting point, °C

100

−45.9 14 34 64.0 31 12.5 85.5 −22 188.5 53.3 10

−129.7 −159.7 −16.6

99.5 40.6 11.6 −6.7 76.5 −23.8 70.5 41.5 +7.5

56.5 164.5 −21.0 28 −104 130.8 73 88.5 −70 103 −55

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-77

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Piperidine Piperonal Propane Propenylbenzene Propionamide Propionic acid anhydride Propionitrile Propiophenone n-Propyl acetate iso-Propyl acetate n-Propyl alcohol (1-propanol) iso-Propyl alcohol (2-propanol) n-Propylamine Propylbenzene Propyl benzoate n-Propyl bromide (1-bromopropane) iso-Propyl bromide (2-bromopropane) n-Propyl n-butyrate isobutyrate iso-Propyl isobutyrate Propyl carbamate n-Propyl chloride (1-chloropropane) iso-Propyl chloride (2-chloropropane) iso-Propyl chloroacetate Propyl chloroglyoxylate Propylene Propylene glycol (1,2-Propanediol) Propylene oxide n-Propyl formate iso-Propyl formate 4,4′-iso-Propylidenebisphenol n-Propyl iodide (1-iodopropane) iso-Propyl iodide (2-iodopropane) n-Propyl levulinate iso-Propyl levulinate Propyl mercaptan (1-propanethiol) 2-iso-Propylnaphthalene iso-Propyl β-naphthyl ketone (2-isobutyronaphthone) 2-iso-Propylphenol 3-iso-Propylphenol 4-iso-Propylphenol Propyl propionate 4-iso-Propylstyrene Propyl isovalerate Pulegone Pyridine Pyrocatechol Pyrocaltechol diacetate (1,2-phenylene diacetate) Pyrogallol Pyrotartaric anhydride Pyruvic acid Quinoline iso-Quinoline Resorcinol Safrole Salicylaldehyde Salicylic acid Sebacic acid Selenophene Skatole Stearaldehyde Stearic acid Stearyl alcohol (1-octadecanol) Styrene Styrene dibromide [(1,2-dibromoethyl) benzene] Suberic acid Succinic anhydride Succinimide Succinyl chloride α-Terpineol Terpenoline

1

5

10

20

87.0 −128.9 17.5 65.0 4.6 20.6 −35.0 50.0 −26.7 −38.3 −15.0 −26.1 −64.4 6.3 54.6 −53.0 −61.8 −1.6 −6.2 −16.3 52.4 −68.3 −78.8 +3.8 9.7 −131.9 45.5 −75.0 −43.0 −52.0 193.0 −36.0 −43.3 59.7 48.0 −56.0 76.0

−7.0 117.4 −115.4 43.8 91.0 28.0 45.3 −13.6 77.9 −5.4 −17.4 +5.0 −7.0 −46.3 31.3 83.8 −33.4 −42.5 +22.1 +16.8 +5.8 77.6 −50.0 −61.1 28.1 32.3 −120.7 70.8 −57.8 −22.7 −32.7 224.2 −13.5 −22.1 86.3 74.5 −36.3 107.9

+3.9 132.0 −108.5 57.0 105.0 39.7 57.7 −3.0 92.2 +5.0 −7.2 14.7 +2.4 −37.2 43.4 98.0 −23.3 −32.8 34.0 28.3 17.0 90.0 −41.0 −52.0 40.2 43.5 −112.1 83.2 −49.0 −12.6 −22.7 240.8 −2.4 −11.7 99.9 88.0 −26.3 123.4

15.8 148.0 −100.9 71.5 119.0 52.0 70.4 +8.8 107.6 16.0 +4.2 25.3 12.7 −27.1 56.8 114.3 −12.4 −22.0 47.0 40.6 29.0 103.2 −31.0 −42.0 53.9 55.6 −104.7 96.4 −39.3 −1.7 −12.1 255.5 +10.0 0.0 114.0 102.4 −15.4 140.3

29.2 165.7 −92.4 87.7 134.8 65.8 85.6 22.0 124.3 28.8 17.0 36.4 23.8 −16.0 71.6 131.8 −0.3 −10.1 61.5 54.3 42.4 117.7 −19.5 −31.0 68.7 68.8 −96.5 111.2 −28.4 +10.8 −0.2 273.0 23.6 +13.2 130.1 118.1 −3.2 159.0

60

100

200

400

760

37.7 177.0 −87.0 97.8 144.3 74.1 94.5 30.1 135.0 37.0 25.1 43.5 30.5 −9.0 81.1 143.3 +7.5 −2.5 70.3 63.0 51.4 126.5 −12.1 −23.5 78.0 77.2 −91.3 119.9 −21.3 18.8 +7.5 282.9 32.1 21.6 140.6 127.8 +4.6 171.4

49.0 191.7 −79.6 111.7 156.0 85.8 107.2 41.4 149.3 47.8 35.7 52.8 39.5 +0.5 94.0 157.4 18.0 +8.0 82.6 73.9 62.3 138.3 −2.5 −13.7 90.3 88.0 −84.1 132.0 −12.0 29.5 17.8 297.0 43.8 32.8 154.0 141.8 15.3 187.6

66.2 214.3 −68.4 132.0 174.2 102.5 127.8 58.2 170.2 64.0 51.7 66.8 53.0 15.0 113.5 180.1 34.0 23.8 101.0 91.8 80.2 155.8 +12.2 +1.3 108.8 104.7 −73.3 149.7 +2.1 45.3 33.6 317.5 61.8 50.0 175.6 161.6 31.5 211.8

85.7 238.5 −55.6 154.7 194.0 122.0 146.0 77.7 194.2 82.0 69.8 82.0 67.8 31.5 135.7 205.2 52.0 41.5 121.7 112.0 100.0 175.8 29.4 18.1 128.0 123.0 −60.9 168.1 17.8 62.6 50.5 339.0 81.8 69.5 198.0 185.2 49.2 238.5

106.0 263.0 −42.1 179.0 213.0 141.1 167.0 97.1 218.0 101.8 89.0 97.8 82.5 48.5 159.2 231.0 71.0 60.0 142.7 133.9 120.5 195.0 46.4 36.5 148.6 150.0 −47.7 188.2 34.5 81.3 68.3 360.5 102.5 89.5 221.2 208.2 67.4 266.0

Temperature, °C

Formula C5H11N C8H6O3 C3H8 C9H10 C3H7NO C3H6O2 C6H10O3 C3H5N C9H10O C5H10O2 C5H10O2 C3H8O C3H8O C3H9N C9H12 C10H12O2 C3H7Br C3H7Br C7H14O2 C7H14O2 C7H14O2 C4H9NO2 C3H7Cl C3H7Cl C5H9ClO2 C5H7ClO3 C3H6 C3H8O2 C3H6O C4H8O2 C4H8O2 C15H16O2 C3H7I C3H7I C8H14O3 C8H14O3 C3H8S C13H14

40

C14H14O C9H12O C9H12O C9H12O C6H12O2 C11H14 C8H16O2 C10H16O C5H5N C6H6O2

133.2 56.6 62.0 67.0 −14.2 34.7 +8.0 58.3 −18.9

165.4 83.8 90.3 94.7 +8.0 62.3 32.8 82.5 +2.5 104.0

181.0 97.0 104.1 108.0 19.4 76.0 45.1 94.0 13.2 118.3

197.7 111.7 119.8 123.4 31.6 91.2 58.0 106.8 24.8 134.0

215.6 127.5 136.2 139.8 45.0 108.0 72.8 121.7 38.0 150.6

227.0 137.7 146.6 149.7 53.8 118.4 82.3 130.2 46.8 161.7

242.3 150.3 160.2 163.3 65.2 132.8 95.0 143.1 57.8 176.0

264.0 170.1 182.0 184.0 82.7 153.9 113.9 162.5 75.0 197.7

288.2 192.6 205.0 206.1 102.0 178.0 135.0 189.8 95.6 221.5

313.0 214.5 228.0 228.2 122.4 202.5 155.9 221.0 115.4 245.5

C10H10O4 C6H6O3 C5H6O3 C3H4O3 C9H7N C9H7N C6H6O2 C10H10O2 C7H6O2 C7H6O3 C10H18O4 C4H4Se C9H9N C18H36O C18H36O2 C18H36O C8H8

98.0 69.7 21.4 59.7 63.5 108.4 63.8 33.0 113.7 183.0 −39.0 95.0 140.0 173.7 150.3 −7.0

129.8 151.7 99.7 45.8 89.6 92.7 138.0 93.0 60.1 136.0 215.7 −16.0 124.2 174.6 209.0 185.6 +18.0

145.7 167.7 114.2 57.9 103.8 107.8 152.1 107.6 73.8 146.2 232.0 −4.0 139.6 192.1 225.0 202.0 30.8

161.8 185.3 130.0 70.8 119.8 123.7 168.0 123.0 88.7 156.8 250.0 +9.1 154.3 210.6 243.4 220.0 44.6

179.8 204.2 147.8 85.3 136.7 141.6 185.3 140.1 105.2 172.2 268.2 24.1 171.9 230.8 263.3 240.4 59.8

191.6 216.3 158.6 94.1 148.1 152.0 195.8 150.3 115.7 182.0 279.8 33.8 183.6 244.2 275.5 252.7 69.5

206.5 232.0 173.8 106.5 163.2 167.6 209.8 165.1 129.4 193.4 294.5 47.0 197.4 260.0 291.0 269.4 82.0

228.7 255.3 196.1 124.7 186.2 190.0 230.8 186.2 150.0 210.0 313.2 66.7 218.8 285.0 316.5 293.5 101.3

253.3 281.5 221.0 144.7 212.3 214.5 253.4 210.0 173.7 230.5 332.8 89.8 242.5 313.8 343.0 320.3 122.5

278.0 309.0 247.4 165.0 237.7 240.5 276.5 233.0 196.5 256.0 352.3 114.3 266.2 342.5 370.0 349.5 145.2

C8H8Br2 C8H14O4 C4H4O3 C4H5NO2 C4H4Cl2O2 C10H18O C10H16

86.0 172.8 92.0 115.0 39.0 52.8 32.3

115.6 205.5 115.0 143.2 65.0 80.4 58.0

129.8 219.5 128.2 157.0 78.0 94.3 70.6

145.2 238.2 145.3 174.0 91.8 109.8 84.8

161.8 254.6 163.0 192.0 107.5 126.0 100.0

172.2 265.4 174.0 203.0 117.2 136.3 109.8

186.3 279.8 189.0 217.4 130.0 150.1 122.7

207.8 300.5 212.0 240.0 149.3 171.2 142.0

230.0 322.8 237.0 263.5 170.0 194.3 163.5

254.0 345.5 261.0 287.5 192.5 217.5 185.0

Melting point, °C −9 37 −187.1 −30.1 79 −22 −45 −91.9 21 −92.5 −127 −85.8 −83 −99.5 −51.6 −109.9 −89.0 −95.2

−122.8 −117 −185 −112.1 −92.9 −98.8 −90 −112

15.5 26 61 −76

−42 105 133 13.6 −15 24.6 110.7 11.2 −7 159 134.5 95 63.5 69.3 58.5 −30.6 142 119.6 125.5 17 35

2-78

PHYSICAL AND CHEMICAL DATA

TABLE 2-10

Vapor Pressures of Organic Compounds, up to 1 atm (Continued) Pressure, mmHg Compound Name

Formula

1,1,1,2-Tetrabromoethane 1,1,2,2-Tetrabromoethane Tetraisobutylene Tetracosane 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene 1,2,4,5-Tetrachlorobenzene 1,1,2,2-Tetrachloro-1,2-difluoroethane 1,1,1,2-Tetrachloroethane 1,1,2,2-Tetrachloroethane 1,2,3,5-Tetrachloro-4-ethylbenzene Tetrachloroethylene 2,3,4,6-Tetrachlorophenol 3,4,5,6-Tetrachloro-1,2-xylene Tetradecane Tetradecylamine Tetradecyltrimethylsilane Tetraethoxysilane 1,2,3,4-Tetraethylbenzene Tetraethylene glycol Tetraethylene glycol chlorohydrin Tetraethyllead Tetraethylsilane Tetralin 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 1,2,4,5-Tetramethylbenzene 2,2,3,3-Tetramethylbutane Tetramethylene dibromide (1,4-dibromobutane) Tetramethyllead Tetramethyltin Tetrapropylene glycol monoisopropyl ether Thioacetic acid (mercaptoacetic acid) Thiodiglycol (2,2′-thiodiethanol) Thiophene Thiophenol (benzenethiol) α-Thujone Thymol Tiglaldehyde Tiglic acid Tiglonitrile Toluene Toluene-2,4-diamine 2-Toluic nitrile (2-tolunitrile) 4-Toluic nitrile (4-tolunitrile) 2-Toluidine 3-Toluidine 4-Toluidine 2-Tolyl isocyanide 4-Tolylhydrazine Tribromoacetaldehyde 1,1,2-Tribromobutane 1,2,2-Tribromobutane 2,2,3-Tribromobutane 1,1,2-Tribromoethane 1,2,3-Tribromopropane Triisobutylamine Triisobutylene 2,4,6-Tritertbutylphenol Trichloroacetic acid Trichloroacetic anhydride Trichloroacetyl bromide 2,4,6-Trichloroaniline 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,3,5-Trichlorobenzene 1,2,3-Trichlorobutane 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene Trichlorofluoromethane 2,4,5-Trichlorophenol 2,4,6-Trichlorophenol

C2H2Br4 C2H2Br4 C16H32 C24H50 C6H2Cl4 C6H2Cl4 C6H2Cl4 C2Cl4F2 C2H2Cl4 C2H2Cl4 C8H6Cl4 C2Cl4 C6H2Cl4O C8H6Cl4 C14H30 C14H31N C17H38Si C8H20O4Si C14H22 C8H18O5 C8H17ClO4 C8H20Pb C8H20Si C10H12 C10H14 C10H14 C10H14 C8H18 C4H8Br2 C4H12Pb C4H12Sn C15H32O5 C2H4O2S C4H10O2S C4H4S C6H6S C10H16O C10H14O C5H8O C5H8O2 C5H7N C7H8 C7H10N2 C8H7N C8H7N C7H9N C7H9N C7H9N C8H7N C7H10N2 C2HBr3O C4H7Br3 C4H7Br3 C4H7Br3 C2H3Br3 C3H5Br3 C12H27N C12H24 C18H30O C2HCl3O2 C4Cl6O3 C2BrCl3O C6H4Cl3N C6H3Cl3 C6H3Cl3 C6H3Cl3 C4H7Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 CCl3F C6H3Cl3O C6H3Cl3O

1

5

10

20

58.0 65.0 63.8 183.8 68.5 58.2

83.3 95.5 93.7 219.6 99.6 89.0

95.7 110.0 108.5 237.6 114.7 104.1

108.5 126.0 124.5 255.3 131.2 121.6

−37.5 −16.3 −3.8 77.0 −20.6 100.0 94.4 76.4 102.6 120.0 16.0 65.7 153.9 110.1 38.4 −1.0 38.0 42.6 40.6 45.0 −17.4

−16.0 +7.4 +20.7 110.0 +2.4 130.3 125.0 106.0 135.8 150.7 40.3 96.2 183.7 141.8 63.6 +23.9 65.3 68.7 65.8 65.0 +3.2

−5.0 19.3 33.0 126.0 13.8 145.3 140.3 120.7 152.0 166.2 52.6 111.6 197.1 156.1 74.8 36.3 79.0 81.8 77.8 74.6 13.5

32.0 −29.0 −51.3 116.6 60.0 42.0 −40.7 18.6 38.3 64.3 −25.0 52.0 −25.5 −26.7 106.5 36.7 42.5 44.0 41.0 42.0 25.2 82.4 18.5 45.0 41.0 38.2 32.6 47.5 32.3 18.0 95.2 51.0 56.2 −7.4 134.0 40.0 38.4

58.8 −6.8 −31.0 147.8 87.7 96.0 −20.8 43.7 65.7 92.8 −1.6 77.8 −2.4 −4.4 137.2 64.0 71.3 69.3 68.0 68.2 51.0 110.0 45.0 73.5 69.0 66.0 58.0 75.8 57.4 44.0 126.1 76.0 85.3 +16.7 157.8 70.0 67.3 63.8 27.2 −32.0 −2.0 −22.8 −67.6 102.1 105.9

72.4 +4.4 −20.6 163.0 101.5 128.0 −10.9 56.0 79.3 107.4 +10.0 90.2 +9.2 +6.4 151.7 77.9 85.8 81.4 82.0 81.8 64.0 123.8 58.0 87.8 83.2 79.8 70.6 90.0 69.8 56.5 142.0 88.2 99.6 29.3 170.0 85.6 81.7 78.0 40.0 −21.9 +8.3 −12.4 −59.0 117.3 120.2

40

60

100

200

400

760

Temperature, °C

+0.5 −52.0 −24.0 −43.8 −84.3 72.0 76.5

+6.7 32.1 46.2 143.7 26.3 161.0 156.0 135.6 170.0 183.5 65.8 127.7 212.3 172.6 88.0 50.0 93.8 95.8 91.0 88.0 24.6

123.2 144.0 142.2 276.3 149.2 140.0 146.0 19.8 46.7 60.8 162.1 40.1 179.1 174.2 152.7 189.0 201.5 81.1 145.8 228.0 190.0 102.4 65.3 110.4 111.5 105.8 104.2 36.8

132.0 155.1 152.6 288.4 160.0 152.0 157.7 28.1 56.0 70.0 175.0 49.2 190.0 185.8 164.0 200.2 213.3 90.7 156.7 237.8 200.5 111.7 74.8 121.3 121.8 115.4 114.8 44.5

144.0 170.0 167.5 305.2 175.7 168.0 173.5 38.6 68.0 83.2 191.6 61.3 205.2 200.5 178.5 215.7 227.8 103.6 172.4 250.0 214.7 123.8 88.0 135.3 135.7 128.3 128.1 54.8

161.5 192.5 190.0 330.5 198.0 193.7 196.0 55.0 87.2 102.2 215.3 79.8 227.2 223.0 201.8 239.8 250.0 123.5 196.0 268.4 236.5 142.0 108.0 157.2 155.7 149.9 149.5 70.2

181.0 217.5 214.6 358.0 225.5 220.0 220.5 73.1 108.2 124.0 243.0 100.0 250.4 248.3 226.8 264.6 275.0 146.2 221.4 288.0 258.2 161.8 130.2 181.8 180.0 173.7 172.1 87.4

200.0 243.5 240.0 386.4 254.0 246.0 245.0 92.0 130.5 145.9 270.0 120.8 275.0 273.5 252.5 291.2 300.0 168.5 248.0 307.8 281.5 183.0 153.0 207.2 204.4 197.9 195.9 106.3

87.6 16.6 −9.3 179.8 115.8 165.0 0.0 69.7 93.7 122.6 23.2 103.8 22.1 18.4 167.9 93.0 101.7 95.1 96.7 95.8 78.2 138.6 72.1 103.2 98.6 94.6 84.2 105.8 83.0 70.0 158.0 101.8 114.3 42.1 182.6 101.8 97.2 93.7 55.0 −10.8 21.6 −1.0 −49.7 134.0 135.8

104.0 30.3 +3.5 197.7 131.8 210.0 +12.5 84.2 110.0 139.8 37.0 119.0 36.7 31.8 185.7 110.0 109.5 110.0 113.5 111.5 94.0 154.1 87.8 120.2 116.0 111.8 100.0 122.8 97.8 86.7 177.4 116.3 131.2 57.2 195.8 119.8 114.8 110.8 71.5 +1.6 35.2 +11.9 −39.0 151.5 152.2

115.1 39.2 11.7 209.0 142.0 240.5 20.1 93.9 120.2 149.8 45.8 127.8 46.0 40.3 196.2 120.8 130.0 119.8 123.8 121.5 104.0 165.0 97.5 131.6 127.0 122.2 110.0 134.0 107.3 96.7 188.0 125.9 141.8 66.7 204.5 131.5 125.7 121.8 82.0 9.5 44.0 20.0 −32.3 162.5 163.5

128.7 50.8 22.8 223.3 154.0 285 30.5 106.6 134.0 164.1 57.7 140.5 58.2 51.9 211.5 135.0 145.2 133.0 136.7 133.7 117.7 178.0 110.2 146.0 141.8 136.3 123.5 148.0 119.7 110.0 203.0 137.8 155.2 79.5 214.6 146.0 140.0 136.0 96.2 20.0 55.7 31.4 −23.0 178.0 177.8

149.8 68.8 39.8 245.0

173.8 89.0 58.5 268.3

197.5 110.0 78.0 292.7

46.5 125.8 154.2 185.5 75.4 158.0 77.8 69.5 232.8 156.0 167.3 153.0 157.6 154.0 137.8 198.0 130.0 167.8 163.5 157.8 143.5 170.0 138.0 130.2 226.2 155.4 176.2 98.4 229.8 168.2 162.0 157.7 118.0 36.2 73.3 48.0 −9.1 201.5 199.0

64.7 146.7 177.8 209.2 95.5 179.2 99.7 89.5 256.0 180.0 193.0 176.2 180.6 176.9 159.9 219.5 151.6 192.0 188.0 182.2 165.4 195.0 157.8 153.0 250.6 175.2 199.8 120.2 246.4 193.5 187.7 183.0 143.0 54.6 93.0 67.0 +6.8 226.5 222.5

84.4 168.0 201.0 231.8 116.4 198.5 122.0 110.6 280.0 205.2 217.6 199.7 203.3 200.4 183.5 242.0 174.0 216.2 213.8 206.5 188.4 220.0 179.0 179.0 276.3 195.6 223.0 143.0 262.0 218.5 213.0 208.4 169.0 74.1 113.9 86.7 23.7 251.8 246.0

Melting point, °C

51.1 46.5 54.5 139 26.5 −68.7 −36 −19.0 69.5 5.5

11.6 −136 −31.0 −6.2 −24.0 79.5 −102.2 −20 −27.5 −16.5 −38.3 51.5 64.5 −95.0 99 −13 29.5 −16.3 −31.5 44.5 65.5

−26 16.5 −22 57 78 52.5 17 63.5 −30.6 −36.7 −73 62 68.5

VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-10

2-79

Vapor Pressures of Organic Compounds, up to 1 atm (Concluded) Pressure, mmHg Compound Name

Tri-2-chlorophenylthiophosphate 1,1,1-Trichloropropane 1,2,3-Trichloropropane 1,1,2-Trichloro-1,2,2-trifluoroethane Tricosane Tridecane Tridecanoic acid Triethoxymethylsilane Triethoxyphenylsilane 1,2,4-Triethylbenzene 1,3,4-Triethylbenzene Triethylborine Triethyl camphoronate citrate Triethyleneglycol Triethylheptylsilane Triethyloctylsilane Triethyl orthoformate phosphate Triethylthallium Trifluorophenylsilane Trimethallyl phosphate 2,3,5-Trimethylacetophenone Trimethylamine 2,4,5-Trimethylaniline 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene 2,2,3-Trimethylbutane Trimethyl citrate Trimethyleneglycol (1,3-propanediol) 1,2,4-Trimethyl-5-ethylbenzene 1,3,5-Trimethyl-2-ethylbenzene 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,4-Trimethyl-3-pentanone Trimethyl phosphate 2,4,5-Trimethylstyrene 2,4,6-Trimethylstyrene Trimethylsuccinic anhydride Triphenylmethane Triphenylphosphate Tripropyleneglycol Tripropyleneglycol monobutyl ether Tripropyleneglycol monoisopropyl ether Tritolyl phosphate Undecane Undecanoic acid 10-Undecenoic acid Undecan-2-ol n-Valeric acid iso-Valeric acid γ-Valerolactone Valeronitrile Vanillin Vinyl acetate 2-Vinylanisole 3-Vinylanisole 4-Vinylanisole Vinyl chloride (1-chloroethylene) cyanide (acrylonitrile) fluoride (1-fluoroethylene) Vinylidene chloride (1,1-dichloroethene) 4-Vinylphenetole 2-Xenyl dichlorophosphate 2,4-Xyaldehyde 2-Xylene (2-xylene) 3-Xylene (3-xylene) 4-Xylene (4-xylene) 2,4-Xylidine 2,6-Xylidine

1

5

10

20

217.2

231.2

246.7

261.7

−7.0 33.7 −49.4 206.3 98.3 166.3 +22.8 98.8 74.2 76.0

+4.2 46.0 −40.3 223.0 104.0 181.0 34.6 112.6 88.5 90.2 −148.0 166.0 144.0 158.1 114.6 120.6 40.5 82.1 51.7 +0.8 149.8 122.3 −73.8 109.0 55.9 50.7 47.4 −18.8 160.4 100.6 84.6 80.5 +3.9 −4.3 +6.9 +7.1 46.4 67.8 91.6 79.7 97.4 197.0 249.8 140.5 147.0 127.3 198.0 73.9 149.0 156.3 112.8 79.8 71.3 79.8 30.0 154.0 −18.0 81.0 83.0 85.7 −83.7 −20.3 −132.2 −51.2 105.6 187.0 99.0 32.1 28.3 27.3 93.0 87.0

16.2 59.3 −30.0 242.0 120.2 195.8 47.2 127.2 104.0 105.8 −140.6 183.6 171.1 174.0 130.3 137.7 53.4 97.8 67.7 12.3 169.8 137.5 −65.0 123.7 69.9 64.5 61.0 −7.5 177.2 115.5 99.7 96.0 16.0 +7.5 19.2 19.3 57.6 83.0 107.1 94.8 113.8 206.8 269.7 155.8 161.8 143.7 213.2 85.6 166.0 172.0 127.5 93.1 84.0 95.2 43.3 170.5 −7.0 94.7 97.2 100.0 −75.7 −9.0 −125.4 −41.7 120.3 205.0 114.0 45.1 41.1 40.1 107.6 102.7

29.9 74.0 −18.5 261.3 137.7 212.4 61.7 143.5 121.7 122.6 −131.4 201.8 190.4 191.3 148.0 155.7 67.5 115.7 85.4 25.4 192.0 154.2 −55.2 139.8 85.4 79.8 76.1 +5.2 194.2 131.0 106.0 113.2 29.5 20.7 33.0 32.9 69.8 100.0 124.2 111.8 131.0 215.5 290.3 173.7 179.8 161.4 229.7 104.4 185.6 188.7 143.7 107.8 98.0 101.9 57.8 188.7 +5.3 110.0 112.5 116.0 −66.8 +3.8 −118.0 −31.1 136.3 223.8 129.7 59.5 55.3 54.4 123.8 120.2

60

100

200

400

760

271.5

283.8

302.8

322.0

341.3

38.3 83.6 −11.2 273.8 148.2 222.0 70.4 153.2 132.2 133.4 −125.2 213.5 202.5 201.5 158.2 168.0 76.0 126.3 95.7 33.2 207.0 165.7 −48.8 149.5 95.3 89.5 85.8 13.3 205.5 141.1 126.3 123.8 38.1 29.1 41.8 41.6 77.3 110.0 135.5 122.3 142.2 221.2 305.2 184.6 190.2 173.2 239.8 115.2 197.2 199.5 153.7 116.6 107.3 122.4 66.9 199.8 13.0 119.8 122.3 126.1 −61.1 11.8 −113.0 −24.0 146.4 236.0 139.8 68.8 64.4 63.5 133.7 131.5

50.0 96.1 −1.7 289.8 162.5 236.0 82.7 167.5 146.8 147.7 −116.0 228.6 217.8 214.6 174.0 184.3 88.0 141.6 112.1 44.2 225.7 179.7 −40.3 162.0 108.8 102.8 98.9 24.4 219.6 153.4 140.3 137.9 49.9 40.7 53.8 53.4 87.6 124.0 149.8 136.8 156.5 228.4 322.5 199.0 204.4 187.8 252.2 128.1 212.5 213.5 167.2 128.3 118.9 136.5 78.6 214.5 23.3 132.3 135.3 139.7 −53.2 22.8 −106.2 −15.0 159.8 251.5 152.2 81.3 76.8 75.9 146.8 146.0

67.7 115.6 +13.5 313.5 185.0 255.2 101.0 188.0 168.3 168.3 −101.0 250.8 242.2 235.2 196.0 208.0 106.0 163.7 136.0 60.1 255.0 201.3 −27.0 182.3 129.0 122.7 118.6 41.2 241.3 172.8 160.3 158.4 67.8 58.1 72.0 71.3 102.2 145.0 171.8 157.8 179.8 239.7 349.8 220.2 224.4 209.7 271.8 149.3 237.8 232.8 187.7 146.0 136.2 157.7 97.7 237.3 38.4 151.0 154.0 159.0 −41.3 38.7 −95.4 −1.0 180.0 275.3 172.3 100.2 95.5 94.6 166.4 168.0

87.5 137.0 30.2 339.8 209.4 276.5 121.8 210.5 193.7 193.2 −81.0 276.0 267.5 256.6 221.0 235.0 125.7 187.0 163.5 78.7 288.5 224.3 −12.5 203.7 152.0 145.4 141.0 60.4 264.2 193.8 184.5 183.5 88.2 78.0 92.7 91.8 118.4 167.8 196.1 182.3 205.5 249.8 379.2 244.3 247.0 232.8 292.7 171.9 262.8 254.0 209.8 165.0 155.2 182.3 118.7 260.0 55.5 172.1 175.8 182.0 −28.0 58.3 −84.0 +14.8 202.8 301.5 194.1 121.7 116.7 115.9 188.3 193.7

108.2 158.0 47.6 366.5 234.0 299.0 143.5 233.5 218.0 217.5 −56.2 301.0 294.0 278.3 247.0 262.0 146.0 211.0 192.1 98.3 324.0 247.5 +2.9 234.5 176.1 169.2 164.7 80.9 287.0 214.2 208.1 208.0 109.8 99.2 114.8 113.5 135.0 192.7 221.2 207.0 231.0 259.2 413.5 267.2 269.5 256.6 313.0 195.8 290.0 275.0 232.0 184.4 175.1 207.5 140.8 285.0 72.5 194.0 197.5 204.5 −13.8 78.5 −72.2 31.7 225.0 328.5 215.5 144.4 139.1 138.3 211.5 217.9

Temperature, °C

Formula C18H12Cl3O3 188.2 PS C3H5Cl3 −28.8 C3H5Cl3 +9.0 C2Cl3F3 −68.0 C23H48 170.0 C13H28 59.4 C13H26O2 137.8 C7H18O3Si −1.5 C12H20O3Si 71.0 C12H18 46.0 C12H18 47.9 C6H15B C15H26O6 C12H20O7 107.0 C6H14O4 114.0 C13H30Si 70.0 C14H32Si 73.7 C7H16O3 +5.5 C6H15O4P 39.6 C6H15Tl +9.3 C6H5F3Si −31.0 C12H21PO4 93.7 C11H14O 79.0 C3H9N −97.1 C9H13N 68.4 C9H12 16.8 C9H12 13.6 C9H12 9.6 C7H16 C9H14O7 106.2 C3H8O2 59.4 C11H16 43.7 C11H16 38.8 C8H18 −29.0 C8H18 −36.5 C8H18 −25.8 C8H18 −26.3 C8H16O 14.7 C3H9O4P 26.0 C11H14 48.1 C11H14 37.5 C7H10O3 53.5 C19H16 169.7 C18H15O4P 193.5 C9H20O4 96.0 C13H28O4 101.5 C12H26O4 82.4 C21H21O4P 154.6 C11H24 32.7 C11H22O2 101.4 C11H20O2 114.0 C11H24O 71.1 C5H10O2 42.2 C5H10O2 34.5 C5H8O2 37.5 C5H9N −6.0 C8H8O3 107.0 C4H6O2 −48.0 C9H10O 41.9 C9H10O 43.4 C9H10O 45.2 C2H3Cl −105.6 C3H3N −51.0 C2H3F −149.3 C2H2Cl2 −77.2 C10H12O 64.0 C12H9Cl2PO 138.2 C9H10O 59.0 C8H10 −3.8 C8H10 −6.9 C8H10 −8.1 52.6 C8H11N C8H11N 44.0

40

150.2 138.7 144.0 99.8 104.8 29.2 67.8 37.6 −9.7 131.0 108.0 −81.7 95.9 42.9 38.3 34.7 146.2 87.2 71.2 67.0 −7.1 −15.0 −3.9 −4.1 36.0 53.7 77.0 65.7 82.6 188.4 230.4 125.7 131.6 112.4 184.2 59.7 133.1 142.8 99.0 67.7 59.6 65.8 +18.1 138.4 −28.0 68.0 69.9 72.0 −90.8 −30.7 −138.0 −60.0 91.7 171.1 85.9 +20.2 +16.8 +15.5 79.8 72.6

Melting point, °C

−77.7 −14.7 −35 47.7 −6.2 41

135

−63.0

−117.1 67 −25.5 −44.1 −44.8 −25.0 78.5

−112.3 −107.3 −101.5 −109.2

93.4 49.4

−25.6 29.5 24.5 −34.5 −37.6 81.5

−153.7 −82 −160.5 −122.5 75 −25.2 −47.9 +13.3

2-80

PHYSICAL AND CHEMICAL DATA

VAPOR PRESSURES OF SOLUTIONS UNITS CONVERSIONS

To convert cubic feet to cubic meters, multiply by 0.02832. To convert bars to pounds-force per square inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

For this subsection, the following units conversions are applicable: °F = 9⁄5°C + 32 To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

TABLE 2-11

Partial Pressures of Water over Aqueous Solutions of HCl* log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. Partial pressure of H2O, mmHg, °C

% HCl

A

B

0°

5°

10°

15°

20°

25°

30°

35°

40°

45°

50°

60°

6 10 14 18 20

8.99156 8.99864 8.97075 8.98014 8.97877

2282 2295 2300 2323 2334

4.18 3.84 3.39 2.87 2.62

6.04 5.52 4.91 4.21 3.83

8.45 7.70 6.95 5.92 5.40

11.7 10.7 9.65 8.26 7.50

15.9 14.6 13.1 11.3 10.3

21.8 20.0 18.0 15.4 14.1

29.1 26.8 24.1 20.6 19.0

39.4 35.5 31.9 27.5 25.1

50.6 47.0 42.1 36.4 33.3

66.2 61.5 55.3 47.9 43.6

86.0 80.0 72.0 62.5 57.0

139 130 116 102 93.5

22 24 26 28 30

9.02708 8.96022 9.01511 8.97611 9.00117

2363 2356 2390 2395 2422

2.33 2.05 1.76 1.50 1.26

3.40 3.04 2.60 2.24 1.90

4.82 4.31 3.71 3.21 2.73

6.75 6.03 5.21 4.54 3.88

9.30 8.30 7.21 6.32 5.41

12.6 11.4 9.95 8.75 7.52

17.1 15.4 13.5 11.8 10.2

22.8 20.4 18.0 15.8 13.7

30.2 27.1 24.0 21.1 18.4

39.8 35.7 31.7 27.9 24.3

52.0 46.7 41.5 36.5 32.0

32 34 36 38 40 42

9.03317 9.07143 9.11815 9.20783 9.33923 9.44953

2453 2487 2526 2579 2647 2709

1.04 0.85 0.68 0.53 0.41 0.31

1.57 1.29 1.03 0.81 0.63 0.48

2.27 1.87 1.50 1.20 0.94 0.72

3.25 2.70 2.19 1.75 1.37 1.06

4.55 3.81 3.10 2.51 2.00 1.56

6.37 5.35 4.41 3.60 2.88 2.30

11.7 9.95 8.33 6.92 5.68 4.60

15.7 13.5 11.4 9.52 7.85 6.45

21.0 18.1 15.4 13.0 10.7 8.90

27.7 24.0 20.4 17.4 14.5 12.1

8.70 7.32 6.08 5.03 4.09 3.28

70°

80°

90°

100° 110°

220 204 185 162 150

333 310 273 248 230

492 463 425 374 345

715 677 625 550 510

960 892 783 729

85.6 77.0 69.0 60.7 53.5

138 124 112 99.0 87.5

211 194 173 154 136

317 290 261 234 207

467 426 387 349 310

670 611 555 499 444

46.5 40.5 34.8 29.6 25.0 21.2

76.5 66.5 57.0 49.1 42.1 35.8

120 104 90.0 77.5 67.3 57.2

184 161 140 120 105 89.2

275 243 212 182 158 135

396 355 311 266 230 195

*Uncertainty, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the uncertainty is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths. International Critical Tables, vol. 3, p. 301.

TABLE 2-12

Partial Pressures of HCl over Aqueous Solutions of HCl*

log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. mmHg, °C

% HCl

A

2 4 6 8 10

11.8037 11.6400 11.2144 11.0406 10.9311

4736 0.0000117 0.000023 0.000044 0.000084 0.000151 0.000275 0.00047 0.00083 0.00140 0.00380 0.0100 0.0245 0.058 0.132 0.280 4471 0.000018 0.000036 .000069 .000131 .00024 .00044 .00077 .00134 .0023 .00385 .0064 .0165 .0405 .095 .21 .46 .93 4202 .000066 .000125 .000234 .000425 .00076 .00131 .00225 .0038 .0062 .0102 .0163 .040 .094 .206 .44 .92 1.78 4042 .000118 .000323 .000583 .00104 .00178 .0031 .00515 .0085 .0136 .022 .0344 .081 .183 .39 .82 1.64 3.10 3908 .00042 .00075 .00134 .00232 .00395 .0067 .0111 .0178 .0282 .045 .069 .157 .35 .73 1.48 2.9 5.4

12 14 16 18 20

10.7900 10.6954 10.6261 10.4957 10.3833

3765 3636 3516 3376 3245

22 24 26 28 30

10.3172 10.2185 10.1303 10.0115 9.8763

3125 .0734 2995 .175 2870 .41 2732 1.0 2593 2.4

.119 .277 .64 1.52 3.57

.187 .43 .98 2.27 5.23

.294 .66 1.47 3.36 7.60

.45 1.00 2.17 4.90 10.6

.68 1.49 3.20 7.05 15.1

1.02 2.17 4.56 9.90 21.0

32 34 36 38 40

9.7523 9.6061 9.5262 9.4670 9.2156

2457 2316 2229 2094 1939

5.7 13.1 29.0 63.0 130

8.3 18.8 41.0 87.0 176

11.8 26.4 56.4 117 233

16.8 36.8 78 158 307

23.5 50.5 105.5 210 399

32.5 68.5 142 277 515

44.5 92 188 360 627

42 44 46

8.9925 1800 253 8.8621 1681 510 940

332 655

430 840

560

709

900

B

0°

.00099 .0024 .0056 .0135 .0316

5°

.00175 .00415 .0095 .0225 .052

10°

.00305 .0071 .016 .037 .084

15°

.0052 .0118 .0265 .060 .132

20°

.0088 .0196 .0428 .095 .205

25°

.0145 .0316 .0685 .148 .32

30°

.0234 .050 .106 .228 .48

35°

.037 .078 .163 .345 .72

40°

45°

50°

60°

70°

80°

90°

100°

110°

.058 .121 .247 .515 1.06

.091 .185 .375 .77 1.55

.136 .275 .55 1.11 2.21

.305 .60 1.17 2.3 4.4

.66 1.25 2.40 4.55 8.5

1.34 2.50 4.66 8.6 15.6

2.65 4.8 8.8 15.7 28.1

5.1 9.0 16.1 28 49

9.3 16.0 28 48 83

1.50 3.14 6.50 13.8 28.6

2.18 4.5 9.2 19.1 39.4

3.14 6.4 12.7 26.4 53

4.42 8.9 17.5 35.7 71

8.6 16.9 32.5 64 124

16.3 31.0 58.5 112 208

29.3 54.5 100 188 340

52 94 169 309 542

90 157 276 493 845

146 253 436 760

60.0 122 246 465 830

81 161 322 598

107 211 416 758

141 273 535 955

238 450 860

390 720

623

970

*Uncertainty, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the uncertainty is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths. International Critical Tables, vol. 3, p. 301.

VAPOR PRESSURES OF SOLUTIONS

FIG. 2-1 Vapor pressures of H3PO4 aqueous: partial pressure of H2O vapor. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

TABLE 2-13 g SO2 / 100 g H2O

Temperature, °C 10

20

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.07 1.07 3.03 5.37 7.93

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

8.00 10.00 15.00 20.00

Vapor pressures of H3PO4 aqueous: weight of H2O in saturated air. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)

FIG. 2-2

Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide* Partial pressures of H2O and SO2, mmHg, °C 0

2.00 3.00 4.00 5.00 6.00

2-81

10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

*Extracted with permission from J. Chem Eng. Data 8, 1963: 333–336. Copyright 1963 American Chemical Society.

186 229 316 404 856

2-82

PHYSICAL AND CHEMICAL DATA

TABLE 2-14 °C 0 10 20 30 40 50 60 70 80 90

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0

60.0

.193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

.836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

70.0

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−02 .170E−01 .281E−01

.197E−04 .490E−04 .115E−04 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .133E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .481 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .682E−01 .101 .147 .208 .291

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.865

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.121 26.274 31.003

2.632 3.395 4.331 5.466 6.831 8.458 10.382 12.640 15.269 18.311

1.606 2.101 2.714 3.467 4.381 5.480 6.788 8.333 10.142 12.242

.913 1.220 1.609 2.096 2.699 3.435 4.326 5.395 6.663 8.155

.401 .542 .724 .952 1.237 1.587 2.012 2.525 3.136 3.857

300 66.934 310 75.495 320 84.705 330 94.567 340 105.083 350 116.251

64.407 72.781 81.816 91.518 101.894 112.946

60.081 68.100 76.792 86.172 96.252 107.043

54.868 62.553 70.947 80.077 89.969 100.646

47.346 54.470 62.337 70.988 80.463 90.802

36.360 42.395 49.164 56.721 65.123 74.426

21.808 25.804 30.343 35.473 41.240 47.692

14.665 17.438 20.591 24.153 28.154 32.622

9.897 11.912 14.227 16.867 19.855 23.217

4.701 5.680 6.806 8.093 9.551 11.193

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-14

2-83

Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .448E−05 .117E−04 .285E−04 .652E−04 .141E−03 .290E−03 .569E−03 .107E−02

.242E−06 .762E−06 .220E−05 .587E−05 .146E−04 .341E−04 .754E−04 .158E−03 .316E−03 .606E−03

.107E−06 .344E−06 .101E−05 .275E−05 .696E−05 .166E−04 .372E−04 .795E−04 .162E−03 .315E−03

.401E−07 .130E−06 .390E−06 .108E−05 .278E−05 .672E−05 .154E−04 .334E−04 .691E−04 .137E−03

.218E−07 .713E−07 .215E−06 .598E−06 .155E−05 .379E−05 .875E−05 .192E−04 .400E−04 .801E−04

.980E−08 .323E−07 .978E−07 .275E−06 .720E−06 .177E−05 .413E−05 .912E−05 .192E−04 .388E−04

.569E−08 .188E−07 .572E−07 .161E−06 .424E−06 .105E−05 .245E−05 .544E−05 .115E−04 .234E−04

.268E−08 .888E−08 .271E−07 .766E−07 .202E−06 .503E−06 .118E−05 .263E−05 .559E−05 .114E−04

.775E−09 .258E−08 .789E−08 .224E−07 .595E−07 .149E−06 .350E−06 .784E−06 .168E−05 .343E−05

.196E−09 .655E−09 .201E−08 .575E−08 .153E−07 .384E−07 .910E−07 .205E−06 .439E−06 .903E−06

100 110 120 130 140 150 160 170 180 190

.194E−02 .338E−02 .571E−02 .938E−02 .150E−01 .233E−01 .354E−01 .526E−01 .766E−01 .110

.112E−02 .198E−02 .341E−02 .569E−02 .923E−02 .146E−01 .225E−01 .340E−01 .502E−01 .729E−01

.590E−03 .107E−02 .186E−02 .315E−02 .519E−02 .832E−02 .130E−01 .199E−01 .298E−01 .438E−01

.261E−03 .479E−03 .851E−03 .146E−02 .245E−02 .399E−02 .633E−02 .983E−02 .149E−01 .222E−01

.154E−03 .285E−03 .511E−03 .886E−03 .149E−02 .245E−02 .393E−02 .614E−02 .941E−02 .141E−01

.752E−04 .141E−03 .254E−03 .445E−03 .757E−03 .125E−02 .202E−02 .319E−02 .492E−02 .744E−02

.455E−04 .855E−04 .155E−03 .278E−03 .467E−03 .776E−03 .126E−02 .199E−02 .309E−02 .469E−02

.223E−04 .420E−04 .766E−04 .135E−03 .232E−03 .387E−03 .629E−03 .999E−03 .155E−02 .236E−02

.674E−05 .128E−04 .233E−04 .414E−04 .711E−04 .119E−03 .194E−03 .309E−03 .482E−03 .735E−03

.178E−05 .339E−05 .623E−05 .111E−04 .191E−04 .321E−04 .526E−04 .840E−04 .131E−03 .201E−03

.631E−01 .894E−01 .125 .171 .232 .310 .409 .534 .689 .880

.325E−01 .467E−01 .660E−01 .918E−01 .126 .170 .227 .300 .391 .505

.208E−01 .300E−01 .427E−01 .598E−01 .825E−01 .112 .151 .200 .263 .341

.110E−01 .161E−01 .230E−01 .325E−01 .451E−01 .618E−01 .835E−01 .111 .147 .192

.698E−02 .102E−01 .147E−01 .208E−01 .290E−01 .398E−01 .540E−01 .723E−01 .957E−01 .125

.352E−02 .516E−02 .743E−02 .105E−01 .147E−01 .202E−01 .274E−01 .366E−01 .485E−01 .634E−01

.110E−02 .161E−02 .232E−02 .329E−02 .460E−02 .633E−02 .858E−02 .115E−01 .152E−01 .199E−01

.300E−03 .442E−03 .638E−03 .906E−03 .127E−02 .174E−02 .237E−02 .317E−02 .420E−02 .548E−02

.248 .316 .400 .502 .624 .770

.162 .208 .264 .331 .413 .511

.820E−01 .105 .133 .167 .208 .256

.257E−01 .328E−01 .415E−01 .520E−01 .646E−01 .795E−01

.708E−02 .905E−02 .114E−01 .143E−01 .178E−01 .218E−01

200 210 220 230 240 250 260 270 280 290

.154 .213 .290 .389 .514 .673 .870 1.112 1.407 1.763

.104 .146 .201 .273 .366 .485 .635 .822 1.052 1.335

300 310 320 330 340 350

2.190 2.696 3.292 3.990 4.801 5.738

1.676 2.088 2.578 3.159 3.843 4.641

1.112 1.394 1.732 2.133 2.608 3.164

.646 .817 1.025 1.274 1.571 1.922

.437 .556 .701 .875 1.083 1.331

2-84

PHYSICAL AND CHEMICAL DATA

TABLE 2-15

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.644E−29 .149E−27 .278E−26 .426E−25 .549E−24 .602E−23 .573E−22 .477E−21 .352E−20 .233E−19

.103E−27 .223E−26 .394E−25 .577E−24 .714E−23 .757E−22 .699E−21 .567E−20 .410E−19 .266E−18

.205E−26 .395E−25 .626E−24 .832E−23 .941E−22 .921E−21 .789E−20 .599E−19 .408E−18 .250E−17

.688E−25 .113E−23 .156E−22 .181E−21 .181E−20 .158E−19 .122E−18 .843E−18 .524E−17 .296E−16

.368E−23 .522E−22 .621E−21 .630E−20 .555E−19 .429E−18 .294E−17 .181E−16 .101E−15 .516E−15

.341E−21 .415E−20 .426E−19 .376E−18 .288E−17 .195E−16 .118E−15 .643E−15 .319E−14 .145E−13

.784E−19 .796E−18 .685E−17 .509E−16 .331E−15 .191E−14 .985E−14 .461E−13 .197E−12 .775E−12

.174E−17 .158E−16 .121E−15 .808E−15 .473E−14 .246E−13 .116E−12 .492E−12 .192E−11 .693E−11

.531E−16 .417E−15 .280E−14 .164E−13 .851E−13 .395E−12 .165E−11 .634E−11 .223E−10 .731E−10

.229E−14 .141E−13 .767E−13 .371E−12 .162E−11 .643E−11 .234E−10 .791E−10 .249E−09 .734E−09

100 110 120 130 140 150 160 170 180 190

.139E−18 .756E−18 .377E−17 .174E−16 .743E−16 .297E−15 .111E−14 .393E−14 .131E−13 .415E−13

.157E−17 .844E−17 .418E−16 .191E−15 .815E−15 .325E−14 .122E−13 .430E−13 .144E−12 .458E−12

.140E−16 .719E−16 .340E−15 .150E−14 .615E−14 .237E−13 .862E−13 .296E−12 .967E−12 .301E−11

.153E−15 .730E−15 .323E−14 .133E−13 .517E−13 .188E−12 .649E−12 .212E−11 .622E−11 .197E−10

.242E−14 .105E−13 .424E−13 .160E−12 .569E−12 .191E−11 .608E−11 .184E−10 .532E−10 .147E−09

.606E−13 .236E−12 .858E−12 .293E−11 .943E−11 .287E−10 .833E−10 .231E−09 .610E−09 .155E−08

.283E−11 .961E−11 .307E−10 .922E−10 .262E−09 .710E−09 .183E−08 .453E−08 .107E−07 .246E−07

.232E−10 .729E−10 .215E−09 .601E−09 .159E−08 .403E−08 .974E−08 .226E−07 .505E−07 .109E−06

.223E−09 .641E−09 .174E−08 .446E−08 .109E−07 .256E−07 .575E−07 .125E−06 .260E−06 .527E−06

.204E−08 .538E−08 .135E−07 .324E−07 .745E−07 .165E−06 .351E−06 .725E−06 .145E−05 .282E−05

200 210 220 230 240 250 260 270 280 290

.125E−12 .362E−12 .100E−11 .265E−11 .678E−11 .167E−10 .399E−10 .920E−10 .206E−09 .449E−09

.139E−11 .404E−11 .112E−10 .301E−10 .777E−10 .193E−09 .466E−09 .109E−08 .247E−08 .545E−08

.893E−11 .254E−10 .695E−10 .183E−09 .465E−09 .114E−08 .272E−08 .628E−08 .141E−07 .308E−07

.561E−10 .154E−09 .405E−09 .103E−08 .253E−08 .602E−08 .139E−07 .312E−07 .683E−07 .145E−06

.391E−09 .100E−08 .246E−08 .587E−08 .135E−07 .303E−07 .660E−07 .140E−06 .288E−06 .580E−06

.379E−08 .894E−08 .204E−07 .450E−07 .965E−07 .201E−06 .408E−06 .807E−06 .156E−05 .295E−05

.542E−07 .116E−06 .240E−06 .482E−06 .944E−06 .180E−05 .336E−05 .612E−05 .109E−04 .191E−04

.228E−06 .462E−06 .911E−06 .175E−05 .328E−05 .600E−05 .108E−04 .189E−04 .326E−04 .553E−04

.103E−05 .198E−05 .368E−05 .668E−05 .119E−04 .206E−04 .352E−04 .590E−04 .973E−04 .158E−03

.534E−05 .986E−05 .178E−04 .314E−04 .543E−04 .923E−04 .154E−03 .253E−03 .408E−03 .649E−03

300 310 320 330 340 350

.953E−09 .197E−08 .397E−08 .782E−08 .151E−07 .285E−07

.117E−07 .245E−07 .502E−07 .100E−06 .196E−06 .376E−06

.657E−07 .136E−06 .277E−06 .551E−06 .107E−05 .204E−05

.302E−06 .614E−06 .122E−05 .237E−05 .452E−05 .846E−05

.114E−05 .220E−05 .414E−05 .766E−05 .139E−04 .246E−04

.546E−05 .990E−05 .176E−04 .308E−04 .529E−04 .893E−04

.329E−04 .556E−04 .923E−04 .151E−03 .243E−03 .387E−03

.921E−04 .151E−03 .245E−03 .391E−03 .617E−03 .963E−03

.253E−03 .398E−03 .621E−03 .956E−03 .145E−02 .219E−02

.102E−02 .158E−02 .242E−02 .367E−02 .550E−02 .815E−02

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-15

2-85

Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded) Weight percent, H2SO4

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

0 10 20 30 40 50 60 70 80 90

.671E−13 .345E−12 .159E−11 .664E−11 .254E−10 .897E−10 .294E−09 .904E−09 .261E−08 .712E−08

.216E−12 .107E−11 .475E−11 .192E−10 .709E−10 .242E−09 .771E−09 .230E−08 .643E−08 .171E−07

.677E−12 .326E−11 .141E−10 .557E−10 .201E−09 .669E−09 .207E−08 .602E−08 .165E−07 .426E−07

.240E−11 .114E−10 .482E−10 .186E−09 .655E−09 .214E−08 .647E−08 .184E−07 .492E−07 .124E−06

.500E−11 .234E−10 .986E−10 .376E−09 .131E−08 .424E−08 .127E−07 .357E−07 .946E−07 .237E−06

.124E−10 .578E−10 .241E−09 .911E−09 .315E−08 .101E−07 .299E−07 .833E−07 .218E−06 .541E−06

.224E−10 .104E−09 .433E−09 .163E−08 .562E−08 .179E−07 .528E−07 .146E−06 .381E−06 .940E−06

.502E−10 .232E−09 .961E−09 .360E−08 .123E−07 .391E−07 .115E−06 .316E−06 .820E−06 .201E−05

.182E−09 .839E−09 .346E−08 .129E−07 .440E−07 .139E−06 .405E−06 .111E−05 .286E−05 .698E−05

.755E−09 .347E−08 .142E−07 .528E−07 .179E−06 .560E−06 .163E−05 .444E−05 .114E−04 .276E−04

100 110 120 130 140 150 160 170 180 190

.184E−07 .456E−07 .108E−06 .244E−06 .533E−06 .112E−05 .229E−05 .453E−05 .870E−05 .163E−04

.430E−07 .103E−06 .238E−06 .526E−06 .112E−05 .230E−05 .459E−05 .886E−05 .166E−04 .304E−04

.105E−06 .247E−06 .555E−06 .120E−05 .250E−05 .504E−05 .983E−05 .186E−04 .343E−04 .615E−04

.300E−06 .689E−06 .152E−05 .321E−05 .656E−05 .129E−04 .247E−04 .459E−04 .829E−04 .146E−03

.565E−06 .128E−05 .280E−05 .586E−05 .118E−04 .231E−04 .438E−04 .806E−04 .144E−03 .252E−03

.127E−05 .287E−05 .619E−05 .128E−04 .257E−04 .497E−04 .932E−04 .170E−03 .301E−03 .520E−03

.220E−05 .494E−05 .106E−04 .219E−04 .435E−04 .837E−04 .156E−03 .283E−03 .499E−03 .859E−03

.470E−05 .105E−04 .224E−04 .459E−04 .910E−04 .174E−03 .324E−03 .586E−03 .103E−02 .177E−02

.162E−04 .359E−04 .764E−04 .156E−03 .308E−03 .588E−03 .109E−02 .196E−02 .343E−02 .587E−02

.638E−04 .141E−03 .298E−03 .606E−03 .119E−02 .226E−02 .416E−02 .746E−02 .130E−01 .222E−01

200 210 220 230 240 250 260 270 280 290

.297E−04 .528E−04 .919E−04 .157E−03 .261E−03 .428E−03 .690E−03 .109E−02 .170E−02 .261E−02

.543E−04 .946E−04 .161E−03 .269E−03 .441E−03 .708E−03 .112E−02 .174E−02 .266E−02 .401E−02

.108E−03 .185E−03 .309E−03 .508E−03 .819E−03 .130E−02 .202E−02 .309E−02 .466E−02 .694E−02

.251E−03 .422E−03 .694E−03 .112E−02 .178E−02 .276E−02 .423E−02 .638E−02 .948E−02 .139E−01

.429E−03 .714E−03 .117E−02 .187E−02 .293E−02 .453E−02 .688E−02 .103E−01 .152E−01 .221E−01

.878E−03 .145E−02 .235E−02 .373E−02 .582E−02 .891E−02 .134E−01 .200E−01 .293E−01 .423E−01

.144E−02 .237E−02 .383E−02 .605E−02 .939E−02 .143E−01 .215E−01 .319E−01 .465E−01 .670E−01

.296E−02 .486E−02 .781E−02 .123E−01 .191E−01 .291E−01 .437E−01 .646E−01 .943E−01 .136

.981E−02 .161E−01 .258E−01 .405E−01 .627E−01 .955E−01 .143 .212 .309 .444

.370E−01 .603E−01 .965E−01 .152 .234 .356 .532 .786 1.144 1.646

300 310 320 330 340 350

.395E−02 .589E−02 .868E−02 .126E−01 .181E−01 .258E−01

.595E−02 .873E−02 .126E−01 .181E−01 .255E−01 .357E−01

.102E−01 .148E−01 .211E−01 .299E−01 .418E−01 .578E−01

.201E−01 .287E−01 .405E−01 .565E−01 .780E−01 .107

.318E−01 .451E−01 .632E−01 .877E−01 .120 .164

.604E−01 .852E−01 .119 .164 .224 .303

.953E−01 .134 .186 .256 .348 .470

.193 .272 .378 .520 .708 .956

.632 .889 1.236 1.703 2.323 3.142

100.0

2.339 3.289 4.575 6.303 8.603 11.640

2-86

PHYSICAL AND CHEMICAL DATA

TABLE 2-16

Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid* Weight percent, H2SO4

°C

10.0

20.0

30.0

40.0

50.0

60.0

70.0

75.0

80.0

85.0

0 10 20 30 40 50 60 70 80 90

.576E−21 .634E−20 .588E−19 .468E−18 .324E−17 .197E−16 .107E−15 .526E−15 .235E−14 .960E−14

.843E−20 .874E−19 .769E−18 .584E−17 .389E−16 .229E−15 .121E−14 .581E−14 .254E−13 .102E−12

.141E−18 .131E−17 .104E−16 .721E−16 .441E−15 .241E−14 .119E−13 .535E−13 .221E−12 .844E−12

.344E−17 .276E−16 .193E−15 .119E−14 .649E−14 .320E−13 .144E−12 .592E−12 .225E−11 .798E−11

.109E−15 .769E−15 .474E−14 .259E−13 .127E−12 .562E−12 .228E−11 .851E−11 .295E−10 .956E−10

.438E−14 .273E−13 .149E−12 .725E−12 .317E−11 .126E−10 .462E−10 .156E−09 .492E−09 .145E−08

.249E−12 .135E−11 .649E−11 .278E−10 .108E−09 .380E−09 .124E−08 .373E−08 .105E−07 .279E−07

.200E−11 .101E−10 .447E−10 .178E−09 .643E−09 .212E−08 .646E−08 .183E−07 .485E−07 .121E−06

.161E−10 .743E−10 .305E−09 .113E−08 .379E−08 .117E−07 .334E−07 .888E−07 .222E−06 .522E−06

.121E−09 .490E−09 .179E−08 .594E−08 .181E−07 .513E−07 .135E−06 .336E−06 .786E−06 .175E−05

100 110 120 130 140 150 160 170 180 190

.353E−13 .127E−12 .418E−12 .129E−11 .375E−11 .103E−10 .272E−10 .682E−10 .164E−09 .378E−09

.381E−12 .132E−11 .432E−11 .132E−10 .385E−10 .106E−09 .279E−09 .702E−09 .170E−08 .394E−08

.300E−11 .997E−11 .312E−10 .924E−10 .259E−09 .694E−09 .178E−08 .436E−08 .103E−07 .234E−07

.264E−10 .824E−10 .243E−09 .678E−09 .181E−08 .460E−08 .112E−07 .264E−07 .599E−07 .131E−06

.291E−09 .835E−09 .227E−08 .589E−08 .146E−07 .346E−07 .789E−07 .174E−06 .369E−06 .760E−06

.402E−08 .106E−07 .264E−07 .631E−07 .144E−06 .316E−06 .670E−06 .137E−05 .271E−05 .521E−05

.698E−07 .166E−06 .375E−06 .814E−06 .169E−05 .340E−05 .659E−05 .124E−04 .225E−04 .400E−04

.287E−06 .644E−06 .138E−05 .285E−05 .565E−05 .108E−04 .200E−04 .359E−04 .627E−04 .107E−03

.117E−05 .249E−05 .508E−05 .995E−05 .188E−04 .343E−04 .608E−04 .104E−03 .175E−03 .286E−03

.371E−05 .752E−05 .147E−04 .277E−04 .503E−04 .889E−04 .152E−03 .255E−03 .416E−03 .663E−03

200 210 220 230 240 250 260 270 280 290

.842E−09 .181E−08 .376E−08 .758E−08 .148E−07 .283E−07 .526E−07 .954E−07 .169E−06 .294E−06

.883E−08 .191E−07 .401E−07 .817E−07 .162E−06 .312E−06 .588E−06 .108E−05 .194E−05 .342E−05

.514E−07 .109E−06 .226E−06 .455E−06 .889E−06 .170E−05 .316E−05 .577E−05 .103E−04 .180E−04

.278E−06 .573E−06 .115E−05 .224E−05 .427E−05 .793E−05 .144E−04 .257E−04 .450E−04 .771E−04

.152E−05 .295E−05 .559E−05 .103E−04 .186E−04 .329E−04 .569E−04 .965E−04 .161E−03 .263E−03

.975E−05 .178E−04 .316E−04 .549E−04 .935E−04 .156E−03 .255E−03 .411E−03 .650E−03 .101E−02

.691E−04 .117E−03 .193E−03 .311E−03 .494E−03 .770E−03 .118E−02 .178E−02 .265E−02 .389E−02

.177E−03 .288E−03 .459E−03 .717E−03 .110E−02 .166E−02 .247E−02 .362E−02 .524E−02 .750E−02

.457E−03 .715E−03 .110E−02 .166E−02 .245E−02 .358E−02 .516E−02 .733E−02 .103E−01 .143E−01

.104E−02 .159E−02 .239E−02 .354E−02 .515E−02 .740E−02 .105E−01 .147E−01 .203E−01 .278E−01

300 310 320 330 340 350

.500E−06 .834E−06 .137E−05 .220E−05 .349E−05 .544E−05

.591E−05 .100E−04 .167E−04 .273E−04 .440E−04 .698E−04

.309E−04 .522E−04 .865E−04 .141E−03 .227E−03 .360E−03

.130E−03 .215E−03 .352E−03 .565E−03 .895E−03 .140E−02

.424E−03 .672E−03 .105E−02 .162E−02 .246E−02 .369E−02

.156E−02 .236E−02 .352E−02 .519E−02 .757E−02 .109E−01

.563E−02 .805E−02 .114E−01 .159E−01 .221E−01 .303E−01

.106E−01 .148E−01 .205E−01 .281E−01 .382E−01 .516E−01

.196E−01 .266E−01 .359E−01 .480E−01 .636E−01 .836E−01

.376E−01 .504E−01 .670E−01 .883E−01 .116 .150

°C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.534E−09 .200E−08 .677E−08 .211E−07 .607E−07 .163E−06 .411E−06 .976E−06 .220E−05 .473E−05

.803E−09 .296E−08 .993E−08 .306E−07 .870E−07 .231E−06 .575E−06 .135E−05 .302E−05 .642E−05

.112E−08 .409E−08 .136E−07 .415E−07 .117E−06 .309E−06 .765E−06 .179E−05 .396E−05 .835E−05

.148E−08 .540E−08 .179E−07 .543E−07 .153E−06 .400E−06 .985E−06 .229E−05 .504E−05 .106E−04

.167E−08 .609E−08 .201E−07 .611E−07 .171E−06 .449E−06 .110E−05 .256E−05 .562E−05 .118E−04

.187E−08 .679E−08 .224E−07 .680E−07 .191E−06 .498E−06 .122E−05 .283E−05 .622E−05 .130E−04

.196E−08 .714E−08 .236E−07 .714E−07 .200E−06 .523E−06 .128E−05 .297E−05 .652E−05 .136E−04

.206E−08 .750E−08 .247E−07 .749E−07 .210E−06 .548E−06 .134E−05 .310E−05 .681E−05 .143E−04

.217E−08 .788E−08 .260E−07 .786E−07 .220E−06 .574E−06 .140E−05 .325E−05 .712E−05 .149E−04

.228E−08 .827E−08 .273E−07 .824E−07 .230E−06 .600E−06 .147E−05 .339E−05 .743E−05 .155E−04

100 110 120 130 140 150 160 170 180 190

.973E−05 .192E−04 .366E−04 .672E−04 .120E−03 .207E−03 .348E−03 .572E−03 .917E−03 .144E−02

.131E−04 .256E−04 .482E−04 .879E−04 .155E−03 .266E−03 .444E−03 .723E−03 .115E−02 .179E−02

.169E−04 .328E−04 .614E−04 .111E−03 .195E−03 .332E−03 .550E−03 .889E−03 .140E−02 .217E−02

.213E−04 .412E−04 .767E−04 .138E−03 .241E−03 .408E−03 .673E−03 .108E−02 .170E−02 .262E−02

.237E−04 .457E−04 .849E−04 .153E−03 .266E−03 .449E−03 .740E−03 .119E−02 .186E−02 .286E−02

.261E−04 .503E−04 .935E−04 .168E−03 .292E−03 .493E−03 .810E−03 .130E−02 .204E−02 .312E−02

.274E−04 .527E−04 .977E−04 .175E−03 .304E−03 .514E−03 .844E−03 .135E−02 .212E−02 .325E−02

.285E−04 .549E−04 .102E−03 .182E−03 .316E−03 .534E−03 .876E−03 .140E−02 .220E−02 .336E−02

.298E−04 .572E−04 .106E−03 .190E−03 .329E−03 .554E−03 .909E−03 .145E−02 .227E−02 .348E−02

.310E−04 .595E−04 .110E−03 .197E−03 .341E−03 .574E−03 .941E−03 .150E−02 .235E−02 .359E−02

200 210 220 230 240 250 260 270 280 290

.221E−02 .333E−02 .494E−02 .719E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01

.273E−02 .408E−02 .601E−02 .869E−02 .124E−01 .174E−01 .240E−01 .329E−01 .444E−01 .592E−01

.329E−02 .490E−02 .715E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01 .676E−01

.395E−02 .585E−02 .850E−02 .122E−01 .171E−01 .238E−01 .326E−01 .441E−01 .589E−01 .778E−01

.431E−02 .637E−02 .924E−02 .132E−01 .186E−01 .257E−01 .352E−01 .475E−01 .633E−01 .835E−01

.470E−02 .693E−02 .100E−01 .143E−01 .201E−01 .278E−01 .380E−01 .513E−01 .683E−01 .900E−01

.488E−02 .720E−02 .104E−01 .149E−01 .209E−01 .289E−01 .394E−01 .531E−01 .706E−01 .930E−01

.505E−02 .744E−02 .108E−01 .153E−01 .215E−01 .297E−01 .405E−01 .545E−01 .725E−01 .954E−01

.522E−02 .768E−02 .111E−01 .158E−01 .221E−01 .305E−01 .416E−01 .560E−01 .744E−01 .978E−01

.538E−02 .791E−02 .114E−01 .162E−01 .227E−01 .314E−01 .427E−01 .574E−01 .762E−01 .100

300 310 320 330 340 350

.678E−01 .892E−01 .116 .150 .192 .243

.782E−01 .102 .132 .170 .216 .272

.888E−01 .115 .149 .190 .240 .301

.102 .132 .169 .214 .270 .337

.109 .141 .180 .228 .287 .358

.117 .151 .193 .245 .307 .383

.121 .156 .199 .252 .317 .394

.124 .160 .204 .258 .328 .402

.127 .164 .209 .263 .330 .410

.130 .167 .213 .269 .386 .417

Weight percent, H2SO4

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, CA, 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

VAPOR PRESSURES OF SOLUTIONS TABLE 2-17

2-87

Total Pressure, bar, of Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4

°C 0 10 20 30 40 50 60 70 80 90

10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664

20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617

30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542

40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434

50.0 .193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298

60.0 .836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161

70.0

75.0

80.0

85.0

.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01

.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−01 .170E−01 .281E−01

.197E−04 .490E−04 .115E−03 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01

.343E−05 .952E−05 .245E−04 .589E−04 .134E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02

.905E−01 .138 .206 .301 .431 .605 .837 1.138 1.525 2.017

.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212

.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673

.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .683E−01 .101 .147 .209 .292

100 110 120 130 140 150 160 170 180 190

.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289

.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658

.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640

.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259

.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397

.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169

200 210 220 230 240 250 260 270 280 290

13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015

13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681

11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715

10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.866

8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926

5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.122 26.275 31.004

2.633 3.396 4.331 5.466 6.832 8.459 10.384 12.642 15.272 18.315

1.606 2.101 2.715 3.468 4.382 5.481 6.791 8.337 10.147 12.250

.913 1.221 1.610 2.098 2.701 3.439 4.332 5.402 6.673 8.170

.402 .544 .726 .956 1.242 1.594 2.023 2.540 3.157 3.886

300 310 320 330 340 350

66.934 75.495 84.705 94.567 105.083 116.251

64.407 72.781 81.816 91.518 101.894 112.947

60.081 68.101 76.792 86.172 96.252 107.043

54.869 62.553 70.947 80.078 89.970 100.647

47.347 54.470 62.338 70.990 80.466 90.806

36.361 42.398 49.168 56.727 65.130 74.437

21.814 25.812 30.355 35.489 41.262 47.723

14.675 17.453 20.611 24.182 28.193 32.674

9.916 11.939 14.264 16.916 19.920 23.303

4.740 5.732 6.876 8.185 9.672 11.351

Weight percent, H2SO4 °C

90.0

92.0

94.0

96.0

97.0

98.0

98.5

99.0

99.5

100.0

0 10 20 30 40 50 60 70 80 90

.518E−06 .159E−05 .449E−05 .117E−04 .385E−04 .653E−04 .141E−03 .291E−03 .571E−03 .107E−02

.243E−06 .765E−06 .221E−05 .590E−05 .147E−04 .344E−04 .759E−04 .159E−03 .319E−03 .612E−03

.109E−06 .348E−06 .102E−05 .279E−05 .708E−05 .169E−04 .380E−04 .813E−04 .166E−03 .324E−03

.416E−07 .136E−06 .407E−06 .113E−05 .293E−05 .712E−05 .164E−04 .357E−04 .742E−04 .148E−03

.235E−07 .774E−07 .235E−06 .659E−06 .173E−05 .425E−05 .987E−05 .218E−04 .458E−04 .921E−04

.117E−07 .391E−07 .121E−06 .344E−06 .914E−06 .228E−05 .538E−05 .120E−04 .257E−04 .524E−04

.768E−08 .261E−07 .812E−07 .234E−06 .630E−06 .159E−05 .379E−05 .856E−05 .184E−04 .390E−04

.479E−08 .166E−07 .528E−07 .155E−06 .425E−06 .109E−05 .264E−05 .605E−05 .132E−04 .277E−04

.313E−08 .113E−07 .373E−07 .114E−06 .323E−06 .861E−06 .216E−05 .514E−05 .117E−04 .253E−04

.323E−08 .124E−07 .435E−07 .141E−06 .425E−06 .120E−05 .319E−05 .804E−05 .193E−04 .441E−04

100 110 120 130 140 150 160 170 180 190

.195E−02 .340E−02 .575E−02 .944E−02 .151E−01 .235E−01 .357E−01 .532E−01 .775E−01 .111

.113E−02 .201E−02 .346E−02 .578E−02 .939E−02 .149E−01 .230E−01 .347E−01 .514E−01 .747E−01

.607E−03 .110E−02 .192E−02 .327E−02 .539E−02 .866E−02 .136E−01 .208E−01 .312E−01 .460E−01

.283E−03 .521E−03 .929E−03 .161E−02 .270E−02 .441E−02 .703E−02 .110E−01 .167E−01 .250E−01

.178E−03 .332E−03 .598E−03 .104E−02 .177E−02 .293E−02 .471E−02 .741E−02 .114E−01 .172E−01

.103E−03 .194E−03 .354E−03 .626E−03 .107E−02 .180E−02 .293E−02 .466E−02 .726E−02 .111E−01

.751E−04 .143E−03 .263E−03 .470E−03 .815E−03 .137E−02 .226E−02 .363E−02 .571E−02 .880E−02

.555E−04 .107E−03 .201E−03 .363E−03 .639E−03 .109E−02 .183E−02 .299E−02 .478E−02 .749E−02

.527E−04 .106E−03 .206E−03 .387E−03 .708E−03 .126E−02 .219E−02 .372E−02 .619E−02 .101E−01

.966E−04 .204E−03 .414E−03 .314E−03 .155E−02 .287E−02 .516E−02 .905E−02 .155E−01 .260E−01

.665E−01 .944E−01 .132 .182 .247 .331 .439 .575 .744 .954

.367E−01 .530E−01 .752E−01 .105 .145 .197 .264 .351 .460 .597

.255E−01 .371E−01 .531E−01 .749E−01 .104 .143 .193 .258 .341 .446

.166E−01 .245E−01 .354E−01 .505E−01 .710E−01 .985E−01 .135 .153 .245 .324

.133E−01 .198E−01 .289E−01 .417E−01 .592E−01 .830E−01 .115 .157 .213 .285

.115E−01 .175E−01 .260E−01 .382E−01 .553E−01 .790E−01 .112 .156 .215 .295

.161E−01 .253E−01 .392E−01 .596E−01 .895E−01 .132 .193 .279 .398 .562

.427E−01 .687E−01 .109 .169 .258 .389 .577 .846 1.225 1.751

200 210 220 230 240 250 260 270 280 290

.156 .216 .295 .396 .525 .688 .881 1.141 1.447 1.817

.107 .150 .207 .282 .379 .503 .660 .856 1.099 1.398

300 310 320 330 340 350

2.261 2.791 3.417 4.153 5.011 6.006

1.761 2.199 2.723 3.347 4.084 4.949

1.211 1.524 1.901 2.353 2.889 3.523

.767 .977 1.234 1.545 1.919 2.366

.578 .742 .944 1.191 1.491 1.852

.425 .553 .713 .911 1.156 1.456

.379 .498 .649 .840 1.078 1.374

.399 .536 .714 .944 1.239 1.614

.785 1.085 1.486 2.018 2.718 3.631

2.476 3.465 4.800 6.586 8.957 12.079

*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.

2-88

PHYSICAL AND CHEMICAL DATA

TABLE 2-18

Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3* mmHg Percentages are weight % HNO3 in solution. 20%

°C

25% H2O

HNO3

HNO3

30% H2O

HNO3

35% H2O

0 5 10 15 20

4.1 5.7 8.0 10.9 15.2

3.8 5.4 7.6 10.3 14.2

3.6 5.0 7.1 9.7 13.2

25 30 35 40 45

20.6 27.6 36.5 47.5 62

19.2 25.7 33.8 44 57.5

17.8 23.8 31.1 41 53

0.09

0.11 .17

40%

HNO3

H2O

45%

HNO3

H2O

3.3 4.6 6.5 8.9 12.0 16.2 21.7 28.3 37.7 48

0.09 .13 .20 .28

0.12 .17 .25 .36 .52

50%

HNO3

H2O

HNO3

3.0 4.2 5.8 8.0 10.8

H2O

0.10 .15

2.6 3.6 5.0 6.9 9.4

0.12 .18 .27

2.1 3.0 4.2 5.8 7.9

14.6 19.5 25.5 33.5 43

.23 .33 .48 .68 .96

12.7 16.9 22.3 29.3 38.0

.39 .56 .80 1.13 1.57

10.7 14.4 19.0 25.0 32.5

49.5 62.5 80 100 126

2.18 2.95 4.05 5.46 7.25

42.5 54 70 88 110

50 55 60 65 70

0.09 .13 .19 .27

80 100 128 162 200

.13 .18 .28 .40 .54

75 94 121 151 187

.25 .35 .51 .71 1.00

69 87 113 140 174

.42 .59 .85 1.18 1.63

63 79 102 127 159

.75 1.04 1.48 2.05 2.80

56 71 90 114 143

1.35 1.83 2.54 3.47 4.65

75 80 85 90 95

.38 .53 .74 1.01 1.37

250 307 378 458 555

.77 1.05 1.44 1.95 2.62

234 287 352 426 517

1.38 1.87 2.53 3.38 4.53

217 267 325 393 478

2.26 3.07 4.15 5.50 7.32

198 243 297 359 436

3.80 5.10 6.83 9.0 11.7

178 218 268 325 394

6.20 8.15 10.7 13.7 17.8

158 195 240 292 355

9.6 12.5 16.3 20.9 26.8

138 170 211 258 315

100 105 110 115 120

1.87 2.50

675 800

3.50 4.65

628 745

6.05 7.90

580 690

530 631 755

15.5 20.0 25.7 32.5

480 573 688 810

23.0 29.2 37.0 46

430 520 625 740

34.2 43.0 54.5 67 84

383 463 560 665 785

°C

HNO3

H2O

HNO3

H2O

HNO3

H2O

HNO3

H2O

0 5 10 15 20

0.14 .21 .31 .45

1.8 2.5 3.5 4.9 6.7

0.19 .28 .41 .59 .84

1.5 2.1 3.0 4.1 5.6

0.41 .60 .86 1.21 1.68

1.3 1.8 2.6 3.5 4.9

0.79 1.12 1.58 2.18 3.00

1.1 1.6 2.2 3.0 4.1

25 30 35 40 45

.66 .93 1.30 1.82 2.50

9.1 12.2 16.1 21.3 28.0

1.21 1.66 2.28 3.10 4.20

7.7 10.3 13.6 18.1 23.7

2.32 3.17 4.26 5.70 7.55

6.6 8.8 11.6 15.5 20.0

4.10 5.50 7.30 9.65 12.6

5.5 7.4 9.8 12.8 16.7

50 55 60 65 70

3.41 4.54 6.15 8.18 10.7

36.3 46 60 76 95

5.68 7.45 9.9 13.0 16.8

31 39 51 64 81

10.0 12.8 16.8 21.7 27.5

26.0 33.0 43.0 54.5 68

16.5 21.0 27.1 34.5 43.3

21.8 27.3 35.3 44.5 56

75 80 85 90 95

13.9 18.0 23.0 29.4 37.3

120 148 182 223 272

21.8 27.5 34.8 43.7 55.0

102 126 156 192 233

35.0 43.5 54.5 67.5 83.5

100 105 110 115 120 125

47 58.5 73 90 110

331 400 485 575 685

69.5 84.5 103 126 156 187

285 345 417 495 590 700

55%

60%

9.7 12.7 16.5

65%

*International Critical Tables, vol. 3, pp. 304–305.

103 124 152 181 218 260

70%

80% HNO3

90%

100%

H2O

HNO3

H2O

HNO3

2 3 4 6 8

1.2 1.7 2.4

5.5 8 11 15 20

10.5 14 18.5 24.5 32

3.2 4 5.5 7 9.5

27 36 47 62 80

1 1.3 1.8 2.4 3

57 77 102 133 170

11 15 22 30 42

41 52 67 85 106

12 15 20 25 31

103 127 157 192 232

4 5 6.5 8 10

215 262 320 385 460 540 625 720 820

86 106 131 160 195

54.5 67.5 83 103 125

70 86 107 130 158

130 158 192 230 278

38 48 60 73 89

282 338 405 480 570

13 16 20 24 29

238 288 345 410 490 580

152 183 221 262 312 372

192 231 278 330 393 469

330 392 465 545 640

108 129 155 185 219

675 790

35 42

VAPOR PRESSURES OF SOLUTIONS TABLE 2-19 Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C* mmHg % HBr 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

20 °C HBr

0.09 0.23 0.71 2.2 6.8 21

25 °C H2O

6.2 4.5 3.3 2.4 1.7 1.3

HBr

50 °C H2O

0.0016 0.0022 0.0033 0.0061 0.011 0.023 0.048 0.10 0.13 0.37 1.1 3.2 9.3 27

HBr

8.2 6.1 4.5 3.3 2.4 1.9

1.3 3.2 7.2 17 40 91

H2O

30.2 24.3 19.3 16.0 13.3 10.4

55 °C HBr

2.0 4.6 10.2 23.0 51 115 260

H2O

38 31 25 21 18 14 11.4

*International Critical Tables, vol. 3, p. 306.

2-89

TABLE 2-22 Total Vapor Pressures of Aqueous Solutions of CH3COOH* Percentages of weight % acetic acid in the solution mmHg °C

25%

50%

75%

20 25 30 35 40

16.3 22.1 29.6 39.4 51.7

15.7 21.4 28.8 38.3 50.2

15.3 20.8 27.8 36.6 48.1

45 50 55 60 65

67.0 87.2 110 141 178

65.0 85.0 107 138 172

62.0 80.1 102 130 162

70 75 80 85 90

223 277 342 419 510

216 269 331 407 497

203 251 310 376 458

95 100

618 743

602 725

550 666

*International Critical Tables, vol. 3, p. 306.

TABLE 2-20 Partial Pressures of HI over Aqueous Solutions of HI at 25°C* mmHg %HI pHI

44 0.00064

46 0.0010

48 0.0022

50 0.0050

52 0.013

54 0.035

56 0.10

*International Critical Tables, vol. 3, p. 306.

TABLE 2-21 Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid For these data reference must be made to the graphs of International Critical Tables, vol. 3, pp. 306–308.

Vapor pressure of aqueous diethylene glycol solutions. (Courtesy of Carbide and Carbon Chemicals Corp.)

FIG. 2-3

2-90 TABLE 2-23

Partial Pressure of H2O over Aqueous Solutions of NH3 (psia) Liquid mole percent NH3 (liquid weight percent NH3)

0

5

10

15

(0)

(4.74)

(9.5)

(14.29)

32 40 50 60 70

0.089 0.122 0.178 0.256 0.363

0.083 0.115 0.168 0.242 0.343

0.077 0.106 0.156 0.225 0.320

0.071 0.097 0.143 0.207 0.294

80 90 100 110 120

0.507 0.699 0.951 1.277 1.695

0.479 0.661 0.899 1.209 1.607

0.448 0.618 0.843 1.135 1.510

130 140 150 160 170

2.226 2.893 3.723 4.747 6.000

2.112 2.748 3.540 4.519 5.717

180 190 200 210 220

7.520 9.350 11.538 14.136 17.201

230 240 250

20.796 24.986 29.844

t, °F

20

25

30

35

40

45

55

60

65

70

80

85

90

95

(23.94)

(28.81)

(33.71)

(38.64)

(43.59)

(48.57)

(53.58)

(58.62)

(63.69)

(68.79)

(73.91)

(79.07)

(84.26)

(89.47)

(94.72)

0.063 0.087 0.129 0.186 0.266

0.055 0.077 0.113 0.164 0.235

0.047 0.065 0.097 0.142 0.204

0.039 0.054 0.081 0.119 0.172

0.031 0.044 0.066 0.098 0.143

0.025 0.035 0.053 0.079 0.116

0.019 0.027 0.041 0.062 0.093

0.014 0.021 0.032 0.049 0.073

0.011 0.016 0.025 0.038 0.058

0.008 0.012 0.019 0.030 0.045

0.006 0.009 0.014 0.023 0.036

0.004 0.007 0.011 0.018 0.028

0.003 0.005 0.008 0.014 0.022

0.002 0.004 0.006 0.010 0.016

0.002 0.002 0.004 0.007 0.011

0.001 0.001 0.002 0.004 0.006

0.413 0.571 0.780 1.052 1.402

0.374 0.518 0.710 0.960 1.283

0.332 0.462 0.634 0.861 1.154

0.289 0.403 0.556 0.758 1.021

0.245 0.345 0.479 0.656 0.889

0.205 0.290 0.405 0.559 0.763

0.168 0.240 0.338 0.470 0.647

0.136 0.196 0.279 0.392 0.544

0.109 0.159 0.228 0.324 0.455

0.087 0.128 0.186 0.268 0.380

0.069 0.103 0.152 0.220 0.316

0.055 0.083 0.123 0.181 0.263

0.043 0.066 0.100 0.148 0.217

0.034 0.052 0.079 0.119 0.176

0.025 0.040 0.061 0.092 0.137

0.018 0.028 0.043 0.065 0.099

0.010 0.015 0.024 0.036 0.056

1.988 2.591 3.343 4.273 5.416

1.850 2.415 3.122 4.000 5.079

1.696 2.221 2.879 3.698 4.709

1.532 2.012 2.618 3.374 4.312

1.361 1.796 2.347 3.039 3.902

1.192 1.582 2.078 2.706 3.493

1.030 1.376 1.821 2.387 3.101

0.881 1.186 1.582 2.090 2.736

0.747 1.016 1.367 1.821 2.405

0.632 0.867 1.177 1.584 2.110

0.532 0.738 1.013 1.376 1.851

0.448 0.628 0.870 1.194 1.622

0.376 0.532 0.746 1.033 1.418

0.313 0.448 0.634 0.887 1.229

0.257 0.371 0.529 0.748 1.047

0.202 0.295 0.425 0.607 0.858

0.147 0.216 0.314 0.453 0.647

0.083 0.124 0.183 0.267 0.386

7.174 8.931 11.035 13.538 16.496

6.807 8.488 10.504 12.910 15.758

6.397 7.994 9.916 12.213 14.941

5.947 7.452 9.270 11.449 14.047

5.465 6.873 8.580 10.635 13.095

4.968 6.275 7.869 9.796 12.115

4.472 5.680 7.160 8.962 11.141

3.995 5.107 6.479 8.160 10.205

3.551 4.573 5.842 7.410 9.331

3.148 4.086 5.262 6.725 8.534

2.787 3.650 4.740 6.110 7.817

2.468 3.262 4.275 5.559 7.175

2.184 2.914 3.856 5.061 6.592

1.928 2.598 3.470 4.598 6.045

1.688 2.297 3.098 4.146 5.504

1.451 1.994 2.718 3.675 4.932

1.201 1.669 2.300 3.147 4.277

0.917 1.290 1.802 2.502 3.455

0.555 0.793 1.129 1.600 2.262

19.971 24.029 28.744

19.111 23.037 27.607

18.162 21.943 26.358

17.124 20.748 24.996

16.020 19.479 23.549

14.886 18.179 22.070

13.760 16.889 20.608

12.679 15.654 19.212

11.672 14.506 17.917

10.754 13.463 16.748

9.930 12.530 15.708

9.192 11.696 14.783

8.522 10.938 13.946

7.889 10.221 13.153

7.255 9.496 12.346

6.573 8.703 11.452

5.777 7.759 10.369

4.751 6.508 8.891

3.196 4.520 6.413

(19.1)

50

75

The values in Tables 2-23 to 2-26 were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002). The primary source for the properties of aqueous ammonia mixtures is R. Tillner-Roth and D. G. Friend, “A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia},” J. Phys. Chem. Ref. Data 27:63–96 (1998).

TABLE 2-24

Mole Percentages of H2O over Aqueous Solutions of NH3 Liquid mole percent NH3 (liquid weight percent NH3)

0

5

10

15

t, °F

(0)

(4.74)

(9.5)

(14.29)

32 40 50 60 70

100 100 100 100 100

32.046 33.233 34.709 36.172 37.619

14.173 15.064 16.192 17.334 18.489

7.263 7.842 8.588 9.359 10.154

80 90 100 110 120

100 100 100 100 100

39.047 40.455 41.840 43.201 44.537

19.653 20.827 22.007 23.194 24.385

130 140 150 160 170

100 100 100 100 100

45.849 47.136 48.398 49.636 50.849

180 190 200 210 220

100 100 100 100 100

230 240 250

100 100 100

20 (19.1)

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

(23.94)

(28.81)

(33.71)

(38.64)

(43.59)

(48.57)

(53.58)

(58.62)

(63.69)

(68.79)

(73.91)

(79.07)

(84.26)

(89.47)

(94.72)

3.959 4.321 4.797 5.299 5.828

2.202 2.427 2.727 3.050 3.396

1.226 1.365 1.554 1.760 1.985

0.679 0.765 0.883 1.015 1.162

0.375 0.428 0.502 0.587 0.683

0.207 0.240 0.287 0.342 0.405

0.116 0.137 0.167 0.203 0.245

0.067 0.080 0.100 0.123 0.152

0.039 0.048 0.061 0.077 0.097

0.024 0.030 0.039 0.050 0.064

0.015 0.019 0.026 0.034 0.044

0.010 0.013 0.017 0.023 0.030

0.007 0.009 0.012 0.016 0.021

0.005 0.006 0.008 0.011 0.015

0.003 0.004 0.005 0.007 0.010

0.001 0.002 0.003 0.004 0.005

10.974 11.816 12.681 13.567 14.475

6.382 6.963 7.571 8.205 8.867

3.765 4.160 4.580 5.027 5.501

2.231 2.497 2.786 3.098 3.434

1.325 1.505 1.703 1.920 2.158

0.791 0.913 1.049 1.201 1.370

0.478 0.561 0.655 0.763 0.884

0.294 0.351 0.418 0.494 0.582

0.186 0.226 0.273 0.328 0.393

0.121 0.149 0.184 0.224 0.273

0.081 0.102 0.127 0.158 0.195

0.056 0.072 0.091 0.114 0.142

0.040 0.051 0.066 0.084 0.106

0.028 0.037 0.048 0.062 0.079

0.020 0.026 0.034 0.044 0.057

0.013 0.017 0.023 0.030 0.039

0.007 0.009 0.012 0.016 0.021

25.580 26.779 27.979 29.182 30.385

15.404 16.353 17.322 18.310 19.317

9.557 10.273 11.017 11.789 12.587

6.002 6.532 7.091 7.680 8.298

3.796 4.184 4.600 5.044 5.517

2.419 2.702 3.010 3.344 3.705

1.558 1.766 1.994 2.246 2.521

1.020 1.173 1.344 1.534 1.746

0.682 0.796 0.925 1.071 1.236

0.467 0.554 0.653 0.767 0.897

0.329 0.396 0.473 0.563 0.668

0.238 0.290 0.352 0.425 0.509

0.177 0.218 0.268 0.327 0.396

0.133 0.166 0.206 0.254 0.312

0.100 0.126 0.158 0.197 0.245

0.073 0.094 0.118 0.149 0.187

0.050 0.064 0.082 0.104 0.132

0.027 0.035 0.045 0.058 0.074

52.038 53.204 54.347 55.468 56.567

31.588 32.792 33.994 35.196 36.395

20.341 21.383 22.442 23.517 24.607

13.413 14.266 15.146 16.052 16.985

8.948 9.628 10.340 11.083 11.858

6.021 6.557 7.124 7.725 8.359

4.094 4.513 4.963 5.445 5.961

2.822 3.151 3.508 3.896 4.316

1.981 2.240 2.526 2.840 3.184

1.421 1.628 1.859 2.116 2.401

1.045 1.212 1.402 1.615 1.854

0.788 0.926 1.083 1.263 1.468

0.609 0.724 0.857 1.011 1.187

0.479 0.576 0.690 0.823 0.977

0.381 0.463 0.561 0.676 0.811

0.302 0.371 0.453 0.552 0.669

0.233 0.288 0.357 0.439 0.539

0.166 0.209 0.261 0.326 0.406

0.094 0.120 0.153 0.195 0.248

57.645 58.702 59.740

37.592 38.787 39.978

25.711 26.830 27.962

17.943 18.927 19.936

12.665 13.503 14.374

9.028 9.732 10.471

6.512 7.098 7.722

4.769 5.258 5.784

3.560 3.970 4.416

2.716 3.064 3.447

2.122 2.421 2.754

1.699 1.960 2.254

1.390 1.621 1.884

1.156 1.363 1.601

0.969 1.155 1.371

0.809 0.975 1.171

0.660 0.806 0.982

0.506 0.629 0.783

0.317 0.406 0.524

The values in Tables 2-23 to 2-26 were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002). The primary source for the properties of aqueous ammonia mixtures is R. Tillner-Roth and D. G. Friend, “A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia},” J. Phys. Chem. Ref. Data 27:63–96 (1998).

2-91

2-92 TABLE 2-25

Partial Pressures of NH3 over Aqueous Solutions of NH3 (psia) Liquid mole percent NH3 (liquid weight percent NH3)

0

5

t, °F (0) (4.74)

10

15

20

25

30

40

45

50

55

60

65

70

75

80

85

90

95

100

(9.5)

(14.29)

(19.1)

(23.94)

(28.81)

(33.71)

35

(38.64)

(43.59)

(48.57)

(53.58)

(58.62)

(63.69)

(68.79)

(73.91)

(79.07)

(84.26)

(89.47)

(94.72)

(100)

21.480 25.798 32.120 39.581 48.307

43.900 48.431 51.828 57.082 63.243 69.520 76.489 83.933 91.745 100.51

52.368 61.665 75.020 90.486 108.26

55.846 65.731 79.931 96.376 115.29

59.055 69.506 84.523 101.93 121.94

62.277 73.322 89.205 107.63 128.85

32 40 50 60 70

0 0 0 0 0

0.177 0.230 0.316 0.426 0.568

0.468 0.600 0.808 1.074 1.410

0.901 1.143 1.522 2.002 2.603

1.533 1.932 2.552 3.330 4.296

2.456 3.078 4.036 5.228 6.696

3.797 4.730 6.154 7.912 10.056

5.710 7.066 9.117 11.625 14.656

8.358 10.267 13.129 16.593 20.742

11.868 14.467 18.328 22.959 28.456

16.279 19.691 24.721 30.702 37.744

27.206 32.478 40.150 49.149 59.615

33.087 39.307 48.315 58.831 71.008

38.745 45.860 56.128 68.074 81.861

80 90 100 110 120

0 0 0 0 0

0.748 0.973 1.250 1.590 2.001

1.831 2.351 2.987 3.758 4.683

3.348 4.261 5.368 6.700 8.285

5.483 6.926 8.663 10.735 13.184

8.483 10.639 13.213 16.258 19.832

12.645 15.741 19.406 23.709 28.718

18.283 22.582 27.630 33.509 40.300

25.664 31.447 38.185 45.971 54.899

34.922 42.460 51.177 61.177 72.569

45.966 58.428 71.689 55.485 70.073 85.514 66.416 83.375 101.23 78.881 98.461 118.98 92.996 115.45 138.90

84.998 100.96 119.04 139.39 162.15

97.654 115.62 135.93 158.73 184.19

109.19 128.99 151.34 176.40 204.35

119.45 140.93 165.15 192.30 222.57

128.56 151.60 177.57 206.69 239.16

136.88 161.39 189.05 220.07 254.71

144.83 170.82 200.19 233.16 270.03

153.13 180.76 212.01 247.19 286.60

130 140 150 160 170

0 0 0 0 0

2.494 5.784 3.082 7.084 3.774 8.604 4.585 10.371 5.527 12.408

10.158 12.352 14.902 17.844 21.216

16.055 19.395 23.250 27.669 32.700

23.989 28.791 34.295 40.562 47.649

34.503 41.135 48.685 57.222 66.816

48.086 65.064 85.455 56.949 76.558 99.944 66.972 89.472 116.12 78.230 103.89 134.09 90.803 119.90 153.93

180 190 200 210 220

0 0 0 0 0

6.612 7.856 9.270 10.869 12.666

25.053 29.393 34.270 39.721 45.779

38.390 44.786 51.933 59.876 68.655

55.614 77.532 64.514 89.432 74.401 102.58 85.326 117.02 97.335 132.82

230 240 250

0 0 0

14.673 31.727 16.905 36.356 19.371 41.449

14.741 17.395 20.397 23.769 27.538

52.477 78.310 110.47 59.843 88.872 124.78 67.906 100.38 140.28

150.00 168.62 188.70

108.87 126.62 146.35 168.16 192.14

134.49 155.67 179.11 204.93 233.19

161.13 185.77 212.96 242.79 275.37

187.48 215.50 246.33 280.08 316.87

212.44 243.66 277.95 315.46 356.26

235.36 269.56 307.13 348.18 392.85

256.14 293.19 333.87 378.36 426.79

275.20 314.99 358.75 406.62 458.82

293.18 335.72 382.56 433.89 489.95

311.05 356.48 406.60 461.65 521.92

330.54 379.36 433.38 492.95 558.45

587.67 659.12 736.52 820.08 909.98

630.24 708.74 794.38 887.64 989.03

104.77 120.18 137.11 155.62 175.75

137.57 156.99 178.21 201.28 226.25

175.73 199.56 225.47 253.53 283.77

218.35 246.89 277.81 311.15 346.95

263.99 297.40 333.47 372.25 413.74

310.76 349.06 390.30 434.50 481.70

356.75 399.82 446.12 495.68 548.49

400.47 448.16 499.38 554.15 612.50

441.24 493.43 549.49 609.47 673.37

479.30 535.98 596.94 662.22 731.88

515.48 576.75 642.74 713.55 789.23

550.92 616.99 688.31 765.01 847.19

197.54 221.04 246.26

253.17 282.06 312.93

316.23 350.91 387.82

385.23 425.98 469.20

457.98 504.92 554.55

531.89 585.01 641.05

604.55 663.80 726.18

674.39 739.77 808.55

741.21 812.89 888.35

805.90 884.22 966.78

869.79 955.18 1045.3

934.86 1028.0 1126.5

1006.3 1109.2 1218.3

1099.1 1218.7 1348.5

The values in Tables 2-23 to 2-26 were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002). The primary source for the properties of aqueous ammonia mixtures is R. Tillner-Roth and D. G. Friend, “A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia},” J. Phys. Chem. Ref. Data 27:63–96 (1998).

TABLE 2-26

Total Vapor Pressures of Aqueous Solutions of NH3 (psia) Liquid mole percent NH3 (liquid weight percent NH3)

0

5

10

t, °F

(0)

(4.74)

(9.5)

32 40 50 60 70

0.089 0.122 0.178 0.256 0.363

0.260 0.345 0.483 0.668 0.911

0.545 0.706 0.964 1.299 1.730

0.971 1.240 1.665 2.209 2.898

80 90 100 110 120

0.507 0.699 0.951 1.277 1.695

1.227 1.633 2.149 2.799 3.608

2.279 2.969 3.830 4.893 6.194

130 140 150 160 170

2.226 4.607 7.773 2.893 5.830 9.674 3.723 7.315 11.947 4.747 9.104 14.644 6.000 11.244 17.824

180 190 200 210 220

7.520 9.350 11.538 14.136 17.201

13.786 16.787 20.305 24.407 29.162

21.548 25.883 30.901 36.679 43.296

230 240 250

20.796 34.644 50.838 24.986 40.934 59.393 29.844 48.115 69.056

15

25

30

35

40

45

55

60

65

70

75

80

85

90

95

100

(23.94)

(28.81)

(33.71)

(38.64)

(43.59)

(48.57)

(53.58)

(58.62)

(63.69)

(68.79)

(73.91)

(79.07)

(84.26)

(89.47)

(94.72)

(100)

1.596 2.020 2.680 3.516 4.562

2.512 3.155 4.149 5.392 6.931

3.844 4.795 6.251 8.053 10.260

5.749 7.120 9.198 11.744 14.828

8.389 10.311 13.195 16.691 20.885

11.893 14.502 18.381 23.038 28.572

16.298 19.718 24.762 30.764 37.837

21.494 25.819 32.152 39.630 48.381

27.217 32.494 40.175 49.187 59.673

33.095 39.319 48.334 58.860 71.053

38.751 45.869 56.143 68.097 81.897

43.904 48.434 51.835 57.087 63.254 69.528 76.507 83.946 91.773 100.53

52.371 61.669 75.026 90.496 108.28

55.848 65.734 79.935 96.383 115.30

59.056 69.507 84.526 101.93 121.95

62.277 73.322 89.205 107.63 128.85

3.761 4.832 6.148 7.751 9.688

5.857 7.445 9.373 11.694 14.467

8.815 11.101 13.847 17.119 20.986

12.934 16.144 19.962 24.467 29.739

18.528 22.927 28.109 34.165 41.189

25.869 31.737 38.590 46.530 55.662

35.090 42.700 51.514 61.647 73.216

46.102 58.537 71.776 55.680 70.232 85.642 66.695 83.603 101.42 79.273 98.785 119.25 93.540 115.91 139.28

85.067 101.06 119.19 139.61 162.47

97.709 115.70 136.05 158.91 184.45

109.23 129.06 151.44 176.55 204.57

119.48 140.98 165.23 192.42 222.75

128.59 151.64 177.63 206.78 239.30

136.90 161.42 189.09 220.14 254.81

144.84 170.84 200.21 233.20 270.09

153.13 180.76 212.01 247.19 286.60

12.008 14.767 18.024 21.844 26.295

17.752 21.616 26.129 31.367 37.409

25.521 30.803 36.913 43.936 51.961

35.864 42.931 51.032 60.262 70.718

49.278 66.094 58.531 77.934 69.050 91.292 80.937 106.28 94.297 123.00

86.336 101.13 117.70 136.18 156.67

109.62 127.64 147.72 169.98 194.54

135.12 156.54 180.29 206.51 235.30

161.66 186.51 213.97 244.16 277.22

187.93 216.13 247.20 281.28 318.49

212.82 244.19 278.70 316.49 357.68

235.67 270.01 307.76 349.07 394.08

256.40 293.56 334.40 379.11 427.84

275.40 315.29 359.17 407.23 459.68

293.33 335.94 382.87 434.34 490.60

311.13 356.60 406.78 461.92 522.31

330.54 379.36 433.38 492.95 558.45

31.450 37.387 44.186 51.934 60.720

44.337 61.079 82.499 52.238 71.387 95.708 61.203 82.981 110.45 71.325 95.961 126.82 82.702 110.43 144.93

588.22 659.91 737.65 821.68 912.24

630.24 708.74 794.38 887.64 989.03

(14.29)

20 (19.1)

70.639 95.434 126.49 81.786 109.62 144.26 94.264 125.38 163.83

164.89 186.80 210.77

50

109.24 125.86 144.27 164.58 186.89

141.57 162.10 184.69 209.44 236.46

179.28 204.13 231.31 260.94 293.10

221.50 250.98 283.07 317.88 355.49

266.78 301.05 338.21 378.36 421.56

313.23 352.32 394.57 440.06 488.88

358.94 402.74 449.98 500.74 555.08

402.40 450.76 502.85 558.75 618.54

442.93 495.73 552.59 613.62 678.88

480.75 537.98 599.66 665.90 736.81

516.68 578.42 645.04 716.70 793.51

551.84 618.28 690.11 767.51 850.64

211.30 237.93 266.87

265.85 297.71 332.14

327.90 365.42 405.74

395.98 439.44 485.95

467.91 517.45 570.25

541.08 596.71 655.83

613.07 674.74 740.12

682.28 749.99 821.71

748.46 822.39 900.70

812.47 892.93 978.24

875.57 962.94 1055.7

939.61 1034.5 1135.4

1009.5 1113.7 1224.7

1099.1 1218.7 1348.5

The values in Tables 2-23 to 2-26 were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002). The primary source for the properties of aqueous ammonia mixtures is R. Tillner-Roth and D. G. Friend, “A Helmholtz Free Energy Formulation of the Thermodynamic Properties of the Mixture {Water + Ammonia},” J. Phys. Chem. Ref. Data 27:63–96 (1998).

2-93

2-94

PHYSICAL AND CHEMICAL DATA

TABLE 2-27 Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate* mmHg %Na2CO3 t, °C

0

5

10

0 10 20 30 40 50 60 70 80 90 100

4.5 9.2 17.5 31.8 55.3 92.5 149.5 239.8 355.5 526.0 760.0

4.5 9.0 17.2 31.2 54.2 90.7 146.5 235 348 516 746

8.8 16.8 30.4 53.0 88.7 143.5 230.5 342 506 731

15

16.3 29.6 57.6 86.5 139.9 225 334 494 715

20

28.8 50.2 84.1 136.1 219 325 482 697

25

27.8 48.4 81.2 131.6 211.5 315 467 676

30

26.4 46.1 77.5 125.7 202.5 301 447 648

*International Critical Tables, vol. 3, p. 372.

TABLE 2-28 Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol* 39.9 °C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 14.99 17.85 21.07 27.31 31.06 40.1 47.0 55.8 68.9 86.0 100.0

54.7 39.2 38.5 37.2 35.8 34.9 32.8 31.5 27.3 20.7 10.1 0

0 66.1 75.5 85.2 100.6 108.8 127.7 141.6 158.4 186.6 225.2 260.7

59.4 °C

Mole fraction CH3OH

PH2O, mmHg

PCH3OH , mmHg

0 22.17 27.40 33.24 39.80 47.08 55.5 69.2 78.5 85.9 100.0

145.4 106.9 102.2 96.6 91.7 84.8 76.9 57.8 43.8 30.1 0

0 210.1 240.2 272.1 301.9 335.6 373.7 439.4 486.6 526.9 609.3

*International Critical Tables, vol. 3, p. 290.

TABLE 2-29 Conc. g NaOH/ 100 g H2O 0 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 700 1000 2000 4000 8000

Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide* mmHg Temperature, °C 0

20

40

60

80

100

120

160

200

250

300

350

4.6 4.4 4.2 3.6 2.9 2.2

17.5 16.9 16.0 13.9 11.3 8.7 6.3 4.4 3.0 2.0 1.3 0.9

55.3 53.2 50.6 44.2 36.6 28.7 20.7 15.5 10.9 7.6 5.2 3.6 1.7

149.5 143.5 137.0 120.5 101.0 81.0 62.5 47.0 34.5 24.5 17.5 12.5 6.3 3.0 1.5

355.5 341.5 325.5 288.5 246.0 202.0 160.5 124.0 94.0 70.5 53.0 38.5 20.5 11.0 6.0 3.5 2.0 0.5 0.1

760.0 730.0 697.0 621.0 537.0 450.0 368.0 294.0 231.0 179.0 138.0 105.0 61.0 35.5 20.5 12.0 7.0 2.0 0.5

1,489 1,430 1,365 1,225 1,070 920 770 635 515 415 330 262 164 102 63 40 25 8 2.7 0.9

4,633 4,450 4,260 3,860 3,460 3,090 2,690 2,340 2,030 1,740 1,490 1,300 915 765 470 340 245 110 50 23 11

11,647 11,200 10,750 9,800 8,950 8,150 7,400 6,750 6,100 5,500 5,000 4,500 3,650 2,980 2,430 1,980 1,620 985 610 380 240 100

29,771 28,600 27,500 25,300 23,300 21,500 19,900 18,400 17,100 15,800 14,700 13,650 11,800 10,300 8,960 7,830 6,870 5,000 3,690 2,750 2,080 1,210 440

64,200 61,800 59,300 54,700 50,800 47,200 44,100 41,200 38,700 36,300 34,200 32,200 28,800 25,900 23,300 21,200 19,200 15,400 12,500 10,300 8,600 6,100 3,300 1,470 150

123,600 118,900 114,100 105,400 98,000 91,600 85,800 80,700 76,000 71,900 68,100 64,600 58,600 53,400 49,000 45,100 41,800 35,000 29,800 25,700 22,400 17,500 11,500 6,800 1,760 120 7

*International Critical Tables, vol. 3, p. 370.

WATER-VAPOR CONTENT OF GASES

2-95

WATER-VAPOR CONTENT OF GASES CHART FOR GASES AT HIGH PRESSURES The accompanying figure is useful in determining the water-vapor content of air at high pressure in contact with liquid water.

Water content of air, °C = (°F − 32) × 5⁄ 9. (Landsbaum, Dadds, and Stutzman. Reprinted from vol. 47, January 1955 issue of Ind. Eng. Chem. [p. 192]. Copyright 1955 by the American Chemical Society and reproduced by permission of the copyright owner.)

FIG. 2-4

2-96

PHYSICAL AND CHEMICAL DATA

DENSITIES OF PURE SUBSTANCES TABLE 2-30 Density (kg/m3) of Saturated Liquid Water from the Triple Point to the Critical Point T, K 273.160* 274 276 278 280 282 284 286 288 290 292 294 296 298 300 302 304 306 308 310 312 314 316 318 320 322 324 326 328 330 332 334 336 338 340 342 344 346 348 350

ρ, kg/m3

T, K

ρ, kg/m3

T, K

ρ, kg/m3

T, K

ρ, kg/m3

T, K

ρ, kg/m3

999.793 999.843 999.914 999.919 999.862 999.746 999.575 999.352 999.079 998.758 998.392 997.983 997.532 997.042 996.513 995.948 995.346 994.711 994.042 993.342 992.610 991.848 991.056 990.235 989.387 988.512 987.610 986.682 985.728 984.750 983.747 982.721 981.671 980.599 979.503 978.386 977.247 976.086 974.904 973.702

352 354 356 358 360 362 364 366 368 370 372 374 376 378 380 382 384 386 388 390 392 394 396 398 400 402 404 406 408 410 412 414 416 418 420 422 424 426 428 430

972.479 971.235 969.972 968.689 967.386 966.064 964.723 963.363 961.984 960.587 959.171 957.737 956.285 954.815 953.327 951.822 950.298 948.758 947.199 945.624 944.030 942.420 940.793 939.148 937.486 935.807 934.111 932.398 930.668 928.921 927.157 925.375 923.577 921.761 919.929 918.079 916.212 914.328 912.426 910.507

432 434 436 438 440 442 444 446 448 450 452 454 456 458 460 462 464 466 468 470 472 474 476 478 480 482 484 486 488 490 492 494 496 498 500 502 504 506 508 510

908.571 906.617 904.645 902.656 900.649 898.624 896.580 894.519 892.439 890.341 888.225 886.089 883.935 881.761 879.569 877.357 875.125 872.873 870.601 868.310 865.997 863.664 861.310 858.934 856.537 854.118 851.678 849.214 846.728 844.219 841.686 839.130 836.549 833.944 831.313 828.658 825.976 823.269 820.534 817.772

512 514 516 518 520 522 524 526 528 530 532 534 536 538 540 542 544 546 548 550 552 554 556 558 560 562 564 566 568 570 572 574 576 578 580 582 584 586 588 590

814.982 812.164 809.318 806.441 803.535 800.597 797.629 794.628 791.594 788.527 785.425 782.288 779.115 775.905 772.657 769.369 766.042 762.674 759.263 755.808 752.308 748.762 745.169 741.525 737.831 734.084 730.283 726.425 722.508 718.530 714.489 710.382 706.206 701.959 697.638 693.238 688.757 684.190 679.533 674.781

592 594 596 598 600 602 604 606 608 610 612 614 616 618 620 622 624 626 628 630 632 634 636 638 640 641 642 643 644 645 646 647 647.096†

669.930 664.974 659.907 654.722 649.411 643.97 638.38 632.64 626.74 620.65 614.37 607.88 601.15 594.16 586.88 579.26 571.25 562.81 553.84 544.25 533.92 522.71 510.42 496.82 481.53 473.01 463.67 453.14 440.73 425.05 402.96 357.34 322

*Triple point †Critical point From Wagner, W., and Pruss, A., “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data 31(2):387–535, 2002.

DENSITIES OF PURE SUBSTANCES TABLE 2-31

2-97

Density (kg/m3) of Mercury from 0 to 350°C* Density, kg /m3

t, °C

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40

13595.08 13570.44 13545.87 13521.36 13496.92

13592.61 13567.98 13543.41 13518.91 13494.48

13590.14 13565.52 13540.96 13516.47 13492.04

13587.68 13563.06 13538.51 13514.02 13489.60

13585.21 13560.60 13536.06 13511.58 13487.16

13582.75 13558.14 13533.61 13509.13 13484.72

13580.29 13555.69 13531.16 13506.69 13482.29

13577.82 13553.23 13528.71 13504.25 13479.85

13575.36 13550.78 13526.26 13501.80 13477.41

13572.90 13548.32 13523.81 13499.36 13474.98

50 60 70 80 90

13472.54 13448.22 13423.96 13399.75 13375.59

13470.11 13445.80 13421.54 13397.34 13373.18

13467.67 13443.37 13419.12 13394.92 13370.77

13465.24 13440.94 13416.69 13392.50 13368.36

13462.81 13438.51 13414.27 13390.08 13365.94

13460.38 13436.09 13411.85 13387.67 13363.53

13457.94 13433.66 13409.43 13385.25 13361.12

13455.51 13431.23 13407.01 13382.84 13358.71

13453.08 13428.81 13404.59 13380.42 13356.30

13450.65 13426.39 13402.17 13378.01 13353.89

100 110 120 130 140

13351.5 13327.4 13303.4 13279.4 13255.4

13349.1 13325.0 13301.0 13277.0 13253.0

13346.7 13322.6 13298.6 13274.6 13250.6

13344.3 13320.2 13296.2 13272.2 13248.2

13341.9 13317.8 13293.8 13269.8 13245.8

13339.4 13315.4 13291.4 13267.4 13243.4

13337.0 13313.0 13288.9 13265.0 13241.0

13334.6 13310.6 13286.6 13262.6 13238.7

13332.2 13308.2 13284.2 13260.2 13236.3

13329.8 13305.8 13281.8 13257.8 13233.9

150 160 170 180 190

13231.5 13207.6 13183.7 13159.8 13136.0

13229.1 13205.2 13181.3 13157.4 13133.6

13226.7 13202.8 13178.9 13155.0 13131.2

13224.3 13200.4 13176.5 13152.6 13128.3

13221.9 13198.0 13174.1 13150.3 13126.4

13219.5 13195.6 13171.7 13147.9 13124.0

13217.1 13193.2 13169.4 13145.5 13121.7

13214.7 13190.8 13167.0 13143.1 13119.3

13212.4 13188.5 13164.6 13140.7 13116.9

13210.0 13186.1 13162.2 13138.3 13114.5

200 210 220 230 240

13112.1 13088.3 13064.5 13040.6 13016.8

13109.7 13085.9 13062.1 13038.3 13014.5

13107.4 13083.5 13059.7 13035.9 13012.1

13105.0 13081.1 13057.3 13033.5 13009.7

13102.6 13078.8 13054.9 13031.1 13007.3

13100.2 13076.4 13052.6 13028.7 13004.9

13097.8 13074.0 13050.2 13026.4 13002.5

13095.4 13071.6 13047.8 13024.0 13000.2

13093.1 13069.2 13045.4 13021.6 12997.8

13090.7 13066.8 13043.0 13019.2 12995.4

250 260 270 280 290

12993.0 12969.2 12945.4 12921.5 12897.7

12990.6 12966.8 12943.0 12919.1 12895.3

12988.3 12964.4 12940.6 12916.7 12892.9

12985.9 12962.0 12938.2 12914.4 12890.5

12983.5 12959.7 12935.8 12912.0 12888.1

12981.1 12957.3 12933.4 12909.6 12885.7

12978.7 12954.9 12931.1 12907.2 12883.3

12976.3 12952.5 12928.7 12904.8 12880.9

12974.0 12950.1 12926.3 12902.4 12878.5

12971.6 12947.7 12923.9 12900.0 12876.2

300 310 320 330 340

12873.8 12849.9 12825.9 12801.9 12777.8

12871.4 12847.5 12823.5 12799.5 12775.4

12869.0 12845.1 12821.1 12797.1 12773.0

12866.6 12842.7 12818.7 12794.7 12770.6

12864.2 12840.3 12816.3 12792.3 12768.2

12861.8 12837.9 12813.9 12789.9 12765.8

12859.4 12835.5 12811.5 12787.5 12763.4

12857.0 12833.1 12809.1 12785.1 12761.0

12854.6 12830.7 12806.7 12782.7 12758.6

12852.2 12828.3 12804.3 12780.2 12756.1

350

12753.7

*From “Mercury—Density and Thermal Expansion at Atmospheric Pressure and Temperatures from 0 to 350 °C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. The density values obtainable from those cited for the specific volume of the saturated liquid in the “Thermodynamic Properties” subsection show minor differences. No attempt was made to adjust either set.

2-98

TABLE 2-32 Cmpd. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Densities of Inorganic and Organic Liquids (mol/dm3) Name

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O

CAS no.

Mol. wt.

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138

C1 1.6994 1.016 1.4486 0.86852 1.2332 1.3064 2.4507 1.3261 1.2414 1.0816 2.8963 3.5383 0.77488 3.8469 0.7371 1.0259 0.83573 0.71587 0.8552 0.43743 0.59867 0.60917 0.70797 0.52257 2.1872 0.8226 1.1908 1.6762 1.187 1.2346 1.0677 0.81696 0.81856 0.98279 0.9682 1.0877 1.1591 1.1448 0.67794 0.50812 0.89458 0.89137 1.3409 1.0361 0.88443 0.87533 2.768 1.7968 2.897 0.99835 1.955 2.23 0.8711 1.3 1.0841 1.817 1.087 1.1202 0.9061

C2 0.26167 0.21845 0.25892 0.25187 0.25886 0.22597 0.27448 0.26124 0.25822 0.2293 0.26733 0.25443 0.26114 0.2881 0.25487 0.26666 0.26326 0.24812 0.26785 0.24833 0.22849 0.26925 0.25982 0.25833 0.29527 0.26632 0.25595 0.26141 0.26114 0.27216 0.27188 0.24755 0.24967 0.26830 0.26244 0.26454 0.27085 0.27154 0.2637 0.25238 0.27463 0.27365 0.27892 0.26731 0.25828 0.24331 0.26212 0.28749 0.27532 0.274 0.27884 0.27645 0.26805 0.26019 0.2581 0.25877 0.26832 0.27669 0.28268

C3 466 761 591.95 606 508.2 545.5 308.3 506 615 535 132.45 405.65 645.6 150.86 824 562.05 689 751 699.35 830 720.15 662 718 773 584.15 670.15 503.8 467 452 425 425.12 680 676 563.1 535.9 419.5 435.5 428.6 575.4 660.5 570.1 554 440 537.2 615.7 582.25 304.21 552 132.92 556.35 227.51 417.15 632.35 460.35 536.4 416.25 503.15 489 705.85

C4

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

0.2913 0.26116 0.2529 0.31172 0.2913 0.28678 0.28752 0.2489 0.30701 0.28939 0.27341 0.2888 0.28234 0.29783 0.28571 0.28394 0.30798 0.2857 0.30523 0.27555 0.23567 0.2632 0.32144 0.27026 0.3295 0.2821 0.29152 0.28402 0.3065 0.28707 0.28688 0.24535 0.22023 0.25488 0.26749 0.2843 0.28116 0.28419 0.29318 0.29373 0.28512 0.2953 0.29661 0.28397 0.248 0.28586 0.2908 0.3226 0.2813 0.287 0.28571 0.2926 0.2799 0.27155 0.2741 0.2833 0.28055 0.27646 0.2707

150.15 353.33 289.81 200.15 178.45 229.32 192.40 185.45 286.15 189.63 59.15 195.41 235.65 83.78 403.00 278.68 258.27 395.45 260.40 321.35 257.85 275.65 243.95 342.20 265.85 242.43 154.55 179.47 136.95 164.25 134.86 220.00 196.15 183.85 158.45 87.80 134.26 167.62 199.65 185.30 157.46 133.02 147.43 176.75 267.95 161.25 216.58 161.11 68.15 250.33 89.56 172.12 227.95 134.80 209.63 175.43 150.35 155.97 285.39

21.499 16.936 17.492 11.643 15.683 20.628 23.692 16.822 14.693 17.265 33.279 43.141 9.668 35.491 8.938 11.422 10.074 8.894 10.011 5.950 9.905 7.065 8.862 6.425 20.109 9.909 15.833 20.640 15.123 14.058 12.620 11.734 11.872 12.035 12.471 14.264 13.894 13.080 8.337 7.026 10.585 10.761 14.901 12.589 11.087 13.047 26.828 19.064 30.180 10.843 21.211 24.242 10.385 17.016 13.702 22.347 13.328 12.855 9.612

466.00 761.00 591.95 606.00 508.20 545.50 308.30 506.00 615.00 535.00 132.45 405.65 645.60 150.86 824.00 562.05 689.00 751.00 699.35 830.00 720.15 662.00 718.00 773.00 584.15 670.15 503.80 467.00 452.00 425.00 425.12 680.00 676.00 563.10 535.90 419.50 435.50 428.60 575.40 660.50 570.10 554.00 440.00 537.20 615.70 582.25 304.21 552.00 132.92 556.35 227.51 417.15 632.35 460.35 536.40 416.25 503.15 489.00 705.85

6.4944 4.6509 5.5948 3.4483 4.7640 5.7813 8.9285 5.0762 4.8075 4.7170 10.8340 13.9070 2.9673 13.3530 2.8921 3.8472 3.1745 2.8852 3.1928 1.7615 2.6201 2.2625 2.7248 2.0229 7.4075 3.0888 4.6525 6.4121 4.5455 4.5363 3.9271 3.3002 3.2786 3.6630 3.6892 4.1117 4.2795 4.2160 2.5709 2.0133 3.2574 3.2573 4.8075 3.8760 3.4243 3.5976 10.5600 6.2500 10.5220 3.6436 7.0112 8.0666 3.2498 4.9963 4.2003 7.0217 4.0511 4.0486 3.2054

2-99

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine

C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N

95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7

108.138 108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133 194.184 88.105 170.207 101.190

1.0861 1.1503 0.58711 1.0743 1.3931 0.88998 0.8243 0.86464 0.92997 1.0897 1.1035 1.7411 0.78578 0.46802 0.41084 0.39348 0.38208 0.43981 0.44289 0.46877 5.2115 0.95523 1.0132 1.1136 0.55941 0.74495 0.74404 0.74858 1.1055 1.2591 1.3897 0.9551 0.89833 0.68184 0.85379 0.9554 0.82227 1.4345 1.173 1.9973 0.6181 0.69213 0.64619 0.89368 0.76327 1.1717 1.5436 0.7565 0.55873 0.52953 0.54405 1.1058 1.5693 0.89615 0.72352 0.47977 1.0214 1.4029 1.1096 0.50824 1.1819 0.52133 0.659

0.30624 0.31861 0.25583 0.20948 0.29255 0.27376 0.26545 0.26888 0.27056 0.28356 0.27035 0.28205 0.27882 0.27146 0.25175 0.2492 0.24645 0.25661 0.27636 0.25875 0.315 0.26364 0.26634 0.24834 0.27243 0.26147 0.26112 0.26276 0.26533 0.27698 0.25678 0.27794 0.26142 0.23796 0.25675 0.26847 0.26314 0.25774 0.22856 0.24653 0.25786 0.26974 0.26881 0.26599 0.26742 0.25895 0.27784 0.27305 0.25143 0.24358 0.25026 0.27866 0.2679 0.23478 0.28629 0.25428 0.26351 0.27991 0.25189 0.26885 0.2813 0.26218 0.26428

697.55 704.65 631 400.15 459.93 553.8 650.1 653 560.4 511.7 507 398 664 674.2 617.7 722.1 688 616.6 696 619.85 38.35 628 650.15 611 584.1 683.95 705 684.75 523 561.6 510 560 572 736.6 496.6 466.7 557.15 386.44 445 351.26 523.1 500.05 576 507.8 543 473.2 437.2 500 591.15 606.15 596.15 615 400.1 649.6 537.3 766 402 503.04 729 772 587 766.8 550

0.30587 0.30104 0.28498 0.20724 0.24913 0.28571 0.28495 0.29943 0.28943 0.25142 0.28699 0.29598 0.31067 0.26869 0.28571 0.28571 0.26125 0.29148 0.27668 0.29479 0.28571 0.29825 0.28571 0.27583 0.29932 0.31526 0.30815 0.30788 0.287 0.30492 0.2902 0.24132 0.2868 0.2062 0.27027 0.2814 0.27369 0.28178 0.28571 0.28153 0.271 0.28571 0.28036 0.28571 0.28571 0.27289 0.2572 0.27408 0.27758 0.26809 0.2658 0.31082 0.2882 0.28091 0.27121 0.30722 0.28421 0.2741 0.3311 0.2612 0.3047 0.31033 0.2766

304.19 307.93 177.14 245.25 182.48 279.69 296.60 242.00 169.67 179.28 138.13 145.59 189.64 267.15 243.51 304.55 280.05 206.89 247.56 229.15 18.73 210.15 282.85 220.60 175.30 248.39 256.15 326.14 176.19 237.49 178.01 200.00 172.71 301.15 223.35 156.85 169.20 154.56 215.00 136.95 176.85 187.65 204.81 159.95 226.10 240.91 180.96 145.19 239.66 223.16 184.99 188.44 131.65 212.72 160.00 274.18 122.93 174.88 291.67 413.80 284.95 300.03 210.15

9.575 9.449 7.939 18.520 14.074 9.380 9.469 10.090 11.160 11.906 13.470 18.658 8.905 5.383 5.393 5.181 5.261 5.733 5.005 5.895 42.945 11.799 11.704 15.358 6.607 9.121 9.166 8.518 13.549 13.462 17.974 10.862 11.526 10.390 10.575 11.487 10.470 18.006 17.424 27.399 8.054 8.067 7.680 11.029 8.843 13.767 16.964 9.031 7.342 7.578 7.626 12.413 18.950 13.954 7.874 6.233 12.898 15.556 14.111 5.538 11.838 6.265 7.993

697.55 704.65 631.00 400.15 459.93 553.80 650.10 653.00 560.40 511.70 507.00 398.00 664.00 674.20 617.70 722.10 688.00 616.60 696.00 619.85 38.35 628.00 650.15 611.00 584.10 683.95 705.00 684.75 523.00 561.60 510.00 560.00 572.00 736.60 496.60 466.70 557.15 386.44 445.00 351.26 523.10 500.05 576.00 507.80 543.00 473.20 437.20 500.00 591.15 606.15 596.15 615.00 400.10 649.60 537.30 766.00 402.00 503.04 729.00 772.00 587.00 766.80 550.00

3.5466 3.6104 2.2949 5.1284 4.7619 3.2509 3.1053 3.2157 3.4372 3.8429 4.0817 6.1730 2.8182 1.7241 1.6320 1.5790 1.5503 1.7139 1.6026 1.8117 16.5440 3.6232 3.8042 4.4842 2.0534 2.8491 2.8494 2.8489 4.1665 4.5458 5.4120 3.4364 3.4363 2.8654 3.3254 3.5587 3.1248 5.5657 5.1321 8.6070 2.3970 2.5659 2.4039 3.3598 2.8542 4.5248 5.5557 2.7706 2.2222 2.1739 2.1739 3.9683 5.8578 3.8170 2.5272 1.8868 3.8761 5.0120 4.4051 1.8904 4.2016 1.9884 2.4936

2-100

TABLE 2-32 Cmpd. no. 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181

Densities of Inorganic and Organic Liquids (mol/dm3) (Continued) Name

Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne

Formula

CAS no.

Mol. wt.

C1

C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10

112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2

170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144

0.33267 0.18166 1.9122 1.6288 0.8996 1.0936 0.70041 0.48864 0.66085 0.63566 0.61587 0.71751 2.0961 0.7842 1.315 1.3462 1.836 1.1343 0.47428 0.55729 0.8185 0.68162 1.3047 0.7405 0.7908 0.58579 4.2895 1.0146 1.6525 2.1854 1.9415 1.2486 1.938 1.1339 7.2475 0.21897 0.59006 0.61259 0.53066 0.55687 0.57114 0.59268 0.58247 0.66016 0.58622 0.67304 0.23289 0.71899 0.70824 0.62833 0.70093 0.67393 0.67816 0.67666 0.76925 0.78045 0.66372 0.84427 0.76277

C2 0.24664 0.23351 0.27937 0.27469 0.25856 0.22636 0.26162 0.23894 0.25707 0.25613 0.26477 0.26903 0.27657 0.20702 0.25125 0.23289 0.26024 0.26168 0.25028 0.2714 0.26929 0.25152 0.2694 0.25563 0.266 0.24246 0.28587 0.27277 0.27099 0.24725 0.22309 0.20352 0.24225 0.24741 0.41865 0.23642 0.25609 0.26211 0.24729 0.24725 0.25534 0.25663 0.25279 0.26657 0.2726 0.26045 0.23659 0.26531 0.26411 0.25598 0.26776 0.25948 0.25634 0.25578 0.26809 0.26065 0.27345 0.27185 0.25248

C3 658 768 305.32 514 523.3 456.15 617.15 698 655 571 609.15 569.5 282.34 593 720 537 469.15 508.4 674.6 583 489 567 499.15 546 500.23 559.95 144.12 560.09 375.31 317.42 408 771 588 490.15 5.2 736 616.8 540.2 677.3 632.3 608.3 606.6 611.4 537.4 645 547 723 591 507.6 660.2 611.3 585.3 587.61 582.82 504 544 623 516.2 549

C4 0.28571 0.28571 0.29187 0.23178 0.278 0.25522 0.28454 0.28421 0.31103 0.27829 0.28054 0.27733 0.29147 0.20254 0.21868 0.23357 0.2696 0.2791 0.25442 0.29538 0.30621 0.3182 0.27866 0.2795 0.292 0.29509 0.28776 0.28291 0.2442 0.27558 0.28571 0.25178 0.24435 0.2612 0.24096 0.28571 0.28384 0.28141 0.28289 0.31471 0.26487 0.27766 0.29818 0.28571 0.29644 0.28388 0.28571 0.27628 0.27537 0.25304 0.24919 0.26552 0.28365 0.27746 0.28571 0.28571 0.29185 0.2771 0.31611

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

263.57 309.58 90.35 159.05 189.60 192.15 178.20 238.45 258.15 175.15 161.84 134.71 104.00 284.29 260.15 195.20 160.65 193.55 235.00 180.00 140.00 204.15 125.26 199.25 145.65 167.55 53.48 230.94 129.95 131.35 181.15 275.60 281.45 187.55 2.20 295.13 229.80 182.57 265.83 239.15 230.00 234.15 238.15 154.12 229.92 192.22 291.31 217.15 177.83 269.25 228.55 223.00 217.35 217.50 133.39 170.05 192.62 141.25 183.65

4.521 2.729 21.640 19.410 11.478 17.588 9.041 7.291 8.220 8.491 7.868 9.018 23.326 15.055 18.310 21.450 23.477 14.006 6.563 6.612 9.924 8.975 16.242 9.632 9.847 8.653 44.888 11.374 19.785 29.526 30.945 25.488 26.806 15.702 37.115 3.219 7.600 7.700 7.221 7.502 7.454 7.575 7.551 8.226 6.728 8.492 3.415 8.724 8.747 8.096 8.456 8.518 8.732 8.763 9.581 10.021 7.773 10.230 10.133

658.00 768.00 305.32 514.00 523.30 456.15 617.15 698.00 655.00 571.00 609.15 569.50 282.34 593.00 720.00 537.00 469.15 508.40 674.60 583.00 489.00 567.00 499.15 546.00 500.23 559.95 144.12 560.09 375.31 317.42 408.00 771.00 588.00 490.15 5.20 736.00 616.80 540.20 677.30 632.30 608.30 606.60 611.40 537.40 645.00 547.00 723.00 591.00 507.60 660.20 611.30 585.30 587.61 582.82 504.00 544.00 623.00 516.20 549.00

1.3490 0.7780 6.8447 5.9296 3.4793 4.8312 2.6772 2.0450 2.5707 2.4818 2.3261 2.6670 7.5789 3.7880 5.2338 5.7804 7.0550 4.3347 1.8950 2.0534 3.0395 2.7100 4.8430 2.8968 2.9729 2.4160 15.0050 3.7196 6.0980 8.8388 8.7028 6.1350 8.0000 4.5831 17.3120 0.9262 2.3041 2.3371 2.1459 2.2523 2.2368 2.3095 2.3042 2.4765 2.1505 2.5841 0.9844 2.7100 2.6816 2.4546 2.6178 2.5972 2.6455 2.6455 2.8694 2.9942 2.4272 3.1056 3.0211

2-101

182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244

Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl -1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether

H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O

302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4

32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122 90.187 46.144 118.176 88.148

1.0516 5.414 2.832 3.342 1.3413 2.5635 2.7672 0.88575 1.2801 0.84266 0.87025 2.9214 2.3267 0.88268 1.13 1.6085 0.97286 1.39 0.53382 0.84623 0.91991 0.72762 0.80828 0.91619 0.93391 1.1157 0.8363 0.75509 0.94575 0.76983 1.0674 0.73109 0.7013 0.70973 0.72836 0.84758 0.88824 0.9109 0.97608 1.2635 0.93767 1.067 1.525 0.84005 0.71687 1.0228 0.97887 0.86567 0.78912 1.9323 0.7761 0.4416 0.72701 0.71004 1.0631 0.92128 1.1446 0.9147 0.96145 0.87496 1.3052 0.64856 0.928

0.16613 0.34893 0.2832 0.2729 0.18589 0.1766 0.27369 0.25736 0.2828 0.217 0.24383 0.28976 0.27073 0.23568 0.2593 0.26436 0.26267 0.21405 0.23274 0.24625 0.27815 0.25244 0.26783 0.26752 0.27275 0.27671 0.27514 0.27183 0.26008 0.26173 0.26257 0.26971 0.266 0.26544 0.27241 0.27037 0.26914 0.276 0.28209 0.27878 0.25035 0.27102 0.2634 0.27638 0.26453 0.20692 0.27017 0.26836 0.25915 0.28018 0.25068 0.2521 0.26754 0.26981 0.27506 0.25442 0.2724 0.2594 0.26536 0.26862 0.26757 0.25877 0.289

653.15 33.19 363.15 324.65 456.65 461.15 373.53 605 471.85 805 662 190.56 512.5 718 506.55 402.4 536 430.05 693 490 460.4 643 577.2 465 470 492 512.74 593 463.2 554.5 442 572.1 686 614 617 532.7 542 526 483 437.8 535.5 533 487.2 497 574.6 488 464.48 553.4 553.1 469.95 566 694 497.7 546.49 407.8 506.2 417.9 530.6 476.25 565 352.5 654 497.1

0.1898 0.2706 0.28571 0.3217 0.28206 0.3733 0.29015 0.26265 0.2972 0.28571 0.28571 0.28881 0.24713 0.27379 0.2764 0.27987 0.2508 0.2275 0.28147 0.29041 0.28667 0.28571 0.23588 0.28164 0.2578 0.30821 0.27553 0.29127 0.30807 0.26879 0.26569 0.29185 0.28571 0.26016 0.2478 0.28258 0.27874 0.26756 0.22529 0.2744 0.29964 0.29364 0.2806 0.27645 0.28918 0.28571 0.28998 0.28364 0.26512 0.28523 0.29773 0.28532 0.28268 0.29974 0.2758 0.27586 0.28172 0.2774 0.30088 0.30259 0.28799 0.31444 0.286

274.69 13.95 185.15 158.97 259.83 189.79 187.68 227.15 177.95 407.95 288.15 90.69 175.47 301.15 175.15 170.45 196.32 179.69 260.75 159.53 113.25 193.00 155.95 135.58 139.39 160.15 157.48 175.30 183.45 187.35 139.05 146.58 285.15 280.15 269.15 130.73 146.62 115.00 182.55 160.00 186.48 167.23 174.15 150.00 189.15 256.15 127.93 180.15 171.64 150.18 224.95 240.00 119.55 176.00 113.54 298.97 132.81 185.65 133.97 160.17 116.34 249.95 164.55

31.934 38.487 27.985 34.854 27.202 60.203 29.130 11.420 13.561 13.533 11.834 28.180 27.915 13.012 14.475 19.031 12.203 25.378 8.220 11.994 10.764 9.992 10.254 11.332 11.216 12.581 9.758 9.006 11.519 9.764 13.626 9.017 8.209 8.293 8.263 10.491 10.980 11.014 10.789 13.995 12.663 12.671 18.811 9.738 8.862 17.666 11.933 10.460 10.352 21.564 10.176 5.938 9.204 8.445 12.574 10.556 13.507 11.678 12.043 10.689 15.791 8.010 9.710

653.15 33.19 363.15 324.65 456.65 461.15 373.53 605.00 471.85 805.00 662.00 190.56 512.50 718.00 506.55 402.40 536.00 430.05 693.00 490.00 460.40 643.00 577.20 465.00 470.00 492.00 512.74 593.00 463.20 554.50 442.00 572.10 686.00 614.00 617.00 532.70 542.00 526.00 483.00 437.80 535.50 533.00 487.20 497.00 574.60 488.00 464.48 553.40 553.10 469.95 566.00 694.00 497.70 546.49 407.80 506.20 417.90 530.60 476.25 565.00 352.50 654.00 497.10

6.3300 15.5160 10.0000 12.2460 7.2156 14.5160 10.1110 3.4417 4.5265 3.8832 3.5691 10.0820 8.5942 3.7452 4.3579 6.0845 3.7037 6.4938 2.2936 3.4365 3.3072 2.8823 3.0179 3.4248 3.4241 4.0320 3.0395 2.7778 3.6364 2.9413 4.0652 2.7107 2.6365 2.6738 2.6738 3.1349 3.3003 3.3004 3.4602 4.5322 3.7454 3.9370 5.7897 3.0395 2.7100 4.9430 3.6232 3.2258 3.0450 6.8966 3.0960 1.7517 2.7174 2.6316 3.8650 3.6211 4.2019 3.5262 3.6232 3.2572 4.8780 2.5063 3.2111

2-102

TABLE 2-32 Cmpd. no. 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303

Densities of Inorganic and Organic Liquids (mol/dm3) (Concluded) Name

Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine

Formula

CAS no.

Mol. wt.

C1

C2

C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N

107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8

58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132 59.110

1.2587 0.6348 7.3718 1.0024 3.2091 2.3736 1.3728 2.781 5.246 0.19199 0.49587 0.46321 0.41582 0.43682 0.41687 0.48661 0.47377 0.52152 0.20448 0.53636 0.5266 0.48251 0.48979 0.50726 0.50006 0.5108 0.55449 0.52577 0.58945 1.0501 3.9143 3.3592 0.25142 0.83871 0.84947 0.73455 0.81754 0.79324 0.90411 0.71811 0.89816 0.65858 0.75345 0.8491 0.92099 0.45554 1.3798 0.63163 0.5393 1.6087 1.3757 1.2457 1.1799 0.61255 1.296 1.0969 1.0224 0.73041 0.9195

0.26433 0.25838 0.3067 0.23655 0.2861 0.2817 0.23793 0.27244 0.3044 0.23337 0.26135 0.25444 0.24284 0.25161 0.24056 0.25722 0.27052 0.25918 0.23474 0.26174 0.25693 0.25196 0.24931 0.25972 0.24851 0.25386 0.25952 0.27234 0.26052 0.215 0.28772 0.29884 0.23837 0.26252 0.26726 0.25636 0.26732 0.25806 0.27207 0.24129 0.26608 0.25367 0.27047 0.2352 0.25419 0.2523 0.31598 0.23373 0.22704 0.26543 0.27453 0.27281 0.2644 0.26769 0.26439 0.25568 0.23452 0.25456 0.23878

C3 437 748.4 44.4 593 126.2 234 588.15 309.57 180.15 758 658 594.6 710.7 670.9 649.5 593.1 681 598.05 747 638.9 568.7 694.26 652.3 629.8 632.7 627.7 566.9 667.3 574 804 154.58 261 708 566.1 469.7 639.16 588.1 561 561.08 560.95 464.8 584.3 598 481.2 519 869 694.25 653 791 394 369.83 536.8 508.3 636 504.4 600.81 564.4 549.73 496.95

C4 0.25819 0.27727 0.2786 0.278 0.2966 0.29529 0.29601 0.2882 0.242 0.28571 0.30736 0.28571 0.30036 0.2498 0.2916 0.28571 0.30284 0.29177 0.28571 0.26348 0.28571 0.26842 0.27824 0.22 0.29942 0.26735 0.28571 0.30063 0.28532 0.28571 0.2924 0.28523 0.28571 0.29444 0.27789 0.25522 0.25348 0.28571 0.30669 0.27996 0.28571 0.28571 0.30583 0.353 0.31077 0.24841 0.32768 0.28571 0.248 0.29895 0.29359 0.23994 0.24653 0.28571 0.29471 0.26857 0.2804 0.27666 0.2461

Tmin, K

Density at Tmin

Tmax, K

Density at Tmax

151.15 333.15 24.56 183.63 63.15 66.46 244.60 182.30 109.50 305.04 255.15 219.66 285.55 268.15 238.15 191.91 253.05 223.15 301.31 246.00 216.38 289.65 257.65 241.55 252.85 255.55 171.45 223.95 193.55 462.65 54.35 80.15 283.07 182.00 143.42 239.15 195.56 200.00 196.29 234.18 108.02 160.75 197.45 167.45 163.83 372.38 314.06 243.15 404.15 136.87 85.47 146.95 185.26 199.00 170.00 252.45 180.26 178.15 188.36

15.691 7.755 61.796 15.556 31.063 26.555 19.632 27.928 44.487 2.889 6.017 6.043 5.759 5.850 6.031 6.372 5.453 6.537 3.042 6.664 6.705 6.311 6.574 6.563 6.648 6.628 7.216 6.099 7.483 16.271 40.770 33.361 3.642 10.534 10.474 9.587 10.061 10.147 10.398 10.102 11.521 9.073 8.858 12.532 12.240 5.985 11.244 9.647 8.222 19.479 16.583 15.206 14.663 7.476 15.929 13.935 16.027 9.794 13.764

437.00 748.40 44.40 593.00 126.20 234.00 588.15 309.57 180.15 758.00 658.00 594.60 710.70 670.90 649.50 593.10 681.00 598.05 747.00 638.90 568.70 694.26 652.30 629.80 632.70 627.70 566.90 667.30 574.00 804.00 154.58 261.00 708.00 566.10 469.70 639.16 588.10 561.00 561.08 560.95 464.80 584.30 598.00 481.20 519.00 869.00 694.25 653.00 791.00 394.00 369.83 536.80 508.30 636.00 504.40 600.81 564.40 549.73 496.95

4.7619 2.4568 24.0360 4.2376 11.2170 8.4260 5.7698 10.2080 17.2340 0.8227 1.8973 1.8210 1.7123 1.7361 1.7329 1.8918 1.7513 2.0122 0.8711 2.0492 2.0500 1.9150 1.9646 1.9531 2.0122 2.0121 2.1366 1.9306 2.2626 4.8842 13.6050 11.2410 1.0550 3.1948 3.1784 2.8653 3.0583 3.0739 3.3231 2.9761 3.3755 2.5962 2.7857 3.6101 3.6232 1.8055 4.3667 2.7024 2.3754 6.0607 5.0111 4.5662 4.4626 2.2883 4.9018 4.2901 4.3595 2.8693 3.8508

304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 344 345 346 347

Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl o-Terphenyl [use Eq. (2)] Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water [use Eq. (2)] m-Xylene o-Xylene p-Xylene

C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

0.57233 1.4403 0.915 1.093 1.0714 1.0923 0.83228 1.1945 0.7397 0.70284 2.106 1.3587 1.4969 0.42685 0.3448 5.7136 0.27248 1.2543 0.67717 1.1628 0.58988 1.2874 0.8792 0.9062 0.29934 0.7035 1.0116 0.6531 0.60394 0.59059 0.6028 0.48195 0.37378 0.36703 0.33113 0.9591 1.2703 1.5115 0.59595 −13.851 0.68902 0.69962 0.67752

0.25171 0.26852 0.26134 0.27762 0.27214 0.26106 0.25385 0.24128 0.2603 0.22268 0.25842 0.2701 0.19013 0.181 0.25116 −0.003474 0.24007 0.28084 0.27772 0.28954 0.27201 0.28194 0.27136 0.25475 0.2433 0.27386 0.25683 0.27002 0.25956 0.27424 0.27446 0.23093 0.21379 0.24876 0.23676 0.2593 0.26041 0.2707 0.24314 0.64038 0.26086 0.26143 0.25887

638.35 364.85 538 517 536.6 626 683 259 636 806 430.75 318.69 490.85 1113 857 693 540.15 720 631.95 568 579.35 591.75 602 675 535.15 433.25 664.5 649.1 543.8 573.5 846 828 639 703.9 519.13 454 432 543.15 −0.00191 617 630.3 616.2

0.29616 0.28775 0.28 0.29781 0.29481 0.20459 0.23658 0.16693 0.3009 0.28571 0.2895 0.2921 0.4359 0.28571 0.29268 0.28571 0.2912 0.2878 0.28674 0.27341 0.30781 0.29241 0.31 0.28571 0.2872 0.2696 0.26268 0.27713 0.2847 0.2741 0.28571 0.29905 0.28571 0.2762 0.27448 0.297 0.2716 0.24856 1.8211E-06 0.27479 0.27365 0.27596

173.55 87.89 180.25 142.61 159.95 213.15 388.85 186.35 242.54 460.65 197.67 223.15 289.95 700.15 329.35 288.15 279.01 164.65 237.38 176.99 373.96 234.94 178.18 236.50 267.76 158.45 156.08 243.15 229.33 165.78 172.22 398.40 354.00 247.57 288.45 180.35 173.15 119.36 178.35 273.16 225.30 247.98 286.41

7.982 18.070 11.590 12.610 12.716 14.363 10.082 15.635 9.109 10.261 25.298 12.631 24.241 8.546 4.553 4.713 3.889 13.998 7.638 12.408 5.724 13.430 10.487 11.478 4.182 8.284 13.144 7.728 7.689 6.915 7.093 7.083 6.452 4.945 4.859 12.287 15.664 18.481 8.824 55.497 8.648 8.623 8.161

638.35 364.85 538.00 517.00 536.60 626.00 683.00 259.00 636.00 806.00 430.75 318.69 490.85 1113.00 857.00 313.19 693.00 540.15 720.00 631.95 568.00 579.35 591.75 602.00 675.00 535.15 433.25 664.50 649.10 543.80 573.50 846.00 828.00 639.00 703.90 519.13 454.00 432.00 543.15 353.15 617.00 630.30 616.20

2.2738 5.3638 3.5012 3.9370 3.9369 4.1841 3.2786 4.9507 2.8417 3.1563 8.1495 5.0304 7.8730 2.3583 1.3728 4.6256 1.1350 4.4662 2.4383 4.0160 2.1686 4.5662 3.2400 3.5572 1.2300 2.5688 3.9388 2.4187 2.3268 2.1536 2.1963 2.0870 1.7484 1.4750 1.3986 3.6988 4.8781 5.5837 2.4511 54.0012 2.6413 2.6761 2.6172

Except for o-terphenyl and water, liquid density ρ is calculated by C4]


= C1/C2[1 + (1−T/C3)

where ρ is in mol/dm and T is in K. The pressure is equal to the vapor pressure for pressures greater than 1 atm and equal to 1 atm when the vapor pressure is less than 1 atm. Equation (2), used for the limited temperature ranges as noted for o-terphenyl and water, is 3


= C1 + C2T + C3T 2 + C4T 3 For water over the entire temperature range of 273.16 to 647.096 K, use
= 17.863 + 58.606τ0.35 − 95.396τ2/3 + 213.89τ − 141.26τ4/3 where τ = 1 − T/647.096.

2-103

All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

2-104

PHYSICAL AND CHEMICAL DATA

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM UNITS AND UNITS CONVERSIONS Most densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32. Compositions are weight percent unless otherwise stated. ADDITIONAL REFERENCES For more detailed data on densities see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; air, moist, vol. 1, p. 71; building stones, vol. 2, p. 52; clays, vol. 2, p. 56; coals, vol. 2, p. 135; compounds, vol. 1, pp. 106, 176, 313, 341; elements, vol. 1, pp. 102, 340; fibers, vol. 2, p. 237; gases and vapors, vol. 3, pp. 3, 345; glass, vol. 2, p. 93; liquids and vitreous solids, vol. 3, p. 22; vol. 1, pp. 102, 340; vol. 2, TABLE 2-33

pp. 456, 463; vol. 3, pp. 20, 35; liquid coolants and saturated vapors are available from WADC-TR-59-598, 1959; plastics are collected in the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co.; solid helium, neon, argon, fluorine, and methane data are given by Johnson (ed.), WADD-TR-60-56, 1960; temperatures of maximum solubility, vol. 3, p. 107; metals, vol. 2, p. 463; oils, fats, and waxes, vol. 2, p. 201; orthobaric, vol. 3, pp. 202, 228, 237, 244; petroleums, vol. 2, pp. 137, 144; plastics, vol. 2, p. 296; porcelains, vol. 2, pp. 68, 75; refrigerating brines, vol. 2, p. 327; rubber, vol. 2, pp. 255, 259; soaps, vol. 5, p. 447; metallic solid solutions, vol. 2, p. 358; solids, vol. 3, pp. 43, 45; vol. 2, p. 456; vol. 3, p. 21; solutions and mixtures, vol. 3, pp. 17, 51, 95, 104, 107, 111, 125, 130; woods, vol. 2, p. 1. Also see the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., 86th ed., etc.

Aluminum Sulfate [Al2(SO4)3]*

%

d 15 4

%

d 15 4

1 2 4 8 12

1.0093 1.0195 1.0404 1.0837 1.1293

16 20 24 26

1.1770 1.2272 1.2803 1.3079

TABLE 2-38 Ammonium Chromate [(NH4)2CrO4]*

*International Critical Tables, vol. 3, p. 70. TABLE 2-34 1 2 4 8 12 16 20 24 28 30

0 °C

5 °C

10 °C

20 °C 25 °C %

d 15 4

0.9943 0.9954 0.9959 0.9958 0.9955 0.9939 0.993 32 0.889 .9906 .9915 .9919 .9917 .9913 .9895 .988 36 .877 .9834 .9840 .9842 .9837 .9832 .9811 .980 40 .865 0.970 .9701 .9701 .9695 .9686 .9677 .9651 .964 45 .849 .958 .9576 .9571 .9561 .9548 .9534 .9501 .948 50 .832 .947 .9461 .9450 .9435 .9420 .9402 .9362 .934 60 .796 .9353 .9335 .9316 .9296 .9275 .9229 70 .755 .9249 .9226 .9202 .9179 .9155 .9101 80 .711 .9150 .9122 .9094 .9067 .9040 .8980 90 .665 .9101 .9070 .9040 .9012 .8983 .8920 100 .618 *International Critical Tables, vol. 3, p. 59.

TABLE 2-35 Ammonium Acetate* (CH3COONH4)

TABLE 2-36 Ammonium Bichromate [(NH4)2Cr2O7]*

d 4t

3.80 10.52 19.75 28.04

20 13 13.7 19.6

1.0219 1.0627 1.1189 1.1707

TABLE 2-39

Ammonium Nitrate (NH4NO3)*

%

0 °C

10 °C

25 °C

40 °C

60 °C

80 °C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 30.0 40.0 50.0

1.0043 1.0088 1.0178 1.0358 1.0539 1.0721 1.0905 1.1090 1.1277 1.1371 1.1862 1.2380

1.0039 1.0082 1.0168 1.0340 1.0515 1.0691 1.0870 1.1051 1.1234 1.1327 1.1810 1.2320

1.0011 1.0051 1.0132 1.0297 1.0464 1.0633 1.0806 1.0982 1.1161 1.1252 1.1727 1.2229

0.9961 1.0000 1.0079 1.0238 1.0400 1.0565 1.0734 1.0907 1.1082 1.1171 1.1640 1.2136

0.9870 .9908 .9985 1.0142 1.0301 1.0462 1.0627 1.0796 1.0968 1.1055 1.1515 1.2006

0.9755 .9793 .9869 1.0024 1.0181 1.0342 1.0506 1.0673 1.0844 1.0931 1.1385 1.1868

*International Critical Tables, vol. 3, p. 59.

%

d425

%

d412

1 2 4 8 12 16 20 24 28 30 35 40 45

0.9992 1.0013 1.0055 1.0136 1.0216 1.0294 1.0368 1.0439 1.0507 1.0540 1.0618 1.0691 1.0760

1 2 4 8 12 16 20

1.0051 1.0108 1.0223 1.0463 1.0715 1.0981 1.1263

*International Critical Tables, vol. 3, p. 70.

*International Critical Tables, vol. 3, p. 62. For data at 16 °C for 3(1)52 percent see Atack, Handbook of Chemical Data, p. 33, Reinhold, New York, 1957. TABLE 2-37

°C

*International Critical Tables, vol. 3, p. 70.

Ammonia (NH3)*

% −15 °C −10 °C −5 °C

%

TABLE 2-40

Ammonium Sulfate [(NH4)2SO4]*

%

0 °C

20 °C

40 °C

80 °C

100 °C

1 2 4 8 12 16 20 24 28 35 40 50

1.0061 1.0124 1.0248 1.0495 1.0740 1.0980 1.1215 1.1448 1.1677 1.2072 1.2350 1.2899

1.0041 1.0101 1.0220 1.0456 1.0691 1.0924 1.1154 1.1383 1.1609 1.2800 1.2277 1.2825

0.9980 1.0039 1.0155 1.0387 1.0619 1.0849 1.1077 1.1304 1.1529 1.1919 1.2196 1.2745

0.9777 .9836 .9953 1.0187 1.0421 1.0653 1.0883 1.1111 1.1338 1.1731 1.2011 1.2568

0.9644 .9705 .9826 1.0066 1.0303 1.0539 1.0772 1.1003 1.1232 1.1629 1.1910 1.2466

*International Critical Tables, vol. 3, p. 60.

Ammonium Chloride (NH4Cl)*

%

0 °C

10 °C

20 °C

30 °C

50 °C

80 °C

100 °C

1 2 4 8 12 16 20 24

1.0033 1.0067 1.0135 1.0266 1.0391 1.0510 1.0625 1.0736

1.0029 1.0062 1.0126 1.0251 1.0370 1.0485 1.0596 1.0705

1.0013 1.0045 1.0107 1.0227 1.0344 1.0457 1.0567 1.0674

0.9987 1.0018 1.0077 1.0195 1.0310 1.0422 1.0532 1.0641

0.9910 .9940 .9999 1.0116 1.0231 1.0343 1.0454 1.0564

0.9749 .9780 .9842 .9963 1.0081 1.0198 1.0312 1.0426

0.9617 .9651 .9718 .9849 .9975 1.0096 1.0213 1.0327

*International Critical Tables, vol. 3, p. 60.

TABLE 2-41

Arsenic Acid (H3AsO4)*

%

d 15 4

%

d 15 4

1 2 6 10 16

1.0057 1.0124 1.0398 1.0681 1.1128

20 30 40 50 60 70

1.1447 1.2331 1.3370 1.4602 1.6070 1.7811

*International Critical Tables, vol. 3, p. 61.

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM TABLE 2-42

Barium Chloride (BaCl2)*

TABLE 2-48

2-105

Chromic Acid (CrO3)*

%

0 °C

20 °C

40 °C

60 °C

80 °C

100 °C

%

d 15 4

%

d 15 4

2 4 8 12 16 20 24 26

1.0181 1.0368 1.0760 1.1178 1.1627 1.2105

1.0159 1.0341 1.0721 1.1128 1.1564 1.2031 1.2531 1.2793

1.0096 1.0275 1.0648 1.1047 1.1478 1.1938 1.2430 1.2688

1.0004 1.0181 1.0551 1.0948 1.1373 1.1828 1.2316 1.2571

0.9890 1.0066 1.0434 1.0827 1.1249 1.1702 1.2186 1.2440

0.9755 .9931 1.0299 1.0692 1.1113 1.1563 1.2045 1.2298

1 2 6 10 16

1.006 1.014 1.045 1.076 1.127

20 26 30 40 50 60

1.163 1.220 1.260 1.371 1.505 1.663

*International Critical Tables, vol. 3, p. 69.

*International Critical Tables, vol. 3, p. 75. TABLE 2-43

Cadmium Nitrate [Cd(NO3)2]*

TABLE 2-49

%

d 18 4

%

d 18 4

2 4 8 12 16

1.0154 1.0326 1.0683 1.1061 1.1468

20 25 30 40 50

1.1904 1.2488 1.3124 1.4590 1.6356

d 18 4

*International Critical Tables, vol. 3, p. 66.

TABLE 2-44 % −5 °C 2 4 8 12 16 20 25 30 35 40

1.0708 1.1083 1.1471 1.1874

%

Violet

Green

Equilibrium mixture of violet and green

1 2 4 8 12 14

1.0076 1.0166 1.0349 1.0724 1.1114 1.1316

1.0071 1.0157 1.0332 1.0691 1.1065

1.0075 1.0165 1.0347 1.0722 1.1111

*International Critical Tables, vol. 3, p. 69.

Calcium Chloride (CaCl2)*

0 °C

Chromium Chloride (CrCl3)*

20 °C 30 °C 40 °C 60 °C 80 °C 100 °C 120 °C† 140°C

1.0171 1.0346 1.0703 1.1072 1.1454 1.1853 1.2376 1.2922

1.0148 1.0316 1.0659 1.1015 1.1386 1.1775 1.2284 1.2816 1.3373 1.3957

1.0120 1.0286 1.0626 1.0978 1.1345 1.1730 1.2236 1.2764 1.3316 1.3895

1.0084 1.0249 1.0586 1.0937 1.1301 1.1684 1.2186 1.2709 1.3255 1.3826

0.9994 1.0158 1.0492 1.0840 1.1202 1.1581 1.2079 1.2597 1.3137 1.3700

0.9881 1.0046 1.0382 1.0730 1.1092 1.1471 1.1965 1.2478 1.3013 1.3571

0.9748 0.9915 1.0257 1.0610 1.0973 1.1352 1.1846 1.2359 1.2893 1.3450

0.9596 0.9765 1.0111 1.0466 1.0835 1.1219

0.9428 0.9601 0.9954 1.0317 1.0691 1.1080

TABLE 2-50

*International Critical Tables, vol. 3, pp. 72–73. †Corrected to atmospheric pressure. TABLE 2-45 Calcium Hydroxide [Ca(OH)2]*

Copper Nitrate [Cu(NO3)2]*

%

d 20 4

%

d 20 4

1 2 4 8

1.007 1.015 1.032 1.069

12 16 20 25

1.107 1.147 1.189 1.248

*International Critical Tables, vol. 3, p. 67.

TABLE 2-46 Calcium Hypochlorite* (CaOCl2)

TABLE 2-51 (CuSO4)*

Copper Sulfate

TABLE 2-52 Cuprous Chloride (CuCl2)*

%

d 15 4

d 425

% total salt

d 15 4

%

0 °C

20 °C

40 °C

%

0 °C

20 °C

40 °C

0.05 .10 .15

0.99979 1.00044 1.00110

0.99773 .99838 .99904

2 4 6 8 10 12

1.0169 1.0345 1.0520 1.0697 1.0876 1.1060

1 4 8 12 16 18

1.0104 1.0429 1.0887 1.1379

1.0086 1.0401 1.084 1.1308 1.180 1.206

1.0024 1.0332 1.0764 1.1222

1 4 8 12 16 20

1.0095 1.0387 1.0788 1.1208 1.1653 1.2121

1.0072 1.036 1.0754 1.1165 1.1595 1.2052

1.002 1.0305 1.0682 1.107 1.151 1.1953

*International Critical Tables, vol. 3, p. 72.

*International Critical Tables, vol. 3, p. 73. CaOCl2 = 89.15% CaCl2 = 7.31% Ca(ClO3)2 = 0.26% Ca(OH)2 = 2.92% TABLE 2-47

Calcium Nitrate [Ca(NO3)2]*

*International Critical Tables, vol. 3, p. 67.

TABLE 2-53

*International Critical Tables, vol. 3, p. 66.

Ferric Chloride (FeCl3)*

%

6 °C

18 °C

25 °C

30 °C

%

0 °C

10 °C

20 °C

30 °C

2* 4 8 12 16 20 25 30 35 40 45 68†

1.0157 1.0316 1.0641 1.0979 1.1330 1.1694 1.2168

1.0137 1.0291 1.0608 1.0937 1.1279 1.1636 1.2106 1.260 1.311 1.365 1.422 1.747

1.0120 1.0272 1.0585 1.0911 1.1250 1.1602 1.2065

1.0105 1.0256 1.0565 1.0887 1.1224 1.1575 1.2032

1.741

1.736

1 2 4 8 12 16 20 25 30 35 40 45 50

1.0086 1.0174 1.0347 1.0703 1.1088 1.1475 1.1870 1.2400 1.2970 1.3605 1.4280

1.0084 1.0168 1.0341 1.0692 1.1071 1.1449 1.1847 1.2380 1.2950 1.3580 1.4235 1.4920 1.5610

1.0068 1.0152 1.0324 1.0669 1.1040 1.1418 1.1820 1.2340 1.2910 1.3530 1.4175 1.4850 1.5510

1.0040 1.0122 1.0292 1.0636 1.1006 1.1386 1.1786 1.2290 1.2850 1.3475 1.4115

*International Critical Tables, vol. 3, pp. 73–74. †Supercooled tetrahydrate (m.p. 41.4°C).

*International Critical Tables, vol. 3, p. 68.

2-106

PHYSICAL AND CHEMICAL DATA

TABLE 2-54 Ferric Sulfate [Fe2(SO4)3]* 17.5

% 1 2 4 8 12 16 20 30 40 50 60

TABLE 2-55 [Fe(NO3)3]*

d4

1.0072 1.0157 1.0327 1.0670 1.1028 1.1409 1.1811 1.3073 1.4487 1.6127 1.7983

Ferric Nitrate 18

d4

%

d4

%

d4

1 2 4 8 12 16 20 25

1.0065 1.0144 1.0304 1.0636 1.0989 1.1359 1.1748 1.2281

5 10 20 30 40 50 60 70 80 90 95 100

1.020 1.040 1.080 1.119 1.159 1.198 1.235 1.258 1.259 1.178 1.089 1.0005

1.017 1.035 1.070 1.101 1.130 1.155

1 2 4 6 8 10 12 14 16 18 20 22 24

1.0022 1.0058 1.0131 1.0204 1.0277 1.0351 1.0425 1.0499 1.0574 1.0649 1.0725 1.0802 1.0880

26 28 30 35 40 45 50 55 60 70 80 90 100

1.0959 1.1040 1.1122 1.1327 1.1536 1.1749 1.1966 1.2188 1.2416 1.2897 1.3406 1.3931 1.4465

TABLE 2-57 Hydrogen Bromide (HBr)*

TABLE 2-62

18 °C

20 °C

%

d4

d4

d4

%

1.0090 1.0380 1.0790 1.1235 1.1690 1.2150

1.00068 1.00275 1.00645 1.0085 1.0375 1.0785 1.1220 1.1675 1.2135

1.0002 1.0022 1.0062 1.0082

1.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 40.0 50.0 60.0 65.0

1.0073 1.0146 1.0295 1.0448 1.0604 1.0764 1.0928 1.1097 1.1272 1.1453 1.1640 1.1832 1.2030 1.2235 1.2446 1.2663 1.3877 1.5305 1.6950 1.7854

1.0068 1.0139 1.0285 1.0435 1.0589 1.0747 1.0910 1.1078 1.1251 1.1430 1.1615 1.1806 1.2003 1.2206 1.2415 1.2630 1.3838 1.5257 1.6892 1.7792

1.0041 1.0111 1.0255 1.0402 1.0552 1.0707 1.0867 1.1032 1.1202 1.1377 1.1557 1.1743 1.1935 1.2134 1.2340 1.2552 1.3736 1.5127 1.6731 1.7613

1 2 4 8 12

TABLE 2-58 Hydrogen Cyanide (HCN)* 15

d4

1 2 4 8 12 16 82 90 100

0.998 0.996 0.993 0.984 0.971 0.956 0.752 0.724 0.691

4

10

25

*International Critical Tables, vol. 3, p. 55.

Hydrogen Chloride (HCl)

%

−5 °C

0 °C

10 °C

20 °C

40 °C

60 °C

80 °C

100 °C

1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

1.0048 1.0104 1.0213 1.0321 1.0428 1.0536 1.0645 1.0754 1.0864 1.0975 1.1087 1.1200 1.1314 1.1426 1.1537 1.1648

1.0052 1.0106 1.0213 1.0319 1.0423 1.0528 1.0634 1.0741 1.0849 1.0958 1.1067 1.1177 1.1287 1.1396 1.1505 1.1613

1.0048 1.0100 1.0202 1.0303 1.0403 1.0504 1.0607 1.0711 1.0815 1.0920 1.1025 1.1131 1.1238 1.1344 1.1449 1.1553

1.0032 1.0082 1.0181 1.0279 1.0376 1.0474 1.0574 1.0675 1.0776 1.0878 1.0980 1.1083 1.1187 1.1290 1.1392 1.1493 1.1593 1.1691 1.1789 1.1885 1.1980

0.9970 1.0019 1.0116 1.0211 1.0305 1.0400 1.0497 1.0594 1.0692 1.0790 1.0888 1.0986 1.1085 1.1183 1.1280 1.1376

0.9881 0.9930 1.0026 1.0121 1.0215 1.0310 1.0406 1.0502 1.0598 1.0694 1.0790 1.0886 1.0982 1.1076 1.1169 1.1260

0.9768 0.9819 0.9919 1.0016 1.0111 1.0206 1.0302 1.0398 1.0494 1.0590 1.0685 1.0780 1.0874 1.0967 1.1058 1.1149

0.9636 0.9688 0.9791 0.9892 0.9992 1.0090 1.0188 1.0286 1.0383 1.0479 1.0574 1.0668 1.0761 1.0853 1.0942 1.1030

*International Critical Tables, vol. 3, p. 54.

*International Critical Tables, vol. 3, p. 54.

Hydrofluosilic Acid (H2SiF6)* 17.5

17.5

%

d4

1.0080 1.0161 1.0324 1.0661 1.1011

d4

16 20 25 30 34

1.1373 1.1748 1.2235 1.2742 1.3162

*O. Söhnel and P. Novotny, Densities of Aqueous Solutions of Inorganic Substances, Elsevier, 1985.

TABLE 2-63

Magnesium Chloride (MgCl2)*

%

0 °C

20 °C

40 °C

60 °C

80 °C

100°C

2 4 8 12 16 20 25 30

1.0168 1.0338 1.0683 1.1035 1.1395 1.1764 1.2246 1.2754

1.0146 1.0311 1.0646 1.0989 1.1342 1.1706 1.2184 1.2688

1.0084 1.0248 1.0580 1.0921 1.1272 1.1635 1.2111 1.2614

0.9995 1.0159 1.0493 1.0836 1.1188 1.1552 1.2031 1.2535

0.9883 1.0050 1.0388 1.0735 1.1092 1.1460 1.1942 1.2451

0.9753 0.9923 1.0269 1.0622 1.0984 1.1359 1.1847 1.2360

*International Critical Tables, vol. 3, p. 71.

TABLE 2-64

*International Critical Tables, vol. 3, p. 61. TABLE 2-59

18

d4

*International Critical Tables, vol. 3, p. 54.

*International Critical Tables, vol. 3, p. 68.

18

%

15 °C

%

20

d4

*International Critical Tables, vol. 3, p. 68.

TABLE 2-56 Ferrous Sulfate (FeSO4)* 0.2 0.4 0.8 1.0 4.0 8.0 12.0 16.0 20.0

0

TABLE 2-61 Hydrogen Peroxide (H2O2)*

%

*International Critical Tables, vol. 3, p. 68.

%

TABLE 2-60 Hydrogen Fluoride (HF)*

Magnesium Sulfate (MgSO4)*

%

0 °C

20 °C

30 °C

40 °C

50 °C

60 °C

80 °C

2 4 8 12 16 20 26

1.0210 1.0423 1.0858 1.1309 1.1777 1.2264 1.3032

1.0186 1.0392 1.0816 1.1256 1.1717 1.2198 1.2961

1.0158 1.0362 1.0782 1.1220 1.1679 1.2159 1.2922

1.0123 1.0326 1.0743 1.1179 1.1637 1.2117 1.2879

1.0081 1.0283 1.0700 1.1135 1.1592 1.2072 1.2836

1.0032 1.0234 1.0650 1.1083

0.9916 1.0118 1.0534 1.0968

*International Critical Tables, vol. 3, p. 72. TABLE 2-65 Nickel Chloride (NiCl2)* 18

TABLE 2-66 Nickel Nitrate [Ni(NO3)2]* 20

TABLE 2-67 Nickel Sulfate (NiSO4 )* 18

%

d4

%

d4

%

d4

1 2 4 8 12 16 20 30

1.0082 1.0179 1.0375 1.0785 1.1217 1.1674 1.2163 1.353

1 2 4 8 12 16 20 30 35

1.0065 1.0150 1.0325 1.0688 1.1070 1.1480 1.191 1.311 1.377

1 2 4 8 12 16 18

1.0091 1.0198 1.0415 1.0852 1.1325 1.1825 1.2090

*International Critical Tables, vol. 3, p. 69.

*International Critical Tables, vol. 3, p. 69.

*International Critical Tables, vol. 3, p. 69.

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM TABLE 2-68

2-107

Nitric Acid (HNO3)*

0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

30 °C

40 °C

50 °C

60 °C

80 °C

100°C

1 2 3 4

1.0058 1.0117 1.0176 1.0236

1.00572 1.01149 1.01730 1.02315

1.00534 1.01099 1.01668 1.02240

1.00464 1.01018 1.01576 1.02137

1.00364 1.00909 1.01457 1.02008

1.00241 1.00778 1.01318 1.01861

1.0009 1.0061 1.0114 1.0168

0.9973 1.0025 1.0077 1.0129

0.9931 0.9982 1.0033 1.0084

0.9882 0.9932 0.9982 1.0033

0.9767 0.9816 0.9865 0.9915

0.9632 0.9681 0.9730 0.9779

5 6 7 8 9

1.0296 1.0357 1.0418 1.0480 1.0543

1.02904 1.03497 1.0410 1.0471 1.0532

1.02816 1.03397 1.0399 1.0458 1.0518

1.02702 1.03272 1.0385 1.0443 1.0502

1.02563 1.03122 1.0369 1.0427 1.0485

1.02408 1.02958 1.0352 1.0409 1.0466

1.0222 1.0277 1.0333 1.0389 1.0446

1.0182 1.0235 1.0289 1.0344 1.0399

1.0136 1.0188 1.0241 1.0295 1.0349

1.0084 1.0136 1.0188 1.0241 1.0294

0.9965 1.0015 1.0066 1.0117 1.0169

0.9829 0.9879 0.9929 0.9980 1.0032

10 11 12 13 14

1.0606 1.0669 1.0733 1.0797 1.0862

1.0594 1.0656 1.0718 1.0781 1.0845

1.0578 1.0639 1.0700 1.0762 1.0824

1.0561 1.0621 1.0681 1.0742 1.0803

1.0543 1.0602 1.0661 1.0721 1.0781

1.0523 1.0581 1.0640 1.0699 1.0758

1.0503 1.0560 1.0618 1.0676 1.0735

1.0455 1.0511 1.0567 1.0624 1.0681

1.0403 1.0458 1.0513 1.0568 1.0624

1.0347 1.0401 1.0455 1.0509 1.0564

1.0221 1.0273 1.0326 1.0379 1.0432

1.0083 1.0134 1.0186 1.0238 1.0289

15 16 17 18 19

1.0927 1.0992 1.1057 1.1123 1.1189

1.0909 1.0973 1.1038 1.1103 1.1168

1.0887 1.0950 1.1014 1.1078 1.1142

1.0865 1.0927 1.0989 1.1052 1.1115

1.0842 1.0903 1.0964 1.1026 1.1088

1.0818 1.0879 1.0940 1.1001 1.1062

1.0794 1.0854 1.0914 1.0974 1.1034

1.0739 1.0797 1.0855 1.0913 1.0972

1.0680 1.0737 1.0794 1.0851 1.0908

1.0619 1.0675 1.0731 1.0787 1.0843

1.0485 1.0538 1.0592 1.0646 1.0700

1.0341 1.0393 1.0444 1.0496 1.0547

20 21 22 23 24

1.1255 1.1322 1.1389 1.1457 1.1525

1.1234 1.1300 1.1366 1.1433 1.1501

1.1206 1.1271 1.1336 1.1402 1.1469

1.1178 1.1242 1.1306 1.1371 1.1437

1.1150 1.1213 1.1276 1.1340 1.1404

1.1123 1.1185 1.1247 1.1310 1.1374

1.1094 1.1155 1.1217 1.1280 1.1343

1.1031 1.1090 1.1150 1.1210 1.1271

1.0966 1.1024 1.1083 1.1142 1.1201

1.0899 1.0956 1.1013 1.1070 1.1127

1.0754 1.0808 1.0862 1.0917 1.0972

1.0598 1.0650 1.0701 1.0753 1.0805

25 26 27 28 29

1.1594 1.1663 1.1733 1.1803 1.1874

1.1569 1.1638 1.1707 1.1777 1.1847

1.1536 1.1603 1.1670 1.1738 1.1807

1.1503 1.1569 1.1635 1.1702 1.1770

1.1469 1.1534 1.1600 1.1666 1.1733

1.1438 1.1502 1.1566 1.1631 1.1697

1.1406 1.1469 1.1533 1.1597 1.1662

1.1332 1.1394 1.1456 1.1519 1.1582

1.1260 1.1320 1.1381 1.1442 1.1503

1.1185 1.1244 1.1303 1.1362 1.1422

1.1027 1.1083 1.1139 1.1195 1.1251

1.0857 1.0910 1.0963 1.1016 1.1069

30 31 32 33 34

1.1945 1.2016 1.2088 1.2160 1.2233

1.1917 1.1988 1.2059 1.2131 1.2203

1.1876 1.1945 1.2014 1.2084 1.2155

1.1838 1.1906 1.1974 1.2043 1.2113

1.1800 1.1867 1.1934 1.2002 1.2071

1.1763 1.1829 1.1896 1.1963 1.2030

1.1727 1.1792 1.1857 1.1922 1.1988

1.1645 1.1708 1.1772 1.1836 1.1901

1.1564 1.1625 1.1687 1.1749 1.1812

1.1482 1.1542 1.1602 1.1662 1.1723

1.1307 1.1363 1.1419 1.1476 1.1533

1.1122 1.1175 1.1228 1.1281 1.1335

35 36 37 38 39

1.2306 1.2375 1.2444 1.2513 1.2581

1.2275 1.2344 1.2412 1.2479 1.2546

1.2227 1.2294 1.2361 1.2428 1.2494

1.2183 1.2249 1.2315 1.2381 1.2446

1.2140 1.2205 1.2270 1.2335 1.2399

1.2098 1.2163 1.2227 1.2291 1.2354

1.2055 1.2119 1.2182 1.2245 1.2308

1.1966 1.2028 1.2089 1.2150 1.2210

1.1876 1.1936 1.1995 1.2054 1.2112

1.1784 1.1842 1.1899 1.1956 1.2013

1.1591 1.1645 1.1699 1.1752 1.1805

1.1390 1.1440 1.1490 1.1540 1.1589

40 41 42 43 44

1.2649 1.2717 1.2786 1.2854 1.2922

1.2613 1.2680 1.2747 1.2814 1.2880

1.2560 1.2626 1.2692 1.2758 1.2824

1.2511 1.2576 1.2641 1.2706 1.2771

1.2463 1.2527 1.2591 1.2655 1.2719

1.2417 1.2480 1.2543 1.2606 1.2669

1.2370 1.2432 1.2494 1.2556 1.2618

1.2270 1.2330 1.2390 1.2450 1.2510

1.2170 1.2229 1.2287 1.2345 1.2403

1.2069 1.2126 1.2182 1.2238 1.2294

1.1858 1.1911 1.1963 1.2015 1.2067

1.1638 1.1687 1.1735 1.1783 1.1831

45 46 47 48 49

1.2990 1.3058 1.3126 1.3194 1.3263

1.2947 1.3014 1.3080 1.3147 1.3214

1.2890 1.2955 1.3021 1.3087 1.3153

1.2836 1.2901 1.2966 1.3031 1.3096

1.2783 1.2847 1.2911 1.2975 1.3040

1.2732 1.2795 1.2858 1.2921 1.2984

1.2680 1.2742 1.2804 1.2867 1.2929

1.2570 1.2630 1.2690 1.2750 1.2811

1.2461 1.2519 1.2577 1.2635 1.2693

1.2350 1.2406 1.2462 1.2518 1.2575

1.2119 1.2171 1.2223 1.2275 1.2328

1.1879 1.1927 1.1976 1.2024 1.2073

50 51 52 53 54

1.3327 1.3391 1.3454 1.3517 1.3579

1.3277 1.3339 1.3401 1.3462 1.3523

1.3215 1.3277 1.3338 1.3399 1.3459

1.3157 1.3218 1.3278 1.3338 1.3397

1.3100 1.3160 1.3219 1.3278 1.3336

1.3043 1.3102 1.3160 1.3218 1.3275

1.2987 1.3045 1.3102 1.3159 1.3215

1.2867 1.2923 1.2978 1.3033 1.3087

1.2748 1.2802 1.2856 1.2909 1.2961

1.2628 1.2680 1.2731 1.2782 1.2833

1.2377 1.2425 1.2473 1.2521 1.2568

1.2118 1.2163 1.2208 1.2252 1.2296

55 56 57 58 59

1.3640 1.3700 1.3759 1.3818 1.3875

1.3583 1.3642 1.3700 1.3757 1.3813

1.3518 1.3576 1.3634 1.3691 1.3747

1.3455 1.3512 1.3569 1.3625 1.3680

1.3393 1.3449 1.3505 1.3560 1.3614

1.3331 1.3386 1.3441 1.3495 1.3548

1.3270 1.3324 1.3377 1.3430 1.3482

1.3141 1.3194 1.3246 1.3298 1.3348

1.3013 1.3064 1.3114 1.3164 1.3213

1.2883 1.2932 1.2981 1.3029 1.3077

1.2615 1.2661 1.2706 1.2751 1.2795

1.2339 1.2382 1.2424 1.2466 1.2507

60 61 62 63 64

1.3931 1.3986 1.4039 1.4091

1.3868 1.3922 1.3975 1.4027 1.4078

1.3801 1.3855 1.3907 1.3958 1.4007

1.3734 1.3787 1.3838 1.3888 1.3936

1.3667 1.3719 1.3769 1.3818 1.3866

1.3600 1.3651 1.3700 1.3748 1.3795

1.3533 1.3583 1.3632 1.3679 1.3725

1.3398 1.3447 1.3494 1.3540

1.3261 1.3308 1.3354 1.3398

1.3124 1.3169 1.3213 1.3255

1.2839 1.2881 1.2922 1.2962

1.2547 1.2587 1.2625 1.2661

%

2-108

PHYSICAL AND CHEMICAL DATA Nitric Acid (HNO3) (Concluded)

TABLE 2-68 0 °C

5 °C

10 °C

15 °C

20 °C

25 °C

30 °C

65 66 67 68 69

1.4128 1.4177 1.4224 1.4271 1.4317

1.4055 1.4103 1.4150 1.4196 1.4241

1.3984 1.4031 1.4077 1.4122 1.4166

1.3913 1.3959 1.4004 1.4048 1.4091

1.3841 1.3887 1.3932 1.3976 1.4019

1.3770 1.3814 1.3857 1.3900 1.3942

70 71 72 73 74

1.4362 1.4406 1.4449 1.4491 1.4532

1.4285 1.4328 1.4371 1.4413 1.4454

1.4210 1.4252 1.4294 1.4335 1.4376

1.4134 1.4176 1.4218 1.4258 1.4298

1.4061 1.4102 1.4142 1.4182 1.4221

1.3983 1.4023 1.4063 1.4103 1.4142

75 76 77 78 79

1.4573 1.4613 1.4652 1.4690 1.4727

1.4494 1.4533 1.4572 1.4610 1.4647

1.4415 1.4454 1.4492 1.4529 1.4565

1.4337 1.4375 1.4413 1.4450 1.4486

1.4259 1.4296 1.4333 1.4369 1.4404

1.4180 1.4217 1.4253 1.4288 1.4323

80 81 82 83 84

1.4764 1.4800 1.4835 1.4869 1.4903

1.4683 1.4718 1.4753 1.4787 1.4820

1.4601 1.4636 1.4670 1.4704 1.4737

1.4521 1.4555 1.4589 1.4622 1.4655

1.4439 1.4473 1.4507 1.4540 1.4572

1.4357 1.4391 1.4424 1.4456 1.4487

85 86 87 88 89

1.4936 1.4968 1.4999 1.5029 1.5058

1.4852 1.4883 1.4913 1.4942 1.4970

1.4769 1.4799 1.4829 1.4858 1.4885

1.4686 1.4716 1.4745 1.4773 1.4800

1.4603 1.4633 1.4662 1.4690 1.4716

1.4518 1.4548 1.4577 1.4605 1.4631

90 91 92 93 94

1.5085 1.5111 1.5136 1.5156 1.5177

1.4997 1.5023 1.5048 1.5068 1.5088

1.4911 1.4936 1.4960 1.4979 1.4999

1.4826 1.4850 1.4873 1.4892 1.4912

1.4741 1.4766 1.4789 1.4807 1.4826

1.4656 1.4681 1.4704 1.4722 1.4741

95 96 97 98 99 100

1.5198 1.5220 1.5244 1.5278 1.5327 1.5402

1.5109 1.5130 1.5152 1.5187 1.5235 1.5310

1.5019 1.5040 1.5062 1.5096 1.5144 1.5217

1.4932 1.4952 1.4974 1.5008 1.5056 1.5129

1.4846 1.4867 1.4889 1.4922 1.4969 1.5040

1.4761 1.4781 1.4802 1.4835 1.4881 1.4952

%

40 °C

50 °C

60 °C

80 °C

100°C

6%

8%

10%

1.0396

1.0534

1.0674

*International Critical Tables, vol. 3, pp. 58–59.

TABLE 2-69 15

%

d4

1 2 4 6 8 10 12 14 16 18 20 22 24 26

1.0050 1.0109 1.0228 1.0348 1.0471 1.0597 1.0726 1.0589 1.0995 1.1135 1.1279 1.1428 1.1581 1.1738

Perchloric Acid (HClO4)* 20

d4

25

50

TABLE 2-71 15

20

50

Potassium Bicarbonate (KHCO3)*

d4

d4

%

d4

d4

d4

°C

1%

2%

4%

1.0020 1.0070 1.0169 1.0270 1.0372 1.0475

0.9933 0.9986 0.9906 1.0205 1.0320 1.0440 1.0560 1.0680 1.0810 1.0940 1.1070 1.1205 1.1345 1.1490

28 30 32 34 36 38 40 45 50 55 60 65 70

1.1900 1.2067 1.2239 1.2418 1.2603 1.2794 1.2991 1.3521 1.4103 1.4733 1.5389 1.6059 1.6736

1.1851 1.2013 1.2183 1.2359 1.2542 1.2732 1.2927 1.3450 1.4018 1.4636 1.5298 1.5986 1.6680

1.1645 1.1800 1.1960 1.2130 1.2310 1.2490 1.2680 1.3180 1.3730 1.4320 1.4950 1.5620 1.6290

0 10 15 20 30 40 50 60 80 100

1.0066 1.0064 1.0058 1.0049 1.0024 0.9990 0.9949 0.9901 0.9786 0.9653

1.0134 1.0132 1.0125 1.0117 1.0092 1.0058 1.0017 0.9969 0.9855 0.9722

1.0270 1.0268 1.0260 1.0252 1.0228 1.0195 1.0154 1.0106 0.9993 0.9860

1.1697

TABLE 2-72 Potassium Bromide (KBr)*

*International Critical Tables, vol. 3, p. 54.

TABLE 2-70

Phosphoric Acid (H3PO4)*

°C

2%

6%

14%

0 10 20 30 40

1.0113 1.0109 1.0092 1.0065 1.0029

1.0339 1.0330 1.0309 1.0279 1.0241

1.0811 1.0792 1.0764 1.0728 1.0685

20%

26%

35%

50%

75%

1.1192 1.1167 1.1567 1.221 1.341 1.1134 1.1529 1.216 1.335 1.579 1.1094 1.1484 1.211 1.329 1.572 1.1048

*International Critical Tables, vol. 3, p. 61.

*International Critical Tables, vol. 3, p. 90.

100%

1.870 1.862

%

d 20 4

1 2 6 12 20 30 40

1.0054 1.0127 1.0426 1.0903 1.1601 1.2593 1.3746

*International Critical Tables, vol. 3, p. 87.

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM TABLE 2-73

Potassium Carbonate (K2CO3)*

TABLE 2-79

2-109

Potassium Nitrate (KNO3)*

%

0 °C

10 °C

20 °C

40 °C

60 °C

80 °C

100°C

%

0 °C

10 °C

20 °C

40 °C

60 °C

80 °C

100 °C

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.0094 1.0189 1.0381 1.0768 1.1160 1.1562 1.1977 1.2405 1.2846 1.3071 1.3646 1.4244 1.4867 1.5517

1.0089 1.0182 1.0369 1.0746 1.1131 1.1530 1.1941 1.2366 1.2804 1.3028 1.3600 1.4195 1.4815 1.5462

1.0072 1.0163 1.0345 1.0715 1.1096 1.1490 1.1898 1.2320 1.2756 1.2979 1.3548 1.4141 1.4759 1.5404

1.0010 1.0098 1.0276 1.0640 1.1013 1.1399 1.1801 1.2219 1.2652 1.2873 1.3440 1.4029 1.4644 1.5285

0.9919 1.0005 1.0180 1.0538 1.0906 1.1290 1.1690 1.2106 1.2538 1.2759 1.3324 1.3913 1.4528 1.5169

0.9803 0.9889 1.0063 1.0418 1.0786 1.1170 1.1570 1.1986 1.2418 1.2640 1.3206 1.3795 1.4408 1.5048

0.9670 0.9756 0.9951 1.0291 1.0663 1.1049 1.1451 1.1869 1.2301 1.2522 1.3089 1.3678 1.4290 1.4928

1 2 4 8 12 16 20 24

1.00654 1.01326 1.02677 1.05419 1.08221

1.00615 1.01262 1.02566 1.05226 1.07963

1.00447 1.01075 1.02344 1.04940 1.07620 1.10392 1.13261 1.16233

0.99825 1.00430 1.01652 1.04152 1.06740 1.09432 1.12240 1.15175

0.9890 0.9949 1.0068 1.0313 1.0567 1.0831 1.1106 1.1391

0.9776 0.9834 0.9951 1.0192 1.0442 1.0703 1.0974 1.1256

0.9641 0.9699 0.9816 1.0056 1.0304 1.0562 1.0831 1.1110

*International Critical Tables, vol. 3, p. 90. TABLE 2-74 Potassium Chromate (K2CrO4)* 15

TABLE 2-75 (KClO3)*

18

Potassium Chlorate

%

d4

d4

°C

1%

2%

3%

4%

1 2 4 8 12 16 20 24 28 30

1.0073 1.0155 1.0321 1.0659 1.1009

1.0066 1.0147 1.0311 1.0647 1.0999 1.1366 1.1748 1.2147 1.2566 1.2784

0 10 20 30 40 60 80 100

1.0061 1.0059 1.0045 1.0020 0.9986 0.9895 0.9781 0.9646

1.0124 1.0122 1.0109 1.0085 1.0051 0.9959 0.9845 0.9709

1.0189 1.0187 1.0174 1.0151 1.0116 1.0024 0.9910 0.9774

1.0256 1.0254 1.0241 1.0218 1.0183 1.0091 0.9977 0.9840

*International Critical Tables, vol. 3, p. 86.

*International Critical Tables, vol. 3, p. 92. TABLE 2-76

Potassium Chloride (KCl)*

%

0 °C

20 °C

25 °C

40 °C

60 °C

80 °C

100°C

1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0

1.00661 1.01335 1.02690 1.05431 1.08222 1.11068 1.13973

1.00462 1.01103 1.02391 1.05003 1.07679 1.10434 1.13280 1.16226

1.00342 1.00977 1.02255 1.04847 1.07506 1.10245 1.13072 1.15995

0.99847 1.00471 1.01727 1.04278 1.06897 1.09600 1.12399 1.15299 1.18304

0.9894 0.9956 1.0080 1.0333 1.0592 1.0861 1.1138 1.1425 1.1723

0.9780 0.9842 0.9966 1.0219 1.0478 1.0746 1.1024 1.1311 1.1609

0.9646 0.9708 0.9634 1.0888 1.0350 1.0619 1.0897 1.1185 1.1483

%

110 °C

120 °C

130 °C

140°C

3.79 7.45 13.62

0.9733 0.9978 1.0388

0.9663 0.9899 1.0313

0.9583 0.9827 1.0238

0.9502 0.9745 1.0159

*International Critical Tables, vol. 3, p. 87. TABLE 2-77 Potassium Chrome Alum [K2Cr2(SO4)4]*

TABLE 2-80 Potassium Dichromate (K2Cr2O7)* 20

d 15 4

%

d 15 4

1 2 6 10 14 20 30 40 50

1.007 1.016 1.052 1.089 1.129 1.193 1.315 1.456 1.615

1.0 2.0 4.0 6.0 8.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 51.7

1.0083 1.0175 1.0359 1.0544 1.0730 1.0918 1.1396 1.1884 1.2387 1.2905 1.3440 1.3991 1.4558 1.5143 1.5355 (sat’d. soln.)

*International Critical Tables, vol. 3, p. 86.

TABLE 2-81 (K2SO4)*

Potassium Sulfate 20

%

d4

%

d4

1 2 4 6 8 10

1.0052 1.0122 1.0264 1.0408 1.0554 1.0703

1 2 4 6 8 10

1.0063 1.0145 1.0310 1.0477 1.0646 1.0817

*International Critical Tables, vol. 3, p. 92. TABLE 2-82 Potassium Sulfite (K2SO3)* 15

*International Critical Tables, vol. 3, p. 88. TABLE 2-83 Sodium Acetate (NaC2H3O2)* 20

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0073 1.0155 1.0322 1.0667 1.1026 1.1402 1.1793 1.2197 1.2404

1 2 4 8 12 18 20 26 28

1.0033 1.0084 1.0186 1.0392 1.0598 1.0807 1.1021 1.1351 1.1462

*International Critical Tables, *International Critical Tables, vol. 3, vol. 3, p. 87. p. 83. TABLE 2-84 Sodium Arsenate (Na3AsO4)* 17

TABLE 2-85 Sodium Bichromate (Na2Cr2O7)* 15

%

d4

%

d4

1 2 4 8 10 12

1.0097 1.0207 1.0431 1.0892 1.1130 1.1373

1 2 4 8 12 16 20 24 28 30 35 40 45 50

1.006 1.013 1.027 1.056 1.084 1.112 1.140 1.166 1.193 1.207 1.244 1.279 1.312 1.342

*International Critical Tables, vol. 3, p. 82.

TABLE 2-78 Potassium Hydroxide (KOH)*

%

*International Critical Tables, vol. 3, p. 92.

*International Critical Tables, vol. 3, p. 89.

*International Critical Tables, vol. 3, p. 86. TABLE 2-86 Sodium Bromide (NaBr)* 20

TABLE 2-87 Sodium Formate (HCOONa)* 25

%

d4

%

d4

1 2 4 8 10 12 20 30 40

1.0060 1.0139 1.0298 1.0631 1.0803 1.0981 1.1745 1.2841 1.4138

1 2 4 8 12 16 20 24 28 30 35 40

1.003 1.009 1.022 1.048 1.074 1.100 1.127 1.155 1.184 1.199 1.236 1.274

*International Critical Tables, vol. 3, p. 80.

*International Critical Tables, vol. 3, p. 83.

2-110

PHYSICAL AND CHEMICAL DATA

TABLE 2-88

Sodium Carbonate (Na2CO3)*

TABLE 2-92

Sodium Hydroxide (NaOH)*

%

0 °C

10 °C

20 °C

30 °C

40 °C

60 °C

80 °C

100 °C

%

0 °C

15 °C

20 °C

40 °C

60 °C

80 °C

100 °C

1 2 4 8 12 14 16 18 20 24 28 30

1.0109 1.0219 1.0439 1.0878 1.1319 1.1543

1.0103 1.0210 1.0423 1.0850 1.1284 1.1506

1.0086 1.0190 1.0398 1.0816 1.1244 1.1463

1.0058 1.0159 1.0363 1.0775 1.1200 1.1417 1.1636 1.1859 1.2086 1.2552 1.3031 1.3274

1.0022 1.0122 1.0323 1.0732 1.1150 1.1365

0.9929 1.0027 1.0223 1.0625 1.1039 1.1251

0.9814 0.9910 1.0105 1.0503 1.0914 1.1125

0.9683 0.9782 0.9980 1.0380 1.0787 1.0996

1 2 4 8 12 16 20 24 28 32 36 40 44 48 50

1.0124 1.0244 1.0482 1.0943 1.1399 1.1849 1.2296 1.2741 1.3182 1.3614 1.4030 1.4435 1.4825 1.5210 1.5400

1.01065 1.02198 1.04441 1.08887 1.13327 1.17761 1.22183 1.26582 1.3094 1.3520 1.3933 1.4334 1.4720 1.5102 1.5290

1.0095 1.0207 1.0428 1.0869 1.1309 1.1751 1.2191 1.2629 1.3064 1.3490 1.3900 1.4300 1.4685 1.5065 1.5253

1.0033 1.0139 1.0352 1.0780 1.1210 1.1645 1.2079 1.2512 1.2942 1.3362 1.3768 1.4164 1.4545 1.4922 1.5109

0.9941 1.0045 1.0254 1.0676 1.1101 1.1531 1.1960 1.2388 1.2814 1.3232 1.3634 1.4027 1.4405 1.4781 1.4967

0.9824 0.9929 1.0139 1.0560 1.0983 1.1408 1.1833 1.2259 1.2682 1.3097 1.3498 1.3889 1.4266 1.4641 1.4827

0.9693 0.9797 1.0009 1.0432 1.0855 1.1277 1.1700 1.2124 1.2546 1.2960 1.3360 1.3750 1.4127 1.4503 1.4690

*International Critical Tables, vol. 3, pp. 82–83. TABLE 2-89

*International Critical Tables, vol. 3, p. 79.

Sodium Chlorate (NaClO3)* 18

18

%

d4

%

d4

1 2 4 6 8 10 12 14 16

1.0053 1.0121 1.0258 1.0397 1.0538 1.0681 1.0827 1.0977 1.1131

18 20 22 24 26 28 30 32 34

1.1288 1.1449 1.1614 1.1782 1.1953 1.2128 1.2307 1.2491 1.2680

TABLE 2-93

*International Critical Tables, vol. 3, p. 80. TABLE 2-90

Sodium Chloride (NaCl)*

%

0 °C

10 °C

25 °C

40 °C

60 °C

80 °C

100 °C

1 2 4 8 12 16 20 24 26

1.00747 1.01509 1.03038 1.06121 1.09244 1.12419 1.15663 1.18999 1.20709

1.00707 1.01442 1.02920 1.05907 1.08946 1.12056 1.15254 1.18557 1.20254

1.00409 1.01112 1.02530 1.05412 1.08365 1.11401 1.14533 1.17776 1.19443

0.99908 1.00593 1.01977 1.04798 1.07699 1.10688 1.13774 1.16971 1.18614

0.9900 0.9967 1.0103 1.0381 1.0667 1.0962 1.1268 1.1584 1.1747

0.9785 0.9852 0.9988 1.0264 1.0549 1.0842 1.1146 1.1463 1.1626

0.9651 0.9719 0.9855 1.0134 1.0420 1.0713 1.1017 1.1331 1.1492

Sodium Nitrate (NaNO3)*

%

0 °C

20 °C

40 °C

60 °C

80 °C

100°C

1 2 4 8 12 16 20 24 28 30 35 40 45

1.0071 1.0144 1.0290 1.0587 1.0891 1.1203 1.1526 1.1860 1.2204 1.2380 1.2834 1.3316

1.0049 1.0117 1.0254 1.0532 1.0819 1.1118 1.1429 1.1752 1.2085 1.2256 1.2701 1.3175 1.3683

0.9986 1.0050 1.0180 1.0447 1.0724 1.1013 1.1314 1.1629 1.1955 1.2122 1.2560 1.3027 1.3528

0.9894 0.9956 1.0082 1.0340 1.0609 1.0892 1.1187 1.1496 1.1816 1.1980 1.2413 1.2875 1.3371

0.9779 0.9840 0.9964 1.0218 1.0481 1.0757 1.1048 1.1351 1.1667 1.1830 1.2258 1.2715 1.3206

0.9644 0.9704 0.9826 1.0078 1.0340 1.0614 1.0901 1.1200 1.1513 1.1674 1.2100 1.2555 1.3044

*International Critical Tables, vol. 3, p. 82.

*International Critical Tables, vol. 3, p. 79. TABLE 2-91 Sodium Chromate (Na2CrO4)*

TABLE 2-94 Sodium Nitrite (NaNO2)*

18

%

d4

%

d4

1 2 4 8 12 16 20 24 26

1.0074 1.0164 1.0344 1.0718 1.1110 1.1518 1.1942 1.2383 1.2611

1 2 4 8 12 16 20

1.0058 1.0125 1.0260 1.0535 1.0816 1.1103 1.1394

*International Critical Tables, vol. 3, p. 82.

*International Critical Tables, vol. 3, p. 86. TABLE 2-95

15

Sodium Silicates* Concentration, % 1

2

4

8

10

14

Na2O/3.9SiO2 Na2O/3.36SiO2 Na2O/2.40SiO2 Na2O/2.44SiO2 Na2O/2.06SiO2 Na2O/1.69SiO2

20

24

30

36

40

45

50

1.445 1.450

1.520

1.594

20

Formula

d4 1.006 1.006 1.007

1.014 1.014 1.016

1.030 1.030 1.034

1.063 1.065 1.071

1.080 1.083 1.090

1.116 1.120 1.130

1.172 1.179

1.211 1.222

1.275 1.290

1.365

1.007 1.007

1.016 1.017

1.035 1.036

1.073 1.077

1.093 1.098

1.134 1.141

1.200 1.210

1.247 1.259

1.309 1.321 1.337

1.387 1.397 1.424

*International Critical Tables, vol. 3, p. 85.

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM TABLE 2-96

Sodium Sulfate (Na2SO4)*

%

0 °C

20 °C

30 °C

40 °C

60 °C

80 °C

100°C

1 2 4 8 12 16 20 24

1.0094 1.0189 1.0381 1.0773 1.1174 1.1585 1.2008 1.2443

1.0073 1.0164 1.0348 1.0724 1.1109 1.1586 1.1915 1.2336

1.0046 1.0135 1.0315 1.0682 1.1062 1.1456 1.1865 1.2292

1.0010 1.0098 1.0276 1.0639 1.1015 1.1406 1.1813 1.2237

0.9919 1.0007 1.0184 1.0544 1.0915 1.1299 1.1696

0.9805 0.9892 1.0068 1.0426 1.0795 1.1176 1.1569

0.9671 0.9758 0.9934 1.0292 1.0661 1.1042

*International Critical Tables, vol. 3, p. 81.

TABLE 2-97 Sodium Sulfide (Na2S)*

TABLE 2-98 Sodium Sulfite (Na2SO3)*

TABLE 2-99 Sodium Thiosulfate (Na2S2O3)*

%

d 18 4

%

d 19 4

%

d 20 4

1 2 4 8 12 16 18

1.0098 1.0211 1.0440 1.0907 1.1388 1.1885 1.2140

1 2 4 8 12 16 18

1.0078 1.0172 1.0363 1.0751 1.1146 1.1549 1.1755

1 2 4 8 12 16 20 24 28 30 35 40

1.0065 1.0148 1.0315 1.0654 1.1003 1.1365 1.1740 1.2128 1.2532 1.2739 1.3273 1.3827

*International Critical Tables, vol. 3, p. 81.

*International Critical Tables, vol. 3, p. 81.

*International Critical Tables, vol. 3, p. 81. TABLE 2-100 Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O) %

d 19 4

1 2 4 8 12 16 20 24 28 30 40 50

1.0052 1.0105 1.0211 1.0423 1.0639 1.0863 1.1087 1.1322 1.1558 1.1676 1.2297 1.2954

TABLE 2-101 Stannic Chloride (SnCl4)*

TABLE 2-102 Stannous Chloride (SnCl2)*

%

d 15 4

%

d 15 4

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65 70

1.007 1.015 1.031 1.064 1.099 1.135 1.173 1.212 1.255 1.278 1.337 1.403 1.475 1.555 1.644 1.742 1.851 1.971

1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65

1.0068 1.0146 1.0306 1.0638 1.0986 1.1353 1.1743 1.2159 1.2603 1.2837 1.3461 1.4145 1.4897 1.5729 1.6656 1.7695 1.8865

*International Critical Tables, vol. 3, p. 63.

*International Critical Tables, vol. 3, p. 63.

2-111

2-112

PHYSICAL AND CHEMICAL DATA

TABLE 2-103

Sulfuric Acid (H2SO4)*

0 °C

10 °C

15 °C

20 °C

25 °C

30 °C

40 °C

50 °C

60 °C

80 °C

100 °C

1 2 3 4

1.0074 1.0147 1.0219 1.0291

1.0068 1.0138 1.0206 1.0275

1.0060 1.0129 1.0197 1.0264

1.0051 1.0118 1.0184 1.0250

1.0038 1.0104 1.0169 1.0234

1.0022 1.0087 1.0152 1.0216

0.9986 1.0050 1.0113 1.0176

0.9944 1.0006 1.0067 1.0129

0.9895 0.9956 1.0017 1.0078

0.9779 0.9839 0.9900 0.9961

0.9645 0.9705 0.9766 0.9827

5 6 7 8 9

1.0364 1.0437 1.0511 1.0585 1.0660

1.0344 1.0414 1.0485 1.0556 1.0628

1.0332 1.0400 1.0469 1.0539 1.0610

1.0317 1.0385 1.0453 1.0522 1.0591

1.0300 1.0367 1.0434 1.0502 1.0571

1.0281 1.0347 1.0414 1.0481 1.0549

1.0240 1.0305 1.0371 1.0437 1.0503

1.0192 1.0256 1.0321 1.0386 1.0451

1.0140 1.0203 1.0266 1.0330 1.0395

1.0022 1.0084 1.0146 1.0209 1.0273

0.9888 0.9950 1.0013 1.0076 1.0140

10 11 12 13 14

1.0735 1.0810 1.0886 1.0962 1.1039

1.0700 1.0773 1.0846 1.0920 1.0994

1.0681 1.0753 1.0825 1.0898 1.0971

1.0661 1.0731 1.0802 1.0874 1.0947

1.0640 1.0710 1.0780 1.0851 1.0922

1.0617 1.0686 1.0756 1.0826 1.0897

1.0570 1.0637 1.0705 1.0774 1.0844

1.0517 1.0584 1.0651 1.0719 1.0788

1.0460 1.0526 1.0593 1.0661 1.0729

1.0338 1.0403 1.0469 1.0536 1.0603

1.0204 1.0269 1.0335 1.0402 1.0469

15 16 17 18 19

1.1116 1.1194 1.1272 1.1351 1.1430

1.1069 1.1145 1.1221 1.1298 1.1375

1.1045 1.1120 1.1195 1.1271 1.1347

1.1020 1.1094 1.1168 1.1243 1.1318

1.0994 1.1067 1.1141 1.1215 1.1290

1.0968 1.1040 1.1113 1.1187 1.1261

1.0914 1.0985 1.1057 1.1129 1.1202

1.0857 1.0927 1.0998 1.1070 1.1142

1.0798 1.0868 1.0938 1.1009 1.1081

1.0671 1.0740 1.0809 1.0879 1.0950

1.0537 1.0605 1.0674 1.0744 1.0814

20 21 22 23 24

1.1510 1.1590 1.1670 1.1751 1.1832

1.1453 1.1531 1.1609 1.1688 1.1768

1.1424 1.1501 1.1579 1.1657 1.1736

1.1394 1.1471 1.1548 1.1626 1.1704

1.1365 1.1441 1.1517 1.1594 1.1672

1.1335 1.1410 1.1486 1.1563 1.1640

1.1275 1.1349 1.1424 1.1500 1.1576

1.1215 1.1288 1.1362 1.1437 1.1512

1.1153 1.1226 1.1299 1.1373 1.1448

1.1021 1.1093 1.1166 1.1239 1.1313

1.0885 1.0957 1.1029 1.1102 1.1176

25 26 27 28 29

1.1914 1.1996 1.2078 1.2160 1.2243

1.1848 1.1929 1.2010 1.2091 1.2173

1.1816 1.1896 1.1976 1.2057 1.2138

1.1783 1.1862 1.1942 1.2023 1.2104

1.1750 1.1829 1.1909 1.1989 1.2069

1.1718 1.1796 1.1875 1.1955 1.2035

1.1653 1.1730 1.1808 1.1887 1.1966

1.1588 1.1665 1.1742 1.1820 1.1898

1.1523 1.1599 1.1676 1.1753 1.1831

1.1388 1.1463 1.1539 1.1616 1.1693

1.1250 1.1325 1.1400 1.1476 1.1553

30 31 32 33 34

1.2326 1.2409 1.2493 1.2577 1.2661

1.2255 1.2338 1.2421 1.2504 1.2588

1.2220 1.2302 1.2385 1.2468 1.2552

1.2185 1.2267 1.2349 1.2432 1.2515

1.2150 1.2232 1.2314 1.2396 1.2479

1.2115 1.2196 1.2278 1.2360 1.2443

1.2046 1.2126 1.2207 1.2289 1.2371

1.1977 1.2057 1.2137 1.2218 1.2300

1.1909 1.1988 1.2068 1.2148 1.2229

1.1771 1.1849 1.1928 1.2008 1.2088

1.1630 1.1708 1.1787 1.1866 1.1946

35 36 37 38 39

1.2746 1.2831 1.2917 1.3004 1.3091

1.2672 1.2757 1.2843 1.2929 1.3016

1.2636 1.2720 1.2805 1.2891 1.2978

1.2599 1.2684 1.2769 1.2855 1.2941

1.2563 1.2647 1.2732 1.2818 1.2904

1.2526 1.2610 1.2695 1.2780 1.2866

1.2454 1.2538 1.2622 1.2707 1.2793

1.2383 1.2466 1.2550 1.2635 1.2720

1.2311 1.2394 1.2477 1.2561 1.2646

1.2169 1.2251 1.2334 1.2418 1.2503

1.2027 1.2109 1.2192 1.2276 1.2361

40 41 42 43 44

1.3179 1.3268 1.3357 1.3447 1.3538

1.3103 1.3191 1.3280 1.3370 1.3461

1.3065 1.3153 1.3242 1.3332 1.3423

1.3028 1.3116 1.3205 1.3294 1.3384

1.2991 1.3079 1.3167 1.3256 1.3346

1.2953 1.3041 1.3129 1.3218 1.3308

1.2880 1.2967 1.3055 1.3144 1.3234

1.2806 1.2893 1.2981 1.3070 1.3160

1.2732 1.2819 1.2907 1.2996 1.3086

1.2589 1.2675 1.2762 1.2850 1.2939

1.2446 1.2532 1.2619 1.2707 1.2796

45 46 47 48 49

1.3630 1.3724 1.3819 1.3915 1.4012

1.3553 1.3646 1.3740 1.3835 1.3931

1.3515 1.3608 1.3702 1.3797 1.3893

1.3476 1.3569 1.3663 1.3758 1.3854

1.3437 1.3530 1.3624 1.3719 1.3814

1.3399 1.3492 1.3586 1.3680 1.3775

1.3325 1.3417 1.3510 1.3604 1.3699

1.3251 1.3343 1.3435 1.3528 1.3623

1.3177 1.3269 1.3362 1.3455 1.3549

1.3029 1.3120 1.3212 1.3305 1.3399

1.2886 1.2976 1.3067 1.3159 1.3253

50 51 52 53 54

1.4110 1.4209 1.4310 1.4412 1.4515

1.4029 1.4128 1.4228 1.4329 1.4431

1.3990 1.4088 1.4188 1.4289 1.4391

1.3951 1.4049 1.4148 1.4248 1.4350

1.3911 1.4009 1.4109 1.4209 1.4310

1.3872 1.3970 1.4069 1.4169 1.4270

1.3795 1.3893 1.3991 1.4091 1.4191

1.3719 1.3816 1.3914 1.4013 1.4113

1.3644 1.3740 1.3837 1.3936 1.4036

1.3494 1.3590 1.3687 1.3785 1.3884

1.3348 1.3444 1.3540 1.3637 1.3735

55 56 57 58 59

1.4619 1.4724 1.4830 1.4937 1.5045

1.4535 1.4640 1.4746 1.4852 1.4959

1.4494 1.4598 1.4703 1.4809 1.4916

1.4453 1.4557 1.4662 1.4768 1.4875

1.4412 1.4516 1.4621 1.4726 1.4832

1.4372 1.4475 1.4580 1.4685 1.4791

1.4293 1.4396 1.4500 1.4604 1.4709

1.4214 1.4317 1.4420 1.4524 1.4629

1.4137 1.4239 1.4342 1.4446 1.4551

1.3984 1.4085 1.4187 1.4290 1.4393

1.3834 1.3934 1.4035 1.4137 1.4240

60 61 62 63 64

1.5154 1.5264 1.5375 1.5487 1.5600

1.5067 1.5177 1.5287 1.5398 1.5510

1.5024 1.5133 1.5243 1.5354 1.5465

1.4983 1.5091 1.5200 1.5310 1.5421

1.4940 1.5048 1.5157 1.5267 1.5378

1.4898 1.5006 1.5115 1.5225 1.5335

1.4816 1.4923 1.5031 1.5140 1.5250

1.4735 1.4842 1.4950 1.5058 1.5167

1.4656 1.4762 1.4869 1.4977 1.5086

1.4497 1.4602 1.4708 1.4815 1.4923

1.4344 1.4449 1.4554 1.4660 1.4766

%

DENSITIES OF AQUEOUS INORGANIC SOLUTIONS AT 1 ATM TABLE 2-103

2-113

Sulfuric Acid (H2SO4) (Concluded)

%

0 °C

10 °C

15 °C

20 °C

25 °C

30 °C

40 °C

50 °C

60 °C

80 °C

100 °C

65 66 67 68 69

1.5714 1.5828 1.5943 1.6059 1.6176

1.5623 1.5736 1.5850 1.5965 1.6081

1.5578 1.5691 1.5805 1.5920 1.6035

1.5533 1.5646 1.5760 1.5874 1.5989

1.5490 1.5602 1.5715 1.5829 1.5944

1.5446 1.5558 1.5671 1.5785 1.5899

1.5361 1.5472 1.5584 1.5697 1.5811

1.5277 1.5388 1.5499 1.5611 1.5724

1.5195 1.5305 1.5416 1.5528 1.5640

1.5031 1.5140 1.5249 1.5359 1.5470

1.4873 1.4981 1.5089 1.5198 1.5307

70 71 72 73 74

1.6293 1.6411 1.6529 1.6648 1.6768

1.6198 1.6315 1.6433 1.6551 1.6670

1.6151 1.6268 1.6385 1.6503 1.6622

1.6105 1.6221 1.6338 1.6456 1.6574

1.6059 1.6175 1.6292 1.6409 1.6526

1.6014 1.6130 1.6246 1.6363 1.6480

1.5925 1.6040 1.6155 1.6271 1.6387

1.5838 1.5952 1.6067 1.6182 1.6297

1.5753 1.5867 1.5981 1.6095 1.6209

1.5582 1.5694 1.5806 1.5919 1.6031

1.5417 1.5527 1.5637 1.5747 1.5857

75 76 77 78 79

1.6888 1.7008 1.7128 1.7247 1.7365

1.6789 1.6908 1.7026 1.7144 1.7261

1.6740 1.6858 1.6976 1.7093 1.7209

1.6692 1.6810 1.6927 1.7043 1.7158

1.6644 1.6761 1.6878 1.6994 1.7108

1.6597 1.6713 1.6829 1.6944 1.7058

1.6503 1.6619 1.6734 1.6847 1.6959

1.6412 1.6526 1.6640 1.6751 1.6862

1.6322 1.6435 1.6547 1.6657 1.6766

1.6142 1.6252 1.6361 1.6469 1.6575

1.5966 1.6074 1.6181 1.6286 1.6390

80 81 82 83 84

1.7482 1.7597 1.7709 1.7815 1.7916

1.7376 1.7489 1.7599 1.7704 1.7804

1.7323 1.7435 1.7544 1.7649 1.7748

1.7272 1.7383 1.7491 1.7594 1.7693

1.7221 1.7331 1.7437 1.7540 1.7639

1.7170 1.7279 1.7385 1.7487 1.7585

1.7069 1.7177 1.7281 1.7382 1.7479

1.6971 1.7077 1.7180 1.7279 1.7375

1.6873 1.6978 1.7080 1.7179 1.7274

1.6680 1.6782 1.6882 1.6979 1.7072

1.6493 1.6594 1.6692 1.6787 1.6878

85 86 87 88 89

1.8009 1.8095 1.8173 1.8243 1.8306

1.7897 1.7983 1.8061 1.8132 1.8195

1.7841 1.7927 1.8006 1.8077 1.8141

1.7786 1.7872 1.7951 1.8022 1.8087

1.7732 1.7818 1.7897 1.7968 1.8033

1.7678 1.7763 1.7842 1.7914 1.7979

1.7571 1.7657 1.7736 1.7809 1.7874

1.7466 1.7552 1.7632 1.7705 1.7770

1.7364 1.7449 1.7529 1.7602 1.7669

1.7161 1.7245 1.7324 1.7397 1.7464

1.6966 1.7050 1.7129 1.7202 1.7269

90 91 92 93 94

1.8361 1.8410 1.8453 1.8490 1.8520

1.8252 1.8302 1.8346 1.8384 1.8415

1.8198 1.8248 1.8293 1.8331 1.8363

1.8144 1.8195 1.8240 1.8279 1.8312

1.8091 1.8142 1.8188 1.8227 1.8260

1.8038 1.8090 1.8136 1.8176 1.8210

1.7933 1.7986 1.8033 1.8074 1.8109

1.7829 1.7883 1.7932 1.7974 1.8011

1.7729 1.7783 1.7832 1.7876 1.7914

1.7525 1.7581 1.7633 1.7681

1.7331 1.7388 1.7439 1.7485

95 96 97 98 99 100

1.8544 1.8560 1.8569 1.8567 1.8551 1.8517

1.8439 1.8457 1.8466 1.8463 1.8445 1.8409

1.8388 1.8406 1.8414 1.8411 1.8393 1.8357

1.8337 1.8355 1.8364 1.8361 1.8342 1.8305

1.8286 1.8305 1.8314 1.8310 1.8292 1.8255

1.8236 1.8255 1.8264 1.8261 1.8242 1.8205

1.8137 1.8157 1.8166 1.8163 1.8145 1.8107

1.8040 1.8060 1.8071 1.8068 1.8050 1.8013

1.7944 1.7965 1.7977 1.7976 1.7958 1.7922

%

d 5.96 4

%

d 13.00 4

d 18.00 4

0.005 .01 .02 .03 .04

1.000 0140 1.000 0576 1.000 1434 1.000 2276 1.000 3104

0.05 0.1 0.2 0.3 0.4

0.999 810 1.000 185 1.000 912 1.001 623 1.002 326

0.999 028 0.999 400 1.000 119 1.000 820 1.001 512

.05 .06 .07 .08 .09

1.000 3920 1.000 4726 1.000 5523 1.000 6313 1.000 7098

0.5 0.6 0.8 1.0 1.2

1.003 023 1.003 716 1.005 090 1.006 452 1.007 807

1.002 197 1.002 877 1.004 227 1.005 570 1.006 909

.10 .15 .20 .25 .30

1.000 7880 1.001 1732 1.001 5514 1.001 9254 1.002 2961

1.4 1.6 1.8 2.0 2.2

1.009 159 1.010 510 1.011 860 1.013 209 1.014 557

1.008 247 1.009 583 1.010 918 1.012 252 1.013 586

.35 .40 .45 .50

1.002 6639 1.003 0292 1.003 3923 1.003 7534

2.4

1.015 904

1.014 919

*International Critical Tables, vol. 3, pp. 56–57.

2-114

PHYSICAL AND CHEMICAL DATA

TABLE 2-104

Zinc Bromide (ZnBr2)*

%

0 °C

20 °C

40 °C

60 °C

80 °C

100 °C

2 4 8 12 16

1.0188 1.0381 1.0777 1.1186 1.1609

1.0167 1.0354 1.0738 1.1135 1.1544

1.0102 1.0285 1.0660 1.1046 1.1445

1.0008 1.0187 1.0554 1.0932 1.1320

0.9890 1.0065 1.0422 1.0789 1.1169

0.9751 0.9921 1.0270 1.0629 1.1000

20 30 40 50 60 65

1.2043 1.3288 1.477 1.661 1.891 2.026

1.1965 1.3170 1.462 1.643 1.869 2.002

1.1855 1.3030 1.445 1.623 1.845 1.976

1.1720 1.2868 1.427 1.602 1.822 1.951

1.1560 1.2688 1.406 1.579 1.797 1.924

1.1382 1.2489 1.385 1.555 1.771 1.898

TABLE 2-106 [Zn(NO3)2]*

Zinc Nitrate

TABLE 2-107 (ZnSO4)*

Zinc Sulfate

%

18 °C

%

18 °C

%

20 °C

2 4 6 8 10 12 14 16

1.0154 1.0322 1.0496 1.0675 1.0859 1.1048 1.1244 1.1445

18 20 25 30 35 40 45 50

1.1652 1.1865 1.2427 1.3029 1.3678 1.4378 1.5134 1.5944

2 4 6 8 10 12 14 16

1.019 1.0403 1.0620 1.0842 1.1071 1.1308 1.1553 1.1806

*International Critical Tables, vol. 3, p. 65.

*International Critical Tables, vol. 3, p. 65.

*International Critical Tables, vol. 3, p. 64. TABLE 2-105

Zinc Chloride (ZnCl2)*

%

0 °C

20 °C

40 °C

60 °C

80 °C

100 °C

2 4 8 12 16

1.0192 1.0384 1.0769 1.1159 1.1558

1.0167 1.0350 1.0715 1.1085 1.1468

1.0099 1.0274 1.0624 1.0980 1.1350

1.0003 1.0172 1.0508 1.0853 1.1212

0.9882 1.0044 1.0369 1.0704 1.1055

0.9739 0.9894 1.0211 1.0541 1.0888

20 30 40 50 60 70

1.1970 1.3062 1.4329 1.5860

1.1866 1.2928 1.4173 1.5681 1.749 1.962

1.1736 1.2778 1.4003 1.5495

1.1590 1.2614 1.3824 1.5300

1.1428 1.2438 1.3637 1.5097

1.1255 1.2252 1.3441 1.4892

*International Critical Tables, vol. 3, p. 64.

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS* From International Critical Tables, vol. 3, pp. 115–129 unless otherwise stated. All compositions are in weight percent in vacuo. All density values are d4t = g /mL in vacuo.

UNITS AND UNITS CONVERSIONS Unless otherwise noted, densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32 TABLE 2-108

*For gasoline and aircraft fuels see Hibbard, NACA Res. Mem. E56I21 (declassified 1958).

Formic Acid (HCOOH)

%

0 °C

15 °C

20 °C

30 °C

%

0 °C

15 °C

20 °C

30 °C

%

0 °C

15 °C

20 °C

30 °C

%

0 °C

15 °C

20 °C

30 °C

0 1 2 3 4

0.9999 1.0028 1.0059 1.0090 1.0120

0.9991 1.0019 1.0045 1.0072 1.0100

0.9982 1.0019 1.0044 1.0070 1.0093

0.9957 0.9980 1.0004 1.0028 1.0053

25 26 27 28 29

1.0706 1.0733 1.0760 1.0787 1.0813

1.0627 1.0652 1.0678 1.0702 1.0726

1.0609 1.0633 1.0656 1.0681 1.0705

1.0540 1.0564 1.0587 1.0609 1.0632

50 51 52 53 54

1.1349 1.1374 1.1399 1.1424 1.1448

1.1225 1.1248 1.1271 1.1294 1.1318

1.1207 1.1223 1.1244 1.1269 1.1295

1.1098 1.1120 1.1142 1.1164 1.1186

75 76 77 78 79

1.1953 1.1976 1.1999 1.2021 1.2043

1.1794 1.1816 1.1837 1.1859 1.1881

1.1769 1.1785 1.1801 1.1818 1.1837

1.1636 1.1656 1.1676 1.1697 1.1717

5 6 7 8 9

1.0150 1.0179 1.0207 1.0237 1.0266

1.0124 1.0151 1.0177 1.0204 1.0230

1.0115 1.0141 1.0170 1.0196 1.0221

1.0075 1.0101 1.0125 1.0149 1.0173

30 31 32 33 34

1.0839 1.0866 1.0891 1.0916 1.0941

1.0750 1.0774 1.0798 1.0821 1.0844

1.0729 1.0753 1.0777 1.0800 1.0823

1.0654 1.0676 1.0699 1.0721 1.0743

55 56 57 58 59

1.1472 1.1497 1.1523 1.1548 1.1573

1.1341 1.1365 1.1388 1.1411 1.1434

1.1320 1.1342 1.1361 1.1381 1.1401

1.1208 1.1230 1.1253 1.1274 1.1295

80 81 82 83 84

1.2065 1.2088 1.2110 1.2132 1.2154

1.1902 1.1924 1.1944 1.1965 1.1985

1.1806 1.1876 1.1896 1.1914 1.1929

1.1737 1.1758 1.1778 1.1798 1.1817

10 11 12 13 14

1.0295 1.0324 1.0351 1.0379 1.0407

1.0256 1.0281 1.0306 1.0330 1.0355

1.0246 1.0271 1.0296 1.0321 1.0345

1.0197 1.0221 1.0244 1.0267 1.0290

35 36 37 38 39

1.0966 1.0993 1.1018 1.1043 1.1069

1.0867 1.0892 1.0916 1.0940 1.0964

1.0847 1.0871 1.0895 1.0919 1.0940

1.0766 1.0788 1.0810 1.0832 1.0854

60 61 62 63 64

1.1597 1.1621 1.1645 1.1669 1.1694

1.1458 1.1481 1.1504 1.1526 1.1549

1.1424 1.1448 1.1473 1.1493 1.1517

1.1317 1.1338 1.1360 1.1382 1.1403

85 86 87 88 89

1.2176 1.2196 1.2217 1.2237 1.2258

1.2005 1.2025 1.2045 1.2064 1.2084

1.1953 1.1976 1.1994 1.2012 1.2028

1.1837 1.1856 1.1875 1.1893 1.1910

15 16 17 18 19

1.0435 1.0463 1.0491 1.0518 1.0545

1.0380 1.0405 1.0430 1.0455 1.0480

1.0370 1.0393 1.0417 1.0441 1.0464

1.0313 1.0336 1.0358 1.0381 1.0404

40 41 42 43 44

1.1095 1.1122 1.1148 1.1174 1.1199

1.0988 1.1012 1.1036 1.1060 1.1084

1.0963 1.0990 1.1015 1.1038 1.1062

1.0876 1.0898 1.0920 1.0943 1.0965

65 66 67 68 69

1.1718 1.1742 1.1766 1.1790 1.1813

1.1572 1.1595 1.1618 1.1640 1.1663

1.1543 1.1565 1.1584 1.1604 1.1628

1.1425 1.1446 1.1467 1.1489 1.1510

90 91 92 93 94

1.2278 1.2297 1.2316 1.2335 1.2354

1.2102 1.2121 1.2139 1.2157 1.2174

1.2044 1.2059 1.2078 1.2099 1.2117

1.1927 1.1945 1.1961 1.1978 1.1994

20 21 22 23 24

1.0571 1.0598 1.0625 1.0652 1.0679

1.0505 1.0532 1.0556 1.0580 1.0604

1.0488 1.0512 1.0537 1.0561 1.0585

1.0427 1.0451 1.0473 1.0496 1.0518

45 46 47 48 49

1.1224 1.1249 1.1274 1.1299 1.1324

1.1109 1.1133 1.1156 1.1179 1.1202

1.1085 1.1108 1.1130 1.1157 1.1185

1.0987 1.1009 1.1031 1.1053 1.1076

70 71 72 73 74

1.1835 1.1858 1.1882 1.1906 1.1929

1.1685 1.1707 1.1729 1.1751 1.1773

1.1655 1.1677 1.1702 1.1728 1.1752

1.1531 1.1552 1.1573 1.1595 1.1615

95 96 97 98 99

1.2372 1.2390 1.2408 1.2425 1.2441

1.2191 1.2208 1.2224 1.2240 1.2257

1.2140 1.2158 1.2170 1.2183 1.2202

1.2008 1.2022 1.2036 1.2048 1.2061

100

1.2456

1.2273

1.2212

1.2073

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-109

2-115

Acetic Acid (CH3COOH)

0 °C

10 °C

15 °C

20 °C

25 °C

30 °C

40 °C

%

0 °C

10 °C

15 °C

20 °C

25 °C

30 °C

40 °C

0 1 2 3 4

0.9999 1.0016 1.0033 1.0051 1.0070

0.9997 1.0013 1.0029 1.0044 1.0060

0.9991 1.0006 1.0021 1.0036 1.0051

0.9982 0.9996 1.0012 1.0025 1.0040

0.9971 0.9987 1.0000 1.0013 1.0027

0.9957 0.9971 0.9984 0.9997 1.0011

0.9922 0.9934 0.9946 0.9958 0.9970

50 51 52 53 54

1.0729 1.0738 1.0748 1.0757 1.0765

1.0654 1.0663 1.0671 1.0679 1.0687

1.0613 1.0622 1.0629 1.0637 1.0644

1.0575 1.0582 1.0590 1.0597 1.0604

1.0534 1.0542 1.0549 1.0555 1.0562

1.0492 1.0499 1.0506 1.0512 1.0518

1.0408 1.0414 1.0421 1.0427 1.0432

5 6 7 8 9

1.0088 1.0106 1.0124 1.0142 1.0159

1.0076 1.0092 1.0108 1.0124 1.0140

1.0066 1.0081 1.0096 1.0111 1.0126

1.0055 1.0069 1.0083 1.0097 1.0111

1.0041 1.0055 1.0068 1.0081 1.0094

1.0024 1.0037 1.0050 1.0063 1.0076

0.9982 0.9994 1.0006 1.0018 1.0030

55 56 57 58 59

1.0774 1.0782 1.0790 1.0798 1.0805

1.0694 1.0701 1.0708 1.0715 1.0722

1.0651 1.0658 1.0665 1.0672 1.0678

1.0611 1.0618 1.0624 1.0631 1.0637

1.0568 1.0574 1.0580 1.0586 1.0592

1.0525 1.0531 1.0536 1.0542 1.0547

1.0438 1.0443 1.0448 1.0453 1.0458

10 11 12 13 14

1.0177 1.0194 1.0211 1.0228 1.0245

1.0156 1.0171 1.0187 1.0202 1.0217

1.0141 1.0155 1.0170 1.0184 1.0199

1.0125 1.0139 1.0154 1.0168 1.0182

1.0107 1.0120 1.0133 1.0146 1.0159

1.0089 1.0102 1.0115 1.0127 1.0139

1.0042 1.0054 1.0065 1.0077 1.0088

60 61 62 63 64

1.0813 1.0820 1.0826 1.0833 1.0838

1.0728 1.0734 1.0740 1.0746 1.0752

1.0684 1.0690 1.0696 1.0701 1.0706

1.0642 1.0648 1.0653 1.0658 1.0662

1.0597 1.0602 1.0607 1.0612 1.0616

1.0552 1.0557 1.0562 1.0566 1.0571

1.0462 1.0466 1.0470 1.0473 1.0477

15 16 17 18 19

1.0262 1.0278 1.0295 1.0311 1.0327

1.0232 1.0247 1.0262 1.0276 1.0291

1.0213 1.0227 1.0241 1.0255 1.0269

1.0195 1.0209 1.0223 1.0236 1.0250

1.0172 1.0185 1.0198 1.0210 1.0223

1.0151 1.0163 1.0175 1.0187 1.0198

1.0099 1.0110 1.0121 1.0132 1.0142

65 66 67 68 69

1.0844 1.0850 1.0856 1.0860 1.0865

1.0757 1.0762 1.0767 1.0771 1.0775

1.0711 1.0716 1.0720 1.0725 1.0729

1.0666 1.0671 1.0675 1.0678 1.0682

1.0621 1.0624 1.0628 1.0631 1.0634

1.0575 1.0578 1.0582 1.0585 1.0588

1.0480 1.0483 1.0486 1.0489 1.0491

20 21 22 23 24

1.0343 1.0358 1.0374 1.0389 1.0404

1.0305 1.0319 1.0333 1.0347 1.0361

1.0283 1.0297 1.0310 1.0323 1.0336

1.0263 1.0276 1.0288 1.0301 1.0313

1.0235 1.0248 1.0260 1.0272 1.0283

1.0210 1.0222 1.0233 1.0244 1.0256

1.0153 1.0164 1.0174 1.0185 1.0195

70 71 72 73 74

1.0869 1.0874 1.0877 1.0881 1.0884

1.0779 1.0783 1.0786 1.0789 1.0792

1.0732 1.0736 1.0738 1.0741 1.0743

1.0685 1.0687 1.0690 1.0693 1.0694

1.0637 1.0640 1.0642 1.0644 1.0645

1.0590 1.0592 1.0594 1.0595 1.0596

1.0493 1.0495 1.0496 1.0497 1.0498

25 26 27 28 29

1.0419 1.0434 1.0449 1.0463 1.0477

1.0375 1.0388 1.0401 1.0414 1.0427

1.0349 1.0362 1.0374 1.0386 1.0399

1.0326 1.0338 1.0349 1.0361 1.0372

1.0295 1.0307 1.0318 1.0329 1.0340

1.0267 1.0278 1.0289 1.0299 1.0310

1.0205 1.0215 1.0225 1.0234 1.0244

75 76 77 78 79

1.0887 1.0889 1.0891 1.0893 1.0894

1.0794 1.0796 1.0797 1.0798 1.0798

1.0745 1.0746 1.0747 1.0747 1.0747

1.0696 1.0698 1.0699 1.0700 1.0700

1.0647 1.0648 1.0648 1.0648 1.0648

1.0597 1.0598 1.0598 1.0598 1.0597

1.0499 1.0499 1.0499 1.0498 1.0497

30 31 32 33 34

1.0491 1.0505 1.0519 1.0532 1.0545

1.0440 1.0453 1.0465 1.0477 1.0489

1.0411 1.0423 1.0435 1.0446 1.0458

1.0384 1.0395 1.0406 1.0417 1.0428

1.0350 1.0361 1.0372 1.0382 1.0392

1.0320 1.0330 1.0341 1.0351 1.0361

1.0253 1.0262 1.0272 1.0281 1.0289

80 81 82 83 84

1.0895 1.0895 1.0895 1.0895 1.0893

1.0798 1.0797 1.0796 1.0795 1.0793

1.0747 1.0745 1.0743 1.0741 1.0738

1.0700 1.0699 1.0698 1.0696 1.0693

1.0647 1.0646 1.0644 1.0642 1.0638

1.0596 1.0594 1.0592 1.0589 1.0585

1.0495 1.0493 1.0490 1.0487 1.0483

35 36 37 38 39

1.0558 1.0571 1.0584 1.0596 1.0608

1.0501 1.0513 1.0524 1.0535 1.0546

1.0469 1.0480 1.0491 1.0501 1.0512

1.0438 1.0449 1.0459 1.0469 1.0479

1.0402 1.0412 1.0422 1.0432 1.0441

1.0371 1.0380 1.0390 1.0399 1.0408

1.0298 1.0306 1.0314 1.0322 1.0330

85 86 87 88 89

1.0891 1.0887 1.0883 1.0877 1.0872

1.0790 1.0787 1.0783 1.0778 1.0773

1.0735 1.0731 1.0726 1.0721 1.0715

1.0689 1.0685 1.0680 1.0675 1.0668

1.0635 1.0630 1.0626 1.0620 1.0613

1.0582 1.0576 1.0571 1.0564 1.0557

1.0479 1.0473 1.0467 1.0460 1.0453

40 41 42 43 44

1.0621 1.0633 1.0644 1.0656 1.0667

1.0557 1.0568 1.0578 1.0588 1.0598

1.0522 1.0532 1.0542 1.0551 1.0561

1.0488 1.0498 1.0507 1.0516 1.0525

1.0450 1.0460 1.0469 1.0477 1.0486

1.0416 1.0425 1.0433 1.0441 1.0449

1.0338 1.0346 1.0353 1.0361 1.0368

90 91 92 93 94

1.0865 1.0857 1.0848 1.0838 1.0826

1.0766 1.0758 1.0749 1.0739 1.0727

1.0708 1.0700 1.0690 1.0680 1.0667

1.0661 1.0652 1.0643 1.0632 1.0619

1.0605 1.0597 1.0587 1.0577 1.0564

1.0549 1.0541 1.0530 1.0518 1.0506

1.0445 1.0436 1.0426 1.0414 1.0401

45 46 47 48 49

1.0679 1.0689 1.0699 1.0709 1.0720

1.0608 1.0618 1.0627 1.0636 1.0645

1.0570 1.0579 1.0588 1.0597 1.0605

1.0534 1.0542 1.0551 1.0559 1.0567

1.0495 1.0503 1.0511 1.0518 1.0526

1.0456 1.0464 1.0471 1.0479 1.0486

1.0375 1.0382 1.0389 1.0395 1.0402

95 96 97 98 99

1.0813 1.0798 1.0780 1.0759 1.0730

1.0714

1.0652 1.0632 1.0611 1.0590 1.0567

1.0605 1.0588 1.0570 1.0549 1.0524

1.0551 1.0535 1.0516 1.0495 1.0468

1.0491 1.0473 1.0454 1.0431 1.0407

1.0386 1.0368 1.0348 1.0325 1.0299

100

1.0697

1.0545

1.0498

1.0440

1.0380

1.0271

%

2-116

PHYSICAL AND CHEMICAL DATA TABLE 2-110

TABLE 2-111

Oxalic Acid (H2C2O4)

%

d 17.5 4

%

d 17.5 4

1 2 4

1.0035 1.0070 1.0140

8 10 12

1.0280 1.0350 1.0420

Methyl Alcohol (CH3OH)*

%

0 °C

10 °C

15.56 °C

20 °C

15 °C

%

0 °C

10 °C

15.56 °C

20 °C

15 °C

%

0 °C

10 °C

15.56 °C

20 °C

15 °C

0 1 2 3 4

0.9999 0.9981 0.9963 0.9946 0.9930

0.9997 0.9980 0.9962 0.9945 0.9929

0.9990 0.9973 0.9955 0.9938 0.9921

0.9982 0.9965 0.9948 0.9931 0.9914

0.99913 0.99727 0.99543 0.99370 0.99198

35 36 37 38 39

0.9534 0.9520 0.9505 0.9490 0.9475

0.9484 0.9469 0.9453 0.9437 0.9420

0.9456 0.9440 0.9422 0.9405 0.9387

0.9433 0.9416 0.9398 0.9381 0.9363

0.94570 0.94404 0.94237 0.94067 0.93894

70 71 72 73 74

0.8869 0.8847 0.8824 0.8801 0.8778

0.8794 0.8770 0.8747 0.8724 0.8699

0.8748 0.8726 0.8702 0.8678 0.8653

0.8715 0.8690 0.8665 0.8641 0.8616

0.87507 0.87271 0.87033 0.86792 0.86546

5 6 7 8 9

0.9914 0.9899 0.9884 0.9870 0.9856

0.9912 0.9896 0.9881 0.9865 0.9849

0.9904 0.9889 0.9872 0.9857 0.9841

0.9896 0.9880 0.9863 0.9847 0.9831

0.99029 0.98864 0.98701 0.98547 0.98394

40 41 42 43 44

0.9459 0.9443 0.9427 0.9411 0.9395

0.9403 0.9387 0.9370 0.9352 0.9334

0.9369 0.9351 0.9333 0.9315 0.9297

0.9345 0.9327 0.9309 0.9290 0.9272

0.93720 0.93543 0.93365 0.93185 0.93001

75 76 77 78 79

0.8754 0.8729 0.8705 0.8680 0.8657

0.8676 0.8651 0.8626 0.8602 0.8577

0.8629 0.8604 0.8579 0.8554 0.8529

0.8592 0.8567 0.8542 0.8518 0.8494

0.86300 0.86051 0.85801 0.85551 0.85300

10 11 12 13 14

0.9842 0.9829 0.9816 0.9804 0.9792

0.9834 0.9820 0.9805 0.9791 0.9778

0.9826 0.9811 0.9796 0.9781 0.9766

0.9815 0.9799 0.9784 0.9768 0.9754

0.98241 0.98093 0.97945 0.97802 0.97660

45 46 47 48 49

0.9377 0.9360 0.9342 0.9324 0.9306

0.9316 0.9298 0.9279 0.9260 0.9240

0.9279 0.9261 0.9242 0.9223 0.9204

0.9252 0.9234 0.9214 0.9196 0.9176

0.92815 0.92627 0.92436 0.92242 0.92048

80 81 82 83 84

0.8634 0.8610 0.8585 0.8560 0.8535

0.8551 0.8527 0.8501 0.8475 0.8449

0.8503 0.8478 0.8452 0.8426 0.8400

0.8469 0.8446 0.8420 0.8394 0.8366

0.85048 0.84794 0.84536 0.84274 0.84009

15 16 17 18 19

0.9780 0.9769 0.9758 0.9747 0.9736

0.9764 0.9751 0.9739 0.9726 0.9713

0.9752 0.9738 0.9723 0.9709 0.9695

0.9740 0.9725 0.9710 0.9696 0.9681

0.97518 0.97377 0.97237 0.97096 0.96955

50 51 52 53 54

0.9287 0.9269 0.9250 0.9230 0.9211

0.9221 0.9202 0.9182 0.9162 0.9142

0.9185 0.9166 0.9146 0.9126 0.9106

0.9156 0.9135 0.9114 0.9094 0.9073

0.91852 0.91653 0.91451 0.91248 0.91044

85 86 87 88 89

0.8510 0.8483 0.8456 0.8428 0.8400

0.8422 0.8394 0.8367 0.8340 0.8314

0.8374 0.8347 0.8320 0.8294 0.8267

0.8340 0.8314 0.8286 0.8258 0.8230

0.83742 0.83475 0.83207 0.82937 0.82667

20 21 22 23 24

0.9725 0.9714 0.9702 0.9690 0.9678

0.9700 0.9687 0.9673 0.9660 0.9646

0.9680 0.9666 0.9652 0.9638 0.9624

0.9666 0.9651 0.9636 0.9622 0.9607

0.96814 0.96673 0.96533 0.96392 0.96251

55 56 57 58 59

0.9191 0.9172 0.9151 0.9131 0.9111

0.9122 0.9101 0.9080 0.9060 0.9039

0.9086 0.9065 0.9045 0.9024 0.9002

0.9052 0.9032 0.9010 0.8988 0.8968

0.90839 0.90631 0.90421 0.90210 0.89996

90 91 92 93 94

0.8374 0.8347 0.8320 0.8293 0.8266

0.8287 0.8261 0.8234 0.8208 0.8180

0.8239 0.8212 0.8185 0.8157 0.8129

0.8202 0.8174 0.8146 0.8118 0.8090

0.82396 0.82124 0.81849 0.81568 0.81285

25 26 27 28 29

0.9666 0.9654 0.9642 0.9629 0.9616

0.9632 0.9618 0.9604 0.9590 0.9575

0.9609 0.9595 0.9580 0.9565 0.9550

0.9592 0.9576 0.9562 0.9546 0.9531

0.96108 0.95963 0.95817 0.95668 0.95518

60 61 62 63 64

0.9090 0.9068 0.9046 0.9024 0.9002

0.9018 0.8998 0.8977 0.8955 0.8933

0.8980 0.8958 0.8936 0.8913 0.8890

0.8946 0.8924 0.8902 0.8879 0.8856

0.89781 0.89563 0.89341 0.89117 0.88890

95 96 97 98 99

0.8240 0.8212 0.8186 0.8158 0.8130

0.8152 0.8124 0.8096 0.8068 0.8040

0.8101 0.8073 0.8045 0.8016 0.7987

0.8062 0.8034 0.8005 0.7976 0.7948

0.80999 0.80713 0.80428 0.80143 0.79859

30 31 32 33 34

0.9604 0.9590 0.9576 0.9563 0.9549

0.9560 0.9546 0.9531 0.9516 0.9500

0.9535 0.9521 0.9505 0.9489 0.9473

0.9515 0.9499 0.9483 0.9466 0.9450

0.95366 0.95213 0.95056 0.94896 0.94734

65 66 67 68 69

0.8980 0.8958 0.8935 0.8913 0.8891

0.8911 0.8888 0.8865 0.8842 0.8818

0.8867 0.8844 0.8820 0.8797 0.8771

0.8834 0.8811 0.8787 0.8763 0.8738

0.88662 0.88433 0.88203 0.87971 0.87739

100

0.8102

0.8009

0.7959

0.7917

0.79577

*It should be noted that the values for 100 percent do not agree with some data available elsewhere, e.g., American Institute of Physics Handbook, McGraw-Hill, New York, 1957. Also, see Atack, Handbook of Chemical Data, Reinhold, New York, 1957. Also, see Tables 2-234 and 2-305 for pure component densities.

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-112

2-117

Ethyl Alcohol (C2H5OH)*

%

10 °C

15 °C

20 °C

25 °C

30 °C

35 °C

40 °C

%

10 °C

15 °C

20 °C

25 °C

30 °C

35 °C

40 °C

0 1 2 3 4

0.99973 785 602 426 258

0.99913 725 542 365 195

0.99823 636 453 275 103

0.99708 520 336 157 0.98984

0.99568 379 194 014 0.98839

0.99406 217 031 0.98849 672

0.99225 034 0.98846 663 485

50 51 52 53 54

0.92126 0.91943 723 502 279

0.91776 555 333 110 0.90885

0.91384 160 0.90936 711 485

0.90985 760 534 307 079

0.90580 353 125 0.89896 667

0.90168 0.89940 710 479 248

0.89750 519 288 056 0.88823

5 6 7 8 9

098 0.98946 801 660 524

032 0.98877 729 584 442

0.98938 780 627 478 331

817 656 500 346 193

670 507 347 189 031

501 335 172 009 0.97846

311 142 0.97975 808 641

55 56 57 58 59

055 0.90831 607 381 154

659 433 207 0.89980 752

258 031 0.89803 574 344

0.89850 621 392 162 0.88931

437 206 0.88975 744 512

016 0.88784 552 319 085

589 356 122 0.87888 653

10 11 12 13 14

393 267 145 026 0.97911

304 171 041 0.97914 790

187 047 0.97910 775 643

043 0.97897 753 611 472

0.97875 723 573 424 278

685 527 371 216 063

475 312 150 0.96989 829

60 61 62 63 64

0.89927 698 468 237 006

523 293 062 0.88830 597

113 0.88882 650 417 183

699 446 233 0.87998 763

278 044 0.87809 574 337

0.87851 615 379 142 0.86905

417 180 0.86943 705 466

15 16 17 18 19

800 692 583 473 363

669 552 433 313 191

514 387 259 129 0.96997

334 199 062 0.96923 782

133 0.96990 844 697 547

0.96911 760 607 452 294

670 512 352 189 023

65 66 67 68 69

0.88774 541 308 074 0.87839

364 130 0.87895 660 424

0.87948 713 477 241 004

527 291 054 0.86817 579

100 0.86863 625 387 148

667 429 190 0.85950 710

227 0.85987 747 407 266

20 21 22 23 24

252 139 024 0.96907 787

068 0.96944 818 689 558

864 729 592 453 312

639 495 348 199 048

395 242 087 0.95929 769

134 0.95973 809 643 476

0.95856 687 516 343 168

70 71 72 73 74

602 365 127 0.86888 648

187 0.86949 710 470 229

0.86766 527 287 047 0.85806

340 100 0.85859 618 376

0.85908 667 426 184 0.84941

470 228 0.84986 743 500

025 0.84783 540 297 053

25 26 27 28 29

665 539 406 268 125

424 287 144 0.95996 844

168 020 0.95867 710 548

0.95895 738 576 410 241

607 442 272 098 0.94922

306 133 0.94955 774 590

0.94991 810 625 438 248

75 76 77 78 79

408 168 0.85927 685 442

0.85988 747 505 262 018

564 322 079 0.84835 590

134 0.84891 647 403 158

698 455 211 0.83966 720

257 013 0.83768 523 277

0.83809 564 319 074 0.82827

30 31 32 33 34

0.95977 823 665 502 334

686 524 357 186 011

382 212 038 0.94860 679

067 0.94890 709 525 337

741 557 370 180 0.93986

403 214 021 0.93825 626

055 0.93860 662 461 257

80 81 82 83 84

197 0.84950 702 453 203

0.84772 525 277 028 0.83777

344 096 0.83848 599 348

0.83911 664 415 164 0.82913

473 224 0.82974 724 473

029 0.82780 530 279 027

578 329 079 0.81828 576

35 36 37 38 39

162 0.94986 805 620 431

0.94832 650 464 273 079

494 306 114 0.93919 720

146 0.93952 756 556 353

790 591 390 186 0.92979

425 221 016 0.92808 597

051 0.92843 634 422 208

85 86 87 88 89

0.83951 697 441 181 0.82919

525 271 014 0.82754 492

095 0.82840 583 323 062

660 405 148 0.81888 626

220 0.81965 708 448 186

0.81774 519 262 003 0.80742

322 067 0.80811 552 291

40 41 42 43 44

238 042 0.93842 639 433

0.93882 682 478 271 062

518 314 107 0.92897 685

148 0.92940 729 516 301

770 558 344 128 0.91910

385 170 0.91952 733 513

0.91992 774 554 332 108

90 91 92 93 94

654 386 114 0.81839 561

227 0.81959 688 413 134

0.81797 529 257 0.80983 705

362 094 0.80823 549 272

0.80922 655 384 111 0.79835

478 211 0.79941 669 393

028 0.79761 491 220 0.78947

45 46 47 48 49

226 017 0.92806 593 379

0.92852 640 426 211 0.91995

472 257 041 0.91823 604

085 0.91868 649 429 208

692 472 250 028 0.90805

291 069 0.90845 621 396

0.90884 660 434 207 0.89979

95 96 97 98 99

278 0.80991 698 399 094

0.80852 566 274 0.79975 670

424 138 0.79846 547 243

0.79991 706 415 117 0.78814

555 271 0.78981 684 382

114 0.78831 542 247 0.77946

670 388 100 0.77806 507

100

0.79784

360

0.78934

506

075

641

203

*For data from −78° to 78°C, see p. 2-142, Table 2N-5, American Institute of Physics Handbook, McGraw-Hill, New York, 1957. See Tables 2-214 and 2-305 for pure component densities.

2-118

PHYSICAL AND CHEMICAL DATA

TABLE 2-113 % alcohol by weight

Densities of Mixtures of C2H5OH and H2O at 20 °C g/mL Tenths of %

0

1

2

3

4

5

6

7

0 1 2 3 4

0.99823 636 453 275 103

804 618 435 257 087

785 599 417 240 070

766 581 399 222 053

748 562 381 205 037

729 544 363 188 020

710 525 345 171 003

692 507 327 154 *987

5 6 7 8 9

0.98938 780 627 478 331

922 765 612 463 316

906 749 597 449 301

890 734 582 434 287

874 718 567 419 273

859 703 553 404 258

843 688 538 389 244

827 673 523 374 229

10 11 12 13 14

187 047 0.97910 775 643

172 033 896 761 630

158 019 883 748 617

144 006 869 735 604

130 *992 855 722 591

117 *978 842 709 578

103 *964 828 696 565

089 *951 815 683 552

15 16 17 18 19

514 387 259 129 0.96997

501 374 246 116 984

488 361 233 103 971

475 349 220 089 957

462 336 207 076 944

450 323 194 063 931

438 310 181 050 917

425 297 168 037 904

412 284 155 024 891

20 21 22 23 24

864 729 592 453 312

850 716 578 439 297

837 702 564 425 283

823 688 551 411 269

810 675 537 396 254

796 661 523 382 240

783 647 509 368 225

769 634 495 354 211

756 620 481 340 196

25 26 27 28 29

168 020 0.95867 710 548

153 005 851 694 532

139 *990 836 678 516

124 *975 820 662 499

109 *959 805 646 483

094 *944 789 630 466

080 *929 773 613 450

30 31 32 33 34

382 212 038 0.94860 679

365 195 020 842 660

349 178 003 824 642

332 161 *985 806 624

315 143 *967 788 605

298 126 *950 770 587

35 36 37 38 39

494 306 114 0.93919 720

475 287 095 899 700

456 268 075 879 680

438 249 056 859 660

419 230 036 840 640

40 41 42 43 44

518 314 107 0.92897 685

498 294 086 876 664

478 273 065 855 642

458 253 044 834 621

45 46 47 48 49

472 257 041 0.91823 604

450 236 019 801 582

429 214 *997 780 560

408 193 *976 758 538

Tenths of % 0

1

2

3

4

5

6

7

50 51 52 53 54

0.91384 160 0.90936 711 485

361 138 914 689 463

339 116 891 666 440

317 093 869 644 417

295 071 846 621 395

272 049 824 598 372

250 026 801 576 349

228 206 183 004 *981 *959 779 756 734 553 531 508 327 304 281

796 642 493 345 201

55 56 57 58 59

258 031 0.89803 574 344

236 008 780 551 321

213 *985 757 528 298

190 *962 734 505 275

167 *939 711 482 252

145 *917 688 459 229

122 *894 665 436 206

099 076 054 *871 *848 *825 643 620 597 413 390 367 183 160 137

075 061 *937 *923 801 788 670 657 539 526

60 61 62 63 64

113 0.88882 650 417 183

090 859 626 393 160

067 836 603 370 136

044 812 580 347 113

400 272 142 010 877

65 66 67 68 69

0.87948 713 477 241 004

925 689 454 218 *981

901 666 430 194 *957

878 642 406 170 *933

854 619 383 147 *909

831 595 359 123 *885

807 572 336 099 *862

784 760 737 548 524 501 312 288 265 075 052 028 *838 *814 *790

742 606 467 326 182

70 71 72 73 74

0.86766 527 287 047 0.85806

742 503 263 022 781

718 479 239 *998 757

694 455 215 *974 733

671 431 191 *950 709

647 407 167 *926 685

623 383 143 *902 661

599 575 551 339 335 311 119 095 071 *878 *854 *830 636 612 588

065 *914 757 597 433

050 035 *898 *883 742 726 581 565 416 400

75 76 77 78 79

564 322 079 0.84835 590

540 297 055 811 566

515 273 031 787 541

491 467 443 419 394 370 346 249 225 200 176 152 128 103 006 *982 *958 *933 *909 *884 *860 762 738 713 689 664 640 615 517 492 467 443 418 393 369

281 108 *932 752 568

264 091 *914 734 550

247 230 074 056 *896 *878 715 697 531 512

80 81 82 83 84

344 096 0.83848 599 348

319 072 823 574 323

294 047 798 549 297

270 245 220 196 171 146 121 022 *997 *972 *947 *923 *898 *873 773 748 723 698 674 649 624 523 498 473 448 423 398 373 272 247 222 196 171 146 120

400 211 017 820 620

382 192 *997 800 599

363 172 *978 780 579

344 325 153 134 *958 *939 760 740 559 539

85 86 87 88 89

095 0.82840 583 323 062

070 815 557 297 035

044 019 *994 *968 *943 *917 *892 *866 789 763 738 712 686 660 635 609 531 505 479 453 427 401 375 349 271 245 219 193 167 140 114 088 009 *983 *956 *930 *903 *877 *850 *824

437 232 023 812 600

417 212 002 791 579

396 191 *981 770 557

376 170 *960 749 536

356 335 149 129 *939 *918 728 707 515 493

90 91 92 93 94

0.81797 529 257 0.80983 705

770 502 230 955 677

744 475 203 928 649

717 448 175 900 621

690 421 148 872 593

386 171 *954 736 516

365 150 *932 714 494

343 128 *910 692 472

322 106 *889 670 450

300 279 085 063 *867 *845 648 626 428 406

95 96 97 98 99

424 138 0.79846 547 243

395 109 816 517 213

367 080 787 487 182

338 051 757 456 151

310 281 253 224 195 166 022 *993 *963 *934 *905 *875 727 698 668 638 608 578 426 396 365 335 305 274 120 089 059 028 *997 *966

100

0.78934

*Indicates change in the first two decimal places.

8

% alcohol by weight

9

673 655 489 471 310 292 137 120 *971 *954 811 658 508 360 215

8

9

021 *998 *975 *951 *928 *905 789 766 743 720 696 673 557 533 510 487 463 440 323 300 277 253 230 206 089 066 042 019 *995 *972

664 394 120 844 565

637 366 093 817 537

610 339 066 789 509

583 312 038 761 480

556 285 010 733 452

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-114 % alcohol by volume at 60°F

Specific Gravity {60°/60 °F [(15.56°/15.56 °C)]} of Mixtures by Volume of C2H5OH and H2O

Tenths of % 0

2-119

1

2

3

4

5

6

7

8

9

% alcohol by volume at 60°F

Tenths of % 0

1

2

3

4

5

6

7

8

9

0 1 2 3 4

1.00000 *985 0.99850 835 703 688 559 545 419 405

*970 820 674 531 391

*955 806 659 516 378

*940 791 645 502 364

*925 776 630 488 350

*910 761 616 474 336

*895 747 602 460 323

*880 732 587 446 309

865 717 573 432 296

50 51 52 53 54

0.93426 230 031 0.92830 626

407 387 368 348 328 309 289 270 250 210 190 171 151 131 111 091 071 051 011 *991 *971 *951 *931 *911 *890 *870 *850 810 789 769 749 728 708 688 667 647 605 585 564 544 523 502 482 461 440

5 6 7 8 9

282 150 022 0.98899 779

269 137 009 887 767

255 124 *997 875 755

242 111 *984 863 743

228 098 *972 851 731

215 085 *960 838 720

202 073 *947 826 708

189 060 *935 814 696

176 163 047 035 *923 *911 803 791 684 672

55 56 57 58 59

419 210 0.91999 784 565

398 189 978 762 543

377 168 956 741 521

357 147 935 719 499

336 126 914 697 477

315 105 892 675 455

294 084 871 653 433

10 11 12 13 14

661 544 430 319 210

649 532 419 308 200

637 521 408 297 190

625 509 396 286 179

614 498 385 275 168

602 487 374 264 157

590 475 363 254 147

579 464 352 243 136

567 452 341 232 125

556 441 330 221 115

60 61 62 63 64

344 120 0.90893 664 434

322 097 870 641 411

299 075 847 618 388

277 052 825 595 365

255 030 802 572 341

232 007 779 549 318

210 *984 756 526 295

15 16 17 18 19

104 0.97998 895 794 694

093 988 885 784 684

083 977 875 774 674

072 967 864 764 664

062 956 854 754 654

051 946 844 744 645

040 936 834 734 635

030 925 824 724 625

019 915 814 714 615

009 905 804 704 605

65 66 67 68 69

202 0.89967 729 489 245

179 943 705 465 220

155 920 681 441 196

132 896 657 416 171

108 872 633 392 147

085 848 609 368 122

061 825 585 343 098

038 801 561 319 073

014 *991 777 753 537 513 295 270 048 024

20 21 22 23 24

596 496 395 293 189

586 486 385 283 179

576 476 375 272 168

566 466 365 262 158

556 456 354 252 147

546 446 344 241 137

536 436 334 231 126

526 425 324 221 116

516 415 313 210 105

506 405 303 200 095

70 71 72 73 74

0.88999 751 499 244 0.87987

974 725 474 218 961

950 700 448 193 935

925 675 423 167 910

900 650 397 141 884

875 625 372 116 858

850 600 346 090 832

825 574 321 064 806

801 549 296 039 780

25 26 27 28 29

084 0.96978 870 760 648

073 967 859 749 637

063 957 848 738 625

052 946 837 727 614

042 935 826 715 603

031 924 815 704 591

020 914 804 693 580

010 903 793 682 568

*999 *988 892 881 782 771 671 659 557 546

75 76 77 78 79

728 465 199 0.86929 656

702 439 172 902 629

676 412 145 875 601

650 386 118 847 574

623 359 092 820 546

597 332 065 793 518

571 306 038 766 491

545 518 492 279 252 226 011 *984 *957 738 711 684 463 435 408

30 31 32 33 34

534 418 296 170 041

522 406 284 157 028

511 394 271 144 015

499 382 259 132 002

488 370 246 119 *988

476 358 234 106 *975

464 346 221 093 *962

453 334 209 080 *948

441 429 321 309 196 183 067 054 *935 *921

80 81 82 83 84

380 100 0.85817 531 240

352 072 789 502 211

324 044 760 473 181

296 269 241 213 185 157 129 015 *987 *959 *931 *902 *874 *846 732 703 674 646 617 588 560 444 415 386 357 328 299 270 152 122 093 063 033 004 *974

35 36 37 38 39

0.95908 770 628 482 332

894 756 614 467 317

881 742 599 452 302

867 728 585 437 286

854 714 570 423 271

840 700 556 408 256

826 685 541 393 240

812 671 526 378 225

784 643 497 347 194

85 86 87 88 89

0.84944 642 336 025 0.83707

914 612 305 *994 675

884 581 274 *962 643

854 551 243 *930 610

824 520 212 *899 578

794 490 181 *867 545

764 459 150 *835 513

734 703 673 428 398 367 119 088 056 *803 *771 *739 480 447 415

40 41 42 43 44

178 020 0.94858 693 524

162 004 842 676 507

147 *988 825 660 490

131 *972 809 643 473

115 *956 792 626 455

100 *940 776 609 438

084 *923 759 592 421

068 *907 743 575 403

052 036 *891 *875 726 710 558 541 386 369

90 91 92 93 94

382 049 0.82705 351 0.81984

349 015 670 315 947

315 *981 635 279 909

282 *947 600 243 871

249 *913 565 206 834

216 *879 529 170 796

183 *845 494 133 757

150 116 083 *810 *776 *741 458 423 387 096 059 022 719 681 642

45 46 47 48 49

351 174 0.93993 808 619

334 156 975 789 600

316 138 956 771 581

298 120 938 752 562

281 102 920 733 543

263 084 901 714 523

245 066 883 695 504

228 048 864 676 485

95 96 97 98 99

603 206 0.80792 356 0.79889

564 165 750 311 841

525 125 707 265 792

486 084 664 219 743

446 042 620 173 693

407 001 577 127 643

367 *960 533 080 593

327 287 247 *918 *876 *834 489 445 401 033 *985 *937 543 492 441

100

389

*Indicates change in first two decimal places.

798 657 512 362 209

210 030 845 657 465

192 011 827 638 446

273 062 849 631 410

252 041 827 610 388

231 020 806 588 366

188 165 143 *962 *939 *916 733 710 687 503 480 457 272 249 225

776 524 270 013 754

2-120

PHYSICAL AND CHEMICAL DATA

TABLE 2-115 n-Propyl Alcohol (C3H7OH) %

0 °C

15 °C

30 °C

%

0 °C

15 °C

30 °C

%

0 °C

15 °C

30 °C

%

0 °C

15 °C

30 °C

%

0 °C

15 °C

30 °C

0 1 2 3 4

0.9999 0.9982 0.9967 0.9952 0.9939

0.9991 0.9974 0.9960 0.9944 0.9929

0.9957 0.9940 0.9924 0.9908 0.9893

20 21 22 23 24

0.9789 0.9776 0.9763 0.9748 0.9733

0.9723 0.9705 0.9688 0.9670 0.9651

0.9643 0.9622 0.9602 0.9583 0.9563

40 41 42 43 44

0.9430 0.9411 0.9391 0.9371 0.9352

0.9331 0.9310 0.9290 0.9269 0.9248

0.9226 0.9205 0.9184 0.9164 0.9143

60 61 62 63 64

0.9033 0.9013 0.8994 0.8974 0.8954

0.8922 0.8902 0.8882 0.8861 0.8841

0.8807 0.8786 0.8766 0.8745 0.8724

80 81 82 83 84

0.8634 0.8614 0.8594 0.8574 0.8554

0.8516 0.8496 0.8475 0.8454 0.8434

0.8394 0.8373 0.8352 0.8332 0.8311

5 6 7 8 9

0.9926 0.9914 0.9904 0.9894 0.9883

0.9915 0.9902 0.9890 0.9877 0.9864

0.9877 0.9862 0.9848 0.9834 0.9819

25 26 27 28 29

0.9717 0.9700 0.9682 0.9664 0.9646

0.9633 0.9614 0.9594 0.9576 0.9556

0.9543 0.9522 0.9501 0.9481 0.9460

45 46 47 48 49

0.9332 0.9311 0.9291 0.9272 0.9252

0.9228 0.9207 0.9186 0.9165 0.9145

0.9122 0.9100 0.9079 0.9057 0.9036

65 66 67 68 69

0.8934 0.8913 0.8894 0.8874 0.8854

0.8820 0.8800 0.8779 0.8759 0.8739

0.8703 0.8682 0.8662 0.8641 0.8620

85 86 87 88 89

0.8534 0.8513 0.8492 0.8471 0.8450

0.8413 0.8393 0.8372 0.8351 0.8330

0.8290 0.8269 0.8248 0.8227 0.8206

10 11 12 13 14

0.9874 0.9865 0.9857 0.9849 0.9841

0.9852 0.9840 0.9828 0.9817 0.9806

0.9804 0.9790 0.9775 0.9760 0.9746

30 31 32 33 34

0.9627 0.9608 0.9589 0.9570 0.9550

0.9535 0.9516 0.9495 0.9474 0.9454

0.9439 0.9418 0.9396 0.9375 0.9354

50 51 52 53 54

0.9232 0.9213 0.9192 0.9173 0.9153

0.9124 0.9104 0.9084 0.9064 0.9044

0.9015 0.8994 0.8973 0.8952 0.8931

70 71 72 73 74

0.8835 0.8815 0.8795 0.8776 0.8756

0.8719 0.8700 0.8680 0.8659 0.8639

0.8600 0.8580 0.8559 0.8539 0.8518

90 91 92 93 94

0.8429 0.8408 0.8387 0.8364 0.8342

0.8308 0.8287 0.8266 0.8244 0.8221

0.8185 0.8164 0.8142 0.8120 0.8098

15 16 17 18 19

0.9833 0.9825 0.9817 0.9808 0.9800

0.9793 0.9780 0.9768 0.9752 0.9739

0.9730 0.9714 0.9698 0.9680 0.9661

35 36 37 38 39

0.9530 0.9511 0.9491 0.9471 0.9450

0.9434 0.9413 0.9392 0.9372 0.9351

0.9333 0.9312 0.9289 0.9269 0.9247

55 56 57 58 59

0.9132 0.9112 0.9093 0.9073 0.9053

0.9023 0.9003 0.8983 0.8963 0.8942

0.8911 0.8890 0.8869 0.8849 0.8828

75 76 77 78 79

0.8736 0.8716 0.8695 0.8675 0.8655

0.8618 0.8598 0.8577 0.8556 0.8536

0.8497 0.8477 0.8456 0.8435 0.8414

95 96 97 98 99

0.8320 0.8296 0.8272 0.8248 0.8222

0.8199 0.8176 0.8153 0.8128 0.8104

0.8077 0.8054 0.8031 0.8008 0.7984

100

0.8194

0.8077

0.7958

TABLE 2-116

Isopropyl Alcohol (C3H7OH)

%

0 °C

15 °C*

15 °C*

20 °C

30 °C

%

0 °C

15 °C*

20 °C

30 °C

%

0 °C

15 °C*

15 °C*

20 °C

30 °C

0 1 2 3 4

0.9999 0.9980 0.9962 0.9946 0.9930

0.9991 0.9973 0.9956 0.9938 0.9922

0.99913 0.9972 0.9954 0.9936 0.9920

0.9982 0.9962 0.9944 0.9926 0.9909

0.9957 0.9939 0.9921 0.9904 0.9887

35 36 37 38 39

0.9557 0.9536 0.9514 0.9493 0.9472

15 °C*

0.9446 0.9424 0.9401 0.9379 0.9356

0.9419 0.9399 0.9377 0.9355 0.9333

0.9338 0.9315 0.9292 0.9269 0.9246

70 71 72 73 74

0.8761 0.8738 0.8714 0.8691 0.8668

0.8639 0.8615 0.8592 0.8568 0.8545

0.86346 0.8611 0.8588 0.8564 0.8541

0.8584 0.8560 0.8537 0.8513 0.8489

0.8511 0.8487 0.8464 0.8440 0.8416

5 6 7 8 9

0.9916 0.9902 0.9890 0.9878 0.9866

0.9906 0.9892 0.9878 0.9864 0.9851

0.9904 0.9890 0.9875 0.9862 0.9849

0.9893 0.9877 0.9862 0.9847 0.9833

0.9871 0.9855 0.9839 0.9824 0.9809

40 41 42 43 44

0.9450 0.9428 0.9406 0.9384 0.9361

0.93333 0.9311 0.9288 0.9266 0.9243

0.9310 0.9287 0.9264 0.9239 0.9215

0.9224 0.9201 0.9177 0.9154 0.9130

75 76 77 78 79

0.8644 0.8621 0.8598 0.8575 0.8551

0.8521 0.8497 0.8474 0.8450 0.8426

0.8517 0.8493 0.8470 0.8446 0.8422

0.8464 0.8439 0.8415 0.8391 0.8366

0.8392 0.8368 0.8344 0.8321 0.8297

10 11 12 13 14

0.9856 0.9846 0.9838 0.9829 0.9821

0.9838 0.9826 0.9813 0.9802 0.9790

0.98362 0.9824 0.9812 0.9800 0.9788

0.9820 0.9808 0.9797 0.9876 0.9776

0.9794 0.9778 0.9764 0.9750 0.9735

45 46 47 48 49

0.9338 0.9315 0.9292 0.9270 0.9247

0.9220 0.9197 0.9174 0.9150 0.9127

0.9191 0.9165 0.9141 0.9117 0.9093

0.9106 0.9082 0.9059 0.9036 0.9013

80 81 82 83 84

0.8528 0.8503 0.8479 0.8456 0.8432

0.8403 0.8379 0.8355 0.8331 0.8307

0.83979 0.8374 0.8350 0.8326 0.8302

0.8342 0.8317 0.8292 0.8268 0.8243

0.8273 0.8248 0.8224 0.8200 0.8175

15 16 17 18 19

0.9814 0.9806 0.9799 0.9792 0.9784

0.9779 0.9768 0.9756 0.9745 0.9730

0.9777 0.9765 0.9753 0.9741 0.9728

0.9765 0.9754 0.9743 0.9731 0.9717

0.9720 0.9705 0.9690 0.9675 0.9658

50 51 52 53 54

0.9224 0.9201 0.9178 0.9155 0.9132

0.91043 0.9081 0.9058 0.9035 0.9011

0.9069 0.9044 0.9020 0.8996 0.8971

0.8990 0.8966 0.8943 0.8919 0.8895

85 86 87 88 89

0.8408 0.8384 0.8360 0.8336 0.8311

0.8282 0.8259 0.8234 0.8209 0.8184

0.8278 0.8254 0.8229 0.8205 0.8180

0.8219 0.8194 0.8169 0.8145 0.8120

0.8151 0.8127 0.8201 0.8078 0.8053

20 21 22 23 24

0.9777 0.9768 0.9759 0.9749 0.9739

0.9719 0.9704 0.9690 0.9675 0.9660

0.97158 0.9703 0.9689 0.9674 0.9659

0.9703 0.9688 0.9669 0.9651 0.9634

0.9642 0.9624 0.9606 0.9587 0.9569

55 56 57 58 59

0.9109 0.9086 0.9063 0.9040 0.9017

0.8988 0.8964 0.8940 0.8917 0.8893

0.8946 0.8921 0.8896 0.8874 0.8850

0.8871 0.8847 0.8823 0.8800 0.8777

90 91 92 93 94

0.8287 0.8262 0.8237 0.8212 0.8186

0.8161 0.8136 0.8110 0.8085 0.8060

0.81553 0.8130 0.8104 0.8079 0.8052

0.8096 0.8072 0.8047 0.8023 0.7998

0.8029 0.8004 0.7979 0.7954 0.7929

25 26 27 28 29

0.9727 0.9714 0.9699 0.9684 0.9669

0.9643 0.9626 0.9608 0.9590 0.9570

0.9642 0.9624 0.9605 0.9586 0.9568

0.9615 0.9597 0.9577 0.9558 0.9540

0.9549 0.9529 0.9509 0.9488 0.9467

60 61 62 63 64

0.8994 0.8970 0.8947 0.8924 0.8901

0.8829 0.8805 0.8781

0.88690 0.8845 0.8821 0.8798 0.8775

0.8825 0.8800 0.8776 0.8751 0.8727

0.8752 0.8728 0.8704 0.8680 0.8656

95 96 97 98 99

0.8160 0.8133 0.8106 0.8078 0.8048

0.8034 0.8008 0.7981 0.7954 0.7926

0.8026 0.7999 0.7972 0.7945 0.7918

0.7973 0.7949 0.7925 0.7901 0.7877

0.7904 0.7878 0.7852 0.7826 0.7799

30 31 32 33 34

0.9652 0.9634 0.9615 0.9596 0.9577

0.9551

0.95493 0.9530 0.9510 0.9489 0.9468

0.9520 0.9500 0.9481 0.9460 0.9440

0.9446 0.9426 0.9405 0.9383 0.9361

65 66 67 68 69

0.8878 0.8854 0.8831 0.8807 0.8784

0.8757 0.8733 0.8710 0.8686 0.8662

0.8752 0.8728 0.8705 0.8682 0.8658

0.8702 0.8679 0.8656 0.8632 0.8609

0.8631 0.8607 0.8583 0.8559 0.8535

100

0.8016

0.7896

0.78913

0.7854

0.7770

*Two different observers; see International Critical Tables, vol. 3, p. 120.

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-117

Glycerol*

Glycerol, % 15 °C

15.5 °C

20 °C

25 °C

30 °C

Density

Glycerol, % 15 °C

Density 15.5 °C

20 °C

25 °C

30 °C

Glycerol, % 15 °C

2-121

Density 20 °C

25 °C

30 °C

100 99 98 97 96

1.26415 1.26160 1.25900 1.25645 1.25385

1.26381 1.26125 1.25865 1.25610 1.25350

1.26108 1.25850 1.25590 1.25335 1.25080

1.15802 1.25545 1.25290 1.25030 1.24770

1.25495 1.25235 1.24975 1.24710 1.24450

65 64 63 62 61

1.17030 1.16755 1.16480 1.16200 1.15925

1.17000 1.16725 1.16445 1.16170 1.15895

1.16750 1.16475 1.16205 1.15930 1.15655

1.16475 1.16200 1.15925 1.15655 1.15380

1.16195 1.15925 1.15650 1.15375 1.15100

30 29 28 27 26

1.07455 1.07195 1.06935 1.06670 1.06410

1.07435 1.07175 1.06915 1.06655 1.06390

1.07270 1.07010 1.06755 1.06495 1.06240

1.07070 1.06815 1.06560 1.06305 1.06055

1.06855 1.06605 1.06355 1.06105 1.05855

95 94 93 92 91

1.25130 1.24865 1.24600 1.24340 1.24075

1.25095 1.24830 1.24565 1.24305 1.24040

1.24825 1.24560 1.24300 1.24035 1.23770

1.24515 1.24250 1.23985 1.23725 1.23460

1.24190 1.23930 1.23670 1.23410 1.23150

60 59 58 57 56

1.15650 1.15370 1.15095 1.14815 1.14535

1.15615 1.15340 1.15065 1.14785 1.14510

1.15380 1.15105 1.14830 1.14555 1.14280

1.15105 1.14835 1.14560 1.14285 1.14015

1.14830 1.14555 1.14285 1.14010 1.13740

25 24 23 22 21

1.06150 1.05885 1.05625 1.05365 1.05100

1.06130 1.05870 1.05610 1.05350 1.05090

1.05980 1.05720 1.05465 1.05205 1.04950

1.05800 1.05545 1.05290 1.05035 1.04780

1.05605 1.05350 1.05100 1.04850 1.04600

90 89 88 87 86

1.23810 1.23545 1.23280 1.23015 1.22750

1.23775 1.23510 1.23245 1.22980 1.22710

1.23510 1.23245 1.22975 1.22710 1.22445

1.23200 1.22935 1.22665 1.22400 1.22135

1.22890 1.22625 1.22360 1.22095 1.21830

55 54 53 52 51

1.14260 1.13980 1.13705 1.13425 1.13150

1.14230 1.13955 1.13680 1.13400 1.13125

1.14005 1.13730 1.13455 1.13180 1.12905

1.13740 1.13465 1.13195 1.12920 1.12650

1.13470 1.13195 1.12925 1.12650 1.12380

20 19 18 17 16

1.04840 1.04590 1.04335 1.04085 1.03835

1.04825 1.04575 1.04325 1.04075 1.03825

1.04690 1.04440 1.04195 1.03945 1.03695

1.04525 1.04280 1.04035 1.03790 1.03545

1.04350 1.04105 1.03860 1.03615 1.03370

85 84 83 82 81

1.22485 1.22220 1.21955 1.21690 1.21425

1.22445 1.22180 1.21915 1.21650 1.21385

1.22180 1.21915 1.21650 1.21380 1.21115

1.21870 1.21605 1.21340 1.21075 1.20810

1.21565 1.21300 1.21035 1.20770 1.20505

50 49 48 47 46

1.12870 1.12600 1.12325 1.12055 1.11780

1.12845 1.12575 1.12305 1.12030 1.11760

1.12630 1.12360 1.12090 1.11820 1.11550

1.12375 1.12110 1.11840 1.11575 1.11310

1.12110 1.11845 1.11580 1.11320 1.11055

15 14 13 12 11

1.03580 1.03330 1.03080 1.02830 1.02575

1.03570 1.03320 1.03070 1.02820 1.02565

1.03450 1.03200 1.02955 1.02705 1.02455

1.03300 1.03055 1.02805 1.02560 1.02315

1.03130 1.02885 1.02640 1.02395 1.02150

80 79 78 77 76

1.21160 1.20885 1.20610 1.20335 1.20060

1.21120 1.20845 1.20570 1.20300 1.20025

1.20850 1.20575 1.20305 1.20030 1.19760

1.20545 1.20275 1.20005 1.19735 1.19465

1.20240 1.19970 1.19705 1.19435 1.19170

45 44 43 42 41

1.11510 1.11235 1.10960 1.10690 1.10415

1.11490 1.11215 1.10945 1.10670 1.10400

1.11280 1.11010 1.10740 1.10470 1.10200

1.11040 1.10775 1.10510 1.10240 1.09975

1.10795 1.10530 1.10265 1.10005 1.09740

10 9 8 7 6

1.02325 1.02085 1.01840 1.01600 1.01360

1.02315 1.02075 1.01835 1.01590 1.01350

1.02210 1.01970 1.01730 1.01495 1.01255

1.02070 1.01835 1.01600 1.01360 1.01125

1.01905 1.01670 1.01440 1.01205 1.00970

75 74 73 72 71

1.19785 1.19510 1.19235 1.18965 1.18690

1.19750 1.19480 1.19205 1.18930 1.18655

1.19485 1.19215 1.18940 1.18670 1.18395

1.19195 1.18925 1.18650 1.18380 1.18110

1.18900 1.18635 1.18365 1.18100 1.17830

40 39 38 37 36

1.10145 1.09875 1.09605 1.09340 1.09070

1.10130 1.09860 1.09590 1.09320 1.09050

1.09930 1.09665 1.09400 1.09135 1.08865

1.09710 1.09445 1.09180 1.08915 1.08655

1.09475 1.09215 1.08955 1.08690 1.08430

5 4 3 2 1

1.01120 1.00875 1.00635 1.00395 1.00155

1.01110 1.00870 1.00630 1.00385 1.00145

1.01015 1.00780 1.00540 1.00300 1.00060

1.00890 1.00655 1.00415 1.00180 0.99945

1.00735 1.00505 1.00270 1.00035 0.99800

70 69 68 67 66

1.18415 1.18135 1.17860 1.17585 1.17305

1.18385 1.18105 1.17830 1.17555 1.17275

1.18125 1.17850 1.17575 1.17300 1.17025

1.17840 1.17565 1.17295 1.17020 1.16745

1.17565 1.17290 1.17020 1.16745 1.16470

35 34 33 32 31

1.08800 1.08530 1.08265 1.07995 1.07725

1.08780 1.08515 1.08245 1.07975 1.07705

1.08600 1.08335 1.08070 1.07800 1.07535

1.08390 1.08125 1.07860 1.07600 1.07335

1.08165 1.07905 1.07645 1.07380 1.07120

0

0.99913 0.99905 0.99823 0.99708 0.99568

*Bosart and Snoddy, Ind. Eng. Chem., 20, (1928): 1378.

TABLE 2-118

Hydrazine (N2H4)*

%

d 15 4

%

d 15 4

1 2 4 8 12 16 20 24 28

1.0002 1.0013 1.0034 1.0077 1.0121 1.0164 1.0207 1.0248 1.0286

30 40 50 60 70 80 90 100

1.0305 1.038 1.044 1.047 1.046 1.040 1.030 1.011

*International Critical Tables, vol. 3, p. 55.

15.5 °C

2-122

PHYSICAL AND CHEMICAL DATA

TABLE 2-119

Densities of Aqueous Solutions of Miscellaneous Organic Compounds*

d, dw, and ds are the density of the solution, pure water, and pure liquid solute, respectively, all in g/mL. ps is the wt % solute. 0.03255 means 2.55 × 10−4. Section A Name

Formula

t, °C

Acetaldehyde Acetamide

C2H4O C2H5NO

Acetone

C3H6O

Acetonitrile Allyl alcohol Benzenepentacarboxylic acid Butyl alcohol (n-)

C2H3N C3H6O C11H6O10 C4H10O

Butyric acid (n-)

C4H8O2

Chloral hydrate

C2H3Cl3O2

Chloroacetic acid

C2H3ClO2

Citric acid (hydrate)

C6H3O7 + H2O

Dichloroacetic acid

C2H2Cl2O2

Diethylamine hydrochloride Ethylamine hydrochloride

C4H12ClN C2H8ClN

Ethylene glycol

C2H6O2

Ethyl ether tartrate Formaldehyde Formamide

C4H10O C8H14O6 CH2O CH3NO

Furfural Isoamyl alcohol

C5H4O2 C5H12O

Isobutyl alcohol

C4H10O



Isobutyric acid

C4H8O2



Isovaleric acid Lactic acid Maleic acid

C5H10O2 C3H6O C4H4O4

Malic acid

C4H6O5

Malonic acid Methyl acetate

C3H4O4 C3H6O2

glucoside (α-)

C7H14O6

Nicotine Nitrophenol (p-)

C10H14N2 C6H5NO3

Oxalic acid

C2H2O4

Phenol

C6H6O

Phenylglycolic acid Picoline (α-) (β-)

C8H8O3 C6H7N C6H7N

Propionic acid

C3H6O2

Pyridine Resorcinol Succinic acid

C5H5N C6H6O2 C4H6O4

Tartaric acid (d, l, or dl)

C4H6O6

*From International Critical Tables, vol. 3, pp. 111–114.

 



    

 



 



18 15 0 4 15 20 25 15 0 25 20 18 25 0 15 30 20 25 18 20 25 21 21 0 15 20 25 15 15 25 20 25 20 15 20 15 18 25 25 25 25 20 25 20 20 0 30 20 15 0 15 17.5 20 25 15 80 25 25 25 18 25 25 18 25 15 17.5 20 30 40 50 60

d = d w + Aps + Bp s2 + Cp3s Range, ps

A

B

0– 30 0– 6 0–100 0–100 0–100 0–100 0–100 0– 16 0– 89 0– 0.6 0– 7.9 0– 10 0– 62 0– 70 0– 78 0– 90 0– 32 0– 86 0– 50 0– 30 0– 97 0– 36 0– 65 0–100 0– 6 0– 5 0– 4.5 0– 95 0– 40 22– 96 0– 8 0– 8 0– 2.5 0– 8 0– 8 0– 9 0– 9 0– 12 0– 5 0– 9 0– 40 0– 40 0– 40 0– 40 0– 20 26– 51 26– 51 0– 60 0– 1.5 0– 4 0– 4 0– 9 0– 4 0– 4 0– 5 0– 65 0– 11 0– 70 0– 60 0– 10 0– 40 0– 60 0– 52 0– 5.5 0– 15 0– 50 0– 50 0– 50 0– 50 0– 50 0– 50

+0.03255 +0.03639 −0.03856 −0.027648 −0.021009 −0.021233 −0.021171 −0.021175 −0.033729 +0.025615 −0.021651 +0.03414 +0.035135 +0.024489 +0.024455 +0.024401 +0.023648 +0.023602 +0.023824 +0.024427 +0.024427 +0.0334 +0.021193 +0.021483 +0.02133 −0.02221 −0.02221 +0.022367 +0.022518 +0.021217 +0.021827 +0.021664 +0.02155 −0.02146 −0.02169 +0.0352 +0.0345 +0.0337 +0.03253 +0.02231 +0.0234 +0.023933 +0.023736 +0.02389 +0.0340 +0.023336 +0.023151 +0.03642 +0.023216 +0.025898 +0.02494 +0.02494 +0.025264 +0.025108 +0.02111 +0.03462 +0.02207 −0.04386 −0.04683 +0.0395 +0.039245 +0.03229 +0.02201 +0.02304 +0.024482 +0.024455 +0.024432 +0.024335 +0.024265 +0.024205 +0.024155

−0.0516 +0.04171 −0.05449 −0.041193 −0.059682 −0.053529 −0.05904 −0.042024 −0.041232 −0.02117 +0.04285 +0.04131 −0.04166 +0.042802 +0.042198 +0.041887 +0.05302 +0.05552 +0.041141 +0.05537 +0.05537 +0.0676 −0.05307 +0.052992 −0.05108 +0.0448 +0.0435 +0.05358 −0.05658 +0.053199 +0.05366 +0.0421 +0.043 +0.056 +0.0438

−0.04282 +0.05186 +0.0575 +0.05957 +0.04175 +0.041066 −0.0574 +0.05996 +0.05975 +0.05454 −0.0455 −0.033185 −0.058 −0.058 −0.031996 −0.031607 −0.04283 −0.0686 +0.0423 −0.051405 −0.0513 −0.04172 −0.0599 −0.05204 +0.05519 +0.04185 +0.04185 +0.041837 +0.04185 +0.04185 +0.04185 +0.04185

C

−0.07588 +0.08272 −0.08624 −0.075327 −0.0856 +0.072984

+0.0611 −0.071291 +0.074366 +0.076549 +0.0722 +0.0717 +0.077534 +0.077534 −0.0747 −0.075248

−0.076005 +0.06542 −0.072529

+0.01544 +0.08978 −0.07687 +0.0441 +0.04254 +0.04208

−0.074167 +0.07361 −0.0828 −0.0819

DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-119

Densities of Aqueous Solutions of Miscellaneous Organic Compounds (Concluded) d = d w + Aps + Bp s2 + Cp3s (Cont.)

Section A Name

t, °C

Formula

Tetraethyl ammonium chloride Thiourea

C8H20ClN CH4N2S

Trichloroacetic acid

C2HCl3O2

Triethylamine hydrochloride

C6H16ClN

Trimethyl carbinol

C4H10O

Urea

CH4N2O

Urethane Valeric acid (n-)

C3H7NO2 C5H10O2

21 15 12.5 20 25 21 20 25 14.8 18 20 25 20 25

 



Section B Name

Formula

Butyl alcohol (n-) Butyric acid (n-) Ethyl ether

C4H10O C4H8O2 C4H10O

Isobutyl alcohol

C4H10O

Isobutyric acid Nicotine Picoline (α-) (β-) Pyridine Trimethyl carbinol

C4H8O2 C10H14N2 C6H7N C6H7N C5H5N C4H10O



C3H6O C4H10O

Chloral hydrate

C2H3Cl3O2

Ethyl tartrate

C7H14O6

Furfural

C5H4O2

Pyridine

C5H5N

A

B

C

0–63 0–7 0–61 10–30 0–94 0–54 0–100 0–100 0–12 0–51 0–35 0–10 0–56 0–3

+0.031884 +0.022995 +0.02499 +0.025053 +0.025051 +0.046 −0.02117 −0.021286 +0.023213 +0.022718 +0.022702 +0.022728 +0.021278 +0.0334

+0.056 +0.05374 +0.04153 +0.041387 +0.056119 +0.05558 −0.041908 −0.04176 −0.044802 +0.051552 +0.053712 −0.041817 −0.05245 −0.0427

+0.07122

t, °C

Range, pw

A

B

20 25 25 0 15 26 20 25 25 25 20

0–20 0–38 0– 1.1 0–14 0–16 0–80 0–40 0–30 0–40 0–40 0–20

+0.022103 +0.021854 +0.0234 +0.022437 +0.02224 +0.021808 +0.02199 +0.022715 +0.021925 +0.021157 +0.022287

−0.04113 −0.042314 +0.0336 −0.04285 −0.04129 −0.042358 −0.04331 −0.04393 −0.04352 −0.05536 +0.05275



+0.061038 −0.0869 +0.07957 +0.07887 +0.051216 +0.072573 −0.072285 +0.051379 −0.073437

d = ds + Apw + Bp 2w + Cp 3w

ds

C

+0.061253 +0.07315 +0.0625 −0.062

dt = do + At + Bt2

ps

do

Range, °C

A

B

76.60 80.95 2.00 10.00 5.00 10.00 25.00 4.62 5.69 6.56 9.34 21.20 29.50 40.40

0.9122 0.8614 1.0094 1.0476 1.0150 1.0270 1.0665 1.0125 1.0140 1.0155 1.0055 1.0115 1.0145 1.0182

0–45 0–43 7–80 7–80 15–80 15–80 15–80 22–74 22–74 22–74 11–73 14–73 12–72 9–74

−0.038 −0.037292 −0.042597 −0.047955 −0.032103 −0.032116 −0.03401 −0.03232 −0.03221 −0.03211 −0.03171 −0.03378 −0.03463 −0.03605

−0.0527 −0.0675 −0.054313 −0.054253 −0.052544 −0.062929 −0.0523 −0.05254 −0.05268 −0.05290 −0.053615 −0.05248 −0.05235 −0.05167

Formula

Allyl alcohol Butyl alcohol (n-)

Range, ps

0.8097 0.9534 0.7077 0.8170 0.8055 0.9425 1.0093 0.9404 0.9515 0.9776 0.7856 Section C

Name

2-123

 



2-124

PHYSICAL AND CHEMICAL DATA

DENSITIES OF MISCELLANEOUS MATERIALS TABLE 2-120

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids*

Water at 4 °C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Sp. gr.

Aver. density lb/ft 3

Sp. gr.

Aver. density lb/ft 3

2.55–2.80 7.7 8.4–8.7 7.4–8.9 8.88

165 481 534 509 554

Timber, Air-dry Apple Ash, black white Birch, sweet, yellow Cedar, white, red

0.66–0.74 0.55 0.64–0.71 0.71–0.72 0.35

44 34 42 44 22

Copper, cast-rolled ore, pyrites German silver Gold, cast-hammered coin (U.S.)

8.8–8.95 4.1–4.3 8.58 19.25–19.35 17.18–17.2

556 262 536 1205 1073

Cherry, wild red Chestnut Cypress Elm, white Fir, Douglas

0.43 0.48 0.45–0.48 0.56 0.48–0.55

27 30 29 35 32

Iridium Iron, gray cast cast, pig wrought spiegeleisen

21.78–22.42 7.03–7.13 7.2 7.6–7.9 7.5

1383 442 450 485 468

balsam Hemlock Hickory Locust Mahogany

0.40 0.45–0.50 0.74–0.80 0.67–0.77 0.56–0.85

25 29 48 45 44

ferro-silicon ore, hematite ore, limonite ore, magnetite slag

6.7–7.3 5.2 3.6–4.0 4.9–5.2 2.5–3.0

437 325 237 315 172

Maple, sugar white Oak, chestnut live red, black

0.68 0.53 0.74 0.87 0.64–0.71

43 33 46 54 42

Lead ore, galena Manganese ore, pyrolusite Mercury

11.34 7.3–7.6 7.42 3.7–4.6 13.6

710 465 475 259 849

white Pine, Norway Oregon red Southern white

0.77 0.55 0.51 0.48 0.61–0.67 0.43

48 34 32 30 38–42 27

8.97 8.9 21.5 10.4–10.6 7.83 7.80 7.70–7.73 7.2–7.5 6.4–7.0 19.22

555 537 1330 656 489 487 481 459 418 1200

Poplar Redwood, California Spruce, white, red Teak, African Indian Walnut, black Willow

0.43 0.42 0.45 0.99 0.66–0.88 0.59 0.42–0.50

27 26 28 62 48 37 28

6.9–7.2 3.9–4.2

440 253

Various Solids Cereals, oats, bulk barley, bulk corn, rye, bulk wheat, bulk Cork

Various Liquids Alcohol, ethyl (100%) methyl (100%) Acid, muriatic, 40% nitric, 91% sulfuric, 87%

0.789 0.796 1.20 1.50 1.80

49 50 75 94 112

0.51 0.62 0.73 0.77 0.22–0.26

26 39 45 48 15

Chloroform Ether Lye, soda, 66% Oils, vegetable mineral, lubricants

1.500 0.736 1.70 0.91–0.94 0.88–0.94

95 46 106 58 57

Cotton, flax, hemp Fats Flour, loose pressed Glass, common

1.47–1.50 0.90–0.97 0.40–0.50 0.70–0.80 2.40–2.80

93 58 28 47 162

0.861–0.867 1.0 0.9584 0.88–0.92 0.125

54 62.428 59.830 56 8

plate or crown crystal dint Hay and straw, bales Leather

2.45–2.72 2.90–3.00 3.2–4.7 0.32 0.86–1.02

161 184 247 20 59

Paper Potatoes, piled Rubber, caoutchouc goods Salt, granulated, piled

0.70–1.15 0.67 0.92–0.96 1.0–2.0 0.77

58 44 59 94 48

Saltpeter Starch Sulfur Wool

1.07 1.53 1.93–2.07 1.32

67 96 125 82

Substance Metals, Alloys, Ores Aluminum, cast-hammered bronze Brass, cast-rolled Bronze, 7.9 to 14% Sn phosphor

Monel metal, rolled Nickel Platinum, cast-hammered Silver, cast-hammered Steel, cold-drawn machine tool Tin, cast-hammered cassiterite Tungsten Zinc, cast-rolled blende

Substance

Turpentine Water, 4°C max. density 100°C ice snow, fresh fallen sea water

1.02–1.03

64

Ashlar Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.3–2.6 2.4–2.7 2.1–2.8 2.4–2.8 2.0–2.6

153 159 153 162 143

Rubble Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone

2.2–2.5 2.3–2.6 2.0–2.7 2.3–2.7 1.9–2.5

147 153 147 156 137

*From Marks’ Standard Handbook for Mechanical Engineers, 10th ed., McGraw-Hill, 1996.

Sp. gr.

Aver. density lb/ft 3

Dry Rubble Masonry Granite, syenite, gneiss Limestone, marble Sandstone, bluestone

1.9–2.3 1.9–2.1 1.8–1.9

130 125 110

Brick Masonry Hard brick Medium brick Soft brick Sand-lime brick

1.8–2.3 1.6–2.0 1.4–1.9 1.4–2.2

128 112 103 112

Concrete Masonry Cement, stone, sand slag, etc. cinder, etc.

2.2–2.4 1.9–2.3 1.5–1.7

144 130 100

0.64–0.72 1.5 0.85–1.00 1.4–1.9 2.08–2.25

40–45 94 53–64 103 94–135

Portland cement Slags, bank slag bank screenings machine slag slag sand

3.1–3.2 1.1–1.2 1.5–1.9 1.5 0.8–0.9

196 67–72 98–117 96 49–55

Earth, etc., Excavated Clay, dry damp plastic and gravel, dry Earth, dry, loose dry, packed moist, loose moist, packed mud, flowing mud, packed Riprap, limestone

1.0 1.76 1.6 1.2 1.5 1.3 1.6 1.7 1.8 1.3–1.4

63 110 100 76 95 78 96 108 115 80–85

1.4 1.7 1.4–1.7 1.6–1.9 1.89–2.16

90 105 90–105 100–120 126

1.28 1.44 0.96 1.00 1.12 1.00

80 90 60 65 70 65

Asbestos Barytes Basait Bauxite Bluestone

2.1–2.8 4.50 2.7–3.2 2.55 2.5–2.6

153 281 184 159 159

Borax Chalk Clay, marl Dolomite Feldspar, orthoclase

1.7–1.8 1.8–2.8 1.8–2.6 2.9 2.5–2.7

109 143 137 181 162

Gneiss Granite Greenstone, trap Gypsum, alabaster Hornblende Limestone Marble Magnesite Phosphate rock, apatite Porphyry

2.7–2.9 2.6–2.7 2.8–3.2 2.3–2.8 3.0 2.1–2.86 2.6–2.86 3.0 3.2 2.6–2.9

175 165 187 159 187 155 170 187 200 172

Substance

Various Building Materials Ashes, cinders Cement, Portland, loose Lime, gypsum, loose Mortar, lime, set Portland cement

Riprap, sandstone Riprap, shale Sand, gravel, dry, loose gravel, dry, packed gravel, wet Excavations in Water Clay River mud Sand or gravel and clay Soil Stone riprap Minerals

SOLUBILITIES TABLE 2-120

2-125

Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids (Concluded)

Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.

Substance

Substance

0.37–0.90 2.5–2.8 2.0–2.6 2.7–2.8 2.6–2.9

40 165 143 171 172

Bituminous Substances Asphaltum Coal, anthracite bituminous lignite peat, turf, dry

1.1–1.5 1.4–1.8 1.2–1.5 1.1–1.4 0.65–0.85

81 97 84 78 47

2.6–2.8 2.6–2.7

169 165

1.5 1.7 1.5 1.3 1.5

96 107 95 82 92

charcoal, pine charcoal, oak coke Graphite Paraffin

0.28–0.44 0.47–0.57 1.0–1.4 1.64–2.7 0.87–0.91

23 33 75 135 56

Sp. gr.

Minerals (Cont.) Pumice, natural Quartz, flint Sandstone Serpentine Shale, slate Soapstone, talc Syenite Stone, Quarried, Piled Basalt, granite, gneiss Greenstone, hornblende Limestone, marble, quartz Sandstone Shale NOTE:

Aver. density lb/ft3

Sp. gr.

Aver. density lb/ft 3

Substance

Sp. gr.

Aver. density lb/ft3

Bituminous Substances (Cont.) Petroleum refined (kerosene) benzine gasoline Pitch Tar, bituminous

0.87 0.78–0.82 0.73–0.75 0.70–0.75 1.07–1.15 1.20

54 50 46 45 69 75

Coal and Coke, Piled Coal, anthracite bituminous, lignite peat, turf charcoal coke

0.75–0.93 0.64–0.87 0.32–0.42 0.16–0.23 0.37–0.51

47–58 40–54 20–26 10–14 23–32

To convert pounds per cubic foot to kilograms per cubic meter, multiply by 16.02. °F = 9⁄5 °C + 32.

TABLE 2-121

Density (kg/m3) of Selected Elements as a Function of Temperature Element symbol

Temperature, K*

Al

Be†

Cr

Cu

Au

Ir

Fe

Pb

Mo

Ni

Pt

Ag

Zn†

50 100 150 200 250

2736 2732 2726 2719 2710

3650 3640 3630 3620 3610

7160 7155 7150 7145 7140

9019 9009 8992 8973 8951

19,490 19,460 19,420 19,380 19,340

22,600 22,580 22,560 22,540 22,520

7910 7900 7890 7880 7870

11,570 11,520 11,470 11,430 11,380

10,260 10,260 10,250 10,250 10,250

8960 8950 8940 8930 8910

21,570 21,550 21,530 21,500 21,470

10,620 10,600 10,575 10,550 10,520

7280 7260 7230 7200 7170

300 400 500 600 800

2701 2681 2661 2639 2591

3600 3580 3555 3530

7135 7120 7110 7080 7040

8930 8885 8837 8787 8686

19,300 19,210 19,130 19,040 18,860

22,500 22,450 22,410 22,360 22,250

7860 7830 7800 7760 7690

11,330 11,230 11,130 11,010 10,430

10,240 10,220 10,210 10,190 10,160

8900 8860 8820 8780 8690

21,450 21,380 21,330 21,270 21,140

10,490 10,430 10,360 10,300 10,160

7135 7070 7000 6935 6430

1000 1200 1400 1600 1800

2365 2305 2255

7000 6945 6890 6760 6700

8568 8458 7920 7750 7600

18,660 18,440 17,230 16,950

22,140 22,030 21,920 21,790 21,660

7650 7620 7520 7420 7320

10,190 9,940

10,120 10,080 10,040 10,000 9,950

8610 8510 8410 8320 7690

21,010 20,870 20,720 20,570 20,400

10,010 9,850 9,170 8,980

6260

21,510

7030

9,900

7450

20,220

2000

7460

NOTE:

Above the horizontal line the condensed phase is solid; below the line, it is liquid. *°R = 9⁄ 5 K. †Polycrystalline form tabulated. Similar tables for an additional 45 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984.

SOLUBILITIES UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert cubic centimeters to cubic feet, multiply by 3.532 × 10−5. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.

To convert grams per liter to pounds per cubic foot, multiply by 6.243 × 10−2. A database containing solubilities originally published in the International Union for Pure and Applied Chemistry (IUPAC)-National Institute of Standards and Technology (NIST) Solubility Data Series is now available at no cost online at http://srdata.nist.gov/solubility.

2-126

TABLE 2-122

Solubilities of Inorganic Compounds in Water at Various Temperatures*

This table shows the grams of anhydrous substance that are soluble in 100 g of water at the temperature in degrees Celsius as indicated; when the name is followed by †, the value is expressed in grams of substance in 100 cm3 of saturated solution. Solid phase gives the hydrated form in equilibrium with the saturated solution. Solid phase

Substance

Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Aluminum chloride sulfate Ammonium aluminum sulfate bicarbonate bromide chloride chloroplatinate chromate chromium sulfate dichromate dihydrogen phosphite hydrogen phosphate iodide magnesium phosphate manganese phosphate nitrate oxalate perchlorate† persulfate sulfate thiocyanate vanadate (meta) Antimonious fluoride sulfide Arsenic oxide Arsenious sulfide

AlCl3 Al2(SO4)3 (NH4)2Al2(SO4)4 NH4HCO3 NH4Br NH4Cl (NH4)2PtCl6 (NH4)2CrO4 (NH4)2Cr2(SO4)4 (NH4)2Cr2O7 NH4H2PO3 (NH4)2HPO4 NH4I NH4MgPO4 NH4MnPO4 NH4NO3 (NH4)2C2O4 NH4ClO4† (NH4)2S2O8 (NH4)2SO4 NH4CNS NH4VO3 SbF3 Sb2S3 As2O5 As2S3

27 28 29

Barium acetate acetate carbonate

Ba(C2H3O2)2 Ba(C2H3O2)2 BaCO3

3H2O 1H2O

30 31 32 33 34 35 36 37 38

chlorate chloride chromate hydroxide iodide iodide nitrate nitrite oxalate

Ba(ClO3)2 BaCl2 BaCrO4 Ba(OH)2 BaI2 BaI2 Ba(NO3)2 Ba(NO2)2 BaC2O4

1H2O 2H2O

3H2O

39 40 41 42 43 44 45 46 47 48 49 50 51

perchlorate sulfate Beryllium sulfate sulfate sulfate Boric acid Boron oxide Bromine Cadmium chloride chloride chloride cyanide hydroxide

Ba(ClO4)2 BaSO4 BeSO4 BeSO4 BeSO4 H3BO3 B2O3 Br2 CdCl2 CdCl2 CdCl2 Cd(CN)2 Cd(OH)2

52 53 54

sulfate Calcium acetate acetate

CdSO4 Ca(C2H3O2)2 Ca(C2H3O2)2

6H2O 18H2O 24H2O

0 °C 31.2 2.1 11.9 60.6 29.4

10 °C 33.5 4.99 15.8 68 33.3 0.7

10.7825°

24II2O 171 6H2O 7H2O 1H2O

154.2 0.023 118.3 2.2 11.56 58.2 70.6 119.8

163.2

3.1 73.0 144

384.7

8H2O 6H2O 2H2O

20 °C 69.8615° 36.4 7.74 21 75.5 37.2

59.5 5.17 × 10−5 at 18° 59

62.1 63 0.00168°

20.34 31.6 0.0002 1.67 170.2 5.0

1H2O

26.95 33.3 0.00028 2.48 185.7 7.0 0.00168°

205.8 1.15 × 10−4

2.0 × 10−4

19014.5° 13115 172.3 0.052 0 192 4.4 20.85 75.4 170 0.48 444.7 0.00017518° 65.8

4H2O 2aH2O 1H2O

2H2O 1H2O

76.48 37.4

40.4 10.94 27 83.2 41.4

40 °C

50 °C

60 °C

70 °C

80 °C

90 °C

100 °C

46.1 14.88

52.2 20.10

59.2 26.70

66.1

73.0

80.8

89.0 109.796°

91.1 45.8

99.2 50.4

107.8 55.2

116.8 60.2

126 65.6

135.6 71.3

145.6 77.3 1.25

190.5 0.036 0 297.0 8.0 30.58

199.6 0.030

208.9 0.040 0 421.0

218.7 0.016 0.005 499.0

228.8 0.019 0.007 580.0

40.4 47.17 26031° 181.4 241.8 5.9 78.0 207.7 0.84 563.6

81.0

69.5

71.2

1.32

344.0 10.3

39.05

740.0

48.19

88.0 1.78

250.3 871.0 57.01

95.3

103.3

75.1

76.7

3.05 73.0

71 0.002218° 33.80 35.7 0.00037 3.89 203.1 9.2 67.5 0.002218° 289.1 2.4 × 10−4

6H2O 4H2O 2H2O 2.66 1.1 4.22 97.59 90.01

30 °C

3.57 1.5 3.4 125.1

5.04 2.2 3.20

135.1

134.5 1.715°

75 0.0024 at 24.2° 41.70 38.2 0.00046 5.59 219.6 11.6

79

77

74

74

49.61 40.7

43.6

66.81 46.4

49.4

8.22

13.12

231.9 14.2

17.1

358.7

426.3

20.94

75 84.84 52.4

104.9 58.8

101.4

247.3 20.3

261.0 27.0 205.8

271.7 34.2 300

0.0024 at 24.2° 2.85 × 10−4 52 43.78 6.60

495.2

562.3

60.67 46.74 8.72 4.0

62 11.54

14.81 6.2

16.73

84.76 23.75 9.5

83 98 30.38

100 110 40.25 15.7

3.13 132.1

76.00 36.0

76.60 34.7

135.3

136.5

140.4

147.0

2.6 × 10−4 at 25° 33.8

78.54 33.2

83.68 32.7

63.13

60.77

31.1

29.7

33.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

1 2 3 4 5 6 7 8 9 10 11

Calcium bicarbonate chloride chloride fluoride hydroxide nitrate nitrate nitrate nitrite nitrite oxalate

Ca(HCO3)2 CaCl2 CaCl2 CaF2 Ca(OH)2 Ca(NO3)2 Ca(NO3)2 Ca(NO3)2 Ca(NO2)2 Ca(NO2)2 CaC2O4

12 13 14 15 16 17 18 19 20 21 22 23 24

sulfate Carbon dioxide, 760 mm ‡ monoxide, 760 mm ‡ Cesium chloride nitrate sulfate Chlorine, 760 mm ‡ Chromic anhydride Cuprio chloride nitrate nitrate sulfate sulfide

CaSO4 CO2 CO CsCl CsNO3 Cs2SO4 Cl2 CrO3 CuCl2 Cu(NO3)2 Cu(NO3)2 CuSO4 CuS

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

Cuprous chloride Ferric chloride Ferrous chloride chloride nitrate sulfate sulfate Hydrobromic acid, 760 mm Hydrochloric acid, 760 mm Iodine Lead acetate bromide carbonate chloride chromate fluoride nitrate sulfate Magnesium bromide chloride hydroxide nitrate sulfate sulfate sulfate Manganous sulfate sulfate sulfate sulfate Mercurous chloride Molybdic oxide Nickel chloride nitrate nitrate sulfate sulfate Nitric oxide, 760 mm Nitrous oxide

CuCl FeCl3 FeCl2 FeCl2 Fe(NO3)2 FeSO4 FeSO4 HBr HCl I2 Pb(C2H3O2)2 PbBr2 PbCO3 PbCl2 PbCrO4 PbF2 Pb(NO3)2 PbSO4 MgBr2 MgCl2 Mg(OH)2 Mg(NO3)2 MgSO4 MgSO4 MgSO4 MnSO4 MnSO4 MnSO4 MnSO4 HgCl MoO3 NiCl2 Ni(NO3)2 Ni(NO3)2 NiSO4 NiSO4 NO N2O

6H2O 2H2O 4H2O 3H2O

16.15 59.5 0.185 102.0

65.0 0.176 115.3

16.60 74.5 0.001618° 0.165 129.3

17.05

17.50

17.95

18.40

102 0.001726° 0.153 152.6

136.8 0.141 195.9 237.5

0.128

0.116

141.7 0.106

147.0 0.094

152.7 0.085

62.07

2H2O

0.1759 0.3346 0.0044 161.4 9.33 167.1 1.46 164.9 70.7 81.8

2H2O 6H2O 3H2O 5H2O

363.6

76.68 6.7 × 10−4 at 13° 0.1928 0.2318 0.0035 174.7 14.9 173.1 0.980 73.76 95.28

14.3

17.4

74.4

81.9 64.5

4H2O

6.8 × 10−4 at 25° 0.1688 0.0028 186.5 23.0 178.7 0.716 77.0 125.1 20.7 3.3 × 10−5 at 18° 1.5225° 91.8

9.5 × 10−4 at 50° 0.2090 0.1257 0.0024 197.3 33.9 184.1 0.562

132.6

80.34

14 × 10−4 at 95° 0.2097 0.0973 0.0021 208.0 47.2 189.9 0.451 174.0 83.8

25

159.8 28.5

33.3

77.3

315.1 82.5

73.0

0.0761 0.0018 218.5 64.4 194.9 0.386 182.1 87.44

0.2047 0.0576 0.0015 229.7 83.8 199.9 0.324

151.9

244.8

0.1966 0.0013 239.5 107.0 205.0 0.274

0.0010 250.0 134.0 210.3 0.219

91.2

99.2

178.8 40

207.8 55

88.7

525.8 100

0.0006 260.1 163.0 214.9 0.125 217.5

71.02 15.65

20.51

83.8 26.5

535.7

210.3

3H2O 0.4554 0.6728

6H2O 6H2O 6H2O 7H2O 6H2O 1H2O 7H2O 5H2O 4H2O 1H2O

38.8 0.0028 91.0 52.8

0.060 48.3 0.0035 94.5 53.5

40.2

67.3 0.04 55.0425° 1.15

63.3 0.056

171.5 59.6 0.078

0.85 0.00011 0.99 7 × 10−6 0.064 56.5 0.0041 96.5 54.5 0.000918°

1.53

1.94

2.36

3.34

4.75

1.20

1.45

1.70

1.98

2.62

3.34

0.068 66 0.0049 99.2

66.55 40.8 53.23

30.9 42.2 60.01 59.5

0.00014 2H2O 6H2O 6H2O 3H2O 7H2O 6H2O

32.9

198 0.029

53.9 79.58

59.5

35.5 44.5

40.8 45.3

62.9 64.5

67.76 66.44

0.0002 0.138 64.2 96.31

48.6

0.264 68.9

75 0.0056 101.6 57.5

0.00984

32 0.00757 0.1705

43.6

130

85

95

104.1

107.5 61.0

115

0.0007 0.476 73.3 122.2

38.8

113.7 66.0

84.74 45.6

68.8

37.3

56.1

120.2 73.0 137.0

50.4

53.5

59.5

64.2 62.9

69.0

74.0 68.3

72.6 58.17

55.0

52.0

48.0

42.5

34.0

0.687 78.3

1.206 82.2

2.055 85.2

163.1 27.22

105.8

165.6 50.9

221.2 82.3

0.1619 0 0 270.5 197.0 220.3 0 206.8 107.9 75.4

105.3 6H2O 7H2O 1H2O

0.077

281.5 358.7

4H2O 2H2O

159

2.106 87.6

169.1

235.1

42.46 0.00618 0.1211

0.00517

0.00440

50.15 0.00376

54.80 0.00324

59.44 0.00267

63.17 0.00199

0.00114

76.7 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

2-127

*By N. A. Lange; abridged from “Table of Solubilities of Inorganic Compounds in Water at Various Temperatures” in Lange’s Handbook of Chemistry, 10th ed., McGraw-Hill, New York, 1961 (except for NaCl, which is from CRC Handbook of Chemistry and Physics, 86th ed., CRC Press, 2005). For tables of the solubility of gases in water at various temperatures, Atack (Handbook of Chemical Data, Reinhold, New York, 1957) gives values at closer temperature intervals, usually 1 or 5 °C, than are tabulated here. For materials marked by ‡, additional data are given in tables subsequent to this one. For the solubility of various hydrocarbons in water at high pressures see J. Chem. Eng. Data, 4, 212 (1959).

2-128

TABLE 2-122

Solubilities of Inorganic Compounds in Water at Various Temperatures (Continued)

Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Potassium acetate acetate alum bicarbonate bisulfate bitartrate carbonate chlorate chloride chromate dichromate ferricyanide hydroxide hydroxide nitrate nitrite perchlorate permanganate persulfate† sulfate thiocyanate Silver cyanide nitrate sulfate Sodium acetate acetate bicarbonate carbonate carbonate chlorate chloride chromate chromate chromate dichromate dichromate dihydrogen phosphate dihydrogen phosphate dihydrogen phosphate hydrogen arsenate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydroxide hydroxide hydroxide hydroxide nitrate nitrite oxalate phosphate, tripyrophosphate sulfate sulfate sulfate sulfide sulfide sulfide sulfite sulfite tetraborate tetraborate vanadate (meta)

Formula KC2H3O2 KC2H3O2 K2SO4⋅Al2(SO4)3 KHCO3 KHSO4 KHC4H4O6 K2CO3 KClO3 KCl K2CrO4 K2Cr2O7 K3Fe(CN)6 KOH KOH KNO3 KNO2 KClO4 KMnO4 K2S2O8† K2SO4 KCNS AgCN AgNO3 Ag2SO4 NaC2H3O2 NaC2H3O2 NaHCO3 Na2CO3 Na2CO3 NaClO8 NaCl Na2CrO4 Na2CrO4 Na2CrO4 Na2Cr2O7 Na2Cr2O7 NaH2PO4 NaH2PO4 NaH2PO4 Na2HAsO4 Na2HPO4 Na2HPO4 Na2HPO4 Na2HPO4 NaOH NaOH NaOH NaOH NaNO3 NaNO2 Na2C2O4 Na3PO4 Na4P2O7 Na2SO4 Na2SO4 Na2SO4 Na2S Na2S Na2S Na2SO3 Na2SO3 Na2B4O7 Na2B4O7 NaVO8

Solid phase 1aH2O aH2O 24H2O

2H2O

2H2O 1H2O

†

3H2O 10H2O 1H2O 10H2O 4H2O

0 °C

10 °C

20 °C

216.7

233.9

255.6

3.0 22.4 36.3 0.32 105.5 3.3 27.6 58.2 5 31 97

4.0 27.7

5.9 33.2 51.4 0.53 110.5 7.4 34.0 61.7 12 43 112

0.40 108 5 31.0 60.0 7 36 103

13.3 278.8 0.75 2.83 1.62 7.35 177.0

20.9

122 0.573 36.3 119 6.9 7

170 0.695 40.8 121 8.15 12.5

31.6 298.4 1.80 6.4 4.49 11.11 217.5 2.2 × 10−5 222 0.796 46.5 123.5 9.6 21.5

79 35.65 31.70

89 35.72 50.17

101 35.89 88.7

2H2O

163.0

2H2O 1H2O

57.9

1.05 4.4 2.60 9.22

30 °C 283.8 8.39 39.1 0.90 113.7 10.5 37.0 63.4 20 50 126 45.8 2.6 9.0 7.19 12.97

40 °C

50 °C

60 °C

70 °C

80 °C

90 °C

337.3 17.00

350 24.75 60.0

364.8 40.0

380.1 71.0

396.3 109.0

1.83 121.2 19.3 42.6 66.8 34

2.46 126.8 24.5 45.5 68.6 43 66

100 °C

323.3 11.70 45.4 67.3 1.32 116.9 14 40.0 65.2 26 60 63.9 334.9 4.4 12.56 9.89 14.76

140 85.5

110.0

48.3 70.4 52

138

4.6 139.8 38.5 51.1 72.1 61

169

147.5 54.0 73.9 70

202

6.5 16.89

9 22.2

11.8

14.8

18

16.50

18.17

19.75

21.4

22.8

1.22

669 1.30

300 0.888 54.5 126 11.1 38.8 50.5 113 36.09

376 0.979 65.5 129.5 12.7

455 1.08 83 134 14.45

525 1.15 139 139.5 16.4

48.5 126 36.37

140 36.69

46.4 155 37.04

88.7

95.96

104

114.6

177.8

133.1

244.8

146

153

172 37.46

45.8 189 37.93

123.0 316.7

124.8 376.2

1.36 161

38.47

121.6 6.95 155.7 57 56.7 75.6 80 82.6104 178 246 412.8 21.8 24.1 952 1.41 170 45.5 230 38.99 125.9 426.3

12H2O 12H2O 7H2O 2H2O

69.9

85.2

106.5

138.2

15.5 3.6

26.5 7.7

37 20.8

47

158.6 7.3 1.67

179.3 65

190.3

207.3 85

225.3

82.9

88.1

92.4

102.9

246.6

51.8 80.2

102.2 4H2O 3aH2O 1H2O

42 51.5 109

119

129

145

174

104 98.4

114 104.1

124

313

12H2O 10H2O 10H2O 7H2O 9H2O 5aH2O 6H2O 7H2O

73 72.1

80 78.0

1.5 3.16 5.0 19.5

4.1 3.95 9.0 30

88 84.5 3.7 11 6.23 19.4 44

15.42

18.8

22.5

20

26.9

36

13.9

96 91.6 20 9.95 40.8

31 13.50

43 17.45

55 21.83

81 30.04

347 180 163.2 6.33 108 40.26

48.8 28.5

46.7

45.3

43.7

42.5

39.82 36.4

42.69 39.1

28.2 10.5

28.8 20.3

28 10H2O 5H2O 2H2O

1.3

1.6

2.7 15.325°

3.9 30.2

68.4

148 132.6

45.73 43.31

51.40 49.14

24.4

31.5

59.23 57.28

28.3 41

52.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

TABLE 2-122

Solubilities of Inorganic Compounds in Water at Various Temperatures (Concluded)

Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Sodium vanadate (meta) Stannous chloride sulfate Strontium acetate acetate chloride chloride nitrate nitrate nitrate sulfate Sulfur dioxide, 760 mm† Thallium sulfate Thorium sulfate sulfate sulfate sulfate Zinc chlorate chlorate nitrate nitrate sulfate sulfate sulfate

Formula NaVO3 SnCl2 SnSO4 Sr(C2H3O2)2 Sr(C2H3O2)2 SrCl2 SrCl2 Sr(NO3)2 Sr(NO3)2 Sr(NO3)2 SrSO4 SO2 Tl2SO4 Th(SO4)2 Th(SO4)2 Th(SO4)2 Th(SO4)2 ZnClO3 ZnClO3 Zn(NO3)2 Zn(NO3)2 ZnSO4 ZnSO4 ZnSO4

Solid phase

0 °C

10 °C

9H2O 8H2O 6H2O 4H2O 6H2O 4H2O 6H2O 3H2O 7H2O 6H2O 1H2O

36.9 43.5

43.61 42.95 47.7

52.7 40.1 0.0113 22.83 2.70 0.74 1.0 1.50 145.0

30 °C

21.1025° 269.815° 19

83.9 4H2O aH2O 6H2O 2H2O 1H2O 4H2O

20 °C

41.6 52.9

40 °C

0.0114 11.29 4.87 1.38 1.62 1.90

60 °C

70 °C

80 °C

32.97

36.9

38.875°

36.24

36.10

85.9

90.5

90 °C

100 °C

18 39.5 58.7

65.3

64.0 70.5 16.21 3.70 0.98 1.25

50 °C

26.23

88.6 0.0114 7.81 6.16 1.995

37.35 72.4

81.8

83.8

97.2

90.1 5.41 2.998

4.5 9.21 5.22

4.04

2.54

2.45

36.4 130.4

100.8 139

93.8

96

98

100

10.92

12.74

14.61

16.53

18.45

6.64 1.63

1.09

86.6

83.7

80.8

152.5 200.3 118.3

94.78

209.2

223.2

273.1

206.9 41.9

47

54.4 70.1

76.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

2-129

2-130

PHYSICAL AND CHEMICAL DATA

TABLE 2-123

Solubility as a Function of Temperature and Henry’s Constant at 25°C for Gases in Water

Name

Formula

A

Acetylene Carbon dioxide Carbon monoxide Ethane Ethylene Helium Hydrogen Methane Nitrogen Oxygen

C2H2 CO2 CO C2H6 C2H4 He H2 CH4 N2 O2

−156.51 −159.854 −171.764 −250.812 −153.027 −105.9768 −125.939 −338.217 −181.587 −171.2542

B 8,160.2 8,741.68 8,296.9 12,695.6 7,965.2 4,259.62 5,528.45 13,282.1 8,632.13 8,391.24

C

D

T range, K

H at 25 °C, atm

21.403 21.6694 23.3376 34.7413 20.5248 14.0094 16.8893 51.9144 24.7981 23.24323

0 −1.10261E-03 0 0 0 0 0 −0.0425831 0 0

274–343 273–353 273–353 275–323 287–346 273–348 273–345 273–523 273–350 273–333

1,330 1,635 58,000 29,400 11,726 142,900 70,800 39,200 84,600 43,400

The constants can be used to calculate solubility by the equation ln x = A + B/T + C ln T + DT, where T is in K and x is the mole fraction of the solute dissolved in water when the solute partial pressure is 1 atm. With the assumption that Henry’s law is valid up to 1 atm, H = 1/x. Values of the constants are from P. G. T. Fogg and W. Gerrard, Solubility of Gases in Liquids, Wiley, 1991, New York, and Solubility Data Series, vol. 1, Helium and Neon, IUPAC, Pergamon Press, Oxford, 1979. For higher-temperature behavior and an up-to-date reference list, see R. Fernandez-Prini, J. L. Alvarez, and A. H. Harvey, J. Phys. Chem. Ref. Data 32(2):903, 2003. To find H at temperatures other than 25 °C, first find the solubility and then take the reciprocal. TABLE 2-124 Henry’s Constant H for Various Compounds in Water at 25°C Compound

CAS no.

Formula

H, atm

Rating

Methane Ethane Propane Butane Pentane Octane Nonane

74828 74840 74986 106978 109660 111659 111842

CH4 C2H6 C3H8 C4H10 C5H12 C8H18 C9H20

36,600 26,700 37,800 51,100 70,000 274,000 329,000

4 3 3 3 3 3 3

74851 115071

C2H4 C3H6

11,700 11,700

3 4

Olefins Ethylene Propylene Aromatics Benzene Toluene o-Xylene Cumene Phenol

Compound

CAS no.

Formula

H, atm

Rating

74873 75003 108907

CH3Cl C2H5Cl C6H5Cl

556 681 204

? 10 10

67561 64175 71238 71363

CH4O C2H6O C3H8O C4H10O

0.272 0.272 0.507 0.482

4 4 3 3

107131 75183 624920 74931 75081 110861

C3H3N C2H6S C2H6S2 CH4S C2H6S C5H5N

5.54 121 68.1 177 161 0.817

3 3 3 3 3 3

Chlorine Containing

Paraffin Hydrocarbons

71432 108883 95476 98828 108952

C6H6 C7H8 C8H10 C9H12 C6H6O

299 354 272 724 0.0394

10 10 10 9 7

Methyl chloride Chloroethane Chlorobenzene Alcohols Methanol Ethanol 1-Propanol 1-Butanol Miscellaneous Acrylonitrile Dimethyl sulfide Dimethyl disulfide Methyl mercaptan Ethyl mercaptan Pyridine

Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AIChE, the DIPPR Acetaldehyde 75,070 C2H4O 5.56 3 Environmental Safety Property Data and Estimations Streering Committee and Propanal 123,386 C3H6O 4.36 4 of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as T. N. Rogers, D. A. Zei, R. L. Rowley, W. V. Ketones Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, Methyl ethyl ketone 78,933 C4H8O 2.59 5 DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). Flammability limits are from B. Esters Lewis and G. Von Elbe, Combustion, Flames and Explosions of Gases, New York: Harcourt Brace Jovanovich (1987). Flash point data are from N. I. Sax, Methyl formate 107313 C2H4O2 13.6 3 Dangerous Properties of Industrial Materials, 6th ed, New York: Van Nostrand Ethyl formate 109944 C3H6O2 13.6 3 Reinhold (1984). Autoignition data are from I. Glassman, Combustion, 3d ed, Methyl acetate 79209 C3H6O2 5.04 3 New York: Academic Press (1996). The ratings reflect DIPPR® ESP’s effort to Butyl acetate 123864 C6H12O2 13.6 3 provide a critical evaluation and quality assessment of each data point with 15 being the highest score possible. The rating is not directly correlated with the estimated experimental uncertainty. The ratings reflect DIPPR Project 911’s effort to provide a critical evaluation and quality assessment of each data point, with 10 being the highest score possible. The rating is not directly correlated with the estimated experimental uncertainty. Henry’s constant is a strong nonlinear function of temperature. A single value measured at one temperature, if used for calculation at a different temperature, can lead to serious errors. Procedures for extrapolation of single-point values over the ambient temperature range (4°C < T < 50°C) are presented in Sec. 22, p. 22–49, under “Estimating Henry’s law constants.” Estimation procedures for the larger range (4°C < T < 200°C) are presented in F. L. Smith and A. H. Harvey, “Avoid Common Pitfalls When Using Henry’s Law,” Chem. Eng. Prog., 103(9), 2007. See also Y.-L. Huang, J. D. Olson, and G. E. Keller II, “Steam Stripping for Removal of Organic Pollutants from Water. 2. Vapor-Liquid Equilibrium Data,” Ind. Eng. Chem. Res., 31, pp. 1759–1768, 1992. (Also see the Supplementary Material, which contains the databank of 404 compounds of environmental interest and other useful property data.). Aldehydes

The H in Tables 2-123 to 2-134 is the proportionality constant in Henry’s law, p = Hx, where x is the mole fraction of the solute in the aqueous liquid phase; p is the partial pressure in atm of the solute in the gas phase; and H is a proportionality constant, generally referred to as Henry’s constant. Values of H often have considerable uncertainty and are strong functions of temperature. To convert values of H at 25°C from atm to atm/(mol/m3), divide by the molar density of water at 25°C, which is 55,342 mol/m3. Henry’s law is valid only for dilute solutions.

Additional values of Henry’s constant can be found in “Environmental Simulation Program”: OLI Systems, Inc., Morris Plains, N.J; “Estimated Henry’s Law Constant,” EPA Online Tools for Site Assessment Calculation (http://www.epa.gov/athens/learn2model/part-two/onsite/ esthenry.htm); “Compilation of Henry’s Law Constants for Inorganic and Organic Species of Potential Importance in Environmental Chemistry,” Rolf Sander, Air Chemistry Department, Max-Planck Institute of Chemistry, Mainz, Germany; “Modeling Atmospheric Chemistry: Interactions between Gas-Phase Species and Liquid Cloud/Aerosol Particles,” Rolf Sander, Surv. Geophys. 20:1–31, 1999 (http://www.henrys-law.org).

SOLUBILITIES TABLE 2-125 Henry’s Constant H for Various Compounds in Water at 25 °C from Infinite Dilution Activity Coefficients

TABLE 2-126

Compound

CAS no.

Formula

H = ∞Pvp, atm

Pentane Hexane Heptane Benzene Toluene o-Xylene Cumene Styrene Formaldehyde Acetaldehyde Propanal Acetone Methyl ethyl ketone Methyl n-propyl ketone Formic acid Methyl acetate Ethyl acetate Butyl acetate Chloroethane 1-Chloropropane Chlorobenzene Methanol Ethanol Pyridine Diethyl ether Thiophene

109660 1100543 142825 71432 108883 95476 98,828 100425 50000 75070 123386 67641 78933 107879 64186 79209 141786 123864 75003 74986 108907 67561 64175 110861 60297 110021

C5H12 C6H14 C7H16 C6H6 C7H8 C8H10 C9H12 C8H8 CH2O C2H4O C3H6O C3H6O C4H8O C5H10O CH2O2 C3H6O2 C4H8O2 C6H12O2 C2H5Cl C3H7Cl C6H5Cl CH4O C2H6O C5H5N C4H10O C4H4S

63700 84600 120000 309 344 267 613 145 14.3 4.54 5.45 2.13 3.11 4.60 0.0404 6.38 8.01 12.3 626 792 219 0.263 0.293 0.544 48.7 160

2-131

Air*

t, °C

0

5

10

15

20

25

30

35

10−4 × H†

4.32

4.88

5.49

6.07

6.64

7.20

7.71

8.23

t, °C

40

45

50

60

70

80

90

100

10−4 × H†

8.70

9.11

9.46

10.1

10.5

10.7

10.8

10.7

*International Critical Tables, vol. 3, p. 257. †H is calculated from the absorption coefficients of O2 and N2, taking into consideration the correction for constant argon content.

TABLE 2-127

Ammonia-Water at 10 and 20 °C* 10 °C

Mass fraction NH3 in liquid 0.0 0.00467 0.00495 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Henry’s constant H at 25 °C is the vapor pressure at 25 °C times the infinite dilution activity coefficient, also at 25 °C. Infinite dilution activity coefficients are from Mitchell and Jurs, J. Chem. Inf. Comput. Sci. 38: 200 (1998). Henry’s constant is a strong nonlinear function of temperature. A single value measured at one temperature, if used for calculation at a different temperature, can lead to serious errors. Procedures for extrapolation of single-point values over the ambient temperature range (4°C < T < 50°C) are presented in Sec. 22, p. 22–49, under “Estimating Henry’s law constants.” Estimation procedures for the larger range (4°C < T < 200°C) are presented in F. L. Smith and A. H. Harvey, “Avoid Common Pitfalls When Using Henry’s Law,” Chem. Eng. Prog., 103(9), 2007. See also Y.-L. Huang, J. D. Olson, and G. E. Keller II, “Steam Stripping for Removal of Organic Pollutants from Water. 2. Vapor-Liquid Equilibrium Data,” Ind. Eng. Chem. Res., 31, pp. 1759–1768, 1992. (Also see the Supplementary Material, which contains the databank of 404 compounds of environmental interest and other useful property data.)

20 °C

Mass fraction NH3 in vapor

P, kPa 1.23 1.37

0.0 0.1

7.07 20.07 47.37 99.84 184.44 292.15 399.03 486.44 554.33 615.05

P, kPa 2.34

0.84164 0.95438 0.98565 0.99544 0.99848 0.99943 0.99975 0.99988 0.99995 1.0

2.60 11.95 32.34 73.85 150.56 269.50 416.63 560.61 678.61 771.87 857.48

Mass fraction NH3 in vapor 0.0 0.1 0.82096 0.94541 0.98199 0.99393 0.99783 0.99913 0.99960 0.99980 0.99991 1.0

*Selected values from R. Tillner-Roth and D. G. Friend, J. Phys. Chem. Ref. Data 27:63 (1998). This reference lists solubilities for temperatures from −70 to 340°C. Densities, enthalpies, and entropies are listed for both the two-phase and single-phase regions for pressures up to 40 MPa.

TABLE 2-128 Carbon Dioxide (CO2)* Liquid mol fraction CO2 × 103 Total pressure, atm 1 2 10 20 30 36

0 °C

10 °C

15 °C

20 °C

25 °C

35 °C

50 °C

75 °C

100 °C

1.445 2.89 12.71 21.23 25.79

0.985 1.946 8.81 15.38 19.80 21.45

0.802 1.587 7.32 13.13 17.49 19.42

0.692 1.374 6.44 11.84 16.22 18.30

0.608 1.207 5.74 10.75 15.05 17.29

0.473 0.943 4.54 8.64 12.80 14.80

0.342 0.683 3.30 6.34 9.10 10.63

0.248 0.495 2.41 4.65 6.78 7.90

0.187 0.373 1.841 3.62 5.35 6.35

*Values selected from G. Houghton, A. M. McLean, and P. D. Ritchie, Chem. Eng. Sci. 6:132–137, 1957.

TABLE 2-129

Carbonyl Sulfide (COS)*

t, °C

0

5

10

15

20

25

30

10−3 × H

0.92

1.17

1.48

1.82

2.19

2.59

3.04

*International Critical Tables, vol. 3, p. 261.

2-132

PHYSICAL AND CHEMICAL DATA

TABLE 2-130 Partial pressure of Cl2, mmHg

Chlorine (Cl2)

TABLE 2-131 Vol % of ClO2 in gas phase

Solubility, g of Cl2 per liter 0 °C

10 °C

20 °C

30 °C

40 °C

50 °C

5 10 30 50 100

0.488 0.679 1.221 1.717 2.79

0.451 0.603 1.024 1.354 2.08

0.438 0.575 0.937 1.210 1.773

0.424 0.553 0.873 1.106 1.573

0.412 0.532 0.821 1.025 1.424

0.398 0.512 0.781 0.962 1.313

150 200 250 300 350

3.81 4.78 5.71

2.73 3.35 3.95 4.54 5.13

2.27 2.74 3.19 3.63 4.06

1.966 2.34 2.69 3.03 3.35

1.754 2.05 2.34 2.61 2.86

1.599 1.856 2.09 2.31 2.53

400 450 500 550 600

5.71 6.26 6.85 7.39 7.97

4.48 4.88 5.29 5.71 6.12

3.69 3.98 4.30 4.60 4.91

3.11 3.36 3.61 3.84 4.08

2.74 2.94 3.14 3.33 3.52

650 700 750 800 900

8.52 9.09 9.65 10.21

6.52 6.90 7.29 7.69 8.46

5.21 5.50 5.80 6.08 6.68

4.32 4.54 4.77 4.99 5.44

3.71 3.89 4.07 4.27 4.62

9.27 10.84 13.23 17.07 21.0

7.27 8.42 10.14 13.02 15.84

5.89 6.81 8.05 10.22 12.32

4.97 5.67 6.70 8.38 10.03

18.73 21.7 24.7 27.7 30.8

14.47 16.62 18.84 20.7 23.3

11.70 13.38 15.04 16.75 18.46

1000 1200 1500 2000 2500

Cl2.8H2O2 separates

3000 3500 4000 4500 5000 Partial pressure of Cl2, mmHg

Solubility, g of Cl2 per liter 60 °C

70 °C

80 °C

90 °C

100 °C

110 °C

5 10 30 50 100

0.383 0.492 0.743 0.912 1.228

0.369 0.470 0.704 0.863 1.149

0.351 0.447 0.671 0.815 1.085

0.339 0.431 0.642 0.781 1.034

0.326 0.415 0.627 0.747 0.987

0.316 0.402 0.598 0.722 0.950

150 200 250 300 350

1.482 1.706 1.914 2.10 2.28

1.382 1.580 1.764 1.932 2.10

1.294 1.479 1.642 1.793 1.940

1.227 1.396 1.553 1.700 1.831

1.174 1.333 1.480 1.610 1.736

1.137 1.276 1.413 1.542 1.661

400 450 500 550 600

2.47 2.64 2.80 2.97 3.13

2.25 2.41 2.55 2.69 2.83

2.08 2.22 2.35 2.47 2.59

1.965 2.09 2.21 2.32 2.43

1.854 1.972 2.08 2.19 2.29

1.773 1.880 1.986 2.09 2.19

650 700 750 800 900

3.29 3.44 3.59 3.75 4.04

2.97 3.10 3.23 3.37 3.63

2.72 2.84 2.96 3.08 3.30

2.55 2.66 2.76 2.87 3.08

2.41 2.50 2.60 2.69 2.89

2.28 2.37 2.47 2.56 2.74

1000 1200 1500 2000 2500

4.36 4.92 5.76 7.14 8.48

3.88 4.37 5.09 6.26 7.40

3.53 3.95 4.58 5.63 6.61

3.28 3.67 4.23 5.17 6.05

3.07 3.43 3.95 4.78 5.59

2.91 3.25 3.74 4.49 5.25

3000 3500 4000 4500 5000

9.83 11.22 12.54 13.88 15.26

8.52 9.65 10.76 11.91 13.01

7.54 8.53 9.52 10.46 11.42

6.92 7.79 8.65 9.49 10.35

6.38 7.16 7.94 8.72 9.48

5.97 6.72 7.42 8.13 8.84

1 3 5 7 10 11 12 13 14 15 16

Chlorine Dioxide (ClO2) Weight of ClO2, grams per liter of solution 0 °C

5 °C

10 °C

15 °C

20 °C

30 °C

40 °C

2.00 6.00 10.0 14.0 20.0

1.50 4.7 7.8 10.9 15.5 17.0 18.6 20.3

1.25 3.85 6.30 8.95 12.8 14.0 15.3 16.6 18.0 19.2 20.3

1.00 3.20 5.25 7.35 10.5 11.7 12.8 13.8 14.9 16.0 17.0

0.90 2.70 4.30 6.15 8.80 9.70 10.55 11.5 12.3 13.2 14.2

0.60 1.95 3.20 4.40 6.30 7.00 7.50 8.20 8.80 9.50 10.1

0.46 1.30 2.25 3.20 4.50 5.00 5.45 5.85 6.35 6.80 7.20

Ishi, Chem. Eng. (Japan), 22:153 (1958).

TABLE 2-132

Hydrogen Chloride (HCl)

Weights of HCl per 100 weights of H2O

Partial pressure of HCl, mmHg

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

0 °C

10 °C

20 °C

30 °C

510 130 29.0 5.7 1.0 0.175 0.0316 0.0056 0.00099 0.000118 0.000018

840 233 56.4 11.8 2.27 0.43 0.084 0.016 0.00305 0.000583 0.000069 0.0000117

399 105.5 23.5 4.90 1.00 0.205 0.0428 0.0088 0.00178 0.00024 0.000044

627 188 44.5 9.90 2.17 0.48 0.106 0.0234 0.00515 0.00077 0.000151

Weights of HCl per 100 weights of H2O

50 °C

80 °C

78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04

535 141 35.7 8.9 2.21 0.55 0.136 0.0344 0.0064 0.00140

623 188 54.5 15.6 4.66 1.34 0.39 0.095 0.0245

Partial pressure of HCl, mm Hg 110 °C

760 253 83 28 9.3 3.10 0.93 0.280

Enthalpy and phase-equilibrium data for the binary system HCl-H2O are given by Van Nuys, Trans. Am. Inst. Chem. Engrs., 39, 663 (1943).

TABLE 2-133

Hydrogen Sulfide (H2S)

t, °C

0

5

10

15

20

25

30

35

10−2 × H

2.68

3.15

3.67

4.23

4.83

5.45

6.09

6.76

t, °C

40

45

50

60

70

80

90

100

10−2 × H

7.45

8.14

8.84

10.3

11.9

13.5

14.4

14.8

International Critical Tables, vol. 3, p. 259.

THERMAL EXPANSION TABLE 2-134

2-133

Partial Vapor Pressure of Sulfur Dioxide over Water, mmHg Temperature, °C

g SO2 / 100 g H2O

0

10

0.01 0.05 0.10 0.15 0.20

0.02 0.38 1.15 2.10 3.17

0.04 0.66 1.91 3.44 5.13

0.25 0.30 0.40 0.50 1.00

4.34 5.57 8.17 10.9 25.8

6.93 8.84 12.8 17.0 39.5

2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00 20.00

20 0.07 1.07 3.03 5.37 7.93 10.6 13.5 19.4 25.6 58.4

30

40

50

60

90

120

0.12 1.68 4.62 8.07 11.8

0.19 2.53 6.80 11.7 17.0

0.29 3.69 9.71 16.5 23.8

0.43 5.24 13.5 22.7 32.6

1.21 12.9 31.7 52.2 73.7

2.82 27.0 63.9 104 145

15.7 19.8 28.3 37.1 83.7

58.6 93.2 129 165 202

88.5 139 192 245 299

129 202 277 353 430

183 285 389 496 602

275 351 542 735

407 517 796

585 741

818

22.5 28.2 40.1 52.3 117

31.4 39.2 55.3 72.0 159

42.8 53.3 74.7 96.8 212

95.8 118 164 211 454

253 393 535 679 824

342 530 720

453 700

955

186 229 316 404 856

Condensed from Rabe, A. E. and Harris, J. F., J. Chem. Eng. Data, 8 (3), 333–336, 1963. Copyright © American Chemical Society and reproduced by permission of the copyright owner.

THERMAL EXPANSION UNITS CONVERSIONS For this subsection, the following units conversion is applicable:

2, p. 93; metals, vol. 2, p. 459; petroleums, vol. 2, p. 145; porcelains, vol. 2, pp. 70, 78; refractory materials, vol. 2, p. 83; solid insulators, vol. 2, p. 310.

°F = 9⁄ 5 °C + 32 THERMAL EXPANSION OF GASES ADDITIONAL REFERENCES The tables given under this subject are reprinted by permission from the Smithsonian Tables. For more detailed data on thermal expansion, see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; alloys, vol. 2, p. 463; building stones, vol. 2, p. 54; carbons, vol. 2, p. 303; elements, vol. 1, p. 102; enamels, vol. 2, p. 115; glass, vol.

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, 1/υ(∂υ/∂T)p, for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (∂p/∂T)v, should be calculated either from the equation of state or from tabulated PVT data.

2-134

PHYSICAL AND CHEMICAL DATA

TABLE 2-135

Linear Expansion of the Solid Elements*

C is the true expansion coefficient at the given temperature; M is the mean coefficient between given temperatures; where one temperature is given, the true coefficient at that temperature is indicated; α and β are coefficients in formula lt = l0(1 + αt + βt2); l0 is length at 0 °C (unless otherwise indicated, when, if x is the reference temperature, lt = lx[1 + α(t − tx) + β(t − tx)2]; lt is length at t °C). Element

Temp., °C

C × 104

Aluminum Aluminum Antimony Arsenic Bismuth Cadmium Cadmium Carbon, diamond graphite Chromium Cobalt Copper Copper Gold Gold Indium Iodine Iridium Iridium Iron, soft cast wrought steel Lead (99.9)

20 300 20 20 20 0 0 50 50

0.224 0.284 0.136 0.05 0.014 0.54 0.20⊥ 0.012 0.06

20 20 200 20

0.123 0.162 0.170 0.140

40

0.417

20

0.065

40 20 20 20

0.1210 0.118 0.119 0.114

Magnesium

100 280 20

0.291 0.343 0.254

Manganese

20

0.233

Molybdenum†

20

0.053

Nickel

20

0.126

Osmium Palladium

40 20

0.066 0.1173

Platinum

20 20

0.0887 0.0893

40 40 0 40 20 20

0.0850 0.0963 0.439 0.0763 0.1846 0.195

Potassium Rhodium Ruthenium Selenium Silicon Silver Sodium Steel, 36.4Ni Tantalum†

20

0.065

Tellurium Thallium Tin

20 40 20 20 27 20‡ 20‡ 20

0.016 0.302 0.214 0.305 0.0444 0.643 0.125⊥ 0.358

Tungsten† Zinc

Temp. range, °C

M × 104

100 500 20

0.235 0.311 0.080⊥

20 −180, −140 −180, −140

0.103⊥ 0.59 0.117⊥

20, 100

0.068

100 300 17, 100 −191, 17

0.166 0.175 0.143 0.132

−190,

0.837

17

20, 100 20, 200

0, 6,

+ 20 100 100 0 100 100 500 100

0.291 0.300 0.240 0.260 0.228 0.159 0.052 0.049 0.055 0.130

0.83 0.0876

0, 100 −3, +18 0, 100

0.660 0.0249 0.197

0.009

500

0.22

20, 20,

100 100

0.526 0.214⊥

20, 6, 0,

500 121 625

0.086 0.121 0.161

0.0064 0.0040

0,

520

0.142

0.0022

0.0636 0.0679

0.0032 0.0011

−17 260 340 0 100 20

0.622 0.031 0.055 0.059 0.0655 0.272⊥

20 100 −100 100 100

0.154⊥ 0.045 0.656 0.639 0.141⊥

0, 0, 0, 100,

750 750 750 240

0.1158 0.1170 0.1118 0.269

0.0053 0.0053 0.0053 0.011

+ 20,

500

0.2480

0.0096

20, 300 −142, 19 19, +305

0.216 0.0515 0.0501

0.0121 0.0057 0.0014

−190, + 20 + 20, +300 500, 1000

0.1308 0.1236 0.1346

0.0166 0.0066 0.0033

−190, 0, −190, 0, 0,

+100 1000 −100 + 80 1000

0.1152 0.1167 0.0875 0.0890 0.0887

0.00517 0.0022 0.00314 0.00121 0.00132

−75, −112

0.0746

−75, −67 0, 875 20, 500 0, 50 260, 500 340, 500 20, 400

0.0182 0.1827 0.1939 0.72 0.144 0.136 0.0646

8, 0, −140, +20, +20,

β × 106

0.11

50 21

−190, 20, 20, −78, 0,

α × 104

0,

0, 80 1070, 1720

0, 100

−100, 20, 0, −190, 0, 25, 25, 0,

Temp. range, °C

0.00479 0.00295

0.0009

95

0.2033

0.0263

−105, +502 + 0, 400

0.0428 0.354

0.00058 0.010

*Smithsonian Tables. For more complete tabulations see Table 142, Smithsonian Physical Tables, 9th ed., 1954; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245. Chemical Rubber Publishing Co.; Goldsmith, and Waterman, WADC-TR-58-476, 1959; Johnson (ed.), WADD-TR-60-56, 1960, etc. †Molybdenum, 300 to 2500 °C; lt = l300[1 + 5.00 × 10−6(t − 300) + 10.5 × 10−10(t − 300)2] Tantalum, 300 to 2800 °C; lt = l300[1 + 6.60 × 10−6(t − 300) + 5.2 × 10−10(t − 300)2] Tungsten, 300 to 2700 °C; lt = l300[1 + 4.44 × 10−6(t − 300) + 4.5 × 10−10(t − 300)2] Beryllium, 20 to 100 °C; 12.3 × 10−6 per °C. Columbium, 0 to 100 °C; 7.2 × 10−6 per °C. Tantalum, 20 to 100 °C; 6.6 × 10−6 per °C. ‡These values for zinc were taken from Grüneisen and Goens, Z. Physik., 29:141 (1924).

THERMAL EXPANSION TABLE 2-136

2-135

Linear Expansion of Miscellaneous Substances*

The coefficient of cubical expansion may be taken as three times the linear coefficient. In the following table, t is the temperature or range of temperature, and C, the coefficient of expansion. t, °C

Substance Amber Bakelite, bleached Brass: Cast Wire Wire 71.5 Cu + 27.7 Zn + 0.3 Sn + 0.5 Pb 71 Cu + 29 Zn Bronze: 3 Cu + 1 Sn 3 Cu + 1 Sn 3 Cu + 1 Sn 86.3 Cu + 9.7 Sn + 4 Zn 97.6 Cu + hard 2.2 Sn + soft 0.2 P Caoutchouc Caoutchouc Celluloid Constantan Duralumin, 94Al

{

Ebonite Fluorspar, CaF2 German silver Gold-platinum, 2 Au + 1 Pt Gold-copper, 2 Au + 1 Cu Glass: Tube Tube Plate Crown (mean) Crown (mean) Flint Jena ther- 16III mometer normal

}

0−30 0−09 20−60

C × 104 0.50 0.61 0.22

0−100 0−100 0−100

0.1875 0.1930 0.1783–0.193

40 0−100

0.1859 0.1906

16.6−100 16.6−350 16.6−957 40 0−80 0−80

0.1844 0.2116 0.1737 0.1782 0.1713 0.1708

16.7−25.3 20−70 4−29 20−100 20−300 25.3−35.4 0−100 0−100 0−100 0−100

0.657–0.686 0.770 1.00 0.1523 0.23 0.25 0.842 0.1950 0.1836 0.1523 0.1552

0−100 0−100 0−100 0−100 50−60 50−60 0−100

0.0833 0.0828 0.0891 0.0897 0.0954 0.0788 0.081

Substance

t, °C

C × 104

Jena thermometer 59III Jena thermometer 59III Gutta percha Ice Iceland spar: Parallel to axis Perpendicular to axis Lead tin (solder) 2 Pb + 1 Sn Limestone Magnalium Manganin Marble Monel metal

0−100 −191–+16 20 −20–−1

0.058 0.424 1.983 0.51

0−80 0−80

0.2631 0.0544

0−100 25−100 12−39 15−100 25−100 25−600 0−16 16−38 38−49

0.2508 0.09 0.238 0.181 0.117 0.14 0.16 1.0662 1.3030 4.7707

40

0.0884

Paraffin Paraffin Paraffin Platinum-iridium, 10 Pt + 1 Ir Platinum-silver, 1 Pt + 2 Ag Porcelain Porcelain Bayeux Quartz: Parallel to axis Parallel to axis Perpend. to axis Quartz glass Quartz glass Quartz glass Rock salt Rubber, hard Rubber, hard Speculum metal Steel, 0.14 C, 34.5 Ni

0−100 20−790 1000−1400

0.1523 0.0413 0.0553

0−80 −190 to + 16 0−80 −190 to + 16 16 to 500 16 to 1000 40 0 −160 0−100 25−100 25−600

0.0797 0.0521 0.1337 −0.0026 0.0057 0.0058 0.4040 0.691 0.300 0.1933 0.037 0.136

Substance

t, °C

Topas: Parallel to lesser horizontal axis 0−100 Parallel to greater horizontal axis 0−100 Parallel to vertical axis 0−100 Tourmaline: Parallel to longitudinal axis 0−100 Parallel to horizontal axis 0−100 Type metal 16.6−254 Vulcanite 0−18 Wedgwood ware 0−100 Wood: Parallel to fiber: Ash 0−100 Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Across the fiber: Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Wax white 10−26 Wax white 26−31 Wax white 31−43 Wax white 43−57

C × 104

0.0832 0.0836 0.0472 0.0937 0.0773 0.1952 0.6360 0.0890 0.0951 0.0257 0.0649 0.0565 0.0361 0.0638 0.0492 0.0541 0.0658 0.614 0.325 0.443 0.404 0.484 0.544 0.341 0.484 2.300 3.120 4.860 15.227

*Smithsonian Tables. For a more complete tabulation see Tables 143, 144. Smithsonian Physical Tables. 9th ed., 1954, also reprinted in American Institute of Physics Handbook, McGraw-Hill, New York, 1957; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245, Chemical Rubber Publishing Co. For data on many solids prior to 1926, see Gruneisen, Handbuch der Physik, vol. 10, pp. 1–52, 1926, translation available as N.A.S.A. RE 2-18-59W, 1959. For eight plastic solids below 300 K, see Scott, Cryogenic Engineering, p. 331, Van Nostrand, Princeton, NJ, 1959. For 11 other materials to 300 K, see Scott, loc. cit., p. 333. For quartz and silica, see Cook, Brit. J. Appl. Phys., 7, 285 (1956).

2-136

PHYSICAL AND CHEMICAL DATA

TABLE 2-137

Volume Expansion of Liquids*

TABLE 2-138

If V0 is the volume at 0°, then at t° the expansion formula is Vt = V0(1 + αt + βt2 + γt3). The table gives values of α, β, and γ, and of C, the true coefficient of volume expansion at 20° for some liquids and solutions. The temperature range of the observation is ∆t. Values for the coefficient of volume expansion of liquids can be derived from the tables of specific volumes of the saturated liquid given as a function of temperature later in this section. C = (dV/dt)/V0 C × 103 at 20°

Liquid

Range

α × 103

β × 106

γ × 108

Acetic acid Acetone Alcohol: Amyl Ethyl, 30% by volume Ethyl, 50% by volume Ethyl, 99.3% by volume Ethyl, 500 atm pressure Ethyl, 3000 atm pressure Methyl Benzene Bromine Calcium chloride: 5.8% solution 40.9% solution Carbon disulfide 500 atm pressure 3000 atm pressure Carbon tetrachloride Chloroform Ether Glycerin Hydrochloric acid, 33.2% solution Mercury Olive oil Pentane Potassium chloride, 24.3% solution Phenol Petroleum, 0.8467 density Sodium chloride, 20.6% solution Sodium sulfate, 24% solution Sulfuric acid: 10.9% solution 100.0% Turpentine Water

16−107 0−54

1.0630 1.3240

0.12636 3.8090

1.0876 1.071 −0.87983 1.487

−15–80 18−39 0−39

0.9001 0.2928 0.7450

0.6573 10.790 1.85

27−46

1.012

2.20

0−40

0.866

0−40 0−61 11−81 0−59

0.524 1.1342 1.17626 1.06218

18−25 17−24 −34–60 0−50 0−50 0−76 0−63 −15–38

0.07878 0.42383 1.13980 0.940 0.581 1.18384 1.10715 1.51324 0.4853

0−33 0−100 0−33

0.4460 0.18182 0.6821 1.4646

0.215 0.0078 1.1405 3.09319

−0.539 1.6084

16−25 36−157

0.2695 0.8340

2.080 0.10732

0.4446

24−120

0.8994

1.396

0.955

0−29

0.3640

1.237

0.414

11−40

0.3599

1.258

0.410

1.18458 0.902 −11.87 0.730 1.12

1.3635 1.27776 1.87714

0.8741 1.199 0.80648 1.237 −0.30854 1.132

4.2742 0.8571 1.37065

0.250 0.458 1.91225 1.218

0.89881 4.66473 2.35918 0.4895

1.35135 1.236 −1.74328 1.273 4.00512 1.656 0.505

0−30 0.2835 2.580 0−30 0.5758 −0.432 −9−106 0.9003 1.9595 0−33 −0.06427 8.5053

0.455 0.18186 0.721 1.608 0.353 1.090

0.387 0.558 −0.44998 0.973 −6.7900 0.207

*Smithsonian Tables, Table 269. For a detailed discussion of mercury data, see Cook, Brit. J. Appl. Phys., 7, 285 (1956). For data on nitrogen and argon, see Johnson (ed.), WADD-TR-60-56, 1960. Bromoform1 7.7 − 50 °C. Vt = 0.34204[1 + 0.00090411(t − 7.7) + 0.0000006766(t − 7.7)2] 0.34204 is the specific volume of bromoform at 7.7 °C. Glycerin2 −62 to 0 °C. Vt = V0(1 + 4.83 × 10−4t − 0.49 × 10−6t2) 0 − 80 °C. Vt = V0(1 + 4.83 × 10−4t + 0.49 × 10−6t2) 3 Mercury 0 − 300 °C. Vt − V0[1 + 10−8(18,153.8t + 0.7548t2 + 0.001533t2 + 0.00000536t4)] 1 Sherman and Sherman, J. Am. Chem. Soc., 50, 1119 (1928). (An obvious error in their equation has been corrected.) 2 Samsoen, Ann. phys., (10) 9, 91 (1928). 3 Harlow, Phil. Mag., (7) 7, 674 (1929).

Volume Expansion of Solids*

If v2 and v1 are the volumes at t2 and t1, respectively, then v2 = v1(1 + C∆t), C being the coefficient of cubical expansion and ∆t the temperature interval. Where only a single temperature is stated, C represents the true coefficient of volume expansion at that temperature. Substance

t or ∆t

C × 104

Antimony Beryl Bismuth Copper† Diamond Emerald Galena Glass, common tube hard Jena, borosilicate 59 III pure silica Gold Ice Iron Lead† Paraffin Platinum Porcelain, Berlin chloride nitrate sulfate Quartz Rock salt Rubber Silver Sodium Stearic acid Sulfur, native Tin Zinc†

0−100 0−100 0−100 0−100 40 40 0−100 0−100 0−100 20−100 0−80 0−100 −20 to −1 0−100 0−100 20 0−100 20 0−100 0−100 20 0−100 50−60 20 0−100 20 33.8−45.4 13.2−50.3 0−100 0−100

0.3167 0.0105 0.3948 0.4998 0.0354 0.0168 0.558 0.276 0.214 0.156 0.0129 0.4411 1.1250 0.3550 0.8399 5.88 0.265 0.0814 1.094 1.967 1.0754 0.3840 1.2120 4.87 0.5831 2.13 8.1 2.23 0.6889 0.8928

*Smithsonian Tables, Table 268. †See additional data below. Aluminum1 100 − 530 °C. V = V0(1 + 2.16 × 10−5t + 0.95 × 10−8t2) 1 Cadmium 130 − 270 °C. V = V0(1 + 8.04 × 10−5t + 5.9 × 10−8t2) 1 Copper 110 − 300 °C. V = V0(1 + 1.62 × 10−5t + 0.20 × 10−8t2) Colophony2 0 − 34 °C. V = V0(1 + 2.21 × 10−4t + 0.31 × 10−6t2) 34 − 150 °C. V = V34[1 + 7.40 × 10−4(t − 34) + 5.91 × 10−6(t − 34)2] 1 Lead 100 − 280 °C. V = V0(1 + 1.60 × 10−5t + 3.2 × 10−8t2) 2 Shellac 0 − 46 °C. V = V0(1 + 2.73 × 10−4t + 0.39 × 10−6t2) 46 − 100 °C. V = V46[1 + 13.10 × 10−4(t − 46) + 0.62 × 10−6(t − 46)2] Silica (vitreous)3 0 − 300 °C. Vt = V0[1 + 10−8(93.6t + 0.7776t2 − 0.003315t2 + 0.000005244t4) Sugar (cane, amorphous)2 0 − 67 °C. Vt = V0(1 + 2.34 × 10−4t + 0.14 × 10−6t2) 67 − 160 °C. Vt = V67[1 + 5.02 × 10−4(t − 67) + 0.43 × 10−6(t − 67)2] Zinc1 120 − 360 °C. Vt = V0(1 + 8.50 × 10−5t + 3.9 × 10−8t2) 1 2 3

Uffelmann, Phil. Mag., (7) 10, 633 (1930). Samsoen, Ann. phys., (10) 9, 83 (1928). Harlow, Phil. Mag., (7) 7, 674 (1929).

JOULE-THOMSON EFFECT

2-137

JOULE-THOMSON EFFECT UNITS CONVERSIONS °F = 9⁄ 5 °C + 32; °R = 9⁄ 5 K

Joule-Thomson coefficients for substances listed in Table 2-184 are given in tables in the Thermodynamic Properties section. For this subsection, the following units conversions are applicable: To convert the Joule-Thomson coefficient, µ, in degrees Celsius per atmosphere to degrees Fahrenheit per atmosphere, multiply by 1.8. TABLE 2-139

To convert bars to pounds-force per square inch, multiply by 14.504; to convert bars to kilopascals, multiply by 1 × 102.

Additional References Available for the Joule-Thomson Coefficient Temp. range, °C

Pressure range, atm Gas Air Ammonia Argon Benzene Butane Carbon dioxide Carbon monoxide Deuterium Dowtherm A Ethane Ethylene Helium Hydrogen

<0

0–300

15, 19, 35

19, 35

39 31

39

7, 8, 37

7, 8, 37

1,* 22, 24 25

17 1,* 22, 24, 25

12, 15, 16 19, 35 28 39 31 26 7, 8, 9, 10 37 17

0–10

10–50

50–200

12, 15, 16 19, 35 28 39 31 26 7, 8, 28 37 17

12, 15, 19 35 39 31 26 7, 8, 37 17 22, 24, 25 1* 46 45

46 45 1, 38 24, 30

1, 38 22, 24, 25 30 6

Methane Mixtures Natural gas Nitrogen

13, 28, 40

13, 40

Nitrous oxide Pentane Propane Steam

26, 34, 44 41 28, 29, 42

34 43 29, 42, 47

>200

38 24, 30

46 45 9, 10 38 24

1, 38 22, 24, 25 30

6 33 13, 40

33 13

6 9, 11 33 9, 10, 13 28, 40 9, 10 26, 34, 44 43 28, 29, 42 45

33 13, 40

34 42, 47

>300

Other references 3, 4, 18 2, 3

31

46 48

13

19

29, 42, 47

29, 47

*See also 14 (generalized chart); 18 (review, to 1919); 20–22; 23 (review, to 1948); 27 (review, to 1905); 32, 36, 41, 50. REFERENCES: 1. Baehr. Z. Elektrochem., 60, 515 (1956). 2. Beattie, J. Math. Phys., 9, 11 (1930). 3. Beattie, Phys. Rev., 35, 643 (1930). 4. Bradley and Hale, Phys. Rev., 29, 258 (1909). 5. Brown and Dean, Bur. Stand. J. Res., 60, 161 (1958). 6. Budenholzer, Sage, et al., Ind. Eng. Chem., 29, 658 (1937). 7. Burnett, Phys. Rev., 22, 590 (1923). 8. Burnett, Univ. Wisconsin Bull. 9(6), 1926. 9. Charnley, Ph.D. thesis. University of Manchester, 1952. 10. Charnley, Isles, et al., Proc. R. Soc. (London), A217, 133 (1953). 11. Charnley, Rowlinson, et al., Proc. R. Soc. (London), A230, 354 (1955). 12. Dalton, Commun. Phys. Lab. Univ. Leiden, no. 109c, 1909. 13. Deming and Deming, Phys. Rev., 48, 448 (1935). 14. Edmister, Pet. Refiner, 28, 128 (1949). 15. Eucken, Clusius, et al., Z. Tech. Phys., 13, 267 (1932). 16. Eumorfopoulos and Rai, Phil. Mag., 7, 961 (1926). 17. Huang, Lin, et al., Z. Phys., 100, 594 (1936). 18. Hoxton, Phys. Rev., 13, 438 (1919). 19. Ishkin and Kaganev, J. Tech. Phys. U.S.S.R., 26, 2323 (1956). 20. Isles, Ph.D. thesis, Leeds University. 21. Jenkin and Pye, Phil. Trans. R. Soc. (London), A213, 67 (1914); A215, 353 (1915). 22. Johnston, J. Am. Chem. Soc., 68, 2362 (1946). 23. Johnston, Trans. Am. Soc. Mech. Eng., 70, 651 (1948). 24. Johnston, Bezman, et al., J. Am. Chem. Soc., 68, 2367 (1946). 25. Johnston, Swanson, et al., J. Am. Chem. Soc., 68, 2373 (1946). 26. Kennedy, Sage, et al., Ind. Eng. Chem., 28, 718 (1936). 27. Kester, Phys. Rev., 21, 260 (1905). 28. Keyes and Collins, Proc. Nat. Acad. Sci., 18, 328 (1932). 29. Kleinschmidt, Mech. Eng., 45, 165 (1923); 48, 155 (1926). 30. Koeppe, Kältetechnik, 8, 275 (1956). 31. Lindsay and Brown, Ind. Eng. Chem., 27, 817 (1935). 32. Noell, dissertation, Munich, 1914, Forschungsdienst, 184, p. 1, 1916. 33. Palienko, Tr. Inst. Ispol’ z. Gaza, Akad. Nauk Ukr. SSR, no. 4, p. 87, 1956. 34. Pattee and Brown, Ind. Eng. Chem., 26, 511, (1934). 35. Roebuck, Proc. Am. Acad. Arts Sci., 60, 537 (1925); 64, 287 (1930). 36. Roebuck, see 49 below, 37. Roebuck and Murrell, Phys. Rev., 55, 240 (1939). 38. Roebuck and Osterberg, Phys. Rev., 37, 110 (1931); 43, 60 (1933). 39. Roebuck and Osterberg, Phys. Rev., 46, 785 (1934). 40. Roebuck and Osterberg, Phys. Rev., 48, 450 (1935). 41. Roebuck, Murrell, et al., J. Am. Chem. Soc., 64, 400 (1942). 42. Sage, unpublished data, California Institute of Technology, 1959. 43. Sage and Lacy, Ind. Eng. Chem., 27, 1484 (1934). 44. Sage, Kennedy, et al., Ind. Eng. Chem., 28, 601 (1936). 45. Sage, Webster, et al., Ind. Eng. Chem., 29, 658 (1937). 46. Ullock, Gaffert, et al., Trans. Am. Inst. Chem. Eng., 32, 73 (1936). 47. Yang, Ind. Eng. Chem., 45, 786 (1953). 48. Zelmanov, J. Phys. U.S.S.R., 3, 43 (1940). 49. Roebuck, recalculated data. 50. Michels et al., van der Waals laboratory publications. Gunn, Cheuh, and Prausnitz, Cryogenics, 6, 324 (1966), review equations relating the inversion temperatures and pressures. The ability of various equations of state to relate these was also discussed by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970); and Juris and Wenzel, Am. Inst. Chem. Eng. J., 18, 684 (1972). Perhaps the most detailed review is that of Hendricks, Peller, and Baron. NASA Tech. Note D 6807, 1972. TABLE 2-140 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc)* Pr

0

0.5

1

1.5

2

2.5

3

4

TrL TrU

0.782 4.984

0.800 4.916

0.818 4.847

0.838 4.777

0.859 4.706

0.880 4.633

0.903 4.550

0.953 4.401

Pr

5

6

7

8

9

10

11

11.79

TrL TrU

1.01 4.23

1.08 4.06

1.16 3.88

1.25 3.68

1.35 3.45

1.50 3.18

1.73 2.86

2.24 2.24

*Calculated from the best three-constant equation recommended by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970). TrL refers to the lower curve, and TrU, to the upper curve.

2-138

PHYSICAL AND CHEMICAL DATA

CRITICAL CONSTANTS ADDITIONAL REFERENCES Other data and estimation techniques for the elements are contained in Gates and Thodos, Am. Inst. Chem. Eng. J., 6 (1960):50–54; and Ohse and von Tippelskirch, High Temperatures—High Pressures, 9 (1977):367–385. For inorganic substances see Mathews, Chem. Rev., TABLE 2-141

72 (1972):71–100; for organics see Kudchaker, Alani, and Zwolinski, Chem. Rev., 68 (1968):659–735; and for fluorocarbons see Advances in Fluorine Chemistry, App. B, Butterworth. Washington, 1963, pp. 173–175. Pages 6–49 and 6–50 of the 84th edition of the Handbook of Chemistry and Physics provide an excellent list of references for critical properties.

Critical Constants and Acentric Factors of Inorganic and Organic Compounds Vc,

Cmpd.

no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

Name Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol o-Cresol p-Cresol Cumene Cyanogen Cyclobutane

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O C7H8O C9H12 C2N2 C4H8

Acentric

CAS no.

Mol. wt.

Tc, K

Pc, MPa

m3/kmol

Zc

factor

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4 95-48-7 106-44-5 98-82-8 460-19-5 287-23-0

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138 108.138 108.138 120.192 52.035 56.106

466 761 591.95 606 508.2 545.5 308.3 506 615 535 132.45 405.65 645.6 150.86 824 562.05 689 751 699.35 830 720.15 662 718 773 584.15 670.15 503.8 467 452 425 425.12 680 676 563.1 535.9 419.5 435.5 428.6 575.4 660.5 570.1 554 440 537.2 615.7 582.25 304.21 552 132.92 556.35 227.51 417.15 632.35 460.35 536.4 416.25 503.15 489 705.85 697.55 704.65 631 400.15 459.93

5.55 6.6 5.786 4 4.701 4.83 6.138 5 5.66 4.48 3.774 11.28 4.25 4.898 5.05 4.895 4.74 4.47 4.215 3.352 4.374 3.11 4.06 3.38 10.3 4.519 6.23 8 4.36 4.32 3.796 5.21 4.02 4.414 4.188 4.02 4.21 4.1 3.09 2.89 3.97 4.06 4.6 4.32 4.06 3.79 7.383 7.9 3.499 4.56 3.745 7.71 4.519 5.27 5.472 6.68 4.58 4.54 4.56 5.01 5.15 3.209 5.98 4.98

0.154 0.215 0.177 0.29 0.209 0.173 0.112 0.197 0.208 0.212 0.09147 0.07247 0.337 0.07459 0.346 0.256 0.315 0.344 0.3132 0.5677 0.382 0.442 0.367 0.497 0.135 0.324 0.215 0.156 0.22 0.221 0.255 0.303 0.305 0.273 0.27 0.241 0.234 0.238 0.389 0.497 0.307 0.307 0.208 0.258 0.293 0.278 0.094 0.16 0.0944 0.276 0.143 0.124 0.308 0.2 0.239 0.143 0.247 0.247 0.312 0.282 0.277 0.434 0.195 0.21

0.221 0.224 0.208 0.23 0.233 0.184 0.268 0.234 0.23 0.214 0.313 0.242 0.267 0.291 0.255 0.268 0.261 0.246 0.227 0.276 0.279 0.25 0.25 0.261 0.286 0.263 0.32 0.321 0.255 0.27 0.274 0.279 0.218 0.258 0.254 0.278 0.272 0.274 0.251 0.262 0.257 0.271 0.262 0.25 0.232 0.218 0.274 0.275 0.299 0.272 0.283 0.276 0.265 0.275 0.293 0.276 0.27 0.276 0.242 0.244 0.244 0.265 0.351 0.273

0.2907 0.4210 0.4665 0.4535 0.3065 0.3379 0.1912 0.3198 0.5383 0.3498 0.2526 0.3502 0.0000 0.5585 0.2103 0.2628 0.6028 0.3662 0.5019 0.3631 0.4332 0.3126 0.4029 0.1290 0.2506 0.2548 0.1922 0.1659 0.1950 0.2002 0.6305 0.7043 0.5883 0.5692 0.1845 0.2019 0.2176 0.4394 0.3941 0.2714 0.2506 0.2470 0.2774 0.6805 0.3714 0.2236 0.1107 0.0482 0.1926 0.1790 0.0688 0.2499 0.1902 0.2219 0.1531 0.2277 0.1986 0.4480 0.4339 0.5072 0.3274 0.2790 0.1847

CRITICAL CONSTANTS TABLE 2-141

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued) Vc,

Cmpd.

no. 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

2-139

Acentric

Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc, MPa

m3/kmol

Zc

factor

Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate

C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2

110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4

84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133 194.184 88.105 170.207 101.190 170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079

553.8 650.1 653 560.4 511.7 507 398 664 674.2 617.7 722.1 688 616.6 696 619.85 38.35 628 650.15 611 584.1 683.95 705 684.75 523 561.6 510 560 572 736.6 496.6 466.7 557.15 386.44 445 351.255 523.1 500.05 576 507.8 543 473.2 437.2 500 591.15 606.15 596.15 615 400.1 649.6 537.3 766 402 503.04 729 772 587 766.8 550 658 768 305.32 514 523.3 456.15 617.15 698 655 571 609.15 569.5 282.34 593 720 537 469.15 508.4

4.08 4.26 4 4.35 4.51 4.8 5.54 3.97 2.6 2.11 2.28 2.308 2.223 2.13 2.37 1.6617 6.03 5.477 7.17 2.46 4.07 4.07 4.07 5.07 5.37 6.08 4.24 4.24 4.27 3.71 3.64 3.96 4.52 4.34 5.784 3.2 2.88 3.02 3.773 3.446 4.87 5.34 3.15 2.938 2.938 2.938 5.36 5.37 4.42 2.91 2.78 3.56 5.53 5.65 2.78 5.208 3.08 3.14 1.82 1.16 4.872 6.137 3.88 5.62 3.609 3.18 3.41 2.95 3.04 3.4 5.041 6.29 8.2 6.85 7.19 4.74

0.308 0.322 0.311 0.291 0.26 0.245 0.162 0.355 0.58 0.617 0.639 0.645 0.584 0.624 0.552 0.060263 0.276 0.2616 0.223 0.487 0.351 0.351 0.351 0.24 0.22 0.185 0.291 0.291 0.349 0.301 0.28 0.318 0.179 0.195 0.123 0.418 0.386 0.416 0.297 0.35 0.221 0.18 0.361 0.45 0.46 0.46 0.252 0.17 0.26199 0.393 0.53 0.258 0.201 0.227 0.529 0.238 0.503 0.402 0.755 1.34 0.1455 0.168 0.286 0.207 0.374 0.489 0.389 0.403 0.43 0.375 0.131 0.264 0.191 0.173 0.140296 0.229

0.273 0.254 0.229 0.272 0.276 0.279 0.271 0.255 0.269 0.254 0.243 0.26 0.253 0.23 0.254 0.314 0.319 0.265 0.315 0.247 0.251 0.244 0.251 0.28 0.253 0.265 0.265 0.259 0.243 0.27 0.263 0.272 0.252 0.229 0.244 0.308 0.267 0.262 0.265 0.267 0.274 0.264 0.274 0.269 0.268 0.273 0.264 0.2744 0.214 0.256 0.231 0.275 0.266 0.212 0.229 0.254 0.243 0.276 0.251 0.243 0.279 0.241 0.255 0.307 0.263 0.268 0.244 0.25 0.258 0.269 0.281 0.337 0.262 0.265 0.25876 0.257

0.2081 0.3690 0.2990 0.2123 0.1949 0.1961 0.1278 0.2641 0.5820 0.4923 0.8126 0.6070 0.4805 0.5874 0.5178 −0.1449 0.1250 0.2067 0.2095 0.4476 0.2790 0.2192 0.2846 0.2339 0.2866 0.1986 0.2529 0.2564 0.9529 0.3039 0.2811 0.2900 0.2751 0.2224 0.2771 0.3883 0.3387 0.4044 0.3277 0.3522 0.2385 0.2999 0.2493 0.2326 0.2324 0.2379 0.2059 0.2002 0.3177 0.2964 0.6568 0.1300 0.1943 0.2806 0.6371 0.2793 0.4389 0.4497 0.5764 0.9069 0.0995 0.6436 0.3664 0.2848 0.3035 0.4771 0.6326 0.4011 0.2455 0.2701 0.0862 0.4724 0.5068 0.2007 0.1974 0.2847

2-140

PHYSICAL AND CHEMICAL DATA

TABLE 2-141

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued) Vc,

Cmpd.

no. 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216

Acentric

Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc, MPa

m3/kmol

Zc

factor

2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol

C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10 H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O

149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2 302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9

144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144 32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185

674.6 583 489 567 499.15 546 500.23 559.95 144.12 560.09 375.31 317.42 408 771 588 490.15 5.2 736 616.8 540.2 677.3 632.3 608.3 606.6 611.4 537.4 645 547 723 591 507.6 660.2 611.3 585.3 587.61 582.82 504 544 623 516.2 549 653.15 33.19 363.15 324.65 456.65 461.15 373.53 605 471.85 805 662 190.564 512.5 718 506.55 402.4 536 430.05 693 490 460.4 643 577.2 465 470 492 512.74 593 463.2 554.5 442 572.1 686 614 617

2.778 2.46 3.41 3.32 5.49 3.362 3.37 3.33 5.172 4.551 5.028 5.875 6.59 7.8 5.81 5.5 0.2275 1.34 3.16 2.74 3.043 3.085 3.001 2.92 2.94 2.92 2.77 3.21 1.4 3.46 3.025 3.308 3.446 3.311 3.287 3.32 3.21 3.53 3.08 3.62 3.53 14.7 1.313 8.552 8.31 5.39 6.48 8.963 3.7 4.54 5.64 4.79 4.599 8.084 4.98 4.75 5.63 4.25 7.46 3.59 3.83 3.38 3.89 3.93 3.447 3.42 4.38 3.371 3.47 4.2 3.473 4.17 3.48 4 3.79 3.79

0.528 0.487 0.329 0.369 0.207 0.345 0.339 0.414 0.066547 0.269 0.164 0.113 0.115 0.163 0.125 0.218 0.0573 1.11 0.434 0.428 0.466 0.444 0.447 0.433 0.434 0.402 0.465 0.387 1.04 0.369 0.371 0.408 0.382 0.385 0.378 0.378 0.348 0.331 0.412 0.322 0.331 0.158 0.064147 0.1 0.081 0.139 0.069 0.0985 0.292 0.221 0.258 0.28 0.0986 0.117 0.267 0.228 0.164 0.27 0.154 0.436 0.291 0.306 0.347 0.329 0.292 0.292 0.248 0.329 0.36 0.275 0.34 0.246 0.369 0.374 0.374 0.374

0.262 0.247 0.276 0.26 0.274 0.256 0.275 0.296 0.287 0.263 0.264 0.252 0.223 0.198 0.149 0.294 0.302 0.244 0.267 0.261 0.252 0.261 0.265 0.251 0.251 0.263 0.24 0.273 0.243 0.26 0.266 0.246 0.259 0.262 0.254 0.259 0.267 0.258 0.245 0.272 0.256 0.428 0.305 0.283 0.249 0.197 0.117 0.284 0.215 0.256 0.217 0.244 0.286 0.222 0.223 0.257 0.276 0.258 0.321 0.272 0.274 0.27 0.252 0.269 0.26 0.256 0.266 0.26 0.253 0.3 0.256 0.279 0.27 0.262 0.278 0.276

0.8067 0.4944 0.3056 0.3891 0.1878 0.3944 0.3473 0.2691 0.0530 0.2472 0.2200 0.1980 0.2818 0.4124 0.3173 0.2015 −0.3900 0.7697 0.4279 0.3495 0.7564 0.5621 0.5628 0.4076 0.4190 0.3432 0.4226 0.3778 0.7174 0.3872 0.3013 0.7299 0.5586 0.5574 0.3846 0.3801 0.2888 0.2183 0.3681 0.3327 0.2214 0.3143 −0.2160 0.0734 0.1315 0.4099 0.3823 0.0942 0.6141 0.2759 0.9418 0.3318 0.0115 0.5658 0.4351 0.3313 0.2115 0.3423 0.2814 0.4205 0.1874 0.2279 0.5894 0.5939 0.2341 0.2870 0.1370 0.3130 0.3229 0.3081 0.3775 0.2252 0.2361 0.2213 0.6805 0.6790

CRITICAL CONSTANTS TABLE 2-141

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued) Vc,

Cmpd.

no. 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292

2-141

Acentric

Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc, MPa

m3/kmol

Zc

factor

Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate

C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO

96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9

84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122 90.187 46.144 118.176 88.148 58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121

532.7 542 526 483 437.8 535.5 533 487.2 497 574.6 488 464.48 553.4 553.1 469.95 566 694 497.7 546.49 407.8 506.2 417.9 530.6 476.25 565 352.5 654 497.1 437 748.4 44.4 593 126.2 234 588.15 309.57 180.15 758 658 594.6 710.7 670.9 649.5 593.1 681 598.05 747 638.9 568.7 694.26 652.3 629.8 632.7 627.7 566.9 667.3 574 804 154.58 261 708 566.1 469.7 639.16 588.1 561 561.08 560.95 464.8 584.3 598 481.2 519 869 694.25 653

3.79 4.13 4.13 3.95 4.4 4.15 4.26 6 3.41 3.27 5.48 3.762 3.8 4.021 7.23 3.68 2.54 3.04 3.042 3.64 3.972 4 4.004 3.801 3.97 4.7 3.36 3.287 4.67 4.05 2.653 5.16 3.4 4.461 6.31 7.245 6.48 1.21 2.73 2.29 2.514 2.527 2.541 2.428 2.31 2.61 1.27 2.96 2.49 2.779 2.783 2.749 2.64 2.704 2.663 2.52 2.88 7.02 5.043 5.57 1.48 3.97 3.37 3.63 3.897 3.7 3.694 3.74 3.56 3.536 3.47 4.17 4.03 2.9 6.13 4.06

0.319 0.303 0.303 0.289 0.221 0.267 0.254 0.172 0.329 0.369 0.202 0.276 0.31 0.328 0.145 0.323 0.572 0.368 0.38 0.259 0.275 0.239 0.282 0.276 0.307 0.205 0.399 0.314 0.21 0.407 0.0417 0.236 0.08921 0.11875 0.173 0.0974 0.058 1.26 0.527 0.551 0.584 0.576 0.577 0.524 0.571 0.497 1.19 0.488 0.486 0.523 0.509 0.512 0.497 0.497 0.464 0.518 0.442 0.205 0.0734 0.089 0.969 0.313 0.313 0.35 0.326 0.326 0.301 0.336 0.293 0.385 0.359 0.277 0.276 0.554 0.229 0.37

0.273 0.278 0.286 0.284 0.267 0.249 0.244 0.255 0.272 0.253 0.273 0.269 0.256 0.28718 0.268 0.253 0.252 0.27 0.254 0.278 0.26 0.275 0.256 0.265 0.259 0.329 0.247 0.25 0.27 0.265 0.3 0.247 0.289 0.272 0.223 0.274 0.251 0.242 0.263 0.255 0.248 0.261 0.271 0.258 0.233 0.261 0.243 0.272 0.256 0.252 0.261 0.269 0.249 0.257 0.262 0.235 0.267 0.215 0.288 0.228 0.244 0.264 0.27 0.239 0.258 0.259 0.238 0.269 0.27 0.28 0.251 0.289 0.258 0.222 0.243 0.277

0.2288 0.2318 0.2296 0.2758 0.2314 0.3234 0.2091 0.2556 0.3078 0.3557 0.3007 0.2656 0.3208 0.2461 0.1582 0.2802 0.7913 0.2791 0.3442 0.1835 0.6152 0.1948 0.3466 0.2770 0.2737 0.1314 0.3230 0.2466 0.2416 0.3020 −0.0396 0.3803 0.0377 0.1200 0.3480 0.1409 0.5829 0.8522 0.5117 0.4435 0.7724 0.5841 0.5911 0.4367 0.5260 0.4710 0.8114 0.4636 0.3996 0.7706 0.5697 0.5807 0.4549 0.4406 0.3921 0.4497 0.4233 0.9176 0.0222 0.2119 0.6863 0.3472 0.2515 0.7052 0.5748 0.5549 0.3433 0.3448 0.2372 0.2685 0.3207 0.2899 0.1752 0.4707 0.4435 0.4123

2-142

PHYSICAL AND CHEMICAL DATA

TABLE 2-141

Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Concluded) Vc,

Cmpd.

no. 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Acentric

Name

Formula

CAS no.

Mol. wt.

Tc, K

Pc, MPa

m3/kmol

Zc

factor

Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C8H4O3 C3H4 C3H8 C3H8O C3H8O C3H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132 59.110 120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

791 394 369.83 536.8 508.3 636 504.4 600.81 564.4 549.73 496.95 638.35 364.85 538 517 536.6 626 683 259 636 806 430.75 318.69 490.85 1113 857 693 540.15 720 631.95 568 579.35 591.75 602 675 535.15 433.25 664.5 649.1 543.8 573.5 846 828 639 703.9 519.13 454 432 543.15 647.096 617 630.3 616.2

4.72 5.25 4.248 5.169 4.765 3.12 4.92 4.668 4.18 3.36 4.74 3.2 4.6 4.02 4.75 4.63 6.1 5.96 3.72 3.84 4.71 7.884 3.76 8.21 3.95 2.99 1.57 5.19 3.65 5.16 2.87 5.69 4.108 4.48 1.68 3.04 4.07 3.454 3.232 2.57 2.82 3.39 3.04 1.95 2.119 3.958 4.86 5.67 3.06 22.064 3.541 3.732 3.511

0.421 0.165 0.2 0.219 0.222 0.437 0.204 0.235 0.229 0.345 0.26 0.44 0.185 0.285 0.254 0.254 0.239 0.291 0.202 0.352 0.317 0.122 0.19852 0.127 0.424 0.731 0.897 0.224 0.408 0.249 0.461 0.219 0.316 0.281 0.826 0.39 0.254 0.414 0.43 0.468 0.455 0.479 0.572 0.685 0.715 0.27 0.205 0.179 0.408 0.0559472 0.375 0.37 0.378

0.302 0.264 0.276 0.254 0.25 0.258 0.239 0.22 0.204 0.254 0.298 0.265 0.281 0.256 0.281 0.264 0.28 0.305 0.349 0.256 0.223 0.269 0.282 0.255 0.181 0.307 0.244 0.259 0.249 0.245 0.28 0.259 0.264 0.252 0.247 0.266 0.287 0.259 0.258 0.266 0.269 0.231 0.253 0.252 0.259 0.248 0.264 0.283 0.276 0.229 0.259 0.264 0.259

0.7025 0.1041 0.1523 0.6209 0.6544 0.3420 0.2559 0.5796 0.3243 0.3889 0.2798 0.3444 0.1376 0.3088 0.2138 0.2318 1.1065 0.4945 0.3858 0.2971 0.9922 0.2454 0.2151 0.4240 1.0591 0.5513 0.6430 0.2254 0.3353 0.1996 0.2450 0.1970 0.2640 0.2591 0.6174 0.3162 0.2062 0.3666 0.3787 0.3035 0.2903 0.8623 0.8972 0.5303 0.6236 0.3513 0.1069 0.1001 0.2815 0.3449 0.3265 0.3101 0.3218

All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for the acentric factor was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

COMPRESSIBILITIES

2-143

COMPRESSIBILITIES physical Properties of Fluid Systems High Accuracy Data. Results can be pasted into a spreadsheet to facilitate calculation of the compressibility factor.

INTRODUCTION The compressibility factor Z can be calculated by using the defining equation Z = PV/(RT), where P is pressure, V is molar volume, R is the gas constant, and T is absolute temperature. Values of P, V, and T for substances listed in Table 2-184 are given in tables in the Thermodynamic Properties section. For the units used in these tables, R is 0.008314472 MPa·dm3/(mol·K). Values at temperatures and pressures other than those in the tables can be generated for many of the substances in Table 2-184 by going to http://webbook. nist.gov and selecting NIST Chemistry WebBook, then Thermo-

TABLE 2-142

UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °R = 9⁄5 K To convert bars to pounds-force per cubic inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.

Composition of Selected Refrigerant Mixtures Composition (mass percent)

Mixture R-410A R-404A R-507A R-407C (Klea 66) R-407A (Klea 60) R-407B (Klea 61)

TABLE 2-143

Tables

R-32

R-125

2-290 2-288 2-296 2-263, 2-264, 2-289 2-143, 2-261 2-144, 2-262

50

50 44 50 25 40 70

23 20 10

R-134a

R-143a

4

52 50

52 40 20

Compressibility Factors for R 407A (Klea 60) Pressure, bar

T, K 250 260 270 280 290 300 310 320 330 340 350 Zdew Tdew, K

1 0.9691 0.9737 0.9773 0.9803 0.9828 0.9848 0.9866 0.9881 0.9893 0.9904 0.9914 0.9593 234.3

5 0.0163 0.0161 0.4268 0.8932 0.9080 0.9199 0.9298 0.9380 0.9449 0.9509 0.9560 0.8809 273.3

10 0.0325 0.0321 0.0318 0.0316 0.0360 0.8253 0.8495 0.8689 0.8847 0.8980 0.9092 0.8107 295.1

15 0.0487 0.0480 0.0476 0.0473 0.0473 0.0476 0.7518 0.7889 0.8173 0.8401 0.8588 0.7502 309.6

20 0.0648 0.0640 0.0633 0.0630 0.0629 0.0632 0.0641 0.5737 0.7386 0.7752 0.8038 0.6936 320.9

25 0.0809 0.0798 0.0790 0.0785 0.0784 0.0787 0.0797 0.0816 0.6279 0.6993 0.7425 0.6381 330.1

30 0.0970 0.0957 0.0947 0.0940 0.0938 0.0942 0.0952 0.0972 0.1011 0.6016 0.6713 0.5813 337.9

Zdew

Pdew

0.9340 0.9136 0.8895 0.8614 0.8290 0.7916 0.7485 0.6983 0.6386 0.5641 0.4564

2.05 3.08 4.46 6.28 8.60 11.53 15.15 19.58 24.96 31.46 39.42

The values in this table were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1).

TABLE 2-144

Compressibility Factors for R 407B (Klea 61) Pressure, bar

T, K 250 260 270 280 290 300 310 320 330 340 350 Zdew Tdew, K

1 0.9703 0.9745 0.9779 0.9808 0.9831 0.9851 0.9868 0.9883 0.9895 0.9906 0.9915 0.9587 230.6

5 0.0180 0.0178 0.8785 0.8961 0.9101 0.9215 0.9310 0.9390 0.9457 0.9515 0.9565 0.8774 269.4

10 0.0359 0.0355 0.0352 0.0351 0.5520 0.8290 0.8522 0.8709 0.8864 0.8993 0.9103 0.8036 291.3

15 0.0538 0.0532 0.0527 0.0526 0.0527 0.0533 0.7569 0.7924 0.8200 0.8422 0.8606 0.7389 305.9

20 0.0716 0.0708 0.0702 0.0699 0.0700 0.0707 0.0721 0.6949 0.7428 0.7784 0.8064 0.6776 317.1

25 0.0894 0.0883 0.0875 0.0872 0.0872 0.0879 0.0895 0.0925 0.6453 0.7042 0.7462 0.6164 326.4

30 0.1071 0.1058 0.1048 0.1043 0.1043 0.1050 0.1067 0.1098 0.1164 0.6108 0.6770 0.5518 334.3

Zdew

Pdew

0.9251 0.9024 0.8757 0.8446 0.8085 0.7666 0.7178 0.6600 0.5890 0.4916

2.40 3.56 5.10 7.10 9.64 12.81 16.71 21.45 27.18 34.12

The values in this table were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1).

2-144

PHYSICAL AND CHEMICAL DATA

TABLE 2-145

Compressibilities of Liquids*

At the constant temperature T, the compressibility β = (1/V 0)(dV/dP). In general as P increases, β decreases rapidly at first and then slowly; the change of β with T is large at low pressures but very small at pressures above 1000 to 2000 megabars. 1 megabar = 0.987 atm = 106 dynes/cm2 based upon the older usage, 1 bar = 1 dyne/cm2.

Substance

Temp., °C

Pressure, megabars

Acetone Acetone Acetone Acetone Amyl alcohol alcohol, iso. alcohol, iso. alcohol, n alcohol, n alcohol, n alcohol, n Benzene Benzene Benzene Bromine Bromine Butyl alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso Carbon bisulfide bisulfide bisulfide bisulfide tetrachloride tetrachloride Chloroform Chloroform Dichloroethylsulfide Dichloroethylsulfide Ethyl acetate acetate

14 20 20 40 14 20 20 20 20 20 40 17 20 20 20 20 18 20 20 20 20 20 16 20 20 20 20 20 20 20 32 32 13 20

23 500 1,000 12,000 23 200 400 500 1,000 12,000 12,000 5 200 400 200 400 8 200 400 500 1,000 12,000 21 500 1,000 12,000 200 400 200 400 1,000 2,000 23 200

Compressibility per megabar β × 106

Substance

111 61 52 9 88 84 70 61 46 8 8 89 77 67 56 51 97 81 64 56 46 8 86 57 48 6 86 73 83 70 34 24 103 90

Ethyl acetate alcohol alcohol alcohol alcohol bromide bromide bromide bromide bromide chloride chloride chloride chloride ether ether ether ether iodide iodide iodide iodide iodide Gallium Glycerol Hexane Hexane Kerosene Kerosene Kerosene Mercury Mercury Mercury Mercury

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

Substance

20 14 20 20 20 20 20 20 20 20 15 20 20 20 25 20 20 20 20 20 20 20 20 30 15 20 20 20 20 20 20 22 22 22

400 23 500 1,000 12,000 200 400 500 1,000 12,000 23 500 1,000 12,000 23 500 1,000 12,000 200 400 500 1,000 12,000 300 5 200 400 500 1,000 12,000 300 500 1,000 12,000

75 100 63 54 8 100 82 70 54 8 151 102 66 8 188 84 61 10 81 69 64 50 8 3.97 22 117 91 55 45 8 3.95 3.97 3.91 2.37

Methyl alcohol alcohol alcohol alcohol alcohol alcohol Nitric acid Oils: Almond Castor Linseed Olive Rapeseed Phosphorus trichloride trichloride trichloride trichloride Propyl alcohol (n) alcohol (n) alcohol (n?) alcohol (n?) alcohol (n?) Toluene Toluene Turpentine Water Water Water Water Water Water Water Xylene, meta meta

Temp., °C

Pressure, megabars

Compressibility per megabar β × 106

15 20 20 20 20 20 0

23 200 400 500 1,000 12,000 17

103 95 80 65 54 8 32

15 15 15 15 20 10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 20 20

5 5 5 5

53 46 51 55 59 71 63 47 8 77 67 65 47 7 74 64 74 49 43 41 39 38 33 9 69 60

250 500 1,000 12,000 200 400 500 1,000 12,000 200 400 13 200 400 500 500 1,000 12,000 200 400

* Smithsonian Tables, Table 106. Scott (Cryogenic Engineering, Van Nostrand, Princeton, N.J., 1959) gives data for liquid nitrogen (p. 283), oxygen (p. 276), and hydrogen (p. 303). For a convenient index to the high-pressure work of Bridgman, see American Institute of Physics Handbook, p. 2-163, McGraw-Hill, New York, 1957.

TABLE 2-146

Compressibilities of Solids

Many data on the compressibility of solids obtained prior to 1926 are contained in Gruneisen, Handbuch der Physik, vol. 10, Springer, Berlin, 1926, pp. 1–52; also available as translation, NASA RE 2-18-59W, 1959. See also Tables 271, 273, 276, 278, and other material in Smithsonian Physical Tables, 9th ed., 1954. For a review of high-pressure work to 1946, see Bridgman, Rev. Mod. Phys., 18, 1 (1946).

LATENT HEATS UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32 To convert calories per gram-mole to British thermal units per

pound-mole, multiply by 1.799; to convert calories per gram to British thermal units per pound, multiply by 1.799. To convert millimeters of mercury to pounds-force per square inch, multiply by 1.934 × 10−2.

LATENT HEATS TABLE 2-147

2-145

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds*

Unless stated otherwise, the values have been taken from the compilations by K. K. Kelley on “Heats of Fusion of Inorganic Compounds,” U.S. Bur. Mines Bull. 393 (1936), and “The Free Energies of Vaporization and Vapor Pressures of Inorganic Substances,” U.S. Bur. Mines Bull. 383 (1935).

Substance Aluminum Al Al2Br6 Al2Cl6 AlF3·3NaF Al2I6 Al2O3 Antimony Sb SbBr3 SbCl3 SbCl5 Sb4O6 Sb4S6 Argon A Arsenic As AsBr3 AsCl3 AsF5 As4O6 Barium Ba BaBr2 BaCl2 BaF2 Ba(NO3)2 Ba3(PO4)2 BaSO4 Beryllium Be Bismuth Bi BiBr3 BiCl3 Bi2O3 Bi2S5 Boron BBr3 BCl3 BF3 B2H6 B3H10 B5H9 B5H11 B10H14 B2H5Br B3N3H6 Bromine Br2 BrF5 Cadmium Cd CdBr2 CdCl2 CdF2 CdI2 CdO CdSO4 Calcium Ca CaBr2 CaCO3 CaCl2 CaF2 Ca(NO3)2 CaO CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaSO4 Carbon C (graphite) CBr4 CCl4 CF4 CH4 C2N2 CNBr CNCl

mp, °C

Heat of fusion,a,b cal/mol

bp at 1 atm, °C

Heat of vaporization,a,b cal/mol

660.0 97.5 192.5 1000 191.0 2045

2,550 5,420 16,960 16,380 7,960 (26,000)

2057 256.4 180.2c

61,020 10,920 26,750c

385.5 3000

15,360

630.5 97 73.4 4 655 546

4,770 3,510 3,030 2,400 (27,000) 11,200

1440

46,670

219 172d 1425

10,360 11,570 17,820

−185.8

1,590

610c

31,000c

122 −52.8 457.2

7,570 4,980 14,300

−189.3 814 31 −16 −80.7 313

290 (6,620) 2,810 2,420 2,800 8,000

704 847 960 1287 595 1730 1350

(1,400)e 6,000 5,370 3,000 (5,980) 18,600 9,700

1280

2,500e

271.3

2,505

224 817 747

2,600 6,800 8,900

−128 −165.5 −119.8 −46.9

480

99.7 −104 −58

7,800

−7.2 −61.3

2,580 1,355

320.9 568 568 1110 387

1,460 (5,000) 5,300 (5,400) 3,660

1000

4,790

851 730 1282 782 1392 561 2707 1550 1392 1512 1297

2,230 4,180 (12,700) 6,100 4,100 5,120 (12,240) 29,400 (18,200) 13,400 6,700

1638

1420 461 441

18,020 17,350

91.3 12.5 −100.9 −92.4 16 58 67 f 16 50.4

7,300 5,680 4,620 3,685 6,470 7,700 8,500 11,600 6,230 7,670

58.0 40.4

7,420 7,470

765

23,870

967

29,860

796 1559c

25,400 53,820c

1487

36,580

3600 90 −24.0

11,000e 1,050 644

−182.5 −27.8 52 −5

224 1,938u

77 −127.9 −161.4 −21.1

2,240

13

*See also subsection “Thermodynamic Properties.”

35,670

7,280 3,110 2,040 5,576u 11,010c 6,300

Substance Carbon (Cont.) CNF CNI CO CO2 COS COCl2 CS2 Cerium Ce Cesium Cs CsBr CsCl CsF CsI CsNO3 Chlorine Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 Chromium Cr CrO2Cl2 Cobalt Co CoCl2 Copper Cu Cu2Br2 Cu2Cl2 CuI Cu2(CN)2 Cu2O CuO Cu2S Fluorine F2 F2O Gallium Ga Germanium Ge GeH4 Ge2H6 Ge3H8 GeHCl3 GeBr4 GeCl4 Ge(CH3)4 Gold Au Helium He Hydrogen H2 HBr HCl HCN HF (HF)6 HI H2O H22O (= D2O) H2O2 HNO3 H3PO2 H3PO3 H3PO4 H4P2O6 H2S H2S2 H2SO4 H2Se H2SeO4 H2Te Indium In

mp, °C

Heat of fusion,a,b cal/mol

−205.0 −57.5 −138.8

200 1,900 1,129 k

−112.0

1,049 l

775

2,120

28.4

500

642 715

3,600 (2,450)

407

3,250

−101.0

1,531m

1550

3,930

1490 727

3,660 7,390

1083.0

3,110

430

4,890

473 1230 1447 1127

(5,400) (13,400) 2,820 5,500

−223 29.8

bp at 1 atm, °C −72.8 141 −191.5 −78.4c −50.2 8.0

5,780c 13,980c 1,444 6,030 c, r 4,423 k 5,990

690 1300 1300 1251 1280

16,320 35,990 35,690 34,330 35,930

−34.1 −101 11.3 2.0 10.9 79

959 −165 −109 −105.6 −71 26.1 −49.5 −88

(8,300)

1063.0

3,030

−271.4 −259.2 −86.9 −114.2 −13.2 −83.0

28 575 476 2,009i 1,094

−50.8 0.0 3.8 −2 −47 17.4 74 42.4 55 −85.5 −87.6 10.5

686 1,436 1,501s 2,520c 600 2,310 3,070 2,520 8,300 568t 1,805 2,360

58 −48.9

3,450 1,670

156.4

781

4,878 m 5,890 6,280 7,100 8,480

2475 117

8,250

1050

27,170

2595 1355 1490 1336

72,810 16,310 11,920 15,940

−188.2 −144.8 1,336

Heat of vaporization,a,b cal/mol

1,640 2,650

2071 −89.1 31.4 110.6 75g 189 84 44 2966

3,580 5,900 7,550 8,000 8,560 7,030 6,460 81,800

−268.4

22

−252.7 −66.7 −85.0 25.7 33.3 51.2

216 4,210 3,860 6,027i 7,460 5,020

100.0 101.4 158

9,729 h,q 9,945 r,q 10,270

−60.3

4,463 t

−41.3

4,880

−2.2

5,650

2-146

PHYSICAL AND CHEMICAL DATA

TABLE 2-147

Substance Iodine I2 ICl(α) ICl(β) IF7 Iron Fe FeCl2 Fe2Cl6 Fe(CO)5 FeO FeS Krypton Kr Lead Pb PbBr2 PbCl2 PbF2 PbI2 PbMoO4 PbO PbS PbSO4 PbWO4 Lithium Li LiBO2 LiBr LiCl LiF LiI LiOH Li2MoO4 LiNO3 Li2SiO3 Li4SiO4 Li2SO4 Li2WO4 Magnesium Mg MgBr2 MgCl2 MgF2 MgO Mg3(PO4)2 MgSiO3 MgSO4 MgZn2 Manganese Mn MnCl2 MnSiO3 MnTiO3 Mercury Hg HgBr2 HgCl2 HgI2 HgSO4 Molybdenum Mo MoF6 MoO3 Neon Ne Nickel Ni NiCl2 Ni(CO)4 Ni2S Ni3S2 Nitrogen N2 NF3 NH3 NH4CNS NH4NO3 N2O NO N2O4 N2O5 NOCl Osmium OsF8 OsO4 (yellow) OsO4 (white) Oxygen O2 O3

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Continued) mp, °C

Heat of fusion,a,b cal/mol

113.0 17.2 13.9

3,650 2,660 2,270

183 4c

7,460c

1530 677 304 −21 1380 1195

3,560 7,800 20,590 3,250 (7,700) 5,000

2735 1026 319 105

84,600 30,210 12,040 9,000

−157

360e

bp at 1 atm, °C

152.9

Heat of vaporization,a,b cal/mol 10,390

2,310e

327.4 488 498 824 412 1065 890 1114 1087 1123

1,224 4,290 5,650 1,860 5,970 (25,800) 2,820 4,150 9,600 (15,200)

1744 914 954 1293 872

42,060 27,700 29,600 38,300 24,850

1472 1281

51,310 (50,000)

179 845 552 614 847 440 462 705

1,100 (5,570) 2,900 3,200 (2,360) (1,420) 2,480 4,200

1372

32,250

1310 1382 1681 1171

35,420 35,960 50,970 40,770

1177 1249 857 742

7,210 7,430 3,040 (6,700)

650 711 712 1221 2642 1184 1524 1127 589

2,160 8,300 8,100 5,900 18,500 (11,300) 14,700 3,500 (8,270)

1220 650 1274 1404

3,450 7,340 (8,200) (7,960)

2152 1190

557 3,960 4,150 4,500 (1,440)

361 319 304 354

13,980 14,080 14,080 14,260

(6,660) 2,500 (2,500)

(4800) 36 1151

(128,000) 6,000

−38.9 241 277 250 850 2622 17 745 −248.5

1107 1418

32,520 32,690

55,150 29,630

77

−246.0

440e

1455

4,200

2730 987c 42.5

87,300 48,360c 7,000

645 790

(2,980) 5,800 −195.8 −129.0 −33.4

1,336 3,000 5,581n

−88.5 −151.7 30 32.4 −6.4

3,950 3,307 7,040 13,800c 6,140

47.4 130

6,840 9,450

−183.0 −111

1,629 2,880

−210.0 −77.7 146 169.6 −90.8 −163.6 −13

56 42 −218.9

172 1,352n (4,700) 1,460 1,563 550 5,540

4,060 2,340 106

Substance Palladium Pd Phosphorus P4 (yellow) P4 (violet) P4 (black) PCl3 PH3 P4O6 P4O10(α) P4O10(β) POCl3 P2S3 Platinum Pt Potassium K KBO2 KBr KCl KCN KCNS K2CO3 K2CrO4 K2Cr2O7 KF KI K2MoO4 KNO3 KOH KPO3 K3PO4 K4P2O7 K2SO4 K2TiO3 K2WO4 Praseodymium Pr Radon Rn Rhenium Re Re2O7 Re2O8 Rubidium Rb RbBr RbCl RbF RbI RbNO3 Selenium Se2 Se6 SeF6 SeO2 SeOCl2 Silicon Si SiCl4 Si2Cl6 Si3Cl8 (SiCl3)2O SiF4 Si2F6 SiF3Cl SiF2Cl2 SiH4 Si2H6 Si3H8 Si4H10 SiH3Br SiH2Br2 SiHCl3 (SiH3)3N (SiH3)2O SiO2 (quartz) SiO2 (cristobalite) Silver Ag AgBr AgCl AgCN AgI AgNO3 Ag2S Ag2SO4 Sodium Na NaBO2

mp, °C

Heat of fusion,a,b cal/mol

1554

4,120

bp at 1 atm, °C

44.2

615

−133.8 23.8 569

270o 3,360 17,080

1.1

3,110

1773.5

4,700

(4400)

(107,000)

63.5 947 742 770 623 179 897 984 398 857 682 922 338 360 817 1340 1092 1074 810 927

574 (5,700) 5,000 6,410 (3,500) 2,250 7,800 6,920 8,770 6,500 4,100 (4,000) 2,840 (2,000) 2,110 8,900 14,000 8,100 (10,600) (4,400)

776

18,920

1383 1407

37,060 38,840

1324

34,690

1327

30,850

932

2,700

−71 (3000) 296 147

15,340 3,800

39.1 677 717 833 638 305

525 3,700 4,400 4,130 2,990 1,340

217

1,220

10

1,010

1427 −67.6 −1

9,470 1,845

−33

280 417c 453c 74.2 −87.7 174 591 358c 105.1 508

Heat of vaporization,a,b cal/mol

8,380

−61.8

4,010

362.4

18,060

679 1352 1381 1408 1304 753 736 −45.8c 317c 168 2290 56.8 139 211.4 135.6 −94.8c −18.9c −70.1 −31.5 −111.6 −14.3 53.1 100 2.4 70.5 31.8 48.7 −15.4 2230

−18.5 −138 −144 −185 −132.5 −117 −93.5 −93.8 −70.0 −126.5 −105.6 −144 1470 1700

3,900

960.5 430 455 350 557 209 842 657

2,700 2,180 3,155 2,750 2,250 2,755 3,360 (4,300)

2212

97.7 966

630 8,660

3,400 2,100

12,520 25,600c 33,100 7,280 3,489 o 10,380 20,670

18,110 37,120 36,920 39,510 35,960 25,490 20,600 6,350c 20,900

6,860 12,340 8,820 6,130c 10,400c 4,460 5,080 2,960 5,110 6,780 8,890 5,650 6,840 6,360 6,850 5,350

60,720

1564

42,520

1506

34,450

914

23,120

LATENT HEATS TABLE 2-147

2-147

Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Concluded)

Substance Sodium (Cont.) NaBr NaCl NaClO3 NaCN NaCNS Na2CO3 NaF NaI Na2MoO4 NaNO3 NaOH aNa2O·aAl2O3·3SiO2 NaPO3 Na4P2O7 Na2S Na2SiO3 Na2Si2O5 Na2SO4 Na2WO4 Strontium Sr SrBr2 SrCl2 SrF2 Sr3(PO4)2 Sulfur S (rhombic) S (monoclinic) S2Cl2 SF6 SO2 SO3(α) SO3(β) SO3(γ) SOBr2 SOCl2 SO2Cl2 Tellurium Te TeCl4 TeF6

mp, °C

Heat of fusion,a,b cal/mol

747 800 255 562 323 854 992 662 687 310 322 1107 988 970 920 1087 884 884 702

6,140 7,220 5,290 (4,400) 4,450 7,000 7,000 5,240 3,600 3,760 2,000 13,150 (5,000) (13,700) (1,200) 10,300 8,460 5,830 5,800

757 643 872 1400 1770

2,190 4,780 4,100 4,260 18,500

112.8 119.2 −75.5 17 32.4 62.2

453

1,769p 2,060 2,890 6,310

3,230

bp at 1 atm, °C

Heat of vaporization,a,b cal/mol

1392 1465

37,950 40,810

1500

37,280

1704

53,260

1378

1384

33,610

444.6

2,200

138 −63.5c −5.0 44.8

8,720 5,600c 5,960p 10,190

139.5 75.4 69.2

9,920 7,600 7,760

1090 392 −38.6c

16,830 6,700c

Substance Thallium Tl TlBr TlCl Tl2CO3 TlI TlNO3 Tl2S Tl2SO4 Tin Sn4 SnBr2 SnBr4 SnCl2 SnCl4 Sn(CH3)4 SnH4 SnI4 Titanium TiBr4 TiCl4 TiO2 Tungsten W WF6 Uranium UF6 Xenon Xe Zinc Zn ZnCl2 Zn(C2H5)2 ZnO ZnS Zirconium ZrBr4 ZrCl4 ZrI4 ZrO2

a

k

b

l

c

m

Values in parentheses are uncertain. For the freezing point or the normal boiling point unless otherwise stated. Sublimation. d Decomposes at about 75 °C; value obtained by extrapolation. e Bichowsky and Rossini, Thermochemistry of the Chemical Substances, Reinhold, New York (1936). f Decomposes before the normal boiling point is reached. g Decomposes at about 40 °C; value obtained by extrapolation. h See also pp. 2-304 through 2-307 on steam table. i Giauque and Ruehrwein, J. Am. Chem. Soc., 61 (1939): 2626. j Giauque and Egan, J. Chem. Phys., 5 (1937): 45.

TABLE 2-148

mp, °C

Alloys 30.5 Pb + 69.5 Sn 36.9 Pb + 63.1 Sn 63.7 Pb + 36.3 Sn 77.8 Pb + 22.2 Sn 1 Pb + 9 Sn 24 Pb + 27.3 Sn + 48.7 Bi 25.8 Pb + 14.7 Sn + 52.4 Bi + 7 Cd Silicates Anorthite (CaAl2Si2O8) Orthoclase (KAlSi2O8) Microcline (KAlSi3O8) Wollastonite (CaSiO8) Malacolite (Ca8MgSi4O12) Diopside (CaMgSi2O4) Olivine (Mg2SiO4) Fayalite (Fe2SiO4) Spermaceti Wax (bees’)

bp at 1 atm, °C

Heat of vaporization,a,b cal/mol

302.5 460 427 273 440 207 449 632

1,030 5,990 4,260 4,400 3,125 2,290 3,000 5,500

1457 819 807

38,810 23,800 24,420

823

25,030

231.8 232 30 247 −33.2

1,720 (1,700) 3,000 3,050 2,190

2270

68,000

−149.8 143.5

(4,300)

38.2 −23 1825

(2,060) 2,240 (11,400)

136

3390 −0.4

(8,400) 1,800

(5900) 17.3

(176,000) 6,350

55.1c

9,990c

−111.5 419.5 283 1975 1645

2715

740 1,595 (5,500)

623 113 78.3 −52.3

20,740 8,330 7,320 4,420

8,350

−108.0

3,110

907 732 118

27,430 28,710 8,960

357c 311c 431c

25,800c 25,290c 29,030c

4,470 (9,000)

20,800

Kemp and Giauque, J. Am. Chem. Soc., 59 (1937): 79. Brown and Manov, J. Am. Chem. Soc., 59 (1937): 500. Giauque and Powell, J. Am. Chem. Soc. 61 (1939): 1970. n Overstreet and Giauque, J. Am. Chem. Soc 59 (1937): 254. o Stephenson and Giauque, J. Chem. Phys., 5 (1937): 149. p Giauque and Stephenson, J. Am. Chem. Soc., 60 (1938): 1389. q Osborne, Stimson, and Ginnings, Bur. Standards J. Research, 23, 197 (1939): 261. r Miles and Menzies, J. Am. Chem. Soc., 58 (1936): 1067. s Long and Kemp, J. Am. Chem. Soc., 58 (1936): 1829. t Giauque and Blue, J. Am. Chem. Soc., 58 (1936): 831. u Ruehrwein and Giauque, J. Am. Chem. Soc., 61 (1939): 2940.

Heats of Fusion of Miscellaneous Materials

Material

Heat of fusion,a,b cal/mol

mp, °C

Heat of fusion, cal/g

183 179 177.5 176.5 236 98.8 75.5

17 15.5 11.6 9.54 28 6.85 8.4

43.9 61.8

100 100 83 100 94 100 130 85 37.0 42.3

2-148

PHYSICAL AND CHEMICAL DATA

TABLE 2-149

Heats of Fusion of Organic Compounds

The values for the hydrocarbons are from the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards, with some from Parks and Huffman, Ind. Eng. Chem., 23, 1138 (1931). The values for the nonhydrocarbon compounds were recalculated from data in International Critical Tables, vol. 5. Hydrocarbon compounds Paraffins Methane Ethane Propane n-Butane 2-Methylpropane n-Pentane 2-Methylbutane 2,2-Dimethylpropane n-Hexane 2-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane n-Heptane 2-Methylhexane 3-Ethylpentane 2,2-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,2,3-Trimethylbutane n-Octane 2-Methylheptane 3-Methylpentane 4-Methylheptane 2,2-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,3,3-Tetramethylbutane n-Nonane n-Decane n-Undecane n-Dodecane Eicosane Pentacosane Tritriacontane Aromatics Benzene Methylbenzene (Toluene) Ethylbenzene o-Xylene m-Xylene p-Xylene n-Propylbenzene Isopropylbenzene 1-Methyl-2-ethylbenzene

Formula

mp, °C

Heat of fusion, cal/g

CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C5H12 C6H14 C6H14 C6H14 C6H14 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C9H20 C10H22 C11H24 C12H26 C20H42 C25H52 C33H68

−182.48 −183.23 −187.65 −138.33 −159.60 −129.723 −159.890 −16.6 −95.320 −153.680 −99.73 −128.41 −90.595 −118.270 −118.593 −123.790 −119.230 −134.46 −24.96 −56.798 −109.04 −120.50 −120.955 −121.18 −91.200 −126.10 −114.960 −90.870 −112.27 −107.365 −100.70 −109.210 +100.69 −53.9 −30.0 −25.9 −9.6 +36.4 +53.3 +71.1

14.03 22.712 19.100 19.167 18.668 27.874 17.076 10.786 36.138 17.407 1.607 2.251 33.513 21.158 22.555 13.982 15.968 16.856 5.250 43.169 21.458 23.795 22.692 24.226 26.903 14.9 23.690 22.657 18.061 19.278 3.204 19.392 14.900 41.2 48.3 34.1 51.3 52.0 53.6 54.0

C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12 C9H12 C9H12

+5.533 −94.991 −94.950 −25.187 −47.872 +13.263 −99.500 −96.028 −80.833

30.100 17.171 20.629 30.614 26.045 38.526 16.97 19.22 21.13 Heat of fusion, cal/g

Formula

mp, °C

Acetic acid Acetone Acrylic acid Allo-cinnamic acid Aminobenzoic acid (o-) (m-) (p-) Amyl alcohol Anethole Aniline Anthraquinone Apiol Azobenzene Azoxybenzene

C2H4O2 C3H6O C3H4O2 C9H8O2 C7H7NO2 C7H7NO2 C7H7NO2 C5H12O C10H12O C6H5NH2 C14H8O2 C12H14O4 C12H10N2 C12H10N2O

16.7 −95.5 12.3 68 145 179.5 188.5 −78.9 22.5 −6.3 284.8 29.5 67.1 36

46.68 23.42 37.03 27.35 35.48 38.03 36.46 26.65 25.80 27.09 37.48 25.80 28.91 21.62

Benzil Benzoic acid Benzophenone Benzylaniline Bromocamphor Bromochlorbenzene (o-) (m-) (p-) Bromoiodobenzene (o-) (m-) (p-) Bromol hydrate Bromophenol (p-) Bromotoluene (p-)

C14H10O2 C7H8O2 C13H10O C13H13N C10H15BrO C6H4BrCl C6H4BrCl C6H4BrCl C6H4BrI C6H4BrI C6H4BrI C2H3Br3O2 C6H5BrO C7H7Br

95.2 122.45 47.85 32.37 78 −12.6 −21.2 64.6 21 9.3 90.1 46 63.5 28

22.15 33.90 23.53 21.86 41.57 15.41 15.29 23.41 12.18 10.27 16.60 16.90 20.50 20.86

Nonhydrocarbon compounds

Hydrocarbon compounds Aromatics—(Cont.) 1-Methyl-3-ethylbenzene 1-Methyl-4-ethylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene Naphthalene Camphene Durene Isodurene Prehnitene p-Cymene n-Butyl benzene tert-Butyl benzene β-Methyl naphthalene Diphenyl Hexamethyl benzene Diphenyl methane Anthracene Phenanthrene Tolane Stilbene Dibenzil Triphenyl methane Alkyl cyclohexanes Cyclohexane Methylcyclohexane Alkyl cyclopentanes Cyclopentane Methylcyclopentane Ethylcyclopentane 1,1-Dimethylcyclopentane cis-1,2-Dimethylcyclopentane trans-1,2-Dimethylcyclopentane trans-1,3-Dimethylcyclopentane Monoolefins Ethene (Ethylene) Propene (Propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutene) 1-Pentene cis-2-pentene trans-2-pentene 2-Methyl-1-butene 3-Methyl-1-butene 2-Methyl-2-butene Acetylenes Acetylene 2-Butyne (dimethylacetylene) Nonhydrocarbon compounds

mp, °C

Heat of fusion, cal/g

C9H12 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C11H10 C12H10 C12H18 C13H12 C14H10 C14H10 C14H10 C14H12 C14H14 C19H16

−95.55 −62.350 −25.375 −43.80 −44.720 +80.0 +51 +79.3 −24.0 −7.7 −68.9 −88.5 −58.1 +34.1 +68.6 +165.5 +25.2 +216.5 +96.3 +60 +124 +51.4 +92.1

15.14 25.29 16.64 24.54 18.97 36.0 57 37.4 23.0 20.0 17.1 19.5 14.9 20.1 28.8 30.4 26.4 38.7 25.0 28.7 40.0 30.7 21.1

C6H12 C7H14

+6.67 −126.58

7.569 16.429

C5H10 C6H12 C7H14 C7H14 C7H14 C7H14 C7H14

−93.80 −142.445 −138.435 −69.73 −53.85 −117.57 −133.680

2.068 19.68 11.10 3.36 3.87 15.68 17.93

C2H4 C3H6 C4H8 C4H8 C4H8 C4H8 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10

−169.15 −185.25 −185.35 −138.91 −105.55 −140.35 −165.27 −151.363 −140.235 −137.560 −168.500 −133.780

28.547 17.054 16.393 31.135 41.564 25.265 16.82 24.239 26.536 26.879 18.009 25.738

C2H2 C4H6

−81.5 −132.23

23.04 40.808

Formula

Formula

mp, °C

Heat of fusion, cal/g

Butyl alcohol (n-) (t-) Butyric acid (n-)

C4H10O C4H10O C4H8O2

−89.2 25.4 −5.7

29.93 21.88 30.04

Capric acid (n-) Caprylic acid (n-) Carbazole Carbon tetrachloride Carvoxime (d-) (l-) (dl-) Cetyl alcohol Chloracetic acid (α-) (β-) Chloral alcoholate hydrate Chloroaniline (p-) Chlorobenzoic acid (o-) (m-) (p-) Chloronitrobenzene (m-) (p-) Cinnamic acid anhydride Cresol (p-) Crotonic acid (α-) (cis-) Cyanamide Cyclohexanol

C10H20O2 C8H16O2 C12H9N CCl4 C10H15NO C10H15NO C10H15NO C16H34O C2H3ClO2 C2H3ClO2 C4H7Cl3O2 C2H3Cl3O2 C6H6ClN C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4ClNO2 C6H4ClNO2 C9H8O2 C18H14O3 C7H8O C4H6O2 C4H6O2 CH2N2 C6H12O

31.99 16.3 243 −22.8 71.5 71 91 49.27 61.2 56 9 47.4 71 140.2 154.25 239.7 44.4 83.5 133 48 34.6 72 71.2 44 25.46

38.87 35.40 42.05 41.57 23.29 23.41 24.61 33.80 31.06 35.12 24.03 33.18 37.15 39.30 36.41 49.21 29.38 31.51 36.50 28.14 26.28 25.32 34.90 49.81 4.19

LATENT HEATS TABLE 2-149

2-149

Heats of Fusion of Organic Compounds (Concluded) Heat of fusion, cal/g

Formula

mp, °C

Dibromobenzene (o-) (m-) (p-) Dibromophenol (2, 4-) Dichloroacetic acid Dichlorobenzene (o-) (m-) (p-) Dihydroxybenzene (o-) (m-) (p-) Di-iodobenzene (o-) (m-) (p-) Dimethyl tartrate (dl-) (d-) pyrone Dinitrobenzene (o-) (m-) (p-) Dinitrotoluene (2, 4-) Dioxane Diphenyl amine

C6H4Br2 C6H4Br2 C6H4Br2 C6H4Br2O C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H6O2 C6H6O2 C6H6O2 C6H4I2 C6H4I2 C6H4I2 C6H10O6 C6H10O6 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C7H6N2O4 C4H8O2 C12H11N

1.8 −6.9 86 12 −4(?) −16.7 −24.8 53.13 104.3 109.65 172.3 23.4 34.2 129 87 49 132 116.93 89.7 173.5 70.14 11.0 52.98

12.78 13.38 20.55 13.97 14.21 21.02 20.55 29.67 49.40 46.20 58.77 10.15 11.54 16.20 35.12 21.50 56.14 32.25 24.70 39.99 26.40 34.85 25.23

Elaidic acid Ethyl acetate alcohol Ethylene dibromide Ethyl ether

C18H34O2 C4H8O2 C2H6O C2H4Br2 C4H10O

44.4 83.8 −114.4 10.012 −116.3

52.08 28.43 25.76 13.52 23.54

Formic acid

CH2O2

8.40

58.89

Glutaric acid Glycerol Glycol, ethylene

C6H8O4 C3H8O3 C2H6O2

97.5 18.07 −11.5

37.39 47.49 43.26

Hydrazo benzene Hydrocinnamic acid Hydroxyacetanilide

C12H12N2 C9H10O2 C8H9NO2

134 48 91.3

22.89 28.14 33.59

Iodotoluene (p-) Isopropyl alcohol ether

C7H7I C3H8O C6H14O

34 −88.5 −86.8

18.75 21.08 25.79

Lauric acid (n-) Levulinic acid

C12H24O2 C5H8O3

43.22 33

43.72 18.97

Menthol (l-) (α) Methyl alcohol Myristic acid Methyl cinnamate fumarate oxalate phenylpropiolate succinate

C10H20O CH4O C14H28O2 C10H10O2 C6H8O4 C4H6O4 C10H8O2 C6H10O4

43.5 −97.8 53.86 36 102 54.35 18 19.5

18.63 23.7 47.49 26.53 57.93 42.64 22.86 35.72

Nonhydrocarbon compounds

Formula

mp, °C

Heat of fusion, cal/g

Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzene Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene Nitrophenol (o-)

C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C6H5NO2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2 C6H5NO3

95.0 120.6 50 71.2 114.0 147.3 5.85 145.8 141.1 239.2 56.7 45.13

38.94 31.30 22.34 27.88 40.97 36.46 22.52 40.06 27.59 52.80 25.44 26.76

Palmitic acid Paraldehyde Pelargic acid (n-) (β-) Pelargonic acid (n-) (α-) Phenol Phenylacetic acid Phenylhydrazine Propyl ether (n)

C16H32O2 C6H12O3 C9H18O2 C9H18O2 C6H6O C8H8O2 C6H8N2 C6H14O

61.82 10.5 12.35 40.92 76.7 19.6 −126.1

39.18 25.02 39.04 30.63 29.03 25.44 36.31 20.66

Quinone

C6H4O2

115.7

40.85

Stearic acid Succinic anhydride Succinonitrile

C18H30O2 C4H4O3 C4H4N2

68.82 119 54.5

47.54 48.74 11.71

Tetrachloroxylene (o-) (p-) Thiophene Thiosinamine Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-) Tribromophenol (2, 4, 6-) Trichloroacetic acid Trinitroglycerol Trinitrotoluene (2, 4, 6-) Tristearin

C8H6Cl4 C8H6Cl4 C4H4S C4H8N2S C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N C6H3Br3O C2HCl3O2 C3H5N3O9 C7H5N3O6 C57H110O6

86 95 −39.4 77 51.5 103.7 108.75 179.6 43.3 93 57.5 12.3 80.83 70.8, 54.5

21.02 22.10 14.11 33.45 27.47 35.40 27.59 39.90 39.90 13.38 8.60 23.02 22.34 45.63

Undecylic acid (α-) (n-) (β-) (n-) Urethane

C11H22O2 C11H22O2 C3H7NO2

28.25 48.7

32.20 42.91 40.85

Veratrol

C8H10O2

22.5

27.45

Xylene dibromide (o-) (m-) dichloride (o-) (m-) (p-)

C8H8Br2 C8H8Br2 C8H8Cl2 C8H8Cl2 C8H8Cl2

Nonhydrocarbon compounds

95 77 55 34 100

24.25 21.45 29.03 26.64 32.73

2-150

TABLE 2-150 Heats of Vaporization of Inorganic and Organic Liquids (J/kmol) Cmpd. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Name Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol

Formula

CAS no.

Mol. wt.

C1 × 1E-07

C2

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138

3.8366 8.107 4.0179 6.352 4.215 4.3511 2.3214 3.8736 4.3756 4.155 0.8474 3.1523 5.8662 0.87308 8.7809 4.5346 6.225 10.19 6.8077 10.523 8.4762 6.228 6.9642 7.635 4 5.552 3.9004 3.169 3.522 3.2632 3.6238 8.9754 9.2247 7.1274 7.9227 3.3774 3.4358 3.3191 5.8276 6.3487 4.9702 4.6432 3.6972 4.6403 6.1947 5.22 2.173 3.496 0.8585 4.3252 1.9311 3.068 5.148 3.524 4.186 2.9745 3.989 3.8871 8.0082

0.40081 0.42 2.6037 0.3986 0.3397 0.34765 0.35938 0.29335 2.2571 0.2733 0.3822 0.3914 0.37127 0.3526 0.1933 0.39053 0.4412 0.478 0.63344 0.87091 0.35251 0.3411 0.44354 0.39182 0.351 0.37694 0.38012 0.3015 0.395 0.3701 0.8337 0.45316 0.42442 0.0483 0.58361 0.5107 0.38004 0.36968 0.38854 0.38222 0.41199 0.399 0.39168 0.3849 1.6524 0.165 0.382 0.2986 0.4921 0.37688 0.94983 0.8458 0.36614 0.3652 0.3584 0.353 0.37956 0.38043 0.45314

C3

C4

−5.0031

2.7069

−4.5116

2.5738

−0.2289

0.2309

0.30877

−0.14162

−0.27365 −0.45568 0.43853

−0.3026

−0.82274

0.39613

0.8966 0.02016 −0.17304

−0.5116 −0.08654 0.05181

−2.8505 0.6692 −0.4339

1.6285 −0.539 0.42213

−0.326

0.2231

−1.0615 −0.9001

0.51894 0.453

Tmin, K

Hv at Tmin × 1E-07

Tmax, K

Hv at Tmax

150.15 353.15 289.81 200.15 178.45 229.32 192.40 185.45 286.15 189.63 59.15 195.41 235.65 83.78 403.00 278.68 258.27 395.45 260.40 321.35 257.85 275.65 243.95 342.20 265.85 242.43 154.55 179.47 136.95 164.25 134.86 220.00 196.15 183.85 158.45 87.80 134.26 167.62 199.65 185.30 157.46 133.02 147.43 176.75 250.00 161.25 216.58 161.11 68.13 250.33 89.56 172.12 227.95 134.80 209.63 175.43 150.35 155.97 285.39

3.2828 6.2386 2.3412 5.4139 3.6390 3.5996 1.6333 3.3881 2.7965 3.6866 0.6759 2.5298 4.9560 0.6561 7.1286 3.4705 5.0597 7.1277 5.3147 7.4895 6.8800 5.1829 5.7930 6.0719 3.2323 4.6875 3.3933 2.7379 3.0540 2.7235 2.8684 7.5185 7.9759 6.3643 6.4607 3.0197 2.9867 2.7630 4.9384 5.5979 4.3505 4.1614 3.1511 3.9797 4.1619 4.7223 1.5202 3.1537 0.6517 3.4528 1.4215 2.2878 4.3707 3.1052 3.5047 2.4520 3.4862 3.3586 6.3326

466.00 761.00 591.95 606.00 508.20 545.50 308.30 506.00 615.00 535.00 132.45 405.65 645.60 150.86 824.00 562.05 689.00 751.00 699.35 830.00 720.15 662.00 718.00 773.00 584.15 670.15 503.80 467.00 452.00 425.00 425.12 680.00 676.00 563.10 535.90 419.50 435.50 428.60 575.40 660.50 570.10 554.00 440.00 537.20 615.70 582.25 304.21 552.00 132.50 556.35 227.51 417.15 632.35 460.35 536.40 416.25 503.15 489.00 705.85

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 915,280 0 0 0 0 0 0 0 0 0 0

2-151

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine

C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N

95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7

108.138 108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133 194.184 88.105 170.207 101.190

7.1979 8.4942 5.766 3.384 3.334 4.4902 9.1791 5.6772 4.4405 3.8911 3.8107 2.7672 5.6067 7.9073 6.6126 13.107 7.9041 6.6985 8.0617 6.9461 0.1657 5.712 5.37 4.82 5.9616 5.6899 6.2117 5.9765 4.2117 4.5507 4.186 4.774 4.675 10.154 4.6133 4.06 4.7659 3.2312 3.4552 2.8081 5.007 4.6117 5.0256 4.3872 4.7999 3.856 4.09 4.1509 5.0402 5.2852 5.1194 4.9825 2.994 5.9217 4.6533 8.1578 2.8365 3.9022 6.629 7.236 5.051 6.8243 5.428

0.40317 0.50234 0.38939 0.3707 0.3395 0.39881 0.6382 0.37431 0.37479 0.36111 0.3543 0.35588 0.38729 0.4129 0.39797 1.0674 −1.36 0.76944 0.41045 0.42109 0.352 0.5255 0.416 0.3771 0.38833 0.35765 0.42845 0.38559 0.36927 0.34444 0.4092 0.39204 0.36529 0.3403 0.42628 0.3868 0.37987 0.37653 0.3499 0.3364 0.4362 0.4 0.29611 0.56226 0.30724 0.3737 0.42005 0.38383 0.4036 0.41607 0.405 0.3958 0.3505 0.37996 0.37577 0.29346 0.35393 0.37731 0.4084 0.2424 0.3791 0.30877 0.3665

−0.97372 4.0854 −0.79975

−0.60662 −0.024545

0.40491 −2.3871 0.42379

0.4202 0.091361

304.19 307.93 177.14 245.25 182.48 279.69 296.60 242.00 169.67 179.28 138.13 145.59 189.64 267.15 243.51 304.55 280.05 206.89 247.56 229.15 18.73 210.15 282.85 220.60 175.30 248.39 256.15 326.14 176.19 237.49 178.01 200.00 172.71 301.15 223.35 156.85 169.20 154.56 215.00 136.95 176.85 187.65 204.81 159.95 226.10 240.91 180.96 145.19 239.66 223.16 184.99 188.44 131.65 212.72 160.00 274.18 122.93 174.88 291.67 413.80 284.95 300.03 210.15

5.7135 6.3649 5.0717 2.3803 2.8083 3.3920 6.2221 4.7739 3.8791 3.3299 3.4046 2.3532 4.9220 6.4201 5.4168 8.7931 8.2959 5.3524 6.7308 5.7192 0.1309 4.6111 4.2346 4.0709 5.1902 4.8419 5.1191 4.6573 3.6189 3.7657 3.5116 4.0147 4.0997 8.4908 3.5761 3.4651 4.1537 2.6659 2.7427 2.3781 4.1823 3.8207 4.4125 3.7528 4.0557 2.9557 3.2678 3.6388 4.0862 4.3662 4.4043 4.3108 2.6032 5.0931 4.0745 7.1632 2.4928 3.3213 5.3804 6.0070 3.9263 5.8546 4.5500

697.55 704.65 631.00 400.15 459.93 553.80 650.10 653.00 560.40 511.70 507.00 398.00 664.00 674.20 617.70 722.10 688.00 616.60 696.00 619.85 38.35 628.00 650.15 611.00 584.10 683.95 705.00 684.75 523.00 561.60 510.00 560.00 572.00 736.60 496.60 466.70 557.15 386.44 445.00 351.26 523.10 500.05 576.00 507.80 543.00 473.20 437.20 500.00 591.15 606.15 596.15 615.00 400.10 649.60 537.30 766.00 402.00 503.04 729.00 772.00 587.00 766.80 550.00

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2-152

TABLE 2-150 Heats of Vaporization of Inorganic and Organic Liquids (J/kmol) (Continued) Cmpd. no. 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181

Name Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne

Formula

CAS no.

Mol. wt.

C1 × 1E-07

C2

C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10

112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2

170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144

7.7337 12.86 2.1091 5.5789 4.933 4.275 5.4805 6.7093 7.898 5.6419 5.3832 4.8287 1.8844 5.7521 8.3518 4.94 3.6652 4.5909 11.184 6.2786 4.258 5.2207 3.844 5.3325 5.438 4.9482 0.88757 4.582 2.7617 2.4708 3.076 7.358 2.3195 4.005 0.012504 10.473 5.956 5.0014 11.274 7.0236 9.6433 6.3357 6.1425 4.9437 6.5473 4.8222 10.156 5.6661 4.4544 9.0746 7.035 11.55 5.6231 5.6232 4.1429 4.808 5.8422 4.574 4.911

0.40681 0.50351 0.60646 0.31245 0.3847 0.5857 0.39524 0.33273 0.39445 0.37985 0.41763 0.37804 0.36485 0.34513 0.42625 0.466 0.37878 0.4123 0.86189 0.39513 0.37221 0.34893 0.37534 0.401 0.60624 0.39871 0.34072 0.3717 0.32162 0.37014 0.2954 0.3564 1.9091 0.3995 1.3038 0.4374 0.36474 0.38795 0.86047 −1.3652 0.783 0.42167 0.39802 0.35428 0.40968 0.33858 0.45726 0.38533 0.39002 0.8926 −0.9575 2.2877 0.38207 0.39972 0.49118 0.436 0.38704 0.3698 0.4392

C3

C4

0.32986 −0.55492

−0.42184 0.32799

−0.332

0.169

−0.47845

0.048646

−5.0003

3.2641

−2.6954

1.7098

−0.40661 3.987 −0.27273

−0.012644 −2.2545 0.038495

0.22149

−0.2353

−0.75172 3.1431 −3.6724

0.34378 −1.8066 2.1326

−0.44821

0.32105

Tmin, K

Hv at Tmin × 1E-07

Tmax, K

Hv at Tmax

263.57 309.58 90.35 159.05 189.60 192.15 178.20 238.45 258.15 175.15 161.84 134.71 104.00 284.29 260.15 195.20 160.65 193.55 235.00 180.00 140.00 204.15 125.26 199.25 145.65 167.55 53.48 230.94 129.95 131.35 181.15 275.70 250.00 196.29 2.20 295.13 229.80 182.57 265.83 239.15 230.00 234.15 238.15 154.12 229.92 192.22 291.31 217.15 177.83 269.25 228.55 223.00 217.35 217.50 133.39 170.05 192.62 141.25 183.65

6.2802 9.5933 1.7879 4.9694 4.1490 3.2955 4.7900 5.8382 6.4816 4.9090 4.7318 4.3603 1.5936 4.5918 6.8989 4.0022 3.1271 3.7679 8.2832 5.4262 3.7556 4.4677 3.4489 4.4449 4.4140 4.2942 0.7578 3.7605 2.4089 2.0276 2.5863 6.2844 1.8865 3.2647 0.0097 8.3699 5.0248 4.2619 7.9579 7.6498 6.9638 5.1579 5.0471 4.3208 5.4656 4.1647 8.0225 4.7495 3.7647 6.4783 7.1509 6.5014 4.7135 4.6655 3.6691 4.0831 5.0630 4.0640 4.1067

658.00 768.00 305.32 514.00 523.30 456.15 617.15 698.00 655.00 571.00 609.15 569.50 282.34 593.00 720.00 537.00 469.15 508.40 674.60 583.00 489.00 567.00 499.15 546.00 500.23 559.95 144.12 560.09 375.31 317.42 408.00 771.00 588.00 490.15 5.20 736.00 616.80 540.20 677.30 632.30 608.30 606.60 611.40 537.40 645.00 547.00 723.00 591.00 507.60 660.20 611.30 585.30 587.61 582.82 504.00 544.00 623.00 516.20 549.00

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2-153

182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244

Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl -1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether

H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O

302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4

32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122 90.187 46.144 118.176 88.148

5.9794 0.10127 2.485 2.2093 3.349 13.451 2.5676 4.0385 4.4041 11.767 4.6095 1.0194 5.0451 7.3402 4.492 3.1889 4.68 3.858 6.8504 4.1233 3.7593 7.48 10.178 3.9091 3.9248 3.648 4.5302 5.3416 3.792 5.3781 3.2835 4.7528 6.477 7.8011 7.8995 4.3595 4.3541 4.209 3.6756 3.53 4.622 4.4842 4.103 4.2678 5.4687 4.2967 3.8501 4.7075 4.5052 3.4448 5.468 10.53 4.2522 5.0002 3.188 7.7646 3.2614 5.008 4.2719 4.8253 2.2656 5.8071 3.872

0.9424 0.698 0.39 0.3466 0.2053 13.36 0.37358 0.82698 0.43325 0.37877 0.23331 0.26087 0.33594 0.38974 0.3685 0.37881 0.349 0.404 0.38852 0.426 0.39173 0.3933 1.3211 0.39866 0.36173 0.3863 0.37779 0.3835 0.3565 0.39523 0.33116 0.39437 0.4853 0.4172 0.42479 0.38507 0.36805 0.36779 0.31266 0.376 0.355 0.41151 0.3825 0.37995 0.40583 0.37922 0.36453 0.33601 0.36493 0.37427 0.4472 0.7454 0.3807 0.3781 0.39006 0.56757 0.38073 0.3959 0.43175 0.38087 0.30269 0.37009 0.044

−1.398 −1.817

−23.383

0.8862 1.447

10.785

−2.033

1.4769

−0.14694

0.22154

−1.2234

0.44836

−0.39297

0.047214

0.448

−0.112

274.69 13.95 185.15 158.97 259.83 277.56 187.68 227.15 177.95 407.95 288.15 90.69 175.47 301.15 175.15 170.45 196.32 179.69 260.75 159.53 113.25 193.00 155.95 135.58 139.39 160.15 157.48 175.30 183.45 187.35 139.05 146.58 299.15 280.15 269.15 130.73 146.62 115.00 182.55 160.00 186.48 167.23 174.15 150.00 189.15 256.15 127.93 180.15 171.64 150.18 224.95 240.00 119.55 176.00 113.54 298.97 132.81 185.65 133.97 160.17 116.34 249.95 164.55

4.5238 0.0913 1.8817 1.7498 2.8176 0.7104 1.9782 3.5534 3.5874 9.0033 4.0342 0.8724 4.3825 5.9384 3.8418 2.5882 3.9913 3.1006 5.7026 3.4864 3.3657 6.5004 7.3738 3.4072 3.4558 3.1332 3.9438 4.6699 3.1681 4.5694 2.8974 4.2291 4.9050 6.0500 6.1926 3.9115 3.8769 3.8439 3.1686 2.9751 3.9704 3.8406 3.4644 3.7232 4.6507 3.2402 3.4235 4.1240 3.9340 2.9825 4.3596 8.1106 3.8300 4.3168 2.8070 4.6771 2.8195 4.2231 3.7041 4.2499 2.0069 4.8591 3.6017

653.15 33.19 363.15 324.65 456.65 461.15 373.53 605.00 471.85 805.00 662.00 190.56 512.50 718.00 506.55 402.40 536.00 430.05 693.00 490.00 460.40 643.00 577.20 465.00 470.00 492.00 512.74 593.00 463.20 554.50 442.00 572.10 686.00 614.00 617.00 532.70 542.00 526.00 483.00 437.80 535.50 533.00 487.20 497.00 574.60 488.00 464.48 553.40 553.10 469.95 566.00 694.00 497.70 546.49 407.80 506.20 417.90 530.60 476.25 565.00 352.50 654.00 497.10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2-154

TABLE 2-150 Heats of Vaporization of Inorganic and Organic Liquids (J/kmol) (Concluded) Cmpd. no. 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302

Name Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate

Formula

CAS no.

Mol. wt.

C1 × 1E-07

C2

C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2

107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4

58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132

3.587 7.0911 0.2389 5.1459 0.74905 1.6402 4.7417 2.3215 2.131 11.674 7.3363 6.037 12.38 7.5429 7.9797 5.9054 7.5239 6.3337 10.969 6.7735 5.518 12.23 7.2468 7.6376 6.5363 6.6142 5.4859 6.8907 5.4046 11.473 0.9008 1.8587 9.6741 5.1478 3.9109 7.3197 7.39 11.111 5.174 5.2359 3.5027 5.0573 5.4315 3.954 4.4158 8.3482 7.306 5.5769 6.916 2.9535 2.9209 6.8988 7.2542 5.8866 4.1492 4 4.9348 5.4327

0.3769 0.46468 0.3494 0.33017 0.40406 0.36494 0.3062 0.384 0.4056 0.45865 0.41735 0.38522 0.69869 −1.5966 −1.0341 0.61039 0.3991 0.3975 0.44327 0.40607 0.38467 0.69294 −1.2464 −0.7612 0.38718 0.58562 0.26207 0.40017 0.35299 0.37238 0.4542 0.30416 0.45399 0.37541 0.38681 1.2093 -0.1464 1.8011 0.39422 0.40465 0.3481 0.45827 0.3972 0.3512 0.44347 0.33172 0.4246 0.30346 0.1755 0.41367 0.78237 0.6458 0.79137 0.38533 0.36751 1.3936 0.41873 0.407

C3

−0.317

C4

0.27343

0.097854 4.6489 3.553 −0.54533

−0.35082 −2.7229 −2.1149 0.30683

0.12287 3.6797 2.7875

−0.36132 −2.0665 −1.6033

−0.40512 0.50642

0.22144 −0.43873

−0.4096

0.3183

−1.9114 1.4751 −2.1801

1.1591 −0.9208 1.0641

−0.19672 −0.22568

0.22394 0.16393

−0.77319 −0.5384 −0.66092

0.39246 0.3317 0.34223

−2.9465

1.794

Tmin, K

Hv at Tmin × 1E-07

Tmax, K

151.15 353.43 24.56 183.63 63.15 66.46 244.60 182.30 109.50 305.04 255.15 219.66 285.55 268.15 238.15 191.91 253.05 223.15 301.31 246.00 216.38 289.65 257.65 241.55 252.85 255.55 171.45 223.95 193.55 462.65 54.36 80.15 283.07 182.00 143.42 239.15 195.56 200.00 196.29 234.18 108.02 160.75 197.45 167.45 163.83 372.38 314.06 243.15 404.15 136.87 85.47 146.95 185.26 199.00 170.00 252.45 180.26 178.15

3.0567 5.2691 0.1803 4.5533 0.6024 1.4519 4.0220 1.6502 1.4578 9.2185 5.9779 5.0545 8.7232 8.2411 8.0370 4.9218 6.2506 5.2606 8.7246 5.5600 4.5898 8.4658 7.6793 7.3759 5.3646 5.2076 4.7927 5.8506 4.6743 8.3393 0.7742 1.6625 7.6728 4.4502 3.3968 5.3813 6.7005 6.6655 4.3663 4.2075 3.2232 4.4343 4.6322 3.4025 3.7321 6.9340 5.6577 4.8418 6.1001 2.4755 2.4787 5.8356 5.5370 5.0941 3.5675 3.0922 4.2005 4.6322

437.00 748.40 44.40 593.00 126.20 234.00 588.15 309.57 180.15 758.00 658.00 594.60 710.70 670.90 649.50 593.10 681.00 598.05 747.00 638.90 568.70 694.26 652.30 629.80 632.70 627.70 566.90 667.30 574.00 804.00 154.58 261.00 708.00 566.10 469.70 639.16 588.10 561.00 561.08 560.95 464.80 584.30 598.00 481.20 519.00 869.00 694.25 653.00 791.00 394.00 369.83 536.80 508.30 636.00 504.40 600.81 564.40 549.73

Hv at Tmax 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C3H9N C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

59.110 120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

4.4488 5.8887 2.5216 4.9687 4.2191 4.4782 8.07 6.49 2.4105 5.726 12.018 3.676 2.571 7.337 8.824 8.7165 9.0539 4.3021 6.8086 5.0642 4.9055 4.5854 4.9507 5.0929 8.4339 4.664 3.305 5.9996 5.9254 4.7711 4.991 10.687 10.686 7.2284 8.7274 4.77 3.649 3.4125 4.5659 5.2053 5.4626 5.5395 5.3819

0.39494 0.38534 0.33721 0.4025 0.41161 0.41073 0.295 0.3112 0.37988 0.4055 0.37149 0.4 0.383 0.5647 0.3224 0.44467 0.36972 0.43054 0.38904 0.40678 0.38756 0.37742 0.38013 0.4257 0.3663 0.354 0.35578 0.35709 0.37949 0.383 0.38 0.40074 0.40607 −1.5834 0.3765 0.4 0.4513 0.36278 0.3199 0.37289 0.37788 0.36695

−0.18399

5.0913

0.22377

−3.2171

0.043 −0.212

0.25795

188.36 173.55 87.89 180.25 142.61 159.95 213.15 388.85 186.35 242.54 460.65 197.67 223.15 289.95 298.15 329.35 279.01 164.65 237.38 176.99 373.96 234.94 178.18 236.50 267.76 158.45 156.08 247.79 229.33 165.78 172.22 398.40 354.00 247.57 288.45 180.35 173.15 119.36 178.35 273.16 225.30 247.98 286.41

3.6857 5.2110 2.3177 4.2162 3.6942 3.8723 7.1374 4.9933 1.4873 4.7128 8.7719 2.8753 1.6208 4.4303 8.8240 7.4548 7.2002 3.7610 5.7314 4.4565 3.1691 3.7484 4.3246 4.2130 6.8015 4.1011 2.8216 5.0818 5.0713 4.1561 4.3530 8.3906 8.5455 5.9240 8.9007 4.0619 2.9876 2.9491 3.9520 4.4733 4.6112 4.5859 4.2787

496.95 638.35 364.85 538.00 517.00 536.60 626.00 683.00 259.00 636.00 806.00 430.75 318.69 490.85 298.15 857.00 693.00 540.15 720.00 631.95 568.00 579.35 591.75 602.00 675.00 535.15 433.25 664.50 649.10 543.80 573.50 846.00 828.00 639.00 703.90 519.13 454.00 432.00 543.15 647.10 617.00 630.30 616.20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 88,240,000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The heat of vaporization Hv is calculated by Hv = C1(1 − Tr) C2+ C3T + C4T + C5T r

2 r

3 r

where Tr = T/Tc, Tc is the critical temperature from Table 2-141, Hv is in J/kmol, and T is in K. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

2-155

2-156

PHYSICAL AND CHEMICAL DATA

SPECIFIC HEATS OF PURE COMPOUNDS UNITS CONVERSIONS

To convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Rankine, multiply by 0.2388.

For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32 °R = 1.8 K

ADDITIONAL REFERENCES

To convert calories per gram-kelvin to British thermal units per pound-degree Rankine, multiply by 1.0; to convert calories per mole-kelvin to British thermal units per pound-mole-degree Rankine, multiply by 1.0. TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds* Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

c l c l c l c c c c l c l c c, sillimanite c, disthene c, andalusite c, mullite c c c

4.80 + 0.00322T 7.00 18.74 + 0.01866T 29.5 13.25 + 0.02800T 31.2 76 19.3 50.5 38.63 + 0.04760T − 449200/T 2 142 16.88 + 0.02266T 28.8 22.08 + 0.008971T − 522500/T 2 40.79 + 0.004763T − 992800/T 2 41.81 + 0.005283T − 1211000/T 2 43.96 + 0.001923T − 1086000/T 2 59.65 + 0.0670T 113.2 + 0.0652T 63.5 235

273–931 931–1273 273–370 370–407 273–465 465–504 288–327 288–326 288–326 273–1273 1273–1373 273–464 464–480 273–1973 273–1573 273–1673 273–1573 273–576 273–575 273–373 288–325

1 5 3 5 3 3 ? ? ? 2 ? 3 5 3 3 2 3 5 3 ? ?

c l c c c c c

5.51 + 0.00178T 7.15 17.2 + 0.0293T 10.3 + 0.0511T 19.1 + 0.0171T 22.6 + 0.0162T 24.2 + 0.0132T

273–903 903–1273 273–370 273–346 273–929 273–1198 273–821

2 5 ? ? ? ? ?

g

4.97

c l c c

5.17 + 0.00234T 31.9 8.37 + 0.0486T 25.8

273–1168 286–371 273–548 293–373

5 ? ? ?

c c c c c, α c, β c c c

17.0 + 0.00334T 28.2 37.3 51 17.26 + 0.0131T 30.0 34 39.8 21.35 + 0.0141T

273–1198 273–307 273–307 289–320 273–1083 1083–1255 273–297 285–371 273–1323

? ? ? ? 5 15 ? ? 5

c c c c

4.698 + 0.001555T − 121000/T 2 8.69 + 0.00365T − 313000/T 2 25.4 20.8

273–1173 273–1175 273–373 273–373

1 5 ? ?

Substance Aluminum1 Al AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·3aH2O AlF3·3NaF AlI3 Al2O3 Al2O3·SiO2 3Al2O3·2SiO2 4Al2O3·3SiO2 Al2(SO4)3 Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 Sb2O3 Sb2O4 Sb2S3 Argon2 A Arsenic As AsCl3 As2O3 As2S3 Barium BaCl2 BaCl2·H2O BaCl2·2H2O Ba(ClO3)2·H2O BaCO3 BaMoO4 Ba(NO3)2 BaSO4 Beryllium3,4 Be BeO BeO·Al2O3 BeSO4

Additional data are contained in the subsection “Thermodynamic Properties.” Data on water are also contained in that subsection. Additional tables for water are found in Eng. Sci. Data Item 68008, 251 Regent Street, London, England, which contains about 5000 values from 1 to 1000 bar, 0 to 1500 °C.

State†

All

0

*From Kelley, U.S. Bur. Mines Bull. 371, 1934. For a revision see Kelley, U.S. Bur. Mines Bull. 477, 1948. Data for many elements and compounds are given by Johnson (ed.), WADD-TR-60-56, 1960, for cryogenic temperatures. Tabulated data for gases can be obtained from many of the references cited in the “Thermodynamic Properties” subsection and other tables in this section. Thinh, Duran, et al., Hydrocarbon Process., 50, 98 (January 1971), review previous equation fits and give newer fits for 408 hydrocarbons and related compounds. Later publications include Duran, Thinh, et al., Hydrocarbon Process., 55, 153 (August 1976); Thompson, J. Chem. Eng. Data, 22(4), 431 (1977); and Passut and Danner, Ind. Eng. Chem. Process Des. Dev., 11, 543 (1972); 13, 193 (1974). † The symbols in this column have the following meaning; c, crystal; l, liquid; g, gas; gls, glass.

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued)

Substance Bismuth4 Bi Bi2O3 Bi2S3 Boron B B2O3 BN Bromine Br2 Cadmium Cd CdO CdS CdSO4·8/3H2O Calcium Ca CaCl2 CaCO3 CaF2 CaMg(CO3)2 CaMoO4 CaO Ca(OH)2 CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaP2O6 CaSO4 CaSO4·2H2O CaWO4 Carbon5 C CH4 CO6 CO2 CS2 Cerium Ce CeO2 Ce2(MoO4)3 Ce2(SO4)3 Ce2(SO4)3·5H2O Cesium Cs CsBr CsCl CsF CsI Chlorine Cl2 Chromium4 Cr CrCl3 Cr2O3 CrSb CrSb2 Cr2(SO4)3 Cobalt4 Co CoAs2·CoS2 CoSb Co2Sn CoS CoSO4·7H2O

State†

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

c l c c

5.38 + 0.00260T 7.60 23.27 + 0.01105T 30.4

273–544 544–1273 273–777 284–372

3 3 2 ?

c gls gls c

1.54 + 0.00440T 5.14 + 0.0320T 30.4 1.61 + 0.00400T

273–1174 273–513 513–623 273–1173

5 3 3 5

g

9.00

300–2000

5

c l c c c

5.46 + 0.002466T 7.13 9.65 + 0.00208T 12.9 + 0.00090T 51.3

273–594 594–973 273–2086 273–1273 293

1 5 ? ? ?

c c c c c c c c c c, anorthite gls c, diopside gls c, wollastonite c, pseudowollastonite gls c c c c

5.31 + 0.00333T 6.29 + 0.00140T 16.9 + 0.00386T 19.68 + 0.01189T − 307600/T 2 14.7 + 0.00380T 40.1 33 10.00 + 0.00484T − 108000/T 2 21.4 63.13 + 0.01500T − 1537000/T 2 67.41 + 0.01048T − 1874000/T 2 54.46 + 0.005746T − 1500000/T 2 51.68 + 0.009724T − 1308000/T 2 27.95 + 0.002056T − 745600/T 2 25.48 + 0.004132T − 488100/T 2 23.16 + 0.009672T − 487100/T 2 39.5 18.52 + 0.02197T − 156800/T 2 46.8 27.9

273–673 673–873 273–1055 273–1033 273–1651 299–372 273–297 273–1173 276–373 273–1673 273–973 273–1573 273–973 273–1573 273–1673 273–973 287–371 273–1373 282–373 292–322

2 2 ? 3 ? ? ? 2 ? 1 1 1 1 1 1 1 ? 5 ? ?

c, graphite c, diamond g g g l

2.673 + 0.002617T − 116900/T 2 2.162 + 0.003059T − 130300/T 2 5.34 + 0.0115T 6.60 + 0.00120T 10.34 + 0.00274T − 195500/T 2 18.4

273–1373 273–1313 273–1200 273–2500 273–1200 293

2 3 2 1a 1a ?

c c c c c

5.88 + 0.00123T 15.1 96 66.4 131.6

273–908 273–373 273–297 273–373 273–319

? ? ? ? ?

c l g c c c c

1.96 + 0.0182T 8.00 4.97 12.6 + 0.00259T 11.7 + 0.00309T 11.3 + 0.00285T 11.6 + 0.00268T

273–301 302 All 273–909 273–752 273–957 273–894

3 3 0 ? ? ? ? 1a

g

8.28 + 0.00056T

273–2000

c l c c c c c

4.84 + 0.00295T 9.70 23 26.0 + 0.00400T 12.3 + 0.00120T 19.2 + 0.00184T 67.4

273–1823 1823–1923 286–319 273–2263 273–1383 273–949 273–373

5 10 ? ? ? ? ?

c l c c c c c

5.12 + 0.00333T 8.40 32.9 11.7 + 0.00156T 15.83 + 0.00950T 10.6 + 0.00251T 96

273–1763 1763–1873 283–373 273–1464 273–903 273–1373 286–303

5 5 ? ? 2 ? ?

2-157

2-158

PHYSICAL AND CHEMICAL DATA TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued)

Substance Copper7 Cu CuAl CuAl2 Cu3Al CuI CuI2 CuO CuO·SiO2·H2O CuS Cu2S CuS·FeS Cu2Sb Cu2Sb Cu2Se Cu3Si CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Fluorine8 F2 Gallium Ga2O3 Ga2(SO4)3 Germanium4 Ge Gold Au AuSb2 Helium9 He Hydrogen10 H H2 HBr HCl HI H 2O H 2S H2S2O7 Indium In Iodine I2 Iridium Ir Iron4 Fe

FeAs2 Fe3C FeCO3 FeO Fe2O3 Fe3O4 Fe2O3·3H2O FeS FeS2 FeSi Fe2SiO4 FeSO4 Fe2(SO4)3 FeSO4·4H2O FeSO4·7H2O Krypton Kr

State†

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

273–1357 1357–1573 273–733 273–773 273–775 273–675 274–328 273–810 293–323 273–1273 273–376 376–1173 292–321 273–573 273–693 273–383 383–488 273–1135 282 282 282 282

1 3 2 2 2 ? ? 2 ? ? 3 2 ? 2 2 5 5 ? ? ? ? ?

c l c c c c c c c c c, α c, β c c c c, α c, β c c c c c

5.44 + 0.001462T 7.50 9.88 + 0.00500T 16.78 + 0.00366T 19.61 + 0.01054T 12.1 + 0.00286T 20.1 10.87 + 0.003576T − 150600/T 2 29 10.6 + 0.00264T 9.38 + 0.0312T 20.9 24 13.73 + 0.01350T 21.79 + 0.00900T 20.85 20.35 20.3 + 0.00587T 24.1 31.3 49.0 67.2

g

6.50 + 0.00100T

300–3000

5

c c

18.2 + 0.0252T 62.4

273–923 273–373

? ?

273–1336 1336–1573 273–628 628–713

2 5 1 ?

c c l c, α c, βγ

5.61 + 0.00144T 7.00 17.12 + 0.00465T 11.47 + 0.01756T

g

4.97

g g g g g l g g c l

4.97 6.62 + 0.00081T 6.80 + 0.00084T 6.70 + 0.00084T 6.93 + 0.00083T See Tables 2-153 and 2-305 8.22 + 0.00015T + 0.00000134T 2 7.20 + 0.00360T 27 58

All 273–2500 273–2000 273–2000 273–2000

0 2 2 1a 2

300–2500 300–600 281 308

? 8 ? ?

9.00

300–2000

5

All

0

c g c

5.50 + 0.00148T

c, α c, β c, γ c, δ l c c c c c c c c, α c, β c c c c c c c

4.13 + 0.00638T 6.12 + 0.00336T 8.40 10.0 8.15 17.8 25.17 + 0.00223T 22.7 12.62 + 0.001492T − 76200/T 2 24.72 + 0.01604T − 423400/T 2 41.17 + 0.01882T − 979500/T 2 47.8 2.03 + 0.0390T 12.05 + 0.00273T 10.7 + 0.01336T 10.54 + 0.00458T 33.57 + 0.01907T − 879700/T 2 22 66.2 63.6 96

g

4.97

273–1873

1

273–1041 1041–1179 1179–1674 1674–1803 1803–1873 283–373 273–1173 293–368 273–1173 273–1097 273–1065 286–373 273–411 411–1468 273–773 273–903 273–1161 293–373 273–373 282 291–319

3 3 5 5 5 ? 10 ? 2 2 2 ? 5 3 ? 2 2 ? ? ? ?

All

0

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued)

Substance Lanthanum La La2O3 La2(MoO4)3 La2(SO4)3 La2(SO4)3·9H2O Lead4 Pb Pb3(AsO4)2 PbB2O4 PbB4O7 PbBr2 PbCl2 2PbCl2·NH4Cl PbCO3 PbCrO4 PbF2 PbI2 PbMoO4 Pb(NO3)2 PbO PbO2 Pb2P2O7 PbS PbSO4 PbS2O3 PbWO4 Lithium Li LiBr LiBr·H2O LiCl LiCl·H2O LiF LiI LiI·H2O LiI·2H2O LiI·3H2O LiNO3 Magnesium4 Mg MgAg Mg4Al3 MgAu Mg2Au Mg3Au MgCl2 MgCl2·6H2O MgCO3 MgCu2 Mg2Cu MgNi2 MgO MgO·Al2O3 MgO·SiO2 6MgO·MgCl2·8B2O3 Mg(OH)2 Mg3Sb2 Mg2Si MgSO4 MgSO4·H2O MgSO4·6H2O MgSO4·7H2O

State†

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

c c c c c

5.91 + 0.00100T 22.6 + 0.00544T 86 66.9 152

273–1009 273–2273 273–307 273–373 273–319

? ? ? ? ?

c l c c c c l c l c c c c c l c c c c c c c c c

5.77 + 0.00202T 6.8 65.5 26.5 41.4 18.13 + 0.00310T 27.4 15.88 + 0.00835T 27.2 53.1 21.1 29.1 16.5 + 0.00412T 18.66 + 0.00293T 32.3 30.4 36.4 10.33 + 0.00318T 12.7 + 0.00780T 48.3 10.63 + 0.00401T 26.4 29 35

273–600 600–1273 286–370 288–371 289–371 273–761 761–860 273–771 771–851 293 286–320 292–323 273–1091 273–648 648–776 292–322 286–320 273–544 273–? 284–371 273–873 293–372 293–373 273–297

2 5 ? ? ? 2 10 2 10 ? ? ? ? 2 20 ? ? 2 ? ? 3 ? ? ?

c g c c c c c c c c c c l

0.68 + 0.0180T 4.97 11.5 + 0.00302T 22.6 11.0 + 0.00339T 23.6 8.20 + 0.00520T 12.5 + 0.00208T 23.6 32.9 43.2 9.17 + 0.0360T 26.8

273–459 All 273–825 278–318 273–887 279–360 273–1117 273–723 277–359 277–345 277–347 273–523 523–575

10 0 ? ? ? ? ? ? ? ? ? 5 5

c l c c c c c c c c c c c c c c, amphibole c, pyroxene gls c, α c, β c c c c c c c

6.20 + 0.00133T − 67800/T 2 7.4 10.58 + 0.00412T 34.4 + 0.0198T 11.3 + 0.00189T 16.2 + 0.00451T 21.2 + 0.00614T 17.3 + 0.00377T 77.1 16.9 14.96 + 0.00776T 15.5 + 0.00652T 15.87 + 0.00692T 10.86 + 0.001197T − 208700/T 2 28 25.60 + 0.004380T − 674200/T 2 23.35 + 0.008062T − 558800/T 2 23.30 + 0.007734T − 542000/T 2 58.7 + 0.408T 107.2 + 0.2876T 18.2 28.2 + 0.00560T 15.4 + 0.00415T 26.7 33 80 89

273–923 923–1048 273–905 273–736 273–1433 273–1073 273–1103 273–991 292–342 290 273–903 273–843 273–903 273–2073 288–319 273–1373 273–773 273–973 273–538 538–623 292–323 273–1234 273–1343 296–372 282 282 291–319

1 10 2 ? ? ? ? ? ? ? 3 ? 2 2 ? 1 1 1 5 5 ? ? ? ? ? ? ?

2-159

2-160

PHYSICAL AND CHEMICAL DATA TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued) Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

c, α c, β c, γ l c c c c c c c c c c

3.76 + 0.00747T 5.06 + 0.00395T 4.80 + 0.00422T 11.0 16.2 + 0.00520T 7.79 + 0.0421T + 0.0000090T 2 7.43 + 0.01038T − 0.00000362T 2 10.33 + 0.0530T − 0.0000257T 2 19.25 + 0.0538T − 0.0000209T 2 1.92 + 0.0471T − 0.0000297T 2 31 10.21 + 0.00656T − 0.00000242T 2 27.5 78

273–1108 1108–1317 1317–1493 1493–1673 273–923 273–773 273–1923 273–1173 273–1773 273–773 291–322 273–1883 293–373 290–319

5 5 5 10 ? ? ? ? ? ? ? ? ? ?

l g g c c c c c, α c, β c c c

6.61 4.97 9.00 11.05 + 0.00370T 15.3 + 0.0103T 25 11.4 + 0.00461T 17.4 + 0.004001T 20.2 11.5 10.9 + 0.00365T 31.0

273–630 All 300–2000 273–798 273–553 285–319 273–563 273–403 403–523 278–371 273–853 273–307

1 0 5 ? ? ? ? 3 3 ? ? ?

c c c

5.69 + 0.00188T − 50300/T 2 15.1 + 0.0121T 19.7 + 0.00315T

273–1773 273–1068 273–729

5 ? ?

g

4.97

c, α c, β l c c c c c c c c

4.26 + 0.00640T 6.99 + 0.000905T 8.55 11.3 + 0.00215T 9.25 + 0.00640T 15.8 + 0.00329T 10.0 + 0.00312T 20.78 + 0.0102T 33.4 82 11.00 + 0.00433T

g g c c, α c, β c c c g

6.50 + 0.00100T 6.70 + 0.00630T 22.8 9.80 + 0.0368T 5.0 + 0.0340T 17.8 31.8 51.6 8.05 + 0.000233T − 156300/T 2

300–3000 300–800 274–328 273–457 457–523 273–328 273–293 275–328 300–5000

3 1a ? 5 5 ? ? ? 2

c

5.686 + 0.000875T

273–1877

1

g

8.27 + 0.000258T − 187700/T

300–5000

1

Substance Manganese Mn

MnCl2 MnCO3 MnO Mn2O3 Mn3O4 MnO2 Mn2O3·H2O MnS MnSO4 MnSO4·5H2O Mercury11 Hg Hg2 HgCl HgCl2 Hg(CN)2 HgI HgI2 HgO HgS Hg2SO4 Molybdenum Mo MoO3 MoS2 Neon12 Ne Nickel4 Ni NiO NiS Ni2Si NiSi Ni3Sn NiSO4 NiSO4·6H2O NiTe Nitrogen13 N2 NH3 NH4Br NH4Cl NH4I NH4NO3 (NH4)2SO4 NO Osmium Os Oxygen14 O2 Palladium Pd Phosphorus P PCl3 P4O10 Platinum4 Pt Potassium K

State†

All 273–626 626–1725 1725–1903 273–1273 273–597 273–1582 273–1273 273–904 293–373 291–325 273–700

2

0 2 5 10 ? 3 ? ? 2 ? ? 2

c

5.41 + 0.00184T

273–1822

2

c, yellow c, red l l c g

5.50 0.21 + 0.0180T 6.6 28.7 15.72 + 0.1092T 73.6

273–317 273–472 317–373 284–371 273–631 631–1371

5 10 10 ? 2 3

c

5.92 + 0.00116T

273–1873

1

c l

5.24 + 0.00555T 7.7

273–336 336–373

5 5

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued)

Substance Potassium—(Cont.) K K2 KAsO3 KBO2 K2B4O7 KBr KCl KClO3 KClO4 2KCl·CuCl2.2H2O 2KCl·PtCl4 2KCl·SnCl4 2KCl·ZnCl2 2KCN·Zn(CN)2 K2CO3 K2CrO4 K2Cr2O7 KF K4Fe(CN)6 K4Fe(CN)6·3H2O KH2AsO4 KH2PO4 KHSO4 KMnO4 KNO3 K2O·Al2O3·3SiO2

K4P2O7 K2SO4 K2S2O3 K2SO4·Al2(SO4)3·24H2O K2SO4·Cr2(SO4)3·24H2O K2SO4·MgSO4·6H2O K2SO4·NiSO4·6H2O K2SO4·ZnSO4·6H2O Prometheum Pr Radon Rn Rhenium Re Rhodium Rh Rubidium Rb RbBr RbCl Rb2CO3 RbF RbI Scandium Sc2O3 Sc2(SO4)3 Selenium Se Silicon Si SiC SiCl4 SiO2

Silver4 Ag

State† g g c c c c c c c c c c c c c c c l c c c c c c c c c l c, orthoclase gls, orthoclase c, microcline gls, microcline c c c c c c c c

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK) 4.97 9.00 25.3 12.6 + 0.0126T 51.3 11.49 + 0.00360T 10.93 + 0.00376T 25.7 26.3 63 55 54.5 43.4 57.4 29.9 35.9 42.80 + 0.0410T 96.9 10.8 + 0.00284T 80.1 114.5 32 28.3 30 28 6.42 + 0.0530T 28.8 29.5 69.26 + 0.00821T − 2331000/T 2 69.81 + 0.01053 − 2403000/T 2 65.65 + 0.01102T − 1748000/T 2 64.83 + 0.01438T − 1641000/T 2 63.1 33.1 37 352 324 106 107 120

Range of temperature, K All 300–2000 290–372 273–1220 290–372 273–543 273–1043 289–371 287–318 292–323 286–319 292–323 279–319 277–319 296–372 289–371 273–671 671–757 273–1129 273–319 273–310 289–319 290–320 292–324 287–318 273–401 401–611 611–683 273–1373 273–1373 273–1373 273–1373 290–371 287–371 293–373 292–322 292–324 292–323 289–319 293–317

Uncertainty, % 0 5 ? ? ? 2 2 ? ? ? ? ? ? ? ? ? 5 5 ? ? ? ? ? ? ? 10 5 10 1a 1a 1a 1a ? ? ? ? ? ? ? ?

c g

4.97

c

6.30 + 0.00053T

273–2273

All

?

c

5.40 + 0.00219T

273–1877

2

c l c c c c c

3.27 + 0.0131T 7.85 11.6 + 0.00255T 11.5 + 0.00249T 28.4 11.3 + 0.00256T 11.6 + 0.00263T

273–312 312–373 273–954 273–987 291–320 273–1048 273–913

2 5 ? ? ? ? ?

c c

21.1 62.0

273–373 273–373

? ?

c l

4.53 + 0.00550T 8.35

273–490 490–570

2 3

c c l c, quartz, α c, quartz, β c, cristobalite, α c, cristobalite, β gls

5.74 + 0.000617T − 101000/T 2 8.89 + 0.00291T − 284000/T 2 32.4 10.87 + 0.008712T − 241200/T 2 10.95 + 0.00550T 3.65 + 0.0240T 17.09 + 0.000454T − 897200/T 2 12.80 + 0.00447T − 302000/T 2

273–1174 273–1629 293–373 273–848 848–1873 273–523 523–1973 273–1973

2 2 ? 1 3a 2a 2 3a

c l

5.60 + 0.00150T 8.2

273–1234 1234–1573

0

1 3

2-161

2-162

PHYSICAL AND CHEMICAL DATA TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Continued)

Substance Silver—(Cont.) Ag3Al Ag2Al AgAl12 AgBr AgCl AgCNO AgI AgNO3 Ag3PO4 Ag2S Ag3Sb Ag2Se Sodium15 Na NaBO2 Na2B4O7 Na2B4O7·10H2O NaBr NaCl NaClO3 NaCNO Na2CO3 NaF Na2HPO4·7H2O Na2HPO4·12H2O NaI NaNO3 Na2O·Al2O3·3SiO2 NaPO3 Na4P2O7 Na2SO4 Na2S2O3 Na2S2O3·5H2O Sodium-potassium alloys15 Strontium SrBr2 SrBr2·H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCO3 SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O SrMoO4 Sr(NO3)2 SrSO4 Sulfur16 S S2 S2Cl2 SO2 Tantalum Ta Tellurium Te Thallium Tl

State†

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

273–902 273–903 273–768 273–703 703–836 273–728 728–806 273–353 273–423 273–433 433–482 482–541 293–325 273–448 448–597 273–694 273–406 406–460

2 2 5 6 5 2 5 ? 6 2 5 5 ? 5 5 5 5 5

273–371 371–451 All 273–1239 289–371 292–323 273–543 273–1074 1073–1205 273–528 528–572 273–353 288–371 273–1261 275–307 275–307 273–936 273–583 583–703 273–1373 273–1173 290–319 290–371 289–371 273–307 273–307

1a 2 0 ? ? ? 2 2 3 3 5 ? ? ? ? ? ? 5 10 1 1 ? ? ? ? ?

c c c c l c l c c, α c, α c, β l c c, α c, β c c, α c, β

22.56 + 0.00570T 16.85 + 0.00450T 58.62 + 0.0575T 8.58 + 0.0141T 14.9 9.60 + 0.00929T 14.05 18.7 8.58 + 0.0141T 18.83 + 0.0160T 25.7 30.2 37.5 18.8 21.8 19.53 + 0.0160T 20.2 20.4

c l g c c c c c l c l c c c c c c c l c, albite gls c c c c c l

5.01 + 0.00536T 7.50 4.97 10.4 + 0.0199T 47.9 147 11.74 + 0.00233T 10.79 + 0.00420T 15.9 9.48 + 0.0468T 31.8 13.1 28.9 10.4 + 0.00289T 86.6 133.4 12.5 + 0.00162T 4.56 + 0.0580T 37.2 63.78 + 0.01171T − 1678000/T 2 61.25 + 0.01768T − 1545000/T 2 22.1 60.7 32.8 34.9 86.2

c c c c c c c c c c c c c c

18.1 + 0.00311T 28.9 82.1 18.2 + 0.00244T 28.7 38.3 21.8 18.6 + 0.00304T 28.5 39.1 84.9 37 38.3 26.2

273–923 277–370 276–327 273–1143 276–365 277–366 281–371 273–783 276–363 275–336 275–333 273–297 290–320 293–369

? ? ? ? ? ? ? ? ? ? ? ? ? ?

c, rhombic c, monoclinic g l g

3.63 + 0.00640T 4.38 + 0.00440T 8.58 + 0.00030T 27.5 7.70 + 0.00530T − 0.00000083T 2

273–368 368–392 300–2500 273–332 300–2500

3 3 5 ? 2a

c

5.91 + 0.00099T

273–1173

2

c

5.19 + 0.00250T

273–600

3

c, α c, β

5.32 + 0.00385T 8.12

273–500 500–576

1 1

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-151

Heat Capacities of the Elements and Inorganic Compounds (Concluded)

Substance Thallium—(Cont.) Tl TlBr TlCl Thorium Th ThO2 Th(SO4)2 Tin4 Sn SnAu SnCl2 SnCl4 SnO SnO2 SnPt SnS SnS2 Titanium Ti TiCl4 TiO2 Tungsten W WO3 Uranium U U3O8 Vanadium V Xenon Xe Zinc4 Zn ZnCl2 ZnO ZnS ZnSb ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O Zirconium ZrO2 ZrO2·SiO2 1

State†

Heat capacity at constant pressure (T = K; 0 °C = 273.1 K), cal/(molK)

Range of temperature, K

Uncertainty, %

l c l c l

7.12 12.53 + 0.00100T 16.0 12.56 + 0.00088T 14.2

576–773 273–733 733–800 273–700 700–803

3 10 10 5 10

c c c

6.40 14.6 + 0.00507T 41.2

273–373 273–1273 273–373

? ? ?

c l c c l c c c c c

5.05 + 0.00480T 6.6 11.79 + 0.00233T 16.2 + 0.00926T 38.4 9.40 + 0.00362T 13.94 + 0.00565T − 252000/T 2 11.49 + 0.00190T 12.1 + 0.00165T 20.5 + 0.00400T

273–504 504–1273 273–581 273–520 286–371 273–1273 273–1373 273–1318 273–1153 273–873

2 10 1 ? ? ? ? 1 ? ?

c l c

8.91 + 0.00114T − 433000/T 2 35.7 11.81 + 0.00754T − 41900/T 2

273–713 285–372 273–713

3 ? 3

c c

5.65 + 0.00866 16.0 + 0.00774T

273–2073 273–1550

1 ?

c c

6.64 59.8

273–372 276–314

? ?

c

5.57 + 0.00097T

273–1993

?

g

4.97

c l c c c c c c c c

5.25 + 0.00270T 7.59 + 0.00055T 15.9 + 0.00800T 11.40 + 0.00145T − 182400/T 2 12.81 + 0.00095T − 194600/T 2 11.5 + 0.00313T 28 34.7 80.8 100.2

273–692 692–1122 273–638 273–1573 273–1173 273–810 293–373 282 282 273–307

1 3 ? 1 5 ? ? ? ? ?

c c

11.62 + 0.01046T − 177700/T 2 26.7

273–1673 297–372

5 ?

All

0

See also Table 2-152. Data to 298 K are also given by Scott, Cryogenic Engineering, Van Nostrand, Princeton, N.J., 1959. For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. Stalder, NACA Tech. Note 4141, 1957 (Fig. 5), gives data from 400 to 2600°R. 4 See also Table 2-152. 5 For data from 400 to 5500 °R see Stalder, NACA Tech. Note 4141, 1975 (Fig. 4). 6 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 7 For data from 400 to 2350 °R see Stalder, NACA Tech. Note 4141, 1957. 8 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 9 For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 10 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 11 See also Table 2-152; Douglas, Ball, et al., Bur. Stand. J. Res., 46 (1951): 334; Busey and Giaque, J. Am. Chem. Soc., 75 (1953): 806; Sheldon, ASME Pap. 49-A-30, 1949. 12 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 13 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 14 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. Ozone: For liquid see Brabets and Waterman, J. Chem. Phys., 28 (1958): 1212. 15 For data on liquid Na-K alloys to 1500 °F and for liquid Na to 1460 °F, see Lubarsky and Kaufman, NACA Rep. 1270, 1956. 16 See also Evans and Wagman, Bur. Stand. J. Res. 49 (1952): 141; Gratch, OTS PB 124957, 1950; Guthrie, Scott et al., J. Am. Chem. Soc., 76 (1954): 1488. 2 3

2-163

2-164

PHYSICAL AND CHEMICAL DATA

TABLE 2-152

Specific Heat [kJ/(kg·K)] of Selected Elements Temperature, K

Symbol

4

6

8

10

20

40

60

80

100

200

250

300

400

600

800

Al Be Bi Cr Co

0.00026 0.00008 0.00054 0.00016 0.00036

0.00050

0.00088

0.214

0.357

0.00541 0.00050 0.00085

0.0089 0.0014 0.0340 0.0021 0.0048

0.0775

0.00220 0.00029 0.00059

0.00140 0.00028 0.01040 0.00081 0.00121

0.0729 0.0107 0.0404

0.092 0.059 0.110

0.102 0.127 0.184

0.481 0.195 0.109 0.190 0.234

0.797 1.109 0.120 0.382 0.376

0.859 1.537 0.121 0.424 0.406

0.902 1.840 0.122 0.450 0.426

0.949 2.191 0.123 0.501 0.451

1.042 2.605 0.142 0.565 0.509

1.134 2.823 0.136 0.611 0.543

Cu Ge Au Ir Fe

0.00011

0.00024

0.00018

0.00047

0.00048 0.00037 0.00126

0.137 0.108 0.084

0.203 0.153 0.100

0.00061

0.00090

0.0076 0.0129 0.0163 0.0021 0.0039

0.059 0.0619 0.0569

0.00038

0.00086 0.00081 0.00255 0.00032 0.00127

0.0276

0.086

0.154

0.254 0.192 0.109 0.090 0.216

0.357 0.286 0.124 0.122 0.384

0.377 0.305 0.127 0.128 0.422

0.386 0.323 0.129 0.131 0.450

0.396 0.343 0.131 0.133 0.491

0.431 0.364 0.136 0.140 0.555

0.448 0.377 0.141 0.146 0.692

Pb Mg Hg Mo Ni

0.00075 0.00034 0.00417 0.00011 0.00054

0.00242 0.00080 0.01420 0.00019 0.00086

0.00747 0.00155 0.01820 0.00032 0.00121

0.01350 0.00172 0.02250 0.00050 0.00178

0.0531 0.0148 0.0515 0.0029 0.0058

0.0944 0.138 0.0895 0.0236 0.0380

0.108 0.336 0.107 0.061 0.103

0.114 0.513 0.116 0.105 0.173

0.118 0.648 0.121 0.140 0.232

0.125 0.929 0.136 0.223 0.383

0.127 0.985 0.141 0.241 0.416

1.129 1.005 0.139 0.248 0.444

0.132 1.082 0.136 0.261 0.490

0.142 1.177 0.135 0.280 0.590

1.263 0.104 0.292 0.530

Pt Ag Sn Zn

0.00019 0.00016 0.00024 0.00011

0.00028 0.00035 0.00127 0.00029

0.00067 0.00093 0.00423 0.00096

0.00112 0.00186 0.00776 0.00250

0.0077 0.0159 0.0400 0.0269

0.0382 0.0778 0.108 0.123

0.069 0.133 0.149 0.205

0.088 0.166 0.173 0.258

0.101 0.187 0.189 0.295

0.127 0.225 0.214 0.366

0.132 0.232 0.220 0.380

0.134 0.236 0.222 0.389

0.136 0.240 0.245 0.404

0.140 0.251 0.257 0.435

0.146 0.264 0.257 0.479

TABLE 2-153

Heat Capacities of Inorganic and Organic Liquids [J/(kmol
K)]

2-165

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

C3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia [use Eq. (2)] Anisole Argon Benzamide Benzene Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol [use Eq. (2)] 1,3-Butanediol [use Eq. (2)] 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide [use Eq. (2)] Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541

115,100 102,300 139,640 36,600 135,600 97,582 −122,020 103,090 55,300 109,900 −214,460 61.289 150,940 134,390 161,440 129,440 162,940 119,780 −5,480 93,383 156,130 −334,997 87,500 100,320 121,770 179,400 121,600 94,364 129,730 135,150 128,860 191,030 55.136 42.152 191,200 426,790 182,050 126,680 112,760 111,850 182,470 232,190 197,890 136,340 65,682 237,700 104,000 −8,304,300 85,600 65.429 −752,700 104,600 63,936 −1,307,500 127,900 124,850 96,910 132,280 69,362

−433 128.7 −320.8 511 −177 −122.2 3,082.7 −247.8 300 −109.75 9,185.1 80,925 93.455 −1,989.4 260.66 −169.5 −344.94 180.34 647.12 242.61 454.49 3,644.21 480 346.89 429.3 −667.11 −9.45 −109.12 −596.54 −311.14 −323.1 −1,675 314,200 324,580 −730.4 −3,694.6 −1,611 −65.47 −104.7 384.52 −13.912 −804.35 −491.54 −300.4 1,329.1 −746.4 174 104,370 −122 28,723 8,966.1 −500.6 46.35 15,338 −345.15 −166.34 −207.9 −153.27 215.01

1.425

C4

C5

0.8985 0.2837 0.34085 −15.895 1.0343 0.35441 −106.12 799.4 0.23602 11.043

0.000689 0.027732

0.41616 −2,651

0.64781 0.85562

−7.77514

1.0701 0.358 0.44032 2.16 0.97007 1.015 12.5 280.19 517.35 2.2998 13.828 11.963 −0.64 0.5214 0.72897 2.7063 1.7219 1.0216 −7.1579 1.829 −433.33 0.5605 −847.39 −30.394 2.2851 −0.1623 −53.974 0.915 0.43209 0.37456 0.50836

0.00591102

−0.0024234 −0.0001523 0.000032 −0.03874 1,413.9 1,449.5 −0.0135 −0.037454 0.002912

4.6121E-05

4.5027E-05

−0.0023017 −0.0012499 0.012755 0.60052 −0.001452 1,959.6 0.034455 0.063483 0.000488

2.008E-06

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

150.15 354.15 289.81 250.00 178.45 229.32 192.40 253.00 286.15 189.63 75.00 203.15 298.15 83.78 403.00 278.68 278.68 258.27 395.45 260.40 321.35 257.85 275.65 243.95 342.20 265.90 293.15 160.00 184.45 136.95 165.00 134.86 220.00 196.15 183.85 158.45 87.80 134.26 167.62 298.15 185.30 157.46 133.02 147.43 176.75 267.95 161.25 220.00 161.11 68.15 250.33 89.56 172.12 227.95 134.80 233.15 175.43 150.35 200.00

0.8221 1.4788 1.2213 1.6435 1.1696 0.8748 0.8021 1.0660 1.4114 1.0183 0.5307 0.7575 1.9978 0.4523 2.6649 1.3251 1.3326 1.6636 2.5042 1.5656 3.0218 1.8905 2.1981 1.8494 2.6868 0.7768 1.4960 0.8818 0.7798 1.1034 1.0333 1.1272 1.5590 0.6251 1.3465 1.3485 1.1015 1.1340 1.0986 2.2649 2.0492 1.6365 1.6003 1.1426 1.4741 1.6902 1.3206 0.7827 0.7577 0.5912 1.2763 0.7810 0.6711 1.3617 0.9800 1.0956 0.7460 1.2073 1.1236

294.00 571.00 391.05 350.00 329.44 354.75 250.00 379.50 375.00 400.00 115.00 401.15 484.20 135.00 563.15 353.24 500.00 442.29 450.00 464.15 640.00 478.60 458.15 472.03 533.37 331.90 495.08 320.00 276.71 290.00 350.00 400.00 670.00 670.00 391.90 372.90 380.00 350.00 274.03 399.26 400.00 390.00 370.00 298.15 300.00 436.42 390.75 290.00 552.00 132.00 388.71 145.10 239.12 360.00 340.00 366.48 373.15 319.67 308.85

1.1097 1.7579 1.5159 2.1545 1.3271 0.9713 0.8853 1.5801 1.6780 1.2271 0.7132 4.1847 2.5153 0.6708 3.0823 1.5040 2.0437 1.9954 2.8572 2.0599 4.4700 2.7617 3.0741 2.6406 3.5075 0.7587 2.0467 1.0453 0.7870 1.2279 1.4148 2.2237 5.2045 5.2437 2.5817 2.7190 1.8103 1.5022 1.2322 2.6537 2.9354 1.9359 1.8844 1.3759 1.6459 2.6031 1.7199 1.6603 1.3125 6.4799 1.6374 0.8007 0.6574 1.8101 1.1632 1.2192 0.9684 1.3523 1.3577

2-166

TABLE 2-153

Heat Capacities of Inorganic and Organic Liquids [J/(kmol
K)] (Continued)

Cmpd. no.

Name

Formula

CAS no.

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

m-Cresol o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane [use Eq. (2)] 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane

C7H8O C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si

108-39-4 95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6

Mol. wt.

C1

108.138 −246,700 108.138 −185,150 108.138 259,980 120.192 61,723 52.035 65,516 56.106 101,920 84.159 −220,600 100.159 −40,000 98.143 6,110.4 82.144 105,850 70.133 122,530 68.117 125,380 42.080 89,952 116.224 177,560 156.265 150,460 142.282 278,620 172.265 219,840 158.281 4,988,500 140.266 417,440 174.347 314,570 138.250 276,900 4.032 187.861 149,400 187.861 200,560 173.835 202,580 130.228 270,720 147.002 114,880 147.002 93,093 147.002 133,950 98.959 126,340 98.959 179,170 84.933 98,968 112.986 144,560 112.986 111,560 105.136 184,200 73.137 101,330 74.122 44,400 90.187 238,520 66.050 67.155 66.050 82,577 52.023 263,980 101.190 98,434 102.175 163,000 114.185 179,270 90.121 187,790 104.148 199,930 54.090 88,153 45.084 −214,870 86.175 129,450 112.213 134,500 112.213 150,130 112.213 155,560 94.199 171,580 46.068 110,100 73.094 147,900 100.202 146,420 194.184 206,560 60.170 131,810

C2 3,256.8 3,148 −1,112.3 494.81 144.7 −215.81 3,118.3 853 600.94 −60 −403.8 −349.7 −196.63 −179.12 586.63 −197.91 140.41 −52,898 −1,616.5 −160.93 −371.23 −231.8 −491.44 −726.3 −259.83 187.25 183.97 −24.84 −94.63 −444.74 −62.941 −53.605 149.44 286 243.18 1,301 −1,038.4 105,580 109.85 −1,791.1 429.04 −4.5 28.37 −313.41 −191.5 124.16 3,787.2 18.5 8.765 −62.38 −145.26 −256.67 −157.47 −106 59.2 325.75

C3 −7.4202 −8.0367 4.9427 0.063229 0.8103 −9.4216 0.68 1.7344 1.143 0.65237 0.76723 1.0737 0.9968 216.35 5.3948 0.95561 1.5774

C4 0.0060467 0.007254 −0.0054367

0.010687

−0.0010975

−0.37538 −0.004348

0.5946 0.9187 1.3377 0.95427 0.2314 0.48191 0.32 0.93009 0.23265 0.30617

−5.5 4.0587 310.21

0.008763 −0.0044691 −490.54

4.3666 0.62 0.5375 1.1023 0.87664 −13.781 0.608 0.81151 0.8851 1.0932 0.5727 0.51853 0.384 0.604

C5

0.016924

0.00023674

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

285.39 304.20 307.93 177.14 200.08 190.00 279.69 296.60 290.00 169.67 179.28 138.13 150.00 189.64 267.15 243.51 304.75 280.05 206.89 247.56 229.15

2.1895 2.3297 2.2740 1.4937 0.9700 0.9017 1.4836 2.1300 1.8038 1.1525 0.9956 0.9888 0.7514 1.7118 3.0718 2.9409 3.5521 3.5373 2.7541 3.3330 2.7466

400.00 400.00 400.00 425.56 300.08 298.15 400.00 434.00 489.75 356.12 322.40 317.38 298.15 431.95 488.15 460.00 543.15 503.00 443.75 512.35 447.15

2.5578 2.5243 2.5794 2.7229 1.1463 1.0961 2.0323 3.3020 3.0042 1.7072 1.3584 1.2953 0.8932 2.4334 4.3682 4.1478 5.9017 5.0169 3.8250 4.8297 4.2629

210.15 282.85 240.00 175.30 248.39 273.15 326.14 176.19 237.49 180.00 180.00 275.00 301.15 223.35 156.92 181.95 154.56 215.00 200.00 275.00 187.65 204.81 159.95 226.10 240.91 180.96 145.19 239.66 223.16 184.99 188.44 131.65 273.82 90.00 274.16 298.15

1.2695 1.3506 1.0532 2.5450 1.6139 1.6061 1.7711 1.1960 1.2601 0.9518 1.4483 1.5266 2.7033 1.5564 1.4698 1.5703 0.9915 1.0619 0.8042 2.1642 1.8399 2.0763 1.6586 2.0145 1.1806 1.1947 1.4495 1.8321 1.8029 1.6610 1.4355 0.9836 1.4767 1.5664 2.9587 1.3181

381.15 410.00 370.10 450.00 400.00 528.75 513.56 330.45 356.59 320.00 361.25 369.52 541.54 328.60 460.00 322.08 359.98 283.65 250.00 357.05 341.45 410.00 337.45 366.15 300.13 298.15 331.13 392.70 402.94 396.58 360.00 250.00 466.44 380.00 360.00 298.15

1.4743 1.5350 1.1701 3.4704 1.8978 2.5506 2.4829 1.3001 1.3885 1.0265 1.6515 1.6678 3.3908 1.8124 3.3202 1.7579 1.6874 1.1374 0.8912 2.5162 2.3375 2.8126 2.0755 2.4734 1.2542 1.3779 2.0224 2.6309 2.6870 2.6989 1.5340 1.0314 1.8200 2.5613 3.2383 1.3181

2-167

117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 149 150 151 152 153 154 155 156 157 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177

Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane [use Eq. (2)] Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Helium-4 Heptadecane Heptanal Heptane [use Eq. (2)] Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene

C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12

75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6

62.134 146,950 78.133 240,300 194.184 190,020 88.105 956,860 170.207 134,160 101.190 49,120 170.335 508,210 282.547 352,720 30.069 44.009 46.068 102,640 88.105 226,230 45.084 121,700 106.165 154,040 150.175 124,500 116.158 56,359 116.158 82,434 112.213 132,360 98.186 178,520 28.053 247,390 60.098 184,440 62.068 35,540 43.068 46,848 44.053 144,710 74.079 80,000 144.211 207,670 130.228 146,040 88.148 106,250 100.159 229,250 62.134 134,670 102.132 76,330 88.148 103,680 163.506 105,150 37.997 −94,585 37.997 1724,400 96.102 −991,200 48.060 85,663 34.033 74,746 30.026 61,900 45.041 63,400 46.026 78,060 68.074 114,370 4.003 387,220 4.003 410,430 240.468 376,970 114.185 222,360 100.202 61.26 130.185 194,570 116.201 2,416,800 116.201 283,127 114.185 270,730 114.185 265,040 98.186 267,950 132.267 236,870 96.170 46,798 226.441 370,350 100.159 117,700 86.175 172,120 116.158 161,980 102.175 1,638,600 102.175 267,628 100.159 208,250 100.159 235,960 84.159 164,640

−380.06 −595 431.04 −5,559.9 447.67 562.24 −1,368.7 807.32 89,718 −139.63 −624.8 38.993 −142.29 370.6 603.02 422.45 72.74 −518.35 −4,428 −150.2 436.78 205.35 −758.87 223.6 −17.907 458.22 292.15 −404.54 −234.39 400.1 726.3 85.318 7,529.9 −59,924 11,734 −118.56 −132.32 28.3 150.6 71.54 −215.69 −465,570 −464,890 347.82 −105.17 314,410 −23.206 −26,105 −1,037.63 −399.89 −375.68 −1,315.9 −158.01 761.13 231.47 329.52 −183.78 44.116 −17,261 −1,033.06 −107.47 −345.94 −200.37

1.2035 1.013

−0.00084787

9.6124 3.1015 0.2122 918.77 −0.030341 1.472

−1,886 0.0020386

0.80539 0.20992 0.64738 2.3255 40.936 0.37044 −0.18486

−0.0016818 −0.1697

2.8261

−0.003064

0.00026816

1.0493 1.1382 0.59656 −2.6047 0.46693 −139.6 537.85 −40.669 0.55459 0.53772

0.72691 211,800 135,100 0.57895 0.65074 1,824.6 0.88395 110.03 3.44064 1.0601 1.0024 6.5242 0.78982 −0.62882 0.68632 0.88734 0.709 71.721 3.35185 0.2062 0.94278 0.8784

0.0040957 1.1301

−0.0033241

0.047333

−42,494

3212.9

−2,547.9 −0.19172

0.00011968

−0.011994

9.3808E-06

−0.12026

0.000071087

0.00070293

174.88 291.67 423.40 284.95 300.03 277.90 263.57 309.58 92.00 159.05 189.60 192.15 178.20 238.45 258.15 285.50 161.84 134.71 104.00 284.29 260.15 250.00 160.65 254.20 235.00 298.15 298.15 204.15 125.26 298.15 145.65 167.55 58.00 53.48 239.99 140.00 140.00 204.00 292.00 281.45 187.55 2.20 1.80 295.13 229.80 182.57 265.83 239.15 230.00 234.15 238.15 154.12 229.92 200.00 291.31 217.15 177.83 269.25 228.55 223.00 217.35 217.50 133.39

1.1276 1.5293 3.7252 1.5306 2.6847 2.0537 3.6292 6.2299 0.6855 0.8787 1.6068 1.2919 1.5426 2.1287 2.1203 2.2015 1.6109 1.4678 0.7012 1.7168 1.3666 0.9819 0.8303 1.3684 2.6141 2.8266 1.9335 1.9410 1.1467 1.9562 1.6686 1.3255 0.5541 0.5798 1.3675 0.7994 0.6676 0.6767 1.0738 0.9820 0.9949 0.1087 0.1135 5.3005 2.3256 1.9989 2.5087 2.3590 2.2649 2.3522 2.3242 1.8150 2.4229 1.7387 4.9602 1.8926 1.6750 2.2526 1.9821 2.0394 2.0185 2.0532 1.5354

310.48 422.15 466.35 374.47 570.00 407.90 330.00 616.93 290.00 390.00 350.21 289.73 409.35 486.55 466.95 428.25 404.95 301.82 252.70 390.41 493.15 329.00 283.85 374.20 510.10 417.15 326.15 386.55 315.25 410.00 320.00 371.05 98.00 56.00 319.99 240.00 220.00 234.00 493.00 380.00 304.50 4.60 2.10 575.30 381.25 520.00 496.15 448.60 432.90 480.00 490.00 366.79 460.00 372.93 560.01 401.45 460.00 478.85 460.00 585.30 460.00 460.00 336.63

1.1959 1.6965 3.9104 2.2277 3.8933 2.7846 3.9429 9.3154 1.2444 1.6450 1.8796 1.3300 2.3075 3.0482 3.3794 3.0185 2.6798 1.8767 0.9758 1.8226 2.0598 1.1441 0.8693 1.6367 4.7157 3.3719 2.0153 2.4295 1.2007 2.4037 2.0358 2.0109 0.5966 0.5535 1.5018 0.8915 0.7166 0.6852 1.3765 1.0525 1.1609 0.2965 0.2995 7.6869 2.7685 4.0657 4.0065 3.8766 4.7873 3.2303 3.2163 2.4096 3.3131 2.4319 7.1521 2.4999 2.7534 3.4568 3.5197 8.1124 2.7087 2.7632 1.9673

2-168

TABLE 2-153

Heat Capacities of Inorganic and Organic Liquids [J/(kmol
K)] (Continued)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236

3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne Hydrazine Hydrogen [use Eq. (2)] Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide [use Eq. (2)] Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane [use Eq. (2)] Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane

C6H10 C6H14S C6H10 C6H10 H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10

928-49-4 111-31-9 693-02-7 764-35-2 302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5

82.144 118.240 82.144 82.144 32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122

C1 82,795 303,320 93,000 94,860 79,815 66.653 57,720 47,300 95,398 62,520 64.666 127,540 −32,469 157,850 146,290 65.708 105,800 62,600 61,260 79,791 275,500 92,520 125,630 135,370 108,300 74,200 247,870 149,510 151,600 81,919 177,850 198,390 105,200 102,930 47,726 131,340 50,578 118,600 118,170 155,920 53,271 46,457 27,030 85,383 132,300 161,240 130,200 92,919 183,650 149,770 143,440 191,170 211,170 115,300 255,100 226,650 142,220 251,890 172,370

C2 283.4 −1,009 326 254.15 50.929 6,765.9 9.9 90 −197.52 −223.02 49,354 −65.35 1,977.1 −41.619 −58.59 38,883 −362.23 243.4 270.9 89.49 −1,147 37.45 279.75 −133.34 146 417.4 −1,145 −247.63 −266.72 181.01 −171.57 −220.35 191.1 129.1 338.4 −63.1 508.59 447.07 447.99 −490 327.92 346.93 413 199.08 200.87 −288.61 −396 324.43 −79.862 −529.82 −154.07 −331.04 −661.97 −263.23 −938.4 15.421 −47.83 −468.32 −1,783.9

C3 3.3885 0.043379 −123.63 0.3883 0.6297 22.493 0.82867 −7.0145 0.42817 0.3582 −257.95 0.9379

C4

C5

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

0.00005805

300.00 192.62 200.00 300.00 274.69 13.95 185.15 165.00 259.83 189.79 187.68 270.00 177.95 407.95 288.15 90.69 175.47 359.00 253.40 200.00 196.32 179.69 260.75 159.53 113.25 321.50 155.95 135.58 139.39 298.15 157.48 175.30 200.00 277.25 250.00 146.58 300.00 300.00 300.00 130.73 200.00 200.00 250.00 160.00 186.48 167.23 174.15 298.15 189.15 256.15 127.93 180.15 171.64 150.18 224.95 240.00 119.55 176.00 113.54

1.6781 2.1495 1.5820 1.7110 0.9708 0.1262 0.5955 0.6215 0.7029 0.4288 0.6733 1.7031 1.4621 2.1213 1.5915 0.5361 0.7112 1.4998 1.2991 0.9769 1.4930 0.9925 1.9857 1.3035 1.2328 2.0839 1.5254 1.3282 1.3207 1.3589 1.6928 1.8315 1.4342 1.8678 1.3233 1.3955 2.0315 2.5272 2.5257 1.2492 1.1885 1.1584 1.3028 1.1566 1.4905 1.3484 0.9793 1.8965 1.9029 1.0263 1.3560 1.6348 1.5808 0.8939 1.6611 2.9128 1.4706 2.0728 0.9961

354.35 430.00 344.48 357.67 653.15 32.00 206.45 185.00 298.85 292.67 370.00 427.65 320.00 603.75 434.15 190.00 400.00 538.50 373.40 249.94 353.35 266.82 472.65 314.56 310.00 481.50 404.15 304.31 311.71 305.40 343.31 510.00 299.49 415.87 325.00 320.00 441.15 438.15 440.15 366.48 348.64 338.05 350.00 280.50 373.15 339.80 304.90 350.00 389.15 366.00 310.00 440.00 357.91 298.15 373.45 518.15 333.41 372.00 380.00

1.8322 2.7639 2.0530 1.8576 1.3158 1.3122 0.5976 0.6395 0.7105 0.5119 4.9183 2.5114 1.6671 2.8880 1.8837 14.9780 1.1097 1.9367 1.6241 1.0216 1.9084 1.0251 2.5785 1.5662 1.7048 2.7518 3.4411 1.5921 1.5673 1.3720 2.0661 2.8394 1.6243 2.6474 1.5771 1.9435 2.7494 3.1448 3.1535 1.8682 1.6760 1.6374 1.7158 1.3638 1.7511 1.5344 1.2195 2.0647 2.4460 1.3668 1.6540 2.3610 1.8641 0.9052 2.4118 5.1864 2.0842 2.4663 2.0725

−0.002762

478.27

−1,623 0.0086913 614.07

2.568 0.63868 −0.292

0.00151

3.4223 0.91849 0.90847 0.74379 0.76096 0.62516 0.8125

2.1383

−0.0015585

−0.061547 −0.9597 0.78179 1.21

0.0019533

0.60769 1.3499 0.7255 0.98445 2.4216 0.60412 2.413 1.0578 0.739 1.2209 14.759

−0.0021383

−0.047909

2-169

237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299

2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane [use Eq. (2)] 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde

C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O

75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6

74.122 56.106 88.105 74.122 90.187 46.144 118.176 88.148 58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079

−925,460 87,680 71,140 144,110 179,850 113,470 76,822 134,300 73,600 29,800 1,034,100 187,740 281,970 101,400 116,270 67,556 −2,979,600 342,570 136,820 383,080 224,336 10,483,000 329,641 254,490 265,350 253,580 399,430 130,650 224,830 205,260 571,370 319,198 300,400 289,980 509,420 240,040 42,642 175,510 175,430 60,046 346,910 112,050 159,080 145,050 201,200 251,596 194,590 193,020 156,100 188,200 213,760 86,200 68,671 103,370 101,720 60,834 145,400 66,230 62.983 158,760 471,710 201,400 99,306

7,894.9 217.1 335.5 −102.09 −264.1 421.6 94.356 184.7 527.5 −138,770 −497.6 −12,281 −682.11 −135.3 54.373 76,602 762.08 531.29 −1,139.8 49.726 −115,220 −1,046.78 −298.06 −46.22 −366.3 374.64 463.61 −186.63 44.392 −4,849 −1,042.21 −426.2 −417.27 −4,279.1 −33.198 886.67 −381.36 −6,152.3 281.16 219.54 257.78 −270.5 28.344 −651.3 −1,028.49 −263.86 −176.43 −456.94 −140.84 −324.4 256.6 246.66 527.03 317.61 215.89 252.4 98.275 113,630 −635 −4,172.1 −450.6 115.73

−17.661 −0.9153

0.013617 0.002266

0.58113 0.79202 −0.0032 7,154 1.0691 248 3.8912 0.345 −652.59 0.20481 2.7101 0.9813 476.87 3.61823 1.1707 0.79154 1.4881 0.58156 0.95891 0.8956 19.725 3.52943 1.1172 1.2218 21.477 0.67889 −0.69315 0.64623 113.92

0.0009795 −162.55 −2.2182

1.3841 0.0074902

1.8879

−0.85381

0.00056246

−0.021532

−0.044462

3.5028E-05

−0.92382

0.0027963

−0.003163

0.00000238

0.65632 0.99537 0.6372 2.275 3.26306 0.76808 0.5669 2.255 0.63581 0.9472

0.29552 633.21 1.969 14.745 1.7053

−873.46 −0.0144

298.96 132.81 300.00 133.97 160.17 298.15 249.95 164.55 151.15 353.43 24.56 183.63 63.15 117.00 244.60 182.30 109.50 305.04 255.15 219.66 285.55 310.00 238.15 191.91 253.05 223.15 301.31 246.00 216.38 289.65 250.00 241.55 252.86 255.55 171.45 240.00 200.00 462.65 54.36 90.00 283.07 200.00 143.42 239.15 200.14 200.00 196.29 234.18 108.02 160.75 197.45 200.00 200.00 372.39 314.06 243.15 404.15 200.00 85.47 146.95 185.26 199.00 200.00

2.2016 1.0568 1.7179 1.4086 1.5787 1.1347 1.8220 1.5410 1.0152 2.1623 0.3666 1.3242 0.5593 0.7486 1.0382 0.7747 0.6229 5.9409 2.7238 2.6348 3.1855 3.5059 2.8555 2.4041 3.0434 2.4594 5.6511 2.4470 2.2934 2.9326 2.5550 2.7338 2.6406 2.6314 2.1327 2.7118 1.9225 1.3740 0.5365 0.8535 4.6165 1.6361 1.4076 1.8827 1.6198 1.7642 1.7239 1.8279 1.2939 1.8199 1.8664 1.3752 1.1800 2.9963 2.0147 1.3080 2.4741 0.8589 0.8488 1.0797 1.1329 1.7926 1.2245

460.00 343.15 390.00 312.20 368.69 298.15 438.65 328.18 278.65 491.14 40.00 387.22 112.00 175.50 473.15 200.00 150.00 603.05 468.15 325.00 528.75 460.00 649.50 420.02 492.95 423.85 589.86 447.15 460.00 512.85 467.10 452.90 500.00 440.65 394.41 472.19 399.35 603.00 142.00 150.00 543.84 376.15 390.00 458.95 389.15 561.00 375.46 375.14 350.00 385.15 399.79 313.33 329.27 500.00 425.00 489.75 557.65 238.65 360.00 400.00 355.30 431.65 328.75

2.9455 1.4596 2.0198 1.6888 1.9014 1.1347 2.6176 1.9954 1.2507 2.8888 0.6980 1.5536 0.7960 1.0154 1.2949 0.7843 1.9909 8.7663 3.8554 2.9890 5.2498 4.6494 11.7608 3.3583 4.3491 3.6566 8.2276 3.3795 3.4189 4.6358 4.1566 5.7113 3.6660 3.4335 2.8235 3.7573 2.8619 1.8052 0.9066 1.0222 6.6042 2.0901 2.0498 2.9228 2.9227 7.0158 2.0380 2.0661 1.7251 2.2827 2.3546 1.6660 1.4989 3.6688 2.3670 2.3745 2.8615 0.8968 2.6079 2.1980 2.0487 3.2463 1.3735

TABLE 2-153

Heat Capacities of Inorganic and Organic Liquids [J/(kmol
K)] (Concluded)

2-170

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

C3

300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propionic acid Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C3H6O2 C3H5N C5H10O2 C3H9N C9H12 C3H6 C4H8O2 C3H2S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

79-09-4 107-12-0 109-60-4 107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

74.079 55.079 102.132 59.110 120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

213,660 118,190 83,400 139,530 174,380 114,140 75,700 138,390 167,330 58,080 45,810 829,380 113,340 244,770 85,743 119,500 258,090

−702.7 −120.98 384.1 78 −101.8 −343.72 326.1 −117.11 −319.1 445.2 368.33 −7,331.5 290.2 −236.96 5.7443

1.6605 0.42075

182,900 353,140 171,730 81,760 123,300 43,326 84,864 140,140 103,350 350,180 111,480 136,050 119,450 178,800 95,275 388,620 40,364 133,530 293,980 129,450 136,300 68,720 −10,320 49,516 276,370 133,860 36,500 −35,500

635.09 29.13 −800.47 455.38 −130.1 630.73 91.725 −152.3 159.3 −104.7 368.13 −288 324.54 −128.47 696.7 −1,439.5 664.46 514.64 −114.98 −3,039.5 −106.17 135 322.8 420.35 −2,090.1 7.8754 1,017.5 1,287.2

C4

C5

0.79 1.0905 0.47059 0.8127 19.203 −0.6051 0.63148

0.0013567

0.86116 2.8934

−0.0025015

0.6229 0.13243 0.695 1.0022 0.9913 0.83741 −1.3765 3.2187 0.96936 27.927 0.75175

8.125 0.52265 −2.63 −2.599

0.0021734

−0.061847

4.3042E-05

−0.014116

9.3701E-06

0.00302 0.002426

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

252.45 180.26 274.70 188.36 173.55 87.89 298.15 142.61 159.95 213.15 388.85 186.35 242.54 460.65 197.67 230.15 303.15

1.4209 1.1005 1.8891 1.5422 1.8051 0.9235 1.7293 1.3126 1.3708 1.5297 1.8904 1.3000 1.6749 2.6961 0.8688 1.1950 2.5809

414.32 370.50 404.70 340.00 432.39 225.45 398.15 350.00 340.87 460.75 683.00 253.15 418.31 604.50 350.00 230.15 303.15

2.0756 1.3112 2.3885 1.6605 2.7806 0.9208 2.0554 1.5505 1.5299 2.6321 2.9738 2.0403 2.2816 3.3228 0.8775 1.1950 2.5809

329.35 279.01 164.65 237.38 176.98 375.41 234.94 178.18 236.50 267.76 200.00 156.08 247.79 229.33 165.78 280.00 398.40 354.00 247.57 289.05 259.56 200.00 200.00 178.35 273.16 217.00 247.98 286.41

3.9207 4.2831 1.0721 1.8986 1.1979 2.8011 1.1372 1.3507 1.4102 3.9400 1.8511 1.1525 1.9987 1.9338 1.8285 2.3791 3.0508 3.1571 3.2493 3.9103 1.5939 0.9572 0.5424 1.2449 0.7615 1.6018 1.7314 1.7697

609.15 526.73 339.12 480.77 394.27 426.00 357.31 500.00 300.00 508.62 361.92 276.02 449.27 350.00 520.00 320.00 475.47 475.00 433.42 520.30 389.35 278.25 400.00 363.85 533.15 540.15 417.58 600.00

5.6977 6.0741 1.3546 3.0069 1.6883 3.1202 1.3455 2.3774 1.5114 5.5619 2.4471 1.3208 2.6526 2.3642 3.9095 2.5757 3.5629 3.7798 4.2624 5.5127 2.0892 1.0628 1.1880 2.0246 0.8939 2.9060 2.2269 3.2520

For the 11 substances, ammonia, 1,2-butanediol, 1,3-butanediol, carbon monoxide, 1,1-difluoroethane, ethane, heptane, hydrogen, hydrogen sulfide, methane, and propane, the liquid heat capacity CpL is calculated with Eq. (2) below. For all other compounds, Eq. (1) is used. For benzene, fluorine, and helium, two sets of constants are given for Eq. (1) that cover different temperature ranges, as shown in the table. (1) CpL = C1 + C2T + C3T 2 + C4T 3 + C4T 4 C12 C32t3 C3C4t4 C42t5 (2) CpL =  + C2 − 2C1C3t − C1C4t2 −  −  −  t 3 2 5 where t = 1 − Tr, Tr = T/Tc, Tc is the critical temperature from Table 2-141, CpL is in J/(kmolK) and T is in K. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. For temperatures less than the normal boiling point, the pressure is 1 atm. Above the normal boiling point, the pressure is the vapor pressure. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-154

Specific Heats of Organic Solids Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Acetic acid Acetone Aminobenzoic acid (o-) (m-) (p-) Aniline Anthracene

C2H4O2 C3H6O C7H7NO2 C7H7NO2 C7H7NO2 C6H7N C14H10

Anthraquinone Apiol Azobenzene

C14H8O2 C12H14O4 C12H10N2

Benzene

C6H6

Benzoic acid Benzophenone

C7H6O2 C13H10O

Betol

C17H12O3

Bromoiodobenzene (o-) (m-) (p-) Bromonaphthalene (β-) Bromophenol

C6H4BrI C6H4BrI C6H4BrI C10H7Br C6H5BrO

Camphene Capric acid Caprylic acid Carbon tetrachloride

C10H16 C10H20O2 C8H16O2 CCl4

Cerotic acid Chloral alcoholate hydrate Chloroacetic acid Chlorobenzoic acid (o-) (m-) (p-) Chlorobromobenzene (o-) (m-) (p-) Crotonic acid Cyamelide Cyanamide Cyanuric acid

C27H54O2 C4H7Cl3O2 C2H3Cl3O2 C2H3ClO2 C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4BrCl C6H4BrCl C6H4BrCl C4H6O2 C3H3N3O3 CH2N2 C3H3N3O3

Dextrin Dextrose

(C6H10O5)x C6H12O6

Dibenzyl Dibromobenzene (o-) (m-) (p-) Dichloroacetic acid Dichlorobenzene (o-) (m-) (p-) Dicyandiamide

C14H14 C6H4Br2 C6H4Br2 C6H4Br2 C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4N4

Temperature, °C −200 to +25 −210 to −80 85 to mp 120 to mp 128 to mp

sp ht, cal/(g°C)

50 100 150 0 to 270 10 28

0.330 + 0.00080t 0.540 + 0.0156t 0.254 + 0.00136t 0.253 + 0.00122t 0.287 + 0.00088t 0.741 0.308 0.350 0.382 0.258 + 0.00069t 0.299 0.330

−250 −225 −200 −150 −100 −50 0 20 to mp −150 −100 −50 0 +20 −150 −100 0 +50 −50 to 0 −75 to −15 −40 to 50 41 32

0.0399 0.0908 0.124 0.170 0.227 0.299 0.375 0.287 + 0.00050t 0.115 0.172 0.220 0.275 0.303 0.129 0.167 0.248 0.308 0.143 + 0.00025t 0.143 0.116 + 0.00032t 0.260 0.263

35 8 −2 −240 −200 −160 −120 −80 −40 15 78 32 60 80 to mp 94 to mp 180 to mp −34 −52 −40 38 to 70 40 20 40

0.380 0.695 0.628 0.013 0.081 0.131 0.162 0.182 0.201 0.387 0.509 0.213 0.363 0.228 + 0.00084t 0.232 + 0.00073t 0.242 + 0.00055t 0.192 0.150 0.150 0.520 + 0.00020t 0.263 0.547 0.318

0 to 90 −250 −200 −100 0 20 28 −36 −25 −50 to +50

0.291 + 0.00096t 0.016 0.077 0.160 0.277 0.300 0.363 0.248 0.134 0.139 + 0.00038t 0.406 0.185 0.186 0.219 + 0.0021t 0.456

−48.5 −52 −50 to +53 0 to 204

2-171

2-172

PHYSICAL AND CHEMICAL DATA TABLE 2-154

Specific Heats of Organic Solids (Continued) Recalculated from International Critical Tables, vol. 5, pp. 101–105

Compound

Formula

Dihydroxybenzene (o-) (m-) (p-)

C6H6O2 C6H6O2 C6H6O2

Di-iodobenzene (o-) (m-) (p-) Dimethyl oxalate Dimethylpyrene Dinitrobenzene (o-) (m-) (p-) Diphenyl Diphenylamine Dulcitol

C6H4I2 C6H4I2 C6H4I2 C4H6O4 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C12H10 C12H11N C6H14O6

Erythritol Ethyl alcohol

C4H10O4 C2H6O (crystalline)

(vitreous)

Temperature, °C

sp. ht., cal/(g°C)

−163 to mp −160 to mp −250 −240 −220 −200 −150 to mp −50 to +15 −52 to −42 −50 to +80 10 to 50 50 −160 to mp −160 to mp 119 to mp 40 26 20

0.278 + 0.00098t 0.269 + 0.00118t 0.025 0.038 0.061 0.081 0.268 + 0.00093t 0.109 + 0.00026t 0.100 + 0.00026t 0.101 + 0.00026t 0.212 + 0.0044t 0.368 0.252 + 0.00083t 0.248 + 0.00077t 0.259 + 0.00057t 0.385 0.337 0.282

60 −190 −180 −160 −140 −130 −190 −180 −175 −170 −190 to −40

0.351 0.232 0.248 0.282 0.318 0.376 0.260 0.296 0.380 0.399 0.366 + 0.00110t

Ethylene glycol

C2H6O2

Formic acid

CH2O2

−22 0

0.387 0.430

Glutaric acid Glycerol

C5H8O4 C3H8O3

20 −265 −260 −250 −220 −200 −100 0

0.299 0.009 0.022 0.047 0.085 0.115 0.217 0.330

Hexachloroethane Hexadecane Hydroxyacetanilide

C2Cl6 C16H34 C8H9NO2

Iodobenzene Isopropyl alcohol

C6H5I C3H8O

Lactose Lauric acid Levoglucosane Levulose

C12H22O11 C12H22O11·H2O C12H24O2 C6H10O5 C6H12O6

Malonic acid Maltose Mannitol Melamine Myristic acid Naphthalene Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene

25 41 to mp

0.174 0.495 0.249 + 0.00154t

40 −200 to −160

0.191 0.051 + 0.00165t

20 20 −30 to +40 40 20

0.287 0.299 0.430 + 0.000027t 0.607 0.275

C3H4O4 C12H22O11 C6H14O6 C3H6N6 C14H28O2

20 20 0 to 100 40 0 to 35

0.275 0.320 0.313 + 0.00025t 0.351 0.381 + 0.00545t

C10H8 C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2

−130 to mp 50 to mp 61 to mp 0 to 50 −160 to mp −160 to mp −160 to mp −163 to mp 66 to mp −160 to mp 0 to 55

0.281 + 0.00111t 0.240 + 0.00147t 0.252 + 0.00128t 0.270 + 0.0031t 0.269 + 0.000920t 0.275 + 0.000946t 0.276 + 0.001000t 0.256 + 0.00085t 0.258 + 0.00091t 0.247 + 0.00077t 0.236 + 0.00215t

SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-154

Specific Heats of Organic Solids (Concluded) Recalculated from International Critical Tables, vol. 5, pp. 101–105 Temperature, °C

sp ht, cal/(g°C)

Oxalic acid

C2H2O4 C2H2O4.2H2O

−200 to +50 −200 −100 0 +50 100

0.259 + 0.00076t 0.117 0.239 0.338 0.385 0.416

Palmitic acid

C16H32O2

Phenol Phthalic acid Picric acid

C6H6O C8H6O4 C6H3N3O7

Propionic acid Propyl alcohol (n-)

C3H6O2 C3H8O

Pyrotartaric acid

C6H8O4

−180 −140 −100 −50 0 +20 14 to 26 20 −100 0 +50 100 120 −33 −200 −175 −150 −130 20

0.167 0.208 0.251 0.306 0.382 0.430 0.561 0.232 0.165 0.240 0.263 0.297 0.332 0.726 0.170 0.363 0.471 0.497 0.301

Quinhydrone

C12H10O4

Quinone

C6H4O2

−250 −225 −200 −100 0 −250 −225 −200 −150 to mp

0.017 0.061 0.098 0.191 0.256 0.031 0.082 0.113 0.282 + 0.00083t

Salol Stearic acid Succinic acid Sucrose Sugar (cane)

C13H10O3 C18H36O2 C4H6O4 C12H22O11 C12H22O11

32 15 0 to 160 20 22 to 51

0.289 0.399 0.248 + 0.00153t 0.299 0.301

Tartaric acid Tartaric acid

C4H6O6 C4H6O6·H2O

Tetrachloroethylene Tetryl

C2Cl4 C7H5N5O8

1 Tetryl + 1 picric acid 1 Tetryl + 2 TNT

C13H8N8O15 C21H15N11O20

Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-)

C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N

Trichloroacetic acid Trimethyl carbinol Trinitrotoluene

C2HCl3O2 C4H10O C7H5N3O6

Trinitroxylene

C8H7N3O6

Triphenylmethane

C19H16

36 −150 −100 −50 0 +50 −40 to 0 −100 −50 0 +100 −100 to +100 −100 0 +50 0 to 49 54 to mp 54 to mp 130 to mp 0 20 40 solid −4 −100 −50 0 +100 −185 to +23 20 to 50 0 to 91

0.287 0.112 0.170 0.231 0.308 0.366 0.198 + 0.00018t 0.182 0.199 0.212 0.236 0.253 + 0.00072t 0.172 0.280 0.325 0.315 + 0.0031t 0.277 + 0.00120t 0.239 + 0.00195t 0.271 + 0.00106t 0.337 0.387 0.440 0.459 0.559 0.170 0.253 0.311 0.385 0.241 0.423 0.189 + 0.0027t

Urea

CH4N2O

Compound

Formula

20

0.320

2-173

2-174 TABLE 2-155 Cmpd. no. 1 7 8 14 16 27 29 31 34 37 38 43 59 60 61 64 67 81 88 95 97 98 99 112 120 125 126 134 145 151 156 157 182 183 190 194 197 217 221 231 236

Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to a Polynomial Cp [J/(kmol⭈K)] Name

Acetaldehyde Acetylene Acrolein Argon Benzene Bromoethane 1,2-Butadiene Butane 1-Butanol cis-2-Butene trans-2-Butene 1-Butyne m-Cresol o-Cresol p-Cresol Cyclobutane Cyclohexanone 1,1-Dibromoethane 1,1-Dichloroethane Diethyl ether 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Dimethyl ether 1,4-Dioxane Ethane Ethanol Ethylcyclopentane Ethyl mercaptan Fluoroethane Furan Helium-4 Hydrazine Hydrogen Isopropyl amine Methanol Methyl acetylene Methylcyclopentane Methylethyl ether Methyl mercaptan 2-Methylpropane

Formula

CAS no.

Mol. wt.

C1

C2

C3

C2H4O C2H2 C3H4O Ar C6H6 C2H5Br C4H6 C4H10 C4H10O C4H8 C4H8 C4H6 C7H8O C7H8O C7H8O C4H8 C6H10O C2H4Br2 C2H4Cl2 C4H10O C2H4F2 C2H4F2 CH2F2 C2H6O C4H8O2 C2H6 C2H6O C7H14 C2H6S C2H5F C4H4O He H4N2 H2 C3H9N CH4O C3H4 C6H12 C3H8O CH4S C4H10

75-07-0 74-86-2 107-02-8 7440-37-1 71-43-2 74-96-4 590-19-2 106-97-8 71-36-3 590-18-1 624-64-6 107-00-6 108-39-4 95-48-7 106-44-5 287-23-0 108-94-1 557-91-5 75-34-3 60-29-7 75-37-6 624-72-6 75-10-5 115-10-6 123-91-1 74-84-0 64-17-5 1640-89-7 75-08-1 353-36-6 110-00-9 7440-59-7 302-01-2 1333-74-0 75-31-0 67-56-1 74-99-7 96-37-7 540-67-0 74-93-1 75-28-5

44.053 26.037 56.063 39.948 78.112 108.965 54.090 58.122 74.122 56.106 56.106 54.090 108.138 108.138 108.138 56.106 98.143 187.861 98.959 74.122 66.050 66.050 52.023 46.068 88.105 30.069 46.068 98.186 62.134 48.060 68.074 4.003 32.045 2.016 59.110 32.042 40.064 84.159 60.095 48.107 58.122

29,705 30,800 30,702 20,786 35,978 27,112 27,400 17,330 25,300 39,760 20,908 25,300 29,002 16,192 29,090 31,863 32,182 20,560 19,560 26,040 29,736 27,581 33,851 25,940 28,345 31,742 32,585 34,710 23,014 30,358 40,860 20,786 32,998 64,979 23,590 30,270 30,810 35,465 23,337 31,520 21,380

127.43 −53.08 80.95 0 −101.69 117.99 177.6 458.16 371.2 108.8 324.73 183.2 158.79 469.81 166 37.226 116.87 285.2 249.01 388 72.364 169.88 −20.966 178.46 88.3 26.567 87.4 304.96 271.36 62.839 −160.3 0 −5.2147 −788.17 310.42 84.64 35.8 147.38 309.03 60.1 271.2

−0.21793 0.384 0.191 0 0.939 0 0 −0.816 −0.461 0 −0.411 0 0.635 −0.479 0.616 0.23616 0.547 −0.332 −0.22187 −0.268 0.228 −0.1581 0.17584 −0.186 0.446 0.12927 0.05 −0.084 −0.4427 0.1067 0.87 0 0.21379 5.8287 −0.274 −0.188 0.27 0.242 −0.285 0 −0.092

C4 × 1E 05

−1845.9

C5 × 1E 10

216400

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

50 50 50 100 50 100 50 50 50 50 50 50 50 50 50 50 50 100 100 50 50 50 50 50 50 50 50 50 50 50 100 100 50 50 50 50 50 50 50 50 50

0.3553 0.2911 0.3523 0.2079 0.3324 0.3891 0.3628 0.3820 0.4271 0.4520 0.3612 0.3446 0.3853 0.3849 0.3893 0.3432 0.3939 0.4576 0.4224 0.4477 0.3392 0.3568 0.3324 0.3440 0.3388 0.3339 0.3708 0.4975 0.3548 0.3377 0.3353 0.2079 0.3327 0.3797 0.3843 0.3403 0.3328 0.4344 0.3808 0.3453 0.3471

200 200 200 1,500 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 1,500 200 250 200 200 200 200 200 200 200

0.4647 0.3554 0.5453 0.2079 0.5320 0.5071 0.6292 0.7632 0.8110 0.6152 0.6941 0.6194 0.8616 0.9099 0.8693 0.4876 0.7744 0.6432 0.6049 0.9292 0.5333 0.5523 0.3669 0.5419 0.6385 0.4223 0.5207 0.9234 0.5958 0.4719 0.4360 0.2079 0.4051 0.2834 0.7471 0.3968 0.4877 0.7462 0.7374 0.4354 0.7194

237 238 243 246 247 248 251 253 289 290 294 295 296 304 310 320 321 322 324 331

2-Methyl-2-propanol 2-Methyl propene alpha-Methyl styrene Naphthalene Neon Nitroethane Nitromethane Nitric oxide 2-Pentyne Phenanthrene Propadiene Propane 1-Propanol Propylbenzene Quinone Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene Thiophene 1,2,4-Trimethylbenzene

C4H10O C4H8 C9H10 C10H8 Ne C2H5NO2 CH3NO2 NO C5H8 C14H10 C3H4 C3H8 C3H8O C9H12 C6H4O2 C4H8O C10H12 C4H8S C4H4S C9H12

75-65-0 115-11-7 98-83-9 91-20-3 7440-01-9 79-24-3 75-52-5 10102-43-9 627-21-4 85-01-8 463-49-0 74-98-6 71-23-8 103-65-1 106-51-4 109-99-9 119-64-2 110-01-0 110-02-1 95-63-6

74.122 56.106 118.176 128.171 20.180 75.067 61.040 30.006 68.117 178.229 40.064 44.096 60.095 120.192 108.095 72.106 132.202 88.171 84.140 120.192

17,080 24,970 37,735 29,120 20,786 33,055 38,782 34,980 24,330 27,700 31,690 26,675 28,800 22,880 29,668 36,970 28,560 41,195 36,765 35,652

381.7 211.8 112.94 82.88 0 89.54 −48.39 −35.32 335.7 210 17.1 147.04 257 538.46 129.07 −12.28 225.1 −88.3 −112.82 323.89

−0.199 0 0.846 0.964 0 0.238 0.413 0.07729 −0.37 1.24 0.282 0 -0.35 -0.546 0.53105 0.444 0.616 0.942 0.862 0.305

−5.7357

0.0014526

50 50 50 50 100 50 50 100 50 50 50 50 50 50 50 50 50 50 50 50

0.3567 0.3556 0.4550 0.3567 0.2079 0.3813 0.3740 0.3217 0.4019 0.4130 0.3325 0.3403 0.4078 0.4844 0.3745 0.3747 0.4136 0.3914 0.3328 0.5261

200 200 200 200 1,500 200 200 1,500 200 200 200 200 200 200 200 200 200 200 200 200

0.8546 0.6733 0.9416 0.8426 0.2079 0.6048 0.4562 0.3586 0.7667 1.1930 0.4639 0.5608 0.6620 1.0873 0.7672 0.5227 0.9822 0.6122 0.4868 1.1263

Constants in this table can be used in the following equation to calculate the ideal gas heat capacity C0p. C0p = C1 + C2T + C3T2 + C4T3 + C5T4 where C0p is in J/(kmol·K) and T is in K. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

2-175

2-176

TABLE 2-156 Cmpd. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions Cp [J/(kmol
K)] Name

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol o-Cresol

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C9H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O

CAS no.

Mol. wt.

C1 × 1E-05

C2 × 1E-05

C3 × 1E-03

C4 × 1E-05

C5

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4 95-48-7

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138 108.138

0.4451 0.342 0.402 0.713 0.5704 0.41914 0.3199 0.48449 0.6059 0.4678 0.28958 0.33427 0.7637 1.9581 0.44767 0.6895 0.77594 0.7186 1.0099 0.84115 0.9521 0.99192 1.0759 0.30113 0.721 0.47191 0.3377 0.575 0.5095 0.7134 1.0478 1.066 0.7454 0.90878 0.64257 0.5765 0.6592 1.1684 1.138 0.92478 0.92367 0.5587 0.89657 1.488 0.6906 0.2937 0.301 0.29108 0.37582 0.92004 0.29142 0.8011 0.4568 0.3942 0.3409 0.621 0.61809 0.7515 0.7988

1.0687 1.294 1.3675 2.222 1.632 0.8876 0.5424 1.2546 1.3703 1.0366 0.0939 0.4898 2.9377 1.7019 2.3085 2.3275 2.6455 2.27 4.4898 3.1428 2.8868 2.9633 4.2105 0.08009 2.064 1.2787 0.715 1.6476 1.705 2.43 2.549 2.575 2.5907 2.5508 2.0618 2.115 2.07 3.769 4.454 2.7795 2.5166 1.6694 2.3731 1.3522 1.9996 0.3454 0.3338 0.08773 0.7054 0.16446 0.09176 2.31 1.2967 0.6573 0.7246 1.843 1.8023 2.09 2.853

1.6141 1.075 1.262 1.6203 1.607 1.5818 1.594 1.3979 1.6475 1.3998 3.012 2.036 1.6051 1.3257 1.4792 1.512 1.7925 1.4669 1.311 1.9539 0.70207 1.5583 1.9041 0.7514 1.6504 1.5957 1.578 1.527 1.5324 1.63 1.8776 1.967 1.6073 1.893 1.6768 1.6299 1.6733 1.956 1.5507 1.6837 1.6109 1.5328 1.9754 1.146 1.5494 1.428 0.896 3.0851 0.5121 1.0764 0.949 2.157 1.5992 0.928 1.723 1.629 1.5438 0.6666 1.4765

0.6135 0.64 0.7003 1.676 0.968 0.5032 0.4325 0.87243 1.0446 0.6536 0.0758 0.2256 2.17 −37.417 1.6836 1.7516 2.2382 1.693 2.8395 2.5743 1.6385 2.2116 4.1785 0.1078 1.687 0.85166 0.4175 0.99 1.337 1.5033 1.875 1.951 1.732 1.852 1.3324 1.2872 1.251 2.818 3.0497 1.5974 1.5641 1.07 1.5866 −678 1.3146 0.264 0.2893 0.084553 0.485 −5083.8 0.1003 2.046 0.859 0.493 0.448 1.2337 1.1893 1.212 2.042

737.8 502 569.7 746.5 731.5 699.8 607.1 633.26 751.49 629.35 1484 882 751.2 41.232 677.66 697.9 835.9 680.77 627.4 850.06 2002.6 719.16 828.81 314.6 765.3 703.87 691.4 677.3 685.6 730.42 833 860.5 712.4 832.13 757.06 739.1 742.2 811.2 708.86 758.68 739.2 656 904.13 6.98 675 588 374.7 1538.2 236.1 2.3486 425 897.6 708.8 399.6 780.5 724 685.93 2214 664.7

200 100 50 200 200 100 200 200 250 200 50 100 300 298.15 200 200 200 200 300 298.15 300 300 200 100 200 200 100 200 200 200 298.15 298.15 200 298.15 250 200 200 300 200 200 200 200 200 298.15 200 50 100 60 100 298 50 200 100 100 150 200 200 200 200

0.4660 0.3448 0.4020 0.7665 0.6049 0.4192 0.3566 0.5467 0.6984 0.5156 0.2896 0.3343 1.1302 1.2745 0.5358 0.7689 0.8126 0.8053 1.8001 1.1198 1.5501 1.4156 1.1481 0.3090 0.7679 0.5089 0.3378 0.6269 0.5756 0.7673 1.2667 1.2679 0.8162 1.1257 0.7571 0.6199 0.7004 1.5358 1.2659 0.9714 0.9763 0.6238 0.9119 1.1533 0.7607 0.2937 0.3100 0.2911 0.4730 0.6106 0.2914 0.8219 0.4569 0.4048 0.3424 0.6674 0.6768 0.8701 0.9158

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500.1 1500.15 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 5000 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

1.2994 1.4997 1.5756 2.5675 1.8820 1.1285 0.7575 1.5620 1.7424 1.3464 0.3496 0.6647 3.0226 3.2501 2.4157 2.6739 2.9712 2.6706 4.9311 3.2880 4.3445 3.2957 4.5557 0.3794 2.4628 1.5121 0.9107 1.9202 1.9555 2.6602 3.0289 3.0311 2.8509 2.8730 2.2898 2.2715 2.2904 3.6724 4.8435 3.1008 2.9615 1.9209 2.6775 2.5905 2.3273 0.6335 0.6148 0.3521 1.0662 1.0465 0.3793 2.5327 1.5112 1.0063 0.9097 2.1126 2.1023 3.2075 3.2163

2-177

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane

C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26

106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3

108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133 194.184 88.105 170.207 101.190 170.335

0.7384 1.081 0.3545 0.44004 0.432 0.9043 0.67384 0.58171 0.416 0.48074 0.338 0.54305 1.9641 1.672 0.24457 1.6984 1.7101 1.931 1.5045 0.3029 0.5927 0.74906 0.391 1.6122 0.7 0.6948 0.6978 0.5521 0.65271 0.3628 0.7145 0.78658 1.208 0.9102 0.8621 0.91273 0.49653 0.51889 0.35489 1.1384 1.093 1.0869 1.1556 1.0113 0.6534 0.5565 0.7772 1.0776 1.1039 1.0991 0.7843 0.5148 0.722 0.85438 1.396 0.61453 0.6037 0.6949 1.174 0.56184 1.0985 1.2114 2.1295

2.908 3.7932 0.5015 2.3074 3.735 2.5771 3.2598 3.1717 3.014 2.5159 1.6894 3.9962 5.1412 5.353 6.546 5.392 5.2089 5.4815 4.3794 0.0975 1.158 1.2725 0.648 4.4777 2.0746 2.0804 2.078 1.205 1.1254 0.6804 1.7344 1.7429 3.066 2.674 2.551 2.41 1.2546 1.2431 0.71002 2.5747 3.683 4.054 1.8305 3.2393 1.6179 1.6384 4.032 4.6718 4.6445 4.6401 1.4364 1.442 1.783 4.5772 4.78 1.7438 1.3747 1.524 5.32 2.7034 4.3412 2.6127 6.633

1.4559 1.7505 1.057 1.6283 1.192 0.7882 1.3955 1.5435 1.4617 1.5803 1.6135 1.3575 1.8989 1.6141 1.0899 1.568 1.7265 1.6085 1.3291 2.515 1.4931 1.981 1.194 1.6831 1.3664 1.3632 1.3635 1.502 1.7376 1.256 1.524 1.7157 2.089 1.719 1.5413 1.6686 1.5394 1.5048 1.5936 0.7384 1.6057 1.7802 0.95919 1.5611 1.7837 1.7341 1.544 1.654 1.6943 1.6679 1.5836 1.6034 1.532 1.5181 2.19 1.3418 1.641 1.6514 2.105 1.5171 1.6222 0.78956 1.7155

2.091 3.0027 0.452 1.5571 1.635 1.3068 2.0209 2.1273 1.8095 1.7454 1.1768 2.5623 4.1278 3.782 4.8642 3.938 3.5935 3.74 2.5557 −0.0275 0.8428 0.9437 0.42 2.918 1.5983 1.594 1.5965 0.8719 0.878 0.4275 1.223 1.2627 2.343 1.7926 1.437 1.652 0.87561 0.76269 0.4622 1.62 2.342 2.9786 0.99605 2.1501 1.0242 1.0899 2.508 3.3397 3.3949 3.3736 0.871 0.7747 1.31 2.974 3.9705 1.0102 0.7988 1.0658 4.1 1.7658 3.6455 1.6903 4.5161

650.42 794.8 396 744.9 530.1 1952.2 677.33 701.62 668.8 718.37 722.8 618.54 862.51 742 424 720.5 782.92 754.75 632.01 368 655.5 845.2 501 781.6 620.16 619.2 619.37 653.5 795.45 548 674.2 765.1 891 794.94 688.9 771.08 694.17 697.51 762 2143 699 791.6 2826.3 689.3 821.4 793.04 649.95 792.5 798.35 781.97 730.65 725.4 762 641.01 900.6 592.09 743.5 722.2 818 700.76 743.62 2394.4 777.5

200 200 100 200 100 200 200 150 100 150 100 300 200 200 298.15 200 298.15 200 298 100 200 200 100 200 200 200 200 200 200 100 150 200 298.15 200 200 200 200 200 200 300 298.15 300 298.15 298.15 200 200 200 200 200 200 200 200 200 200 300 200 200 200 298.15 200 300 300 200

0.8707 1.1480 0.3648 0.4903 0.4366 0.9648 0.7802 0.5978 0.4165 0.4918 0.3381 1.2644 2.0192 1.7967 2.5232 1.8502 2.2304 2.0434 2.1938 0.3020 0.6442 0.7635 0.3929 1.6841 0.8245 0.8198 0.8228 0.6061 0.6722 0.3637 0.7268 0.8217 1.4197 0.9502 0.9316 0.9567 0.5373 0.5536 0.3681 1.5995 1.5669 1.5102 1.2777 1.4638 0.6721 0.5812 0.9363 1.1535 1.1777 1.1820 0.8155 0.5436 0.7594 1.0550 1.7481 0.7095 0.6298 0.7355 1.6816 0.6403 1.7298 1.5900 2.2442

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500.1 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1000.15 1500 1200 1500 1500

3.2102 4.1808 0.8100 2.3234 3.6516 3.8251 3.4743 3.2132 2.9298 2.5619 1.7213 3.7236 6.0539 6.0932 6.1099 6.2186 5.8745 6.4613 5.2794 0.3425 1.5673 1.7041 0.9599 5.2145 2.5161 2.5161 2.5175 1.5615 1.5743 0.9543 2.1609 2.1894 3.4674 3.0519 2.9244 2.8724 1.5424 1.5510 0.9419 4.1941 4.0535 4.3093 3.0678 3.6669 1.9148 1.8585 4.0353 4.9543 4.9243 4.9275 1.9523 1.6581 2.2596 4.5983 4.4740 2.0944 1.6949 1.9255 4.1139 2.8174 4.5143 4.2484 7.4325

2-178

TABLE 2-156 Cmpd. no. 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183

Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions Cp [J/(kmol
K)] (Continued ) Name

Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne Hydrazine Hydrogen

Formula

CAS no.

Mol. wt.

C1 × 1E-05

C2 × 1E-05

C3 × 1E-03

C4 × 1E-05

C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10 H4N2 H2

112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2 302-01-2 1333-74-0

282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144 32.045 2.016

3.2481 0.40326 0.492 0.9981 0.594 0.7844 1.0944 1.0455 1.115 1.1059 0.82052 0.3338 0.7286 0.63012 0.343 0.3346 0.537 1.5777 1.634 1.0953 1.24 0.5576 0.937 1.132 0.85105 0.29122 0.62653 0.44373 0.33289 0.3327 0.3822 0.3381 0.3727 2.7878 1.404 1.2015 1.3135 1.2215 1.4569 1.2768 1.2507 1.1851 1.442 1.0712 2.6283 1.232 1.044 1.1622 1.0625 1.2615 1.094 1.1237 1.0434 0.9376 1.2662 0.9129 1.036 0.38711 0.27617

11.09 1.3422 1.4577 2.0931 1.618 3.399 4.1794 2.3148 3.391 4.6306 4.0342 0.9479 1.8436 1.4584 1.427 1.2116 1.886 4.4017 4.5119 3.0032 3.2 1.3617 2.829 2.94 1.0378 0.10132 2.1646 1.3119 0.73989 0.49542 0.93 0.7593 1.6606 9.5247 2.5907 4.001 2.3317 3.991 2.8252 3.381 2.148 3.6362 4.1603 3.0258 8.9733 2.2146 3.523 2.0708 3.521 3.5964 1.807 2.936 3.0749 3.015 3.7294 2.5577 3.009 0.8576 0.0956

1.636 1.6555 1.6628 2.0226 1.812 1.559 0.88375 0.71 1.6705 1.6628 1.567 1.596 1.688 1.673 1.638 1.6084 1.207 1.7494 1.7532 1.7988 1.967 1.5221 1.648 1.827 0.59737 1.453 1.564 1.6422 1.8639 1.8666 1.845 1.1925 1.5112 1.6935 0.8315 1.6766 0.67567 1.58 0.81695 1.3831 0.6912 1.7359 1.6603 1.5273 1.6912 0.84 1.6946 0.68661 1.5835 1.8445 0.689 1.401 1.7459 1.9057 1.6574 1.529 2.116 1.7228 2.466

7.45 0.73223 0.939 1.803 1.078 2.426 −1.609 1.471 2.518 3.299 2.6697 0.551 1.199 0.97296 1.037 0.8241 0.864 3.2378 3.1032 2.1311 2.346 0.8073 2.155 2.055 0.94745 0.094101 1.7278 0.85441 0.46079 0.28075 0.69 0.318 1.3145 6.6651 1.312 2.74 1.824 2.835 1.766 1.888 1.619 2.5048 2.6572 2.0975 6.264 1.219 2.369 1.5355 2.462 2.594 1.474 1.601 2.0728 1.986 2.308 1.737 2.106 0.56635 0.0376

C5 726.27 752.87 744.7 928.05 820 702 1183.1 2061.6 733.6 781.1 715.52 740.8 767.3 773.65 744.7 737.3 496 792.34 809.75 817.35 896 687.5 724.7 852 2122.7 662.91 724.29 738.77 891.16 934.9 850 550 686 744.57 2201 756.4 1846 717.7 2537.2 650.3 1759.3 785.73 759.39 689.62 744.41 2205 761.6 1932.5 715.75 819.17 1772 650.5 793.53 817 757.8 683 902.4 733.53 567.6

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

200 200 200 200 200 200 300 300 298 200 200 60 300 300 150 50 100 298.15 298.15 298.15 298.15 200 300 298.15 167 50 200 200 50 50 150 50 200 200 200 200 300 200 298.15 200 150 298.15 200 200 200 200 200 298.15 200 298.15 200 150 298 300 200 200 300 200 250

3.5235 0.4256 0.5224 1.0126 0.6139 0.8912 1.4598 1.5102 1.5583 1.1875 0.9272 0.3338 0.9178 0.7800 0.3480 0.3346 0.5412 2.0279 2.0360 1.3620 1.4479 0.5970 1.3377 1.3538 0.8926 0.2912 0.6914 0.4726 0.3329 0.3327 0.3833 0.3381 0.4376 3.0034 1.4479 1.2828 1.8497 1.3330 1.8136 1.3968 1.2688 1.5434 1.5191 1.1721 2.8312 1.2672 1.1117 1.6107 1.1607 1.5829 1.1815 1.1443 1.3301 1.1909 1.3340 1.0004 1.2215 0.4070 0.2843

1500 1500 1500 1500 1500 1500 1500 1200.15 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1200 1200 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500

12.2110 1.4562 1.6576 2.6594 1.8528 3.6147 4.2540 3.6330 3.6213 4.9184 4.1472 1.0987 2.2016 1.8095 1.5178 1.3297 2.1485 5.1201 4.8744 3.2289 3.4234 1.6729 3.0569 3.4535 2.2349 0.3812 2.4736 1.5008 0.9024 0.7113 1.1203 0.9933 1.7940 10.4160 4.2863 4.4283 4.2941 4.5346 4.6604 4.1386 3.8446 4.0836 4.7831 3.5985 9.8182 3.7314 3.8620 3.7636 3.9726 4.0672 3.3207 3.5874 3.4819 3.1889 4.2483 3.0371 3.1894 1.0571 0.3225

2-179

184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246

Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl -1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene

HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C2H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8

10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3

80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122 90.187 46.144 118.176 88.148 58.079 128.171

0.2912 0.29157 0.30125 0.29134 0.33288 0.74694 0.68545 0.49522 0.7251 0.33298 0.39252 0.6116 0.555 0.4478 0.1206 0.41 0.9396 0.671 0.746 1.8458 0.92165 0.87026 0.81924 0.7906 0.82051 1.0785 0.8274 0.894 0.59895 0.9227 0.7959 0.92279 0.92279 0.66456 0.69411 0.6422 0.7283 0.68681 0.784 0.75083 0.506 0.7284 1.227 0.474 0.89232 1.5914 0.99247 0.4146 0.864 1.7483 0.903 0.94326 0.6549 0.7704 0.6125 0.7765 0.92151 0.93775 0.46149 0.78548 0.9779 0.60865 0.6805

0.0953 0.09048 0.3171 0.093252 0.26086 2.4356 2.1876 1.8718 2.089 0.79933 0.879 2.029 1.782 1.0917 2.3766 1.0578 2.559 2.222 3.265 1.743 3.3371 2.5556 2.6038 1.656 3.0869 2.7388 2.1377 2.91 1.1636 4.115 2.596 2.6709 2.6709 3.507 3.0209 3.0711 1.0307 1.9959 2.1032 1.9577 1.219 3.1713 2.195 1.226 2.4765 1.764 2.7275 0.8307 1.811 4.9288 3.801 3.5965 2.4776 2.539 2.066 2.442 2.3943 2.6178 1.2781 3.5969 3.091 1.5965 3.5494

2.142 2.0938 1.6102 2.905 0.9134 1.715 1.5831 1.2958 1.8516 2.0869 1.9165 1.7683 1.26 1.5508 1.0543 1.708 0.825 1.421 1.545 1.22 1.8365 1.7757 1.7593 1.6926 1.3864 1.5885 1.755 1.57 1.565 1.6504 0.6213 0.68784 0.68784 1.5892 1.6903 1.6387 1.5429 1.5534 1.5488 1.6424 1.637 1.352 0.842 2.188 1.696 1.2076 2.003 1.589 0.7543 1.7384 1.602 1.3533 1.587 1.5502 1.545 1.714 1.6936 1.7291 1.4565 1.4342 1.643 1.619 1.4262

0.0157 −0.00107 0.2179 1.95E−03 −0.17979 1.8484 1.3855 1.4852 1.6483 0.41602 0.53654 1.3302 0.853 0.675 1.8186 0.6836 1.36 1.194 1.923 −56.11 2.4645 1.7636 1.7195 1.2167 1.7886 1.9067 1.5149 2.073 0.81581 2.9006 2.288 1.9847 1.9847 2.3526 2.1209 2.1298 0.7811 1.1168 1.1855 1.1949 0.894 1.8948 1.191 0.85983 1.5598 −407.4 1.8974 0.4612 0.8 3.5897 2.453 2.0569 1.575 1.669 1.2057 1.818 1.4896 1.6236 0.79115 2.5336 2.099 0.93783 2.5984

1400 120 626 1.33E+03 949.4 757.75 691.76 569.96 798.43 991.96 896.7 835.5 562 658.2 418.8 735 3000 614.7 666.7 31.2 757.99 807.82 800.93 788.4 613.87 749.6 782 678.3 690.39 779.48 1698.6 1732.4 1732.4 727.13 781.56 750.25 668.94 692.04 693 749.19 743 585.14 2460 1008.2 791.4 10.503 849.64 716.7 2160 788.01 691.6 599.92 706.99 679.3 676 716 797.79 783.23 643.23 651.69 731.191 739.55 650.1

50 50 100 50 100 298.15 200 300 298.15 50 200 300 298 200 298.15 150 300 150 200 300 298.15 200 200 298.15 300 273.15 200 298 200 200 300 300 300 200 200 200 200 200 200 273.16 250 300 298.15 298.15 200 300 273 200 298.15 298.15 200 300 200 200 200 300 298 298.15 200 200 298 300 200

0.2912 0.2914 0.3014 0.2913 0.3329 1.0427 0.7510 0.9790 0.9475 0.3330 0.3980 0.7698 0.8489 0.4882 0.9908 0.4136 1.2586 0.6931 0.8546 1.2793 1.3135 0.9060 0.8559 0.9632 1.3300 1.3173 0.8646 1.3461 0.6380 0.9953 1.5302 1.5099 1.5099 0.7510 0.7464 0.7083 0.7717 0.7396 0.8397 0.9004 0.5888 1.3200 1.4755 0.5195 0.9280 1.1291 1.1377 0.4329 1.1621 2.2567 1.0192 1.5600 0.7218 0.8567 0.6763 1.1242 1.1251 1.1728 0.5141 0.9445 1.3522 0.7748 0.8454

1500 1500 1500 1500 1500 1200 1500 1200 1200.1 1500 1500 1500 1500 1500 1200.1 1500 1200 1500 1500 1500 1500 1500 1500 1500.15 1200 1200 1500 1200 1500 1500 1200 1200 1200 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500.15 1500 1500 1500 1500 1500 1500 1500 1500 1200 1500 1500 1500 1200 1200 1500 1500 1500 1500 1500 1500

0.3479 0.3406 0.5522 0.3224 0.5143 2.5383 2.4540 2.0517 2.2057 0.8890 1.0533 2.2209 2.0754 1.3293 2.1663 1.2388 3.3569 2.5028 3.3792 3.2262 3.4856 2.8923 2.8709 2.1502 3.1994 3.1687 2.5255 3.0766 1.5593 4.3180 4.1359 4.1467 4.1467 3.5495 3.1496 3.1549 1.5893 2.2931 2.4816 2.3178 1.5109 3.1987 3.6532 1.3595 2.8696 2.9991 2.9952 1.0781 2.8637 5.7177 3.9617 3.7409 2.6656 2.8508 2.2814 2.5276 2.6391 2.9904 1.5253 3.8592 3.4779 1.8871 3.7359

2-180 TABLE 2-156 Cmpd. no. 248 249 250 251 252 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303

Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions Cp [J/(kmol
K)] (Concluded) Name

Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine

Formula

CAS no.

Mol. wt.

C1 × 1E-05

C2 × 1E-05

C3 × 1E-03

C4 × 1E-05

C5

Tmin, K

Cp at Tmin × 1E-05

Tmax, K

Cp at Tmax × 1E-05

C2H5NO2 N2 F3N CH3NO2 N2O C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N

79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8

75.067 28.013 71.002 61.040 44.013 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132 59.110

0.54619 0.29105 0.33284 0.42267 0.29338 3.1062 1.7347 1.5175 0.1266 1.54 1.8197 1.5352 1.7646 1.6289 2.9502 1.6088 1.3554 1.4082 1.3805 1.6383 1.3901 1.4952 1.3599 1.5981 1.2307 0.25751 0.29103 0.33483 2.4679 1.0743 0.8805 2.836 0.906 1.0853 0.90053 0.96896 0.82523 1.1327 1.0974 0.753 0.70737 0.9374 0.434 0.59683 0.7364 0.426 0.5192 0.619 0.73145 1.0563 0.7174 0.6959 0.5357 1.7994 0.76078

1.6492 0.086149 0.49837 1.0842 0.3236 10.575 4.5115 4.915 6.011 4.936 3.5542 4.6844 5.044 3.9708 10.034 4.218 4.431 4.3436 4.459 3.1897 3.806 4.4103 4.1605 4.6063 3.4942 1.1734 0.1004 0.29577 8.4212 2.8363 3.011 1.08 3.062 3.0747 2.7085 2.4907 2.5943 2.947 3.2959 2.0905 2.2229 4.758 2.445 2.5533 2.544 1.1194 1.9245 2.0213 2.0313 4.3397 1.914 1.7778 1.4617 1.753 2.1049

1.4803 1.7016 0.7093 1.4885 1.1238 0.76791 1.712 1.6448 1.0815 1.578 0.81514 1.7288 1.6182 1.8928 0.77107 1.9126 1.6356 1.4662 1.5751 0.81595 1.3717 0.80211 1.7317 1.6295 1.528 2.7969 2.5265 1.5217 1.6865 1.9549 1.6502 2.107 1.6054 1.8672 1.6592 1.4177 1.7291 1.7418 1.6761 1.5307 1.557 1.382 1.152 1.2397 1.0852 1.5772 1.6265 1.6293 1.9375 1.6098 2.0144 1.7098 1.553 1.196 1.7256

1.0635 0.0010347 0.23264 0.68603 0.2177 −4.5661 3.3256 3.47 4.5946 3.588 2.1974 3.2304 3.3857 3.2136 -4.3012 3.278 3.054 2.7687 3.2016 1.9814 2.2573 −2.0958 2.8675 3.0301 2.4617 0.65788 0.09356 0.27151 5.8537 2.0146 1.892 −3.56 2.115 2.2271 1.8012 1.301 1.768 2.0987 1.9486 1.378 1.3125 3.485 1.512 1.5519 0.808 0.7546 1.168 1.2956 1.4815 3.181 1.1708 1.2654 0.91197 −4.12 1.3936

666.94 909.79 372.91 683.57 479.4 912.03 810.96 749.6 418.2 721.11 2508.8 783.67 755.48 855.52 916.73 869 746.4 659.38 718.8 2521.3 660.96 981.95 784.47 756.28 694.81 878.91 1153.8 680.35 743.6 890.44 747.6 283 717.97 825.4 743.96 646.7 778.7 795.78 757.67 672.8 690.78 627.4 507 576.78 573 680.8 723.6 727.4 843.37 729.66 930.6 763.78 678.2 108.2 789.03

200 50 100 200 100 200 200 200 298.15 200 298.15 298.15 200 298.15 200 200 200 298.15 200 298.15 150 200 298.15 200 200 298.15 50 100 200 200 200 298.15 200 298.15 200 200 298.15 298 200 200 200 200 100 298.15 298.15 200 200 200 298.15 300 200 298.15 200 298.15 200

0.6062 0.2911 0.3404 0.4571 0.2948 3.3533 1.8005 1.6257 2.2953 1.6777 2.2720 2.0014 1.8658 1.9693 3.1800 1.6504 1.4529 2.0652 1.5055 2.0428 1.4162 1.5775 1.7723 1.6881 1.3448 0.3201 0.2910 0.3349 2.6586 1.0960 0.9404 1.3824 0.9890 1.3539 0.9591 1.0536 1.0856 1.4202 1.1547 0.8276 0.7700 1.1959 0.4401 1.1054 1.0745 0.4646 0.5632 0.6665 0.8966 1.6392 0.7266 0.8938 0.5832 1.3594 0.7933

1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1000.1 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1000.15 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500

1.9237 0.3484 0.8092 1.3280 0.5828 11.6130 5.4439 5.5407 5.5267 5.6606 5.8526 5.2776 5.9082 4.7924 11.0160 4.9286 4.9764 5.0411 5.0965 5.2565 4.6547 4.9067 4.6807 5.3549 4.1604 0.6502 0.3653 0.5928 9.2209 3.2404 3.2927 3.2952 3.4133 3.4701 3.0797 3.0358 2.8897 3.4994 3.6956 2.4754 2.5052 5.0645 2.6045 2.8390 2.6737 1.3376 2.0556 2.2458 2.2760 4.6527 2.1149 2.1248 1.7235 3.2024 2.4353

304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

0.96885 0.43852 0.871 0.73815 0.7474 2.0114 0.6487 0.3681 0.893 0.71806 0.33375 0.35256 0.33408 0.945 2.0719 2.3082 0.46905 0.8145 0.51848 1.1352 0.40399 0.5814 0.66554 2.1496 1.2766 0.7107 1.052 1.0106 1.139 0.982 2.0367 2.154 1.9529 1.859 0.536 0.55978 0.42364 0.84894 0.33363 0.7568 0.8521 0.7512

3.7954 1.506 2.447 1.9529 1.9523 0.8082 2.1227 0.71245 2.1503 2.2669 0.25864 1.227 0.49677 2.526 6.2668 7.8678 2.5314 4.395 2.4535 5.6331 1.627 2.863 1.1257 7.3045 2.5559 1.5051 3.79 3.8314 5.286 5.402 1.8181 2.4432 6.0998 5.869 2.119 1.2141 0.8735 1.1471 0.2679 3.3924 3.2954 3.397

1.5168 1.3988 1.9254 1.5954 1.631 1.8656 1.3491 0.65201 0.772 1.2739 0.9328 0.67938 0.87322 0.829 2.4044 1.6823 1.5998 1.471 1.5018 1.6211 1.4562 1.4406 1.5454 1.6695 0.80937 0.79662 1.4814 1.501 1.594 1.531 1.2089 1.1126 1.7087 1.5718 1.198 1.6102 1.6492 1.38 2.6105 1.496 1.4944 1.4928

2.6618 0.74754 1.888 1.2356 1.2112 −2.4404 1.514 0.46721 0.999 1.7342 0.1088 0.78407 0.28563 0.5 6.345 5.4486 1.7051 3.065 1.6871 3.3829 1.322 1.898 0.97196 4.9998 1.4829 0.84537 2.331 2.395 3.351 3.493 0.79777 0.58651 4.1302 4.326 1.147 0.89079 0.6556 0.9 0.08896 2.247 2.115 2.247

694.3 616.46 821.3 730.5 750.92 279.98 614.8 286.03 2442 537.65 423.7 351.27 393.74 2010 967.71 743.1 740.64 666.4 665.31 681.9 648.81 650.43 717.04 741.02 2231.7 2187.6 667.3 678.3 677.94 639.9 1060.8 950.59 775.4 722.7 510 710.4 739.07 644.61 1169 675.9 675.8 675.1

200 130 298.15 200 200 298.15 200 100 100 300 100 100 100 298.15 298.15 200 200 200 200 200 200 200 298.15 200 200 200 200 200 200 200 298.15 298.15 200 200 100 200 200 298.15 100 200 200 200

1.0927 0.4436 1.1022 0.7825 0.7848 1.0218 0.7711 0.4182 0.8931 1.3370 0.3354 0.3872 0.3408 1.2478 2.4763 2.4864 0.5259 0.9881 0.6147 1.3069 0.4886 0.7016 0.8496 2.3156 1.3278 0.7439 1.1832 1.1354 1.3139 1.2194 2.1054 2.2726 2.0594 2.0232 0.5404 0.5967 0.4457 1.0788 0.3336 0.8759 0.9643 0.8710

1500 1500 1500 1500 1500 1000.15 1500 1500 1500 1200 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 1500 2273.15 1500 1500 1500

4.1613 1.6817 2.7484 2.3287 2.3216 2.1175 2.4969 1.0537 3.2416 2.5823 0.5695 1.5397 0.7967 3.4444 6.6947 8.6225 2.5538 4.5348 2.5679 5.5784 1.8098 3.0029 1.6433 8.0251 4.2046 2.4322 4.1983 4.1854 5.3769 5.3754 3.7585 4.3560 6.8342 6.7834 2.3750 1.5590 1.1423 1.8595 0.5276 3.5920 3.5965 3.5923

Constants in this table can be used in the following equation to calculate the ideal gas heat capacity C0p.









2 2 C3/T C5/T C0p = C1 + C2  + C4  sinh(C3/T) cosh(C5/T) where C0p is in J/(kmol·K) and T is in K. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

2-181

2-182

PHYSICAL AND CHEMICAL DATA

TABLE 2-157

Cp/Cv: Ratios of Specific Heats of Gases at 1 atm Pressure*

Compound Acetaldehyde Acetic acid Acetylene

Formula C2H4O C2H4O2 C2H2

Air

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

30 136 15 −71 925 17 −78 −118 15 15 −180 0–100

1.14 1.15 1.26 1.31 1.36 1.403 1.408 1.415 1.320 1.670 1.715 1.67

Ammonia Argon

NH3 Ar

Benzene Bromine

C6H6 Br2

90 20–350

1.10 1.32

Carbon dioxide

CO2

disulfide monoxide

CS2 CO

1.299 1.37 1.21 1.402 1.433 1.355 1.15 1.256 1.315

Chlorine Chloroform Cyanogen Cyclohexane

Cl2 CHCl3 (CN)2 C6H12

15 −75 100 15 −180 15 100 15 80

Dichlorodifluormethane

CCl2F2

25

1.139

Ethane

C2H6

100 15 −82 90 35 80 100 15 −91

1.157 1.200 1.28 1.13 1.08 1.086 1.201 1.253 1.345

−180 80 15 −76 −181 20 15 100 65 140 210

1.667 1.066 1.407 1.441 1.607 1.42 1.41 1.40 1.31 1.28 1.24

Ethyl alcohol ether

C2H6O C4H10O

Ethylene

C2H4

Helium Hexane (n-) Hydrogen

He C6H14 H2

bromide chloride

HBr HCl

cyanide

HCN

Compound

Formula

Hydrogen (Cont.) iodide sulfide

HI H2S

Iodine Isobutane

I2 C4H10

Krypton

Kr

Mercury Methane

Hg CH4

Methyl acetate alcohol ether Methylal

C3H6O2 CH4O C2H6O C3H8O2

Neon Nitric oxide

Ne NO

Nitrogen

N2

Nitrous oxide

N2O

Oxygen

O2

Pentane (n-) Phosphorus Potassium

C5H12 P K

Sodium Sulfur dioxide

Na SO2

Xenon

Xe

Temperature, °C

Ratio of specific heats, (γ) = Cp /Cv

20–100 15 −45 −57

1.40 1.332 1.350 1.356

185 15

1.30 1.110

19

1.672

360 600 300 15 −80 −115 15 77 6–30 13 40

1.67 1.113 1.196 1.310 1.339 1.347 1.14 1.237 1.11 1.06 1.09

19 15 −45 −80 15 −181 100 15 −30 −70

1.667 1.400 1.39 1.38 1.402 1.433 1.28 1.303 1.31 1.34

15 −76 −181

1.398 1.405 1.439

86 300 850

1.071 1.17 1.77

750–920 15

1.68 1.290

19

1.678

*For compounds that appear in Table 2-184, values are from E. W. Lemmon, M. O. McLinden, and D. G. Friend, “Thermophysical Properties of Fluid Systems” in NIST Chemistry WebBook, NIST Standard Reference Database Number 69, Eds. P. J. Linstrom and W. G. Mallard, June 2005, National Institute of Standards and Technology, Gaithersburg, Md. (http://webbook.nist.gov). Values for other compounds are from International Critical Tables, vol. 5, pp. 80–82.

SPECIFIC HEATS OF AQUEOUS SOLUTIONS

2-183

SPECIFIC HEATS OF AQUEOUS SOLUTIONS UNITS CONVERSIONS

ADDITIONAL REFERENCES

For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32 To convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0.

Most of the tables below are from International Critical Tables, vol. 5, pp. 115–116, 122–125.

TABLE 2-158

TABLE 2-165

Acetic Acid (at 38 °C)

Mole % acetic acid Cal/(g⋅°C)

TABLE 2-159

0 1.0

6.98 0.911

30.9 0.73

54.5 0.631

100 0.535

Ammonia Specific heat, cal/(g⋅°C)

Mole % NH3

2.4 °C

20.6 °C

41 °C

61 °C

0 10.5 20.9 31.2 41.4

1.01 0.98 0.96 0.956 0.985

1.0 0.995 0.99 1.0

0.995 1.06 1.03

1.0 1.02

TABLE 2-160

TABLE 2-161

100 0.497

5 °C

20 °C

40 °C

5.88 12.3 27.3 45.8 69.6 100

1.02 0.975 0.877 0.776 0.681 0.576

1.0 0.982 0.917 0.811 0.708 0.60

0.995 0.98 0.92 0.83 0.726 0.617

TABLE 2-166

95 0.52

90.5 0.53

82.3 0.56

75.2 0.581

Copper Sulfate

Composition CuSO4 + 50H2O CuSO4 + 200H2O CuSO4 + 400H2O TABLE 2-162

Specific heat, cal/(g⋅°C) Mole % CH3OH

Aniline (at 20 °C)

Mol % aniline Cal/(g⋅°C)

Methyl Alcohol

Temperature

Specific heat, cal/(g⋅°C)

12 to 15 °C 12 to 14 °C 13 to 17 °C

0.848 0.951 0.975

Nitric Acid

% HNO3 by Weight

Specific heat at 20 °C, cal/(g⋅°C)

0 10 20 30 40 50 60 70 80 90

1.000 0.900 0.810 0.730 0.675 0.650 0.640 0.615 0.575 0.515

Ethyl Alcohol Specific heat, cal/(g⋅°C)

Mole % C2H5OH 4.16 11.5 37.0 61.0 100.0 TABLE 2-163

3 °C

23 °C

41 °C

1.05 1.02 0.805 0.67 0.54

1.02 1.03 0.86 0.727 0.577

1.02 1.03 0.875 0.748 0.621

Glycerol Specific heat, cal/(g⋅°C)

Mole % C3H5(OH)3

15 °C

32 °C

2.12 4.66 11.5 22.7 43.9 100.0

0.961 0.929 0.851 0.765 0.67 0.555

0.960 0.924 0.841 0.758 0.672 0.576

TABLE 2-164

Hydrochloric Acid Specific heat, cal/(g⋅°C)

Mole % HCl 0.0 9.09 16.7 20.0 25.9

0 °C

10 °C

20 °C

40 °C

60 °C

1.00 0.72 0.61 0.58 0.55

0.72 0.605 0.575

0.74 0.631 0.591

0.75 0.645 0.615

0.78 0.67 0.638 0.61

TABLE 2-167

Phosphoric Acid*

%H2PO4

Cp at 21.3 °C cal/(g⋅°C)

%H3PO4

Cp at 21.3 °C cal/(g⋅°C)

2.50 3.80 5.33 8.81 10.27 14.39 16.23 19.99 22.10 24.56 25.98 28.15 29.96 32.09 33.95 36.26 38.10 40.10 42.08 44.11 46.22 48.16 49.79

0.9903 0.9970 0.9669 0.9389 0.9293 0.8958 0.8796 0.8489 0.8300 0.8125 0.8004 0.7856 0.7735 0.7590 0.7432 0.7270 0.7160 0.7024 0.6877 0.6748 0.6607 0.6475 0.6370

50.00 52.19 53.72 56.04 58.06 60.23 62.10 64.14 66.13 68.14 69.97 69.50 71.88 73.71 75.79 77.69 79.54 80.00 82.00 84.00 85.98 88.01 89.72

0.6350 0.6220 0.6113 0.5972 0.5831 0.5704 0.5603 0.5460 0.5349 0.5242 0.5157 0.5160 0.5046 0.4940 0.4847 0.4786 0.4680 0.4686 0.4593 0.4500 0.4419 0.4359 0.4206

*Z. Physik. Chem., A167, 42 (1933).

2-184

PHYSICAL AND CHEMICAL DATA

TABLE 2-168

Potassium Chloride

TABLE 2-172

Sodium Chloride

Specific heat, cal/(g°C)

Specific heat, cal/(g°C)

Mole % KCl

6 °C

20 °C

33 °C

40 °C

Mole % NaCl

0.99 3.85 5.66 7.41

0.945 0.828 0.77

0.947 0.831 0.775 0.727

0.947 0.835 0.778

0.947 0.837 0.775

0.249 0.99 2.44 9.09

TABLE 2-169

Potassium Hydroxide (at 19 °C)

Mole % KOH Cal/(g°C)

TABLE 2-170

0 1.0

0.497 0.975

1.64 0.93

TABLE 2-173 4.76 0.814

9.09 0.75

Normal Propyl Alcohol

Mole % NaOH Cal/(g . °C)

TABLE 2-174

5 °C

20 °C

40 °C

1.55 5.03 11.4 23.1 41.2 73.0 100.0

1.03 1.07 1.035 0.877 0.75 0.612 0.534

1.02 1.06 1.032 0.90 0.78 0.645 0.57

1.01 1.03 0.99 0.91 0.815 0.708 0.621

TABLE 2-171 % Na2CO3 by weight 0.000 1.498 2.000 2.901 4.000 5.000 6.000 8.000 10.000 13.790 13.840 20.000 25.000

Sodium Carbonate* Temperature, °C 17.6

30.0

76.6

98.0

0.9992 0.9807

0.9986

1.0098

1.0084

0.9786 0.9597

33 °C

57 °C

0.96 0.91 0.805

0.99 0.97 0.915 0.81

0.97 0.915 0.81

0.923 0.82

0 1.0

0.5 0.985

1.0 0.97

9.09 0.835

16.7 0.80

28.6 0.784

37.5 0.782

Sulfuric Acid*

%H2SO4

Cp at 20 °C, cal/(g°C)

%H2SO4

Cp at 20 °C, cal/(g°C)

0.34 0.68 1.34 2.65 3.50 5.16 9.82 15.36 21.40 22.27 23.22 24.25 25.39 26.63 28.00 29.52 30.34 31.20 33.11

0.9968 0.9937 0.9877 0.9762 0.9688 0.9549 0.9177 0.8767 0.8339 0.8275 0.8205 0.8127 0.8041 0.7945 0.7837 0.7717 0.7647 0.7579 0.7422

35.25 37.69 40.49 43.75 47.57 52.13 57.65 64.47 73.13 77.91 81.33 82.49 84.48 85.48 89.36 91.81 94.82 97.44 100.00

0.7238 .7023 .6770 .6476 .6153 .5801 .5420 .5012 .4628 .4518 .4481 .4467 .4408 .4346 .4016 .3787 .3554 .3404 .3352

*Vinal and Craig, Bur. Standards J. Research, 24, 475 (1940).

0.9594 0.9428

20 °C

Sodium Hydroxide (at 20 °C)

Specific heat, cal/(g°C) Mole % C3H7OH

6 °C

0.9761 0.9392

0.9183 0.9086 0.8924

*J. Chem. Soc. 3062–3079 (1931).

0.9452 0.8881 0.8631

0.8936 0.8615

TABLE 2-175 0.8911

Zinc Sulfate

Composition

Temperature

ZnSO4 + 50H2O ZnSO4 + 200H2O

20 to 52 °C 20 to 52 °C

Specific heat, cal/(g°C) 0.842 0.952

HEATS AND FREE ENERGIES OF FORMATION

2-185

SPECIFIC HEATS OF MISCELLANEOUS MATERIALS TABLE 2-176 and Solids

Specific Heats of Miscellaneous Liquids

Material

Specific heat, cal/(g⋅°C) 0.2 (100 °C); 0.274 (1500 °C) 0.186 (100 °C) 0.25 0.22 0.3 to 0.4 About 0.2 0.168 (26 to 76 °C) 0.314 (40 to 892 °C) 0.387 (56 to 1450 °C) 0.204

Alumina Alundum Asbestos Asphalt Bakelite Brickwork Carbon (gas retort) (see under Graphite) Cellulose Cement, Portland Clinker Charcoal (wood) Chrome brick Clay Coal tar oils Coal tars Coke Concrete Cryolite Diamond Fireclay brick Fluorspar Gasoline Glass (crown) (flint) (pyrex) (silicate) wool Granite Graphite Gypsum Kerosene Limestone Litharge Magnesia Magnesite brick Marble Porcelain, fired Berlin Porcelain, green Berlin Porcelain, fired earthenware Porcelain, green earthenware

0.32 0.186 0.242 0.17 0.224 0.26 to 0.37 0.34 (15 to 90 °C) 0.35 (40 °C); 0.45 (200 °C) 0.265 (21 to 400 °C) 0.359 (21 to 800 °C) 0.403 (21 to 1300 °C) 0.156 (70 to 312 °F); 0.219 (72 to 1472 °F) 0.253 (16 to 55 °C) 0.147 0.198 (100 °C); 0.298 (1500 °C) 0.21 (30 °C) 0.53 0.16 to 0.20 0.117 0.20 0.188 to 0.204 (0 to 100 °C) 0.24 to 0.26 (0 to 700 °C) 0.157 0.20 (20 to 100 °C) 0.165 (26 to 76 °C); 0.390 (56 to 1450 °C) 0.259 (16 to 46 °C) 0.47 0.217 0.055 0.234 (100 °C); 0.188 (1500 °C) 0.222 (100 °C); 0.195 (1500 °C) 0.21 (18 °C) 0.189 (60 °C) 0.185 (60 °C) 0.186 (60 °C) 0.181 (60 °C)

TABLE 2-176 Specific Heats of Miscellaneous Liquids and Solids (Concluded) Material

Specific heat, cal/(g⋅°C)

Pyrex glass Pyrites (copper) Pyrites (iron) Pyroxylin plastics Quartz Rubber (vulcanized) Sand Silica Silica brick Silicon carbide brick Silk Steel Stone Stoneware (common) Turpentine Wood (Oak) Woods, miscellaneous Wool Zirconium oxide

TABLE 2-177

0.20 0.131 (30 °C) 0.136 (30 °C) 0.34 to 0.38 0.17 (0 °C); 0.28 (350 °C) 0.415 0.191 0.316 0.202 (100 °C); 0.195 (1500 °C) 0.202 (100 °C) 0.33 0.12 about 0.2 0.188 (60 °C) 0.42 (18 °C) 0.570 0.45 to 0.65 0.325 0.11 (100 °C); 0.179 (1500 °C)

Oils (Animal, Vegetable, Mineral Oils) 15  Cp[cal/(g⋅°C)] = A/d 4 + B(t − 15)

where d = density, g/cm3. °F = 9⁄5 °C + 32; to convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0; to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. Oils

A

Castor Citron Fatty drying nondrying semidrying oils (except castor) Naphthene base Olive Paraffin base Petroleum oils

0.500

PROPERTIES OF FORMATION AND COMBUSTION REACTIONS UNITS CONVERSIONS °F = 9⁄5 °C + 32; to convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

B

0.0007 (0.438 at 54 °C) 0.440 0.0007 0.450 0.0007 0.445 0.0007 0.450 0.0007 0.405 0.0009 (0.47 at 7 °C) 0.425 0.0009 0.415 0.0009

2-186

PHYSICAL AND CHEMICAL DATA

TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds

The values given in the following table for the heats and free energies of formation of inorganic compounds are derived from (a) Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York, 1936; (b) Latimer, “Oxidation States of the Elements and Their Potentials in Aqueous Solution,” PrenticeHall, New York, 1938; (c) the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards; and (d) the tables of Selected Values of Chemical Thermodynamic Properties of the National Bureau of Standards. The reader is referred to the preceding books and tables for additional details as to methods of calculation, standard states, and so on.

Compound Aluminum Al AlBr3 Al4C3 AlCl3 AlF3 AlI3 AlN Al(NH4)(SO4)2 Al(NH4)(SO4)2·12H2O Al(NO3)3·6H2O Al(NO3)3·9H2O Al2O3 Al(OH)3 Al2O3·SiO2 Al2O3·SiO2 Al2O3·SiO2 3Al2O3·2SiO2 Al2S3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 SbCl5 SbF3 SbI3 Sb2O3 Sb2O4 Sb2O5 Sb2S3 Arsenic As AsBr3 AsCl3 AsF3 AsH3 AsI3 As2O3 As2O5 As2S3 Barium Ba BaBr2 BaCl2 Ba(ClO3)2 Ba(ClO4)2 Ba(CN)2 Ba(CNO)2 BaCN2 BaCO3 BaCrO4

State† c c aq c c aq, 600 c aq c aq c c c c c c, corundum c c, sillimanite c, disthene c, andalusite c, mullite c c aq c c c c c l c c c, I, orthorhombic c, II, octahedral c c c, black

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol 0.00 −123.4 −209.5 −30.8 −163.8 −243.9 −329 −360.8 −72.8 −163.4 −57.7 −561.19 −1419.36 −680.89 −897.59 −399.09 −304.8 −648.7 −642.4 −642.0 −1874 −121.6 −820.99 −893.9 −1268.15 −2120 0.00 −59.9 −91.3 −104.8 −216.6 −22.8 −165.4 −166.6 −213.0 −230.0 −38.2

c c l l g c c c c amorphous

0.00 −45.9 −80.2 −223.76 43.6 −13.6 −154.1 −217.9 −20 −34.76

c c aq, 400 c aq, 300 c aq, 1600 c aq, 800 c c aq c c, witherite c

0.00 −180.38 −185.67 −205.25 −207.92 −176.6 −170.0 −210.2

*For footnotes see end of table.

−48 −212.1 −63.6 −284.2 −342.2

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 0.00 −189.2 −29.0 −209.5 −312.6 −152.5 −50.4 −486.17 −1179.26 −526.32 −376.87 −272.9

−739.53 −759.3 −1103.39

Compound Barium (Cont.) BaF2 BaH2 Ba(HCO3)2 BaI2 Ba(IO3)2 BaMoO4 Ba3N2 Ba(NO2)2 Ba(NO3)2 BaO Ba(OH)2 BaO·SiO2 Ba3(PO4)2 BaPtCl6 BaS BaSO3 BaSO4 BaWO4 Beryllium Be BeBr2

0.00 −77.8

−146.0 −186.6 −196.1 −36.9 0.00 −70.5 −212.27 37.7 −134.8 −183.9 −20 0.00 −183.0 −196.5 −134.4 −155.3 −180.7 −271.4

BeCl2 BeI2 Be3N2 BeO Be(OH)2 BeS BeSO4 Bismuth Bi BiCl3 BiI3 BiO Bi2O3 Bi(OH)3 Bi2S3 Bi2(SO4)3 Boron B BBr3 BCl3 BF3 B2H6 BN B2O3 B(OH)3 B2S3 Bromine Br2 BrCl

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

c aq, 1600 c aq c aq, 400 c aq c c c aq c aq, 600 c c aq, 400 c c c c c c c

−287.9 −284.6 −40.8 −459 −144.6 −155.17 −264.5 −237.50 −370 −90.7 −184.5 −179.05 −236.99 −227.74 −133.0 −225.9 −237.76 −363 −992 −284.9 −111.2 −282.5 −340.2 −402

c c aq c aq c aq c c c c c aq

0.00 −79.4 −142 −112.6 −163.9 −39.4 −112 −134.5 −145.3 −215.6 −56.1 −281

c c aq c aq c c c c c

0.00 −90.5 −101.6 −24 −27 −49.5 −137.1 −171.1 −43.9 −607.1

c l g g g g c c gls c c

0.00 −52.7 −44.6 −94.5 −265.2 7.5 −32.1 −302.0 −297.6 −260.0 −56.6

l g g

0.00 7.47 3.06

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −265.3 −31.5 −414.4 −158.52 −198.35

−150.75 −189.94

−209.02

−313.4 0.00 −127.9 −141.4 −103.4 −122.4 −138.3

−254.8 0.00 −76.4

−43.2 −117.9 −39.1 0.00 −50.9 −90.8 −261.0 19.9 −27.2 −282.9 −280.3 −229.4 0.00 0.931 −0.63

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Cadmium Cd CdBr2 CdCl2 Cd(CN)2 CdCO3 CdI2 Cd3N2 Cd(NO3)2 CdO Cd(OH)2 CdS CdSO4 Calcium Ca CaBr2 CaC2 CaCl2 CaCN2 Ca(CN)2 CaCO3 CaCO3·MgCO3 CaC2O4 Ca(C2H3O2)2 CaF2 CaH2 CaI2 Ca3N2 Ca(NO3)2 Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O CaO Ca(OH)2 CaO·SiO2 CaS CaSO4 CaSO4·aH2O CaSO4·2H2O CaWO4 Carbon C CO CO2 Cerium Ce CeN Cesium Cs CsBr CsCl

2-187

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

c c aq, 400 c aq, 400 c c c aq, 400 c aq, 400 c c c c aq, 400

0.00 −75.8 −76.6 −92.149 −96.44 36.2 −178.2 −48.40 −47.46 39.8 −115.67 −62.35 −135.0 −34.5 −222.23 −232.635

c c aq, 400 c c aq c c aq c, calcite c, aragonite c c c aq c aq c c aq, 400 c c aq, 400 c c c c c aq, 800 c, II, wollastonite c, I, pseudowollastonite c c, insoluble form c, soluble form α c, soluble form β c c c

0.00 −162.20 −187.19 −14.8 −190.6 −209.15 −85 −43.3 −289.5 −289.54 −558.8 −332.2 −356.3 −364.1 −290.2 −286.5 −46 −128.49 −156.63 −103.2 −224.05 −228.29 −367.95 −439.05 −509.43 −151.7 −235.58 −239.2 −377.9 −376.6 −114.3 −338.73 −336.58 −335.52 −376.13 −479.33 −387

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 0.00 −70.7 −67.6 −81.889 −81.2 −163.2 −43.22 −71.05 −55.28 −113.7 −33.6 −194.65 0.00 −181.86 −16.0 −179.8 −195.36 −54.0 −270.8 −270.57

−311.3 −264.1 −35.7 −157.37 −88.2 −177.38 −293.57 −351.58 −409.32 −144.3 −213.9 −207.9 −357.5 −356.6 −113.1 −311.9 −309.8 −308.8 −425.47

c, graphite c, diamond g g

0.00 0.453 −26.416 −94.052

0.00 0.685 −32.808 −94.260

c c

0.00 −78.2

0.00 −70.8

0.00 −97.64 −91.39 −106.31 −102.01

0.00

c c aq, 500 c aq, 400

−94.86 −101.61

Compound Cesium (Cont.) Cs2CO3 CsF CsH CsHCO3 CsI CsNH2 CsNO3 Cs2O CsOH Cs2S Cs2SO4 Chlorine Cl2 ClF ClO ClO2 ClO3 Cl2O Cl2O7 Chromium Cr CrBr3 Cr3C2 Cr4C CrCl2 CrF2 CrF3 CrI2 CrO3 Cr2O3 Cr2(SO4)3 Cobalt Co CoBr2 Co3C CoCl2 CoCO3 CoF2 CoI2 Co(NO3)2 CoO Co3O4 Co(OH)2 Co(OH)3 CoS Co2S3 CoSO4 Columbium Cb Cb2O5 Copper Cu CuBr CuBr2 CuCl CuCl2

State† c c aq, 400 c c aq, 2000 c aq, 400 c c aq, 400 c c aq, 200 c c aq g g g g g g g c aq c c c aq c c c aq c c aq

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol −271.88 −131.67 −140.48 −12 −230.6 −226.6 −83.91 −75.74 −28.2 −121.14 −111.54 −82.1 −100.2 −117.0 −87 −344.86 −340.12 0.00 −25.7 33 24.7 37 18.20 63 0.00 −21.008 −16.378 −103.1 −152 −231 −63.7 −139.3 −268.8

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

−135.98 −7.30 −210.56 −82.61 −96.53 −107.87 −316.66 0.00 29.5 22.40 0.00 −122.7 −21.20 −16.74 −93.8 −102.1

−64.1 −249.3 −626.3

c c aq c c aq, 400 c aq c aq c aq c c c c c c c aq, 400

0.00 −55.0 −73.61 9.49 −76.9 −95.58 −172.39 −172.98 −24.2 −43.15 −102.8 −114.9 −57.5 −196.5 −131.5 −177.0 −22.3 −40.0 −216.6

c c

0.00 −462.96

0.00

0.00 −26.7 −34.0 −42.4 −31.4 −48.83 −64.7

0.00 −23.8

c c c aq c c aq, 400

0.00 −61.96 7.08 −66.6 −75.46 −155.36 −144.2 −37.4 −65.3 −108.9 −142.0 −19.8 −188.9

−33.25 −24.13

2-188

PHYSICAL AND CHEMICAL DATA

TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Copper (Cont.) CuClO4 Cu(ClO3)2 Cu(ClO4)2 CuI CuI2 Cu3N Cu(NO3)2 CuO Cu2O Cu(OH)2 CuS Cu2S CuSO4 Cu2SO4 Erbium Er Er(OH)3 Fluorine F2 F2O Gallium Ga GaBr3 GaCl3 GaN Ga2O Ga2O3 Germanium Ge Ge3N4 GeO2 Gold Au AuBr AuBr3 AuCl AuCl3 AuI Au2O3 Au(OH)3 Hafnium Hf HfO2 Hydrogen H3AsO3 H3AsO4 HBr HBrO HBrO3 HCl HCN HClO HClO3 HClO4 HC2H3O2 H2C2O4 HCOOH

State† aq aq, 400 aq c c aq c c aq, 200 c c c c c c aq, 800 c aq

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol −28.3 −17.8 −4.8 −11.9 17.78 −73.1 −83.6 −38.5 −43.00 −108.9 −11.6 −18.97 −184.7 −200.78 −179.6

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 1.34 15.4 −5.5 −16.66 −8.76 −36.6 −31.9 −38.13 −85.5 −11.69 −20.56 −158.3 −160.19 −152.0

c c

0.00 −326.8

0.00

g g

0.00 5.5

0.00 9.7

Compound Hydrogen (Cont.) H2CO3 HF HI HIO HIO3 HN3 HNO3 HNO3·H2O HNO3·3H2O H2O H2O2 H3PO2 H3PO3 H3PO4 H2S

c c c c c c

0.00 −92.4 −125.4 −26.2 −84.3 −259.9

0.00

c c c

0.00 −15.7 −128.6

0.00

c c c aq c c aq c c c

0.00 −3.4 −14.5 −11.0 −8.3 −28.3 −32.96 0.2 11.0 −100.6

0.00

H2S2 H2SO3 H2SO4 H2Se H2SeO3 H2SeO4

24.47

H2SiO3 H4SiO4 H2Te H2TeO3

4.21 −0.76 18.71

H2TeO4 Indium In InBr3 InCl3

c c

0.00 −271.1

0.00 −258.2

aq c aq g aq, 400 aq aq g aq, 400 g aq, 100 aq, 400 aq aq, 660 l aq, 400 c aq, 300 l aq, 200

−175.6 −214.9 −214.8 −8.66 −28.80 −25.4 −11.51 −22.063 −39.85 31.1 24.2 −28.18 −23.4 −31.4 −116.2 −116.74 −196.7 −194.6 −97.8 −98.0

−153.04 −183.93 −12.72 −24.58 −19.90 5.00 −22.778 −31.330 27.94 26.55 −19.11 −0.25 −10.70 −93.56 −96.8 −165.64 −82.7 −85.1

InI3 InN In2O3 Iodine I2 IBr ICl ICl3 I2O5 Iridium Ir IrCl IrCl2 IrCl3 IrF6 IrO2 Iron Fe FeBr2

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

aq g aq, 200 g aq, 400 aq c aq g g l aq, 400 l l g l l aq, 200 c aq c aq c aq, 400 g aq, 2000 l aq, 200 l aq, 400 g aq c aq c aq, 400 c c g c aq aq

−167.19 −64.2 −75.75 6.27 −13.47 −38 −56.77 −54.8 70.3 −31.99 −41.35 −49.210 −112.91 −252.15 −57.7979 −68.3174 −45.16 −45.80 −145.5 −145.6 −232.2 −232.2 −306.2 −309.32 −4.77 −9.38 −3.6 −146.88 −193.69 −212.03 20.5 18.1 −126.5 −122.4 −130.23 −143.4 −267.8 −340.6 36.9 −145.0 −145.0 −165.6

c c aq c aq c aq c c

0.00 −97.2 −112.9 −128.5 −145.6 −56.5 −67.2 −4.8 −222.47

c g g g c c

0.00 14.88 10.05 4.20 −21.8 −42.5

0.00 4.63 1.24 −1.32 −6.05

c c c c l c

0.00 −20.5 −40.6 −60.5 −130 −40.14

0.00 −16.9 −32.0 −46.5

0.00 −57.15 −78.7

0.00

c, α c aq, 540

−149.0 −64.7 0.365 −12.35 −23.33 −32.25 78.50 −17.57 −19.05 −78.36 −193.70 −54.6351 −56.6899 −28.23 −31.47 −120.0 −204.0 −270.0 −7.85 −128.54 17.0 18.4 −101.36 −247.9 33.1 −115.7

0.00 −97.2 −117.5 −60.5

−69.47

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Iron (Cont.) FeBr3 Fe3C Fe(CO)5 FeCO3 FeCl2 FeCl3 FeF2 FeI2 FeI3 Fe4N Fe(NO3)2 Fe(NO3)3 FeO Fe2O3 Fe3O4 Fe(OH)2 Fe(OH)3 FeO·SiO2 Fe2P FeSi FeS FeS2 FeSO4 Fe2(SO4)3 FeTiO3 Lanthanum La LaCl3 La3H8 LaN La2O3 LaS2 La2S3 La2(SO4)3 Lead Pb PbBr2 PbCO3 Pb(C2H3O2)2 PbC2O4 PbCl2 PbF2 PbI2 Pb(NO3)2 PbO PbO2 Pb3O4 Pb(OH)2 PbS PbSO4 Lithium Li LiBr LiBrO3 Li2C2 LiCN LiCNO

2-189

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

aq c l c, siderite c aq c aq, 2000 aq, 1200 c aq aq c aq aq, 800 c c c c c c c c c c, pyrites c, marcasite c aq, 400 aq, 400 c, ilmenite

−95.5 5.69 −187.6 −172.4 −81.9 −100.0 −96.4 −128.5 −177.2 −24.2 −47.7 −49.7 −2.55 −118.9 −156.5 −64.62 −198.5 −266.9 −135.9 −197.3 −273.5 −13 −19.0 −22.64 −38.62 −33.0 −221.3 −236.2 −653.3 −295.51

c c aq c c c c c aq

0.00 −253.1 −284.7 −160 −72.0 −539 −148.3 −351.4 −972

c c aq c, cerussite c aq, 400 c c aq c c c aq, 400 c, red c, yellow c c c c c

0.00 −66.24 −56.4 −167.6 −232.6 −234.2 −205.3 −85.68 −82.5 −159.5 −41.77 −106.88 −99.46 −51.72 −50.86 −65.0 −172.4 −123.0 −22.38 −218.5

c c aq, 400 aq c aq aq

0.00 −83.75 −95.40 −77.9 −13.0 −31.4 −101.2

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −76.26 4.24 −154.8 −72.6 −83.0 −96.5 −151.7 −45 −39.5 0.862 −72.8 −81.3 −59.38 −179.1 −242.3 −115.7 −166.3

−23.23 −35.93 −195.5 −196.4 −533.4 −277.06 0.00

Compound Lithium (Cont.) LiC2H3O2 Li2CO3 LiCl LiClO3 LiClO4 LiF LiH LiHCO3 LiI LiIO3 Li3N LiNO3 Li2O Li2O2 LiOH LiOH·H2O Li2O·SiO2 Li2Se Li2SO4 Li2SO4·H2O Magnesium Mg Mg(AsO4)2 MgBr2

−64.6

0.00 −62.06 −54.97 −150.0 −184.40 −75.04 −68.47 −148.1 −41.47 −58.3 −45.53 −43.88 −52.0 −142.2 −102.2 −21.98 −192.9 0.00 −95.28 −65.70 −31.35 −94.12

Mg(CN)2 MgCN2 Mg(C2H3O2)2 MgCO3 MgCl2 MgCl2·H2O MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgF2 MgI2 MgMoO4 Mg3N2 Mg(NO3)2 Mg(NO3)2·2H2O Mg(NO3)2·6H2O MgO MgO·SiO2 Mg(OH)2 MgS MgSO4 MgTe MgWO4 Manganese Mn MnBr2 Mn3C

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

aq c aq, 1900 c aq, 278 aq aq c aq, 400 c aq, 2000 c aq, 400 aq c c aq, 400 c c aq c aq, 400 c gls c aq c aq, 400 c

−183.9 −289.7 −293.1 −97.63 −106.45 −87.5 −106.3 −145.57 −144.85 −22.9 −231.1 −65.07 −80.09 −121.3 −47.45 −115.350 −115.88 −142.3 −151.9 −159 −116.58 −121.47 −188.92 −374 −84.9 −95.5 −340.23 −347.02 −411.57

c c aq c aq, 400 aq c aq c c aq, 400 c c c c c c aq, 400 c c c aq, 400 c c c c c, ppt. c, brucite c aq c aq, 400 c c

0.00 −731.3 −749 −123.9 −167.33 −39.7 −61 −344.6 −261.7 −153.220 −189.76 −230.970 −305.810 −453.820 −597.240 −263.8 −86.8 −136.79 −329.9 −115.2 −188.770 −209.927 −336.625 −624.48 −143.84 −347.5 −221.90 −223.9 −84.2 −108 −304.94 −325.4 −25 −345.2

c, α c aq c

0.00 −91 −106 1.1

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −160.00 −269.8 −267.58 −102.03 −70.95 −81.4 −136.40 −210.98 −83.03 −102.95 −37.33 −96.95 −138.0 −106.44 −108.29

−105.64 −314.66 −375.07 0.00 −630.14 −156.94 −29.08 −286.38 −241.7 −143.77 −205.93 −267.20 −387.98 −505.45 −132.45 −100.8 −140.66 −160.28 −496.03 −136.17 −326.7 −200.17 −193.3 −277.7 −283.88

0.00 −97.8 1.26

2-190

PHYSICAL AND CHEMICAL DATA

TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Manganese (Cont.) Mn(C2H3O2)2 MnCO3 MnC2O4 MnCl2 MnF2 MnI2 Mn5N2 Mn(NO3)2 Mn(NO3)2.6H2O MnO MnO2 Mn2O3 Mn3O4 MnO.SiO2 Mn(OH)2 Mn(OH)3 Mn3(PO4)2 MnSe MnS MnSO4 Mn2(SO4)3 Mercury Hg HgBr HgBr2 Hg(C2H3O2)2 HgCl2 HgCl Hg2Cl2 Hg(CN)2 HgC2O4 HgH HgI2 HgI Hg2I2 Hg(NO3)2 Hg2(NO3)2 HgO Hg2O HgS HgSO4 Hg2SO4 Molybdenum Mo Mo2C Mo2N MoO2 MoO3 MoS2 MoS3 Nickel Ni NiBr2 Ni3C Ni(C2H3O2)2 Ni(CN)2 NiCl2

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

c aq c c c aq, 400 aq, 1200 c aq c c aq, 400 c c c c c c c c c c c, green c aq, 400 c aq

−270.3 −282.7 −211 −240.9 −112.0 −128.9 −206.1 −49.8 −76.2 −57.77 −134.9 −148.0 −557.07 −92.04 −124.58 −229.5 −331.65 −301.3 −163.4 −221 −736 −26.3 −47.0 −254.18 −265.2 −635 −657

l g c aq c aq c aq g c c aq, 1110 c g c, red g c aq aq c, red c, yellow ppt. c c, black c c

0.00 23 −40.68 −38.4 −196.3 −192.5 −53.4 −50.3 19 −63.13 62.8 66.25 −159.3 57.1 −25.3 33 −28.88 −56.8 −58.5 −21.6 −20.8 −21.6 −10.7 −166.6 −177.34

c c c c c c c

0.00 4.36 −8.3 −130 −180.39 −56.27 −61.48

c c aq c aq aq c

0.00 −53.4 −72.6 9.2 −249.6 230.9 −75.0

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

Compound

State†

Nickel (Cont.) −227.2 −192.5 −102.2 −180.0 −73.3 −46.49 −101.1 −441.2 −86.77 −111.49 −209.9 −306.22 −282.1 −143.1 −190 −27.5 −48.0 −228.41

NiF2 NiI2 Ni(NO3)2 NiO Ni(OH)2 Ni(OH)3 NiS NiSO4 Nitrogen N2 NF3 NH3 NH4Br NH4C2H3O2 NH4CN NH4CNS (NH4)2CO3 (NH4)2C2O4

0.00 18 −38.8 −9.74 −139.2 −42.2 −23.25 14

NH4Cl NH4ClO4 (NH4)2CrO4 NH4F NH4I NH4NO3

52.25 −24.0 23 −26.53 −13.09 −15.65 −13.94 −12.80 −8.80 −149.12 0.00 2.91 −118.0 −162.01 −54.19 −57.38 0.00 −60.7 8.88 −190.1 66.3

NH4OH (NH4)2S (NH4)2SO4 N2H4 N2H4·H2O N2H4·H2SO4 N2O NO NO2 N2O4 N2O5 NOBr NOCl Osmium Os OsO4 Oxygen O2 O3 Palladium Pd PdO Phosphorus P P

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

aq, 400 c aq c aq c aq, 200 c c c c c aq, 200

−94.34 −157.5 −171.6 −22.4 −42.0 −101.5 −113.5 −58.4 −129.8 −163.2 −20.4 −216 −231.3

g g g aq, 200 c aq c aq, 400 c aq c aq aq c aq c aq, 400 c aq c aq c aq c aq c aq, 500 aq aq, 400 c aq, 400 l l c g g g g c l g

0.00 −27 −10.96 −19.27 −64.57 −60.27 −148.1 −148.58 −0.7 3.6 −17.8 −12.3 −223.4 −266.3 −260.6 −75.23 −71.20 −69.4 −63.2 −276.9 −271.3 −111.6 −110.2 −48.43 −44.97 −87.40 −80.89 −87.59 −55.21 −281.74 −279.33 12.06 −57.96 −232.2 19.55 21.600 7.96 2.23 −10.0 11.6 12.8

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −74.19 −142.9 −36.2 −64.0 −51.7 −105.6

−187.6 0.00 −3.903 −43.54 −108.26 20.4 4.4 −164.1 −196.2 −48.59 −21.1 −209.3 −84.7 −31.3

−14.50 −215.06 −214.02

24.82 20.719 12.26 23.41 19.26 16.1

c c g

0.00 −93.6 −80.1

0.00 −70.9 −68.1

g g

0.00 33.88

0.00 38.86

c c

0.00 −20.40

0.00

c, white (“yellow”) c, red (“violet”) g

0.00 −4.22 150.35

0.00 −1.80 141.88

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound Phosphorus (Cont.) P2 P4 PBr3 PBr5 PCl3 PCl5 PH3 PI3 P2O5 POCl3 Platinum Pt PtBr4 PtCl2 PtCl4 PtI4 Pt(OH)2 PtS PtS2 Potassium K K3AsO3 K3AsO4 KH2AsO4 KBr KBrO3 KC2H3O2 KCl KClO3 KClO4 KCN KCNO KCNS K2CO3 K2C2O4 K2CrO4 K2Cr2O7 KF K3Fe(CN)6 K4Fe(CN)6 KH KHCO3 KI KIO3 KIO4 KMnO4 K2MoO4

2-191

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

g g l c g l g g c c g

33.82 13.2 −45 −60.6 −70.0 −76.8 −91.0 2.21 −10.9 −360.0 −138.4

c c aq c c aq c c c c

0.00 −40.6 −50.7 −34 −62.6 −82.3 −18 −87.5 −20.18 −26.64

c aq aq c c aq, 400 c aq, 1667 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 180 c aq c aq c c aq, 2000 c aq, 500 c aq, 400 aq c aq, 400 aq, 880

0.00 −323.0 −390.3 −271.2 −94.06 −89.19 −81.58 −71.68 −173.80 −177.38 −104.348 −100.164 −93.5 −81.34 −103.8 −101.14 −28.1 −25.3 −99.6 −94.5 −47.0 −41.07 −274.01 −280.90 −319.9 −315.5 −333.4 −328.2 −488.5 −472.1 −134.50 −138.36 −48.4 −34.5 −131.8 −119.9 −10 −229.8 −224.85 −78.88 −73.95 −121.69 −115.18 −98.1 −192.9 −182.5 −364.2

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 24.60 5.89 −65.2 −63.3 −73.2 −1.45 −127.2 0.00

Compound Potassium (Cont.) KNH2 KNO2 KNO3 K2O K2O·Al2O3·SiO2 K2O·Al2O3·SiO2 KOH K3PO3 K3PO4 KH2PO4 K2PtCl4 K2PtCl6

−67.9 −18.55 −24.28

K2Se K2SeO4 K2S

0.00 −355.7 −236.7 −90.8 −92.0 −60.30 −156.73 −97.76 −98.76 −69.30 −72.86 −28.08 −90.85 −44.08

K2SO3 K2SO4 K2SO4·Al2(SO4)3 K2SO4·Al2(SO4)3· 24H2O K2S2O6 Rhenium Re ReF6 Rhodium Rh RhO Rh2O Rh2O3 Rubidium Rb RbBr

−264.04

RbCN Rb2CO3

−293.1

RbCl

−306.3 −440.9 −133.13

−5.3 −207.71 −77.37 −79.76 −101.87 −99.68 −169.1 −168.0 −342.9

RbF RbHCO3 RbI RbNH2 RbNO3 Rb2O Rb2O2 RbOH Ruthenium Ru RuS2 Selenium Se

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

c aq c aq, 400 c c, leucite gls c, adularia c, microcline gls c aq, 400 aq aq c c aq c aq, 9400 c aq aq c aq, 400 c aq c aq, 400 c

−28.25 −86.0 −118.08 −109.79 −86.2 −1379.6 −1368.2 −1784.5 −1784.5 −1747 −102.02 −114.96 −397.5 −478.7 −362.7 −254.7 −242.6 −299.5 −286.1 −74.4 −83.4 −267.1 −121.5 −110.75 −267.7 −269.7 −342.65 −336.48 −1178.38

c c

−2895.44 −418.62

−2455.68

c g

0.00 −274

0.00

c c c c

0.00 −21.7 −22.7 −68.3

0.00

0.00 −95.82 −45.0 −90.54 −25.9 −273.22 −282.61 −105.06 −53.6 −101.06 −133.23 −139.31 −230.01 −225.59 −81.04 −31.2 −74.57 −27.74 −119.22 −110.52 −82.9 −107 −101.3 −115.8

0.00

c c g aq, 500 aq c aq, 220 c g aq, ∞ c aq, 400 c aq, 2000 c g aq, 400 c c aq, 400 c c c aq, 200 c c c, I, hexagonal

−75.9 −94.29 −93.68

−105.0 −443.3 −326.1 −226.5 −263.6 −99.10 −240.0 −111.44 −251.3 −314.62 −310.96 −1068.48

−52.50 −93.38 −263.78 −98.48 −57.9 −100.13 −134.5 −209.07 −40.5 −81.13 −95.05

−106.39

0.00 −46.99

0.00 −44.11

0.00

0.00

2-192

PHYSICAL AND CHEMICAL DATA

TABLE 2-178

Compound

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

Selenium (Cont.) Se2Cl2 SeF6 SeO2 Silicon Si SiBr4 SiC SiCl4 SiF4 SiH4 SiI4 Si3N4 SiO2

Silver Ag AgBr Ag2C2 AgC2H3O2 AgCN Ag2CO3 Ag2C2O4 AgCl AgF AgI AgIO3 AgNO2 AgNO3 Ag2O Ag2S Ag2SO4 Sodium Na Na3AsO3 Na3AsO4 NaBr NaBrO NaBrO3 NaC2H3O2 NaCN NaCNO NaCNS Na2CO3 NaCO2NH2 Na2C2O4 NaCl NaClO3 NaClO4

Compound

State†

Sodium (Cont.) c, II, red, monoclinic l g c

0.2 −22.06 −246 −56.33

−13.73 −222

c l c l g g g c c c, cristobalite, 1600° form c, cristobalite, 1100° form c, quartz c, tridymite

0.00 −93.0 −28 −150.0 −142.5 −370 −14.8 −29.8 −179.25 −202.62

0.00

c c c c aq c c c c c aq, 400 c c c aq c aq, 6500 c c c aq

0.00 −23.90 84.5 −95.9 −91.7 33.8 −119.5 −158.7 −30.11 −48.7 −53.1 −15.14 −42.02 −11.6 −2.9 −29.4 −24.02 −6.95 −5.5 −170.1 −165.8

c aq, 500 c aq, 500 c aq, 400 aq aq, 400 c aq, 400 c aq, 200 c aq c aq, 400 c aq, 1000 c c aq, 600 c aq, 400 c aq, 400 c

0.00 −314.61 −366 −381.97 −86.72 −86.33 −78.9 −68.89 −170.45 −175.450 −22.47 −22.29 −96.3 −91.7 −39.94 −38.23 −269.46 −275.13 −142.17 −313.8 −309.92 −98.321 −97.324 −83.59 −78.42 −101.12

−27.4 −133.9 −133.0 −360 −9.4 −154.74

Na2Cr2O7 NaF NaH NaHCO3 NaI NaIO3 Na2MoO4 NaNO2 NaNO3

−202.46 −203.35 −203.23

Na2CrO4

−190.4 0.00 −23.02 −70.86 38.70 −103.0 −25.98 −47.26 −16.17 −24.08 3.76 9.99 −7.66 −7.81 −2.23 −7.6 −146.8 −139.22 0.00 −341.17 −87.17 −57.59 −152.31 −23.24 −86.00 −39.24 −249.55 −251.36 −283.42 −91.894 −93.92 −62.84

Na2O Na2O2 Na2O·SiO2 Na2O·Al2O3·3SiO2 Na2O·Al2O3·4SiO2 NaOH Na3PO3 Na3PO4 Na2PtCl4 Na2PtCl6 Na2Se Na2SeO4 Na2S Na2SO3 Na2SO4 Na2SO4·10H2O Na2WO4 Strontium Sr SrBr2 Sr(C2H3O2)2 Sr(CN)2 SrCO3 SrCl2 SrF2 Sr(HCO3)2 SrI2 Sr3N2 Sr(NO3)2 SrO SrO·SiO2 SrO2 Sr2O Sr(OH)2 Sr3(PO4)2 SrS

aq, 476 c aq, 800 aq, 1200 c aq, 400 c c aq c aq, ∞ aq, 400 c aq c aq c aq, 400 c c c c, natrolite c c aq, 400 aq, 1000 c aq, 400 aq c aq c aq, 440 c aq, 800 c aq, 400 c aq, 800 c aq, 1100 c c aq c c aq, 400 c aq aq c c aq, 400 c aq c aq, 400 c c aq, 400 c gls c c c aq, 800 c aq c

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol

−97.66 −319.8 −323.0 −465.9 −135.94 −135.711 −14 −226.0 −222.1 −69.28 −71.10 −112.300 −364 −358.7 −86.6 −83.1 −111.71 −106.880 −99.45 −119.2 −383.91 −1180 −1366 −101.96 −112.193 −389.1 −457 −471.9 −237.2 −272.1 −280.9 −59.1 −78.1 −254 −261.5 −89.8 −105.17 −261.2 −264.1 −330.50 −330.82 −1033.85 −391 −381.5

−73.29

0.00 −171.0 −187.24 −358.0 −364.4 −59.5 −290.9 −197.84 −209.20 −289.0 −459.1 −136.1 −156.70 −91.4 −233.2 −228.73 −140.8 −364 −153.3 −153.6 −228.7 −239.4 −980 −985 −113.1

−296.58 −431.18 −129.0 −128.29 −9.30 −202.66 −202.87 −74.92 −94.84 −333.18 −71.04 −87.62 −88.84 −90.06 −105.0 −361.49 −90.60 −100.18 −428.74 −216.78

−89.42 −230.30 −101.76 −240.14 −241.58 −302.38 −301.28 −870.52 −345.18 0.00 −182.36 −311.80 −54.50 −271.9 −195.86 −413.76 −157.87 −76.5 −185.70 −133.7 −139.0 −208.27 −881.54

HEATS AND FREE ENERGIES OF FORMATION TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Continued )

Compound

State†

Strontium (Cont.) SrSO4 SrWO4 Sulfur S

S2 S6 S8 S2Br2 SCl4 S2Cl2 S2Cl4 SF6 SO SO2 SO3

SO2Cl2 Tantalum Ta TaN Ta2O5 Tellurium Te TeBr4 TeCl4 TeF6 TeO2 Thallium Tl TlBr TlCl TlCl3 TlF TlI TlNO3 Tl2O Tl2O3 TlOH Tl2S Tl2SO4 Thorium Th ThBr4 ThC2 ThCl4 ThI4 Th3N4 ThO2 Th(OH)4 Th(SO4)2

2-193

aq c aq, 400 c

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol −120.4 −345.3 −345.0 −393 0.00 −0.071 0.257

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −109.78 −309.30 0.00 0.023 0.072 0.071 43.57 19.36 13.97 12.770

53.25 31.02 27.78 27.090 −4 −13.7 −14.2 −24.1 −262 19.02 −70.94 −94.39 −103.03 −105.09 −105.92 −109.34 −82.04 −89.80

−237 12.75 −71.68 −88.59 −88.28 −88.22 −88.34 −88.98 −74.06 −75.06

c c c

0.00 −51.2 −486.0

0.00 −45.11 −453.7

c c c g c

0.00 −49.3 −77.4 −315 −77.56

0.00

c c aq c aq c aq aq c aq c aq c c c aq c c aq, 800

0.00 −41.5 −28.0 −49.37 −38.4 −82.4 −91.0 −77.6 −31.1 −12.7 −58.2 −48.4 −43.18 −120 −57.44 −53.9 −22 −222.8 −214.1

c c aq c c aq aq c c c, “soluble” c aq

0.00 −281.5 −352.0 −45.1 −335 −392 −292.0 −309.0 −291.6 −336.1 −632 −668.1

−5.90

−57.4 −292 −64.66 0.00 −39.43 −32.34 −44.46 −39.09 −44.25 −73.46 −31.3 −20.09 −36.32 −34.01 −45.54 −45.35 −197.79 −191.62 0.00

SnBr2

SnCl2 SnCl4 SnI2 SnO SnO2 Sn(OH)2 Sn(OH)4 SnS Titanium Ti TiC TiCl4 TiN TiO2 Tungsten W WO2 WO3 WS2 Uranium U UC2 UCl3 UCl4 U3N4 UO2 UO2(NO3)2·6H2O UO3 U3O8 Vanadium V VCl2 VCl3 VCl4 VN V2O2 V2O3 V2O4 V2O5 Zinc Zn ZnSb ZnBr2 Zn(C2H3O2)2 Zn(CN)2 ZnCO3 ZnCl2

−295.31

ZnF2 ZnI2

−322.32 −246.33 −282.3 −280.1

Zn(NO3)2 ZnO ZnO·SiO2 Zn(OH)2 ZnS ZnSO4

−549.2

c, II, tetragonal c, III, “gray,” cubic c aq c aq c aq l aq c aq c c c c c

0.00 0.6 −61.4 −60.0 −94.8 −110.6 −83.6 −81.7 −127.3 −157.6 −38.9 −33.3 −67.7 −138.1 −136.2 −268.9 −18.61

−68.94 −110.4 −124.67

c c l c c, III, rutil amorphous

0.00 −110 −181.4 −80.0 −225.0 −214.1

0.00 −109.2 −165.5 −73.17 −211.9 −201.4

c c c c

0.00 −130.5 −195.7 −84

0.00 −118.3 −177.3

c c c c c c c c c

0.00 −29 −213 −251 −274 −256.6 −756.8 −291.6 −845.1

0.00

c c l l c c c c c

0.00 −147 −187 −165 −41.43 −195 −296 −342 −373

c c c aq, 400 c aq, 400 c c c aq, 400 aq c aq aq, 400 c, hexagonal c c, rhombic c, wurtzite c aq, 400

0.00 −3.6 −77.0 −93.6 −259.4 −269.4 17.06 −192.9 −99.9 −115.44 −192.9 −50.50 −61.6 −134.9 −83.36 −282.6 −153.66 −45.3 −233.4 −252.12

Compound Tin Sn

SnBr4

c, rhombic c, monoclinic l, λ l, λµ equilibrium g g g g l l l l g g g g l c, α c, β c, γ g l

State†

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 0.00 1.1 −55.43 −97.66

−30.95 −60.75 −123.6 −115.95 −226.00

−249.6 −242.2 −617.8

0.00

−35.08 −277 −316 −342 0.00 −3.88 −72.9 −214.4 −173.5 −88.8 −166.6 −49.93 −87.7 −76.19 −44.2 −211.28

2-194

PHYSICAL AND CHEMICAL DATA

TABLE 2-178

Heats and Free Energies of Formation of Inorganic Compounds (Concluded)

Compound Zirconium Zr ZrC ZrCl4 ZrN

State† c c c c

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol 0.00 −29.8 −268.9 −82.5

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol 0.00 −34.6 −75.9

Compound Zirconium (Cont.) ZrO2 Zr(OH)4 ZrO(OH)2

State† c, monoclinic c c

Heat of formation‡§ ∆H (formation) at 25 °C, kcal/mol −258.5 −411.0 −337

Free energy of formation¶ ∆F (formation) at 25 °C, kcal/mol −244.6 −307.6

† The physical state is indicated as follows: c, crystal (solid); l, liquid; g, gas; gls, glass or solid supercooled liquid; aq, in aqueous solution. A number following the symbol aq applies only to the values of the heats of formation (not to those of free energies of formation); and indicates the number of moles of water per mole of solute; when no number is given, the solution is understood to be dilute. For the free energy of formation of a substance in aqueous solution, the concentration is always that of the hypothetical solution of unit molality. ‡ The increment in heat content, ∆H, is the reaction of forming the given substance from its elements in their standard states. When ∆H is negative, heat is evolved in the process, and, when positive, heat is absorbed. § The heat of solution in water of a given solid, liquid, or gaseous compound is given by the difference in the value for the heat of formation of the given compound in the solid, liquid, or gaseous state and its heat of formation in aqueous solution. The following two examples serve as an illustration of the procedure: (1) For NaCl(c) and NaCl(aq, 400H2O), the values of ∆H(formation) are, respectively, −98.321 and −97.324 kcal/mol. Subtraction of the first value from the second gives ∆H = 0.998 kcal/mol for the reaction of dissolving crystalline sodium chloride in 400 mol of water. When this process occurs at a constant pressure of 1 atm, 0.998 kgcal of energy are absorbed. (2) For HCl(g) and HCl(aq, 400H2O), the values for ∆H(formation) are, respectively, −22.06 and −39.85 kcal/mol. Subtraction of the first from the second gives ∆H = −17.79 kcal/mol for the reaction of dissolving gaseous hydrogen chloride in 400 mol of water. At a constant pressure of 1 atm, 17.79 kcal of energy are evolved in this process.  The increment in the free energy, ∆F, is the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are: for a gas, fugacity (approximately equal to the pressure) of 1 atm; for a pure liquid or solid, the substance at a pressure of 1 atm; for a substance in aqueous solution, the hypothetical solution of unit molality, which has all the properties of the infinitely dilute solution except the property of concentration. ¶ The free energy of solution of a given substance from its normal standard state as a solid, liquid, or gas to the hypothetical one molal state in aqueous solution may be calculated in a manner similar to that described in footnote § for calculating the heat of solution.

TABLE 2-179

Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K

2-195

Cmpd. no.

Name

Formula

CAS no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6

Mol. wt.

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy of formation, J/kmol × 1E-07

Ideal gas entropy, J/(kmolK) × 1E-05

Standard net enthalpy of combustion, J/kmol × 1E-09

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541

−16.64 −23.83 −46.11 −57.25 −21.57 7.404 22.82 −8.18 −35.591 18.37 0 −4.5898 −6.79 0 −10.09 8.288 11.15 −29.41 21.57 5.68 −9.025 −11.5 9.33 17.849 3.091 10.5018 −6.36 −3.77 16.23 10.924 −12.579 −44.58 −43.32 −27.51 −29.29 −0.05 −0.74 −1.1 −48.56 −1.314 −8.78 −9.66 16.52 −20.7 −47.58 3.40578 −39.351 11.69 −11.053 −9.581 −92.21 0 5.109 −11.226 −10.29 −8.196 −13.318 −14.477

−13.33 −15.96 −40.3 −47.34 −15.13 9.1868 21.068 −5.68 −30.6 19.37 0 −1.64 2.27 0 −0.211 12.96 14.76 −21.42 25.78 17.3 −0.254 3.37 16.3 27.63 0.314 13.8532 −2.582 −2.819 19.86 14.972 −1.67 −30.44 −29.18 −15.07 −16.7 7.041 6.536 6.32 −31.26 14.54 1.139 0.512 20.225 −11.63 −36 10.8658 −39.437 6.68 −13.715 −5.354 −87.76 0 9.829 −6.0499 −7.01 −5.844 −5.261 −6.136

2.642 2.722 2.825 3.899 2.954 2.4329 2.0081 2.97 3.15 2.753 1.99 1.9266 3.61 1.54737 3.641 2.693 3.369 3.69 3.21 4.4 3.713 4.39 3.607 3.9367 2.4535 3.24386 2.873 2.458 2.93 2.7889 3.0991 4.065 4.065 3.618 3.566 3.074 3.012 2.965 4.425 4.3949 3.752 3.667 2.9039 3.4365 3.601 3.25432 2.13677 2.379 1.97556 3.0991 2.62 2.22972 3.1403 2.7578 2.956 2.3418 3.1547 3.0594

−1.1045 −1.0741 −0.7866 −1.675 −1.659 −1.19043 −1.257 −1.5468 −1.32717 −1.69 0 −0.31683 −3.6072 0 −3.39877 −3.136 −3.4474 −3.0951 −3.5238 −6.2876 −3.56 −4.83 −4.06 −6.248 −3.01917 −1.285 −0.70542 −2.4617 −2.409 −2.65732 −2.2678 −2.2824 −2.454 −2.446 −2.5408 −2.5339 −2.53 −3.28 −5.5644 −2.9554 −2.949 −2.4647 −2.3035 −2.008 −2.4148 −1.0769 −0.283 −0.2653 0.5286 −2.976 −1.2849 −0.38 −0.67538 −1.867 −1.863

2-196

TABLE 2-179

Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K (Continued)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

m-Cresol o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane

C7H8O C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si

108-39-4 95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6

108.138 108.138 108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy of formation, J/kmol × 1E-07

−13.23 −12.857 −12.535 0.4 30.9072 2.85 −12.33 −28.62 −22.61 −0.46 −7.703 3.23 5.33 −9.602 −33.17 −24.946 −59.43 −39.85 −12.21 −21.09 4.1 0 −4.08 −3.89

−4.019 −3.543 −3.166 13.79 29.7598 11.22 3.191 −10.95 −9.028 10.77 3.885 11.05 10.44 4.886 −6.739 3.318 −30.5 −10.02 12.27 6.165 25.16 0 −1.181 −1.054

−33.34 2.57 3.02 2.25 −12.941 −12.979 −9.552 −15.08 −16.28 −40.847 −7.142 −25.21 −8.356 −49.7 −44.77 −45.23 −14.38 −31.92 −31.14 −38.97 −38.42 14.57 −1.845 −17.68 −18.1 −17.2172 −17.9996 −2.42 −18.41 −19.17 −19.41 −60.5 −9.47

−8.827 7.79 8.29 7.67 −7.259 −7.3945 −6.896 −6.52 −8.018 −22.574 7.308 −12.21 1.774 −43.9485 −39.19 −42.4747 6.42 −12.48 −12.37 −23.8 −20.11 18.49 6.839 −0.3125 3.52293 4.12124 3.44761 1.516 −11.28 −8.84 0.5717 −46.7749 −1.925

Ideal gas entropy, J/(kmolK) × 1E-05

Standard net enthalpy of combustion, J/kmol × 1E-09

3.5604 3.5259 3.5075 3.86 2.41463 2.64396 2.97276 3.277 3.3426 3.10518 2.929 2.91267 2.37378 3.646 5.7912 5.457 5.99 5.971 5.433 6.116 5.263 1.4486 3.276 3.297 2.92964 5.014 3.4353 3.4185 3.3674 3.0501 3.0828 2.7018 3.448 3.548 4.29 3.522 3.423 3.681 2.824 2.88194 2.4658 4.12 3.989 4.27 3.726 4.038 2.833 2.7296 3.6592 3.65012 3.7451 3.70912 3.35291 2.667 3.26 4.1455 6.6 2.9953

−3.52783 −3.528 −3.52256 −4.951 −1.0961 −2.5678 −3.656 −3.4639 −3.299 −3.532 −3.0709 −2.9393 −1.9593 −3.968 −5.959 −6.29422 −5.72 −6.116 −6.1809 −6.6161 −6.1037 −0.24625 −1.16 −1.1769 −4.94691 −2.825 −2.826 −2.802 −1.1104 −1.105 −0.51388 −1.72 −1.707 −2.4105 −2.8003 −2.5035 −2.9607 −0.773662 −0.823 −0.183031 −3.99 −3.70261 −4.095 −2.394 −2.996 −2.4189 −1.6146 −3.84761 −4.8639 −4.87084 −4.86436 −2.0441 −1.3284 −1.78871 −4.46075 −4.4662 −2.569

2-197

117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179

Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan

C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S

75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9

62.134 78.133 194.184 88.105 170.207 101.190 170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240

−3.724 −15.046 −64.4 −31.58 5.2 −11.6 −29.072 −45.646 −8.382 −23.495 −44.45 −4.715 2.992 −32.6 −53.78 −48.55 −17.15 −12.69 5.251 −1.73 −39.22 12.3428 −5.263 −38.83 −55.95 −33.37 −28.58 −28.61 −4.63 −46.36 −27.22 −59.54 0 −11.6566 −26.44 −23.43 −10.86 −19.22 −40.55 −3.48 0 −39.445 −26.94 −18.765 −53.62 −33.68 −35.54 −30.1 −30.0453 −6.289 −14.95 10.3 −37.417 −24.86 −16.694 −51.19 −31.62 −33.46 −27.9826 −27.76 −4.167 10.6 −12.92

0.7302 −8.1441 −47.4 −18.16 17.5 8.68 4.981 11.57 −3.192 −16.785 −32.8 3.616 13.073 −19.05 −35.9 −31.22 3.955 4.48 6.844 10.3 −30.18 17.7987 −1.323 −30.31 −32.5 −9.042 −12.64 −13.3 −0.4814 −31.93 −11.52 −51.01 0 −6.9036 −21.23 −21.04 −10.26 −14.71 −37.78 0.08225 0 9.083 −9.191 0.8165 −33.4 −12.55 −14.25 −12.25 −11.96 9.482 3.622 22.7 8.216 −10.005 −0.006634 −33.8 −13.39 −15.06 −13.0081 −12.6 8.7 19.9 2.759

2.8585 3.0627 5.5 3.0012 4.13 4.29 6.2415 9.3787 2.2912 2.8064 3.597 2.848 3.6063 4.55 4.23 4.417 3.826 3.783 2.192 3.21833 3.04891 2.5062 2.4299 3.282 5.1 5.076 3.8 4.069 2.961 4.025 3.881 4.07 2.02682 3.02629 2.644 2.22734 2.1866 2.4857 2.487 2.6714 1.26044 8.2023 4.6138 4.2798 4.8 4.795 4.74 4.58 4.486 4.252 4.939 4.085 7.8102 4.2214 3.8874 4.41 4.402 4.349 4.17856 4.092 3.863 3.76 4.546

−1.7443 −1.6054 −4.4115 −2.1863 −5.8939 −4.0189 −7.51368 −12.3908 −1.42864 −1.235 −2.061 −1.5874 −4.3448 −4.41 −3.21203 −3.284 −4.87051 −4.2839 −1.323 −1.691 −1.0527 −1.481 −1.218 −1.50696 −4.448 −4.943 −3.103 −3.4863 −1.7366 −2.674 −3.12 −1.671 −2.81451 −1.127 −0.5219 −0.5268 −0.5021 −0.2115 −1.9959 0 −10.5618 −4.136 −4.46473 −3.839 −4.285 −4.282 −4.098 −4.09952 −4.3499 −4.7865 −4.2717 −9.95145 −3.52 −3.8551 −3.23 −3.675 −3.67 −3.49 −3.492 −3.7397 −3.64 −4.1762

2-198

TABLE 2-179

Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K (Continued)

Cmpd. no.

Name

Formula

CAS no.

180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237

1-Hexyne 2-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl -1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol

C6H10 C6H10 H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O

693-02-7 764-35-2 302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0

Mol. wt.

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy of formation, J/kmol × 1E-07

Ideal gas entropy, J/(kmolK) × 1E-05

Standard net enthalpy of combustion, J/kmol × 1E-09

82.144 82.144 32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122

12.37 10.5 9.5353 0 −3.629 −9.231 13.5143 −27.33 −2.063 −48.41 −8.38 −76.68 −36.8 −7.452 −20.094 −24 −41.19 18.49 −33.3 −2.297 −28.79 12.908 −15.37 −49.8 −30.3 −3.53 −4.18 26 −25.81 −10.2 13.8 −45.07 −21.5 −15.48 −33.2 −32.7 −35.26 −10.62 −0.38 0.74 −40.2 −21.64 −23.9 −5.96 −35.24 −26.6 −28.64 −6.24 −25.2 −26.26 −8.96 −2.29 −36 −57.95 −17.455 −27.8 −13.499 −31.24

21.85 19.9 15.917 0 −5.334 −9.53 12.4725 −27.54 −3.344 −36.21 3.192 −67 −28.8 −5.049 −16.232 −13.5 −32.42 19.384 −25.7 3.207 −18.1 19.75 −1.405 −34.99 −14.1 6.668 6.045 30.25 −10.17 2.691 20.72 −30.53 −16.61 2.733 −12.9 −12.68 −15.24 3.63 10.38 11.38 −34.83 −11.71 −14.7 1.147 −29.5 −10.7 −13.51 0.0244 −12.18 −13.93 1.4509 −0.98 −25.4 −31.8 −0.5338 −9.35 −2.144 −17.76

3.694 3.72 2.3861 1.30571 1.98591 1.86786 2.01719 1.7367 2.056 3.412 3.124 3.7 3.5 1.8627 2.3988 3.2 3.198 2.4836 3.66 2.433 4.14 3.2151 3.4374 3.9 3.869 3.395 3.386 2.78 3.901 4.118 3.189 3.988 2.98277 3.433 3.75 3.853 3.853 3.399 3.264 3.305 3.287 3.0881 3.394 3.332 2.852 3.81 4.129 1.955 3.416 3.699 3.59 2.55 4.01 5.533 3.8089 4.32 2.955 3.263

−3.661 −3.64 −0.5342 −0.24182 −0.06904 −0.0286 −0.62329 0.1524 −0.518 −2.0004 −2.1566 −0.7732 −1.93 −0.80262 −0.6382 −1.71 −1.461 −1.8487 −1.9303 −0.97508 −3.772 −3.032 −3.23954 −2.622 −3.062 −3.1159 −3.1088 −2.93 −3.12818 −3.5723 −3.046 −2.686 −1.693 −4.25714 −4.058 −4.0574 −4.0318 −3.6741 −3.534 −3.5464 −1.357 −1.9314 −2.268 −2.354 −0.8924 −3.122 −3.4762 −1.06 −2.5311 −2.877 −2.957 −1.1517 −2.54 −5.056 −3.84915 −3.739 −2.64895 −2.4239

2-199

238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid

C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2

115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4

56.106 88.105 74.122 90.187 46.144 118.176 88.148 58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079

−1.71 −42.75 −23.82 −8.23 −2.91 11.83 −28.32 −10.8 15.058 0 −10.21 0 −13.2089 −7.47 8.205 9.025 −43.579 −31.09 −22.874 −57.73 −37.79 −39.71 −10.35 −19.08 6.17 −41.512 −29.02 −20.875 −55.6 −35.73 −37.62 −32.16 −33.9 −8.194 −17.01 8.23 −72.37 0 14.2671 −35.311 −22.78 −14.676 −49.13 −29.57 −31.37 −25.92 −25.79 −2.162 −11.3 −10.84 14.44 12.89 20.12 −9.6399 −1.454 −37.14 19.05 −10.468 −25.46 −27.21 4.677 −18.63 −47.99

5.808 −31.1 −11.1 1.793 1.853 21.73 −11.7 −4.73 22.408 0 −0.6125 0 −9.06 −0.6934 10.416 8.657 10.74 −7.553 2.498 −31.7 −10.86 −12.61 11.23 5.28 24.34 9.91 −8.377 1.6 −32.5 −11.7 −13.43 −11.38 −12.81 10.57 4.457 23.5 −66.14 0 16.3164 7.426 −10.71 −0.8813 −34.7 −14.23 −15.88 −13.83 −13.44 7.837 1.814 1.94408 21.03 19.45 30.219 −3.2637 4.87212 −30.7001 20.08 −2.439 −15.99 −17.52 20.85 −12.46 −38.5

2.9309 3.596 3.52 3.717 2.565 3.725 3.578 3.08 3.3315 1.46219 3.168 1.915 2.6062 2.751 2.1985 2.106 8.9866 5.3988 5.064 5.59 5.579 5.523 5.041 5.724 4.8699 8.5945 5.0063 4.6723 5.2 5.187 5.132 4.962 4.879 4.637 5.331 4.478 3.433 2.05043 2.38823 7.4181 3.8289 3.4945 4.02 4.01 3.958 3.786 3.7 3.462 4.05 4.154 3.298 3.3084 3.945 3.1481 3.527 3.995 2.439 2.702 3.226 3.175 4.233 3.044 2.949

−2.5242 −2.078 −2.51739 −2.962 −1.999 −4.8214 −3.105 −1.77431 −4.9809 0 −1.25 −0.6432 −0.0820482 −0.0902489 −11.7812 −5.35 −5.68455 −5.061 −5.506 −5.506 −5.5716 −6.006 −5.493 −11.1715 −4.74 −5.07415 −4.448 −4.895 −4.894 −4.6984 −4.711 −4.961 −5.3962 −4.88145 −0.1989 0 −0.142671 −9.34237 −2.91 −3.24494 −2.617 −3.064 −3.058 −2.87956 −2.8804 −3.13037 −3.564 −3.5641 −3.051 −3.0291 −6.8282 −2.921 −3.298 −3.1715 −1.8563 −2.04311 −1.844 −1.834 −5.232 −1.6857 −1.395

2-200 TABLE 2-179

Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K (Concluded)

Cmpd. no.

Name

Formula

CAS no.

301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C3H5N C5H10O2 C3H9N C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

107-12-0 109-60-4 107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

Mol. wt.

Ideal gas enthalpy of formation, J/kmol × 1E-07

Ideal gas Gibbs energy of formation, J/kmol × 1E-07

Ideal gas entropy, J/(kmolK) × 1E-05

Standard net enthalpy of combustion, J/kmol × 1E-09

55.079 102.132 59.110 120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

5.18 −46.48 −7.05 0.79 2.023 −40.76 −7.59 −6.75 −42.15 −12.29 −161.494 14.74 −82.29 −29.684 −122.047 −39.572 −71.79 27.66 −33.244 −18.418 2.661 −3.376 −22.56 11.544 5.017 −14.2 −31.177 −9.58 −2.431 −0.95 −1.38 −22.401 −21.845 6.24 4.34 −27.043 −41.9 −31.49 30.46 2.845 −48.116 −24.1814 1.732 1.908 1.803

9.74949 −32.04 4.17 13.76 6.264 −29.36 −0.218 0.2583 −30.4 −6.92 −157.27 21.39 −69.73 −30.012 −111.653 −37.095 −59.9 42.3 6.599 −7.969 16.71 4.59 2.239 12.67 12.22 −8.097 5.771 11.41 9.899 12.61 11.71 1.394 1.828 26.79 28.44 4.116 −9.177 −22.79 30.6 4.195 −42.5514 −22.859 11.876 12.2 12.14

2.8614 4.023 3.242 4.0014 2.67 3.678 3.243 3.365 3.52 3.205 2.82651 3.451 4.034 2.481 2.91625 2.5651 4.48 5.263 7.0259 2.9729 3.6964 3.1 3.893 2.784 3.2099 3.371 6.6337 4.054 2.87 3.805 3.961 4.2296 4.2702 4.435 4.607 5.8493 6.363 3.28 2.794 2.7354 3.73966 1.88724 3.5854 3.5383 3.52165

−1.8007 −2.672 −2.165 −4.95415 −1.9262 −2.041 −2.3398 −2.3458 −1.6476 −2.658 0.7055 −4.219 −1.3591 0.924 0.1422 −3.0576 −9.053 −8.73282 −2.325 −5.3575 −2.76549 −5.0639 −2.4352 −3.734 −0.9685 −8.1229 −4.0405 −2.2449 −4.934 −4.9307 −5.06528 −5.06876 −2.6867 −3.2959 −6.9036 −6.726 −1.95 −2.362 −1.178 −1.544 −4.3318 −4.333 −4.333

The compounds are considered to be formed from the elements in their standard states at 298.15 K and 101,325 Pa. These include C (graphite) and S (rhombic). Enthalpy of combustion is the net value for the compound in its standard state at 298.15 K and 101,325 Pa. Products of combustion are taken to be CO2 (gas), H2O (gas), Cl2 (gas), Br2 (gas), I2 (gas), SO2 (gas), N2 (gas), P4O10 (crystalline), SiO2 (crystobalite), and Al2O3 (crystal, alpha).

Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007).

PROPERTIES OF FORMATION AND COMBUSTION REACTIONS TABLE 2-180

2-201

Ideal Gas Sensible Enthalpies, hT - h298 (kJ/kmol), of Combustion Products

Temperature, K

CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

−2858 −1692 −1110 −529 0

−3414 −2079 −1383 −665 0

−2040 −1209 −793 −377 0

−2976 −1756 −1150 −546 0

−2774 −1656 −1091 −522 0

−2040 −1209 −793 −378 0

−2951 −1743 −1142 −543 0

−3495 −2104 −1392 −672 0

−2857 −1692 −1110 −528 0

−3553 −2164 −1438 −692 0

−2186 −1285 −840 −398 0

−2868 −1703 −1118 −533 0

−3736 −2258 −1496 −718 0

−3282 −1948 −1279 −609 0

300 320 340 360 380

54 638 1221 1805 2389

69 823 1594 2382 3184

38 454 870 1285 1701

55 654 1251 1847 2442

53 630 1209 1791 2373

38 454 870 1286 1701

55 652 1248 1845 2442

68 816 1571 2347 3130

54 636 1219 1802 2386

72 854 1654 2470 3302

41 478 913 1346 1777

54 643 1234 1828 2425

74 881 1702 2538 3387

62 735 1410 2088 2769

400 420 440 460 480

2975 3563 4153 4643 5335

4003 4835 5683 6544 7416

2117 2532 2948 3364 3779

3035 3627 4219 4810 5401

2959 3544 4131 4715 5298

2117 2533 2949 3364 3780

3040 3638 4240 4844 5450

3927 4735 5557 6392 7239

2971 3557 4143 4731 5320

4149 5010 5884 6771 7670

2207 2635 3063 3490 3918

3025 3629 4236 4847 5463

4250 5126 6015 6917 7831

3452 4139 4829 5523 6222

500 550 600 650 700

5931 7428 8942 10477 12023

8305 10572 12907 15303 17754

4196 5235 6274 7314 8353

5992 7385 8943 10423 11902

5882 6760 8811 10278 11749

4196 5235 6274 7314 8353

6059 7592 9144 10716 12307

8099 10340 12555 14882 17250

5911 7395 8894 10407 11937

8580 10897 13295 15744 18243

4343 5402 6462 7515 8570

6084 7653 9244 10859 12499

8758 11123 13544 16022 18548

6925 8699 10501 12321 14192

750 800 850 900 950

13592 15177 16781 18401 20031

20260 22806 25398 28030 30689

9392 10431 11471 12510 13550

13391 14880 16384 17888 19412

13223 14702 16186 17676 19175

9329 10431 11471 12510 13550

13919 15548 17195 18858 20537

19671 22136 24641 27179 29749

13481 15046 16624 18223 19834

20791 23383 26014 28681 31381

9620 10671 11718 12767 13812

14158 15835 17531 19241 20965

21117 23721 26369 29023 31714

16082 18002 19954 21938 23954

1000 1100 1200 1300 1400

21690 25035 28430 31868 35343

33397 38884 44473 50148 55896

14589 16667 18746 20824 22903

20935 24024 27160 30342 33569

20680 23719 26797 29918 33082

14589 16667 18746 20824 22903

22229 25653 29120 32626 36164

32344 37605 42946 48351 53808

21463 24760 28109 31503 34936

34110 39647 45274 50976 56740

14860 16950 19039 21126 23212

22703 26212 29761 33344 36957

34428 39914 45464 51069 56718

26000 30191 34506 38942 43493

1500 1600 1700 1800 1900

38850 42385 45945 49526 53126

61705 67569 73480 79431 85419

24982 27060 29139 31217 33296

36839 40151 43502 46889 50310

36290 39541 42835 46169 49541

24982 27060 29139 31218 33296

39729 43319 46929 50557 54201

59309 64846 70414 76007 81624

38405 41904 45429 48978 52548

62557 68420 74320 80254 86216

25296 27381 29464 31547 33630

40599 44266 47958 51673 55413

62404 68123 73870 79642 85436

48151 52908 57758 62693 67706

2000 2100 2200 2300 2400

56744 60376 64021 67683 71324

91439 97488 103562 109660 115779

35375 37453 39532 41610 43689

53762 57243 60752 64285 67841

52951 56397 59876 63387 66928

35375 37454 39534 41614 43695

57859 61530 65212 68904 72606

87259 92911 98577 104257 109947

56137 59742 63361 66995 70640

92203 98212 104240 110284 116344

35713 37796 39878 41962 44045

59175 62961 66769 70600 74453

91250 97081 102929 108792 114669

72790 77941 83153 88421 93741

2500 2600 2700 2800 2900

74985 78673 82369 86074 89786

121917 128073 134246 140433 146636

45768 47846 49925 52004 54082

71419 75017 78633 82267 85918

70498 74096 77720 81369 85043

45777 47860 49945 52033 54124

76316 80034 83759 87491 91229

115648 121357 127075 132799 138530

74296 77963 81639 85323 89015

122417 128501 134596 140701 146814

46130 48216 50303 52391 54481

78328 82224 86141 90079 94036

120559 126462 132376 138302 144238

99108 104520 109973 115464 120990

3000 3500 4000 4500 5000

93504 112185 130989 149895 168890

152852 184109 215622 247354 279283

56161 66554 75947 87340 97733

89584 108119 126939 145991 165246

88740 107555 126874 146660 166876

56218 66769 77532 88614 100111

94973 113768 132671 151662 170730

144267 173020 201859 230756 259692

92715 111306 130027 148850 167763

152935 183636 214453 245348 276299

56574 67079 77675 88386 99222

98013 118165 188705 159572 180749

150184 180057 210145 240427 270893

126549 154768 183552 212764 242313

Converted and usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.).

2-202

PHYSICAL AND CHEMICAL DATA

TABLE 2-181 Temperature, K

Ideal Gas Entropies s°, kJ/(kmol
K), of Combustion Products CO

CO2

H

OH

H2

N

NO

NO2

N2

N2O

O

O2

SO2

H2O

200 240 260 280 298.15

186.0 191.3 193.7 195.3 197.7

200.0 206.0 208.8 211.5 213.8

106.4 110.1 111.8 113.3 114.7

171.6 177.1 179.5 181.8 183.7

119.4 124.5 126.8 129.2 130.7

145.0 148.7 150.4 151.9 153.3

198.7 204.1 206.6 208.8 210.8

225.9 232.2 235.0 237.7 240.0

180.0 185.2 187.6 189.8 191.6

205.6 211.9 214.8 217.5 220.0

152.2 156.2 158.0 159.7 161.1

193.5 198.7 201.1 203.3 205.1

233.0 239.9 242.8 245.8 248.2

175.5 181.4 184.1 186.6 188.8

300 320 340 360 380

197.8 199.7 201.5 203.2 204.7

214.0 216.5 218.8 221.0 223.2

114.8 116.2 117.4 118.6 119.7

183.9 185.9 187.7 189.4 191.0

130.9 132.8 134.5 136.2 137.7

153.4 154.8 156.0 157.2 158.3

210.9 212.9 214.7 216.4 218.0

240.3 242.7 245.0 247.2 249.3

191.8 193.7 195.5 197.2 198.7

220.2 222.7 225.2 227.5 229.7

161.2 162.6 163.9 165.2 166.3

205.3 207.2 209.0 210.7 212.5

248.5 251.1 253.6 256.0 258.2

189.0 191.2 193.3 195.2 197.1

400 420 440 460 480

206.2 207.7 209.0 210.4 211.6

225.3 227.3 229.3 231.2 233.1

120.8 121.8 122.8 123.7 124.6

192.5 194.0 195.3 196.6 197.9

139.2 140.6 141.9 143.2 144.5

159.4 160.4 161.4 162.3 163.1

219.5 221.0 222.3 223.7 225.0

251.3 253.2 255.1 257.0 258.8

200.2 201.5 202.9 204.2 205.5

231.9 234.0 236.0 238.0 239.9

167.4 168.4 169.4 170.4 171.3

213.8 215.3 216.7 218.0 219.4

260.4 262.5 264.6 266.6 268.5

198.8 200.5 202.0 203.6 205.1

500 550 600 650 700

212.8 215.7 218.3 220.8 223.1

234.9 239.2 243.3 247.1 250.8

125.5 127.5 129.3 131.0 132.5

199.1 201.8 204.4 206.8 209.0

145.7 148.6 151.1 153.4 155.6

164.0 166.0 167.8 169.4 171.0

226.3 229.1 231.9 234.4 236.8

260.6 264.7 268.8 272.6 276.0

206.7 209.4 212.2 214.6 216.9

241.8 246.2 250.4 254.3 258.0

172.2 174.2 176.1 177.7 179.3

220.7 223.7 226.5 229.1 231.5

270.5 274.9 279.2 283.1 286.9

206.5 210.5 213.1 215.9 218.7

750 800 850 900 950

225.2 227.3 229.2 231.1 232.8

255.4 257.5 260.6 263.6 266.5

133.9 135.2 136.4 137.7 138.8

211.1 213.0 214.8 216.5 218.1

157.6 159.5 161.4 163.1 164.7

172.5 173.8 175.1 176.3 177.4

239.0 241.1 243.0 245.0 246.8

279.3 282.5 285.5 288.4 291.3

219.0 221.0 223.0 224.8 226.5

261.5 264.8 268.0 271.1 274.0

180.7 182.1 183.4 184.6 185.7

233.7 235.9 237.9 239.9 241.8

290.4 293.8 297.0 300.1 303.0

221.3 223.8 226.2 228.5 230.6

1000 1100 1200 1300 1400

234.5 237.7 240.7 243.4 246.0

269.3 274.5 279.4 283.9 288.2

139.9 141.9 143.7 145.3 146.9

219.7 222.7 225.4 228.0 230.3

166.2 169.1 171.8 174.3 176.6

178.5 180.4 182.2 183.9 185.4

248.4 251.8 254.8 257.6 260.2

293.9 298.9 303.6 307.9 311.9

228.2 231.3 234.2 236.9 239.5

276.8 282.1 287.0 291.5 295.8

186.8 188.8 190.6 192.3 193.8

243.6 246.9 250.0 252.9 255.6

305.8 311.0 315.8 320.3 324.5

232.7 236.7 240.5 244.0 247.4

1500 1600 1700 1800 1900

248.4 250.7 252.9 254.9 256.8

292.2 296.0 299.6 303.0 306.2

148.3 149.6 150.9 152.1 153.2

232.6 234.7 236.8 238.7 240.6

178.8 180.9 182.9 184.8 186.7

186.9 188.2 189.5 190.7 191.8

262.7 265.0 267.2 269.3 271.3

315.7 319.3 322.7 325.9 328.9

241.9 244.1 246.3 248.3 250.2

299.8 303.6 307.2 310.6 313.8

195.3 196.6 197.9 199.1 200.2

258.1 260.4 262.7 264.8 266.8

328.4 332.1 335.6 338.9 342.0

250.6 253.7 256.6 259.5 262.2

2000 2100 2200 2300 2400

258.7 260.5 262.2 263.8 265.4

309.3 312.2 315.1 317.8 320.4

154.3 155.3 156.3 157.2 158.1

242.3 244.0 245.7 247.2 248.7

188.4 190.1 191.7 193.3 194.8

192.9 193.9 194.8 195.8 196.7

273.1 274.9 276.6 278.3 279.8

331.8 334.5 337.2 339.7 342.1

252.1 253.8 255.5 257.1 258.7

316.9 319.8 322.6 325.3 327.9

201.3 202.3 203.2 204.2 205.0

268.7 270.6 272.4 274.1 275.7

345.0 347.9 350.6 353.2 355.7

264.8 267.3 269.7 272.0 274.3

2500 2600 2700 2800 2900

266.9 268.3 269.7 271.0 272.3

322.9 325.3 327.6 329.9 332.1

158.9 159.7 160.5 161.3 162.0

250.2 251.6 253.0 254.3 255.6

196.2 197.7 199.0 200.3 201.6

197.5 198.3 199.1 199.9 200.6

281.4 282.8 284.2 285.6 286.9

344.5 346.7 348.9 350.9 352.9

260.2 261.6 263.0 264.3 265.6

330.4 332.7 335.0 337.3 339.4

205.9 206.7 207.5 208.3 209.0

277.3 278.8 280.3 281.7 283.1

358.1 360.4 362.6 364.8 366.9

276.5 278.6 380.7 282.7 284.6

3000 3500 4000 4500 5000

273.6 279.4 284.4 288.8 292.8

334.2 343.8 352.2 359.7 366.4

162.7 165.9 168.7 171.1 173.3

256.8 262.5 267.6 272.1 276.1

202.9 208.7 213.8 218.5 222.8

201.3 204.6 207.4 210.1 212.5

288.2 294.0 299.0 303.5 307.5

354.9 363.8 371.5 378.3 384.4

266.9 272.6 277.6 282.1 286.0

341.5 350.9 359.2 366.5 373.0

209.7 212.9 215.8 218.3 220.6

284.4 290.7 296.2 301.1 305.5

368.9 378.1 386.1 393.3 399.7

286.5 295.2 302.9 309.8 316.0

Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.). Equilibrium constants can be calculated by combining ∆h°f values from Table 2-179, hT − h298 from Table 2-180, and s° values from the above, using the formula ln kp = −∆G/(RT), where ∆G = ∆h°f + (hT − h298) − T °. s

HEATS OF SOLUTION

2-203

HEATS OF SOLUTION TABLE 2-182

Heats of Solution of Inorganic Compounds in Water

Heat evolved, in kilocalories per gram formula weight, on solution in water at 18 °C. Computed from data in Bichowsky and Rossini, Thermochemistry of Chemical Substances, Reinhold, New York, 1936. Substance

Dilution*

Formula

Heat, kcal/mol

Aluminum bromide chloride

aq 600 600 aq aq aq aq aq aq aq aq ∞ aq 600 aq ∞ aq ∞ 800 aq aq aq aq aq

AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·aH2O AlF3·3aH2O AlI3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O NH4Br NH4Cl (NH4)2CrO4 (NH4)2Cr2O7 NH4I NH4NO3 NH4BO3·H2O (NH4)2SO4 NH4HSO4 (NH4)2SO3 (NH4)2SO3·H2O SbF3 SbI3 H3AsO4

+85.3 +77.9 +13.2 +31 +19.0 −1.7 +89.0 +126 +56.2 +6.7 −4.45 −3.82 −5.82 −12.9 −3.56 −6.47 −9.0 −2.75 +0.56 −1.2 −4.13 −1.7 −0.8 −0.4

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq aq aq aq aq aq aq

Ba(BrO3)2·H2O BaBr2 BaBr2·H2O BaBr2·2H2O Ba(ClO3)2 Ba(ClO3)2·H2O BaCl2 BaCl2·H2O BaCl2.2H2O Ba(CN)2 Ba(CN)2·H2O Ba(CN)2·2H2O Ba(IO3)2 Ba(IO3)2·H2O BaI2 BaI2·H2O BaI2·2H2O BaI2·2aH2O BaI2·7H2O Ba(NO3)2 Ba(ClO4)2 Ba(ClO4)2·3H2O BaS BeBr2 BeCl2 BeI2 BeSO4 BeSO4·H2O BeSO4·2H2O BeSO4·4H2O BiI3 H3BO3

−15.9 +5.3 −0.8 −3.87 −6.7 −10.6 +2.4 −2.17 −4.5 +1.5 −2.4 −4.9 −9.1 −11.3 +10.5 +2.7 +0.14 −0.58 −6.61 −10.2 −2.8 −10.5 +7.2 +62.6 +51.1 +72.6 +18.1 +13.5 +7.9 +1.1 +3 −5.4

400 400 400 400 400 400 400 400 400 400 ∞ ∞

CdBr2 CdBr2·4H2O CdCl2 CdCl2·H2O CdCl2·2aH2O Cd(NO3)2·H2O Cd(NO3)2·4H2O CdSO4 CdSO4·H2O CdSO4·2wH2O Ca(C2H3O2)2 Ca(C2H3O2)2·H2O

+0.4 −7.3 +3.1 +0.6 −3.00 +4.17 −5.08 +10.69 +6.05 +2.51 +7.6 +6.5

fluoride iodide sulfate Ammonium bromide chloride chromate dichromate iodide nitrate perborate sulfate sulfate, acid sulfite Antimony fluoride iodide Arsenic acid Barium bromate bromide chlorate chloride cyanide iodate iodide

nitrate perchlorate sulfide Beryllium bromide chloride iodide sulfate

Bismuth iodide Boric acid Cadmium bromide chloride nitrate sulfate Calcium acetate

Substance Calcium—(Cont.) bromide

Dilution*

Formula

Heat, kcal/mol

∞ ∞ ∞ ∞ ∞ ∞ ∞ 400 ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq ∞ ∞ ∞ aq

+24.86 −0.9 +4.9 +12.3 +12.5 +2.4 −4.11 +0.7 +28.0 +1.8 +4.1 +0.7 −3.2 −4.2 −7.99 −0.6 −1 +5.1 +3.6 −0.18 +18.6 +5.3 +2.0 +5.7 +18.4 −1.25 +18.5 +9.8 −2.9 +18.8 +15.0 −1.4 −3.6 +2.4 +0.5 +10.3 −2.6 −10.7 +15.9 +9.3 +3.65 −2.85 +11.6

Cuprous sulfate

aq

CaBr2 CaBr2·6H2O CaCl2 CaCl2·H2O CaCl2·2H2O CaCl2·4H2O CaCl2·6H2O Ca(CHO2)2 CaI2 CaI2·8H2O Ca(NO3)2 Ca(NO3)2·H2O Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O Ca(H2PO4)2·H2O CaHPO4·2H2O CaSO4 CaSO4·aH2O CaSO4·2H2O CrCl2 CrCl2·3H2O CrCl2·4H2O CrI2 CoBr2 CoBr2·6H2O CoCl2 CoCl2·2H2O CoCl2·6H2O CoI2 CoSO4 CoSO4·6H2O CoSO4·7H2O Cu(C2H3O2)2 Cu(CHO2)2 Cu(NO3)2 Cu(NO3)2·3H2O Cu(NO3)2·6H2O CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Cu2SO4

Ferric chloride

1000 1000 1000 800 aq 400 400 400 aq 400 400 400 400

FeCl3 FeCl3·2aH2O FeCl3·6H2O Fe(NO3)3·9H2O FeBr2 FeCl2 FeCl2·2H2O FeCl2·4H2O FeI2 FeSO4 FeSO4·H2O FeSO4·4H2O FeSO4·7H2O

+31.7 +21.0 +5.6 −9.1 +18.0 +17.9 +8.7 +2.7 +23.3 +14.7 +7.35 +1.4 −4.4

400 400 aq aq aq 400 ∞ ∞ ∞ ∞ ∞

Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O PbBr2 PbCl2 Pb(CHO2)2 Pb(NO3)2 LiBr LiBr·H2O LiBr·2H2O LiBr·3H2O LiCl

+1.4 −5.9 −10.1 −3.4 −6.9 −7.61 +11.54 +5.30 +2.05 −1.59 +8.66

chloride

formate iodide nitrate

phosphate, monodibasic sulfate Chromous chloride iodide Cobaltous bromide chloride iodide sulfate Cupric acetate formate nitrate sulfate

nitrate Ferrous bromide chloride iodide sulfate

Lead acetate bromide chloride formate nitrate Lithium bromide

chloride

aq aq aq 400 400 400 aq 400 400 400 aq aq 200 200 200 800

*The numbers represent moles of water used to dissolve 1 g formula weight of substance; ∞ means “infinite dilution”; and aq means “aqueous solution of unspecified dilution.”

2-204

PHYSICAL AND CHEMICAL DATA

TABLE 2-182

Heats of Solution of Inorganic Compounds in Water (Continued)

Substance Lithium—(Cont.)

fluoride hydroxide iodide

nitrate sulfate Magnesium bromide chloride

iodide nitrate phosphate sulfate

sulfide Manganic nitrate sulfate Manganous acetate bromide chloride formate iodide

sulfate Mercuric acetate bromide chloride nitrate Mercurous nitrate Nickel bromide Nickel chloride

iodide nitrate sulfate

Dilution*

Formula

Heat, kcal/mol

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

LiCl·H2O LiCl·2H2O LiCl·3H2O LiF LiOH LiOH·fH2O LiOH·H2O LiI LiI·aH2O LiI·H2O LiI·2H2O LiI·3H2O LiNO3 LiNO3·3H2O Li2SO4 Li2SO4·H2O

+4.45 +1.07 −1.98 −0.74 +4.74 +4.39 +9.6 +14.92 +10.08 +6.93 +3.43 −0.17 +0.466 −7.87 +6.71 +3.77

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq ∞ ∞ ∞ ∞ ∞ ∞ aq 400 400 400 aq aq aq aq aq aq 400 400 400 aq aq aq aq aq aq aq 400 400 400 aq aq aq aq aq

MgBr2 MgBr2·H2O MgBr2·6H2O MgCl2 MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgI2 Mg(NO3)2·6H2O Mg3(PO4)2 MgSO4 MgSO4·H2O MgSO4·2H2O MgSO4·4H2O MgSO4·6H2O MgSO4·7H2O MgS Mn(NO3)2 Mn(NO3)2·3H2O Mn(NO3)2·6H2O Mn2(SO4)3 Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O MnBr2 MnBr2·H2O MnBr2·4H2O MnCl2 MnCl2·2H2O MnCl2·4H2O Mn(CHO2)2 Mn(CHO2)2·2H2O MnI2 MnI2·H2O MnI2·2H2O MnI2·4H2O MnI2·6H2O MnSO4 MnSO4·H2O MnSO4·7H2O Hg(C2H3O2)2 HgBr2 HgCl2 Hg(NO3)2·aH2O Hg2(NO3)2·2H2O

+43.7 +35.9 +19.8 +36.3 +20.8 +10.5 +3.4 +50.2 −3.7 +10.2 +21.1 +14.0 +11.7 +4.9 +0.55 −3.18 +25.8 +12.9 −3.9 −6.2 +22 +12.2 +1.6 +15 +14.4 +16.1 +16.0 +8.2 +1.5 +4.3 −2.9 +26.2 +24.1 +22.7 +19.9 +21.2 +13.8 +11.9 −1.7 −4.0 −2.4 −3.3 −0.7 −11.5

aq aq 800 800 800 800 aq 200 200 200 200

NiBr2 NiBr2·3H2O NiCl2 NiCl2·2H2O NiCl2·4H2O NiCl2·6H2O NiI2 Ni(NO3)2 Ni(NO3)2·6H2O NiSO4 NiSO4·7H2O

+19.0 +0.2 +19.23 +10.4 +4.2 −1.15 +19.4 +11.8 −7.5 +15.1 −4.2

Substance

Dilution*

Phosphoric acid, orthopyroPotassium acetate aluminum sulfate

∞ 400 aq aq aq ∞ 800 ∞ aq aq ∞ aq ∞

+2.79 −0.1 +25.9 +4.65 +3.55 +48.5 +26.6 −10.1 −5.1 −10.13 −5.13 +6.58 +4.25 −0.43 −10.31 −4.404 −4.9 +55 +42 +33 +7 −9.5 −3.0 −17.8 +3.96 −1.85 −6.05 +0.86 +1.21 +12.91 +4.27 +3.48 +0.86 −6.93 −5.23 −8.633 −4.6 −7.5 −12.94 −10.4 +4.7 −11.0 −10.22 −6.32 −3.10 −11.0 +1.8 +1.37 −6.08 −13.0 −4.5

aq 200 ∞ ∞ 500 500 1800 900 900 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 800 800 800 200 200

AgC2H3O2 AgNO3 NaC2H3O2 NaC2H3O2·3H2O Na3AsO4 Na3AsO4·12H2O NaHCO3 Na2B4O7 Na2B4O7·10H2O NaBr NaBr·2H2O Na2CO3 Na2CO3·H2O Na2CO3·7H2O Na2CO3·10H2O NaClO3 NaCl Na2CrO4 Na2CrO4·4H2O Na2CrO4·10H2O NaCN NaCN·aH2O

−5.4 −4.4 +4.085 −4.665 +15.6 −12.61 −4.1 +10.0 −16.8 −0.58 −4.57 +5.57 +2.19 −10.81 −16.22 −5.37 −1.164 +2.50 −7.52 −16.0 −0.37 −0.92

2000 ∞ ∞ ∞

chlorate chloride chromate chrome sulfate

∞ ∞ 2185 600

cyanide dichromate fluoride

200 1600 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 400

hydroxide

iodate iodide nitrate oxalate perchlorate permanganate phosphate, dihydrogen pyrosulfite sulfate sulfate, acid sulfide sulfite thiocyanate thionate, dithiosulfate Silver acetate nitrate Sodium acetate arsenate bicarbonate borate, tetrabromide carbonate

chlorate chloride chromate cyanide

Heat, kcal/mol

H3PO4 H3PO4·aH2O H4P2O7 H4P2O7·1aH2O KC2H3O2 KAl(SO4)2 KAl(SO4)2·3H2O KAl(SO4)2·12H2O KHCO3 KBrO3 KBr K2CO3 K2CO3·aH2O K2CO3·1aH2O KClO3 KCl K2CrO4 KCr(SO4)2 KCr(SO4)2·H2O KCr(SO4)2·2H2O KCr(SO4)2·6H2O KCr(SO4)2·12H2O KCN K2Cr2O7 KF KF·2H2O KF·4H2O KHS KHS·dH2O KOH KOH·eH2O KOH·H2O KOH·7H2O KIO3 KI KNO3 K2C2O4 K2C2O4·H2O KClO4 KMnO4 KH2PO4 K2S2O5 K2S2O5·aH2O K2SO4 KHSO4 K2S K2SO3 K2SO3·H2O KCNS K2S2O6 K2S2O3

400 400 aq aq ∞ 600 600

bicarbonate bromate bromide carbonate

hydrosulfide

Formula

HEATS OF SOLUTION TABLE 2-182

Heats of Solution of Inorganic Compounds in Water (Concluded)

Substance

Dilution*

Formula

Heat, kcal/mol

200 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 600 ∞ aq ∞ 1600 1600 1600 1600 1600 1600 600 600 800 800 1600 1600 1200 1200 ∞ ∞ 800 800 ∞ ∞ ∞ ∞ ∞ ∞ ∞

NaCN·2H2O NaF NaHS NaHS·2H2O NaOH NaOH·aH2O NaOH·wH2O NaOH·eH2O NaOH·H2O NaI NaI·2H2O NaPO3 NaNO3 NaNO2 NaClO4 Na2HPO4 Na3PO4 Na3PO4·12H2O Na2HPO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O NaH2PO3 NaH2PO3·2aH2O Na2HPO3 Na2HPO3·5H2O Na4P2O7 Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O Na2SO4 Na2SO4·10H2O NaHSO4 NaHSO4·H2O Na2S Na2S·4aH2O Na2S·5H2O Na2S·9H2O Na2SO3 Na2SO3·7H2O NaCNS

−4.41 −0.27 +4.62 −1.49 +10.18 +8.17 +7.08 +6.48 +5.17 +1.57 −3.89 +3.97 −5.05 −3.6 −4.15 +5.21 +13 −15.3 −0.82 −12.04 −23.18 +0.90 −5.29 +9.30 −4.54 +11.9 −11.7 −2.2 −14.0 +0.28 −18.74 +1.74 +0.15 +15.2 +0.09 −6.54 −16.65 +2.8 −11.1 −1.83

Sodium—(Cont.) fluoride hydrosulfide Sodium hydroxide

iodide metaphosphate nitrate nitrite perchlorate phosphate di triphosphate di diphosphite, monodipyrophosphate disulfate sulfate, acid sulfide

sulfite thiocyanate NOTE:

2-205

Substance Sodium—(Cont.) thionate, diSodium thiosulfate Stannic bromide Stannous bromide iodide Strontium acetate bromide

chloride

iodide

nitrate sulfate Sulfuric acid, pyroZinc acetate bromide chloride iodide nitrate sulfate

To convert kilocalories per mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.

Dilution*

Formula

Heat, kcal/mol

aq aq aq aq aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

Na2S2O6 Na2S2O6·2H2O Na2S2O3 Na2S2O3·5H2O SnBr4 SnBr2 SnI2 Sr(C2H3O2)2 Sr(C2H3O2)2·aH2O SrBr2 SrBr2·H2O SrBr2·2H2O SrBr2·4H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCl2·6H2O SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O Sr(NO3)2 Sr(NO3)2·4H2O SrSO4 H2S2O7

−5.80 −11.86 +2.0 −11.30 +15.5 −1.6 −5.8 +6.2 +5.9 +16.4 +9.25 +6.5 +0.4 −6.1 +11.54 +6.4 +2.95 −7.1 +20.7 +12.65 +10.4 −4.5 −4.8 −12.4 +0.5 −18.08

400 400 400 400 400 aq 400 400 400 400 400 400

Zn(C2H3O2)2 Zn(C2H3O2)2·H2O Zn(C2H3O2)2·2H2O ZnBr2 ZnCl2 ZnI2 Zn(NO3)2·3H2O Zn(NO3)2·6H2O ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O

+9.8 +7.0 +3.9 +15.0 +15.72 +11.6 −5 −6.0 +18.5 +10.0 −0.8 −4.3

2-206

PHYSICAL AND CHEMICAL DATA TABLE 2-183 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature) Recalculated and rearranged from International Critical Tables, vol. 5, pp. 148–150. cal/mol = Btu/(lb⋅mol) × 1.799.

Solute Acetic acid (solid), C2H4O2 Acetylacetone, C5H8O2 Acetylurea, C3H6N2O2 Aconitic acid, C6H6O6 Ammonium benzoate, C7H9NO2 picrate succinate (n-) Aniline, hydrochloride, C6H8ClN Barium picrate Benzoic acid, C7H6O2 Camphoric acid, C10H16O4 Citric acid, C6H8O7 Dextrin, C12H20O10 Fumaric acid, C4H4O4 Hexamethylenetetramine, C6H12N4 Hydroxybenzamide (m-), C7H7NO2 (m-), (HCl) (o-), C7H7NO2 (p-) Hydroxybenzoic acid (o-), C7H6O3 (p-), C7H6O3 Hydroxybenzyl alcohol (o-), C7H8O2 Inulin, C36H62O31 Isosuccinic acid, C4H6O4 Itaconic acid, C5H6O4 Lactose, C12H22O11·H2O Lead picrate (2H2O) Magnesium picrate (8H2O) Maleic acid, C4H4O4 Malic acid, C4H6O5 Malonic acid, C3H4O4 Mandelic acid, C8H2O3 Mannitol, C6H14O6 Menthol, C10H20O Nicotine dihydrochloride, C10H16Cl2N2 Nitrobenzoic acid (m-), C7H5NO4 (o-), C7H5NO4 (p-), C7H5NO4 Nitrophenol (m-), C6H5NO3 (o-), C6H5NO3 (p-), C6H5NO3

Heat of solution, cal/mol solute* −2,251 −641 −6,812 −4,206 −2,700 −8,700 −3,489 −2,732 −4,708 −6,501 −502 −5,401 268 −5,903 4,780 −4,161 −7,003 −4,340 −5,392 −6,350 −5,781 −3,203 −96 −3,420 −5,922 −3,705 −7,098 −13,193 14,699 −15,894 −4,441 −3,150 −4,493 −3,090 −5,260 0 6,561 −5,593 −5,306 −8,891 −5,210 −6,310 −4,493

Solute Oxalic acid, C2H2O4 (2H2O) Phenol (solid), C6H6O Phthalic acid, C8H6O4 Picric acid, C6H3N3O7 Piperic acid, C12H10O4 Piperonylic acid, C8H6O4 Potassium benzoate citrate tartrate (n-) (0.5 H2O) Pyrogallol, C6H6O3 Pyrotartaric acid Quinone Raffinose, C18H32O16 (5H2O) Resorcinol, C6H6O2 Silver malonate (n-) Sodium citrate (tri-) picrate potassium tartrate (4H2O) succinate (n-) (6H2O) tartrate (n-) (2H2O) Strontium picrate (6H2O) Succinic acid, C4H6O4 Succinimide, C4H5NO2 Sucrose, C12H22O11 Tartaric acid (d-) Thiourea, CH4N2S Urea, CH4N2O acetate formate nitrate oxalate Vanillic acid Vanillin Zinc picrate (8H2O)

Heat of solution, cal/mol solute* −2,290 −8,485 −2,605 −4,871 −7,098 −10,492 −9,106 −1,506 2,820 −5,562 −3,705 −5,019 −3,991 −9,703 −3,960 −9,799 5,270 −6,441 −1,817 −12,342 2,390 −10,994 −1,121 −5,882 7,887 −14,412 −6,405 −4,302 −1,319 −3,451 −5,330 −3,609 −8,795 −7,194 −10,803 −17,806 −5,160 −5,210 −11,496 −15,894

*+ denotes heat evolved, and − denotes heat absorbed. The data in the International Critical Tables were calculated by E. Anderson.

THERMODYNAMIC PROPERTIES

2-207

THERMODYNAMIC PROPERTIES EXPLANATION OF TABLES The following subsection presents information on the thermodynamic properties of a number of fluids. In some cases transport properties are also included. Properties for the compounds listed in Table 2-184 were generated by using the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). Megan Friend’s help in generating these tables is acknowledged and gratefully appreciated. The number of digits provided in these tables was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format. Properties for many of these compounds can also be generated by going to http://webbook.nist.gov and selecting NIST Chemistry WebBook, then Thermophysical Properties of Fluid Systems High Accuracy Data. This site allows the user to generate tables of thermodynamic properties. The user can select the units as well as the temperatures and/or pressure increments for which properties are to be generated. The resulting table can be copied into a spreadsheet. Because of this capability, properties for the compounds listed in Table 2-184 are not tabulated at as many temperatures and pressures as might otherwise be the case. Notation cp = isobaric specific heat cv = isochoric specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy µ = viscosity v = specific volume f = subscript denoting saturated liquid g = subscript denoting saturated vapor UNITS CONVERSIONS For this subsection, the following units conversions are applicable: cp, specific heat: To convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Fahrenheit, multiply by 0.23885. e, internal energy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. g, gravity acceleration: To convert meters per second squared to feet per second squared, multiply by 3.2808.

h, enthalpy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. k, thermal conductivity: To convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779. p, pressure: To convert bars to kilopascals, multiply by 1 × 102; to convert bars to pounds-force per square inch, multiply by 14.504; and to convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. s, entropy: to convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Rankine, multiply by 0.23885. t, temperature: °F = 9⁄5 °C + 32. T, absolute temperature: °R = 9⁄ 5 K. u, internal energy: to convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. µ, viscosity: to convert pascal-seconds to pound-force–seconds per square foot, multiply by 0.020885; to convert pascal-seconds to cp, multiply by 1000. v, specific volume: to convert cubic meters per kilogram to cubic feet per pound, multiply by 16.018. ρ, density: to convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428.

ADDITIONAL REFERENCES Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971. D’Ans and Lax, Handbook for Chemists and Physicists (in German), 3 vols., Springer-Verlag, Berlin. Engineering Data Book, Natural Gas Processors Suppliers Association, Tulsa, Okla. Ganic, Hartnett, and Rohsenow, Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. Gray, American Institute of Physics Handbook, 3d ed., McGraw-Hill, New York, 1972. Kay and Laby, Tables of Physical and Chemical Constants, Longman, London, various editions and dates. Landolt-Börnstein Tables, many volumes and dates, Springer-Verlag, Berlin. Lange, Handbook of Chemistry, McGraw-Hill, New York, various editions and dates. Partington, Advanced Treatise on Physical Chemistry, 5 vols., Longman, London, 1950. Raznjevic, Handbook of Thermodynamic Tables and Charts, McGraw-Hill, New York, 1976 and other editions. Reynolds, Thermodynamic Properties in SI, Department of Mechanical Engineering, Stanford University, 1979. Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Selected Values of Properties of Chemical Compounds and Selected Values of the Properties of Hydrocarbons and Related Compounds, Thermodynamics Research Center, NIST, Boulder, Colo., looseleaf, periodic publication. Vargaftik, Tables of the Thermophysical Properties of Gases and Liquids, Wiley, New York, 1975. Vargaftik, Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978. Weast, Handbook of Chemistry and Physics, Chemical Rubber Co., Boca Raton, FL, annually.

2-208

PHYSICAL AND CHEMICAL DATA TABLE 2-184 List of Substances for Which Thermodynamic Property Tables Were Generated from NIST Standard Reference Database 23 Table no.

Name

2-185 2-187 2-189 2-190 2-193 2-195 2-196 2-197 2-198 2-199 2-200 2-203 2-207 2-208 2-209 2-210 2-212 2-213 2-214 2-215 2-216 2-219 2-220 2-221 2-223 2-224 2-226 2-227 2-228 2-229 2-233 2-234 2-235 2-236 2-238 2-239 2-241 2-242 2-243 2-244 2-245 2-246 2-248 2-249 2-250 2-251 2-252 2-255 2-256 2-257 2-258 2-271 2-272 2-274 2-275 2-276 2-277 2-278 2-279 2-280 2-281 2-282 2-284 2-285 2-288 2-289 2-290 2-296 2-300 2-301 2-303 2-305 2-307

Acetone Air Ammonia Argon Benzene Butane 1-Butene cis-2-Butene trans-2-butene Carbon dioxide Carbon monoxide Carbonyl sulfide Cyclohexane Decane Deuterium oxide 2,2-Dimethylpropane Dodecane Ethane Ethanol Ethylene Fluorine Helium Heptane Hexane Normal hydrogen para-Hydrogen Hydrogen sulfide Isobutane Isobutene Krypton Methane Methanol 2-Methylbutane 2-Methylpentane Neon Nitrogen Nitrogen trifluoride Nitrous oxide Nonane Octane Oxygen Pentane Propane Propylene R-11 R-12 R-13 R-22 R-23 R-32 R-41 R-113 R-114 R-116 R-123 R-124 R-125 R-134a R-141b R-142b R-143a R-152a R-218 R227ea R-404A R-407C R-410A R-507A Sulfur dioxide Sulfur hexafluoride Toluene Water Xenon

Chemical formula

Alternate name

C3H6O NH3 Ar C6H6 C4H10 C4H8 C4H8 C4H8 CO2 CO COS C6H12 C10H22 D2O C5H14 C12H26 C2H6 C2H6O C2H4 F2 He C7H16 C6H14 H2 H2 H2S C4H10 C4H8 Kr CH4 CH4O C5H12 C6H14 Ne N2 NF3 N2O C9H20 C8H18 O2 C5H12 C3H8 C3H6 CCl3F CCl2F2 CClF3 CHClF2 CHF3 CH2F2 CH3F C2Cl3F3 C2Cl2F4 C2F6 C2HF5 C2HClF4 C2HF5 C2H2F4 C2H3Cl2F C2H3ClF2 C2H3F3 C2H4F2 C3F8 C3HF7

SO2 SF6 C7H8 H2O Xe

Heavy water Neopentane

2-methyl propene

Isopentane Isohexane

Trichlorofluoromethane Dichlorodifluoromethane Chlorotrifluoromethane Chlorodifluoromethane Trifluoromethane Difluoromethane Fluoromethane 1,1,2-Trichlorotrifluoroethane 1,2-Dichlorotetrafluoroethane Hexafluoroethane 2,2-Dichloro-1,1,1-trifluoroethane 1-Chloro-1,2,2,2-tetrafluoroethane Pentafluoroethane 1,1,1,2-Tetrafluoroethane 1,1-Dichloro-1-fluoroethane 1-Chloro-1,1-difluoroethane 1,1,1-Trifluoroethane 1,1-Difluoroethane Octafluoropropane 1,1,1,2,3,3,3-Heptafluoropropane

THERMODYNAMIC PROPERTIES TABLE 2-185

2-209

Thermodynamic Properties of Acetone

Temperature K

Pressure MPa

178.50 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 508.10

2.3265E-06 2.8743E-06 1.9454E-05 9.6588E-05 0.00037556 0.0012008 0.0032765 0.0078514 0.016899 0.033259 0.060720 0.10404 0.16891 0.26188 0.39033 0.56235 0.78681 1.0733 1.4324 1.8759 2.4172 3.0725 3.8632 4.6924

178.50 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 508.10

2.3265E-06 2.8743E-06 1.9454E-05 9.6588E-05 0.00037556 0.0012008 0.0032765 0.0078514 0.016899 0.033259 0.060720 0.10404 0.16891 0.26188 0.39033 0.56235 0.78681 1.0733 1.4324 1.8759 2.4172 3.0725 3.8632 4.6924

200.00 250.00 300.00 328.84

0.10000 0.10000 0.10000 0.10000

328.84 350.00 400.00 450.00 500.00 550.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

200.00 250.00 300.00 350.00 400.00 416.48

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

416.48 450.00 500.00 550.00

1.0000 1.0000 1.0000 1.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

0.063601 0.063715 0.064868 0.066048 0.067264 0.068525 0.069840 0.071218 0.072673 0.074217 0.075867 0.077643 0.079569 0.081677 0.084008 0.086616 0.089578 0.093001 0.097051 0.10200 0.10832 0.11706 0.13145 0.21277

0.47366 0.64687 2.3835 4.1282 5.8823 7.6487 9.4311 11.234 13.060 14.915 16.802 18.725 20.687 22.693 24.746 26.852 29.015 31.243 33.546 35.938 38.445 41.117 44.096 49.249

0.47366 0.64687 2.3835 4.1282 5.8823 7.6488 9.4314 11.234 13.062 14.918 16.807 18.733 20.701 22.714 24.779 26.900 29.085 31.343 33.685 36.130 38.707 41.476 44.604 50.247

637,900. 520,660. 83,324. 18,065. 4,973.1 1,656.0 642.89 282.74 137.74 72.996 41.482 24.979 15.782 10.377 7.0503 4.9192 3.5050 2.5368 1.8547 1.3611 0.99393 0.71168 0.48154 0.21277

36.689 36.764 37.528 38.314 39.121 39.947 40.790 41.649 42.522 43.406 44.302 45.207 46.119 47.033 47.946 48.849 49.733 50.582 51.376 52.083 52.648 52.968 52.771 49.249

38.173 38.260 39.149 40.059 40.989 41.936 42.897 43.869 44.849 45.834 46.821 47.806 48.784 49.751 50.698 51.615 52.490 53.305 54.033 54.636 55.050 55.154 54.631 50.247

0.065254 0.069389 0.074210 0.077500

2.9626 8.8328 14.913 18.575

2.9691 8.8397 14.921 18.583

45.137 46.843 50.998 55.474 60.316 65.522

Enthalpy kJ/mol

Entropy kJ/(mol⋅K)

Cv kJ/(mol⋅K)

Cp kJ/(mol⋅K)

Sound speed m/s

0.0080825 0.0090488 0.018316 0.026935 0.035003 0.042602 0.049806 0.056674 0.063259 0.069601 0.075739 0.081702 0.087517 0.093209 0.098798 0.10431 0.10975 0.11516 0.12056 0.12599 0.13150 0.13720 0.14341 0.15437

0.082500 0.082598 0.083407 0.084076 0.084758 0.085541 0.086468 0.087553 0.088794 0.090180 0.091697 0.093329 0.095063 0.096886 0.098794 0.10078 0.10286 0.10504 0.10736 0.10986 0.11265 0.11600 0.12077

0.11544 0.11550 0.11604 0.11660 0.11731 0.11825 0.11946 0.12094 0.12270 0.12474 0.12704 0.12962 0.13249 0.13568 0.13924 0.14328 0.14794 0.15350 0.16042 0.16967 0.18350 0.20893 0.28551

1765.7 1757.0 1672.3 1591.8 1514.4 1439.4 1366.3 1294.8 1224.5 1155.2 1086.7 1018.8 951.24 883.84 816.36 748.57 680.21 610.99 540.51 468.19 392.99 312.66 221.66 0

0.21928 0.21801 0.20686 0.19803 0.19103 0.18546 0.18104 0.17754 0.17479 0.17266 0.17102 0.16980 0.16892 0.16831 0.16791 0.16768 0.16754 0.16745 0.16734 0.16711 0.16664 0.16569 0.16367 0.15437

0.050120 0.050280 0.051928 0.053740 0.055800 0.058169 0.060883 0.063945 0.067329 0.070988 0.074863 0.078895 0.083030 0.087227 0.091459 0.095718 0.10001 0.10438 0.10887 0.11357 0.11865 0.12436 0.13126

0.058440 0.058600 0.060265 0.062119 0.064267 0.066795 0.069763 0.073198 0.077094 0.081429 0.086172 0.091302 0.096822 0.10277 0.10927 0.11649 0.12481 0.13483 0.14772 0.16583 0.19480 0.25197 0.42947

172.60 173.29 179.95 186.29 192.29 197.94 203.19 207.99 212.26 215.93 218.90 221.08 222.35 222.60 221.70 219.53 215.94 210.76 203.80 194.82 183.50 169.39 151.36 0

0.021248 0.047436 0.069594 0.081247

0.083638 0.086143 0.090180 0.093199

0.11621 0.11902 0.12473 0.12941

47.730 49.643 54.255 59.166 64.436 70.066

0.16988 0.17552 0.18783 0.19939 0.21049 0.22122

0.078579 0.079533 0.085418 0.092823 0.10033 0.10753

0.090892 0.090386 0.094849 0.10175 0.10903 0.11612

220.94 229.44 246.85 262.23 276.40 289.72

2.9486 8.8130 14.885 21.312 28.263 30.714

3.0138 8.8824 14.959 21.392 28.351 30.806

0.021178 0.047357 0.069499 0.089316 0.10788 0.11389

0.083649 0.086152 0.090182 0.095644 0.10213 0.10452

0.11619 0.11896 0.12460 0.13326 0.14605 0.15210

1649.7 1396.0 1162.0 936.35 707.25 627.32

50.387 54.081 59.388 64.832

53.120 57.281 63.156 69.107

0.16747 0.17709 0.18947 0.20081

0.10335 0.10087 0.10402 0.10950

0.13228 0.11921 0.11743 0.12100

212.13 233.76 256.99 275.55

JouleThomson K/MPa

Saturated Properties 15.723 15.695 15.416 15.141 14.867 14.593 14.319 14.041 13.760 13.474 13.181 12.880 12.568 12.243 11.904 11.545 11.163 10.753 10.304 9.8043 9.2319 8.5423 7.6072 4.7000 1.5677E-06 1.9207E-06 1.2001E-05 5.5355E-05 0.00020108 0.00060385 0.0015555 0.0035368 0.0072603 0.013699 0.024107 0.040034 0.063362 0.096367 0.14184 0.20329 0.28530 0.39420 0.53918 0.73472 1.0061 1.4051 2.0767 4.7000

−0.43351 −0.43308 −0.42849 −0.42274 −0.41520 −0.40545 −0.39322 −0.37827 −0.36033 −0.33907 −0.31399 −0.28437 −0.24915 −0.20678 −0.15495 −0.090162 −0.0069455 0.10371 0.25760 0.48516 0.85357 1.5474 3.3240 14.310 3845.4 3637.4 2139.7 1312.0 834.10 547.82 370.79 258.27 184.97 136.14 102.93 79.878 63.590 51.884 43.343 37.032 32.325 28.797 26.154 24.184 22.717 21.551 20.240 14.310

Single-Phase Properties 15.325 14.411 13.475 12.903 0.038565 0.035712 0.030709 0.027083 0.024272 0.022008 15.333 14.423 13.491 12.483 11.308 10.852 0.36582 0.31254 0.26538 0.23391

25.930 28.002 32.563 36.923 41.200 45.437 0.065220 0.069336 0.074123 0.080107 0.088431 0.092149 2.7336 3.1996 3.7681 4.2751

1645.6 1391.1 1155.7 1024.0

−0.42678 −0.39768 −0.33922 −0.28685 81.384 58.339 30.192 18.173 12.201 8.8355 −0.42708 −0.39848 −0.34115 −0.24033 −0.042437 0.074613 29.536 20.211 12.984 9.1542

2-210

PHYSICAL AND CHEMICAL DATA

TABLE 2-185 Temperature K

Thermodynamic Properties of Acetone (Concluded ) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Cv kJ/(mol⋅K)

Cp kJ/(mol⋅K)

Sound speed m/s

JouleThomson K/MPa

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

15.367 14.471 13.560 12.588 11.490 10.123 7.8139 1.7344

0.065073 0.069106 0.073747 0.079439 0.087035 0.098782 0.12798 0.57657

2.8871 8.7271 14.762 21.128 27.958 35.450 44.435 60.563

3.2125 9.0726 15.130 21.525 28.393 35.944 45.075 63.446

0.020868 0.047011 0.069085 0.088784 0.10711 0.12488 0.14406 0.17943

0.083704 0.086197 0.090197 0.095584 0.10186 0.10898 0.11961 0.12191

0.11609 0.11871 0.12408 0.13214 0.14320 0.16059 0.23343 0.17820

1667.9 1417.7 1189.0 972.15 759.27 538.79 262.33 205.69

−0.42837 −0.40187 −0.34909 −0.25988 −0.10136 0.26123 2.3418 10.650

15.410 14.528 13.641 12.709 11.683 10.491 8.9733 6.6600

0.064894 0.068831 0.073307 0.078687 0.085592 0.095320 0.11144 0.15015

2.8125 8.6237 14.616 20.916 27.629 34.864 42.815 52.079

3.4614 9.3120 15.349 21.703 28.485 35.818 43.930 53.581

0.020488 0.046589 0.068589 0.088163 0.10626 0.12352 0.14060 0.15896

0.083781 0.086264 0.090234 0.095554 0.10166 0.10827 0.11552 0.12442

0.11598 0.11843 0.12351 0.13100 0.14066 0.15332 0.17314 0.22174

1689.9 1443.6 1220.9 1013.1 815.03 622.74 433.48 255.34

−0.42983 −0.40569 −0.35775 −0.27983 −0.15336 0.080235 0.63674 2.7218

Enthalpy kJ/mol

Entropy kJ/(mol⋅K)

Single-Phase Properties 200.00 250.00 300.00 350.00 400.00 450.00 500.00 550.00 200.00 250.00 300.00 350.00 400.00 450.00 500.00 550.00

10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000

250.00 300.00 350.00 400.00 450.00 500.00 550.00

100.00 100.00 100.00 100.00 100.00 100.00 100.00

15.320 14.657 14.023 13.409 12.813 12.234 11.674

0.065276 0.068228 0.071312 0.074574 0.078044 0.081739 0.085664

7.2620 12.852 18.631 24.654 30.941 37.489 44.286

13.790 19.675 25.763 32.112 38.745 45.663 52.852

0.040421 0.061873 0.080632 0.097579 0.11320 0.12777 0.14147

0.088285 0.092127 0.097243 0.10299 0.10892 0.11478 0.12045

0.11631 0.11946 0.12424 0.12980 0.13553 0.14112 0.14639

1791.8 1616.6 1466.4 1337.4 1226.9 1133.0 1053.8

−0.43634 −0.42000 −0.39555 −0.36734 −0.33807 −0.30922 −0.28171

450.00 500.00 550.00

500.00 500.00 500.00

15.616 15.306 15.012

0.064037 0.065335 0.066615

27.237 33.413 39.856

59.256 66.081 73.163

0.097266 0.11164 0.12514

0.11562 0.12123 0.12669

0.13393 0.13909 0.14416

2201.1 2129.8 2067.5

−0.39010 −0.37710 −0.36510

The values in this table were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.1% in the saturated liquid density between 280 and 310 K, 0.5% in density in the liquid phase below 380 K, and 1% in density elsewhere, including all states at pressures above 100 MPa. The uncertainties in vapor pressure are 0.5% above 270 K (0.25% between 290 and 390 K), and the uncertainties in heat capacities and speeds of sound are 1%. These uncertainties (in caloric properties and sound speeds) may be higher at pressures above the saturation pressure and at temperatures above 320 K in the liquid phase and at supercritical conditions. TABLE 2-186

Saturated Acetylene*

Temperature, K

Pressure, bar

162.0 169.3 173.9 180.0 184.3

vcond, m3/kg

vg, m3/kg

hcond, kJ/kg

hg, kJ/kg

scond, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.101 0.203 0.304 0.507 0.709

5.081 2.644 1.805 1.116 0.810

158 173 182 194 203

983 994 999 1007 1011

2.967 3.039 3.095 3.161 3.216

8.062 7.889 7.797 7.672 7.596

189.1 192.4t

1.013 1.283

0.5780 0.4617

214 221

1015 1018

3.272 3.312

7.511 7.455

192.4t 200.9 209.4

1.283 2.027 3.040

0.00164 0.00165 0.00169

0.4617 0.3011 0.2074

378 411 445

1018 1027 1035

4.127 4.296 4.461

7.455 7.362 7.280

221.5 230.4 240.7 253.2 263.0

5.066 7.093 10.13 15.20 20.27

0.00174 0.00179 0.00186 0.00195 0.00204

0.1264 0.0907 0.0635 0.0420 0.0309

493 528 565 602 628

1046 1052 1058 1061 1061

4.684 4.837 4.990 5.133 5.231

7.180 7.111 7.037 6.947 6.878

271.6 278.9 284.9 290.4 300.0

25.33 30.40 35.46 40.53 50.66

0.00213 0.00223 0.00232 0.00242 0.00270

0.0240 0.0193 0.0159 0.0133 0.0093

654 680 704 727 778

1060 1057 1051 1041 1017

5.326 5.414 5.494 5.576 5.737

6.822 6.767 6.716 6.658 6.534

307.8 308.7c

60.80 62.47

0.00335 0.00434

0.0061 0.0043

850 908

968 908

5.965 6.158

6.351 6.158

*Values recalculated into SI units from those of Din. Thermodynamic Functions of Gases, vol. 2, Butterworth, London, 1956. Above the solid line the condensed phase is solid; below the line it is liquid. t = triple point; c = critical point.

TABLE 2-187

Thermodynamic Properties of Air

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

59.75 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

0.005265 0.005546 0.006797 0.008270 0.009994 0.012000 0.014320 0.016988 0.020042 0.023520 0.027461 0.031908 0.036905 0.042498 0.048733 0.055659 0.063326 0.071786 0.081091 0.091294 0.10245 0.11462 0.12785 0.14221 0.15775 0.17453 0.19262 0.21207 0.23295 0.25531 0.27922 0.30475 0.33196 0.36091 0.39166 0.42429 0.45886 0.49543 0.53408 0.57486 0.61786 0.66313 0.71074 0.76077 0.81329 0.86836 0.92606 0.98645 1.0496 1.1156 1.1845 1.2564 1.3314 1.4095 1.4908 1.5753

33.067 33.031 32.888 32.745 32.601 32.457 32.312 32.166 32.020 31.873 31.725 31.576 31.427 31.277 31.126 30.974 30.821 30.668 30.513 30.357 30.200 30.042 29.883 29.722 29.560 29.397 29.232 29.066 28.898 28.729 28.558 28.385 28.210 28.033 27.854 27.673 27.489 27.304 27.115 26.924 26.730 26.533 26.333 26.130 25.923 25.713 25.499 25.281 25.058 24.831 24.598 24.361 24.118 23.868 23.613 23.350

0.030242 0.030275 0.030406 0.030539 0.030674 0.030810 0.030949 0.031089 0.031231 0.031375 0.031521 0.031669 0.031820 0.031972 0.032127 0.032285 0.032445 0.032608 0.032773 0.032941 0.033112 0.033287 0.033464 0.033645 0.033829 0.034017 0.034209 0.034404 0.034604 0.034808 0.035017 0.035230 0.035449 0.035672 0.035901 0.036137 0.036378 0.036625 0.036880 0.037142 0.037411 0.037688 0.037975 0.038270 0.038575 0.038891 0.039217 0.039556 0.039908 0.040273 0.040653 0.041050 0.041464 0.041896 0.042350 0.042826

−1.0619 −1.0481 −0.99308 −0.93803 −0.88298 −0.82792 −0.77286 −0.71777 −0.66267 −0.60755 −0.55239 −0.49720 −0.44196 −0.38669 −0.33135 −0.27597 −0.22051 −0.16499 −0.10939 −0.05371 0.002063 0.057934 0.11391 0.17000 0.22621 0.28255 0.33903 0.39566 0.45245 0.50940 0.56653 0.62386 0.68138 0.73912 0.79709 0.85529 0.91375 0.97248 1.0315 1.0908 1.1505 1.2104 1.2708 1.3315 1.3926 1.4542 1.5162 1.5787 1.6417 1.7053 1.7695 1.8343 1.8997 1.9659 2.0329 2.1007

−1.0617 −1.0480 −0.99287 −0.93778 −0.88267 −0.82755 −0.77241 −0.71725 −0.66205 −0.60681 −0.55152 −0.49619 −0.44079 −0.38533 −0.32979 −0.27417 −0.21846 −0.16265 −0.10673 −0.05070 0.005456 0.061749 0.11819 0.17478 0.23155 0.28849 0.34562 0.40296 0.46051 0.51829 0.57631 0.63459 0.69315 0.75199 0.81115 0.87062 0.93044 0.99063 1.0512 1.1122 1.1736 1.2354 1.2978 1.3606 1.4240 1.4880 1.5525 1.6177 1.6836 1.7502 1.8176 1.8858 1.9549 2.0250 2.0960 2.1682

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

JouleThomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.034011 0.033955 0.033731 0.033512 0.033298 0.033089 0.032884 0.032683 0.032486 0.032294 0.032105 0.031920 0.031739 0.031562 0.031388 0.031217 0.031050 0.030886 0.030725 0.030568 0.030413 0.030262 0.030113 0.029968 0.029826 0.029686 0.029550 0.029417 0.029286 0.029158 0.029033 0.028911 0.028792 0.028676 0.028563 0.028453 0.028346 0.028241 0.028140 0.028042 0.027948 0.027856 0.027768 0.027684 0.027603 0.027525 0.027452 0.027383 0.027317 0.027256 0.027200 0.027149 0.027103 0.027062 0.027028 0.027000

0.055064 0.055062 0.055060 0.055062 0.055069 0.055081 0.055098 0.055120 0.055148 0.055181 0.055220 0.055266 0.055317 0.055376 0.055441 0.055514 0.055594 0.055682 0.055779 0.055884 0.055998 0.056122 0.056256 0.056400 0.056556 0.056723 0.056902 0.057094 0.057300 0.057521 0.057757 0.058009 0.058278 0.058566 0.058874 0.059202 0.059553 0.059928 0.060329 0.060757 0.061216 0.061707 0.062232 0.062796 0.063401 0.064052 0.064753 0.065508 0.066323 0.067206 0.068163 0.069205 0.070341 0.071585 0.072951 0.074459

1030.3 1028.3 1020.3 1012.2 1004.0 995.77 987.48 979.13 970.72 962.24 953.70 945.10 936.43 927.70 918.90 910.04 901.11 892.11 883.05 873.91 864.71 855.44 846.09 836.67 827.18 817.61 807.96 798.24 788.44 778.56 768.59 758.55 748.42 738.20 727.90 717.51 707.03 696.46 685.80 675.05 664.20 653.26 642.22 631.08 619.84 608.50 597.06 585.51 573.85 562.09 550.21 538.21 526.10 513.86 501.48 488.97

−0.40785 −0.40743 −0.40565 −0.40375 −0.40173 −0.39958 −0.39729 −0.39485 −0.39227 −0.38952 −0.38660 −0.38352 −0.38024 −0.37677 −0.37310 −0.36922 −0.36511 −0.36076 −0.35616 −0.35130 −0.34616 −0.34074 −0.33500 −0.32894 −0.32254 −0.31577 −0.30862 −0.30107 −0.29308 −0.28464 −0.27572 −0.26628 −0.25629 −0.24573 −0.23455 −0.22270 −0.21016 −0.19686 −0.18275 −0.16779 −0.15189 −0.13501 −0.11705 −0.09794 −0.07758 −0.05588 −0.03271 −0.00795 0.018543 0.046927 0.077386 0.11012 0.14538 0.18342 0.22456 0.26917

171.43 171.02 169.40 167.78 166.16 164.53 162.91 161.28 159.65 158.01 156.37 154.73 153.09 151.44 149.79 148.14 146.49 144.83 143.16 141.50 139.83 138.15 136.48 134.80 133.11 131.42 129.78 128.11 126.44 124.76 123.07 121.38 119.69 118.00 116.30 114.61 112.91 111.21 109.51 107.81 106.11 104.41 102.71 101.01 99.316 97.623 95.933 94.247 92.565 90.888 89.216 87.551 85.893 84.242 82.599 80.965

376.64 371.92 353.83 336.91 321.09 306.27 292.39 279.38 267.17 255.71 244.94 234.81 225.28 216.31 207.85 199.88 192.35 185.23 178.51 172.14 166.11 160.39 154.96 149.80 144.90 140.23 135.78 131.54 127.50 123.63 119.93 116.38 112.98 109.72 106.59 103.58 100.68 97.879 95.179 92.571 90.048 87.605 85.236 82.937 80.703 78.529 76.412 74.347 72.331 70.361 68.432 66.542 64.688 62.867 61.075 59.311

Saturated Properties

2-211

−0.01536 −0.01513 −0.01422 −0.01333 −0.01245 −0.01158 −0.01073 −0.00989 −0.00906 −0.00824 −0.00744 −0.00664 −0.00586 −0.00508 −0.00432 −0.00357 −0.00282 −0.00209 −0.00136 −0.00064 6.86E-05 0.000772 0.001467 0.002156 0.002838 0.003513 0.004181 0.004844 0.005501 0.006153 0.006799 0.007440 0.008077 0.008708 0.009336 0.009960 0.010579 0.011195 0.011808 0.012418 0.013025 0.013630 0.014232 0.014833 0.015431 0.016029 0.016625 0.017221 0.017816 0.018411 0.019006 0.019602 0.020200 0.020799 0.021400 0.022004

2-212

TABLE 2-187 Temperature K

Thermodynamic Properties of Air (Continued ) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

2.1695 2.2392 2.3100 2.3821 2.4554 2.5303 2.6069 2.6854 2.7662 2.8496 2.9363 3.0269 3.1227 3.2253 3.3379 3.4661 3.6243 3.8680 4.4004

2.2415 2.3161 2.3922 2.4697 2.5490 2.6302 2.7135 2.7992 2.8878 2.9796 3.0753 3.1759 3.2827 3.3976 3.5243 3.6695 3.8497 4.1302 4.7627

4.8774 4.8825 4.9025 4.9225 4.9424 4.9621 4.9817 5.0012 5.0205 5.0397 5.0587 5.0774 5.0960 5.1144 5.1326 5.1505 5.1682 5.1856 5.2028 5.2196 5.2362 5.2525 5.2684 5.2841 5.2994 5.3143 5.3289 5.3431 5.3569 5.3703 5.3832 5.3958 5.4079 5.4195 5.4307 5.4413 5.4514 5.4610

5.3730 5.3800 5.4081 5.4361 5.4639 5.4915 5.5189 5.5461 5.5731 5.5998 5.6263 5.6525 5.6784 5.7040 5.7292 5.7541 5.7786 5.8027 5.8264 5.8497 5.8726 5.8949 5.9169 5.9383 5.9591 5.9795 5.9993 6.0185 6.0372 6.0552 6.0726 6.0893 6.1054 6.1207 6.1354 6.1492 6.1624 6.1747

Entropy kJ/(molK)

Sound speed m/s

JouleThomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.076131 0.077996 0.080090 0.082459 0.085163 0.088280 0.091919 0.096227 0.10142 0.10781 0.11589 0.12645 0.14089 0.16186 0.19519 0.25624 0.40151 1.0148

476.31 463.48 450.49 437.29 423.88 410.23 396.30 382.04 367.40 352.31 336.67 320.36 303.21 285.00 265.37 243.75 219.07 189.12 0

0.31767 0.37057 0.42848 0.49214 0.56243 0.64047 0.72765 0.82574 0.93703 1.0646 1.2125 1.3865 1.5951 1.8510 2.1752 2.6058 3.2246 4.2808 6.3978

79.340 77.724 76.119 74.523 72.938 71.363 69.798 68.243 66.700 65.170 63.658 62.176 60.751 59.445 58.409 58.054 59.591 67.802

57.571 55.852 54.152 52.467 50.794 49.130 47.469 45.809 44.141 42.460 40.755 39.013 37.215 35.332 33.316 31.072 28.384 24.467

0.029217 0.029225 0.029261 0.029302 0.029348 0.029399 0.029455 0.029518 0.029587 0.029663 0.029746 0.029836 0.029934 0.030040 0.030155 0.030278 0.030410 0.030552 0.030703 0.030865 0.031037 0.031220 0.031415 0.031621 0.031840 0.032072 0.032317 0.032577 0.032851 0.033141 0.033447 0.033770 0.034111 0.034472 0.034853 0.035256 0.035681 0.036132

154.83 155.14 156.38 157.60 158.81 159.99 161.16 162.30 163.42 164.53 165.60 166.66 167.69 168.70 169.69 170.65 171.58 172.49 173.37 174.23 175.05 175.85 176.62 177.36 178.07 178.75 179.40 180.02 180.61 181.17 181.69 182.19 182.65 183.08 183.47 183.84 184.17 184.46

Cv kJ/(molK)

Cp kJ/(molK)

0.022611 0.023223 0.023840 0.024462 0.025092 0.025731 0.026380 0.027041 0.027717 0.028412 0.029131 0.029880 0.030668 0.031512 0.032436 0.033492 0.034804 0.036863 0.041603

0.026979 0.026965 0.026961 0.026966 0.026982 0.027010 0.027053 0.027113 0.027194 0.027300 0.027438 0.027618 0.027855 0.028171 0.028607 0.029242 0.030266 0.032343

0.096708 0.096323 0.094825 0.093392 0.092020 0.090705 0.089445 0.088235 0.087074 0.085959 0.084887 0.083855 0.082862 0.081906 0.080983 0.080094 0.079235 0.078406 0.077604 0.076828 0.076076 0.075348 0.074643 0.073957 0.073292 0.072645 0.072016 0.071403 0.070806 0.070224 0.069655 0.069099 0.068556 0.068024 0.067503 0.066991 0.066489 0.065995

0.020805 0.020809 0.020825 0.020843 0.020864 0.020886 0.020911 0.020938 0.020968 0.021000 0.021035 0.021072 0.021113 0.021156 0.021201 0.021250 0.021302 0.021356 0.021414 0.021474 0.021538 0.021605 0.021674 0.021747 0.021822 0.021901 0.021983 0.022068 0.022155 0.022246 0.022340 0.022436 0.022536 0.022638 0.022744 0.022852 0.022964 0.023078

Saturated Properties 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 132.63 59.75 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

1.6633 1.7546 1.8495 1.9479 2.0499 2.1557 2.2653 2.3787 2.4960 2.6173 2.7427 2.8721 3.0055 3.1431 3.2845 3.4295 3.5770 3.7228 3.7858 0.002432 0.002584 0.003274 0.004111 0.005120 0.006325 0.007756 0.009442 0.011416 0.013713 0.016372 0.019431 0.022933 0.026921 0.031443 0.036547 0.042282 0.048702 0.055859 0.063810 0.072611 0.082321 0.093001 0.10471 0.11751 0.13147 0.14665 0.16312 0.18094 0.20018 0.22091 0.24320 0.26712 0.29273 0.32011 0.34934 0.38047 0.41359

23.080 22.801 22.514 22.217 21.908 21.588 21.253 20.903 20.534 20.144 19.727 19.278 18.788 18.242 17.616 16.863 15.869 14.198 10.448 0.004907 0.005192 0.006475 0.008005 0.009817 0.011948 0.014438 0.017326 0.020659 0.024481 0.028841 0.033789 0.039379 0.045664 0.052702 0.060550 0.069268 0.078918 0.089564 0.10127 0.11410 0.12813 0.14343 0.16006 0.17811 0.19765 0.21875 0.24150 0.26598 0.29228 0.32048 0.35068 0.38298 0.41747 0.45426 0.49345 0.53517 0.57953

0.043328 0.043857 0.044417 0.045011 0.045645 0.046323 0.047052 0.047841 0.048700 0.049643 0.050691 0.051871 0.053225 0.054818 0.056765 0.059300 0.063015 0.070432 0.095715 203.80 192.59 154.45 124.93 101.86 83.693 69.263 57.715 48.406 40.849 34.673 29.595 25.394 21.899 18.975 16.515 14.437 12.671 11.165 9.8746 8.7639 7.8043 6.9721 6.2475 5.6145 5.0595 4.5715 4.1408 3.7597 3.4214 3.1203 2.8516 2.6111 2.3954 2.2014 2.0265 1.8686 1.7255

58.283 57.634 55.151 52.832 50.666 48.640 46.742 44.963 43.293 41.724 40.248 38.858 37.548 36.313 35.146 34.043 32.999 32.010 31.072 30.183 29.337 28.534 27.769 27.041 26.346 25.684 25.051 24.447 23.869 23.316 22.786 22.278 21.791 21.324 20.876 20.445 20.031 19.632

5.2938 5.3199 5.4244 5.5291 5.6340 5.7391 5.8444 5.9500 6.0559 6.1621 6.2688 6.3759 6.4835 6.5917 6.7005 6.8099 6.9202 7.0312 7.1431 7.2560 7.3700 7.4851 7.6014 7.7192 7.8384 7.9591 8.0817 8.2060 8.3324 8.4610 8.5919 8.7254 8.8616 9.0008 9.1433 9.2893 9.4390 9.5929

4.2197 4.2382 4.3119 4.3855 4.4590 4.5324 4.6057 4.6788 4.7519 4.8248 4.8976 4.9703 5.0429 5.1154 5.1878 5.2602 5.3325 5.4048 5.4771 5.5494 5.6217 5.6940 5.7664 5.8389 5.9116 5.9844 6.0574 6.1307 6.2043 6.2781 6.3524 6.4272 6.5024 6.5782 6.6547 6.7318 6.8098 6.8887

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 132.63

0.44878 0.48609 0.52562 0.56742 0.61159 0.65820 0.70732 0.75903 0.81341 0.87055 0.93052 0.9934 1.0593 1.1282 1.2004 1.2757 1.3545 1.4366 1.5223 1.6115 1.7045 1.8013 1.9020 2.0067 2.1156 2.2287 2.3462 2.4682 2.5949 2.7266 2.8633 3.0055 3.1536 3.3084 3.4712 3.6462 3.7858

0.62667 0.67671 0.72980 0.78609 0.84575 0.90895 0.97587 1.0467 1.1217 1.2011 1.2852 1.3742 1.4684 1.5682 1.6740 1.7862 1.9053 2.0318 2.1664 2.3097 2.4625 2.6259 2.8009 2.9889 3.1913 3.4103 3.6481 3.9078 4.1934 4.5101 4.8653 5.2697 5.7405 6.3074 7.0343 8.1273 10.448

1.5957 1.4777 1.3702 1.2721 1.1824 1.1002 1.0247 0.95535 0.89147 0.83254 0.77810 0.72772 0.68102 0.63767 0.59737 0.55985 0.52486 0.49217 0.46160 0.43296 0.40608 0.38082 0.35702 0.33457 0.31335 0.29323 0.27412 0.25590 0.23847 0.22173 0.20554 0.18976 0.17420 0.15854 0.14216 0.12304 0.095715

5.4701 5.4785 5.4864 5.4936 5.5002 5.5060 5.5112 5.5156 5.5193 5.5221 5.5240 5.5250 5.5251 5.5241 5.5221 5.5188 5.5143 5.5085 5.5012 5.4924 5.4819 5.4695 5.4550 5.4383 5.4190 5.3969 5.3715 5.3424 5.3089 5.2701 5.2248 5.1713 5.1069 5.0268 4.9209 4.7566 4.4004

6.1862 6.1968 6.2066 6.2154 6.2233 6.2302 6.2360 6.2408 6.2444 6.2469 6.2481 6.2480 6.2465 6.2436 6.2391 6.2330 6.2252 6.2156 6.2039 6.1901 6.1740 6.1554 6.1341 6.1097 6.0819 6.0504 6.0147 5.9740 5.9277 5.8746 5.8133 5.7417 5.6563 5.5513 5.4143 5.2053 4.7627

5.6800 9.8544 14.072 18.500 23.201 28.145 33.282 38.568 43.966 49.453

6.4941 12.348 18.231 24.323 30.686 37.293 44.094 51.042 58.104 65.253

1.2007 1.5924

0.065510 0.065031 0.064560 0.064094 0.063633 0.063177 0.062726 0.062277 0.061832 0.061389 0.060947 0.060506 0.060065 0.059623 0.059180 0.058735 0.058286 0.057833 0.057375 0.056910 0.056437 0.055955 0.055461 0.054954 0.054432 0.053890 0.053326 0.052735 0.052112 0.051448 0.050732 0.049950 0.049076 0.048067 0.046830 0.045064 0.041603

0.023196 0.023317 0.023441 0.023568 0.023698 0.023833 0.023970 0.024112 0.024258 0.024408 0.024563 0.024722 0.024887 0.025058 0.025234 0.025418 0.025608 0.025807 0.026015 0.026232 0.026461 0.026701 0.026956 0.027226 0.027514 0.027823 0.028155 0.028516 0.028910 0.029344 0.029827 0.030371 0.030994 0.031726 0.032619 0.033814

0.036610 0.037116 0.037654 0.038225 0.038834 0.039483 0.040176 0.040918 0.041714 0.042570 0.043492 0.044490 0.045573 0.046751 0.048038 0.049450 0.051005 0.052727 0.054644 0.056790 0.059209 0.061956 0.065102 0.068738 0.072988 0.078015 0.084052 0.091426 0.10063 0.11241 0.12801 0.14959 0.18134 0.23261 0.32992 0.59804

184.72 184.95 185.14 185.30 185.42 185.51 185.55 185.57 185.54 185.48 185.38 185.24 185.07 184.85 184.60 184.30 183.97 183.59 183.17 182.71 182.21 181.66 181.08 180.45 179.78 179.06 178.31 177.51 176.68 175.81 174.91 173.96 172.98 171.93 170.79 169.40 0

19.249 18.879 18.523 18.180 17.848 17.528 17.218 16.918 16.628 16.346 16.072 15.805 15.546 15.292 15.044 14.800 14.561 14.324 14.090 13.856 13.623 13.388 13.151 12.909 12.661 12.405 12.137 11.854 11.553 11.229 10.874 10.480 10.033 9.5119 8.8740 7.9854 6.3978

9.7513 9.9145 10.083 10.257 10.438 10.626 10.821 11.024 11.237 11.459 11.693 11.939 12.198 12.473 12.764 13.074 13.406 13.762 14.145 14.559 15.008 15.499 16.039 16.635 17.298 18.042 18.884 19.849 20.968 22.288 23.877 25.841 28.367 31.807 37.001 46.996

6.9686 7.0495 7.1317 7.2153 7.3003 7.3870 7.4755 7.5659 7.6586 7.7537 7.8514 7.9521 8.0560 8.1634 8.2749 8.3907 8.5114 8.6375 8.7696 8.9086 9.0552 9.2104 9.3755 9.5518 9.7412 9.9456 10.168 10.411 10.681 10.982 11.324 11.720 12.191 12.775 13.553 14.798

0.080463 0.11269 0.12770 0.13794 0.14593 0.15255 0.15823 0.16320 0.16762 0.17160

0.021087 0.020796 0.021504 0.022817 0.024150 0.025246 0.026091 0.026734 0.027229 0.027619

0.030116 0.029149 0.029830 0.031137 0.032467 0.033562 0.034406 0.035049 0.035544 0.035934

198.24 347.36 446.40 523.89 589.60 648.15 701.76 751.59 798.38 842.62

17.423 2.2510 0.50305 −0.12430 −0.41124 −0.56194 −0.64963 −0.70457 −0.74078 −0.76547

9.4692 26.384 39.944 51.755 62.543 72.680 82.381 91.781 100.97 110.01

7.1068 18.537 27.090 34.176 40.394 46.051 51.325 56.325 61.127 65.783

1.2383 1.6321

0.013532 0.017351

0.027868 0.027368

0.061355 0.065680

658.25 582.97

−0.14308 −0.00232

104.97 93.879

88.326 73.903

5.5251 9.8022 14.046 18.485 23.190 28.138 33.278 38.565 43.964 49.451

6.2479 12.289 18.218 24.326 30.698 37.311 44.114 51.065 58.128 65.278

0.060461 0.093372 0.10851 0.11877 0.12677 0.13340 0.13908 0.14405 0.14847 0.15245

0.024739 0.020859 0.021526 0.022830 0.024159 0.025253 0.026096 0.026738 0.027233 0.027622

0.044597 0.029563 0.029954 0.031194 0.032498 0.033582 0.034419 0.035057 0.035550 0.035939

185.23 348.45 448.46 525.96 591.54 649.96 703.44 753.17 799.86 844.02

15.779 2.1789 0.47425 −0.13809 −0.41899 −0.56686 −0.65304 −0.70711 −0.74278 −0.76711

11.965 26.684 40.110 51.868 62.628 72.748 82.438 91.830 101.01 110.05

7.9625 18.672 27.179 34.242 40.446 46.094 51.361 56.357 61.155 65.808

1.0983 9.5710

1.2820 12.042

0.012483 0.079244

0.028034 0.021131

0.058181 0.031423

710.56 355.63

−0.21837 1.8817

111.13 28.389

96.436 19.420

Single-Phase Properties 100 300 500 700 900 1100 1300 1500 1700 1900 100 106.22

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 1

2-213

108.1 300 500 700 900 1100 1300 1500 1700 1900

1 1 1 1 1 1 1 1 1 1

100 300

5 5

0.12283 0.040103 0.024046 0.017175 0.013359 0.010931 0.009249 0.008016 0.007073 0.006329 26.593 25.232 1.3836 0.40205 0.23974 0.17119 0.13319 0.10902 0.092279 0.079999 0.070604 0.063185 27.222 2.0232

8.1414 24.936 41.586 58.223 74.855 91.486 108.12 124.75 141.38 158.00 0.037604 0.039632 0.72278 2.4873 4.1711 5.8415 7.5079 9.1727 10.837 12.500 14.163 15.827 0.036735 0.49426

TABLE 2-187 2-214

Temperature K

Thermodynamic Properties of Air (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

JouleThomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Single Phase Properties 500 700 900 1100 1300 1500 1700 1900

5 5 5 5 5 5 5 5

1.1814 0.84321 0.65711 0.53874 0.45667 0.39636 0.35015 0.31361

0.84642 1.1859 1.5218 1.8562 2.1898 2.5229 2.8559 3.1887

13.935 18.417 23.146 28.107 33.256 38.550 43.954 49.445

18.167 24.347 30.755 37.388 44.205 51.165 58.234 65.389

0.094907 0.10529 0.11334 0.11999 0.12568 0.13066 0.13509 0.13907

0.021621 0.022885 0.024197 0.025282 0.026119 0.026757 0.027249 0.027636

0.030478 0.031434 0.032632 0.033664 0.034473 0.035095 0.035577 0.035958

458.30 535.45 600.34 658.10 711.01 760.23 806.49 850.28

0.36370 −0.19118 −0.44905 −0.58606 −0.66646 −0.71716 −0.75073 −0.77366

40.969 52.433 63.045 73.076 82.707 92.057 101.21 110.22

27.606 34.545 40.682 46.287 51.523 56.497 61.278 65.917

100 300 500 700 900 1100 1300 1500 1700 1900

10 10 10 10 10 10 10 10 10 10

27.863 4.0370 2.3157 1.6542 1.2922 1.0618 0.90165 0.78374 0.69321 0.62149

0.035889 0.24771 0.43183 0.60452 0.77388 0.94184 1.1091 1.2759 1.4426 1.6090

0.99444 9.2885 13.802 18.336 23.092 28.070 33.231 38.532 43.943 49.438

1.3533 11.766 18.120 24.382 30.831 37.489 44.321 51.292 58.368 65.528

0.011382 0.072612 0.088894 0.099422 0.10752 0.11420 0.11990 0.12489 0.12932 0.13330

0.028284 0.021441 0.021733 0.022952 0.024243 0.025317 0.026146 0.026780 0.027268 0.027653

0.055716 0.033664 0.031078 0.031710 0.032786 0.033760 0.034537 0.035139 0.035608 0.035981

763.47 369.50 471.81 547.83 611.64 668.47 720.60 769.17 814.87 858.18

−0.27969 1.5212 0.25100 −0.24405 −0.47890 −0.60517 −0.67990 −0.72730 −0.75881 −0.78039

117.77 31.116 42.260 53.257 63.641 73.538 83.082 92.372 101.48 110.45

105.78 20.637 28.194 34.944 40.985 46.531 51.728 56.673 61.432 66.054

100 300 500 700 900 1100 1300 1500 1700 1900

100 100 100 100 100 100 100 100 100 100

33.161 21.138 15.089 11.803 9.7481 8.3307 7.2877 6.4847 5.8456 5.3239

0.030156 0.047309 0.066273 0.084722 0.10258 0.12004 0.13722 0.15421 0.17107 0.18783

0.24746 7.0356 12.371 17.367 22.408 27.580 32.880 38.287 43.779 49.340

3.2631 11.767 18.999 25.840 32.667 39.584 46.602 53.708 60.886 68.123

0.001378 0.049067 0.067619 0.079134 0.087711 0.09465 0.10051 0.10559 0.11009 0.11411

0.031980 0.023981 0.023117 0.023855 0.024903 0.025831 0.026565 0.027131 0.027569 0.027915

0.048218 0.038366 0.034686 0.034011 0.034331 0.034845 0.035323 0.035723 0.036049 0.036317

1192.4 818.47 772.41 790.14 821.78 857.40 894.00 930.40 966.13 1001.0

−0.47290 −0.49747 −0.55640 −0.62591 −0.67702 −0.71435 −0.74281 −0.76506 −0.78264 −0.79653

179.20 86.312 71.549 73.572 79.057 85.797 93.151 100.84 108.75 116.78

252.46 53.642 42.159 43.339 46.948 51.158 55.511 59.875 64.208 68.504

300 500 700 900 1100 1300 1500 1700 1900

500 500 500 500 500 500 500 500 500

34.106 29.826 26.714 24.283 22.305 20.651 19.243 18.027 16.963

0.029320 0.033528 0.037433 0.041180 0.044833 0.048423 0.051966 0.055473 0.058952

6.2145 11.583 16.768 22.008 27.358 32.814 38.354 43.961 49.623

20.875 28.348 35.484 42.598 49.775 57.025 64.337 71.698 79.098

0.033155 0.052311 0.064323 0.073261 0.080460 0.086515 0.091746 0.096353 0.10047

0.028875 0.026614 0.026496 0.026991 0.027539 0.028000 0.02836 0.02864 0.02886

0.039265 0.036111 0.035494 0.035702 0.036073 0.036415 0.036693 0.036911 0.037085

1678.8 1573.6 1514.8 1482.8 1468.3 1465.1 1469.3 1478.5 1491.1

−0.57656 −0.65015 −0.67879 −0.68796 −0.69130 −0.69354 −0.69594 −0.69875 −0.70188

208.23 178.50 161.67 151.95 146.88 144.95 145.84 148.48 152.39

181.12 120.62 97.470 86.531 81.387 79.411 79.312 80.393 82.251

300 500 700 900 1100 1300 1500 1700 1900

1000 1000 1000 1000 1000 1000 1000 1000 1000

40.130 36.567 33.895 31.736 29.916 28.338 26.946 25.701 24.577

0.024919 0.027347 0.029503 0.031510 0.033427 0.035288 0.037111 0.038909 0.040688

6.8286 12.271 17.554 22.890 28.327 33.857 39.461 45.123 50.830

31.747 39.618 47.057 54.399 61.754 69.145 76.573 84.032 91.519

0.024761 0.044944 0.057468 0.066695 0.074073 0.080246 0.085561 0.090229 0.094392

0.032271 0.029334 0.028754 0.028917 0.029215 0.029476 0.029675 0.029821 0.029928

0.041510 0.037843 0.036801 0.036702 0.036858 0.037051 0.037224 0.037369 0.037491

2208.5 2104.7 2033.9 1984.7 1951.3 1929.3 1915.7 1908.3 1905.8

−0.50493 −0.57316 −0.60504 −0.61882 −0.62560 −0.62968 −0.63251 −0.63465 −0.63632

274.96 247.30 230.60 219.72 212.46 207.70 204.81 203.41 203.25

337.76 219.41 174.51 149.43 133.76 123.58 116.94 112.74 110.27

This table was generated for a standard three-component dry air containing mole fractions 0.7812 nitrogen, 0.2096 oxygen, and 0.0092 argon. The values in this table were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., Jacobsen, R. T, Penoncello, S. G., and Friend, D. G., “Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at Pressures to 2000 MPa,” J. Phys. Chem. Ref. Data 29(3):331–385, 2000. The source for viscosity and thermal conductivity is Lemmon, E. W., and Jacobsen, R. T., “Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air,” Int. J. Thermophys. 25:21–69, 2004. Properties at the freezing point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. In the range from the solidification point to 873 K at pressures to 70 MPa, the estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated speed of sound values is 0.2% and that for calculated heat capacities is 1%. At temperatures above 873 K and 70 MPa, the estimated uncertainty of calculated density values is 0.5%, increasing to 1.0% at 2000 K and 2000 MPa. For viscosity, the uncertainty is 1% in the dilute gas. The uncertainty is around 2% between 270 and 300 K and increases to 5% outside of this region. There are very few measurements between 130 and 270 K for air to validate this claim, and the uncertainties may be even higher in this supercritical region. For thermal conductivity, the uncertainty for the dilute gas is 2% with increasing uncertainties near the triple points. The uncertainties range from 3% between 140 and 300 K to 5% at the triple point and at high temperatures. The uncertainties above 100 MPa are not known due to a lack of experimental data.

−100

0

10. 8.

60

(R-729) reference state: h = 0.0 kJ/kg, s = 0.00 kJ/(kg·K) for ideal gas at 0 K

0.

0.

0.

Air

100

200 0.

300

.

400

150.

200

30

40

50

−200 20.

500

3

100.

g/m ρ = 80. k

700.

10. 8.

40.

6.

30.

c.p.

110

100

90

70

60.

4.

20.

T = 80 K

2.

800.

900.

6. 4.

600 20.

15.

120

2.

10.

120

500

480

440

420

400

380

360

340

320

300

280

260

240

220

200

160

140

6.

80

0 6.6

0 6.4

0

)

g·K

/(k

00

J 0k

3.0

0.4

0.2

20

7.

7.

s=

7.4

300

400

0.1 0.08

0.40

0.06 0.04

0.20 0.15 0 7.8

200

0.60

0.30

0

100

0.6

1.0

7.6

0

4.0

1.5

0.02

−100

1. 0.8

2.0

70

0.01 −200

6.0

0.80

6.2

or

100

saturated vap

0.9

0.8

0.7

2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00

0.04

0.5

80

0.1 0.08 0.06

0.6

0.3 x = 0.4

0.2

0.1

ted li

0.2

quid

90

180

0.4

T = 460 K

8.0

T = 100 K

0.6

satura

Pressure (MPa)

110

1. 0.8

0.02

0.10

500

0.01 600

Enthalpy (kJ/kg)

2-215

FIG. 2-5 Pressure-enthalpy diagram for dry air. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Lemmon, E. W., Jacobsen, R. T., Penoncello, S. G., and Friend, D. G., “Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at Pressures to 2000 MPa,” J. Phys. Chem. Ref. Data 29:331–385, 2000.

2-216

PHYSICAL AND CHEMICAL DATA

TABLE 2-188

Air

Other tables include Stewart, R. B., S. G. Penoncello, et al., University of Idaho CATS report, 85-5, 1985 (0.1–700 bar, 85–750 K), and Lemmon, E. W., Jacobsen, R. T., Penoncello, S. G., and Friend, D. G., Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at Pressures to 2000 MPa, J. Phys. Chem. Ref. Data, 29(3): 331–385, 2000. Tables including reactions with hydrocarbons include Gordon, S., NASA Techn. Paper 1907, 4 vols., 1982. See also Gupta, R. N., K-P. Lee, et al., NASA RP 1232, 1990 (89 pp.) and RP 1260, 1991 (75 pp.). Analytic expressions for high temperatures were given by Matsuzaki, R., Jap. J. Appl. Phys., 21, 7 (1982): 1009–1013 and Japanese National Aerospace Laboratory report NAL TR 671, 1981 (45 pp.). Functions from 1500 to 15,000 K were tabulated by Hilsenrath, J. and M. Klein, AEDC-TR-65-58 = AD 612 301, 1965 (333 pp.). Tables from 10000 to 10,000,000 K were authored by Gilmore, F. R., Lockheed rept. 3-27-67-1, vol 1., 1967 (340 pp.), also published as Radiative Properties of Air, IFI/Plenum, New York, 1969 (648 pp.). Saturation and superheat tables and a chart to 7000 psia, 660°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, Ga, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. AIR, MOIST An ASHRAE publication, Thermodynamic Properties of Dry Air and Water and S. I. Psychrometric Charts, 1983 (360 pp.), extensively reviews moist air properties. Gandiduson, P., Chem. Eng., Oct. 29, 1984 gives on page 118 a nomograph from 50 to 120°F, while equations in SI units were given by Nelson, B., Chem. Eng. Progr. 76, 5 (May 1980): 83–85. Liley, P. E., 2000 Solved Problems in M.E. Thermodynamics, McGraw-Hill, New York, 1989, gives four simple equations with which most calculations can be made. Devres, Y.O., Appl. Energy 48 (1994): 1–18 gives equations with which three known properties can be used to determine four others. Klappert, M. T. and G. F. Schilling, Rand RM-4244-PR = AD 604 856, 1984 (40 pp.) gives tables from 100 to 270 K, while programs from −60 to 2°F are given by Sando, F. A., ASHRAE Trans., 96, 2 (1990): 299–308. Viscosity references include Kestin, J. and J. H. Whitelaw, Int. J. Ht. Mass Transf. 7, 11 (1964): 1245–1255; Studnokov, E. L., Inz.-Fiz. Zhur. 19, 2 (1970): 338–340; Hochramer, D. and F. Munczak, Setzb. Ost. Acad. Wiss II 175, 10 (1966): 540–550. For thermal conductivity see, for instance, Mason, E. A. and L. Monchick, Humidity and Moisture Control in Science and Industry, Reinhold, New York, 1965 (257–272).

TABLE 2-189

Thermodynamic Properties of Ammonia

Temperature K

Pressure MPa

195.50 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 405.40

0.0060912 0.0086509 0.017739 0.033790 0.060407 0.10223 0.16494 0.25531 0.38107 0.55092 0.77436 1.0617 1.4240 1.8728 2.4205 3.0802 3.8660 4.7929 5.8778 7.1402 8.6045 10.305 11.339

43.035 42.754 42.111 41.442 40.748 40.032 39.293 38.533 37.748 36.939 36.101 35.230 34.320 33.363 32.350 31.264 30.087 28.788 27.321 25.606 23.465 20.232 13.212

195.50 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 405.40

0.0060912 0.0086509 0.017739 0.033790 0.060407 0.10223 0.16494 0.25531 0.38107 0.55092 0.77436 1.0617 1.4240 1.8728 2.4205 3.0802 3.8660 4.7929 5.8778 7.1402 8.6045 10.305 11.339

0.0037635 0.0052305 0.010249 0.018721 0.032214 0.052667 0.082417 0.12421 0.18126 0.25729 0.35664 0.48448 0.64702 0.85202 1.1094 1.4325 1.8399 2.3598 3.0375 3.9558 5.2979 7.6973 13.212

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.00000 0.32333 1.0480 1.7825 2.5265 3.2793 4.0403 4.8093 5.5862 6.3712 7.1651 7.9691 8.7850 9.6153 10.463 11.333 12.232 13.169 14.158 15.224 16.424 17.969 20.640

0.00014154 0.32353 1.0484 1.7833 2.5279 3.2818 4.0445 4.8160 5.5963 6.3861 7.1866 7.9993 8.8265 9.6714 10.538 11.432 12.361 13.335 14.373 15.503 16.790 18.478 21.499

23.661 23.770 24.006 24.233 24.450 24.655 24.846 25.021 25.179 25.317 25.435 25.528 25.595 25.632 25.634 25.595 25.505 25.350 25.107 24.734 24.144 23.047 20.640

25.279 25.424 25.737 26.038 26.325 26.596 26.847 27.077 27.281 27.459 27.606 27.720 27.796 27.830 27.816 27.746 27.606 27.381 27.042 26.539 25.768 24.386 21.499

Entropy kJ/(molK)

Sound speed m/s

Cv kJ/(molK)

Cp kJ/(molK)

0.00000 0.0016351 0.0051707 0.0085874 0.011894 0.015098 0.018205 0.021222 0.024154 0.027010 0.029797 0.032525 0.035203 0.037843 0.040458 0.043065 0.045682 0.048339 0.051075 0.053961 0.057149 0.061223 0.068559

0.049972 0.049837 0.049521 0.049207 0.048906 0.048613 0.048327 0.048047 0.047774 0.047511 0.047266 0.047044 0.046856 0.046715 0.046636 0.046642 0.046767 0.047064 0.047619 0.048589 0.050319 0.054109

0.071565 0.071988 0.072971 0.073950 0.074883 0.075764 0.076608 0.077448 0.078328 0.079296 0.080412 0.081747 0.083390 0.085465 0.088145 0.091701 0.096576 0.10357 0.11435 0.13314 0.17550 0.38707

2124.2 2080.2 1992.7 1913.7 1839.2 1766.9 1695.6 1624.5 1553.1 1481.0 1407.8 1333.2 1256.7 1177.9 1096.5 1011.8 923.38 830.62 732.78 628.75 515.88 384.58 0

0.12931 0.12714 0.12273 0.11884 0.11536 0.11224 0.10942 0.10684 0.10447 0.10227 0.10021 0.098259 0.096395 0.094589 0.092817 0.091046 0.089242 0.087355 0.085316 0.083003 0.080169 0.075992 0.068559

0.026510 0.026650 0.027053 0.027583 0.028245 0.029043 0.029978 0.031050 0.032253 0.033581 0.035028 0.036584 0.038244 0.040004 0.041868 0.043844 0.045954 0.048233 0.050744 0.053589 0.056957 0.061281

0.035130 0.035345 0.035961 0.036783 0.037836 0.039142 0.040728 0.042623 0.044859 0.047476 0.050530 0.054099 0.058302 0.063320 0.069443 0.077150 0.087280 0.10141 0.12286 0.16000 0.24170 0.59477

354.12 357.91 365.94 373.38 380.19 386.30 391.66 396.20 399.86 402.59 404.30 404.95 404.45 402.70 399.61 395.05 388.86 380.83 370.69 357.96 341.67 318.22 0

Therm. Joule-Thomson cond. K/MPa mW/(mK)

Viscosity µPas

Saturated Properties 0.023237 0.023389 0.023747 0.024130 0.024541 0.024980 0.025450 0.025952 0.026491 0.027072 0.027700 0.028385 0.029138 0.029973 0.030912 0.031986 0.033237 0.034737 0.036602 0.039054 0.042616 0.049426 0.075690 265.71 191.19 97.573 53.415 31.043 18.987 12.133 8.0506 5.5168 3.8867 2.8040 2.0641 1.5455 1.1737 0.90139 0.69810 0.54350 0.42377 0.32922 0.25279 0.18875 0.12992 0.075690

−0.23362 −0.22917 −0.21883 −0.20813 −0.19712 −0.18561 −0.17327 −0.15963 −0.14414 −0.12612 −0.10470 −0.078790 −0.046923 −0.0070718 0.043673 0.10967 0.19774 0.31928 0.49497 0.76738 1.2455 2.3557 5.0513 171.13 152.55 120.01 96.215 78.430 64.852 54.280 45.905 39.175 33.701 29.207 25.489 22.391 19.794 17.599 15.728 14.112 12.690 11.400 10.172 8.9038 7.3513 5.0513

818.99 803.14 768.02 733.17 698.80 665.09 632.16 600.07 568.85 538.50 508.99 480.25 452.23 424.83 397.96 371.51 345.32 319.25 293.07 266.57 239.65 216.00

559.57 507.28 414.98 346.68 294.94 254.85 223.08 197.34 176.06 158.12 142.74 129.33 117.49 106.91 97.325 88.555 80.430 72.796 65.493 58.315 50.877 41.802

19.636 19.684 19.860 20.132 20.503 20.978 21.560 22.258 23.079 24.034 25.138 26.408 27.872 29.568 31.559 33.945 36.900 40.752 46.149 54.556 70.114 113.54

6.8396 6.9515 7.2115 7.4846 7.7679 8.0587 8.3552 8.6558 8.9595 9.2664 9.5771 9.8938 10.220 10.561 10.927 11.330 11.792 12.346 13.053 14.025 15.527 18.529

2-217

2-218

TABLE 2-189

Thermodynamic Properties of Ammonia (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

200.00 239.56 239.56 300.00 400.00 500.00 600.00 700.00 200.00 298.05 298.05 300.00 400.00 500.00 600.00 700.00 200.00 300.00 362.03 362.03 400.00 500.00 600.00 700.00 200.00 300.00 398.32 398.32 400.00 500.00 600.00 700.00 300.00 400.00 500.00 600.00 700.00 300.00 400.00 500.00 600.00 700.00 300.00 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 100.00 100.00 100.00 100.00 100.00 500.00 500.00 500.00 500.00 500.00 1000.0 1000.0 1000.0 1000.0 1000.0

42.756 40.064 0.051595 0.040502 0.030171 0.024091 0.020060 0.017188 42.774 35.403 0.45697 0.45215 0.31157 0.24426 0.20197 0.17248 42.852 35.450 28.505 2.4828 1.8706 1.3046 1.0412 0.87563 42.947 35.714 20.945 7.1390 6.5455 2.8656 2.1650 1.7835 38.995 33.105 27.067 21.518 17.303 45.670 42.416 39.515 36.909 34.550 49.944 47.551 45.362 43.378 41.556

0.023388 0.024960 19.382 24.690 33.144 41.509 49.849 58.179 0.023379 0.028246 2.1883 2.2117 3.2095 4.0940 4.9513 5.7977 0.023336 0.028209 0.035081 0.40277 0.53459 0.76650 0.96040 1.1420 0.023284 0.028000 0.047744 0.14008 0.15278 0.34897 0.46190 0.56069 0.025644 0.030207 0.036945 0.046473 0.057794 0.021896 0.023576 0.025307 0.027094 0.028943 0.020022 0.021030 0.022045 0.023053 0.024064

0.32270 3.2461 24.646 26.378 29.297 32.514 36.096 40.068 0.31651 7.8111 25.512 25.592 29.019 32.359 35.994 39.994 0.28942 7.8852 13.365 25.309 27.540 31.630 35.527 39.662 0.25644 7.7848 17.655 23.303 23.801 30.616 34.920 39.241 6.5830 13.432 20.212 26.825 33.074 4.7114 10.633 16.367 22.007 27.680 4.1818 9.8612 15.432 20.911 26.418

0.32504 3.2486 26.584 28.847 32.612 36.665 41.081 45.885 0.33989 7.8393 27.700 27.804 32.229 36.453 40.945 45.792 0.40611 8.0263 13.540 27.323 30.213 35.462 40.329 45.373 0.48928 8.0648 18.132 24.704 25.329 34.106 39.539 44.848 9.1474 16.453 23.907 31.472 38.854 15.660 22.421 29.021 35.554 42.152 24.204 30.891 37.477 43.964 50.481

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.071983 0.075726 0.039079 0.036849 0.038883 0.042280 0.046083 0.050015 0.071938 0.081465 0.053356 0.052493 0.041627 0.043338 0.046628 0.050341 0.071739 0.080899 0.10538 0.10501 0.061581 0.048779 0.049210 0.051836 0.071495 0.079960 0.30653 0.46915 0.30552 0.057806 0.052806 0.053796 0.072740 0.073557 0.075495 0.075193 0.072317 0.067831 0.066802 0.065418 0.065476 0.066615 0.065784 0.066677 0.065150 0.064819 0.065697

2080.3 1770.0 386.05 434.39 497.93 550.96 597.69 640.16 2081.5 1347.9 404.91 407.16 488.94 546.79 595.60 639.15 2086.8 1361.2 811.17 378.95 441.81 528.14 586.79 635.17 2093.5 1394.2 409.04 323.12 336.28 505.64 577.38 631.50 1774.7 1378.2 1081.8 918.11 861.52 2597.1 2353.2 2176.9 2044.6 1943.8 3230.2 2997.6 2842.8 2728.6 2639.0

−0.22921 −0.18613 65.377 27.493 10.681 5.5276 3.2702 2.0841 −0.22959 −0.084271 26.163 25.620 10.494 5.4884 3.2544 2.0746 −0.23126 −0.089577 0.34968 12.419 9.6373 5.2830 3.1693 2.0254 −0.23328 −0.10159 2.0704 7.6606 7.7633 4.9335 3.0278 1.9491 −0.19551 −0.11309 0.049919 0.23722 0.32753 −0.25055 −0.25064 −0.25260 −0.24722 −0.23682 −0.25989 −0.25431 −0.26084 −0.26235 −0.25820

803.24 666.56 20.955 25.100 37.215 53.119 68.607 78.312 804.23 485.81 26.145 26.308 38.087 53.750 69.123 78.751 808.60 487.57 313.94 41.693 45.730 57.294 71.791 80.941 814.02 496.50 218.73 101.04 95.455 63.922 76.053 84.235 622.86 431.98 305.65 234.79 196.04 989.00 804.05 674.00 582.63 511.57 1324.0 1138.9 996.49 887.73 797.25

507.47 256.42 8.0459 10.161 13.971 17.863 21.682 25.391 509.28 131.82 9.8313 9.9115 13.927 17.877 21.717 25.434 517.30 132.49 71.291 12.475 14.036 18.073 21.941 25.662 527.29 136.36 43.632 17.793 17.230 18.722 22.393 26.035 193.71 96.237 60.386 46.188 41.237 376.31 188.46 120.77 91.251 77.538 554.62 274.91 174.11 129.02 107.14

Single-Phase Properties 0.0016320 0.014960 0.11237 0.12080 0.13162 0.14065 0.14869 0.15609 0.0016010 0.031996 0.098633 0.098979 0.11177 0.12119 0.12937 0.13684 0.0014649 0.032243 0.048887 0.086956 0.094581 0.10634 0.11521 0.12298 0.0012980 0.031903 0.060394 0.076892 0.078458 0.098525 0.10844 0.11663 0.027511 0.048523 0.065147 0.078942 0.090326 0.018023 0.037482 0.052215 0.064127 0.074295 0.011750 0.030984 0.045686 0.057514 0.067559

0.049842 0.048626 0.029005 0.028021 0.030417 0.033897 0.037731 0.041678 0.049890 0.047085 0.036271 0.035866 0.031641 0.034312 0.037928 0.041791 0.050097 0.047090 0.047152 0.048722 0.038466 0.036193 0.038798 0.042289 0.050342 0.047164 0.053149 0.060447 0.057611 0.038603 0.039862 0.042896 0.048894 0.046636 0.045999 0.046723 0.048331 0.052877 0.051527 0.050431 0.050614 0.051816 0.055176 0.054864 0.053323 0.052940 0.053649

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Tillner-Roth, R., Harms-Watzenberg, F., and Baehr, H. D.,“Eine neue Fundamentalgleichung fuer Ammoniak,” DKV-Tagungsbericht, 20:167–181, 1993. The source for viscosity is Fenghour, A., Wakeham, W. A., Vesovic, V., Watson, J. T. R., Millat, J., and Vogel, E., “The Viscosity of Ammonia,” J. Phys. Chem. Ref. Data 24:1649–1667, 1995. The source for thermal conductivity is Tufeu, R., Ivanov, D. Y., Garrabos, Y., and Le Neindre, B., “Thermal Conductivity of Ammonia in a Large Temperature and Pressure Range Including the Critical Region,” Ber. Bunsenges. Phys. Chem. 88:422–427, 1984. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.2% in density, 2% in heat capacity, and 2% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.2%. The uncertainty varies from 0.5% for the viscosity of the dilute gas phase at moderate temperatures to about 5% for the viscosity at high pressures and temperatures. The uncertainty in thermal conductivity is 2%.

1200

0.

0.

1400

0.

0.

1600

.

1800

2000

2200 3

0. k ρ = 10 80.

150

20.

60. c.p.

40.

20

15.

80

10. 8.0

2.

4.0 T = 20 °C

20

300

280

260

20 40 60 80 100 120 140 160 180 200 220

saturated

0.9

0.8

0.7

0.6

x=

sat

0.2

0.5

0.4

0.3

0.2

ura

ted

0.1

liqu

id

vapor

0

240

0.6 0.4

1. 0.8

2.0

0.6

1.5

0.4

1.0 0.80 0.60

60

00

40

60

0.02

K)

g·

8.

7.

0 7.2

0 6.8

0

0 6.4

6.0

0

0 5.6

5.2

4.80

4.40

4.00

3.60

3.20

2.80

2.40

2.00

1.60

1.20

0.80

0.40

0.06

s

=

(k J/

k

0.1 0.08

0.20

0.06

80

8.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0.15

0.10 0.080

8.

0.2

0.30 40

0.04

2.

3.0

0.40

0.1 0.08

4.

6.0

40

0.01 −200

6.

20.

60

Pressure (MPa)

10. 8.

30.

100

1. 0.8

2400 40.

g/m

.

200

25

30

35

40

45

50

0.

0.

. 550

600

1000

120

0

40

60

6. 4.

800

650.

700.

10. 8.

40

(R-717) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

T = 20 °C

20.

.

Ammonia

600

120

400

100

200

80

0

60

−200 40.

0.060

2200

0.04

0.02

0.01 2400

Enthalpy (kJ/kg) 2-219

FIG. 2-6 Pressure-enthalpy diagram for ammonia. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Tillner-Roth, R., Harms-Watzenberg, F., and Baehr, H. D., “Eine neue Fundamentalgleichung für Ammoniak, DKV-Tagungsbericht 20(II):167–181, 1993.

Enthalpy-concentration diagram for aqueous ammonia. From Thermodynamic and Physical Properties NH3 –H2O, Int. Inst. Refrigeration, Paris, France, 1994 (88 pp.). Reproduced by permission. In order to determine equilibrium compositions, draw a vertical from any liquid composition on any boiling line (the lowest plots) to intersect the appropriate auxiliary curve (the intermediate curves). A horizontal then drawn from this point to the appropriate dew line (the upper curves) will establish the vapor composition. The Int. Inst. Refrigeration publication also gives extensive P-v-x tables from −50 to 316°C. Other sources include Park, Y. M. and Sonntag, R. E., ASHRAE Trans., 96, 1 (1990): 150–159 (x, h, s, tables, 360 to 640 K); Ibrahim, O. M. and S. A. Klein, ASHRAE Trans., 99, 1 (1993): 1495–1502 (Eqs., 0.2 to 110 bar, 293 to 413 K); Smolen, T. M., D. B. Manley, et al., J. Chem. Eng. Data, 36 (1991): 202–208 (p-x correlation, 0.9 to 450 psia, 293–413 K); Ruiter, J. P., Int. J. Refrig., 13 (1990): 223–236 gives ten subroutines for computer calculations. FIG. 2-7

TABLE 2-190

Thermodynamic Properties of Argon

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

83.806 85.000 87.000 89.000 91.000 93.000 95.000 97.000 99.000 101.00 103.00 105.00 107.00 109.00 111.00 113.00 115.00 117.00 119.00 121.00 123.00 125.00 127.00 129.00 131.00 133.00 135.00 137.00 139.00 141.00 143.00 145.00 147.00 149.00 150.69

0.068891 0.078897 0.098131 0.12078 0.14723 0.17785 0.21305 0.25323 0.29882 0.35023 0.40789 0.47224 0.54371 0.62276 0.70982 0.80535 0.90981 1.0237 1.1473 1.2814 1.4262 1.5823 1.7503 1.9305 2.1237 2.3303 2.5509 2.7862 3.0369 3.3037 3.5876 3.8896 4.2111 4.5541 4.8630

35.465 35.284 34.977 34.667 34.353 34.035 33.713 33.386 33.054 32.715 32.371 32.020 31.662 31.296 30.921 30.538 30.144 29.739 29.322 28.891 28.446 27.983 27.502 26.999 26.471 25.913 25.320 24.683 23.993 23.233 22.379 21.385 20.163 18.446 13.407

0.028197 0.028342 0.028590 0.028846 0.029109 0.029381 0.029662 0.029953 0.030254 0.030567 0.030892 0.031230 0.031584 0.031953 0.032340 0.032747 0.033174 0.033626 0.034104 0.034613 0.035155 0.035735 0.036361 0.037039 0.037778 0.038591 0.039495 0.040513 0.041678 0.043041 0.044686 0.046762 0.049597 0.054213 0.074586

−4.8531 −4.8000 −4.7110 −4.6220 −4.5328 −4.4433 −4.3534 −4.2632 −4.1725 −4.0812 −3.9893 −3.8968 −3.8034 −3.7092 −3.6141 −3.5180 −3.4207 −3.3222 −3.2223 −3.1208 −3.0175 −2.9123 −2.8049 −2.6949 −2.5820 −2.4655 −2.3449 −2.2192 −2.0871 −1.9468 −1.7953 −1.6277 −1.4335 −1.1822 −0.53575

−4.8512 −4.7977 −4.7082 −4.6185 −4.5285 −4.4380 −4.3471 −4.2556 −4.1634 −4.0705 −3.9767 −3.8820 −3.7862 −3.6893 −3.5912 −3.4916 −3.3905 −3.2878 −3.1831 −3.0764 −2.9674 −2.8558 −2.7413 −2.6234 −2.5018 −2.3756 −2.2442 −2.1063 −1.9605 −1.8046 −1.6350 −1.4458 −1.2247 −0.93533 −0.17304

0.053110 0.053740 0.054774 0.055787 0.056778 0.057752 0.058708 0.059649 0.060575 0.061489 0.062391 0.063283 0.064165 0.065039 0.065906 0.066767 0.067623 0.068477 0.069328 0.070179 0.071031 0.071886 0.072747 0.073616 0.074496 0.075392 0.076309 0.077253 0.078235 0.079268 0.080375 0.081593 0.083001 0.084835 0.089789

83.806 85.000 87.000 89.000 91.000 93.000 95.000 97.000 99.000 101.00 103.00 105.00 107.00 109.00 111.00 113.00 115.00 117.00 119.00 121.00 123.00 125.00

0.068891 0.078897 0.098131 0.12078 0.14723 0.17785 0.21305 0.25323 0.29882 0.35023 0.40789 0.47224 0.54371 0.62276 0.70982 0.80535 0.90981 1.0237 1.1473 1.2814 1.4262 1.5823

9.8526 8.7019 7.1253 5.8895 4.9104 4.1266 3.4935 2.9774 2.5533 2.2021 1.9092 1.6632 1.4553 1.2786 1.1274 0.99752 0.88525 0.78776 0.70269 0.62813 0.56248 0.50443

1.0103 1.0212 1.0388 1.0554 1.0710 1.0855 1.0988 1.1108 1.1216 1.1309 1.1387 1.1449 1.1494 1.1521 1.1528 1.1515 1.1480 1.1421 1.1336 1.1223 1.1079 1.0901

1.6891 1.7078 1.7380 1.7668 1.7939 1.8194 1.8431 1.8648 1.8845 1.9021 1.9174 1.9303 1.9406 1.9483 1.9531 1.9549 1.9534 1.9485 1.9398 1.9271 1.9101 1.8883

0.13115 0.13028 0.12887 0.12753 0.12626 0.12504 0.12387 0.12275 0.12167 0.12062 0.11962 0.11864 0.11769 0.11676 0.11585 0.11497 0.11409 0.11323 0.11238 0.11153 0.11069 0.10984

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.021956 0.021764 0.021461 0.021178 0.020912 0.020662 0.020424 0.020198 0.019982 0.019777 0.019580 0.019393 0.019214 0.019044 0.018883 0.018731 0.018589 0.018458 0.018337 0.018229 0.018134 0.018056 0.017995 0.017955 0.017941 0.017957 0.018011 0.018109 0.018264 0.018501 0.018882 0.019546 0.020842 0.023975

0.044570 0.044571 0.044620 0.044720 0.044869 0.045063 0.045301 0.045583 0.045909 0.046282 0.046705 0.047181 0.047715 0.048313 0.048984 0.049737 0.050584 0.051541 0.052626 0.053865 0.055288 0.056938 0.058870 0.061161 0.063916 0.067291 0.071519 0.076964 0.084237 0.094443 0.10982 0.13580 0.18993 0.37941

862.43 854.24 840.43 826.48 812.39 798.14 783.72 769.14 754.37 739.40 724.21 708.81 693.16 677.26 661.08 644.60 627.79 610.64 593.11 575.16 556.75 537.83 518.34 498.21 477.36 455.69 433.10 409.47 384.62 358.17 329.41 297.06 258.79 209.33 0

−0.40500 −0.40120 −0.39386 −0.38530 −0.37551 −0.36447 −0.35214 −0.33847 −0.32336 −0.30671 −0.28840 −0.26827 −0.24614 −0.22181 −0.19503 −0.16549 −0.13284 −0.096667 −0.056465 −0.011619 0.038624 0.095199 0.15928 0.23236 0.31637 0.41386 0.52823 0.66427 0.82889 1.0326 1.2928 1.6405 2.1416 2.9836 5.2503

133.63 131.86 128.91 125.97 123.06 120.16 117.28 114.44 111.63 108.82 106.03 103.26 100.50 97.768 95.055 92.363 89.695 87.049 84.428 81.832 79.260 76.711 74.185 71.679 69.190 66.713 64.243 61.773 59.297 56.816 54.352 52.009 50.224 51.152

290.17 279.46 262.70 247.29 233.09 219.98 207.87 196.65 186.24 176.58 167.58 159.18 151.32 143.96 137.04 130.52 124.35 118.50 112.93 107.61 102.49 97.565 92.791 88.143 83.595 79.119 74.689 70.276 65.850 61.374 56.802 52.060 46.982 41.002

0.012972 0.013015 0.013092 0.013176 0.013267 0.013364 0.013469 0.013580 0.013700 0.013828 0.013964 0.014108 0.014263 0.014427 0.014602 0.014789 0.014989 0.015203 0.015432 0.015679 0.015945 0.016234

0.022172 0.022310 0.022563 0.022845 0.023159 0.023506 0.023892 0.024318 0.024789 0.025310 0.025887 0.026526 0.027235 0.028024 0.028904 0.029890 0.030998 0.032251 0.033674 0.035305 0.037186 0.039377

168.12 169.08 170.64 172.12 173.52 174.85 176.11 177.29 178.39 179.41 180.36 181.22 182.01 182.71 183.33 183.87 184.33 184.70 184.98 185.17 185.27 185.28

35.712 34.706 33.127 31.668 30.315 29.056 27.883 26.785 25.755 24.786 23.873 23.009 22.189 21.410 20.667 19.955 19.272 18.615 17.979 17.361 16.758 16.167

Saturated Properties

2-221

0.10150 0.11492 0.14035 0.16979 0.20365 0.24233 0.28625 0.33587 0.39165 0.45412 0.52379 0.60126 0.68714 0.78213 0.88696 1.0025 1.1296 1.2694 1.4231 1.5920 1.7778 1.9824

5.3590 5.4485 5.6006 5.7559 5.9147 6.0776 6.2452 6.4180 6.5969 6.7825 6.9760 7.1783 7.3908 7.6149 7.8524 8.1054 8.3764 8.6685 8.9853 9.3315 9.7128 10.136

6.8558 6.9622 7.1414 7.3219 7.5039 7.6877 7.8736 8.0617 8.2524 8.4461 8.6432 8.8440 9.0492 9.2592 9.4749 9.6969 9.9262 10.164 10.411 10.670 10.942 11.228

2-222 TABLE 2-190

Thermodynamic Properties of Argon (Concluded)

Temperature K

Pressure MPa

127.00 129.00 131.00 133.00 135.00 137.00 139.00 141.00 143.00 145.00 147.00 149.00 150.69

1.7503 1.9305 2.1237 2.3303 2.5509 2.7862 3.0369 3.3037 3.5876 3.8896 4.2111 4.5541 4.8630

100.00 200.00 300.00 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

100.00 116.60

1.0000 1.0000

116.60 200.00 300.00 400.00 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

100.00 200.00 300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.45288 0.40690 0.36570 0.32861 0.29505 0.26451 0.23652 0.21066 0.18647 0.16342 0.14073 0.11646 0.074586

1.0685 1.0428 1.0122 0.97615 0.93360 0.88331 0.82353 0.75170 0.66386 0.55308 0.40516 0.17894 −0.53575

1.8612 1.8283 1.7888 1.7419 1.6862 1.6203 1.5418 1.4476 1.3328 1.1887 0.99779 0.70930 −0.17304

1.2122 2.4833 3.7351 4.9842 6.2324 7.4803 8.7279

2.0249 4.1414 6.2279 8.3099 10.390 12.470 14.550

0.13179 0.14649 0.15495 0.16094 0.16559 0.16938 0.17258

0.030345 0.033533

−4.1375 −3.3421

−4.1072 −3.3086

1.2403 0.61937 0.40331 0.30073 0.24013 0.19997 0.17137

0.80628 1.6145 2.4795 3.3252 4.1644 5.0007 5.8354

1.1435 2.3849 3.6807 4.9476 6.2054 7.4592 8.7110

33.343 3.5342 2.0596 1.5022 1.1914 0.98986 0.84774

0.029992 0.28295 0.48554 0.66567 0.83938 1.0102 1.1796

33.779 8.4545 4.1955 2.9896 2.3554 1.9532 1.6722

0.029604 0.11828 0.23835 0.33449 0.42455 0.51198 0.59801

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

0.041958 0.045036 0.048764 0.053370 0.059209 0.066854 0.077279 0.092257 0.11542 0.15563 0.24158 0.54500

185.18 184.98 184.68 184.27 183.74 183.03 182.09 180.81 179.05 176.57 172.74 165.39 133.87

0.012807 0.012497 0.012479 0.012475 0.012474 0.012473 0.012473

0.021852 0.020918 0.020834 0.020810 0.020801 0.020796 0.020793

0.060927 0.068305

0.019909 0.018483

1.9497 3.9995 6.1603 8.2728 10.370 12.460 14.546

0.11340 0.12686 0.13563 0.14171 0.14639 0.15020 0.15342

−4.1962 1.8892 3.4362 4.7860 6.0868 7.3671 8.6370

−4.0463 3.3039 5.8640 8.1144 10.284 12.418 14.535

−4.2615 1.1163 3.1284 4.5884 5.9428 7.2556 8.5473

−3.9654 2.2991 5.5119 7.9333 10.188 12.375 14.527

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

15.585 15.007 14.430 13.851 13.262 12.655 12.019 11.341 10.606 9.7892 8.8407 7.6233 0

10.611 11.150 11.768 12.487 13.337 14.363 15.633 17.255 19.424 22.530 27.572 38.688

11.533 11.860 12.212 12.596 13.018 13.489 14.022 14.638 15.370 16.274 17.475 19.366

184.16 263.21 322.67 372.65 416.64 456.40 492.95

25.048 7.4099 3.6153 1.9970 1.1105 0.55603 0.17948

6.4504 12.539 17.837 22.516 26.726 30.573 34.134

8.2341 15.998 22.741 28.704 34.077 38.997 43.556

0.045870 0.051339

751.59 614.12

−0.32095 −0.10424

0.015159 0.012725 0.012545 0.012507 0.012493 0.012487 0.012483

0.031986 0.022195 0.021266 0.021025 0.020928 0.020879 0.020851

184.63 261.59 323.44 374.06 418.25 458.06 494.61

18.745 7.2269 3.5287 1.9489 1.0808 0.53628 0.16568

0.060330 0.11098 0.12145 0.12793 0.13277 0.13667 0.13993

0.020096 0.013847 0.012825 0.012643 0.012578 0.012547 0.012530

0.044750 0.030481 0.023284 0.021972 0.021480 0.021239 0.021104

777.34 257.20 328.37 381.06 425.81 465.68 502.09

−0.35058 6.1700 3.1276 1.7370 0.95180 0.45113 0.10636

114.41 16.132 19.721 23.882 27.811 31.477 34.910

193.32 18.205 23.850 29.440 34.619 39.420 43.899

0.059648 0.10142 0.11472 0.12170 0.12674 0.13073 0.13404

0.020331 0.015240 0.013145 0.012803 0.012681 0.012621 0.012587

0.043670 0.048544 0.025892 0.023107 0.022129 0.021662 0.021400

805.91 267.70 338.42 391.51 436.14 475.69 511.73

−0.37991 4.1966 2.6066 1.4795 0.79870 0.35092 0.036761

118.60 23.573 22.192 25.505 29.044 32.479 35.757

205.82 23.429 25.583 30.479 35.348 39.971 44.338

Saturated Properties 2.2081 2.4576 2.7345 3.0431 3.3893 3.7806 4.2280 4.7471 5.3629 6.1190 7.1057 8.5869 13.407

0.10899 0.10813 0.10725 0.10635 0.10542 0.10445 0.10343 0.10233 0.10113 0.099762 0.098120 0.095873 0.089789

0.016549 0.016892 0.017269 0.017684 0.018152 0.018697 0.019352 0.020167 0.021213 0.022619 0.024694 0.028532

Single-Phase Properties

100.00 200.00 300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000 10.000 10.000

0.12304 0.060310 0.040115 0.030069 0.024050 0.020041 0.017177 32.955 29.821

8.1275 16.581 24.928 33.257 41.580 49.899 58.217

110.85 87.579 8.6079 12.995 18.142 22.749 26.915 30.733 34.272

183.08 119.65 10.115 16.256 22.901 28.817 34.163 39.065 43.612

200.00 300.00 400.00 500.00 600.00 700.00

100.00 100.00 100.00 100.00 100.00 100.00

30.368 24.152 19.710 16.607 14.372 12.695

0.032930 0.041404 0.050736 0.060216 0.069578 0.078769

−1.9234 0.48428 2.5103 4.2850 5.9046 7.4268

1.3696 4.6247 7.5839 10.307 12.862 15.304

0.077319 0.090561 0.099092 0.10518 0.10984 0.11361

0.017598 0.015534 0.014579 0.014048 0.013716 0.013492

0.034137 0.031005 0.028287 0.026288 0.024915 0.023969

200.00 300.00 400.00 500.00 600.00 700.00

500.00 500.00 500.00 500.00 500.00 500.00

41.639 38.382 35.672 33.368 31.379 29.643

0.024016 0.026054 0.028033 0.029969 0.031868 0.033734

−2.6867 −0.59281 1.3125 3.0950 4.7903 6.4198

9.3214 12.434 15.329 18.079 20.724 23.287

0.063085 0.075743 0.084083 0.090226 0.095050 0.099002

0.022923 0.019980 0.018316 0.017262 0.016535 0.016000

0.032604 0.029872 0.028142 0.026929 0.026007 0.025273

1519.6 1443.8 1393.0 1358.5 1334.9 1318.8

−0.60932 −0.67083 −0.71809 −0.75700 −0.79113 −0.82207

218.05 184.82 161.66 145.02 132.94 124.11

504.85 259.42 183.29 151.69 135.48 125.84

44.836 42.715 40.856 39.201 37.711

0.022304 0.023411 0.024476 0.025510 0.026518

−0.13952 1.7386 3.5106 5.2086 6.8515

22.164 25.150 27.987 30.718 33.369

0.068356 0.076958 0.083293 0.088276 0.092364

0.022820 0.020840 0.019528 0.018596 0.017896

0.030815 0.029020 0.027790 0.026881 0.026165

1866.0 1823.7 1791.9 1767.8 1749.3

−0.61332 −0.65733 −0.69220 −0.72129 −0.74683

267.67 243.17 223.98 208.59 196.06

666.74 410.53 297.24 239.91 207.86

300.00 400.00 500.00 600.00 700.00

1000.0 1000.0 1000.0 1000.0 1000.0

854.48 733.39 690.42 683.08 690.94 705.52

−0.51219 −0.45856 −0.45296 −0.49513 −0.55858 −0.62622

99.994 70.273 59.077 54.952 53.635 53.660

120.12 75.589 61.649 57.655 57.185 58.260

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Tegeler, Ch., Span, R., and Wagner, W., “A New Equation of State for Argon Covering the Fluid Region for Temperatures from the Melting Line to 700 K at Pressures up to 1000 MPa,” J. Phys. Chem. Ref. Data 28(3):779–850, 1999. The source for viscosity and thermal conductivity is Lemmon, E. W., and Jacobsen, R. T., “Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air,” Int. J. Thermophys. 25:21–69, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainty in density is less than 0.02% for pressures up to 12 MPa and temperatures up to 340 K with the exception of the critical region and less than 0.03% for pressures up to 30 MPa and temperatures between 235 and 520 K. Elsewhere, the uncertainty in density is generally within 0.2%. In the region with densities up to one-half the critical density and for temperatures between 90 and 450 K, the estimated uncertainty of calculated speeds of sound is in general less than 0.02%. In the liquid and supercritical regions, the uncertainty is less than 1%. The uncertainty in heat capacities is within 0.3% for the vapor and 2% for the liquid. The formulation gives reasonable extrapolation behavior up to very high pressures (50 GPa) and temperatures (17,000 K). For viscosity, the uncertainty is 0.5% in the dilute gas. Away from the dilute gas (pressures greater than 1 MPa and in the liquid), the uncertainties are as low as 1% between 270 and 300 K at pressures less than 100 MPa, and increase outside that range. The uncertainties are around 2% at temperatures of 180 K and higher. Below this and away from the critical region, the uncertainties steadily increase to around 5% at the triple points of the fluids. The uncertainties in the critical region are higher. For thermal conductivity, the uncertainty for the dilute gas is 2% with increasing uncertainties near the triple point. For the nondilute gas, the uncertainty is 2% for temperatures greater than 170 K. The uncertainty is 3% at temperatures less than the critical point and 5% in the critical region, except for states very near the critical point.

2-223

2-224

TABLE 2-191

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System

Liquid mole fraction N2/(N2 +O2)

Ar

N2

Vapor mole fraction O2

Ar

N2

O2

p0 (kPa)

Relative volatility Temperature K

N2/Ar

N2/O2

Ar/O2

Ar

Activity coefficent

Enthalpy kJ/kmol

Isobaric specific heat, kJ/(kmolK)

N2

O2

Ar

N2

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 101.325 kPa (1 atm) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.00 0.99 0.98 0.97 0.96 0.95 0.93 0.90 0.80 0.60 0.40 0.20 0.10

0.00000 0.01590 0.03151 0.04684 0.06189 0.07667 0.10548 0.14692 0.27211 0.48104 0.65969 0.82774 0.91247

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.0000 0.9841 0.9685 0.9532 0.9381 0.9233 0.8945 0.8531 0.7279 0.5190 0.3403 0.1723 0.0875

90.188 90.128 90.070 90.013 89.956 89.901 89.794 89.640 89.183 88.470 87.948 87.566 87.420

— — — — — — — — — — — — —

— — — — — — — — — — — — —

1.606 1.600 1.594 1.589 1.583 1.578 1.567 1.550 1.495 1.390 1.292 1.201 1.158

136.010 135.210 134.430 133.680 132.930 132.200 130.810 128.810 123.040 114.430 108.420 104.180 102.590

366.29 364.42 362.62 360.86 359.10 357.41 354.14 349.47 335.88 315.46 301.10 290.90 287.07

101.33 100.69 100.08 99.49 98.89 98.32 97.22 95.65 91.10 84.35 79.66 76.35 75.12

1.196 1.192 1.188 1.183 1.179 1.175 1.167 1.156 1.120 1.065 1.028 1.006 1.001

— — — — — — — — — — — — —

1.000 1.000 1.001 1.001 1.001 1.002 1.002 1.004 1.012 1.039 1.082 1.143 1.181

−4268 −4271 −4275 −4278 −4282 −4285 −4292 −4302 −4336 −4409 −4491 −4586 −4638

2550 2536 2523 2510 2497 2484 2460 2425 2320 2147 2004 1873 1808

54.378 54.269 54.160 54.052 53.944 53.837 53.622 53.302 52.253 50.224 48.281 46.417 45.515

31.061 30.929 30.801 30.674 30.550 30.429 30.192 29.851 28.822 27.089 25.578 24.123 23.378

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09 0.08 0.06 0.04 0.02 0.01

0.90 0.89 0.88 0.87 0.86 0.86 0.84 0.81 0.72 0.54 0.36 0.18 0.09

0.0000 0.0122 0.0242 0.0361 0.0479 0.0596 0.0826 0.1164 0.2230 0.4179 0.6027 0.7916 0.8923

0.3150 0.3108 0.3066 0.3025 0.2985 0.2945 0.2866 0.2751 0.2390 0.1749 0.1167 0.0598 0.0306

0.6850 0.6771 0.6692 0.6614 0.6536 0.6459 0.6308 0.6086 0.5380 0.4072 0.2807 0.1486 0.0771

87.546 87.529 87.512 87.495 87.479 87.463 87.432 87.389 87.270 87.121 87.077 87.132 87.201

2.573 2.579 2.584 2.589 2.595 2.600 2.611 2.627 2.680 2.790 2.904 3.023 3.085

4.138 4.131 4.124 4.117 4.110 4.103 4.089 4.068 3.999 3.866 3.742 3.625 3.569

1.608 1.602 1.596 1.590 1.584 1.578 1.566 1.549 1.492 1.386 1.288 1.199 1.157

103.960 103.780 103.590 103.410 103.230 103.060 102.720 102.260 100.980 99.401 98.938 99.517 100.250

290.37 289.93 289.48 289.03 288.61 288.19 287.38 286.26 283.17 279.34 278.22 279.63 281.40

76.18 76.04 75.89 75.75 75.61 75.48 75.22 74.86 73.87 72.64 72.28 72.73 73.30

1.193 1.188 1.184 1.180 1.176 1.172 1.164 1.153 1.119 1.065 1.029 1.008 1.002

1.099 1.097 1.095 1.093 1.092 1.090 1.086 1.082 1.069 1.057 1.062 1.084 1.101

1.012 1.013 1.013 1.013 1.014 1.014 1.015 1.017 1.025 1.052 1.093 1.150 1.185

−4235 −4238 −4241 −4244 −4248 −4251 −4257 −4266 −4300 −4378 −4469 −4575 −4633

2471 2461 2452 2443 2434 2425 2408 2382 2302 2157 2023 1889 1818

54.388 54.281 54.175 54.068 53.962 53.856 53.645 53.330 52.291 50.269 48.319 46.440 45.527

30.764 30.669 30.576 30.484 30.392 30.302 30.123 29.861 29.029 27.486 25.986 24.404 23.542

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.18 0.16 0.12 0.08 0.04 0.02

0.80 0.79 0.78 0.78 0.77 0.76 0.74 0.72 0.64 0.48 0.32 0.16 0.08

0.0000 0.0098 0.0195 0.0292 0.0388 0.0484 0.0674 0.0957 0.1879 0.3685 0.5541 0.7583 0.8729

0.5104 0.5050 0.4995 0.4941 0.4888 0.4835 0.4729 0.4573 0.4069 0.3108 0.2158 0.1150 0.0599

0.4896 0.4853 0.4810 0.4767 0.4724 0.4682 0.4597 0.4470 0.4052 0.3206 0.2302 0.1267 0.0671

85.511 85.519 85.526 85.534 85.542 85.551 85.568 85.594 85.695 85.947 86.280 86.717 86.987

2.606 2.611 2.616 2.621 2.626 2.631 2.641 2.656 2.707 2.812 2.921 3.033 3.090

4.170 4.162 4.154 4.146 4.139 4.131 4.115 4.092 4.017 3.878 3.750 3.630 3.572

1.600 1.594 1.588 1.582 1.576 1.570 1.558 1.541 1.484 1.379 1.284 1.197 1.156

83.506 83.580 83.645 83.719 83.793 83.876 84.033 84.274 85.216 87.602 90.832 95.211 97.996

240.35 240.53 240.69 240.88 241.06 241.27 241.66 242.25 244.59 250.48 258.43 269.15 275.93

60.37 60.42 60.47 60.53 60.59 60.65 60.77 60.96 61.68 63.52 66.01 69.39 71.55

1.188 1.184 1.180 1.176 1.172 1.169 1.161 1.150 1.117 1.066 1.030 1.009 1.003

1.076 1.074 1.073 1.071 1.070 1.069 1.066 1.063 1.054 1.048 1.057 1.082 1.101

1.027 1.027 1.028 1.028 1.029 1.029 1.030 1.032 1.040 1.066 1.104 1.157 1.188

−4176 −4179 −4183 −4186 −4190 −4193 −4200 −4211 −4250 −4340 −4444 −4563 −4628

2409 2403 2396 2389 2383 2376 2363 2344 2282 2161 2038 1903 1827

54.481 54.374 54.267 54.161 54.054 53.948 53.736 53.420 52.376 50.337 48.366 46.464 45.539

30.819 30.742 30.665 30.588 30.512 30.436 30.285 30.061 29.328 27.881 26.371 24.673 23.701

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.40 0.40 0.39 0.39 0.38 0.38 0.37 0.36 0.32 0.24 0.16 0.08 0.04

0.60 0.59 0.59 0.58 0.58 0.57 0.56 0.54 0.48 0.36 0.24 0.12 0.06

0.0000 0.0070 0.0139 0.0209 0.0279 0.0349 0.0490 0.0702 0.1424 0.2971 0.4762 0.6988 0.8364

0.7339 0.7285 0.7230 0.7176 0.7121 0.7067 0.6957 0.6792 0.6237 0.5066 0.3743 0.2133 0.1153

0.2661 0.2646 0.2630 0.2615 0.2600 0.2585 0.2553 0.2506 0.2340 0.1963 0.1496 0.0880 0.0483

82.557 82.590 82.623 82.656 82.689 82.722 82.789 82.890 83.238 83.994 84.870 85.936 86.573

2.639 2.644 2.649 2.654 2.659 2.664 2.673 2.688 2.738 2.842 2.947 3.052 3.101

4.138 4.131 4.123 4.116 4.108 4.101 4.087 4.066 3.998 3.871 3.753 3.637 3.577

1.568 1.562 1.557 1.551 1.545 1.540 1.529 1.513 1.460 1.362 1.273 1.192 1.153

— — — — — — — — — 70.40 77.76 87.50 93.75

179.50 180.11 180.72 181.33 181.95 182.56 183.81 185.71 192.37 207.45 226.01 250.22 265.58

42.12 42.29 42.47 42.65 42.82 43.00 43.37 43.92 45.87 50.35 55.96 63.43 68.26

— — — — — — — — — 1.069 1.034 1.012 1.004

1.036 1.035 1.034 1.033 1.033 1.032 1.031 1.029 1.027 1.031 1.049 1.079 1.100

1.067 1.067 1.067 1.068 1.068 1.068 1.069 1.071 1.077 1.097 1.129 1.171 1.196

−4012 −4017 −4022 −4028 −4033 −4038 −4049 −4066 −4123 −4250 −4389 −4538 −4616

2320 2316 2313 2309 2305 2301 2294 2282 2243 2158 2056 1926 1844

54.860 54.749 54.637 54.526 54.415 54.304 54.083 53.752 52.662 50.537 48.489 46.520 45.566

31.014 30.957 30.899 30.841 30.783 30.724 30.607 30.431 29.830 28.539 27.042 25.171 24.005

0.60 0.60 0.60 0.60

0.00 0.01 0.02 0.03

0.60 0.59 0.59 0.58

0.40 0.40 0.39 0.39

0.0000 0.0054 0.0108 0.0163

0.8586 0.8538 0.8489 0.8441

0.1414 0.1408 0.1402 0.1396

80.441 80.485 80.530 80.574

2.657 2.661 2.666 2.671

4.047 4.041 4.036 4.030

1.523 1.519 1.514 1.509

— — — —

143.61 144.30 145.00 145.69

31.97 32.16 32.35 32.54

— — — —

1.010 1.009 1.009 1.009

1.121 1.121 1.120 1.121

−3821 −3828 −3836 −3843

2256 2254 2252 2249

55.455 55.334 55.214 55.093

31.192 31.145 31.097 31.050

2-225

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.58 0.57 0.56 0.54 0.48 0.36 0.24 0.12 0.06

0.38 0.38 0.37 0.36 0.32 0.24 0.16 0.08 0.04

0.0218 0.0273 0.0384 0.0554 0.1145 0.2487 0.4170 0.6475 0.8025

0.8392 0.8343 0.8244 0.8094 0.7573 0.6402 0.4949 0.2979 0.1665

0.1390 0.1384 0.1372 0.1352 0.1282 0.1112 0.0881 0.0546 0.0310

80.619 80.664 80.755 80.893 81.368 82.415 83.658 85.216 86.175

2.676 2.681 2.690 2.705 2.756 2.860 2.967 3.068 3.111

4.024 4.018 4.007 3.991 3.938 3.839 3.743 3.640 3.581

1.504 1.499 1.489 1.475 1.429 1.342 1.262 1.186 1.151

— — — — — — — 80.82 89.80

146.40 147.12 148.56 150.78 158.60 176.90 200.64 233.67 255.91

32.74 32.94 33.34 33.95 36.14 41.36 48.32 58.30 65.21

— — — — — — — 1.015 1.006

1.008 1.008 1.008 1.007 1.008 1.019 1.041 1.077 1.099

1.121 1.121 1.121 1.121 1.123 1.135 1.155 1.186 1.204

−3851 −3859 −3874 −3898 −3978 −4148 −4327 −4512 −4604

2247 2245 2240 2233 2207 2147 2066 1945 1858

54.973 54.853 54.614 54.257 53.083 50.811 48.644 46.584 45.595

31.002 30.954 30.857 30.709 30.196 29.035 27.586 25.615 24.292

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.80 0.79 0.78 0.78 0.77 0.76 0.74 0.72 0.64 0.48 0.32 0.16 0.08

0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.18 0.16 0.12 0.08 0.04 0.02

0.0000 0.0044 0.0089 0.0133 0.0179 0.0224 0.0317 0.0458 0.0958 0.2139 0.3708 0.6029 0.7712

0.9404 0.9362 0.9319 0.9276 0.9233 0.9190 0.9102 0.8967 0.8492 0.7375 0.5897 0.3716 0.2139

0.0596 0.0594 0.0592 0.0590 0.0588 0.0586 0.0581 0.0575 0.0550 0.0486 0.0396 0.0255 0.0149

78.768 78.820 78.871 78.923 78.974 79.027 79.132 79.292 79.848 81.090 82.596 84.549 85.793

2.673 2.677 2.682 2.687 2.691 2.696 2.705 2.720 2.769 2.874 2.982 3.082 3.121

3.942 3.938 3.934 3.930 3.926 3.921 3.913 3.901 3.863 3.793 3.723 3.639 3.584

1.475 1.471 1.467 1.463 1.459 1.455 1.447 1.434 1.395 1.320 1.249 1.181 1.148

— — — — — — — — — — — 74.99 86.14

119.30 120.01 120.70 121.41 122.12 122.85 124.31 126.55 134.60 153.99 180.22 219.07 246.87

25.42 25.60 25.79 25.97 26.16 26.35 26.74 27.34 29.51 34.85 42.32 53.85 62.39

— — — — — — — — — — — 1.018 1.008

0.998 0.998 0.998 0.998 0.998 0.997 0.997 0.997 0.999 1.011 1.036 1.074 1.098

1.189 1.188 1.187 1.187 1.186 1.186 1.185 1.183 1.179 1.178 1.185 1.201 1.212

−3620 −3630 −3640 −3650 −3660 −3670 −3691 −3721 −3825 −4039 −4261 −4483 −4591

2205 2204 2202 2201 2200 2198 2195 2191 2175 2133 2069 1960 1872

56.237 56.103 55.970 55.836 55.704 55.571 55.307 54.914 53.625 51.154 48.829 46.656 45.625

31.347 31.306 31.266 31.225 31.183 31.142 31.058 30.929 30.476 29.417 28.030 26.011 24.563

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.90 0.89 0.88 0.87 0.86 0.86 0.84 0.81 0.72 0.54 0.36 0.18 0.09

0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09 0.08 0.06 0.04 0.02 0.01

0.0000 0.0040 0.0081 0.0122 0.0164 0.0206 0.0291 0.0421 0.0886 0.1999 0.3513 0.5828 0.7563

0.9722 0.9683 0.9643 0.9603 0.9562 0.9521 0.9438 0.9310 0.8857 0.7772 0.6299 0.4049 0.2364

0.0278 0.0277 0.0276 0.0275 0.0274 0.0273 0.0271 0.0268 0.0257 0.0229 0.0189 0.0124 0.0073

78.036 78.090 78.144 78.199 78.254 78.309 78.420 78.589 79.177 80.497 82.110 84.233 85.608

2.683 2.687 2.692 2.696 2.701 2.705 2.714 2.728 2.776 2.880 2.988 3.088 3.125

3.891 3.888 3.884 3.881 3.877 3.874 3.867 3.857 3.825 3.768 3.711 3.638 3.585

1.450 1.447 1.443 1.439 1.436 1.432 1.425 1.414 1.378 1.308 1.242 1.178 1.147

— — — — — — — — — — — 72.35 84.404

109.71 110.4 111.09 111.79 112.5 113.22 114.67 116.9 124.93 144.49 171.42 212.4 242.58

22.91 23.088 23.267 23.45 23.634 23.82 24.198 24.783 26.908 32.209 39.784 51.833 61.055

— — — — — — — — — — — 1.020 1.009

0.998 0.997 0.997 0.997 0.997 0.997 0.996 0.996 0.998 1.009 1.034 1.073 1.097

1.228 1.227 1.226 1.225 1.224 1.223 1.221 1.219 1.211 1.202 1.201 1.209 1.216

−3519.2 −3530.4 −3541.6 −3552.8 −3564.1 −3575.3 −3598 −3632.1 −3747.3 −3983.9 −4226.4 −4468.2 −4584.8

2182.4 2181.5 2180.5 2179.5 2178.5 2177.5 2175.3 2172 2159.4 2124.8 2068.5 1965.6 1877.8

56.7 56.6 56.4 56.3 56.1 56.0 55.7 55.3 53.9 51.3 48.9 46.7 45.6

31.418 31.380 31.342 31.303 31.264 31.225 31.146 31.024 30.594 29.578 28.223 26.192 24.692

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.97 0.96 0.95 0.94 0.93 0.92 0.90 0.87 0.78 0.58 0.39 0.19 0.10

0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.02 0.01 0.01 0.00

0.0000 0.0038 0.0077 0.0116 0.0155 0.0195 0.0275 0.0399 0.0842 0.1912 0.3389 0.5694 0.7462

0.9921 0.9883 0.9844 0.9806 0.9767 0.9727 0.9647 0.9524 0.9084 0.8022 0.6557 0.4270 0.2516

0.0080 0.0079 0.0079 0.0079 0.0079 0.0078 0.0078 0.0077 0.0074 0.0066 0.0055 0.0036 0.0022

77.555 77.610 77.666 77.722 77.779 77.836 77.950 78.125 78.734 80.104 81.786 84.018 85.481

2.691 2.695 2.700 2.704 2.708 2.712 2.721 2.735 2.781 2.884 2.992 3.092 3.128

3.857 3.854 3.851 3.848 3.844 3.841 3.835 3.826 3.798 3.750 3.702 3.636 3.585

1.433 1.430 1.426 1.423 1.420 1.416 1.409 1.399 1.366 1.300 1.237 1.176 1.146

— — — — — — — — — — — 70.593 83.23

103.74 104.41 105.09 105.78 106.49 107.2 108.62 110.85 118.85 138.44 165.73 207.94 239.66

21.375 21.546 21.722 21.898 22.079 22.261 22.629 23.204 25.294 30.551 38.159 50.496 60.153

— — — — — — — — — — — 1.022 1.009

0.999 0.999 0.998 0.998 0.998 0.998 0.998 0.997 0.998 1.009 1.033 1.072 1.097

1.257 1.256 1.254 1.253 1.252 1.251 1.249 1.245 1.235 1.219 1.212 1.215 1.219

−3448.5 −3460.5 −3472.5 −3484.6 −3496.6 −3508.7 −3533 −3569.5 −3692.7 −3944.8 −4202.1 −4457.7 −4580.2

2167.6 2166.9 2166.1 2165.3 2164.5 2163.7 2162.1 2159.4 2149.1 2119.2 2068.1 1969.5 1881.9

57.0 56.9 56.7 56.6 56.5 56.3 56.0 55.6 54.2 51.5 49.0 46.7 45.7

31.465 31.429 31.392 31.355 31.318 31.280 31.203 31.086 30.670 29.681 28.347 26.313 24.780

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.00 0.99 0.98 0.97 0.96 0.95 0.93 0.90 0.80 0.60 0.40 0.20 0.10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.0000 0.0037 0.0075 0.0113 0.0151 0.0190 0.0269 0.0390 0.0824 0.1877 0.3338 0.5639 0.7420

1.0000 0.9963 0.9925 0.9887 0.9849 0.9810 0.9731 0.9610 0.9176 0.8123 0.6662 0.4361 0.2580

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

77.355 77.411 77.468 77.525 77.582 77.640 77.756 77.933 78.550 79.941 81.650 83.928 85.426

2.695 2.699 2.703 2.708 2.712 2.716 2.725 2.738 2.784 2.885 2.994 3.094 3.130

— — — — — — — — — — — — —

— — — — — — — — — — — — —

— — — — — — — — — — — 69.867 82.725

101.33 102 102.68 103.37 104.06 104.77 106.2 108.41 116.38 135.99 163.38 206.1 238.41

20.762 20.932 21.106 21.282 21.459 21.64 22.006 22.574 24.647 29.883 37.493 49.945 59.766

— — — — — — — — — — — 1.022 1.010

1.000 1.000 0.999 0.999 0.999 0.999 0.998 0.998 0.999 1.009 1.033 1.072 1.097

— — — — — — — — — — — — —

−3418.2 −3430.5 −3442.9 −3455.3 −3467.7 −3480.2 −3505.1 −3542.7 −3669.3 −3927.9 −4191.7 −4453.2 −4578.2

2161.5 2160.8 2160.1 2159.4 2158.7 2158 2156.5 2154.1 2144.7 2116.8 2067.7 1971 1883.6

57.2 57.0 56.9 56.7 56.6 56.4 56.2 55.7 54.3 51.6 49.0 46.7 45.7

31.485 31.449 31.413 31.377 31.340 31.303 31.227 31.112 30.701 29.723 28.398 26.364 24.817

0.00 0.00

0.00 0.01

0.00 0.00

1.00 0.99

0.0000 0.0138

0.0000 0.0000

1.0000 0.9862

105.910 105.850

— —

1.120 1.117

— —

1.000 1.001

−3392.4 −3397.5

2857 2843.2

56.8 56.7

33.744 33.638

Pressure, 405.3 kPa (4 atm) — —

1.392 1.388

503.85 501.72

1146.8 1142.6

405.33 403.52

2-226

TABLE 2-191

Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Concluded)

Liquid mole fraction N2/(N2 +O2)

Ar

N2

Vapor mole fraction O2

Ar

N2

O2

p0 (kPa)

Relative volatility Temperature K

N2/Ar

N2/O2

Ar/O2

Ar

Activity coefficent

Enthalpy kJ/kmol

Isobaric specific heat, kJ/(kmolK)

N2

O2

Ar

N2

O2

Liquid

Vapor

Liquid

Vapor

Pressure, 405.3 kPa (4 atm) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.98 0.97 0.96 0.95 0.93 0.90 0.80 0.60 0.40 0.20 0.10

0.0275 0.0410 0.0543 0.0675 0.0933 0.1310 0.2484 0.4556 0.6412 0.8190 0.9084

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.9725 0.9590 0.9457 0.9326 0.9067 0.8690 0.7516 0.5445 0.3588 0.1810 0.0916

105.790 105.740 105.680 105.630 105.520 105.370 104.900 104.150 103.590 103.190 103.030

— — — — — — — — — — —

— — — — — — — — — — —

1.385 1.381 1.378 1.374 1.367 1.357 1.322 1.255 1.191 1.131 1.102

499.6 497.83 495.72 493.96 490.12 484.91 468.86 444.04 426.16 413.71 408.8

1138.3 1134.8 1130.5 1127 1119.3 1108.8 1076.5 1026.3 989.9 964.49 954.46

401.72 400.22 398.43 396.94 393.68 389.27 375.67 354.7 339.63 329.15 325.03

1.115 1.112 1.110 1.107 1.102 1.095 1.074 1.040 1.016 1.003 1.001

— — — — — — — — — — —

1.001 1.001 1.002 1.002 1.004 1.005 1.014 1.037 1.071 1.115 1.142

−3402.6 −3407.8 −3412.9 −3418 −3428.3 −3443.6 −3495.1 −3601.7 −3716.7 −3842.6 −3910.1

2829.5 2816.1 2802.8 2789.7 2764 2726.7 2611.3 2411.9 2238.3 2076.7 1997.2

56.5 56.4 56.3 56.2 56.0 55.7 54.6 52.5 50.5 48.6 47.6

33.534 33.431 33.329 33.229 33.031 32.744 31.846 30.249 28.788 27.354 26.620

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09 0.08 0.06 0.04 0.02 0.01

0.90 0.89 0.88 0.87 0.86 0.86 0.84 0.81 0.72 0.54 0.36 0.18 0.09

0.0000 0.0117 0.0233 0.0348 0.0462 0.0575 0.0799 0.1129 0.2186 0.4159 0.6047 0.7958 0.8955

0.2399 0.2369 0.2339 0.2309 0.2280 0.2251 0.2193 0.2109 0.1841 0.1352 0.0899 0.0456 0.0232

0.7601 0.7515 0.7429 0.7343 0.7259 0.7174 0.7008 0.6762 0.5973 0.4489 0.3054 0.1586 0.0813

103.430 103.400 103.380 103.360 103.340 103.320 103.280 103.220 103.060 102.840 102.740 102.760 102.820

2.045 2.048 2.051 2.054 2.057 2.060 2.066 2.075 2.105 2.166 2.229 2.294 2.328

2.840 2.837 2.833 2.830 2.827 2.823 2.817 2.807 2.774 2.709 2.648 2.591 2.563

1.389 1.385 1.381 1.378 1.374 1.370 1.363 1.352 1.317 1.251 1.188 1.129 1.101

421.14 420.21 419.59 418.97 418.34 417.72 416.49 414.63 409.72 403.04 400.02 400.62 402.43

979.68 977.77 976.5 975.23 973.97 972.7 970.17 966.38 956.33 942.64 936.47 937.7 941.41

335.41 334.62 334.1 333.57 333.05 332.53 331.49 329.93 325.8 320.19 317.66 318.16 319.68

1.129 1.127 1.124 1.121 1.118 1.116 1.111 1.104 1.081 1.046 1.021 1.006 1.002

0.992 0.992 0.990 0.989 0.988 0.987 0.985 0.983 0.975 0.968 0.972 0.986 0.997

1.021 1.022 1.022 1.022 1.022 1.023 1.024 1.026 1.032 1.052 1.082 1.122 1.146

−3351.1 −3356.1 −3361.1 −3366.1 −3371.1 −3376.2 −3386.3 −3401.7 −3454.5 −3568 −3693.2 −3830.8 −3904.2

2779.8 2769.2 2758.7 2748.2 2737.9 2727.6 2707.4 2677.6 2583 2409.5 2247.1 2086.1 2003.1

56.8 56.7 56.5 56.4 56.3 56.2 56.0 55.7 54.6 52.6 50.6 48.6 47.6

33.931 33.841 33.752 33.663 33.575 33.488 33.315 33.059 32.238 30.684 29.163 27.585 26.748

0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.18 0.16 0.12 0.08 0.04 0.02

0.80 0.79 0.78 0.78 0.77 0.76 0.74 0.72 0.64 0.48 0.32 0.16 0.08

0.0000 0.0101 0.0201 0.0301 0.0400 0.0499 0.0696 0.0989 0.1948 0.3822 0.5719 0.7738 0.8829

0.4159 0.4114 0.4069 0.4025 0.3981 0.3937 0.3849 0.3720 0.3300 0.2496 0.1706 0.0889 0.0457

0.5841 0.5785 0.5730 0.5674 0.5619 0.5564 0.5455 0.5291 0.4753 0.3682 0.2575 0.1373 0.0713

101.340 101.340 101.340 101.350 101.350 101.350 101.360 101.380 101.440 101.640 101.940 102.350 102.610

2.061 2.064 2.067 2.070 2.072 2.075 2.081 2.089 2.118 2.176 2.237 2.299 2.330

2.848 2.845 2.841 2.837 2.834 2.830 2.823 2.812 2.777 2.711 2.649 2.592 2.564

1.382 1.378 1.375 1.371 1.367 1.364 1.356 1.346 1.311 1.246 1.185 1.127 1.100

359.58 359.58 359.58 359.85 359.85 359.85 360.13 360.69 362.36 367.98 376.52 388.44 396.13

853.02 853.02 853.02 853.59 853.59 853.59 854.17 855.32 858.79 870.43 888.1 912.66 928.48

283.81 283.81 283.81 284.04 284.04 284.04 284.27 284.74 286.13 290.82 297.97 307.94 314.4

1.138 1.135 1.132 1.129 1.127 1.124 1.119 1.111 1.089 1.052 1.026 1.009 1.004

0.988 0.987 0.986 0.985 0.984 0.984 0.982 0.979 0.973 0.968 0.973 0.987 0.998

1.043 1.043 1.044 1.043 1.044 1.045 1.045 1.046 1.052 1.069 1.095 1.129 1.150

−3293.1 −3298.4 −3303.7 −3309.1 −3314.5 −3319.9 −3330.8 −3347.5 −3405.2 −3530.2 −3668.1 −3818.6 −3898.3

2714.9 2706.6 2698.3 2690.1 2681.9 2673.7 2657.6 2633.6 2555.4 2404.5 2253.6 2094.6 2008.8

56.9 56.8 56.6 56.5 56.4 56.3 56.1 55.8 54.7 52.6 50.6 48.6 47.6

34.186 34.106 34.026 33.946 33.867 33.787 33.630 33.395 32.622 31.093 29.517 27.809 26.873

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.40 0.40 0.39 0.39 0.38 0.38 0.37 0.36 0.32 0.24 0.16 0.08 0.04

0.60 0.59 0.59 0.58 0.58 0.57 0.56 0.54 0.48 0.36 0.24 0.12 0.06

0.0000 0.0079 0.0157 0.0236 0.0315 0.0394 0.0552 0.0790 0.1595 0.3285 0.5155 0.7330 0.8588

0.6538 0.6484 0.6429 0.6375 0.6321 0.6267 0.6159 0.5996 0.5449 0.4320 0.3092 0.1691 0.0891

0.3462 0.3438 0.3414 0.3389 0.3364 0.3340 0.3290 0.3214 0.2956 0.2396 0.1752 0.0979 0.0521

97.997 98.032 98.067 98.101 98.136 98.172 98.242 98.350 98.721 99.531 100.460 101.570 102.200

2.082 2.084 2.087 2.090 2.092 2.095 2.100 2.108 2.136 2.192 2.249 2.307 2.335

2.832 2.829 2.825 2.822 2.818 2.815 2.808 2.798 2.765 2.704 2.647 2.592 2.564

1.360 1.357 1.354 1.350 1.347 1.344 1.337 1.327 1.295 1.234 1.177 1.124 1.098

275.25 276.05 276.85 277.63 278.43 279.26 280.88 283.38 292.12 311.88 335.75 366.00 384.05

675.50 677.21 678.92 680.58 682.30 684.07 687.51 692.86 711.44 753.27 803.38 866.34 903.62

213.91 214.57 215.23 215.87 216.53 217.21 218.54 220.61 227.81 244.15 263.96 289.17 304.26

1.156 1.153 1.150 1.148 1.145 1.142 1.137 1.130 1.106 1.067 1.037 1.015 1.007

0.981 0.980 0.979 0.978 0.978 0.977 0.976 0.974 0.970 0.968 0.975 0.989 0.999

1.093 1.093 1.093 1.093 1.093 1.093 1.093 1.094 1.096 1.105 1.121 1.143 1.157

−3143.7 −3150.5 −3157.4 −3164.4 −3171.3 −3178.3 −3192.4 −3213.9 −3287.5 −3444.6 −3613.8 −3793.2 −3886.1

2610.3 2605.1 2599.8 2594.5 2589.2 2584.0 2573.4 2557.4 2503.3 2389.2 2261.2 2109.2 2019.4

57.346 57.232 57.118 57.004 56.891 56.778 56.551 56.212 55.091 52.889 50.745 48.673 47.668

34.697 34.629 34.560 34.492 34.423 34.354 34.217 34.009 33.304 31.817 30.162 28.233 27.117

0.60 0.60 0.60 0.60 0.60 0.60

0.00 0.01 0.02 0.03 0.04 0.05

0.60 0.59 0.59 0.58 0.58 0.57

0.40 0.40 0.39 0.39 0.38 0.38

0.0000 0.0064 0.0129 0.0194 0.0259 0.0325

0.8073 0.8019 0.7966 0.7912 0.7857 0.7803

0.1927 0.1916 0.1906 0.1895 0.1883 0.1872

95.398 95.451 95.503 95.556 95.610 95.663

2.095 2.097 2.100 2.102 2.105 2.108

2.793 2.790 2.787 2.784 2.781 2.778

1.333 1.330 1.327 1.324 1.321 1.318

220.63 221.66 222.67 223.70 224.76 225.80

557.21 559.46 561.68 563.94 566.25 568.53

169.24 170.08 170.90 171.74 172.59 173.44

1.180 1.177 1.174 1.172 1.169 1.166

0.979 0.978 0.978 0.977 0.976 0.976

1.154 1.153 1.153 1.152 1.152 1.151

−2969.3 −2978.2 −2987.2 −2996.2 −3005.3 −3014.3

2527.7 2524.3 2520.9 2517.5 2514.0 2510.5

58.120 57.994 57.869 57.743 57.618 57.493

35.181 35.120 35.057 34.995 34.932 34.869

0.60 0.60 0.60 0.60 0.60 0.60 0.60

0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.56 0.54 0.48 0.36 0.24 0.12 0.06

0.37 0.36 0.32 0.24 0.16 0.08 0.04

0.0457 0.0657 0.1350 0.2878 0.4690 0.6962 0.8358

0.7694 0.7528 0.6957 0.5707 0.4239 0.2417 0.1303

0.1850 0.1815 0.1694 0.1415 0.1070 0.0622 0.0339

95.771 95.934 96.496 97.721 99.133 100.820 101.810

2.113 2.121 2.148 2.203 2.260 2.314 2.339

2.773 2.765 2.738 2.688 2.641 2.592 2.565

1.312 1.304 1.275 1.220 1.169 1.120 1.096

227.93 231.17 242.61 269.02 302.05 345.35 372.80

573.19 580.27 605.18 662.15 732.50 823.43 880.41

175.17 177.81 187.15 208.79 236.01 271.95 294.86

1.160 1.152 1.127 1.084 1.049 1.021 1.010

0.975 0.974 0.971 0.970 0.977 0.991 1.000

1.150 1.149 1.146 1.145 1.149 1.158 1.164

−3032.6 −3060.2 −3154.0 −3349.9 −3554.9 −3766.5 −3873.6

2503.5 2492.8 2455.3 2369.9 2263.2 2121.1 2029.2

57.243 56.870 55.639 53.238 50.933 48.741 47.694

34.742 34.550 33.887 32.433 30.730 28.628 27.351

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.80 0.79 0.78 0.78 0.77 0.76 0.74 0.72 0.64 0.48 0.32 0.16 0.08

0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.18 0.16 0.12 0.08 0.04 0.02

0.0000 0.0055 0.0110 0.0165 0.0221 0.0277 0.0390 0.0563 0.1171 0.2562 0.4302 0.6627 0.8139

0.9165 0.9114 0.9063 0.9012 0.8961 0.8909 0.8804 0.8643 0.8082 0.6801 0.5204 0.3076 0.1696

0.0835 0.0831 0.0827 0.0823 0.0819 0.0815 0.0806 0.0793 0.0747 0.0637 0.0494 0.0297 0.0165

93.251 93.315 93.380 93.445 93.510 93.576 93.708 93.910 94.605 96.135 97.925 100.110 101.420

2.106 2.108 2.111 2.113 2.116 2.118 2.123 2.131 2.157 2.212 2.268 2.321 2.343

2.744 2.742 2.740 2.738 2.736 2.734 2.729 2.723 2.704 2.667 2.631 2.590 2.565

1.303 1.301 1.298 1.296 1.293 1.291 1.286 1.278 1.254 1.206 1.160 1.116 1.095

182.01 183.08 184.17 185.27 186.37 187.50 189.76 193.27 205.71 235.21 273.62 326.60 361.80

471.46 473.86 476.32 478.78 481.25 483.77 488.84 496.68 524.32 589.09 672.00 784.23 857.63

138.01 138.88 139.75 140.64 141.53 142.43 144.26 147.09 157.14 181.11 212.57 256.36 285.67

1.212 1.209 1.205 1.202 1.199 1.196 1.190 1.181 1.153 1.104 1.062 1.028 1.013

0.985 0.984 0.984 0.983 0.983 0.982 0.981 0.980 0.976 0.975 0.981 0.994 1.002

1.226 1.225 1.224 1.223 1.221 1.220 1.218 1.215 1.205 1.189 1.178 1.174 1.172

−2783.0 −2794.3 −2805.5 −2816.8 −2828.1 −2839.4 −2862.1 −2896.3 −3011.9 −3249.2 −3492.7 −3738.8 −3860.8

2458.2 2456.1 2454.0 2451.8 2449.7 2447.5 2443.0 2436.0 2410.8 2348.3 2261.3 2130.7 2038.1

59.157 59.014 58.871 58.729 58.586 58.444 58.161 57.739 56.353 53.683 51.165 48.820 47.724

35.652 35.593 35.533 35.474 35.414 35.354 35.232 35.047 34.405 32.970 31.234 28.996 27.577

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.90 0.89 0.88 0.87 0.86 0.86 0.84 0.81 0.72 0.54 0.36 0.18 0.09

0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.09 0.08 0.06 0.04 0.02 0.01

0.0000 0.0051 0.0102 0.0153 0.0205 0.0258 0.0364 0.0526 0.1099 0.2429 0.4131 0.6471 0.8034

0.9607 0.9558 0.9509 0.9459 0.9409 0.9359 0.9256 0.9100 0.8548 0.7267 0.5631 0.3384 0.1884

0.0393 0.0391 0.0389 0.0387 0.0385 0.0384 0.0380 0.0374 0.0354 0.0304 0.0238 0.0145 0.0082

92.292 92.361 92.431 92.501 92.571 92.642 92.784 93.001 93.751 95.408 97.360 99.771 101.230

2.111 2.114 2.116 2.119 2.121 2.123 2.128 2.136 2.161 2.216 2.272 2.324 2.346

2.719 2.717 2.716 2.714 2.712 2.710 2.707 2.702 2.686 2.655 2.625 2.589 2.565

1.288 1.286 1.283 1.281 1.279 1.276 1.272 1.265 1.242 1.198 1.156 1.114 1.094

166.50 167.59 168.69 169.79 170.91 172.04 174.33 177.86 190.51 220.83 261.03 317.92 356.53

436.43 438.89 441.39 443.90 446.42 448.99 454.16 462.14 490.50 557.64 644.98 765.99 846.69

125.58 126.45 127.33 128.22 129.11 130.02 131.85 134.69 144.86 169.40 202.23 249.16 281.27

1.231 1.227 1.224 1.221 1.218 1.214 1.208 1.199 1.169 1.115 1.069 1.031 1.015

0.991 0.991 0.990 0.989 0.989 0.988 0.987 0.985 0.981 0.978 0.983 0.995 1.002

1.267 1.265 1.264 1.262 1.261 1.259 1.256 1.251 1.237 1.212 1.194 1.181 1.176

−2687.7 −2700.1 −2712.5 −2724.9 −2737.3 −2749.8 −2774.7 −2812.4 −2939.0 −3197.3 −3460.7 −3724.5 −3854.3

2426.8 2425.2 2423.6 2422.0 2420.3 2418.6 2415.2 2409.7 2389.6 2337.1 2259.1 2134.7 2042.2

59.774 59.619 59.466 59.312 59.159 59.007 58.703 58.250 56.769 53.939 51.297 48.865 47.740

35.885 35.827 35.768 35.709 35.650 35.590 35.469 35.286 34.648 33.216 31.467 29.170 27.687

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

0.97 0.96 0.95 0.94 0.93 0.92 0.90 0.87 0.78 0.58 0.39 0.19 0.10

0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.02 0.01 0.01 0.00

0.0000 0.0048 0.0097 0.0146 0.0196 0.0246 0.0347 0.0503 0.1053 0.2344 0.4019 0.6366 0.7962

0.9887 0.9839 0.9791 0.9742 0.9693 0.9643 0.9543 0.9389 0.8845 0.7567 0.5911 0.3591 0.2014

0.0113 0.0113 0.0112 0.0112 0.0111 0.0111 0.0110 0.0108 0.0102 0.0088 0.0070 0.0043 0.0024

91.656 91.728 91.801 91.874 91.948 92.022 92.171 92.397 93.182 94.921 96.979 99.538 101.100

2.116 2.118 2.120 2.123 2.125 2.128 2.132 2.140 2.165 2.219 2.274 2.326 2.347

2.702 2.700 2.699 2.697 2.696 2.694 2.691 2.687 2.673 2.647 2.621 2.588 2.565

1.277 1.275 1.273 1.271 1.268 1.266 1.262 1.256 1.235 1.193 1.152 1.112 1.093

156.79 157.87 158.97 160.07 161.20 162.33 164.62 168.15 180.86 211.56 252.78 312.06 352.96

414.28 416.75 419.26 421.78 424.35 426.93 432.16 440.17 468.87 537.25 627.20 753.64 839.27

117.83 118.68 119.56 120.44 121.34 122.24 124.08 126.90 137.09 161.88 195.47 244.29 278.29

1.245 1.242 1.238 1.235 1.231 1.228 1.221 1.212 1.180 1.123 1.074 1.034 1.016

0.997 0.996 0.996 0.995 0.994 0.993 0.992 0.990 0.985 0.981 0.984 0.995 1.003

1.298 1.296 1.294 1.292 1.290 1.288 1.284 1.279 1.261 1.230 1.205 1.187 1.179

−2620.4 −2633.6 −2646.8 −2660.0 −2673.3 −2686.5 −2713.1 −2753.1 −2887.4 −3160.7 −3437.9 −3714.4 −3849.7

2405.9 2404.6 2403.3 2402.0 2400.6 2399.3 2396.5 2392.0 2375.1 2329.1 2257.3 2137.3 2045.0

60.245 60.082 59.920 59.758 59.596 59.436 59.115 58.639 57.084 54.131 51.395 48.897 47.752

36.049 35.990 35.932 35.873 35.813 35.754 35.634 35.450 34.814 33.381 31.622 29.289 27.763

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.00 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90

1.00 0.99 0.98 0.97 0.96 0.95 0.93 0.90 0.80 0.60 0.40 0.20 0.10

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.0000 0.0047 0.0095 0.0143 0.0192 0.0241 0.0341 0.0493 0.1035 0.2310 0.3973 0.6322 0.7931

1.0000 0.9953 0.9905 0.9857 0.9808 0.9759 0.9659 0.9507 0.8965 0.7690 0.6027 0.3678 0.2069

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

91.391 91.465 91.539 91.613 91.688 91.763 91.915 92.146 92.945 94.718 96.818 99.439 101.040

2.118 2.120 2.122 2.125 2.127 2.129 2.134 2.141 2.166 2.220 2.275 2.327 2.347

— — — — — — — — — — — — —

— — — — — — — — — — — — —

152.87 153.96 155.05 156.15 157.27 158.39 160.69 164.24 176.95 207.79 249.36 309.59 351.32

405.30 407.79 410.30 412.81 415.38 417.95 423.20 431.27 460.07 528.92 619.80 748.43 835.85

114.70 115.57 116.44 117.31 118.21 119.10 120.94 123.77 133.95 158.82 192.67 242.25 276.92

1.252 1.248 1.245 1.241 1.238 1.234 1.228 1.217 1.185 1.126 1.076 1.035 1.017

1.000 0.999 0.998 0.998 0.997 0.996 0.995 0.993 0.987 0.982 0.985 0.996 1.003

— — — — — — — — — — — — —

−2591.4 −2605.0 −2618.5 −2632.1 −2645.7 −2659.3 −2686.6 −2727.6 −2865.3 −3144.8 −3428.1 −3710.1 −3847.7

2397.1 2396.0 2394.8 2393.6 2392.4 2391.2 2388.6 2384.6 2369.0 2325.6 2256.4 2138.3 2046.2

60.458 60.291 60.124 59.959 59.793 59.629 59.301 58.814 57.225 54.217 51.438 48.912 47.757

36.119 36.060 36.002 35.943 35.883 35.824 35.704 35.520 34.883 33.450 31.687 29.339 27.795

2-227

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source is Lemmon, E. W., Jacobsen, R. T., Penoncello, S. G., and Friend, D. G., “Thermodynamic Properties of Air and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at Pressures to 2000 MPa,” J. Phys. Chem. Ref. Data 29(3):331–385, 2000. p° is the vapor pressure.

2-228

PHYSICAL AND CHEMICAL DATA

TABLE 2-192

Thermodynamic Properties of the International Standard Atmosphere* T, K

P, bar

ρ, kg/m3

g, m/s2

M

a, m/s

µ, Pa⋅s

k, W/(m⋅K)

λ, m

0 1,000 2,000 3,000 4,000

288.15 281.65 275.15 268.66 262.17

1.01325 0.89876 0.79501 0.70121 0.61660

1.2250 1.1117 1.0066 0.90925 0.81935

9.80665 9.8036 9.8005 9.7974 9.7943

28.964 28.964 28.964 28.964 28.964

340.29 336.43 332.53 328.58 324.59

1.79.−5 1.76.−5 1.73.−5 1.69.−5 1.66.−5

2.54.−5 2.49.−5 2.43.−5 2.38.−5 2.33.−5

6.63.−8 7.31.−8 8.07.−8 8.94.−8 9.92.−8

0 1,000 2,999 2,999 3,997

5,000 6,000 7,000 8,000 9,000

255.68 249.19 242.70 236.22 229.73

0.54048 0.47217 0.41105 0.35651 0.30800

0.73643 0.66011 0.59002 0.52579 0.46706

9.7912 9.7882 9.7851 9.7820 9.7789

28.964 28.964 28.964 28.964 28.964

320.55 316.45 312.31 308.11 303.85

1.63.−5 1.59.−5 1.56.−5 1.53.−5 1.49.−5

2.28.−5 2.22.−5 2.17.−5 2.12.−5 2.06.−5

1.10.−7 1.23.−7 1.38.−7 1.55.−7 1.74.−7

4,996 5,994 6,992 7,990 8,987

10,000 15,000 20,000 25,000 30,000

223.25 216.65 216.65 221.55 226.51

0.26499 0.12111 0.05529 0.02549 0.01197

0.41351 0.19476 0.08891 0.04008 0.01841

9.7759 9.7605 9.7452 9.7300 9.7147

28.964 28.964 28.964 28.964 28.964

299.53 295.07 295.07 298.39 301.71

1.46.−5 1.42.−5 1.42.−5 1.45.−5 1.48.−5

2.01.−5 1.95.−5 1.95.−5 1.99.−5 2.04.−5

1.97.−7 4.17.−7 9.14.−7 2.03.−6 4.42.−6

9,984 14,965 19,937 24,902 29,859

40,000 50,000 60,000 70,000 80,000

250.35 270.65 247.02 219.59 198.64

2.87.−3 8.00.−4 2.20.−4 5.22.−5 1.05.−5

4.00.−3 1.03.−3 3.10.−4 8.28.−5 1.85.−5

9.6844 9.6542 9.6241 9.5942 9.5644

28.964 28.964 28.964 28.964 28.964

317.19 329.80 315.07 297.06 282.54

1.60.−5 1.70.−5 1.58.−5 1.44.−5 1.32.−5

2.23.−5 2.40.−5 2.21.−5 1.98.−5 1.80.−5

2.03.−5 7.91.−5 2.62.−4 9.81.−4 4.40.−3

39,750 49,610 59,439 69,238 79,006

90,000 100,000 150,000 200,000 250,000

186.87 195.08 634.39 854.56 941.33

1.84.−6 3.20.−7 4.54.−9 8.47.−10 2.48.−10

3.43.−6 5.60.−7 2.08.−9 2.54.−10 6.07.−11

9.5348 9.5052 9.3597 9.2175 9.0785

28.95 28.40 24.10 21.30 19.19

2.37.−2 0.142 33 240 890

88,744 98,451 146,542 193,899 240,540

300,000 400,000 500,000 600,000 800,000

976.01 995.83 999.24 999.85 999.99

8.77.−11 1.45.−11 3.02.−12 8.21.−13 1.70.−13

1.92.−11 2.80.−12 5.22.−13 1.14.−13 1.14.−14

8.9427 8.6799 8.4286 8.1880 7.7368

17.73 15.98 14.33 11.51 5.54

2600 1.6.+4 7.7.+4 2.8.+5 1.4.+6

286,480 376,320 463,540 548,252 710,574

1,000,000

1000.00

7.51.−14

3.56.−15

7.3218

3.94

3.1.+6

864,071

Z, m

H, m

*Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and the U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, µ = viscosity, k = thermal conductivity, λ = mean free path, ρ = density, and H = geopotential altitude. The notation 1.79.−5 signifies 1.79 × 10−5.

TABLE 2-193

Thermodynamic Properties of Benzene

Temperature K

Pressure MPa

278.7 280 295 310 325 340 355 370 385 400 415 430 445 460 475 490 505 520 535 550 562.05

0.0047988 0.0051472 0.010954 0.021374 0.038777 0.066128 0.10696 0.16531 0.24565 0.35284 0.49204 0.66869 0.88851 1.1575 1.4820 1.8689 2.3256 2.8608 3.4853 4.2145 4.9012

278.7 280 295 310 325 340 355 370 385 400 415 430 445 460 475 490 505 520 535 550 562.05

0.0047988 0.0051472 0.010954 0.021374 0.038777 0.066128 0.10696 0.16531 0.24565 0.35284 0.49204 0.66869 0.88851 1.1575 1.4820 1.8689 2.3256 2.8608 3.4853 4.2145 4.9012

300.00 350.00 352.81

0.10000 0.10000 0.10000

352.81 400.00 450.00 500.00 550.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.45084 −0.45037 −0.43580 −0.41192 −0.38479 −0.35639 −0.32675 −0.29499 −0.25976 −0.21937 −0.17166 −0.11372 −0.04137 0.05173 0.17604 0.35018 0.61096 1.0427 1.8878 4.2186 15.682

Saturated Properties 11.441 11.424 11.227 11.024 10.816 10.605 10.389 10.170 9.9444 9.7123 9.4716 9.2204 8.9560 8.6747 8.3719 8.0406 7.6701 7.2419 6.7170 5.9774 3.9561 0.0020778 0.0022187 0.0044939 0.0083797 0.014587 0.023968 0.037514 0.056356 0.081780 0.11526 0.15852 0.21364 0.28321 0.37058 0.48038 0.61931 0.79794 1.0349 1.3697 1.9242 3.9561

0.087404 0.087532 0.089073 0.090715 0.092455 0.094297 0.096251 0.098331 0.10056 0.10296 0.10558 0.10845 0.11166 0.11528 0.11945 0.12437 0.13038 0.13809 0.14888 0.16730 0.25278 481.28 450.72 222.52 119.34 68.556 41.723 26.657 17.744 12.228 8.6760 6.3082 4.6807 3.5310 2.6984 2.0817 1.6147 1.2532 0.96626 0.73008 0.51970 0.25278

−10.278 −10.105 −8.1218 −6.1160 −4.0576 −1.9360 0.25114 2.5029 4.8180 7.1959 9.6372 12.144 14.721 17.372 20.108 22.940 25.887 28.980 32.284 35.991 41.513

−10.278 −10.105 −8.1208 −6.1140 −4.0540 −1.9297 0.26143 2.5192 4.8427 7.2322 9.6892 12.217 14.820 17.506 20.285 23.172 26.190 29.375 32.802 36.696 42.752

−0.03257 −0.03196 −0.02506 −0.01842 −0.01194 −0.00556 0.000737 0.006950 0.013084 0.019144 0.025138 0.031076 0.036970 0.042839 0.048703 0.054589 0.060537 0.066611 0.072938 0.079902 0.090526

0.095741 0.095210 0.092757 0.094410 0.097865 0.10197 0.10619 0.11029 0.11418 0.11788 0.12142 0.12487 0.12827 0.13170 0.13524 0.13896 0.14300 0.14755 0.15298 0.16038

0.13301 0.13274 0.13249 0.13531 0.13934 0.14373 0.14815 0.15251 0.15684 0.16124 0.16581 0.17070 0.17609 0.18223 0.18948 0.19847 0.21044 0.22839 0.26197 0.37221

1369.2 1366.0 1316.2 1251.1 1180.0 1108.1 1037.7 969.46 903.49 839.45 776.89 715.33 654.26 593.15 531.41 468.41 403.32 335.01 261.60 179.15 0

22.307 22.394 23.427 24.522 25.676 26.881 28.135 29.431 30.765 32.132 33.526 34.942 36.370 37.803 39.228 40.625 41.966 43.200 44.221 44.709 41.513

24.617 24.714 25.864 27.073 28.334 29.640 30.986 32.365 33.769 35.194 36.630 38.072 39.508 40.927 42.313 43.642 44.880 45.965 46.766 46.899 42.752

0.092630 0.092396 0.090149 0.088632 0.087716 0.087296 0.087285 0.087613 0.088218 0.089048 0.090056 0.091203 0.092449 0.093754 0.095076 0.096364 0.097548 0.098514 0.099038 0.098452 0.090526

0.067199 0.067633 0.072601 0.077506 0.082363 0.087187 0.091992 0.096792 0.10160 0.10643 0.11129 0.11620 0.12117 0.12624 0.13144 0.13684 0.14252 0.14867 0.15562 0.16433

0.075641 0.076082 0.081164 0.086243 0.091352 0.096528 0.10181 0.10724 0.11287 0.11877 0.12502 0.13176 0.13918 0.14759 0.15753 0.17002 0.18715 0.21415 0.26827 0.45840

182.14 182.47 186.19 189.61 192.67 195.28 197.37 198.86 199.67 199.73 198.94 197.20 194.42 190.45 185.11 178.19 169.36 158.20 144.00 125.56 0

−0.022832 −0.0013533 −0.00017807

0.092996 0.10479 0.10558

0.13320 0.14668 0.14751

0.087264 0.10072 0.11482 0.12870 0.14233 0.15567

0.091290 0.10432 0.11715 0.12898 0.13981 0.14967

0.10103 0.11349 0.12601 0.13766 0.14838 0.15817

69.790 68.760 58.188 49.702 42.885 37.396 32.960 29.366 26.447 24.076 22.158 20.623 19.421 18.522 17.915 17.610 17.642 18.090 19.073 20.568 15.682

Single-Phase Properties 11.161 10.462 10.421

2-229

0.035232 0.030711 0.027110 0.024299 0.022032 0.020161

0.089602 0.095585 0.095958 28.383 32.562 36.887 41.154 45.388 49.600

−7.4606 −0.48554 −0.072586

−7.4516 −0.47598 −0.062991

27.949 32.595 38.153 44.322 51.054 58.300

30.787 35.851 41.842 48.437 55.592 63.260

1296.2 1061.0 1047.8 197.10 210.66 223.76 235.93 247.41 258.31

−0.42861 −0.33684 −0.33119 33.552 20.922 13.727 9.5972 7.0666 5.4319

2-230 TABLE 2-193

Thermodynamic Properties of Benzene (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

300.00 350.00 400.00 450.00 451.58

1.0000 1.0000 1.0000 1.0000 1.0000

11.170 10.476 9.7282 8.8655 8.8350

0.089523 0.095456 0.10279 0.11280 0.11319

−7.4897 −0.52680 7.1526 15.593 15.873

−7.4002 −0.43135 7.2554 15.706 15.987

451.58 500.00 550.00 600.00

1.0000 1.0000 1.0000 1.0000

3.1340 3.7002 4.2144 4.6952

36.999 43.395 50.362 57.755

40.133 47.096 54.577 62.450

300.00 350.00 400.00 450.00 500.00 550.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

11.213 10.537 9.8210 9.0240 8.0418 6.4075 1.5540

0.089185 0.094904 0.10182 0.11082 0.12435 0.15607 0.64348

−7.6160 −0.70378 6.8987 15.189 24.331 35.199 53.945

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.022929 −0.0014714 0.019036 0.038922 0.039545

0.092978 0.10480 0.11785 0.12940 0.12977

0.13308 0.14651 0.16097 0.17801 0.17867

1301.8 1068.0 846.37 634.37 627.51

−0.43004 −0.33971 −0.22423 −0.013693 −0.0036091

0.093016 0.10766 0.12192 0.13561

0.12338 0.13222 0.14175 0.15089

0.14272 0.14607 0.15344 0.16150

192.83 214.43 231.44 245.85

−7.1700 −0.22926 7.4078 15.743 24.953 35.979 57.162

−0.023353 −0.0019818 0.018394 0.038012 0.057399 0.078369 0.11554

0.092913 0.10485 0.11775 0.12877 0.13990 0.15609 0.16178

0.13259 0.14581 0.15956 0.17438 0.19602 0.27120 0.21373

1325.8 1097.9 886.30 692.84 496.64 246.43 176.29

−0.43605 −0.35136 −0.25035 −0.088322 0.27193 2.1959 9.3898

Single-Phase Properties

0.31908 0.27026 0.23728 0.21298

18.991 11.641 7.9458 5.8511

300.00 350.00 400.00 450.00 500.00 550.00 600.00

10.000 10.000 10.000 10.000 10.000 10.000 10.000

11.263 10.608 9.9256 9.1886 8.3418 7.2764 5.7109

0.088783 0.094266 0.10075 0.10883 0.11988 0.13743 0.17510

−7.7671 −0.91178 6.6097 14.760 23.600 33.351 44.567

−6.8793 0.030879 7.6172 15.848 24.799 34.726 46.318

−0.023867 −0.0025897 0.017651 0.037025 0.055872 0.074776 0.094920

0.092869 0.10497 0.11774 0.12832 0.13831 0.14918 0.16240

0.13203 0.14508 0.15816 0.17132 0.18753 0.21149 0.25868

1354.1 1132.3 930.56 752.83 586.00 420.33 257.52

−0.44280 −0.36378 −0.27601 −0.15003 0.073722 0.56941 2.1761

350.00 400.00 450.00 500.00 550.00 600.00

50.000 50.000 50.000 50.000 50.000 50.000

11.061 10.527 10.001 9.4723 8.9322 8.3797

0.090412 0.094991 0.099985 0.10557 0.11196 0.11934

−2.2210 4.9365 12.570 20.624 29.094 37.984

2.2996 9.6861 17.570 25.902 34.691 43.951

−0.0066440 0.013066 0.031627 0.049177 0.065924 0.082032

0.10685 0.11931 0.12871 0.13669 0.14419 0.15154

0.14195 0.15301 0.16219 0.17116 0.18047 0.18989

1337.7 1173.3 1041.8 932.63 839.96 761.25

−0.41928 −0.37184 −0.32474 −0.26918 −0.20274 −0.12687

350.00 400.00 450.00 500.00 550.00 600.00

75.000 75.000 75.000 75.000 75.000 75.000

11.280 10.798 10.331 9.8716 9.4118 8.9496

0.088652 0.092613 0.096793 0.10130 0.10625 0.11174

−2.8344 4.2071 11.699 19.570 27.813 36.435

3.8145 11.153 18.958 27.167 35.782 44.815

−0.0087087 0.010874 0.029250 0.046542 0.062956 0.078670

0.10831 0.12070 0.12977 0.13730 0.14435 0.15124

0.14116 0.15181 0.16021 0.16819 0.17644 0.18489

1431.4 1277.8 1157.1 1058.5 975.53 905.02

−0.43585 −0.39637 −0.36122 −0.32227 −0.27775 −0.22901

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Polt, A., Platzer, B., and Maurer, G., “Parameter der thermischen Zustandsgleichung von Bender fuer 14 mehratomige reine Stoffe,” Chem. Tech. (Leipzig), 44(6):216–224, 1992. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density for benzene are 1% in the vapor phase, 0.3% in the liquid phase up to 400 K (with lower uncertainties at lower temperatures), 1% in the liquid phase between 400 and 500 K, and 2% and rising at temperatures above 500 K. Near the saturation line at temperatures below 350 K, the liquid-phase uncertainty decreases to 0.05%. The uncertainties in vapor pressures are 0.15% at temperatures below 380 K, and 0.5% at higher temperatures. The uncertainties in heat capacities and sound speeds are 2% in the vapor phase and 5% in the liquid phase.

THERMODYNAMIC PROPERTIES TABLE 2-194

2-231

Saturated Bromine* vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

260 280 300 320 340

0.042 0.124 0.310 0.680 1.330

3.106.−4 3.168.−4 3.232.−4 3.311.−4 3.385.−4

3.195 1.169 0.5002 0.2425 0.1309

−147.2 −138.9 −131.6 −124.2 −112.3

51.8 56.2 60.6 64.8 71.1

0.903 0.933 0.956 0.978 1.004

1.669 1.629 1.597 1.570 1.539

0.486 0.479 0.475 0.473 0.471

13.4 11.5 9.3 7.8 6.7

0.131 0.127 0.122 0.118 0.114

360 380 400 420 440

2.384 4.010 6.390 9.730 14.25

3.464.−4 3.550.−4 3.647.−4 3.752.−4 3.885.−4

0.0767 0.0477 0.0311 0.0211 0.0148

−108.6 −100.6 −93.4 −85.8 −77.7

73.1 76.9 80.6 84.0 87.1

1.026 1.048 1.063 1.084 1.103

1.531 1.515 1.501 1.488 1.477

0.470 0.471 0.475 0.480 0.489

5.7 5.0 4.5 4.0 3.7

0.109 0.104 0.099 0.094 0.089

460 480 500 520 540

20.17 27.75 37.21 48.81 62.80

4.023.−4 4.179.−4 4.378.−4 4.623.−4 4.938.−4

0.0107 0.00786 0.00589 0.00445 0.00337

−69.0 −59.7 −49.3 −37.7 −24.0

89.9 92.2 94.0 95.0 94.8

1.122 1.142 1.161 1.183 1.207

1.467 1.457 1.448 1.438 1.428

0.503 0.527 0.595 0.710 0.860

3.3 3.1 2.8 2.6 2.5

0.084 0.079 0.073 0.066 0.059

79.41 98.90 103.4

5.368.−4 6.250.−4 8.475.−4

0.00251 0.00167 0.00085

−7.1 18.8 64.8

92.5 82.5 64.8

1.237 1.280 1.356

1.414 1.390 1.356

1.063 2.31 ∞

2.3 2.2 2.1

0.050 0.035 ∞

T, K

560 580 584.2c

P, bar

*Reproduced or converted from a tabulation by Seshadri, Viswanath, and Kuloor, Ind. J. Technol., 6 (1970): 191–198. c = critical point.

TABLE 2-195

Thermodynamic Properties of Butane

2-232

Temperature K

Pressure MPa

134.9 140 155 170 185 200 215 230 245 260 275 290 305 320 335 350 365 380 395 410 425.13

6.6566E-07 1.6922E-06 0.00001767 0.00011616 0.00054113 0.0019390 0.0056671 0.014106 0.030885 0.060978 0.11065 0.18734 0.29946 0.45624 0.66761 0.94418 1.2974 1.7399 2.2868 2.9578 3.796

134.9 140 155 170 185 200 215 230 245 260 275 290 305 320 335 350 365 380 395 410 425.13

6.6566E-07 1.6922E-06 0.00001767 0.00011616 0.00054113 0.0019390 0.0056671 0.014106 0.030885 0.060978 0.11065 0.18734 0.29946 0.45624 0.66761 0.94418 1.2974 1.7399 2.2868 2.9578 3.796

150.00 200.00 250.00 272.31

0.10000 0.10000 0.10000 0.10000

272.31 300.00 350.00 400.00 450.00 500.00 550.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

150.00 200.00

1.0000 1.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−5.2207 −4.6347 −2.9051 −1.1617 0.59963 2.3840 4.1972 6.0455 7.9351 9.8722 11.863 13.914 16.031 18.222 20.494 22.858 25.328 27.928 30.703 33.782 39.364

−5.2207 −4.6347 −2.9051 −1.1617 0.59967 2.3842 4.1977 6.0467 7.9379 9.8780 11.874 13.933 16.062 18.271 20.568 22.967 25.485 28.150 31.018 34.237 40.331

−0.57143 −0.56883 −0.55989 −0.54850 −0.53413 −0.51637 −0.49482 −0.46909 −0.43868 −0.40288 −0.36055 −0.30993 −0.24825 −0.17103 −0.07074 0.066051 0.26565 0.58704 1.1936 2.7451 15.851

176.56 174.84 169.24 163.00 156.32 149.36 142.26 135.13 128.09 121.19 114.50 108.06 101.92 96.103 90.627 85.498 80.706 76.219 71.976 67.979

2304.3 1887.3 1195.3 844.68 634.79 496.50 399.50 328.38 274.41 232.32 198.71 171.28 148.46 129.09 112.34 97.559 84.220 71.859 59.958 47.596

22.482 22.771 23.648 24.567 25.525 26.519 27.550 28.614 29.709 30.834 31.984 33.156 34.343 35.541 36.739 37.923 39.073 40.147 41.076 41.673 39.364

23.604 23.935 24.937 25.980 27.062 28.179 29.328 30.507 31.711 32.934 34.170 35.414 36.657 37.890 39.100 40.270 41.370 42.353 43.130 43.475 40.331

−3.4846 2.3816 8.5729 11.502

-3.4765 2.3902 8.5822 11.512

−0.014834 0.018891 0.046489 0.057711

0.083837 0.085632 0.091432 0.095158

0.11544 0.11985 0.12863 0.13415

31.776 34.241 39.149 44.681 50.834 57.578 64.875

33.948 36.663 42.009 47.970 54.548 61.714 69.432

0.14010 0.14959 0.16605 0.18195 0.19743 0.21252 0.22723

0.085131 0.091498 0.10399 0.11669 0.12893 0.14043 0.15112

0.095267 0.10103 0.11299 0.12545 0.13755 0.14897 0.15960

−3.4986 2.3598

−3.4180 2.4460

−0.014927 0.018782

0.083887 0.085677

0.11540 0.11977

Saturated Properties 12.645 12.563 12.323 12.082 11.840 11.596 11.349 11.097 10.840 10.575 10.301 10.016 9.7171 9.4002 9.0607 8.6916 8.2822 7.8141 7.2504 6.4885 3.9228 5.935E-07 1.4538E-06 1.3712E-05 8.2203E-05 0.00035209 0.0011686 0.0031862 0.0074503 0.015434 0.029042 0.050611 0.082943 0.12941 0.19420 0.28270 0.40241 0.56471 0.78888 1.1133 1.6409 3.9228

0.079082 0.079598 0.081151 0.082767 0.084458 0.086236 0.088115 0.090113 0.092253 0.094562 0.097075 0.099837 0.10291 0.10638 0.11037 0.11505 0.12074 0.12797 0.13792 0.15412 0.25492 1684900 687860 72930 12165 2840.2 855.70 313.85 134.22 64.792 34.433 19.759 12.056 7.7271 5.1494 3.5374 2.4850 1.7708 1.2676 0.89820 0.60943 0.25492

−0.02703 −0.02277 −0.01103 −0.00030 0.009630 0.018903 0.027645 0.035954 0.043912 0.051585 0.059030 0.066292 0.073412 0.080426 0.087371 0.094284 0.10121 0.10822 0.11543 0.12318 0.13735

0.083783 0.083802 0.083864 0.084118 0.084688 0.085627 0.086949 0.088641 0.090679 0.093026 0.095645 0.098497 0.10155 0.10478 0.10817 0.11173 0.11552 0.11966 0.12445 0.13061

0.11467 0.11492 0.11573 0.11677 0.11813 0.11986 0.12199 0.12455 0.12755 0.13098 0.13488 0.13925 0.14418 0.14976 0.15620 0.16389 0.17360 0.18719 0.21028 0.27186

1826.8 1793.1 1699.3 1610.1 1523.4 1438.2 1354.1 1270.8 1188.2 1106.4 1025.2 944.51 864.09 783.62 702.63 620.37 535.75 447.13 352.08 246.50 0

0.18665 0.18130 0.16859 0.15936 0.15267 0.14788 0.14453 0.14230 0.14094 0.14026 0.14011 0.14037 0.14094 0.14174 0.14269 0.14372 0.14473 0.14559 0.14609 0.14572 0.13735

0.055993 0.057048 0.059941 0.062670 0.065393 0.068227 0.071254 0.074521 0.078047 0.081835 0.085872 0.090138 0.094611 0.099274 0.10412 0.10910 0.11417 0.11962 0.12600 0.13401

0.064308 0.065363 0.068258 0.070995 0.073744 0.076640 0.079787 0.083261 0.087120 0.091404 0.096154 0.10142 0.10730 0.11394 0.12161 0.13080 0.14259 0.16009 0.19220 0.28133

148.87 151.48 158.89 165.94 172.61 178.87 184.64 189.81 194.27 197.86 200.46 201.90 202.04 200.69 197.62 192.54 185.07 174.66 160.51 141.36 0

361.89 315.59 218.48 157.83 117.96 90.683 71.432 57.513 47.254 39.578 33.766 29.330 25.936 23.353 21.423 20.061 19.268 19.058 19.446 20.469 15.851

4.8545 5.0913 5.8353 6.6513 7.5389 8.4974 9.5258 10.624 11.792 13.034 14.357 15.775 17.311 18.999 20.889 23.058 25.626 28.813 33.129 40.394

3.3208 3.4522 3.8372 4.2205 4.6014 4.9792 5.3531 5.7226 6.0877 6.4502 6.8133 7.1826 7.5668 7.9782 8.4346 8.9610 9.5964 10.407 11.532 13.373

Single-Phase Properties 12.404 11.597 10.753 10.351 0.046045 0.041289 0.034963 0.030400 0.026924 0.024176 0.021945 12.411 11.608

0.080622 0.086227 0.092993 0.096608 21.718 24.220 28.601 32.894 37.142 41.363 45.569 0.080576 0.086150

1730.2 1438.8 1161.4 1039.7 200.07 211.28 229.21 245.29 260.14 274.06 287.25 1733.9 1443.9

−0.56316 −0.51649 −0.42760 −0.36868 34.694 24.802 14.912 9.8194 6.9019 5.0916 3.8945 −0.56354 −0.51758

171.22 149.40 125.81 115.68 14.113 16.747 22.115 28.275 35.230 42.982 51.533 171.48 149.79

1370.9 496.90 259.42 204.22 6.7480 7.4522 8.6907 9.8987 11.083 12.248 13.395 1381.3 500.58

250.00 300.00 350.00 352.62 352.62 400.00 450.00 500.00 550.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

10.770 9.8412 8.6953 8.6234 0.42728 0.34181 0.28958 0.25369 0.22679

0.092855 0.10161 0.11500 0.11596 2.3404 2.9256 3.4533 3.9418 4.4094

8.5389 15.271 22.851 23.281 38.127 43.808 50.185 57.063 64.449

8.6318 15.373 22.966 23.397 40.468 46.734 53.638 61.005 68.858

0.046353 0.070896 0.094264 0.095491 0.14390 0.16057 0.17683 0.19234 0.20730

0.091477 0.10054 0.11173 0.11237 0.10997 0.11928 0.13036 0.14132 0.15171

0.12845 0.14209 0.16378 0.16540 0.13262 0.13410 0.14250 0.15221 0.16189

1168.5 899.57 621.53 605.80 191.42 221.49 244.24 262.94 279.30

−0.43053 −0.27749 0.063603 0.095319 19.881 11.545 7.5723 5.3868 4.0325

126.34 104.51 85.560 84.637 23.473 29.230 36.235 44.067 52.702

150.00 200.00 250.00 300.00

5.0000 5.0000 5.0000 5.0000

12.442 11.653 10.838 9.9548

0.080374 0.085815 0.092266 0.10045

−3.5595 2.2656 8.3942 15.037

−3.1577 2.6947 8.8555 15.539

−0.015338 0.018306 0.045767 0.070104

0.084107 0.085873 0.091674 0.10072

0.11525 0.11943 0.12773 0.14034

1749.9 1466.4 1199.3 943.54

−0.56513 −0.52214 −0.44237 −0.31083

172.63 151.52 128.64 107.50

350.00 400.00 450.00 500.00 550.00

5.0000 5.0000 5.0000 5.0000 5.0000

8.9238 7.5216 3.1209 1.7051 1.3443

0.11206 0.13295 0.32042 0.58647 0.74389

22.411 30.873 44.259 54.030 62.232

22.971 31.537 45.861 56.962 65.951

0.092982 0.11581 0.14921 0.17272 0.18986

0.11170 0.12435 0.14234 0.14611 0.15448

0.15798 0.18921 0.35080 0.18483 0.17804

694.05 430.26 147.05 208.77 244.93

−0.067667 0.64313 11.475 6.6929 4.4692

89.620 75.035 55.862 51.142 57.838

106.76 65.426 19.322 14.775 14.928

261.81 157.29 97.696 95.139 9.0627 10.156 11.323 12.475 13.611 1428.2 516.90 272.30 165.96

150.00 200.00 250.00 300.00 350.00 400.00 450.00 500.00 550.00

10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000 10.000

12.480 11.707 10.919 10.082 9.1484 8.0346 6.5680 4.6089 3.2339

0.080129 0.085415 0.091586 0.099189 0.10931 0.12446 0.15225 0.21697 0.30923

−3.6330 2.1534 8.2254 14.776 21.971 29.964 38.976 49.153 58.940

−2.8317 3.0075 9.1412 15.768 23.064 31.209 40.499 51.323 62.032

−0.015840 0.017730 0.045071 0.069202 0.091668 0.11339 0.13524 0.15803 0.17846

0.084373 0.086111 0.091916 0.10096 0.11182 0.12357 0.13606 0.14784 0.15662

0.11508 0.11906 0.12697 0.13868 0.15374 0.17293 0.20093 0.22343 0.20486

1769.4 1493.4 1235.4 993.02 766.31 554.16 360.83 243.50 242.94

−0.56694 −0.52718 −0.45489 −0.34276 −0.16435 0.17567 1.0509 2.9827 3.2916

174.03 153.61 131.40 110.99 94.017 81.081 71.907 65.910 65.615

1488.3 537.30 285.19 176.33 116.79 77.782 48.326 27.075 19.386

150.00 200.00 250.00 300.00 350.00 400.00 450.00 500.00 550.00

30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000 30.000

12.622 11.905 11.197 10.483 9.7548 9.0036 8.2322 7.4563 6.7039

0.079224 0.083996 0.089313 0.095388 0.10251 0.11107 0.12147 0.13411 0.14917

−3.9002 1.7557 7.6534 13.956 20.772 28.147 36.070 44.490 53.337

−1.5235 4.2756 10.333 16.817 23.848 31.479 39.714 48.514 57.812

−0.017740 0.015602 0.042606 0.066223 0.087875 0.10824 0.12762 0.14616 0.16387

0.085345 0.086999 0.092823 0.10187 0.11265 0.12399 0.13519 0.14590 0.15594

0.11453 0.11796 0.12491 0.13486 0.14654 0.15873 0.17053 0.18124 0.19043

1842.9 1591.7 1361.4 1153.4 971.05 816.42 691.11 595.43 527.27

−0.57242 −0.54200 −0.48867 −0.41613 −0.32840 −0.22551 −0.10687 0.023732 0.15444

179.28 161.37 141.44 123.09 108.07 97.051 90.031 86.541 85.903

1748.0 619.22 335.06 214.53 150.45 110.91 83.811 64.099 49.561

150.00 200.00 250.00 300.00 350.00 400.00 450.00 500.00 550.00

65.000 65.000 65.000 65.000 65.000 65.000 65.000 65.000 65.000

12.844 12.196 11.576 10.979 10.399 9.8373 9.2935 8.7710 8.2737

0.077859 0.081993 0.086384 0.091087 0.096160 0.10165 0.10760 0.11401 0.12087

−4.2847 1.2046 6.9077 12.978 19.516 26.564 34.118 42.152 50.632

0.77614 6.5342 12.523 18.898 25.767 33.172 41.112 49.563 58.488

−0.020732 0.012378 0.039078 0.062302 0.083456 0.10322 0.12191 0.13971 0.15671

0.086776 0.088354 0.094214 0.10329 0.11406 0.12536 0.13648 0.14709 0.15706

0.11392 0.11690 0.12318 0.13220 0.14267 0.15351 0.16402 0.17389 0.18299

1957.6 1737.3 1537.0 1359.9 1208.7 1082.5 979.42 896.76 831.73

−0.57756 −0.55581 −0.51712 −0.46758 −0.41511 −0.36425 −0.31694 −0.27399 −0.23593

187.43 173.22 156.34 140.34 126.97 116.89 110.18 106.55 105.59

2300.4 766.46 419.12 275.14 199.77 154.25 123.77 101.78 85.075

2-233

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, NIST Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Buecker, D., and Wagner, W., “Reference Equations of State for the Thermodynamic Properties of Fluid Phase n-Butane and Isobutane,” J. Phys. Chem. Ref. Data 35(2): 929–1019, 2006. The source for viscosity is Vogel, E., Kuechenmeister, C., and Bich, E., “Viscosity for n-Butane in the Fluid Region,” High Temp.—High Pressures 31(2):173–186, 1999. The source for thermal conductivity is Perkins, R. A, Ramires, M. L. V., Nieto de Castro, C. A., and Cusco, L., “Measurement and Correlation of the Thermal Conductivity of Butane from 135 K to 600 K at Pressures to 70 MPa,” J. Chem. Eng. Data 47(5):1263–1271, 2002. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single–phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.02% at temperatures below 340 K and pressures below 12 MPa (both liquid and vapor states), 0.1% at temperatures below 270 K and pressures above 12 MPa, 0.2% between 340 and 515 K at pressures less than 0.6 MPa, and 0.4% elsewhere. In the critical region, deviations in pressure are 0.5%. At temperatures above 500 K, the uncertainties in density increase up to 1%. Uncertainties in heat capacities are typically 1%, rising to 5% in the critical region and at pressures above 30 MPa. Uncertainties in the speed of sound are typically 0.5%, rising to 1% at temperatures below 200 K and to 4% in a large area around the critical point. The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to 3.0% over the rest of the fluid surface. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%.

2-234 TABLE 2-196

Thermodynamic Properties of 1-Butene

Temperature K

Pressure MPa

87.800 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 419.29

5.95E-13 1.2730E-05 9.3335E-05 0.00046966 0.0017787 0.0054125 0.013876 0.031052 0.062301 0.11440 0.19537 0.31425 0.48093 0.70602 1.0009 1.3777 1.8505 2.4356 3.1556 4.0057

87.800 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 419.29

5.95E-13 1.2730E-05 9.3335E-05 0.00046966 0.0017787 0.0054125 0.013876 0.031052 0.062301 0.11440 0.19537 0.31425 0.48093 0.70602 1.0009 1.3777 1.8505 2.4356 3.1556 4.0057

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

0.068578 0.074746 0.076351 0.078017 0.079756 0.081585 0.083519 0.085582 0.087798 0.090204 0.092841 0.095769 0.099067 0.10285 0.10730 0.11269 0.11956 0.12901 0.14444 0.23585

−19.753 −13.108 −11.509 −9.8961 −8.2633 −6.6054 −4.9174 −3.1944 −1.4319 0.37503 2.2316 4.1435 6.1173 8.1613 10.286 12.507 14.848 17.357 20.167 25.194 10.763 13.339 14.060 14.822 15.621 16.456 17.323 18.219 19.139 20.081 21.039 22.009 22.985 23.957 24.912 25.827 26.665 27.351 27.680 25.194

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−19.753 −13.108 −11.509 −9.8961 −8.2632 −6.6050 −4.9162 −3.1918 −1.4264 0.38535 2.2497 4.1736 6.1650 8.2339 10.393 12.662 15.069 17.672 20.622 26.139

−0.12145 −0.064173 −0.054011 −0.044655 −0.035943 −0.027753 −0.019990 −0.012577 −0.0054542 0.0014312 0.0081238 0.014663 0.021087 0.027432 0.033738 0.040056 0.046456 0.053070 0.060243 0.073219

0.080229 0.071962 0.072351 0.073193 0.074385 0.075854 0.077551 0.079443 0.081507 0.083732 0.086107 0.088628 0.091289 0.094094 0.097052 0.10019 0.10358 0.10741 0.11236

0.10913 0.10631 0.10701 0.10813 0.10964 0.11149 0.11367 0.11619 0.11904 0.12227 0.12593 0.13010 0.13492 0.14061 0.14758 0.15666 0.16976 0.19258 0.25490

2086.5 1744.9 1649.6 1555.6 1463.4 1373.4 1285.6 1199.6 1115.4 1032.5 950.65 869.48 788.58 707.49 625.58 541.98 455.28 362.75 258.01 0

−0.55371 −0.55469 −0.54558 −0.53276 −0.51623 −0.49587 −0.47138 −0.44220 −0.40748 −0.36587 −0.31542 −0.25308 −0.17406 −0.070421 0.071832 0.27973 0.61278 1.2326 2.7886 14.832

11.493 14.586 15.432 16.317 17.238 18.192 19.174 20.177 21.195 22.223 23.253 24.279 25.290 26.274 27.212 28.078 28.823 29.354 29.425 26.139

0.23442 0.12046 0.10927 0.10097 0.094833 0.090329 0.087076 0.084791 0.083257 0.082311 0.081820 0.081681 0.081801 0.082098 0.082489 0.082878 0.083133 0.083025 0.081979 0.073219

0.035901 0.046819 0.049589 0.052497 0.055552 0.058761 0.062133 0.065668 0.069362 0.073208 0.077198 0.081324 0.085583 0.089980 0.094533 0.099287 0.10434 0.10992 0.11666

0.044215 0.055138 0.057921 0.060867 0.064008 0.067378 0.071013 0.074946 0.079208 0.083840 0.088901 0.094492 0.10078 0.10808 0.11696 0.12863 0.14599 0.17800 0.27163

126.59 161.79 168.95 175.70 182.01 187.84 193.09 197.63 201.32 204.02 205.56 205.77 204.47 201.43 196.37 188.91 178.55 164.58 145.93 0

Saturated Properties 14.582 13.379 13.097 12.818 12.538 12.257 11.973 11.685 11.390 11.086 10.771 10.442 10.094 9.7227 9.3197 8.8735 8.3641 7.7513 6.9234 4.2400 8.1441E-13 1.0208E-05 6.8053E-05 0.00031411 0.0010999 0.0031172 0.0074985 0.015861 0.030305 0.053404 0.088232 0.13847 0.20868 0.30478 0.43505 0.61215 0.85778 1.2160 1.8085 4.2400

1.2279E+12 97966. 14694. 3183.5 909.21 320.80 133.36 63.049 32.998 18.725 11.334 7.2216 4.7920 3.2811 2.2986 1.6336 1.1658 0.82239 0.55294 0.23585

6453.8 519.37 334.05 224.29 156.42 112.91 84.169 64.687 51.184 41.642 34.789 29.808 26.172 23.538 21.690 20.504 19.927 19.957 20.477 14.832

Single-Phase Properties 100.00 200.00 266.51

0.10000 0.10000 0.10000

266.51 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

100.00 200.00 300.00 344.96

1.0000 1.0000 1.0000 1.0000

344.96 400.00 500.00

1.0000 1.0000 1.0000

100.00 200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000

14.342 12.446 11.158 0.047098 0.041199 0.030377 0.024172 14.347 12.457 10.465 9.3208 0.43467 0.33780 0.25311

0.069727 0.080347 0.089625 21.232 24.273 32.920 41.370 0.069701 0.080273 0.095554 0.10729 2.3006 2.9603 3.9508

−18.432 −7.7161 −0.050006

−18.425 −7.7081 −0.041044

−0.10737 −0.033172 −0.00015346

0.076348 0.074849 0.083200

0.10761 0.11021 0.12149

2041.3 1433.8 1051.7

−0.55898 −0.51000 −0.37627

19.860 22.407 31.321 42.371

21.983 24.834 34.613 46.508

0.082486 0.092558 0.12055 0.14700

0.072300 0.078346 0.099614 0.12081

0.082725 0.087850 0.10831 0.12931

203.49 217.19 251.29 280.25

43.586 27.387 10.192 5.2895

−18.441 −7.7377 4.1009 10.280

−18.371 −7.6574 4.1965 10.388

−0.10745 −0.033280 0.014521 0.033722

0.076343 0.074873 0.088626 0.097044

0.10760 0.11013 0.12974 0.14756

2045.4 1439.7 878.06 625.79

24.909 30.561 41.920

27.210 33.521 45.871

0.082488 0.099463 0.12695

0.094521 0.10210 0.12152

0.11694 0.11588 0.13206

196.39 229.62 269.53

21.694 11.604 5.5503

−0.55906 −0.51117 −0.26069 0.071399

14.371 12.507 10.593 7.7924 1.6432

0.069585 0.079955 0.094404 0.12833 0.60858

−18.478 −7.8313 3.8689 18.376 39.334

−18.131 −7.4315 4.3409 19.018 42.377

−0.10783 −0.033753 0.013736 0.055643 0.10830

0.076326 0.074982 0.088648 0.10867 0.12539

0.10752 0.10983 0.12795 0.17944 0.15726

2063.2 1464.8 924.82 398.89 221.46

−0.55939 −0.51602 −0.29848 0.92138 6.5598

100.00 200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000 10.000

14.400 12.566 10.734 8.4254 4.4457

0.069444 0.079577 0.093162 0.11869 0.22494

−18.524 −7.9428 3.6108 17.389 35.004

−17.830 −7.1471 4.5424 18.576 37.254

−0.10830 −0.034325 0.012845 0.053002 0.094450

0.076316 0.075127 0.088724 0.10763 0.12805

0.10743 0.10949 0.12625 0.15809 0.20133

2084.7 1494.9 976.91 531.56 235.89

−0.55976 −0.52136 −0.33426 0.27589 3.6822

100.00 200.00 300.00 400.00 500.00

30.000 30.000 30.000 30.000 30.000

14.511 12.781 11.176 9.5263 7.8006

0.068913 0.078241 0.089478 0.10497 0.12820

−18.695 −8.3397 2.8019 15.547 30.094

−16.628 −5.9925 5.4862 18.696 33.940

−0.11011 −0.036441 0.0099119 0.047777 0.081710

0.076383 0.075766 0.089276 0.10751 0.12649

0.10716 0.10853 0.12244 0.14231 0.16189

2164.7 1602.5 1142.6 802.89 578.80

−0.56070 −0.53694 −0.41541 −0.21113 0.077488

200.00 300.00 400.00 500.00

70.000 70.000 70.000 70.000

13.133 11.778 10.537 9.3918

0.076142 0.084903 0.094902 0.10648

−8.9642 1.7245 13.811 27.536

−3.6342 7.6677 20.454 34.989

−0.040078 0.0055841 0.042252 0.074613

0.077144 0.090645 0.10876 0.12760

0.10763 0.11961 0.13655 0.15389

1777.6 1376.9 1099.3 914.99

−0.55246 −0.47674 −0.37969 −0.28997

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Ihmels, E. C., “Thermodynamic Properties of the Butenes. Part II. Short Fundamental Equations of State,” Fluid Phase Equilibria 228–229C:173–187, 2005. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of densities calculated by the equation of state (based on a coverage factor of 2) are 0.1% in the liquid phase at temperatures above 270 K (rising to 0.5% in density at temperatures below 200 K), 0.2% at temperatures above the critical temperature and at pressures above 10 MPa, and 0.5% in the vapor phase, including supercritical conditions below 10 MPa. The uncertainty in vapor pressure is 0.25% above 200 K. The uncertainty in heat capacities is 0.5% at saturated liquid conditions, rising to 5% at much higher pressures and at temperatures above 350 K.

2-235

2-236 TABLE 2-197

Thermodynamic Properties of cis-2-Butene

Temperature K

Pressure MPa

134.30 140.00 155.00 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 435.75

2.6365E-07 8.0149E-07 9.6316E-06 7.0502E-05 0.00035720 0.0013683 0.0042205 0.010977 0.024918 0.050686 0.094280 0.16294 0.26496 0.40952 0.60651 0.86653 1.2009 1.6221 2.1441 2.7840 3.5654 4.2360

134.30 140.00 155.00 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 435.75

2.6365E-07 8.0149E-07 9.6316E-06 7.0502E-05 0.00035720 0.0013683 0.0042205 0.010977 0.024918 0.050686 0.094280 0.16294 0.26496 0.40952 0.60651 0.86653 1.2009 1.6221 2.1441 2.7840 3.5654 4.2360

Density mol/dm3

Volume dm3/mol

Int. Energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Saturated Properties 14.084 13.976 13.696 13.419 13.143 12.868 12.591 12.312 12.030 11.742 11.446 11.140 10.822 10.488 10.134 9.7540 9.3387 8.8746 8.3381 7.6790 6.7389 4.2440 2.3611E-07 6.8856E-07 7.4740E-06 4.9890E-05 0.00023239 0.00082449 0.0023717 0.0057914 0.012427 0.024061 0.042902 0.071600 0.11331 0.17184 0.25199 0.36018 0.50560 0.70265 0.97689 1.3825 2.0863 4.2440

0.071002 0.071549 0.073014 0.074523 0.076087 0.077715 0.079421 0.081218 0.083127 0.085168 0.087369 0.089765 0.092403 0.095344 0.098675 0.10252 0.10708 0.11268 0.11993 0.13023 0.14839 0.23563 4235300. 1452300. 133800. 20044. 4303.0 1212.9 421.65 172.67 80.467 41.562 23.309 13.966 8.8256 5.8195 3.9684 2.7764 1.9779 1.4232 1.0237 0.72331 0.47931 0.23563

−16.205 −15.556 −13.866 −12.194 −10.529 −8.8626 −7.1860 −5.4919 −3.7731 −2.0229 −0.23513 1.5965 3.4781 5.4164 7.4188 9.4943 11.655 13.918 16.313 18.901 21.868 26.307

−16.205 −15.556 −13.866 −12.194 −10.529 −8.8625 −7.1857 −5.4910 −3.7710 −2.0186 −0.22689 1.6111 3.5026 5.4554 7.4786 9.5831 11.783 14.101 16.570 19.263 22.397 27.305

−0.081920 −0.077189 −0.065723 −0.055424 −0.046038 −0.037376 −0.029294 −0.021677 −0.014438 −0.0075050 −0.00082003 0.0056653 0.011993 0.018199 0.024318 0.030386 0.036443 0.042540 0.048756 0.055246 0.062490 0.073584

0.079059 0.078138 0.076346 0.075319 0.074926 0.075072 0.075681 0.076690 0.078045 0.079699 0.081613 0.083753 0.086090 0.088602 0.091273 0.094094 0.097073 0.10024 0.10367 0.10763 0.11310

0.11421 0.11344 0.11195 0.11114 0.11096 0.11135 0.11228 0.11371 0.11560 0.11795 0.12075 0.12400 0.12777 0.13211 0.13720 0.14328 0.15087 0.16100 0.17620 0.20480 0.30217

1931.8 1897.3 1805.9 1714.3 1623.4 1533.6 1445.0 1357.7 1271.7 1186.9 1103.2 1020.4 938.24 856.41 774.55 692.22 608.84 523.58 435.01 340.18 231.16 0

13.340 13.603 14.312 15.048 15.811 16.601 17.417 18.259 19.124 20.011 20.917 21.839 22.775 23.718 24.662 25.597 26.506 27.364 28.124 28.692 28.785 26.307

14.457 14.767 15.601 16.461 17.348 18.261 19.197 20.154 21.129 22.117 23.114 24.115 25.113 26.101 27.069 28.003 28.882 29.673 30.319 30.706 30.494 27.305

0.14638 0.13940 0.12439 0.11314 0.10465 0.098239 0.093417 0.089825 0.087196 0.085325 0.084057 0.083264 0.082846 0.082717 0.082798 0.083015 0.083288 0.083518 0.083563 0.083155 0.081542 0.073584

0.045782 0.046445 0.048209 0.050082 0.052140 0.054433 0.056990 0.059826 0.062937 0.066305 0.069907 0.073716 0.077706 0.081857 0.086156 0.090605 0.095223 0.10007 0.10526 0.11109 0.11843

0.054097 0.054760 0.056526 0.058410 0.060499 0.062862 0.065554 0.068614 0.072067 0.075928 0.080211 0.084945 0.090186 0.096045 0.10272 0.11059 0.12039 0.13368 0.15445 0.19596 0.34488

153.35 156.40 164.10 171.37 178.22 184.64 190.57 195.95 200.67 204.61 207.65 209.64 210.44 209.88 207.78 203.91 197.97 189.62 178.35 163.51 144.08 0

−0.50947 −0.51181 −0.51506 −0.51402 −0.50857 −0.49860 −0.48403 −0.46470 −0.44035 −0.41046 −0.37418 −0.33016 −0.27626 −0.20906 −0.12307 −0.0089736 0.14999 0.38688 0.77697 1.5369 3.6646 14.802 1022.9 846.64 534.19 352.76 241.97 171.51 125.19 93.915 72.325 57.138 46.280 38.405 32.629 28.363 25.214 22.923 21.324 20.319 19.863 19.920 20.169 14.802

Single-Phase Properties 150.00 250.00 276.53

0.10000 0.10000 0.10000

276.53 350.00 450.00

0.10000 0.10000 0.10000

150.00 250.00 350.00 356.43

1.0000 1.0000 1.0000 1.0000

356.43 450.00

1.0000 1.0000

150.00 250.00 350.00 450.00

5.0000 5.0000 5.0000 5.0000

13.790 11.936 11.415 0.045328 0.034963 0.026927 13.797 11.951 9.7615 9.5810 0.41725 0.28985

0.072517 0.083783 0.087603 22.062 28.602 37.137 0.072480 0.083676 0.10244 0.10437 2.3967 3.4501

−14.429 −3.1958 −0.050523

−14.421 −3.1874 −0.041763

−0.069410 −0.012105 −0.00015051

0.076853 0.078567 0.081821

0.11236 0.11633 0.12106

1836.9 1243.8 1094.7

−0.51448 −0.43122 −0.37008

21.010 26.707 36.211

23.216 29.567 39.925

0.083956 0.10426 0.13019

0.070286 0.084207 0.10536

0.080672 0.093226 0.11396

207.90 235.49 266.57

45.358 18.349 8.0313

−14.442 −3.2258 9.4809 10.409

−14.369 −3.1421 9.5833 10.513

−0.069498 −0.012226 0.030348 0.032980

0.076879 0.078590 0.094087 0.095350

0.11234 0.11619 0.14310 0.14630

1840.9 1250.7 694.58 656.67

25.991 35.598

28.388 39.048

0.083130 0.10966

0.092560 0.10690

0.11449 0.11882

201.64 250.49

22.160 8.7484

−0.51475 −0.43363 −0.013023 0.052242

13.827 12.016 9.9638 3.4526

0.072321 0.083219 0.10036 0.28964

−14.500 −3.3549 9.1178 29.287

−14.138 −2.9388 9.6196 30.736

−0.069886 −0.012747 0.029293 0.080884

0.076995 0.078700 0.093949 0.12364

0.11223 0.11561 0.13889 0.49664

1858.7 1280.3 758.54 144.63

−0.51590 −0.44345 −0.10935 13.395

150.00 250.00 350.00 450.00

10.000 10.000 10.000 10.000

13.864 12.094 10.171 7.3847

0.072127 0.082686 0.098318 0.13542

−14.569 −3.5069 8.7400 23.767

−13.848 −2.6800 9.7232 25.121

−0.070362 −0.013371 0.028171 0.066644

0.077144 0.078849 0.093912 0.11375

0.11212 0.11500 0.13550 0.17964

1880.2 1315.2 825.19 387.53

−0.51720 −0.45403 −0.18764 1.0233

150.00 250.00 350.00 450.00

20.000 20.000 20.000 20.000

13.936 12.236 10.498 8.4814

0.071757 0.081725 0.095255 0.11791

−14.703 −3.7846 8.1362 22.018

−13.268 −2.1501 10.041 24.376

−0.071287 −0.014539 0.026317 0.062213

0.077453 0.079171 0.094047 0.11264

0.11192 0.11403 0.13149 0.15577

1921.2 1379.1 932.72 586.98

−0.51938 −0.47093 −0.28055 0.13495

250.00 350.00 450.00

50.000 50.000 50.000

12.592 11.161 9.7684

0.079414 0.089596 0.10237

−4.4655 6.9055 19.807

−0.017580 0.022259 0.056195

0.080215 0.094937 0.11311

0.11231 0.12655 0.14428

1538.7 1160.5 893.45

−0.50151 −0.39521 −0.26115

−0.49483 11.385 24.926

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Ihmels, E. C., “Thermodynamic Properties of the Butenes. Part II. Short Fundamental Equations of State,” Fluid Phase Equilibria 228–229C:173–187, 2005. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in densities calculated using the equation of state are 0.1% in the liquid phase at temperatures above 270 K (rising to 0.5% at temperatures below 200 K), 0.2% at temperatures above the critical temperature and at pressures above 10 MPa, and 0.5% in the vapor phase, including supercritical conditions below 10 MPa. The uncertainty in the vapor phase may be higher than 0.5% in some regions. The uncertainty in vapor pressure is 0.2% between 220 and 310 K and 0.5% above 310 K, and the uncertainty in heat capacities is 0.5% at saturated liquid conditions, rising to 5% at much higher pressures and at temperatures above 300 K.

2-237

2-238 TABLE 2-198

Thermodynamic Properties of trans-2-Butene

Temperature K

Pressure MPa

167.60 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 428.61

0.000074817 0.000098940 0.00047141 0.0017229 0.0051236 0.012948 0.028726 0.057367 0.10513 0.17951 0.28905 0.44317 0.65207 0.92670 1.2790 1.7221 2.2718 2.9483 3.7851 4.0191

167.60 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 428.61

0.000074817 0.000098940 0.00047141 0.0017229 0.0051236 0.012948 0.028726 0.057367 0.10513 0.17951 0.28905 0.44317 0.65207 0.92670 1.2790 1.7221 2.2718 2.9483 3.7851 4.0191

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.53710 −0.53489 −0.52040 −0.50389 −0.48446 −0.46138 −0.43391 −0.40117 −0.36199 −0.31463 −0.25645 −0.18323 −0.088064 0.041045 0.22658 0.51577 1.0273 2.1709 7.0491 14.836

Saturated Properties 13.141 13.097 12.827 12.556 12.284 12.008 11.728 11.441 11.146 10.840 10.520 10.183 9.8227 9.4330 9.0030 8.5154 7.9376 7.1898 5.8610 4.2130 0.000053703 0.000070020 0.00030677 0.0010387 0.0028818 0.0068419 0.014361 0.027327 0.048069 0.079385 0.12463 0.18794 0.27462 0.39199 0.55094 0.76954 1.0825 1.5757 2.6760 4.2130

0.076099 0.076351 0.077960 0.079642 0.081408 0.083277 0.085267 0.087404 0.089718 0.092251 0.095056 0.098206 0.10181 0.10601 0.11107 0.11743 0.12598 0.13909 0.17062 0.23736 18,621. 14,282. 3259.8 962.76 347.00 146.16 69.632 36.594 20.803 12.597 8.0236 5.3209 3.6414 2.5511 1.8151 1.2995 0.92378 0.63463 0.37369 0.23736

−12.322 −12.060 −10.409 −8.7337 −7.0328 −5.3024 −3.5381 −1.7351 0.11181 2.0081 3.9596 5.9728 8.0553 10.217 12.471 14.840 17.363 20.137 23.657 26.217

−12.322 −12.060 −10.409 −8.7336 −7.0324 −5.3013 −3.5357 −1.7301 0.12124 2.0247 3.9871 6.0163 8.1216 10.315 12.613 15.042 17.650 20.547 24.303 27.171

−0.056582 −0.055030 −0.045722 −0.037018 −0.028818 −0.021038 −0.013608 −0.0064657 0.00044042 0.0071551 0.013718 0.020164 0.026529 0.032849 0.039171 0.045556 0.052111 0.059086 0.067765 0.074373

0.075471 0.075651 0.076779 0.077989 0.079349 0.080901 0.082660 0.084626 0.086786 0.089124 0.091618 0.094251 0.097011 0.099897 0.10292 0.10614 0.10969 0.11399 0.12174

0.10906 0.10931 0.11086 0.11250 0.11434 0.11646 0.11890 0.12171 0.12491 0.12854 0.13267 0.13740 0.14294 0.14962 0.15815 0.17004 0.18946 0.23383 0.63363

1653.9 1639.3 1549.4 1461.5 1375.6 1291.2 1208.2 1126.4 1045.4 965.18 885.29 805.47 725.30 644.30 561.78 476.68 387.01 288.25 165.30 0

13.917 14.051 14.913 15.805 16.724 17.667 18.632 19.616 20.617 21.632 22.656 23.685 24.709 25.716 26.685 27.583 28.347 28.822 28.287 26.217

15.310 15.465 16.450 17.464 18.502 19.560 20.632 21.716 22.804 23.893 24.976 26.043 27.083 28.080 29.006 29.821 30.445 30.693 29.702 27.171

0.10829 0.10688 0.099458 0.093969 0.089946 0.087053 0.085037 0.083710 0.082924 0.082564 0.082533 0.082747 0.083130 0.083605 0.084083 0.084447 0.084505 0.083831 0.080468 0.074373

0.056121 0.056517 0.058961 0.061448 0.064084 0.066934 0.070029 0.073369 0.076939 0.080711 0.084658 0.088756 0.092992 0.097368 0.10191 0.10670 0.11191 0.11800 0.12690

0.064451 0.064850 0.067333 0.069906 0.072698 0.075804 0.079283 0.083170 0.087492 0.092287 0.097621 0.10363 0.11056 0.11889 0.12962 0.14503 0.17158 0.23687 0.83276

168.85 169.97 176.77 183.16 189.08 194.41 199.05 202.86 205.70 207.43 207.89 206.90 204.25 199.66 192.81 183.21 170.22 152.90 129.63 0

333.88 313.10 214.55 152.47 111.85 84.452 65.512 52.156 42.578 35.618 30.512 26.754 24.009 22.059 20.773 20.089 19.999 20.460 20.057 14.836

Single-Phase Properties 200.00 273.69

0.10000 0.10000

273.69 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 353.43

1.0000 1.0000 1.0000

353.43 400.00 500.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

−0.037029 −0.00015647

0.077991 0.086590

0.11249 0.12461

22.709 25.044 35.016 46.932

0.082975 0.091118 0.11968 0.14618

0.076618 0.080917 0.10071 0.12004

0.087096 0.090604 0.10946 0.12856

205.49 216.41 251.03 280.24

−8.7565 3.2597 10.724

−8.6769 3.3536 10.831

−0.037132 0.011401 0.034294

0.078010 0.090766 0.10058

0.11242 0.13089 0.15138

1467.8 920.75 625.59

2.3574 2.9288 3.9367

25.942 30.880 42.308

28.299 33.808 46.245

0.083718 0.098365 0.12606

0.098392 0.10374 0.12089

0.12109 0.11833 0.13164

198.31 227.44 268.79

21.710 12.735 6.0235

12.615 10.766 8.1724 1.7031

0.079271 0.092881 0.12236 0.58716

−8.8455 3.0452 17.547 39.420

−8.4492 3.5096 18.158 42.355

−0.037581 0.010676 0.052539 0.10676

0.078096 0.090769 0.10973 0.12565

0.11215 0.12936 0.17328 0.16251

1492.3 964.95 451.39 215.66

−0.50911 −0.31451 0.61704 7.3063

12.557 11.172 0.045873 0.041314 0.030400 0.024180 12.568 10.652 9.3386 0.42419 0.34144 0.25402

−8.7360 −0.051921

−8.7280 −0.042970

20.529 22.623 31.727 42.796

0.079566 0.093883 0.10708

0.079634 0.089507 21.799 24.205 32.895 41.357

1462.2 1052.5

−0.50400 −0.36572 43.299 30.137 11.036 5.7119 −0.50499 −0.28393 0.077370

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

12.671 10.895 8.6867 4.7944

0.078919 0.091783 0.11512 0.20858

−8.9520 2.8039 16.708 34.462

−8.1628 3.7218 17.859 36.547

−0.038127 0.0098449 0.050312 0.091756

0.078213 0.090820 0.10892 0.12831

0.11184 0.12787 0.15802 0.21038

1521.6 1014.7 568.97 241.07

−0.51368 −0.34416 0.18397 3.5123

200.00 300.00 400.00 500.00

25.000 25.000 25.000 25.000

12.827 11.215 9.5005 7.5935

0.077961 0.089166 0.10526 0.13169

−9.2436 2.2040 15.305 30.258

−7.2946 4.4332 17.937 33.551

−0.039668 0.0076959 0.046401 0.081157

0.078610 0.091136 0.10862 0.12604

0.11112 0.12501 0.14563 0.16587

1602.1 1139.2 778.83 531.13

−0.52436 −0.40118 −0.17082 0.22774

200.00 300.00 400.00 500.00

50.000 50.000 50.000 50.000

13.054 11.620 10.240 8.9015

0.076608 0.086061 0.097652 0.11234

−9.6567 1.4532 13.991 28.100

−5.8263 5.7562 18.873 33.717

−0.041984 0.0048127 0.042428 0.075482

0.079350 0.091871 0.10919 0.12638

0.11038 0.12263 0.13997 0.15648

1718.0 1299.8 994.88 788.67

−0.53554 −0.44882 −0.32579 −0.19336

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Ihmels, E. C., “Thermodynamic Properties of the Butenes. Part II. Short Fundamental Equations of State,” Fluid Phase Equilibria 228–229C:173–187, 2005. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in densities calculated using the equation of state are 0.1% in the liquid phase at temperatures above 270 K (rising to 0.5% at temperatures below 200 K), 0.2% at temperatures above the critical temperature and at pressures above 10 MPa, and 0.5% in the vapor phase, including supercritical conditions below 10 MPa. The uncertainty in the vapor phase may be higher than 0.5% in some regions. The uncertainty in vapor pressure is 0.3% above 200 K, and the uncertainty in heat capacities is 0.5% at saturated liquid conditions, rising to 5% at much higher pressures and at temperatures above 250 K.

2-239

2-240 TABLE 2-199 Temperature K

Thermodynamic Properties of Carbon Dioxide Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

180.63 176.15 169.67 163.28 156.98 150.75 144.58 138.47 132.40 126.35 120.31 114.25 108.17 102.03 95.810 89.546 83.558 80.593

256.70 242.01 222.19 204.23 187.88 172.96 159.30 146.74 135.14 124.40 114.40 105.02 96.174 87.731 79.548 71.409 62.936 53.107

11.014 11.301 11.745 12.221 12.736 13.297 13.917 14.610 15.396 16.306 17.381 18.687 20.325 22.468 25.424 29.821 37.215 53.689

10.951 11.135 11.409 11.689 11.976 12.272 12.579 12.902 13.245 13.614 14.017 14.469 14.987 15.601 16.361 17.357 18.792 21.306

Saturated Properties 216.59 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 304.13

0.51796 0.59913 0.73509 0.89291 1.0747 1.2825 1.5185 1.7850 2.0843 2.4188 2.7909 3.2033 3.6589 4.1607 4.7123 5.3177 5.9822 6.7131 7.3773

26.777 26.497 26.078 25.646 25.201 24.742 24.264 23.767 23.246 22.697 22.114 21.491 20.817 20.077 19.247 18.284 17.100 15.434 10.625

0.037345 0.037740 0.038347 0.038992 0.039680 0.040418 0.041213 0.042075 0.043018 0.044059 0.045219 0.046531 0.048037 0.049808 0.051957 0.054693 0.058480 0.064793 0.094118

3.5030 3.7943 4.2235 4.6554 5.0908 5.5303 5.9749 6.4256 6.8836 7.3505 7.8282 8.3190 8.8266 9.3560 9.9154 10.519 11.197 12.036 13.928

3.5223 3.8169 4.2517 4.6902 5.1334 5.5821 6.0375 6.5007 6.9733 7.4571 7.9544 8.4681 9.0024 9.5633 10.160 10.810 11.547 12.471 14.622

0.022943 0.024279 0.026209 0.028110 0.029986 0.031840 0.033678 0.035505 0.037326 0.039148 0.040979 0.042829 0.044711 0.046643 0.048657 0.050805 0.053196 0.056151 0.063094

0.042895 0.042682 0.042383 0.042103 0.041843 0.041605 0.041393 0.041212 0.041079 0.041029 0.041109 0.041351 0.041750 0.042270 0.042900 0.043734 0.045175 0.049288

0.085960 0.086338 0.087024 0.087886 0.088954 0.090263 0.091866 0.093831 0.096251 0.099258 0.10306 0.10798 0.11457 0.12385 0.13790 0.16176 0.21098 0.38279

975.85 951.21 915.16 879.09 842.88 806.38 769.44 731.78 693.01 652.58 610.07 565.46 519.14 471.54 422.75 371.95 315.91 245.67 0

−0.14430 −0.13180 −0.11104 −0.086994 −0.059053 −0.026454 0.011808 0.057087 0.11121 0.17663 0.25672 0.35639 0.48324 0.64959 0.87650 1.2037 1.7218 2.7258 5.8665

216.59 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 304.13

0.51796 0.59913 0.73509 0.89291 1.0747 1.2825 1.5185 1.7850 2.0843 2.4188 2.7909 3.2033 3.6589 4.1607 4.7123 5.3177 5.9822 6.7131 7.3773

0.31268 0.35941 0.43766 0.52878 0.63442 0.75654 0.89743 1.0599 1.2472 1.4637 1.7149 2.0080 2.3535 2.7663 3.2702 3.9074 4.7654 6.1028 10.625

3.1982 2.7824 2.2849 1.8912 1.5762 1.3218 1.1143 0.94353 0.80180 0.68320 0.58314 0.49800 0.42490 0.36150 0.30579 0.25593 0.20985 0.16386 0.094118

17.286 17.329 17.387 17.438 17.481 17.515 17.538 17.550 17.549 17.532 17.498 17.441 17.359 17.241 17.078 16.848 16.509 15.935 13.928

18.943 18.996 19.067 19.127 19.175 19.210 19.230 19.234 19.220 19.185 19.125 19.037 18.913 18.746 18.519 18.209 17.764 17.035 14.622

0.094138 0.093276 0.092055 0.090878 0.089736 0.088622 0.087526 0.086439 0.085352 0.084254 0.083133 0.081972 0.080750 0.079437 0.077987 0.076319 0.074270 0.071364 0.063094

0.027691 0.028120 0.028782 0.029488 0.030241 0.031042 0.031899 0.032827 0.033844 0.034955 0.036164 0.037482 0.038949 0.040628 0.042629 0.045155 0.048677 0.054908

0.039992 0.040943 0.042489 0.044244 0.046248 0.048555 0.051242 0.054421 0.058244 0.062912 0.068721 0.076168 0.086123 0.10020 0.12177 0.15906 0.23904 0.52463

222.78 223.15 223.49 223.57 223.40 222.96 222.24 221.22 219.87 218.19 216.15 213.75 210.96 207.72 203.94 199.45 193.84 185.33 0

26.174 25.084 23.617 22.288 21.077 19.969 18.950 18.005 17.117 16.277 15.476 14.704 13.947 13.185 12.387 11.509 10.459 9.0093 5.8665

Single-Phase Properties 250.00 450.00 650.00 850.00 1050.0

0.10000 0.10000 0.10000 0.10000 0.10000

0.048542 0.026758 0.018506 0.014148 0.011452

250.00 450.00 650.00 850.00 1050.0

1.0000 1.0000 1.0000 1.0000 1.0000

0.53250 0.27038 0.18527 0.14131 0.11430

250.00 287.43

5.0000 5.0000

287.43 450.00 650.00 850.00 1050.0

5.0000 5.0000 5.0000 5.0000 5.0000

250.00 450.00 650.00 850.00 1050.0

10.000 10.000 10.000 10.000 10.000

18.448 24.664 32.199 40.636 49.704

20.509 28.401 37.602 47.705 58.436

0.11415 0.13712 0.15397 0.16750 0.17883

0.026766 0.034775 0.040192 0.043944 0.046573

0.035428 0.043148 0.048529 0.052271 0.054895

247.79 324.41 385.01 437.11 483.65

17.399 4.0212 1.6551 0.78058 0.34646

12.950 29.346 45.466 60.295 73.843

12.565 21.901 29.873 36.707 42.692

1.8779 3.6985 5.3976 7.0767 8.7487

18.023 24.546 32.133 40.591 49.671

19.901 28.244 37.530 47.668 58.419

0.093263 0.11771 0.13473 0.14830 0.15965

0.029361 0.034954 0.040239 0.043965 0.046585

0.042504 0.043866 0.048779 0.052397 0.054970

235.08 322.89 385.36 438.06 484.84

17.606 3.9880 1.6311 0.76632 0.33777

13.584 29.620 45.651 60.435 73.956

12.691 21.954 29.907 36.732 42.712

0.041563 0.053196

6.2824 10.202

6.4902 10.468

0.034925 0.049681

0.041321 0.043268

0.090937 0.14775

762.21 398.39

142.22 92.760

153.15 75.598

0.28090 0.70647 1.0755 1.4237 1.7650

16.977 24.000 31.842 40.395 49.524

18.381 27.533 37.219 47.513 58.349

0.077209 0.10313 0.12091 0.13469 0.14613

0.043774 0.035769 0.040445 0.044055 0.046637

0.13705 0.047478 0.049898 0.052945 0.055297

201.86 317.50 387.59 442.64 490.31

11.974 3.8034 1.5263 0.70611 0.30129

27.323 31.164 46.589 61.117 74.494

16.808 22.429 30.157 36.899 42.836

24.459 2.9910 1.8632 1.3930 1.1205

0.040885 0.33433 0.53671 0.71790 0.89248

6.0862 23.276 31.482 40.155 49.347

6.4950 26.619 36.849 47.334 58.271

0.034120 0.095787 0.11461 0.12866 0.14021

0.041488 0.036785 0.040693 0.044164 0.046701

0.087624 0.052935 0.051293 0.053603 0.055685

804.05 314.60 391.91 449.04 497.48

−0.034849 3.4705 1.3965 0.63635 0.25964

147.52 33.917 48.005 62.093 75.242

162.47 23.679 30.687 37.224 43.066

24.060 18.798 3.5600 1.4155 0.92982 0.70241 0.56658

20.601 37.372 54.037 70.683 87.321

0.015208 1.0195

250.00 450.00 650.00 850.00 1050.0

100.00 100.00 100.00 100.00 100.00

28.075 19.246 13.677 10.636 8.7929

0.035619 0.051959 0.073117 0.094022 0.11373

4.3002 16.560 27.132 37.076 46.995

7.8621 21.756 34.444 46.478 58.368

0.026023 0.067062 0.090445 0.10660 0.11916

0.043569 0.040841 0.043108 0.045620 0.047676

0.073521 0.066107 0.061252 0.059534 0.059512

1227.6 753.30 646.36 646.61 668.90

−0.27302 −0.11128 −0.054084 −0.13292 −0.21482

206.28 106.65 86.093 87.259 94.022

287.05 83.996 58.868 54.445 55.058

450.00 650.00 850.00 1050.0

500.00 500.00 500.00 500.00

28.922 25.661 23.144 21.126

0.034576 0.038969 0.043208 0.047334

13.014 23.302 33.551 43.903

30.302 42.786 55.155 67.570

0.050604 0.073576 0.090166 0.10328

0.047702 0.048419 0.049676 0.050818

0.063434 0.061885 0.061912 0.062247

1576.4 1404.7 1320.1 1278.7

−0.38514 −0.40369 −0.41098 −0.41674

239.59 197.25 177.31 168.50

303.64 191.14 145.07 123.33

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., and Wagner, W., “A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa,” J. Phys. Chem. Ref. Data 25(6):1509–1596, 1996. The source for viscosity is Fenghour, A., Wakeham, W. A., and Vesovic, V., “The Viscosity of Carbon Dioxide,” J. Phys. Chem. Ref. Data 27:31–44, 1998. The source for thermal conductivity is Vesovic, V., Wakeham, W. A., Olchowy, G. A., Sengers, J. V., Watson, J. T. R., and Millat, J., “The Transport Properties of Carbon Dioxide,” J. Phys. Chem. Ref. Data 19:763–808, 1990. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. At pressures up to 30 MPa and temperatures up to 523 K, the estimated uncertainty ranges from 0.03% to 0.05% in density, 0.03% (in the vapor) to 1% in the speed of sound (0.5% in the liquid), and 0.15% (in the vapor) to 1.5% (in the liquid) in heat capacity. Special interest has been focused on the description of the critical region and the extrapolation behavior of the formulation (to the limits of chemical stability). The uncertainty in viscosity ranges from 0.3% in the dilute gas near room temperature to 5% at the highest pressures. The uncertainty in thermal conductivity is less than 5%.

2-241

2-242

TABLE 2-200 Temperature K

Thermodynamic Properties of Carbon Monoxide Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.032971 0.033262 0.033588 0.033924 0.034270 0.034628 0.034999 0.035383 0.035782 0.036197 0.036630 0.037082 0.037556 0.038052 0.038574 0.039125 0.039708 0.040326 0.040985 0.041689 0.042446 0.043263 0.044151 0.045124 0.046197 0.047395 0.048749 0.050307 0.052141 0.054377 0.057259 0.061393 0.092166

−0.81158 −0.70065 −0.58046 −0.46058 −0.34088 −0.22127 −0.10165 0.018099 0.13806 0.25835 0.37906 0.50030 0.62218 0.74482 0.86835 0.99289 1.1186 1.2457 1.3742 1.5045 1.6368 1.7713 1.9085 2.0487 2.1925 2.3405 2.4938 2.6536 2.8221 3.0024 3.2010 3.4328 4.2912

−0.81106 −0.69995 −0.57950 −0.45927 −0.33915 −0.21900 −0.098716 0.021834 0.14277 0.26421 0.38629 0.50915 0.63291 0.75773 0.88377 1.0112 1.1402 1.2710 1.4039 1.5390 1.6768 1.8175 1.9616 2.1097 2.2624 2.4205 2.5853 2.7583 2.9420 3.1403 3.3608 3.6210 4.6137

−0.010820 −0.0092140 −0.0075210 −0.0058785 −0.0042823 −0.0027285 −0.0012138 0.00026503 0.0017110 0.0031269 0.0045153 0.0058787 0.0072195 0.0085399 0.0098422 0.011129 0.012402 0.013663 0.014916 0.016162 0.017404 0.018646 0.019891 0.021142 0.022406 0.023688 0.024996 0.026343 0.027744 0.029230 0.030854 0.032745 0.040039

5.1252 5.1600 5.1971 5.2334 5.2688 5.3031 5.3363 5.3682 5.3988 5.4280 5.4556 5.4816 5.5058 5.5280 5.5482 5.5661 5.5816 5.5945 5.6044 5.6112 5.6145 5.6138 5.6088

5.6859 5.7343 5.7859 5.8361 5.8849 5.9320 5.9775 6.0210 6.0625 6.1019 6.1388 6.1733 6.2050 6.2338 6.2595 6.2819 6.3007 6.3157 6.3265 6.3327 6.3339 6.3295 6.3191

0.084499 0.082704 0.080887 0.079194 0.077613 0.076131 0.074739 0.073426 0.072185 0.071007 0.069885 0.068813 0.067785 0.066796 0.065840 0.064912 0.064007 0.063120 0.062248 0.061385 0.060526 0.059665 0.058797

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.035351 0.034805 0.034248 0.033724 0.033232 0.032768 0.032329 0.031915 0.031522 0.031150 0.030798 0.030463 0.030146 0.029846 0.029562 0.029294 0.029043 0.028809 0.028592 0.028395 0.028218 0.028066 0.027941 0.027850 0.027800 0.027803 0.027874 0.028038 0.028333 0.028826 0.029646 0.031097

0.060430 0.060226 0.060064 0.059961 0.059917 0.059930 0.060002 0.060132 0.060324 0.060578 0.060899 0.061291 0.061760 0.062314 0.062962 0.063716 0.064590 0.065604 0.066781 0.068153 0.069759 0.071656 0.073916 0.076648 0.080005 0.084225 0.089692 0.097070 0.10762 0.12411 0.15392 0.22603

998.20 980.50 961.22 941.89 922.49 903.01 883.44 863.76 843.95 824.00 803.89 783.60 763.12 742.41 721.45 700.22 678.68 656.78 634.50 611.77 588.54 564.73 540.25 515.01 488.86 461.63 433.11 403.00 370.88 336.15 297.82 254.03 0

−0.36906 −0.36553 −0.36074 −0.35489 −0.34794 −0.33981 −0.33041 −0.31966 −0.30742 −0.29356 −0.27794 −0.26034 −0.24056 −0.21834 −0.19335 −0.16523 −0.13353 −0.097704 −0.057078 −0.010824 0.042104 0.10304 0.17371 0.25641 0.35427 0.47167 0.61495 0.79382 1.0239 1.3325 1.7728 2.4703 6.1475

180.28 175.49 170.45 165.55 160.76 156.06 151.45 146.89 142.40 137.96 133.57 129.23 124.94 120.69 116.51 112.38 108.31 104.30 100.36 96.482 92.679 88.948 85.290 81.702 78.180 74.716 71.296 67.896 64.476 60.972 57.261 53.107

274.18 252.15 232.14 215.32 201.01 188.69 177.96 168.52 160.13 152.60 145.77 139.52 133.75 128.38 123.34 118.57 114.02 109.66 105.45 101.35 97.342 93.404 89.510 85.641 81.774 77.888 73.954 69.940 65.797 61.448 56.748 51.348

0.021089 0.021155 0.021238 0.021333 0.021441 0.021563 0.021699 0.021850 0.022017 0.022199 0.022397 0.022611 0.022842 0.023089 0.023352 0.023633 0.023931 0.024248 0.024586 0.024945 0.025329 0.025741 0.026186

0.029785 0.029947 0.030153 0.030394 0.030672 0.030993 0.031360 0.031777 0.032250 0.032783 0.033383 0.034057 0.034813 0.035661 0.036615 0.037690 0.038906 0.040288 0.041869 0.043694 0.045820 0.048326 0.051322

167.25 169.22 171.27 173.22 175.07 176.80 178.42 179.92 181.29 182.54 183.66 184.65 185.51 186.22 186.80 187.23 187.52 187.67 187.66 187.51 187.20 186.73 186.11

40.804 38.426 36.126 34.080 32.250 30.604 29.116 27.763 26.527 25.392 24.345 23.377 22.477 21.638 20.853 20.118 19.426 18.773 18.154 17.565 17.001 16.458 15.930

Saturated Properties 68.160 70.000 72.000 74.000 76.000 78.000 80.000 82.000 84.000 86.000 88.000 90.000 92.000 94.000 96.000 98.000 100.00 102.00 104.00 106.00 108.00 110.00 112.00 114.00 116.00 118.00 120.00 122.00 124.00 126.00 128.00 130.00 132.86

0.015537 0.021053 0.028718 0.038447 0.050599 0.065559 0.083738 0.10556 0.13148 0.16196 0.19748 0.23852 0.28559 0.33919 0.39983 0.46805 0.54438 0.62934 0.72348 0.82736 0.94154 1.0666 1.2031 1.3517 1.5130 1.6877 1.8765 2.0802 2.2997 2.5360 2.7904 3.0647 3.4982

68.160 70.000 72.000 74.000 76.000 78.000 80.000 82.000 84.000 86.000 88.000 90.000 92.000 94.000 96.000 98.000 100.00 102.00 104.00 106.00 108.00 110.00 112.00

0.015537 0.021053 0.028718 0.038447 0.050599 0.065559 0.083738 0.10556 0.13148 0.16196 0.19748 0.23852 0.28559 0.33919 0.39983 0.46805 0.54438 0.62934 0.72348 0.82736 0.94154 1.0666 1.2031

30.330 30.064 29.773 29.478 29.180 28.878 28.573 28.262 27.947 27.626 27.300 26.967 26.627 26.280 25.924 25.559 25.184 24.798 24.399 23.987 23.560 23.114 22.649 22.161 21.646 21.099 20.513 19.878 19.179 18.390 17.464 16.288 10.850 0.027707 0.036656 0.048780 0.063796 0.082130 0.10424 0.13059 0.16171 0.19810 0.24036 0.28906 0.34486 0.40845 0.48058 0.56209 0.65388 0.75700 0.87260 1.0020 1.1468 1.3088 1.4903 1.6938

36.091 27.281 20.500 15.675 12.176 9.5935 7.6573 6.1841 5.0478 4.1605 3.4595 2.8997 2.4483 2.0808 1.7791 1.5293 1.3210 1.1460 0.99799 0.87198 0.76404 0.67102 0.59039

6.6865 6.8845 7.1009 7.3188 7.5382 7.7592 7.9820 8.2067 8.4335 8.6627 8.8944 9.1291 9.3672 9.6091 9.8555 10.107 10.366 10.632 10.909 11.198 11.502 11.828 12.181

4.6366 4.7768 4.9329 5.0934 5.2589 5.4300 5.6076 5.7922 5.9847 6.1860 6.3968 6.6182 6.8512 7.0969 7.3566 7.6317 7.9239 8.2350 8.5675 8.9238 9.3073 9.7221 10.173

114.00 116.00 118.00 120.00 122.00 124.00 126.00 128.00 130.00 132.86

1.3517 1.5130 1.6877 1.8765 2.0802 2.2997 2.5360 2.7904 3.0647 3.4982

100.00 200.00 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000 0.10000

100.00 108.96

1.0000 1.0000

108.96 200.00 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000 1.0000

100.00 200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000

1.9228 2.1815 2.4754 2.8123 3.2027 3.6629 4.2194 4.9212 5.8832 10.850

0.52008 0.45841 0.40397 0.35558 0.31224 0.27301 0.23700 0.20320 0.16998 0.092166

5.5986 5.5827 5.5598 5.5286 5.4872 5.4324 5.3595 5.2594 5.1113 4.2912

6.3016 6.2762 6.2416 6.1959 6.1367 6.0602 5.9605 5.8264 5.6322 4.6137

0.057914 0.057008 0.056070 0.055084 0.054034 0.052892 0.051613 0.050117 0.048216 0.040039

5.7653 7.8674 9.9522 12.045 14.169

6.5785 9.5259 12.446 15.371 18.328

0.080014 0.10048 0.11231 0.12073 0.12733

0.039586 0.042829

1.1047 1.7009

1.1443 1.7437

0.71782 1.6200 2.4867 3.3334 4.1732

5.6147 7.7647 9.8936 12.005 14.140

25.864 3.4130 2.0232 1.4824 1.1786

0.038663 0.29299 0.49426 0.67458 0.84845

0.026671 0.027203 0.027794 0.028462 0.029229 0.030133 0.031233 0.032636 0.034579

0.054966 0.059493 0.065263 0.072864 0.083320 0.098585 0.12291 0.16759 0.27599

185.33 184.38 183.27 181.99 180.52 178.84 176.93 174.68 171.86 0

15.411 14.894 14.372 13.833 13.265 12.648 11.956 11.140 10.100 6.1475

12.569 13.005 13.507 14.101 14.826 15.747 16.981 18.777 21.845

10.667 11.213 11.821 12.509 13.301 14.234 15.373 16.840 18.936

0.021118 0.020812 0.020833 0.021028 0.021479

0.030153 0.029239 0.029191 0.029364 0.029807

201.29 288.05 353.12 407.29 454.00

17.820 5.3111 2.5186 1.2653 0.56244

10.075 19.227 26.605 33.106 39.272

6.9147 12.897 17.731 21.870 25.540

0.012262 0.017998

0.029062 0.028142

0.064114 0.070627

685.44 577.22

−0.14414 0.070176

112.87 90.884

6.3325 9.3847 12.380 15.338 18.313

0.060114 0.080819 0.092976 0.10149 0.10812

0.025522 0.020996 0.020895 0.021064 0.021505

0.046966 0.030510 0.029646 0.029598 0.029948

186.99 286.20 354.42 409.43 456.39

16.739 5.1924 2.4256 1.2088 0.52786

11.655 19.474 26.760 33.222 39.364

0.99666 7.2656 9.6364 11.834 14.015

1.1900 8.7305 12.108 15.207 18.258

0.011154 0.064994 0.078767 0.087691 0.094498

0.029263 0.021878 0.021174 0.021224 0.021618

0.060925 0.038000 0.031673 0.030585 0.030535

737.92 285.27 362.95 420.15 467.56

−0.21740 4.3757 2.0288 0.98413 0.39254

152.30 22.190 27.812 33.871 39.837

112.19 15.094 18.716 22.588 26.139

Single-Phase Properties 0.12298 0.060293 0.040104 0.030062 0.024045 25.261 23.349 1.3931 0.61727 0.40214 0.29999 0.23962

8.1315 16.586 24.935 33.265 41.588

113.83 95.450 9.5017 13.192 17.918 22.024 25.676

100.00 200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000 10.000

26.482 7.4298 4.0263 2.9079 2.3052

0.037761 0.13459 0.24837 0.34389 0.43381

0.88669 6.5960 9.3290 11.634 13.870

1.2643 7.9419 11.813 15.073 18.208

0.0099878 0.056188 0.072068 0.081462 0.088461

0.029539 0.022832 0.021511 0.021420 0.021755

0.058409 0.048831 0.034036 0.031689 0.031188

792.04 307.84 379.01 435.68 482.48

−0.27800 2.8854 1.5731 0.74980 0.25486

200.46 34.772 30.972 35.414 40.797

110.33 19.114 19.862 23.298 26.682

100.00 200.00 300.00 400.00 500.00

50.000 50.000 50.000 50.000 50.000

29.422 20.591 14.766 11.439 9.3865

0.033988 0.048566 0.067725 0.087418 0.10654

0.39153 4.5424 7.7949 10.518 13.024

2.0910 6.9707 11.181 14.889 18.350

0.0040257 0.038097 0.055259 0.065951 0.073681

0.031398 0.025036 0.023212 0.022620 0.022659

0.052094 0.045541 0.039083 0.035519 0.033937

1066.8 706.01 609.73 604.51 622.30

−0.43831 −0.28689 −0.21242 −0.27153 −0.36911

567.46 256.88 139.18 94.476 76.899

99.463 50.929 34.086 31.319 32.167

31.474 24.888 20.200 16.970 14.662

0.031772 0.040181 0.049505 0.058928 0.068206

0.095937 3.9022 7.0625 9.8487 12.449

3.2732 7.9203 12.013 15.741 19.269

−0.00053857 0.031951 0.048608 0.059353 0.067230

0.033037 0.026437 0.024358 0.023555 0.023434

0.050530 0.043352 0.038799 0.036051 0.034683

1282.4 987.69 866.60 822.33 808.91

−0.47725 −0.50923 −0.54516 −0.59581 −0.64123

100.00 200.00 300.00 400.00 500.00

100.00 100.00 100.00 100.00 100.00

1005.7 536.84 331.69 229.18 173.58

90.560 73.380 54.648 45.748 42.504

2-243

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850,2006. The source for viscosity and thermal conductivity is Version 9.08 of the NIST14 database. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The equation of state is valid from the triple point to 500 K with pressures to 100 MPa. At higher pressures, the deviations from the equation increase rapidly, and it is not recommended to use the equation above 100 MPa. The uncertainties in the equation are 0.3% in density (approaching 1% near the critical point), 0.2% in vapor pressure, and 2% in heat capacities. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6%.

2-244

PHYSICAL AND CHEMICAL DATA

Temperature-entropy diagram for carbon monoxide. Pressure P, in atmospheres; density ρ, in grams per cubic centimeter; enthalpy H, in joules per gram. (From Hust and Stewart, NBS Tech. Note 202, 1963.)

FIG. 2-8

THERMODYNAMIC PROPERTIES TABLE 2-201

2-245

Thermophysical Properties of Saturated Carbon Tetrachloride hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf , 10−6 Pa·s

kf , W/(m⋅K)

Pr

280 290 300 310 320

0.064 0.105 0.165 0.251 0.370

0.000 0.000 0.000 0.000 0.000

619 625 633 641 649

2.414 1.495 0.971 0.669 0.463

205.5 212.9 220.9 228.8 236.9

420.7 425.7 430.9 436.1 441.3

1.018 1.042 1.068 1.095 1.121

1.787 1.775 1.768 1.764 1.760

0.835 0.844 0.853 0.863 0.874

1042 892 774 679 603

0.1043 0.1020 0.0998 0.0975 0.0952

8.34 7.38 6.62 6.01 5.54

330 340 350 360 370

0.531 0.743 1.017 1.361 1.795

0.000 0.000 0.000 0.000 0.000

657 666 674 684 694

0.3306 0.2407 0.1802 0.1370 0.1053

246.0 254.5 263.1 271.8 280.8

446.4 451.5 456.6 461.7 466.6

1.149 1.174 1.199 1.224 1.248

1.756 1.754 1.752 1.751 1.751

0.885 0.897 0.910 0.924 0.939

539 486 441 402 368

0.0930 0.0907 0.0884 0.0861 0.0839

5.13 4.81 4.54 4.31 4.12

380 390 400 410 420

2.327 2.970 3.735 4.642 5.700

0.000 0.000 0.000 0.000 0.000

704 715 727 739 753

0.0820 0.0651 0.0525 0.0426 0.0350

289.7 298.1 307.9 317.1 326.0

471.5 475.8 481.2 485.8 490.4

1.272 1.295 1.319 1.341 1.363

1.750 1.751 1.752 1.753 1.754

0.954 0.970 0.987 1.010 1.034

338 311 287 265 246

0.0816 0.0794 0.0771 0.0749 0.0726

3.95 3.80 3.67 3.57 3.50

430 440 450 460 470

6.927 8.342 9.958 11.792 13.869

0.000 0.000 0.000 0.000 0.000

766 780 796 801 834

0.02899 0.02413 0.02020 0.01692 0.01425

335.2 344.3 353.6 363.1 372.8

494.9 499.2 503.4 507.3 511.1

1.384 1.405 1.426 1.446 1.467

1.756 1.757 1.759 1.760 1.761

1.060 1.094 1.141 1.207 1.240

227 211 195 180 167

0.0704 0.0682 0.0660 0.0638 0.0666

3.42 3.38 3.37 3.36 3.36

480 490 500 510 520

16.21 18.83 21.77 25.02 28.68

0.000 0.000 0.000 0.000 0.000

856 880 858 945 987

0.01205 0.01011 0.00858 0.00722 0.00607

382.6 392.0 402.5 412.9 424.3

514.6 517.5 520.2 522.6 524.2

1.487 1.507 1.526 1.546 1.568

1.762 1.763 1.762 1.761 1.760

1.278 1.320 1.375 1.44 1.52

156 145 133

0.0594 0.0511 0.0549

3.36 3.35 3.35

530 540 550 556.4c

32.71 37.18 44.12 45.60

0.001 0.001 0.001 0.001

041 121 248 792

0.00500 0.00400 0.00309 0.00179

436.4 448.3 463.4 494.4

524.5 522.7 518.2 494.4

1.590 1.614 1.638 1.692

1.756 1.749 1.738 1.692

T, K

vf, m3/kg

P, bar

vg, m3/kg

c = critical point. Base points: hf = 200 at 273.15 K = 0°C = hA − 300 kJ/kg; sf = 1.000 at 273.15 K = 0°C = sA − 4.000 kJ/(kg⋅K). Values mostly rounded and converted from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, vol. 9, Hemisphere, Washington, DC, 1987 (243 pp.). Some irregularities exist in these data.

TABLE 2-202

Saturated Carbon Tetrafluoride (R14)* µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.887 0.887 0.890 0.896 0.904

3.56

0.136 0.128 0.119 0.111 0.104

6.696 6.662 6.629 6.607 6.583

0.922 0.975 1.031 1.104 1.203

3.28 3.03 2.80 2.59 2.39

0.097 0.089 0.081 0.072 0.064

6.558 6.536 6.490 6.371

1.334 1.506 1.73 ∞

2.19 2.01 1.85

0.057 0.049 0.042 ∞

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

100 110 120 130 140

0.0089 0.0286 0.0924 0.2986 0.6901

5.370.−4 5.515.−4 5.668.−4 5.834.−4 6.018.−4

10.77 3.648 1.228 0.4051 0.1855

495.8 502.7 510.4 518.8 527.7

648.4 652.9 657.1 661.1 664.8

5.487 5.556 5.624 5.691 5.757

7.003 6.919 6.847 6.786 6.736

150 160 170 180 190

1.4074 2.598 4.426 7.067 10.702

6.225.−4 6.460.−4 6.733.−4 7.055.−4 7.449.−4

0.0951 0.0532 0.0318 0.0200 0.0131

537.2 549.4 557.6 568.2 579.3

668.3 671.4 674.0 676.1 677.4

5.822 5.885 5.947 6.007 6.066

200 210 220 227.5c

15.531 21.794 29.269 37.45

7.957.−4 8.674.−4 9.931.−4 1.598.−3

0.0087 0.0058 0.0036 0.0016

591.0 603.5 618.5 646.9

677.8 676.4 671.4 646.9

6.124 6.182 6.233 6.371

T, K

*P, v, h, and s values interpolated, extrapolated, and converted from Oguchi, Reito, 52 (1977): 869–889. c = critical point. The notation 5.370.−4 signifies 5.370 × 10−4. Equations and constants approximated to ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2–3 (1991). The 1993 ASHRAE Handbook—Fundamentals (S.I. ed.) contains a saturation table from −140 to −45.65 °C and an enthalpy–log-pressure diagram from 0.1 to 300 bar, −140 to 300 °C. For properties to 1000 bar from 90 to 420 K, see Rublo, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36 (1991): 171–184. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Chari, Ph.D. thesis, University of Michigan, 1960, presents saturation-temperature tables in fps units for 1°F increments from −270 to −51°F. Thermodynamic and transport properties, equations, and computer code and tables at constant entropy from 89 to 845 K are given by Hunt, J. L. and Boney, L. R., NASA TN D-7181, 1973 (105 pp.), largely based upon the Chari data.

2-246 TABLE 2-203

Thermodynamic Properties of Carbonyl Sulfide

Temperature K

Pressure MPa

134.30 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 378.77

0.000064435 0.00014316 0.00049244 0.0014232 0.0035714 0.0079826 0.016210 0.030380 0.053219 0.088035 0.13868 0.20947 0.30513 0.43069 0.59148 0.79300 1.0409 1.3412 1.6998 2.1232 2.6180 3.1915 3.8523 4.6109 5.4827 6.3688

134.30 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 378.77

0.000064435 0.00014316 0.00049244 0.0014232 0.0035714 0.0079826 0.016210 0.030380 0.053219 0.088035 0.13868 0.20947 0.30513 0.43069 0.59148 0.79300 1.0409 1.3412 1.6998 2.1232 2.6180 3.1915 3.8523 4.6109 5.4827 6.3688

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.036049 −0.032962 −0.027934 −0.023315 −0.019029 −0.015019 −0.011239 −0.0076537 −0.0042337 −0.00095536 0.0022014 0.0052538 0.0082168 0.011104 0.013927 0.016699 0.019432 0.022138 0.024832 0.027531 0.030256 0.033041 0.035938 0.039054 0.042698 0.050522

0.050295 0.049071 0.047317 0.045963 0.044919 0.044118 0.043507 0.043047 0.042708 0.042464 0.042299 0.042198 0.042150 0.042147 0.042186 0.042262 0.042376 0.042533 0.042738 0.043007 0.043365 0.043857 0.044569 0.045699 0.047851

0.074835 0.073691 0.072140 0.071067 0.070377 0.070000 0.069885 0.069996 0.070307 0.070804 0.071479 0.072335 0.073381 0.074640 0.076145 0.077949 0.080132 0.082812 0.086180 0.090546 0.096471 0.10507 0.11899 0.14652 0.23509

1449.7 1423.0 1376.1 1329.2 1282.3 1235.4 1188.5 1141.6 1094.7 1047.8 1000.8 953.72 906.38 858.76 810.74 762.21 713.03 663.05 612.06 559.82 505.95 449.91 390.71 326.44 252.51 0

−0.47647 −0.48236 −0.48929 −0.49217 −0.49125 −0.48674 −0.47875 −0.46727 −0.45221 −0.43335 −0.41029 −0.38249 −0.34913 −0.30911 −0.26092 −0.20244 −0.13069 −0.041371 0.071920 0.21913 0.41667 0.69379 1.1092 1.8033 3.2373 8.9233

0.12786 0.12252 0.11442 0.10767 0.10200 0.097198 0.093108 0.089601 0.086575 0.083945 0.081643 0.079610 0.077800 0.076171 0.074687 0.073316 0.072030 0.070799 0.069593 0.068379 0.067117 0.065750 0.064185 0.062245 0.059425 0.050522

0.022770 0.023105 0.023740 0.024427 0.025159 0.025933 0.026745 0.027592 0.028471 0.029382 0.030323 0.031293 0.032292 0.033320 0.034376 0.035463 0.036585 0.037748 0.038964 0.040251 0.041639 0.043181 0.044969 0.047203 0.050441

0.031089 0.031429 0.032079 0.032795 0.033580 0.034438 0.035375 0.036399 0.037521 0.038751 0.040105 0.041602 0.043268 0.045137 0.047258 0.049700 0.052570 0.056029 0.060340 0.065951 0.073691 0.085260 0.10478 0.14532 0.28170

159.29 162.31 167.40 172.22 176.77 181.06 185.07 188.78 192.15 195.15 197.76 199.92 201.62 202.81 203.46 203.53 202.98 201.77 199.84 197.14 193.61 189.17 183.76 177.26 169.47 0

Saturated Properties 22.518 22.330 22.002 21.674 21.346 21.016 20.683 20.345 20.002 19.653 19.295 18.927 18.548 18.155 17.747 17.320 16.870 16.394 15.885 15.335 14.733 14.060 13.287 12.352 11.072 7.4100 0.000057710 0.00012301 0.00039506 0.0010711 0.0025330 0.0053579 0.010340 0.018492 0.031038 0.049400 0.075184 0.11018 0.15639 0.21605 0.29173 0.38642 0.50374 0.64823 0.82581 1.0446 1.3165 1.6600 2.1078 2.7278 3.7183 7.4100

0.044409 0.044783 0.045451 0.046138 0.046847 0.047583 0.048349 0.049151 0.049994 0.050884 0.051827 0.052834 0.053914 0.055080 0.056348 0.057737 0.059276 0.060998 0.062953 0.065210 0.067877 0.071123 0.075261 0.080962 0.090320 0.13495 17,328. 8,129.4 2,531.3 933.62 394.79 186.64 96.715 54.078 32.218 20.243 13.301 9.0759 6.3942 4.6285 3.4278 2.5879 1.9852 1.5427 1.2109 0.95730 0.75961 0.60241 0.47442 0.36660 0.26894 0.13495

−6.2965 −5.8733 −5.1446 −4.4289 −3.7220 −3.0204 −2.3213 −1.6224 −0.92144 −0.21676 0.49332 1.2104 1.9359 2.6716 3.4192 4.1805 4.9579 5.7542 6.5728 7.4185 8.2981 9.2217 10.207 11.289 12.572 15.239

−6.2965 −5.8733 −5.1446 −4.4289 −3.7219 −3.0201 −2.3206 −1.6209 −0.91878 −0.21228 0.50050 1.2214 1.9524 2.6954 3.4525 4.2263 5.0196 5.8360 6.6798 7.5570 8.4758 9.4487 10.497 11.662 13.067 16.099

14.600 14.730 14.963 15.201 15.443 15.689 15.938 16.187 16.436 16.684 16.927 17.166 17.397 17.619 17.830 18.027 18.207 18.365 18.497 18.596 18.651 18.647 18.555 18.321 17.782 15.239

15.716 15.894 16.209 16.529 16.853 17.179 17.505 17.830 18.151 18.466 18.772 19.067 19.348 19.613 19.858 20.079 20.273 20.434 20.556 20.629 20.640 20.570 20.383 20.011 19.256 16.099

230.42 199.85 158.07 127.26 104.13 86.482 72.846 62.184 53.756 47.024 41.595 37.175 33.545 30.541 28.040 25.945 24.185 22.698 21.437 20.357 19.412 18.546 17.674 16.625 14.934 8.9233

Single-Phase Properties 150.00 200.00 222.70

0.10000 0.10000 0.10000

222.70 250.00 300.00 350.00

0.10000 0.10000 0.10000 0.10000

150.00 200.00 250.00 288.48

1.0000 1.0000 1.0000 1.0000

288.48 300.00 350.00

1.0000 1.0000 1.0000

150.00 200.00 250.00 300.00 350.00

5.0000 5.0000 5.0000 5.0000 5.0000

−5.1456 −1.6235 −0.025288

−5.1411 −1.6186 −0.020175

16.750 17.590 19.215 20.954

18.550 19.628 21.683 23.845

0.045423 0.049103 0.053843 0.059031

−5.1547 −1.6382 1.9164 4.8385

2.0650 2.1944 2.7038

22.068 20.446 18.707 16.644 13.583

22.003 20.347 19.557 0.055571 0.049051 0.040528 0.034596 22.015 20.365 18.572 16.940 0.48426 0.45571 0.36985

0.045448 0.049148 0.051133

−0.027941 −0.0076594 −0.000090227

0.072138 0.069990 0.070969

0.083293 0.087860 0.095346 0.10201

0.029634 0.030973 0.033493 0.035716

0.039104 0.040014 0.042188 0.044263

195.90 207.32 226.21 243.37

−5.1093 −1.5891 1.9703 4.8975

−0.028002 −0.0077331 0.0081385 0.019018

0.047344 0.043071 0.042163 0.042357

0.072112 0.069917 0.073223 0.079771

1379.4 1145.9 910.91 720.56

18.180 18.640 20.566

20.245 20.834 23.270

0.072221 0.074222 0.081735

0.036412 0.036112 0.036823

0.052100 0.050249 0.048032

203.11 209.62 233.01

0.045315 0.048909 0.053456 0.060080 0.073622

−5.1945 −1.7019 1.8087 5.5592 10.000

−4.9679 −1.4574 2.0760 5.8596 10.369

−0.028269 −0.0080546 0.0077029 0.021479 0.035328

0.047452 0.043169 0.042245 0.042516 0.044139

0.072006 0.069609 0.072395 0.080087 0.10909

1392.3 1163.4 935.94 701.33 422.84

−0.49145 −0.47508 −0.37256 −0.11429 0.86866

17.995 20.387 24.674 28.905

1376.4 1141.9 1035.1

−0.48934 −0.46739 −0.42755

0.047320 0.043049 0.042413

45.441 32.301 19.802 13.574 −0.48974 −0.46887 −0.35287 −0.14261 24.434 21.627 14.214

150.00 200.00 250.00 300.00 350.00

10.000 10.000 10.000 10.000 10.000

22.132 20.543 18.864 16.938 14.430

0.045183 0.048678 0.053012 0.059038 0.069300

−5.2426 −1.7780 1.6837 5.3293 9.3887

−4.7907 −1.2912 2.2138 5.9197 10.082

−0.028596 −0.0084436 0.0071898 0.020687 0.033492

0.047587 0.043292 0.042351 0.042551 0.043442

0.071886 0.069265 0.071522 0.077434 0.091456

1408.1 1184.3 965.13 746.52 517.30

−0.49341 −0.48206 −0.39347 −0.18740 0.36632

150.00 200.00 250.00 300.00 350.00

20.000 20.000 20.000 20.000 20.000

22.256 20.726 19.146 17.418 15.414

0.044932 0.048249 0.052230 0.057411 0.064877

−5.3340 −1.9196 1.4600 4.9523 8.6515

−4.4353 −0.95458 2.5046 6.1005 9.9491

−0.029231 −0.0091833 0.0062486 0.019350 0.031203

0.047854 0.043536 0.042574 0.042705 0.043268

0.071679 0.068683 0.070163 0.074036 0.080401

1438.4 1223.7 1018.0 821.57 634.40

−0.49677 −0.49387 −0.42634 −0.28388 −0.00046299

200.00 250.00 300.00 350.00

50.000 50.000 50.000 50.000

21.201 19.828 18.430 16.985

0.047167 0.050433 0.054259 0.058876

−2.2782 0.93091 4.1637 7.4493

0.080117 3.4526 6.8767 10.393

−0.011163 0.0038870 0.016368 0.027205

0.044246 0.043245 0.043321 0.043748

0.067514 0.067745 0.069343 0.071325

1327.4 1148.2 986.75 844.44

−0.51767 −0.48534 −0.42082 −0.32515

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The resulting equation has uncertainties of 0.1% in density in the liquid below 450 K, 1% in density at temperatures between 450 and 500 K, 3% in density at temperatures above 500 K, 0.5% in density in the vapor phase and at supercritical conditions below 10 MPa and 450 K, 0.5% in vapor pressure, and 2% in isobaric heat capacity.

2-247

2-248

PHYSICAL AND CHEMICAL DATA

TABLE 2-204 T, K 301.6m 400 500 600 700 800 900 1000 1200 1500

Saturated Cesium* P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.66.−9 3.83.−6 3.11.−4 5.65.−3 0.0440

5.444.−4 5.615.−4 5.800.−4 5.999.−4 6.215.−4

7.01.+7 6.54.+4 1001 65.63 9.671

74.6 98.5 122.0 144.9 167.0

637.6 651.9 666.1 678.4 688.9

0.696 0.765 0.817 0.859 0.893

2.563 2.148 1.905 1.748 1.638

0.245 0.240 0.232 0.224 0.219

0.2029 0.6620 1.693 6.790 27.6

6.443.−4 6.689.−4 6.954.−4 7.628.−4 8.84.−4

188.7 210.6 233.2 281.1 358.8

698.3 707.3 716.4 736.1 772.2

0.922 0.975 0.972 1.015 1.072

1.559 1.500 1.455 1.394 1.345

0.217 0.222 0.231 0.248 0.275

2.353 0.796 0.335 0.097 0.029

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.66.−9 signifies 2.66 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1550 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 327 psia, 1300−2700 °R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-205

Thermophysical Properties of Saturated Chlorine

T, °C

P, bar

vf , m3/kg

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

sf , kJ/(kg·K)

sg , kJ/(kg·K)

cpf , kJ/(kg·K)

cpg , kJ/(kg·K)

µf , 10−6 Pa·s

µg , 10−6 Pa·s

kf ,W/(m· K)

kg,W/(m· K)

Prf

Prg

−50 −40 −30 −20 −10

0.475 0.773 1.203 1.802 2.608

0.000 0.000 0.000 0.000 0.000

623 634 645 656 668

0.5448 0.3481 0.2314 0.1593 0.1134

221.5 231.0 240.6 250.3 260.0

518.2 522.2 526.1 529.9 533.9

1.7650 1.8074 1.8480 1.8869 1.9243

3.0946 3.0562 3.0223 2.9921 2.9649

0.9454 0.9474 0.9496 0.9520 0.9547

0.476 0.484 0.497 0.513 0.532

565 520 483 452 422

10.3 10.8 11.4 11.9 12.4

0.1684 0.1650 0.1613 0.1573 0.1527

0.0061 0.0065 0.0069 0.0074 0.0078

3.17 2.99 2.85 2.74 2.64

0.809 0.815 0.820 0.826 0.841

0 10 20 30 40

3.664 5.014 6.702 8.774 11.27

0.000 0.000 0.000 0.000 0.000

681 695 710 726 744

0.0829 0.0619 0.0471 0.0364 0.0286

269.7 279.4 289.2 299.0 308.8

537.4 540.5 543.3 545.7 548.0

1.9604 1.9953 2.0291 2.0622 2.0946

2.9402 2.9177 2.8924 2.8777 2.8593

0.9579 0.9618 0.9667 0.9728 0.9816

0.554 0.579 0.607 0.638 0.674

393 368 348 333 318

13.0 13.5 14.1 14.7 15.2

0.1478 0.1427 0.1378 0.1327 0.1282

0.0083 0.0088 0.0093 0.0099 0.0104

2.55 2.48 2.45 2.44 2.43

0.864 0.888 0.918 0.950 0.985

50 60 70 80 90

14.25 17.76 21.85 26.65 32.17

0.000 0.000 0.000 0.000 0.000

763 784 808 834 865

0.02276 0.01827 0.01481 0.01202 0.00972

318.6 329.1 340.0 351.4 364.1

549.8 551.2 552.1 552.5 552.4

2.1264 2.1578 2.1892 2.2207 2.2528

2.8417 2.8245 2.8074 2.7900 2.7714

0.9968 1.022 1.054 1.124 1.253

0.720 0.786 0.885 1.017 1.205

304 290 278 267 256

15.8 16.4 17.1 17.9 18.7

0.1230 0.1171 0.1122 0.1050 0.0986

0.0110 0.0117 0.0126 0.0137 0.0149

2.46 2.53 2.61 2.85 3.26

1.034 1.107 1.201 1.331 1.510

100 110 120 130 140

38.44 45.54 53.57 62.68 72.84

0.000 0.000 0.001 0.001 0.001

901 956 016 121 335

0.00789 0.00639 0.00508 0.00392 0.00282

377.8 391.3 407.1 426.1 451.1

551.0 548.8 543.7 535.0 517.3

2.2860 2.3207 2.3590 2.4032 2.4595

2.7502 2.7317 2.7064 2.6733 2.6198

1.418 1.632 1.891

1.434 1.696 1.960

247 238 230

19.5 20.6 22.2

0.0916 0.0850 0.0775

0.0163 0.0178 0.0195

3.82 4.57 5.61

1.700 1.96 2.23

144c

77.10

0.001 77

0.00177

483.1

483.1

2.5365

2.5365

c = critical point. Values interpolated and converted from Martin, J. J., 1977 (private communication), and from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983. Values of Ziegler, Chem.-Ing.Tech., 22 (1950): 229, apparently were also used in Landolt-Bornstein, IVa, (1967): 238–239, and in Ullmans Enzyklopädie der technische Chemie, 9, Verlag Chemie, Weinheim, 1975 (317–372).

2-249

PHYSICAL AND CHEMICAL DATA 10 Chlorine

8 6

200°C 180

4

2.

6

160

140 2.

7

2 120

100°C 2.

8

10 8 80 6

•

9

/kg kJ 0 . 3 60

2.

4

Satu rate d va por

Pressure, bar

2-250

2

K

0

3. t En

ro

py

40

20

1

3.

1 2

3.

0.8 0°C 0.6 –20

3.3

0.4

–40 3.4

0.2 –60

0.1

FIG. 2-9

500 Enthalpy–log-pressure diagram for chlorine.

3.5

550 Enthalpy, kJ/kg

600

THERMODYNAMIC PROPERTIES TABLE 2-206 T, K

P, bar

2-251

Saturated Chloroform (R20) vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg·K)

sg, kJ/(kg·K)

280 300 320 340 360

0.115 0.293 0.620 1.224 2.255

0.000 0.000 0.000 0.000 0.000

660 678 695 715 739

1.689 0.714 0.358 0.190 0.107

−46.0 −32.6 −13.4 5.2 23.3

219.5 230.6 241.1 252.1 263.0

−0.165 −0.105 −0.041 0.015 0.065

0.798 0.773 0.754 0.741 0.731

380 400 420 440 460

3.830 6.039 9.058 13.39 18.80

0.000 0.000 0.000 0.000 0.000

765 795 822 871 921

0.0653 0.0425 0.0288 0.0195 0.0137

41.7 61.4 82.8 106.1 131.6

273.7 284.2 294.2 303.6 311.2

0.114 0.165 0.217 0.270 0.325

480 500 520 530 536.6c

26.00 34.66 44.68 50.44 54.72

0.000 0.001 0.001 0.001 0.002

980 059 193 328 00

0.00962 0.00673 0.00467 0.00359 0.00200

157.4 186.2 219.6 242.7 284.1

316.5 320.8 321.3 315.7 284.1

0.380 0.436 0.499 0.540 0.602

cpf, kJ/(kg·K)

µf, 10−6 Pa·s

kf, W/(m·K)

Prf

1.03

748 587 468 381 319

0.120 0.114 0.109 0.103 0.095

3.35

0.725 0.722 0.721 0.719 0.716

1.07 1.11 1.15 1.21 1.32

273 237 206 177 155

0.0921 0.0863 0.0808 0.0750 0.0694

3.17 3.04 2.93 2.86 2.95

0.711 0.706 0.694 0.678 0.602

1.43 1.59

129.6 105.5 81.2 67.7

0.0641 0.0584 0.0518 0.0461

2.89 2.87

c = critical point. hf = sf = 0 at n.b.p., 334.5 K. P, v, h, and s interpolated from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, U.S.S.R. N.S.R.D.S. series, vol. 9., Hemisphere, 1987.

2-252

TABLE 2-207

Thermodynamic Properties of Cyclohexane

Temperature K

Pressure MPa

Density mol/dm3

279.47 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 440.00 450.00 460.00 470.00 480.00 490.00 500.00 510.00 520.00 530.00 540.00 550.00 553.64

0.0052538 0.0089097 0.014139 0.021670 0.032188 0.046481 0.065433 0.090023 0.12131 0.16044 0.20862 0.26711 0.33727 0.42046 0.51814 0.63180 0.76300 0.91333 1.0845 1.2781 1.4961 1.7402 2.0124 2.3150 2.6505 3.0222 3.4352 3.8928 4.0750

9.4045 9.2862 9.1736 9.0604 8.9462 8.8309 8.7142 8.5958 8.4756 8.3533 8.2285 8.1009 7.9701 7.8357 7.6970 7.5533 7.4037 7.2473 7.0828 6.9085 6.7225 6.5223 6.3048 6.0655 5.7961 5.4765 5.0307 4.1555 3.2438

279.47 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 440.00 450.00 460.00 470.00 480.00 490.00 500.00 510.00 520.00 530.00 540.00 550.00 553.64

0.0052538 0.0089097 0.014139 0.021670 0.032188 0.046481 0.065433 0.090023 0.12131 0.16044 0.20862 0.26711 0.33727 0.42046 0.51814 0.63180 0.76300 0.91333 1.0845 1.2781 1.4961 1.7402 2.0124 2.3150 2.6505 3.0222 3.4352 3.8928 4.0750

0.0022692 0.0037161 0.0057156 0.0085063 0.012293 0.017306 0.023800 0.032056 0.042386 0.055130 0.070668 0.089425 0.11188 0.13860 0.17021 0.20749 0.25135 0.30293 0.36362 0.43522 0.52006 0.62133 0.74350 0.89342 1.0826 1.3338 1.7017 2.3597 3.2438

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

1403.6 1314.7 1245.3 1185.7 1132.8 1084.4 1039.4 996.64 955.45 915.28 875.71 836.40 797.11 757.60 717.72 677.33 636.33 594.66 552.32 509.35 465.88 422.08 378.12 333.83 287.99 236.86 171.37 108.39 0

−0.50892 −0.47482 −0.44719 −0.42239 −0.39928 −0.37708 −0.35516 −0.33304 −0.31028 −0.28642 −0.26103 −0.23356 −0.20340 −0.16978 −0.13174 −0.087993 −0.036876 0.023882 0.097405 0.18807 0.30221 0.44929 0.64479 0.91793 1.3377 2.1175 4.2415 10.542 19.224

Saturated Properties 0.10633 0.10769 0.10901 0.11037 0.11178 0.11324 0.11476 0.11634 0.11799 0.11971 0.12153 0.12344 0.12547 0.12762 0.12992 0.13239 0.13507 0.13798 0.14119 0.14475 0.14875 0.15332 0.15861 0.16487 0.17253 0.18260 0.19878 0.24065 0.30828 440.68 269.10 174.96 117.56 81.348 57.785 42.017 31.195 23.593 18.139 14.151 11.183 8.9379 7.2152 5.8751 4.8195 3.9785 3.3011 2.7501 2.2977 1.9228 1.6095 1.3450 1.1193 0.92368 0.74972 0.58765 0.42378 0.30828

−12.076 −10.567 −9.0626 −7.4959 −5.8733 −4.1995 −2.4777 −0.71047 1.1006 2.9544 4.8503 6.7880 8.7677 10.790 12.855 14.964 17.119 19.321 21.572 23.876 26.236 28.655 31.140 33.699 36.350 39.135 42.215 46.176 48.978

−12.075 −10.566 −9.0611 −7.4935 −5.8697 −4.1942 −2.4702 −0.70000 1.1149 2.9736 4.8756 6.8210 8.8100 10.843 12.922 15.048 17.222 19.447 21.726 24.061 26.458 28.922 31.459 34.081 36.807 39.687 42.898 47.113 50.235

−0.038084 −0.032787 −0.027686 −0.022550 −0.017398 −0.012248 −0.0071081 −0.0019852 0.0031169 0.0081966 0.013253 0.018287 0.023301 0.028296 0.033275 0.038241 0.043199 0.048153 0.053108 0.058071 0.063050 0.068054 0.073094 0.078189 0.083372 0.088731 0.094586 0.10213 0.10770

0.094354 0.10358 0.11089 0.11715 0.12261 0.12749 0.13193 0.13604 0.13992 0.14362 0.14720 0.15069 0.15412 0.15751 0.16088 0.16425 0.16762 0.17101 0.17442 0.17788 0.18140 0.18499 0.18868 0.19249 0.19650 0.20083 0.20611 0.21549

0.13918 0.14714 0.15372 0.15959 0.16495 0.16991 0.17461 0.17911 0.18349 0.18781 0.19210 0.19642 0.20079 0.20528 0.20990 0.21472 0.21980 0.22522 0.23106 0.23747 0.24464 0.25284 0.26259 0.27496 0.29284 0.32631 0.43566 0.97375

19.476 20.430 21.380 22.370 23.397 24.459 25.555 26.684 27.842 29.029 30.242 31.480 32.739 34.019 35.315 36.625 37.945 39.270 40.597 41.918 43.228 44.519 45.779 46.995 48.145 49.186 50.020 50.334 48.978

21.791 22.828 23.854 24.917 26.015 27.145 28.305 29.492 30.704 31.939 33.194 34.467 35.754 37.053 38.359 39.670 40.980 42.285 43.579 44.855 46.105 47.319 48.486 49.586 50.593 51.452 52.039 51.984 50.235

0.083095 0.082366 0.082030 0.082001 0.082241 0.082719 0.083406 0.084277 0.085309 0.086481 0.087775 0.089174 0.090660 0.092220 0.093839 0.095502 0.097195 0.098905 0.10062 0.10231 0.10398 0.10560 0.10715 0.10859 0.10988 0.11093 0.11151 0.11099 0.10770

0.088889 0.093915 0.098579 0.10318 0.10775 0.11233 0.11692 0.12155 0.12623 0.13097 0.13577 0.14063 0.14557 0.15059 0.15569 0.16088 0.16615 0.17153 0.17701 0.18259 0.18828 0.19406 0.19990 0.20573 0.21142 0.21669 0.22086 0.22221

0.097282 0.10238 0.10715 0.11190 0.11667 0.12149 0.12641 0.13145 0.13663 0.14199 0.14756 0.15338 0.15949 0.16596 0.17286 0.18030 0.18840 0.19737 0.20746 0.21903 0.23261 0.24895 0.26922 0.29534 0.33088 0.38411 0.48310 0.96007

173.20 175.72 177.99 180.09 182.00 183.70 185.14 186.30 187.14 187.65 187.78 187.51 186.80 185.62 183.94 181.73 178.96 175.58 171.56 166.87 161.44 155.22 148.10 139.91 130.29 118.65 104.10 87.229 0

37.820 39.495 39.306 38.189 36.613 34.853 33.070 31.353 29.750 28.282 26.956 25.772 24.723 23.803 23.004 22.319 21.739 21.259 20.872 20.572 20.351 20.204 20.128 20.128 20.237 20.538 21.201 22.656 19.224

Single-Phase Properties −9.0661 −0.090837

−9.0552 −0.079147

−0.027698 −0.00022335

0.11091 0.13740

0.15371 0.18063

1245.8 982.29

−0.44737 −0.32529

27.080 33.301 49.598 69.610 92.840

29.907 36.546 53.708 74.568 98.639

0.084616 0.10224 0.14037 0.17831 0.21535

0.12316 0.14262 0.18205 0.21697 0.24653

0.13322 0.15196 0.19081 0.22554 0.25502

186.62 200.08 224.91 246.67 266.50

30.786 18.560 8.2156 4.7321 3.1661

0.10888 0.12522 0.13962

−9.1021 8.7122 20.490

−8.9932 8.8374 20.630

−0.027818 0.023162 0.050740

0.11106 0.15416 0.17279

0.15360 0.20043 0.22821

1250.6 803.90 572.63

0.33339 0.27733 0.21435 0.17821

2.9995 3.6058 4.6653 5.6112

39.963 48.369 68.879 92.287

42.963 51.975 73.544 97.898

0.099799 0.11868 0.15793 0.19542

0.17438 0.18654 0.21804 0.24682

0.20248 0.20332 0.22955 0.25703

173.56 198.76 233.28 258.70

21.046 11.678 5.2822 3.2649

9.2252 8.0768 6.5774 1.8230 1.0620

0.10840 0.12381 0.15204 0.54855 0.94165

−9.2570 8.3977 30.362 63.154 89.416

−8.7150 9.0168 31.122 65.897 94.125

−0.028339 0.022365 0.071493 0.13428 0.17787

0.11167 0.15438 0.18830 0.23018 0.24921

0.15315 0.19856 0.24756 0.32421 0.27380

1272.4 843.04 459.37 141.27 220.29

−0.45730 −0.23916 0.29542 9.9456 3.9618

300.00 353.45

0.10000 0.10000

9.1745 8.5546

353.45 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000

0.035368 0.030817 0.024334 0.020172 0.017244

300.00 400.00 455.22

1.0000 1.0000 1.0000

9.1840 7.9861 7.1625

455.22 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000

300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000

0.10900 0.11690 28.274 32.450 41.095 49.575 57.992

−0.44930 −0.20919 0.060445

300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000

9.2742 8.1782 6.8741 4.9059 2.6733

0.10783 0.12228 0.14547 0.20384 0.37407

−9.4403 8.0451 29.472 55.388 84.812

−8.3620 9.2679 30.927 57.426 88.552

−0.028964 0.021455 0.069619 0.11776 0.16576

0.11229 0.15456 0.18805 0.22275 0.25026

0.15264 0.19678 0.23749 0.30052 0.30095

1300.3 888.19 554.89 265.77 217.61

−0.46622 −0.26775 0.059727 1.5290 2.8239

300.00 400.00 500.00 600.00 700.00

25.000 25.000 25.000 25.000 25.000

9.4073 8.4285 7.3968 6.2973 5.1531

0.10630 0.11864 0.13519 0.15880 0.19406

−9.9312 7.1716 27.819 51.613 78.272

−7.2736 10.138 31.199 55.583 83.123

−0.030689 0.019116 0.065971 0.11034 0.15273

0.11356 0.15480 0.18805 0.21982 0.24808

0.15139 0.19342 0.22750 0.25993 0.28977

1386.2 1006.6 739.78 540.01 414.69

−0.48739 −0.32157 −0.16655 0.046982 0.35809

400.00 500.00 600.00 700.00

75.000 75.000 75.000 75.000

8.9762 8.2410 7.5637 6.9477

0.11141 0.12134 0.13221 0.14393

5.2440 25.023 47.767 73.174

13.599 34.124 57.683 83.969

0.013443 0.059111 0.10198 0.14245

0.15529 0.18956 0.22144 0.24950

0.18906 0.22091 0.24976 0.27541

1308.3 1084.3 931.19 824.46

−0.39015 −0.31328 −0.25806 −0.21348

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Penoncello, S. G., Goodwin, A. R. H., and Jacobsen, R. T., “A Thermodynamic Property Formulation for Cyclohexane,” Int. J. Thermophys. 16(2):519–531, 1995. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 2% in heat capacity, and 1% in the speed of sound, except in the critical region.

2-253

2-254 TABLE 2-208 Temperature K

Thermodynamic Properties of Decane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.18497 0.18621 0.19012 0.19417 0.19838 0.20279 0.20744 0.21239 0.21770 0.22343 0.22968 0.23659 0.24434 0.25317 0.26349 0.27594 0.29166 0.31306 0.34638 0.42098 0.60976

−69.252 −67.383 −61.516 −55.461 −49.202 −42.725 −36.021 −29.084 −21.908 −14.493 -6.8339 1.0694 9.2198 17.622 26.283 35.219 44.457 54.055 64.149 75.300 82.386

−69.252 −67.383 −61.516 −55.461 −49.202 −42.725 −36.020 −29.082 −21.905 −14.485 −6.8192 1.0950 9.2624 17.690 26.388 35.375 44.687 54.391 64.646 76.096 83.668

1,441,800. 724,920. 112,280. 23,769. 6,439.4 2,124.2 821.18 361.00 176.20 93.649 53.355 32.164 20.290 13.268 8.9134 6.0979 4.2045 2.8784 1.8993 1.0853 0.60976

−15.410 −14.150 −10.121 −5.8536 −1.3417 3.4167 8.4178 13.654 19.113 24.781 30.642 36.677 42.864 49.176 55.580 62.026 68.438 74.676 80.422 84.493 82.386

−13.386 −12.071 −7.8760 −3.4432 1.2331 6.1533 11.312 16.697 22.295 28.087 34.053 40.168 46.408 52.738 59.118 65.490 71.762 77.770 83.148 86.546 83.668

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.20307 −0.19549 −0.17292 −0.15129 −0.13043 −0.11018 −0.090462 −0.071189 −0.052306 −0.033764 −0.015527 0.0024370 0.020158 0.037667 0.054997 0.072193 0.089321 0.10649 0.12397 0.14276 0.15491

0.22393 0.22704 0.23712 0.24780 0.25893 0.27034 0.28191 0.29354 0.30512 0.31659 0.32790 0.33901 0.34991 0.36059 0.37108 0.38143 0.39176 0.40231 0.41372 0.42875

0.28635 0.28898 0.29787 0.30773 0.31832 0.32946 0.34100 0.35282 0.36483 0.37700 0.38931 0.40181 0.41460 0.42793 0.44218 0.45818 0.47770 0.50540 0.55950 0.87382

1468.5 1438.4 1349.4 1265.6 1186.0 1109.9 1036.6 965.68 896.48 828.54 761.40 694.57 627.57 559.85 490.80 419.70 345.68 267.65 184.05 91.074 0

−0.48339 −0.47764 −0.45826 −0.43680 −0.41360 −0.38881 −0.36228 −0.33365 −0.30225 −0.26699 −0.22626 −0.17752 −0.11677 −0.037250 0.073195 0.23865 0.51353 1.0476 2.4223 9.9501 34.686

144.08 142.30 136.88 131.57 126.38 121.32 116.40 111.63 107.04 102.62 98.409 94.401 90.612 87.051 83.726 80.640 77.794 75.192 72.934 73.097

2433.6 2066.6 1350.0 957.69 718.75 561.90 452.88 373.57 313.65 266.90 229.40 198.56 172.63 150.36 130.83 113.31 97.183 81.781 66.154 47.430

0.026360 0.025751 0.025744 0.028079 0.032268 0.037934 0.044772 0.052538 0.061027 0.070070 0.079524 0.089267 0.099191 0.10919 0.11917 0.12901 0.13855 0.14751 0.15533 0.15989 0.15491

0.19210 0.19573 0.20739 0.21963 0.23228 0.24518 0.25823 0.27134 0.28446 0.29754 0.31053 0.32343 0.33621 0.34890 0.36154 0.37421 0.38707 0.40046 0.41513 0.43359

0.20042 0.20405 0.21571 0.22797 0.24065 0.25363 0.26684 0.28022 0.29376 0.30748 0.32142 0.33567 0.35042 0.36600 0.38308 0.40300 0.42896 0.47018 0.56612 1.2127

121.84 123.41 128.09 132.58 136.85 140.87 144.56 147.82 150.52 152.52 153.66 153.78 152.66 150.07 145.68 139.04 129.53 116.14 97.344 71.032 0

Saturated Properties 243.50 250.00 270.00 290.00 310.00 330.00 350.00 370.00 390.00 410.00 430.00 450.00 470.00 490.00 510.00 530.00 550.00 570.00 590.00 610.00 617.70

1.4042E-06 2.8673E-06 1.9993E-05 0.00010141 0.00039985 0.0012883 0.0035240 0.0084305 0.018060 0.035300 0.063919 0.10855 0.17465 0.26846 0.39696 0.56801 0.79054 1.0751 1.4353 1.8918 2.1014

5.4064 5.3702 5.2597 5.1502 5.0409 4.9312 4.8206 4.7082 4.5936 4.4757 4.3538 4.2266 4.0927 3.9499 3.7952 3.6240 3.4286 3.1943 2.8870 2.3754 1.6400

243.50 250.00 270.00 290.00 310.00 330.00 350.00 370.00 390.00 410.00 430.00 450.00 470.00 490.00 510.00 530.00 550.00 570.00 590.00 610.00 617.70

1.4042E-06 2.8673E-06 1.9993E-05 0.00010141 0.00039985 0.0012883 0.0035240 0.0084305 0.018060 0.035300 0.063919 0.10855 0.17465 0.26846 0.39696 0.56801 0.79054 1.0751 1.4353 1.8918 2.1014

6.9358E-07 1.3795E-06 8.9067E-06 4.2071E-05 0.00015529 0.00047076 0.0012178 0.0027701 0.0056755 0.010678 0.018742 0.031091 0.049284 0.075371 0.11219 0.16399 0.23784 0.34741 0.52650 0.92143 1.6400

320.14 287.01 208.26 154.49 116.99 90.369 71.168 57.142 46.790 39.094 33.355 29.089 25.969 23.781 22.413 21.855 22.255 24.065 28.492 37.614 34.686

6.4788 6.7500 7.6856 8.7681 9.9889 11.339 12.808 14.385 16.056 17.808 19.629 21.506 23.438 25.431 27.522 29.799 32.477 36.104 42.347 59.648

4.3408 4.4520 4.7935 5.1337 5.4722 5.8086 6.1419 6.4713 6.7963 7.1172 7.4356 7.7555 8.0842 8.4344 8.8280 9.3033 9.9327 10.874 12.566 17.240

Single-Phase Properties −67.388 −36.029 −0.23039

−67.369 −36.008 −0.20685

−0.19551 −0.090485 −0.00046204

0.22705 0.28192 0.33722

0.28898 0.34098 0.39976

1438.9 1037.4 705.42

−0.47770 −0.36251 −0.18612

142.33 116.44 95.037

35.686 36.739 71.828 111.67

39.166 40.251 76.277 117.00

0.087670 0.090090 0.16222 0.23014

0.32134 0.32313 0.37514 0.41928

0.33333 0.33498 0.38484 0.42830

153.83 154.68 176.60 194.21

29.697 28.767 12.764 7.0015

21.198 21.529 32.572 44.613

0.18606 0.20711 0.23575 0.29042 0.30715

−67.431 −36.104 0.92464 44.348 51.697

−67.245 −35.897 1.1604 44.638 52.004

−0.19569 −0.090699 0.0021144 0.089120 0.10233

0.22712 0.28198 0.33902 0.39165 0.39972

0.28892 0.34075 0.40097 0.47560 0.49744

1443.8 1044.7 706.99 353.85 286.94

−0.47822 −0.36461 −0.18764 0.47267 0.87812

142.62 116.88 95.062 78.107 75.797

2093.8 458.58 201.71 98.236 85.478

0.31644 0.21839

3.1602 4.5790

73.196 109.33

76.356 113.91

0.14542 0.20734

0.39714 0.42517

0.45776 0.44558

119.81 167.63

23.453 8.8875

35.085 44.104

10.603 11.626

5.3914 4.8586 4.3034 3.6402 2.6391

0.18548 0.20582 0.23237 0.27471 0.37891

−67.618 −36.424 0.32894 42.808 91.156

−66.691 −35.395 1.4908 44.181 93.050

−0.19644 −0.091624 0.00076852 0.086242 0.16772

0.22744 0.28223 0.33912 0.39052 0.43712

0.28869 0.33985 0.39795 0.45616 0.52513

1465.1 1075.7 757.95 467.92 213.89

−0.48040 −0.37303 −0.22390 0.098466 1.6230

143.89 118.77 97.851 82.872 75.400

2204.8 481.47 215.38 114.20 55.304

4.8938 4.3700 3.7932 3.1242

0.20434 0.22883 0.26363 0.32009

−36.797 −0.32038 41.527 88.002

−34.754 1.9680 44.163 91.203

−0.092723 −0.00073280 0.083766 0.16225

0.28255 0.33929 0.39008 0.43362

0.33892 0.39532 0.44754 0.49145

1112.2 813.41 562.29 369.51

−0.38180 −0.25552 −0.060225 0.31317

121.04 101.03 87.304 81.597

510.17 231.67 129.37 76.394

250.00 350.00 446.75

0.10000 0.10000 0.10000

5.3706 4.8213 4.2477

446.75 450.00 550.00 650.00

0.10000 0.10000 0.10000 0.10000

0.028735 0.028470 0.022477 0.018766

250.00 350.00 450.00 550.00 565.17

1.0000 1.0000 1.0000 1.0000 1.0000

5.3745 4.8284 4.2417 3.4433 3.2558

565.17 650.00

1.0000 1.0000

250.00 350.00 450.00 550.00 650.00

5.0000 5.0000 5.0000 5.0000 5.0000

350.00 450.00 550.00 650.00

10.000 10.000 10.000 10.000

0.18620 0.20741 0.23542 34.801 35.125 44.491 53.287

2069.3 453.43 203.20 7.7032 7.7604 9.4815 11.158

350.00 450.00 550.00 650.00

100.00 100.00 100.00 100.00

5.3085 4.9747 4.6758 4.4066

0.18838 0.20102 0.21387 0.22693

−41.033 −6.2415 33.447 77.332

−22.195 13.860 54.834 100.03

−0.10704 −0.016697 0.065371 0.14078

0.28785 0.34377 0.39337 0.43497

0.33463 0.38610 0.43213 0.47043

1562.8 1360.6 1216.3 1111.9

−0.43121 −0.37226 −0.32986 −0.30079

152.54 139.48 130.26 124.76

1088.0 498.37 308.65 219.90

550.00 650.00

300.00 300.00

5.3265 5.1406

0.18774 0.19453

28.160 71.521

84.482 129.88

0.047128 0.12288

0.40016 0.44112

0.43412 0.47265

1858.1 1766.6

−0.34462 −0.32003

192.00 188.95

698.31 480.95

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W,. and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850, 2006. The source for viscosity is Huber, M. L., Laesecke, A., and Xiang, H. W, “Viscosity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-decane,” Fluid Phase Equilibria 224:263–270, 2004. The source for thermal conductivity is Huber, M. L., and Perkins, R. A., “Thermal Conductivity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-Decane,” Fluid Phase Equilibria 227:47–55, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.05% in the saturated liquid density between 290 and 320 K, 0.2% in the liquid phase at temperatures to 400 K (with somewhat higher uncertainties above 100 MPa, up to 0.5%), 1% in the liquid phase up to 500 MPa, and 2% at higher temperatures as well as in the vapor phase. Vapor pressures have an uncertainty of 0.2%, and the uncertainties in liquid heat capacities and liquid sound speeds are 1%. The uncertainty in heat capacities may be higher at pressures above 10 MPa. The estimated uncertainty in viscosity is 1% along the saturated liquid line, 2% in compressed liquid to 200 MPa, 5% in vapor and supercritical regions. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%.

2-255

2-256 TABLE 2-209

Thermodynamic Properties of Deuterium Oxide (Heavy Water)

Temperature K

Pressure MPa

Density mol/dm3

276.97 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 580.00 600.00 620.00 640.00 643.89

0.00066103 0.00082243 0.0030641 0.0094511 0.025012 0.058391 0.12292 0.23743 0.42676 0.72190 1.1598 1.7833 2.6406 3.7850 5.2758 7.1787 9.5679 12.530 16.171 20.654 21.660

55.198 55.214 55.126 54.780 54.254 53.593 52.817 51.937 50.957 49.875 48.687 47.386 45.961 44.390 42.641 40.660 38.357 35.563 31.867 24.976 17.875

276.97 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 580.00 600.00 620.00 640.00 643.89

0.00066103 0.00082242 0.0030641 0.0094511 0.025012 0.058391 0.12292 0.23743 0.42676 0.72190 1.1598 1.7833 2.6406 3.7850 5.2758 7.1787 9.5679 12.530 16.171 20.654 21.660

0.00028721 0.00035350 0.0012303 0.0035636 0.0089021 0.019720 0.039608 0.073418 0.12741 0.20947 0.32939 0.49953 0.73576 1.0594 1.5008 2.1062 2.9537 4.1942 6.2033 11.096 17.875

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−8.2294 −7.9735 −6.2759 −4.5798 −2.8930 −1.2168 0.45077 2.1139 3.7786 5.4521 7.1417 8.8551 10.601 12.390 14.236 16.162 18.203 20.425 22.980 26.745 29.926

−8.2294 −7.9735 −6.2758 −4.5796 −2.8926 −1.2157 0.45310 2.1185 3.7870 5.4666 7.1655 8.8927 10.658 12.475 14.360 16.339 18.453 20.777 23.487 27.572 31.137

36.006 36.083 36.589 37.091 37.585 38.064 38.521 38.946 39.329 39.660 39.925 40.116 40.219 40.219 40.096 39.815 39.318 38.492 37.081 33.747 29.926

38.307 38.409 39.079 39.743 40.394 41.025 41.624 42.180 42.679 43.106 43.447 43.686 43.808 43.792 43.612 43.224 42.557 41.479 39.688 35.608 31.137

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.025460 −0.024541 −0.018685 −0.013212 −0.0080990 −0.0033082 0.0011999 0.0054654 0.0095268 0.013420 0.017176 0.020824 0.024390 0.027902 0.031394 0.034909 0.038511 0.042313 0.046577 0.052801 0.058280

0.084185 0.084523 0.084432 0.082466 0.079894 0.077173 0.074503 0.071990 0.069689 0.067622 0.065790 0.064181 0.062780 0.061580 0.060585 0.059829 0.059403 0.059524 0.060792 0.066781

0.084334 0.084574 0.084957 0.084598 0.084075 0.083580 0.083248 0.083196 0.083502 0.084212 0.085360 0.087004 0.089266 0.092393 0.096848 0.10351 0.11428 0.13431 0.18616 0.82671

1324.3 1332.9 1394.9 1436.7 1455.5 1455.6 1440.4 1413.3 1376.7 1331.9 1279.7 1219.9 1152.3 1076.5 991.90 897.91 793.19 674.99 537.16 359.72 0

−0.22218 −0.21843 −0.19991 −0.18770 −0.17806 −0.16908 −0.15944 −0.14815 −0.13444 −0.11766 −0.097022 −0.071339 −0.038642 0.0043364 0.063056 0.14707 0.27477 0.48803 0.91327 2.2860 3.7063

564.56 569.58 596.96 615.88 628.04 634.49 635.99 633.11 626.31 616.01 602.53 586.08 566.82 544.81 520.11 492.75 462.70 430.22 396.40 420.92

2085.6 1868.6 1046.6 688.90 497.65 381.94 306.06 253.36 215.11 186.33 164.01 146.23 131.71 119.53 109.01 99.548 90.597 81.477 70.992 53.746

0.14256 0.14111 0.13250 0.12530 0.11922 0.11403 0.10955 0.10562 0.10213 0.098964 0.096048 0.093309 0.090688 0.088127 0.085564 0.082917 0.080069 0.076817 0.072707 0.065358 0.058280

0.025895 0.025952 0.026375 0.026887 0.027487 0.028182 0.029002 0.030006 0.031275 0.032903 0.034981 0.037576 0.040723 0.044416 0.048619 0.053289 0.058406 0.064014 0.070293 0.077762

0.034265 0.034329 0.034822 0.035451 0.036239 0.037225 0.038484 0.040139 0.042357 0.045350 0.049371 0.054720 0.061780 0.071116 0.083703 0.10149 0.12892 0.17822 0.29956 1.5264

389.85 391.88 404.89 417.19 428.74 439.47 449.26 457.95 465.33 471.15 475.13 476.97 476.36 472.96 466.44 456.44 442.43 423.35 395.96 343.38 0

Saturated Properties 0.018117 0.018111 0.018140 0.018255 0.018432 0.018659 0.018933 0.019254 0.019624 0.020050 0.020539 0.021103 0.021758 0.022528 0.023452 0.024594 0.026071 0.028119 0.031381 0.040039 0.055943 3481.8 2828.9 812.82 280.61 112.33 50.709 25.247 13.621 7.8485 4.7740 3.0359 2.0019 1.3591 0.94392 0.66632 0.47479 0.33856 0.23842 0.16120 0.090120 0.055943

394.78 363.66 220.57 143.76 100.20 74.063 57.443 46.206 38.108 31.903 26.904 22.747 19.244 16.294 13.819 11.743 9.9759 8.4154 6.9363 5.1241 3.7063

16.529 16.753 18.277 19.890 21.616 23.487 25.539 27.812 30.356 33.231 36.529 40.398 45.081 50.974 58.649 68.218 79.570 98.667 133.97 285.74

9.6017 9.6779 10.228 10.840 11.492 12.169 12.856 13.544 14.226 14.897 15.558 16.213 16.871 17.547 18.267 19.071 20.029 21.290 23.259 28.595

Single-Phase Properties −6.2760 −0.032221

−6.2742 −0.030336

38.391 39.139 42.023 45.017 48.166 51.481

41.454 42.427 46.162 49.996 53.980 58.128

0.018132 0.019246 0.020374

−6.2770 2.1086 6.5927

0.28575 0.25193 0.20474 0.17382 0.15139 55.254 52.073 46.085 42.947

3.4996 3.9693 4.8843 5.7529 6.6052 0.018098 0.019204 0.021699 0.023284

1.4172 1.1342 0.91389 0.77903

300.00 374.20

0.10000 0.10000

374.20 400.00 500.00 600.00 700.00 800.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

300.00 400.00 453.53

1.0000 1.0000 1.0000

55.152 51.959 49.083

453.53 500.00 600.00 700.00 800.00 300.00 400.00 500.00 536.66

1.0000 1.0000 1.0000 1.0000 1.0000 5.0000 5.0000 5.0000 5.0000

536.66 600.00 700.00 800.00

5.0000 5.0000 5.0000 5.0000

300.00 400.00 500.00 583.19

10.000 10.000 10.000 10.000

583.19 600.00 700.00 800.00

10.000 10.000 10.000 10.000

300.00 400.00 500.00 600.00 700.00 800.00

50.000 50.000 50.000 50.000 50.000 50.000

300.00 700.00 800.00

100.00 100.00 100.00

55.129 53.053

−0.018686 −0.000080962

0.084951 0.083320

0.11078 0.11330 0.12163 0.12861 0.13475 0.14029

0.028748 0.028403 0.029147 0.030636 0.032285 0.033959

0.038084 0.037410 0.037685 0.039048 0.040652 0.042306

−6.2589 2.1279 6.6131

−0.018689 0.0054522 0.015974

0.084361 0.071968 0.066357

0.084894 0.083157 0.084938

1397.1 1415.4 1297.4

−0.19988 −0.14835 −0.10418

597.49 633.54 607.22

1046.3 253.63 170.66

39.847 41.482 44.748 48.001 51.363 −6.2816 2.0813 10.558 13.923

43.347 45.451 49.632 53.754 57.969 −6.1911 2.1773 10.667 14.039

0.096969 0.10139 0.10902 0.11537 0.12100 −0.018705 0.0053836 0.024304 0.030810

0.034254 0.032240 0.031708 0.032710 0.034141 0.084079 0.071858 0.062722 0.060736

0.047940 0.043385 0.041087 0.041565 0.042787 0.084647 0.082958 0.088880 0.095981

474.06 503.79 556.10 600.64 640.67 1405.7 1426.2 1163.3 1006.7

28.417 19.634 10.106 6.1060 4.1567 −0.19974 −0.14937 −0.042258 0.051819

35.408 38.239 48.345 60.651 74.254 599.61 635.76 568.91 524.42

15.345 17.445 21.886 26.233 30.468 1045.2 255.02 132.48 110.68

0.70563 0.88170 1.0942 1.2836

40.127 43.329 47.210 50.818

43.655 47.738 52.681 57.237

0.085995 0.093213 0.10085 0.10693

0.047884 0.038071 0.034830 0.034981

0.081310 0.054713 0.046443 0.045153

467.76 524.33 584.32 631.66

57.219 53.862 63.474 76.389

18.142 21.508 26.310 30.769

0.018057 0.019152 0.021579 0.026351

−6.2874 2.0479 10.471 18.544

−6.1068 2.2394 10.687 18.807

−0.018725 0.0052992 0.024128 0.039101

0.083735 0.071726 0.062611 0.059377

0.084346 0.082718 0.088119 0.11663

1415.9 1439.3 1185.7 775.32

−0.19956 −0.15059 −0.049352 0.30140

0.32053 0.36059 0.50864 0.61793

39.212 40.719 46.072 50.085

42.417 44.325 51.158 56.265

0.079584 0.082812 0.093421 0.10025

0.059266 0.051641 0.038049 0.036102

0.13479 0.097980 0.054970 0.048671

439.77 468.72 562.64 621.09

9.7157 9.0626 5.9486 4.1215

56.338 53.254 48.076 40.498 26.703 12.014

0.017750 0.018778 0.020801 0.024693 0.037449 0.083233

−6.3323 1.8073 9.8899 18.559 29.678 42.333

−5.4448 2.7462 10.930 19.794 31.551 46.495

−0.018905 0.0046700 0.022920 0.039048 0.057059 0.077124

0.081303 0.070872 0.062103 0.057208 0.055751 0.045204

0.082295 0.081058 0.083801 0.096053 0.15684 0.10482

1484.2 1527.2 1327.9 997.60 588.37 569.23

−0.19786 −0.15849 −0.088396 0.10313 1.0407 2.4839

622.24 659.16 603.54 493.46 331.10 169.04

1038.3 269.56 145.01 98.391 62.248 40.739

57.439 35.855 26.322

0.017410 0.027890 0.037991

−6.3859 25.888 34.906

−4.6449 28.677 38.706

−0.019168 0.050728 0.064110

0.078994 0.052858 0.045160

0.080433 0.095839 0.10162

1546.7 944.96 785.38

−0.19571 0.21724 0.69848

645.56 435.02 323.29

1041.5 84.790 66.016

55.380 52.214 46.341 37.950 3.1198 2.7732 1.9660 1.6183

30.631 32.883 41.395 49.784 58.134 66.469

1395.1 1446.2

−0.19991 −0.16237

0.084425 0.075265

0.032646 0.030411 0.024157 0.020087 0.017202 0.015045

0.018139 0.018849

446.53 462.35 515.85 562.30 604.21 642.79

61.573 46.430 19.445 10.077 6.0901 4.1486

14.202 9.9010 6.1037 4.1656

597.01 636.03 24.923 27.024 36.485 47.629 60.182 73.869

602.23 638.50 573.22 457.66 81.935 72.595 69.164 80.100

1046.5 325.15 12.656 13.658 17.747 21.980 26.224 30.405

1043.9 256.74 134.06 89.173 20.204 21.210 26.519 31.206

2-257

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Hill, P. G., MacMillan, R. D. C., and Lee, V., “A Fundamental Equation of State for Heavy Water,” J. Phys. Chem. Ref. Data, 11(1):1–14, 1982. The source for viscosity is International Association for the Properties of Water and Steam, “Viscosity and Thermal Conductivity of Heavy Water Substance,” Physical Chemistry of Aqueous Systems: Proceedings of the 12th International Conference on the Properties of Water and Steam, Orlando, Fla., Sept. 11–16, 1994, A107-A138. The source for thermal conductivity is International Association for the Properties of Water and Steam, “Viscosity and Thermal Conductivity of Heavy Water Substance,” Physical Chemistry of Aqueous Systems: Proceedings of the 12th International Conference on the Properties of Water and Steam, Orlando, Fla., Sept. 11–16, 1994, A107-A138. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. For a discussion of the uncertainties associated with the equation of state and thermal conductivity entries of this table, please see the source references given above. The uncertainty in viscosity is 1% in the liquid below 474 K, 2% in the liquid at higher temperatures and in the vapor, and 5% between 623 and 723 K at pressures between 16 and 50 MPa. The uncertainty in viscosity is 2% in the liquid below 623 K and in the vapor below 573 K, 5% elsewhere in the liquid and vapor, and 10% in the critical region (623 to 723 K and 21.66 to 50 MPa).

2-258 TABLE 2-210 Temperature K

Thermodynamic Properties of 2,2-Dimethylpropane (Neopentane) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.11492 0.11554 0.11745 0.11946 0.12160 0.12389 0.12636 0.12903 0.13194 0.13515 0.13872 0.14273 0.14733 0.15270 0.15913 0.16712 0.17764 0.19306 0.22442 0.30581

−4.0448 −3.5333 −2.0036 −0.43520 1.1729 2.8218 4.5127 6.2470 8.0264 9.8530 11.729 13.659 15.647 17.699 19.824 22.038 24.368 26.878 29.850 32.506

−4.0407 −3.5286 −1.9963 −0.42422 1.1889 2.8444 4.5439 6.2892 8.0826 9.9266 11.825 13.781 15.802 17.894 20.069 22.345 24.755 27.373 30.525 33.484

18.019 18.336 19.288 20.267 21.271 22.298 23.346 24.411 25.490 26.580 27.676 28.773 29.862 30.932 31.969 32.946 33.817 34.476 34.498 32.506

20.107 20.447 21.461 22.498 23.554 24.625 25.710 26.803 27.901 28.999 30.091 31.170 32.226 33.246 34.211 35.088 35.822 36.287 35.972 33.484

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.014961 −0.012981 −0.0072081 −0.0015039 0.0041396 0.0097305 0.015276 0.020784 0.026263 0.031720 0.037165 0.042610 0.048067 0.053555 0.059099 0.064738 0.070543 0.076679 0.083875 0.090610

0.11106 0.11212 0.11525 0.11839 0.12153 0.12467 0.12781 0.13096 0.13411 0.13728 0.14047 0.14369 0.14696 0.15032 0.15380 0.15752 0.16164 0.16662 0.17440

0.14987 0.15116 0.15504 0.15903 0.16315 0.16742 0.17188 0.17655 0.18151 0.18683 0.19263 0.19910 0.20651 0.21536 0.22655 0.24199 0.26680 0.32099 0.65083

1061.0 1043.4 992.29 941.76 891.68 841.93 792.35 742.82 693.20 643.34 593.09 542.29 490.74 438.21 384.34 328.55 269.76 205.53 129.02 0

−0.45488 −0.44532 −0.41587 −0.38403 −0.34919 −0.31054 −0.26700 −0.21713 −0.15892 −0.089543 −0.0048767 0.10132 0.23893 0.42456 0.68847 1.0924 1.7855 3.2489 8.3551 19.128

0.079147 0.079232 0.079672 0.080361 0.081260 0.082334 0.083553 0.084890 0.086318 0.087815 0.089355 0.090912 0.092458 0.093957 0.095360 0.096596 0.097537 0.097901 0.096542 0.090610

0.097392 0.098838 0.10307 0.10727 0.11146 0.11562 0.11978 0.12394 0.12810 0.13228 0.13648 0.14073 0.14505 0.14948 0.15407 0.15892 0.16422 0.17039 0.17895

0.10663 0.10818 0.11278 0.11744 0.12220 0.12709 0.13214 0.13741 0.14299 0.14898 0.15554 0.16293 0.17158 0.18223 0.19633 0.21707 0.25302 0.33783 0.86890

176.08 176.77 178.55 179.94 180.88 181.34 181.26 180.58 179.24 177.18 174.30 170.51 165.68 159.65 152.23 143.15 132.06 118.40 100.88 0

Saturated Properties 256.60 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 433.74

0.035401 0.041178 0.062611 0.091973 0.13106 0.18184 0.24636 0.32680 0.42545 0.54466 0.68688 0.85466 1.0507 1.2778 1.5390 1.8380 2.1788 2.5668 3.0111 3.1963

8.7017 8.6547 8.5146 8.3711 8.2236 8.0715 7.9140 7.7503 7.5792 7.3993 7.2090 7.0060 6.7873 6.5488 6.2842 5.9837 5.6295 5.1798 4.4559 3.2700

256.60 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 433.74

0.035401 0.041178 0.062611 0.091973 0.13106 0.18184 0.24636 0.32680 0.42545 0.54466 0.68688 0.85466 1.0507 1.2778 1.5390 1.8380 2.1788 2.5668 3.0111 3.1963

0.016951 0.019508 0.028804 0.041222 0.057423 0.078145 0.10423 0.13664 0.17651 0.22521 0.28448 0.35651 0.44433 0.55219 0.68654 0.85798 1.0863 1.4175 2.0429 3.2700

58.993 51.262 34.717 24.259 17.415 12.797 9.5944 7.3187 5.6655 4.4402 3.5152 2.8049 2.2506 1.8110 1.4566 1.1655 0.92056 0.70547 0.48949 0.30581

42.543 40.673 35.912 32.069 28.950 26.412 24.346 22.672 21.330 20.277 19.485 18.940 18.642 18.607 18.876 19.520 20.655 22.431 24.389 19.128

Single-Phase Properties 300.00 350.00 400.00 450.00 500.00 550.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

0.041573 0.035109 0.030488 0.026981 0.024215 0.021973

300.00 350.00 367.56

1.0000 1.0000 1.0000

8.0927 7.2257 6.8424

367.56 400.00 450.00 500.00 550.00

1.0000 1.0000 1.0000 1.0000 1.0000

300.00 350.00 400.00 450.00 500.00 550.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

300.00 350.00 400.00 450.00 500.00 550.00

10.000 10.000 10.000 10.000 10.000 10.000

24.054 28.482 32.800 37.063 41.296 45.510

22.482 28.668 35.683 43.493 52.060 61.352

24.888 31.516 38.963 47.199 56.190 65.903

0.087934 0.10834 0.12820 0.14758 0.16652 0.18502

0.11446 0.13169 0.14811 0.16371 0.17858 0.19273

0.12428 0.14083 0.15694 0.17238 0.18715 0.20124

186.71 203.23 217.95 231.47 244.10 256.02

24.744 14.420 9.4234 6.6348 4.9220 3.7932

0.12357 0.13839 0.14615

2.7638 11.687 15.155

2.8874 11.825 15.301

0.0095365 0.037043 0.046731

0.12470 0.14045 0.14616

0.16698 0.19204 0.20459

852.01 599.54 503.42

−0.31857 −0.017539 0.20165

0.42120 0.35635 0.29713 0.25832 0.22988

2.3741 2.8062 3.3655 3.8712 4.3501

29.597 34.611 42.712 51.444 60.842

31.971 37.418 46.077 55.315 65.192

0.092083 0.10628 0.12667 0.14613 0.16494

0.14399 0.15145 0.16541 0.17958 0.19337

0.16932 0.16863 0.17859 0.19109 0.20397

166.96 188.60 211.98 230.27 245.89

18.691 12.099 7.6415 5.3700 4.0038

8.1890 7.4104 6.4044 4.6105 1.9914 1.4527

0.12211 0.13495 0.15614 0.21690 0.50215 0.68839

2.4999 11.213 21.099 33.057 47.248 57.986

3.1104 11.887 21.879 34.142 49.759 61.428

0.0086423 0.035660 0.062302 0.091100 0.12411 0.14637

0.12485 0.14039 0.15611 0.17461 0.18590 0.19664

0.16516 0.18654 0.21533 0.30156 0.24909 0.22704

897.98 671.51 444.29 198.33 166.16 205.52

−0.35167 −0.13585 0.37314 3.5214 7.2775 4.5988

8.2958 7.5920 6.7750 5.7764 4.5715 3.4249

0.12054 0.13172 0.14760 0.17312 0.21875 0.29198

2.2072 10.739 20.216 30.691 42.114 53.968

3.4126 12.056 21.692 32.422 44.301 56.887

0.0076277 0.034237 0.059940 0.085190 0.11020 0.13419

0.12506 0.14048 0.15568 0.17074 0.18530 0.19832

0.16345 0.18250 0.20328 0.22624 0.24770 0.25208

949.31 743.85 555.11 391.52 276.41 234.55

−0.38267 −0.22237 0.041740 0.55851 1.5460 2.4058

400.00 450.00 500.00 550.00

100.00 100.00 100.00 100.00

8.5990 8.2554 7.9290 7.6203

0.11629 0.12113 0.12612 0.13123

15.633 24.735 34.489 44.867

27.262 36.848 47.101 57.990

0.045388 0.067954 0.089546 0.11029

0.15862 0.17287 0.18665 0.19993

0.18487 0.19848 0.21152 0.22395

1257.1 1172.6 1103.1 1046.3

−0.42302 −0.38739 −0.35738 −0.33233

550.00

200.00

8.6161

0.11606

42.742

65.955

0.10250

0.20230

0.22337

1444.9

−0.36818

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density in the equation of state range from 0.2% in the liquid phase at pressures less than 10 MPa to 1% in the liquid phase at higher pressures (up to 200 MPa) and at temperatures above the critical point (up to 550 K). The uncertainty in density in the vapor phase is 0.5%. Uncertainties in other properties are 0.1% for the vapor pressure, 2% in liquid-phase heat capacities, 0.5% in vapor-phase heat capacities, 1% for liquid-phase sound speeds, and 0.02% for vapor-phase sound speeds.

2-259

2-260

PHYSICAL AND CHEMICAL DATA

TABLE 2-211

Saturated Diphenyl* hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0 13.0 30.0 47.2 65.0

444.2 444.2 446.7 449.7 454.5

0.000 0.036 0.084 0.130 0.178

1.298 1.266 1.236 1.213 1.200

1.760 1.782 1.813 1.844 1.875

15.0 13.5 11.7 10.3 9.1

0.139 0.138 0.136 0.135 0.133

82.7 99.3 139.9 180.3 222.7

462.7 461.2 499.0 532.4 569.7

0.224 0.273 0.358 0.451 0.545

1.194 1.202 1.228 1.267 1.378

1.906 1.936 1.998 2.060 2.122

8.1 7.3 6.0 5.0 4.3

0.132 0.130 0.127 0.125 0.122

0.9594 0.4452 0.3652 0.2261 0.1447

267.6 314.9 361.5 404.5 457.2

611.6 651.8 687.8 723.8 762.7

0.652 0.746 0.824 0.915 1.032

1.367 1.424 1.477 1.529 1.582

2.184 2.246 2.308 2.370 2.432

3.7 3.3 2.7 2.4 2.2

0.119 0.116 0.113 0.110 0.107

1.258.−3 1.291.−3 1.326.−3 1.366.−3 1.412.−3

0.0977 0.0685 0.0504 0.0381 0.0301

522.3 563.7 630.4 689.1 745.9

801.7 842.4 886.4 930.9 977.1

1.125 1.223 1.316 1.375 1.457

1.635 1.688 1.740 1.748 1.791

2.494 2.556 2.618 2.680 2.741

1.90 1.71 1.54 1.39 1.24

0.105 0.102 0.099 0.096 0.093

2.803 2.865 2.93 3.00

1.10 0.97

0.090 0.087

T, K

P, bar

vf, m3/kg

vg, m3/kg

343 350 360 370 380

0.0010 0.0016 0.0029 0.0049 0.0064

1.010.−3 1.014.−3 1.021.−3 1.030.−3 1.037.−3

252.5 156.1 85.0 49.9 29.9

390 400 420 440 460

0.0129 0.0200 0.0432 0.0879 0.1694

1.046.−3 1.054.−3 1.072.−3 1.092.−3 1.112.−3

480 500 520 540 560

0.3112 0.5218 0.8375 1.290 1.941

1.132.−3 1.154.−3 1.177.−3 1.204.−3 1.230.−3

580 600 620 640 660

2.818 3.926 5.408 7.328 9.572

18.3 11.7 5.84 3.021 1.652

hf, kJ/kg

680 700 720 740 760

12.05 15.21 19.14 23.93 28.71

1.465.−3 1.529.−3 1.56.−3 1.70.−3 1.95.−3

0.0236 0.0186 0.0147 0.0113 0.0085

802.8 860.1 917.5 975.2 1033.1

1024.9 1073.1 1116.7 1152.8 1182.5

1.585 1.663 1.746 1.822 1.901

1.856 1.951 2.003 2.058 2.099

780 800

34.83 42.46

2.16.−3 3.18.−3

0.0058 0.0032

1091.2 1148.4

1163.0 1148.4

1.977 2.047

2.107 2.047

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 557, 1967 tables based on Technical Data on Fuel, British National Committee, World Energy Conference, London.

TABLE 2-212

Thermodynamic Properties of Dodecane

Temperature K

Pressure MPa

263.60 270.00 290.00 310.00 330.00 350.00 370.00 390.00 410.00 430.00 450.00 470.00 490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00 658.10

6.2621E-07 1.2617E-06 8.8545E-06 4.5799E-05 0.00018556 0.00061679 0.0017432 0.0043088 0.0095271 0.019190 0.035739 0.062299 0.10268 0.16136 0.24347 0.35486 0.50211 0.69279 0.93585 1.2426 1.6298 1.8176

4.5291 4.5008 4.4131 4.3261 4.2393 4.1522 4.0641 3.9747 3.8833 3.7893 3.6920 3.5907 3.4842 3.3713 3.2503 3.1189 2.9736 2.8091 2.6161 2.3747 2.0078 1.3300

263.60 270.00 290.00 310.00 330.00 350.00 370.00 390.00 410.00 430.00 450.00 470.00 490.00 510.00 530.00 550.00 570.00 590.00 610.00 630.00 650.00 658.10

6.2621E-07 1.2617E-06 8.8545E-06 4.5799E-05 0.00018556 0.00061679 0.0017432 0.0043088 0.0095271 0.019190 0.035739 0.062299 0.10268 0.16136 0.24347 0.35486 0.50211 0.69279 0.93585 1.2426 1.6298 1.8176

2.8572E-07 5.6205E-07 3.6724E-06 1.7772E-05 6.7670E-05 0.00021228 0.00056862 0.0013382 0.0028309 0.0054855 0.0098881 0.016797 0.027184 0.042311 0.063865 0.094219 0.13692 0.19777 0.28746 0.43012 0.71569 1.3300

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−96.621 −94.309 −86.960 −79.402 −71.612 −63.574 −55.276 −46.712 −37.875 −28.762 −19.371 −9.6999 0.25270 10.490 21.016 31.839 42.975 54.448 66.310 78.676 92.046 101.36

−96.621 −94.309 −86.960 −79.402 −71.612 −63.574 −55.276 −46.711 −37.872 −28.757 −19.361 −9.6825 0.28217 10.538 21.091 31.953 43.144 54.695 66.667 79.199 92.858 102.72

−33.699 −32.120 −26.993 −21.566 −15.834 −9.7983 −3.4632 3.1615 10.063 17.224 24.628 32.254 40.078 48.075 56.215 64.460 72.758 81.036 89.170 96.902 103.34 101.36

−31.507 −29.875 −24.582 −18.989 −13.092 −6.8927 −0.39757 6.3814 13.428 20.723 28.242 35.962 43.855 51.889 60.027 68.226 76.425 84.539 92.425 99.791 105.62 102.72

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.26003 −0.25136 −0.22511 −0.19991 −0.17556 −0.15192 −0.12887 −0.10633 −0.084231 −0.062532 −0.041187 −0.020162 0.00057539 0.021053 0.041300 0.061352 0.081249 0.10105 0.12086 0.14087 0.16193 0.17683

0.28923 0.29275 0.30440 0.31679 0.32968 0.34285 0.35616 0.36947 0.38269 0.39574 0.40859 0.42118 0.43351 0.44557 0.45738 0.46897 0.48040 0.49179 0.50334 0.51558 0.53081

0.35985 0.36265 0.37246 0.38355 0.39559 0.40831 0.42150 0.43503 0.44878 0.46268 0.47672 0.49091 0.50530 0.52003 0.53534 0.55164 0.56974 0.59128 0.62041 0.67253 0.91317

1427.2 1398.7 1313.5 1233.4 1157.5 1085.1 1015.6 948.42 883.07 819.18 756.37 694.32 632.70 571.20 509.51 447.33 384.30 320.04 253.82 183.58 100.02 0

−0.45503 −0.45038 −0.43398 −0.41526 −0.39464 −0.37230 −0.34817 −0.32197 −0.29316 −0.26089 −0.22391 −0.18039 −0.12755 −0.061022 0.026306 0.14688 0.32430 0.60884 1.1288 2.3418 8.0937 40.219

143.93 142.27 137.24 132.45 127.86 123.43 119.12 114.92 110.81 106.77 102.79 98.863 94.976 91.129 87.324 83.568 79.876 76.283 72.856 69.764 68.073

2892.5 2447.1 1578.5 1112.9 832.98 650.53 524.02 431.83 361.82 306.78 262.21 225.20 193.79 166.63 142.76 121.46 102.19 84.516 68.050 52.302 35.652

−0.013011 −0.012718 −0.010011 −0.0050291 0.0017708 0.010028 0.019454 0.029808 0.040891 0.052536 0.064598 0.076955 0.089500 0.10213 0.11477 0.12730 0.13964 0.15163 0.16308 0.17355 0.18156 0.17683

0.24443 0.24903 0.26382 0.27906 0.29454 0.31008 0.32559 0.34098 0.35619 0.37119 0.38596 0.40048 0.41475 0.42878 0.44259 0.45625 0.46984 0.48353 0.49764 0.51290 0.53205

0.25274 0.25734 0.27214 0.28739 0.30288 0.31848 0.33408 0.34965 0.36515 0.38060 0.39603 0.41151 0.42715 0.44316 0.45987 0.47792 0.49854 0.52443 0.56276 0.64061 1.0334

Entropy kJ/(molK)

Saturated Properties 0.22079 0.22218 0.22660 0.23115 0.23589 0.24084 0.24605 0.25159 0.25751 0.26390 0.27085 0.27850 0.28701 0.29662 0.30766 0.32063 0.33629 0.35598 0.38225 0.42111 0.49805 0.75188 3,499,900. 1,779,200. 272,300. 56,267. 14,778. 4,710.8 1,758.6 747.29 353.24 182.30 101.13 59.534 36.786 23.635 15.658 10.614 7.3034 5.0563 3.4787 2.3249 1.3973 0.75188

115.35 116.70 120.83 124.81 128.63 132.26 135.66 138.74 141.40 143.51 144.95 145.55 145.16 143.58 140.61 135.98 129.36 120.31 108.12 91.557 67.007 49.653

330.02 296.37 215.32 160.20 121.85 94.623 74.946 60.510 49.785 41.738 35.662 31.061 27.591 25.017 23.184 22.013 21.503 21.771 23.181 26.817 37.545 0

6.7106 7.0413 8.1425 9.3434 10.640 12.030 13.512 15.090 16.769 18.561 20.479 22.543 24.775 27.206 29.871 32.819 36.118 39.870 44.272 49.828 60.081

3.8271 3.9297 4.2511 4.5733 4.8958 5.2183 5.5399 5.8599 6.1775 6.4920 6.8034 7.1122 7.4202 7.7302 8.0477 8.3808 8.7440 9.1635 9.6956 10.493 12.293

2-261

2-262 TABLE 2-212 Temperature K

Thermodynamic Properties of Dodecane (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.22883 0.25445 0.28651

−83.216 −42.339 −0.30755

−83.193 −42.313 −0.27889

39.638 44.299 89.425 139.71

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Single-Phase Properties −0.21241 −0.095254 −0.00056937

0.31053 0.37610 0.43283

0.37786 0.44184 0.50449

1273.5 916.43 636.11

−0.42497 −0.30827 −0.13079

134.86 112.91 95.191

43.413 48.185 94.254 145.42

0.088800 0.098451 0.18229 0.26108

0.41397 0.42081 0.47769 0.52517

0.42628 0.43268 0.48757 0.53435

145.21 147.79 167.29 183.12

27.759 25.222 12.534 7.4958

24.647 25.670 35.525 46.039

−83.277 −42.444 5.1293 60.165 69.090

−83.048 −42.190 5.4196 60.531 69.480

−0.21262 −0.095518 0.010431 0.11067 0.12541

0.31061 0.37615 0.43951 0.49726 0.50605

0.37775 0.44147 0.51125 0.60093 0.62919

1279.2 924.90 616.23 296.10 238.23

−0.42580 −0.31138 −0.11178 0.75711 1.3130

135.18 113.40 93.839 74.943 72.109

3.1841 4.7249

90.990 136.66

94.175 141.39

0.16559 0.23754

0.50099 0.53113

0.57515 0.55438

104.78 152.31

23.748 9.1496

45.410 51.144

4.3910 3.9683 3.5099 2.9472 2.1498

0.22774 0.25200 0.28491 0.33930 0.46517

−83.541 −42.890 4.2808 57.896 117.69

−82.402 −41.630 5.7054 59.593 120.02

−0.21351 −0.096648 0.0087025 0.10677 0.19979

0.31099 0.37639 0.43938 0.49475 0.54281

0.37733 0.44002 0.50646 0.57114 0.63863

1303.7 960.72 674.60 420.49 215.10

−0.42924 −0.32355 −0.16525 0.19333 1.5765

136.59 115.50 97.177 81.179 70.846

1398.2 423.55 201.12 99.888 42.370

4.4115 4.0039 3.5776 3.0995 2.5593

0.22668 0.24976 0.27952 0.32263 0.39074

−83.856 −43.403 3.3852 56.144 113.82

−81.590 −40.906 6.1804 59.370 117.72

−0.21459 −0.097972 0.0068314 0.10365 0.19351

0.31147 0.37674 0.43944 0.49390 0.53945

0.37689 0.43856 0.50252 0.55974 0.60482

1333.1 1001.9 735.90 517.52 359.64

−0.43298 −0.33581 −0.20845 −0.0053841 0.33979

138.29 117.98 100.85 86.682 77.506

1484.7 452.04 221.12 119.96 65.788

300.00 400.00 488.89

0.10000 0.10000 0.10000

4.3700 3.9300 3.4902

488.89 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000

0.026496 0.025735 0.020711 0.017494

300.00 400.00 500.00 600.00 614.58

1.0000 1.0000 1.0000 1.0000 1.0000

4.3739 3.9374 3.4447 2.7332 2.5662

0.22863 0.25398 0.29030 0.36588 0.38968

614.58 700.00

1.0000 1.0000

0.31406 0.21164

300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000

300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000

37.741 38.857 48.284 57.163

1315.3 395.13 195.41 7.4030 7.5935 9.2766 10.929 1330.4 400.40 183.76 77.601 64.404 9.8451 11.188

400.00 500.00 600.00 700.00

100.00 100.00 100.00 100.00

4.3951 4.1426 3.9126 3.7019

0.22753 0.24140 0.25559 0.27013

−48.840 −4.0985 46.019 100.67

−26.088 20.041 71.578 127.69

−0.11422 −0.011523 0.082301 0.16871

0.38400 0.44564 0.49884 0.54304

0.43214 0.48959 0.53969 0.58105

1472.8 1297.7 1169.1 1073.9

−0.39899 −0.35002 −0.31375 −0.28792

149.79 140.67 134.18 129.56

967.99 510.30 334.45 242.80

600.00 700.00

300.00 300.00

4.4838 4.3441

0.22302 0.23020

40.123 94.208

107.03 163.27

0.062648 0.14926

0.50752 0.55080

0.54106 0.58241

1807.8 1725.5

−0.33126 −0.31036

190.55 190.65

737.48 531.88

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E.W., and Huber, M.L., “Thermodynamic Properties of n-Dodecane,” Energy & Fuels, 18:960–967, 2004. The source for viscosity and thermal conductivity is Huber, M. L., Laesecke, A., and Perkins, R. A., “Transport Properties of n-Dodecane,” Energy & Fuels 18: 968–975, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties (where the uncertainties can be considered as estimates of a combined expanded uncertainty with a coverage factor of 2) of density values calculated using the equation of state in the liquid phase (including at saturation) are 0.2% for pressures less than 200 MPa and 0.5% for higher pressures. The uncertainty for heat capacities is 1%, and that for sound speeds is 0.5%. The estimated uncertainties of vapor pressures calculated using the Maxwell criterion are 0.2% for temperatures above 350 K and approach 5% as the temperature decreases to the triple point temperature. These estimated uncertainties for calculated properties are consistent with the experimental accuracies of the various available experimental data. The estimated uncertainty in viscosity is 0.5% along the saturated-liquid line, 2% in compressed liquid to 200 MPa, 5% in vapor and supercritical regions. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%.

TABLE 2-213

Thermodynamic Properties of Ethane

Temperature K

Pressure MPa

90.368 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 305.32

1.1421E-06 1.1081E-05 7.4287E-05 0.0003523 0.0012839 0.0038136 0.009638 0.021405 0.042819 0.078638 0.13459 0.21723 0.33380 0.49205 0.70018 0.96679 1.3008 1.7118 2.2100 2.8067 3.5159 4.3573 4.8722

90.368 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 305.32

1.1421E-06 1.1081E-05 7.4287E-05 0.0003523 0.0012839 0.0038136 0.0096380 0.021405 0.042819 0.078638 0.13459 0.21723 0.33380 0.49205 0.70018 0.96679 1.3008 1.7118 2.2100 2.8067 3.5159 4.3573 4.8722

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−6.5907 −5.9245 −5.2401 −4.5559 −3.8686 −3.1764 −2.4784 −1.7736 −1.0608 −0.3389 0.3938 1.1391 1.8991 2.6766 3.4747 4.2973 5.1501 6.0406 6.9809 7.9919 9.1194 10.525 12.490

−6.5907 −5.9245 −5.2401 −4.5559 −3.8685 −3.1762 −2.4779 −1.7724 −1.0585 −0.33459 0.40133 1.1515 1.9188 2.7064 3.5184 4.3598 5.2374 6.1606 7.1438 8.2124 9.4204 10.957 13.200

10.542 10.803 11.080 11.362 11.648 11.938 12.231 12.525 12.817 13.104 13.384 13.654 13.914 14.160 14.389 14.597 14.776 14.915 14.999 14.998 14.846 14.314 12.490

11.293 11.634 11.994 12.359 12.728 13.099 13.471 13.841 14.205 14.559 14.898 15.221 15.523 15.801 16.050 16.264 16.434 16.545 16.578 16.495 16.215 15.458 13.200

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.55899 −0.56656 −0.56429 −0.55668 −0.54585 −0.53246 −0.51640 −0.49723 −0.47424 −0.44655 −0.41304 −0.37225 −0.32220 −0.26011 −0.18185 −0.08101 0.05295 0.23862 0.51141 0.94945 1.7713 3.9421 9.6957

255.62 247.83 239.12 229.91 220.35 210.57 200.71 190.84 181.03 171.34 161.84 152.56 143.47 134.60 125.95 117.51 109.29 101.24 93.309 85.434 77.574 71.489

1280.8 873.22 633.38 485.78 388.17 319.79 269.55 231.16 200.83 176.18 155.68 138.27 123.23 110.05 98.339 87.799 78.190 69.305 60.936 52.839 44.603 34.970

Saturated Properties 21.668 21.316 20.951 20.584 20.214 19.840 19.461 19.074 18.680 18.275 17.858 17.426 16.976 16.504 16.006 15.475 14.901 14.270 13.559 12.728 11.684 10.094 6.8569 1.5201E-06 1.3327E-05 8.1234E-05 0.00035326 0.0011893 0.0032857 0.0077732 0.016263 0.030843 0.054053 0.088865 0.13870 0.20749 0.29989 0.42157 0.57983 0.78456 1.0502 1.3998 1.8748 2.5679 3.8079 6.8569

0.046151 0.046913 0.04773 0.048581 0.049469 0.050402 0.051385 0.052426 0.053534 0.054720 0.055998 0.057386 0.058907 0.060590 0.062476 0.064622 0.067112 0.070079 0.073750 0.078565 0.085590 0.099071 0.14584 657,870 75,033 12,310 2,830.8 840.82 304.35 128.65 61.489 32.422 18.500 11.253 7.2100 4.8195 3.3346 2.3721 1.7246 1.2746 0.9522 0.71441 0.53338 0.38943 0.26261 0.14584

−0.04975 −0.04275 −0.03623 −0.03027 −0.02477 −0.01964 −0.01483 −0.01028 −0.00596 −0.00183 0.002133 0.005957 0.009667 0.013287 0.016839 0.020347 0.023839 0.027349 0.030924 0.034644 0.038677 0.043620 0.050820

0.048264 0.046324 0.045172 0.044453 0.043962 0.043607 0.043351 0.043183 0.043100 0.043108 0.043210 0.043411 0.043717 0.044131 0.044661 0.045314 0.046109 0.047076 0.048269 0.049743 0.051927 0.057488

0.069935 0.068639 0.068351 0.068544 0.068959 0.069499 0.070139 0.070888 0.071768 0.072812 0.074056 0.075548 0.077345 0.079529 0.082221 0.085612 0.090024 0.096059 0.10496 0.11989 0.15219 0.30137

2008.7 1938.4 1866.4 1794.4 1722.0 1649.1 1575.5 1501.3 1426.3 1350.5 1273.7 1196.0 1117.3 1037.3 956.09 873.25 788.33 700.52 608.92 512.38 405.70 274.91 0

0.14815 0.13284 0.12045 0.11069 0.10289 0.096610 0.091501 0.087308 0.083831 0.080912 0.078433 0.076302 0.074447 0.072806 0.071324 0.069948 0.068624 0.067290 0.065865 0.064224 0.062107 0.058625 0.050820

0.026809 0.027384 0.027997 0.028651 0.029380 0.030154 0.030876 0.031513 0.032173 0.033015 0.034120 0.035457 0.036937 0.038486 0.040077 0.041742 0.043548 0.045586 0.047971 0.050986 0.055181 0.062820

0.035124 0.035699 0.036318 0.036992 0.037770 0.038623 0.039449 0.040235 0.041138 0.042375 0.044070 0.046220 0.048771 0.051718 0.055176 0.059419 0.064956 0.072712 0.084634 0.10591 0.15523 0.39989

180.93 189.86 198.61 206.89 214.69 222.01 228.84 235.12 240.73 245.54 249.41 252.26 254.02 254.65 254.05 252.14 248.79 243.81 237.02 228.10 216.50 200.51 0

249.88 199.02 177.19 158.49 129.38 96.433 70.230 54.075 45.339 40.255 36.410 32.914 29.687 26.853 24.485 22.563 21.017 19.757 18.700 17.721 16.582 14.497 9.6957

2.9082 3.4563 4.0447 4.6568 5.2959 5.9661 6.6725 7.4216 8.2209 9.0790 10.005 11.011 12.109 13.318 14.662 16.179 17.930 20.023 22.667 26.327 32.319 47.465

3.0427 3.3157 3.6034 3.8961 4.1937 4.4956 4.8011 5.1100 5.4223 5.7388 6.0609 6.3908 6.7319 7.0889 7.4687 7.8814 8.3424 8.8764 9.5257 10.373 11.617 14.023

2-263

2-264

TABLE 2-213

Thermodynamic Properties of Ethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

100.00 200.00 300.00 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

21.317 0.061716 0.040388 0.030155 0.024084 0.020056

100.00 200.00 241.10

1.0000 1.0000 1.0000

21.329 17.457 15.414

241.10 300.00 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000 1.0000

100.00 200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.046910 16.203 24.760 33.162 41.521 49.861

−5.9253 13.768 17.702 22.785 29.137 36.675

−5.9206 15.388 20.178 26.102 33.289 41.661

−0.042756 0.083336 0.10263 0.11958 0.13557 0.15080

0.046885 0.057283 0.064876

−5.9325 1.1164 4.3894

−5.8856 1.1737 4.4543

1.6672 2.3012 3.2290 4.1060 4.9639

14.618 17.331 22.568 28.986 36.562

21.381 17.610 10.907 1.7653 1.2769 1.0245

0.046771 0.056786 0.091680 0.56647 0.78316 0.97605

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.046328 0.034553 0.044475 0.057153 0.069620 0.080870

0.068636 0.043810 0.053048 0.065586 0.078001 0.089227

1938.8 257.90 312.23 355.24 393.08 427.63

−0.56660 30.845 10.722 5.0759 2.7932 1.6883

247.88 10.723 21.218 35.965 53.769 73.338

873.92 6.3683 9.4081 12.213 14.778 17.135

−0.042828 0.0058431 0.020731

0.046362 0.043441 0.045394

0.068612 0.075331 0.086038

1942.4 1204.0 864.04

−0.56691 −0.37737 −0.068172

248.30 153.30 116.60

880.24 139.33 86.702

16.285 19.632 25.797 33.092 41.526

0.069802 0.082230 0.099896 0.11613 0.13147

0.041932 0.045482 0.057476 0.069783 0.080969

0.059951 0.057061 0.067069 0.078784 0.089708

251.85 296.92 348.76 390.24 426.64

22.377 11.381 5.1250 2.7821 1.6715

16.358 22.177 36.459 54.084 73.562

−5.9638 1.0063 10.087 21.484 28.293 36.055

−5.7300 1.2902 10.546 24.316 32.209 40.936

−0.043144 0.0052852 0.042046 0.083790 0.10138 0.11726

0.046517 0.043595 0.053743 0.059017 0.070476 0.081384

0.068510 0.074345 0.17141 0.076481 0.082676 0.091919

1957.7 1242.5 359.63 322.27 380.40 424.37

−0.56823 −0.40076 2.3430 5.2035 2.6757 1.5738

250.17 156.99 73.067 39.974 55.976 74.801

909.15 144.70 39.648 13.905 15.995 18.124

Single-Phase Properties

0.59982 0.43455 0.30970 0.24355 0.20145

7.9293 9.6518 12.417 14.954 17.289

100.00 200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000 10.000

21.444 17.786 12.680 4.1977 2.6794 2.0757

0.046634 0.056225 0.078863 0.23822 0.37322 0.48177

−6.0016 0.88047 9.0258 19.826 27.390 35.421

−5.5352 1.4427 9.8145 22.208 31.122 40.239

−0.043532 0.0046352 0.038217 0.073976 0.093896 0.11050

0.046712 0.043786 0.050799 0.060880 0.071226 0.081836

0.068393 0.073343 0.10280 0.096368 0.088180 0.094716

1976.2 1286.7 579.32 310.38 377.77 427.41

−0.56975 −0.42471 0.58968 4.4129 2.3866 1.4041

252.48 161.36 86.388 48.699 59.523 76.885

947.29 151.27 52.867 17.942 17.954 19.504

100.00 200.00 300.00 400.00 500.00 600.00

30.000 30.000 30.000 30.000 30.000 30.000

21.682 18.371 14.784 10.812 7.6781 5.8933

0.046121 0.054434 0.067640 0.092490 0.13024 0.16968

−6.1384 0.46960 7.6392 15.788 24.381 33.201

−4.7548 2.1026 9.6684 18.563 28.288 38.291

−0.045002 0.0024068 0.032933 0.058427 0.080115 0.098343

0.047466 0.044528 0.050618 0.061395 0.072527 0.082911

0.068003 0.070795 0.081850 0.094994 0.098504 0.10204

2044.8 1433.6 909.38 585.71 497.87 505.75

−0.57447 −0.48623 −0.18133 0.40872 0.71195 0.59060

261.51 177.27 112.64 81.974 78.720 88.635

1127.3 177.04 77.610 42.216 30.510 27.539

200.00 300.00 400.00 500.00 600.00

70.000 70.000 70.000 70.000 70.000

19.230 16.537 14.033 11.869 10.144

0.052001 0.060469 0.071259 0.084254 0.098582

0.098974 6.4673 13.756 21.871 30.754

3.5411 10.700 18.744 27.769 37.654

0.0010269 0.027905 0.050977 0.071077 0.089078

0.045892 0.051766 0.062135 0.073299 0.083730

0.068554 0.075512 0.085472 0.094785 0.10274

1648.3 1238.5 970.70 827.80 764.56

−0.54018 −0.42067 −0.27243 −0.15903 −0.10326

204.45 145.70 115.84 107.39 111.32

230.70 113.52 71.766 53.331 44.741

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Buecker, D., and Wagner, W., “A Reference Equation of State for the Thermodynamic Properties of Ethane for Temperatures from the Melting Line to 675 K and Pressures up to 900 MPa,” J. Phys. Chem. Ref. Data, 35(1):205–206, 2006. The source for viscosity and thermal conductivity is Friend, D.G., Ingham, H., and Ely, J.F., “Thermophysical Properties of Ethane,” J. Phys. Chem. Ref. Data, 35(1):205–266, 2006. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.02% to 0.04% in density from the melting line up to temperatures of 520 K and pressures of 30 MPa. The uncertainties increase to 0.3% at higher temperatures and to 1% at higher pressures. The uncertainty in speed of sound ranges from 0.02% in the gaseous phase to 0.15% in the liquid phase. Above 450 K, the uncertainties increase to 0.3% at lower pressures and to 1% at higher pressures. At pressures above 40 MPa at all temperatures, the uncertainties are 1% up to 100 MPa, and 5% at higher pressures. The uncertainties in heat capacities range from 2% in the vapor and liquid regions below 450 K and 30 MPa to 5% at high pressures. The uncertainties in vapor pressure are 0.01% above 170 K and 10 MPa below 170 K. The uncertainty in viscosity is 2%. The uncertainty in thermal conductivity is 2%.

TABLE 2-214 Thermodynamic Properties of Ethanol Temperature K

Pressure MPa

250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 370.00 385.00 400.00 415.00 430.00 445.00 460.00 475.00 490.00 505.00 513.90

0.00027007 0.00089527 0.0025823 0.0066146 0.015298 0.032394 0.063544 0.11663 0.20205 0.33279 0.52446 0.79509 1.1649 1.6559 2.2916 3.0963 4.0954 5.3159 6.1480

250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 370.00 385.00 400.00 415.00 430.00 445.00 460.00 475.00 490.00 505.00 513.90

0.00027007 0.00089527 0.0025823 0.0066146 0.015298 0.032394 0.063544 0.11663 0.20205 0.33279 0.52446 0.79509 1.1649 1.6559 2.2916 3.0963 4.0954 5.3159 6.1480

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

6.9274 8.3792 9.9424 11.630 13.445 15.385 17.444 19.615 21.892 24.268 26.740 29.307 31.970 34.737 37.629 40.684 44.002 47.926 53.880

6.9275 8.3793 9.9426 11.631 13.446 15.387 17.448 19.622 21.905 24.290 26.775 29.362 32.054 34.862 37.810 40.943 44.374 48.480 54.906

0.037330 0.042968 0.048704 0.054574 0.060574 0.066684 0.072875 0.079123 0.085403 0.091699 0.098000 0.10430 0.11061 0.11695 0.12335 0.12991 0.13684 0.14485 0.15723

49.039 49.932 50.851 51.792 52.749 53.717 54.684 55.640 56.573 57.469 58.312 59.087 59.774 60.348 60.777 61.004 60.916 60.144 53.880

51.116 52.134 53.174 54.234 55.307 56.383 57.450 58.494 59.500 60.451 61.329 62.115 62.785 63.312 63.654 63.747 63.453 62.328 54.906

0.21409 0.20808 0.20310 0.19899 0.19561 0.19282 0.19053 0.18862 0.18701 0.18562 0.18438 0.18322 0.18208 0.18088 0.17954 0.17792 0.17578 0.17228 0.15723

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.076657 0.083798 0.091653 0.099433 0.10670 0.11322 0.11893 0.12381 0.12792 0.13130 0.13405 0.13625 0.13798 0.13934 0.14041 0.14134 0.14234 0.14382

0.093612 0.10028 0.10829 0.11678 0.12524 0.13340 0.14115 0.14847 0.15543 0.16215 0.16883 0.17576 0.18341 0.19262 0.20504 0.22469 0.26508 0.41790

1325.0 1260.8 1202.8 1149.2 1098.1 1048.0 997.94 947.31 895.56 842.31 787.16 729.67 669.25 605.07 535.80 459.19 371.03 264.74 0

−0.44553 −0.41423 −0.37872 −0.34323 −0.30910 −0.27615 −0.24356 −0.21011 −0.17428 −0.13410 −0.086812 −0.028333 0.047976 0.15384 0.31228 0.57597 1.0976 2.5369 8.6373

178.12 173.58 169.56 165.87 162.38 159.01 155.69 152.39 149.09 145.78 142.47 139.18 135.93 132.78 129.85 127.41 126.33 129.43

3140.9 2182.0 1564.4 1152.5 869.40 669.49 524.87 417.88 337.04 274.76 225.91 186.93 155.35 129.37 107.62 88.972 72.213 55.104

0.058885 0.060795 0.062753 0.064753 0.066816 0.068988 0.071336 0.073932 0.076838 0.080106 0.083774 0.087876 0.092450 0.097544 0.10323 0.10966 0.11709 0.12644

0.067215 0.069146 0.071149 0.073238 0.075464 0.077921 0.080736 0.084059 0.088058 0.092930 0.098936 0.10646 0.11610 0.12898 0.14727 0.17612 0.23200 0.42053

226.86 233.03 238.89 244.41 249.49 254.02 257.88 260.92 263.02 264.03 263.82 262.27 259.21 254.44 247.61 238.10 224.59 203.70 0

Cv kJ/(molK)

Saturated Properties 17.911 17.642 17.376 17.106 16.828 16.537 16.231 15.905 15.557 15.181 14.774 14.331 13.843 13.298 12.676 11.941 11.007 9.5842 5.9910 0.00012998 0.00040670 0.0011115 0.0027080 0.0059814 0.012150 0.022975 0.040873 0.069039 0.11160 0.17385 0.26261 0.38683 0.55876 0.79629 1.1286 1.6143 2.4339 5.9910

0.055831 0.056681 0.057551 0.058460 0.059426 0.060469 0.061610 0.062872 0.064281 0.065871 0.067684 0.069779 0.072241 0.075202 0.078889 0.083745 0.090848 0.10434 0.16692 7693.7 2458.8 899.69 369.28 167.18 82.305 43.525 24.466 14.485 8.9606 5.7521 3.8080 2.5851 1.7897 1.2558 0.88602 0.61945 0.41086 0.16692

149.30 111.11 87.283 71.858 61.180 53.164 46.697 41.226 36.486 32.353 28.756 25.644 22.967 20.681 18.747 17.136 15.831 14.728 8.6373

14.936 15.737 16.612 17.566 18.602 19.731 20.969 22.341 23.886 25.659 27.741 30.251 33.369 37.377 42.735 50.248 61.578 82.512

7.2715 7.7433 8.2114 8.6756 9.1353 9.5902 10.040 10.486 10.929 11.372 11.820 12.283 12.774 13.318 13.961 14.786 15.982 18.148

2-265

2-266 TABLE 2-214 Thermodynamic Properties of Ethanol (Concluded) Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

164.74 153.26

1047.2 443.11

Single-Phase Properties 300.00 351.05

0.10000 0.10000

351.05 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

300.00 400.00 423.85

1.0000 1.0000 1.0000

423.85 500.00 600.00

1.0000 1.0000 1.0000

300.00 400.00 500.00 501.39

5.0000 5.0000 5.0000 5.0000

501.39 600.00

5.0000 5.0000

300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000

12.219 19.033

12.225 19.040

0.056554 0.077475

0.10193 0.12261

0.11962 0.14658

1132.5 960.72

−0.33179 −0.21908

55.390 59.207 67.925 77.796

58.222 62.477 72.058 82.775

0.18909 0.20043 0.22176 0.24127

0.073221 0.080640 0.092910 0.10403

0.083127 0.089997 0.10162 0.11252

260.21 279.09 312.39 341.55

42.587 28.685 11.830 5.6356

0.058707 0.067589 0.071181

12.202 26.715 30.866

12.261 26.783 30.938

0.056497 0.097937 0.10802

0.10191 0.13400 0.13732

0.11954 0.16857 0.18015

1137.9 791.85 694.41

3.0216 3.9114 4.8846

59.504 67.014 77.311

62.526 70.925 82.195

0.18255 0.20078 0.22131

0.090516 0.096953 0.10581

0.11184 0.11008 0.11605

260.65 300.64 337.33

24.014 12.007 5.6301

0.058443 0.066842 0.097846 0.099875

12.129 26.516 46.419 46.876

12.421 26.851 46.908 47.375

0.056249 0.097435 0.14179 0.14272

0.10185 0.13359 0.14311 0.14340

0.11922 0.16658 0.32410 0.35152

1161.4 829.44 308.88 292.31

−0.33665 −0.11211 1.7596 2.0063

2.1809 1.1372

0.45852 0.87939

60.445 74.966

62.737 79.363

0.17336 0.20395

0.12389 0.11419

0.34099 0.13659

209.80 314.06

15.000 5.6703

75.676 61.725

17.454 18.972

17.203 15.147 11.521 2.8001

0.058131 0.066020 0.086800 0.35713

12.041 26.293 44.752 71.266

12.623 26.953 45.620 74.837

0.055950 0.096860 0.13830 0.19172

0.10179 0.13313 0.14031 0.12599

0.11885 0.16456 0.22204 0.18744

1189.5 872.36 464.50 273.66

−0.34096 −0.13414 0.60618 5.6926

170.07 149.80 130.15 84.190

1111.8 252.40 80.680 23.411

17.016 15.993 0.035314 0.030577 0.024191 0.020086 17.034 14.795 14.049 0.33095 0.25567 0.20473 17.111 14.961 10.220 10.013

0.058768 0.062527 28.317 32.704 41.338 49.786

−0.33273 −0.089821 0.013963

21.965 26.374 37.865 52.622 165.24 142.87 137.25 32.003 39.539 53.583 167.43 146.07 127.42 128.00

10.369 11.853 14.768 17.543 1053.2 227.32 167.52 12.568 14.859 17.678 1079.6 238.82 61.882 59.510

300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00

18.389 17.030 15.408 13.601

0.054380 0.058722 0.064899 0.073523

10.984 24.075 39.356 55.055

16.422 29.947 45.846 62.407

0.051802 0.090466 0.12589 0.15608

0.10149 0.12901 0.13221 0.12575

0.11571 0.15081 0.16433 0.16553

1558.1 1348.2 1166.1 1015.1

−0.37198 −0.25352 −0.17199 −0.082822

207.54 195.29 188.35 187.31

1611.3 435.09 192.15 109.49

300.00 400.00 500.00 600.00

200.00 200.00 200.00 200.00

19.244 18.138 16.878 15.505

0.051963 0.055134 0.059250 0.064494

10.349 22.905 37.295 51.902

20.742 33.931 49.145 64.801

0.048505 0.086238 0.12014 0.14869

0.10196 0.12678 0.12868 0.12066

0.11495 0.14539 0.15623 0.15566

1830.4 1660.3 1525.6 1422.7

−0.37578 −0.28090 −0.22946 −0.19099

238.67 228.57 224.49 226.40

2085.4 591.02 269.30 148.43

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Dillon, H.E., and Penoncello, S.G., “A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol,” Int. J. Thermophys., 25(2):321–335, 2004. The source for viscosity is Kiselev, S. B., Ely, J. F., Abdulagatov, I. M., and Huber, M. L.,”Generalized SAFT-DFT/DMT Model for the Thermodynamic, Interfacial, and Transport Properties of Associating Fluids: Application for n-Alkanols,” Ind. Eng. Chem. Res., 44:6916–6927, 2005. The source for thermal conductivity is unpublished, 2004; however, the fit uses functional form found in Marsh, K., Perkins, R., and Ramires, M.L.V., “Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa,” J. Chem. Eng. Data, 47(4):932–940, 2002. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.2% in density, 3% in heat capacities, 1% in speed of sound, and 0.5% in vapor pressure and saturation densities. The estimated uncertainty in the liquid phase along the saturation boundary is approximately 3%, increasing to 10% at pressures to 100 MPa, and is estimated at 10% in the vapor phase. The estimated uncertainty in the liquid phase is approximately 5% and is estimated as 10% in the vapor phase.

THERMODYNAMIC PROPERTIES

Enthalpy-concentration diagram for aqueous ethyl alcohol. Reference states: Enthalpies of liquid water and ethyl alcohol at 0 °C are zero. NOTE: In order to interpolate equilibrium compositions, a vertical may be erected from any liquid composition on the boiling line and its intersection with the auxiliary line determined. A horizontal from this intersection will establish the equilibrium vapor composition on the dew line. (Bosnjakovic, Technische Thermodynamik, T. Steinkopff, Leipzig, 1935.) FIG. 2-10

2-267

2-268 TABLE 2-215 Temperature K

Thermodynamic Properties of Ethylene Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−4.4352 −4.3661 −3.6839 −3.0030 −2.3250 −1.6496 −0.97554 −0.30104 0.37612 1.0585 1.7492 2.4516 3.1699 3.9095 4.6774 5.4842 6.3476 7.3042 8.4637 10.545

−4.4352 −4.3661 −3.6838 −3.0029 −2.3247 −1.6488 −0.97364 −0.29724 0.38311 1.0706 1.7688 2.4820 3.2155 3.9758 4.7715 5.6157 6.5297 7.5568 8.8262 11.206

10.621 10.647 10.895 11.142 11.385 11.624 11.855 12.078 12.289 12.486 12.667 12.829 12.966 13.074 13.144 13.162 13.105 12.923 12.460 10.545

11.486 11.520 11.850 12.179 12.502 12.818 13.122 13.412 13.683 13.933 14.155 14.346 14.501 14.610 14.664 14.645 14.522 14.232 13.587 11.206

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

−0.033074 −0.032412 −0.026206 −0.020529 −0.015311 −0.010484 −0.0059884 −0.0017710 0.0022141 0.0060075 0.0096456 0.013162 0.016589 0.019960 0.023314 0.026699 0.030191 0.033937 0.038375 0.046609

0.045503 0.045394 0.044202 0.042929 0.041693 0.040562 0.039568 0.038729 0.038051 0.037536 0.037184 0.036995 0.036970 0.037115 0.037446 0.037996 0.038851 0.040322 0.043922

0.068155 0.068183 0.068194 0.067951 0.067669 0.067475 0.067448 0.067649 0.068127 0.068937 0.070151 0.071867 0.074242 0.077537 0.082229 0.089292 0.10110 0.12549 0.21287

1766.6 1760.2 1694.4 1626.0 1556.2 1485.3 1413.2 1339.9 1265.0 1188.5 1110.0 1029.3 946.13 860.08 770.62 676.97 577.38 467.22 334.73 0

−0.50257 −0.50171 −0.49455 −0.48779 −0.47928 −0.46747 −0.45105 −0.42866 −0.39877 −0.35944 −0.30804 −0.24081 −0.15205 −0.032706 0.13270 0.37286 0.74844 1.4173 2.9732 8.7988

270.65 269.25 255.56 242.07 228.83 215.95 203.50 191.54 180.09 169.21 158.78 148.78 139.15 129.81 120.64 111.48 102.11 92.171 81.576

0.12003 0.11888 0.10888 0.10093 0.094519 0.089286 0.084954 0.081316 0.078216 0.075532 0.073165 0.071037 0.069079 0.067225 0.065409 0.063551 0.061532 0.059128 0.055687 0.046609

0.024972 0.024975 0.025028 0.025131 0.025307 0.025577 0.025956 0.026454 0.027081 0.027844 0.028751 0.029816 0.031060 0.032511 0.034212 0.036243 0.038789 0.042250 0.048136

0.033295 0.033300 0.033376 0.033533 0.033814 0.034263 0.034922 0.035833 0.037045 0.038617 0.040637 0.043243 0.046651 0.051237 0.057696 0.067503 0.084393 0.12086 0.26071

202.67 203.65 213.03 221.86 230.09 237.62 244.36 250.18 254.99 258.69 261.18 262.38 262.18 260.50 257.19 252.09 244.94 235.28 221.57 0

Entropy kJ/(molK)

Viscosity µPas

Saturated Properties 103.99 105.00 115.00 125.00 135.00 145.00 155.00 165.00 175.00 185.00 195.00 205.00 215.00 225.00 235.00 245.00 255.00 265.00 275.00 282.35

0.00012196 0.00014568 0.00069745 0.0025267 0.0073921 0.018309 0.039755 0.077693 0.13944 0.23344 0.36901 0.55614 0.80534 1.1276 1.5342 2.0376 2.6509 3.3898 4.2752 5.0417

103.99 105.00 115.00 125.00 135.00 145.00 155.00 165.00 175.00 185.00 195.00 205.00 215.00 225.00 235.00 245.00 255.00 265.00 275.00 282.35

0.00012196 0.00014568 0.00069745 0.0025267 0.0073921 0.018309 0.039755 0.077693 0.13944 0.23344 0.36901 0.55614 0.80534 1.1276 1.5342 2.0376 2.6509 3.3898 4.2752 5.0417

23.334 23.288 22.834 22.375 21.909 21.435 20.952 20.456 19.945 19.417 18.865 18.286 17.671 17.011 16.291 15.488 14.560 13.419 11.793 7.6368 0.00014109 0.00016690 0.00073000 0.0024363 0.0066178 0.015333 0.031379 0.058231 0.099997 0.16142 0.24802 0.36638 0.52471 0.73386 1.0092 1.3747 1.8719 2.5886 3.7947 7.6368

0.042856 0.042940 0.043793 0.044692 0.045643 0.046652 0.047729 0.048885 0.050137 0.051502 0.053008 0.054687 0.056589 0.058784 0.061382 0.064566 0.068682 0.074519 0.084795 0.13095 7087.6 5991.6 1369.9 410.46 151.11 65.217 31.869 17.173 10.000 6.1950 4.0319 2.7294 1.9058 1.3627 0.99086 0.72745 0.53422 0.38631 0.26352 0.13095

154.90 150.43 114.92 90.667 73.502 60.977 51.588 44.379 38.733 34.236 30.602 27.626 25.155 23.074 21.287 19.706 18.213 16.604 14.315 8.7988

6.8012 6.7393 6.6223 7.0116 7.5850 8.2013 8.8125 9.4170 10.036 10.700 11.446 12.313 13.346 14.607 16.186 18.250 21.166 26.045 39.168

685.73 662.48 488.07 378.67 305.49 253.84 215.68 186.36 163.03 143.88 127.73 113.76 101.40 90.217 79.891 70.142 60.685 51.112 40.397 0.77270 1.0011 2.7342 3.8262 4.5578 5.0886 5.5112 5.8799 6.2279 6.5765 6.9410 7.3341 7.7689 8.2612 8.8342 9.5251 10.404 11.632 13.765

Single-Phase Properties 150.00 169.16

0.10000 0.10000

169.16 225.00 300.00 375.00 450.00

0.10000 0.10000 0.10000 0.10000 0.10000

150.00 221.33

1.0000 1.0000

221.33 225.00 300.00 375.00 450.00

1.0000 1.0000 1.0000 1.0000 1.0000

150.00 225.00 281.98

5.0000 5.0000 5.0000

281.98 300.00 375.00 450.00

5.0000 5.0000 5.0000 5.0000

150.00 225.00 300.00 375.00 450.00

10.000 10.000 10.000 10.000 10.000

−1.3137 −0.020110

−1.3090 −0.015171

12.167 13.722 16.100 18.990 22.436

13.527 15.568 18.580 22.099 26.173

0.047119 0.057938

−1.3281 3.6350

1.5378 1.5821 2.3510 3.0336 3.6904

21.333 17.364 9.0494

21.197 20.246

−0.0082061 −0.000089238

0.067428 0.067811

0.079970 0.090386 0.10190 0.11234 0.12222

0.026699 0.028730 0.034782 0.042202 0.049547

0.036297 0.037508 0.043294 0.050626 0.057930

252.31 291.07 330.82 364.16 394.34

−1.2810 3.6929

−0.0083026 0.018726

0.040075 0.037041

0.067329 0.076196

1456.3 892.05

13.038 13.173 15.822 18.803 22.297

14.576 14.755 18.173 21.836 25.987

0.067898 0.068701 0.081821 0.092696 0.10277

0.031952 0.031799 0.035459 0.042478 0.049700

0.049381 0.048297 0.046096 0.051961 0.058724

261.30 265.10 320.09 358.72 391.57

23.800 22.637 10.271 5.7053 3.5291

0.046876 0.057591 0.11050

−1.3902 3.7130 9.9242

−1.1558 4.0009 10.477

−0.0087214 0.019071 0.044044

0.040199 0.037097 0.061633

0.066921 0.073970 3.8740

1483.4 926.49 189.76

−0.47016 −0.13327 7.3487

214.70 135.08 145.71

231.18 93.288 27.667

6.2408 3.1277 1.8590 1.4297

0.16024 0.31972 0.53792 0.69947

11.247 13.977 17.877 21.650

12.048 15.576 20.567 25.147

0.049617 0.061893 0.076837 0.087965

0.066988 0.040822 0.043817 0.050366

5.9539 0.089599 0.060244 0.062753

191.76 263.71 337.12 381.96

10.151 10.910 5.6244 3.3730

182.63 30.862 35.453 45.106

19.004 13.344 14.287 16.277

21.464 17.739 11.510 4.3583 3.0293

0.046589 0.056372 0.086881 0.22945 0.33011

−1.4634 3.5022 9.6517 16.483 20.798

−0.99754 4.0659 10.521 18.778 24.099

−0.0092238 0.018094 0.042563 0.067434 0.080391

0.040364 0.037190 0.041889 0.045508 0.051131

0.066475 0.070996 0.11956 0.077265 0.068674

1515.1 996.58 420.01 328.58 379.55

−0.47905 −0.22081 1.9372 4.7029 2.9790

219.60 140.93 79.635 45.198 50.075

229.03 96.609 37.621 17.402 18.014

21.223 17.260 0.65028 0.63206 0.42535 0.32964 0.27097

13.602 18.456 24.805 31.096 37.364

1449.9 1309.0

−0.46008 −0.41725

0.040048 0.038427

0.073519 0.054182 0.040315 0.032159 0.026764

0.047176 0.049393

41.871 20.438 10.015 5.6845 3.5485 −0.46205 −0.080890

209.74 186.72 9.6709 12.903 20.558 31.002 42.200 210.67 133.22 14.112 14.323 21.362 31.533 42.581

233.39 176.06 6.0256 7.8384 10.381 12.827 15.092 232.98 94.208 8.0724 8.1765 10.589 13.017 15.284

150.00 225.00 300.00 375.00 450.00

100.00 100.00 100.00 100.00 100.00

23.148 20.862 18.775 16.889 15.219

0.043200 0.047935 0.053261 0.059210 0.065708

−2.3062 1.8396 5.9492 10.287 14.942

2.0139 6.6331 11.275 16.208 21.513

−0.015964 0.0090426 0.026831 0.041488 0.054370

0.043582 0.040468 0.043293 0.048765 0.054851

0.063328 0.060888 0.063490 0.068211 0.073224

1891.7 1581.0 1328.5 1143.7 1020.9

−0.54092 −0.54008 −0.48350 −0.41039 −0.34600

294.42 200.24 154.23 131.89 121.24

199.89 126.05 98.312 81.449 70.072

150.00 225.00 300.00 375.00 450.00

300.00 300.00 300.00 300.00 300.00

25.298 23.718 22.336 21.120 20.042

0.039530 0.042162 0.044772 0.047350 0.049894

−2.9624 0.79998 4.5667 8.6213 13.065

8.8964 13.448 17.998 22.826 28.033

−0.024942 −0.00029268 0.017143 0.031488 0.044131

0.048504 0.044752 0.047124 0.052222 0.057988

0.062644 0.059805 0.062135 0.066814 0.072043

2343.7 2151.3 1984.6 1847.1 1740.6

−0.54689 −0.57336 −0.55352 −0.51701 −0.48199

432.13 275.32 209.21 178.33 163.32

161.49 155.50 157.18 156.74 153.81

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Smukala, J., Span, R., and Wagner, W., “A New Equation of State for Ethylene Covering the Fluid Region for Temperatures from the Melting Line to 450 K at Pressures up to 300 MPa,” J. Phys. Chem. Ref. Data, 29(5):1053–1122, 2000. The source for viscosity is Holland, P.M., Eaton, B.E., and Hanley, H.J.M., “A Correlation of the Viscosity and Thermal Conductivity Data of Gaseous and Liquid Ethylene,” J. Phys. Chem. Ref. Data, 12(4):917–932, 1983. The source for thermal conductivity is Holland, P.M., Eaton, B.E., and Hanley, H.J.M., “A Correlation of the Viscosity and Thermal Conductivity Data of Gaseous and Liquid Ethylene,” J. Phys. Chem. Ref. Data, 12(4):917–932, 1983. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density of the equation of state range from 0.02% in the liquid and most of the vapor phase to 0.1% for supercritical states. At p 100 MPa, the uncertainty in density is 0.5%. The uncertainty in heat capacity is 3% in the liquid phase, 0.2% in the vapor phase, and as high as 5% in the supercritical region at higher pressures. For the speed of sound, the uncertainty is 0.05 to 0.1% in the vapor phase, rising to 3% in the liquid phase. The uncertainty in vapor pressure is less than 0.05% above 140 K. The uncertainty in viscosity is 5%, increasing to 10% in the dense liquid. The uncertainty in thermal conductivity is 5%, increasing to 10% in the dense liquid. 2-269

2-270

TABLE 2-216

Thermodynamic Properties of Fluorine

Temperature K

Pressure MPa

Density mol/dm3

53.481 55.000 60.000 65.000 70.000 75.000 80.000 85.000 90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 144.41

0.00023881 0.00038159 0.0014872 0.0046126 0.011981 0.027062 0.054668 0.10090 0.17296 0.27894 0.42751 0.62778 0.88912 1.2212 1.6342 2.1389 2.7475 3.4739 4.3357 5.2394

44.917 44.657 43.829 43.005 42.169 41.315 40.437 39.531 38.590 37.609 36.582 35.499 34.351 33.119 31.777 30.280 28.549 26.416 23.427 15.603

53.481 55.000 60.000 65.000 70.000 75.000 80.000 85.000 90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 144.41

0.00023881 0.00038159 0.0014872 0.0046126 0.011981 0.027062 0.054668 0.10090 0.17296 0.27894 0.42751 0.62778 0.88912 1.2212 1.6342 2.1389 2.7475 3.4739 4.3357 5.2394

0.00053725 0.00083488 0.0029857 0.0085667 0.020743 0.044003 0.084093 0.14789 0.24329 0.37919 0.56568 0.81460 1.1406 1.5630 2.1098 2.8245 3.7825 5.1364 7.3237 15.603

84.922

0.10000

39.546

84.922 100.00 150.00 200.00 250.00 300.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

100.00 111.80

1.0000 1.0000

111.80 150.00

1.0000 1.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.026073 −0.024487 −0.019663 −0.015187 −0.011024 −0.0071357 −0.0034817 −0.000024789 0.0032663 0.0064180 0.0094534 0.012394 0.015261 0.018078 0.020877 0.023700 0.026617 0.029765 0.033502 0.041160

0.036802 0.035214 0.034254 0.033385 0.032745 0.032241 0.031737 0.031196 0.030634 0.030073 0.029532 0.029024 0.028561 0.028159 0.027847 0.027681 0.027791 0.028512 0.030963

0.057963 0.055732 0.055680 0.056098 0.056274 0.056496 0.056860 0.057394 0.058117 0.059051 0.060243 0.061773 0.063791 0.066559 0.070566 0.076800 0.087576 0.10995 0.18235

1041.6 1040.8 1043.5 1019.2 973.15 921.41 871.16 823.90 779.02 735.47 692.27 648.60 603.67 556.65 506.54 452.05 391.41 321.84 238.35 0

−0.30422 −0.31810 −0.31768 −0.31059 −0.30322 −0.29375 −0.28123 −0.26509 −0.24474 −0.21935 −0.18762 −0.14746 −0.095557 −0.026510 0.068825 0.20691 0.42042 0.78572 1.5459 4.3227

0.020800 0.020805 0.020831 0.020886 0.020984 0.021139 0.021361 0.021653 0.022013 0.022432 0.022900 0.023406 0.023951 0.024547 0.025234 0.026096 0.027303 0.029246 0.033084

0.029129 0.029139 0.029204 0.029341 0.029589 0.029984 0.030562 0.031349 0.032368 0.033644 0.035219 0.037174 0.039668 0.043016 0.047844 0.055492 0.069276 0.099896 0.21077

127.97 129.76 135.46 140.83 145.85 150.47 154.65 158.34 161.52 164.12 166.12 167.46 168.12 168.05 167.26 165.81 163.76 161.03 156.46 0

76.144 71.404 58.461 48.604 40.910 34.772 29.788 25.696 22.322 19.545 17.279 15.449 13.988 12.827 11.886 11.069 10.250 9.2577 7.8082 4.3227

Saturated Properties 0.022263 0.022393 0.022816 0.023253 0.023714 0.024204 0.024730 0.025296 0.025913 0.026589 0.027336 0.028170 0.029112 0.030194 0.031470 0.033025 0.035027 0.037855 0.042685 0.064090 1861.3 1197.8 334.92 116.73 48.208 22.726 11.892 6.7617 4.1103 2.6372 1.7678 1.2276 0.87675 0.63979 0.47398 0.35404 0.26438 0.19469 0.13654 0.064090

−1.7783 −1.6923 −1.4151 −1.1355 −0.85459 −0.57283 −0.28975 −0.0046707 0.28316 0.57448 0.87013 1.1711 1.4785 1.7943 2.1213 2.4642 2.8314 3.2397 3.7349 4.7226 5.3412 5.3726 5.4753 5.5765 5.6750 5.7697 5.8592 5.9423 6.0181 6.0855 6.1438 6.1916 6.2274 6.2487 6.2511 6.2277 6.1664 6.0431 5.7921 4.7226

−1.7783 −1.6923 −1.4151 −1.1354 −0.85430 −0.57217 −0.28840 −0.0021183 0.28764 0.58190 0.88182 1.1887 1.5044 1.8312 2.1728 2.5349 2.9276 3.3712 3.9200 5.0584 5.7857 5.8296 5.9734 6.1149 6.2526 6.3847 6.5093 6.6246 6.7290 6.8212 6.8995 6.9623 7.0070 7.0300 7.0256 6.9850 6.8928 6.7195 6.3841 5.0584

0.11536 0.11228 0.10348 0.096356 0.090504 0.085622 0.081489 0.077936 0.074837 0.072094 0.069630 0.067380 0.065284 0.063285 0.061318 0.059301 0.057119 0.054567 0.051103 0.041160

Single-Phase Properties

0.14668 0.12275 0.080628 0.060267 0.048155 0.040106 36.645 33.918 1.2803 0.85139

0.025287 6.8177 8.1468 12.403 16.593 20.766 24.934

−0.0091317

−0.0066030

5.9411 6.2721 7.3358 8.3977 9.4869 10.617

6.6228 7.0868 8.5761 10.057 11.564 13.110

−0.000077307

0.031205

0.057385

824.62

−0.26537

0.077988 0.083019 0.095107 0.10363 0.11035 0.11598

0.021648 0.021268 0.021022 0.021401 0.022142 0.023021

0.031335 0.030345 0.029539 0.029803 0.030505 0.031368

158.29 173.11 213.56 246.33 274.27 298.96

25.754 17.864 7.4423 4.1219 2.6600 1.8799

0.027289 0.029483

0.86105 1.5911

0.88834 1.6206

0.0093623 0.016279

0.029558 0.028409

0.060025 0.064682

696.10 587.03

−0.19109 −0.073051

0.78106 1.1746

6.2369 7.1946

7.0180 8.3692

0.064557 0.075007

0.024158 0.021931

0.040751 0.032606

168.18 207.71

13.540 7.3873

200.00 250.00 300.00

1.0000 1.0000 1.0000

0.61498 0.48577 0.40251

1.6261 2.0586 2.4844

8.3181 9.4313 10.573

100.00 143.33

5.0000 5.0000

37.065 19.879

0.026980 0.050304

143.33 150.00 200.00 250.00 300.00

5.0000 5.0000 5.0000 5.0000 5.0000

10.490 6.5097 3.3899 2.5221 2.0427

0.095333 0.15362 0.29499 0.39650 0.48954

5.3751 6.2260 7.9204 9.1743 10.377

0.80142 4.2229

9.9441 11.490 13.057

0.084080 0.090978 0.096691

0.021681 0.022225 0.023032

0.030928 0.031046 0.031678

244.27 273.76 299.28

4.1390 2.6581 1.8692

0.0087568 0.037196

0.029727 0.036294

0.058693 0.63033

721.70 169.23

−0.21256 2.8620

5.8518 6.9941 9.3953 11.157 12.824

0.046806 0.054653 0.068706 0.076582 0.082665

0.039553 0.027354 0.022993 0.022698 0.023148

1.0196 0.084441 0.037593 0.033813 0.033134

148.40 177.58 237.67 273.06 301.36

6.0573 6.7256 3.8562 2.4478 1.6957

0.93632 4.4744

100.00 150.00 200.00 250.00 300.00

10.000 10.000 10.000 10.000 10.000

37.536 24.141 7.6450 5.2188 4.1242

0.026641 0.041423 0.13080 0.19162 0.24247

0.73479 3.9349 7.3273 8.8378 10.136

1.0012 4.3491 8.6354 10.754 12.561

0.0080653 0.034822 0.060060 0.069563 0.076160

0.029857 0.027090 0.024489 0.023418 0.023411

0.057385 0.094016 0.050206 0.037864 0.035025

752.43 324.06 240.08 278.39 307.87

−0.23402 0.86131 3.1116 2.0656 1.4058

100.00 150.00 200.00 250.00 300.00

15.000 15.000 15.000 15.000 15.000

37.961 27.008 12.412 7.9757 6.1790

0.026343 0.037026 0.080569 0.12538 0.16184

0.67509 3.5725 6.7026 8.4961 9.9020

1.0702 4.1279 7.9111 10.377 12.330

0.0074311 0.032052 0.053898 0.064977 0.072112

0.029870 0.026615 0.025220 0.024073 0.023725

0.056345 0.070563 0.060759 0.042125 0.036972

783.34 415.88 263.46 289.64 318.16

-0.25114 0.33099 2.0529 1.7085 1.1770

100.00 150.00 200.00 250.00 300.00

20.000 20.000 20.000 20.000 20.000

38.348 28.645 16.458 10.648 8.1626

0.026077 0.034910 0.060762 0.093913 0.12251

0.62089 3.3586 6.1944 8.1664 9.6749

1.1424 4.0568 7.4096 10.045 12.125

0.0068429 0.030382 0.049659 0.061490 0.069094

0.029763 0.026518 0.025474 0.024527 0.024015

0.055495 0.062828 0.062710 0.045421 0.038840

815.20 480.50 304.19 307.96 331.68

−0.26507 0.11826 1.2436 1.3200 0.98352

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is de Reuck, K.M., “International Thermodynamic Tables of the Fluid State—11 Fluorine,” International Union of Pure and Applied Chemistry, Pergamon Press, Oxford, 1990. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.2% in density, 2% in heat capacity, and 1.5% in the speed of sound, except in the critical region.

TABLE 2-217

Flutec

Proprietary name for a series of fluorocarbons produced by the Imperial Smelting Corp., Avonmouth, Bristol, UK. Bulletins of thermodynamic properties include PP1 (C6F14), PP2 (C7F14), PP3 (C8F16), PP5 (C10F18), PP9 (C11F20), and PP50, usually for 0.1–100 kg/m2, 0–500°C. See also Green, S. W., Chem. & Ind. (1969): 63–67.

TABLE 2-218

Halon

A series of fire-extinguishing fluids. Halon 1211 is produced by ICI, and Halon 1301, by duPont, the latter issuing a bulletin with thermodynamic properties and a diagram for the range 0.6–600 psia, −160–460°F.

2-271

2-272 TABLE 2-219 Temperature K

Thermodynamic Properties of Helium Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Saturated Properties 2.1768 2.2000 2.3500 2.5000 2.6500 2.8000 2.9500 3.1000 3.2500 3.4000 3.5500 3.7000 3.8500 4.0000 4.1500 4.3000 4.4500 4.6000 4.7500 4.9000 5.0500 5.1953

0.0048565 0.0051477 0.0073079 0.010001 0.013298 0.017270 0.021983 0.027502 0.033890 0.041209 0.049518 0.058879 0.069351 0.080998 0.093886 0.10809 0.12368 0.14075 0.15942 0.17983 0.20225 0.22637

36.537 36.523 36.396 36.217 35.992 35.727 35.423 35.082 34.705 34.291 33.838 33.342 32.799 32.203 31.545 30.813 29.986 29.032 27.895 26.456 24.384 17.399

0.027370 0.027380 0.027475 0.027611 0.027784 0.027990 0.028230 0.028504 0.028814 0.029162 0.029553 0.029993 0.030489 0.031053 0.031700 0.032454 0.033349 0.034445 0.035849 0.037798 0.041010 0.057475

−0.030158 −0.029596 −0.026858 −0.025011 −0.023514 −0.022106 −0.020660 −0.019120 −0.017461 −0.015675 −0.013757 −0.011706 −0.0095155 −0.0071756 −0.0046702 −0.0019735 0.00095551 0.0041842 0.0078334 0.012155 0.017863 0.034500

−0.030025 −0.029455 −0.026658 −0.024734 −0.023145 −0.021623 −0.020040 −0.018336 −0.016485 −0.014473 −0.012294 −0.0099402 −0.0074010 −0.0046603 −0.0016940 0.0015343 0.0050800 0.0090322 0.013548 0.018953 0.026157 0.047510

−0.0087120 −0.0084552 −0.0072478 −0.0064839 −0.0059016 −0.0053835 −0.0048790 −0.0043676 −0.0038422 −0.0033010 −0.0027443 −0.0021719 −0.0015832 −0.00097628 −0.00034763 0.00030858 0.0010017 0.0017471 0.0025725 0.0035365 0.0048066 0.0087388

0.025164 0.023062 0.014101 0.010092 0.0085050 0.0080743 0.0081637 0.0084575 0.0088040 0.0091348 0.0094238 0.0096652 0.0098629 0.010024 0.010158 0.010273 0.010378 0.010481 0.010592 0.010726 0.010920

0.025289 0.023216 0.014502 0.010808 0.0095913 0.0095802 0.010142 0.010969 0.011921 0.012944 0.014037 0.015223 0.016553 0.018104 0.019993 0.022416 0.025720 0.030604 0.038735 0.055286 0.10769

216.83 216.59 215.68 216.03 216.74 216.57 215.08 212.50 209.15 205.24 200.91 196.21 191.15 185.70 179.82 173.44 166.48 158.80 150.22 140.44 128.80 0

−1.0411 −1.1290 −1.7580 −2.2958 −2.5162 −2.4447 −2.2326 −1.9844 −1.7406 −1.5095 −1.2873 −1.0669 −0.84045 −0.59987 −0.33639 −0.03948 0.30472 0.71637 1.2278 1.8978 2.8635 5.6142

13.522 13.631 14.294 14.894 15.442 15.946 16.408 16.829 17.209 17.548 17.844 18.098 18.307 18.474 18.599 18.688 18.753 18.811 18.892 19.051 19.490

3.5963 3.6153 3.7039 3.7456 3.7541 3.7391 3.7074 3.6637 3.6113 3.5525 3.4890 3.4218 3.3516 3.2785 3.2025 3.1231 3.0392 2.9490 2.8488 2.7312 2.5752

2.1768 2.2000 2.3500 2.5000 2.6500 2.8000 2.9500 3.1000 3.2500 3.4000 3.5500 3.7000 3.8500 4.0000 4.1500 4.3000 4.4500 4.6000 4.7500 4.9000 5.0500 5.1953

0.0048565 0.0051477 0.0073079 0.010001 0.013298 0.017270 0.021983 0.027502 0.033890 0.041209 0.049518 0.058879 0.069351 0.080998 0.093886 0.10809 0.12368 0.14075 0.15942 0.17983 0.20225 0.22637

0.28619 0.30076 0.40515 0.52873 0.67363 0.84193 1.0357 1.2573 1.5091 1.7941 2.1157 2.4783 2.8871 3.3494 3.8747 4.4768 5.1759 6.0046 7.0212 8.3467 10.326 17.399

3.4942 3.3249 2.4682 1.8913 1.4845 1.1878 0.96551 0.79536 0.66264 0.55738 0.47265 0.40351 0.34637 0.29856 0.25808 0.22337 0.19320 0.16654 0.14243 0.11981 0.096844 0.057475

0.045977 0.046200 0.047628 0.049016 0.050355 0.051635 0.052848 0.053985 0.055037 0.055993 0.056842 0.057570 0.058158 0.058586 0.058825 0.058834 0.058554 0.057896 0.056698 0.054637 0.050816 0.034500

0.062946 0.063316 0.065665 0.067931 0.070096 0.072147 0.074072 0.075859 0.077494 0.078962 0.080247 0.081328 0.082179 0.082769 0.083055 0.082977 0.082449 0.081335 0.079403 0.076183 0.070403 0.047510

0.033998 0.033714 0.032039 0.030582 0.029283 0.028106 0.027023 0.026018 0.025074 0.024180 0.023324 0.022495 0.021684 0.020881 0.020074 0.019249 0.018388 0.017465 0.016437 0.015216 0.013568 0.0087388

0.013890 0.013900 0.013945 0.013954 0.013937 0.013897 0.013840 0.013767 0.013682 0.013588 0.013485 0.013376 0.013262 0.013144 0.013022 0.012898 0.012770 0.012637 0.012496 0.012341 0.012146

0.024259 0.024319 0.024692 0.025049 0.025413 0.025800 0.026230 0.026720 0.027292 0.027973 0.028799 0.029817 0.031095 0.032736 0.034897 0.037844 0.042056 0.048501 0.059461 0.081908 0.15172

83.225 83.564 85.668 87.618 89.415 91.063 92.568 93.934 95.167 96.271 97.250 98.108 98.847 99.469 99.978 100.38 100.67 100.88 101.05 101.28 101.95 0

27.717 27.045 23.375 20.607 18.480 16.819 15.504 14.450 13.597 12.902 12.329 11.856 11.460 11.127 10.841 10.587 10.349 10.104 9.8116 9.3953 8.6477 5.6142

3.9767 4.0381 4.4135 4.7651 5.1055 5.4423 5.7797 6.1203 6.4659 6.8180 7.1823 7.5561 7.9453 8.3558 8.7965 9.2815 9.8336 10.490 11.319 12.465 14.402

0.53761 0.54487 0.59162 0.63825 0.68502 0.73220 0.77999 0.82863 0.87831 0.92926 0.98171 1.0359 1.0922 1.1509 1.2125 1.2776 1.3473 1.4230 1.5075 1.6065 1.7364

Single-Phase Properties 1.2679 4.3860 7.5039 10.622 13.740 16.857

2.1006 7.2972 12.494 17.690 22.886 28.083

0.089342 0.11538 0.12659 0.13383 0.13919 0.14344

0.012475 0.012472 0.012472 0.012472 0.012472 0.012472

0.020791 0.020785 0.020785 0.020786 0.020786 0.020786

589.24 1101.2 1441.5 1715.7 1951.7 2162.0

−0.54790 −0.61403 −0.57336 −0.54223 −0.51882 −0.50057

73.713 173.53 252.40 321.87 385.47 444.91

9.7778 22.154 32.215 41.153 49.382 57.105

0.84359 2.9212 4.9988 7.0768 9.1550 11.233

1.2672 4.3875 7.5054 10.623 13.741 16.859

2.1108 7.3086 12.504 17.700 22.896 28.092

0.070191 0.096243 0.10744 0.11468 0.12004 0.12430

0.012506 0.012477 0.012474 0.012473 0.012473 0.012472

0.020837 0.020782 0.020783 0.020784 0.020784 0.020784

596.88 1104.7 1444.0 1717.6 1953.3 2163.5

−0.54470 −0.61441 −0.57410 −0.54296 −0.51949 −0.50117

74.600 174.19 253.00 322.43 386.00 445.41

9.8922 22.180 32.231 41.164 49.391 57.112

5.6058 1.6858 0.99204 0.70266 0.54393 0.44368

0.17839 0.59320 1.0080 1.4232 1.8385 2.2539

1.2641 4.3938 7.5120 10.630 13.747 16.864

2.1560 7.3598 12.552 17.745 22.939 28.134

0.056777 0.082880 0.094074 0.10131 0.10667 0.11092

0.012635 0.012500 0.012484 0.012479 0.012477 0.012475

0.021029 0.020769 0.020771 0.020775 0.020777 0.020779

630.65 1120.3 1454.9 1726.3 1960.6 2169.9

−0.53640 −0.61554 −0.57711 −0.54605 −0.52237 −0.50380

79.465 176.72 255.23 324.52 387.99 447.32

10.374 22.297 32.299 41.212 49.428 57.143

100.00 350.00 600.00 850.00 1100.0 1350.0

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

0.12010 0.034351 0.020042 0.014148 0.010933 0.0089085

100.00 350.00 600.00 850.00 1100.0 1350.0

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.1854 0.34233 0.20005 0.14131 0.10923 0.089021

100.00 350.00 600.00 850.00 1100.0 1350.0

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

8.3265 29.111 49.896 70.681 91.467 112.25

100.00 350.00 600.00 850.00 1100.0 1350.0

10.000 10.000 10.000 10.000 10.000 10.000

10.505 3.3086 1.9638 1.3957 1.0823 0.88379

0.095194 0.30224 0.50921 0.71649 0.92394 1.1315

1.2606 4.4013 7.5200 10.637 13.754 16.871

2.2125 7.4237 12.612 17.802 22.994 28.186

0.050975 0.077139 0.088326 0.095556 0.10091 0.10516

0.012775 0.012527 0.012497 0.012487 0.012482 0.012479

0.021226 0.020755 0.020756 0.020763 0.020769 0.020772

672.30 1140.1 1468.8 1737.3 1969.9 2178.0

−0.53355 −0.61584 −0.58023 −0.54953 −0.52572 −0.50691

85.417 179.43 257.42 326.55 389.94 449.21

10.926 22.435 32.383 41.272 49.475 57.181

100.00 350.00 600.00 850.00 1100.0 1350.0

50.000 50.000 50.000 50.000 50.000 50.000

35.384 14.387 9.0593 6.6048 5.1933 4.2772

0.028261 0.069507 0.11038 0.15140 0.19256 0.23380

1.2602 4.4519 7.5781 10.695 13.811 16.926

2.6732 7.9272 13.097 18.265 23.438 28.615

0.037457 0.063904 0.075052 0.082253 0.087587 0.091828

0.013484 0.012710 0.012587 0.012543 0.012521 0.012508

0.021607 0.020725 0.020666 0.020682 0.020701 0.020716

962.59 1299.0 1584.0 1829.4 2047.7 2245.9

−0.53310 −0.59388 −0.58658 −0.56519 −0.54419 −0.52586

127.36 197.39 268.59 334.90 397.07 455.79

14.393 23.353 32.979 41.713 49.825 57.471

51.724 24.886 16.512 12.359 8.2113

0.019333 0.040183 0.060563 0.080910 0.12178

1.3032 4.5034 7.6393 10.759 16.988

3.2366 8.5217 13.696 18.850 29.166

0.031667 0.058239 0.069398 0.076579 0.086124

0.013965 0.012884 0.012683 0.012605 0.012542

0.021327 0.020817 0.020630 0.020615 0.020653

1228.1 1482.2 1727.9 1948.4 2335.4

−0.51530 −0.55155 −0.57040 −0.56473 -0.53766

197.79 217.91 281.89 342.93 458.27

18.001 24.221 33.590 42.188 57.800

100.00 350.00 600.00 850.00 1350.0

100.00 100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is McCarty, R.D., and Arp, V. D., “A New Wide Range Equation of State for Helium,” Adv. Cryo. Eng. 35:1465–1475, 1990. The source for viscosity is Arp, V. D., McCarty, R. D., and Friend, D. G., “Thermophysical Properties of Helium-4 from 0.8 to 1500 K with Pressures to 2000 MPa,” NIST Technical Note 1334, Boulder, Colo., 1998. The source for thermal conductivity is Hands, B. A., and Arp, V. D., “A Correlation of Thermal Conductivity Data for Helium,” Cryogenics, 21(12):697–703, 1981. Properties at the triple point and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state range from 1% at low temperatures (<20 K) to 0.1% at temperatures between 200 and 400 K, and from 3% in the speed of sound in the liquid phase to 0.1% in the speed of sound between 100 and 500 K. The uncertainty in heat capacities is about 5%. The uncertainty in viscosity is 10%. The uncertainty in thermal conductivity is 5%, except at low temperatures where it increases to 10%.

2-273

2-274 TABLE 2-220 Temperature K

Thermodynamic Properties of Heptane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

1669.5 1656.1 1550.9 1452.3 1358.8 1269.3 1183.0 1099.5 1018.0 938.09 859.20 780.76 702.10 622.51 541.08 456.64 367.46 270.69 161.32 0

−0.52328 −0.52236 −0.51267 −0.49878 −0.48063 −0.45837 −0.43224 −0.40232 −0.36834 −0.32951 −0.28429 −0.22999 −0.16200 −0.072112 0.055234 0.25318 0.60537 1.3922 4.2679 26.539

155.74 155.56 153.03 149.21 144.65 139.70 134.59 129.47 124.41 119.45 114.61 109.90 105.28 100.73 96.175 91.522 86.595 81.058 74.208

Viscosity µPas

Saturated Properties 182.55 185.00 205.00 225.00 245.00 265.00 285.00 305.00 325.00 345.00 365.00 385.00 405.00 425.00 445.00 465.00 485.00 505.00 525.00 540.13

1.7549E-07 2.5788E-07 4.0249E-06 3.5822E-05 0.00021055 0.00090411 0.0030469 0.0084921 0.020357 0.043242 0.083287 0.14809 0.24655 0.38874 0.58581 0.85019 1.1961 1.6406 2.2070 2.7311

7.7457 7.7246 7.5549 7.3882 7.2229 7.0579 6.8917 6.7233 6.5515 6.3747 6.1914 5.9995 5.7965 5.5785 5.3401 5.0723 4.7587 4.3628 3.7651 2.3153

182.55 185.00 205.00 225.00 245.00 265.00 285.00 305.00 325.00 345.00 365.00 385.00 405.00 425.00 445.00 465.00 485.00 505.00 525.00 540.13

1.7549E-07 2.5788E-07 4.0249E-06 3.5822E-05 0.00021055 0.00090411 0.0030469 0.0084921 0.020357 0.043242 0.083287 0.14809 0.24655 0.38874 0.58581 0.85019 1.1961 1.6406 2.2070 2.7311

1.1562E-07 1.6766E-07 2.3614E-06 1.9151E-05 0.00010342 0.00041102 0.0012912 0.0033793 0.0076646 0.015529 0.028777 0.049694 0.081188 0.12711 0.19291 0.28714 0.42516 0.64076 1.0396 2.3153

0.12910 0.12946 0.13236 0.13535 0.13845 0.14169 0.14510 0.14874 0.15264 0.15687 0.16151 0.16668 0.17252 0.17926 0.18726 0.19715 0.21014 0.22921 0.26559 0.43191

−41.621 −41.139 −37.178 −33.158 −29.056 −24.852 −20.526 −16.058 −11.436 −6.6463 −1.6791 3.4744 8.8226 14.375 20.147 26.159 32.455 39.129 46.501 55.157

−41.621 −41.139 −37.178 −33.158 −29.056 −24.852 −20.525 −16.057 −11.433 −6.6396 −1.6656 3.4991 8.8651 14.445 20.256 26.327 32.706 39.506 47.087 56.336

−0.15398 −0.15136 −0.13103 −0.11232 −0.094861 −0.078369 −0.062631 −0.047486 −0.032809 −0.018509 −0.0045150 0.0092301 0.022774 0.036160 0.049436 0.062666 0.075947 0.089485 0.10393 0.12095

0.14886 0.14924 0.15264 0.15666 0.16146 0.16706 0.17337 0.18029 0.18765 0.19534 0.20321 0.21119 0.21920 0.22720 0.23517 0.24315 0.25123 0.25971 0.26971

0.19662 0.19687 0.19936 0.20287 0.20746 0.21311 0.21972 0.22715 0.23527 0.24397 0.25320 0.26293 0.27327 0.28444 0.29693 0.31186 0.33203 0.36700 0.48246

8,649,000. 5,964,600. 423,480. 52,218. 9,669.6 2,433.0 774.44 295.92 130.47 64.395 34.750 20.123 12.317 7.8674 5.1838 3.4827 2.3521 1.5607 0.96195 0.43191

1.0688 1.3477 3.7004 6.1891 8.8178 11.592 14.516 17.589 20.809 24.169 27.660 31.271 34.984 38.780 42.627 46.478 50.254 53.786 56.608 55.157

2.5867 2.8859 5.4048 8.0596 10.854 13.792 16.875 20.102 23.465 26.954 30.555 34.251 38.021 41.838 45.663 49.439 53.067 56.346 58.731 56.336

0.088185 0.086612 0.076690 0.070867 0.068037 0.067457 0.068598 0.071069 0.074569 0.078862 0.083760 0.089105 0.094763 0.10061 0.10653 0.11237 0.11793 0.12283 0.12611 0.12095

0.11339 0.11424 0.12103 0.12796 0.13529 0.14313 0.15149 0.16033 0.16957 0.17912 0.18889 0.19880 0.20878 0.21881 0.22889 0.23906 0.24945 0.26036 0.27248

0.12171 0.12255 0.12935 0.13628 0.14363 0.15153 0.16002 0.16910 0.17874 0.18891 0.19961 0.21088 0.22285 0.23587 0.25064 0.26877 0.29447 0.34231 0.50359

127.51 128.33 134.83 140.99 146.83 152.32 157.38 161.89 165.69 168.60 170.42 170.92 169.84 166.88 161.58 153.35 141.27 123.97 99.655 0

478.91 454.07 301.84 208.67 148.90 109.17 82.064 63.190 49.846 40.301 33.422 28.458 24.914 22.470 20.951 20.312 20.689 22.542 26.764 26.539

4.8076 4.9177 5.8582 6.8855 8.0159 9.2615 10.631 12.128 13.757 15.521 17.424 19.473 21.682 24.071 26.681 29.592 32.980 37.355 45.104

3857.1 3523.8 1889.1 1175.0 805.42 589.93 452.87 359.75 293.16 243.54 205.28 174.90 150.12 129.39 111.58 95.804 81.246 66.925 50.757 0.93944 1.6197 3.8170 4.4411 4.8725 5.2813 5.6934 6.1135 6.5434 6.9849 7.4417 7.9194 8.4272 8.9798 9.6020 10.339 11.280 12.649 15.245

Single-Phase Properties −38.176 −17.194 −0.13101

−38.163 −17.180 −0.11470

−0.13596 −0.051241 −0.00030834

0.15175 0.17852 0.20564

0.19864 0.22520 0.25610

1576.9 1120.8 835.31

−0.51551 −0.41037 −0.26890

28.747 34.506 57.027 83.290

31.670 37.702 61.114 88.236

0.085345 0.10099 0.15307 0.20242

0.19189 0.20404 0.24438 0.27936

0.20297 0.21430 0.25347 0.28808

170.72 179.24 203.88 224.61

31.736 23.441 10.262 5.6176

18.032 20.824 31.544 43.213

0.13154 0.14759 0.17045 0.20269

−38.202 −17.244 7.3729 29.049

−38.071 −17.096 7.5433 29.251

−0.13609 −0.051406 0.019169 0.068829

0.15181 0.17857 0.21722 0.24689

0.19861 0.22503 0.26994 0.32029

1581.3 1127.5 732.61 415.88

−0.51586 −0.41229 −0.19065 0.38883

154.03 131.16 107.01 89.279

2203.0 385.02 158.14 88.930

0.34452 0.30146 0.22064

2.9026 3.3172 4.5322

48.251 54.943 82.131

51.154 58.260 86.663

0.11501 0.12960 0.18133

0.24385 0.25108 0.28189

0.27933 0.27563 0.29560

148.28 166.14 207.02

20.343 14.530 6.3945

31.091 33.576 44.625

10.743 11.093 12.599

7.6229 6.8141 5.9526 4.8210 2.2376

0.13118 0.14675 0.16799 0.20743 0.44691

−38.316 −17.456 6.9316 35.998 72.724

−37.661 −16.722 7.7715 37.035 74.958

−0.13667 −0.052121 0.018047 0.083125 0.15188

0.15208 0.17879 0.21734 0.25622 0.29509

0.19848 0.22435 0.26714 0.32159 0.43674

1600.2 1156.4 783.30 423.74 134.50

−0.51732 −0.42010 −0.23205 0.32892 7.5010

154.89 132.76 109.78 89.492 63.671

2331.7 405.07 169.17 84.960 27.947

200.00 300.00 371.09

0.10000 0.10000 0.10000

7.5975 6.7666 6.1340

371.09 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

0.034210 0.031296 0.024471 0.020221

200.00 300.00 400.00 474.30

1.0000 1.0000 1.0000 1.0000

7.6022 6.7756 5.8668 4.9336

474.30 500.00 600.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

0.13162 0.14778 0.16303 29.231 31.953 40.864 49.453

153.84 130.79 113.16

2174.6 380.51 195.32 7.5845 8.1460 10.052 11.883

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

7.6479 6.8595 6.0454 5.0885 3.8337

0.13075 0.14578 0.16542 0.19652 0.26084

−38.454 −17.705 6.4514 34.818 67.259

−37.147 −16.247 8.1056 36.783 69.867

−0.13737 −0.052975 0.016799 0.080607 0.14081

0.15241 0.17908 0.21753 0.25576 0.29039

0.19835 0.22363 0.26465 0.30908 0.35093

1623.3 1190.5 838.61 532.87 303.32

−0.51893 −0.42839 −0.26850 0.040826 0.87871

155.94 134.68 112.88 95.612 82.499

2498.6 430.15 182.05 98.261 55.622

300.00 400.00 500.00 600.00

50.000 50.000 50.000 50.000

7.1493 6.5292 5.9397 5.3846

0.13987 0.15316 0.16836 0.18572

−19.254 3.9425 30.741 60.741

−12.260 11.600 39.159 70.026

−0.058701 0.0096885 0.071034 0.12723

0.18121 0.21935 0.25677 0.28945

0.22068 0.25730 0.29309 0.32327

1412.7 1139.8 940.60 800.70

−0.46360 −0.37509 −0.29668 −0.23327

147.62 130.89 119.90 114.54

635.00 269.35 158.19 109.31

7.4106 6.8927 6.4248 6.0005

0.13494 0.14508 0.15565 0.16665

−20.560 2.1360 28.397 57.894

−0.064263 0.0037009 0.064512 0.12021

0.18354 0.22143 0.25865 0.29113

0.21976 0.25527 0.29041 0.32068

1622.2 1385.0 1216.1 1095.6

−0.47764 −0.40643 −0.35005 −0.31028

160.40 146.71 137.75 133.53

909.65 364.81 209.24 143.14

300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00

−7.0660 16.644 43.961 74.559

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., and Wagner, W., “Equations of State for Technical Applications. II. Results for Nonpolar Fluids,” Int. J. Thermophys., 24(1):41–109, 2003. The source for viscosity is NIST14, Version 9.08. The source for thermal conductivity is NIST14, Version 9.08. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are approximately 0.2% (to 0.5% at high pressures) in density, 1% (in the vapor phase) to 2% in heat capacity, 1% (in the vapor phase) to 2% in the speed of sound, and 0.2% in vapor pressure, except in the critical region. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6%.

2-275

2-276 TABLE 2-221 Temperature K

Thermodynamic Properties of Hexane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−30.053 −29.702 −27.247 −24.744 −22.190 −19.583 −16.916 −14.184 −11.380 −8.4998 −5.5378 −2.4895 0.64962 3.8834 7.2160 10.652 14.195 17.854 21.638 25.562 29.656 33.983 38.752 45.698

−30.053 −29.702 −27.247 −24.744 −22.190 −19.583 −16.916 −14.183 −11.379 −8.4969 −5.5324 −2.4800 0.66541 3.9085 7.2544 10.709 14.278 17.970 21.799 25.782 29.956 34.393 39.331 46.822

6.5712 6.7793 8.2570 9.8013 11.411 13.086 14.827 16.633 18.505 20.440 22.438 24.494 26.606 28.769 30.977 33.223 35.497 37.783 40.061 42.299 44.438 46.363 47.752 45.698

8.0497 8.2759 9.8783 11.547 13.281 15.079 16.941 18.865 20.850 22.893 24.989 27.135 29.324 31.549 33.804 36.077 38.356 40.621 42.848 44.994 46.992 48.702 49.742 46.822

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.11799 −0.11602 −0.10292 −0.090556 −0.078815 −0.067597 −0.056819 −0.046408 −0.036304 −0.026456 −0.016823 −0.0073691 0.0019333 0.011109 0.020180 0.029166 0.038087 0.046965 0.055828 0.064717 0.073697 0.082910 0.092809 0.10743

0.11891 0.11944 0.12299 0.12642 0.12992 0.13361 0.13753 0.14172 0.14616 0.15084 0.15571 0.16074 0.16590 0.17114 0.17645 0.18178 0.18713 0.19251 0.19791 0.20338 0.20902 0.21509 0.22251

0.16166 0.16212 0.16527 0.16850 0.17196 0.17575 0.17992 0.18448 0.18943 0.19474 0.20040 0.20638 0.21269 0.21932 0.22633 0.23378 0.24184 0.25078 0.26113 0.27400 0.29208 0.32426 0.42766

1638.9 1626.3 1542.5 1463.6 1388.4 1316.2 1246.3 1178.5 1112.3 1047.6 984.03 921.37 859.40 797.86 736.47 674.93 612.85 549.78 485.05 417.67 346.06 267.23 174.59 0

−0.55189 −0.54996 −0.53626 −0.52146 −0.50504 −0.48667 −0.46617 −0.44339 −0.41819 −0.39031 −0.35939 −0.32478 −0.28553 −0.24020 −0.18658 −0.12130 −0.038945 0.069646 0.22117 0.44954 0.83549 1.6275 4.0963 21.950

155.43 155.64 155.51 153.36 149.89 145.59 140.79 135.74 130.60 125.50 120.53 115.73 111.15 106.82 102.74 98.910 95.319 91.941 88.736 85.638 82.537 79.234 75.402

2392.6 2235.0 1474.2 1048.0 786.13 613.59 493.58 406.45 340.96 290.31 250.18 217.72 190.99 168.61 149.58 133.16 118.78 105.97 94.348 83.551 73.177 62.632 50.415

0.096281 0.094966 0.087463 0.082256 0.078835 0.076827 0.075951 0.075993 0.076780 0.078177 0.080072 0.082372 0.085000 0.087889 0.090978 0.094213 0.097538 0.10090 0.10422 0.10741 0.11033 0.11272 0.11384 0.10743

0.095572 0.096248 0.10079 0.10523 0.10975 0.11444 0.11936 0.12457 0.13005 0.13579 0.14178 0.14798 0.15435 0.16085 0.16746 0.17415 0.18092 0.18775 0.19467 0.20173 0.20903 0.21678 0.22558

0.10389 0.10456 0.10910 0.11356 0.11809 0.12282 0.12783 0.13317 0.13886 0.14492 0.15136 0.15817 0.16537 0.17297 0.18106 0.18976 0.19931 0.21014 0.22312 0.24003 0.26537 0.31367 0.46871

136.57 137.36 142.71 147.84 152.75 157.43 161.83 165.91 169.59 172.79 175.43 177.42 178.64 179.01 178.39 176.64 173.59 169.01 162.60 153.94 142.42 127.17 106.85 0

Saturated Properties 177.83 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 507.82

1.2771E-06 1.7416E-06 1.2006E-05 6.0531E-05 0.00023840 0.00077035 0.0021200 0.0051148 0.011066 0.021865 0.040031 0.068721 0.11169 0.17326 0.25821 0.37176 0.51954 0.70759 0.94243 1.2313 1.5827 2.0071 2.5196 3.0429

8.8394 8.8166 8.6607 8.5066 8.3535 8.2006 8.0472 7.8927 7.7363 7.5774 7.4151 7.2487 7.0772 6.8995 6.7141 6.5194 6.3129 6.0916 5.8506 5.5825 5.2736 4.8943 4.3496 2.7059

177.83 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 507.82

1.2771E-06 1.7416E-06 1.2006E-05 6.0531E-05 0.00023840 0.00077035 0.0021200 0.0051148 0.011066 0.021865 0.040031 0.068721 0.11169 0.17326 0.25821 0.37176 0.51954 0.70759 0.94243 1.2313 1.5827 2.0071 2.5196 3.0429

8.6377E-07 1.1637E-06 7.4054E-06 3.4674E-05 0.00012750 0.00038658 0.0010029 0.0022917 0.0047185 0.0089152 0.015688 0.026023 0.041099 0.062314 0.091349 0.13027 0.18173 0.24933 0.33826 0.45679 0.61964 0.85826 1.2664 2.7059

0.11313 0.11342 0.11546 0.11756 0.11971 0.12194 0.12427 0.12670 0.12926 0.13197 0.13486 0.13796 0.14130 0.14494 0.14894 0.15339 0.15840 0.16416 0.17092 0.17913 0.18963 0.20432 0.22990 0.36957 1,157,700. 859,330. 135,040. 28,840. 7,842.9 2,586.8 997.06 436.35 211.93 112.17 63.742 38.427 24.332 16.048 10.947 7.6763 5.5026 4.0108 2.9563 2.1892 1.6138 1.1651 0.78962 0.36957

381.55 363.60 265.13 198.42 151.75 118.24 93.691 75.403 61.606 51.090 43.011 36.767 31.926 28.172 25.277 23.080 21.471 20.386 19.809 19.784 20.448 22.098 25.172 21.950

5.2567 5.3651 6.1371 6.9555 7.8286 8.7644 9.7697 10.851 12.012 13.258 14.592 16.018 17.540 19.162 20.892 22.739 24.719 26.858 29.203 31.843 34.979 39.160 46.755

3.0564 3.2844 4.1334 4.5617 4.9114 5.2472 5.5832 5.9228 6.2669 6.6161 6.9712 7.3333 7.7043 8.0869 8.4853 8.9054 9.3563 9.8524 10.417 11.090 11.951 13.186 15.416

Single-Phase Properties 200.00 300.00 341.45

0.10000 0.10000 0.10000

8.6098 7.5784 7.1184

341.45 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

0.037028 0.030895 0.024344 0.020168

200.00 300.00

1.0000 1.0000

8.6159 7.5906

400.00 438.24

1.0000 1.0000

438.24 500.00 600.00 200.00 300.00 400.00 500.00 600.00

0.11615 0.13195 0.14048 27.006 32.368 41.078 49.583

−26.420 −8.5042 −0.10255

−26.409 −8.4910 −0.088504

−0.098737 −0.026471 −0.00025850

0.12414 0.15084 0.16467

0.16632 0.19472 0.21117

1516.3 1048.3 874.04

−0.53154 −0.39056 −0.29532

178.43 196.30 221.55 243.38

32.965 17.891 8.1885 4.6052

155.01 125.55 112.22

26.101 35.725 54.980 77.506

28.801 38.962 59.088 82.464

0.084352 0.11177 0.15654 0.19908

0.15282 0.17421 0.20940 0.23990

0.16363 0.18376 0.21824 0.24850

0.11606 0.13174

−26.445 −8.5541

−26.329 −8.4223

−0.098862 −0.026637

0.12419 0.15088

0.16628 0.19451

6.4028 5.7953

0.15618 0.17255

12.928 22.473

13.084 22.645

0.034934 0.057745

0.18534 0.19908

0.23836 0.26364

642.57 470.76

−0.081596 0.26228

96.993 88.060

124.96 91.958

1.0000 1.0000 1.0000

0.36099 0.27702 0.21383

2.7701 3.6098 4.6765

40.550 53.661 76.704

43.320 57.271 81.381

0.10492 0.13470 0.17859

0.19618 0.21315 0.24143

0.22634 0.22970 0.25320

160.93 196.29 230.76

19.754 10.036 5.0002

29.744 36.713 49.365

10.551 11.711 13.575

5.0000 5.0000 5.0000 5.0000 5.0000

8.6424 7.6426 6.5335 4.8882 1.5349

0.11571 0.13085 0.15306 0.20457 0.65149

−26.553 −8.7667 12.434 38.358 71.648

−25.975 −8.1124 13.199 39.381 74.906

−0.099407 −0.027355 0.033675 0.091842 0.15663

0.12440 0.15106 0.18536 0.22066 0.24956

0.16610 0.19368 0.23424 0.30042 0.30605

1541.5 1087.5 702.66 311.48 173.73

−0.53416 −0.40464 −0.15548 1.0544 6.3809

157.01 128.58 100.50 81.341 59.195

1411.9 312.23 136.08 64.368 20.153

1521.0 1055.7

−0.53204 −0.39338

17.170 23.256 35.216 48.247

1308.5 290.66 196.89

155.38 126.12

7.6155 8.9061 11.054 13.104 1327.2 294.62

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

8.6744 7.7032 6.6684 5.3956 3.6515

0.11528 0.12982 0.14996 0.18533 0.27386

−26.683 −9.0138 11.917 36.755 65.462

−25.530 −7.7156 13.416 38.608 68.201

−0.10007 −0.028204 0.032326 0.088363 0.14220

0.12466 0.15128 0.18546 0.21933 0.24998

0.16591 0.19280 0.23093 0.27365 0.31536

1566.5 1124.9 765.76 455.87 248.16

−0.53649 −0.41638 −0.21470 0.25439 1.7834

159.01 131.56 104.41 88.518 79.582

1521.9 334.21 148.74 80.764 43.840

300.00 400.00 500.00 600.00

50.000 50.000 50.000 50.000

8.0787 7.3187 6.5956 5.9215

0.12378 0.13664 0.15162 0.16888

−10.511 9.3999 32.487 58.351

−4.3222 16.232 40.068 66.794

−0.033766 0.025137 0.078194 0.12685

0.15301 0.18689 0.21979 0.24844

0.18931 0.22225 0.25371 0.27993

1362.7 1090.0 895.54 763.82

−0.46412 −0.36559 −0.27952 −0.21072

152.57 128.45 115.10 111.13

513.27 231.13 140.55 100.20

8.4063 7.7789 7.2120 6.7006

0.11896 0.12855 0.13866 0.14924

−11.737 7.6799 30.244 55.635

0.15835 20.535 44.110 70.559

−0.039032 0.019368 0.071845 0.11999

0.15494 0.18864 0.22138 0.24989

0.18811 0.21997 0.25089 0.27732

1582.2 1345.0 1178.8 1062.8

−0.48273 −0.40550 −0.34615 −0.30501

175.12 152.26 138.03 132.17

753.68 319.21 188.34 132.10

300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., and Wagner, W., “Equations of State for Technical Applications. II. Results for Nonpolar Fluids,” Int. J. Thermophys. 24(1):41–109, 2003. The source for viscosity and thermal conductivity is NIST14, Version 9.08. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are approximately 0.2% (to 0.5% at high pressures) in density, 1% (in the vapor phase) to 2% in heat capacity, 1% (in the vapor phase) to 2% in the speed of sound, and 0.2% in vapor pressure, except in the critical region. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6 %.

2-277

2-278

PHYSICAL AND CHEMICAL DATA

TABLE 2-222 Temperature, K 386.6 390 400 410 420 440 460 480 500 520

Saturated Hydrazine vf, m3/kg

vg, m3/kg

h f, kJ/kg

hg, kJ/kg

s f, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

1.013 1.135 1.560 2.102 2.786

0.001 053 0.001 060 0.001 081 0.001 104 0.001 127

0.9833 0.8850 0.6579 0.4994 0.3850

−105.9 −104.8 −101.4 −97.6 −93.9

65.4 66.0 68.2 70.6 73.0

0.5994 0.6029 0.6120 0.6211 0.6300

1.0426 1.0409 1.0360 1.0314 1.0275

4.732 7.610 11.76 17.42 29.59

0.001 178 0.001 235 0.001 299 0.001 374 0.001 460

0.2355 0.1500 0.1005 0.0690 0.0407

−86.1 −76.9 −67.1 −57.3 −47.8

77.6 82.1 86.6 90.8 94.6

0.6492 0.6707 0.6916 0.7124 0.7320

1.0212 1.0163 1.0118 1.0086 1.0058

Pressure, bar

540 560 580 600 620

34.75 47.09 62.44 81.17 102.7

0.001 563 0.001 681 0.001 835 0.002 045 0.002 320

0.0353 0.0263 0.0196 0.0142 0.0106

−36.0 −25.2 −12.4 5.2 23.2

97.7 101.2 103.6 104.2 103.6

0.7566 0.7762 0.8002 0.8335 0.8671

1.0042 1.0020 1.0002 0.9988 0.9967

640 653 c

128.1 146.9

0.002 86 0.004 33

0.0074 0.0043

45.9 83.7

98.1 83.7

0.9035 0.9715

0.9906 0.9715

Converted from E. F. Fricke, Republic Aviation Co. rept. F-5028-101. c = critical point.

TABLE 2-223

Thermodynamic Properties of Normal Hydrogen

Temperature K

Pressure MPa

13.957 14.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.190

0.0077031 0.0078936 0.013436 0.021534 0.032848 0.048078 0.067960 0.093249 0.12472 0.16314 0.20932 0.26406 0.32818 0.40250 0.48788 0.58524 0.69554 0.81989 0.95964 1.1168 1.3301

38.148 38.129 37.701 37.261 36.802 36.321 35.812 35.274 34.702 34.092 33.439 32.738 31.979 31.152 30.242 29.225 28.067 26.706 25.017 22.637 14.940

13.957 14.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.190

0.0077031 0.0078936 0.013436 0.021534 0.032848 0.048078 0.067960 0.093249 0.12472 0.16314 0.20932 0.26406 0.32818 0.40250 0.48788 0.58524 0.69554 0.81989 0.95964 1.1168 1.3301

0.067540 0.069018 0.11050 0.16764 0.24349 0.34126 0.46437 0.61652 0.80187 1.0251 1.2919 1.6089 1.9848 2.4307 2.9618 3.6003 4.3810 5.3643 6.6763 8.6823 14.940

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.026214 0.026226 0.026524 0.026838 0.027172 0.027533 0.027923 0.028350 0.028817 0.029333 0.029905 0.030546 0.031271 0.032101 0.033067 0.034217 0.035629 0.037444 0.039973 0.044175 0.066934

−0.10434 −0.10367 −0.088896 −0.074446 −0.059414 −0.043440 −0.026375 −0.0081516 0.011274 0.031947 0.053929 0.077308 0.10222 0.12884 0.15744 0.18843 0.22245 0.26061 0.30524 0.36302 0.53004

−0.10414 −0.10346 −0.088539 −0.073868 −0.058521 −0.042116 −0.024477 −0.0055080 0.014868 0.036732 0.060188 0.085375 0.11248 0.14176 0.17357 0.20846 0.24723 0.29132 0.34360 0.41236 0.61907

−0.0059480 −0.0059000 −0.0048799 −0.0039471 −0.0030355 −0.0021219 −0.0011983 −0.00026211 0.00068790 0.0016528 0.0026344 0.0036357 0.0046610 0.0057167 0.0068122 0.0079614 0.0091865 0.010527 0.012063 0.014035 0.020012

14.806 14.489 9.0494 5.9651 4.1069 2.9303 2.1535 1.6220 1.2471 0.97549 0.77406 0.62153 0.50383 0.41141 0.33763 0.27775 0.22826 0.18642 0.14978 0.11518 0.066934

0.68715 0.68764 0.69864 0.70899 0.71875 0.72783 0.73614 0.74359 0.75005 0.75541 0.75951 0.76218 0.76318 0.76224 0.75895 0.75276 0.74277 0.72747 0.70374 0.66274 0.53004

0.80120 0.80201 0.82024 0.83745 0.85365 0.86871 0.88249 0.89484 0.90558 0.91455 0.92154 0.92630 0.92853 0.92783 0.92368 0.91531 0.90154 0.88031 0.84748 0.79136 0.61907

0.058918 0.058777 0.055705 0.053010 0.050622 0.048480 0.046537 0.044755 0.043103 0.041554 0.040085 0.038674 0.037303 0.035950 0.034594 0.033206 0.031749 0.030160 0.028317 0.025879 0.020012

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.011064 0.010957 0.0096961 0.0096482 0.010003 0.010462 0.010915 0.011323 0.011677 0.011978 0.012235 0.012457 0.012655 0.012840 0.013025 0.013224 0.013460 0.013764 0.014198 0.014926

0.015654 0.015547 0.014420 0.014709 0.015557 0.016642 0.017842 0.019120 0.020476 0.021935 0.023539 0.025351 0.027465 0.030024 0.033265 0.037610 0.043909 0.054194 0.074872 0.14185

1361.1 1359.6 1318.5 1271.6 1226.6 1185.5 1147.3 1110.7 1074.7 1038.2 1000.5 960.99 919.10 874.29 826.00 773.58 716.22 652.73 581.16 497.24 0

−1.4137 −1.4230 −1.5204 −1.4695 −1.3623 −1.2409 −1.1194 −1.0003 −0.88232 −0.76268 −0.63795 −0.50414 −0.35648 −0.18882 0.0073109 0.24446 0.54283 0.93827 1.5038 2.4292 5.3208

76.293 76.650 84.106 90.079 94.784 98.405 101.10 103.01 104.24 104.87 104.98 104.60 103.79 102.53 100.83 98.654 95.935 92.547 88.221 82.176

25.463 25.310 22.215 19.784 17.815 16.182 14.799 13.607 12.565 11.641 10.811 10.057 9.3625 8.7151 8.1034 7.5160 6.9409 6.3620 5.7518 5.0391

0.013157 0.013129 0.012872 0.012907 0.012992 0.013083 0.013178 0.013280 0.013392 0.013514 0.013650 0.013802 0.013973 0.014167 0.014392 0.014655 0.014971 0.015358 0.015854 0.016535

0.021964 0.021944 0.021898 0.022199 0.022618 0.023121 0.023724 0.024449 0.025329 0.026401 0.027724 0.029376 0.031482 0.034234 0.037960 0.043253 0.051322 0.065054 0.093486 0.18606

304.61 305.17 316.15 325.05 333.00 340.22 346.75 352.59 357.75 362.25 366.11 369.34 371.95 373.96 375.38 376.19 376.39 375.97 374.91 373.31 0

Saturated Properties

31.943 31.808 28.572 25.724 23.407 21.522 19.961 18.642 17.507 16.513 15.629 14.829 14.091 13.396 12.726 12.061 11.374 10.624 9.7362 8.5059 5.3208

10.375 10.431 11.624 12.681 13.681 14.669 15.675 16.716 17.806 18.956 20.180 21.493 22.916 24.477 26.218 28.202 30.535 33.407 37.226 43.200

0.66345 0.66695 0.74268 0.81064 0.87421 0.93555 0.99611 1.0569 1.1186 1.1819 1.2472 1.3151 1.3863 1.4619 1.5433 1.6331 1.7362 1.8628 2.0375 2.3378

2-279

2-280

TABLE 2-223

Thermodynamic Properties of Normal Hydrogen (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.81207 1.7949 3.0224 4.4642 5.9947 7.5545

1.0088 2.6261 4.4785 6.5442 8.6984 10.882

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Single-Phase Properties 25.000 100.00 175.00 250.00 325.00 400.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

25.000 31.268

1.0000 1.0000

31.268 100.00 175.00 250.00 325.00 400.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

25.000 100.00 175.00 250.00 325.00 400.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

0.50823 0.12030 0.068680 0.048077 0.036986 0.030054

1.9676 8.3127 14.560 20.800 27.037 33.273

0.049309 0.079163 0.092882 0.10269 0.11022 0.11626

0.012734 0.014263 0.018150 0.020003 0.020681 0.020865

0.022519 0.022637 0.026480 0.028323 0.028998 0.029180

403.66 808.92 1026.9 1209.1 1371.7 1519.7

12.894 1.4058 0.13575 −0.22980 −0.38965 −0.47650

20.761 68.334 117.11 160.59 197.72 234.06

1.3142 4.1896 6.1845 7.9025 9.4561 10.892

0.030538 0.040861

0.089693 0.31894

0.12023 0.35980

0.0041410 0.012531

0.012580 0.014353

0.025394 0.084709

985.14 560.14

−0.51115 1.7031

106.80 86.829

9.9923 5.5759

0.14049 0.83027 1.4653 2.0926 2.7176 3.3418

0.69511 1.7679 3.0101 4.4576 5.9910 7.5525

0.83559 2.5982 4.4754 6.5501 8.7086 10.894

0.027747 0.059751 0.073667 0.083515 0.091062 0.097113

0.016014 0.014331 0.018190 0.020028 0.020699 0.020878

0.10713 0.023244 0.026659 0.028401 0.029039 0.029204

374.52 817.03 1035.8 1217.3 1379.1 1526.4

9.4548 1.3036 0.11718 −0.23428 −0.39039 −0.47605

38.524 70.413 118.31 161.46 198.43 234.65

2.0997 4.2550 6.2213 7.9283 9.4759 10.908

35.661 5.9683 3.3132 2.3268 1.7990 1.4680

0.028042 0.16755 0.30183 0.42978 0.55587 0.68120

0.046611 1.6549 2.9582 4.4292 5.9750 7.5440

0.18682 2.4927 4.4673 6.5781 8.7543 10.950

0.0021443 0.045314 0.059998 0.070022 0.077631 0.083710

0.012376 0.014583 0.018352 0.020136 0.020776 0.020937

0.020610 0.025613 0.027370 0.028723 0.029211 0.029304

1223.4 865.94 1077.3 1254.0 1412.0 1556.2

−0.90198 0.86369 0.032971 −0.25678 −0.39563 −0.47537

119.36 80.395 123.58 165.19 201.36 237.09

13.101 4.5875 6.3871 8.0420 9.5631 10.979

32.746 24.474 7.1182 1.2044 0.68243 0.47788 0.36797 0.29924

25.000 100.00 175.00 250.00 325.00 400.00

10.000 10.000 10.000 10.000 10.000 10.000

37.930 11.417 6.3697 4.5028 3.5006 2.8687

0.026364 0.087588 0.15699 0.22209 0.28567 0.34859

0.020221 1.5346 2.9000 4.3966 5.9563 7.5339

0.28386 2.4105 4.4699 6.6175 8.8130 11.020

0.00059913 0.038585 0.053931 0.064133 0.071811 0.077921

0.012222 0.014838 0.018532 0.020260 0.020867 0.021006

0.018499 0.027423 0.028063 0.029065 0.029402 0.029419

1402.1 955.43 1133.3 1300.6 1453.0 1593.1

−1.0762 0.39679 −0.068718 −0.28786 −0.40519 −0.47687

131.12 94.196 130.39 169.92 205.05 240.15

16.625 5.1692 6.6199 8.1898 9.6733 11.067

100.00 175.00 250.00 325.00 400.00

50.000 50.000 50.000 50.000 50.000

31.993 22.700 17.524 14.304 12.107

0.031257 0.044053 0.057066 0.069911 0.082595

1.1768 2.6415 4.2297 5.8539 7.4773

2.7397 4.8442 7.0830 9.3494 11.607

0.023964 0.039587 0.050225 0.058153 0.064404

0.016349 0.019545 0.020999 0.021434 0.021458

0.026254 0.029321 0.030163 0.030202 0.029985

1710.1 1632.6 1690.7 1784.8 1887.2

−0.57213 −0.46415 −0.46214 −0.48443 −0.51016

192.47 189.52 211.67 237.63 267.11

10.534 9.1377 9.7772 10.827 11.965

33.019 27.257 23.228 20.261

0.030286 0.036688 0.043051 0.049356

2.5589 4.1604 5.8083 7.4555

5.5875 7.8292 10.113 12.391

0.033643 0.044291 0.052281 0.058588

0.020316 0.021603 0.021923 0.021864

0.029140 0.030346 0.030469 0.030246

2128.4 2125.4 2170.6 2235.8

−0.52750 −0.50698 −0.51215 −0.52481

282.17 303.19 327.83 356.83

13.079 12.218 12.546 13.289

175.00 250.00 325.00 400.00

100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Younglove, B. A., “Thermophysical Properties of Fluids. I. Argon, Ethylene, Parahydrogen, Nitrogen, Nitrogen Trifluoride, and Oxygen,” J. Phys. Chem. Ref. Data, Suppl. 1, 11: 1–11, 1982. The source for viscosity is McCarty, R. D., and Weber, L. A., “Thermophysical Properties of Parahydrogen from the Freezing Liquid Line to 5000 R for Pressures to 10,000 psia,” N.B.S. Tech. Note 617, 1972. The source for thermal conductivity is McCarty, R. D., and Weber, L. A., “Thermophysical Properties of Parahydrogen from the Freezing Liquid Line to 5000 R for Pressures to 10,000 psia,” N.B.S. Tech. Note 617, 1972. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.1% in the liquid phase, 0.25% in the vapor phase, and 0.2% in the supercritical region. The uncertainty in heat capacity is 3%, and the uncertainty in speed of sound is 2% in the liquid phase and 1% elsewhere. The uncertainty in viscosity ranges from 4% to 15%. The uncertainty in thermal conductivity below 100 K is estimated to be 3% below 150 atm and up to 10% below 700 atm. For temperatures around 100 K at low densities, the uncertainty is about 1%. Above 100 K, the uncertainty is estimated to be on the order of 10%.

TABLE 2-224

Thermodynamic Properties of para-Hydrogen

Temperature K

Pressure MPa

13.800 14.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 32.938

0.0070373 0.0078936 0.013436 0.021534 0.032848 0.048078 0.067960 0.093249 0.12472 0.16314 0.20932 0.26406 0.32818 0.40250 0.48788 0.58524 0.69554 0.81989 0.95964 1.1168 1.2838

38.214 38.129 37.701 37.261 36.802 36.321 35.812 35.274 34.702 34.092 33.439 32.738 31.979 31.152 30.242 29.225 28.067 26.706 25.017 22.637 15.556

13.800 14.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 32.938

0.0070373 0.0078936 0.013436 0.021534 0.032848 0.048078 0.067960 0.093249 0.12472 0.16314 0.20932 0.26406 0.32818 0.40250 0.48788 0.58524 0.69554 0.81989 0.95964 1.1168 1.2838

0.062340 0.069018 0.11050 0.16764 0.24349 0.34126 0.46437 0.61652 0.80187 1.0251 1.2919 1.6089 1.9848 2.4307 2.9618 3.6003 4.3810 5.3643 6.6763 8.6823 15.556

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.026168 0.026226 0.026524 0.026838 0.027172 0.027533 0.027923 0.028350 0.028817 0.029333 0.029905 0.030546 0.031271 0.032101 0.033067 0.034217 0.035629 0.037444 0.039973 0.044175 0.064284

−0.10640 −0.10334 −0.088758 −0.074351 −0.059344 −0.043395 −0.026352 −0.0081471 0.011263 0.031923 0.053894 0.077266 0.10217 0.12879 0.15739 0.18839 0.22242 0.26061 0.30526 0.36307 0.51279

−0.10621 −0.10313 −0.088401 −0.073773 −0.058452 −0.042071 −0.024454 −0.0055035 0.014857 0.036709 0.060154 0.085332 0.11243 0.14171 0.17353 0.20842 0.24721 0.29131 0.34362 0.41240 0.59532

−0.0060986 −0.0058783 −0.0048719 −0.0039418 −0.0030317 −0.0021196 −0.0011972 −0.00026188 0.00068738 0.0016517 0.0026329 0.0036338 0.0046588 0.0057145 0.0068101 0.0079596 0.0091852 0.010526 0.012063 0.014036 0.019385

16.041 14.489 9.0494 5.9651 4.1069 2.9303 2.1535 1.6220 1.2471 0.97549 0.77406 0.62153 0.50383 0.41141 0.33763 0.27775 0.22826 0.18642 0.14978 0.11518 0.064284

0.68576 0.68798 0.69878 0.70909 0.71881 0.72787 0.73616 0.74359 0.75004 0.75539 0.75948 0.76213 0.76314 0.76219 0.75891 0.75272 0.74275 0.72747 0.70375 0.66278 0.51279

0.79865 0.80235 0.82037 0.83754 0.85372 0.86876 0.88251 0.89484 0.90557 0.91453 0.92151 0.92626 0.92848 0.92778 0.92363 0.91527 0.90151 0.88031 0.84749 0.79141 0.59532

0.059471 0.058799 0.055713 0.053015 0.050625 0.048482 0.046538 0.044755 0.043102 0.041553 0.040083 0.038672 0.037301 0.035948 0.034592 0.033204 0.031747 0.030159 0.028317 0.025880 0.019385

4.1229 10.397 16.641 22.879 29.116

1.1351 2.4062 4.2197 5.9434 7.5556

1.5474 3.4459 5.8837 8.2313 10.467

0.064284 0.086764 0.10202 0.11200 0.11919

1.0703 2.3861 4.2097 5.9379 7.5525

1.4520 3.4298 5.8845 8.2390 10.478

0.043845 0.067460 0.082821 0.092837 0.10004

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.010937 0.010548 0.0096208 0.0096210 0.0099778 0.010438 0.010895 0.011306 0.011663 0.011966 0.012226 0.012450 0.012652 0.012841 0.013030 0.013235 0.013476 0.013785 0.014225 0.014957

0.015530 0.015138 0.014345 0.014682 0.015532 0.016618 0.017822 0.019103 0.020462 0.021923 0.023529 0.025344 0.027461 0.030025 0.033270 0.037620 0.043924 0.054215 0.074899 0.14188

1376.3 1367.4 1320.2 1272.2 1227.2 1186.0 1147.7 1111.1 1075.0 1038.4 1000.7 961.13 919.17 874.28 825.90 773.39 715.93 652.36 580.72 496.76 0

−1.4263 −1.4615 −1.5284 −1.4722 −1.3645 −1.2427 −1.1207 −1.0012 −0.88292 −0.76309 −0.63820 −0.50428 −0.35652 −0.18881 0.0073097 0.24439 0.54264 0.93791 1.5033 2.4286 5.2019

75.249 76.869 84.107 90.003 94.698 98.339 101.06 103.00 104.25 104.89 105.01 104.64 103.82 102.56 100.85 98.669 95.941 92.543 88.209 82.155

26.035 25.310 22.215 19.784 17.815 16.182 14.799 13.607 12.565 11.641 10.811 10.057 9.3625 8.7151 8.1034 7.5160 6.9409 6.3620 5.7518 5.0391

0.012705 0.012720 0.012797 0.012879 0.012967 0.013059 0.013158 0.013263 0.013378 0.013503 0.013641 0.013795 0.013969 0.014168 0.014397 0.014666 0.014986 0.015380 0.015881 0.016566

0.021484 0.021535 0.021823 0.022172 0.022593 0.023097 0.023703 0.024433 0.025315 0.026390 0.027714 0.029369 0.031478 0.034235 0.037965 0.043264 0.051337 0.065075 0.093513 0.18609

305.17 307.13 316.53 325.20 333.13 340.35 346.87 352.69 357.84 362.33 366.17 369.39 371.98 373.96 375.33 376.10 376.25 375.77 374.65 372.98 0

0.012710 0.022470 0.024087 0.022011 0.021169

0.021301 0.030820 0.032413 0.030330 0.029486

583.04 841.25 1054.3 1250.9 1418.7

0.012853 0.022526 0.024121 0.022033 0.021186

0.024464 0.031195 0.032545 0.030392 0.029519

575.43 848.73 1062.6 1258.8 1425.9

Saturated Properties

33.288 32.412 28.670 25.755 23.433 21.544 19.978 18.654 17.516 16.521 15.634 14.832 14.092 13.396 12.725 12.058 11.370 10.621 9.7335 8.5044 5.2019

10.454 10.649 11.625 12.604 13.595 14.603 15.637 16.705 17.816 18.981 20.214 21.529 22.951 24.507 26.241 28.217 30.541 33.404 37.214 43.179

0.65046 0.66695 0.74268 0.81064 0.87421 0.93555 0.99611 1.0569 1.1186 1.1819 1.2472 1.3151 1.3863 1.4619 1.5433 1.6331 1.7362 1.8628 2.0375 2.3378

4.8322 0.59482 −0.023305 −0.27924 −0.41834

38.576 106.18 155.64 183.10 212.22

2.4717 4.9004 6.7798 8.4358 9.9461

4.4711 0.55405 −0.032981 −0.28185 −0.41854

42.291 107.83 156.70 183.91 212.88

2.6080 4.9523 6.8120 8.4592 9.9645

Single-Phase Properties

2-281

50.000 125.00 200.00 275.00 350.00

0.10000 0.10000 0.10000 0.10000 0.10000

0.24255 0.096180 0.060094 0.043708 0.034345

50.000 125.00 200.00 275.00 350.00

1.0000 1.0000 1.0000 1.0000 1.0000

2.6201 0.95806 0.59708 0.43458 0.34179

0.38166 1.0438 1.6748 2.3010 2.9257

2-282

TABLE 2-224

Thermodynamic Properties of para-Hydrogen (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

5.0000 5.0000 5.0000 5.0000 5.0000

16.715 4.6789 2.9000 2.1189 1.6731

0.059826 0.21372 0.34482 0.47194 0.59770

0.73484 2.3021 4.1677 5.9145 7.5394

1.0340 3.3707 5.8918 8.2742 10.528

0.024551 0.053441 0.069234 0.079371 0.086623

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

0.013420 0.022747 0.024261 0.022129 0.021255

0.041376 0.032653 0.033076 0.030649 0.029661

692.39 888.35 1100.2 1293.6 1457.5

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

1.6074 0.37288 −0.078236 −0.29566 −0.42110

79.882 115.33 161.31 187.32 215.63

4.3382 5.1978 6.9555 8.5624 10.045

Single-Phase Properties 50.000 125.00 200.00 275.00 350.00 50.000 125.00 200.00 275.00 350.00

10.000 10.000 10.000 10.000 10.000

26.141 8.9624 5.5879 4.1087 3.2606

0.038254 0.11158 0.17896 0.24339 0.30669

0.53828 2.2107 4.1201 5.8874 7.5240

0.92082 3.3264 5.9097 8.3212 10.591

0.017729 0.047053 0.063249 0.073512 0.080815

0.013420 0.022979 0.024419 0.022240 0.021338

0.031369 0.033916 0.033613 0.030928 0.029822

1020.9 952.54 1149.8 1337.5 1497.0

−0.00090690 0.16557 −0.13543 −0.31595 −0.42707

109.43 125.40 167.23 191.64 219.08

6.6673 5.5904 7.1499 8.6948 10.147

50.000 125.00 200.00 275.00 350.00

50.000 50.000 50.000 50.000 50.000

41.635 28.231 20.660 16.295 13.485

0.024018 0.035422 0.048403 0.061368 0.074155

0.33890 1.8792 3.8955 5.7445 7.4386

1.5398 3.6503 6.3157 8.8129 11.146

0.0075571 0.032377 0.049075 0.059704 0.067214

0.014831 0.024288 0.025325 0.022913 0.021862

0.021072 0.034391 0.034876 0.031923 0.030536

1903.0 1571.6 1590.0 1701.1 1813.9

−0.85893 −0.41268 −0.39075 −0.44355 −0.48674

205.72 207.03 219.10 229.75 249.51

18.078 9.5718 9.2598 10.106 11.203

38.364 30.845 25.763 22.144

0.026066 0.032420 0.038815 0.045159

1.8103 3.8148 5.6828 7.4012

4.4169 7.0568 9.5643 11.917

0.026596 0.043119 0.053790 0.061362

0.025170 0.026034 0.023474 0.022320

0.033554 0.034871 0.032151 0.030806

2093.0 2056.2 2114.9 2185.5

−0.46785 −0.43974 −0.48043 −0.50940

308.69 320.11 324.21 340.22

15.278 12.564 12.252 12.765

125.00 200.00 275.00 350.00

100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Younglove, B. A., “Thermophysical Properties of Fluids. I. Argon, Ethylene, Parahydrogen, Nitrogen, Nitrogen Trifluoride, and Oxygen,” J. Phys. Chem. Ref. Data, Suppl. 1, 11: 1–11, 1982. The source for viscosity is McCarty, R. D., and Weber, L. A., “Thermophysical Properties of Parahydrogen from the Freezing Liquid Line to 5000 R for Pressures to 10,000 psia,” N.B.S. Tech. Note 617, 1972. The source for thermal conductivity is McCarty, R. D., and Weber, L. A., “Thermophysical Properties of Parahydrogen from the Freezing Liquid Line to 5000 R for Pressures to 10,000 psia,” N.B.S. Tech. Note 617, 1972. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.1% in the liquid phase, 0.25% in the vapor phase, and 0.2% in the supercritical region. The uncertainty in heat capacity is 3%, and the uncertainty in speed of sound is 2% in the liquid phase and 1% elsewhere. The uncertainty in viscosity ranges from 4% to 15%. The uncertainty in thermal conductivity below 100 K is estimated to be 3% below 150 atm and up to 10% below 700 atm. For temperatures around 100 K at low densities, the uncertainty is about 1%. Above 100 K, the uncertainty is estimated to be on the order of 10%.

TABLE 2-225 T, K 273 300 350 400 450

Saturated Hydrogen Peroxide*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

0.0004 0.0031 0.0564 0.4521 2.143

0.00068 0.00069 0.00072 0.00076 0.00081

1672 235 15.1 2.12 0.487

−5577 −5510 −5376 −5238 −5091

−4027 −3995 −3933 −3878 −3820

2.990 3.224 3.631 4.032 4.346

8.662 8.269 7.758 7.440 7.172

1.45 1.48 1.54 1.61 1.68

18.0 11.3 4.3 2.2 1.3

0.483 0.481 0.474 0.464 0.453

1.75 1.82 1.90

500 550 600 650 700

7.126 18.56 40.75 79.27 141.7

0.00088 0.00095 0.00107 0.00125 0.00171

0.155 0.0605 0.0268 0.0125 0.0048

−4945 −4794 −4635 −4463 −4195

−3777 −3745 −3731 −3746 −3860

4.656 4.941 5.209 5.485 5.682

6.992 6.846 6.720 6.582 6.339

708.5c

155.3

0.00284

0.0028

−4012

−4012

5.732

5.732

0.89 0.65 0.50

0.443 0.431 0.416

*Values reproduced or converted from a tabulation by Tsykalo and Tabachnikov in V. A. Rabinovich (ed.), Thermophysical Properties of Gases and Liquids, Standartov, Moscow, 1968; NBS-NSF transl. TT 69-55091, 1970. The reader may be reminded that very pure hydrogen peroxide is very difficult to obtain owing to its decomposition or instability. c = critical point. The FMC Corp., Philadelphia, PA tech. bull. 67, 1969 (100 pp.) contains an enthalpy-pressure diagram to 3000 psia, 1100 K.

TABLE 2-226 Temperature K

Thermodynamic Properties of Hydrogen Sulfide Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.034345 0.034479 0.035082 0.035717 0.036390 0.037107 0.037875 0.038702 0.039599 0.040580 0.041662 0.042868 0.044230 0.045791 0.047618 0.049818 0.052573 0.056256 0.061837 0.074429 0.098135

−1.7210 −1.5628 −0.87841 −0.19759 0.48138 1.1602 1.8406 2.5245 3.2135 3.9100 4.6161 5.3348 6.0696 6.8248 7.6068 8.4246 9.2932 10.241 11.335 12.903 14.470

−1.7202 −1.5619 −0.87664 −0.19446 0.48661 1.1686 1.8534 2.5434 3.2408 3.9481 4.6685 5.4053 6.1629 6.9469 7.7650 8.6283 9.5548 10.578 11.779 13.538 15.353

16.328 16.382 16.611 16.832 17.043 17.244 17.431 17.604 17.761 17.899 18.016 18.108 18.171 18.199 18.183 18.109 17.957 17.684 17.192 16.046 14.470

17.876 17.947 18.250 18.541 18.818 19.079 19.320 19.539 19.735 19.902 20.039 20.139 20.198 20.207 20.156 20.027 19.793 19.403 18.736 17.266 15.353

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.34039 −0.33923 −0.33253 −0.32287 −0.30988 −0.29305 −0.27174 −0.24509 −0.21197 −0.17084 −0.11959 −0.055218 0.026636 0.13260 0.27324 0.46655 0.74618 1.1841 1.9714 3.9324 6.3885

254.24 251.74 240.93 230.26 219.81 209.52 199.43 189.56 179.91 170.48 161.26 152.24 143.40 134.71 126.16 117.71 109.36 101.19 93.864 92.754

439.13 428.67 385.68 346.75 311.74 280.37 252.29 227.14 204.60 184.32 166.02 149.44 134.32 120.43 107.58 95.533 84.050 72.784 61.060 46.102

Saturated Properties 187.70 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 373.10

0.023259 0.027106 0.050340 0.087474 0.14366 0.22485 0.33767 0.48934 0.68751 0.94022 1.2558 1.6429 2.1103 2.6672 3.3233 4.0889 4.9755 5.9969 7.1713 8.5294 8.9987

29.116 29.003 28.505 27.998 27.480 26.949 26.403 25.838 25.253 24.642 24.002 23.327 22.609 21.838 21.000 20.073 19.021 17.776 16.172 13.436 10.190

187.70 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 373.10

0.023259 0.027106 0.050340 0.087474 0.14366 0.22485 0.33767 0.48934 0.68751 0.94022 1.2558 1.6429 2.1103 2.6672 3.3233 4.0889 4.9755 5.9969 7.1713 8.5294 8.9987

0.015024 0.017314 0.030704 0.051165 0.080932 0.12253 0.17879 0.25286 0.34834 0.46937 0.62086 0.80887 1.0411 1.3280 1.6843 2.1323 2.7096 3.4881 4.6442 6.9933 10.190

66.559 57.758 32.569 19.545 12.356 8.1613 5.5933 3.9547 2.8707 2.1305 1.6107 1.2363 0.96050 0.75300 0.59373 0.46898 0.36906 0.28669 0.21532 0.14299 0.098135

−0.0085877 −0.0077504 −0.0042394 −0.00091743 0.0022415 0.0052596 0.0081564 0.010949 0.013654 0.016285 0.018857 0.021385 0.023885 0.026373 0.028873 0.031414 0.034044 0.036848 0.040035 0.044599 0.049374

0.044390 0.044124 0.043042 0.042067 0.041188 0.040393 0.039677 0.039030 0.038449 0.037930 0.037470 0.037070 0.036732 0.036462 0.036273 0.036191 0.036265 0.036600 0.037471 0.040079

0.068835 0.068707 0.068273 0.068029 0.067975 0.068115 0.068461 0.069032 0.069859 0.070989 0.072490 0.074466 0.077082 0.080603 0.085498 0.092666 0.10410 0.12534 0.17963 0.63367

1437.8 1425.8 1373.4 1321.0 1268.4 1215.6 1162.3 1108.5 1053.9 998.54 942.08 884.34 825.04 763.84 700.23 633.51 562.59 485.59 398.86 292.76 0

0.095815 0.094930 0.091395 0.088301 0.085567 0.083129 0.080933 0.078934 0.077092 0.075375 0.073752 0.072193 0.070669 0.069149 0.067594 0.065956 0.064157 0.062064 0.059360 0.054674 0.049374

0.025347 0.025386 0.025586 0.025837 0.026142 0.026502 0.026917 0.027388 0.027914 0.028496 0.029136 0.029838 0.030608 0.031458 0.032407 0.033485 0.034746 0.036293 0.038364 0.041755

0.034000 0.034078 0.034487 0.035021 0.035698 0.036537 0.037563 0.038807 0.040312 0.042139 0.044378 0.047166 0.050723 0.055410 0.061879 0.071400 0.086837 0.11617 0.19265 0.80649

245.84 247.20 252.82 257.96 262.58 266.64 270.10 272.91 275.05 276.47 277.15 277.05 276.12 274.34 271.64 268.00 263.35 257.65 250.84 242.80 0

55.730 53.868 46.796 41.090 36.435 32.601 29.412 26.737 24.476 22.550 20.897 19.466 18.212 17.097 16.081 15.116 14.142 13.053 11.629 9.0701 6.3885

10.628 10.775 11.429 12.107 12.816 13.566 14.365 15.227 16.166 17.202 18.360 19.675 21.197 22.997 25.187 27.946 31.600 36.820 45.513 70.939

8.0025 8.1053 8.5566 9.0159 9.4844 9.9634 10.455 10.961 11.485 12.031 12.604 13.213 13.867 14.582 15.380 16.300 17.405 18.833 20.940 25.604

2-283

2-284

TABLE 2-226

Thermodynamic Properties of Hydrogen Sulfide (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.035080 0.035888

−0.87902 −0.021243

−0.87551 −0.017654

16.888 19.164 21.830 24.642 27.626 30.789

18.615 21.640 25.146 28.794 32.611 36.607

0.035053 0.040795

−0.89011 4.0550

−0.85506 4.0958

2.0080 2.2968 3.2255 4.0993 4.9547 5.8015

17.925 18.775 21.626 24.507 27.525 30.710

19.933 21.072 24.852 28.606 32.480 36.511

0.034935 0.043749 0.052654

−0.93837 5.9433 9.3164

−0.76369 6.1620 9.5797

0.36675 0.55412 0.77288 0.96476 1.1466

17.952 20.549 23.863 27.065 30.353

19.786 23.319 27.728 31.889 36.086

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

0.043042 0.041830

0.068269 0.067997

1373.6 1307.4

−0.33258 −0.31984

240.95 227.54

0.087559 0.099486 0.10956 0.11770 0.12465 0.13081

0.025911 0.025979 0.027268 0.028923 0.030708 0.032534

0.035183 0.034563 0.035693 0.037297 0.039059 0.040873

259.21 309.73 356.35 396.06 431.20 463.05

39.791 16.968 8.9467 5.5432 3.7250 2.6185

−0.0042980 0.016821

0.043058 0.037830

0.068210 0.071266

1377.6 986.97

−0.33351 −0.16116

0.075033 0.079019 0.089907 0.098281 0.10534 0.11155

0.028623 0.027626 0.027708 0.029100 0.030799 0.032589

0.042564 0.039427 0.037234 0.038036 0.039492 0.041157

276.67 296.60 351.09 393.73 430.30 462.94

22.188 17.369 9.0111 5.5366 3.7023 2.5947

−0.0045411 0.023458 0.034113

0.043127 0.036740 0.036269

0.067957 0.075047 0.10449

1394.7 855.61 560.70

0.064108 0.073747 0.083609 0.091196 0.097665

0.034782 0.030043 0.029940 0.031216 0.032838

0.087361 0.048271 0.041996 0.041590 0.042472

263.22 325.86 384.33 427.30 463.26

Viscosity µPas

Single-Phase Properties 200.00 212.60

0.10000 0.10000

212.60 300.00 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

200.00 272.07

1.0000 1.0000

272.07 300.00 400.00 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

200.00 300.00 340.26

5.0000 5.0000 5.0000

340.26 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000

28.506 27.865 0.057900 0.040389 0.030157 0.024088 0.020059 0.017187 28.528 24.513 0.49800 0.43539 0.31003 0.24394 0.20183 0.17237 28.625 22.858 18.992 2.7266 1.8047 1.2939 1.0365 0.87211

17.271 24.759 33.160 41.515 49.853 58.182

−0.0042425 −0.000082766

−0.33745 −0.017492 0.75500 14.116 9.1749 5.4471 3.5816 2.4840

12.288 17.999 24.990 32.218 39.592 47.091 241.31 168.56 17.430 19.015 25.609 32.691 39.980 47.423 242.90 146.15 109.14

385.79 337.30 9.1366 12.954 17.172 21.094 24.714 28.082 387.91 180.39 12.147 13.337 17.465 21.319 24.893 28.227 397.26 139.57 83.760

31.710 29.786 35.063 41.773 48.924

17.437 19.070 22.459 25.773 28.935

200.00 300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000 10.000

28.741 23.238 5.0473 2.8037 2.1399 1.7663

0.034793 0.043033 0.19812 0.35667 0.46730 0.56617

−0.99643 5.7496 18.370 22.959 26.466 29.903

-0.64850 6.1800 20.351 26.526 31.139 35.564

−0.0048367 0.022795 0.062081 0.076030 0.084452 0.091275

0.043212 0.036779 0.034651 0.031080 0.031755 0.033155

0.067668 0.072377 0.10189 0.048875 0.044597 0.044212

1415.5 902.78 291.29 375.68 426.01 465.45

−0.34197 −0.077399 8.2243 5.1487 3.3812 2.3347

244.82 150.49 44.719 38.963 44.210 50.878

408.90 148.08 23.639 24.438 27.165 30.013

300.00 400.00 500.00 600.00 700.00

75.000 75.000 75.000 75.000 75.000

26.050 21.973 17.947 14.519 11.974

0.038388 0.045510 0.055720 0.068874 0.083511

4.3332 9.9713 15.404 20.474 25.148

7.2123 13.384 19.583 25.640 31.412

0.017506 0.035260 0.049092 0.060142 0.069045

0.037754 0.035381 0.034762 0.034994 0.035721

0.061705 0.061962 0.061649 0.059227 0.056291

1276.9 983.51 786.35 688.11 654.55

−0.33612 −0.18247 0.057516 0.27314 0.36585

187.18 134.39 103.90 87.712 82.684

232.59 124.22 81.074 62.531 54.563

27.794 24.751 21.937 19.449 17.335

0.035979 0.040403 0.045585 0.051416 0.057687

3.5100 8.6429 13.539 18.248 22.811

8.9069 14.703 20.377 25.960 31.464

0.013888 0.030575 0.043238 0.053420 0.061906

0.038777 0.036402 0.035779 0.036044 0.036804

0.058983 0.057226 0.056273 0.055409 0.054711

1538.4 1302.6 1132.1 1019.1 949.06

−0.40376 −0.37006 −0.31802 −0.26874 −0.23292

214.24 165.64 135.75 118.38 110.03

311.25 174.85 119.57 93.323 79.581

300.00 400.00 500.00 600.00 700.00

150.00 150.00 150.00 150.00 150.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E.W., McLinden, M.O., and Huber, M.L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3): 785–850, 2006. The source for viscosity and thermal conductivity is NIST14, Version 9.08. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.1% in the liquid phase below the critical temperature, 0.4% in the vapor phase, 1% at supercritical temperatures up to 500 K, and 2.5% at higher temperatures. Uncertainties will be higher near the critical point, and may be lower than 0.5% between 400 and 500 K. The uncertainty in vapor pressure is 0.25%, and the uncertainty in heat capacities is estimated to be 1%. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6%.

THERMODYNAMIC PROPERTIES

FIG. 2-11 Enthalpy-concentration diagram for aqueous hydrogen chloride at 1 atm. Reference states: enthalpy of liquid water at 0°C is zero; enthalpy of pure saturated HCl vapor at 1 atm (−85.03°C) is 8000 kcal/mol. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. Saturation values may be read at the ends of the tie lines. [Van Nuys, Trans. Am. Inst. Chem. Eng., 39, 663 (1943).]

2-285

2-286 TABLE 2-227

Thermodynamic Properties of Isobutane

Temperature K

Pressure MPa

113.73 115 130 145 160 175 190 205 220 235 250 265 280 295 310 325 340 355 370 385 400 407.81

2.2891E-08 3.1711E-08 8.8064E-07 1.1510E-05 8.8176E-05 0.00045673 0.0017628 0.0054294 0.014023 0.031511 0.06335 0.11640 0.19876 0.31952 0.48858 0.71658 1.0148 1.3957 1.8727 2.4620 3.1856 3.629

113.73 115 130 145 160 175 190 205 220 235 250 265 280 295 310 325 340 355 370 385 400 407.81

2.2891E-08 3.1711E-08 8.8064E-07 1.1510E-05 8.8176E-05 0.00045673 0.0017628 0.0054294 0.014023 0.031511 0.06335 0.11640 0.19876 0.31952 0.48858 0.71658 1.0148 1.3957 1.8727 2.4620 3.1856 3.629

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−6.5316 −6.4068 −4.9101 −3.3714 −1.7912 −0.16946 1.4945 3.2024 4.9572 6.7622 8.6216 10.540 12.522 14.575 16.705 18.920 21.234 23.665 26.245 29.045 32.319 35.910

−6.5316 −6.4068 −4.9101 −3.3714 −1.7911 −0.16942 1.4946 3.2029 4.9585 6.7652 8.6277 10.551 12.543 14.608 16.758 19.001 21.354 23.839 26.494 29.404 32.862 36.846

20.462 20.517 21.194 21.928 22.716 23.555 24.442 25.374 26.348 27.360 28.406 29.481 30.581 31.700 32.830 33.960 35.075 36.146 37.123 37.906 38.148 35.910

21.408 21.473 22.275 23.133 24.046 25.009 26.018 27.070 28.159 29.279 30.423 31.584 32.754 33.925 35.086 36.222 37.313 38.323 39.190 39.793 39.712 36.846

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.03951 −0.03842 −0.02619 −0.01499 −0.00463 0.005061 0.014182 0.022832 0.031092 0.039028 0.046697 0.054149 0.061427 0.068569 0.075615 0.082601 0.089574 0.096591 0.10375 0.11123 0.11975 0.12937

0.068243 0.068392 0.070255 0.072110 0.073940 0.075811 0.077792 0.079933 0.082269 0.084817 0.087581 0.090556 0.093728 0.097085 0.10062 0.10432 0.10821 0.11234 0.11686 0.12220 0.13078

0.098159 0.098388 0.10118 0.10397 0.10673 0.10951 0.11237 0.11540 0.11865 0.12216 0.12600 0.13021 0.13486 0.14006 0.14598 0.15290 0.16139 0.17260 0.18958 0.22324 0.36901

1999.8 1988.6 1863.4 1750.1 1645.3 1546.1 1450.9 1358.6 1268.6 1180.3 1093.4 1007.7 922.77 838.36 753.97 669.03 582.74 494.00 401.01 300.29 184.38 0

−0.68375 −0.68180 −0.65814 −0.63434 −0.61032 −0.58545 −0.55901 −0.53024 −0.49839 −0.46255 −0.42158 −0.37389 −0.31712 −0.24757 −0.15928 −0.04193 0.12359 0.37690 0.81512 1.7502 4.9259 16.282

157.92 157.70 154.39 150.06 144.95 139.28 133.23 126.94 120.56 114.21 107.94 101.85 95.986 90.398 85.118 80.169 75.561 71.293 67.356 63.794 61.898

8767.2 8096.1 3698.4 2060.1 1302.4 897.76 657.25 502.16 395.73 319.15 262.04 218.23 183.83 156.24 133.64 114.69 98.394 83.919 70.526 57.352 42.498

0.20615 0.20401 0.18292 0.16780 0.15686 0.14894 0.14325 0.13926 0.13655 0.13483 0.13388 0.13352 0.13361 0.13405 0.13474 0.13559 0.13651 0.13739 0.13806 0.13822 0.13688 0.12937

0.042812 0.043157 0.047087 0.050787 0.054332 0.057806 0.061291 0.064859 0.068567 0.072455 0.076548 0.080855 0.085375 0.090101 0.09503 0.10017 0.10541 0.11078 0.11703 0.12502 0.13683

0.051127 0.051472 0.055402 0.059103 0.062655 0.066151 0.06969 0.07337 0.077281 0.081501 0.086099 0.091145 0.096722 0.10295 0.11003 0.11832 0.12842 0.14204 0.16455 0.21348 0.43911

139.39 140.07 147.92 155.36 162.43 169.14 175.44 181.26 186.48 190.96 194.52 197.02 198.29 198.13 196.37 192.73 186.90 178.45 166.77 150.88 128.90 0

Saturated Properties 12.738 12.717 12.473 12.229 11.984 11.736 11.485 11.231 10.971 10.705 10.431 10.148 9.8523 9.5416 9.2114 8.8560 8.4662 8.0276 7.5139 6.8650 5.8674 3.8798 2.4207E-08 3.3165E-08 8.1474E-07 9.5478E-06 6.6294E-05 0.0003141 0.0011181 0.0032007 0.0077418 0.016419 0.031405 0.055358 0.091469 0.14360 0.21659 0.31684 0.45354 0.64121 0.90588 1.3046 2.0369 3.8798

0.078508 0.078635 0.080171 0.081772 0.083447 0.085209 0.087069 0.089043 0.091150 0.093415 0.095866 0.098544 0.10150 0.10480 0.10856 0.11292 0.11812 0.12457 0.13309 0.14567 0.17043 0.25775 41,310,000 30,152,000 1,227,400 104,740 15,084 3,183.6 894.34 312.43 129.17 60.904 31.842 18.064 10.933 6.9638 4.6171 3.1561 2.2049 1.5596 1.1039 0.76649 0.49095 0.25775

577.37 553.27 348.08 233.14 164.07 120.18 91.024 70.957 56.748 46.450 38.843 33.140 28.823 25.545 23.074 21.254 20.032 19.459 19.526 20.215 21.293 16.282

2.2717 2.3426 3.2114 4.1386 5.1239 6.1666 7.2652 8.4173 9.6201 10.872 12.174 13.532 14.961 16.487 18.151 20.022 22.207 24.900 28.498 34.050 46.452

2.8476 2.8811 3.2763 3.6690 4.0588 4.4452 4.8275 5.2047 5.5763 5.9424 6.3045 6.6660 7.0328 7.4136 7.8217 8.2772 8.8129 9.4859 10.409 11.844 14.761

Single-Phase Properties 200.00 261.07

0.10000 0.10000

261.07 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 339.34

1.0000 1.0000 1.0000

339.34 400.00 500.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

11.317 10.223

2.6343 10.040

0.019983 0.052213

0.079206 0.089756

0.11436 0.12906

29.197 32.513 42.834 55.743

31.278 34.944 46.127 59.881

0.13356 0.14664 0.17863 0.20922

0.079705 0.089474 0.11631 0.14101

0.089774 0.098798 0.12501 0.14952

196.48 212.04 245.50 274.09

34.485 21.436 8.6878 4.5740

13.170 17.082 28.880 43.232

0.088272 0.10574 0.11786

2.6017 15.229 21.129

2.6900 15.335 21.247

0.019864 0.070771 0.089265

0.079268 0.098270 0.10803

0.11426 0.14150 0.16097

1395.0 819.00 586.60

−0.54183 −0.23012 0.11482

129.47 89.092 75.758

2.2394 2.9675 3.9573

35.027 42.086 55.288

37.266 45.054 59.246

0.13647 0.15757 0.18916

0.10518 0.11833 0.14174

0.12792 0.13200 0.15229

187.21 224.30 263.78

20.073 9.9934 4.7852

22.102 29.559 44.232

11.379 9.5950 6.7797 1.6118

0.087884 0.10422 0.14750 0.62041

2.4992 14.958 31.029 52.732

2.9387 15.479 31.767 55.834

0.019346 0.069851 0.11631 0.17060

0.079538 0.098458 0.12520 0.14556

0.11385 0.13915 0.20584 0.17644

1417.9 870.18 328.20 219.06

−0.54814 −0.28002 1.3689 5.5914

131.11 92.042 64.364 50.270

574.91 160.04 56.352 15.871

11.329 9.4575 8.4843 0.44655 0.33698 0.25270

20.817 24.305 32.925 41.375

1389.7 1030.1

−0.54030 −0.38717

2.6255 10.031

0.048038 0.041143 0.030372 0.024169

0.088361 0.097817

129.10 103.43

547.68 228.69 6.5710 7.5436 9.9294 12.195 552.65 149.88 99.071 8.7869 10.129 12.391

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

11.438 9.7443 7.5550 4.0999

0.087424 0.10262 0.13236 0.24391

2.3778 14.662 29.780 48.735

3.2521 15.689 31.103 51.174

0.018721 0.068827 0.11292 0.15751

0.079860 0.098710 0.12349 0.14786

0.11339 0.13705 0.17456 0.21137

1445.4 926.11 486.23 239.25

−0.55506 −0.32494 0.34096 3.1085

133.10 95.422 70.562 61.481

603.11 172.11 72.797 28.154

200.00 300.00 400.00 500.00

15.000 15.000 15.000 15.000

11.496 9.8755 7.9692 5.5655

0.086989 0.10126 0.12548 0.17968

2.2631 14.403 29.065 46.521

3.5679 15.922 30.948 49.217

0.018121 0.067905 0.11092 0.15156

0.080166 0.098964 0.12324 0.14693

0.11300 0.13552 0.16619 0.19670

1471.7 975.70 583.15 343.01

−0.56110 −0.35793 0.062447 1.1026

135.03 98.548 75.048 67.712

631.76 183.72 84.081 41.020

200.00 300.00 400.00 500.00

30.000 30.000 30.000 30.000

11.654 10.198 8.6989 7.1701

0.085807 0.098056 0.11496 0.13947

1.9531 13.768 27.757 43.945

4.5274 16.709 31.205 48.129

0.016439 0.065552 0.10708 0.14475

0.081000 0.099691 0.12354 0.14649

0.11206 0.13266 0.15745 0.18019

1544.8 1100.2 775.70 572.71

−0.57512 −0.41943 −0.21676 0.033041

140.47 106.90 85.299 78.479

720.69 216.98 109.96 66.356

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Buecker, D., and Wagner, W., “Reference Equations of State for the Thermodynamic Properties of Fluid Phase n-Butane and Isobutane,” J. Phys. Chem. Ref. Data 35(2):929–1019, 2006. The source for viscosity is Vogel, E., Kuechenmeister, C., and Bich, E., “Viscosity Correlation for Isobutane over Wide Ranges of the Fluid Region,” Int. J. Thermophys, 21(2):343–356, 2000. The source for thermal conductivity is Perkins, R. A., “Measurement and Correlation of the Thermal Conductivity of Isobutane from 114 K to 600 K at Pressures to 70 MPa,” J. Chem. Eng. Data 47(5):1272–1279, 2002. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.02% at temperatures below 340 K and pressures below 12 MPa (both liquid and vapor states), 0.3% at temperatures below 300 K and pressures above 12 MPa, 0.1% in the vapor phase between 340 and 450 K, and 0.5% elsewhere. In the critical region, deviations in pressure are 0.5%. Uncertainties in heat capacities are typically 1–2%, rising to 5% in the critical region and at temperatures below 200 K. Uncertainties in the speed of sound are typically 1–2%, rising to 5% at temperatures below 200 K and in the critical region. The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K to 3.0% over the rest of the fluid surface. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%.

2-287

2-288 TABLE 2-228

Thermodynamic Properties of Isobutene (2-Methyl 1-Propene)

Temperature K

Pressure MPa

132.40 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 418.09

6.7619E-07 1.1116E-06 1.3438E-05 9.7764E-05 0.00048920 0.0018451 0.0055971 0.014316 0.031980 0.064075 0.11754 0.20057 0.32243 0.49324 0.72386 1.0259 1.4121 1.8963 2.4955 3.2319 4.0157

132.40 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 418.09

6.7619E-07 1.1116E-06 1.3438E-05 9.7764E-05 0.00048920 0.0018451 0.0055971 0.014316 0.031980 0.064075 0.11754 0.20057 0.32243 0.49324 0.72386 1.0259 1.4121 1.8963 2.4955 3.2319 4.0157

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.56574 −0.56403 −0.55349 −0.54130 −0.52677 −0.50932 −0.48847 −0.46371 −0.43442 −0.39974 −0.35836 −0.30827 −0.24634 −0.16750 −0.063276 0.081482 0.29633 0.64712 1.3162 3.0714 15.166

Saturated Properties 13.667 13.619 13.343 13.070 12.796 12.522 12.246 11.966 11.681 11.389 11.088 10.774 10.446 10.098 9.7251 9.3181 8.8642 8.3412 7.7041 6.8249 4.1700 6.1426E-07 9.9037E-07 1.0775E-05 7.1284E-05 0.00032720 0.0011411 0.0032245 0.0077400 0.016347 0.031202 0.054953 0.090771 0.14247 0.21479 0.31394 0.44866 0.63242 0.88850 1.2646 1.8955 4.1700

0.073170 0.073428 0.074943 0.076514 0.078149 0.079860 0.081662 0.083571 0.085610 0.087805 0.090191 0.092812 0.095729 0.099027 0.10283 0.10732 0.11281 0.11989 0.12980 0.14652 0.23981 1,628,000. 1,009,700. 92,803. 14,028. 3,056.2 876.38 310.13 129.20 61.174 32.049 18.197 11.017 7.0191 4.6557 3.1853 2.2289 1.5812 1.1255 0.79073 0.52758 0.23981

−15.196 −14.919 −13.315 −11.692 −10.049 −8.3822 −6.6870 −4.9587 −3.1926 −1.3838 0.47283 2.3824 4.3506 6.3837 8.4899 10.680 12.969 15.383 17.975 20.892 25.759

−15.196 −14.919 −13.315 −11.692 −10.049 −8.3821 −6.6865 −4.9575 −3.1899 −1.3782 0.48343 2.4010 4.3814 6.4326 8.5643 10.790 13.128 15.611 18.299 21.365 26.722

−0.078577 −0.076509 −0.065239 −0.054931 −0.045402 −0.036508 −0.028133 −0.020185 −0.012587 −0.0052767 0.0017979 0.0086815 0.015413 0.022030 0.028568 0.035068 0.041581 0.048183 0.055015 0.062470 0.075104

0.074156 0.074179 0.074543 0.075233 0.076215 0.077475 0.079001 0.080776 0.082776 0.084977 0.087351 0.089877 0.092533 0.095306 0.098189 0.10119 0.10433 0.10770 0.11149 0.11647

0.10623 0.10642 0.10755 0.10881 0.11028 0.11203 0.11407 0.11645 0.11917 0.12224 0.12570 0.12958 0.13397 0.13900 0.14491 0.15215 0.16163 0.17548 0.20028 0.27285

1842.9 1827.0 1734.4 1641.6 1549.8 1459.4 1370.6 1283.4 1197.6 1113.1 1029.6 946.77 864.37 781.98 699.15 615.32 529.74 441.23 347.41 242.00 0

12.185 12.304 13.018 13.778 14.581 15.424 16.303 17.214 18.155 19.122 20.111 21.118 22.138 23.164 24.185 25.186 26.144 27.018 27.725 28.046 25.759

13.286 13.427 14.265 15.150 16.076 17.041 18.039 19.064 20.112 21.176 22.250 23.328 24.401 25.460 26.491 27.473 28.377 29.152 29.699 29.751 26.722

0.13654 0.13346 0.11863 0.10775 0.099740 0.093867 0.089606 0.086578 0.084503 0.083170 0.082415 0.082110 0.082147 0.082436 0.082890 0.083425 0.083938 0.084292 0.084244 0.083176 0.075104

0.045424 0.046004 0.049224 0.052312 0.055381 0.058529 0.061825 0.065309 0.068992 0.072865 0.076909 0.081099 0.085414 0.089840 0.094375 0.099036 0.10387 0.10900 0.11468 0.12162

0.053739 0.054319 0.057543 0.060645 0.063755 0.066994 0.070461 0.074224 0.078326 0.082797 0.087668 0.092989 0.098854 0.10544 0.11307 0.12238 0.13473 0.15340 0.18881 0.29879

152.35 153.69 161.19 168.31 175.06 181.39 187.22 192.45 196.96 200.59 203.21 204.65 204.73 203.27 200.03 194.71 186.95 176.23 161.89 142.88 0

932.58 849.42 514.47 329.24 220.54 153.60 110.74 82.434 63.248 49.962 40.590 33.877 29.016 25.486 22.947 21.185 20.076 19.567 19.640 20.140 15.166

Single-Phase Properties 150.00 250.00 265.81

0.10000 0.10000 0.10000

265.81 350.00 450.00 550.00

0.10000 0.10000 0.10000 0.10000

150.00 250.00 343.85

1.0000 1.0000 1.0000

343.85 350.00 450.00 550.00

1.0000 1.0000 1.0000 1.0000

150.00 250.00 350.00 450.00 550.00

5.0000 5.0000 5.0000 5.0000 5.0000

13.344 11.488 11.173 0.047251 0.034911 0.026911 0.021943 13.352 11.506 9.3507 0.43682 0.42121 0.28769 0.22650

0.074938 0.087047 0.089503 21.164 28.645 37.160 45.572 0.074895 0.086914 0.10694 2.2893 2.3741 3.4759 4.4150

−13.316 −1.9936 −0.050895

−13.309 −1.9849 −0.041944

−0.065249 −0.0076921 −0.00015725

0.074545 0.084223 0.086672

0.10754 0.12117 0.12469

1734.9 1141.5 1052.8

19.833 26.949 37.274 49.523

21.949 29.814 40.990 54.081

0.082576 0.10818 0.13616 0.16237

0.075763 0.092639 0.11313 0.13123

0.086265 0.10158 0.12171 0.13969

202.59 234.72 266.02 293.54

−13.330 −2.0267 10.509

−13.255 −1.9398 10.616

−0.065342 −0.0078249 0.034571

0.074562 0.084239 0.10096

0.10751 0.12099 0.15153

1739.5 1149.4 621.79

25.111 25.767 36.712 49.150

27.400 28.142 40.188 53.565

0.083383 0.085521 0.11575 0.14255

0.098674 0.098979 0.11442 0.13168

0.12159 0.11981 0.12599 0.14162

195.20 199.99 251.31 285.82

−0.55354 −0.41214 −0.37070 42.897 15.195 6.9323 4.0208 −0.55391 −0.41526 0.068431 21.295 19.480 7.4906 4.1331

13.386 11.581 9.4333 2.4761 1.3207

0.074707 0.086351 0.10601 0.40387 0.75720

−13.390 −2.1681 10.992 32.462 47.230

−13.017 −1.7364 11.522 34.481 51.016

−0.065747 −0.0083971 0.035956 0.092351 0.12568

0.074643 0.084319 0.10188 0.12422 0.13385

0.10737 0.12027 0.14839 0.22025 0.15463

1759.5 1182.8 663.68 169.25 255.06

−0.55549 −0.42781 −0.014643 10.828 4.4151

150.00 250.00 350.00 450.00 550.00

10.000 10.000 10.000 10.000 10.000

13.426 11.668 9.6916 6.6358 3.1171

0.074480 0.085703 0.10318 0.15070 0.32081

−13.463 −2.3329 10.547 26.495 44.413

−12.718 −1.4758 11.579 28.002 47.621

−0.066244 −0.0090755 0.034624 0.075649 0.11510

0.074751 0.084439 0.10172 0.12143 0.13595

0.10722 0.11952 0.14363 0.19240 0.17757

1783.6 1221.8 741.89 325.60 248.73

−0.55725 −0.44099 −0.13006 1.5013 3.4960

150.00 250.00 350.00 450.00 550.00

20.000 20.000 20.000 20.000 20.000

13.504 11.827 10.075 8.0205 5.7872

0.074050 0.084554 0.099256 0.12468 0.17279

−13.601 −2.6298 9.8719 24.414 40.594

−12.120 −0.93871 11.857 26.907 44.050

−0.067205 −0.010332 0.032532 0.070229 0.10458

0.074987 0.084720 0.10175 0.11990 0.13615

0.10695 0.11836 0.13862 0.16238 0.17749

1829.3 1292.1 862.17 539.83 377.20

−0.56021 −0.46137 −0.25114 0.19834 0.90256

250.00 350.00 450.00 550.00

50.000 50.000 50.000 50.000

12.214 10.805 9.4377 8.1524

0.081874 0.092548 0.10596 0.12266

−3.3402 8.5711 22.086 37.030

0.75353 13.199 27.384 43.163

−0.013539 0.028177 0.063736 0.095351

0.085706 0.10258 0.12031 0.13651

0.11640 0.13309 0.15031 0.16471

1463.7 1105.6 858.20 704.72

−0.49631 −0.38284 −0.25457 −0.13437

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W, and Ihmels, E. C., “Thermodynamic Properties of the Butenes. Part II. Short Fundamental Equations of State,” Fluid Phase Equilibria, 228–229C:173–187, 2005. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in densities calculated using the equation of state are 0.1% in the liquid phase at temperatures above 270 K (rising to 0.5% at temperatures below 200 K), 0.2% at temperatures above the critical temperature and at pressures above 10 MPa, and 0.5% in the vapor phase, including supercritical conditions below 10 MPa. The uncertainty in the vapor phase may be higher than 0.5% in some regions. The uncertainty in vapor pressure is 0.5% above 250 K, and the uncertainty in heat capacities is 0.5% at saturated liquid conditions, rising to 5% at much higher pressures and at temperatures above 250 K.

2-289

2-290 TABLE 2-229 Temperature K

Thermodynamic Properties of Krypton Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

0.034250 0.034705 0.035270 0.035867 0.036502 0.037180 0.037906 0.038689 0.039539 0.040469 0.041494 0.042636 0.043925 0.045404 0.047140 0.049240 0.051903 0.055571 0.061645 0.092166

−0.17537 0.008003 0.22606 0.44593 0.66802 0.89279 1.1207 1.3524 1.5885 1.8297 2.0770 2.3317 2.5952 2.8697 3.1584 3.4661 3.8015 4.1824 4.6620 5.8827

12.752 9.3246 6.6272 4.8380 3.6142 2.7542 2.1351 1.6796 1.3380 1.0771 0.87464 0.71513 0.58764 0.48428 0.39926 0.32818 0.26752 0.21409 0.16360 0.092166

7.9735 8.0103 8.0498 8.0846 8.1140 8.1375 8.1544 8.1640 8.1655 8.1578 8.1395 8.1089 8.0637 8.0007 7.9154 7.8004 7.6434 7.4192 7.0576 5.8827

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.17285 0.011593 0.23136 0.45351 0.67859 0.90720 1.1400 1.3778 1.6213 1.8716 2.1300 2.398 2.6775 2.9713 3.2833 3.6194 3.9901 4.4167 4.9622 6.3920

−0.00146 9.61E-05 0.001877 0.003602 0.005280 0.006916 0.008518 0.010092 0.011644 0.013181 0.014711 0.016241 0.017781 0.019345 0.020950 0.022623 0.024412 0.026413 0.028917 0.035568

0.021225 0.020889 0.020519 0.020175 0.019855 0.019558 0.019283 0.019030 0.018799 0.018594 0.018417 0.018276 0.018179 0.018142 0.018193 0.018377 0.018781 0.019602 0.021432

0.043334 0.043577 0.043953 0.044431 0.045022 0.045743 0.046614 0.047664 0.048934 0.050480 0.052386 0.054776 0.057846 0.061924 0.067602 0.076087 0.090283 0.11946 0.21813

698.97 681.44 660.52 639.35 617.89 596.06 573.82 551.09 527.76 503.74 478.86 452.95 425.75 396.95 366.08 332.48 295.17 252.47 200.85 0

−0.50800 −0.49215 −0.47002 −0.44379 −0.41290 −0.37662 −0.33403 −0.28396 −0.22485 −0.15466 −0.07060 0.03121 0.15636 0.31329 0.51538 0.78550 1.1669 1.7554 2.8313 6.4431

8.9108 8.9748 9.0444 9.1066 9.1606 9.2055 9.2405 9.2644 9.2761 9.2742 9.2567 9.2216 9.1657 9.0850 8.9736 8.8224 8.6156 8.3219 7.8541 6.3920

0.077003 0.074790 0.072381 0.070164 0.068109 0.066190 0.064384 0.062670 0.061030 0.059447 0.057903 0.056379 0.054856 0.053310 0.051708 0.050007 0.048132 0.045939 0.043024 0.035568

0.013270 0.013453 0.013699 0.013976 0.014283 0.014618 0.014982 0.015372 0.015789 0.016235 0.016713 0.017230 0.017796 0.018426 0.019147 0.019998 0.021054 0.022465 0.024624

0.022596 0.023062 0.023713 0.024485 0.025395 0.026463 0.027716 0.029193 0.030948 0.033060 0.035646 0.038887 0.043072 0.048693 0.056647 0.068771 0.089467 0.13256 0.27498

136.12 138.00 140.03 141.85 143.45 144.82 145.98 146.92 147.64 148.13 148.39 148.42 148.21 147.75 147.01 145.97 144.56 142.62 139.66 0

49.758 45.644 41.448 37.834 34.693 31.940 29.510 27.351 25.418 23.675 22.088 20.627 19.260 17.955 16.679 15.388 14.023 12.486 10.545 6.4431

Saturated Properties 115.77 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 209.48

0.073503 0.10344 0.15007 0.21125 0.28957 0.38778 0.50867 0.65513 0.83007 1.0365 1.2774 1.5560 1.8754 2.2390 2.6505 3.1140 3.6340 4.2163 4.8690 5.5255

29.197 28.814 28.353 27.880 27.396 26.896 26.381 25.847 25.291 24.710 24.100 23.455 22.766 22.024 21.213 20.309 19.267 17.995 16.222 10.850

115.77 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 209.48

0.073503 0.10344 0.15007 0.21125 0.28957 0.38778 0.50867 0.65513 0.83007 1.0365 1.2774 1.5560 1.8754 2.2390 2.6505 3.1140 3.6340 4.2163 4.8690 5.5255

0.078418 0.10724 0.15089 0.20670 0.27668 0.36308 0.46836 0.59537 0.74740 0.92843 1.1433 1.3983 1.7017 2.0649 2.5046 3.0471 3.7380 4.6709 6.1126 10.850

Single-Phase Properties 200.00 300.00 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

0.060557 0.040171 0.030088 0.024059 0.020044 0.017179

200.00 300.00 400.00 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.64884 0.40911 0.30268 0.24094 0.20035 0.17155

200.00 205.94

5.0000 5.0000

205.94 300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

200.00 300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000 10.000

18.695 15.764

16.513 24.893 33.236 41.565 49.889 58.210 1.5412 2.4443 3.3039 4.1505 4.9913 5.8291

9.0557 10.313 11.564 12.813 14.062 15.310

10.707 12.803 14.888 16.970 19.051 21.131

0.086165 0.094664 0.10066 0.10531 0.10910 0.11231

0.012553 0.012488 0.012478 0.012475 0.012474 0.012474

0.021091 0.020880 0.020832 0.020813 0.020804 0.020799

181.32 222.65 257.24 287.64 315.11 340.36

15.987 7.9817 4.8486 3.1571 2.0967 1.3720

8.8431 10.207 11.493 12.761 14.020 15.276

10.384 12.651 14.797 16.911 19.012 21.105

0.065940 0.075164 0.081341 0.086060 0.089890 0.093117

0.013359 0.012634 0.012536 0.012509 0.012498 0.012492

0.024440 0.021770 0.021250 0.021058 0.020965 0.020912

176.18 221.94 257.69 288.54 316.19 341.52

15.667 7.8257 4.7410 3.0825 2.0440 1.3340

0.053490 0.063436

4.0257 4.7766

4.2932 5.0938

0.025582 0.029518

0.018694 0.022035

0.092820 0.27057

286.26 189.55

1.3105 3.1528

0.15373 0.44908 0.64511 0.82624 1.0015 1.1738

6.9575 9.6909 11.172 12.528 13.839 15.130

7.7261 11.936 14.397 16.659 18.847 20.999

0.042300 0.060067 0.067171 0.072221 0.076212 0.079530

0.025196 0.013335 0.012800 0.012660 0.012603 0.012572

0.34754 0.026859 0.023237 0.022154 0.021667 0.021402

138.88 220.60 260.80 293.14 321.36 346.89

10.086 6.9488 4.2506 2.7636 1.8241 1.1768

20.764 4.9580 3.1738 2.4249 1.9842 1.6870

0.048160 0.20169 0.31508 0.41238 0.50399 0.59277

3.5349 8.9533 10.764 12.241 13.620 14.954

4.0165 10.970 13.915 16.365 18.660 20.881

0.022941 0.051900 0.060436 0.065913 0.070099 0.073525

0.017485 0.014221 0.013132 0.012848 0.012733 0.012672

0.059683 0.035983 0.025901 0.023506 0.022506 0.021979

386.03 225.22 267.23 300.18 328.56 354.04

0.41868 5.4452 3.6190 2.3913 1.5769 1.0028

6.5051 2.2268 1.5501 1.2103 0.99847 0.85191

200.00 300.00 400.00 500.00 600.00 700.00

100.00 100.00 100.00 100.00 100.00 100.00

27.903 23.193 19.351 16.428 14.234 12.558

0.035839 0.043116 0.051678 0.060871 0.070256 0.079631

1.8128 4.6453 7.1023 9.2300 11.121 12.850

5.3966 8.9569 12.270 15.317 18.146 20.814

0.012020 0.026484 0.036035 0.042844 0.048008 0.052122

0.019196 0.016847 0.015591 0.014841 0.014355 0.014021

0.036607 0.034479 0.031754 0.029280 0.027397 0.026021

779.27 643.59 570.49 537.49 525.89 525.42

−0.62535 −0.55209 −0.49661 −0.48427 −0.50576 −0.54812

200.00 300.00 400.00 500.00 600.00 700.00

200.00 200.00 200.00 200.00 200.00 200.00

30.673 27.185 24.274 21.877 19.905 18.266

0.032602 0.036785 0.041196 0.045709 0.050239 0.054748

1.2743 3.8223 6.1068 8.1787 10.086 11.868

7.7946 11.179 14.346 17.321 20.134 22.818

0.0070011 0.020757 0.029882 0.036526 0.041659 0.045799

0.020666 0.018195 0.016813 0.015938 0.015341 0.014910

0.035096 0.032708 0.030662 0.028881 0.027435 0.026300

976.34 867.80 798.01 754.77 728.59 713.29

−0.69885 −0.72858 −0.75899 −0.79823 −0.84213 −0.88605

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W, and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The equation of state is valid from the triple point to 750 K with pressures to 200 MPa, although the uncertainties increase substantially above 100 MPa. The uncertainties in density are typically 0.2% below 100 MPa, increasing to 1% at pressures up to 200 MPa. The uncertainty in vapor pressure is 0.2%, and the uncertainties in speed of sound are 0.01% in the vapor phase (including supercritical conditions) at low pressures, 1% below 20 MPa in the liquid phase, and 3% below 100 MPa at other state points. The limited amount of heat capacity data shows that the uncertainty is 1% near the triple point, and uncertainties in heat capacities at other states are probably within 2%, at least at pressures up to 20 MPa.

2-291

2-292

PHYSICAL AND CHEMICAL DATA TABLE 2-230 T, K 453.7m 500 600 700 800 900 1000 1200 1400 1500

Saturated Lithium*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

1.78.−13 8.21.−12 4.18.−9 3.51.−7 9.57.−6

1.912.−3 1.946.−3 1.988.−3 2.028.−3 2.070.−3

2.40.+7 9.94.+5

1703 1905 2334 2697 3174

24259 24390 24674 24869 25162

6.776 7.199 7.983 8.633 9.192

56.492 52.169 45.216 40.307 36.678

4.30 4.34 4.23 4.19 4.17

1.24.−4 9.60.−4 0.0204 0.1794 0.4269

2.114.−3 2.160.−3 2.262.−3 2.370.−3 2.433.−3

8.55.+4 1.22.+4 669.3 86.06 38.17

3590 4006 4835 5668 6088

25341 25477 25654 25778 25845

9.682 10.120 10.876 11.518 11.808

33.850 31.591 28.225 25.882 24.979

4.16 4.16 4.14 4.19 4.20

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 1.78.−13 signifies 1.78 × 10−13. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 14 bar, 2200 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 140 psia, 2100–3600 °R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-231

Lithium Bromide—Water Solutions

Ruiter, J. P., Rev. Int. Froid = Int. J. Refrig., 13 (1990): 223–236 gives subroutines for computer calculations. See also ASHRAE Handbook—Fundamentals.

THERMODYNAMIC PROPERTIES TABLE 2-232

2-293

Saturated Mercury*

T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

203.15 213.15 223.15 233.15 243.15

2.298 ⋅ 10−11 1.288 ⋅ 10−10 6.169 ⋅ 10−10 2.580 ⋅ 10−9 9.573 ⋅ 10−9

7.26239 7.27570 7.28900 7.30231 7.31563

3.665 ⋅ 109 6.862 ⋅ 108 1.499 ⋅ 108 3.746 ⋅ 107 1.053 ⋅ 107

33.131 34.567 35.997 37.422 38.842

342.637 343.674 344.710 345.746 346.782

309.506 309.107 308.713 308.324 307.940

0.32434 0.33124 0.33780 0.34404 0.35001

1.84787 1.78142 1.72123 1.66647 1.61647

253.15 263.15 273.15 283.15 293.15

3.198 ⋅ 10−8 9.736 ⋅ 10−8 2.728 ⋅ 10−7 7.101 ⋅ 10−7 1.729 ⋅ 10−6

7.32896 7.34229 7.35563 7.36898 7.38234

3.281 ⋅ 106 1.120 ⋅ 106 4.150 ⋅ 105 1.653 ⋅ 105 7.026 ⋅ 104

40.258 41.668 43.074 44.476 45.874

347.819 348.855 349.891 350.927 351.964

307.561 307.187 306.817 306.451 306.090

0.35571 0.36118 0.36642 0.37146 0.37631

1.57065 1.52852 1.48967 1.45375 1.42045

303.15 313.15 323.15 333.15 343.15

3.968 ⋅ 10−6 8.626 ⋅ 10−6 1.786 ⋅ 10−5 3.356 ⋅ 10−5 6.724 ⋅ 10−5

7.39572 7.40911 7.42252 7.43594 7.44938

3.167 ⋅ 104 1.505 ⋅ 104 7.501 ⋅ 103 3.905 ⋅ 103 2.115 ⋅ 103

47.268 48.659 50.046 51.430 52.810

353.000 354.036 355.072 356.108 357.145

305.732 305.377 305.026 304.678 304.335

0.38099 0.38550 0.38986 0.39408 0.39816

1.38951 1.36068 1.33378 1.30862 1.28505

353.15 363.15 373.15 383.15

1.232 ⋅ 10−4 2.182 ⋅ 10−4 3.745 ⋅ 10−4 6.247 ⋅ 10−4

7.46285 7.47633 7.48984 7.50337

1.188 ⋅ 103 6.899 ⋅ 102 413.0 254.2

54.188 55.563 56.936 58.306

358.181 359.217 360.253 361.289

303.993 303.654 303.317 302.983

0.40212 0.40596 0.40969 0.41331

1.26292 1.24213 1.22255 1.20408

393.15 403.15 413.15 423.15 433.15

1.015 ⋅ 10−3 1.608 ⋅ 10−3 2.491 ⋅ 10−3 3.778 ⋅ 10−3 5.618 ⋅ 10−3

7.51693 7.53052 7.55415 7.55780 7.57148

153.6 103.9 68.75 46.43 31.96

59.674 61.039 62.403 63.765 65.125

362.326 363.362 364.397 365.433 366.469

302.652 302.323 301.994 301.668 301.344

0.41684 0.42027 0.42361 0.42687 0.43004

1.18665 1.17017 1.15456 1.13978 1.12575

443.15 453.15 463.15 473.15 483.15

8.204 ⋅ 10−3 1.178 ⋅ 10−2 1.664 ⋅ 10−2 2.315 ⋅ 10−2 3.177 ⋅ 10−2

7.58520 7.59897 7.61277 7.62662 7.64051

66.484 67.842 69.198 70.553 71.908

367.504 368.539 369.574 370.609 371.642

301.020 300.697 300.376 300.056 299.734

0.43314 0.43617 0.43913 0.44203 0.44486

1.11242 1.09975 1.08768 1.07619 1.06524

493.15 503.15 513.15 523.15 533.15

4.304 ⋅ 10−2 5.758 ⋅ 10−2 7.614 ⋅ 10−2 9.959 ⋅ 10−2 0.12892

7.65444 7.66843 7.68247 7.69656 7.71071

4.748 3.621 2.793 2.176 1.7132

73.261 74.614 75.967 77.319 78.671

372.676 373.708 374.740 375.771 376.800

299.415 299.094 298.773 298.452 298.129

0.44763 0.45035 0.45301 0.45562 0.45818

1.05478 1.04479 1.03525 1.02611 1.01737

543.15 553.15 563.15 573.15 583.15

0.16527 0.20993 0.26435 0.33015 0.40910

7.72491 7.73918 7.75351 7.7679 7.7823

1.3613 1.0912 0.88213 0.71874 0.59002

80.023 81.375 82.728 84.080 85.434

377.829 378.855 379.880 380.904 381.925

297.806 297.480 297.152 296.824 296.491

0.46069 0.46316 0.46558 0.46796 0.47030

1.00899 1.00095 0.99324 0.98584 0.97893

593.15 603.15 613.15 623.15 633.15

0.50320 0.61460 0.74567 0.89896 1.0772

7.7969 7.8115 7.8262 7.8409 7.8558

0.48779 0.40600 0.34008 0.28660 0.24291

86.788 88.143 89.499 90.856 92.215

382.944 383.960 384.973 385.984 386.991

296.156 295.817 295.474 295.128 294.776

0.47260 0.47487 0.47709 0.47929 0.48145

0.97190 0.96532 0.95899 0.95289 0.94702

643.15 653.15 663.15 673.15 683.15

1.2834 1.5207 1.9725 2.1024 2.454

7.8707 7.8858 7.9008 7.9160 7.9313

0.20702 0.17735 0.15269 0.13207 0.11476

93.575 94.937 96.300 97.666 99.033

387.994 388.994 389.989 390.980 391.966

294.419 294.057 293.689 293.314 292.933

0.48358 0.48568 0.48774 0.48978 0.49180

0.94135 0.93589 0.93061 0.92552 0.92059

693.15 703.15 713.15 723.15 733.15

2.852 3.299 3.801 4.362 4.986

7.9467 7.9622 7.9778 7.9935 8.0094

0.10014 0.08775 0.07719 0.06815 0.06039

100.403 101.775 103.150 104.528 105.908

392.947 393.923 394.893 395.858 396.816

292.544 292.148 291.743 291.330 290.908

0.49378 0.49574 0.49768 0.49959 0.50148

0.91583 0.91123 0.90677 0.90245 0.89827

743.15 753.15 763.15 773.15 783.15

5.679 6.446 7.292 8.222 9.242

8.0252 8.0413 8.0574 8.074 8.090

0.05369 0.04789 0.04285 0.03846 0.03462

107.292 108.679 110.069 111.463 112.861

397.767 398.711 399.649 400.579 401.501

290.475 290.032 289.580 289.116 288.640

0.50335 0.50519 0.50702 0.50882 0.51061

0.89422 0.89029 0.88647 0.88277 0.87917

793.15 803.15 813.15 823.15 833.15

10.358 11.576 12.901 14.340 15.899

8.106 8.123 8.140 8.157 8.174

0.03124 0.02827 0.02565 0.02333 0.02126

114.262 115.668 117.078 118.492 119.911

402.415 403.321 404.218 405.106 405.985

288.153 287.653 287.140 286.614 286.074

0.51238 0.51412 0.51586 0.51757 0.51927

0.87568 0.87228 0.86898 0.86576 0.86263

22.39 15.95 11.54 8.469 6.301

*From Vukalovich, Ivanov, Fokin, and Yakovlev, Thermophysical Properties of Mercury, Standartov, Moscow, 1971. For the saturated liquid the specific volume at 203.15 K is 7.26239 × 10−5 m3/kg, etc. All the tabular values for 203.15 K, 213.15 K, 223.15 K, and 233.15 K represent a metastable equilibrium between the subcooled liquid and the saturated vapor. Saturation and superheat tables and a diagram to 100 bar, 1600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 1 to 8200 psia and 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961.

2-294

PHYSICAL AND CHEMICAL DATA

TABLE 2-232

Saturated Mercury* (Concluded)

T, K

P, bar

vf × 105, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

hfg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

843.15 853.15 863.15 873.15 883.15

17.584 19.403 21.36 23.46 25.72

8.191 8.209 8.226 8.244 8.262

0.019426 0.017785 0.016317 0.015000 0.013815

121.335 122.763 124.197 125.636 127.080

406.855 407.715 408.565 409.405 410.235

285.520 284.952 284.368 283.769 283.155

0.52095 0.52262 0.52427 0.52591 0.52753

0.85959 0.85662 0.85372 0.85090 0.84815

893.15 903.15 913.15 923.15 933.15

28.14 30.72 33.47 36.41 39.53

8.280 8.298 8.316 8.335 8.353

0.012748 0.011784 0.010911 0.010120 0.009401

128.530 129.986 131.448 132.915 134.389

411.054 411.861 412.658 413.444 414.218

282.524 281.875 281.210 280.529 279.829

0.52914 0.53074 0.53232 0.53389 0.53545

0.84546 0.84284 0.84028 0.83777 0.83533

943.15 953.15 963.15 973.15 983.15

42.85 46.36 50.09 54.03 58.20

8.372 8.391 8.410 8.430 8.450

0.008746 0.008150 0.007604 0.007105 0.006648

135.869 137.356 138.850 140.350 141.858

414.980 415.731 416.469 417.195 417.909

279.111 278.375 277.619 276.845 276.051

0.53700 0.53854 0.54006 0.54158 0.54308

0.83294 0.83060 0.82831 0.82606 0.82387

993.15 1003.15 1013.15 1023.15 1033.15

62.59 67.22 72.10 77.22 82.60

8.468 8.488 8.508 8.529 8.550

0.006228 0.005842 0.005487 0.005159 0.004856

143.372 144.894 146.424 147.961 149.506

418.610 419.298 419.974 420.636 421.286

275.238 274.404 273.550 272.675 271.780

0.54458 0.54607 0.54754 0.54901 0.55047

0.82172 0.81961 0.81754 0.81552 0.81353

1043.15 1053.15 1063.15 1073.15

88.25 94.17 100.37 106.85

8.570 8.590 8.612 8.632

0.004576 0.004317 0.004077 0.003854

151.059 152.619 154.188 155.766

421.923 422.546 423.156 423.752

270.864 269.927 268.968 267.986

0.55192 0.55336 0.55479 0.55621

0.81158 0.80966 0.80778 0.80593

THERMODYNAMIC PROPERTIES

Enthalpy-log-pressure diagram for mercury. (Drawn from tabular data in footnote reference to Table 2-232.)

FIG. 2-12

2-295

2-296 TABLE 2-233 Temperature K

Thermodynamic Properties of Methane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.035534 0.036554 0.037143 0.037768 0.038431 0.039138 0.039896 0.040714 0.041600 0.042569 0.043636 0.044825 0.046165 0.047702 0.049500 0.051667 0.054394 0.058078 0.063825 0.079902 0.098628

−1.1526 −0.64728 −0.37306 −0.096585 0.18242 0.46425 0.74927 1.0379 1.3307 1.6284 1.9317 2.2418 2.5602 2.8887 3.2304 3.5895 3.9734 4.3965 4.8955 5.7074 6.2136

−1.1522 −0.64602 −0.37097 −0.093257 0.18750 0.47174 0.75999 1.0529 1.3511 1.6557 1.9676 2.2884 2.6199 2.9647 3.3262 3.7098 4.1244 4.5873 5.1420 6.0685 6.6672

−0.011389 −0.0060856 −0.0034096 −0.00083691 0.0016441 0.0040439 0.0063722 0.0086383 0.010851 0.013020 0.015154 0.017264 0.019362 0.021462 0.023584 0.025755 0.028021 0.030467 0.033313 0.038000 0.041109

63.981 23.782 15.116 10.038 6.9171 4.9183 3.5915 2.6825 2.0424 1.5803 1.2393 0.98256 0.78568 0.63206 0.51014 0.41163 0.33038 0.26139 0.19945 0.12816 0.098628

6.8310 7.0469 7.1582 7.2654 7.3680 7.4652 7.5562 7.6403 7.7165 7.7837 7.8406 7.8856 7.9166 7.9306 7.9238 7.8898 7.8184 7.6893 7.4515 6.7850 6.2136

7.5793 7.8644 8.0104 8.1501 8.2825 8.4067 8.5215 8.6257 8.7180 8.7970 8.8608 8.9074 8.9340 8.9369 8.9109 8.8482 8.7357 8.5480 8.2217 7.3641 6.6672

0.084885 0.079019 0.076413 0.074103 0.072036 0.070168 0.068464 0.066891 0.065421 0.064029 0.062694 0.061391 0.060098 0.058789 0.057431 0.055982 0.054371 0.052471 0.049961 0.044819 0.041109

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.034776 0.033908 0.033500 0.033115 0.032749 0.032400 0.032069 0.031757 0.031469 0.031206 0.030974 0.030780 0.030631 0.030541 0.030531 0.030634 0.030920 0.031554 0.033085 0.041746

0.054029 0.054681 0.055135 0.055656 0.056253 0.056941 0.057741 0.058684 0.059809 0.061169 0.062840 0.064932 0.067613 0.071156 0.076044 0.083218 0.094816 0.11699 0.17822 1.5082

1538.6 1452.0 1403.9 1354.7 1304.6 1253.5 1201.3 1148.1 1093.6 1037.7 980.17 920.85 859.39 795.43 728.42 657.52 581.27 497.01 398.59 250.31 0

−0.48191 −0.45812 −0.44202 −0.42328 −0.40145 −0.37589 −0.34578 −0.31006 −0.26735 −0.21579 −0.15286 −0.075032 0.022798 0.14836 0.31398 0.54087 0.86918 1.3866 2.3397 5.2488 6.8877

211.24 199.67 193.03 186.18 179.21 172.15 165.04 157.91 150.78 143.65 136.54 129.43 122.32 115.19 108.01 100.73 93.324 85.799 78.733 96.970

204.52 155.78 136.86 121.34 108.39 97.432 88.031 79.868 72.699 66.333 60.620 55.437 50.682 46.266 42.105 38.115 34.196 30.193 25.773 18.982

0.025243 0.025487 0.025652 0.025842 0.026056 0.026295 0.026560 0.026854 0.027182 0.027549 0.027965 0.028439 0.028989 0.029636 0.030412 0.031374 0.032615 0.034338 0.037087 0.045796

0.033851 0.034425 0.034853 0.035378 0.036016 0.036786 0.037714 0.038836 0.040203 0.041885 0.043985 0.046657 0.050144 0.054849 0.061496 0.071527 0.088273 0.12151 0.21701 2.2590

249.13 260.09 265.31 270.01 274.17 277.76 280.76 283.13 284.86 285.93 286.31 285.97 284.88 283.01 280.30 276.66 271.99 266.04 258.03 238.55 0

47.921 37.826 33.883 30.662 28.004 25.790 23.928 22.347 20.993 19.819 18.789 17.870 17.035 16.255 15.500 14.732 13.896 12.892 11.492 8.4951 6.8877

8.8517 10.015 10.669 11.350 12.062 12.811 13.604 14.449 15.355 16.334 17.402 18.581 19.904 21.423 23.225 25.477 28.545 33.392 43.706 119.40

Saturated Properties 90.694 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 155.00 160.00 165.00 170.00 175.00 180.00 185.00 190.00 190.56

0.011696 0.034376 0.056377 0.088130 0.13221 0.19143 0.26876 0.36732 0.49035 0.64118 0.82322 1.0400 1.2950 1.5921 1.9351 2.3283 2.7765 3.2852 3.8617 4.5186 4.5992

28.142 27.357 26.923 26.478 26.021 25.551 25.065 24.562 24.038 23.491 22.917 22.309 21.661 20.964 20.202 19.355 18.384 17.218 15.668 12.515 10.139

90.694 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 155.00 160.00 165.00 170.00 175.00 180.00 185.00 190.00 190.56

0.011696 0.034376 0.056377 0.088130 0.13221 0.19143 0.26876 0.36732 0.49035 0.64118 0.82322 1.0400 1.2950 1.5921 1.9351 2.3283 2.7765 3.2852 3.8617 4.5186 4.5992

0.015630 0.042048 0.066154 0.099622 0.14457 0.20332 0.27844 0.37278 0.48962 0.63279 0.80691 1.0177 1.2728 1.5821 1.9603 2.4294 3.0268 3.8257 5.0137 7.8027 10.139

3.6388 3.9976 4.1951 4.3964 4.6019 4.8123 5.0285 5.2517 5.4833 5.7254 5.9806 6.2526 6.5462 6.8688 7.2313 7.6515 8.1609 8.8251 9.8238 12.455

Single-Phase Properties −0.64803 −0.012738

−0.64438 −0.0089413

7.2969 9.5570 12.175 15.151 18.673 22.795

8.1908 11.209 14.665 18.475 22.831 27.784

0.036493 0.044610

−0.65829 2.1878

1.0220 1.5537 2.4524 3.3108 4.1567 4.9971

27.586 5.4706 2.1799 1.5333 1.2013 0.99281

100.00 111.51

0.10000 0.10000

111.51 200.00 300.00 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

100.00 149.14

1.0000 1.0000

149.14 200.00 300.00 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.97852 0.64363 0.40776 0.30205 0.24058 0.20012

100.00 200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

100.00 200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000 10.000

27.360 26.341

−0.0060931 −0.000079677

0.054672 0.055828

0.073456 0.093427 0.10741 0.11834 0.12803 0.13705

0.025904 0.025259 0.027479 0.032300 0.038196 0.044179

0.035558 0.033784 0.035869 0.040652 0.046533 0.052509

271.33 369.98 449.74 510.56 561.86 608.04

29.808 9.2893 4.3216 2.2395 1.2245 0.68722

−0.62179 2.2325

−0.0061960 0.016902

0.033950 0.030810

0.054562 0.064535

1459.6 931.21

−0.46060 −0.089695

7.8788 9.3582 12.072 15.083 18.623 22.755

8.9007 10.912 14.524 18.393 22.780 27.752

0.061614 0.073276 0.087922 0.099023 0.10879 0.11784

0.028353 0.025879 0.027621 0.032360 0.038227 0.044198

0.046147 0.036730 0.036721 0.041056 0.046766 0.052659

286.08 357.81 447.04 510.57 562.99 609.73

18.022 9.5001 4.2699 2.2001 1.1998 0.67124

18.368 23.028 35.152 50.558 68.902 89.200

0.036250 0.18279 0.45874 0.65221 0.83240 1.0072

−0.70190 7.8197 11.590 14.779 18.401 22.581

−0.52065 8.7337 13.884 18.040 22.563 27.617

−0.0066393 0.051495 0.072954 0.084897 0.094971 0.10417

0.034116 0.032029 0.028262 0.032614 0.038361 0.044277

0.054117 0.11667 0.041234 0.042903 0.047789 0.053309

1490.0 291.29 439.25 513.11 569.49 618.13

−0.46993 8.9784 3.9428 2.0089 1.0870 0.60013

204.45 40.612 38.480 52.693 70.509 90.498

165.28 10.828 12.194 14.872 17.410 19.768

27.802 16.593 4.6859 3.1002 2.3887 1.9619

0.035969 0.060268 0.21340 0.32256 0.41863 0.50971

−0.75239 5.1551 10.942 14.401 18.132 22.371

−0.39270 5.7578 13.077 17.627 22.318 27.468

−0.0071652 0.034542 0.065137 0.078246 0.088698 0.098073

0.034314 0.030129 0.028995 0.032902 0.038516 0.044371

0.053642 0.085085 0.048165 0.045220 0.049007 0.054070

1525.7 567.92 444.53 522.58 580.99 630.63

−0.47979 1.0266 3.2606 1.7355 0.94125 0.51200

209.07 84.234 44.730 55.941 72.781 92.268

174.83 29.399 13.896 15.766 18.011 20.217

27.403 22.416

8.9395 16.524 24.901 33.243 41.572 49.895

1452.6 1339.7

−0.45829 −0.41705

0.033911 0.033003

0.11186 0.060518 0.040158 0.030082 0.024055 0.020042

0.036549 0.037963

199.74 184.09 11.561 21.941 34.552 50.127 68.564 88.921 200.62 130.66

155.91 117.20 4.4579 7.8096 11.245 14.272 16.976 19.431 157.63 56.297 6.2043 8.0145 11.367 14.357 17.040 19.483

200.00 300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00 100.00

25.496 21.266 17.881 15.305 13.357

0.039222 0.047024 0.055926 0.065340 0.074869

3.0510 7.0865 11.121 15.405 20.074

6.9732 11.789 16.713 21.939 27.561

0.020596 0.040126 0.054276 0.065922 0.076160

0.032058 0.031823 0.035273 0.040312 0.045724

0.048512 0.048281 0.050523 0.054139 0.058364

1541.0 1267.5 1115.8 1044.8 1018.4

−0.51619 −0.44889 −0.37484 −0.32811 −0.30439

188.05 137.68 120.38 120.87 130.36

80.392 47.835 37.584 33.590 32.111

200.00 300.00 400.00 500.00 600.00

500.00 500.00 500.00 500.00 500.00

33.003 30.786 28.929 27.331 25.934

0.030301 0.032482 0.034567 0.036588 0.038559

2.3322 5.9505 9.7401 13.934 18.612

17.482 22.192 27.024 32.228 37.892

0.0061671 0.025271 0.039152 0.050747 0.061061

0.037832 0.037006 0.039890 0.044407 0.049344

0.047821 0.047114 0.049933 0.054280 0.059017

2664.2 2500.0 2360.3 2250.3 2168.1

−0.53926 −0.55416 −0.52806 −0.49035 −0.45514

429.60 358.93 312.36 285.41 272.14

205.24 106.90 78.768 66.669 60.413

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Setzmann, U., and Wagner, W., “A New Equation of State and Tables of Thermodynamic Properties for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 1000 MPa,” J. Phys. Chem. Ref. Data 20(6):1061–1151, 1991. The source for viscosity is Younglove, B. A., and Ely, J. F., “Thermophysical Properties of Fluids. II. Methane, Ethane, Propane, Isobutane and Normal Butane,” J. Phys. Chem. Ref. Data 16:577–798, 1987. The source for thermal conductivity is Friend, D. G., Ely, J. F., and Ingham, H., “Tables for the Thermophysical Properties of Methane,” NIST Tech. Note 1325, 1989. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.03% for pressures below 12 MPa and temperatures below 350 K and up to 0.07% for pressures less than 50 MPa. For the speed of sound, the uncertainty ranges from 0.03% (in the vapor phase) to 0.3% depending on temperature and pressure. Heat capacities may be generally calculated within an uncertainty of 1%. The uncertainty in viscosity is 2%, except in the critical region which is 5%. The uncertainty in thermal conductivity of the dilute gas between 130 and 625 K is 2.5%. For temperatures below 130 K, the uncertainty is less than 10%. Excluding the dilute gas, the uncertainty is 2% between 110 and 725 K at pressures up to 70 MPa, except near the critical point which has an uncertainty of 5% or greater. For the vapor at lower temperatures and the dense liquid near the triple point, an uncertainty of 10% is possible. 2-297

2-298 TABLE 2-234 Temperature K

Thermodynamic Properties of Methanol Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Saturated Properties 175.61 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 510.00 513.38

1.8635E-07 3.7619E-07 3.2175E-06 1.9841E-05 9.4330E-05 0.00036348 0.0011791 0.0033166 0.0082787 0.018682 0.038692 0.074453 0.13447 0.22992 0.37483 0.58617 0.88399 1.2914 1.8349 2.5433 3.4456 4.5713 5.9794 7.7496 8.2159

175.61 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 480.00 495.00 510.00 513.38

1.8635E-07 3.7619E-07 3.2175E-06 1.9841E-05 9.4330E-05 0.00036348 0.0011791 0.0033166 0.0082787 0.018682 0.038692 0.074453 0.13447 0.22992 0.37483 0.58617 0.88399 1.2914 1.8349 2.5433 3.4456 4.5713 5.9794 7.7496 8.2159

28.230 28.096 27.629 27.163 26.703 26.250 25.802 25.360 24.922 24.484 24.041 23.590 23.124 22.638 22.123 21.571 20.973 20.315 19.579 18.741 17.759 16.553 14.880 11.689 8.7852 1.2764E-07 2.5140E-07 1.9855E-06 1.1378E-05 5.0556E-05 0.00018304 0.00056065 0.0014959 0.0035581 0.0076845 0.015300 0.028438 0.049870 0.083267 0.13344 0.20674 0.31179 0.46071 0.67055 0.96219 1.3555 1.9102 2.9050 5.1706 8.7852

0.035423 0.035592 0.036194 0.036815 0.037449 0.038096 0.038756 0.039432 0.040125 0.040844 0.041595 0.042390 0.043244 0.044174 0.045203 0.046358 0.047681 0.049226 0.051075 0.053360 0.056310 0.060411 0.067203 0.085547 0.11383 7,834,400. 3,977,700. 503,660. 87,892. 19,780. 5,463.4 1,783.7 668.48 281.05 130.13 65.359 35.164 20.052 12.009 7.4940 4.8370 3.2073 2.1706 1.4913 1.0393 0.73775 0.52352 0.34423 0.19340 0.11383

−12.440 −12.130 −11.067 −10.001 −8.9277 −7.8395 −6.7318 −5.5991 −4.4351 −3.2329 −1.9858 −0.68700 0.66961 2.0901 3.5806 5.1475 6.7983 8.5423 10.392 12.364 14.488 16.820 19.521 23.297 25.917

−12.440 −12.130 −11.067 −10.001 −8.9277 −7.8395 −6.7318 −5.5990 −4.4347 −3.2322 −1.9842 −0.68385 0.67543 2.1003 3.5975 5.1746 6.8404 8.6058 10.485 12.500 14.682 17.096 19.923 23.960 26.852

−0.049524 −0.047781 −0.042108 −0.036846 −0.031908 −0.027226 −0.022750 −0.018434 −0.014239 −0.010129 −0.0060725 −0.0020451 0.0019747 0.0060049 0.010061 0.014159 0.018314 0.022546 0.026879 0.031347 0.036009 0.040977 0.046590 0.054351 0.059911

0.056728 0.056689 0.056604 0.057072 0.057992 0.059275 0.060916 0.062917 0.065250 0.067864 0.070693 0.073674 0.076752 0.079881 0.083033 0.086189 0.089346 0.092514 0.095722 0.099032 0.10257 0.10666 0.11250 0.12653

0.070390 0.070750 0.070897 0.071215 0.072004 0.073141 0.074617 0.076487 0.078803 0.081584 0.084823 0.088505 0.092616 0.097164 0.10219 0.10776 0.11405 0.12133 0.13007 0.14120 0.15679 0.18345 0.25717 1.1088

1625.1 1590.2 1496.4 1425.3 1363.2 1304.6 1248.2 1194.1 1142.6 1093.5 1046.3 1000.2 954.32 907.64 859.31 808.48 754.41 696.47 634.17 567.09 494.36 412.12 308.94 192.83 0

28.219 28.353 28.810 29.259 29.698 30.123 30.534 30.932 31.321 31.703 32.077 32.442 32.789 33.108 33.385 33.601 33.736 33.767 33.687 33.541 33.439 33.258 32.267 29.688 25.917

29.679 29.850 30.430 31.003 31.564 32.109 32.637 33.149 33.648 34.134 34.606 35.060 35.485 35.869 36.194 36.436 36.571 36.570 36.423 36.184 35.981 35.652 34.325 31.187 26.852

0.19032 0.18544 0.17069 0.15841 0.14806 0.13923 0.13164 0.12508 0.11938 0.11442 0.11009 0.10627 0.10287 0.099808 0.096984 0.094317 0.091723 0.089128 0.086505 0.083978 0.081813 0.079634 0.075685 0.068520 0.059911

0.031874 0.032397 0.035224 0.040104 0.047248 0.056324 0.066572 0.077055 0.086920 0.095581 0.10279 0.10860 0.11331 0.11736 0.12125 0.12546 0.13033 0.13587 0.14101 0.14238 0.13589 0.12618 0.12608 0.13259

0.040287 0.040854 0.043954 0.049389 0.057480 0.067973 0.080135 0.093000 0.10564 0.11740 0.12798 0.13749 0.14644 0.15559 0.16605 0.17917 0.19663 0.21986 0.24709 0.26502 0.25879 0.27959 0.42448 1.9096

239.95 242.62 251.06 258.49 265.30 271.89 278.43 284.90 291.19 297.15 302.63 307.47 311.48 314.48 316.20 316.34 314.53 310.36 303.71 295.26 285.25 267.83 247.46 212.65 0

−0.40884 −0.40373 −0.39791 −0.39361 −0.38674 −0.37793 −0.36733 −0.35457 −0.33915 −0.32073 −0.29904 −0.27385 −0.24479 −0.21121 −0.17199 −0.12529 −0.068163 0.0042669 0.10021 0.23444 0.43762 0.79465 1.6506 4.6061 6.7425 1187400. 857090. 293110. 105090. 39363. 15552. 6557.5 2971.8 1449.9 759.96 426.34 254.84 161.53 108.02 75.802 55.467 42.002 32.587 25.532 19.860 15.568 13.904 12.099 9.5115 6.7425

Single-Phase Properties 200.00 300.00 337.30

0.10000 0.10000 0.10000

337.30 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 400.00 409.75

1.0000 1.0000 1.0000 1.0000

409.75 500.00 600.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 484.95

5.0000 5.0000 5.0000 5.0000

484.95 500.00 600.00

5.0000 5.0000 5.0000

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

27.474 24.486 23.366 0.037626 0.030452 0.024157 0.020089

0.036398 0.040839 0.042798

−10.709 −3.2300 −0.030266

−0.040317 −0.010133 −0.000089518

0.056702 0.067862 0.075163

0.070943 0.081580 0.090451

1471.5 1094.1 977.93

−0.39677 −0.32081 −0.26023

32.613 36.075 40.921 46.476

35.271 39.359 45.060 51.454

0.10457 0.11581 0.12851 0.14014

0.11100 0.044972 0.051823 0.059065

0.14187 0.054208 0.060380 0.067441

309.54 349.19 387.15 420.71

−10.720 −3.2472 6.2298 7.3401

−10.684 −3.2064 6.2770 7.3883

−0.040354 −0.010176 0.016901 0.019645

0.056724 0.067848 0.088257 0.090347

0.070932 0.081541 0.11177 0.11623

1475.1 1100.0 775.46 736.51

33.758 40.335 46.300

36.586 44.303 51.178

0.090904 0.10818 0.12070

0.13203 0.056676 0.061344

0.20333 0.068069 0.070369

313.48 376.08 413.09

38.678 13.330 4.5635

0.036283 0.040592 0.046640 0.062203

−10.752 −3.3039 6.0896 17.655

−10.571 −3.1010 6.3228 17.966

−0.040517 −0.010367 0.016546 0.042725

0.056820 0.067795 0.087676 0.10826

0.070883 0.081377 0.11029 0.19836

1490.9 1125.5 818.58 381.15

−0.39825 −0.32568 −0.11504 0.98684

2.1711 1.7679 1.1389

0.46060 0.56566 0.87808

33.047 35.907 45.247

35.350 38.735 49.638

0.078574 0.085457 0.10553

0.12499 0.098975 0.072927

0.32263 0.17315 0.087489

260.06 301.43 379.21

13.826 12.155 5.1009

27.648 24.779 21.717 15.932 2.6640

0.036169 0.040357 0.046048 0.062765 0.37537

−10.791 −3.3713 5.9321 19.374 43.262

−10.430 −2.9677 6.3925 20.002 47.015

−0.040716 −0.010598 0.016141 0.046226 0.096406

0.056935 0.067746 0.087087 0.10760 0.088868

0.070820 0.081196 0.10880 0.18959 0.12122

1509.9 1155.3 865.91 424.68 343.60

−0.39966 −0.32990 −0.13884 0.83939 5.0965

−7.8460 −0.44914 8.4277 19.458 32.525

−0.043691 −0.013840 0.011565 0.036085 0.059862

0.057827 0.067889 0.082823 0.095694 0.10406

0.068992 0.079818 0.098951 0.12152 0.13787

1772.1 1515.8 1334.7 1164.1 977.50

−0.41976 −0.35799 −0.26099 −0.14407 −0.023905

10.949 19.430 29.089 40.322

−0.022106 0.0022308 0.023726 0.044161

0.070293 0.077541 0.084883 0.092281

0.080627 0.089761 0.10419 0.12017

2316.0 2194.8 2123.2 2074.1

−0.34762 −0.30897 −0.24751 −0.19279

27.491 24.514 21.193 20.772 0.35352 0.25202 0.20501 27.561 24.635 21.441 16.076

26.577 32.839 41.396 49.779

−10.713 −3.2341 −0.034546

0.036376 0.040793 0.047185 0.048143 2.8287 3.9680 4.8778

200.00 300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00 100.00

28.911 26.630 24.493 22.020 19.139

0.034588 0.037552 0.040827 0.045413 0.052250

−11.305 −4.2043 4.3449 14.917 27.300

300.00 400.00 500.00 600.00

500.00 500.00 500.00 500.00

30.547 29.154 27.670 26.003

0.032736 0.034300 0.036140 0.038457

−5.4195 2.2795 11.020 21.094

202.71 40.941 12.933 4.3382 −0.39705 −0.32177 −0.090229 −0.047179

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is de Reuck, K. M., and Craven, R. J. B., “Methanol, International Thermodynamic Tables of the Fluid State—12,” IUPAC, Blackwell Scientific Publications, London, 1993. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are generally 0.1% in density and 2% in the speed of sound, except in the critical region and high pressures.

2-299

2-300 TABLE 2-235 Temperature K

Thermodynamic Properties of 2-Methyl Butane (Isopentane) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−26.072 −25.774 −23.896 −22.028 −20.148 −18.240 −16.293 −14.298 −12.247 −10.136 −7.9595 −5.7124 −3.3908 −0.99085 1.4914 4.0602 6.7200 9.4764 12.336 15.310 18.410 21.662 25.110 28.871 35.298

−26.072 −25.774 −23.896 −22.028 −20.148 −18.240 −16.293 −14.298 −12.247 −10.136 −7.9583 −5.7097 −3.3856 −0.98128 1.5078 4.0869 6.7616 9.5389 12.428 15.440 18.593 21.916 25.463 29.370 36.331

7.8069 7.9134 8.6432 9.4580 10.355 11.333 12.390 13.524 14.732 16.011 17.357 18.766 20.234 21.756 23.326 24.939 26.587 28.263 29.953 31.643 33.305 34.899 36.342 37.430 35.298

8.7435 8.8695 9.7241 10.664 11.686 12.788 13.969 15.227 16.557 17.955 19.416 20.935 22.503 24.115 25.763 27.438 29.132 30.831 32.522 34.183 35.782 37.268 38.543 39.362 36.331

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. Cond. mW/(mK)

Viscosity µPas

−0.13287 −0.13026 −0.11491 −0.10131 −0.08897 −0.07757 −0.06690 −0.05679 −0.04714 −0.03786 −0.02888 −0.02016 −0.01163 −0.00329 0.004922 0.013015 0.021017 0.028954 0.036852 0.044740 0.052658 0.060668 0.068880 0.077578 0.092549

0.088268 0.087968 0.087279 0.088069 0.089805 0.092192 0.095056 0.098282 0.10179 0.10553 0.10944 0.11351 0.11771 0.12200 0.12638 0.13084 0.13535 0.13992 0.14455 0.14926 0.15409 0.15915 0.16468 0.17145

0.12675 0.12625 0.12456 0.12473 0.12613 0.12840 0.13133 0.13478 0.13865 0.14289 0.14744 0.15230 0.15744 0.16288 0.16864 0.17476 0.18133 0.18849 0.19649 0.20576 0.21716 0.23260 0.25764 0.31907

2024.6 2007.2 1899.0 1796.0 1698.3 1605.5 1517.0 1432.2 1350.7 1271.9 1195.4 1120.7 1047.6 975.50 904.21 833.33 762.50 691.33 619.38 546.13 470.92 392.75 309.77 217.58 0

−0.61677 −0.61938 −0.62810 −0.62623 −0.61691 −0.60215 −0.58328 −0.56116 −0.53631 −0.50892 −0.47898 −0.44620 −0.41003 −0.36957 −0.32346 −0.26964 −0.20494 −0.12436 −0.01964 0.12385 0.33435 0.67407 1.3120 2.9280 18.924

177.46 177.30 174.59 169.97 164.30 158.07 151.59 145.02 138.51 132.11 125.89 119.87 114.07 108.49 103.13 98.008 93.105 88.417 83.934 79.645 75.537 71.596 67.829 64.403

18091 15987 7701.7 4126.6 2438.0 1564.5 1074.5 779.51 591.10 464.45 375.48 310.55 261.53 223.37 192.84 167.77 146.69 128.56 112.60 98.258 85.057 72.595 60.422 47.740

0.17618 0.17099 0.14371 0.12415 0.10999 0.099732 0.092376 0.087227 0.083784 0.081674 0.080615 0.080389 0.080825 0.081787 0.083164 0.084865 0.086812 0.088933 0.091161 0.093423 0.095630 0.097660 0.099299 0.10003 0.092549

0.044856 0.045769 0.051515 0.057098 0.062541 0.067888 0.073180 0.078451 0.083733 0.089051 0.094425 0.099871 0.10539 0.11100 0.11669 0.12246 0.12831 0.13425 0.14029 0.14648 0.15286 0.15958 0.16690 0.17558

0.053170 0.054083 0.059829 0.065412 0.070856 0.076206 0.081510 0.086811 0.092156 0.09759 0.10316 0.10890 0.11485 0.12105 0.12757 0.13447 0.14187 0.14999 0.15916 0.17005 0.18403 0.20431 0.24064 0.34061

124.05 125.14 131.91 138.36 144.53 150.45 156.12 161.53 166.66 171.43 175.77 179.56 182.71 185.07 186.54 186.97 186.20 184.05 180.31 174.70 166.85 156.26 142.23 123.63 0

Saturated Properties 112.65 115 130 145 160 175 190 205 220 235 250 265 280 295 310 325 340 355 370 385 400 415 430 445 460.35

8.9347E-11 1.9071E-10 1.1761E-08 2.8119E-07 3.4585E-06 2.6222E-05 0.00013839 0.00055325 0.0017804 0.0048228 0.011375 0.023974 0.046069 0.082015 0.13700 0.21693 0.32835 0.47831 0.67436 0.92461 1.2378 1.6238 2.0941 2.6640 3.3782

112.65 115 130 145 160 175 190 205 220 235 250 265 280 295 310 325 340 355 370 385 400 415 430 445 460.35

8.9527E-11 1.9071E-10 1.1761E-08 2.8119E-07 3.4585E-06 2.6222E-05 0.00013839 0.00055325 0.0017804 0.0048228 0.011375 0.023974 0.046069 0.082015 0.13700 0.21693 0.32835 0.47831 0.67436 0.92461 1.2378 1.6238 2.0941 2.6640 3.3782

10.936 10.903 10.698 10.500 10.306 10.116 9.9281 9.7405 9.5525 9.3629 9.1708 8.9750 8.7745 8.5679 8.3537 8.1302 7.8950 7.6451 7.3765 7.0835 6.7572 6.3830 5.9327 5.3333 3.2710 9.5585E-11 1.9975E-10 1.0881E-08 2.3324E-07 2.5998E-06 1.8023E-05 8.7629E-05 0.00032489 0.00097556 0.0024804 0.0055234 0.011056 0.020302 0.034759 0.056212 0.086783 0.12904 0.18620 0.26252 0.36398 0.49975 0.68534 0.95152 1.3789 3.2710

0.091442 0.091719 0.093478 0.095242 0.097028 0.09885 0.10072 0.10266 0.10468 0.10680 0.10904 0.11142 0.11397 0.11671 0.11971 0.12300 0.12666 0.13080 0.13557 0.14117 0.14799 0.15667 0.16856 0.18750 0.30572 1.05E+10 5.01E+09 91,905,000 4,287,500 384,650 55,485 11,412 3,077.9 1,025.1 403.15 181.05 90.449 49.256 28.769 17.790 11.523 7.7495 5.3705 3.8092 2.7474 2.0010 1.4591 1.0510 0.72521 0.30572

1474.9 867.62 543.86 358.94 247.17 176.38 129.80 98.138 76.055 60.317 48.903 40.504 34.252 29.560 26.024 23.372 21.417 20.043 19.189 18.851 19.092 20.076 22.072 18.924

2.3120 2.3996 3.0011 3.6746 4.4196 5.2372 6.1289 7.0964 8.1416 9.2673 10.477 11.776 13.169 14.664 16.270 17.997 19.860 21.877 24.077 26.507 29.248 32.469 36.596 43.254

2.775 2.827 3.1624 3.5046 3.853 4.2074 4.5669 4.931 5.2989 5.6698 6.0432 6.4187 6.7963 7.1769 7.5619 7.9543 8.3586 8.7821 9.2355 9.7357 10.311 11.011 11.947 13.438

Single-Phase Properties 150.00 225.00 300.00 300.60

0.10000 0.10000 0.10000 0.10000

300.60 375.00 450.00

0.10000 0.10000 0.10000

150.00 225.00 300.00 375.00 388.92

1.0000 1.0000 1.0000 1.0000 1.0000

388.92 450.00

1.0000 1.0000

150.00 225.00 300.00 375.00 450.00

5.0000 5.0000 5.0000 5.0000 5.0000

150.00 225.00 300.00 375.00 450.00

10.000 10.000 10.000 10.000 10.000

10.435 9.4905 8.4975 8.4890 0.041839 0.032734 0.027025 10.441 9.5002 8.5157 7.2951 7.0019 0.39567 0.30367

0.095828 0.10537 0.11768 0.11780 23.901 30.549 37.003 0.095776 0.10526 0.11743 0.13708 0.14282 2.5273 3.2931

−21.406 −11.554 −0.17289 −0.074609

−21.396 −11.543 −0.16112 −0.062829

−0.097086 −0.044025 −0.00053568 −0.00020836

0.088566 0.10302 0.12345 0.12363

0.12508 0.14001 0.16476 0.16499

1763.3 1324.8 951.69 948.83

−0.62386 −0.52760 −0.35493 −0.35312 32.344 14.520 8.0746

168.19 136.40 106.68 106.46

22.336 31.687 42.874

24.726 34.742 46.575

0.082257 0.11194 0.14064

0.11311 0.13702 0.16045

0.12344 0.14615 0.16919

185.73 210.31 231.26

−21.422 −11.582 −0.22558 13.279 16.107

−21.326 −11.477 −0.10815 13.417 16.250

−0.097192 −0.044150 −0.00071178 0.039387 0.046804

0.088617 0.10306 0.12349 0.14609 0.15050

0.12507 0.13992 0.16444 0.19894 0.20848

1767.4 1330.6 960.79 600.15 526.70

32.081 41.914

34.609 45.207

0.094009 0.11931

0.14812 0.16299

0.17332 0.17764

172.88 207.91

19.050 9.5995

27.189 34.291

−0.62411 −0.52893 −0.36015 0.013112 0.17073

15.250 23.271 32.656 168.39 136.76 107.23 82.749 78.555

3428.5 543.85 212.48 211.22 7.3199 9.1730 10.981 3436.2 547.08 214.82 108.44 94.715 9.8772 11.376

10.466 9.5418 8.5922 7.4791 5.7922

0.095550 0.10480 0.11638 0.13371 0.17265

−21.491 −11.704 −0.44721 12.787 28.764

−21.013 −11.180 0.13471 13.456 29.627

−0.097661 −0.044697 −0.0014609 0.038048 0.077218

0.088840 0.10325 0.12363 0.14600 0.16985

0.12500 0.13954 0.16321 0.19336 0.24769

1785.1 1355.9 999.20 670.90 327.62

−0.62511 −0.53441 −0.38047 −0.10299 1.0932

169.25 138.31 109.58 86.612 68.533

3470.1 561.26 224.97 119.86 59.364

10.496 9.5915 8.6795 7.6607 6.3848

0.095275 0.10426 0.11521 0.13054 0.15662

−21.575 −11.848 −0.69963 12.293 27.389

−20.622 −10.805 0.45251 13.598 28.955

−0.098235 −0.045356 −0.0023313 0.036668 0.073910

0.089117 0.10348 0.12382 0.14603 0.16843

0.12494 0.13913 0.16197 0.18923 0.22130

1806.7 1386.1 1043.2 742.26 471.81

−0.62619 −0.54037 −0.40072 −0.18850 0.27965

170.30 140.18 112.31 90.691 75.593

3511.8 578.68 237.18 132.45 78.440

−0.054491 −0.012648 0.024366 0.058454

0.10727 0.12720 0.14895 0.17040

0.13669 0.15576 0.17686 0.19748

1787.0 1539.5 1352.4 1210.1

−0.57796 −0.50239 −0.43356 −0.37778

165.84 145.78 130.89 121.48

860.73 420.70 290.41 234.19

0.15705 0.17779

0.17792 0.19805

2411.2 2302.8

−0.44269 −0.40062

211.59 205.82

864.12 747.58

225.00 300.00 375.00 450.00

100.00 100.00 100.00 100.00

10.235 9.6137 9.0393 8.5029

0.097700 0.10402 0.11063 0.11761

−13.576 −3.2651 8.5444 21.892

−3.8061 7.1368 19.607 33.653

375.00 450.00

500.00 500.00

10.851 10.550

0.092161 0.094785

5.2760 18.067

51.356 65.459

0.0033611 0.037589

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. The source for viscosity and thermal conductivity is NIST14, Version 9.08. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties are approximately 0.2% in density at temperatures up to 320 K, 0.5% in density at higher temperatures, 2% in heat capacity above 250 K, 4% in heat capacity at lower temperatures, 0.1% in the vapor phase speed of sound, 3% in the liquid-phase speed of sound, and 0.4% in vapor pressure at temperatures above 200 K. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6%

2-301

2-302 TABLE 2-236 Temperature K

Thermodynamic Properties of 2-Methyl Pentane (Isohexane) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. Energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.59545 −0.59577 −0.60440 −0.60052 −0.58671 −0.56534 −0.53846 −0.50751 −0.47322 −0.43565 −0.39411 −0.34721 −0.29249 −0.22585 −0.14022 −0.022615 0.15352 0.45116 1.0612 2.9251 22.418

Saturated Properties 119.60 120.00 140.00 160.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 497.70

7.3432E-12 8.6393E-12 2.6628E-09 1.6516E-07 3.6662E-06 4.0345E−05 0.00026966 0.0012504 0.0044049 0.012572 0.030459 0.064855 0.12457 0.22023 0.36396 0.56922 0.85081 1.2253 1.7125 2.3384 3.0426

9.3708 9.3662 9.1438 8.9315 8.7257 8.5239 8.3237 8.1236 7.9216 7.7163 7.5058 7.2881 7.0608 6.8210 6.5643 6.2850 5.9735 5.6136 5.1716 4.5506 2.7150

119.60 120.00 140.00 160.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 497.70

7.3432E-12 8.6393E-12 2.6628E-09 1.6516E-07 3.6662E-06 4.0345E-05 0.00026966 0.0012504 0.0044049 0.012572 0.030459 0.064855 0.12457 0.22023 0.36396 0.56922 0.85081 1.2253 1.7125 2.3384 3.0426

7.7171E-12 8.8145E-12 2.2870E-09 1.2415E-07 2.4497E-06 2.4265E-05 0.00014751 0.00062783 0.0020480 0.0054602 0.012471 0.025273 0.046680 0.080240 0.13054 0.20388 0.30983 0.46493 0.70383 1.1281 2.7150

0.10671 0.10677 0.10936 0.11196 0.11460 0.11732 0.12014 0.12310 0.12624 0.12960 0.13323 0.13721 0.14163 0.14661 0.15234 0.15911 0.16740 0.17814 0.19337 0.21975 0.36832 1.2958E+11 1.1345E+11 43,7250,000. 8,054,900. 4,08,210. 41,212. 6,779.4 1,592.8 488.27 183.14 80.187 39.568 21.422 12.463 7.6606 4.9048 3.2276 2.1509 1.4208 0.88647 0.36832

−36.470 −36.409 −33.376 −30.354 −27.297 −24.169 −20.943 −17.596 −14.114 −10.484 −6.6984 −2.7483 1.3732 5.6736 10.162 14.852 19.761 24.925 30.414 36.430 44.813

−36.470 −36.409 −33.376 −30.354 −27.297 −24.169 −20.943 −17.596 −14.113 −10.483 −6.6944 −2.7394 1.3908 5.7059 10.218 14.942 19.903 25.143 30.745 36.944 45.934

−0.17002 −0.16951 −0.14612 −0.12595 −0.10795 −0.091478 −0.076105 −0.061550 −0.047617 −0.034172 −0.021115 −0.0083705 0.0041212 0.016411 0.028548 0.040580 0.052569 0.064603 0.076849 0.089761 0.10771

0.10838 0.10835 0.10816 0.10991 0.11321 0.11773 0.12318 0.12928 0.13582 0.14266 0.14969 0.15683 0.16404 0.17128 0.17854 0.18581 0.19312 0.20057 0.20838 0.21745

0.15299 0.15291 0.15094 0.15162 0.15435 0.15866 0.16418 0.17061 0.17772 0.18535 0.19342 0.20190 0.21081 0.22027 0.23047 0.24183 0.25521 0.27266 0.30051 0.37420

2019.9 2016.9 1877.8 1750.1 1631.2 1519.7 1414.3 1314.2 1218.4 1126.1 1036.6 948.94 862.49 776.48 690.10 602.46 512.43 418.45 317.90 204.95 0

3.2114 3.2365 4.5677 6.0465 7.6734 9.4535 11.391 13.489 15.742 18.144 20.686 23.355 26.139 29.022 31.987 35.009 38.050 41.048 43.875 46.190 44.813

4.2058 4.2342 5.7317 7.3769 9.1700 11.116 13.220 15.480 17.893 20.447 23.128 25.921 28.807 31.767 34.775 37.801 40.796 43.683 46.308 48.263 45.934

0.17008 0.16919 0.13321 0.10987 0.094644 0.084949 0.079177 0.076268 0.075483 0.076291 0.078294 0.081194 0.084758 0.088803 0.093173 0.097727 0.10231 0.10674 0.11068 0.11334 0.10771

0.062593 0.062751 0.070285 0.077610 0.085136 0.092992 0.10116 0.10956 0.11814 0.12686 0.13569 0.14461 0.15360 0.16265 0.17176 0.18095 0.19027 0.19983 0.20995 0.22144

0.070908 0.071065 0.078600 0.085925 0.093451 0.10131 0.10950 0.11797 0.12670 0.13570 0.14499 0.15463 0.16471 0.17537 0.18690 0.19986 0.21550 0.23692 0.27434 0.38625

114.33 114.51 122.90 130.73 138.07 144.98 151.49 157.59 163.19 168.11 172.17 175.13 176.74 176.73 174.77 170.42 163.11 151.97 135.67 112.14 0

1878.9 1852.6 971.14 557.20 341.59 220.67 148.98 104.56 76.030 57.168 44.401 35.601 29.461 25.167 22.214 20.301 19.294 19.227 20.404 23.602 22.418

Single-Phase Properties 150.00 250.00 332.95

0.10000 0.10000 0.10000

9.0371 8.0239 7.1422

332.95 350.00 450.00 550.00

0.10000 0.10000 0.10000 0.10000

0.037947 0.035789 0.027168 0.022045

150.00 250.00 350.00 428.63

1.0000 1.0000 1.0000 1.0000

9.0415 8.0324 6.9610 5.8255

428.63 450.00 550.00

1.0000 1.0000 1.0000

150.00 250.00 350.00 450.00 550.00

5.0000 5.0000 5.0000 5.0000 5.0000

150.00 250.00 350.00 450.00 550.00

10.000 10.000 10.000 10.000 10.000

0.11065 0.12463 0.14001 26.353 27.941 36.808 45.361

−31.868 −15.876 −0.099839

−31.857 −15.864 −0.085838

−0.13572 −0.054530 −0.00025709

0.10883 0.13251 0.16149

0.15099 0.17408 0.20761

1813.1 1266.4 892.88

−0.60387 −0.49091 −0.31289

25.145 27.784 45.429 66.610

27.780 30.578 49.110 71.146

0.083438 0.091632 0.13801 0.18212

0.15042 0.15693 0.19438 0.22795

0.16110 0.16712 0.20338 0.23660

176.34 181.93 209.65 232.80

31.381 25.747 10.406 5.5017

0.11060 0.12450 0.14366 0.17166

−31.886 −15.912 3.4284 21.956

−31.775 −15.787 3.5720 22.127

−0.13584 −0.054671 0.010082 0.057750

0.10889 0.13256 0.16769 0.19631

0.15098 0.17397 0.21498 0.26204

1816.8 1272.3 829.36 472.47

−0.60402 −0.49228 −0.26842 0.26084

0.36934 0.33044 0.23887

2.7075 3.0263 4.1863

39.354 43.784 65.703

42.062 46.810 69.889

0.10426 0.11507 0.16131

0.19435 0.19990 0.22981

0.22372 0.22188 0.24260

158.83 173.64 216.16

19.140 14.396 6.1837

9.0608 8.0692 7.0426 5.7124 2.3237

0.11037 0.12393 0.14199 0.17506 0.43035

−31.962 −16.063 3.1052 26.615 58.312

−31.410 −15.443 3.8152 27.490 60.464

−0.13635 −0.055282 0.0091444 0.068380 0.13398

0.10915 0.13278 0.16784 0.20366 0.24134

0.15096 0.17352 0.21301 0.26396 0.37655

1833.1 1297.7 873.72 476.76 137.16

−0.60459 −0.49791 −0.29846 0.22550 8.5257

9.0844 8.1131 7.1333 5.9777 4.3857

0.11008 0.12326 0.14019 0.16729 0.22801

−32.055 −16.241 2.7454 25.695 52.778

−30.954 −15.009 4.1473 27.368 55.059

−0.13698 −0.056017 0.0080780 0.066210 0.12162

0.10949 0.13305 0.16805 0.20340 0.23673

0.15093 0.17304 0.21116 0.25395 0.29917

1853.0 1328.1 923.39 576.50 305.87

−0.60519 −0.50401 −0.32667 −0.0027791 1.0063

250.00 350.00 450.00 550.00

100.00 100.00 100.00 100.00

8.6765 8.0271 7.4403 6.9067

0.11525 0.12458 0.13440 0.14479

−18.375 −0.67131 20.331 44.426

−6.8492 11.787 33.771 58.905

−0.066142 −0.0037487 0.051327 0.10166

0.13735 0.17182 0.20635 0.23764

0.17026 0.20310 0.23620 0.26573

1729.6 1445.5 1247.4 1108.2

−0.54229 −0.44895 −0.37634 −0.32474

450.00 550.00

500.00 500.00

9.0895 8.7910

0.11002 0.11375

16.069 39.474

71.078 96.350

0.028505 0.079120

0.21492 0.24537

0.23767 0.26717

2315.0 2204.0

−0.39042 −0.35059

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M.O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.2% in density in the liquid phase, 0.5% in density in the vapor phase, 1% in density above the critical temperature, 0.2% in vapor pressure between 280 and 350 K, 0.5% in vapor pressure at higher temperatures, 2% in heat capacities, and 1% in the speed of sound.

2-303

2-304

PHYSICAL AND CHEMICAL DATA

TABLE 2-237

Saturated Methyl Chloride* µf, 10−4 Pa⋅s

kf, W/(m⋅K)

1.483 1.486 1.489 1.492 1.496

4.44 4.27 4.11 3.96 3.82

0.241 0.236 0.232 0.228 0.224

5.896 5.866 5.845 5.822 5.786

1.500 1.504 1.508 1.513 1.518

3.69 3.57 3.46 3.35 3.25

0.219 0.215 0.211 0.207 0.202

4.050 4.080 4.110 4.139 4.168

5.762 5.740 5.720 5.699 5.680

1.523 1.528 1.533 1.539 1.546

3.16 3.08 3.00 2.92 2.85

0.198 0.194 0.190 0.186 0.182

824.4 826.8 829.0 831.2 833.2

4.197 4.225 4.253 4.280 4.308

5.662 5.644 5.628 5.612 5.597

1.554 1.565 1.574 1.583 1.594

2.78 2.72 2.66 2.61 2.56

0.177 0.173 0.169 0.165 0.160

461.2 469.3 477.4 485.6 493.8

835.2 837.0 838.8 840.5 841.9

4.334 4.361 4.388 4.414 4.440

5.581 5.567 5.553 5.540 5.527

1.605 1.617 1.631 1.644 1.658

2.51 2.46 2.42 2.37 2.33

0.156 0.152 0.148 0.143 0.139

0.0386 0.0343 0.0282 0.0228 0.0186

502.1 510.4 518.8 538.3 562.9

843.3 844.5 846.4 847.5 847.6

4.465 4.491 4.542 4.592 4.643

5.516 5.504 5.481 5.457 5.434

2.30 2.27 2.12 1.99 1.87

0.135 0.131 0.124 0.117 0.110

13.47.−4 14.11.−4 14.67.−4 15.66.−4 16.48.−4

0.0151 0.0117 0.0096 0.0075 0.0063

581.6 602.8 622.9 643.6 663.2

845.9 842.6 837.4 826.4 819.1

4.694 4.747 4.805 4.870 4.904

5.398 5.382 5.358 5.323 5.289

1.77 1.67 1.59 1.51

0.103 0.095 0.086 0.075

17.97.−4 21.10.−4 27.40.−4

0.0052 0.0038 0.0027

677.3 714.1 749.3

807.1 778.6 749.3

4.954 5.025 5.116

5.256 5.200 5.116

T, K

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

175 180 185 190 195

0.0117 0.0165 0.0233 0.0327 0.0462

8.84.−4 8.91.−4 8.97.−4 9.04.−4 9.10.−4

27.90 19.85 14.12 10.12 7.208

274.5 280.9 287.5 294.5 301.7

764.3 767.7 771.0 774.3 777.5

3.529 3.570 3.603 3.647 3.684

6.328 6.274 6.222 6.172 6.124

1.469 1.472 1.475 1.477 1.480

200 205 210 215 220

0.0653 0.0919 0.1315 0.181 0.243

9.17.−4 9.25.−4 9.33.−4 9.40.−4 9.48.−4

5.137 3.835 2.656 1.975 1.505

309.0 316.3 323.7 331.0 338.4

780.7 783.9 787.0 790.1 793.2

3.722 3.756 3.791 3.825 3.859

6.080 6.038 5.998 5.961 5.928

225 230 235 240 245

0.319 0.417 0.539 0.688 0.866

9.56.−4 9.65.−4 9.73.−4 9.81.−4 9.89.−4

1.168 0.911 0.718 0.572 0.462

345.7 353.1 360.5 368.0 375.6

796.3 799.3 802.3 805.3 808.2

3.892 3.925 3.957 3.988 4.019

250 255 260 265 270

1.076 1.328 1.627 1.970 2.364

9.98.−4 10.08.−4 10.18.−4 10.27.−4 10.36.−4

0.377 0.311 0.257 0.215 0.1807

383.2 390.7 398.3 406.0 413.7

811.1 814.0 816.8 819.4 822.0

275 280 285 290 295

2.830 3.347 3.936 4.612 5.361

10.46.−4 10.57.−4 10.68.−4 10.79.−4 10.91.−4

0.1524 0.1301 0.1115 0.0960 0.0830

421.5 429.4 437.3 445.2 453.2

300 305 310 315 320

6.189 7.110 8.111 9.243 10.47

11.03.−4 11.15.−4 11.27.−4 11.40.−4 11.55.−4

0.0723 0.0632 0.0556 0.0489 0.0433

325 330 340 350 360

11.78 13.27 16.52 20.53 25.29

11.70.−4 11.86.−4 12.17.−4 12.54.−4 12.97.−4

370 380 390 400 405

30.74 36.99 44.05 52.29 56.6

410 415 416c

61.5 67.4 69.0

*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 677, 1967 tables by Steinle/Dienemann. c = critical point. The notation 8.84.−4 signifies 8.84 × 10−4.

TABLE 2-238 Temperature K

Thermodynamic Properties of Neon Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol −0.099873 −0.081689 −0.042983 −0.0055930 0.031967 0.070340 0.10979 0.15041 0.19218 0.23507 0.27908 0.32426 0.37073 0.41874 0.46869 0.52120 0.57729 0.63879 0.70940 0.79805 0.92402 1.0877

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.0038092 −0.0030751 −0.0015561 −0.00014422 0.0012227 0.0025706 0.0039099 0.0052441 0.0065735 0.0078977 0.0092172 0.010534 0.011853 0.013181 0.014529 0.015914 0.017363 0.018922 0.020686 0.022883 0.026030 0.030313

0.018679 0.018439 0.018193 0.018006 0.017803 0.017589 0.017371 0.017156 0.016952 0.016768 0.016614 0.016498 0.016426 0.016404 0.016438 0.016536 0.016715 0.017014 0.017522 0.018471 0.020210

0.043081 0.040416 0.037847 0.037540 0.038200 0.039303 0.040595 0.041956 0.043347 0.044788 0.046343 0.048115 0.050251 0.052967 0.056597 0.061719 0.069475 0.082509 0.10852 0.17680 0.44810

674.66 652.08 617.51 596.00 579.65 564.56 548.99 532.26 514.19 494.83 474.28 452.65 429.96 406.17 381.08 354.35 325.39 293.27 256.54 213.78 170.57 0

−0.22893 −0.25101 −0.27322 −0.27053 −0.25492 −0.23297 −0.20788 −0.18090 −0.15212 −0.12096 −0.086349 −0.046782 −0.00027396 0.055871 0.12549 0.21441 0.33222 0.49632 0.74180 1.1433 1.8283 2.9448

0.068586 0.067767 0.065882 0.064031 0.062250 0.060549 0.058928 0.057377 0.055888 0.054449 0.053047 0.051670 0.050303 0.048930 0.047531 0.046079 0.044539 0.042855 0.040930 0.038566 0.035154 0.030313

0.035186 0.029819 0.022043 0.018031 0.016001 0.015021 0.014604 0.014497 0.014566 0.014745 0.015000 0.015316 0.015688 0.016118 0.016614 0.017189 0.017863 0.018664 0.019634 0.020823 0.022208

0.047919 0.041791 0.033098 0.028839 0.026943 0.026349 0.026517 0.027183 0.028231 0.029633 0.031418 0.033669 0.036531 0.040243 0.045207 0.052136 0.062405 0.079029 0.10997 0.18428 0.55080

113.01 115.48 121.12 126.17 130.23 133.28 135.55 137.25 138.54 139.52 140.26 140.78 141.11 141.24 141.18 140.93 140.49 139.83 138.95 137.87 136.83 0

27.054 23.404 18.114 15.446 14.023 13.110 12.393 11.758 11.166 10.606 10.073 9.5602 9.0641 8.5780 8.0944 7.6033 7.0917 6.5404 5.9193 5.1760 4.1789 2.9448

Saturated Properties 24.562 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 42.000 43.000 44.000 44.492

0.043464 0.050920 0.071611 0.098173 0.13159 0.17287 0.22307 0.28324 0.35446 0.43782 0.53443 0.64543 0.77202 0.91543 1.0770 1.2581 1.4603 1.6855 1.9355 2.2121 2.5168 2.6786

62.059 61.645 60.761 59.903 59.032 58.129 57.186 56.198 55.162 54.078 52.941 51.749 50.493 49.162 47.736 46.187 44.462 42.471 40.021 36.640 31.327 23.882

0.016114 0.016222 0.016458 0.016694 0.016940 0.017203 0.017487 0.017794 0.018128 0.018492 0.018889 0.019324 0.019805 0.020341 0.020948 0.021651 0.022491 0.023545 0.024987 0.027292 0.031921 0.041873

24.562 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 33.000 34.000 35.000 36.000 37.000 38.000 39.000 40.000 41.000 42.000 43.000 44.000 44.492

0.043464 0.050920 0.071611 0.098173 0.13159 0.17287 0.22307 0.28324 0.35446 0.43782 0.53443 0.64543 0.77202 0.91543 1.0770 1.2581 1.4603 1.6855 1.9355 2.2121 2.5168 2.6786

0.22026 0.25406 0.34598 0.46111 0.60299 0.77543 0.98260 1.2292 1.5204 1.8625 2.2630 2.7308 3.2778 3.9193 4.6762 5.5788 6.6734 8.0363 9.8091 12.303 16.541 23.882

4.5402 3.9361 2.8903 2.1687 1.6584 1.2896 1.0177 0.81356 0.65772 0.53690 0.44190 0.36619 0.30508 0.25515 0.21385 0.17925 0.14985 0.12443 0.10195 0.081282 0.060455 0.041873

1.4817 1.4898 1.5046 1.5159 1.5247 1.5318 1.5372 1.5411 1.5435 1.5443 1.5432 1.5401 1.5347 1.5265 1.5150 1.4994 1.4784 1.4500 1.4107 1.3530 1.2537 1.0877

−0.099172 −0.080863 −0.041805 −0.0039541 0.034196 0.073314 0.11370 0.15545 0.19860 0.24317 0.28917 0.33673 0.38602 0.43736 0.49125 0.54844 0.61014 0.67848 0.75776 0.85843 1.0044 1.1999 1.6790 1.6902 1.7116 1.7288 1.7430 1.7547 1.7642 1.7716 1.7767 1.7794 1.7794 1.7765 1.7702 1.7601 1.7453 1.7249 1.6972 1.6597 1.6080 1.5328 1.4058 1.1999

2-305

2-306

TABLE 2-238 Temperature K

Thermodynamic Properties of Neon (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 100.00 200.00 300.00 400.00 500.00 600.00 700.00

0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

0.12034 0.060106 0.040072 0.030057 0.024047 0.020040 0.017178

100.00 200.00 300.00 400.00 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.2090 0.59828 0.39903 0.29952 0.23978 0.19992 0.17143

100.00 200.00 300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

6.1145 2.9287 1.9580 1.4747 1.1837 0.98897 0.84934

100.00 200.00 300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000 10.000 10.000

8.3098 16.637 24.955 33.270 41.585 49.899 58.214

2.4474 3.6964 4.9441 6.1915 7.4388 8.6861 9.9333

3.2784 5.3601 7.4396 9.5185 11.597 13.676 15.755

0.091975 0.10641 0.11484 0.12082 0.12546 0.12925 0.13245

0.012479 0.012475 0.012473 0.012473 0.012473 0.012472 0.012472

0.020851 0.020800 0.020791 0.020789 0.020787 0.020787 0.020786

262.24 370.87 454.13 524.31 586.15 642.06 693.47

1.6033 0.12447 −0.30667 −0.49467 −0.59137 −0.64538 −0.67670

0.82716 1.6715 2.5061 3.3387 4.1705 5.0020 5.8334

2.4215 3.6864 4.9392 6.1891 7.4379 8.6861 9.9339

3.2486 5.3579 7.4453 9.5278 11.608 13.688 15.767

0.072572 0.087212 0.095677 0.10167 0.10631 0.11010 0.11331

0.012547 0.012502 0.012491 0.012485 0.012482 0.012480 0.012478

0.021444 0.020926 0.020841 0.020813 0.020800 0.020794 0.020791

264.06 373.32 456.29 526.19 587.81 643.54 694.81

1.5285 0.11185 −0.30952 −0.49518 −0.59127 −0.64516 −0.67651

0.16355 0.34145 0.51074 0.67810 0.84478 1.0112 1.1774

2.3075 3.6436 4.9181 6.1787 7.4338 8.6860 9.9366

3.1252 5.3508 7.4718 9.5693 11.658 13.742 15.824

0.058098 0.073620 0.082226 0.088261 0.092921 0.096721 0.099930

0.012814 0.012619 0.012565 0.012539 0.012523 0.012513 0.012506

0.024056 0.021453 0.021051 0.020917 0.020858 0.020828 0.020810

275.27 384.50 465.95 534.57 595.18 650.12 700.76

1.1885 0.054603 −0.32325 −0.49810 −0.59116 −0.64435 −0.67575

12.113 5.6967 3.8255 2.8938 2.3306 1.9519 1.6795

0.082559 0.17554 0.26141 0.34556 0.42908 0.51231 0.59540

2.1736 3.5933 4.8930 6.1661 7.4286 8.6858 9.9399

2.9992 5.3487 7.5070 9.6217 11.719 13.809 15.894

0.051158 0.067620 0.076381 0.082467 0.087148 0.090958 0.094172

0.013095 0.012752 0.012653 0.012603 0.012573 0.012553 0.012539

0.026878 0.022026 0.021291 0.021040 0.020927 0.020868 0.020835

296.54 399.17 478.16 545.08 604.43 658.38 708.23

0.80676 −0.014628 −0.34148 −0.50266 −0.59158 −0.64363 −0.67492

728.18 673.52 694.50 729.70 767.75 805.72 842.83

−0.37331 −0.43751 −0.49312 −0.54535 −0.58985 −0.62513 −0.65210

100.00 200.00 300.00 400.00 500.00 600.00 700.00

100.00 100.00 100.00 100.00 100.00 100.00 100.00

50.564 34.632 26.457 21.512 18.185 15.781 13.955

0.019777 0.028875 0.037796 0.046486 0.054990 0.063366 0.071657

1.4006 3.1327 4.6205 6.0160 7.3645 8.6852 9.9877

3.3783 6.0202 8.4001 10.665 12.863 15.022 17.153

0.028294 0.046773 0.056445 0.062965 0.067874 0.071810 0.075096

0.016249 0.014308 0.013708 0.013422 0.013245 0.013121 0.013028

0.028809 0.024694 0.023102 0.022262 0.021756 0.021432 0.021216

300.00 400.00 500.00 600.00 700.00

500.00 500.00 500.00 500.00 500.00

59.131 53.366 48.708 44.864 41.633

0.016912 0.018738 0.020530 0.022290 0.024020

4.6368 6.0941 7.5120 8.9046 10.278

13.093 15.463 17.777 20.049 22.288

0.042072 0.048897 0.054063 0.058207 0.061658

0.015948 0.015200 0.014770 0.014481 0.014265

0.024091 0.023386 0.022915 0.022544 0.022235

1306.5 1293.9 1294.1 1301.3 1312.5

−0.47221 −0.49185 −0.50859 −0.52455 −0.53993

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Katti, R. S., Jacobsen, R. T., Stewart, R. B., and Jahangiri, M., “Thermodynamic Properties for Neon for Temperatures from the Triple Point to 700 K at Pressures to 700 MPa,” Adv. Cryo. Eng. 31:1189–1197, 1986. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 2% in heat capacity, and 1% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.2%.

TABLE 2-239 Temperature K

Thermodynamic Properties of Nitrogen Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

−4.2230 −4.1194 −4.0071 −3.8946 −3.7819 −3.6689 −3.5556 −3.4419 −3.3278 −3.2132 −3.0980 −2.9822 −2.8657 −2.7483 −2.6301 −2.5107 −2.3902 −2.2683 −2.1449 −2.0196 −1.8923 −1.7625 −1.6298 −1.4938 −1.3537 −1.2086 −1.0571 −0.89741 −0.72635 −0.53833 −0.32093 −0.031475 0.51527

−4.2226 −4.1188 −4.0063 −3.8935 −3.7804 −3.6669 −3.5530 −3.4385 −3.3235 −3.2078 −3.0913 −2.9739 −2.8555 −2.7360 −2.6152 −2.4930 −2.3691 −2.2434 −2.1156 −1.9854 −1.8525 −1.7163 −1.5765 −1.4323 −1.2828 −1.1271 −0.96336 −0.78944 −0.60161 −0.39322 −0.14962 0.17937 0.81891

0.067951 0.069569 0.071270 0.072924 0.074535 0.076105 0.077637 0.079133 0.080597 0.082030 0.083436 0.084815 0.086172 0.087507 0.088823 0.090123 0.091408 0.092682 0.093946 0.095204 0.096459 0.097715 0.098977 0.10025 0.10154 0.10285 0.10420 0.10561 0.10710 0.10873 0.11059 0.11310 0.11807

1.2945 1.3299 1.3675 1.4042 1.4400 1.4747 1.5082 1.5404 1.5713 1.6007 1.6284 1.6544 1.6784 1.7005 1.7203 1.7377 1.7525 1.7645 1.7733 1.7788 1.7804 1.7778 1.7703

1.8147 1.8639 1.9160 1.9668 2.0162 2.0639 2.1099 2.1539 2.1957 2.2353 2.2725 2.3070 2.3386 2.3672 2.3925 2.4143 2.4324 2.4463 2.4557 2.4603 2.4595 2.4528 2.4394

0.16355 0.16161 0.15966 0.15786 0.15618 0.15461 0.15314 0.15176 0.15046 0.14923 0.14806 0.14694 0.14587 0.14485 0.14385 0.14289 0.14195 0.14103 0.14012 0.13922 0.13832 0.13742 0.13651

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.032951 0.032591 0.032207 0.031831 0.031463 0.031106 0.030760 0.030427 0.030105 0.029795 0.029499 0.029215 0.028944 0.028687 0.028444 0.028215 0.028001 0.027804 0.027624 0.027464 0.027327 0.027214 0.027133 0.027088 0.027089 0.027149 0.027290 0.027545 0.027981 0.028755 0.030317 0.034680

0.056033 0.056121 0.056231 0.056360 0.056512 0.056690 0.056899 0.057142 0.057425 0.057752 0.058130 0.058566 0.059068 0.059647 0.060315 0.061088 0.061983 0.063026 0.064246 0.065684 0.067392 0.069443 0.071937 0.075021 0.078914 0.083966 0.090771 0.10044 0.11531 0.14140 0.20028 0.46831

995.28 976.36 956.04 935.83 915.66 895.49 875.28 855.00 834.61 814.07 793.36 772.44 751.28 729.84 708.09 685.99 663.50 640.57 617.14 593.17 568.58 543.30 517.24 490.29 462.32 433.19 402.67 370.43 335.85 297.68 253.32 195.48 0

−0.40419 −0.39833 −0.39135 −0.38364 −0.37508 −0.36560 −0.35506 −0.34334 −0.33029 −0.31574 −0.29951 −0.28135 −0.26099 −0.23813 −0.21237 −0.18326 −0.15025 −0.11264 −0.069613 −0.020100 0.037239 0.10414 0.18288 0.27654 0.38936 0.52741 0.69974 0.92076 1.2154 1.6317 2.2811 3.5308 6.0831

173.24 169.51 165.49 161.47 157.47 153.46 149.47 145.48 141.50 137.55 133.61 129.66 125.72 121.77 117.83 113.89 109.95 106.02 102.08 98.144 94.208 90.272 86.337 82.404 78.472 74.544 70.626 66.728 62.883 59.196 56.121 56.435

311.59 282.07 254.55 230.85 210.32 192.43 176.75 162.94 150.71 139.82 130.07 121.31 113.38 106.18 99.602 93.568 88.004 82.847 78.042 73.543 69.306 65.292 61.464 57.786 54.224 50.740 47.290 43.824 40.270 36.509 32.310 26.935

0.021007 0.021059 0.021123 0.021196 0.021278 0.021370 0.021472 0.021585 0.021709 0.021845 0.021994 0.022157 0.022334 0.022528 0.022738 0.022967 0.023217 0.023489 0.023787 0.024113 0.024471 0.024860 0.025284

0.029647 0.029788 0.029969 0.030180 0.030427 0.030712 0.031039 0.031413 0.031839 0.032323 0.032873 0.033496 0.034204 0.035008 0.035925 0.036973 0.038177 0.039568 0.041185 0.043081 0.045326 0.048012 0.051276

161.11 163.20 165.37 167.43 169.39 171.23 172.95 174.55 176.03 177.38 178.60 179.68 180.63 181.43 182.10 182.62 182.99 183.21 183.28 183.18 182.93 182.51 181.93

40.718 38.268 35.907 33.803 31.922 30.231 28.707 27.328 26.074 24.931 23.884 22.923 22.035 21.212 20.446 19.730 19.057 18.421 17.815 17.236 16.676 16.132 15.600

Saturated Properties

2-307

63.151 65.000 67.000 69.000 71.000 73.000 75.000 77.000 79.000 81.000 83.000 85.000 87.000 89.000 91.000 93.000 95.000 97.000 99.000 101.00 103.00 105.00 107.00 109.00 111.00 113.00 115.00 117.00 119.00 121.00 123.00 125.00 126.19

0.012520 0.017404 0.024300 0.033213 0.044527 0.058656 0.076043 0.097152 0.12247 0.15251 0.18780 0.22886 0.27626 0.33055 0.39230 0.46210 0.54052 0.62817 0.72566 0.83358 0.95259 1.0833 1.2264 1.3826 1.5526 1.7371 1.9370 2.1533 2.3869 2.6391 2.9116 3.2069 3.3958

63.151 65.000 67.000 69.000 71.000 73.000 75.000 77.000 79.000 81.000 83.000 85.000 87.000 89.000 91.000 93.000 95.000 97.000 99.000 101.00 103.00 105.00 107.00

0.012520 0.017404 0.024300 0.033213 0.044527 0.058656 0.076043 0.097152 0.12247 0.15251 0.18780 0.22886 0.27626 0.33055 0.39230 0.46210 0.54052 0.62817 0.72566 0.83358 0.95259 1.0833 1.2264

30.957 30.685 30.387 30.085 29.779 29.468 29.153 28.832 28.506 28.175 27.837 27.492 27.139 26.779 26.409 26.030 25.640 25.238 24.822 24.390 23.941 23.471 22.978 22.457 21.902 21.306 20.658 19.943 19.134 18.187 16.997 15.210 11.184 0.024070 0.032594 0.044300 0.059031 0.077273 0.099542 0.12638 0.15838 0.19613 0.24030 0.29157 0.35069 0.41846 0.49576 0.58355 0.68291 0.79504 0.92134 1.0634 1.2231 1.4027 1.6049 1.8331

0.032303 0.032589 0.032909 0.033239 0.033581 0.033935 0.034302 0.034683 0.035080 0.035493 0.035924 0.036375 0.036847 0.037343 0.037865 0.038417 0.039002 0.039623 0.040288 0.041000 0.041769 0.042605 0.043520 0.044530 0.045658 0.046935 0.048407 0.050144 0.052262 0.054985 0.058834 0.065747 0.089414 41.546 30.680 22.573 16.940 12.941 10.046 7.9124 6.3140 5.0986 4.1614 3.4297 2.8515 2.3897 2.0171 1.7137 1.4643 1.2578 1.0854 0.94038 0.81759 0.71291 0.62309 0.54553

5.6209 5.8164 6.0298 6.2457 6.4645 6.6870 6.9138 7.1458 7.3839 7.6295 7.8837 8.1483 8.4251 8.7163 9.0247 9.3533 9.7060 10.087 10.503 10.960 11.467 12.035 12.679

4.3763 4.5123 4.6601 4.8088 4.9585 5.1096 5.2621 5.4164 5.5727 5.7313 5.8924 6.0565 6.2238 6.3948 6.5700 6.7499 6.9353 7.1270 7.3260 7.5334 7.7509 7.9804 8.2245

TABLE 2-239 2-308

Temperature K

Thermodynamic Properties of Nitrogen (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

1.7573 1.7377 1.7102 1.6730 1.6234 1.5572 1.4665 1.3343 1.1039 0.51527

2.4183 2.3884 2.3479 2.2946 2.2251 2.1341 2.0119 1.8376 1.5417 0.81891

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

0.055332 0.060528 0.067435 0.077010 0.091003 0.11312 0.15295 0.24490 0.66512

181.19 180.28 179.15 177.75 176.01 173.87 171.17 167.43 160.26 0

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

15.075 14.546 13.996 13.409 12.767 12.045 11.203 10.148 8.6030 6.0831

13.419 14.284 15.315 16.580 18.186 20.329 23.424 28.604 41.535

8.4867 8.7716 9.0860 9.4395 9.8474 10.336 10.953 11.813 13.326

9.3806 44.840 70.075 92.344

6.9581 29.577 44.199 56.398

Saturated Properties 109.00 111.00 113.00 115.00 117.00 119.00 121.00 123.00 125.00 126.19

1.3826 1.5526 1.7371 1.9370 2.1533 2.3869 2.6391 2.9116 3.2069 3.3958

100.00 600.00 1100.0 1600.0

0.10000 0.10000 0.10000 0.10000

100.00 103.75

1.0000 1.0000

103.75 600.00 1100.0 1600.0

1.0000 1.0000 1.0000 1.0000

100.00 600.00 1100.0 1600.0

5.0000 5.0000 5.0000 5.0000

2.0916 2.3860 2.7240 3.1162 3.5786 4.1370 4.8380 5.7846 7.3244 11.184

0.47811 0.41911 0.36711 0.32091 0.27944 0.24172 0.20670 0.17287 0.13653 0.089414

0.13557 0.13461 0.13360 0.13253 0.13138 0.13009 0.12860 0.12675 0.12400 0.11807

0.025750 0.026284 0.026924 0.027721 0.028723 0.029997 0.031683 0.034185 0.039278

Single-Phase Properties

100.00 600.00 1100.0 1600.0 600.00 1100.0 1600.0 600.00 1100.0 1600.0

10.000 10.000 10.000 10.000 500.00 500.00 500.00 1000.0 1000.0 1000.0

0.12268 0.020037 0.010930 0.0075152 24.658 23.768 1.4754 0.19960 0.10899 0.074993

8.1514 49.908 91.489 133.06 0.040554 0.042073 0.67778 5.0099 9.1755 13.335

2.0396 12.573 24.284 37.272

2.8547 17.564 33.433 50.579

0.15950 0.21217 0.23131 0.24414

0.021049 0.021796 0.024932 0.026815

0.030012 0.030118 0.033248 0.035130

201.64 496.27 660.05 788.94

16.082 0.021483 −0.65654 −0.81543

−2.0907 −1.8441

−2.0501 −1.8020

0.094493 0.096928

0.027546 0.027281

0.064564 0.068113

609.42 559.22

−0.054514 0.060996

100.58 92.738

76.255 67.783

1.7800 12.554 24.277 37.270

2.4577 17.564 33.452 50.605

0.13799 0.19300 0.21216 0.22499

0.024612 0.021812 0.024938 0.026820

0.046272 0.030198 0.033267 0.035138

182.79 498.66 662.07 790.64

16.471 0.0061465 −0.65940 −0.81612

11.671 44.992 70.155 92.399

7.8351 29.626 44.221 56.411

25.436 0.98084 0.53797 0.37146

0.039314 1.0195 1.8588 2.6921

−2.2176 12.469 24.247 37.259

−2.0210 17.567 33.541 50.720

0.093188 0.17948 0.19875 0.21161

0.027713 0.021881 0.024969 0.026839

0.059868 0.030539 0.033350 0.035170

673.24 509.60 671.08 798.18

−0.17096 −0.057679 −0.67112 −0.81886

108.13 45.797 70.555 92.663

84.510 29.882 44.330 56.476

26.188 1.9183 1.0590 0.73435

0.038186 0.52130 0.94433 1.3618

−2.3398 12.368 24.211 37.246

−1.9580 17.581 33.654 50.864

0.091882 0.17355 0.19296 0.20583

0.028004 0.021965 0.025006 0.026863

0.056646 0.030926 0.033447 0.035209

734.22 523.87 682.37 807.57

−0.25658 −0.12928 −0.68394 −0.82170

115.90 46.995 71.130 93.033

93.648 30.284 44.493 56.570

27.434 21.868 18.335

0.036451 0.045729 0.054541

10.778 23.840 37.584

29.003 46.705 64.855

0.13791 0.15935 0.17295

0.026493 0.027586 0.028647

0.035336 0.035848 0.036665

1574.4 1501.4 1506.6

−0.70223 −0.72394 −0.73166

177.40 149.37 147.47

103.10 79.801 79.226

34.270 29.362 25.920

0.029180 0.034057 0.038580

11.714 25.065 38.999

40.894 59.122 77.579

0.13093 0.15303 0.16685

0.029169 0.029373 0.029977

0.036905 0.036577 0.037212

2107.2 1985.1 1942.0

−0.61888 −0.65271 −0.65439

278.97 232.49 214.36

208.00 129.86 110.35

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., Lemmon, E. W., Jacobsen, R. T., Wagner, W., and Yokozeki, A., “A Reference Quality Thermodynamic Property Formulation for Nitrogen,” J. Phys. Chem. Ref. Data 29(6):1361–1433, 2000. See also Int. J. Thermophys. 14(4):1121–1132, 1998. The source for viscosity is Lemmon, E. W., and Jacobsen, R. T., “Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air,” Int. J. Thermophys. 25:21–69, 2004. The source for thermal conductivity is Lemmon, E. W., and Jacobsen, R. T., “Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air,” Int. J. Thermophys. 25:21–69, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density of the equation of state is 0.02% from the triple point up to temperatures of 523 K and pressures up to 12 MPa and from temperatures of 240 to 523 K at pressures less than 30 MPa. In the range from 270 to 350 K at pressures less than 12 MPa, the uncertainty in density is 0.01%. The uncertainty at very high pressures (>1 GPa) is 0.6% in density. The uncertainty in pressure in the critical region is estimated to be 0.02%. In the gaseous and supercritical region, the speed of sound can be calculated with a typical uncertainty of 0.005% to 0.1%. At liquid states and at high pressures, the uncertainty increases to 0.5% to 1.5%. For pressures up to 30 MPa, the estimated uncertainty for heat capacities ranges from 0.3% at gaseous and gaslike supercritical states up to 0.8% at liquid states and at certain gaseous and supercritical states at low temperatures. The uncertainty is 2% for pressures up to 200 MPa and larger at higher pressures. The estimated uncertainties of vapor pressure, saturated-liquid density, and saturated-vapor density are in general 0.02% for each property. The formulation yields a reasonable extrapolation behavior up to the limits of chemical stability of nitrogen. For viscosity, the uncertainty is 0.5% in dilute gas. Away from the dilute gas (pressures greater than 1 MPa and in the liquid), the uncertainties are as low as 1% between 270 and 300 K at pressures less than 100 MPa, and increase outside that range. The uncertainties are around 2% at temperatures of 180 K and higher. Below this and away from the critical region, the uncertainties steadily increase to around 5% at the triple points of the fluids. The uncertainties in the critical region are higher. For thermal conductivity, the uncertainty for the dilute gas is 2% with increasing uncertainties near the triple point. For the nondilute gas, the uncertainty is 2% for temperatures greater than 150 K. The uncertainty is 3% at temperatures less than the critical point and 5% in the critical region, except for states very near the critical point.

10. 8.

0

Nitrogen

0.

100

0.

−100

(R-728) reference state: h = 0.0 kJ/kg, s = 0.00 kJ/(kg·K) for ideal gas at 0 K

50

60

−200 20.

0.

200 0.

.

400

150.

200

30

40

300

500 3

/m ρ = 80. kg

100.

700.

800.

6. 4.

600 20.

60.

10. 8.

40.

6.

30. c.p.

4.

120

110

100

90

2.

T = 80 K

70

20. 15.

2.

10.

110

100

80

0.04

200

180

160

140

4.0

0.6

500

480

T = 460 K

440

420

400

380

360

320

300

280

260

1. 0.8

0.4

2.0

1.5

0.2

1.0 0.80

6.

60

0 6.4

0 6.2

0 6.0

0 5.8

0.06

6.0

3.0

)

2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 5.20 5.40 5.60

0.1 0.08

120

or

0.9

saturated vap

0.8

0.7

0.6

0.5

x = 0.4

0.3

0.2

0.1

liquid rated satu

0.2

90

240

0.4

220

0.6

340

T = 100 K

freezing line

Pressure (MPa)

8.0

1. 0.8

80

J/

00

7.

6.

·K (kg

s=

k .20

7

0 7.4

70

0.60

0.1 0.08

0.40

0.06

0.30 0

7.6

0.15

0.02

0.04

0.20 0

7.8

0.02

0.10

triple point = 63.15 K

0.01 −200

solid + vapor region

−100

0

100

200

300

400

500

0.01 600

Enthalpy (kJ/kg) 2-309

FIG. 2-13 Pressure-enthalpy diagram for nitrogen. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Span, R., Lemmon, E. W., Jacobsen, R. T., Wagner, W., and Yokozeki, A,. “A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa.,” J. Phys. Chem. Ref. Data 29:1361–1433, 2000.

2-310

PHYSICAL AND CHEMICAL DATA TABLE 2-240

Saturated Nitrogen Tetroxide vg, m3/kg

Mf

Mg

299.32 309.57 326.66 337.43 345.45

0.000 0.000 0.000 0.000 0.000

694 711 733 749 762

0.2996 0.1630 0.0876 0.0608 0.0469

91.857 91.886 91.766 91.625 91.488

79.157 76.503 73.538 71.748 70.480

10 15 20 30 40

351.88 364.09 373.17 386.57 396.52

0.000 0.000 0.000 0.000 0.000

774 800 822 863 903

0.0382 0.0262 0.0199 0.0133 0.0098

91.346 90.979 90.601 89.823 89.018

69.483 67.742 66.547 64.997 64.099

50 60 80 100

404.50 411.20 422.07 430.76

0.000 0.000 0.001 0.001

945 993 129 577

0.00761 0.00607 0.00394 0.00209

88.191 87.344 85.602 83.817

63.532 63.181 62.959 63.366

Pressure, bar 1.0133 2 4 6 8

Temperature, K

vf, m3/kg

Condensed from McCarty, R. D., H-U. Steurer, et al., NBS IR 86 - 3054, 1986 (106 pp.). M = mol wt for the reaction N2O4 → 2NO2 → 2NO + O2. No derived thermodynamic functions were tabulated due to unduly large differences in literature values, but 92 references are given.

TABLE 2-241

Thermodynamic Properties of Nitrogen Trifluoride

Temperature K

Pressure MPa

90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 155.00 160.00 165.00 170.00 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 234.00

0.00015612 0.00040547 0.00094625 0.0020164 0.0039755 0.0073310 0.012759 0.021119 0.033457 0.051008 0.075184 0.10757 0.14989 0.20401 0.27193 0.35573 0.45760 0.57979 0.72462 0.89450 1.0918 1.3192 1.5791 1.8742 2.2073 2.5815 3.0003 3.4682 3.9918 4.4607

90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 155.00 160.00 165.00 170.00 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00

0.00015612 0.00040547 0.00094625 0.0020164 0.0039755 0.0073310 0.012759 0.021119 0.033457 0.051008 0.075184 0.10757 0.14989 0.20401 0.27193 0.35573 0.45760 0.57979 0.72462 0.89450 1.0918 1.3192 1.5791 1.8742 2.2073 2.5815

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.033490 −0.029648 −0.026000 −0.022531 −0.019227 −0.016074 −0.013057 −0.010163 −0.0073784 −0.0046923 −0.0020929 0.00042986 0.0028858 0.0052840 0.0076328 0.0099402 0.012214 0.014461 0.016690 0.018907 0.021121 0.023341 0.025577 0.027843 0.030157 0.032548 0.035068 0.037830 0.041203 0.049280

0.045791 0.045221 0.044514 0.043768 0.043048 0.042394 0.041825 0.041346 0.040957 0.040653 0.040428 0.040274 0.040186 0.040159 0.040187 0.040269 0.040404 0.040591 0.040834 0.041136 0.041505 0.041953 0.042494 0.043150 0.043953 0.044950 0.046220 0.047920 0.050506

0.070957 0.071116 0.071131 0.071067 0.070980 0.070913 0.070898 0.070958 0.071113 0.071379 0.071771 0.072303 0.072988 0.073841 0.074876 0.076114 0.077577 0.079300 0.081326 0.083719 0.086571 0.090023 0.094298 0.099779 0.10717 0.11795 0.13575 0.17267 0.30655

1081.4 1049.3 1018.1 986.86 955.57 924.52 894.02 864.29 835.40 807.28 779.79 752.68 725.72 698.67 671.28 643.36 614.73 585.25 554.78 523.23 490.50 456.52 421.22 384.52 346.33 306.49 264.69 220.19 171.03 0

−0.44772 −0.44233 −0.43718 −0.43184 −0.42594 −0.41921 −0.41143 −0.40241 −0.39193 −0.37972 −0.36549 −0.34889 −0.32951 −0.30688 −0.28044 −0.24950 −0.21319 −0.17042 −0.11971 −0.059078 0.014249 0.10427 0.21696 0.36170 0.55413 0.82239 1.2231 1.8903 3.2539 8.0982

0.11824 0.11213 0.10681 0.10217 0.098099 0.094513 0.091341 0.088521 0.086004 0.083746 0.081709 0.079863 0.078179 0.076634 0.075207 0.073880 0.072636 0.071459 0.070337 0.069254 0.068196 0.067150 0.066097 0.065016 0.063879 0.062642

0.025384 0.025553 0.025759 0.026007 0.026301 0.026641 0.027032 0.027474 0.027968 0.028515 0.029116 0.029770 0.030477 0.031236 0.032049 0.032913 0.033831 0.034801 0.035825 0.036905 0.038044 0.039245 0.040516 0.041867 0.043315 0.044890

0.033709 0.033889 0.034117 0.034400 0.034747 0.035166 0.035666 0.036255 0.036942 0.037735 0.038644 0.039680 0.040854 0.042183 0.043684 0.045384 0.047313 0.049515 0.052050 0.055001 0.058491 0.062704 0.067933 0.074677 0.083847 0.097325

118.26 121.38 124.38 127.25 129.99 132.58 135.01 137.27 139.35 141.23 142.91 144.37 145.59 146.58 147.30 147.77 147.95 147.85 147.44 146.73 145.68 144.29 142.53 140.38 137.79 134.71

Saturated Properties 25.006 24.719 24.427 24.132 23.831 23.526 23.217 22.902 22.583 22.259 21.931 21.596 21.256 20.908 20.552 20.186 19.809 19.418 19.011 18.584 18.135 17.659 17.148 16.595 15.987 15.305 14.514 13.541 12.148 7.9200

2-311

0.00020869 0.00051363 0.0011394 0.0023144 0.0043618 0.0077094 0.012896 0.020570 0.031490 0.046520 0.066627 0.092885 0.12648 0.16872 0.22107 0.28514 0.36279 0.45613 0.56766 0.70033 0.85776 1.0445 1.2665 1.5317 1.8518 2.2450

0.039990 0.040455 0.040938 0.041439 0.041962 0.042506 0.043073 0.043664 0.044281 0.044925 0.045599 0.046304 0.047046 0.047829 0.048658 0.049539 0.050483 0.051499 0.052602 0.053808 0.055141 0.056630 0.058316 0.060260 0.062552 0.065340 0.068899 0.073850 0.082320 0.12626 4791.8 1946.9 877.68 432.07 229.26 129.71 77.545 48.615 31.756 21.496 15.009 10.766 7.9063 5.9269 4.5236 3.5071 2.7564 2.1923 1.7616 1.4279 1.1658 0.95740 0.78960 0.65287 0.54001 0.44543

−3.8551 −3.4999 −3.1442 −2.7887 −2.4336 −2.0790 −1.7245 −1.3701 −1.0151 −0.65928 −0.30195 0.057452 0.41959 0.78514 1.1549 1.5295 1.9100 2.2972 2.6920 3.0957 3.5095 3.9350 4.3742 4.8297 5.3052 5.8067 6.3449 6.9434 7.6794 9.3687

−3.8551 −3.4999 −3.1442 −2.7886 −2.4335 −2.0786 −1.7240 −1.3691 −1.0137 −0.65699 −0.29853 0.062433 0.42664 0.79490 1.1681 1.5472 1.9331 2.3270 2.7301 3.1438 3.5697 4.0097 4.4663 4.9426 5.4433 5.9754 6.5516 7.1995 8.0080 9.9320

9.0527 9.1794 9.3064 9.4337 9.5609 9.6879 9.8143 9.9397 10.064 10.186 10.305 10.422 10.536 10.645 10.750 10.850 10.943 11.031 11.110 11.181 11.241 11.289 11.323 11.340 11.333 11.296

9.8008 9.9688 10.137 10.305 10.472 10.639 10.804 10.966 11.126 11.282 11.434 11.580 11.721 11.854 11.980 12.097 12.205 12.302 12.386 12.458 12.514 12.552 12.570 12.563 12.525 12.446

125.93 109.58 96.037 84.718 75.177 67.076 60.154 54.206 49.068 44.607 40.717 37.310 34.315 31.671 29.330 27.250 25.397 23.743 22.263 20.936 19.746 18.678 17.718 16.854 16.072 15.354

2-312

TABLE 2-241

Thermodynamic Properties of Nitrogen Trifluoride (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

220.00 225.00 230.00 234.00

3.0003 3.4682 3.9918 4.4607

2.7432 3.4146 4.4712 7.9200

0.36453 0.29286 0.22365 0.12626

11.213 11.055 10.725 9.3687

100.00 143.95

0.10000 0.10000

0.040934 0.046153

−3.1452 −0.018242

−3.1411 −0.013626

143.95 200.00 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000 0.10000

10.398 12.215 16.237 21.208 26.869

11.550 13.856 18.723 24.530 31.025

100.00 187.76

1.0000 1.0000

0.040901 0.054527

−3.1540 3.3230

187.76 200.00 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000 1.0000

1.2755 1.4245 2.4068 3.2887 4.1476

100.00 200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000

24.532 17.659 2.4334 1.5833 1.2132

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.046643 0.048690 0.051363

0.11971 0.16599 0.32814

131.07 126.78 121.80 0

14.662 13.900 12.709 8.0982

0.044502 0.040301

0.071124 0.072179

1018.9 758.35

−0.43728 −0.35259

0.080236 0.093729 0.11332 0.12997 0.14444

0.029628 0.034723 0.045284 0.053558 0.059251

0.039451 0.043508 0.053741 0.061943 0.067609

144.08 169.06 203.45 232.46 258.38

37.990 15.650 5.5743 2.8668 1.7661

−3.1131 3.3775

−0.026098 0.020131

0.044398 0.041331

0.071055 0.085231

1026.3 505.29

11.215 11.738 16.042 21.082 26.771

12.491 13.162 18.449 24.371 30.918

0.068667 0.072132 0.093526 0.11051 0.12510

0.037527 0.037614 0.045687 0.053632 0.059244

0.056853 0.053386 0.055540 0.062664 0.067998

146.19 154.62 199.37 231.33 258.62

0.040763 0.056627 0.41095 0.63161 0.82425

−3.1921 4.1435 15.064 20.512 26.341

−2.9883 4.4266 17.119 23.670 30.462

−0.026483 0.024397 0.076882 0.095734 0.11087

0.043955 0.041665 0.047559 0.053907 0.059206

0.070771 0.087072 0.067409 0.066129 0.069735

1057.9 477.43 183.97 228.50 260.85

−0.44174 0.053822 5.3277 2.5958 1.5696

Saturated Properties 12.307 12.071 11.618 9.9320

0.061229 0.059479 0.056898 0.049280

Single-Phase Properties 24.430 21.667 0.086803 0.060926 0.040232 0.030102 0.024060 24.449 18.339 0.78400 0.70199 0.41548 0.30407 0.24110

11.520 16.413 24.856 33.220 41.562

−0.026009 −0.000094174

−0.43816 −0.020346 20.263 16.645 5.5586 2.8289 1.7349

100.00 200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000 10.000

24.630 18.214 5.9501 3.2770 2.4256

0.040600 0.054902 0.16806 0.30515 0.41227

−3.2377 3.8853 13.534 19.798 25.826

−2.8317 4.4343 15.214 22.849 29.949

−0.026951 0.023042 0.066202 0.088300 0.10414

0.043446 0.041251 0.049590 0.054171 0.059167

0.070455 0.081506 0.093780 0.070777 0.071809

1094.6 536.12 185.51 231.76 266.86

−0.44552 −0.078962 3.9475 2.1918 1.3296

100.00 200.00 300.00 400.00 500.00

25.000 25.000 25.000 25.000 25.000

24.895 19.339 12.709 7.8701 5.7388

0.040168 0.051709 0.078686 0.12706 0.17425

−3.3631 3.3539 10.815 17.964 24.506

−2.3589 4.6466 12.782 21.141 28.863

−0.028280 0.020118 0.052928 0.077032 0.094275

0.042128 0.041959 0.049110 0.054897 0.059344

0.069766 0.074322 0.086464 0.079284 0.076236

1191.1 642.55 338.34 285.01 303.16

−0.45304 −0.26070 0.50765 0.90188 0.66132

100.00 200.00 300.00 400.00 500.00

50.000 50.000 50.000 50.000 50.000

25.271 20.560 15.995 12.237 9.6861

0.039572 0.048639 0.062520 0.081722 0.10324

−3.5465 2.8054 9.4909 16.326 23.086

−1.5679 5.2373 12.617 20.412 28.248

−0.030335 0.016816 0.046643 0.069057 0.086544

0.040341 0.044958 0.052084 0.057087 0.060630

0.069490 0.070043 0.076746 0.078504 0.078201

1322.6 734.96 484.09 397.17 380.22

−0.45498 −0.38097 −0.15530 0.026532 0.066751

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Younglove, B. A., “Thermophysical Properties of Fluids. I. Argon, Ethylene, Parahydrogen, Nitrogen, Nitrogen Trifluoride, and Oxygen,” J. Phys. Chem. Ref. Data Suppl. 1, 11: 1–11, 1982. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.25% in the liquid phase and 0.3% in the vapor and supercritical regions. The uncertainty in speed of sound and heat capacity is 5%.

TABLE 2-242

Thermodynamic Properties of Nitrous Oxide

Temperature K

Pressure MPa

Density mol/dm3

182.33 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 309.52

0.087837 0.10325 0.13782 0.18085 0.23367 0.29767 0.37431 0.46509 0.57160 0.69544 0.83828 1.0018 1.1878 1.3979 1.6341 1.8982 2.1920 2.5177 2.8772 3.2728 3.7068 4.1820 4.7012 5.2681 5.8874 6.5663 7.2447

28.113 27.935 27.599 27.257 26.910 26.557 26.197 25.829 25.453 25.069 24.674 24.268 23.849 23.417 22.968 22.502 22.015 21.505 20.966 20.393 19.777 19.106 18.361 17.505 16.466 15.022 10.270

182.33 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 309.52

0.087837 0.10325 0.13782 0.18085 0.23367 0.29767 0.37431 0.46509 0.57160 0.69544 0.83828 1.0018 1.1878 1.3979 1.6341 1.8982 2.1920 2.5177 2.8772 3.2728 3.7068 4.1820 4.7012 5.2681 5.8874 6.5663 7.2447

0.059336 0.068951 0.090180 0.11615 0.14754 0.18508 0.22958 0.28190 0.34298 0.41387 0.49575 0.58993 0.69791 0.82145 0.96261 1.1239 1.3082 1.5196 1.7627 2.0443 2.3734 2.7632 3.2349 3.8252 4.6102 5.8106 10.270

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.035570 0.035797 0.036234 0.036688 0.037161 0.037656 0.038173 0.038716 0.039288 0.039891 0.040529 0.041207 0.041930 0.042705 0.043538 0.044440 0.045423 0.046501 0.047697 0.049037 0.050564 0.052339 0.054464 0.057125 0.060732 0.066569 0.097371

−0.18142 0.020258 0.39840 0.77745 1.1577 1.5396 1.9234 2.3094 2.6982 3.0899 3.4852 3.8844 4.2881 4.6969 5.1115 5.5326 5.9614 6.3990 6.8471 7.3078 7.7841 8.2804 8.8039 9.3670 9.9960 10.768 12.745

−0.17830 0.023954 0.40340 0.78408 1.1664 1.5508 1.9377 2.3274 2.7206 3.1177 3.5192 3.9257 4.3379 4.7566 5.1826 5.6170 6.0609 6.5161 6.9843 7.4682 7.9715 8.4993 9.0600 9.6680 10.354 11.205 13.450

14.931 14.984 15.082 15.176 15.268 15.356 15.440 15.521 15.596 15.666 15.730 15.789 15.840 15.883 15.917 15.941 15.953 15.952 15.935 15.899 15.839 15.749 15.618 15.429 15.145 14.661 12.745

16.411 16.481 16.610 16.733 16.852 16.965 17.071 17.170 17.262 17.346 17.421 17.487 17.541 17.584 17.614 17.630 17.629 17.609 17.567 17.500 17.401 17.262 17.071 16.806 16.422 15.791 13.450

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.00096900 0.00012922 0.0021464 0.0041159 0.0060420 0.0079284 0.0097789 0.011597 0.013385 0.015148 0.016888 0.018607 0.020311 0.022000 0.023681 0.025355 0.027028 0.028704 0.030391 0.032097 0.033833 0.035615 0.037468 0.039439 0.041620 0.044293 0.051414

0.042295 0.042134 0.041852 0.041596 0.041365 0.041155 0.040967 0.040797 0.040645 0.040510 0.040392 0.040290 0.040203 0.040132 0.040078 0.040041 0.040025 0.040030 0.040064 0.040134 0.040251 0.040437 0.040727 0.041190 0.041979 0.043564

0.075603 0.075669 0.075851 0.076111 0.076452 0.076875 0.077388 0.077995 0.078706 0.079533 0.080489 0.081595 0.082875 0.084362 0.086100 0.088150 0.090597 0.093564 0.097233 0.10189 0.10803 0.11651 0.12910 0.14997 0.19222 0.32907

1149.6 1132.4 1100.2 1068.2 1036.3 1004.5 972.70 940.94 909.16 877.35 845.46 813.47 781.31 748.95 716.31 683.32 649.86 615.81 580.97 545.11 507.90 468.86 427.31 382.17 331.57 271.24 0

−0.26723 −0.26287 −0.25377 −0.24334 −0.23146 −0.21798 −0.20272 −0.18545 −0.16592 −0.14379 −0.11869 −0.090113 −0.057462 −0.019968 0.023355 0.073794 0.13307 0.20351 0.28843 0.39261 0.52333 0.69225 0.91943 1.2430 1.7475 2.6810 6.2530

0.090017 0.089089 0.087443 0.085908 0.084469 0.083117 0.081842 0.080634 0.079485 0.078387 0.077332 0.076314 0.075325 0.074359 0.073408 0.072464 0.071519 0.070564 0.069588 0.068575 0.067508 0.066361 0.065093 0.063636 0.061848 0.059328 0.051414

0.025545 0.025844 0.026427 0.027038 0.027674 0.028331 0.029006 0.029694 0.030395 0.031105 0.031824 0.032552 0.033289 0.034037 0.034798 0.035577 0.036377 0.037205 0.038072 0.038988 0.039971 0.041046 0.042252 0.043653 0.045376 0.047733

0.034993 0.035439 0.036334 0.037309 0.038365 0.039506 0.040736 0.042062 0.043494 0.045048 0.046741 0.048600 0.050660 0.052967 0.055585 0.058600 0.062136 0.066370 0.071572 0.078164 0.086856 0.098934 0.11700 0.14723 0.20883 0.40691

211.93 212.91 214.62 216.14 217.48 218.62 219.56 220.30 220.82 221.13 221.20 221.04 220.64 219.98 219.07 217.88 216.41 214.64 212.56 210.15 207.39 204.24 200.67 196.63 192.02 186.66 0

Saturated Properties

2-313

16.853 14.503 11.089 8.6098 6.7780 5.4030 4.3558 3.5474 2.9156 2.4162 2.0171 1.6951 1.4329 1.2174 1.0388 0.88979 0.76438 0.65808 0.56730 0.48917 0.42135 0.36189 0.30912 0.26142 0.21691 0.17210 0.097371

44.590 42.793 39.645 36.768 34.146 31.765 29.606 27.653 25.888 24.295 22.858 21.560 20.387 19.325 18.361 17.483 16.679 15.939 15.251 14.603 13.981 13.366 12.734 12.043 11.210 10.014 6.2530

2-314 TABLE 2-242

Thermodynamic Properties of Nitrous Oxide (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

200.00 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

0.061319 0.040304 0.030129 0.024075

200.00 234.95

1.0000 1.0000

234.95 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

200.00 292.69

5.0000 5.0000

292.69 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.092110 0.10698 0.11873 0.12862

0.025918 0.030437 0.034363 0.037563

0.035025 0.038943 0.042764 0.045926

221.52 267.85 306.03 339.54

0.0059677 0.018590

0.041381 0.040291

0.076317 0.081583

1040.5 813.80

0.076324 0.086931 0.099190 0.10926

0.032545 0.031268 0.034610 0.037680

0.048580 0.041833 0.043833 0.046488

221.04 260.01 303.02 338.49

21.573 10.924 5.5924 3.3701

0.0055897 0.038510

0.041467 0.040948

0.075665 0.13883

1061.6 403.56

−0.24314 1.0778

Single-Phase Properties

26.940 24.272 0.58888 0.42412 0.30696 0.24260 27.091 17.918

16.308 24.811 33.190 41.537 0.037120 0.041200 1.6981 2.3578 3.2578 4.1220 0.036913 0.055811

15.382 18.214 21.467 25.072 1.1429 3.8803 15.788 17.944 21.310 24.963 1.0679 9.1008

17.013 20.695 24.786 29.226 1.1800 3.9215 17.486 20.302 24.568 29.085 1.2525 9.3799

31.472 10.733 5.6159 3.4059 −0.23345 −0.090429

3.5344 3.1047 1.6773 1.2538

0.28294 0.32209 0.59621 0.79755

15.525 16.114 20.558 24.469

16.940 17.725 23.539 28.457

0.064339 0.066990 0.083920 0.094897

0.042976 0.039518 0.035787 0.038199

0.13106 0.091593 0.049876 0.049186

198.56 212.60 290.66 334.89

12.373 11.486 5.4199 3.1966

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

27.269 18.407 3.7981 2.6054

0.036672 0.054327 0.26329 0.38381

0.97935 9.1891 19.470 23.835

1.3461 9.7324 22.102 27.673

0.0051379 0.038771 0.075432 0.087896

0.041579 0.040304 0.037387 0.038831

0.074951 0.11033 0.061591 0.052973

1086.5 462.21 279.96 333.21

−0.25384 0.72534 4.9171 2.9341

200.00 300.00 400.00 500.00

25.000 25.000 25.000 25.000

27.751 20.902 11.546 6.9433

0.036035 0.047843 0.086610 0.14402

0.74199 8.0747 16.070 21.962

1.6429 9.2708 18.235 25.563

0.0038961 0.034710 0.060433 0.076883

0.041933 0.039937 0.040060 0.040346

0.073298 0.082004 0.087395 0.063761

1154.2 654.64 352.41 357.12

−0.27902 0.10041 1.7332 1.8422

300.00 400.00 500.00

50.000 50.000 50.000

22.869 17.051 12.463

0.043727 0.058649 0.080236

7.1665 13.791 19.764

9.3528 16.723 23.776

0.031190 0.052389 0.068153

0.040453 0.040588 0.041531

0.072893 0.073379 0.067192

828.55 568.05 477.36

−0.14288 0.23352 0.52378

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.1% in density in the liquid and vapor phases between 220 and 300 K, 0.25% at temperatures above 300 K and at temperatures below 220 K, and 0.5% in the critical region, except very close to the critical point. The uncertainty in vapor pressure is 0.2%, that for heat capacities is 3%, and that for the speed of sound in the vapor phase is 0.05% above 220 K. The uncertainty in the liquid phase is not known but is estimated to be within 5%.

THERMODYNAMIC PROPERTIES

FIG. 2-14 Mollier diagram for nitrous oxide. (Fig. 9, Univ. Texas Rep., Cont. DAI-23072-ORD-685, June 1, 1956, by Couch and Kobe. Reproduced by permission.) Some irregularity in the compressibility factors from 80 to 160 atm, 50 to 100 °C exists (Couch, private communication, 1967). See Couch et al., J. Chem. Eng. Data, 6, (1961) for PVT data.

2-315

2-316 TABLE 2-243 Temperature K

Thermodynamic Properties of Nonane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−60.675 −60.599 −55.459 −50.195 −44.779 −39.190 −33.413 −27.437 −21.252 −14.851 −8.2313 −1.3870 5.6854 12.991 20.537 28.335 36.408 44.798 53.602 73.819

−60.675 −60.599 −55.459 −50.195 −44.779 −39.190 −33.413 −27.436 −21.249 −14.846 −8.2208 −1.3679 5.7183 13.045 20.622 28.464 36.600 45.080 54.015 75.080

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.19174 −0.19139 −0.16904 −0.14797 −0.12791 −0.10863 −0.089996 −0.071883 −0.054209 −0.036909 −0.019932 −0.0032367 0.013213 0.029452 0.045513 0.061438 0.077283 0.093141 0.10920 0.144498

0.19748 0.19757 0.20402 0.21164 0.22016 0.22932 0.23894 0.24886 0.25894 0.26908 0.27921 0.28927 0.29920 0.30899 0.31863 0.32814 0.33756 0.34703 0.35685 —

0.25433 0.25440 0.25984 0.26681 0.27497 0.28402 0.29376 0.30400 0.31464 0.32557 0.33677 0.34824 0.36002 0.37228 0.38530 0.39965 0.41650 0.43881 0.47629 —

1544.5 1543.0 1449.8 1361.7 1277.9 1197.9 1121.1 1046.8 974.60 903.98 834.45 765.54 696.76 627.57 557.35 485.36 410.60 331.60 245.76 —

−0.50071 −0.50053 −0.48613 −0.46790 −0.44671 −0.42315 −0.39747 −0.36960 −0.33918 −0.30546 −0.26723 −0.22257 −0.16849 −0.10004 −0.0087139 0.12153 0.32444 0.68436 1.4810 31.599

153.02 152.90 145.29 138.35 131.98 126.11 120.68 115.63 110.92 106.50 102.33 98.374 94.594 90.947 87.384 83.848 80.262 76.518 72.450 —

4037.3 3978.4 1914.0 1194.3 842.44 638.13 505.83 413.37 344.96 292.08 249.76 214.92 185.56 160.30 138.18 118.43 100.46 83.674 67.309 —

0.047770 0.047676 0.043246 0.041911 0.042975 0.045920 0.050349 0.055950 0.062471 0.069707 0.077488 0.085673 0.094142 0.10279 0.11150 0.12018 0.12868 0.13679 0.14413 0.144498

0.16161 0.16174 0.17103 0.18119 0.19199 0.20323 0.21478 0.22652 0.23839 0.25032 0.26226 0.27416 0.28601 0.29779 0.30950 0.32118 0.33293 0.34492 0.35757 —

0.16993 0.17006 0.17935 0.18951 0.20033 0.21161 0.22326 0.23519 0.24737 0.25979 0.27245 0.28541 0.29878 0.31276 0.32778 0.34467 0.36530 0.39454 0.44973 —

122.37 122.45 127.73 132.76 137.56 142.11 146.36 150.21 153.55 156.23 158.10 158.99 158.70 157.00 153.58 148.05 139.82 127.99 111.07 —

Saturated Properties 219.70 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 594.55

4.4449E-07 4.6229E-07 4.8933E-06 3.3858E-05 0.00016888 0.00065182 0.0020515 0.0054791 0.012806 0.026830 0.051367 0.091247 0.15227 0.24110 0.36527 0.53313 0.75413 1.0392 1.4023 2.2820

6.0520 6.0501 5.9249 5.8013 5.6785 5.5557 5.4321 5.3071 5.1799 5.0496 4.9154 4.7760 4.6299 4.4754 4.3096 4.1286 3.9259 3.6898 3.3948 1.8100

219.70 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 594.55

4.4449E-07 4.6229E-07 4.8933E-06 3.3858E-05 0.00016888 0.00065182 0.0020515 0.0054791 0.012806 0.026830 0.051367 0.091247 0.15227 0.24110 0.36527 0.53313 0.75413 1.0392 1.4023 2.2820

2.4333E-07 2.5273E-07 2.4522E-06 1.5664E-05 7.2578E-05 0.00026168 0.00077364 0.0019519 0.0043356 0.0086887 0.016023 0.027625 0.045117 0.070581 0.10681 0.15781 0.22994 0.33476 0.49815 1.8100

0.16524 0.16529 0.16878 0.17237 0.17610 0.18000 0.18409 0.18843 0.19305 0.19803 0.20344 0.20938 0.21599 0.22345 0.23204 0.24221 0.25472 0.27102 0.29457 0.55249 4,109,700. 3,956,800. 407,790. 63,840. 13,778. 3,821.5 1,292.6 512.31 230.65 115.09 62.412 36.200 22.165 14.168 9.3627 6.3369 4.3490 2.9872 2.0074 0.55249

−9.8808 −9.8323 −6.5064 −2.9862 0.74180 4.6851 8.8457 13.220 17.802 22.580 27.541 32.671 37.952 43.363 48.876 54.456 60.045 65.548 70.762 73.819

−8.0542 −8.0032 −4.5110 −0.82471 3.0688 7.1760 11.497 16.027 20.756 25.668 30.747 35.974 41.327 46.779 52.296 57.834 63.325 68.652 73.577 75.080

363.94 361.95 254.36 183.36 135.32 102.10 78.695 61.932 49.767 40.846 34.258 29.384 25.803 23.233 21.504 20.545 20.405 21.346 24.126 31.599

6.2016 6.2133 7.0659 8.0682 9.2127 10.493 11.902 13.436 15.091 16.864 18.757 20.773 22.918 25.205 27.655 30.306 33.233 36.613 40.989 —

4.0513 4.0567 4.4125 4.7673 5.1206 5.4722 5.8211 6.1663 6.5069 6.8425 7.1737 7.5025 7.8332 8.1731 8.5341 8.9361 9.4142 10.039 10.990 —

Single-Phase Properties −52.849 −39.196 −24.378 −8.2368 −0.19510

−52.832 −39.178 −24.359 −8.2164 −0.17406

−0.15838 −0.10865 −0.063016 −0.019946 −0.00041018

0.20771 0.22933 0.25389 0.27921 0.29097

0.26315 0.28401 0.30925 0.33675 0.35022

1405.7 1198.6 1011.2 834.97 753.80

33.563 41.125 56.284

36.880 44.697 60.317

0.087103 0.10500 0.13790

0.27619 0.28978 0.31440

0.28766 0.30044 0.32419

159.03 165.87 177.26

0.17042 0.17978 0.19037 0.20292 0.21878 0.24114

−52.889 −39.250 −24.450 −8.3375 9.1743 28.203

−52.719 −39.070 −24.260 −8.1346 9.3930 28.444

−0.15854 −0.10883 −0.063223 −0.020199 0.021057 0.061172

0.20778 0.22940 0.25395 0.27927 0.30414 0.32810

0.26309 0.28387 0.30900 0.33627 0.36516 0.39811

1410.5 1204.4 1018.5 844.45 674.11 496.26

−0.47809 −0.42451 −0.35745 −0.27280 −0.14850 0.096651

142.11 126.55 113.76 102.96 93.482 84.482

1501.9 645.13 380.90 252.79 174.91 120.06

5.8875 5.5884 5.2884 4.9781 4.6466 4.2772

0.16985 0.17894 0.18909 0.20088 0.21521 0.23380

−53.064 −39.480 −24.758 −8.7590 8.5689 27.246

−52.214 −38.585 −23.813 −7.7546 9.6450 28.415

−0.15925 −0.10961 −0.064113 −0.021267 0.019689 0.059215

0.20809 0.22969 0.25422 0.27950 0.30429 0.32797

0.26282 0.28333 0.30802 0.33447 0.36159 0.38941

1431.5 1229.8 1049.7 884.19 727.45 574.69

−0.48055 −0.42952 −0.36725 −0.29292 −0.19528 −0.043276

143.55 128.25 115.82 105.52 96.800 89.174

1587.0 673.50 397.43 265.35 186.34 132.45

5.9111 5.6194 5.3297 5.0347 4.7272 4.3995

0.16917 0.17796 0.18763 0.19862 0.21154 0.22730

−53.273 −39.753 −25.116 −9.2354 7.9189 26.321

−51.582 −37.973 −23.240 −7.2492 10.034 28.594

−0.16011 −0.11054 −0.065168 −0.022500 0.018184 0.057270

0.20848 0.23005 0.25456 0.27981 0.30452 0.32803

0.26253 0.28275 0.30700 0.33273 0.35854 0.38371

1456.9 1260.0 1086.3 929.33 784.82 650.34

−0.48327 −0.43494 −0.37743 −0.31235 −0.23486 −0.13338

145.29 130.31 118.27 108.48 100.47 93.894

1699.9 709.84 418.13 280.67 199.70 145.66

250.00 300.00 350.00 400.00 423.42

0.10000 0.10000 0.10000 0.10000 0.10000

5.8635 5.5564 5.2447 4.9160 4.7515

423.42 450.00 500.00

0.10000 0.10000 0.10000

0.030145 0.027993 0.024798

250.00 300.00 350.00 400.00 450.00 500.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

5.8679 5.5624 5.2530 4.9280 4.5708 4.1469

250.00 300.00 350.00 400.00 450.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

250.00 300.00 350.00 400.00 450.00 500.00

10.000 10.000 10.000 10.000 10.000 10.000

0.17055 0.17997 0.19067 0.20342 0.21046 33.173 35.724 40.326

−0.47750 −0.42329 −0.35500 −0.26752 −0.21411 28.690 22.301 14.792

141.78 126.15 113.28 102.36 97.717 21.130 23.838 29.272

1483.4 638.82 377.18 249.91 209.56 7.5587 8.0448 8.9428

300.00 350.00 400.00 450.00 500.00

100.00 100.00 100.00 100.00 100.00

6.0143 5.8100 5.6178 5.4358 5.2628

0.16627 0.17212 0.17800 0.18396 0.19001

−43.037 −29.086 −13.998 2.2421 19.589

−26.410 −11.875 3.8029 20.639 38.590

−0.12342 −0.078654 −0.036820 0.0028129 0.040621

0.23562 0.25980 0.28476 0.30915 0.33224

0.27987 0.30194 0.32520 0.34808 0.36974

1662.5 1534.1 1426.4 1335.5 1258.2

−0.46881 −0.43419 −0.40174 −0.37331 −0.34915

158.86 149.98 143.57 139.19 136.49

1638.6 851.44 557.39 406.92 316.22

500.00

300.00

5.9601

0.16778

15.017

65.352

0.023367

0.33895

0.37096

1885.2

−0.36365

184.31

733.45

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. The source for viscosity is Huber, M. L., Laesecke, A., and Xiang, H. W., “Viscosity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-Decane,” Fluid Phase Equilibria 224:263–270, 2004. The source for thermal conductivity is Huber, M. L., and Perkins, R. A., “Thermal Conductivity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-Decane,” Fluid Phase Equilibria 227:47–55, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation are 0.05% in the saturated-liquid density between 280 and 335 K and 0.2% in density in the liquid phase below 430 K and 10 MPa. The uncertainty increases to 0.3% up to 100 MPa and 0.5% up to 800 MPa. In the vapor phase and at supercritical state points, the uncertainty in density is 1%, whereas in the liquid phase between 430 K and the critical point it is 0.5% in density. Other uncertainties are 0.2% in vapor pressure between 300 and 430 K, 0.5% in vapor pressure at higher temperatures, 2% in heat capacities below 550 K, 5% at higher temperatures, and 1% in the liquid-phase speed of sound below 430 K. The estimated uncertainty in viscosity is 1.0% along the saturated-liquid line, 5% elsewhere. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%.

2-317

2-318 TABLE 2-244 Temperature K

Thermodynamic Properties of Octane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

−47.586 −46.751 −42.103 −37.353 −32.477 −27.455 −22.268 −16.904 −11.351 −5.6010 0.35204 6.5150 12.895 19.502 26.352 33.473 40.921 48.832 57.752 64.527

−47.586 −46.751 −42.103 −37.353 −32.477 −27.454 −22.267 −16.901 −11.345 −5.5904 0.37164 6.5490 12.951 19.591 26.489 33.677 41.223 49.282 58.476 65.741

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.15718 −0.15335 −0.13313 −0.11412 −0.096062 −0.078739 −0.062005 −0.045746 −0.029879 −0.014338 0.00092888 0.015963 0.030805 0.045494 0.060080 0.074631 0.089269 0.10426 0.12068 0.13332

0.18017 0.18100 0.18613 0.19222 0.19917 0.20684 0.21505 0.22363 0.23245 0.24139 0.25035 0.25926 0.26810 0.27685 0.28552 0.29419 0.30300 0.31238 0.32413

0.22965 0.23031 0.23473 0.24044 0.24730 0.25511 0.26370 0.27290 0.28260 0.29271 0.30321 0.31416 0.32572 0.33823 0.35240 0.36979 0.39457 0.44291 0.68855

1496.9 1479.7 1388.4 1301.6 1218.5 1138.6 1061.2 985.87 912.09 839.36 767.20 695.05 622.34 548.39 472.36 393.17 309.23 217.99 115.40 0

−0.50088 −0.49876 −0.48455 −0.46633 −0.44452 −0.41949 −0.39136 −0.35994 −0.32455 −0.28395 −0.23602 −0.17723 −0.10167 0.0013050 0.15251 0.39829 0.86399 2.0228 7.6023 28.218

153.28 151.81 144.08 136.89 130.16 123.83 117.86 112.19 106.78 101.59 96.614 91.804 87.139 82.591 78.136 73.755 69.457 65.431 64.209

2275.3 2055.6 1275.2 875.51 644.21 498.03 399.06 328.21 275.06 233.59 200.14 172.37 148.75 128.19 109.92 93.322 77.803 62.572 45.346

0.062519 0.061430 0.057431 0.056201 0.057097 0.059649 0.063500 0.068366 0.074020 0.080278 0.086985 0.094013 0.10124 0.10857 0.11587 0.12299 0.12970 0.13551 0.13877 0.13332

0.14278 0.14422 0.15246 0.16128 0.17068 0.18058 0.19087 0.20147 0.21227 0.22318 0.23415 0.24512 0.25607 0.26700 0.27795 0.28899 0.30032 0.31238 0.32642

0.15110 0.15253 0.16078 0.16962 0.17906 0.18906 0.19955 0.21047 0.22178 0.23348 0.24557 0.25818 0.27154 0.28610 0.30284 0.32404 0.35598 0.42387 0.78365

Saturated Properties 216.37 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 569.32

1.9889E-06 3.0719E-06 2.5565E-05 0.00014507 0.00061336 0.0020600 0.0057644 0.013932 0.029907 0.058269 0.10483 0.17652 0.28132 0.42812 0.62676 0.88820 1.2251 1.6533 2.1958 2.4978

6.6864 6.6606 6.5193 6.3793 6.2396 6.0991 5.9569 5.8120 5.6633 5.5097 5.3495 5.1809 5.0012 4.8068 4.5922 4.3477 4.0556 3.6731 3.0341 2.0564

216.37 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 569.32

1.9889E-06 3.0719E-06 2.5565E-05 0.00014507 0.00061336 0.0020600 0.0057644 0.013932 0.029907 0.058269 0.10483 0.17652 0.28132 0.42812 0.62676 0.88820 1.2251 1.6533 2.1958 2.4978

1.1056E-06 1.6794E-06 1.2813E-05 6.7137E-05 0.00026380 0.00082859 0.0021820 0.0049961 0.010232 0.019165 0.033420 0.055042 0.086662 0.13183 0.19572 0.28672 0.42068 0.63573 1.0851 2.0564

0.14956 0.15014 0.15339 0.15676 0.16027 0.16396 0.16787 0.17206 0.17657 0.18150 0.18693 0.19302 0.19995 0.20804 0.21776 0.23000 0.24657 0.27225 0.32959 0.48629 904,510. 595,460. 78,047. 14,895. 3,790.7 1,206.9 458.30 200.15 97.731 52.177 29.922 18.168 11.539 7.5854 5.1093 3.4877 2.3771 1.5730 0.92161 0.48629

−1.8491 −1.3282 1.6370 4.7709 8.0822 11.576 15.253 19.108 23.136 27.323 31.658 36.123 40.698 45.358 50.064 54.757 59.333 63.555 66.584 64.527

−0.050104 0.50093 3.6323 6.9318 10.407 14.062 17.894 21.897 26.058 30.363 34.794 39.330 43.945 48.605 53.266 57.855 62.245 66.156 68.608 65.741

129.10 130.14 135.72 141.02 146.04 150.71 154.94 158.59 161.53 163.56 164.51 164.17 162.27 158.49 152.37 143.24 130.05 111.06 83.717 0

314.66 294.38 207.62 150.42 111.63 84.719 65.728 52.133 42.292 35.112 29.858 26.034 23.316 21.503 20.514 20.401 21.459 24.546 31.412 28.218

6.1594 6.3194 7.2936 8.4195 9.6907 11.102 12.651 14.338 16.164 18.138 20.268 22.571 25.072 27.811 30.863 34.377 38.702 44.855 57.602

4.2034 4.2717 4.6468 5.0206 5.3924 5.7614 6.1263 6.4865 6.8419 7.1948 7.5505 7.9191 8.3171 8.7715 9.3258 10.056 11.113 12.885 17.102

Single-Phase Properties −27.460 −0.16168

−27.444 −0.14303

−0.078758 −0.00035827

0.20684 0.24959

0.25509 0.30230

1139.3 773.32

−0.41966 −0.24045

123.88 97.030

31.284 31.684 57.485 87.490

34.413 34.830 61.548 92.422

0.086401 0.087444 0.14688 0.20307

0.23322 0.23402 0.27938 0.31885

0.24453 0.24527 0.28878 0.32771

164.48 164.97 189.53 209.46

30.243 29.707 12.431 6.6034

20.080 20.249 31.224 43.584

0.16374 0.18638 0.22959 0.23532

−27.511 0.25030 33.432 36.111

−27.347 0.43668 33.661 36.346

−0.078927 0.00067378 0.074548 0.079880

0.20690 0.25038 0.29416 0.29733

0.25495 0.30262 0.36900 0.37746

1145.7 778.09 396.85 363.62

−0.42119 −0.24368 0.38286 0.52889

124.30 97.295 73.948 72.196

503.48 202.65 93.722 87.646

0.32874 0.22849

3.0419 4.3766

56.425 85.870

59.467 90.247

0.29302 0.32280

0.33370 0.33940

139.04 186.79

20.620 8.0078

35.810 45.491

10.386 11.637

6.1393 5.4311 4.5686 3.0593

0.16288 0.18413 0.21888 0.32688

−27.730 −0.17211 32.275 70.832

−26.915 0.74852 33.369 72.466

−0.079664 −0.00039889 0.072174 0.14322

0.20714 0.25054 0.29359 0.33497

0.25435 0.30044 0.35339 0.44331

1173.4 823.37 499.49 188.23

−0.42743 −0.27213 0.076618 2.6621

126.15 100.17 79.631 64.841

525.55 213.73 105.97 42.657

300.00 398.30

0.10000 0.10000

6.0999 5.3634

398.30 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

0.031955 0.031787 0.024611 0.020274

300.00 400.00 500.00 507.20

1.0000 1.0000 1.0000 1.0000

6.1073 5.3654 4.3556 4.2495

507.20 600.00

1.0000 1.0000

300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000

0.16394 0.18645 31.294 31.459 40.633 49.325

0.12547 0.18117

498.56 202.73 7.5200 7.5528 9.4433 11.290

300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000

6.1772 5.5037 4.7443 3.7971

0.16188 0.18170 0.21078 0.26336

−27.987 −0.64047 31.275 67.577

−26.368 1.1765 33.383 70.211

−0.080546 −0.0016150 0.070056 0.13708

0.20745 0.25077 0.29343 0.33156

0.25373 0.29842 0.34556 0.38999

1206.2 873.81 588.81 358.22

−0.43408 −0.29834 −0.072942 0.45738

128.36 103.45 84.985 73.121

553.60 227.34 118.55 61.952

300.00 400.00 500.00 600.00

50.000 50.000 50.000 50.000

6.4220 5.8987 5.4043 4.9374

0.15572 0.16953 0.18504 0.20254

−29.605 −3.1776 27.307 61.436

−21.819 5.2989 36.559 71.563

−0.086527 −0.0087968 0.060791 0.12452

0.20976 0.25276 0.29481 0.33146

0.25121 0.29206 0.33234 0.36666

1422.7 1159.5 965.74 826.07

−0.46254 −0.38084 −0.30867 −0.24994

143.51 123.35 110.66 103.74

798.49 335.22 198.43 134.83

6.6452 6.2050 5.8089 5.4489

0.15048 0.16116 0.17215 0.18352

−30.984 −5.0534 24.893 58.505

−15.936 11.063 42.108 76.857

−0.092407 −0.015013 0.054099 0.11736

0.21232 0.25503 0.29686 0.33331

0.25061 0.29032 0.32988 0.36412

1629.4 1399.1 1233.7 1113.8

−0.47349 −0.40628 −0.35233 −0.31375

158.61 141.31 131.06 125.82

1163.7 478.53 291.34 208.31

300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., and Wagner, W., “Equations of State for Technical Applications. II. Results for Nonpolar Fluids,” Int. J. Thermophys. 24(1):41–109, 2003. The source for viscosity is Huber, M. L., Laesecke, A., and Xiang, H. W., “Viscosity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-Decane,” Fluid Phase Equilibria 224:263–270, 2004. The source for thermal conductivity is Huber, M. L., and Perkins, R. A., “Thermal Conductivity Correlations for Minor Constituent Fluids in Natural Gas: n-Octane, n-Nonane and n-Decane,” Fluid Phase Equilibria 227:47–55, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are approximately 0.2% (to 0.5% at high pressures) in density, 1% (in the vapor phase) to 2% in heat capacity, 1% (in the vapor phase) to 2% in the speed of sound, and 0.2% in vapor pressure, except in the critical region. The estimated uncertainty in viscosity is 0.5% along the saturated-liquid line, 2% in compressed liquid to 200 MPa, 5% in vapor and supercritical regions. Uncertainty in thermal conductivity is 3%, except in the supercritical region and dilute gas which have an uncertainty of 5%.

2-319

2-320 TABLE 2-245

Thermodynamic Properties of Oxygen

Temperature K

Pressure MPa

Density mol/dm3

54.361 55.000 60.000 65.000 70.000 75.000 80.000 85.000 90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 154.58

0.00014628 0.00017857 0.00072582 0.0023349 0.0062623 0.014547 0.030123 0.056831 0.099350 0.16308 0.25400 0.37853 0.54340 0.75559 1.0223 1.3509 1.7491 2.2250 2.7878 3.4477 4.2186 5.0428

40.816 40.734 40.064 39.367 38.656 37.936 37.203 36.457 35.692 34.905 34.092 33.245 32.360 31.426 30.434 29.367 28.203 26.907 25.415 23.599 21.110 13.630

54.361 55.000 60.000 65.000 70.000 75.000 80.000 85.000 90.000 95.000 100.00 105.00 110.00 115.00 120.00 125.00 130.00 135.00 140.00 145.00 150.00 154.58

0.00014628 0.00017857 0.00072582 0.0023349 0.0062623 0.014547 0.030123 0.056831 0.099350 0.16308 0.25400 0.37853 0.54340 0.75559 1.0223 1.3509 1.7491 2.2250 2.7878 3.4477 4.2186 5.0428

0.00032370 0.00039060 0.0014561 0.0043291 0.010804 0.023509 0.045891 0.082138 0.13710 0.21627 0.32579 0.47267 0.66506 0.91283 1.2284 1.6285 2.1366 2.7893 3.6487 4.8412 6.7170 13.630

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.024500 0.024549 0.024960 0.025402 0.025869 0.026360 0.026879 0.027430 0.028017 0.028649 0.029333 0.030079 0.030903 0.031820 0.032858 0.034051 0.035457 0.037165 0.039347 0.042375 0.047372 0.073368

−6.1954 −6.1613 −5.8938 −5.6258 −5.3573 −5.0889 −4.8202 −4.5510 −4.2806 −4.0084 −3.7337 −3.4556 −3.1732 −2.8853 −2.5904 −2.2867 −1.9711 −1.6394 −1.2839 −0.88908 −0.41330 0.66752

−6.1954 −6.1612 −5.8938 −5.6257 −5.3572 −5.0885 −4.8194 −4.5495 −4.2778 −4.0038 −3.7263 −3.4442 −3.1564 −2.8612 −2.5568 −2.2407 −1.9091 −1.5567 −1.1742 −0.74298 −0.21346 1.0375

0.066946 0.067571 0.072225 0.076516 0.080495 0.084199 0.087667 0.090931 0.094023 0.096967 0.099787 0.10250 0.10513 0.10770 0.11022 0.11271 0.11520 0.11773 0.12035 0.12319 0.12654 0.13442

3089.2 2560.2 686.75 230.99 92.556 42.536 21.791 12.175 7.2938 4.6239 3.0695 2.1156 1.5036 1.0955 0.81405 0.61407 0.46803 0.35852 0.27407 0.20656 0.14888 0.073368

1.1195 1.1327 1.2355 1.3377 1.4393 1.5397 1.6377 1.7320 1.8209 1.9031 1.9772 2.0421 2.0966 2.1391 2.1678 2.1801 2.1722 2.1380 2.0670 1.9383 1.6938 0.66752

1.5714 1.5898 1.7339 1.8770 2.0189 2.1584 2.2941 2.4239 2.5455 2.6571 2.7569 2.8430 2.9136 2.9668 3.0000 3.0097 2.9908 2.9357 2.8311 2.6505 2.3219 1.0375

0.20982 0.20850 0.19935 0.19194 0.18587 0.18083 0.17659 0.17297 0.16984 0.16708 0.16462 0.16238 0.16032 0.15838 0.15652 0.15471 0.15289 0.15100 0.14896 0.14659 0.14345 0.13442

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.038252 0.037651 0.034835 0.033469 0.032532 0.031745 0.031030 0.030365 0.029745 0.029169 0.028636 0.028146 0.027703 0.027311 0.026976 0.026712 0.026536 0.026485 0.026634 0.027189 0.028982

0.053541 0.053489 0.053548 0.053668 0.053697 0.053719 0.053808 0.054012 0.054361 0.054880 0.055599 0.056557 0.057816 0.059469 0.061666 0.064659 0.068905 0.075327 0.086099 0.10778 0.17484

1123.4 1126.9 1127.4 1101.7 1066.3 1027.5 987.43 946.87 905.90 864.40 822.19 779.06 734.77 689.03 641.52 591.86 539.50 483.69 423.10 355.20 273.80 0

−0.37992 −0.37886 −0.37011 −0.36312 −0.35686 −0.34972 −0.34056 −0.32856 −0.31302 −0.29316 −0.26804 −0.23637 −0.19639 −0.14551 −0.079899 0.0063780 0.12309 0.28750 0.53357 0.93865 1.7389 5.0628

201.92 201.02 193.94 186.82 179.70 172.58 165.44 158.27 151.05 143.81 136.55 129.25 121.92 114.57 107.23 99.912 92.634 85.404 78.217 71.056 64.190

773.62 747.53 578.07 457.94 371.79 308.66 261.22 224.62 195.64 172.12 152.56 135.93 121.52 108.81 97.426 87.086 77.571 68.687 60.223 51.869 42.900

0.021241 0.021297 0.021815 0.022310 0.022565 0.022513 0.022239 0.021896 0.021624 0.021515 0.021605 0.021894 0.022361 0.022978 0.023726 0.024597 0.025604 0.026794 0.028269 0.030276 0.033574

0.029631 0.029698 0.030320 0.030934 0.031294 0.031336 0.031177 0.031019 0.031053 0.031420 0.032204 0.033461 0.035245 0.037647 0.040839 0.045146 0.051204 0.060349 0.075824 0.10781 0.21201

140.32 141.11 147.03 152.65 158.07 163.33 168.36 173.06 177.30 180.99 184.06 186.44 188.14 189.13 189.41 188.96 187.75 185.74 182.82 178.78 172.82 0

Saturated Properties

507.90 480.26 284.62 156.71 87.254 52.570 35.817 27.728 23.649 21.338 19.753 18.446 17.250 16.118 15.045 14.029 13.062 12.120 11.155 10.071 8.6358 5.0628

4.4204 4.4842 4.9840 5.4863 5.9925 6.5051 7.0277 7.5654 8.1241 8.7113 9.3362 10.010 10.748 11.571 12.509 13.607 14.940 16.641 18.977 22.582 29.666

4.0962 4.1481 4.5528 4.9555 5.3557 5.7533 6.1486 6.5423 6.9355 7.3301 7.7281 8.1324 8.5467 8.9760 9.4273 9.9112 10.445 11.061 11.823 12.881 14.721

Single-Phase Properties 100.00 300.00 500.00 700.00 900.00

0.10000 0.10000 0.10000 0.10000 0.10000

100.00 119.62

1.0000 1.0000

119.62 300.00 500.00 700.00 900.00

1.0000 1.0000 1.0000 1.0000 1.0000

100.00 154.36

5.0000 5.0000

154.36 300.00 500.00 700.00 900.00

5.0000 5.0000 5.0000 5.0000 5.0000

0.12316 0.040116 0.024050 0.017177 0.013360

2.0355 6.2338 10.604 15.357 20.438

2.8474 8.7265 14.762 21.179 27.923

0.17297 0.20531 0.22069 0.23147 0.23994

0.020885 0.021078 0.022781 0.024672 0.026045

0.029925 0.029435 0.031108 0.032992 0.034363

188.37 329.72 421.27 493.31 555.60

18.479 2.6530 0.75388 0.10517 −0.18735

0.029276 0.032774

−3.7444 −2.6131

−3.7151 −2.5803

0.099680 0.11003

0.028683 0.027000

0.055399 0.061476

826.85 645.19

−0.27181 −0.085501

0.83209 2.4791 4.1649 5.8360 7.5029

2.1662 6.1772 10.576 15.340 20.426

2.9983 8.6563 14.741 21.176 27.929

0.15666 0.18598 0.20149 0.21230 0.22078

0.023665 0.021148 0.022802 0.024682 0.026051

0.040564 0.029887 0.031240 0.033052 0.034395

189.41 329.90 422.68 494.87 557.14

15.124 2.6066 0.73726 0.098376 −0.19062

12.433 26.894 41.288 54.139 66.001

34.497 16.011

0.028988 0.062457

−3.7983 0.35374

−3.6533 0.66602

0.099132 0.13204

0.028935 0.038878

0.054458 3.5718

850.39 163.89

−0.28978 4.2044

140.71 75.954

160.92 29.668

11.160 2.0616 1.1908 0.84728 0.65931

0.089610 0.48505 0.83975 1.1802 1.5167

1.0294 5.9227 10.454 15.264 20.373

1.4774 8.3480 14.653 21.165 27.956

0.13729 0.17177 0.18787 0.19881 0.20734

0.041906 0.021448 0.022894 0.024726 0.026076

4.2513 0.032003 0.031815 0.033309 0.034537

158.85 332.25 429.36 501.98 564.07

6.0016 2.3730 0.66261 0.068114 −0.20519

72.313 28.797 42.362 54.901 66.593

20.574 21.766 31.267 39.261 46.305

34.158 30.512 1.2018 0.40337 0.24010 0.17135 0.13328

8.1192 24.928 41.579 58.216 74.849

9.0852 26.485 41.046 53.966 65.867 137.23 107.79

7.7121 20.652 30.486 38.653 45.806 153.89 98.249 9.3921 20.846 30.630 38.766 45.899

100.00 300.00 500.00 700.00 900.00

10.000 10.000 10.000 10.000 10.000

34.885 4.2056 2.3538 1.6705 1.3010

0.028665 0.23778 0.42484 0.59861 0.76866

−3.8593 5.6024 10.306 15.171 20.307

−3.5726 7.9802 14.554 21.157 27.993

0.098498 0.16499 0.18182 0.19292 0.20150

0.029235 0.021790 0.022999 0.024776 0.026104

0.053516 0.034749 0.032491 0.033613 0.034706

877.07 339.35 438.67 511.24 572.92

−0.30803 2.0332 0.56900 0.030534 −0.22339

144.82 31.466 43.708 55.839 67.321

169.49 23.153 32.074 39.873 46.804

100.00 300.00 500.00 700.00 900.00

25.000 25.000 25.000 25.000 25.000

35.884 10.393 5.6243 3.9923 3.1222

0.027867 0.096215 0.17780 0.25048 0.32028

−4.0109 4.7194 9.8920 14.907 20.117

−3.3142 7.1247 14.337 21.169 28.124

0.096845 0.15490 0.17346 0.18495 0.19369

0.030037 0.022521 0.023256 0.024901 0.026174

0.051627 0.040917 0.034167 0.034397 0.035155

945.24 390.80 472.62 541.32 600.66

−0.34532 1.0167 0.30658 −0.076019 −0.27597

155.97 41.851 47.943 58.651 69.464

194.38 29.605 34.705 41.714 48.271

100.00 300.00 500.00 700.00 900.00

75.000 75.000 75.000 75.000 75.000

38.263 21.603 13.760 10.201 8.1749

0.026135 0.046289 0.072675 0.098029 0.12233

−4.3340 3.1884 8.8798 14.192 19.571

−2.3739 6.6601 14.330 21.544 28.745

0.092788 0.14315 0.16284 0.17498 0.18403

0.031906 0.023601 0.023725 0.025126 0.026293

0.049123 0.041272 0.036534 0.035903 0.036153

1115.1 645.54 619.75 657.04 701.72

−0.39472 −0.18640 −0.20732 −0.31840 −0.40609

184.96 75.261 64.149 68.835 76.863

274.96 53.378 45.084 48.269 53.163

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Schmidt, R., and Wagner, W., “A New Form of the Equation of State for Pure Substances and Its Application to Oxygen,” Fluid Phase Equilibria, 19:175–200, 1985. The source for viscosity and thermal conductivity is Lemmon, E. W., and Jacobsen, R. T., “Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air,” Int. J. Thermophys. 25:21–69, 2004. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 2% in heat capacity, and 1% in the speed of sound, except in the critical region. For viscosity, the uncertainty is 1% in the dilute gas at temperatures above 200 K, and 5% in the dilute gas at lower temperatures. The uncertainty is around 2% between 270 and 300 K, and increases to 5% outside of this region. The uncertainty may be higher in the liquid near the triple point. The uncertainty for the dilute gas is 2% with increasing uncertainties near the triple point. For thermal conductivity, the uncertainties range from 3% between 270 and 300 K to 5% elsewhere. The uncertainties above 100 MPa are not known due to a lack of experimental data.

2-321

−100

0

0.

0. 60

50

0.

.

0.

0.

.

400

500 20. 3

kg/m ρ = 100. 80.

150.

200

30

40

300

60. 140

1000.

c.p.

40.

150

30.

10. 8. 6. 4.

20.

130

120

110

2.

200

140

100

80

4.

1100.

1200.

6.

70

800

900

(R-732) reference state: h = 0.0 kJ/kg, s = 0.00 kJ/(kg·K) for ideal gas at 0 K

T = 90 K

10. 8.

.

Oxygen

100

150

2-322

−200 20.

15.

130

2.

8.0

480

500 3.0

40

20

6.

6.

0 6.0

5.8

0

0

1. 0.8 0.6 0.4

0.2

1.0 0.80

5.6

5.40

5.20

5.00

T = 460 K

440

420

400

340

320

300

280

240

220

200

160

140

120

0.9

saturated va

0.8

0.7 4.80

4.60

4.40

4.20

3.80

4.00

3.40

3.60

3.00

3.20

4.0

1.5

0.60

)

2.80

6.0

2.0

90

2.60

0.06

0.6

por

0.1 0.08

0.5

0.3

satu

x = 0.4

0.2

0.2

0.1

liquid

100

180

0.4

260

T = 110 K

380

0.6

360

1. 0.8

rated

Pressure (MPa)

10. 120

g·K

/(k

0

6 6.

s=

0.04

0 6.8

kJ

0.40 0.30

0.1 0.08 0.06 0.04

0

7.0

80

0.20

0.02

.20

7

0.15

0.02

0.10

0.01 −200

−100

0

100

200

300

400

0.01 500

Enthalpy (kJ/kg) Pressure-enthalpy diagram for oxygen. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O. and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Schmidt, R., and Wagner, W., “A New Form of the Equation of State for Pure Substances and Its Application to Oxygen,” Fluid Phase Equilibria 19:175–200, 1985.

FIG. 2-15

THERMODYNAMIC PROPERTIES

2-323

FIG. 2-16 Enthalpy-concentration diagram for oxygen-nitrogen mixture at 1 atm. Reference states: Enthalpies of liquid oxygen and liquid nitrogen at the normal boiling point of nitrogen are zero. (Dodge, Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1944.) Wilson, Silverberg, and Zellner, AFAPL TDR 64-64 (AD 603151), 1964, p. 314, present extensive vapor-liquid equilibrium data for the three-component system argon-nitrogen-oxygen as well as for binary systems including oxygen-nitrogen. Calculations for this mixture are also available with the NIST REFPROP software.

2-324 TABLE 2-246 Temperature K

Thermodynamic Properties of Pentane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−25.092 −24.164 −22.032 −19.892 −17.737 −15.558 −13.348 −11.099 −8.8042 −6.4555 −4.0461 −1.5694 0.98088 3.6106 6.3259 9.1332 12.040 15.057 18.197 21.482 24.952 28.698 33.125 36.504

−25.092 −24.164 −22.032 −19.892 −17.737 −15.557 −13.347 −11.099 −8.8031 −6.4532 −4.0414 −1.5608 0.99557 3.6346 6.3635 9.1900 12.123 15.176 18.364 21.715 25.275 29.150 33.811 37.552

8.3431 8.8107 9.9311 11.110 12.343 13.627 14.962 16.347 17.780 19.260 20.787 22.357 23.968 25.617 27.297 29.002 30.721 32.439 34.133 35.764 37.264 38.471 38.730 36.504

9.5360 10.058 11.303 12.607 13.964 15.372 16.829 18.333 19.882 21.472 23.101 24.763 26.454 28.168 29.896 31.629 33.352 35.046 36.682 38.214 39.557 40.520 40.328 37.552

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.11487 −0.10855 −0.095008 −0.082594 −0.071091 −0.060326 −0.050163 −0.040489 −0.031215 −0.022266 −0.013582 −0.0051134 0.0031816 0.011338 0.019386 0.027354 0.035271 0.043168 0.051083 0.059073 0.067234 0.075780 0.085678 0.093548

0.10392 0.10408 0.10469 0.10558 0.10680 0.10837 0.11032 0.11267 0.11539 0.11847 0.12186 0.12552 0.12940 0.13348 0.13770 0.14204 0.14649 0.15103 0.15570 0.16056 0.16578 0.17187 0.18139

0.14205 0.14201 0.14231 0.14312 0.14441 0.14621 0.14853 0.15137 0.15473 0.15857 0.16289 0.16766 0.17288 0.17857 0.18478 0.19161 0.19928 0.20814 0.21894 0.23325 0.25525 0.30145 0.60959

1829.9 1788.3 1696.9 1610.3 1527.4 1447.4 1369.7 1293.8 1219.4 1146.3 1074.0 1002.5 931.46 860.59 789.61 718.18 645.89 572.27 496.72 418.39 335.90 246.17 138.75 0

−0.54905 −0.54859 −0.54514 −0.53850 −0.52866 −0.51549 −0.49891 −0.47884 −0.45516 −0.42770 −0.39609 −0.35971 −0.31748 −0.26769 −0.20766 −0.13303 −0.036610 0.094152 0.28299 0.58010 1.1123 2.3196 7.5108 19.135

175.03 172.92 167.37 161.20 154.70 148.08 141.47 134.95 128.57 122.40 116.43 110.69 105.17 99.882 94.813 89.958 85.304 80.839 76.544 72.402 68.402 64.610 63.369

3709.1 2750.8 1587.7 1048.4 757.19 581.46 466.22 385.57 326.15 280.49 244.17 214.42 189.45 168.04 149.32 132.67 117.61 103.78 90.837 78.483 66.354 53.842 38.530

0.12648 0.11959 0.10702 0.097955 0.091475 0.086957 0.083956 0.082144 0.081276 0.081162 0.081654 0.082633 0.084003 0.085681 0.087596 0.089684 0.091880 0.094117 0.096313 0.098358 0.10007 0.10105 0.099694 0.093548

0.070619 0.072591 0.076717 0.080473 0.084066 0.087665 0.091392 0.095331 0.099526 0.10400 0.10874 0.11373 0.11894 0.12435 0.12993 0.13567 0.14155 0.14759 0.15383 0.16037 0.16742 0.17552 0.18671

0.078934 0.080906 0.085033 0.088791 0.092395 0.096021 0.099805 0.10385 0.10823 0.11297 0.11811 0.12367 0.12966 0.13613 0.14317 0.15092 0.15968 0.17000 0.18300 0.20125 0.23193 0.30440 0.82173

135.94 138.80 145.17 151.28 157.11 162.67 167.92 172.81 177.25 181.16 184.42 186.93 188.56 189.17 188.63 186.76 183.35 178.13 170.76 160.78 147.55 130.14 106.55 0

Saturated Properties 143.47 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 469.70

7.6322E-08 2.6809E-07 3.1471E-06 2.3256E-05 0.00012116 0.00048191 0.0015504 0.0042112 0.0099767 0.021139 0.040858 0.073168 0.12293 0.19575 0.29786 0.43606 0.61766 0.85052 1.1432 1.5050 1.9472 2.4836 3.1355 3.3710

143.47 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 469.70

7.6322E-08 2.6809E-07 3.1471E-06 2.3256E-05 0.00012116 0.00048191 0.0015504 0.0042112 0.0099767 0.021139 0.040858 0.073168 0.12293 0.19575 0.29786 0.43606 0.61766 0.85052 1.1432 1.5050 1.9472 2.4836 3.1355 3.3710

10.566 10.482 10.292 10.105 9.9190 9.7339 9.5483 9.3613 9.1719 8.9791 8.7818 8.5788 8.3688 8.1499 7.9200 7.6765 7.4155 7.1319 6.8175 6.4595 6.0340 5.4866 4.5754 3.2156 6.3981E-08 2.1496E-07 2.2940E-06 1.5540E-05 7.4749E-05 0.00027623 0.00083048 0.0021198 0.0047454 0.0095553 0.017654 0.030405 0.049446 0.076727 0.11460 0.16601 0.23477 0.32623 0.44834 0.61427 0.84898 1.2118 1.9618 3.2156

0.094640 0.095399 0.097163 0.098964 0.10082 0.10273 0.10473 0.10682 0.10903 0.11137 0.11387 0.11657 0.11949 0.12270 0.12626 0.13027 0.13485 0.14022 0.14668 0.15481 0.16573 0.18226 0.21856 0.31099 15,630,000. 4,652,000. 435,920. 64,349. 13,378. 3,620.1 1,204.1 471.75 210.73 104.65 56.645 32.889 20.224 13.033 8.7259 6.0239 4.2594 3.0654 2.2305 1.6279 1.1779 0.82522 0.50974 0.31099

542.15 453.68 311.80 222.93 164.55 124.69 96.614 76.365 61.487 50.401 42.052 35.715 30.879 27.184 24.375 22.275 20.767 19.785 19.316 19.406 20.186 21.884 24.088 19.135

4.0211 4.3338 5.0759 5.8544 6.6764 7.5505 8.4862 9.4926 10.578 11.752 13.022 14.394 15.879 17.484 19.221 21.106 23.164 25.433 27.980 30.941 34.634 40.108 55.658

3.2806 3.4236 3.7558 4.0934 4.4362 4.7837 5.1356 5.4918 5.8525 6.2179 6.5890 6.9672 7.3547 7.7547 8.1719 8.6130 9.0883 9.6135 10.215 10.939 11.885 13.312 16.549

Single-Phase Properties 200.00 300.00 308.83

0.10000 0.10000 0.10000

9.8581 8.5793 8.4562

308.83 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

0.040733 0.030611 0.024257 0.020135

200.00 300.00 398.07

1.0000 1.0000 1.0000

9.8656 8.5958 6.9671

398.07 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

0.10144 0.11656 0.11826 24.550 32.668 41.224 49.665

−17.016 −1.5709 −0.077672

−17.006 −1.5592 −0.065846

−0.067441 −0.0051184 −0.00021257

0.10728 0.12552 0.12778

0.14495 0.16765 0.17068

1501.0 1002.8 960.65

−0.52472 −0.35984 −0.33565

152.53 110.71 107.41

690.14 214.50 199.24

23.300 35.260 51.273 70.070

25.755 38.527 55.396 75.037

0.083398 0.11949 0.15700 0.19274

0.11677 0.14472 0.17449 0.20055

0.12714 0.15376 0.18314 0.20906

188.00 217.34 243.86 267.28

32.712 12.986 6.4310 3.8205

15.254 24.749 37.084 50.416

7.1940 9.3018 11.487 13.525

0.10136 0.11634 0.14353

−17.038 −1.6210 16.730

−16.937 −1.5046 16.874

−0.067555 −0.0052858 0.047422

0.10733 0.12555 0.15353

0.14489 0.16737 0.21364

1505.8 1011.2 531.90

−0.52539 −0.36413 0.18628

152.82 111.23 78.507

0.38746 0.38317 0.26375 0.20980

2.5809 2.6098 3.7914 4.7663

33.355 33.666 50.465 69.531

35.936 36.276 54.257 74.297

0.095310 0.096162 0.13622 0.17270

0.15092 0.15126 0.17624 0.20129

0.17655 0.17612 0.18896 0.21181

174.46 175.97 226.09 257.70

19.468 18.915 7.2197 3.9909

26.763 26.951 38.498 51.523

9.8980 8.6652 7.1612 3.2553 1.2834

0.10103 0.11540 0.13964 0.30719 0.77916

−17.137 −1.8329 16.501 42.207 66.610

−16.632 −1.2559 17.199 43.743 70.506

−0.068053 −0.0060014 0.046820 0.10528 0.15448

0.10753 0.12571 0.15388 0.19073 0.20500

0.14467 0.16627 0.20601 0.42935 0.23291

1527.0 1047.2 606.55 135.39 220.81

−0.52817 −0.38106 0.017312 9.4810 4.4347

154.08 113.46 82.387 59.214 57.890

726.41 228.42 107.19 26.017 17.494

696.76 217.09 96.722 9.9253 9.9649 12.017 13.960

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

9.9373 8.7452 7.3792 5.4581 3.0774

0.10063 0.11435 0.13552 0.18321 0.32495

−17.255 −2.0763 15.896 37.622 62.405

−16.249 −0.93280 17.251 39.455 65.655

−0.068660 −0.0068390 0.045232 0.094589 0.14230

0.10778 0.12592 0.15383 0.18299 0.20754

0.14442 0.16515 0.20020 0.24567 0.26294

1552.6 1088.8 687.58 349.25 229.05

−0.53127 −0.39826 −0.10747 0.88491 2.8865

155.61 116.08 86.892 69.546 67.434

763.99 242.27 120.04 57.792 28.573

200.00 300.00 400.00 500.00 600.00

50.000 50.000 50.000 50.000 50.000

10.209 9.2247 8.2838 7.3733 6.5265

0.097953 0.10840 0.12072 0.13562 0.15322

−18.043 −3.5039 13.310 32.748 54.478

−13.146 1.9164 19.346 39.529 62.139

−0.072988 −0.012155 0.037795 0.082724 0.12389

0.10968 0.12755 0.15501 0.18248 0.20695

0.14331 0.16079 0.18829 0.21473 0.23672

1731.1 1343.9 1056.0 853.23 723.60

−0.54573 −0.46478 −0.35370 −0.25198 −0.17131

166.40 132.95 109.81 97.812 94.951

1086.9 346.81 195.02 130.14 95.489

10.479 9.6305 8.8688 8.1734 7.5451

0.095431 0.10384 0.11275 0.12235 0.13254

−18.760 −4.6330 11.680 30.587 51.860

−9.2168 5.7508 22.955 42.822 65.113

−0.077502 −0.017033 0.032278 0.076502 0.11708

0.11180 0.12937 0.15665 0.18394 0.20826

0.14305 0.15917 0.18547 0.21138 0.23383

1911.9 1570.5 1325.1 1153.2 1035.8

−0.55097 −0.49017 −0.40837 −0.34418 −0.29936

177.42 148.83 128.55 117.64 114.07

1550.8 473.34 270.23 187.06 142.50

200.00 300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Span, R., and Wagner, W., “Equations of State for Technical Applications. II. Results for Nonpolar Fluids,” Int. J. Thermophys. 24(1):41–109, 2003. The source for viscosity and thermal conductivity is NIST14, Version 9.08. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are approximately 0.2% (to 0.5% at high pressures) in density, 1% (in the vapor phase) to 2% in heat capacity, 1% (in the vapor phase) to 2% in the speed of sound, and 0.2% in vapor pressure, except in the critical region. For viscosity, estimated uncertainty is 2%. For thermal conductivity, estimated uncertainty, except near the critical region, is 4–6%.

2-325

2-326

PHYSICAL AND CHEMICAL DATA TABLE 2-247 T, K 336.4m 400 500 600 700 800 1000 1200 1400 1500

Saturated Potassium*

P, bar

vf, m3/kg

1.37.−9 1.84.−7 3.13.−5 9.26.−4 0.01022

0.001208 0.001229 0.001266 0.001304 0.001346

0.06116 0.7322 3.913 12.44 20.0

0.001389 0.001488 0.001605 0.001742 0.001816

vg, m3/kg

h f, kJ/kg

h g, kJ/kg

sf, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

c pf, kJ/(kg⋅K)

4.64.+6 3.39.+4 3164 142.3

93.8 145.5 225.1 302.7 379.4

2327 2342 2390 2433 2468

1.928 2.068 2.246 2.388 2.506

8.567 7.559 6.576 5.937 5.490

0.822 0.805 0.785 0.771 0.762

455.5 609.7 773.5 948.0 1040.0

2498 2552 2610 2679 2718

2.608 2.780 2.929 3.063 3.123

5.161 4.722 4.459 4.299 4.209

0.761 0.792 0.846 0.899 0.924

26.75 2.691 0.584 0.207 0.132

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972; and Hemisphere, Washington, 1975. m = melting point. The notation 1.37.−9 signifies 1.37 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1650 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 250 psia, 1300 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR61-96, 1961. An extensive review of properties of the solid and the saturated liquid is given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994):385–497.

FIG. 2-17 Mollier diagram for potassium. Basis: enthalpy = 0.0 cal/g atom at 298 K; entropy = 15.8 cal/(g atom·K) at 298 K. (Aerojet-General Rep. AGN8194, vol. 2, 1967. Reproduced by permission.)

TABLE 2-248 Temperature K

Thermodynamic Properties of Propane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−7.4323 −6.1638 −4.8753 −3.5682 −2.2418 −0.89395 0.47907 1.8816 3.3189 4.7964 6.3203 7.8975 9.5360 11.246 13.041 14.943 16.996 19.317 23.653

−7.4323 −6.1638 −4.8753 −3.5682 −2.2418 −0.89372 0.47981 1.8836 3.3234 4.8056 6.3375 7.9275 9.5854 11.324 13.159 15.120 17.257 19.707 24.503

15.887 16.399 16.947 17.527 18.137 18.775 19.437 20.120 20.819 21.528 22.242 22.954 23.656 24.336 24.977 25.543 25.971 26.098 23.653

16.718 17.356 18.028 18.732 19.466 20.226 21.007 21.801 22.602 23.401 24.189 24.956 25.690 26.376 26.989 27.488 27.791 27.699 24.503

−7.4332 1.4085 4.3719

−7.4270 1.4157 4.3795

21.326 25.482 33.084 42.616 −7.4410 1.3862 11.833 11.844

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

−0.048110 −0.036293 −0.025763 −0.016249 −0.0075455 0.00050603 0.0080327 0.015137 0.021903 0.028399 0.034686 0.040815 0.046833 0.052790 0.058740 0.064759 0.070985 0.077769 0.090537

0.058376 0.058304 0.058442 0.058710 0.059141 0.059774 0.060636 0.061740 0.063087 0.064672 0.066484 0.068514 0.070755 0.073209 0.075898 0.078882 0.082328 0.087215

0.083833 0.085262 0.086525 0.087764 0.089109 0.090658 0.092484 0.094646 0.097193 0.10018 0.10368 0.10780 0.11272 0.11880 0.12672 0.13817 0.15818 0.21184

2030.4 1930.7 1827.2 1724.5 1623.4 1523.6 1424.7 1326.6 1229.1 1131.9 1034.8 937.63 839.90 741.08 640.30 536.11 425.33 300.04 0

−0.62852 −0.61354 −0.59904 −0.58338 −0.56521 −0.54342 −0.51702 −0.48497 −0.44601 −0.39841 −0.33958 −0.26538 −0.16892 −0.037850 0.15178 0.45297 1.0100 2.3857 12.890

203.23 196.90 189.40 181.11 172.30 163.19 153.97 144.76 135.72 126.91 118.40 110.26 102.52 95.209 88.324 81.819 75.581 69.491

3780.3 1822.7 1080.5 729.42 534.62 412.89 329.94 269.81 224.28 188.66 160.10 136.69 117.11 100.38 85.735 72.510 60.014 47.137

0.19339 0.16822 0.15042 0.13755 0.12813 0.12119 0.11607 0.11230 0.10953 0.10753 0.10609 0.10507 0.10435 0.10381 0.10335 0.10282 0.10197 0.10028 0.090537

0.032981 0.035381 0.037630 0.039780 0.041911 0.044098 0.046406 0.048888 0.051579 0.054504 0.057677 0.061104 0.064794 0.068763 0.072895 0.077315 0.083072 0.091334

0.041296 0.043696 0.045946 0.048104 0.050262 0.052519 0.054975 0.057727 0.060867 0.064488 0.068703 0.073667 0.079635 0.087064 0.096756 0.11131 0.13963 0.22264

153.65 163.64 172.99 181.78 190.03 197.68 204.61 210.64 215.59 219.25 221.39 221.82 220.30 216.53 210.16 200.74 187.44 168.95 0

−0.048119 0.012800 0.026576

0.058377 0.061348 0.064197

0.083831 0.093874 0.099283

23.175 27.937 36.389 46.761

0.10803 0.12602 0.15020 0.17327

0.053648 0.065918 0.085775 0.10431

0.063403 0.074796 0.094327 0.11276

218.35 249.37 286.19 318.30

−7.3796 1.4577 11.924 11.934

−0.048197 0.012688 0.054770 0.054806

0.058391 0.061383 0.074077 0.074092

0.083810 0.093756 0.12118 0.12123

2034.6 1366.3 707.81 707.18

Saturated Properties 100.00 115.00 130.00 145.00 160.00 175.00 190.00 205.00 220.00 235.00 250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 369.83

2.5330E-08 1.0677E-06 1.7600E-05 0.00015328 0.00084980 0.0033874 0.010547 0.027195 0.060583 0.12030 0.21798 0.36693 0.58173 0.87805 1.2726 1.7837 2.4320 3.2432 4.2477

100.00 115.00 130.00 145.00 160.00 175.00 190.00 205.00 220.00 235.00 250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 369.83

2.5330E-08 1.0677E-06 1.7600E-05 0.00015328 0.00084980 0.0033874 0.010547 0.027195 0.060583 0.12030 0.21798 0.36693 0.58173 0.87805 1.2726 1.7837 2.4320 3.2432 4.2477

100.00 200.00 230.74

0.10000 0.10000 0.10000

230.74 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

100.00 200.00 300.00 300.09

1.0000 1.0000 1.0000 1.0000

16.287 15.941 15.596 15.251 14.904 14.554 14.198 13.834 13.459 13.070 12.663 12.234 11.775 11.277 10.726 10.097 9.3403 8.3162 5.0000 3.0465E-08 1.1166E-06 1.6284E-05 0.00012717 0.00063940 0.0023346 0.0067219 0.016180 0.033974 0.064224 0.11195 0.18328 0.28598 0.43050 0.63231 0.91726 1.3363 2.0253 5.0000

0.061397 0.062730 0.064117 0.065568 0.067094 0.068710 0.070434 0.072288 0.074301 0.076511 0.078968 0.081740 0.084925 0.088673 0.093229 0.099036 0.10706 0.12025 0.20000 32,825,000. 895,550. 61,410. 7,863.4 1,564.0 428.34 148.77 61.804 29.435 15.570 8.9325 5.4561 3.4968 2.3229 1.5815 1.0902 0.74835 0.49375 0.20000

420.53 267.23 181.79 130.32 97.395 75.331 59.995 49.008 40.931 34.865 30.235 26.669 23.923 21.832 20.354 19.511 19.120 18.854 12.890

2.4171 3.2182 4.0849 5.0168 6.0131 7.0721 8.1917 9.3706 10.611 11.922 13.323 14.845 16.544 18.503 20.863 23.881 28.187 36.060

2.9792 3.3431 3.7167 4.0976 4.4832 4.8708 5.2578 5.6429 6.0262 6.4107 6.8027 7.2123 7.6551 8.1548 8.7494 9.5073 10.574 12.373

Single-Phase Properties 16.288 13.958 13.182 0.054083 0.040726 0.030257 0.024125

2-327

16.295 13.974 11.101 11.097

0.061394 0.071646 0.075860 18.490 24.554 33.050 41.452 0.061368 0.071560 0.090085 0.090112

2030.8 1359.8 1159.5

−0.62854 −0.49655 −0.41295 36.423 16.357 7.1977 3.9113 −0.62871 −0.49871 0.017030 0.018141

203.26 147.87 129.38 11.542 18.513 30.995 46.363 203.49 148.40 92.871 92.828

3784.5 288.12 197.97 6.3010 8.1962 10.819 13.290 3822.7 290.36 95.307 95.215

2-328 TABLE 2-248

Thermodynamic Properties of Propane (Continued)

Temperature K

Pressure MPa

Density mol/dm3

300.09 400.00 500.00

1.0000 1.0000 1.0000

0.49153 0.32145 0.24776

100.00 200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Single-Phase Properties 2.0344 3.1109 4.0362

24.559 32.638 42.322

26.593 35.749 46.359

0.10366 0.12993 0.15353

0.070157 0.086666 0.10468

0.090035 0.097825 0.11438

214.68 272.57 311.60

21.259 7.6356 3.9684

19.249 31.915 47.555

8.3433 10.996 13.477

16.326 14.047 11.363 2.5448 1.4059

0.061251 0.071191 0.088002 0.39296 0.71129

−7.4751 1.2900 11.506 29.499 40.868

−7.1688 1.6459 11.946 31.463 44.425

−0.048542 0.012202 0.053658 0.10822 0.13725

0.058457 0.061542 0.074005 0.093912 0.10638

0.083721 0.093272 0.11643 0.16341 0.12446

2051.3 1394.2 779.30 197.49 284.88

−0.62944 −0.50758 −0.10225 10.349 4.1092

204.51 150.70 97.139 42.438 52.411

3995.6 300.33 103.08 14.948 15.207

100.00 200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000 10.000

16.365 14.133 11.627 7.5861 3.2486

0.061108 0.070758 0.086007 0.13182 0.30783

−7.5164 1.1767 11.173 24.108 38.754

−6.9053 1.8843 12.033 25.426 41.832

−0.048966 0.011619 0.052499 0.090675 0.12746

0.058554 0.061744 0.074065 0.093138 0.10818

0.083620 0.092746 0.11280 0.16714 0.14277

2071.2 1427.0 851.72 338.94 275.78

−0.63025 −0.51720 −0.19531 1.7653 3.4683

205.77 153.49 101.80 68.250 58.851

4219.8 312.82 111.73 41.091 19.828

100.00 200.00 300.00 400.00 500.00

50.000 50.000 50.000 50.000 50.000

16.644 14.696 12.850 11.043 9.3527

0.060081 0.068047 0.077822 0.090552 0.10692

−7.8016 0.46479 9.6226 20.208 32.246

−4.7976 3.8672 13.514 24.735 37.592

−0.052119 0.0076714 0.046601 0.078767 0.10739

0.059685 0.063347 0.075522 0.092492 0.10930

0.083087 0.090345 0.10376 0.12069 0.13596

2203.4 1638.0 1203.5 901.34 721.97

−0.63391 −0.56103 −0.42987 −0.26933 −0.11844

214.99 173.15 129.30 103.15 94.269

6389.5 416.61 167.38 95.902 63.347

−0.055583

0.061388

0.082816

2328.3

−0.63418

225.01

0.065107 0.077244 0.094108 0.11078

0.089250 0.10131 0.11714 0.13224

1833.0 1467.7 1214.6 1052.7

−0.58053 −0.49303 −0.40159 −0.33205

193.78 155.37 129.74 118.57

100.00

100.00

16.946

0.059010

−8.0684

−2.1674

200.00 300.00 400.00 500.00

100.00 100.00 100.00 100.00

15.221 13.700 12.331 11.114

0.065700 0.072994 0.081095 0.089974

−0.13496 8.6070 18.712 30.308

6.4350 15.906 26.822 39.306

0.0038087 0.042047 0.073340 0.10113

10396. 561.59 229.66 138.80 98.032

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Buecker, D., and Wagner, W., “Reference Equations of State for the Thermodynamic Properties of Fluid Phase n-Butane and Isobutane,” J. Phys. Chem. Ref. Data 35(2):929–1019, 2006. The source for viscosity is Vogel, E., Kuechenmeister, C., Bich, E., and Laesecke, A., “Reference Correlation of the Viscosity of Propane,” J. Phys. Chem. Ref. Data 27(5):947–970, 1998. The source for thermal conductivity is Marsh, K., Perkins, R., and Ramires, M. L. V., “Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa,” J. Chem. Eng. Data 47(4):932–940, 2002. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. Typical uncertainties in density are 0.02% in the liquid phase, 0.05% in the vapor phase and at supercritical temperatures, and 0.1% in the critical region, except very near the critical point, where the uncertainty in pressure is 0.1%. For vapor pressures, the uncertainty is 0.02% above 180 K, 0.05% above 1 Pa (115 K), and dropping to 0.001 mPa at the triple point. The uncertainty in heat capacity (isobaric, isochoric, and saturated) is 0.5% at temperatures above 125 K and 2% at temperatures below 125 K for the liquid, and is 0.5% for all vapor states. The uncertainty in the liquid-phase speed of sound is 0.5%, and that for the vapor phase is 0.05%. The uncertainties are higher for all properties very near the critical point except pressure (saturated vapor/liquid and single phase). The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to about 2.5% from 100 to 475 K up to about 30 MPa, and to about 4% outside this range. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%.

TABLE 2-249 Temperature K

Thermodynamic Properties of Propylene Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.039412 −0.029431 −0.020215 −0.011635 −0.0036693 0.0037429 0.010683 0.017233 0.023468 0.029455 0.035251 0.040906 0.046469 0.051993 0.057541 0.063218 0.069261 0.076741 0.084569

0.047146 0.048479 0.051603 0.053564 0.054606 0.055253 0.055851 0.056582 0.057524 0.058697 0.060098 0.061715 0.063540 0.065577 0.067855 0.070450 0.073575 0.078178

0.074152 0.076350 0.080066 0.082659 0.084443 0.085952 0.087538 0.089404 0.091663 0.094401 0.097711 0.10174 0.10676 0.11327 0.12246 0.13754 0.17154 0.39594

1984.2 1860.6 1740.6 1637.2 1542.5 1450.8 1359.1 1266.3 1171.8 1075.9 978.69 880.57 781.63 681.71 580.13 475.08 362.18 229.03 0

−0.63335 −0.61019 −0.57672 −0.55180 −0.53067 −0.50839 −0.48173 −0.44843 −0.40645 −0.35329 −0.28530 −0.19657 −0.077005 0.092061 0.34911 0.78805 1.7115 4.9073 12.042

0.18156 0.15964 0.14383 0.13221 0.12357 0.11710 0.11221 0.10851 0.10570 0.10354 0.10186 0.10053 0.099433 0.098452 0.097460 0.096280 0.094563 0.090988 0.084569

0.031371 0.033039 0.034573 0.036090 0.037686 0.039423 0.041343 0.043468 0.045811 0.048376 0.051168 0.054191 0.057457 0.060989 0.064833 0.069077 0.073899 0.079791

0.039685 0.041354 0.042891 0.044423 0.046064 0.047910 0.050046 0.052548 0.055492 0.058970 0.063112 0.068131 0.074408 0.082685 0.094593 0.11448 0.15900 0.42332

162.00 172.27 181.89 190.90 199.27 206.90 213.67 219.38 223.85 226.88 228.25 227.73 225.06 219.92 211.93 200.59 185.21 164.25 0

Saturated Properties 105.00 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 365.57

1.8242E-07 4.8828E-06 5.9371E-05 0.00041697 0.0019723 0.0069671 0.019747 0.047272 0.099223 0.18775 0.32701 0.53269 0.82165 1.2118 1.7225 2.3751 3.1956 4.2202 4.6646

105.00 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 365.57

1.8242E-07 4.8828E-06 5.9371E-05 0.00041697 0.0019723 0.0069671 0.019747 0.047272 0.099223 0.18775 0.32701 0.53269 0.82165 1.2118 1.7225 2.3751 3.1956 4.2202 4.6646

100.00 200.00 225.17

0.10000 0.10000 0.10000

225.17 300.00 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000

100.00 200.00 292.39

1.0000 1.0000 1.0000

292.39 300.00

1.0000 1.0000

17.799 17.383 16.973 16.567 16.162 15.755 15.343 14.922 14.487 14.035 13.559 13.054 12.509 11.911 11.240 10.455 9.4631 7.8236 5.3086 2.0896E-07 4.8940E-06 5.2900E-05 0.00033450 0.0014403 0.0046776 0.012310 0.027633 0.054946 0.099531 0.16779 0.26768 0.40967 0.60872 0.88863 1.2929 1.9214 3.1801 5.3086

0.056183 0.057526 0.058917 0.060361 0.061873 0.063470 0.065175 0.067016 0.069027 0.071251 0.073750 0.076607 0.079945 0.083956 0.088971 0.095644 0.10567 0.12782 0.18837 4,785,700. 204,330. 18,904. 2,989.5 694.32 213.78 81.237 36.188 18.200 10.047 5.9598 3.7358 2.4410 1.6428 1.1253 0.77343 0.52045 0.31446 0.18837

−6.2877 −5.1660 −3.9917 −2.7699 −1.5160 −0.23811 1.0625 2.3884 3.7442 5.1356 6.5692 8.0523 9.5938 11.205 12.904 14.722 16.734 19.290 21.862

−6.2877 −5.1660 −3.9917 −2.7699 −1.5159 −0.23766 1.0638 2.3916 3.7510 5.1490 6.5933 8.0931 9.6595 11.307 13.057 14.949 17.071 19.829 22.741

16.041 16.525 17.032 17.560 18.109 18.676 19.258 19.850 20.446 21.042 21.630 22.202 22.749 23.254 23.693 24.022 24.137 23.631 21.862

16.914 17.522 18.154 18.807 19.479 20.166 20.862 21.560 22.252 22.928 23.579 24.192 24.754 25.245 25.632 25.859 25.801 24.958 22.741

260.32 186.64 139.37 107.24 84.470 67.837 55.441 46.072 38.921 33.424 29.182 25.909 23.398 21.502 20.107 19.116 18.381 17.115 12.042

Single-Phase Properties 17.939 15.206 14.482 0.055346 0.040648 0.030232 0.024118 0.020074 17.947 15.224 12.222

2-329

0.49938 0.47670

−6.6629 1.4997 3.7598

−6.6573 1.5063 3.7667

−0.043075 0.012896 0.023538

0.049534 0.056075 0.057536

0.076050 0.088113 0.091692

20.453 24.309 30.772 38.745 48.088

22.260 26.769 34.080 42.891 53.070

0.10567 0.12289 0.14382 0.16342 0.18194

0.045839 0.056666 0.072240 0.086809 0.099688

0.055529 0.065495 0.080757 0.095223 0.10806

223.89 258.13 295.63 328.33 357.96

0.055720 0.065684 0.081820

−6.6707 1.4781 10.378

−6.6149 1.5437 10.460

−0.043152 0.012788 0.049191

0.049534 0.056080 0.064516

0.076021 0.087997 0.10973

2014.7 1335.9 732.56

2.0025 2.0978

23.004 23.500

25.006 25.598

0.098942 0.10094

0.059161 0.059932

0.078164 0.077330

222.86 228.89

0.055745 0.065766 0.069051 18.068 24.601 33.078 41.463 49.815

2009.3 1328.9 1170.7

−0.61954 −0.47165 −0.40591 38.850 16.866 7.2207 3.7462 2.2036 −0.61988 −0.47384 −0.0016360 22.395 20.287

2-330

TABLE 2-249 Temperature K

Thermodynamic Properties of Propylene (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

400.00 500.00 600.00

1.0000 1.0000 1.0000

0.31858 0.24704 0.20332

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

Enthalpy kJ/mol

3.1389 4.0480 4.9183

30.390 38.516 47.935

33.529 42.564 52.854

15.306 12.206 2.2829 1.3832 1.0713

0.065334 0.081924 0.43803 0.72298 0.93348

1.3850 10.889 27.896 37.351 47.197

15.402 12.515 7.5283 3.1462 2.2419

0.064925 0.079905 0.13283 0.31784 0.44605

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.12371 0.14381 0.16254

0.073331 0.087329 0.099980

0.084055 0.096722 0.10888

283.73 322.07 354.28

7.6465 3.8350 2.2229

1.7117 11.299 30.087 40.966 51.865

0.012317 0.050918 0.10377 0.12811 0.14796

0.056114 0.065369 0.080080 0.089729 0.10124

0.087522 0.10822 0.12610 0.10566 0.11304

1365.9 752.67 221.38 297.48 342.05

−0.48279 −0.041773 9.5490 4.0173 2.2088

1.2752 10.553 22.740 35.603 46.180

1.9245 11.352 24.068 38.781 50.641

0.011753 0.049747 0.085918 0.11902 0.14065

0.056174 0.065258 0.081711 0.092261 0.10253

0.087009 0.10422 0.16868 0.12166 0.11891

1401.2 828.84 302.55 287.51 340.40

−0.49245 −0.15061 2.6128 3.5544 1.9833

Single-Phase Properties

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

200.00 300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00 100.00

16.616 14.839 13.274 11.892 10.680

0.060184 0.067390 0.075333 0.084087 0.093635

−0.021002 8.0040 17.017 27.127 38.256

5.9974 14.743 24.550 35.536 47.620

0.0040990 0.039443 0.067574 0.092040 0.11404

0.058338 0.067573 0.081108 0.094320 0.10601

0.083715 0.092245 0.10402 0.11555 0.12591

1815.4 1457.4 1192.7 1020.4 913.65

−0.55416 −0.48492 −0.40335 −0.33132 −0.27115

200.00 300.00 400.00 500.00 600.00

200.00 200.00 200.00 200.00 200.00

17.488 16.028 14.825 13.819 12.963

0.057182 0.062389 0.067452 0.072363 0.077142

−0.77627 6.8870 15.619 25.539 36.558

10.660 19.365 29.110 40.011 51.987

−0.0018754 0.033304 0.061259 0.085539 0.10734

0.060831 0.070484 0.084735 0.098618 0.11085

0.083362 0.091727 0.10329 0.11458 0.12471

2084.9 1796.1 1568.7 1403.8 1282.8

−0.55991 −0.51205 −0.45996 −0.42037 −0.39117

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Angus, S., Armstrong, B., and de Reuck, K. M., “International Thermodynamic Tables of the Fluid State—7 Propylene,” International Union of Pure and Applied Chemistry, Pergamon Press, Oxford, 1980. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are generally 0.1% in density (except in the critical region), 1% in the heat capacity in the vapor phase, and 2–5% in the heat capacity in the liquid phase.

TABLE 2-250

Thermodynamic Properties of R-11, Trichlorofluoromethane

Temperature K

Pressure MPa

162.68 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 471.11

6.5101E-06 9.0030E-06 5.8433E-05 0.00027433 0.0010018 0.0030014 0.0076770 0.017281 0.035048 0.065240 0.11311 0.18478 0.28718 0.42787 0.61496 0.85703 1.1631 1.5427 2.0061 2.5648 3.2329 4.0318 4.4076

162.68 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 420.00 435.00 450.00 465.00 471.11

6.5101E-06 9.0030E-06 5.8433E-05 0.00027433 0.0010018 0.0030014 0.0076770 0.017281 0.035048 0.065240 0.11311 0.18478 0.28718 0.42787 0.61496 0.85703 1.1631 1.5427 2.0061 2.5648 3.2329 4.0318 4.4076

200.00 296.49

0.10000 0.10000

296.49 300.00 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s 1224.2 1221.1 1178.8 1119.8 1057.5 996.42 937.75 881.67 827.91 776.09 725.77 676.56 628.04 579.81 531.47 482.57 432.58 380.86 326.51 268.20 203.77 132.04 0

Joule-Thomson K/MPa

Saturated Properties 12.874 12.841 12.618 12.392 12.164 11.933 11.699 11.462 11.219 10.971 10.715 10.449 10.172 9.8812 9.5733 9.2442 8.8882 8.4966 8.0545 7.5340 6.8663 5.7930 4.0330 4.8133E-06 6.5631E-06 3.9056E-05 0.00016934 0.00057483 0.0016103 0.0038729 0.0082428 0.015895 0.028297 0.047202 0.074674 0.11315 0.16555 0.23549 0.32766 0.44843 0.60714 0.81886 1.1115 1.5503 2.3952 4.0330

0.077673 0.077878 0.079249 0.080694 0.082210 0.083801 0.085475 0.087246 0.089131 0.091150 0.093329 0.095702 0.098308 0.10120 0.10446 0.10818 0.11251 0.11769 0.12415 0.13273 0.14564 0.17262 0.24796 207,760. 152,370. 25,605. 5,905.3 1,739.7 621.01 258.21 121.32 62.911 35.340 21.186 13.391 8.8379 6.0405 4.2464 3.0519 2.2300 1.6471 1.2212 0.89966 0.64504 0.41751 0.24796

15.035 15.280 16.886 18.531 20.204 21.899 23.613 25.346 27.099 28.875 30.677 32.508 34.371 36.271 38.213 40.203 42.247 44.359 46.558 48.882 51.423 54.535 57.789

15.035 15.280 16.886 18.531 20.204 21.899 23.614 25.347 27.102 28.881 30.687 32.525 34.399 36.315 38.277 40.295 42.378 44.541 46.808 49.223 51.894 55.231 58.881

0.079300 0.080791 0.090108 0.098885 0.10715 0.11494 0.12232 0.12932 0.13600 0.14240 0.14856 0.15452 0.16030 0.16594 0.17145 0.17687 0.18223 0.18756 0.19292 0.19840 0.20422 0.21124 0.21886

0.074722 0.074630 0.075084 0.076137 0.077185 0.078127 0.078986 0.079799 0.080594 0.081391 0.082199 0.083025 0.083873 0.084749 0.085663 0.086633 0.087686 0.088870 0.090270 0.092058 0.094658 0.099494

0.10506 0.10557 0.10851 0.11069 0.11231 0.11364 0.11489 0.11621 0.11767 0.11933 0.12124 0.12343 0.12595 0.12888 0.13232 0.13648 0.14171 0.14871 0.15899 0.17674 0.21832 0.43941

44.924 45.039 45.809 46.618 47.463 48.339 49.242 50.168 51.115 52.075 53.045 54.015 54.979 55.928 56.851 57.736 58.569 59.331 59.992 60.498 60.727 60.210 57.789

46.276 46.411 47.305 48.238 49.206 50.203 51.224 52.265 53.320 54.381 55.441 56.490 57.517 58.512 59.462 60.352 61.163 61.872 62.442 62.805 62.813 61.894 58.881

0.27134 0.26946 0.25910 0.25123 0.24525 0.24074 0.23736 0.23488 0.23310 0.23188 0.23108 0.23060 0.23036 0.23028 0.23030 0.23035 0.23039 0.23035 0.23014 0.22962 0.22849 0.22557 0.21886

0.049536 0.050011 0.052998 0.055869 0.058662 0.061395 0.064063 0.066646 0.069122 0.071480 0.073723 0.075868 0.077940 0.079972 0.081998 0.084057 0.086191 0.088457 0.090937 0.093775 0.097282 0.10241

0.057858 0.058335 0.061344 0.064262 0.067140 0.070004 0.072855 0.075684 0.078487 0.081281 0.084113 0.087058 0.090222 0.093753 0.097861 0.10286 0.10929 0.11813 0.13148 0.15496 0.20995 0.51972

107.23 107.92 112.26 116.42 120.39 124.15 127.67 130.89 133.75 136.17 138.07 139.37 140.01 139.89 138.95 137.09 134.19 130.10 124.58 117.29 107.74 95.245 0

−0.60355 −0.59915 −0.57445 −0.55514 −0.53850 −0.52214 −0.50429 −0.48363 −0.45904 −0.42946 −0.39364 −0.34999 −0.29629 −0.22926 −0.14389 −0.032121 0.11986 0.33760 0.67402 1.2582 2.4973 6.3140 15.427 1885.7 1701.5 905.28 508.88 301.60 188.62 124.75 87.418 64.864 50.774 41.663 35.564 31.344 28.342 26.165 24.577 23.443 22.687 22.277 22.202 22.379 21.779 15.427

Single-Phase Properties 12.317 10.776

2-331

0.042088 0.041523 0.030462 0.024206 0.020110

0.081186 0.092803 23.760 24.083 32.827 41.312 49.728

19.084 30.252

19.092 30.262

0.10168 0.14714

0.076484 0.082009

0.11128 0.12077

1099.8 737.44

−0.54951 −0.40266

52.817 53.078 60.756 69.066 77.853

55.193 55.486 64.039 73.197 82.825

0.23123 0.23221 0.25678 0.27719 0.29474

0.073207 0.073294 0.079843 0.085650 0.089636

0.083443 0.083384 0.088628 0.094219 0.098111

137.67 138.61 161.80 181.31 198.74

43.472 41.094 15.373 9.4764 6.5190

2-332 TABLE 2-250

Thermodynamic Properties of R-11, Trichlorofluoromethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

200.00 300.00 382.43

1.0000 1.0000 1.0000

12.325 10.733 9.0718

0.081134 0.093174 0.11023

19.066 30.637 41.208

19.147 30.730 41.318

382.43 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000

2.6083 2.8311 3.8853 4.8256

58.156 59.743 68.469 77.413

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

12.359 10.810 8.8674 2.1555 1.2189

0.080912 0.092509 0.11277 0.46393 0.82040

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.10159 0.14843 0.17953

0.076374 0.082161 0.087142

0.11122 0.12101 0.13891

1106.2 731.85 457.99

−0.55026 −0.39806 0.036901

60.765 62.574 72.354 82.239

0.23038 0.23501 0.25684 0.27486

0.085101 0.083835 0.086291 0.090004

0.10582 0.10080 0.097662 0.10006

135.79 143.19 172.67 194.29

18.988 30.464 43.167 64.141 75.225

19.393 30.926 43.731 66.460 79.327

0.10120 0.14785 0.18456 0.23416 0.25776

0.075954 0.082048 0.087727 0.094709 0.091726

0.11098 0.12006 0.13909 0.18205 0.11326

1133.5 757.72 448.07 120.07 173.08

−0.55346 −0.41608 0.070413 14.043 7.0186

Single-Phase Properties

0.38339 0.35322 0.25738 0.20723

23.965 19.717 10.081 6.6365

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

12.399 10.899 9.1187 6.3465 3.0124

0.080649 0.091755 0.10966 0.15757 0.33196

18.894 30.263 42.641 56.993 71.945

19.700 31.181 43.737 58.568 75.265

0.10072 0.14716 0.18319 0.21613 0.24670

0.075562 0.082018 0.087335 0.093373 0.093817

0.11072 0.11907 0.13358 0.17100 0.13897

1164.8 787.03 500.70 231.94 167.77

−0.55716 −0.43502 −0.076836 1.9364 5.3068

200.00 300.00 400.00 500.00 600.00

20.000 20.000 20.000 20.000 20.000

12.475 11.057 9.4912 7.6205 5.5967

0.080163 0.090438 0.10536 0.13123 0.17868

18.716 29.905 41.844 54.651 67.668

20.320 31.714 43.951 57.275 71.241

0.099796 0.14590 0.18104 0.21072 0.23619

0.075119 0.082196 0.087334 0.091707 0.094708

0.11027 0.11753 0.12773 0.13823 0.13794

1220.3 838.26 578.74 378.80 265.18

−0.56373 −0.46461 −0.23372 0.31683 1.2664

200.00 300.00 400.00 500.00 600.00

30.000 30.000 30.000 30.000 30.000

12.543 11.197 9.7739 8.2190 6.6706

0.079723 0.089311 0.10231 0.12167 0.14991

18.552 29.591 41.235 53.466 65.781

20.943 32.270 44.304 57.116 70.278

0.098918 0.14476 0.17932 0.20788 0.23188

0.075012 0.082559 0.087676 0.091885 0.095040

0.10990 0.11638 0.12450 0.13097 0.13106

1268.1 882.50 638.80 464.35 357.26

−0.56938 −0.48678 −0.31979 −0.028029 0.34752

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Jacobsen, R. T., Penoncello, S. G., and Lemmon, E. W., “A Fundamental Equation for Trichlorofluoromethane (R-11),” Fluid Phase Equilibria 80:45–56, 1992. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density for the liquid, and 0.25% for the vapor, 2% in heat capacity, and 1% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.2%.

50

180

11

170

160

50

10

11

.

00

10

0. 95 . 0 90 0. 85 0. 80

. 00

7

190

00 12

150

130

.

00

.

.

. 50 12

13

120

110

100

90

13

00

50

.

.

. 00

70

.

450

c.p.

500

1. 0.8

550 20.

10. 8.

0. 60 500. 3 . 400 g/m 00. k ρ= 3 200.

6. 4.

150.

180 170 160 150 140 130 120 110

40

30

20

10

0

10

400

100. 80.

40. 30.

100 90

0.6

20.

80

15.

70

0.4 60

T = 40 °C 30

0.1 0.08

d va rate

0.9

0.8

0.7

0.6

·K

0.01 150

200

0.4

0.2

4.0 3.0

0.1 0.08

2.0

0.06 0.04

250

300

kg

1.0

J/(

5 2.0

0k

0.80

350

400

450

0.60

0.02

2.

10

s=

− 20

2.0

1.9

5

0 1.9

5 1.8

1.8

0

5 1.7

0 1.7

1.65

1.60

1.55

1.50

1.45

1.40

1.35

1.30

1.25

1.20

1.15

1.10

1.05

1.00

0.95

− 10

0.02

0.6

)

sa

0

6.0

1.5

satu

x=

0.5

0.4

0.3

0.2

0.1

ate

dl iqu

id

10

tur

0.04

por

20

0.06

1. 0.8

10. 8.0

50

0.2

2.

60.

30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 T = 200 °C 210 220

Pressure (MPa)

2.

T = −20 °C

4.

350

1500.

1550.

6.

14

145

(trichlorofluoromethane) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

50

10. 8.

0.

R-11

300

140

250

80

200

60

150 20.

500

0.01 550

Enthalpy (kJ/kg) Pressure-enthalpy diagram for Refrigerant 11. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Jacobsen, R. T., Penoncello, S. G., and Lemmon, E. W., “A Fundamental Equation for Trichlorofluoromethane (R-11),” Fluid Phase Equilibria 80:45–56, 1992.

FIG. 2-18

2-333

2-334 TABLE 2-251 Temperature K

Thermodynamic Properties of R-12, Dichlorodifluoromethane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

1310.0 1290.2 1215.7 1143.1 1072.5 1003.9 937.07 872.00 808.46 746.23 685.03 624.56 564.41 504.07 442.82 379.59 312.77 239.96 158.26 0

−0.53048 −0.53554 −0.54646 −0.54643 −0.53813 −0.52327 −0.50270 −0.47654 −0.44424 −0.40455 −0.35528 −0.29292 −0.21173 −0.10203 0.053692 0.29048 0.68965 1.4874 3.7100 13.369

Saturated Properties 116.10 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 385.12

2.4255E-07 5.8432E-07 1.0092E-05 9.1829E-05 0.00053030 0.0021945 0.0070692 0.018793 0.043015 0.087479 0.16186 0.27747 0.44693 0.68394 1.0032 1.4203 1.9528 2.6209 3.4527 4.1362

116.10 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 385.12

2.4255E-07 5.8432E-07 1.0092E-05 9.1829E-05 0.00053030 0.0021945 0.0070692 0.018793 0.043015 0.087479 0.16186 0.27747 0.44693 0.68394 1.0032 1.4203 1.9528 2.6209 3.4527 4.1362

15.125 15.039 14.712 14.387 14.064 13.738 13.409 13.075 12.733 12.380 12.013 11.629 11.223 10.787 10.312 9.7818 9.1671 8.4050 7.3032 4.6728 2.5127E-07 5.8564E-07 8.9910E-06 7.3643E-05 0.00038683 0.0014695 0.0043835 0.010883 0.023468 0.045368 0.080547 0.13381 0.21108 0.32006 0.47149 0.68182 0.97985 1.4275 2.2257 4.6728

0.066114 0.066492 0.067972 0.069505 0.071106 0.072790 0.074574 0.076482 0.078538 0.080776 0.083241 0.085990 0.089104 0.092703 0.096973 0.10223 0.10909 0.11898 0.13693 0.21401 3,979,800. 1,707,500. 111,220. 13,579. 2,585.1 680.49 228.13 91.884 42.611 22.042 12.415 7.4735 4.7376 3.1244 2.1209 1.4667 1.0206 0.70055 0.44930 0.21401

8.0204 8.4220 9.9342 11.419 12.897 14.381 15.881 17.403 18.953 20.533 22.149 23.803 25.502 27.250 29.059 30.943 32.930 35.078 37.570 41.164

8.0204 8.4220 9.9342 11.419 12.897 14.381 15.882 17.405 18.956 20.540 22.162 23.827 25.541 27.314 29.157 31.088 33.143 35.390 38.042 42.049

0.033619 0.037021 0.048899 0.059328 0.068717 0.077328 0.085332 0.092852 0.099978 0.10678 0.11331 0.11961 0.12574 0.13172 0.13762 0.14348 0.14940 0.15556 0.16248 0.17270

0.069219 0.068095 0.065219 0.063937 0.063633 0.063942 0.064638 0.065582 0.066683 0.067882 0.069143 0.070443 0.071775 0.073152 0.074611 0.076246 0.078279 0.081286 0.087150

0.10352 0.10239 0.099608 0.098564 0.098622 0.099406 0.10070 0.10237 0.10436 0.10664 0.10925 0.11226 0.11579 0.12012 0.12572 0.13360 0.14625 0.17199 0.26475

33.279 33.415 33.968 34.567 35.208 35.888 36.602 37.342 38.102 38.873 39.647 40.415 41.165 41.887 42.562 43.166 43.655 43.941 43.738 41.164

34.244 34.413 35.091 35.814 36.579 37.381 38.214 39.069 39.935 40.801 41.657 42.488 43.283 44.024 44.689 45.249 45.648 45.777 45.289 42.049

0.25949 0.25361 0.23524 0.22196 0.21224 0.20511 0.19986 0.19601 0.19322 0.19120 0.18976 0.18873 0.18799 0.18742 0.18693 0.18639 0.18565 0.18441 0.18181 0.17270

0.034581 0.035365 0.038409 0.041427 0.044365 0.047211 0.049977 0.052681 0.055348 0.058002 0.060669 0.063373 0.066138 0.068991 0.071972 0.075158 0.078715 0.083065 0.089754

0.042896 0.043680 0.046725 0.049750 0.052712 0.055621 0.058515 0.061451 0.064497 0.067735 0.071271 0.075247 0.079875 0.085504 0.092789 0.10314 0.12016 0.15653 0.30373

99.513 100.95 106.27 111.28 116.02 120.49 124.63 128.34 131.52 134.05 135.80 136.64 136.46 135.10 132.40 128.14 122.00 113.54 102.14 0

532.61 464.92 288.39 190.47 132.65 96.659 73.304 57.635 46.836 39.210 33.704 29.653 26.640 24.405 22.790 21.703 21.085 20.831 20.225 13.369

Single-Phase Properties 125.00 225.00 243.09

0.10000 0.10000 0.10000

243.09 325.00 425.00 525.00

0.10000 0.10000 0.10000 0.10000

125.00 225.00 314.87

1.0000 1.0000 1.0000

314.87 325.00 425.00 525.00

1.0000 1.0000 1.0000 1.0000

125.00 225.00 325.00 425.00 525.00

5.0000 5.0000 5.0000 5.0000 5.0000

125.00 225.00 325.00 425.00 525.00

10.000 10.000 10.000 10.000 10.000

8.9296 18.951 20.863

8.9363 18.959 20.871

0.041166 0.099970 0.10814

0.066915 0.066686 0.068138

0.10121 0.10435 0.10715

1265.3 808.74 733.55

−0.54064 −0.44443 −0.39528

39.033 44.257 51.527 59.527

40.979 46.920 55.039 63.880

0.19086 0.21187 0.23360 0.25225

0.058549 0.067874 0.076661 0.082795

0.068435 0.076721 0.085212 0.091242

134.48 156.65 179.16 198.90

37.929 16.166 8.3282 5.1283

0.066945 0.078422 0.096933

8.9186 18.922 29.043

8.9856 19.001 29.140

0.041078 0.099842 0.13757

0.066988 0.066739 0.074598

0.10119 0.10418 0.12566

1267.0 813.10 443.35

2.1279 2.2587 3.3167 4.2430

42.556 43.348 51.055 59.210

44.684 45.606 54.372 63.453

0.18693 0.18982 0.21333 0.23250

0.071945 0.071678 0.077527 0.083185

0.092716 0.089701 0.088660 0.092926

132.43 137.48 171.39 195.20

14.968 12.827 10.252 2.2637 1.3295

0.066807 0.077958 0.097546 0.44176 0.75217

8.8706 18.799 29.898 47.937 57.642

9.2046 19.189 30.385 50.146 61.402

0.040691 0.099289 0.14025 0.19225 0.21615

0.067308 0.066963 0.075544 0.083408 0.085038

0.10110 0.10352 0.12368 0.14054 0.10337

1274.3 831.96 450.15 131.42 180.93

−0.54205 −0.45954 0.017671 10.771 5.2339

15.006 12.917 10.544 6.6383 3.0541

0.066639 0.077418 0.094842 0.15064 0.32743

8.8124 18.655 29.472 42.586 55.399

9.4788 19.429 30.420 44.093 58.673

0.040215 0.098631 0.13887 0.17528 0.20629

0.067691 0.067222 0.075613 0.084440 0.087008

0.10099 0.10281 0.11858 0.16445 0.12146

1283.4 854.41 501.05 204.25 177.27

−0.54334 −0.47253 −0.12043 2.4031 4.2532

14.931 12.734 12.306 0.051383 0.037543 0.028469 0.022973 14.938 12.752 10.316 0.46996 0.44273 0.30150 0.23568

0.066977 0.078531 0.081264 19.462 26.636 35.126 43.529

−0.54091 −0.44741 0.052069 22.802 19.919 8.7630 5.1839

225.00 325.00 425.00 525.00

100.00 100.00 100.00 100.00

13.992 12.599 11.358 10.262

0.071470 0.079369 0.088042 0.097444

17.017 26.406 36.302 46.498

24.164 34.343 45.106 56.243

0.090059 0.12742 0.15626 0.17978

0.069779 0.077370 0.084005 0.088827

0.098455 0.10504 0.10984 0.11260

1155.1 937.26 788.21 690.80

−0.55347 −0.49945 −0.45110 −0.41225

225.00 325.00 425.00 525.00

200.00 200.00 200.00 200.00

14.702 13.559 12.582 11.738

0.068017 0.073751 0.079477 0.085195

16.054 25.016 34.535 44.429

29.657 39.766 50.431 61.468

0.083541 0.12065 0.14923 0.17254

0.070631 0.077764 0.084238 0.088994

0.097995 0.10414 0.10881 0.11168

1397.8 1220.9 1091.3 999.41

−0.56210 −0.52944 −0.50685 −0.49445

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Marx, V., Pruss, A., and Wagner, W., “Neue Zustandsgleichungen fuer R 12, R 22, R 11 und R 113. Beschreibung des thermodynamishchen Zustandsverhaltens bei Temperaturen bis 525 K und Druecken bis 200 MPa,” Duesseldorf: VDI Verlag, Series 19 (Waermetechnik/Kaeltetechnik), No. 57, 1992. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.2% below the critical point temperature and increase to 1% in and above the critical region. The uncertainties for vapor pressures are 0.2% above 200 K and greater than 1% below 200 K. The uncertainties in heat capacities and sound speeds are 1% each.

2-335

.

.

00

100

80

90

11

12

400 0.

0.

.

−10

100 90 80 70

500

. 600 . 3 500 g/m 00. k ρ= 4 300.

700

80

90

450

110

.

00

50

50

60

12

13

40

30

13

00

50

.

.

0. 140

145

.

00

10

1500.

0

350

c.p.

−20

1550.

300

−30

2.

−50

4.

−40

6.

(dichlorodifluoromethane) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

−70 1600. T = −60 °C

10. 8.

0.

R-12

250

70

200

20

150

10

2-336

100 20.

550 20.

10. 8. 6.

200. 150.

4.

100. 80.

2.

60.

60

40.

1. 0.8

30.

40 30

0.6

20.

20

1. 0.8 0.6

15. 10

0.4

0.4 10.

0

8.0

−10

0.2

−20 −10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 T = 180 °C 190 200

Pressure (MPa)

50

T = −20 °C

ted va

0.9

0.8

0.7

− 50

0.06 0.04

·K )

200

kg

0.80

J/(

250

300

350

400

0

0.02

2.1

5 2.0

0k 2.0

5

0.60 0.40

s=

1.9

0 1.9

5 1.8

0 1.8

5 1.7

0 1.7

1.65

1.60

1.50 1.55

1.45

1.35 1.40

1.30

1.20

1.25

1.15

1.10

1.00

1.05

0.90

0.95

0.85

0.80

150

0.1 0.08

1.0

− 70

0.01 100

3.0

1.5

− 60

0.75

0.02

0.2

4.0

2.0

satura

x=

0.5

0.4

0.3

0.2

0.1

sat

0.04

0.6

− 40

id ura

ted

liqu

0.06

por

− 30

0.1 0.08

6.0

450

500

0.01 550

Enthalpy (kJ/kg) FIG. 2-19 Pressure-enthalpy diagram for Refrigerant 12. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Marx, V., Pruβ, A., and Wagner, W., “Neue Zustandsgleichungen für R 12, R 22, R 11 und R 113. Beschreibung des thermodynamischen Zustandsverhaltens bei Temperaturen bis 525 K und Drücken bis 200 MPa,” VDI-Fortschritt-Ber. Series 19, No. 57, Düsseldorf: VDI Verlag, 1992.

TABLE 2-252

Thermodynamic Properties of R-13, Chlorotrifluoromethane

Temperature K

Pressure MPa

92.000 100.00 110.00 120.00 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 302.00

3.2889E-07 2.6891E-06 2.3101E-05 0.00013272 0.00056195 0.0018797 0.0052264 0.012543 0.026741 0.051763 0.092557 0.15496 0.24553 0.37145 0.54033 0.76025 1.0397 1.3878 1.8145 2.3316 2.9538 3.7065 3.8790

92.000 100.00 110.00 120.00 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 302.00

3.2889E-07 2.6891E-06 2.3101E-05 0.00013272 0.00056195 0.0018797 0.0052264 0.012543 0.026741 0.051763 0.092557 0.15496 0.24553 0.37145 0.54033 0.76025 1.0397 1.3878 1.8145 2.3316 2.9538 3.7065 3.8790

100.00 175.00 191.43

0.10000 0.10000 0.10000

191.43 250.00 325.00 400.00

0.10000 0.10000 0.10000 0.10000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

918.05 1032.1 1078.3 1072.1 1039.3 992.15 937.71 879.99 821.49 763.71 707.45 653.03 600.48 549.57 499.97 451.26 402.91 354.31 304.57 252.29 194.59 122.38 0

−0.61098 −0.56374 −0.53784 −0.52512 −0.51544 −0.50447 −0.49023 −0.47186 −0.44883 −0.42058 −0.38628 −0.34462 −0.29355 −0.22995 −0.14899 −0.043013 0.10085 0.30583 0.61895 1.1529 2.2680 6.3031 11.123

Saturated Properties 17.840 17.596 17.288 16.976 16.660 16.339 16.012 15.679 15.337 14.987 14.626 14.254 13.866 13.461 13.033 12.577 12.083 11.537 10.919 10.183 9.2260 7.4440 5.5800 4.2996E-07 3.2343E-06 2.5261E-05 0.00013307 0.00052045 0.0016191 0.0042135 0.0095248 0.019245 0.035532 0.060985 0.098644 0.15204 0.22536 0.32372 0.45374 0.62459 0.84999 1.1527 1.5762 2.2289 3.7166 5.5800

0.056054 0.056830 0.057844 0.058907 0.060023 0.061202 0.062451 0.063780 0.065200 0.066725 0.068370 0.070158 0.072118 0.074289 0.076727 0.079511 0.082762 0.086674 0.091587 0.098198 0.10839 0.13434 0.17921 2,325,800. 309,190. 39,587. 7,514.8 1,921.4 617.64 237.33 104.99 51.961 28.144 16.398 10.137 6.5771 4.4373 3.0891 2.2039 1.6010 1.1765 0.86755 0.63445 0.44866 0.26906 0.17921

3.4760 4.1217 4.9721 5.8413 6.7158 7.5918 8.4698 9.3523 10.242 11.143 12.057 12.988 13.939 14.913 15.914 16.947 18.018 19.137 20.319 21.591 23.021 25.005 26.462

3.4760 4.1217 4.9721 5.8413 6.7159 7.5919 8.4701 9.3531 10.244 11.146 12.063 12.999 13.956 14.940 15.955 17.007 18.104 19.258 20.485 21.820 23.341 25.503 27.158

0.0038039 0.010530 0.018633 0.026196 0.033195 0.039687 0.045744 0.051439 0.056835 0.061982 0.066925 0.071701 0.076341 0.080874 0.085328 0.089732 0.094118 0.098526 0.10301 0.10769 0.11280 0.11981 0.12522

0.059797 0.059177 0.057632 0.055848 0.054335 0.053311 0.052818 0.052815 0.053227 0.053972 0.054972 0.056159 0.057477 0.058882 0.060340 0.061829 0.063338 0.064870 0.066451 0.068161 0.070241 0.074099

0.077526 0.083231 0.086329 0.087308 0.087538 0.087670 0.087987 0.088595 0.089527 0.090788 0.092386 0.094338 0.096688 0.099513 0.10294 0.10719 0.11266 0.12009 0.13115 0.15049 0.19837 0.74301

22.976 23.192 23.477 23.781 24.101 24.435 24.782 25.137 25.499 25.864 26.231 26.597 26.959 27.315 27.659 27.987 28.289 28.552 28.753 28.850 28.740 27.886 26.462

23.741 24.023 24.392 24.778 25.181 25.596 26.022 26.454 26.888 27.321 27.749 28.168 28.574 28.963 29.329 29.663 29.954 30.185 30.327 30.329 30.066 28.884 27.158

0.22408 0.20954 0.19518 0.18400 0.17523 0.16829 0.16276 0.15832 0.15474 0.15184 0.14948 0.14755 0.14595 0.14462 0.14347 0.14246 0.14152 0.14055 0.13947 0.13808 0.13599 0.13108 0.12522

0.026152 0.027668 0.029536 0.031384 0.033228 0.035092 0.037002 0.038978 0.041033 0.043168 0.045375 0.047639 0.049946 0.052282 0.054636 0.057009 0.059411 0.061872 0.064450 0.067265 0.070591 0.075537

0.034466 0.035983 0.037856 0.039718 0.041601 0.043546 0.045603 0.047816 0.050218 0.052835 0.055688 0.058800 0.062214 0.066006 0.070318 0.075407 0.081759 0.090357 0.10345 0.12745 0.19031 0.91719

98.239 101.74 105.92 109.91 113.70 117.29 120.64 123.73 126.48 128.85 130.77 132.18 133.00 133.16 132.58 131.16 128.80 125.37 120.73 114.75 107.22 98.019 0

846.97 607.63 414.42 291.57 210.73 155.99 118.01 91.120 71.760 57.628 47.196 39.425 33.599 29.214 25.919 23.463 21.667 20.399 19.555 19.007 18.455 15.921 11.123

Single-Phase Properties 17.598 15.165 14.574

2-337

0.065539 0.048881 0.037259 0.030170

0.056826 0.065942 0.068616 15.258 20.458 26.839 33.145

4.1207 10.689 12.189

4.1264 10.696 12.196

0.010520 0.059427 0.067617

0.059127 0.053557 0.055131

0.083227 0.090103 0.092642

1033.2 792.98 699.56

−0.56370 −0.43564 −0.38082

26.284 29.189 33.487 38.384

27.809 31.235 36.171 41.699

0.14918 0.16476 0.18198 0.19726

0.045695 0.052574 0.061422 0.068694

0.056115 0.061514 0.069986 0.077154

131.00 150.16 170.52 188.46

45.943 17.864 8.3711 5.2121

2-338

TABLE 2-252

Thermodynamic Properties of R-13, Chlorotrifluoromethane (Concluded)

Temperature K

Pressure MPa

100.00 175.00 248.71

1.0000 1.0000 1.0000

248.71 250.00 325.00 400.00

1.0000 1.0000 1.0000 1.0000

100.00 175.00

5.0000 5.0000

250.00 325.00 400.00

5.0000 5.0000 5.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

17.609 15.188 12.149

0.056790 0.065843 0.082311

4.1118 10.666 17.878

4.1686 10.731 17.960

1.6671 1.6848 2.5108 3.2128

28.252 28.338 33.099 38.116

17.658 15.284

0.056633 0.065427

12.436 3.5611 1.8042

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.010431 0.059291 0.093552

0.058689 0.053474 0.063142

0.083187 0.089928 0.11187

1042.6 800.56 409.14

−0.56328 −0.43907 0.079502

29.919 30.023 35.610 41.329

0.14164 0.14205 0.16162 0.17744

0.059099 0.059028 0.062521 0.068870

0.080842 0.080216 0.073958 0.078791

129.16 129.97 162.24 184.36

21.866 21.418 8.9819 5.3350

4.0724 10.564

4.3555 10.891

0.010032 0.058704

0.056929 0.053220

0.083080 0.089221

1081.6 831.21

−0.56120 −0.45278

0.080411 0.28081 0.55425

17.696 29.869 36.726

18.098 31.273 39.497

0.092801 0.13759 0.16061

0.062047 0.071896 0.070112

0.10644 0.18862 0.090875

461.89 120.24 170.26

−0.057256 10.597 5.4004

Single-Phase Properties

0.59985 0.59355 0.39827 0.31126

100.00 175.00 250.00 325.00 400.00

10.000 10.000 10.000 10.000 10.000

17.716 15.396 12.772 8.8242 4.2213

0.056446 0.064954 0.078295 0.11332 0.23689

4.0235 10.446 17.377 25.758 34.655

4.5880 11.096 18.160 26.891 37.023

0.0095294 0.058009 0.091462 0.12181 0.14997

0.055111 0.053119 0.061300 0.069005 0.071938

0.083086 0.088480 0.10203 0.13617 0.11434

1124.6 863.64 518.33 240.23 173.36

−0.55799 −0.46709 −0.17104 1.3924 3.9992

100.00 175.00 250.00 325.00 400.00

15.000 15.000 15.000 15.000 15.000

17.772 15.499 13.041 9.9265 6.3827

0.056268 0.064521 0.076680 0.10074 0.15667

3.9751 10.336 17.115 24.846 32.962

4.8192 11.304 18.265 26.357 35.312

0.0090235 0.057352 0.090334 0.11854 0.14333

0.053642 0.053200 0.061058 0.068084 0.072431

0.083232 0.087860 0.099258 0.11665 0.11694

1162.0 890.77 562.82 322.91 213.68

−0.55407 −0.47907 −0.24338 0.49723 1.9494

175.00 250.00 325.00 400.00

35.000 35.000 35.000 35.000

15.857 13.811 11.718 9.7082

0.063065 0.072404 0.085341 0.10301

9.9615 16.358 23.258 30.364

12.169 18.892 26.245 33.969

0.055005 0.086896 0.11257 0.13395

0.054657 0.062146 0.069359 0.074599

0.086150 0.093881 0.10153 0.10349

962.40 679.17 492.75 383.00

−0.51269 −0.38660 −0.18758 0.022671

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Magee, J. W., Outcalt, S. L., and Ely, J. F., “Molar Heat Capacity C(v), Vapor Pressure, and (p, Rho, T) Measurements from 92 to 350 K at Pressures to 35 MPa and a New Equation of State for Chlorotrifluoromethane (R13),” Int. J. Thermophys. 21(5):1097–1121, 2000. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.15% in density and 2% in heat capacity, except in the critical region. The uncertainty in vapor pressure is 0.1%.

THERMODYNAMIC PROPERTIES TABLE 2-253

2-339

Saturated Refrigerant 13B1, Bromotrifluoromethane*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

170 180 190 200 210

0.059 0.127 0.250 0.455 0.777

4.594.−4 4.677.−4 4.765.−4 4.860.−4 4.961.−4

1.6015 0.7840 0.4190 0.2407 0.1467

−40.90 −34.75 −28.51 −22.17 −15.68

90.95 94.37 97.83 101.32 104.82

−0.2033 −0.1682 −0.1345 −0.1020 −0.0704

0.5723 0.5491 0.5305 0.5154 0.5033

0.597 0.618 0.634 0.648 0.663

9.54 7.60 6.20 5.13 4.33

0.101 0.096 0.091 0.086 0.082

215.4 220 230 240 250

1.013 1.254 1.933 2.863 4.096

5.020.−4 5.071.−4 5.190.−4 5.321.−4 5.466.−4

0.1147 0.0940 0.0628 0.0433 0.0308

−12.09 −9.02 −2.19 4.83 12.03

106.70 108.28 111.68 114.99 118.16

−0.0536 −0.0396 −0.0094 0.0202 0.0494

0.4978 0.4936 0.4857 0.4793 0.4739

0.670 0.676 0.690 0.703 0.721

3.97 3.71 3.22 2.83 2.51

0.079 0.077 0.073 0.068 0.063

260 270 280 290 300

5.690 7.703 10.20 13.25 16.91

5.627.−4 5.809.−4 6.018.−4 6.264.−4 6.562.−4

0.0224 0.0166 0.0124 0.0094 0.0072

19.44 27.06 34.94 43.11 51.68

121.16 123.93 126.41 128.51 130.09

0.0781 0.1064 0.1345 0.1625 0.1908

0.4693 0.4652 0.4612 0.4570 0.4522

0.742 0.767 0.800 0.842 0.891

2.25 2.04 1.84 1.69 1.57

0.059 0.054 0.049 0.045 0.040

310 320 330 340.2c

21.28 26.44 32.48 39.64

6.940.−4 7.458.−4 8.295.−4 1.344.−3

0.0055 0.0041 0.0030 0.0013

60.81 70.80 82.42 108.70

130.97 130.76 128.59 108.70

0.2197 0.2503 0.2845 0.3605

0.4460 0.4376 0.4245 0.3605

0.951 1.09 1.29 ∞

1.45 1.26 0.99 0.35

0.035 0.030 0.026 ∞

T, K

*Values reproduced or converted from Table 4, p. 17.83, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 4.594.−4 signifies 4.594 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 35 bar, −80 to 220 °C. For tables and a chart to 500 psia, 480 °F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

Refrigerant 14 (tetrafluoromethane) See Carbon Tetrafluoride (Table 2-202).

TABLE 2-254

Refrigerant 20 See Chloroform (Table 2-206).

Saturated Refrigerant 21, Dichlorofluoromethane

Temperature, K

Pressure, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

250 260 270 280 290

0.2415 0.3953 0.6200 0.9364 1.3682

0.000 0.000 0.000 0.000 0.000

677 687 698 709 722

0.8292 0.5247 0.3455 0.2355 0.1654

16.6 26.5 36.6 46.7 57.1

274.8 279.9 284.9 290.0 295.0

0.0687 0.1076 0.1454 0.1824 0.2186

1.1015 1.0820 1.0653 1.0511 1.0389

300 310 320 330 340

1.9417 2.6849 3.6279 4.8022 6.2409

0.000 0.000 0.000 0.000 0.000

735 748 763 778 794

0.1192 0.0879 0.0661 0.0505 0.0391

67.7 78.4 89.5 100.7 112.3

300.0 304.8 309.5 314.1 318.4

0.2543 0.2894 0.3242 0.3586 0.3927

1.0286 1.0196 1.0119 1.0051 0.9989

vf, m3/kg

sg, kJ⋅(kg⋅K)

350 360 370 380 390

7.978 10.049 12.489 15.337 18.630

0.000 0.000 0.000 0.000 0.000

812 830 850 870 893

0.0307 0.0243 0.0194 0.0155 0.0125

124.1 136.2 148.6 161.2 173.9

322.4 326.1 329.3 331.9 333.8

0.4266 0.4602 0.4935 0.5264 0.5587

0.9932 0.9877 0.9820 0.9758 0.9688

400 410 420 430 440

22.41 26.72 31.60 37.10 43.26

0.000 0.000 0.000 0.001 0.001

918 944 972 002 034

0.01011 0.00820 0.00672 0.00564 0.00491

186.4 198.3 208.7 216.4 221.1

334.8 334.7 333.7 332.4 332.3

0.5896 0.6180 0.6418 0.6587 0.6682

0.9605 0.9506 0.9394 0.9286 0.9208

Reproduced and rounded from unpublished Center for Applied Thermodynamic Studies, Moscow ID report, 1981. For a thermodynamic diagram to 350 bar, 370 °C, see Rombusch, U. K., Allgem. Warme., 11, 3 (1962).

2-340 TABLE 2-255

Thermodynamic Properties of R-22, Chlorodifluoromethane

Temperature K

Pressure MPa

115.73 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 369.30

3.7947E-07 9.9588E-07 1.7187E-05 0.00015627 0.00089946 0.0037009 0.011835 0.031218 0.070909 0.14319 0.26329 0.44888 0.71966 1.0970 1.6039 2.2661 3.1130 4.1837 4.9900

115.73 120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 369.30

3.7947E-07 9.9588E-07 1.7187E-05 0.00015627 0.00089946 0.0037009 0.011835 0.031218 0.070909 0.14319 0.26329 0.44888 0.71966 1.0970 1.6039 2.2661 3.1130 4.1837 4.9900

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.44463 −0.44526 −0.44448 −0.43872 −0.43084 −0.42063 −0.40608 −0.38505 −0.35555 −0.31561 −0.26263 −0.19248 −0.097827 0.035333 0.23608 0.57205 1.2334 3.0745 10.366

Saturated Properties 19.907 19.777 19.325 18.873 18.420 17.963 17.500 17.028 16.542 16.036 15.506 14.944 14.341 13.686 12.956 12.114 11.069 9.5229 6.0582 3.9436E-07 9.9813E-07 1.5313E-05 0.00012533 0.00065634 0.0024808 0.0073561 0.018161 0.038991 0.075182 0.13342 0.22208 0.35201 0.53822 0.80363 1.1882 1.7777 2.8529 6.0582

0.050235 0.050564 0.051747 0.052985 0.054289 0.055670 0.057141 0.058726 0.060453 0.062359 0.064493 0.066919 0.069730 0.073070 0.077181 0.082547 0.090340 0.10501 0.16506 2,535,700. 1,001,900. 65,305. 7,979.0 1,523.6 403.09 135.94 55.062 25.647 13.301 7.4950 4.5029 2.8409 1.8580 1.2443 0.84159 0.56253 0.35052 0.16506

2.5595 2.9559 4.3400 5.7179 7.0951 8.4715 9.8483 11.230 12.622 14.034 15.472 16.945 18.462 20.034 21.676 23.418 25.325 27.613 30.901

2.5595 2.9559 4.3400 5.7179 7.0951 8.4717 9.8490 11.232 12.627 14.043 15.489 16.975 18.513 20.114 21.800 23.605 25.606 28.053 31.725

0.0065813 0.0099451 0.020814 0.030493 0.039243 0.047227 0.054574 0.061399 0.067804 0.073877 0.079692 0.085308 0.090782 0.096166 0.10152 0.10696 0.11267 0.11931 0.12907

0.061918 0.061567 0.060123 0.059099 0.058356 0.057679 0.057097 0.056707 0.056579 0.056737 0.057171 0.057856 0.058767 0.059887 0.061218 0.062814 0.064858 0.068488

0.092976 0.092700 0.091960 0.091824 0.091789 0.091751 0.091898 0.092431 0.093482 0.095120 0.097391 0.10037 0.10424 0.10941 0.11688 0.12929 0.15550 0.25950

1410.9 1388.4 1312.3 1239.0 1166.5 1095.0 1024.5 954.55 884.79 814.96 744.97 674.69 603.91 532.11 458.37 381.15 298.16 201.90 0

27.807 27.929 28.373 28.840 29.329 29.836 30.357 30.885 31.411 31.928 32.428 32.900 33.335 33.717 34.021 34.207 34.184 33.661 30.901

28.769 28.927 29.495 30.087 30.699 31.328 31.966 32.603 33.229 33.833 34.401 34.922 35.380 35.755 36.017 36.114 35.935 35.127 31.725

0.23305 0.22637 0.20715 0.19295 0.18230 0.17421 0.16800 0.16317 0.15937 0.15634 0.15386 0.15178 0.14996 0.14830 0.14666 0.14486 0.14261 0.13896 0.12907

0.028465 0.028872 0.030386 0.031990 0.033655 0.035388 0.037218 0.039177 0.041300 0.043615 0.046138 0.048881 0.051851 0.055064 0.058574 0.062516 0.067254 0.074178

0.036779 0.037186 0.038703 0.040316 0.042018 0.043849 0.045881 0.048204 0.050920 0.054144 0.058023 0.062763 0.068713 0.076534 0.087659 0.10579 0.14389 0.29995

119.91 121.91 128.58 134.79 140.59 145.98 150.87 155.15 158.70 161.35 162.96 163.38 162.45 159.98 155.73 149.36 140.39 127.92 0

398.80 367.18 269.83 197.57 146.30 110.28 84.902 66.865 53.872 44.348 37.240 31.852 27.725 24.549 22.102 20.201 18.613 16.641 10.366

Single-Phase Properties 150.00 232.06

0.10000 0.10000

232.06 250.00 350.00 450.00 550.00

0.10000 0.10000 0.10000 0.10000 0.10000

150.00 250.00 296.57

1.0000 1.0000 1.0000

296.57 350.00 450.00 550.00

1.0000 1.0000 1.0000 1.0000

150.00 250.00 350.00 450.00 550.00

5.0000 5.0000 5.0000 5.0000 5.0000

5.7167 13.284

5.7220 13.290

0.030484 0.070699

0.059102 0.056618

0.091820 0.094177

1239.3 851.94

−0.43876 −0.33819

31.656 32.442 37.325 43.093 49.620

33.517 34.465 40.211 46.822 54.186

0.15786 0.16180 0.18105 0.19762 0.21238

0.042364 0.043860 0.053191 0.061672 0.068475

0.052366 0.053391 0.061845 0.070141 0.076877

160.06 166.39 196.16 221.08 243.30

49.032 38.154 13.439 6.7638 4.0604

0.052952 0.063648 0.072248

5.7053 14.961 19.669

5.7582 15.024 19.741

0.030408 0.077665 0.094938

0.059134 0.057020 0.059612

0.091780 0.096318 0.10807

1241.8 773.25 548.68

2.0425 2.6523 3.6110 4.5027

33.635 36.754 42.774 49.400

35.677 39.407 46.385 53.903

0.14867 0.16025 0.17776 0.19283

0.054305 0.055369 0.062289 0.068737

0.074523 0.068138 0.072300 0.077971

160.70 184.85 216.22 241.16

25.205 14.183 6.8582 4.0624

18.931 15.837 11.141 1.6422 1.1832

0.052823 0.063142 0.089759 0.60893 0.84520

5.6555 14.822 25.585 41.131 48.377

5.9197 15.138 26.034 44.175 52.603

0.030074 0.077105 0.11341 0.16067 0.17760

0.059277 0.057135 0.064765 0.065284 0.069855

0.091614 0.095206 0.14356 0.087278 0.083655

1253.0 797.37 317.33 194.56 232.93

−0.44109 −0.30700 1.0314 7.0507 3.9600

18.874 16.306 0.053734 0.049441 0.034652 0.026819 0.021901 18.885 15.711 13.841 0.48960 0.37703 0.27693 0.22209

0.052982 0.061325 18.610 20.226 28.859 37.287 45.660

−0.43921 −0.28642 0.00029448

150.00 250.00 350.00 450.00 550.00

10.000 10.000 10.000 10.000 10.000

18.987 15.982 12.008 4.2433 2.5432

0.052667 0.062569 0.083275 0.23566 0.39321

5.5953 14.663 24.782 38.423 47.019

6.1220 15.289 25.615 40.780 50.951

0.029665 0.076450 0.11098 0.14893 0.16944

0.059460 0.057274 0.063647 0.068870 0.071068

0.091420 0.094054 0.11843 0.12461 0.092204

1266.7 825.41 412.24 184.82 229.21

−0.44331 −0.32844 0.39255 5.5220 3.5059

150.00 250.00 350.00 450.00 550.00

30.000 30.000 30.000 30.000 30.000

19.198 16.469 13.518 10.241 7.3810

0.052089 0.060719 0.073976 0.097648 0.13548

5.3741 14.132 23.279 33.086 42.738

6.9367 15.953 25.499 36.016 46.802

0.028113 0.074181 0.10621 0.13259 0.15425

0.060251 0.057799 0.063170 0.069234 0.073417

0.090769 0.091018 0.10053 0.10864 0.10533

1318.0 921.00 603.47 392.35 317.51

−0.45090 −0.38542 −0.13764 0.40072 0.89994

150.00 250.00 350.00 450.00 550.00

60.000 60.000 60.000 60.000 60.000

19.480 17.029 14.650 12.381 10.405

0.051336 0.058724 0.068258 0.080770 0.096108

5.0916 13.533 22.111 31.083 40.152

8.1717 17.056 26.206 35.929 45.919

0.026007 0.071434 0.10216 0.12657 0.14662

0.061560 0.058519 0.063912 0.070220 0.075020

0.089983 0.088686 0.094730 0.099099 0.10030

1384.9 1034.6 764.59 589.21 492.11

−0.45992 −0.42947 −0.31532 −0.17799 −0.055349

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Kamei, A., Beyerlein, S. W., and Jacobsen, R. T., “Application of Nonlinear Regression in the Development of a Wide Range Formulation for HCFC-22,” Int. J. Thermophys. 16:1155–1164, 1995. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 1% in heat capacity, and 0.3% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.2%.

2-341

00

0.

450

.

600

500

550

400.

.

500

3

10. 8.

90

200.

0

6.

150.

100. 80.

80 70

40.

40

15.

10 0

10. 8.0

T = −20 °C

0.2

ted va

0.9

0.8

0.7

0.6

/(k

0.80

0.2

0.1 0.08 0.06 0.04

0.60

2.

20 2.

40

kJ

−70

0.02

0.4

3.0

s=

2.2

0

0 2.1

0

0 1.9

0 1.8

1.70

1.60

1.50

1.40

1.30

1.20

1.10

1.00

0.90

0.80

0.70

−60

0.6

1.0

g·

0.04

4.0

1.5 K)

−50

1. 0.8

6.0

2.0

satura

x=

sat

0.06

0.5

0.4

0.3

0.2

0.1

id

−40

ura

ted

liqu

0.1 0.08

por

−30

−20 −10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 T = 180 °C 190 200

−10

2.0

Pressure (MPa)

20.

20

0.4

2.

30.

30

0.6

4.

60.

60 50

1. 0.8

600 20.

g/m

0. k ρ = 30

c.p.

−10

−20

0.

70

80

90

10

80

70

400

0.

.

.

. 10

11

50

40

350

50

00

50 11

12

125

00

.

.

0.

300

1350.

1400.

−30

−40

−50

T = −60 °C

−80

4.

1450.

1500.

6.

2.

130 10

(chlorodifluoromethane) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

−70

10. 8.

0.

R-22

250

60

200

30

150

20

2-342

100 20.

0.40

0.02

0.30

0.01 100

−80

150

200

250

300

350

400

450

500

550

0.01 600

Enthalpy (kJ/kg) FIG. 2-20 Pressure-enthalpy diagram for Refrigerant 22. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Kamei, A., Beyerlein, S. W., and Jacobsen, R. T., “Application of Nonlinear Regression in the Development of a Wide Range Formulation for HCFC-22,” Int. J. Thermophysics 16:1155–1164, 1995.

TABLE 2-256

Thermodynamic Properties of R-23, Trifluoromethane

Temperature K

Pressure MPa

118.02 120.00 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 299.29

5.8041E-05 8.1883E-05 0.00038808 0.0014276 0.0043019 0.011055 0.024979 0.050823 0.094862 0.16485 0.26991 0.42035 0.62755 0.90390 1.2628 1.7190 2.2887 2.9913 3.8516 4.8317

118.02 120.00 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 299.29

5.8041E-05 8.1883E-05 0.00038808 0.0014276 0.0043019 0.011055 0.024979 0.050823 0.094862 0.16485 0.26991 0.42035 0.62755 0.90390 1.2628 1.7190 2.2887 2.9913 3.8516 4.8317

150.00 190.90

0.10000 0.10000

190.90 250.00 350.00 450.00

0.10000 0.10000 0.10000 0.10000

150.00 242.93

1.0000 1.0000

242.93 250.00 350.00 450.00

1.0000 1.0000 1.0000 1.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

−0.25719 −0.088512 0.75355 1.5903 2.4276 3.2675 4.1116 4.9618 5.8204 6.6899 7.5735 8.4748 9.3983 10.350 11.336 12.370 13.469 14.670 16.074 19.029

−0.25719 −0.088509 0.75356 1.5904 2.4278 3.2680 4.1127 4.9642 5.8250 6.6981 7.5873 8.4969 9.4325 10.401 11.411 12.477 13.621 14.885 16.386 19.672

19.268 19.320 19.581 19.845 20.109 20.372 20.632 20.886 21.133 21.369 21.589 21.791 21.968 22.112 22.214 22.256 22.209 22.017 21.529 19.029

20.249 20.317 20.661 21.005 21.348 21.688 22.020 22.342 22.648 22.935 23.196 23.426 23.617 23.759 23.839 23.835 23.712 23.403 22.727 19.672

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

268.59 261.17 229.32 204.76 185.36 169.64 156.60 145.55 135.98 127.54 119.92 112.91 106.33 100.01 93.771 87.445 80.778 73.344 64.064

2055.0 1889.2 1285.4 927.74 702.09 551.48 446.01 369.03 310.79 265.32 228.80 198.71 173.29 151.31 131.85 114.17 97.624 81.444 64.222

Saturated Properties 24.308 24.217 23.746 23.263 22.771 22.272 21.763 21.243 20.709 20.158 19.584 18.982 18.345 17.660 16.913 16.081 15.123 13.958 12.367 7.5200 5.9176E-05 8.2116E-05 0.00035955 0.0012302 0.0034701 0.0083994 0.017990 0.034921 0.062608 0.10526 0.16798 0.25707 0.38050 0.54888 0.77721 1.0885 1.5223 2.1579 3.2161 7.5200

0.041139 0.041293 0.042112 0.042987 0.043915 0.044900 0.045950 0.047074 0.048288 0.049608 0.051061 0.052680 0.054512 0.056625 0.059125 0.062186 0.066126 0.071643 0.080863 0.13298 16,899. 12,178. 2,781.2 812.86 288.17 119.06 55.587 28.636 15.972 9.5007 5.9532 3.8900 2.6281 1.8219 1.2866 0.91866 0.65691 0.46341 0.31093 0.13298

−0.0049376 −0.0035202 0.0032204 0.0094218 0.015198 0.020619 0.025736 0.030596 0.035238 0.039699 0.044011 0.048206 0.052315 0.056370 0.060409 0.064481 0.068658 0.073078 0.078112 0.088893

0.057095 0.055755 0.052298 0.051128 0.050605 0.050302 0.050114 0.050024 0.050032 0.050140 0.050350 0.050666 0.051093 0.051645 0.052347 0.053248 0.054447 0.056175 0.059136

0.085477 0.084936 0.083783 0.083657 0.083836 0.084188 0.084721 0.085477 0.086504 0.087863 0.089628 0.091906 0.094862 0.098763 0.10409 0.11180 0.12412 0.14787 0.21909

1211.2 1208.8 1168.8 1111.2 1051.8 994.03 938.04 883.23 829.00 774.87 720.45 665.40 609.38 552.02 492.89 431.39 366.67 297.30 220.12 0

0.16881 0.16652 0.15635 0.14810 0.14134 0.13574 0.13107 0.12714 0.12378 0.12088 0.11834 0.11607 0.11399 0.11203 0.11012 0.10816 0.10603 0.10350 0.099978 0.088893

0.026617 0.026770 0.027678 0.028813 0.030153 0.031671 0.033344 0.035155 0.037094 0.039149 0.041312 0.043578 0.045950 0.048442 0.051088 0.053955 0.057163 0.060947 0.065857

0.034991 0.035155 0.036154 0.037439 0.039008 0.040857 0.042992 0.045436 0.048225 0.051416 0.055095 0.059399 0.064561 0.070995 0.079472 0.091578 0.11107 0.14938 0.26628

135.67 136.72 141.80 146.52 150.89 154.87 158.43 161.51 164.02 165.89 167.03 167.35 166.76 165.16 162.43 158.44 153.01 145.86 136.34 0

−0.37494 −0.37515 −0.37102 −0.36298 −0.35278 −0.34018 −0.32445 −0.30465 −0.27966 −0.24798 −0.20758 −0.15554 −0.087415 0.0038879 0.13049 0.31494 0.60438 1.1166 2.2579 8.4094 1947.8 1684.9 854.63 472.27 282.89 182.40 125.40 90.972 68.962 54.182 43.846 36.378 30.839 26.639 23.387 20.801 18.652 16.699 14.524 8.4094

3.7962 3.9025 4.4400 4.9794 5.5230 6.0746 6.6400 7.2268 7.8452 8.5073 9.2284 10.028 10.931 11.970 13.197 14.687 16.578 19.150 23.215

5.3460 5.4612 6.0411 6.6166 7.1868 7.7511 8.3091 8.8611 9.4081 9.9521 10.497 11.048 11.616 12.216 12.873 13.633 14.579 15.906 18.223

Single-Phase Properties 22.773 20.660 0.065782 0.048799 0.034513 0.026771 22.792 17.448

2-343

0.60881 0.57474 0.35949 0.27164

0.043911 0.048402 15.202 20.492 28.974 37.355

2.4262 5.8986

2.4306 5.9034

0.015189 0.035649

0.050608 0.050037

0.083827 0.086612

1052.1 824.11

−0.35289 −0.27709 67.381 22.325 8.0459 4.0690

21.155 23.373 27.705 32.978

22.675 25.422 30.603 36.713

0.12350 0.13604 0.15338 0.16869

0.037275 0.038440 0.048023 0.057058

0.048496 0.047375 0.056526 0.065464

164.22 188.56 220.23 247.21

0.043874 0.057313

2.4134 10.635

2.4573 10.692

0.015103 0.057555

0.050638 0.051834

0.083752 0.10015

1055.5 534.88

1.6425 1.7399 2.7818 3.6813

22.147 22.544 27.401 32.791

23.790 24.284 30.182 36.472

0.11147 0.11347 0.13337 0.14914

0.049200 0.046933 0.048882 0.057361

0.073220 0.067137 0.059316 0.066629

164.48 169.72 214.59 245.15

−0.35386 0.036629 25.601 23.136 8.0837 4.0345

185.41 135.18 7.9031 11.046 16.386 21.735 185.86 98.175 12.308 12.612 17.536 22.720

702.64 306.22 9.4574 12.547 17.318 21.621 707.90 145.37 12.401 12.737 17.376 21.629

2-344 TABLE 2-256

Thermodynamic Properties of R-23, Trifluoromethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

5.0000 5.0000 5.0000 5.0000

22.876 17.302 2.2512 1.4480

0.043714 0.057797 0.44420 0.69062

2.3577 11.094 25.765 31.931

2.5763 11.383 27.986 35.384

0.014729 0.059422 0.11523 0.13386

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.050778 0.051903 0.053415 0.058685

0.083431 0.098833 0.081317 0.072566

1070.0 539.89 190.19 237.91

−0.35795 0.027079 7.7729 3.7699

187.83 97.307 23.888 27.113

731.40 141.75 19.400 22.593

Single-Phase Properties 150.00 250.00 350.00 450.00 150.00 250.00 350.00 450.00

10.000 10.000 10.000 10.000

22.976 17.719 6.4822 3.1053

0.043523 0.056437 0.15427 0.32203

2.2910 10.828 22.835 30.818

2.7262 11.392 24.378 34.039

0.014275 0.058319 0.10116 0.12572

0.050969 0.051625 0.058836 0.060182

0.083065 0.094482 0.14411 0.081111

1086.9 590.19 186.97 235.46

−0.36259 −0.061719 4.7798 3.2093

190.23 101.24 39.852 33.390

761.17 153.45 30.808 25.446

150.00 250.00 350.00 450.00

50.000 50.000 50.000 50.000

23.670 19.633 15.604 12.055

0.042247 0.050935 0.064086 0.082953

1.8477 9.5803 17.587 25.724

3.9600 12.127 20.792 29.871

0.011072 0.052740 0.081836 0.10464

0.052531 0.052051 0.057678 0.064073

0.080990 0.083522 0.089445 0.091553

1193.2 826.45 563.57 437.72

−0.38760 −0.29649 −0.085583 0.13382

207.85 121.46 92.030 77.100

1020.9 226.71 113.25 73.803

20.957 17.954 15.405

0.047717 0.055698 0.064915

8.7360 16.172 23.879

13.508 21.741 30.370

0.048428 0.076094 0.097761

0.053244 0.059172 0.065748

0.080401 0.084434 0.087934

1002.1 778.97 652.70

−0.36685 −0.30018 −0.24430

137.85 111.99 100.34

307.24 167.69 119.14

250.00 350.00 450.00

100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Penoncello, S. G., Lemmon, E. W., Jacobsen, R. T., and Shan, Z., “A Fundamental Equation for Triflurormethane (R-23),” J. Phys. Chem. Ref. Data, 32(4):1473–1499, 2003. The source for viscosity and thermal conductivity is Shan, Z., Penoncello, S. G., and Jacobsen, R. T., “A Generalized Model for Viscosity and Thermal Conductivity of Trifluoromethane (R-23),” ASHRAE Trans. 106:1–11, 2000. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 0.5% in heat capacities and speed of sound, and 0.2% in vapor pressures.Uncertainties in the critical region will be higher. The uncertainty in viscosity is 1%. The uncertainty in thermal conductivity is 2%.

TABLE 2-257

Thermodynamic Properties of R-32, Difluoromethane

Temperature K

Pressure MPa

136.34 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 351.26

4.8000E-05 8.3535E-05 0.00032474 0.0010410 0.0028536 0.0068782 0.014904 0.029545 0.054344 0.093819 0.15345 0.23965 0.35967 0.52157 0.73415 1.0069 1.3501 1.7749 2.2934 2.9194 3.6686 4.5614 5.6311 5.7826

136.34 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 351.26

4.8000E-05 8.3535E-05 0.00032474 0.0010410 0.0028536 0.0068782 0.014904 0.029545 0.054344 0.093819 0.15345 0.23965 0.35967 0.52157 0.73415 1.0069 1.3501 1.7749 2.2934 2.9194 3.6686 4.5614 5.6311 5.7826

150.00 221.24

0.10000 0.10000

221.24 225.00 300.00 375.00

0.10000 0.10000 0.10000 0.10000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

−0.99220 −0.68946 0.13324 0.95053 1.7640 2.5753 3.3862 4.1983 5.0135 5.8337 6.6608 7.4969 8.3443 9.2056 10.084 10.983 11.908 12.866 13.867 14.930 16.088 17.428 19.453 20.836

−0.99220 −0.68946 0.13325 0.95057 1.7641 2.5756 3.3868 4.1995 5.0158 5.8377 6.6675 7.5077 8.3609 9.2303 10.120 11.034 11.979 12.963 13.998 15.107 16.328 17.760 19.977 21.546

21.981 22.076 22.335 22.593 22.850 23.103 23.350 23.588 23.816 24.032 24.234 24.421 24.590 24.738 24.860 24.952 25.006 25.011 24.950 24.797 24.503 23.943 22.380 20.836

23.115 23.239 23.581 23.921 24.258 24.589 24.910 25.219 25.513 25.788 26.042 26.272 26.474 26.643 26.775 26.862 26.894 26.858 26.735 26.491 26.068 25.316 23.365 21.546

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

−0.0054608 −0.0032696 0.0024067 0.0076815 0.012613 0.017251 0.021635 0.025800 0.029778 0.033594 0.037271 0.040830 0.044291 0.047671 0.050989 0.054264 0.057517 0.060775 0.064076 0.067477 0.071088 0.075179 0.081343 0.085769

0.055447 0.054980 0.053793 0.052740 0.051818 0.051021 0.050345 0.049783 0.049333 0.048988 0.048747 0.048604 0.048559 0.048610 0.048761 0.049019 0.049399 0.049934 0.050685 0.051776 0.053487 0.056594 0.066340

0.082847 0.082588 0.081975 0.081513 0.081215 0.081087 0.081137 0.081373 0.081803 0.082443 0.083313 0.084442 0.085874 0.087676 0.089947 0.092852 0.096659 0.10185 0.10938 0.12140 0.14404 0.20457 1.2085

1414.4 1395.1 1342.3 1289.9 1237.6 1185.7 1133.8 1082.1 1030.4 978.59 926.62 874.35 821.65 768.33 714.18 658.88 602.05 543.11 481.27 415.41 343.84 263.77 163.70 0

−0.33760 −0.33728 −0.33542 −0.33191 −0.32650 −0.31891 −0.30886 −0.29600 −0.27988 −0.25996 −0.23550 −0.20551 −0.16864 −0.12292 −0.065518 0.0079177 0.10428 0.23517 0.42163 0.70602 1.1876 2.1660 5.4955 8.0731

242.91 241.74 237.64 232.45 226.39 219.64 212.37 204.70 196.75 188.62 180.39 172.14 163.92 155.78 147.75 139.86 132.11 124.48 116.94 109.42 101.80 94.166 97.067

0.17135 0.16765 0.15872 0.15125 0.14493 0.13955 0.13492 0.13090 0.12738 0.12428 0.12151 0.11901 0.11674 0.11464 0.11268 0.11079 0.10895 0.10709 0.10516 0.10305 0.10060 0.097403 0.091024 0.085769

0.025987 0.026110 0.026507 0.027014 0.027667 0.028505 0.029560 0.030843 0.032341 0.034016 0.035821 0.037709 0.039648 0.041621 0.043631 0.045693 0.047840 0.050119 0.052598 0.055390 0.058707 0.063103 0.071998

0.034319 0.034451 0.034889 0.035477 0.036272 0.037336 0.038728 0.040483 0.042613 0.045105 0.047943 0.051127 0.054696 0.058741 0.063434 0.069063 0.076110 0.085424 0.098649 0.11948 0.15836 0.26199 1.9028

169.60 171.76 177.47 182.88 187.97 192.69 197.00 200.85 204.20 206.99 209.19 210.73 211.57 211.65 210.90 209.26 206.64 202.95 198.02 191.66 183.49 172.68 154.59 0

0.0024002 0.034057

0.053795 0.048953

0.081971 0.082538

1342.7 972.15

0.12392 0.12467 0.13708 0.14749

0.034234 0.033681 0.035327 0.041082

0.045439 0.044513 0.044188 0.049630

207.30 209.39 241.95 267.64

Saturated Properties 27.473 27.302 26.835 26.364 25.889 25.409 24.921 24.424 23.916 23.394 22.858 22.303 21.726 21.124 20.491 19.820 19.102 18.323 17.460 16.477 15.299 13.740 10.732 8.1501 4.2353E-05 7.1788E-05 0.00026061 0.00078411 0.0020270 0.0046295 0.0095503 0.018112 0.032028 0.053428 0.084890 0.12949 0.19093 0.27370 0.38340 0.52726 0.71503 0.96054 1.2848 1.7233 2.3442 3.3211 5.7166 8.1501

0.036399 0.036627 0.037265 0.037930 0.038626 0.039357 0.040127 0.040944 0.041814 0.042745 0.043749 0.044838 0.046028 0.047340 0.048802 0.050454 0.052350 0.054577 0.057273 0.060691 0.065364 0.072779 0.093180 0.12270 23,611. 13,930. 3,837.2 1,275.3 493.35 216.01 104.71 55.213 31.223 18.717 11.780 7.7224 5.2375 3.6537 2.6083 1.8966 1.3985 1.0411 0.77830 0.58029 0.42658 0.30111 0.17493 0.12270

881.12 769.01 541.12 391.73 291.02 221.02 170.81 133.79 106.00 84.924 68.870 56.605 47.194 39.922 34.246 29.758 26.148 23.187 20.693 18.510 16.477 14.312 10.637 8.0731

6.9492 6.9554 7.0006 7.0875 7.2166 7.3887 7.6049 7.8668 8.1765 8.5374 8.9546 9.4365 9.9965 10.656 11.449 12.431 13.691 15.376 17.748 21.309 27.173 38.601 87.141

Single-Phase Properties 26.836 23.329

2-345

0.056727 0.055592 0.040576 0.032244

0.037263 0.042866 17.628 17.988 24.645 31.013

0.13226 5.9359 24.058 24.191 26.760 29.631

0.13599 5.9402 25.821 25.989 29.224 32.733

−0.33547 −0.25719 82.688 76.114 25.024 12.117

237.67 187.60 8.5859 8.7199 12.643 18.907

2-346 TABLE 2-257

Thermodynamic Properties of R-32, Difluoromethane (Concluded)

Temperature K

Pressure MPa

150.00 225.00 279.77

1.0000 1.0000 1.0000

279.77 300.00 375.00

1.0000 1.0000 1.0000

150.00 225.00 300.00 344.33

5.0000 5.0000 5.0000 5.0000

344.33 375.00

5.0000 5.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

26.851 23.159 19.836

0.037243 0.043179 0.050414

0.12350 6.2265 10.962

0.16075 6.2697 11.013

1.9100 2.1692 2.9484

24.951 25.950 29.227

0.037154 0.042923 0.053448 0.078078

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

0.0023417 0.035360 0.054190

0.053807 0.048867 0.049012

0.081935 0.082690 0.092777

1345.7 957.47 660.15

−0.33584 −0.25084 0.0060372

237.89 185.10 140.04

26.861 28.120 32.175

0.11084 0.11518 0.12727

0.045646 0.041298 0.042549

0.068922 0.057916 0.053538

209.31 223.83 259.50

0.085198 6.1394 12.637 18.130

0.27097 6.3541 12.905 18.520

0.0020846 0.034970 0.060001 0.077304

0.053863 0.048922 0.049679 0.059015

0.081784 0.082027 0.096911 0.28240

1358.9 978.81 586.17 224.84

0.25071 0.43051

23.524 26.928

24.777 29.080

0.095475 0.10755

0.065786 0.050835

0.39574 0.090625

166.61 218.44

Single-Phase Properties

0.52357 0.46100 0.33917 26.915 23.298 18.710 12.808 3.9887 2.3228

29.848 23.889 11.929 −0.33741 −0.26135 0.13431 2.9944 13.148 10.800

12.406 13.334 19.059 238.83 187.71 128.72 91.412 48.235 26.239

150.00 225.00 300.00 375.00

10.000 10.000 10.000 10.000

26.993 23.461 19.196 10.448

0.037047 0.042625 0.052094 0.095714

0.038702 6.0369 12.346 21.112

0.40917 6.4632 12.867 22.069

0.0017692 0.034504 0.058997 0.085980

0.053932 0.048996 0.049538 0.057265

0.081608 0.081303 0.092105 0.21222

1375.0 1004.1 639.69 231.14

−0.33924 −0.27287 0.031716 3.5190

239.97 190.81 134.49 77.284

150.00 225.00 300.00 375.00

30.000 30.000 30.000 30.000

27.285 24.030 20.517 16.472

0.036650 0.041614 0.048739 0.060708

−0.13357 5.6813 11.542 17.786

0.96593 6.9297 13.004 19.607

0.00056834 0.032836 0.056105 0.075712

0.054209 0.049307 0.049670 0.052941

0.081027 0.079197 0.083697 0.093069

1436.1 1094.1 789.57 536.98

−0.34524 −0.30661 −0.15824 0.21392

244.02 201.89 152.78 112.94

225.00 300.00 375.00

70.000 70.000 70.000

24.916 22.090 19.309

0.040135 0.045270 0.051788

5.1400 10.583 16.127

7.9495 13.752 19.752

0.030109 0.052355 0.070195

0.049916 0.050281 0.053544

0.076915 0.078370 0.081767

1240.7 986.17 788.81

−0.34341 −0.28333 −0.18583

219.54 179.29 147.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Tillner-Roth, R., and Yokozeki, A., “An International Standard Equation of State for Difluoromethane (R-32) for Temperatures from the Triple Point at 136.34 K to 435 K and Pressures up to 70 MPa,” J. Phys. Chem. Ref. Data 26(6):1273–1328, 1997. Validated equations for the viscosity are not currently available for this fluid. The source for thermal conductivity is unpublished; however, the fit uses the functional form found in Marsh, K., Perkins, R., and Ramires, M. L. V., “Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa,” J. Chem. Eng. Data 47(4):932–940, 2002. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. For the equation of state, typical uncertainties are 0.05% for density, 0.02% for the vapor pressure, and 0.5% to 1% for the heat capacity and speed of sound in the liquid phase. In the vapor phase, the uncertainty in the speed of sound is 0.02%. For thermal conductivity, the estimated uncertainty of the correlation is 5%, except for the dilute gas and points approaching critical where the uncertainty rises to 10%.

0

100

200

300

400

500

600

700

800 20.

20. 0.

0.

0.

3

.

/m 00. kg ρ= 2 150.

300

40

50

70

70

80

900

60

50

40

100 30

20

0.

.

0.

0. 105

10

−10

0.

60

10. 8.

100. 80.

c.p.

1150.

1200.

60.

60

−20

−30

−40

50

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0

0.6

−40

0.4

4.0

0.2

1.0

0.1 0.08

0.80

0.06

K) g· /(k

20 3.

3.

0.30

3.

30

s

400

500

600

0.04

0.40

=

300

0.6

6.0

0.60

kJ 10

00 3.

0 2.9

0 2.8

0 2.7

0

0

0 2.6

2.5

2.4

2.30

2.20

2.00

2.10

1.80

1.90

1.70

1.50

1.60

1.40

1.30

1.20

1.00

1.10

0.90

0.70

0.80

0.60

0.50

0.40

160

140

120

100

80

60

20

vapor saturated

0.9

0.8

0.7

0.6

0.5

.4 x=0

0.3

0.2

0.1

d liq rate satu

−60

0.01 200

8.0

1.5

−80

100

1. 0.8

3.0

−70

0.02

10.

2.0

−50

0.04

0

0.2

−20

T = −30 °C

40

−20

0.4

T = 180 °C

−10

uid

Pressure (MPa)

15.

10

0

2.

30

1. 0.8

0.06

4.

40.

40

20

0.1 0.08

6.

200

2.

T = −50 °C

−70

−80

4.

1250.

1300.

6.

−60

10. 8.

0

1100

.

R-32 (difluoromethane) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

700

0.02

0.20

0.01 800

Enthalpy (kJ/kg) 2-347

FIG. 2-21 Pressure-enthalpy diagram for Refrigerant 32. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Tillner-Roth, R., and Yokozeki, A., “An International Standard Equation of State for Difluoromethane (R-32) for Temperatures from the Triple Point at 136.34 K to 435 K and Pressures up to 70 MPa,” J. Phys. Chem. Ref. Data 26(6):1273–1328, 1997.

2-348 TABLE 2-258

Thermodynamic Properties of R-41, Fluoromethane

Temperature K

Pressure MPa

129.82 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 317.28

0.00034504 0.00035370 0.0012519 0.0036677 0.0092373 0.020585 0.041505 0.077054 0.13355 0.21852 0.34056 0.50928 0.73518 1.0296 1.4048 1.8740 2.4519 3.1548 4.0027 5.0232 5.9062

129.82 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 317.28

0.00034504 0.00035370 0.0012519 0.0036677 0.0092373 0.020585 0.041505 0.077054 0.13355 0.21852 0.34056 0.50928 0.73518 1.0296 1.4048 1.8740 2.4519 3.1548 4.0027 5.0232 5.9062

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

−0.019920 −0.019820 −0.014492 −0.0095735 −0.0050057 −0.00073794 0.0032736 0.0070672 0.010676 0.014131 0.017458 0.020682 0.023828 0.026921 0.029987 0.033058 0.036178 0.039414 0.042906 0.047088 0.055113

0.050428 0.050396 0.048727 0.047265 0.045989 0.044881 0.043923 0.043101 0.042407 0.041832 0.041372 0.041022 0.040783 0.040657 0.040652 0.040786 0.041098 0.041677 0.042763 0.045267

0.072206 0.072196 0.071584 0.071018 0.070568 0.070278 0.070180 0.070303 0.070678 0.071342 0.072345 0.073761 0.075695 0.078315 0.081902 0.086969 0.094564 0.10725 0.13357 0.23071

1413.8 1413.1 1371.7 1325.5 1276.1 1224.2 1170.6 1115.6 1059.3 1001.8 943.19 883.36 822.19 759.47 694.85 627.81 557.39 481.85 397.63 295.99 0

0.12874 0.12858 0.12051 0.11382 0.10819 0.10339 0.099254 0.095640 0.092444 0.089580 0.086979 0.084582 0.082334 0.080186 0.078086 0.075974 0.073776 0.071373 0.068535 0.064565 0.055113

0.025226 0.025228 0.025399 0.025715 0.026225 0.026955 0.027900 0.029030 0.030304 0.031684 0.033143 0.034668 0.036262 0.037939 0.039727 0.041668 0.043822 0.046290 0.049264 0.053237

0.033592 0.033595 0.033848 0.034322 0.035097 0.036230 0.037744 0.039636 0.041897 0.044538 0.047606 0.051200 0.055502 0.060820 0.067681 0.077052 0.090909 0.11407 0.16236 0.33830

205.35 205.49 213.06 220.17 226.74 232.71 238.00 242.53 246.25 249.08 250.96 251.81 251.54 250.07 247.30 243.07 237.22 229.42 219.09 204.65 0

Joule-Thomson K/MPa

Saturated Properties 29.650 29.640 29.080 28.514 27.942 27.362 26.770 26.165 25.544 24.901 24.233 23.534 22.796 22.010 21.164 20.239 19.207 18.018 16.564 14.501 9.3000 0.00031992 0.00032749 0.0010777 0.0029538 0.0070026 0.014779 0.028398 0.050558 0.084573 0.13444 0.20496 0.30200 0.43294 0.60737 0.83844 1.1452 1.5578 2.1293 2.9728 4.4455 9.3000

0.033727 0.033738 0.034388 0.035070 0.035788 0.036548 0.037355 0.038218 0.039148 0.040159 0.041266 0.042492 0.043868 0.045434 0.047251 0.049410 0.052065 0.055501 0.060372 0.068961 0.10753 3125.8 3053.5 927.93 338.55 142.80 67.662 35.214 19.779 11.824 7.4384 4.8791 3.3112 2.3098 1.6464 1.1927 0.87319 0.64192 0.46963 0.33638 0.22495 0.10753

−3.7550 −3.7420 −3.0231 −2.3102 −1.6024 −0.89848 −0.19668 0.50493 1.2085 1.9164 2.6312 3.3561 4.0946 4.8508 5.6304 6.4408 7.2930 8.2059 9.2189 10.456 12.765

−3.7549 −3.7420 −3.0230 −2.3100 −1.6021 −0.89772 −0.19513 0.50787 1.2137 1.9251 2.6453 3.3778 4.1268 4.8976 5.6968 6.5334 7.4207 8.3810 9.4606 10.802 13.400

14.466 14.470 14.716 14.957 15.190 15.411 15.620 15.813 15.988 16.144 16.278 16.388 16.470 16.519 16.527 16.484 16.374 16.167 15.803 15.090 12.765

15.544 15.550 15.878 16.199 16.509 16.804 17.081 17.337 17.567 17.769 17.940 18.075 18.168 18.214 18.203 18.121 17.948 17.649 17.149 16.220 13.400

−0.35314 −0.35309 −0.35022 −0.34606 −0.33979 −0.33071 −0.31808 −0.30109 −0.27874 −0.24979 −0.21256 −0.16475 −0.10306 −0.022535 0.084654 0.23174 0.44318 0.77036 1.3453 2.6641 7.4905 365.81 363.97 278.51 217.09 170.82 135.12 107.41 85.993 69.518 56.880 47.167 39.654 33.780 29.121 25.360 22.257 19.624 17.292 15.070 12.533 7.4905

Single-Phase Properties 150.00 194.60

0.10000 0.10000

194.60 225.00 300.00 375.00

0.10000 0.10000 0.10000 0.10000

150.00 225.00 249.10

1.0000 1.0000 1.0000

249.10 300.00 375.00

1.0000 1.0000 1.0000

150.00 225.00 300.00 309.79

5.0000 5.0000 5.0000 5.0000

309.79 375.00

5.0000 5.0000

−0.0095803 0.0087476

0.047264 0.042767

0.071013 0.070442

17.446 18.604 21.380 24.398

0.094125 0.099660 0.11030 0.11926

0.029600 0.027396 0.029496 0.034142

0.040629 0.036814 0.038140 0.042610

244.34 265.51 305.33 336.84

−2.3206 2.9761 4.7819

−2.2855 3.0179 4.8272

−0.0096430 0.019009 0.026644

0.047252 0.041178 0.040664

0.070965 0.072808 0.078045

1330.7 918.60 765.18

1.6961 2.2815 2.9971

16.516 18.437 21.029

18.212 20.718 24.026

0.080377 0.089572 0.099411

0.037784 0.032042 0.034956

0.060289 0.044524 0.044943

250.26 291.25 329.85

29.499 17.553 9.3564

0.034919 0.041392 0.058628 0.068696

−2.3615 2.8706 9.0202 10.426

−2.1869 3.0775 9.3133 10.769

−0.0099181 0.018535 0.042218 0.046985

0.047205 0.041150 0.042199 0.045181

0.070762 0.071655 0.11693 0.22600

1351.1 952.55 436.81 298.41

−0.34938 −0.21829 1.0494 2.6188

4.4014 2.0056

0.22720 0.49861

15.113 19.699

16.249 22.192

0.064673 0.082450

0.053137 0.038898

0.32991 0.060151

205.02 299.64

12.597 8.3579

28.517 25.882 0.064499 0.054701 0.040421 0.032197 28.539 23.924 22.083 0.58959 0.43831 0.33366 28.637 24.159 17.057 14.557

−2.3112 0.82800

−2.3077 0.83187

15.896 16.776 18.906 21.292

0.035039 0.041800 0.045284

0.035067 0.038637 15.504 18.281 24.739 31.059

1326.0 1089.8

−0.34613 −0.29155 77.876 44.413 18.192 9.5450 −0.34675 −0.19408 −0.030727

150.00 225.00 300.00 375.00

10.000 10.000 10.000 10.000

28.756 24.428 18.494 5.4445

0.034776 0.040937 0.054071 0.18367

−2.4107 2.7503 8.4188 17.496

−2.0629 3.1596 8.9595 19.333

−0.010253 0.017985 0.040105 0.070759

0.047153 0.041136 0.041153 0.043934

0.070531 0.070486 0.090910 0.10163

1375.8 991.68 556.97 274.07

−0.35235 −0.24309 0.45639 6.1896

150.00 225.00 300.00 375.00

30.000 30.000 30.000 30.000

29.189 25.306 20.991 15.862

0.034259 0.039516 0.047639 0.063045

−2.5895 2.3574 7.3266 12.699

−1.5617 3.5429 8.7557 14.590

−0.011513 0.016117 0.036076 0.053392

0.047004 0.041195 0.040629 0.043123

0.069806 0.067503 0.072733 0.082994

1467.3 1123.5 794.05 538.73

−0.36156 −0.30724 −0.067863 0.53694

225.00 300.00 375.00

70.000 70.000 70.000

26.558 23.298 20.079

0.037653 0.042923 0.049804

1.8062 6.3044 10.864

4.4419 9.3090 14.350

0.013266 0.031929 0.046913

0.041529 0.041001 0.043364

0.064822 0.065644 0.068927

1323.8 1056.2 851.76

−0.36575 −0.28880 −0.15494

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data, 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.1% in density (except near the critical point), 0.25% in vapor-pressure, 1% in heat capacities, 0.2% in the vapor-phase speed of sound, and 3% in the liquid speed of sound. The liquid speed of sound uncertainty is an estimate and cannot be verified without experimental information. The uncertainties above 290 K in vapor pressure may be as high as 0.5%.

2-349

2-350

PHYSICAL AND CHEMICAL DATA

TABLE 2-259

Saturated R-401A (SUVA MP 39)

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf , W/(m·K)

Prf

−40 −30 −20 −10 0

0.733 1.155 1.748 2.553 3.615

0.533 0.871 1.361 2.043 2.965

0.000 0.000 0.000 0.000 0.000

712 728 744 762 781

0.3778 0.2391 0.1576 0.1075 0.0755

154.0 164.9 176.2 188.6 200.0

385.0 390.6 396.3 401.8 407.3

0.8188 0.8647 0.9099 0.9577 1.0000

1.8244 1.8059 1.7907 1.7781 1.7675

1.078 1.109 1.137 1.165 1.197

351 323 291 266 241

0.1209 0.1154 0.1107 0.1057 0.1012

3.13 3.06 2.99 2.93 2.85

10 20 30 40 50

4.984 6.712 8.857 11.475 14.628

4.177 5.733 7.697 10.133 13.112

0.000 0.000 0.000 0.000 0.000

803 826 851 878 909

0.0544 0.0399 0.0298 0.0225 0.0172

212.7 225.3 238.3 252.0 266.4

412.6 417.6 422.2 426.5 430.1

1.0454 1.0884 1.1316 1.1752 1.2194

1.7587 1.7510 1.7439 1.7372 1.7304

1.233 1.277 1.329 1.392 1.468

221 202 186 170 157

0.0967 0.0922 0.0877 0.0830 0.0781

2.82 2.80 2.83 2.85 2.95

60 70 80 90 100

18.378 22.79 27.92 33.83 40.53

16.711 21.01 26.12 32.13 39.22

0.000 0.000 0.001 0.001 0.001

944 988 028 084 140

0.01313 0.01005 0.00764 0.00570 0.00403

281.6 297.9 315.9 336.2 361.4

433.0 434.9 435.4 433.5 426.9

1.2647 1.3118 1.3616 1.4163 1.4820

1.7228 1.7138 1.7022 1.6858 1.6584

1.564 1.652 1.802 1.958 2.16

143 131 122 115 110

0.0737 0.0684 0.0631 0.0577 0.0533

3.04 3.16 3.48 3.90 4.46

108.0c

46.04

46.04

0.001 96

0.00196

397

397

Temp., °C

vf, m3/kg

c = critical point. SUVA MP 39 = R401A = CHClF2 (R22) 53% wt + CH3CHF2 (R 152a) 13% wt + CHClFCF3 (R124) 34% wt, near-azeotropic blend. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-260

R-401A (SUVA MP 39) at Atmospheric Pressure

Temp., °C

−27.01

−20

0

20

40

60

80

100

120

140

v (m3/kg) h (kJ/kg) s (kJ/kg⋅K) cp (kJ/kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.2102 351.7 1.8009 0.648 10.17 0.00878 0.750 0.9829

0.2167 396.9 1.8193 0.669 10.43 0.00921 0.758 0.9852

0.2351 410.4 1.8706 0.698 11.18 0.01041 0.750 0.9906

0.2534 424.5 1.9204 0.727 11.93 0.01161 0.749 0.9949

0.2715 439.2 1.9689 0.757 12.68 0.01282 0.749 0.9979

0.2896 454.4 2.0161 0.787 13.42 0.01404 0.748 1.0005

0.3076 470.3 2.0623 0.811 14.17 0.01536 0.748 1.0025

0.3256 486.6 2.1073 0.836 14.89 0.01668 0.748 1.0043

0.3435 503.5 2.1513 0.859 15.61 0.01796 0.747 1.0056

0.3613 521.2 2.1943 0.883 16.32 0.01929 0.747 1.0060

For composition see footnote to Table 2-259. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.

THERMODYNAMIC PROPERTIES TABLE 2-261 Pressure, bar

Thermodynamic Properties of Saturated R-407A (Klea 60) Tf , K

Tg , K

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

227.3 236.1 242.8 248.3 253.0

234.0 242.5 249.1 254.5 259.1

0.000 0.000 0.000 0.000 0.000

7118 7263 7381 7483 7573

0.2097 0.1433 0.1093 0.0885 0.0744

−7.80 3.92 12.89 20.27 26.57

229.64 235.02 239.07 242.35 245.08

0.9965 0.9833 0.9744 0.9679 0.9629

4 5 6 8 10

260.7 267.3 272.9 282.1 289.8

266.8 273.1 278.5 287.5 295.0

0.000 0.000 0.000 0.000 0.000

7735 7880 8012 8254 8480

0.0564 0.0442 0.0384 0.0286 0.0228

37.23 46.12 53.84 67.02 78.23

249.54 253.07 255.24 260.70 263.86

0.9552 0.9496 0.9450 0.9378 0.9318

12.5 15 17.5 20 22.5

297.9 304.8 311.0 316.5 321.4

302.8 309.5 315.4 320.7 325.5

0.000 0.000 0.000 0.000 0.000

8750 9017 9290 9613 9884

0.01802 0.01481 0.01247 0.01069 0.00928

90.50 101.51 111.64 121.18 130.31

266.95 269.12 270.58 271.46 271.79

0.9257 0.9190 0.9128 0.9065 0.8999

25 27.5 30

326.1 330.4 334.5

329.8 333.9 337.6

0.001 023 0.001 063 0.001 115

0.00828 0.00717 0.00635

139.17 147.89 156.58

271.63 270.97 269.81

0.8927 0.8850 0.8765

1 1.5 2 2.5 3

vf , m3/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

hf = sf = 0 at 233.15 K = −40 °C. Converted and interpolated from Thermodynamic Properties of Klea 60 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble point temperature; Tg = dew point temperature.

TABLE 2-262 Pressure, bar

Thermodynamic Properties of Saturated R-407B (Klea 61) Tf , K

Tg , K

vg , m3/kg

hf , kJ/kg

hg , kJ/kg

225.6 234.3 241.8 246.4 251.1

230.0 238.5 245.0 250.4 254.9

0.000 0.000 0.000 0.000 0.000

6852 6994 7110 7211 7301

0.1800 0.1230 0.0937 0.0758 0.0637

−9.45 2.52 9.72 16.59 22.47

191.64 196.90 200.88 204.10 206.80

0.8433 0.8341 0.8282 0.8245 0.8215

4 5 6 8 10

258.9 265.4 270.9 280.2 287.8

262.6 269.0 274.4 283.4 290.9

0.000 0.000 0.000 0.000 0.000

7463 7607 7740 7985 8214

0.04831 0.03888 0.03249 0.02435 0.01936

32.43 40.76 48.00 59.82 70.98

211.22 214.74 217.65 222.21 225.63

0.8172 0.8141 0.8123 0.8080 0.8048

12.5 15 17.5 20 22.5

295.8 302.8 308.8 314.3 319.3

298.7 305.5 311.4 316.7 321.5

0.000 0.000 0.000 0.000 0.000

8491 8768 9053 9353 9680

0.01528 0.01251 0.01049 0.00896 0.00774

82.59 93.02 102.67 111.79 120.55

228.80 231.08 232.64 233.60 233.99

0.8010 0.7971 0.7929 0.7882 0.7829

25 27.5 30

323.9 328.1 332.1

325.9 330.0 333.7

0.001 005 0.001 048 0.001 102

0.00674 0.00590 0.00518

129.11 137.62 146.21

233.85 233.16 231.84

0.7769 0.7700 0.7619

1 1.5 2 2.5 3

vf , m3/kg

sf , kJ/(kg⋅K)

sg , kJ/(kg⋅K)

Converted and interpolated from Thermodynamic Properties of Klea 61 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40°C.

2-351

2-352

PHYSICAL AND CHEMICAL DATA

30 0.9 0

5

0.70 0.75

0 .9

20

R 60

0.80

0

425

1.0

15

0.90

400

1.1 0

10

1.0

5

kJ/ kg

•

K

0.85

8

5

375

350

1.2

6

0

1.1

Pressure, bar

0.95

1.3

0

325

1.2

5

4

1.

35

300 K

40

2

5

1.

275

1

FIG. 2-22

1.4

1.5

250 Enthalpy–log-pressure diagram for R-407A (Klea 60).

300 Enthalpy (h), kJ/kg

350

400

THERMODYNAMIC PROPERTIES

2-353

30 0.80

0.70 0.75

0.8 5

0.80

20

450

0.9 0

0.85 0.90 15

0.9

5

0.95

425

J/k

g

•

K

400

1.0

0k

10

5 1.0

rated

0

375

350

1.1

6

Satu

Pressure, bar

vapo

r

8

1.1

5

4

1.2

0

325

1.2

5

300

30

2

1.3

5

1.

275 1.5 250 K

1 200 FIG. 2-23

250 Enthalpy–log-pressure diagram for R-407B (Klea 61).

300 Enthalpy (h), kJ/kg

350

2-354

PHYSICAL AND CHEMICAL DATA

TABLE 2-263

Saturated R-404A (SUVA HP 62)

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

−50 −40 −30 −20 −10

0.852 1.367 2.095 3.087 4.404

0.821 1.325 2.041 3.018 4.321

0.000 0.000 0.000 0.000 0.000

761 779 799 820 843

0.2244 0.1434 0.0953 0.0656 0.0463

133.1 145.6 159.9 172.8 186.1

337.3 343.8 350.3 356.5 362.6

0.7318 0.7862 0.8460 0.8975 0.9487

1.6487 1.6380 1.6301 1.6245 1.6202

0.0970

1.220 1.260 1.302

370 318 276 238 207

0.0868 0.0834 0.0801

3.88 3.60 3.37

0 10 20 30 40

6.111 8.278 10.977 14.287 18.292

6.013 8.165 10.851 14.150 18.148

0.000 0.000 0.000 0.000 0.001

868 898 933 977 037

0.03338 0.02444 0.01809 0.01348 0.01003

200.0 214.5 229.9 246.2 263.8

368.3 373.6 378.3 382.2 385.0

1.0000 1.0515 1.1038 1.1574 1.2130

1.6188 1.6138 1.6106 1.6065 1.6005

1.351 1.412 1.489 1.592 1.753

181 158 138 122 106

0.0767 0.0733 0.0698 0.0663 0.0624

3.19 3.04 2.94 2.93 2.98

50 60 70 72.1c

23.08 28.75 35.58 37.32

22.94 28.63

0.001 122 0.001 261

386.1 384.2 375.9 361

1.5910 1.5742

2.09

91 76 61

0.0583 0.0535

3.26

0.002 06

283.2 305.8 339.8 361

1.2723 1.3389

37.32

0.00739 0.00527 0.00285 0.00206

Temp., °C

vf, m3/kg

Prf

c = critical point. SUVA HP 62 = CHF2CF3 (R125) 44% wt + CH3CF3 (R143a) 52% wt + CH2FCF3 (R134a) 4% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-264

R-404A (SUVA HP 62) at Atmospheric Pressure

Temp., °C

−45.63

−40

−20

0

20

40

60

80

100

120

v (m3/kg) h (kJ/kg) s (kJ/kg⋅K) cp (kJ/kg⋅K) µ (10−6 Pa⋅s) k (W/m⋅K) Pr Z

0.1866 336.0 1.6599 0.732 9.47 0.00860 0.806 0.9755

0.1921 344.4 1.6636 0.738 9.68 0.00932 0.767 0.9800

0.2100 359.9 1.7274 0.781 10.45 0.01059 0.771 0.9867

0.2278 376.2 1.7891 0.821 11.22 0.01186 0.777 0.9919

0.2455 393.1 1.8491 0.860 11.99 0.01313 0.785 0.9961

0.2630 410.9 1.9076 0.897 12.76 0.01440 0.795 0.9989

0.2805 429.3 1.9646 0.933 13.53 0.01568 0.805 1.0014

0.2980 448.4 2.0203 0.967 14.30 0.01695 0.816 1.0037

0.3153 468.2 2.0747 1.000 15.07 0.01827 0.827 1.0050

0.3325 488.7 2.1278 1.032 15.84 0.01949 0.839 1.0060

v,h, and s from DuPont bull. T—HP62—SI, June 1993 (17 pp.). cp and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

THERMODYNAMIC PROPERTIES TABLE 2-265

2-355

Saturated R-401B (SUVA MP 66)

Pf, bar

Pg, bar

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

−40 −30 −20 −10 0

0.788 1.239 1.872 2.726 3.850

0.585 0.952 1.479 2.212 3.198

0.000 0.000 0.000 0.000 0.000

710 725 740 758 778

0.3498 0.2224 0.1471 0.1008 0.0710

153.8 164.8 176.0 188.6 200.0

386.0 391.6 397.1 402.6 407.8

0.8184 0.8643 0.9095 0.9577 1.0000

1.8291 1.8100 1.7940 1.7807 1.7694

1.078 1.109 1.137 1.165 1.197

349 313 282 257 236

0.1209 0.1154 0.1106 0.1057 0.1012

3.11 3.01 2.90 2.83 2.79

10 20 30 40 50

5.297 7.120 9.379 12.133 15.444

4.491 6.146 8.229 10.808 13.955

0.000 0.000 0.000 0.000 0.000

801 827 858 895 939

0.05124 0.03771 0.02818 0.02131 0.01625

212.6 225.1 238.2 251.9 266.3

412.9 417.7 422.1 426.1 429.4

1.0450 1.0879 1.1311 1.1747 1.2190

1.7598 1.7512 1.7433 1.7357 1.7278

1.233 1.277 1.329 1.392 1.468

217 198 181 168 151

0.0967 0.0922 0.0877 0.0830 0.0781

2.77 2.74 2.74 2.82 2.84

60 70 80 90 100

19.378 24.00 29.37 35.55 42.30

17.750 22.28 27.64 33.96

0.000 0.001 0.001 0.001

994 066 164 313

0.01244 0.00951 0.00721 0.00534

281.6 298.1 316.3 337.2

431.9 433.4 433.2 430.4

1.2645 1.3120 1.3625 1.4187

1.7191 1.7088 1.6956 1.6768

1.564 1.652 1.802

139 127 116

0.0737 0.0684 0.0631 0.0577 0.0533

2.95 3.07 3.31

106.1c

46.82

46.82

0.001 95

0.00195

389

389

Temp., °C

vf, m3/kg

c = critical point. SUVA MP 66 = R-401B = CHClF2 (R22) 61% wt + CH3CHF2 (R152a) 11% wt + CHClFCF3 (R124) 28% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts are approximate.

TABLE 2-266 Temp., °C 3

v (m /kg) h (kJ/kg) s (kJ/kg⋅K) cp (kJ/kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

R-401B (SUVA MP 66) at Atmospheric Pressure −28.63b

−20

0

20

40

60

80

100

120

140

0.2086 392.2 1.8081 0.641 9.78 0.00817 0.767 0.9652

0.2177 397.9 1.8299 0.652 10.43 0.00921 0.738 0.9730

0.2362 411.2 1.8804 0.688 11.18 0.01041 0.737 0.9783

0.2545 425.1 1.9295 0.716 11.93 0.01161 0.736 0.9822

0.2727 439.6 1.9772 0.744 12.68 0.01282 0.735 0.9852

0.2908 454.6 2.0237 0.771 13.42 0.01404 0.735 0.9876

0.3089 470.1 2.0690 0.796 14.17 0.01536 0.734 0.9896

0.3269 486.2 2.1132 0.822 14.89 0.01668 0.734 0.9912

0.3449 502.7 2.1564 0.844 15.61 0.01796 0.733 0.9925

0.3629 519.4 2.1986 0.866 16.32 0.01929 0.733 0.9937

v, h, and s from DuPont bull. T—MP 66—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 10, Jan. 1993 (27 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts. b = normal boiling point.

TABLE 2-267 Temp., °C

Saturated R-402A (SUVA HP 80) vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

1.6327 1.6206 1.6110 1.6034 1.5972

0.0970

1.193 1.217 1.236

377 317 283 247 215

0.0880 0.0849 0.0813

3.84 3.54 3.27

1.0000 1.0461 1.0927 1.1403 1.1897

1.5919 1.5870 1.5820 1.5762 1.5690

1.253 1.286 1.340 1.412 1.512

188 165 146 128 113

0.0778 0.0743 0.0708 0.0672 0.0634

3.03 2.86 2.76 2.69 2.70

1.2420 1.2998

1.5589 1.5433

1.64 1.81

98 83 68

0.0593 0.0551

2.71 2.79

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

679 695 713 733 757

0.2033 0.1303 0.0869 0.0598 0.0423

139.6 150.8 163.1 174.9 187.6

334.1 339.9 345.6 351.1 356.4

0.7578 0.8070 0.8584 0.9053 0.9541

785 819 860 911 977

0.03060 0.02248 0.01671 0.01250 0.00936

200.0 213.0 226.7 241.2 256.8

361.3 365.9 369.8 373.1 375.4

24.04 29.97

0.001 070 0.001 212

0.00696 0.00505

273.9 293.6

376.2 374.6

41.35

0.001 850

0.00185

340

340

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.962 1.520 2.305 3.370 4.776

0.872 1.403 2.156 3.188 4.560

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40

6.588 8.877 11.720 15.195 19.388

6.336 8.592 11.404 14.855 19.034

0.000 0.000 0.000 0.000 0.000

50 60 70 75.5c

24.39 30.30 41.35

Prf

c = critical point. SUVA HP 80 = R-402A = CHF2CF3 (R125) 60% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 38% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values, read from charts, may be approximate.

2-356

PHYSICAL AND CHEMICAL DATA

TABLE 2-268

R-402A (SUVA HP 80) at Atmospheric Pressure

Temp., °C

−46.95b

−40

−20

0

20

40

60

80

100

120

v (m3/kg) h (kJ/kg) s (kJ/kg⋅K) cp (kJ/kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

0.1768 335.9 1.6286 0.648 9.42 0.00888 0.687 0.9673

0.1827 340.5 1.6490 0.654 9.69 0.00932 0.680 0.9697

0.1996 354.3 1.7055 0.687 10.45 0.01059 0.678 0.9758

0.2164 368.6 1.7599 0.721 11.22 0.01186 0.681 0.9804

0.2331 383.5 1.8124 0.749 11.99 0.01313 0.685 0.9840

0.2497 398.7 1.8633 0.779 12.75 0.01440 0.690 0.9868

0.2663 414.9 1.9128 0.807 13.52 0.01568 0.696 0.9892

0.2828 431.4 1.9610 0.836 14.29 0.01695 0.703 0.9910

0.2992 448.5 2.0081 0.863 15.06 0.01822 0.713 0.9923

0.3155 466.1 2.0541 0.890 15.82 0.01949 0.722 0.9932

b = normal boiling pt. v, h, and s from DuPont bull. T—HP 80—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-269 Temp., °C

Saturated R-402B (SUVA HP 81) vf, m3/kg

cpf, kJ/(kg⋅K)

µf, 10−6 Pa·s

kf, W/(m⋅K)

Prf

1.7122 1.6957 1.6820 1.6706 1.6611

1.178 1.191 1.204

383 333 290 253 223

0.1031 0.0983 0.0941 0.0900 0.0863

3.63 3.35 3.11

1.0000 1.0450 1.0905 1.1367 1.1842 1.2339

1.6528 1.6451 1.6376 1.6299 1.6211 1.6104

1.221 1.288 1.313 1.37 1.75 2.07

195 173 151 137 122 106

0.0818 0.0790 0.0753 0.0715 0.0676 0.0633

2.91 2.82 2.63 2.49 3.16 3.47

1.2873 1.3164

1.5961 1.5866

91 75

0.0586 0.0544

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

687 702 719 739 761

0.2425 0.1548 0.1028 0.0707 0.0499

140.3 151.4 163.3 174.9 187.8

351.7 357.2 362.7 368.0 373.0

0.7606 0.8092 0.8589 0.9054 0.9550

787 817 854 899 955 030

0.03610 0.02656 0.01980 0.01490 0.01125 0.00848

200.0 212.7 226.0 240.1 255.1 271.4

377.8 382.2 386.0 389.3 391.5 392.8

28.03 34.60

0.001 136 0.001 307

0.00632 0.00456

289.5 299.6

392.2 390.9

44.45

0.001 88

0.00188

351

351

Pf, bar

Pg, bar

−50 −40 −30 −20 −10

0.883 1.403 2.135 3.132 4.451

0.787 1.273 1.967 2.923 4.198

0.000 0.000 0.000 0.000 0.000

0 10 20 30 40 50

6.153 8.307 10.984 14.261 18.216 22.93

5.852 7.959 10.591 13.827 17.750 22.45

0.000 0.000 0.000 0.000 0.000 0.001

60 70 80 82.6c

28.50 35.01 44.45

c = critical point. SUVA HP 81 = R402B (38/2/60) = CHF2CF3 (R125) 38% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 60% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.

TABLE 2-270 Temp., °C 3

v (m /kg) h (kJ/kg) s (kJ/kg⋅K) cp (kJ/kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z

R-402B (SUVA HP 81) at Atmospheric Pressure −44.87b

−40

−20

0

20

40

60

80

100

120

0.1903 354.7 1.7032 1.187 10.16 0.00739 1.632 0.9622

0.1960 357.7 1.7169 1.177 10.33 0.00768 1.583 0.9703

0.2142 370.8 1.7711 1.169 11.10 0.00902 1.439 0.9766

0.2322 384.6 1.8232 1.159 11.86 0.01036 1.327 0.9811

0.2500 398.8 1.8735 1.149 12.62 0.01170 1.239 0.9843

0.2678 413.6 1.9222 1.143 13.39 0.01304 1.174 0.9870

0.2856 428.9 1.9696 1.134 14.15 0.01438 1.124 0.9894

0.3032 444.7 2.0158 1.128 14.78 0.01572 1.061 0.9909

0.3209 461.0 2.0607 1.124 15.54 0.01706 1.024 0.9926

0.3386 477.7 2.1047 1.120 16.30 0.01840 0.992 0.9940

b = normal boiling point. v, h, and s from DuPont bull. T—HP 81—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values, read from charts, may be approximate. Material used by permission of DuPont Fluoroproducts.

TABLE 2-271 Temperature K

Thermodynamic Properties of R-113, 1,1,2-Trichlorotrifluoroethane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Saturated Properties

2-357

236.93 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 440.00 450.00 460.00 470.00 480.00 487.21

0.0018714 0.0022999 0.0043168 0.0076484 0.012885 0.020766 0.032185 0.048190 0.069977 0.098880 0.13636 0.18400 0.24349 0.31660 0.40521 0.51127 0.63682 0.78398 0.95499 1.1522 1.3781 1.6354 1.9271 2.2568 2.6288 3.0498 3.3923

9.0987 9.0614 8.9395 8.8172 8.6941 8.5702 8.4450 8.3183 8.1897 8.0588 7.9252 7.7884 7.6479 7.5029 7.3529 7.1968 7.0336 6.8619 6.6797 6.4845 6.2728 6.0390 5.7742 5.4624 5.0712 4.5104 2.9887

236.93 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 440.00 450.00 460.00 470.00 480.00 487.21

0.0018714 0.0022999 0.0043168 0.0076484 0.012885 0.020766 0.032185 0.048190 0.069977 0.098880 0.13636 0.18400 0.24349 0.31660 0.40521 0.51127 0.63682 0.78398 0.95499 1.1522 1.3781 1.6354 1.9271 2.2568 2.6288 3.0498 3.3923

0.00095199 0.0011554 0.0020850 0.0035597 0.0057917 0.0090358 0.013590 0.019793 0.028029 0.038727 0.052364 0.069475 0.090662 0.11662 0.14813 0.18615 0.23182 0.28656 0.35216 0.43104 0.52654 0.64356 0.78990 0.97939 1.2415 1.6684 2.9887

0.10991 0.11036 0.11186 0.11342 0.11502 0.11668 0.11841 0.12022 0.12211 0.12409 0.12618 0.12840 0.13076 0.13328 0.13600 0.13895 0.14217 0.14573 0.14971 0.15421 0.15942 0.16559 0.17318 0.18307 0.19719 0.22171 0.33460

31.483 31.985 33.627 35.279 36.945 38.626 40.323 42.036 43.767 45.516 47.284 49.071 50.879 52.708 54.560 56.437 58.340 60.271 62.235 64.236 66.280 68.379 70.548 72.817 75.247 78.016 82.401

31.484 31.986 33.627 35.280 36.947 38.629 40.327 42.042 43.776 45.528 47.301 49.095 50.911 52.751 54.616 56.508 58.430 60.386 62.378 64.414 66.500 68.650 70.881 73.230 75.765 78.692 83.536

0.16386 0.16596 0.17266 0.17914 0.18543 0.19154 0.19750 0.20331 0.20898 0.21453 0.21998 0.22531 0.23055 0.23571 0.24079 0.24579 0.25074 0.25564 0.26050 0.26533 0.27015 0.27500 0.27990 0.28494 0.29024 0.29621 0.30602

0.11672 0.11694 0.11797 0.11931 0.12084 0.12245 0.12410 0.12576 0.12740 0.12901 0.13060 0.13215 0.13367 0.13517 0.13665 0.13813 0.13961 0.14112 0.14266 0.14428 0.14601 0.14794 0.15016 0.15288 0.15638 0.16109

0.16341 0.16366 0.16467 0.16593 0.16735 0.16891 0.17056 0.17231 0.17414 0.17606 0.17807 0.18020 0.18247 0.18491 0.18755 0.19047 0.19374 0.19749 0.20191 0.20730 0.21416 0.22344 0.23712 0.25998 0.30706 0.47179

907.79 897.63 863.61 828.94 794.26 759.92 726.09 692.84 660.17 628.04 596.38 565.13 534.20 503.50 472.94 442.41 411.80 380.97 349.77 317.99 285.39 251.65 216.41 179.46 141.14 101.80 0

−0.45956 −0.45692 −0.44732 −0.43622 −0.42364 −0.40951 −0.39367 −0.37590 −0.35588 −0.33321 −0.30739 −0.27774 −0.24340 −0.20323 −0.15569 −0.098686 −0.029233 0.057005 0.16660 0.31000 0.50481 0.78295 1.2078 1.9196 3.2778 6.5781 19.849

1050.4 865.51 479.61 280.92 172.66 110.67 73.586 50.523 35.677 25.822 19.097 14.394 11.030 8.5752 6.7508 5.3719 4.3136 3.4897 2.8396 2.3200 1.8992 1.5539 1.2660 1.0210 0.80548 0.59937 0.33460

61.277 61.580 62.580 63.602 64.643 65.701 66.774 67.860 68.957 70.061 71.172 72.286 73.400 74.513 75.621 76.720 77.808 78.878 79.926 80.944 81.922 82.844 83.689 84.417 84.942 85.005 82.401

63.243 63.570 64.650 65.750 66.868 67.999 69.143 70.295 71.453 72.615 73.776 74.934 76.086 77.228 78.356 79.467 80.555 81.614 82.638 83.617 84.539 85.386 86.129 86.721 87.059 86.833 83.536

0.29790 0.29756 0.29675 0.29634 0.29625 0.29644 0.29686 0.29748 0.29826 0.29918 0.30020 0.30131 0.30248 0.30370 0.30495 0.30621 0.30747 0.30871 0.30991 0.31105 0.31210 0.31304 0.31379 0.31426 0.31427 0.31317 0.30602

0.098915 0.099759 0.10244 0.10502 0.10751 0.10994 0.11231 0.11463 0.11691 0.11917 0.12140 0.12362 0.12583 0.12804 0.13024 0.13246 0.13469 0.13695 0.13926 0.14165 0.14414 0.14679 0.14969 0.15299 0.15703 0.16277

0.10733 0.10819 0.11094 0.11361 0.11623 0.11883 0.12142 0.12402 0.12666 0.12935 0.13214 0.13503 0.13807 0.14130 0.14477 0.14857 0.15281 0.15765 0.16333 0.17026 0.17911 0.19114 0.20894 0.23897 0.30270 0.54016

106.58 107.21 109.17 111.03 112.77 114.38 115.82 117.08 118.14 118.97 119.57 119.89 119.93 119.66 119.05 118.07 116.69 114.87 112.57 109.72 106.25 102.05 97.003 90.902 83.480 74.333 0

82.183 77.881 66.150 57.141 50.111 44.541 40.065 36.421 33.419 30.923 28.833 27.074 25.594 24.353 23.322 22.480 21.816 21.322 21.000 20.858 20.915 21.204 21.779 22.716 24.091 25.627 19.849

2-358 TABLE 2-271

Thermodynamic Properties of R-113, 1,1,2-Trichlorotrifluoroethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

250.00 320.34

0.10000 0.10000

8.9405 8.0543

320.34 325.00 400.00 475.00

0.10000 0.10000 0.10000 0.10000

0.039138 0.038492 0.030653 0.025598

250.00 325.00 400.00 412.41

1.0000 1.0000 1.0000 1.0000

8.9494 8.0093 6.8729 6.6340

412.41 475.00

1.0000 1.0000

250.00 325.00 400.00 475.00

5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties

250.00 325.00 400.00 475.00

10.000 10.000 10.000 10.000

33.623 45.576

33.634 45.588

0.17265 0.21472

0.11797 0.12907

0.16466 0.17612

864.02 626.96

−0.44747 −0.33239

70.099 70.661 80.120 90.324

72.654 73.259 83.382 94.230

0.29921 0.30109 0.32908 0.35391

0.11925 0.11993 0.13100 0.14017

0.12945 0.12998 0.14005 0.14891

119.00 120.05 135.10 148.01

30.846 29.068 14.858 9.5374

0.11174 0.12485 0.14550 0.15074

33.589 46.338 60.238 62.713

33.701 46.463 60.383 62.864

0.17251 0.21708 0.25555 0.26166

0.11802 0.12985 0.14107 0.14304

0.16456 0.17671 0.19705 0.20311

867.87 618.26 384.09 342.18

−0.44882 −0.32646 0.047064 0.19754

0.36984 0.28704

2.7039 3.4839

80.174 89.325

82.878 92.809

0.31019 0.33262

0.13983 0.14241

0.16487 0.15696

111.93 133.62

20.948 11.300

8.9881 8.0819 7.0485 5.5199

0.11126 0.12373 0.14187 0.18116

33.441 46.082 59.694 74.959

33.998 46.701 60.403 75.865

0.17191 0.21629 0.25417 0.28950

0.11826 0.13008 0.14058 0.15167

0.16413 0.17538 0.19113 0.23166

884.22 644.07 433.49 217.18

−0.45442 −0.34900 −0.086367 1.2219

9.0343 8.1644 7.2197 6.0609

0.11069 0.12248 0.13851 0.16499

33.266 45.791 59.149 73.507

34.373 47.016 60.534 75.157

0.17119 0.21536 0.25274 0.28621

0.11861 0.13042 0.14040 0.14885

0.16368 0.17407 0.18683 0.20446

903.13 672.87 481.58 313.71

−0.46053 −0.37126 −0.18229 0.32498

970.17 867.10 788.68

−0.48114 −0.45624 −0.42986

0.11185 0.12416 25.551 25.979 32.623 39.065

325.00 400.00 475.00

100.00 100.00 100.00

9.0570 8.5204 8.0191

0.11041 0.11737 0.12470

42.816 54.955 67.454

53.857 66.691 79.924

0.20444 0.23995 0.27026

0.13658 0.14351 0.14864

0.16811 0.17396 0.17872

475.00

200.00

8.7846

0.11384

65.196

87.963

0.26222

0.14928

0.17693

1035.9

−0.46436

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Marx, V., Pruss, A., and Wagner, W., “Neue Zustandsgleichungen fuer R 12, R 22, R 11 und R 113. Beschreibung des thermodynamishchen Zustandsverhaltens bei Temperaturen bis 525 K und Druecken bis 200 MPa,” Duesseldorf: VDI Verlag, Series 19 (Waermetechnik/Kaeltetechnik), No. 57, 1992. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density is 0.2%, that for vapor pressure is 0.3%, and that for the isobaric heat capacity is 2%. The uncertainties are higher in and above the critical region.

TABLE 2-272

Thermodynamic Properties of R-114, 1,2-Dichlorotetrafluoroethane

Temperature K

Pressure MPa

Density mol/dm3

275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 345.00 350.00 355.00 360.00 365.00 370.00 375.00 380.00 385.00 390.00 395.00 400.00 405.00 410.00 415.00 418.83

0.094764 0.11455 0.13740 0.16362 0.19353 0.22746 0.26574 0.30872 0.35677 0.41024 0.46953 0.53501 0.60708 0.68615 0.77265 0.86701 0.96967 1.0811 1.2018 1.3323 1.4730 1.6246 1.7878 1.9630 2.1511 2.3530 2.5694 2.8015 3.0507 3.2516

8.9110 8.8268 8.7415 8.6548 8.5667 8.4771 8.3858 8.2927 8.1976 8.1004 8.0007 7.8984 7.7932 7.6847 7.5726 7.4565 7.3358 7.2098 7.0779 6.9389 6.7916 6.6345 6.4651 6.2804 6.0759 5.8442 5.5731 5.2381 4.7774 3.3932

275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 345.00 350.00 355.00 360.00 365.00 370.00 375.00 380.00 385.00 390.00 395.00 400.00

0.094764 0.11455 0.13740 0.16362 0.19353 0.22746 0.26574 0.30872 0.35677 0.41024 0.46953 0.53501 0.60708 0.68615 0.77265 0.86701 0.96967 1.0811 1.2018 1.3323 1.4730 1.6246 1.7878 1.9630 2.1511 2.3530

0.043032 0.051392 0.060964 0.071869 0.084238 0.098213 0.11395 0.13161 0.15138 0.17347 0.19810 0.22553 0.25605 0.28999 0.32774 0.36975 0.41655 0.46878 0.52722 0.59285 0.66690 0.75097 0.84722 0.95861 1.0895 1.2467

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.11222 0.11329 0.11440 0.11554 0.11673 0.11796 0.11925 0.12059 0.12199 0.12345 0.12499 0.12661 0.12832 0.13013 0.13205 0.13411 0.13632 0.13870 0.14129 0.14412 0.14724 0.15073 0.15468 0.15922 0.16459 0.17111 0.17943 0.19091 0.20932 0.29471

34.479 35.301 36.128 36.960 37.798 38.641 39.491 40.348 41.211 42.082 42.960 43.846 44.739 45.642 46.553 47.475 48.407 49.350 50.306 51.276 52.261 53.265 54.291 55.343 56.428 57.557 58.749 60.042 61.532 64.391

34.490 35.314 36.143 36.979 37.820 38.668 39.523 40.385 41.255 42.133 43.019 43.913 44.817 45.731 46.655 47.591 48.539 49.500 50.476 51.468 52.478 53.510 54.567 55.655 56.782 57.960 59.211 60.577 62.171 65.349

23.239 19.458 16.403 13.914 11.871 10.182 8.7760 7.5984 6.6059 5.7648 5.0480 4.4340 3.9055 3.4484 3.0512 2.7045 2.4007 2.1332 1.8967 1.6868 1.4995 1.3316 1.1803 1.0432 0.91784 0.80213

55.626 56.123 56.622 57.121 57.620 58.119 58.617 59.113 59.608 60.100 60.589 61.074 61.554 62.029 62.497 62.957 63.408 63.849 64.277 64.690 65.086 65.461 65.810 66.126 66.401 66.621

57.828 58.352 58.875 59.397 59.917 60.435 60.949 61.459 61.965 62.465 62.959 63.446 63.925 64.395 64.855 65.302 65.736 66.155 66.557 66.938 67.295 67.624 67.920 68.174 68.376 68.509

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.17204 0.17500 0.17792 0.18082 0.18368 0.18652 0.18933 0.19212 0.19488 0.19763 0.20035 0.20306 0.20575 0.20843 0.21109 0.21375 0.21640 0.21904 0.22169 0.22434 0.22700 0.22967 0.23237 0.23511 0.23790 0.24078 0.24379 0.24704 0.25078 0.25828

0.11622 0.11691 0.11763 0.11838 0.11917 0.11997 0.12079 0.12162 0.12247 0.12333 0.12419 0.12507 0.12595 0.12684 0.12775 0.12866 0.12959 0.13054 0.13152 0.13252 0.13356 0.13465 0.13581 0.13705 0.13842 0.13997 0.14180 0.14408 0.14726

0.16411 0.16513 0.16623 0.16741 0.16866 0.16998 0.17138 0.17286 0.17441 0.17606 0.17781 0.17967 0.18167 0.18383 0.18617 0.18875 0.19162 0.19485 0.19854 0.20284 0.20795 0.21417 0.22201 0.23227 0.24644 0.26750 0.30245 0.37230 0.57876

636.17 617.82 599.54 581.32 563.14 545.01 526.89 508.80 490.70 472.58 454.45 436.26 418.03 399.72 381.32 362.81 344.17 325.38 306.40 287.21 267.77 248.03 227.95 207.45 186.45 164.86 142.55 119.37 95.263 0

−0.32927 −0.31514 −0.29975 −0.28301 −0.26478 −0.24491 −0.22318 −0.19936 −0.17315 −0.14418 −0.11201 −0.076088 −0.035744 0.0098734 0.061838 0.12152 0.19072 0.27180 0.36794 0.48354 0.62482 0.80081 1.0252 1.3196 1.7200 2.2906 3.1570 4.5962 7.3420 18.568

0.25690 0.25728 0.25769 0.25812 0.25859 0.25908 0.25958 0.26010 0.26063 0.26117 0.26171 0.26225 0.26279 0.26332 0.26384 0.26435 0.26484 0.26531 0.26575 0.26615 0.26651 0.26681 0.26705 0.26720 0.26725 0.26715

0.10567 0.10683 0.10799 0.10916 0.11032 0.11150 0.11268 0.11387 0.11508 0.11630 0.11754 0.11880 0.12008 0.12139 0.12274 0.12412 0.12555 0.12703 0.12856 0.13016 0.13185 0.13363 0.13552 0.13755 0.13975 0.14217

0.11538 0.11680 0.11826 0.11977 0.12133 0.12295 0.12465 0.12643 0.12831 0.13031 0.13245 0.13475 0.13725 0.13998 0.14299 0.14634 0.15013 0.15444 0.15945 0.16534 0.17244 0.18118 0.19228 0.20692 0.22720 0.25728

116.26 116.64 116.94 117.15 117.27 117.29 117.20 117.00 116.68 116.25 115.69 115.00 114.17 113.19 112.05 110.75 109.28 107.63 105.77 103.71 101.42 98.885 96.087 92.999 89.593 85.834

24.935 24.301 23.725 23.203 22.730 22.303 21.919 21.576 21.272 21.005 20.776 20.582 20.423 20.301 20.216 20.169 20.160 20.193 20.269 20.392 20.565 20.790 21.072 21.411 21.805 22.243

Saturated Properties

2-359

2-360

TABLE 2-272

Thermodynamic Properties of R-114, 1,2-Dichlorotetrafluoroethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

405.00 410.00 415.00 418.83

2.5694 2.8015 3.0507 3.2516

1.4416 1.6966 2.0686 3.3932

300.00 350.00 400.00 450.00 500.00

0.10000 0.10000 0.10000 0.10000 0.10000

0.041312 0.035003 0.030432 0.026945 0.024188

300.00 350.00 356.40

1.0000 1.0000 1.0000

8.4984 7.4651 7.3010

356.40 400.00 450.00 500.00

1.0000 1.0000 1.0000 1.0000

300.00 350.00 400.00 450.00 500.00

5.0000 5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

0.69369 0.58941 0.48342 0.29471

66.763 66.782 66.570 64.391

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

0.14487 0.14792 0.15144

0.30666 0.40278 0.67085

81.678 77.066 71.914 0

Joule-Thomson K/MPa

Saturated Properties 68.545 68.433 68.045 65.349

0.26684 0.26620 0.26493 0.25828

22.687 23.035 22.963 18.568

Single-Phase Properties 24.206 28.569 32.860 37.113 41.342

58.335 64.107 70.284 76.817 83.661

60.756 66.964 73.570 80.529 87.796

0.26666 0.28578 0.30341 0.31980 0.33510

0.11051 0.11931 0.12703 0.13376 0.13956

0.11990 0.12828 0.13580 0.14241 0.14813

122.03 132.80 142.49 151.42 159.78

19.386 12.959 9.4836 7.3132 5.8366

0.11767 0.13396 0.13697

38.583 47.454 48.670

38.701 47.588 48.807

0.18633 0.21369 0.21714

0.11994 0.12863 0.12986

0.16950 0.18840 0.19249

551.87 364.95 338.92

−0.25320 0.11384 0.21213

0.43061 0.34682 0.29218 0.25510

2.3223 2.8833 3.4226 3.9201

63.533 69.357 76.104 83.069

65.855 72.240 79.526 86.989

0.26498 0.28188 0.29904 0.31477

0.12596 0.12912 0.13466 0.14003

0.15128 0.14474 0.14726 0.15132

108.84 126.86 141.14 152.73

20.165 11.867 8.1166 6.1412

8.6001 7.6832 6.4329 3.3217 1.7533

0.11628 0.13015 0.15545 0.30105 0.57034

38.303 46.919 56.400 69.251 79.658

38.884 47.570 57.177 70.756 82.509

0.18538 0.21213 0.23775 0.26953 0.29444

0.11983 0.12793 0.13659 0.14905 0.14423

0.16738 0.18096 0.20814 0.34888 0.18709

584.86 420.00 251.68 94.479 121.74

−0.28910 −0.048046 0.75207 9.2719 7.0501

300.00 350.00 400.00 450.00 500.00

10.000 10.000 10.000 10.000 10.000

8.7118 7.8877 6.9197 5.6865 4.2030

0.11479 0.12678 0.14452 0.17586 0.23793

37.991 46.398 55.329 64.892 74.962

39.139 47.666 56.774 66.650 77.341

0.18430 0.21057 0.23488 0.25812 0.28064

0.11975 0.12746 0.13488 0.14158 0.14594

0.16542 0.17597 0.18894 0.20710 0.21449

621.41 473.47 338.62 225.09 163.22

−0.32232 −0.15701 0.18601 1.0459 2.5802

300.00 350.00 400.00 450.00 500.00

15.000 15.000 15.000 15.000 15.000

8.8106 8.0508 7.2160 6.2819 5.2683

0.11350 0.12421 0.13858 0.15919 0.18981

37.713 45.970 54.628 63.643 72.897

39.415 47.834 56.707 66.030 75.744

0.18332 0.20926 0.23294 0.25489 0.27536

0.11973 0.12720 0.13421 0.14025 0.14487

0.16394 0.17290 0.18200 0.19084 0.19667

654.14 517.56 398.49 301.49 235.39

−0.34729 −0.22334 −0.017272 0.34360 0.89448

300.00 350.00 400.00 450.00 500.00

20.000 20.000 20.000 20.000 20.000

8.8998 8.1881 7.4361 6.6421 5.8257

0.11236 0.12213 0.13448 0.15056 0.17165

37.460 45.603 54.086 62.835 71.746

39.707 48.046 56.775 65.846 75.179

0.18241 0.20811 0.23141 0.25277 0.27243

0.11973 0.12704 0.13385 0.13972 0.14442

0.16278 0.17079 0.17821 0.18438 0.18846

683.99 555.75 446.47 358.35 294.31

−0.36679 −0.26854 −0.12632 0.081032 0.35805

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Platzer, B., Polt, A., and Maurer, G., Thermophysical Properties of Refrigerants, Springer-Verlag, Berlin, 1990. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density is 0.2% up to 400 K and 1% at higher temperatures. The vapor pressure uncertainty is 1.5%. In the liquid phase, the uncertainty in isobaric heat capacity is 3%.

THERMODYNAMIC PROPERTIES TABLE 2-273 Temp., °F −100 −80 −60 −40 −20

Saturated Refrigerant 115, Chloropentafluoroethane* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

2.327 4.573 8.306 14.13 22.74

0.00966 0.00986 0.01009 0.01033 0.01060

10.57 5.624 3.218 1.953 1.245

−13.07 −8.78 −4.43 0.00 4.50

45.83 48.39 50.96 53.53 56.07

−0.0335 −0.0219 −0.0108 0.0000 0.0104

0.1302 0.1286 0.1278 0.1275 0.1277

0 20 40 60 80

34.94 51.59 73.65 102.1 138.1

0.01090 0.01123 0.01161 0.01204 0.01255

0.8257 0.5657 0.3979 0.2857 0.2081

9.09 13.76 18.54 23.45 28.54

58.56 61.00 63.35 65.60 67.71

0.0206 0.0305 0.0401 0.0496 0.0591

0.1282 0.1290 0.1298 0.1308 0.1317

100 120 140 160 170

182.7 237.3 303.2 382.0 427.0

0.01316 0.01393 0.01496 0.01664 0.01838

0.1530 0.1125 0.0817 0.0567 0.0444

33.85 39.50 45.67 52.76 56.56

69.63 71.24 72.36 72.42 71.33

0.0686 0.0782 0.0884 0.0996 0.1055

0.1325 0.1330 0.1329 0.1314 0.1290

175.89 c

457.6

0.0261

0.0261

64.30

64.30

0.1175

0.1175

*Unpublished data of General Chemicals Division, Allied Chemical Company. Used by permission. c = critical temperature. No material in SI units appears in the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Tables and a chart to 50 ata, 200 °C are given by Mathias, H. and H. J. Loffler, Techn. Univ. Berlin rept., 1966 (42 pp.). A chart to 1500 psia, 500 °F was given by Mears, W. H., E. Rosenthal, et al., J. Chem. Eng. Data, 11, 3 (1966): 338–343.

2-361

2-362 TABLE 2-274

Thermodynamic Properties of R-116, Hexafluoroethane

Temperature K

Pressure MPa

173.10 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 293.03

0.026084 0.029802 0.041661 0.057021 0.076558 0.10101 0.13116 0.16784 0.21194 0.26438 0.32611 0.39814 0.48150 0.57725 0.68650 0.81042 0.95020 1.1071 1.2826 1.4781 1.6952 1.9360 2.2027 2.4983 2.8276 3.0477

173.10 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 293.03

0.026084 0.029802 0.041661 0.057021 0.076558 0.10101 0.13116 0.16784 0.21194 0.26438 0.32611 0.39814 0.48150 0.57725 0.68650 0.81042 0.95020 1.1071 1.2826 1.4781 1.6952 1.9360 2.2027 2.4983 2.8276 3.0477

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.081274 0.081649 0.082664 0.083720 0.084824 0.085979 0.087192 0.088469 0.089818 0.091248 0.092769 0.094395 0.096140 0.098026 0.10007 0.10232 0.10480 0.10756 0.11070 0.11431 0.11857 0.12376 0.13043 0.13986 0.15692 0.22502

13.031 13.266 13.887 14.516 15.153 15.797 16.450 17.111 17.781 18.461 19.151 19.851 20.563 21.286 22.023 22.774 23.541 24.325 25.130 25.959 26.818 27.715 28.667 29.710 30.973 33.029

13.033 13.268 13.891 14.521 15.159 15.806 16.461 17.126 17.800 18.485 19.181 19.889 20.609 21.343 22.092 22.857 23.640 24.444 25.272 26.128 27.019 27.955 28.954 30.059 31.417 33.715

28.993 29.113 29.433 29.755 30.079 30.406 30.733 31.061 31.389 31.716 32.041 32.364 32.682 32.995 33.302 33.599 33.885 34.157 34.409 34.636 34.830 34.975 35.047 34.992 34.631 33.029

30.408 30.541 30.893 31.247 31.601 31.954 32.306 32.656 33.003 33.345 33.683 34.014 34.336 34.649 34.950 35.237 35.507 35.755 35.977 36.164 36.306 36.387 36.375 36.207 35.669 33.715

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.073325 0.074672 0.078175 0.081622 0.085018 0.088366 0.091671 0.094937 0.098169 0.10137 0.10454 0.10769 0.11082 0.11394 0.11705 0.12015 0.12326 0.12638 0.12952 0.13269 0.13593 0.13926 0.14274 0.14651 0.15106 0.15879

0.083104 0.083413 0.084236 0.085077 0.085933 0.086803 0.087686 0.088580 0.089485 0.090402 0.091330 0.092270 0.093224 0.094193 0.095179 0.096188 0.097226 0.098301 0.099429 0.10063 0.10195 0.10345 0.10525 0.10768 0.11183

0.12322 0.12375 0.12518 0.12670 0.12829 0.12996 0.13173 0.13361 0.13560 0.13773 0.14002 0.14249 0.14520 0.14818 0.15152 0.15531 0.15970 0.16491 0.17129 0.17942 0.19040 0.20651 0.23354 0.29195 0.54643

653.29 645.09 623.59 602.22 580.95 559.75 538.61 517.48 496.36 475.20 453.97 432.64 411.18 389.53 367.64 345.46 322.91 299.89 276.28 251.91 226.54 199.81 171.13 139.42 102.12 0

−0.38383 −0.37807 −0.36202 −0.34452 −0.32540 −0.30443 −0.28132 −0.25573 −0.22722 −0.19525 −0.15913 −0.11798 −0.070666 −0.015673 0.049034 0.12626 0.21998 0.33604 0.48338 0.67638 0.93994 1.3211 1.9215 3.0109 5.6531 13.224

0.17370 0.17338 0.17263 0.17203 0.17155 0.17118 0.17090 0.17069 0.17056 0.17049 0.17046 0.17047 0.17051 0.17056 0.17063 0.17068 0.17072 0.17073 0.17069 0.17056 0.17033 0.16992 0.16924 0.16807 0.16572 0.15879

0.067328 0.067932 0.069527 0.071130 0.072742 0.074365 0.076000 0.077648 0.079312 0.080992 0.082691 0.084412 0.086158 0.087932 0.089742 0.091594 0.093497 0.095465 0.097518 0.099682 0.10200 0.10454 0.10743 0.11094 0.11584

0.076383 0.077064 0.078887 0.080765 0.082706 0.084719 0.086816 0.089011 0.091322 0.093772 0.096389 0.099211 0.10229 0.10569 0.10951 0.11388 0.11901 0.12520 0.13296 0.14316 0.15741 0.17918 0.21720 0.30201 0.66697

106.90 107.29 108.24 109.07 109.78 110.35 110.78 111.05 111.15 111.07 110.80 110.33 109.63 108.69 107.50 106.02 104.24 102.13 99.661 96.789 93.477 89.681 85.354 80.445 74.869 0

43.932 42.424 38.839 35.743 33.062 30.736 28.716 26.960 25.435 24.112 22.967 21.983 21.143 20.435 19.850 19.381 19.024 18.777 18.637 18.605 18.677 18.837 19.040 19.127 18.438 13.224

Saturated Properties 12.304 12.247 12.097 11.945 11.789 11.631 11.469 11.303 11.134 10.959 10.779 10.594 10.401 10.201 9.9926 9.7735 9.5423 9.2968 9.0336 8.7482 8.4341 8.0803 7.6671 7.1502 6.3727 4.4440 0.018437 0.020874 0.028519 0.038215 0.050323 0.065236 0.083387 0.10525 0.13135 0.16228 0.19868 0.24133 0.29110 0.34902 0.41636 0.49465 0.58587 0.69256 0.81820 0.96766 1.1482 1.3717 1.6592 2.0572 2.7232 4.4440

54.238 47.907 35.064 26.168 19.872 15.329 11.992 9.5011 7.6132 6.1624 5.0331 4.1437 3.4353 2.8652 2.4018 2.0216 1.7069 1.4439 1.2222 1.0334 0.87090 0.72904 0.60271 0.48610 0.36721 0.22502

Single-Phase Properties 175.00 194.81

0.10000 0.10000

194.81 250.00 325.00 400.00

0.10000 0.10000 0.10000 0.10000

175.00 250.00 251.65

1.0000 1.0000 1.0000

251.65 325.00 400.00

1.0000 1.0000 1.0000

250.00 325.00 400.00

5.0000 5.0000 5.0000

13.263 15.773

13.271 15.782

0.074658 0.088242

0.083417 0.086771

0.12373 0.12990

645.46 560.54

−0.37835 −0.30525

30.394 34.892 42.103 50.430

31.941 36.932 44.784 53.744

0.17119 0.19368 0.22106 0.24582

0.074304 0.087457 0.10389 0.11740

0.084642 0.096533 0.11253 0.12590

110.33 126.48 144.51 160.19

30.817 13.742 6.5149 3.7890

0.081484 0.10473 0.10567

13.232 23.534 23.797

13.314 23.638 23.903

0.074480 0.12323 0.12428

0.083472 0.097224 0.097576

0.12349 0.15951 0.16132

650.25 323.69 315.37

−0.38185 0.21614 0.25543

0.61920 0.40237 0.31155

1.6150 2.4853 3.2097

33.976 41.594 50.096

35.591 44.080 53.305

0.17073 0.20033 0.22584

0.094138 0.10529 0.11796

0.12091 0.11782 0.12829

103.59 135.84 156.26

18.931 7.1157 3.8772

9.9221 3.7219 1.7961

0.10078 0.26868 0.55676

23.073 37.500 48.386

23.577 38.843 51.170

0.12134 0.17386 0.20825

0.097206 0.11497 0.12060

0.14935 0.24037 0.14444

376.03 101.35 144.06

12.249 11.637 0.064625 0.049033 0.037293 0.030174 12.272 9.5480 9.4631

0.081637 0.085935 15.474 20.394 26.815 33.141

0.0088070 7.5484 3.8425

250.00 325.00 400.00

10.000 10.000 10.000

10.261 7.1151 3.9471

0.097459 0.14055 0.25335

22.648 34.117 46.132

23.622 35.522 48.666

0.11954 0.16098 0.19742

0.097376 0.11174 0.12255

0.14319 0.17544 0.16440

425.10 207.64 156.43

−0.12130 1.0712 2.4329

250.00 325.00 400.00

20.000 20.000 20.000

10.747 8.5892 6.4948

0.093053 0.11643 0.15397

22.031 32.498 43.572

23.892 34.826 46.651

0.11682 0.15497 0.18769

0.097852 0.11132 0.12291

0.13725 0.15339 0.16013

498.98 328.94 246.48

−0.24812 0.093523 0.50074

250.00 325.00 400.00

50.000 50.000 50.000

11.642 10.197 8.8968

0.085893 0.098067 0.11240

20.902 30.607 40.984

25.197 35.511 46.604

0.11135 0.14736 0.17805

0.099289 0.11250 0.12396

0.13120 0.14332 0.15192

647.29 516.00 437.11

−0.37511 −0.28102 −0.20643

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation are 0.5% in density for liquid and vapor states and 1% in density or pressure for supercritical states. For vapor pressure, the uncertainty is 0.3%, that for vapor-phase speed of sounds is 0.2%, and the uncertainty for heat capacities is 5%.

2-363

2-364 TABLE 2-275

Thermodynamic Properties of R-123, 2,2-Dichloro-1,1,1-Trifluoroethane

Temperature K

Pressure MPa

166.00 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 440.00 455.00 456.83

4.2021E-06 7.5115E-06 5.1172E-05 0.00024954 0.00093965 0.0028868 0.0075380 0.017260 0.035500 0.066848 0.11700 0.19264 0.30136 0.45147 0.65201 0.91268 1.2440 1.6577 2.1672 2.7898 3.5547 3.6619

166.00 170.00 185.00 200.00 215.00 230.00 245.00 260.00 275.00 290.00 305.00 320.00 335.00 350.00 365.00 380.00 395.00 410.00 425.00 440.00 455.00 456.83

4.2021E-06 7.5115E-06 5.1172E-05 0.00024954 0.00093965 0.0028868 0.0075380 0.017260 0.035500 0.066848 0.11700 0.19264 0.30136 0.45147 0.65201 0.91268 1.2440 1.6577 2.1672 2.7898 3.5547 3.6619

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

15.111 15.678 17.800 19.919 22.050 24.204 26.388 28.607 30.863 33.158 35.492 37.867 40.286 42.751 45.270 47.853 50.514 53.281 56.204 59.412 63.738 65.873

15.111 15.678 17.800 19.919 22.050 24.204 26.389 28.609 30.867 33.165 35.504 37.888 40.319 42.804 45.349 47.967 50.677 53.510 56.527 59.873 64.496 66.891

0.081219 0.084599 0.096559 0.10757 0.11785 0.12753 0.13673 0.14552 0.15395 0.16208 0.16993 0.17753 0.18492 0.19212 0.19917 0.20612 0.21300 0.21990 0.22695 0.23446 0.24446 0.24965

47.940 48.198 49.206 50.269 51.386 52.550 53.755 54.996 56.264 57.553 58.855 60.163 61.468 62.760 64.026 65.252 66.412 67.468 68.348 68.866 67.881 65.873

49.320 49.612 50.744 51.931 53.171 54.456 55.779 57.131 58.504 59.887 61.270 62.645 63.999 65.319 66.591 67.794 68.899 69.857 70.581 70.847 69.293 66.891

0.28730 0.28421 0.27463 0.26763 0.26259 0.25906 0.25669 0.25522 0.25445 0.25422 0.25440 0.25489 0.25560 0.25645 0.25737 0.25829 0.25913 0.25977 0.26002 0.25940 0.25500 0.24965

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.096271 0.096537 0.096881 0.097283 0.098162 0.099479 0.10110 0.10291 0.10483 0.10680 0.10879 0.11080 0.11283 0.11491 0.11704 0.11927 0.12167 0.12433 0.12747 0.13159 0.13975

0.14206 0.14181 0.14120 0.14151 0.14272 0.14455 0.14676 0.14919 0.15176 0.15443 0.15722 0.16020 0.16350 0.16729 0.17186 0.17772 0.18579 0.19804 0.21999 0.27613 1.2865

1243.8 1227.9 1167.0 1104.3 1040.9 977.54 914.89 853.42 793.34 734.70 677.42 621.30 566.09 511.47 457.07 402.48 347.23 290.69 231.85 168.40 90.823 0

−0.47555 −0.47661 −0.47672 −0.47027 −0.45811 −0.44149 −0.42124 −0.39765 −0.37048 −0.33894 −0.30164 −0.25640 −0.19988 −0.12681 −0.028618 0.10963 0.31613 0.65103 1.2687 2.7240 10.053 16.566

115.89 114.18 109.51 105.59 101.48 97.035 92.395 87.716 83.129 78.719 74.533 70.585 66.863 63.340 59.972 56.709 53.486 50.222 46.810 43.102 39.324

4953.8 4180.7 2571.4 1795.0 1337.4 1036.8 825.41 669.88 551.59 459.38 386.07 326.86 278.37 238.12 204.30 175.46 150.45 128.23 107.74 87.306 58.622

0.064141 0.065216 0.069152 0.072951 0.076634 0.080226 0.083755 0.087245 0.090715 0.094177 0.097639 0.10111 0.10459 0.10810 0.11165 0.11529 0.11908 0.12313 0.12767 0.13325 0.14227

0.072457 0.073532 0.077473 0.081290 0.085017 0.088699 0.092390 0.096144 0.10001 0.10405 0.10832 0.11290 0.11790 0.12353 0.13011 0.13821 0.14902 0.16522 0.19466 0.27310 1.6671

100.97 102.08 106.14 110.02 113.72 117.20 120.41 123.27 125.72 127.66 129.01 129.68 129.58 128.62 126.67 123.57 119.13 113.03 104.85 93.850 78.287 0

Saturated Properties 11.580 11.520 11.298 11.079 10.861 10.640 10.415 10.185 9.9497 9.7073 9.4567 9.1962 8.9238 8.6365 8.3301 7.9984 7.6316 7.2132 6.7113 6.0457 4.6936 3.5964 3.0446E-06 5.3144E-06 3.3272E-05 0.00015013 0.00052637 0.0015145 0.0037253 0.0080830 0.015852 0.028647 0.048445 0.077628 0.11908 0.17640 0.25423 0.35900 0.50026 0.69385 0.97042 1.4081 2.5178 3.5964

0.086355 0.086807 0.088511 0.090257 0.092076 0.093989 0.096017 0.098182 0.10051 0.10302 0.10575 0.10874 0.11206 0.11579 0.12005 0.12502 0.13103 0.13863 0.14900 0.16541 0.21305 0.27805 328,450. 188,170. 30,055. 6,660.7 1,899.8 660.29 268.44 123.72 63.083 34.907 20.642 12.882 8.3975 5.6690 3.9334 2.7855 1.9990 1.4412 1.0305 0.71016 0.39718 0.27805

335.67 305.67 219.47 162.12 122.83 95.252 75.497 61.102 50.463 42.504 36.493 31.923 28.442 25.804 23.846 22.466 21.617 21.309 21.605 22.561 21.950 16.566

1.6736 1.9014 2.7552 3.6080 4.4590 5.3073 6.1533 6.9987 7.8477 8.7073 9.5872 10.500 11.463 12.497 13.627 14.889 16.332 18.039 20.168 23.176 30.870

5.5300 5.7017 6.3386 6.9646 7.5791 8.1817 8.7712 9.3465 9.9065 10.450 10.979 11.494 12.002 12.515 13.054 13.656 14.390 15.384 16.913 19.744 29.346

Single-Phase Properties 200.00 300.00 300.61

0.10000 0.10000 0.10000

300.61 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 384.30

1.0000 1.0000 1.0000

384.30 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

19.917 34.709 34.805

19.926 34.720 34.815

0.10756 0.16734 0.16766

0.097300 0.10812 0.10820

0.14151 0.15627 0.15639

1104.5 696.38 694.05

−0.47033 −0.31484 −0.31326

58.473 68.902 80.750 93.743

60.866 72.178 84.879 98.716

0.25432 0.28673 0.31502 0.34022

0.096626 0.11170 0.12437 0.13502

0.10704 0.12066 0.13300 0.14352

128.68 150.93 169.33 185.60

38.084 14.728 7.9263 5.0504

0.090194 0.10462 0.12662

19.896 34.660 48.607

19.986 34.764 48.733

0.10745 0.16717 0.20809

0.097447 0.10822 0.11994

0.14145 0.15597 0.17975

1105.5 701.57 386.73

−0.47094 −0.31949 0.16009

105.80 76.273 55.784

2.5301 2.7480 3.8625 4.8220

65.592 67.553 80.024 93.232

68.122 70.301 83.887 98.054

0.25855 0.26410 0.29441 0.32022

0.11636 0.11698 0.12571 0.13546

0.14096 0.13715 0.13778 0.14581

122.45 129.70 160.11 181.06

22.169 18.389 8.3865 5.1181

15.282 16.135 21.585 27.069

13.849 14.368 17.431 19.994

11.118 9.6324 7.7754 2.1822 1.2117

0.089944 0.10382 0.12861 0.45826 0.82529

19.803 34.450 50.791 74.934 90.757

20.253 34.969 51.434 77.225 94.883

0.10699 0.16646 0.21368 0.27035 0.30270

0.098069 0.10865 0.12180 0.13698 0.13749

0.14121 0.15478 0.17810 0.23468 0.16051

1110.5 723.64 388.06 111.01 161.94

−0.47345 −0.33792 0.14142 11.061 5.1664

106.62 77.874 55.285 30.182 30.009

1902.1 436.12 160.58 28.291 24.445

11.080 9.5412 9.5309 0.041795 0.030530 0.024219 0.020111 11.087 9.5586 7.8974 0.39525 0.36390 0.25890 0.20738

0.090251 0.10481 0.10492 23.926 32.754 41.290 49.724

105.61 75.903 75.734 9.3269 15.085 20.780 26.460

1797.1 408.75 405.88 10.826 14.262 17.224 19.696 1815.8 413.78 167.94

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

11.156 9.7177 8.0432 5.5005 2.8283

0.089639 0.10290 0.12433 0.18180 0.35357

19.691 34.208 50.152 68.419 87.359

20.588 35.237 51.395 70.237 90.895

0.10642 0.16564 0.21200 0.25388 0.29164

0.098779 0.10912 0.12141 0.13402 0.13939

0.14095 0.15357 0.17093 0.21202 0.18482

1117.0 749.33 447.83 205.61 163.33

−0.47622 −0.35684 −0.033773 1.7914 3.7808

107.63 79.793 58.469 41.785 35.942

2017.3 464.14 181.18 78.103 39.777

200.00 300.00 400.00 500.00 600.00

20.000 20.000 20.000 20.000 20.000

11.229 9.8704 8.4163 6.7515 4.9944

0.089054 0.10131 0.11882 0.14812 0.20022

19.480 33.778 49.229 65.791 83.034

21.261 35.804 51.605 68.753 87.039

0.10532 0.16412 0.20949 0.24770 0.28102

0.099998 0.10993 0.12134 0.13081 0.13822

0.14055 0.15175 0.16430 0.17889 0.18342

1130.7 795.84 534.29 348.69 257.14

−0.48062 −0.38528 −0.19523 0.27303 0.96765

109.63 83.416 63.653 49.919 41.677

2276.1 521.04 217.09 115.74 74.025

200.00 300.00 400.00 500.00 600.00

40.000 40.000 40.000 40.000 40.000

11.368 10.125 8.9071 7.7025 6.5412

0.087968 0.098764 0.11227 0.12983 0.15288

19.099 33.072 47.981 63.586 79.686

22.618 37.023 52.472 68.779 85.801

0.10325 0.16152 0.20590 0.24225 0.27327

0.10179 0.11120 0.12199 0.13053 0.13791

0.14011 0.14952 0.15901 0.16697 0.17293

1160.1 875.49 653.37 503.02 409.64

−0.48571 −0.42067 −0.32210 −0.17413 0.011715

113.48 90.039 72.037 60.777 52.389

2947.0 640.89 282.73 165.05 115.87

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Younglove, B. A., and McLinden, M. O., “An International Standard Equation of State for the Thermodynamic Properties of Refrigerant 123 (2,2-Dichloro-1,1,1-Trifluoroethane),” J. Phys. Chem. Ref. Data 23:731–779, 1994. The source for viscosity is Tanaka, Y, and Sotani, T., “Transport Properties (Thermal Conductivity and Viscosity),” in McLinden, M. O., Ed., R123—Thermodynamic and Physical Properties, Paris: International Institute of Refrigeration, 1995. See also Int. J. Thermophys. 17(2):293–328, 1996. The source for thermal conductivity is Laesecke, A., Perkins, R. A., and Howley, J. B., “An Improved Correlation for the Thermal Conductivity of HCFC123 (2,2-Dichloro-1,1,1-Trifluoroethane),” Int. J. Refrigeration 19:231–238, 1996. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 1.5% in heat capacity, and 2% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.1%. Uncertainties are greater below 180 K. The uncertainty in viscosity is 5%. The uncertainty in thermal conductivity is 2%.

2-365

2-366 FIG. 2-24

Enthalpy–log-pressure diagram for Refrigerant 123.

TABLE 2-276

Thermodynamic Properties of R-124, 2-Chloro-1,1,1,2-Tetrafluoroethane

Temperature K

Pressure MPa

120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 395.43

2.6739E-08 8.0309E-07 1.1223E-05 9.0881E-05 0.00049265 0.0019732 0.0062626 0.016573 0.037986 0.077612 0.14453 0.24955 0.40500 0.62447 0.92279 1.3163 1.8234 2.4663 3.2770 3.6243

120.00 135.00 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 395.43

2.6739E-08 8.0309E-07 1.1223E-05 9.0881E-05 0.00049265 0.0019732 0.0062626 0.016573 0.037986 0.077612 0.14453 0.24955 0.40500 0.62447 0.92279 1.3163 1.8234 2.4663 3.2770 3.6243

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

1293.6 1225.8 1157.4 1089.0 1021.3 954.41 888.65 824.09 760.75 698.57 637.46 577.23 517.54 457.87 397.44 335.05 269.18 198.17 115.48 0

−0.48257 −0.47832 −0.47012 −0.45872 −0.44440 −0.42707 −0.40640 −0.38173 −0.35205 −0.31583 −0.27079 −0.21337 −0.13780 −0.034029 0.11699 0.35590 0.78492 1.7419 5.4469 14.607

Saturated Properties 13.576 13.288 13.003 12.718 12.433 12.145 11.853 11.554 11.248 10.931 10.601 10.253 9.8843 9.4868 9.0506 8.5590 7.9793 7.2348 6.0066 4.1033 2.6799E-08 7.1548E-07 8.9989E-06 6.6260E-05 0.00032946 0.0012200 0.0036077 0.0089652 0.019455 0.037965 0.068150 0.11457 0.18300 0.28114 0.42003 0.61719 0.90480 1.3577 2.2903 4.1033

0.073661 0.075255 0.076906 0.078626 0.080432 0.082339 0.084369 0.086547 0.088904 0.091481 0.094332 0.097528 0.10117 0.10541 0.11049 0.11684 0.12532 0.13822 0.16648 0.24371 37,314,000. 1,397,700. 111,120. 15,092. 3,035.3 819.64 277.19 111.54 51.400 26.340 14.674 8.7285 5.4645 3.5569 2.3808 1.6202 1.1052 0.73653 0.43662 0.24371

6.7015 8.6014 10.510 12.437 14.388 16.367 18.378 20.426 22.512 24.641 26.816 29.043 31.327 33.677 36.106 38.634 41.302 44.202 47.714 50.813

6.7015 8.6014 10.510 12.437 14.388 16.367 18.379 20.427 22.515 24.648 26.830 29.067 31.368 33.743 36.208 38.788 41.531 44.543 48.259 51.696

0.027106 0.042024 0.055431 0.067673 0.078987 0.089547 0.099484 0.10890 0.11787 0.12648 0.13477 0.14280 0.15061 0.15826 0.16580 0.17332 0.18092 0.18889 0.19827 0.20684

0.088123 0.087495 0.087696 0.088451 0.089589 0.091000 0.092610 0.094368 0.096239 0.098199 0.10024 0.10235 0.10454 0.10685 0.10933 0.11213 0.11553 0.12005 0.12812

0.12662 0.12684 0.12778 0.12919 0.13095 0.13299 0.13527 0.13778 0.14055 0.14361 0.14702 0.15090 0.15545 0.16102 0.16825 0.17860 0.19587 0.23450 0.46802

36.727 37.530 38.388 39.300 40.262 41.269 42.318 43.400 44.507 45.631 46.762 47.892 49.009 50.100 51.145 52.111 52.940 53.497 53.233 50.813

37.725 38.653 39.636 40.671 41.757 42.887 44.054 45.248 46.459 47.675 48.883 50.070 51.222 52.321 53.342 54.244 54.955 55.314 54.664 51.696

0.28563 0.26462 0.24960 0.23879 0.23104 0.22555 0.22175 0.21922 0.21764 0.21678 0.21645 0.21649 0.21679 0.21724 0.21772 0.21811 0.21821 0.21761 0.21470 0.20684

0.051678 0.055405 0.059044 0.062605 0.066113 0.069600 0.073111 0.076698 0.080420 0.084321 0.088420 0.092708 0.097166 0.10179 0.10663 0.11180 0.11757 0.12466 0.13608

0.059992 0.063720 0.067361 0.070933 0.074474 0.078043 0.081721 0.085612 0.089837 0.094522 0.099789 0.10578 0.11271 0.12102 0.13162 0.14664 0.17189 0.23071 0.61311

92.125 97.256 102.10 106.70 111.05 115.13 118.89 122.21 124.99 127.10 128.40 128.75 128.00 125.97 122.47 117.17 109.67 99.240 84.410 0

997.50 558.90 338.42 218.53 149.08 106.83 80.088 62.512 50.458 41.824 35.421 30.583 26.921 24.195 22.252 21.001 20.405 20.431 20.196 14.607

2-367

2-368

TABLE 2-276

Thermodynamic Properties of R-124, 2-Chloro-1,1,1,2-Tetrafluoroethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

150.00 225.00 260.87

0.10000 0.10000 0.10000

13.004 11.556 10.804

0.076901 0.086537 0.092562

260.87 300.00 375.00 450.00

0.10000 0.10000 0.10000 0.10000

150.00 225.00 300.00 333.28

1.0000 1.0000 1.0000 1.0000

333.28 375.00 450.00

1.0000 1.0000 1.0000

150.00 225.00 300.00 375.00 450.00

5.0000 5.0000 5.0000 5.0000 5.0000

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 10.509 20.423 25.487

10.516 20.431 25.496

0.055420 0.10889 0.12976

0.087698 0.094369 0.098988

0.12777 0.13777 0.14489

1157.7 824.54 674.53

−0.47016 −0.38192 −0.29941

46.073 49.579 57.006 65.395

48.150 52.011 60.088 69.114

0.21660 0.23038 0.25436 0.27627

0.085902 0.092039 0.10524 0.11804

0.096509 0.10148 0.11403 0.12661

127.71 138.33 155.52 170.43

39.096 22.359 10.936 6.3680

0.076859 0.086432 0.10093 0.11175

10.494 20.392 31.280 36.649

10.570 20.479 31.381 36.760

0.055319 0.10875 0.15045 0.16744

0.087712 0.094382 0.10452 0.10991

0.12773 0.13761 0.15494 0.17017

1160.8 829.31 523.28 384.03

−0.47053 −0.38397 −0.14686 0.15954

2.1867 2.7250 3.5074

51.364 56.110 64.848

53.551 58.835 68.356

0.21782 0.23277 0.25590

0.10772 0.10870 0.11949

0.13441 0.12428 0.13091

121.47 140.23 162.32

21.922 12.660 6.7329

13.042 11.630 10.055 7.8072 2.0764

0.076674 0.085982 0.099451 0.12809 0.48161

10.428 20.262 30.991 43.322 61.347

10.811 20.692 31.488 43.963 63.755

0.054877 0.10817 0.14947 0.18645 0.23451

0.087777 0.094440 0.10448 0.11692 0.12778

0.12756 0.13696 0.15210 0.19006 0.18398

1174.3 849.95 558.86 276.48 124.73

−0.47207 −0.39231 −0.19712 0.71939 8.2236

0.048164 0.041129 0.032447 0.026891 13.011 11.570 9.9082 8.9488 0.45731 0.36697 0.28511

20.763 24.314 30.820 37.187

150.00 225.00 300.00 375.00 450.00

10.000 10.000 10.000 10.000 10.000

13.080 11.702 10.213 8.3628 5.4895

0.076452 0.085459 0.097912 0.11958 0.18217

10.348 20.109 30.675 42.388 56.064

11.113 20.963 31.654 43.584 57.886

0.054334 0.10747 0.14838 0.18379 0.21844

0.087856 0.094513 0.10449 0.11566 0.12834

0.12736 0.13626 0.14957 0.17088 0.21040

1190.9 874.54 597.87 359.58 179.08

−0.47382 −0.40129 −0.24225 0.22172 2.3542

150.00 225.00 300.00 375.00 450.00

20.000 20.000 20.000 20.000 20.000

13.153 11.832 10.473 8.9796 7.2897

0.076029 0.084516 0.095482 0.11136 0.13718

10.197 19.827 30.150 41.285 53.276

11.717 21.518 32.059 43.513 56.020

0.053282 0.10615 0.14651 0.18054 0.21090

0.088011 0.094657 0.10459 0.11536 0.12652

0.12702 0.13514 0.14630 0.15949 0.17382

1222.7 920.41 664.60 462.43 319.21

−0.47677 −0.41563 −0.30098 −0.080024 0.35516

150.00 225.00 300.00 375.00 450.00

40.000 40.000 40.000 40.000 40.000

13.287 12.057 10.865 9.6799 8.5132

0.075260 0.082941 0.092042 0.10331 0.11746

9.9211 19.345 29.347 39.971 51.192

12.932 22.662 33.028 44.103 55.890

0.051293 0.10380 0.14350 0.17641 0.20504

0.088302 0.094922 0.10483 0.11550 0.12642

0.12653 0.13366 0.14289 0.15243 0.16186

1282.4 1002.2 772.84 600.05 479.46

−0.48099 −0.43482 −0.36270 −0.26440 −0.14030

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is de Vries, B., Tillner-Roth, R., and Baehr, H. D., “Thermodynamic Properties of HCFC 124,” 19th Int. Congress of Refrigeration, The Hague, The Netherlands, International Institute of Refrigeration, IVa:582–589, 1995. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.05% in density, 1% in heat capacity, and 1% in the speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.1%.

TABLE 2-277

Thermodynamic Properties of R-125, Pentafluoroethane

Temperature K

Pressure MPa

172.52 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 339.17

0.0029140 0.0056285 0.012328 0.024602 0.045417 0.078505 0.12833 0.20004 0.29934 0.43250 0.60624 0.82782 1.1050 1.4463 1.8610 2.3600 2.9579 3.6179

172.52 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 339.17

0.0029140 0.0056285 0.012328 0.024602 0.045417 0.078505 0.12833 0.20004 0.29934 0.43250 0.60624 0.82782 1.1050 1.4463 1.8610 2.3600 2.9579 3.6179

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

10.457 11.389 12.646 13.919 15.210 16.523 17.858 19.219 20.607 22.025 23.478 24.971 26.511 28.112 29.793 31.595 33.632 37.417

10.457 11.389 12.647 13.921 15.214 16.529 17.869 19.235 20.632 22.063 23.532 25.048 26.619 28.259 29.994 31.869 34.017 38.174

0.058837 0.064124 0.070919 0.077448 0.083750 0.089856 0.095792 0.10158 0.10725 0.11282 0.11830 0.12374 0.12916 0.13461 0.14015 0.14593 0.15231 0.16438

31.863 32.307 32.913 33.530 34.157 34.788 35.421 36.052 36.678 37.292 37.890 38.460 38.988 39.453 39.828 40.054 39.964 37.417

33.293 33.795 34.477 35.167 35.860 36.552 37.237 37.911 38.568 39.202 39.805 40.363 40.858 41.267 41.554 41.651 41.355 38.174

0.19120 0.18860 0.18582 0.18368 0.18207 0.18087 0.18000 0.17940 0.17900 0.17874 0.17857 0.17844 0.17826 0.17797 0.17744 0.17649 0.17454 0.16438

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.081329 0.082012 0.083102 0.084327 0.085644 0.087029 0.088472 0.089971 0.091529 0.093153 0.094843 0.096594 0.098430 0.10043 0.10274 0.10571 0.11043

0.12417 0.12500 0.12647 0.12825 0.13028 0.13254 0.13505 0.13785 0.14102 0.14468 0.14903 0.15440 0.16135 0.17099 0.18593 0.21395 0.29625

932.57 893.63 843.11 793.91 745.67 698.07 650.89 603.92 556.99 510.02 462.91 415.46 367.36 318.17 267.31 213.55 153.34 0

−0.38374 −0.37406 −0.35818 −0.33901 −0.31627 −0.28935 −0.25723 −0.21839 −0.17062 −0.11056 −0.032921 0.070972 0.21606 0.43036 0.77370 1.4029 2.9184 12.361

116.02 112.52 107.79 103.06 98.331 93.653 89.019 84.443 79.940 75.520 71.187 66.940 62.772 58.667 54.597 50.534 46.661

1152.4 957.54 768.40 631.00 527.00 445.76 380.67 327.41 283.01 245.39 213.02 184.73 159.60 136.86 115.81 95.602 74.602

0.059815 0.061648 0.064126 0.066646 0.069223 0.071864 0.074575 0.077362 0.080230 0.083141 0.086003 0.088869 0.092050 0.095933 0.10077 0.10679 0.11481

0.068285 0.070217 0.072893 0.075712 0.078713 0.081939 0.085437 0.089271 0.093526 0.098283 0.10368 0.11025 0.11918 0.13255 0.15449 0.19725 0.32843

116.43 118.54 121.15 123.47 125.44 127.01 128.11 128.68 128.64 127.93 126.44 124.10 120.81 116.42 110.71 103.29 93.550 0

90.257 77.516 64.018 53.589 45.456 39.066 34.014 29.998 26.787 24.232 22.293 20.938 20.043 19.438 19.016 18.738 18.404 12.361

Saturated Properties 14.086 13.885 13.613 13.336 13.052 12.762 12.461 12.150 11.824 11.481 11.117 10.724 10.295 9.8162 9.2637 8.5923 7.6744 4.7790 0.0020381 0.0037809 0.0078784 0.015031 0.026661 0.044514 0.070679 0.10763 0.15835 0.22645 0.31661 0.43510 0.59084 0.79742 1.0777 1.4778 2.1269 4.7790

0.070990 0.072020 0.073461 0.074988 0.076615 0.078360 0.080247 0.082305 0.084572 0.087098 0.089954 0.093245 0.097130 0.10187 0.10795 0.11638 0.13030 0.20925 490.65 264.49 126.93 66.529 37.508 22.465 14.148 9.2907 6.3153 4.4159 3.1585 2.2983 1.6925 1.2540 0.92787 0.67670 0.47016 0.20925

5.2349 5.7185 6.3724 7.0353 7.7081 8.3929 9.0929 9.8136 10.563 11.356 12.213 13.169 14.286 15.680 17.586 20.574 26.607

7.4339 7.7624 8.1999 8.6344 9.0657 9.4944 9.9221 10.353 10.791 11.246 11.732 12.266 12.884 13.638 14.635 16.104 18.766

2-369

2-370

TABLE 2-277

Thermodynamic Properties of R-125, Pentafluoroethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

200.00 224.79

0.10000 0.10000

13.337 12.619

224.79 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

200.00 286.46

1.0000 1.0000

286.46 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

Single-Phase Properties 13.916 17.160

13.924 17.168

0.077436 0.092721

0.084327 0.087714

0.12823 0.13371

794.34 675.42

−0.33920 −0.27468

35.092 41.155 50.746 61.864

36.881 43.613 54.054 66.012

0.18042 0.20616 0.23608 0.26271

0.073155 0.086960 0.10398 0.11764

0.083578 0.095910 0.11252 0.12607

127.60 149.16 172.25 192.24

36.498 13.603 5.6807 3.1093

0.074878 0.095673

13.888 25.960

13.963 26.055

0.077295 0.12724

0.084329 0.097767

0.12805 0.15865

799.42 384.49

−0.34145 0.15862

103.50 64.240

1.8842 2.0720 3.1466 4.0661

38.807 40.159 50.310 61.591

40.691 42.231 53.456 65.657

0.17833 0.18359 0.21585 0.24302

0.090862 0.092286 0.10520 0.11806

0.11564 0.11224 0.11626 0.12764

122.09 129.90 165.47 189.33

20.318 16.539 5.9354 3.1109

13.866 14.732 22.513 31.336

12.653 13.270 17.404 21.014

13.432 10.214 2.1222 1.3333

0.074450 0.097901 0.47120 0.75004

13.768 27.606 47.739 60.288

14.140 28.095 50.095 64.038

0.076686 0.13288 0.19593 0.22710

0.084364 0.099404 0.11164 0.12001

0.12732 0.15790 0.15240 0.13643

820.94 379.39 136.55 180.53

−0.35061 0.16755 6.6581 2.9952

105.29 62.727 27.340 33.551

671.00 155.81 22.244 23.530

0.055877 0.040689 0.030228 0.024108 13.355 10.452 0.53072 0.48261 0.31780 0.24593

0.074979 0.079245 17.897 24.576 33.082 41.479

103.09 91.425 8.7263 14.156 22.115 30.917

631.60 412.85 9.6994 13.041 17.070 20.691 638.78 168.19

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

13.522 10.597 5.5436 2.8438

0.073953 0.094369 0.18039 0.35165

13.626 27.096 43.724 58.555

14.366 28.039 45.528 62.072

0.075959 0.13109 0.18103 0.21813

0.084459 0.098728 0.11417 0.12198

0.12655 0.14971 0.19389 0.14925

845.71 441.85 165.42 183.47

−0.36040 −0.0032571 2.8474 2.4316

107.42 67.164 40.583 37.529

712.07 177.37 46.568 29.745

200.00 300.00 400.00 500.00

30.000 30.000 30.000 30.000

13.835 11.489 9.0272 6.8239

0.072281 0.087039 0.11078 0.14654

13.138 25.838 39.705 54.141

15.306 28.449 43.028 58.538

0.073355 0.12645 0.16830 0.20288

0.085197 0.098551 0.11217 0.12339

0.12439 0.13883 0.15193 0.15662

927.73 597.35 391.05 307.14

−0.38765 −0.23171 0.060756 0.34144

115.19 79.855 59.824 53.649

889.29 245.90 112.78 67.694

300.00 400.00 500.00

60.000 60.000 60.000

12.259 10.465 8.9121

0.081575 0.095559 0.11221

24.732 37.854 51.703

29.626 43.587 58.436

0.12196 0.16206 0.19516

0.10001 0.11339 0.12470

0.13426 0.14453 0.15193

735.32 563.68 471.23

−0.32748 −0.23451 −0.15582

93.938 75.022 67.810

338.04 173.26 113.08

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Jacobsen, R. T., “A New Functional Form and New Fitting Techniques for Equations of State with Application to Pentafluoroethane (HFC-125),” J. Phys. Chem. Ref. Data 34(1):69–108, 2005. The source for viscosity is Huber, M. L., and Laesecke, A.,“Correlation for the Viscosity of Pentafluoroethane (R125) from the Triple Point to 500 K at Pressures up to 60 MPa,” Ind. Eng. Chem. Res., 45(12):4447–4453, 2006. The source for thermal conductivity is Perkins, R., and Huber, M. L., “Measurement and Correlation of the Thermal Conductivity of Pentafluoroethane (R125) from 190 K to 512 K at Pressures to 70 MPa,” J. Chem. Eng. Data 51(3):898–904, 2006. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density is 0.1% at temperatures from the triple point to 400 K at pressures up to 60 MPa, except in the critical region, where an uncertainty of 0.2% in pressure is generally attained. In the limited region between 340 and 400 K and at pressures from 4 to 10 MPa, as well as for all states above 400 K, the uncertainty in density increases to 0.5%. At temperatures below 330 K and pressures below 30 MPa, the uncertainty in density in the liquid phase may be as low as 0.04%. In the vapor and supercritical region, speed of sound data are represented within 0.05% at pressures below 1 MPa. The estimated uncertainty for heat capacities is 0.5%, and the estimated uncertainty for the speed of sound in the liquid phase is 0.5% for T > 250 K. The estimated uncertainties of vapor pressures and saturated liquid densities calculated using the Maxwell criterion are 0.1% for each property, and the estimated uncertainty for saturated vapor densities is 0.2%. The uncertainty in density increases as the critical point is approached, while the accompanying uncertainty in calculated pressures is 0.2%. The viscosity correlation has an estimated uncertainty of 3.0% along the saturation boundary in the liquid phase, and 0.8% in the vapor. For thermal conductivity, the estimated uncertainty of the correlation is 3%, except for the dilute gas and points approaching critical, where the uncertainty rises to 5%.

FIG. 2-25

Enthalpy–log-pressure diagram for Refrigerant 125.

2-371

2-372 TABLE 2-278

Thermodynamic Properties of R-134a, 1,1,1,2-Tetrafluoroethane

Temperature K

Pressure MPa

169.85 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 374.21

0.00038956 0.00039617 0.0011275 0.0028170 0.0063130 0.012910 0.024433 0.043287 0.072481 0.11561 0.17684 0.26082 0.37271 0.51805 0.70282 0.93340 1.2166 1.5599 1.9715 2.4611 3.0405 3.7278 4.0591

169.85 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 374.21

0.00038956 0.00039617 0.0011275 0.0028170 0.0063130 0.012910 0.024433 0.043287 0.072481 0.11561 0.17684 0.26082 0.37271 0.51805 0.70282 0.93340 1.2166 1.5599 1.9715 2.4611 3.0405 3.7278 4.0591

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

7.2907 7.3088 8.5179 9.7328 10.957 12.194 13.444 14.710 15.992 17.293 18.613 19.956 21.322 22.716 24.141 25.603 27.108 28.667 30.297 32.029 33.932 36.283 38.947

7.2907 7.3088 8.5179 9.7330 10.958 12.195 13.446 14.713 15.997 17.301 18.627 19.976 21.352 22.759 24.201 25.685 27.219 28.816 30.495 32.293 34.289 36.797 39.756

0.042100 0.042207 0.049117 0.055686 0.061966 0.067999 0.073815 0.079441 0.084899 0.090209 0.095389 0.10046 0.10543 0.11032 0.11516 0.11996 0.12475 0.12956 0.13446 0.13952 0.14496 0.15159 0.15938

32.764 32.772 33.287 33.821 34.371 34.934 35.508 36.090 36.675 37.261 37.844 38.420 38.986 39.538 40.069 40.573 41.038 41.451 41.785 41.994 41.973 41.323 38.947

34.175 34.184 34.781 35.395 36.023 36.662 37.308 37.956 38.602 39.242 39.870 40.482 41.073 41.636 42.166 42.653 43.083 43.438 43.687 43.775 43.576 42.617 39.756

0.20038 0.20029 0.19502 0.19075 0.18729 0.18451 0.18228 0.18050 0.17909 0.17797 0.17709 0.17640 0.17586 0.17542 0.17504 0.17469 0.17432 0.17387 0.17326 0.17232 0.17075 0.16731 0.15938

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.080831 0.080824 0.080732 0.081114 0.081784 0.082633 0.083595 0.084636 0.085734 0.086879 0.088067 0.089298 0.090576 0.091908 0.093303 0.094777 0.096352 0.098067 0.10001 0.10241 0.10601 0.11372

0.12079 0.12079 0.12112 0.12193 0.12303 0.12434 0.12582 0.12746 0.12927 0.13126 0.13348 0.13597 0.13883 0.14216 0.14615 0.15108 0.15740 0.16598 0.17863 0.20012 0.24863 0.52085

1120.0 1119.2 1068.3 1017.7 967.61 918.33 869.85 822.11 775.00 728.39 682.14 636.12 590.17 544.15 497.89 451.23 404.00 355.90 306.37 254.06 196.05 127.23 0

−0.38145 −0.38136 −0.37370 −0.36352 −0.35119 −0.33678 −0.32011 −0.30082 −0.27839 −0.25204 −0.22073 −0.18299 −0.13675 −0.079015 −0.0052732 0.091533 0.22306 0.41006 0.69376 1.1714 2.1419 5.1434 11.931

145.24 145.15 139.12 133.32 127.74 122.36 117.17 112.14 107.27 102.53 97.922 93.414 88.995 84.644 80.341 76.063 71.781 67.464 63.075 58.581 54.062 51.767

2153.6 2139.7 1479.1 1106.2 867.31 702.27 582.15 491.22 420.20 363.25 316.57 277.54 244.34 215.64 190.46 168.04 147.78 129.20 111.81 95.095 78.146 57.956

0.051318 0.051354 0.053742 0.056118 0.058489 0.060874 0.063296 0.065783 0.068357 0.071031 0.073812 0.076698 0.079686 0.082776 0.085974 0.089297 0.092780 0.096484 0.10052 0.10510 0.11074 0.11928

0.059719 0.059756 0.062208 0.064682 0.067201 0.069802 0.072534 0.075455 0.078618 0.082078 0.085888 0.090115 0.094850 0.10023 0.10650 0.11404 0.12355 0.13638 0.15548 0.18870 0.26594 0.70016

Saturated Properties 15.594 15.590 15.331 15.069 14.804 14.535 14.262 13.984 13.699 13.406 13.104 12.791 12.465 12.121 11.758 11.368 10.945 10.478 9.9483 9.3237 8.5279 7.2558 5.0171 0.00027611 0.00028055 0.00075481 0.0017896 0.0038201 0.0074704 0.013574 0.023188 0.037603 0.058360 0.087278 0.12651 0.17865 0.24685 0.33512 0.44874 0.59505 0.78498 1.0363 1.3818 1.8973 2.8805 5.0171

0.064126 0.064142 0.065228 0.066362 0.067550 0.068798 0.070116 0.071512 0.072999 0.074593 0.076311 0.078179 0.080227 0.082499 0.085050 0.087965 0.091364 0.095439 0.10052 0.10725 0.11726 0.13782 0.19932 3621.7 3564.4 1324.8 558.79 261.77 133.86 73.669 43.125 26.593 17.135 11.458 7.9043 5.5976 4.0511 2.9840 2.2285 1.6805 1.2739 0.96498 0.72368 0.52707 0.34717 0.19932

126.79 126.84 130.05 133.11 135.98 138.63 141.01 143.06 144.73 145.98 146.75 146.99 146.63 145.61 143.88 141.33 137.86 133.33 127.57 120.33 111.25 99.370 0

373.57 370.78 234.43 160.10 116.94 90.215 72.584 60.236 51.130 44.137 38.613 34.169 30.561 27.621 25.230 23.301 21.768 20.578 19.687 19.033 18.448 17.050 11.931

3.0801 3.0921 3.8934 4.6952 5.4978 6.3018 7.1080 7.9176 8.7324 9.5551 10.389 11.241 12.118 13.035 14.011 15.081 16.303 17.780 19.711 22.525 27.365 40.137

6.8294 6.8353 7.2319 7.6253 8.0147 8.3993 8.7786 9.1524 9.5209 9.8853 10.247 10.611 10.980 11.363 11.771 12.219 12.735 13.358 14.164 15.300 17.140 21.336

Single-Phase Properties 200.00 246.79

0.10000 0.10000

246.79 275.00 350.00 425.00

0.10000 0.10000 0.10000 0.10000

200.00 275.00 312.54

1.0000 1.0000 1.0000

312.54 350.00 425.00

1.0000 1.0000 1.0000

200.00 275.00 350.00 425.00

5.0000 5.0000 5.0000 5.0000

10.955 16.873

10.962 16.880

0.061955 0.088519

0.081787 0.086506

0.12301 0.13060

968.03 743.31

−0.35132 −0.26099

37.073 39.124 45.154 52.096

39.037 41.348 48.032 55.610

0.17830 0.18716 0.20860 0.22818

0.070161 0.073781 0.086248 0.098386

0.080931 0.083445 0.095065 0.10695

145.63 154.76 175.31 192.97

46.198 29.047 12.552 6.7852

0.067479 0.079009 0.088775

10.933 20.597 25.980

11.001 20.676 26.069

0.061846 0.10281 0.12117

0.081812 0.089915 0.095165

0.12291 0.13695 0.15252

972.08 619.10 439.31

−0.35256 −0.16746 0.12101

128.11 91.627 74.978

2.0729 2.5555 3.3352

40.695 44.290 51.597

42.768 46.846 54.933

0.17460 0.18694 0.20785

0.090164 0.090315 0.099891

0.11623 0.10603 0.11116

140.54 159.63 185.14

22.877 13.885 7.0297

15.374 17.989 23.806

12.343 13.936 16.917

14.880 12.804 9.8674 2.0736

0.067202 0.078103 0.10134 0.48225

10.839 20.385 31.397 48.647

11.175 20.776 31.904 51.058

0.061371 0.10203 0.13765 0.18734

0.081929 0.089864 0.10066 0.10791

0.12246 0.13495 0.17178 0.15381

989.55 651.41 320.01 148.25

−0.35768 −0.19924 0.59952 7.9048

129.56 94.015 63.012 28.574

915.11 277.35 109.32 20.974 966.05 295.02 128.79 46.711

14.805 13.501 0.050898 0.044972 0.034753 0.028455 14.819 12.657 11.264 0.48242 0.39132 0.29983

0.067543 0.074068 19.647 22.236 28.775 35.143

127.78 104.04 9.2899 11.540 17.537 23.539

868.18 380.27 9.7687 10.906 13.823 16.650 876.60 262.84 162.71

200.00 275.00 350.00 425.00

10.000 10.000 10.000 10.000

14.954 12.967 10.478 6.1370

0.066874 0.077121 0.095440 0.16295

10.727 20.149 30.642 43.563

11.395 20.920 31.597 45.193

0.060796 0.10115 0.13537 0.17038

0.082085 0.089868 0.099573 0.11141

0.12196 0.13304 0.15486 0.20870

1010.3 687.36 400.60 177.89

−0.36339 −0.22964 0.21924 3.0434

131.31 96.744 68.919 44.888

200.00 275.00 350.00 425.00

30.000 30.000 30.000 30.000

15.216 13.479 11.662 9.7202

0.065720 0.074190 0.085750 0.10288

10.326 19.398 29.071 39.385

12.298 21.624 31.644 42.471

0.058683 0.098210 0.13038 0.15838

0.082769 0.090220 0.098885 0.10808

0.12047 0.12865 0.13885 0.14967

1084.1 801.47 582.52 425.63

−0.38053 −0.30014 −0.15240 0.10364

137.79 105.87 82.955 67.154

1211.0 364.87 183.26 107.03

275.00 350.00 425.00

70.000 70.000 70.000

14.181 12.797 11.494

0.070517 0.078141 0.087004

18.373 27.492 37.066

23.310 32.961 43.157

0.093839 0.12484 0.15121

0.091314 0.099542 0.10829

0.12519 0.13226 0.13963

962.41 787.10 661.39

−0.35619 −0.30093 −0.23655

119.84 99.868 86.640

521.91 277.09 181.77

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Tillner-Roth, R., and Baehr, H. D., “An International Standard Formulation of the Thermodynamic Properties of 1,1,1,2-Tetrafluoroethane (HFC-134a) for temperatures from 170 K to 455 K at Pressures up to 70 MPa,” J. Phys. Chem. Ref. Data 23:657–729, 1994. The source for viscosity is Huber, M. L., Laesecke, A., and Perkins, R. A., “Model for the Viscosity and Thermal Conductivity of Refrigerants, Including a New Correlation for the Viscosity of R134a,” Ind. Eng. Chem. Res. 42:3163–3178, 2003. The source for thermal conductivity is Perkins, R. A., Laesecke, A., Howley, J., Ramires, M. L. V., Gurova, A. N., and Cusco, L.,”Experimental Thermal Conductivity Values for the IUPAC Round-Robin Sample of 1,1,1,2-Tetrafluoroethane (R134a),” NISTIR, 2000. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. Typical uncertainties are 0.05% for density, 0.02% for vapor pressure, 0.5% to 1% for heat capacity, 0.05% for vapor speed of sound, and 1% for liquid speed of sound, except in the critical region. The uncertainty in viscosity is 1.5% along the saturated-liquid line, 3% in the liquid phase, 0.5% in the dilute gas, 3% to 5% in the vapor phase, and 5% in the supercritical region, rising to 8% at pressures above 40 MPa. Below 200 K, the uncertainty is 8%. The uncertainty in thermal conductivity is 5%.

2-373

2.

300

.

.

450

0.

80

.

700

500

.

600

550

kg/m 400.

10. 8.

300.

150.

4.

100. 80.

80

0

−10

−20

6.

200. c.p.

90

−30

600 20.

3

500. ρ=

100

90

80

0.

90

10

10

11

70

60

400

00

50

00

50 11

12 50

40

.

.

. 00

50

350

1350.

1400.

T = −40 °C

−60

4.

−50

1450.

6.

12

130

(1,1,1,2-tetrafluoroethane) reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

10

10. 8.

0.

R-134a

250 .

200

30

150

20

2-374

100 20.

70

60.

2.

60

40. 30.

40

1. 0.8

30

0.6

20.

20

15.

10

0.4

10.

−40

) kg

·K

200

250

J/( 40 2.

300

0.4

350

400

450

500

180

0.2

4.0

2.0

0.1 0.08

1.5

0.06

1.0

0.04

0.60

2.3 s=

−60

150

0.6

0.80

0k

0 2.2

0 2.1

2.0

0

0 1.9

1.80

1.70

1.60

1.50

1.40

1.30

1.20

1.10

1.00

0.90

0.80

0.70

−50

0.01 100

170

150

T = 160 °C

140

130

120

110

ted v

0.04

0.02

6.0

3.0

satura

0.8

0.7

0.6

0.5

0.4 x=

0.3

0.2

iqu

0.1

dl sa

tur

ate

0.06

0.9

−30

id

0.1 0.08

apor

−20

100

T = −10 °C

0.2

1. 0.8

8.0

0

0 10 20 30 40 50 60 70 80 90

Pressure (MPa)

50

550

0.02

0.40

0.01 600

Enthalpy (kJ/kg) FIG. 2-26 Pressure-enthalpy diagram for Refrigerant 134a. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of Tillner-Roth, R., and Baehr, H. D., “An International Standard Formulation of the Thermodynamic Properties of 1,1,1,2-Tetrafluoroethane (HFC-134a) Covering Temperatures from 170 K to 455 K at Pressures up to 70 MPa,” J. Phys. Chem. Ref. Data 23(5):657–729, 1994.

TABLE 2-279 Temperature K

Thermodynamic Properties of R-141b, 1,1-Dichloro-1-Fluoroethane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

1362.2 1336.2 1264.8 1196.3 1130.5 1067.1 1005.9 946.56 888.79 832.34 776.95 722.38 668.39 614.74 561.17 507.38 452.97 397.38 339.70 278.17 208.92 118.23 0

−0.46011 −0.46486 −0.47325 −0.47511 −0.47131 −0.46244 −0.44884 −0.43061 −0.40759 −0.37931 −0.34494 −0.30314 −0.25184 −0.18789 −0.10633 0.00079658 0.14720 0.35834 0.68768 1.2701 2.5762 8.4848 15.683

Saturated Properties 169.68 175.00 190.00 205.00 220.00 235.00 250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 370.00 385.00 400.00 415.00 430.00 445.00 460.00 475.00 477.50

6.4927E-06 1.3392E-05 7.9884E-05 0.00035111 0.0012182 0.0035074 0.0086986 0.019117 0.038053 0.069799 0.11960 0.19353 0.29839 0.44157 0.63091 0.87470 1.1817 1.5611 2.0234 2.5805 3.2487 4.0574 4.2117

169.68 175.00 190.00 205.00 220.00 235.00 250.00 265.00 280.00 295.00 310.00 325.00 340.00 355.00 370.00 385.00 400.00 415.00 430.00 445.00 460.00 475.00 477.50

6.4927E-06 1.3392E-05 7.9884E-05 0.00035111 0.0012182 0.0035074 0.0086986 0.019117 0.038053 0.069799 0.11960 0.19353 0.29839 0.44157 0.63091 0.87470 1.1817 1.5611 2.0234 2.5805 3.2487 4.0574 4.2117

12.560 12.475 12.241 12.009 11.779 11.549 11.318 11.083 10.845 10.602 10.351 10.092 9.8211 9.5371 9.2365 8.9148 8.5660 8.1809 7.7444 7.2267 6.5517 5.2532 3.9210 4.6024E-06 9.2047E-06 5.0581E-05 0.00020615 0.00066719 0.0018021 0.0042156 0.0087866 0.016681 0.029348 0.048516 0.076209 0.11480 0.16710 0.23659 0.32780 0.44696 0.60344 0.81289 1.1059 1.5609 2.6472 3.9210

0.079620 0.080158 0.081696 0.083271 0.084897 0.086588 0.088358 0.090224 0.092205 0.094324 0.096609 0.099093 0.10182 0.10485 0.10827 0.11217 0.11674 0.12224 0.12913 0.13838 0.15263 0.19036 0.25504 217,280. 108,640. 19,770. 4,850.8 1,498.8 554.92 237.22 113.81 59.949 34.074 20.612 13.122 8.7112 5.9846 4.2267 3.0507 2.2373 1.6572 1.2302 0.90427 0.64064 0.37776 0.25504

9.9164 10.635 12.622 14.573 16.505 18.434 20.369 22.319 24.291 26.290 28.323 30.393 32.505 34.664 36.876 39.148 41.489 43.912 46.442 49.122 52.069 55.987 58.424

9.9164 10.635 12.622 14.573 16.506 18.434 20.370 22.321 24.294 26.297 28.334 30.412 32.535 34.710 36.945 39.246 41.627 44.103 46.703 49.480 52.565 56.760 59.498

0.054895 0.059062 0.069959 0.079842 0.088942 0.097423 0.10540 0.11298 0.12022 0.12717 0.13389 0.14041 0.14677 0.15299 0.15909 0.16512 0.17110 0.17706 0.18308 0.18925 0.19586 0.20453 0.21020

0.094555 0.093422 0.091221 0.090109 0.089760 0.089963 0.090575 0.091495 0.092648 0.093975 0.095432 0.096986 0.098610 0.10029 0.10201 0.10377 0.10557 0.10745 0.10945 0.11171 0.11464 0.12111

0.13580 0.13422 0.13103 0.12928 0.12859 0.12869 0.12943 0.13069 0.13238 0.13446 0.13691 0.13972 0.14292 0.14659 0.15085 0.15592 0.16222 0.17053 0.18264 0.20353 0.25500 0.92342

42.292 42.585 43.445 44.353 45.303 46.291 47.312 48.362 49.437 50.531 51.641 52.763 53.890 55.016 56.134 57.234 58.300 59.311 60.231 60.993 61.427 60.578 58.424

43.703 44.040 45.025 46.056 47.129 48.237 49.376 50.538 51.718 52.910 54.106 55.302 56.489 57.659 58.801 59.902 60.944 61.898 62.720 63.326 63.508 62.110 59.498

0.25401 0.24995 0.24050 0.23342 0.22814 0.22424 0.22143 0.21946 0.21816 0.21739 0.21703 0.21700 0.21722 0.21763 0.21817 0.21877 0.21939 0.21994 0.22033 0.22037 0.21965 0.21579 0.21020

0.054464 0.055738 0.059180 0.062459 0.065653 0.068824 0.072021 0.075271 0.078582 0.081944 0.085336 0.088731 0.092106 0.095442 0.098733 0.10198 0.10522 0.10848 0.11185 0.11548 0.11977 0.12640

0.062782 0.064059 0.067515 0.070830 0.074095 0.077393 0.080793 0.084345 0.088081 0.092019 0.096171 0.10056 0.10521 0.11023 0.11578 0.12213 0.12983 0.13996 0.15495 0.18176 0.25134 1.1603

117.92 119.57 124.11 128.46 132.61 136.53 140.16 143.44 146.30 148.66 150.46 151.61 152.02 151.61 150.27 147.85 144.19 139.04 132.06 122.74 110.25 93.316 0

881.00 725.60 439.35 282.90 192.43 137.32 102.08 78.530 62.188 50.500 41.951 35.598 30.835 27.254 24.582 22.633 21.289 20.490 20.229 20.562 21.544 21.336 15.683

2-375

2-376 TABLE 2-279

Thermodynamic Properties of R-141b, 1,1-Dichloro-1-Fluoroethane (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

200.00 300.00 304.82

0.10000 0.10000 0.10000

12.087 10.519 10.438

0.082735 0.095063 0.095799

13.923 26.963 27.617

13.931 26.973 27.627

0.076634 0.12943 0.13160

304.82 400.00 500.00

0.10000 0.10000 0.10000

51.257 59.875 70.152

53.693 63.154 74.282

200.00 300.00 391.53

1.0000 1.0000 1.0000

0.082685 0.094914 0.11407

13.904 26.923 40.158

391.53 400.00 500.00

1.0000 1.0000 1.0000

2.6612 2.7746 3.8663

200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

12.127 10.606 8.7814 2.5394

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.090387 0.094448 0.094917

0.12973 0.13523 0.13602

1219.1 813.87 795.96

−0.47521 −0.36865 −0.35757

0.21711 0.24405 0.26884

0.084163 0.096356 0.10833

0.094713 0.10524 0.11691

149.91 173.74 194.56

44.615 15.111 8.0045

13.987 27.018 40.272

0.076539 0.12930 0.16773

0.090431 0.094476 0.10455

0.12969 0.13502 0.15848

1222.5 819.45 483.80

57.703 58.638 69.520

60.364 61.413 73.387

0.21904 0.22169 0.24841

0.10339 0.10298 0.10961

0.12527 0.12252 0.12123

146.42 150.55 184.02

0.082464 0.094284 0.11388 0.39380

13.822 26.751 41.014 63.911

14.234 27.222 41.584 65.880

0.076124 0.12872 0.16989 0.22278

0.090629 0.094606 0.10538 0.12220

0.12954 0.13416 0.15604 0.29767

1237.0 843.29 502.40 113.94

−0.47789 −0.38665 0.0060632 14.372

12.166 10.688 9.0042 6.4723

0.082198 0.093559 0.11106 0.15451

13.722 26.550 40.515 56.831

14.544 27.486 41.625 58.376

0.075616 0.12803 0.16859 0.20581

0.090880 0.094784 0.10531 0.11637

0.12937 0.13326 0.15122 0.19130

1254.5 871.09 554.81 265.53

−0.48033 −0.40188 −0.10586 1.3996

Single-Phase Properties

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

0.041044 0.030497 0.024216 12.094 10.536 8.7668 0.37577 0.36041 0.25865

24.364 32.790 41.296

−0.47573 −0.37223 0.058567 21.978 19.770 8.7395

300.00 400.00 500.00

100.00 100.00 100.00

11.645 10.676 9.7763

0.085873 0.093670 0.10229

24.317 36.738 49.945

32.904 46.105 60.174

0.11935 0.15726 0.18863

0.098509 0.10834 0.11752

0.12788 0.13645 0.14460

1200.2 995.82 852.98

−0.49651 −0.44654 −0.39437

400.00 500.00

400.00 400.00

12.505 11.986

0.079970 0.083429

33.644 46.201

65.632 79.572

0.14209 0.17316

0.11569 0.12401

0.13564 0.14302

1606.9 1498.0

−0.48581 −0.46416

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The equation has uncertainties of 0.2% in density between 180 and 400 K at pressures to 100 MPa, and 0.5% in density at higher pressures. The uncertainty in density may be higher as temperatures approach 400 K. Vapor pressures are represented with an uncertainty of 0.2% from 270 to 400 K. The uncertainty in speed of sound is 0.01% in the vapor phase and 0.5% in the liquid phase. There are no heat capacity data to verify the equation of state; however, the uncertainties are estimated to be within 2%.

TABLE 2-280

Thermodynamic Properties of R-142b, 1-Chloro-1,1-Difluoroethane

Temperature K

Pressure MPa

142.72 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 410.26

3.6327E-06 1.1727E-05 9.0545E-05 0.00047371 0.0018484 0.0057557 0.015025 0.034108 0.069220 0.12829 0.22078 0.35741 0.54995 0.81110 1.1545 1.5950 2.1496 2.8397 3.6991 4.0548

142.72 150.00 165.00 180.00 195.00 210.00 225.00 240.00 255.00 270.00 285.00 300.00 315.00 330.00 345.00 360.00 375.00 390.00 405.00 410.26

3.6327E-06 1.1727E-05 9.0545E-05 0.00047371 0.0018484 0.0057557 0.015025 0.034108 0.069220 0.12829 0.22078 0.35741 0.54995 0.81110 1.1545 1.5950 2.1496 2.8397 3.6991 4.0548

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.49370 −0.49255 −0.48684 −0.47689 −0.46285 −0.44466 −0.42204 −0.39443 −0.36085 −0.31977 −0.26876 −0.20400 −0.11928 −0.0039877 0.16142 0.41716 0.86137 1.8155 5.3582 14.009

Saturated Properties 14.439 14.281 13.963 13.650 13.339 13.028 12.715 12.396 12.069 11.732 11.380 11.010 10.617 10.193 9.7272 9.2033 8.5896 7.8118 6.5381 4.4380

2-377

3.0614E-06 9.4031E-06 6.6018E-05 0.00031681 0.0011428 0.0033148 0.0081206 0.017432 0.033721 0.060064 0.10020 0.15871 0.24138 0.35595 0.51364 0.73250 1.0462 1.5347 2.5436 4.4380

0.069257 0.070022 0.071619 0.073262 0.074968 0.076756 0.078649 0.080672 0.082856 0.085239 0.087873 0.090824 0.094190 0.098109 0.10280 0.10866 0.11642 0.12801 0.15295 0.22533 326,650. 106,350. 15,147. 3,156.4 875.03 301.68 123.14 57.365 29.655 16.649 9.9797 6.3006 4.1428 2.8094 1.9469 1.3652 0.95585 0.65159 0.39314 0.22533

5.0138 5.8147 7.4687 9.1343 10.818 12.526 14.263 16.033 17.841 19.692 21.590 23.538 25.545 27.617 29.765 32.009 34.381 36.963 40.117 43.151

5.0138 5.8147 7.4687 9.1343 10.818 12.526 14.264 16.036 17.847 19.703 21.609 23.571 25.597 27.696 29.884 32.182 34.632 37.327 40.683 44.065

0.026013 0.031486 0.041995 0.051656 0.060640 0.069077 0.077064 0.084681 0.091990 0.099042 0.10588 0.11255 0.11908 0.12551 0.13189 0.13828 0.14477 0.15160 0.15973 0.16786

0.067327 0.068270 0.070168 0.072051 0.073955 0.075895 0.077877 0.079897 0.081949 0.084027 0.086128 0.088252 0.090406 0.092605 0.094876 0.097271 0.099910 0.10313 0.10871

0.11000 0.11005 0.11057 0.11158 0.11299 0.11477 0.11689 0.11931 0.12205 0.12513 0.12859 0.13253 0.13714 0.14271 0.14984 0.15984 0.17613 0.21217 0.44141

1457.9 1403.7 1301.7 1209.7 1125.1 1046.0 971.28 899.90 831.11 764.28 698.87 634.41 570.43 506.46 441.95 376.09 307.44 232.74 141.73 0

32.685 33.008 33.711 34.461 35.253 36.082 36.944 37.831 38.737 39.657 40.582 41.506 42.419 43.306 44.149 44.916 45.551 45.920 45.483 43.151

33.872 34.256 35.083 35.956 36.870 37.819 38.794 39.787 40.790 41.793 42.786 43.758 44.697 45.585 46.397 47.094 47.605 47.771 46.938 44.065

0.22821 0.22109 0.20935 0.20067 0.19424 0.18952 0.18609 0.18365 0.18196 0.18086 0.18019 0.17984 0.17972 0.17972 0.17975 0.17970 0.17937 0.17838 0.17518 0.16786

0.043630 0.045288 0.048631 0.051923 0.055225 0.058589 0.062056 0.065643 0.069344 0.073133 0.076978 0.080851 0.084737 0.088646 0.092615 0.096730 0.10116 0.10632 0.11368

0.051946 0.053606 0.056964 0.060296 0.063688 0.067227 0.070991 0.075042 0.079422 0.084167 0.089331 0.095026 0.10147 0.10911 0.11883 0.13261 0.15586 0.21004 0.55965

118.57 121.19 126.42 131.39 136.07 140.40 144.30 147.66 150.35 152.25 153.23 153.15 151.86 149.19 144.89 138.64 129.97 118.18 102.00 0

826.84 618.59 363.33 230.77 156.62 112.18 83.865 64.873 51.637 42.170 35.289 30.249 26.559 23.894 22.043 20.877 20.328 20.306 19.783 14.009

2-378

TABLE 2-280 Temperature K

Thermodynamic Properties of R-142b, 1-Chloro-1,1-Difluoroethane (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/molK

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 150.00 225.00 263.70

0.10000 0.10000 0.10000

263.70 300.00 375.00 450.00

0.10000 0.10000 0.10000 0.10000

150.00 225.00 300.00 338.73

1.0000 1.0000 1.0000 1.0000

338.73 375.00 450.00

1.0000 1.0000 1.0000

150.00 225.00 300.00 375.00 450.00

5.0000 5.0000 5.0000 5.0000 5.0000

5.8131 14.260 18.910

5.8201 14.268 18.918

0.031476 0.077053 0.096110

0.068275 0.077880 0.083152

0.11005 0.11688 0.12379

1404.1 971.75 792.14

−0.49259 −0.42221 −0.33806

39.269 41.979 48.175 55.179

41.372 44.411 51.257 58.897

0.18126 0.19205 0.21237 0.23092

0.071534 0.076432 0.087964 0.098211

0.082128 0.085864 0.096734 0.10678

151.56 162.77 182.58 199.94

45.775 26.250 12.432 7.4502

0.069982 0.078553 0.090630 0.10073

5.7989 14.234 23.498 28.857

5.8689 14.312 23.589 28.957

0.031381 0.076937 0.11241 0.12923

0.068317 0.077905 0.088238 0.093915

0.11003 0.11676 0.13215 0.14661

1407.4 976.69 640.59 469.03

2.2648 2.7214 3.5002

43.804 47.311 54.646

46.069 50.033 58.146

0.17974 0.19087 0.21058

0.090944 0.091549 0.099383

0.11443 0.10695 0.11074

146.90 164.80 190.29

22.728 14.567 7.9246

14.321 12.791 11.170 8.9978 2.2231

0.069826 0.078178 0.089523 0.11114 0.44982

5.7370 14.121 23.263 33.787 50.992

6.0861 14.512 23.710 34.343 53.241

0.030965 0.076430 0.11162 0.14314 0.18867

0.068506 0.078025 0.088194 0.098897 0.10800

0.10997 0.11629 0.13015 0.15862 0.17517

1421.7 997.91 676.41 373.38 139.23

−0.49417 −0.43157 −0.25069 0.42121 10.314

14.282 12.716 11.875 0.047556 0.041113 0.032446 0.026895 14.289 12.730 11.034 9.9277 0.44153 0.36746 0.28570

0.070018 0.078641 0.084211 21.028 24.323 30.820 37.182

−0.49289 −0.42404 −0.21142 0.084297

150.00 225.00 300.00 375.00 450.00

10.000 10.000 10.000 10.000 10.000

14.360 12.864 11.321 9.4491 6.5247

0.069636 0.077739 0.088335 0.10583 0.15326

5.6618 13.987 23.001 33.097 45.004

6.3581 14.765 23.885 34.155 46.536

0.030454 0.075821 0.11072 0.14120 0.17118

0.068749 0.078190 0.088215 0.098194 0.10853

0.10991 0.11578 0.12826 0.14723 0.18939

1439.0 1023.0 716.03 451.41 220.16

−0.49558 −0.43976 −0.28769 0.11527 2.1126

225.00 300.00 375.00 450.00

30.000 30.000 30.000 30.000

13.119 11.787 10.406 8.9530

0.076223 0.084836 0.096097 0.11169

13.519 22.183 31.547 41.538

15.805 24.728 34.430 44.889

0.073606 0.10777 0.13659 0.16199

0.078954 0.088691 0.097974 0.10623

0.11435 0.12401 0.13464 0.14384

1110.6 840.46 635.15 484.32

−0.46323 −0.37159 −0.22643 −0.010974

225.00 300.00 375.00 450.00

60.000 60.000 60.000 60.000

13.434 12.284 11.180 10.119

0.074437 0.081409 0.089443 0.098828

12.957 21.322 30.259 39.706

17.423 26.207 35.626 45.636

0.070756 0.10439 0.13238 0.15670

0.080223 0.089774 0.098881 0.10699

0.11326 0.12130 0.12975 0.13689

1217.3 975.81 801.25 675.38

−0.48274 −0.42589 −0.35484 −0.27949

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density are 0.3% in the liquid phase below 370 K, 1% at higher temperatures in the liquid and supercritical regions, and 0.5% in the vapor phase. Uncertainties for other properties are 0.5% for vapor pressure, 2% for heat capacities and liquid sound speeds, and 0.2% for vapor sound speeds.

TABLE 2-281 Temperature K

Thermodynamic Properties of R-143a, 1,1,1-Trifluoroethane Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.43936 −0.42914 −0.41402 −0.39585 −0.37472 −0.35034 −0.32211 −0.28906 −0.24979 −0.20231 −0.14368 −0.069548 0.026895 0.15683 0.34002 0.61491 1.0681 1.9469 4.3897 12.397

Saturated Properties 161.34 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 345.86

0.0010749 0.0025084 0.0059324 0.012629 0.024624 0.044602 0.075908 0.12252 0.18902 0.28049 0.40251 0.56112 0.76276 1.0144 1.3234 1.6983 2.1483 2.6850 3.3250 3.7618

161.34 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 345.86

0.0010749 0.0025084 0.0059324 0.012629 0.024624 0.044602 0.075908 0.12252 0.18902 0.28049 0.40251 0.56112 0.76276 1.0144 1.3234 1.6983 2.1483 2.6850 3.3250 3.7618

15.832 15.583 15.291 14.994 14.692 14.384 14.069 13.745 13.410 13.062 12.698 12.314 11.904 11.461 10.975 10.426 9.7846 8.9829 7.7913 5.1285 0.00080362 0.0017832 0.0039967 0.0081006 0.015110 0.026311 0.043269 0.067850 0.10227 0.14916 0.21175 0.29409 0.40144 0.54109 0.72367 0.96617 1.2991 1.7898 2.6696 5.1285

0.063163 0.064174 0.065399 0.066693 0.068062 0.069519 0.071077 0.072753 0.074570 0.076556 0.078751 0.081208 0.084004 0.087249 0.091119 0.095914 0.10220 0.11132 0.12835 0.19499 1244.4 560.78 250.20 123.45 66.180 38.007 23.111 14.738 9.7783 6.7041 4.7225 3.4004 2.4910 1.8481 1.3818 1.0350 0.76974 0.55871 0.37458 0.19499

4.4138 5.2969 6.3240 7.3629 8.4164 9.4869 10.576 11.685 12.817 13.973 15.156 16.368 17.615 18.903 20.239 21.638 23.125 24.750 26.688 29.429

4.4138 5.2971 6.3244 7.3637 8.4181 9.4900 10.581 11.694 12.831 13.995 15.188 16.414 17.680 18.991 20.360 21.801 23.344 25.048 27.114 30.163

0.026403 0.031735 0.037606 0.043223 0.048626 0.053849 0.058915 0.063848 0.068665 0.073385 0.078027 0.082607 0.087149 0.091675 0.096221 0.10083 0.10559 0.11065 0.11659 0.12527

0.068393 0.068179 0.068405 0.068990 0.069825 0.070836 0.071969 0.073190 0.074475 0.075809 0.077186 0.078605 0.080078 0.081625 0.083293 0.085172 0.087455 0.090641 0.096654

0.10179 0.10225 0.10325 0.10460 0.10621 0.10803 0.11005 0.11227 0.11474 0.11750 0.12066 0.12435 0.12879 0.13438 0.14180 0.15244 0.16980 0.20591 0.35704

1058.1 1016.7 969.14 921.61 874.04 826.46 778.88 731.28 683.65 635.90 587.94 539.61 490.70 440.95 389.93 337.04 281.21 220.39 149.32 0

25.521 25.895 26.340 26.796 27.262 27.736 28.213 28.693 29.170 29.641 30.102 30.548 30.972 31.366 31.715 32.000 32.183 32.184 31.732 29.429

26.859 27.302 27.824 28.355 28.892 29.431 29.968 30.498 31.018 31.521 32.003 32.456 32.872 33.240 33.544 33.758 33.837 33.684 32.977 30.163

0.16552 0.16118 0.15705 0.15370 0.15100 0.14881 0.14704 0.14560 0.14444 0.14349 0.14270 0.14202 0.14141 0.14081 0.14017 0.13940 0.13838 0.13682 0.13384 0.12527

0.044397 0.046371 0.048691 0.051040 0.053424 0.055867 0.058395 0.061026 0.063766 0.066612 0.069560 0.072606 0.075756 0.079035 0.082489 0.086214 0.090400 0.095488 0.10298

0.052938 0.055037 0.057550 0.060156 0.062886 0.065796 0.068954 0.072428 0.076287 0.080611 0.085515 0.091184 0.097932 0.10632 0.11743 0.13359 0.16076 0.22018 0.47999

137.57 140.62 143.91 146.92 149.60 151.89 153.71 155.02 155.74 155.81 155.15 153.70 151.36 148.04 143.59 137.85 130.56 121.34 109.29 0

385.09 262.77 176.43 124.70 92.835 72.442 58.764 49.133 42.049 36.657 32.456 29.136 26.498 24.406 22.765 21.499 20.526 19.682 18.259 12.397

2-379

2-380 TABLE 2-281

Thermodynamic Properties of R-143a, 1,1,1-Trifluoroethane (Concluded)

Temperature K

Pressure MPa

200.00 225.63

0.10000 0.10000

225.63 300.00 400.00 500.00 600.00

0.10000 0.10000 0.10000 0.10000 0.10000

200.00 289.48

1.0000 1.0000

289.48 300.00 400.00 500.00 600.00

1.0000 1.0000 1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00 600.00

5.0000 5.0000 5.0000 5.0000 5.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 8.4143 11.198

8.4211 11.205

0.048616 0.061709

0.069829 0.072648

0.10620 0.11128

874.45 752.07

−0.37493 −0.30416

28.483 33.380 41.260 50.551 61.004

30.268 35.833 44.566 54.698 65.987

0.14619 0.16745 0.19247 0.21503 0.23558

0.059863 0.070758 0.086121 0.099065 0.10950

0.070868 0.079793 0.094695 0.10751 0.11790

154.52 179.93 207.36 231.13 252.55

52.958 18.414 7.5341 4.1010 2.5191

0.067956 0.087067

8.3891 18.835

8.4570 18.922

0.048489 0.091441

0.069866 0.081543

0.10604 0.13406

879.27 443.55

−0.37736 0.14902

1.8765 2.0285 3.1249 4.0544 4.9391

31.346 32.269 40.778 50.244 60.781

33.223 34.298 43.903 54.298 65.720

0.14084 0.14449 0.17212 0.19527 0.21607

0.078861 0.077834 0.087343 0.099519 0.10975

0.10583 0.099455 0.098681 0.10927 0.11891

148.24 155.53 198.53 227.30 251.11

24.503 20.787 7.7199 4.0976 2.4913

14.806 11.380 2.2504 1.3639 1.0491

0.067539 0.087876 0.44436 0.73318 0.95318

8.2811 19.812 38.004 48.802 59.789

8.6188 20.251 40.225 52.468 64.554

0.047943 0.094764 0.15165 0.17903 0.20106

0.070021 0.082741 0.093496 0.10135 0.11075

0.10541 0.13222 0.13815 0.11872 0.12364

900.01 452.11 159.25 213.49 246.86

−0.38724 0.11636 8.1026 3.8903 2.2996

14.694 13.888 0.056043 0.040759 0.030247 0.024114 0.020066 14.715 11.485 0.53292 0.49298 0.32001 0.24665 0.20246

0.068054 0.072006 17.843 24.534 33.061 41.469 49.836

200.00 300.00 400.00 500.00 600.00

10.000 10.000 10.000 10.000 10.000

14.913 11.776 6.3531 3.0122 2.1596

0.067054 0.084916 0.15740 0.33199 0.46305

8.1545 19.382 33.679 46.933 58.580

8.8250 20.231 35.253 50.252 63.211

0.047292 0.093258 0.13608 0.16976 0.19339

0.070195 0.082583 0.095653 0.10303 0.11178

0.10473 0.12585 0.17697 0.13206 0.12945

924.61 514.06 196.98 211.16 248.28

−0.39776 −0.037998 2.9315 3.0876 1.9304

200.00 300.00 400.00 500.00 600.00

50.000 50.000 50.000 50.000 50.000

15.598 13.358 11.268 9.4018 7.8978

0.064113 0.074862 0.088747 0.10636 0.12662

7.3673 17.612 28.859 40.839 53.306

10.573 21.355 33.296 46.157 59.637

0.042937 0.086486 0.12077 0.14943 0.17399

0.070934 0.083360 0.096003 0.10668 0.11548

0.10170 0.11389 0.12454 0.13213 0.13720

1089.7 794.12 590.09 478.93 431.69

−0.44295 −0.33796 −0.20571 −0.077157 −0.0052131

14.343 12.767 11.435 10.314

0.069721 0.078330 0.087453 0.096952

16.500 27.298 38.923 51.227

23.472 35.131 47.668 60.923

0.081539 0.11502 0.14296 0.16711

0.084059 0.097186 0.10821 0.11716

0.11166 0.12126 0.12921 0.13566

1008.9 833.69 723.22 656.57

−0.40055 −0.35302 −0.31534 −0.28902

300.00 400.00 500.00 600.00

100.00 100.00 100.00 100.00

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Jacobsen, R. T., “An International Standard Formulation for the Thermodynamic Properties of 1,1,1-Trifluoroethane (HFC-143a) for Temperatures from 161 to 450 K and Pressures to 50 MPa,” J. Phys. Chem. Ref. Data 29(4):521–552, 2000. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainties of properties calculated using the equation of state are 0.1% in density, 0.5% in heat capacities, 0.02% in the speed of sound for the vapor at pressures less than 1 MPa, 0.5% in speed of sound elsewhere, and 0.1% in vapor pressure, except in the critical region.

TABLE 2-282

Thermodynamic Properties of R-152a, 1,1-Difluoroethane

Temperature K

Pressure MPa

154.56 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 386.41

6.4139E-05 0.00012983 0.00041463 0.0011411 0.0027751 0.0060859 0.012233 0.022836 0.040024 0.066451 0.10530 0.16024 0.23541 0.33537 0.46506 0.62978 0.83519 1.0873 1.3925 1.7577 2.1907 2.7000 3.2967 3.9966 4.5168

154.56 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 386.41

6.4139E-05 1.2983E-04 0.00041463 0.0011411 0.0027751 0.0060859 0.012233 0.022836 0.040024 0.066451 0.10530 0.16024 0.23541 0.33537 0.46506 0.62978 0.83519 1.0873 1.3925 1.7577 2.1907 2.7000 3.2967 3.9966 4.5168

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.43262 −0.42405 −0.41381 −0.40568 −0.39686 −0.38620 −0.37317 −0.35753 −0.33902 −0.31735 −0.29203 −0.26235 −0.22731 −0.18547 −0.13478 −0.072262 0.0064939 0.10833 0.24433 0.43380 0.71349 1.1636 2.0014 4.1151 11.292

Saturated Properties 18.061 17.910 17.630 17.350 17.067 16.781 16.491 16.198 15.899 15.594 15.281 14.960 14.628 14.283 13.924 13.546 13.147 12.719 12.257 11.748 11.176 10.510 9.6829 8.4686 5.5715

2-381

4.9919E-05 9.7623E-05 0.00029357 0.00076374 0.0017625 0.0036817 0.0070756 0.012679 0.021415 0.034408 0.052990 0.078721 0.11343 0.15926 0.21879 0.29518 0.39242 0.51573 0.67231 0.87269 1.1336 1.4847 1.9912 2.8726 5.5715

0.055369 0.055836 0.056720 0.057638 0.058594 0.059592 0.060638 0.061737 0.062897 0.064128 0.065440 0.066846 0.068363 0.070011 0.071819 0.073821 0.076065 0.078621 0.081587 0.085118 0.089475 0.095147 0.10328 0.11808 0.17949 20,032. 10,244. 3,406.3 1,309.3 567.37 271.61 141.33 78.873 46.696 29.063 18.871 12.703 8.8162 6.2791 4.5705 3.3877 2.5483 1.9390 1.4874 1.1459 0.88218 0.67352 0.50220 0.34812 0.17949

0.91117 1.4460 2.4420 3.4453 4.4524 5.4638 6.4814 7.5077 8.5448 9.5949 10.660 11.741 12.841 13.962 15.105 16.273 17.470 18.699 19.968 21.284 22.662 24.126 25.730 27.655 30.732

0.91117 1.4460 2.4421 3.4454 4.4525 5.4641 6.4822 7.5091 8.5474 9.5992 10.667 11.752 12.857 13.985 15.138 16.319 17.533 18.785 20.081 21.434 22.858 24.383 26.070 28.127 31.543

0.0073929 0.010794 0.016832 0.022566 0.028011 0.033199 0.038164 0.042938 0.047548 0.052017 0.056365 0.060607 0.064759 0.068835 0.072848 0.076812 0.080741 0.084652 0.088567 0.092513 0.096532 0.10070 0.10515 0.11043 0.11914

0.065711 0.066998 0.067926 0.068069 0.068043 0.068097 0.068309 0.068689 0.069217 0.069868 0.070618 0.071447 0.072342 0.073294 0.074300 0.075363 0.076490 0.077694 0.078997 0.080433 0.082059 0.083973 0.086395 0.090038

0.097583 0.098932 0.10007 0.10053 0.10090 0.10143 0.10218 0.10317 0.10439 0.10583 0.10747 0.10932 0.11140 0.11374 0.11641 0.11951 0.12318 0.12765 0.13333 0.14093 0.15192 0.16999 0.20774 0.35969

1400.9 1359.2 1293.7 1236.1 1182.6 1131.3 1081.2 1031.8 982.80 933.98 885.23 836.44 787.50 738.29 688.71 638.61 587.86 536.30 483.73 429.89 374.37 316.40 254.38 183.93 0

26.412 26.619 27.009 27.412 27.825 28.247 28.676 29.110 29.546 29.984 30.419 30.851 31.277 31.694 32.099 32.488 32.857 33.199 33.506 33.764 33.953 34.036 33.933 33.381 30.732

27.697 27.949 28.422 28.906 29.399 29.900 30.405 30.911 31.415 31.915 32.406 32.887 33.352 33.800 34.224 34.622 34.985 35.308 35.577 35.779 35.886 35.855 35.589 34.772 31.543

0.18069 0.17644 0.16965 0.16401 0.15931 0.15538 0.15208 0.14931 0.14697 0.14500 0.14332 0.14189 0.14067 0.13960 0.13866 0.13782 0.13704 0.13629 0.13552 0.13470 0.13375 0.13256 0.13088 0.12792 0.11914

0.037824 0.038592 0.040032 0.041518 0.043061 0.044675 0.046371 0.048158 0.050042 0.052027 0.054111 0.056291 0.058564 0.060926 0.063373 0.065904 0.068522 0.071233 0.074054 0.077013 0.080160 0.083590 0.087501 0.092461

0.046148 0.046923 0.048389 0.049922 0.051550 0.053297 0.055192 0.057260 0.059528 0.062017 0.064754 0.067769 0.071103 0.074811 0.078977 0.083731 0.089270 0.095924 0.10426 0.11534 0.13140 0.15806 0.21461 0.44168

154.04 156.44 160.70 164.78 168.64 172.27 175.63 178.66 181.32 183.57 185.35 186.60 187.28 187.33 186.70 185.34 183.16 180.10 176.06 170.91 164.49 156.55 146.71 134.10 0

336.54 294.63 233.22 186.99 151.71 124.45 103.17 86.396 73.068 62.400 53.805 46.838 41.163 36.517 32.701 29.557 26.966 24.832 23.083 21.662 20.515 19.584 18.738 17.420 11.292

2-382 TABLE 2-282 Temperature K

Thermodynamic Properties of R-152a, 1,1-Difluoroethane (Concluded) Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 200.00 248.83

0.10000 0.10000

248.83 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 316.76

1.0000 1.0000 1.0000

316.76 400.00 500.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 500.00

5.0000 5.0000 5.0000 5.0000

5.4618 10.535

5.4678 10.541

0.033189 0.055862

0.068102 0.070526

0.10142 0.10726

1131.7 890.93

−0.38632 −0.29520

30.369 33.323 40.103 48.286

32.350 35.766 43.405 52.430

0.14351 0.15598 0.17786 0.19794

0.053862 0.060341 0.074770 0.088343

0.064421 0.069611 0.083404 0.096818

185.16 204.37 235.25 261.80

54.718 27.531 10.509 5.4858

0.059530 0.073703 0.077752

5.4436 16.249 18.296

5.5031 16.323 18.374

0.033097 0.076732 0.083384

0.068150 0.075359 0.077294

0.10133 0.11920 0.12609

1135.5 642.81 553.13

2.1165 3.0733 4.0246

33.092 39.505 47.922

35.208 42.579 51.947

0.13653 0.15720 0.17807

0.070342 0.076729 0.088892

0.093615 0.088940 0.098930

181.20 222.89 256.19

25.479 10.836 5.4940

16.867 13.783 3.1526 1.4336

0.059288 0.072554 0.31720 0.69753

5.3645 16.011 34.823 46.169

5.6609 16.373 36.409 49.657

0.032698 0.075926 0.13126 0.16117

0.068345 0.075345 0.092011 0.091345

0.10098 0.11645 0.26333 0.11166

1152.1 684.64 148.13 232.93

−0.39242 −0.13417 12.808 5.3800

16.782 15.318 0.050481 0.040939 0.030290 0.024131 16.798 13.568 12.861 0.47247 0.32539 0.24847

0.059586 0.065282 19.809 24.427 33.014 41.441

−0.38750 −0.078450 0.072197

200.00 300.00 400.00 500.00

10.000 10.000 10.000 10.000

16.949 14.014 9.5224 3.4436

0.059000 0.071358 0.10502 0.29039

5.2700 15.752 28.480 43.674

5.8600 16.466 29.530 46.578

0.032215 0.075036 0.11233 0.15056

0.068554 0.075376 0.087211 0.094041

0.10058 0.11401 0.16027 0.13591

1172.7 729.99 310.79 219.92

−0.39792 −0.18420 1.2458 4.4385

200.00 300.00 400.00 500.00

30.000 30.000 30.000 30.000

17.248 14.709 11.934 8.9569

0.057977 0.067986 0.083794 0.11165

4.9338 14.971 26.048 38.069

6.6732 17.010 28.561 41.418

0.030434 0.072215 0.10535 0.13400

0.069111 0.075740 0.085262 0.094199

0.099355 0.10883 0.12249 0.13295

1253.5 869.51 576.62 407.30

−0.41462 −0.29079 −0.0089214 0.48690

200.00 300.00 400.00 500.00

60.000 60.000 60.000 60.000

17.625 15.424 13.309 11.312

0.056738 0.064836 0.075135 0.088399

4.5216 14.174 24.599 35.736

7.9258 18.064 29.107 41.039

0.028097 0.069098 0.10080 0.12739

0.069524 0.076558 0.086520 0.095954

0.098187 0.10570 0.11507 0.12324

1367.5 1020.4 770.60 613.84

−0.42991 −0.35644 −0.24663 −0.11541

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Outcalt, S. L., and McLinden, M. O., “A Modified Benedict-Webb-Rubin Equation of State for the Thermodynamic Properties of R152a (1,1-Difluoroethane),” J. Phys. Chem. Ref. Data 25(2):605–636, 1996. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 2% in heat capacity, and 0.05% in the vapor speed of sound, except in the critical region. The uncertainty in vapor pressure is 0.1%.

THERMODYNAMIC PROPERTIES TABLE 2-283 Temp., °F

Saturated Refrigerant 216a, 1,3-Dichloro-1,1,2,2,3,3-Hexafluoropropane Pressure, lb/in2 abs.

Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−40 −20 0 20 40

0.339 0.713 1.382 2.497 4.247

0.00927 0.00942 0.00958 0.00974 0.00992

59.957 29.749 15.986 9.184 5.582

0.000 4.778 9.541 14.298 19.056

62.415 65.276 68.208 71.199 74.239

0.0000 0.0111 0.0217 0.0318 0.0415

0.1487 0.1487 0.1493 0.1504 0.1520

60 80 100 120 140

6.862 10.612 15.797 22.753 31.845

0.01010 0.01030 0.01050 0.01073 0.01097

3.558 2.361 1.6215 1.1462 0.8304

23.821 28.598 33.391 38.205 43.049

77.319 80.429 83.559 86.701 89.845

0.0509 0.0599 0.0686 0.0770 0.0852

0.1538 0.1559 0.1582 0.1607 0.1632

160 180 200 220 240

43.468 58.046 76.033 97.913 124.21

0.01124 0.01153 0.01186 0.01223 0.01266

0.6142 0.4623 0.3529 0.2725 0.2121

47.930 52.861 57.857 62.939 68.132

92.981 96.099 99.186 102.225 105.196

0.0931 0.1009 0.1085 0.1161 0.1235

0.1658 0.1685 0.1712 0.1739 0.1765

260 280 300 320 340

155.50 192.40 235.63 286.03 344.81

0.01317 0.01378 0.01458 0.01570 0.01764

0.1660 0.1300 0.1013 0.0776 0.0565

73.474 79.015 84.835 91.089 98.234

108.066 110.789 113.282 115.373 116.538

0.1309 0.1384 0.1460 0.1539 0.1628

0.1790 0.1813 0.1834 0.1851 0.1856

355.98c

399.45

0.02771

0.0277

110.248

110.248

0.1773

0.1773

*From published data, Chemicals Division, Union Carbide Corporation. Used by permission. The paper describing these data is by Shank, ASHRAE J., 1(1965): 94–101. c = critical temperature. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale from −118.59 to 113.26 °C. The thermodynamic diagram from 0.1 to 30 bar extends to 180 °C. For tables and a diagram to 400 psia, 360 °F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Thermal conductivity data as a function of pressure and temperature are reported by Krauss, R. and K. Stephan, Proc. 12th Symp. Thermophys. Props., Boulder, CO, 1994.

2-383

2-384 TABLE 2-284

Thermodynamic Properties of R-218, Octafluoropropane

Temperature K

Pressure MPa

125.45 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 345.02

2.0186E-06 4.8985E-06 2.7215E-05 0.00011605 0.00040065 0.0011659 0.0029499 0.0066482 0.013606 0.025681 0.045264 0.075274 0.11911 0.18060 0.26395 0.37369 0.51466 0.69206 0.91150 1.1793 1.5030 1.8924 2.3634 2.6402

125.45 130.00 140.00 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 345.02

2.0186E-06 4.8985E-06 2.7215E-05 1.1605E-04 0.00040065 0.0011659 0.0029499 0.0066482 0.013606 0.025681 0.045264 0.075274 0.11911 0.18060 0.26395 0.37369 0.51466 0.69206 0.91150 1.1793 1.5030 1.8924 2.3634 2.6402

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.48474 −0.48168 −0.47276 −0.46146 −0.44819 −0.43313 −0.41623 −0.39731 −0.37604 −0.35194 −0.32433 −0.29229 −0.25451 −0.20911 −0.15332 −0.082915 0.0089029 0.13369 0.31279 0.59020 1.0733 2.1079 5.6878 17.096

Saturated Properties 10.687 10.598 10.407 10.219 10.033 9.8482 9.6632 9.4771 9.2890 9.0979 8.9027 8.7022 8.4950 8.2795 8.0537 7.8151 7.5602 7.2843 6.9802 6.6366 6.2324 5.7189 4.9092 3.3400 1.9353E-06 4.5321E-06 2.3384E-05 9.3093E-05 0.00030151 0.00082688 0.0019804 0.0042437 0.0082953 0.015024 0.025536 0.041158 0.063461 0.094301 0.13591 0.19105 0.26332 0.35768 0.48153 0.64702 0.87704 1.2252 1.8924 3.3400

0.093572 0.094353 0.096088 0.097856 0.099670 0.10154 0.10349 0.10552 0.10765 0.10992 0.11233 0.11491 0.11772 0.12078 0.12417 0.12796 0.13227 0.13728 0.14326 0.15068 0.16045 0.17486 0.20370 0.29940 516,720. 220,650. 42,765. 10,742. 3,316.6 1,209.4 504.96 235.65 120.55 66.559 39.161 24.297 15.758 10.604 7.3580 5.2343 3.7977 2.7958 2.0767 1.5456 1.1402 0.81620 0.52844 0.29940

12.346 13.024 14.528 16.054 17.607 19.189 20.800 22.444 24.121 25.832 27.580 29.365 31.190 33.056 34.966 36.923 38.933 41.000 43.134 45.351 47.677 50.177 53.102 56.252

12.346 13.024 14.528 16.054 17.607 19.189 20.801 22.445 24.122 25.835 27.585 29.374 31.204 33.077 34.998 36.971 39.001 41.095 43.265 45.528 47.918 50.508 53.583 57.042

0.057736 0.063044 0.074187 0.084719 0.094739 0.10433 0.11354 0.12242 0.13102 0.13937 0.14750 0.15544 0.16320 0.17082 0.17832 0.18571 0.19302 0.20029 0.20754 0.21483 0.22226 0.23002 0.23893 0.24883

0.086723 0.088773 0.093214 0.097514 0.10165 0.10561 0.10942 0.11310 0.11667 0.12014 0.12355 0.12690 0.13022 0.13351 0.13680 0.14010 0.14343 0.14683 0.15035 0.15405 0.15812 0.16294 0.17021

0.14866 0.14939 0.15145 0.15392 0.15668 0.15964 0.16276 0.16603 0.16944 0.17301 0.17675 0.18069 0.18489 0.18940 0.19432 0.19980 0.20607 0.21355 0.22299 0.23603 0.25690 0.30134 0.52335

1157.7 1119.0 1041.4 972.09 909.32 851.65 797.99 747.52 699.61 653.72 609.45 566.44 524.41 483.09 442.24 401.64 361.06 320.24 278.86 236.46 192.26 144.68 89.750 0

37.874 38.208 38.983 39.808 40.680 41.595 42.548 43.536 44.553 45.596 46.661 47.744 48.843 49.952 51.067 52.183 53.291 54.381 55.437 56.433 57.322 57.996 58.081 56.252

38.917 39.289 40.147 41.055 42.009 43.005 44.038 45.102 46.193 47.305 48.433 49.573 50.720 51.867 53.009 54.139 55.246 56.316 57.330 58.256 59.035 59.540 59.330 57.042

0.26954 0.26508 0.25718 0.25139 0.24725 0.24442 0.24263 0.24167 0.24138 0.24161 0.24227 0.24326 0.24452 0.24598 0.24759 0.24929 0.25104 0.25277 0.25442 0.25589 0.25700 0.25739 0.25584 0.24883

0.072287 0.074826 0.080190 0.085274 0.090119 0.094776 0.099300 0.10374 0.10814 0.11253 0.11693 0.12135 0.12577 0.13020 0.13463 0.13906 0.14352 0.14802 0.15262 0.15742 0.16259 0.16855 0.17647

0.080604 0.083144 0.088516 0.093623 0.098514 0.10326 0.10792 0.11258 0.11730 0.12213 0.12712 0.13232 0.13776 0.14351 0.14970 0.15649 0.16420 0.17340 0.18521 0.20207 0.23055 0.29532 0.62247

78.648 79.920 82.654 85.299 87.846 90.278 92.569 94.686 96.589 98.233 99.570 100.55 101.12 101.22 100.79 99.737 97.968 95.356 91.730 86.859 80.422 72.007 61.241 0

854.14 682.22 432.76 288.12 200.28 144.70 108.17 83.339 65.928 53.394 44.174 37.278 32.063 28.095 25.083 22.826 21.188 20.088 19.493 19.423 19.958 21.201 22.532 17.096

Single-Phase Properties 150.00 225.00 236.07

0.10000 0.10000 0.10000

236.07 300.00 375.00

0.10000 0.10000 0.10000

150.00 225.00 300.00 303.52

1.0000 1.0000 1.0000 1.0000

303.52 375.00

1.0000 1.0000

150.00 225.00 300.00 375.00

5.0000 5.0000 5.0000 5.0000

16.052 28.465 30.468

16.061 28.477 30.479

0.084701 0.15148 0.16017

0.097521 0.12523 0.12892

0.15391 0.17867 0.18321

972.51 588.11 540.83

−0.46154 −0.30918 −0.27015

48.410 56.919 68.272

50.269 59.362 71.361

0.24400 0.27801 0.31363

0.12403 0.14104 0.16055

0.13558 0.15028 0.16927

100.95 116.42 131.00

33.943 12.416 6.1732

0.097772 0.11333 0.14308 0.14568

16.028 28.413 43.116 43.905

16.125 28.527 43.259 44.051

0.084540 0.15125 0.20748 0.21010

0.097582 0.12522 0.15030 0.15163

0.15385 0.17828 0.22248 0.22705

976.23 594.72 280.68 264.07

−0.46230 −0.31433 0.30296 0.39586

1.8718 2.8080

55.797 67.521

57.669 70.329

0.25497 0.29246

0.15428 0.16284

0.19038 0.17689

90.170 120.35

19.406 7.1384

10.262 8.9050 7.2996 4.0113

0.097443 0.11230 0.13699 0.24930

15.923 28.196 42.444 60.519

16.411 28.757 43.129 61.765

0.083839 0.15027 0.20518 0.26018

0.097864 0.12525 0.14888 0.17483

0.15361 0.17677 0.20895 0.31884

992.31 622.21 346.02 101.99

−0.46544 −0.33410 0.039275 4.7668

10.220 8.8041 8.5773 0.053789 0.040933 0.032372 10.228 8.8236 6.9891 6.8645 0.53424 0.35613

0.097848 0.11358 0.11659 18.591 24.430 30.891

150.00 225.00 300.00 375.00

10.000 10.000 10.000 10.000

10.304 8.9969 7.5606 5.6952

0.097050 0.11115 0.13227 0.17559

15.799 27.949 41.847 57.659

16.769 29.060 43.170 59.415

0.082988 0.14913 0.20307 0.25128

0.098234 0.12538 0.14819 0.16841

0.15336 0.17531 0.20166 0.23156

1011.4 653.07 403.66 219.93

−0.46885 −0.35341 −0.10223 0.67650

150.00 225.00 300.00 375.00

20.000 20.000 20.000 20.000

10.382 9.1565 7.9189 6.5981

0.096321 0.10921 0.12628 0.15156

15.565 27.517 40.989 55.836

17.492 29.701 43.515 58.867

0.081360 0.14709 0.19992 0.24552

0.099023 0.12580 0.14793 0.16668

0.15301 0.17326 0.19500 0.21331

1046.6 706.41 486.73 339.02

−0.47434 −0.38080 −0.22897 0.012472

150.00 225.00 300.00 375.00

30.000 30.000 30.000 30.000

10.454 9.2933 8.1774 7.0690

0.095654 0.10760 0.12229 0.14146

15.351 27.147 40.353 54.800

18.220 30.375 44.022 59.044

0.079818 0.14527 0.19747 0.24210

0.099855 0.12636 0.14819 0.16662

0.15280 0.17191 0.19173 0.20796

1078.7 751.98 549.34 416.45

−0.47844 −0.39925 −0.28972 −0.15627

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density is 0.2% for the liquid phase and 0.5% for the vapor phase. Above the critical temperature, the uncertainties are estimated to be 1% in density and 0.5% in pressure. Calculated vapor pressures have an uncertainty of 0.5%. The uncertainties for heat capacities and sound speeds are 1%.

2-385

2-386 TABLE 2-285

Thermodynamic Properties of R-227ea, 1,1,1,2,3,3,3-Heptafluoropropane

Temperature K

Pressure MPa

146.35 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 375.95

7.0312E-06 1.2811E-05 5.6086E-05 0.00019958 0.00059967 0.0015672 0.0036466 0.0076967 0.014961 0.027114 0.046286 0.075058 0.11644 0.17384 0.25102 0.35208 0.48145 0.64388 0.84450 1.0889 1.3836 1.7358 2.1552 2.6553 2.9991

146.35 150.00 160.00 170.00 180.00 190.00 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 375.95

7.0312E-06 1.2811E-05 5.6086E-05 0.00019958 0.00059967 0.0015672 0.0036466 0.0076967 0.014961 0.027114 0.046286 0.075058 0.11644 0.17384 0.25102 0.35208 0.48145 0.64388 0.84450 1.0889 1.3836 1.7358 2.1552 2.6553 2.9991

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

1068.8 1048.3 994.14 942.81 893.95 847.26 802.44 759.23 717.37 676.63 636.81 597.69 559.10 520.85 482.77 444.69 406.42 367.76 328.49 288.32 246.84 203.37 156.53 102.78 0

−0.42071 −0.42085 −0.41876 −0.41349 −0.40546 −0.39497 −0.38217 −0.36707 −0.34954 −0.32930 −0.30590 −0.27867 −0.24663 −0.20842 −0.16205 −0.10460 −0.031613 0.064014 0.19426 0.38093 0.66802 1.1593 2.1701 5.2930 16.489

Saturated Properties 11.086 11.020 10.841 10.663 10.487 10.310 10.133 9.9539 9.7727 9.5883 9.3999 9.2065 9.0068 8.7997 8.5834 8.3560 8.1148 7.8563 7.5756 7.2652 6.9135 6.4988 5.9733 5.1649 3.4100 5.7786E-06 1.0273E-05 4.2170E-05 0.00014128 0.00040123 0.00099473 0.0022037 0.0044452 0.0082891 0.014469 0.023890 0.037635 0.056982 0.083437 0.11880 0.16526 0.22559 0.30344 0.40387 0.53442 0.70732 0.94497 1.2970 1.9293 3.4100

0.090205 0.090746 0.092245 0.093780 0.095359 0.096992 0.098689 0.10046 0.10233 0.10429 0.10638 0.10862 0.11103 0.11364 0.11650 0.11967 0.12323 0.12729 0.13200 0.13764 0.14465 0.15387 0.16741 0.19361 0.29326 173,050. 97,345. 23,714. 7,078.0 2,492.3 1,005.3 453.78 224.96 120.64 69.111 41.858 26.571 17.550 11.985 8.4176 6.0510 4.4328 3.2956 2.4761 1.8712 1.4138 1.0582 0.77102 0.51831 0.29326

12.227 12.821 14.446 16.072 17.705 19.352 21.017 22.703 24.413 26.148 27.912 29.706 31.531 33.390 35.286 37.220 39.197 41.221 43.299 45.442 47.666 50.002 52.517 55.458 59.193

12.227 12.821 14.446 16.072 17.705 19.352 21.017 22.704 24.414 26.151 27.917 29.714 31.544 33.410 35.315 37.262 39.256 41.303 43.411 45.592 47.866 50.269 52.877 55.972 60.073

0.063811 0.067824 0.078309 0.088164 0.097501 0.10641 0.11495 0.12317 0.13112 0.13884 0.14635 0.15367 0.16083 0.16784 0.17474 0.18153 0.18824 0.19488 0.20149 0.20810 0.21476 0.22158 0.22873 0.23696 0.24773

0.11121 0.11136 0.11222 0.11357 0.11528 0.11723 0.11934 0.12157 0.12386 0.12618 0.12851 0.13085 0.13318 0.13549 0.13781 0.14011 0.14243 0.14477 0.14716 0.14964 0.15227 0.15517 0.15867 0.16399

0.16308 0.16271 0.16238 0.16286 0.16396 0.16554 0.16750 0.16976 0.17228 0.17502 0.17797 0.18114 0.18454 0.18822 0.19225 0.19673 0.20183 0.20782 0.21515 0.22467 0.23817 0.26018 0.30696 0.51614

41.158 41.423 42.179 42.976 43.814 44.689 45.600 46.543 47.515 48.513 49.533 50.570 51.622 52.684 53.752 54.818 55.878 56.922 57.938 58.909 59.808 60.587 61.146 61.164 59.193

42.374 42.670 43.509 44.389 45.308 46.265 47.255 48.275 49.320 50.387 51.470 52.565 53.666 54.768 55.864 56.949 58.012 59.044 60.029 60.946 61.764 62.424 62.807 62.540 60.073

0.26981 0.26682 0.25995 0.25474 0.25085 0.24805 0.24613 0.24494 0.24433 0.24421 0.24448 0.24507 0.24591 0.24695 0.24813 0.24942 0.25076 0.25211 0.25342 0.25463 0.25564 0.25630 0.25632 0.25471 0.24773

0.072002 0.073562 0.077861 0.082177 0.086488 0.090782 0.095053 0.099298 0.10352 0.10771 0.11188 0.11601 0.12010 0.12416 0.12818 0.13216 0.13612 0.14009 0.14409 0.14817 0.15244 0.15704 0.16228 0.16893

0.080321 0.081883 0.086194 0.090535 0.094896 0.099275 0.10368 0.10813 0.11263 0.11721 0.12189 0.12670 0.13166 0.13684 0.14232 0.14822 0.15475 0.16227 0.17136 0.18320 0.20027 0.22897 0.29185 0.56287

89.345 90.352 93.045 95.643 98.141 100.53 102.77 104.85 106.73 108.37 109.71 110.73 111.36 111.55 111.26 110.42 108.94 106.75 103.73 99.740 94.584 88.022 79.757 69.452 0

630.83 540.51 363.36 253.26 182.56 135.75 103.86 81.549 65.553 53.830 45.075 38.434 33.337 29.395 26.339 23.978 22.183 20.864 19.969 19.477 19.399 19.775 20.625 21.487 16.489

Single-Phase Properties 150.00 225.00 256.43

0.10000 0.10000 0.10000

256.43 300.00 375.00 450.00

0.10000 0.10000 0.10000 0.10000

150.00 225.00 300.00 326.58

1.0000 1.0000 1.0000 1.0000

326.58 375.00 450.00

1.0000 1.0000 1.0000

150.00 225.00 300.00 375.00 450.00

5.0000 5.0000 5.0000 5.0000 5.0000

12.819 25.274 30.875

12.828 25.284 30.886

0.067809 0.13499 0.15829

0.11137 0.12502 0.13235

0.16271 0.17360 0.18330

1048.6 697.31 572.84

−0.42089 −0.34002 −0.25870

51.245 56.684 67.081 78.645

53.272 59.112 70.162 82.364

0.24559 0.26660 0.29941 0.32903

0.11864 0.12925 0.14667 0.16079

0.12987 0.13881 0.15550 0.16939

111.18 122.16 137.79 151.34

35.009 17.772 8.4581 5.1086

0.090682 0.10312 0.12284 0.13559

12.799 25.234 39.137 44.701

12.890 25.337 39.260 44.837

0.067676 0.13482 0.18804 0.20584

0.11146 0.12507 0.14240 0.14878

0.16267 0.17337 0.20096 0.22108

1051.5 702.21 412.50 302.19

2.0587 2.7173 3.5127

58.583 66.118 78.039

60.642 68.835 81.552

0.25423 0.27764 0.30854

0.14676 0.15025 0.16198

0.17873 0.16647 0.17375

101.23 123.32 144.14

19.600 10.030 5.4173

11.058 9.7613 8.3150 6.0063 1.9568

0.090430 0.10245 0.12026 0.16649 0.51103

12.712 25.066 38.729 54.845 74.454

13.164 25.578 39.330 55.677 77.009

0.067092 0.13406 0.18665 0.23506 0.28716

0.11182 0.12533 0.14230 0.15968 0.16853

0.16253 0.17247 0.19592 0.25736 0.21820

1064.0 723.03 454.01 188.73 117.53

−0.42305 −0.35330 −0.12586 1.3729 5.8791

11.021 9.6823 9.0790 0.049336 0.041177 0.032454 0.026888 11.028 9.6973 8.1405 7.3754 0.48574 0.36801 0.28468

0.090739 0.10328 0.11014 20.269 24.286 30.813 37.191

−0.42131 −0.34268 −0.045381 0.30867

225.00 300.00 375.00 450.00

10.000 10.000 10.000 10.000

9.8358 8.4921 6.7736 4.4095

0.10167 0.11776 0.14763 0.22679

24.870 38.308 53.368 70.066

25.887 39.485 54.845 72.334

0.13316 0.18518 0.23077 0.27322

0.12567 0.14237 0.15755 0.17015

0.17153 0.19196 0.21921 0.23938

747.22 496.71 286.63 162.15

−0.36439 −0.18923 0.34861 2.0189

225.00 300.00 375.00 450.00

30.000 30.000 30.000 30.000

10.091 8.9874 7.8569 6.7284

0.099103 0.11127 0.12728 0.14862

24.209 37.111 51.038 65.707

27.182 40.449 54.856 70.165

0.13000 0.18079 0.22359 0.26078

0.12709 0.14335 0.15736 0.16895

0.16914 0.18477 0.19887 0.20846

829.42 619.36 467.67 368.25

−0.39369 −0.30379 −0.17451 −0.017558

225.00 300.00 375.00 450.00

60.000 60.000 60.000 60.000

10.391 9.4675 8.5978 7.7868

0.096240 0.10562 0.11631 0.12842

23.457 35.965 49.374 63.495

29.231 42.302 56.353 71.201

0.12610 0.17614 0.21789 0.25396

0.12922 0.14519 0.15896 0.17036

0.16752 0.18109 0.19315 0.20230

926.46 743.76 617.08 530.91

−0.41551 −0.36285 −0.30676 −0.25596

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is an interim equation from Lemmon, E. W., personal communication (NIST, Boulder, CO, 2006). Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.2% in density, except in a small region close to the critical point, 0.2% in vapor pressure between 250 and 360 K, 0.4% in vapor pressure outside this region, 1% in heat capacities (with increasing uncertainties in the critical region and at higher temperatures), 0.1% in the vapor-phase speed of sound, and 3% in the liquid-phase speed of sound.

2-387

2-388

PHYSICAL AND CHEMICAL DATA TABLE 2-286

Saturated Refrigerant 245cb 1,1,1,2,2-Pentafluoropropane*

P, bar

v f, m3/kg

vg, m3/kg

172 180 190 200 210

0.0034 0.0076 0.0190 0.0425 0.0870

6.46.−4 6.57.−4 6.70.−4 6.83.−4 6.97.−4

31.49 14.63 6.20 2.91 1.48

220 230 240 250 260

0.1654 0.2946 0.4958 0.7946 1.2204

7.11.−4 7.25.−4 7.40.−4 7.55.−4 7.72.−4

270 280 290 300 310

1.806 2.584 3.600 4.888 6.491

320 330 340 350 360 370 375 380.1c

T, K

h f, kJ/kg

h g, kJ/kg

s f, kJ/(kg⋅K)

s g, kJ/(kg⋅K)

−63.4 −55.9 −46.2 −36.0 −25.7

133.8 138.7 145.1 151.7 158.5

−0.3131 −0.2707 −0.2182 −0.1666 −0.1157

0.8327 0.8099 0.7885 0.7725 0.7612

0.822 0.475 0.292 0.192 0.125

−14.8 −3.6 8.0 19.9 32.3

165.4 172.5 179.6 186.8 194.0

−0.0654 −0.0156 0.0337 0.0824 0.1305

0.7539 0.7500 0.7487 0.7497 0.7525

7.89.−4 8.08.−4 8.30.−4 8.53.−4 8.80.−4

0.0862 0.0611 0.0443 0.0327 0.0246

44.9 57.9 71.1 84.6 98.4

201.1 208.3 215.3 222.2 228.9

0.1781 0.2249 0.2711 0.3161 0.3614

0.7567 0.7621 0.7683 0.7751 0.7822

8.456 10.83 13.67 17.04 21.02

9.11.−4 9.48.−4 9.93.−4 0.00105 0.00113

0.0186 0.0143 0.0111 0.0084 0.0063

112.6 127.1 142.1 157.2 174.7

235.3 241.4 246.9 251.5 254.8

0.4057 0.4497 0.4937 0.5382 0.5844

0.7893 0.7960 0.8018 0.8060 0.8071

25.71 28.46 31.37

0.00125 0.00137 0.00204

0.0045 0.0036 0.0020

193.6 205.2 231.8

255.2 252.5 231.8

0.6349 0.6649 0.7341

0.8013 0.7953 0.7341

*Values converted from tables of Shank, Thermodynamic Properties of UCON 245 Refrigerant, Union Carbide Corporation, New York, 1966. See also Shank, J. Chem. Eng. Data, 12, 474–480 (1967). c = critical point. The notation 6.46.−4 signifies 6.46 × 10−4.

TABLE 2-287 Refrigerant RC 318, Octafluorocyclobutane* P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.0216 0.0449 0.0875 0.1608 0.2810

5.507.−4 5.593.−4 5.683.−4 5.778.−4 5.879.−4

3.810 1.931 1.038 0.588 0.349

353.5 361.0 369.2 377.6 386.4

498.0 500.1 502.2 504.4 510.9

3.909 3.947 3.984 4.022 4.060

4.560 4.564 4.569 4.574 4.578

0.98 1.00

11.7 9.55

0.088 0.085

250 260 270 280 290

0.466 0.741 1.133 1.672 2.392

5.988.−4 6.106.−4 6.234.−4 6.375.−4 6.529.−4

0.2166 0.1401 0.0938 0.0647 0.0458

395.6 405.2 415.1 425.8 436.2

517.4 524.0 530.7 537.3 543.9

4.097 4.133 4.172 4.210 4.247

4.584 4.592 4.599 4.609 4.618

1.02 1.03 1.05 1.07 1.09

7.90 6.63 5.64 4.85 4.22

0.082 0.078 0.075 0.071 0.068

300 310 320 330 340

3.325 4.522 6.007 7.826 10.018

6.694.−4 6.893.−4 7.115.−4 7.365.−4 7.666.−4

0.0332 0.0245 0.0184 0.0139 0.0106

447.3 458.7 470.5 482.7 495.2

550.4 556.9 563.3 569.4 575.4

4.284 4.322 4.359 4.396 4.433

4.626 4.638 4.648 4.659 4.669

1.12 1.15 1.18 1.23 1.27

3.70 3.20 2.94 2.66 2.33

0.065 0.061 0.058 0.054 0.051

350 360 370 380 388.5c

12.632 15.71 19.33 23.59 27.83

8.034.−4 8.508.−4 9.172.−4 1.031.−3 1.613.−3

0.0082 0.0062 0.0047 0.0033 0.0016

508.1 521.5 535.6 551.4 577.2

581.0 585.8 589.9 591.5 577.2

4.469 4.507 4.544 4.585 4.651

4.678 4.685 4.691 4.691 4.651

1.32 1.39

2.00

0.048

T, K

*Values of P, v, h, and s interpolated, extrapolated, and converted from tables of Oguchi, Reito, 52 (1977): 869–889. Values of cp, µ, and k interpolated and converted from tables in Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. Saturation and superheat tables and a diagram to 80 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Cipollone, R., ASHRAE Trans., 97, 2 (1991): 262–267.

TABLE 2-288

Thermodynamic Properties of R-404A

Temperature K

Pressure MPa

200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 345.00 345.27

0.022649 0.030989 0.041658 0.055101 0.071804 0.092293 0.11713 0.14693 0.18232 0.22397 0.27258 0.32888 0.39363 0.46763 0.55168 0.64664 0.75338 0.87280 1.0059 1.1536 1.3169 1.4970 1.6950 1.9122 2.1499 2.4096 2.6932 3.0027 3.3414 3.7150 3.7348

200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00

0.021264 0.029285 0.039592 0.052629 0.068883 0.088879 0.11318 0.14240 0.17718 0.21817 0.26610 0.32169 0.38571 0.45896 0.54225 0.63645 0.74245 0.86115 0.99353 1.1406 1.3034 1.4830

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.070377 0.071131 0.071905 0.072703 0.073527 0.074380 0.075265 0.076185 0.077145 0.078147 0.079197 0.080301 0.081465 0.082697 0.084006 0.085402 0.086899 0.088513 0.090266 0.092182 0.094295 0.096652 0.099314 0.10237 0.10595 0.11027 0.11570 0.12302 0.13448 0.17413 0.20243

10.353 10.948 11.544 12.144 12.746 13.353 13.963 14.578 15.199 15.824 16.456 17.094 17.738 18.390 19.049 19.717 20.394 21.081 21.780 22.490 23.215 23.956 24.717 25.500 26.310 27.157 28.054 29.026 30.147 32.108 32.875

10.355 10.950 11.547 12.148 12.751 13.359 13.972 14.590 15.213 15.842 16.478 17.120 17.770 18.429 19.096 19.772 20.460 21.159 21.870 22.597 23.339 24.101 24.885 25.695 26.538 27.423 28.365 29.396 30.597 32.755 33.631

29.920 30.185 30.451 30.718 30.987 31.256 31.525 31.795 32.063 32.330 32.596 32.859 33.119 33.375 33.626 33.872 34.111 34.342 34.563 34.773 34.968 35.147

31.555 31.853 32.152 32.451 32.749 33.046 33.340 33.633 33.922 34.208 34.489 34.765 35.034 35.296 35.550 35.794 36.028 36.249 36.455 36.645 36.814 36.960

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.058867 0.061803 0.064678 0.067498 0.070269 0.072995 0.075679 0.078326 0.080939 0.083520 0.086073 0.088600 0.091104 0.093589 0.096056 0.098510 0.10095 0.10339 0.10583 0.10826 0.11071 0.11317 0.11566 0.11818 0.12075 0.12341 0.12619 0.12918 0.13261 0.13874 0.14126

0.076939 0.077522 0.078116 0.078719 0.079326 0.079940 0.080558 0.081183 0.081815 0.082456 0.083107 0.083770 0.084446 0.085137 0.085846 0.086574 0.087326 0.088104 0.088914 0.089761 0.090653 0.091603 0.092625 0.093744 0.094998 0.096455 0.098241 0.10064 0.10445 0.11650

0.11881 0.11915 0.11965 0.12028 0.12103 0.12188 0.12282 0.12386 0.12499 0.12621 0.12754 0.12899 0.13057 0.13229 0.13419 0.13630 0.13866 0.14133 0.14438 0.14793 0.15211 0.15715 0.16339 0.17139 0.18212 0.19751 0.22197 0.26871 0.40392 8.2559

859.89 831.56 804.42 778.20 752.69 727.72 703.16 678.91 654.88 631.01 607.23 583.50 559.77 535.99 512.13 488.15 464.02 439.69 415.13 390.28 365.11 339.56 313.56 287.00 259.73 231.53 201.95 170.20 134.59 89.976 0

−0.34161 −0.33384 −0.32460 −0.31394 −0.30185 −0.28830 −0.27318 −0.25636 −0.23766 −0.21683 −0.19356 −0.16749 −0.13814 −0.10493 −0.067104 −0.023728 0.026418 0.084928 0.15392 0.23626 0.33594 0.45865 0.61280 0.81141 1.0757 1.4433 1.9881 2.8807 4.6353 10.564 12.409

0.16521 0.16408 0.16307 0.16217 0.16138 0.16068 0.16006 0.15952 0.15903 0.15861 0.15823 0.15789 0.15759 0.15732 0.15707 0.15684 0.15661 0.15639 0.15616 0.15592 0.15566 0.15536

0.058696 0.059968 0.061245 0.062526 0.063815 0.065113 0.066424 0.067750 0.069095 0.070463 0.071855 0.073276 0.074728 0.076214 0.077737 0.079300 0.080909 0.082567 0.084282 0.086062 0.087917 0.089863

0.068032 0.069509 0.071021 0.072573 0.074172 0.075826 0.077546 0.079344 0.081232 0.083226 0.085343 0.087604 0.090033 0.092660 0.095524 0.098671 0.10217 0.10609 0.11056 0.11574 0.12186 0.12929

138.13 139.28 140.34 141.29 142.12 142.84 143.43 143.88 144.18 144.33 144.32 144.14 143.77 143.22 142.46 141.49 140.29 138.85 137.16 135.19 132.92 130.34

88.073 77.215 68.305 60.948 54.835 49.721 45.413 41.761 38.642 35.962 33.645 31.630 29.871 28.327 26.970 25.773 24.718 23.789 22.972 22.256 21.633 21.094

Saturated Properties 14.209 14.059 13.907 13.755 13.600 13.444 13.286 13.126 12.963 12.796 12.627 12.453 12.275 12.092 11.904 11.709 11.508 11.298 11.078 10.848 10.605 10.346 10.069 9.7686 9.4384 9.0688 8.6431 8.1285 7.4362 5.7429 4.9400

2-389

0.013010 0.017550 0.023271 0.030378 0.039095 0.049667 0.062359 0.077463 0.095292 0.11619 0.14055 0.16879 0.20137 0.23885 0.28183 0.33102 0.38725 0.45152 0.52501 0.60922 0.70599 0.81772

76.866 56.979 42.971 32.919 25.579 20.134 16.036 12.909 10.494 8.6063 7.1149 5.9247 4.9659 4.1867 3.5483 3.0210 2.5823 2.2148 1.9047 1.6414 1.4165 1.2229

2-390 TABLE 2-288

Thermodynamic Properties of R-404A (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

310.00 315.00 320.00 325.00 330.00 335.00 340.00 345.00 345.27

1.6806 1.8975 2.1351 2.3950 2.6789 2.9893 3.3299 3.7109 3.7348

0.94761 1.1001 1.2815 1.5019 1.7781 2.1438 2.6882 4.2113 4.9400

1.0553 0.90903 0.78032 0.66583 0.56239 0.46645 0.37199 0.23746 0.20243

35.304 35.435 35.530 35.578 35.558 35.429 35.084 33.615 32.875

226.65

0.10000

0.074669

13.554

13.561

227.41 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

31.386 36.596 45.121 55.103

289.79

1.0000

0.090189

290.23 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

300.00 400.00 500.00

5.0000 5.0000 5.0000

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.091922 0.094123 0.096513 0.099162 0.10220 0.10585 0.11074 0.12022

0.13856 0.15062 0.16713 0.19141 0.23111 0.30867 0.53035 8.6291

127.41 124.10 120.38 116.21 111.51 106.19 100.03 90.307 0

20.630 20.234 19.889 19.571 19.233 18.763 17.851 14.130 12.409

0.073887

0.080143

0.12218

719.55

−0.28348

33.188 39.050 48.428 59.250

0.16038 0.18269 0.20956 0.23365

0.065742 0.076875 0.092844 0.10612

0.076645 0.085907 0.10141 0.11456

143.14 166.24 191.81 213.92

47.558 16.998 6.9229 3.7455

21.750

21.840

0.10572

0.088879

0.14425

416.16

1.8916 2.0323 3.1292 4.0571

34.573 35.486 44.644 54.804

36.464 37.518 47.773 58.861

0.15615 0.15972 0.18922 0.21391

0.084363 0.083854 0.094142 0.10659

0.11078 0.10554 0.10545 0.11631

137.07 143.42 183.69 210.41

10.994 2.2256 1.3561

0.090955 0.44932 0.73741

22.770 41.867 53.389

23.225 44.113 57.076

0.10919 0.16875 0.19774

0.089725 0.10069 0.10859

0.14193 0.14547 0.12579

427.56 148.00 198.35

0.11428 7.6582 3.5826

Saturated Properties (Concluded) 37.078 37.159 37.196 37.173 37.065 36.824 36.323 34.496 33.631

0.15501 0.15459 0.15408 0.15343 0.15257 0.15136 0.14946 0.14379 0.14126

Single-Phase Properties 13.392 0.055492 0.040750 0.030243 0.024113 11.088 0.52866 0.49205 0.31957 0.24648

18.021 24.540 33.066 41.471

0.15078 22.936 19.634 7.1401 3.7496

300.00 400.00 500.00

10.000 10.000 10.000

11.371 6.1241 2.9622

0.087944 0.16329 0.33758

22.323 37.557 51.539

23.203 39.190 54.915

0.10763 0.15323 0.18852

0.089307 0.10301 0.11046

0.13525 0.18569 0.13925

489.01 184.17 198.11

−0.035312 2.8522 2.8527

300.00 400.00 500.00

25.000 25.000 25.000

12.107 9.2730 6.6326

0.082594 0.10784 0.15077

21.419 34.286 47.758

23.484 36.982 51.527

0.10432 0.14304 0.17548

0.089393 0.10166 0.11215

0.12713 0.14227 0.14633

614.12 381.56 292.53

−0.22002 0.20064 0.65319

300.00 400.00 500.00

50.000 50.000 50.000

12.867 10.834 9.0225

0.077719 0.092300 0.11083

20.472 32.538 45.317

24.358 37.153 50.859

0.10057 0.13731 0.16786

0.090168 0.10251 0.11339

0.12262 0.13302 0.14051

753.19 559.24 455.37

−0.32391 −0.19415 −0.070101

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., “Pseudo Pure-Fluid Equations of State for the Refrigerant Blends R-410A, R-404A, R-507A, and R-407C,” Int. J. Thermophys. 24(4):991–1006, 2003. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated heat capacities and speed of sound values is 0.5%. Uncertainties of bubble and dew point pressures are 0.5%.

TABLE 2-289

Thermodynamic Properties of R-407C

Temperature K

Pressure MPa

200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 359.35

0.019158 0.035795 0.062640 0.10366 0.16353 0.24755 0.36157 0.51193 0.70540 0.94916 1.2507 1.6182 2.0599 2.5851 3.2038 3.9255 4.6317

200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 359.35

0.011312 0.022624 0.041929 0.072846 0.11979 0.18793 0.28317 0.41203 0.58173 0.80008 1.0757 1.4179 1.8375 2.3470 2.9627 3.7100 4.6317

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.31996 −0.30662 −0.28934 −0.26790 −0.24161 −0.20937 −0.16951 −0.11964 −0.056172 0.026372 0.13683 0.29038 0.51547 0.87274 1.5203 3.0499 10.947

Saturated Properties 17.036 16.697 16.352 15.999 15.637 15.264 14.877 14.472 14.045 13.591 13.102 12.567 11.969 11.278 10.435 9.2661 5.2600 0.0068643 0.013151 0.023450 0.039384 0.062913 0.096374 0.14256 0.20484 0.28739 0.39560 0.53670 0.72101 0.96439 1.2939 1.7642 2.5260 5.2600

0.058698 0.059892 0.061156 0.062503 0.063949 0.065512 0.067218 0.069099 0.071198 0.073576 0.076322 0.079573 0.083552 0.088671 0.095832 0.10792 0.19011 145.68 76.041 42.644 25.391 15.895 10.376 7.0147 4.8820 3.4796 2.5278 1.8632 1.3869 1.0369 0.77283 0.56682 0.39588 0.19011

8.8272 9.9359 11.051 12.175 13.312 14.464 15.632 16.822 18.035 19.278 20.555 21.877 23.255 24.711 26.287 28.110 32.145

8.8283 9.9380 11.055 12.182 13.323 14.480 15.657 16.857 18.085 19.348 20.651 22.006 23.427 24.940 26.594 28.534 33.025

0.050593 0.056002 0.061189 0.066188 0.071026 0.075728 0.080314 0.084805 0.089223 0.093590 0.097931 0.10228 0.10668 0.11119 0.11596 0.12137 0.13372

0.070988 0.071320 0.071817 0.072410 0.073074 0.073803 0.074597 0.075463 0.076412 0.077458 0.078624 0.079950 0.081509 0.083453 0.086157 0.090943

0.11073 0.11118 0.11203 0.11319 0.11465 0.11641 0.11853 0.12111 0.12427 0.12822 0.13331 0.14013 0.14989 0.16541 0.19551 0.28993

956.60 903.06 851.40 801.12 751.77 703.01 654.53 606.09 557.44 508.32 458.46 407.51 354.98 300.07 241.20 174.57 0

30.051 30.504 30.957 31.409 31.857 32.298 32.728 33.145 33.544 33.918 34.259 34.556 34.790 34.931 34.916 34.578 32.145

31.699 32.224 32.745 33.259 33.761 34.248 34.715 35.157 35.568 35.940 36.263 36.523 36.696 36.745 36.595 36.047 33.025

0.16726 0.16412 0.16151 0.15933 0.15749 0.15593 0.15460 0.15343 0.15240 0.15144 0.15051 0.14956 0.14852 0.14727 0.14560 0.14299 0.13372

0.048920 0.050967 0.053143 0.055439 0.057839 0.060328 0.062897 0.065542 0.068266 0.071085 0.074027 0.077140 0.080507 0.084283 0.088801 0.095065

0.057805 0.060200 0.062854 0.065784 0.069010 0.072563 0.076500 0.080917 0.085971 0.091920 0.099203 0.10861 0.12170 0.14208 0.18030 0.28700

149.59 152.36 154.78 156.79 158.33 159.36 159.80 159.60 158.69 156.99 154.41 150.83 146.11 140.03 132.24 122.09 0

109.12 88.122 72.006 59.560 49.904 42.378 36.483 31.840 28.163 25.236 22.898 21.024 19.516 18.284 17.206 15.951 10.947

2-391

2-392 TABLE 2-289

Thermodynamic Properties of R-407C (Concluded)

Temperature K

Pressure MPa

200.00 229.25

0.10000 0.10000

236.25 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

291.84

1.0000

297.47 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

300.00 400.00 500.00

5.0000 5.0000 5.0000

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Single-Phase Properties 8.8253 12.091

8.8312 12.097

0.050583 0.065819

0.070990 0.072363

0.11072 0.11310

956.99 804.85

−0.32010 −0.26966

31.690 35.535 42.554 50.849

33.574 37.991 45.862 54.997

0.15814 0.17467 0.19722 0.21756

0.056928 0.063341 0.076588 0.088895

0.067764 0.072378 0.085147 0.097330

157.81 179.00 205.99 229.27

53.242 20.041 7.7925 4.0471

0.074050

19.510

19.584

0.094388

0.077662

0.12906

499.23

2.0105 2.0465 3.1425 4.0637

34.177 34.384 42.101 50.576

36.187 36.431 45.244 54.639

0.15075 0.15156 0.17694 0.19786

0.073268 0.072744 0.078067 0.089416

0.097199 0.095419 0.089213 0.099001

155.15 156.77 198.26 225.88

13.412 2.1880 1.3504

0.074559 0.45703 0.74050

20.240 39.458 49.289

20.613 41.743 52.992

0.096862 0.15675 0.18193

0.078093 0.086188 0.091753

0.12762 0.12964 0.10811

507.10 161.10 213.00

0.027202 8.5257 3.9036

17.038 16.026 0.053062 0.040722 0.030231 0.024109 13.504 0.49738 0.48865 0.31821 0.24608

0.058692 0.062399 18.846 24.557 33.079 41.479

0.044233 23.442 22.566 7.9701 4.0390

300.00 400.00 500.00

10.000 10.000 10.000

13.740 7.1029 2.9957

0.072780 0.14079 0.33381

19.898 34.426 47.547

20.626 35.834 50.885

0.095679 0.13888 0.17282

0.077756 0.090433 0.094263

0.12301 0.20031 0.12254

558.94 184.71 207.67

−0.063246 3.3408 3.3327

300.00 400.00 500.00

25.000 25.000 25.000

14.443 10.899 7.3363

0.069238 0.091752 0.13631

19.146 30.990 43.479

20.877 33.284 46.886

0.092972 0.12855 0.15889

0.077634 0.087056 0.096319

0.11624 0.13223 0.13592

672.43 399.92 289.22

−0.19834 0.28513 0.97116

300.00 400.00 500.00

50.000 50.000 50.000

15.220 12.648 10.260

0.065703 0.079064 0.097468

18.302 29.255 40.787

21.587 33.209 45.660

0.089730 0.12310 0.15086

0.078160 0.087179 0.096753

0.11176 0.12077 0.12761

802.78 579.57 457.54

−0.28843 −0.12975 0.047330

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., “Pseudo Pure-Fluid Equations of State for the Refrigerant Blends R-410A, R-404A, R-507A, and R-407C,” Int. J. Thermophys. 24(4):991–1006, 2003. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated heat capacities and speed of sound values is 0.5%. Uncertainties of bubble and dew point pressures are 0.5%.

50

100

150

200

250

300

350

400

450

500

550

600

20. 0. 90

10

11

00

00

.

.

0. 120

125

130

0.

0.

70

80

.

600

.

500

400.

3

kg/m

4.

70

60

50

40

30

20

0

80

100. 80.

60

−20

−30

6.

150.

c.p.

60.

50

2. 40.

40

30.

30

1. 0.8

20

20.

10

15.

0.6

1. 0.8 0.6

0

10.

0.4

−10 −20 −10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 T = 160 °C 170 180

8.0

T = −20 °C

0.2

por ted va

kg

·K

)

0.80

2.5

0

J/( 0k

0.1 0.08 0.06 0.04

0.60

s=

2.4

0 2.3

0 2.2

0

0

0.2

3.0

1.0

2.0

1.90

4.0

1.5

satura

0.9 1.80

1.70

1.60

1.50

1.30

1.20

−70

1.40

0.02

0.4

6.0

2.0

−60 1.10

1.00

0.90

0.80

0.70

0.04 0.60

0.7

−50

0.8

0.06

0.6

x=

0.5

0.4

0.3

0.2

−40

sat

ura

ted

0.1 0.08

0.1

liqu

id

−30

2.1

Pressure (MPa)

10. 8.

200. 10

1350.

−10

00. ρ= 3

70

−50

−70

2.

T = −60 °C

−80

4.

1400.

1450.

6.

−40

10. 8.

0.

0.

R-407C [R-32/125/134a (23/25/52)] reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

650 20.

0.40

0.02

0.30

0.01 50

100

150

200

250

300

350

400

450

500

550

600

0.01 650

Enthalpy (kJ/kg) 2-393

FIG. 2-27 Pressure-enthalpy diagram for Refrigerant 407C. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the mixture model of Lemmon, E. W., and Jacobsen, R. T., “Equations of State for Mixtures of R-32, R-125, R-134a, R-143a, and R-152a,” J. Phys. Chem. Ref. Data 33:593–620, 2004.

2-394 TABLE 2-290

Thermodynamic Properties of R-410A

Temperature K

Pressure MPa

200.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 344.49

0.029160 0.053727 0.071143 0.092819 0.11946 0.15182 0.19070 0.23697 0.29152 0.35531 0.42933 0.51461 0.61223 0.72330 0.84899 0.99048 1.1490 1.3260 1.5226 1.7404 1.9809 2.2456 2.5364 2.8550 3.2037 3.5848 4.0009 4.4556 4.9012

200.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00

0.029010 0.053489 0.070844 0.092447 0.11900 0.15125 0.19000 0.23611 0.29049 0.35407 0.42786 0.51287 0.61019 0.72092 0.84622 0.98729 1.1454 1.3218 1.5179 1.7351

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.051256 0.052375 0.052962 0.053571 0.054202 0.054858 0.055542 0.056255 0.057002 0.057786 0.058611 0.059482 0.060406 0.061388 0.062439 0.063567 0.064785 0.066109 0.067559 0.069160 0.070948 0.072969 0.075293 0.078025 0.081343 0.085578 0.091491 0.10161 0.15813

7.0380 8.0188 8.5112 9.0052 9.5012 9.9997 10.501 11.006 11.514 12.026 12.543 13.065 13.593 14.127 14.669 15.218 15.776 16.344 16.924 17.516 18.123 18.747 19.392 20.064 20.772 21.531 22.376 23.414 25.988

7.0395 8.0217 8.5149 9.0101 9.5077 10.008 10.512 11.019 11.530 12.047 12.568 13.096 13.630 14.172 14.722 15.281 15.851 16.432 17.026 17.636 18.263 18.911 19.583 20.287 21.032 21.837 22.742 23.867 26.763

26.495 26.835 27.002 27.167 27.329 27.488 27.645 27.798 27.947 28.092 28.232 28.367 28.496 28.619 28.733 28.839 28.935 29.019 29.090 29.144

28.125 28.530 28.726 28.919 29.107 29.290 29.468 29.640 29.806 29.965 30.116 30.258 30.392 30.515 30.626 30.725 30.809 30.876 30.925 30.951

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.040995 0.045781 0.048098 0.050370 0.052600 0.054791 0.056948 0.059073 0.061169 0.063240 0.065289 0.067318 0.069331 0.071331 0.073321 0.075304 0.077284 0.079266 0.081254 0.083253 0.085270 0.087314 0.089398 0.091537 0.093762 0.096123 0.098732 0.10194 0.11022

0.062260 0.062050 0.062014 0.062020 0.062066 0.062151 0.062271 0.062426 0.062615 0.062837 0.063092 0.063380 0.063701 0.064057 0.064451 0.064884 0.065363 0.065893 0.066483 0.067147 0.067901 0.068773 0.069800 0.071046 0.072616 0.074717 0.077843 0.083650

0.097942 0.098396 0.098729 0.099138 0.099628 0.10020 0.10088 0.10165 0.10253 0.10353 0.10466 0.10594 0.10738 0.10902 0.11088 0.11300 0.11543 0.11825 0.12156 0.12550 0.13029 0.13630 0.14413 0.15493 0.17109 0.19853 0.25685 0.46832

929.01 879.84 855.20 830.52 805.81 781.06 756.26 731.41 706.48 681.45 656.31 631.02 605.55 579.88 553.95 527.72 501.14 474.14 446.66 418.60 389.87 360.33 329.82 298.10 264.83 229.46 190.98 147.49 0

−0.30179 −0.28524 −0.27544 −0.26446 −0.25217 −0.23841 −0.22300 −0.20574 −0.18637 −0.16459 −0.14006 −0.11232 −0.080861 −0.045000 −0.0039006 0.043515 0.098651 0.16337 0.24022 0.33275 0.44607 0.58788 0.77028 1.0135 1.3544 1.8665 2.7232 4.4554 9.7623

0.14644 0.14345 0.14212 0.14087 0.13972 0.13864 0.13762 0.13667 0.13577 0.13492 0.13411 0.13333 0.13259 0.13187 0.13116 0.13047 0.12978 0.12908 0.12837 0.12764

0.042482 0.044604 0.045719 0.046862 0.048026 0.049206 0.050400 0.051603 0.052814 0.054033 0.055260 0.056497 0.057747 0.059014 0.060302 0.061618 0.062969 0.064364 0.065814 0.067335

0.052236 0.055055 0.056590 0.058205 0.059899 0.061674 0.063533 0.065483 0.067535 0.069705 0.072011 0.074481 0.077148 0.080057 0.083265 0.086846 0.090901 0.095568 0.10104 0.10760

164.41 167.03 168.16 169.16 170.03 170.75 171.32 171.73 171.97 172.04 171.93 171.62 171.11 170.39 169.45 168.27 166.84 165.16 163.20 160.94

Saturated Properties 19.510 19.093 18.881 18.667 18.449 18.229 18.005 17.776 17.543 17.305 17.062 16.812 16.555 16.290 16.016 15.732 15.436 15.127 14.802 14.459 14.095 13.704 13.282 12.816 12.294 11.685 10.930 9.8413 6.3240 0.017797 0.031567 0.041089 0.052763 0.066925 0.083936 0.10420 0.12814 0.15625 0.18905 0.22714 0.27117 0.32190 0.38018 0.44702 0.52357 0.61123 0.71170 0.82707 0.95997

56.190 31.678 24.338 18.953 14.942 11.914 9.5972 7.8039 6.4000 5.2895 4.4026 3.6877 3.1066 2.6303 2.2371 1.9100 1.6360 1.4051 1.2091 1.0417

113.67 90.100 80.508 72.155 64.889 58.571 53.077 48.299 44.137 40.508 37.336 34.560 32.123 29.979 28.087 26.412 24.924 23.598 22.410 21.338

305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 344.49

1.9749 2.2390 2.5291 2.8472 3.1955 3.5766 3.9935 4.4504 4.9012

221.45

0.10000

221.53 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

280.32

1.0000

280.42 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

300.00 400.00 500.00

5.0000 5.0000 5.0000

1.1138 1.2933 1.5048 1.7576 2.0668 2.4582 2.9848 3.7974 6.3240

0.89779 0.77322 0.66456 0.56894 0.48384 0.40681 0.33503 0.26334 0.15813

29.178 29.189 29.170 29.112 29.002 28.817 28.510 27.951 25.988

30.951 30.920 30.850 30.732 30.548 30.272 29.848 29.123 26.763

0.12688 0.12606 0.12517 0.12418 0.12305 0.12169 0.11995 0.11740 0.11022

0.068945 0.070671 0.072551 0.074643 0.077041 0.079915 0.083629 0.089197

0.11566 0.12589 0.13943 0.15831 0.18670 0.23464 0.33370 0.65947

158.36 155.44 152.13 148.40 144.16 139.30 133.59 126.39 0

20.365 19.470 18.632 17.826 17.018 16.153 15.123 13.641 9.7623

0.051020

0.062030

0.099271

823.36

−0.26104 69.827 19.643 7.7467 4.0758

Single-Phase Properties 18.604 0.056810 0.040605 0.030202 0.024099

0.053751

9.1541

27.217 31.028 36.670 43.331

28.977 33.491 39.981 47.480

0.14051 0.15794 0.17654 0.19323

0.047215 0.050980 0.061554 0.071249

0.058714 0.059877 0.070067 0.079663

169.44 198.35 227.37 252.59

0.063641

15.253

15.317

0.075429

0.064913

0.11314

526.05

1.8849 2.1460 3.1769 4.0808

28.848 30.151 36.304 43.106

30.733 32.297 39.481 47.187

0.13041 0.13580 0.15648 0.17364

0.061731 0.057548 0.062713 0.071665

0.087169 0.075210 0.073258 0.081012

168.16 180.65 220.92 249.79

14.870 1.9755 1.3185

0.067248 0.50621 0.75845

17.202 34.349 42.072

17.539 36.880 45.864

0.082188 0.13813 0.15821

0.066139 0.068570 0.073521

0.11773 0.097588 0.087959

472.56 192.34 239.51

0.17344 7.6786 3.7957 0.036775 5.0121 3.3106

15.713 0.53054 0.46599 0.31478 0.24505

17.603 24.628 33.111 41.495

9.1488

0.046760 26.279 20.254 7.7768 4.0343

300.00 400.00 500.00

10.000 10.000 10.000

15.342 5.7949 2.8642

0.065180 0.17257 0.34914

16.830 30.845 40.710

17.482 32.570 44.202

0.080897 0.12363 0.14982

0.065435 0.074099 0.075667

0.11125 0.16518 0.098492

533.86 182.45 233.93

300.00 400.00 500.00

25.000 25.000 25.000

16.289 11.685 7.4115

0.061392 0.085582 0.13493

16.058 26.530 37.197

17.592 28.670 40.570

0.078110 0.10987 0.13644

0.065000 0.072157 0.078574

0.10273 0.11830 0.11515

658.09 379.59 291.31

−0.14678 0.50709 1.2724

300.00 400.00 500.00

50.000 50.000 50.000

17.287 14.049 11.128

0.057845 0.071182 0.089864

15.231 24.722 34.526

18.123 28.281 39.019

0.074923 0.10409 0.12804

0.065499 0.072480 0.079657

0.097459 0.10533 0.10880

792.80 566.13 449.76

−0.26163 −0.063564 0.13566

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., “Pseudo Pure-Fluid Equations of State for the Refrigerant Blends R-410A, R-404A, R-507A, and R-407C,” Int. J. Thermophys. 24(4):991–1006, 2003. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated heat capacities and speed of sound values is 0.5%. Uncertainties of bubble and dew point pressures are 0.5%.

2-395

2-396

PHYSICAL AND CHEMICAL DATA

TABLE 2-291

Saturated Refrigerant 500*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.1219 0.2258 0.3936 0.6511 1.0291

6.966.−4 7.090.−4 7.222.−4 7.361.−4 7.509.−4

1.360 0.766 0.457 0.286 0.187

−29.56 −21.03 −12.17 −2.97 6.58

185.87 191.25 196.63 201.96 207.23

−0.1363 −0.0948 −0.0536 −0.0130 0.0277

0.9408 0.9161 0.8955 0.8782 0.8638

1.044 1.018 0.997 0.987 0.987

6.11 5.15 4.42 3.85 3.42

0.113 0.109 0.106 0.102 0.098

250 260 270 280 290

1.5632 2.2932 3.2624 4.5172 6.1064

7.668.−4 7.839.−4 8.024.−4 8.226.−4 8.450.−4

0.1261 0.0879 0.0628 0.0459 0.0342

16.50 26.78 37.44 48.48 59.91

212.40 217.45 222.35 227.06 231.56

0.0680 0.1082 0.1481 0.1878 0.2275

0.8517 0.8415 0.8329 0.8257 0.8194

0.997 1.017 1.048 1.089 1.140

3.04 2.74 2.48 2.26 2.08

0.094 0.090 0.086 0.082 0.078

300 310 320 330 340

8.0809 10.49 13.40 16.86 20.93

8.699.−4 8.981.−4 9.306.−4 9.690.−4 1.016.−3

0.0259 0.0198 0.0154 0.0119 0.0093

71.76 84.05 96.83 110.17 124.20

235.79 239.69 243.19 246.14 248.36

0.2671 0.3067 0.3464 0.3864 0.4271

0.8139 0.8088 0.8038 0.7985 0.7922

1.201 1.273 1.355 1.447 1.550

1.92 1.77 1.63 1.48 1.34

0.074 0.070 0.066 0.062 0.058

350 360 370 378.6c

25.70 31.25 37.72 44.26

1.077.−3 1.162.−3 1.307.−3 2.012.−3

0.0072 0.0055 0.0040 0.0020

139.18 155.66 175.59 219.50

249.47 248.71 244.26 219.50

0.4689 0.5135 0.5650 0.6729

0.7841 0.7721 0.7509 0.6729

1.663 1.919 2.07 ∞

T, K

*Values reproduced and converted from Table 12, p. 17.99, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.966.−4 signifies 6.966 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 105.60 °C. The thermodynamic diagram from 0.1 to 70 bar extends to 240 °C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 80 bar, 560 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 480 °F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). Specific heat and viscosity appear in Thermophysical Properties of Refrigerants, ASHRAE, 1993.

TABLE 2-292

Saturated Refrigerant 502*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

200 210 220 230 240

0.2274 0.4098 0.6965 1.1251 1.7392

6.381.−4 6.507.−4 6.640.−4 6.783.−4 6.938.−4

0.646 0.374 0.228 0.146 0.0969

−29.04 −20.83 −12.15 −2.99 6.66

153.34 158.42 163.49 168.50 173.42

−0.1337 −0.0937 −0.0534 −0.0128 0.0280

0.7782 0.7599 0.7449 0.7328 0.7228

1.018 1.036 1.055 1.075 1.097

5.72 4.88 4.23 3.71 3.28

0.103 0.099 0.095 0.091 0.087

250 260 270 280 290

2.5867 3.7188 5.1893 7.0530 9.3660

7.105.−4 7.289.−4 7.492.−4 7.720.−4 7.979.−4

0.0665 0.0470 0.0340 0.0251 0.0188

16.78 27.36 38.36 49.77 61.55

178.20 182.81 187.21 191.35 195.16

0.0691 0.1102 0.1514 0.1923 0.2330

0.7148 0.7082 0.7027 0.6980 0.6937

1.120 1.144 1.170 1.197 1.225

2.94 2.65 2.41 2.18 1.99

0.083 0.079 0.075 0.072 0.068

T, K

300 310 320 330 340

12.19 15.57 19.60 24.35 29.95

8.280.−4 8.637.−4 9.081.−4 9.666.−4 1.053.−3

0.0143 0.0109 0.0084 0.0064 0.0048

73.68 86.17 99.06 112.53 127.13

198.56 201.43 203.57 204.62 203.71

0.2734 0.3134 0.3532 0.3933 0.4351

0.6896 0.6852 0.6798 0.6723 0.6604

1.254 1.285 1.317 1.351 1.386

1.79 1.59 1.40 1.23 1.07

0.064 0.060 0.056 0.052 0.048

350 355.3c

36.62 40.75

1.220.−3 1.786.−3

0.0033 0.0018

145.44 174.00

197.82 174.00

0.4859 0.5634

0.6355 0.5634

1.422

0.93

0.044

*Values reproduced and converted from Table 13, p. 17.101, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.381.−4 signifies 6.381 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 82.2 °C. The thermodynamic diagram from 0.1 to 80 bar extends to 180 °C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecaryk, K., and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 20 bar, 515 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 400 °F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.

THERMODYNAMIC PROPERTIES TABLE 2-293

2-397

Saturated Refrigerant 503*

P, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

µf, 10−4 Pa⋅s

kf, W/(m⋅K)

150 160 170 180 190

0.0750 0.1798 0.3828 0.7395 1.3187

6.384.−4 6.478.−4 6.585.−4 6.700.−4 6.850.−4

1.894 0.837 0.414 0.224 0.130

−89.60 −79.73 −69.55 −59.08 −48.36

111.02 115.40 119.70 123.84 127.77

−0.4694 −0.4057 −0.3441 −0.2844 −0.2267

0.8681 0.8139 0.7691 0.7318 0.7003

0.482 0.554 0.620 0.682 0.747

6.12 5.05 4.16 3.43 2.94

0.128 0.123 0.116 0.111 0.105

200 210 220 230 240

2.1999 3.4713 5.2281 7.5713 10.61

7.014.−4 7.204.−4 7.426.−4 7.687.−4 8.001.−4

0.0803 0.0520 0.0350 0.0242 0.0172

−37.45 −26.36 −15.10 − 3.65 8.07

131.45 134.84 137.87 140.49 142.58

−0.1710 −0.1173 −0.0656 −0.0155 0.0334

0.6735 0.6503 0.6298 0.6112 0.5939

0.817 0.896 0.988 1.017 1.227

2.56 2.25 1.98 1.73 1.52

0.099 0.094 0.088 0.082 0.076

250 260 270 280 290

14.46 19.25 25.13 32.27 40.87

8.386.−4 8.874.−4 9.526.−4 1.050.−3 1.264.−3

0.0124 0.0090 0.0064 0.0045 0.0028

20.22 33.10 47.22 63.64 86.41

143.98 144.38 143.23 139.25 127.51

0.0817 0.1305 0.1816 0.2384 0.3131

0.5767 0.5585 0.5373 0.5085 0.4548

1.382 1.57 1.79 2.03 2.35

1.33 1.17 1.03 0.91

0.070 0.065 0.059 0.054

292.6c

43.57

1.773.−3

0.0018

110.20

110.20

0.3864

0.3864

∞

T, K

∞

*P, v, h, and s values reproduced and converted from Table 14, p. 17.103, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. cp, µ, and k values interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.384.−4 signifies 6.384 × 10−4. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 460 °F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale for saturation conditions from −125 to 19.50 °C. The thermodynamic diagram from 0.1 to 80 bar extends to 220 °C. TABLE 2-294

Saturated Refrigerant 504* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Liquid

Vapor

Liquid

Vapor

Liquid

Vapor

−120 −100 −80 −60 −40

2.964 6.042 11.34 19.85 32.76

0.01095 0.01119 0.01146 0.01175 0.01206

15.31 7.874 4.372 2.585 1.609

−21.48 −16.39 −11.12 −5.65 0.00

86.69 89.31 91.84 94.25 96.50

−0.0565 −0.0420 −0.0277 −0.0137 0.0000

0.2609 0.2519 0.2435 0.2362 0.2299

−20 0 20 40 60

51.44 77.41 112.3 158.0 216.2

0.01242 0.01282 0.01328 0.01379 0.01443

1.045 0.7029 0.4859 0.3431 0.2458

5.85 11.91 18.22 24.81 31.78

98.58 100.45 102.09 103.44 104.41

0.0135 0.0269 0.0401 0.0533 0.0667

0.2244 0.2195 0.2150 0.2107 0.2065

80 100 120 140 150

289.2 379.1 488.3 618.1 692.2

0.01522 0.01629 0.01783 0.02083 0.02597

0.1773 0.1274 0.0893 0.0578 0.0394

39.25 47.43 56.78 69.97 76.96

104.85 104.49 102.72 97.70 89.76

0.0804 0.0948 0.1107 0.1322 0.1432

0.2020 0.1968 0.1899 0.1784 0.1642

*Unpublished data of Allied Chemical Company, 1970. Used by permission. TABLE 2-295

Thermodynamic Properties of Refrigerant 507*

Temp., K

Pressure, bar

230.5 240 250 260 270 280

1.013 1.59 2.42 3.54 4.95 6.70

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

0.000 0.000 0.000 0.000 0.000 0.000

vf, m3/kg 574 602 627 658 695 738

0.1280 0.0826 0.0546 0.0377 0.0270 0.0198

−3.1 10.3 22.6 37.6 51.6 64.7

143.3 150.2 154.5 159.0 163.8 169.0

−0.015 0.042 0.092 0.149 0.202 0.250

0.620 0.623 0.619 0.617 0.618 0.620

290 300 310 320 330

8.85 11.52 14.74 18.76 23.65

0.000 0.000 0.000 0.001 0.001

787 839 903 006 221

0.0148 0.0112 0.0084 0.0062 0.0042

77.2 89.4 101.6 115.7 135.5

174.6 180.3 185.4 188.6 189.3

0.295 0.336 0.378 0.422 0.481

0.634 0.640 0.648 0.649 0.641

340 341.5c

29.57 32.67

0.001 618 0.001 97

0.0025 0.0020

161.7 172.7

179.9 172.7

0.557 0.590

0.611 0.590

*Azeotropic mixture of R152a and R218. hf = sf = 0 at 233.15 K = −40 °C. Interpolated, extrapolated and converted from Lavrenchenko, G. K., M. G. Khmelnuk, et al., Int. J. Refrig., 17, 7 (1994): 461. Some values are tentative. This source also gives a ln P–h diagram from 0.6 to 30 bar, −50 to 70 °C. Differences exist between the published diagram and tables. c = critical point.

2-398 TABLE 2-296

Thermodynamic Properties of R-507A

Temperature K

Pressure MPa

200.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 315.00 320.00 325.00 330.00 335.00 340.00 343.77

0.023233 0.042731 0.056515 0.073637 0.094634 0.12008 0.15060 0.18683 0.22945 0.27919 0.33679 0.40302 0.47868 0.56461 0.66165 0.77071 0.89272 1.0286 1.1794 1.3462 1.5300 1.7322 1.9539 2.1967 2.4622 2.7523 3.0697 3.4178 3.7049

200.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00

0.023222 0.042726 0.056512 0.073634 0.094628 0.12007 0.15057 0.18678 0.22938 0.27908 0.33664 0.40281 0.47840 0.56424 0.66119 0.77015 0.89202 1.0278 1.1784 1.3450 1.5287

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.070772 0.072318 0.073126 0.073961 0.074825 0.075722 0.076655 0.077628 0.078646 0.079714 0.080837 0.082022 0.083277 0.084612 0.086039 0.087571 0.089225 0.091024 0.092996 0.095177 0.097618 0.10039 0.10358 0.10735 0.11195 0.11785 0.12607 0.14013 0.20145

10.553 11.749 12.351 12.956 13.566 14.179 14.798 15.421 16.051 16.686 17.328 17.976 18.632 19.296 19.968 20.650 21.343 22.046 22.763 23.495 24.243 25.011 25.803 26.626 27.487 28.405 29.412 30.620 32.908

10.555 11.752 12.355 12.962 13.573 14.188 14.809 15.436 16.069 16.708 17.355 18.009 18.672 19.344 20.025 20.718 21.422 22.140 22.873 23.623 24.392 25.185 26.006 26.862 27.763 28.729 29.799 31.099 33.654

29.992 30.526 30.796 31.067 31.338 31.610 31.881 32.152 32.421 32.688 32.953 33.214 33.471 33.724 33.970 34.209 34.440 34.661 34.869 35.063 35.239

31.624 32.225 32.525 32.825 33.124 33.420 33.715 34.005 34.292 34.575 34.851 35.121 35.384 35.638 35.882 36.115 36.334 36.539 36.726 36.892 37.033

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.059915 0.065751 0.068583 0.071366 0.074104 0.076802 0.079463 0.082089 0.084685 0.087253 0.089796 0.092316 0.094817 0.097301 0.099772 0.10223 0.10469 0.10714 0.10960 0.11207 0.11455 0.11707 0.11962 0.12224 0.12494 0.12779 0.13089 0.13460 0.14193

0.077250 0.078412 0.079028 0.079658 0.080295 0.080939 0.081589 0.082245 0.082907 0.083579 0.084260 0.084954 0.085661 0.086385 0.087128 0.087895 0.088688 0.089513 0.090377 0.091290 0.092263 0.093316 0.094477 0.095794 0.097351 0.099317 0.10208 0.10695

0.11931 0.12015 0.12081 0.12158 0.12247 0.12345 0.12453 0.12569 0.12696 0.12833 0.12982 0.13144 0.13321 0.13517 0.13735 0.13978 0.14254 0.14571 0.14941 0.15379 0.15912 0.16577 0.17441 0.18622 0.20363 0.23262 0.29323 0.52176

851.53 796.42 770.28 744.80 719.84 695.27 671.01 646.97 623.07 599.27 575.51 551.75 527.94 504.05 480.04 455.86 431.49 406.87 381.97 356.72 331.07 304.91 278.14 250.55 221.86 191.54 158.62 121.00 0

−0.34068 −0.32336 −0.31246 −0.30008 −0.28620 −0.27072 −0.25349 −0.23433 −0.21297 −0.18910 −0.16232 −0.13214 −0.097937 −0.058916 −0.014071 0.037902 0.098716 0.17067 0.25689 0.36179 0.49171 0.65619 0.87027 1.1591 1.5687 2.1936 3.2672 5.5872 12.382

0.16527 0.16325 0.16240 0.16166 0.16100 0.16043 0.15992 0.15947 0.15907 0.15872 0.15841 0.15814 0.15788 0.15765 0.15744 0.15723 0.15702 0.15680 0.15656 0.15630 0.15600

0.059362 0.061910 0.063188 0.064473 0.065766 0.067072 0.068393 0.069735 0.071100 0.072492 0.073915 0.075372 0.076865 0.078399 0.079978 0.081605 0.083287 0.085030 0.086845 0.088742 0.090740

0.068766 0.071764 0.073318 0.074919 0.076576 0.078300 0.080105 0.082005 0.084017 0.086159 0.088453 0.090927 0.093613 0.096551 0.099795 0.10341 0.10750 0.11218 0.11764 0.12415 0.13211

137.04 139.19 140.11 140.92 141.60 142.16 142.57 142.84 142.95 142.90 142.67 142.26 141.66 140.85 139.83 138.57 137.08 135.32 133.29 130.95 128.30

85.105 65.962 58.876 53.008 48.113 44.001 40.521 37.557 35.013 32.816 30.909 29.244 27.785 26.502 25.372 24.376 23.500 22.730 22.057 21.472 20.967

Saturated Properties 14.130 13.828 13.675 13.521 13.365 13.206 13.045 12.882 12.715 12.545 12.371 12.192 12.008 11.819 11.623 11.419 11.208 10.986 10.753 10.507 10.244 9.9616 9.6544 9.3151 8.9322 8.4856 7.9323 7.1361 4.9640 0.014226 0.025154 0.032678 0.041875 0.052996 0.066313 0.082125 0.10075 0.12256 0.14794 0.17732 0.21120 0.25014 0.29478 0.34585 0.40424 0.47097 0.54731 0.63483 0.73552 0.85197

70.296 39.755 30.602 23.881 18.869 15.080 12.177 9.9251 8.1593 6.7597 5.6396 4.7348 3.9977 3.3924 2.8914 2.4738 2.1233 1.8271 1.5752 1.3596 1.1737

310.00 315.00 320.00 325.00 330.00 335.00 340.00 343.77

1.7307 1.9522 2.1948 2.4602 2.7503 3.0678 3.4166 3.7049

226.14

0.10000

226.14 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

288.99

1.0000

289.02 300.00 400.00 500.00

1.0000 1.0000 1.0000 1.0000

300.00 400.00 500.00

5.0000 5.0000 5.0000

0.98768 1.1475 1.3389 1.5733 1.8717 2.2784 2.9318 4.9640

1.0125 0.87142 0.74689 0.63559 0.53427 0.43891 0.34109 0.20145

35.391 35.515 35.601 35.634 35.589 35.411 34.936 32.908

37.144 37.217 37.241 37.198 37.058 36.758 36.101 33.654

0.075025

13.705

13.712

31.400 36.736 45.323 55.366

0.090648

0.15565 0.15521 0.15467 0.15397 0.15303 0.15166 0.14932 0.14193

0.092861 0.095142 0.097636 0.10043 0.10368 0.10772 0.11346

0.14217 0.15545 0.17401 0.20216 0.25052 0.35438 0.74186

125.29 121.90 118.07 113.77 108.90 103.35 96.734 0

20.534 20.163 19.838 19.531 19.179 18.634 17.407 12.382

0.074720

0.080441

0.12268

714.23

−0.28283

33.191 39.190 48.629 59.514

0.16087 0.18376 0.21080 0.23504

0.066061 0.077477 0.093474 0.10672

0.076961 0.086497 0.10204 0.11516

141.74 165.13 190.54 212.51

47.115 16.625 6.8071 3.6966

21.903

21.994

0.10665

0.089344

0.14503

411.87

1.8815 2.0390 3.1312 4.0580

34.618 35.647 44.851 55.070

36.500 37.686 47.983 59.128

0.15684 0.16087 0.19047 0.21530

0.084683 0.084205 0.094741 0.10718

0.11121 0.10551 0.10601 0.11688

135.69 142.77 182.55 209.06

10.905 2.2078 1.3533

0.091702 0.45293 0.73892

23.040 42.120 53.669

23.498 44.385 57.364

0.11052 0.17014 0.19916

0.090271 0.10115 0.10913

0.14306 0.14483 0.12621

420.12 147.77 197.37

0.12774 7.5193 3.5221

Single-Phase Properties 13.329 0.055821 0.040743 0.030241 0.024113 11.032 0.53150 0.49044 0.31937 0.24643

17.915 24.544 33.067 41.472

0.15510 22.873 19.202 7.0158 3.6962

300.00 400.00 500.00

10.000 10.000 10.000

11.289 6.0317 2.9478

0.088585 0.16579 0.33924

22.583 37.890 51.842

23.469 39.548 55.234

0.10892 0.15478 0.18999

0.089813 0.10349 0.11096

0.13611 0.18529 0.13942

482.39 182.13 197.45

−0.028276 2.8871 2.8080

300.00 400.00 500.00

25.000 25.000 25.000

12.033 9.2040 6.5873

0.083106 0.10865 0.15181

21.667 34.588 48.091

23.744 37.304 51.887

0.10556 0.14446 0.17699

0.089890 0.10218 0.11266

0.12779 0.14278 0.14661

608.04 377.87 290.93

−0.21781 0.20389 0.64689

300.00 400.00 500.00

50.000 50.000 50.000

12.796 10.770 8.9711

0.078149 0.092847 0.11147

20.711 32.830 45.655

24.618 37.472 51.229

0.10177 0.13868 0.16935

0.090684 0.10306 0.11396

0.12322 0.13357 0.14097

746.86 554.88 452.46

−0.32316 −0.19407 −0.072291

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., “Pseudo Pure-Fluid Equations of State for the Refrigerant Blends R-410A, R-404A, R-507A, and R-407C,” Int. J. Thermophys. 24(4):991–1006, 2003. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the critical point temperature are given in the last entry of the saturation tables. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperaturepressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The estimated uncertainty of density values calculated with the equation of state is 0.1%. The estimated uncertainty of calculated heat capacities and speed of sound values is 0.5%. Uncertainties of bubble and dew point pressures are 0.5%.

2-399

2-400

PHYSICAL AND CHEMICAL DATA TABLE 2-297

Saturated Rubidium*

T, K

P, bar

vf, m3/kg

312.7m 400 500 600 700

2.46.−9 1.69.−6 1.73.−4 0.0037 0.0317

6.75.−4 6.98.−4 7.22.−4 7.46.−4 7.73.−4

0.1584 1.467 6.466 18.6 28.5

8.10.−4 8.65.−4 9.40.−4 1.03.−3 1.08.−3

800 1000 1200 1400 1500

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

2.3.+5 2790 156.6 20.75

118.7 151.6 188.8 225.4 261.3

1036 1057 1078 1096 1111

0.998 1.091 1.174 1.241 1.296

3.932 3.355 2.953 2.692 2.511

0.379 0.375 0.369 0.362 0.357

4.662 0.605 0.159

296.8 367.6 440.1

1124 1150 1179

1.343 1.422 1.490

2.378 2.205 2.104

0.353 0.360 0.385

*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.46.−9 signifies 2.46 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 40 bar, 1600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 320 psia, 1200 to 2700 °R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.

TABLE 2-298 Temp., °C

Thermophysical Properties of Saturated Seawater

Pressure, bar

v, (m3/kg)103

cp, kJ/(kg⋅K)

µ, Ns/m2

k, W/(m⋅K)

NPr

105κ, 1/bar

0 1 2 3 4

0.005993 0.006438 0.006916 0.007427 0.007970

1.000158 1.000099 1.000057 1.000033 1.000025

4.000 4.000 4.000 4.000 4.001

0.001884 0.001827 0.001772 0.001720 0.001669

0.560 0.563 0.565 0.567 0.569

13.46 12.98 12.55 12.13 11.74

5.06 5.02 4.98 4.95 4.92

5 6 7 8 9

0.008548 0.009163 0.009816 0.010511 0.011248

1.000033 1.000057 1.000096 1.000149 1.000261

4.001 4.001 4.002 4.002 4.002

0.001620 0.001574 0.001529 0.001486 0.001445

0.571 0.574 0.576 0.578 0.580

11.35 10.97 10.62 10.29 9.97

4.89 4.86 4.83 4.80 4.78

10 11 12 13 14

0.01203 0.01286 0.01374 0.01467 0.01566

1.000298 1.000392 1.000500 1.000620 1.000727

4.003 4.003 4.003 4.004 4.004

0.001405 0.001367 0.001330 0.001294 0.001259

0.582 0.584 0.586 0.588 0.590

9.70 9.37 9.09 8.81 8.54

4.76 4.74 4.72 4.70 4.68

15 16 17 18 19

0.01671 0.01781 0.01898 0.02022 0.02153

1.000899 1.001055 1.001224 1.001404 1.001595

4.005 4.005 4.006 4.006 4.007

0.001226 0.001195 0.001165 0.001136 0.001107

0.592 0.594 0.595 0.597 0.599

8.29 8.06 7.82 7.62 7.41

4.66 4.65 4.63 4.62 4.60

20 21 22 23 24

0.02291 0.02437 0.02591 0.02753 0.02924

1.001796 1.002009 1.002232 1.002465 1.002708

4.007 4.007 4.008 4.008 4.009

0.001080 0.001054 0.001029 0.001005 0.000981

0.600 0.602 0.604 0.605 0.607

7.21 7.02 6.82 6.66 6.48

4.59 4.57 4.56 4.55 4.54

25 26 27 28 29

0.03104 0.03294 0.03494 0.03705 0.03926

1.002961 1.003224 1.003496 1.003778 1.004069

4.009 4.009 4.010 4.010 4.011

0.000958 0.000936 0.000915 0.000895 0.000875

0.608 0.609 0.611 0.612 0.614

6.31 6.16 6.01 5.86 5.72

4.53 4.52 4.51 4.50 4.49

30

0.04159

1.004369

4.011

0.000855

0.615

5.58

4.48

κ = (−1/V)(∂v/∂p)T ⋅ 105. Thus, at 0 °C, the compressibility is 5.06 × 10−5/bar. For further information see, for instance, Bromley, LeR. A., J. Chem. Eng. Data, 12, 2 (1967): 202–206; 13, 1 (1968): 60–62 and 13, 3: 399–402; 15, 2 (1970): 246–253; and A.I.Ch.E.J., 20, 2 (1974): 326–335. Thermal conductivity data sources include Castelli, V. J., E. M. Stanley, et al., Deep Sea Res., 211 (1974): 311–318; Levy, F. L., Int. J. Refrig., 5, 3 (1982): 155–159. For velocity of sound, see, for instance, U.S. Naval Oceanographic Office SP 58, 1962 (50 pp.). More recent information is contained in UNESCO technical papers. See Marine Science No. 38, 1981 (6 pp.) and No. 44, 1983 (53 pp.). For sea ice properties, see Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372.

TABLE 2-299 Temp., K

Saturated Sodium

Pressure, bar

vf, m3/kg

vg, m3/kg

hf, kJ/kg

hg, kJ/kg

sf, kJ/(kg⋅K)

sg, kJ/(kg⋅K)

cpf, kJ/(kg⋅K)

cpg, kJ/(kg⋅K)

µf, 10−6 Pa·s

µg, 10−6 Pa·s

kf, W/(m⋅K)

kg, W/(m⋅K)

Prg

1.59.−10 1.80.−9 8.99.−7 5.57.−5 0.00105

0.001 0.001 0.001 0.001 0.001

078 088 115 144 174

8.54.+9 8.08.+8 1.99.+6 38022 2320

207 247 382 514 642

4739 4757 4817 4872 4921

2.259 2.920 3.222 3.462 3.661

14.475 14.195 12.092 10.745 10.631

1.383 1.372 1.334 1.301 1.277

0.86 1.25 1.80 2.28

688 599 415 321 264

800 900 1000 1100 1154.7

0.00941 0.05147 0.1995 0.6016 1.013

0.001 0.001 0.001 0.001 0.001

208 242 280 323 347

291.5 58.8 16.6 5.95 3.89

769 895 1020 1146 1215

4966 5007 5044 5079 5097

3.830 3.978 4.110 4.230 4.290

9.076 8.547 8.134 7.805 7.652

1.260 1.252 1.252 1.261 1.271

2.59 2.72 2.70 2.62 2.56

227 201 181 166 159

19.6 20.6 23.0 25.3 26.5

62.9 58.3 54.2 50.5 48.7

0.0343 0.0406 0.0455 0.0492 0.0522

0.0045 0.0043 0.0042 0.0042 0.0041

1.48 1.38 1.36 1.35 1.30

27.5 29.9 32.2 34.6 37.1

47.2 44.0 41.1 38.2 35.4

0.0547 0.0570 0.0592

0.0041 0.0042 0.0044 0.0046 0.0050

1.26 1.27 1.30

1200 1300 1400 1500 1600

1.50 3.26 6.30 11.13 18.28

0.001 0.001 0.001 0.001 0.001

366 416 471 531 597

2.54 1.24 0.676 0.400 0.253

1273 1402 1534 1671 1812

5111 5140 5168 5193 5217

4.340 4.444 4.542 4.636 4.727

7.538 7.319 7.138 6.984 6.855

1.279 1.305 1.340 1.384 1.437

2.51 2.43 2.39 2.36 2.34

153 143 135 128 122

1700 1800 1900 2000 2100

28.28 41.61 58.70 79.91 105.5

0.001 0.001 0.001 0.001 0.002

675 761 862 984 174

0.168 0.117 0.084 0.063 0.0472

1959 2113 2274 2444 2625

5238 5256 5268 5273 5265

4.816 4.904 4.992 5.079

6.745 6.650 6.568 6.494

1.500 1.574 1.661 1.764 1.926

2.41 2.46 2.53 2.66 2.91

117 112 108 104

2200 2300 2400 2500 2503.7c

135.7 170.6 210.3 254.7 256.4

0.002 0.002 0.002 0.004 0.004

320 584 985 19 57

0.0361 0.0275 0.0203 0.0098 0.0046

2822 3047 3331 3965 4294

5241 5188 5078 4617 4294

2.190 2.690 4.012 39.3

3.40 4.47 8.03 417.

89.4 87.2 80.1 73.7 68.0

Prf

371 400 500 600 700

32.6 29.7 26.6 23.2

0.0106 0.0094 0.0069 0.0057 0.0050

0.0054 0.0059 0.0067 0.0079

c = critical point. sf values converted from Cordfunke, E. H. P. and R. J. M. Konings, Thermochemical Data for Reactor Materials and Fission Products, North Holland Elsevier, NY, 1990. sg determined as sf + (hg − hf)/T. µg and kg values estimated by P. E. Liley. All other values are from Fink, J. K. and L. Leibowitz, Argonne Nat. Lab Rept. ANL/RE-95-2, 1995. The Fink and Leibowitz work also appeared in High Temp. Materials Sci., 35, 65–103, 1996. Saturation and superheat tables and a diagram to 14 bar, 1700 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram for 0.1–150 psia, 1500–2700 °R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961.

2-401

2-402

PHYSICAL AND CHEMICAL DATA

2500 K 2400

6500 2300 2200 2100 2000 1900 1800

ar

1700 5

10 b

1600 1 ba

r

2

h, kJ/kg

6000

0.5

1500

0.2

1400

0.1

1300

5500

0.0

5

1200

0.0

2

1100

0.0

1

1000

5000 7.0

r

8.0

9.0 Entropy (s), kJ/kg • K

bar

800

01

vapo

0.0

te d

02

tura

0.0

Sa

0.0

05

900

700

10.0

Mollier Diagram for Sodium. Drawn from the Vargaftik et al. values in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, UK, 1985. These values are identical with those of Vargaftik, N. B., Handbook of Thermophysical Properties of Gases and Liquids, Moscow, 1972, and the Hemisphere translation, p. 19. An apparent discontinuity exists between the superheat values and the saturation values, not reproduced here. For a Mollier diagram in f.p.s. units from 0.1 to 150 psia, 1500 to 2700°R, see Fig. 3-36, p. 3-232 of the 6th edition of this handbook. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., Chase, M. W. et al., J. Phys. Chem. Ref. Data, 23(3), 385–497, 1994.

FIG. 2-28

THERMODYNAMIC PROPERTIES

Enthalpy-concentration diagram for aqueous sodium hydroxide at 1 atm. Reference states: enthalpy of liquid water at 32 °F and vapor pressure is zero; partial molal enthalpy of infinitely dilute NaOH solution at 64 °F and 1 atm is zero. [McCabe, Trans. Am. Inst. Chem. Eng., 31, 129 (1935).]

FIG. 2-29

2-403

2-404 TABLE 2-300

Thermodynamic Properties of Sulfur Dioxide

Temperature K

Pressure MPa

197.70 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 430.64

0.0016602 0.0020260 0.0045390 0.0093340 0.017835 0.031988 0.054309 0.087910 0.13649 0.20431 0.29614 0.41725 0.57327 0.77025 1.0145 1.3128 1.6720 2.0994 2.6028 3.1904 3.8713 4.6557 5.5562 6.5903 7.7908 7.8753

197.70 200.00 210.00 220.00 230.00 240.00 250.00 260.00 270.00 280.00 290.00 300.00 310.00 320.00 330.00 340.00 350.00 360.00 370.00 380.00 390.00 400.00 410.00 420.00 430.00 430.64

0.0016602 0.0020260 0.0045390 0.0093340 0.017835 0.031988 0.054309 0.087910 0.13649 0.20431 0.29614 0.41725 0.57327 0.77025 1.0145 1.3128 1.6720 2.0994 2.6028 3.1904 3.8713 4.6557 5.5562 6.5903 7.7908 7.8753

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

−0.024986 −0.023967 −0.019681 −0.015610 −0.011730 −0.0080208 −0.0044649 −0.0010454 0.0022525 0.0054424 0.0085368 0.011547 0.014485 0.017362 0.020187 0.022972 0.025730 0.028473 0.031220 0.033991 0.036818 0.039753 0.042899 0.046536 0.053143 0.056233

0.056222 0.056094 0.055527 0.054955 0.054393 0.053849 0.053331 0.052843 0.052388 0.051969 0.051587 0.051243 0.050938 0.050675 0.050455 0.050281 0.050157 0.050089 0.050089 0.050173 0.050373 0.050753 0.051457 0.052928 0.058931

0.088123 0.088031 0.087673 0.087396 0.087214 0.087137 0.087177 0.087344 0.087651 0.088112 0.088748 0.089582 0.090647 0.091988 0.093668 0.095777 0.098448 0.10189 0.10644 0.11271 0.12189 0.13673 0.16539 0.24772 3.7896

1361.6 1350.1 1301.1 1253.3 1206.4 1160.3 1114.8 1069.7 1024.9 980.26 935.65 890.99 846.15 801.04 755.54 709.53 662.87 615.37 566.77 516.72 464.59 409.36 349.05 279.41 182.70 0

−0.31925 −0.31864 −0.31528 −0.31064 −0.30450 −0.29665 −0.28683 −0.27473 −0.26000 −0.24217 −0.22070 −0.19485 −0.16370 −0.12603 −0.080167 −0.023832 0.046220 0.13476 0.24913 0.40128 0.61221 0.92296 1.4276 2.4089 5.8716 7.4962

0.11899 0.11774 0.11277 0.10841 0.10458 0.10120 0.098195 0.095507 0.093090 0.090902 0.088908 0.087077 0.085379 0.083789 0.082284 0.080838 0.079427 0.078025 0.076601 0.075116 0.073516 0.071714 0.069543 0.066577 0.059582 0.056233

0.028340 0.028464 0.029049 0.029713 0.030462 0.031293 0.032204 0.033186 0.034230 0.035327 0.036465 0.037638 0.038838 0.040062 0.041310 0.042587 0.043900 0.045264 0.046699 0.048236 0.049922 0.051835 0.054121 0.057117 0.062123

0.036808 0.036952 0.037650 0.038475 0.039438 0.040551 0.041819 0.043246 0.044838 0.046602 0.048552 0.050709 0.053111 0.055813 0.058901 0.062502 0.066814 0.072150 0.079031 0.088395 0.10210 0.12442 0.16795 0.29279 4.6423

182.19 183.16 187.23 191.08 194.70 198.07 201.15 203.92 206.35 208.42 210.09 211.33 212.12 212.43 212.22 211.47 210.13 208.17 205.54 202.18 198.02 192.95 186.81 179.31 168.96 0

Saturated Properties 25.290 25.205 24.835 24.463 24.088 23.709 23.324 22.932 22.532 22.122 21.701 21.267 20.818 20.352 19.865 19.354 18.814 18.239 17.621 16.949 16.204 15.355 14.340 12.987 9.8261 8.1950 0.0010120 0.0012211 0.0026096 0.0051346 0.0094158 0.016257 0.026653 0.041793 0.063061 0.092046 0.13055 0.18062 0.24460 0.32522 0.42570 0.54995 0.70287 0.89084 1.1225 1.4103 1.7733 2.2444 2.8887 3.8804 6.6270 8.1950

0.039541 0.039674 0.040265 0.040878 0.041514 0.042179 0.042875 0.043608 0.044382 0.045204 0.046081 0.047021 0.048035 0.049136 0.050340 0.051670 0.053153 0.054828 0.056750 0.059001 0.061713 0.065124 0.069734 0.077001 0.10177 0.12203 988.16 818.93 383.20 194.76 106.20 61.512 37.519 23.928 15.858 10.864 7.6600 5.5366 4.0883 3.0748 2.3491 1.8184 1.4227 1.1225 0.89086 0.70908 0.56391 0.44556 0.34618 0.25770 0.15090 0.12203

−5.7212 −5.5186 −4.6402 −3.7650 −2.8921 −2.0207 −1.1497 −0.27785 0.59591 1.4729 2.3545 3.2423 4.1378 5.0430 5.9601 6.8916 7.8408 8.8116 9.8095 10.842 11.921 13.066 14.317 15.782 18.415 19.585

−5.7211 −5.5185 −4.6400 −3.7646 −2.8914 −2.0194 −1.1473 −0.27402 0.60197 1.4821 2.3682 3.2619 4.1653 5.0809 6.0112 6.9594 7.9296 8.9267 9.9572 11.030 12.160 13.369 14.704 16.290 19.207 20.546

21.102 21.164 21.435 21.702 21.967 22.226 22.480 22.726 22.964 23.191 23.407 23.611 23.799 23.969 24.120 24.247 24.345 24.409 24.430 24.396 24.289 24.079 23.705 23.009 20.801 19.585

22.743 22.824 23.174 23.520 23.861 24.194 24.518 24.830 25.128 25.411 25.676 25.921 26.142 26.338 26.503 26.634 26.724 26.765 26.748 26.658 26.472 26.153 25.628 24.707 21.976 20.546

299.66 280.77 214.32 166.88 132.33 106.70 87.372 72.572 61.089 52.073 44.917 39.185 34.550 30.772 27.666 25.092 22.939 21.118 19.558 18.192 16.958 15.777 14.529 12.934 9.1885 7.4962

Single-Phase Properties 200.00 262.84

0.10000 0.10000

262.84 300.00 400.00 500.00

0.10000 0.10000 0.10000 0.10000

200.00 300.00 329.46

1.0000 1.0000 1.0000

329.46 400.00 500.00

1.0000 1.0000 1.0000

200.00 300.00 400.00 403.99

5.0000 5.0000 5.0000 5.0000

403.99 500.00

5.0000 5.0000

25.206 22.819

−5.5197 −0.030118

−5.5158 −0.025736

22.794 24.043 27.458 31.151

24.916 26.497 30.765 35.297

0.039653 0.046976 0.050272

−5.5302 3.2251 5.9101

2.3827 3.1281 4.0483

−0.023973 −0.000097639

0.088028 0.087416

0.094795 0.10042 0.11269 0.12279

0.033476 0.032826 0.035338 0.038292

0.043681 0.041994 0.043891 0.046720

204.65 219.46 252.48 280.65

−5.4905 3.2721 5.9604

−0.024025 0.011490 0.020035

0.056110 0.051245 0.050466

0.087995 0.089444 0.093567

1353.1 894.23 758.01

24.112 27.007 30.891

26.495 30.135 34.939

0.082363 0.092405 0.10312

0.041242 0.037220 0.038821

0.058722 0.048421 0.048357

212.25 244.07 276.82

27.819 15.238 8.3664

0.039570 0.046686 0.064770 0.066773

−5.5759 3.1113 13.003 13.548

−5.3780 3.3448 13.327 13.882

−0.024255 0.011108 0.039592 0.040973

0.056174 0.051267 0.050649 0.050981

0.087853 0.088566 0.13367 0.14563

1365.0 915.79 417.04 386.08

−0.32064 −0.21095 0.87444 1.0922

2.4748 1.3950

0.40407 0.71686

23.955 29.591

25.975 33.176

0.070906 0.087109

0.052690 0.041506

0.13790 0.058653

190.64 259.84

15.297 8.2323

25.219 21.287 19.892 0.41969 0.31968 0.24701 25.271 21.420 15.439 14.976

21.213 24.532 33.068 41.465

1350.4 1057.0

−0.31868 −0.27084

0.056096 0.052710

0.047141 0.040764 0.030240 0.024117

0.039672 0.043823

69.019 39.646 15.140 8.3690 −0.31905 −0.19703 −0.082891

300.00 400.00 500.00

10.000 10.000 10.000

21.575 16.326 3.4161

0.046350 0.061251 0.29273

2.9776 12.327 27.477

3.4411 12.940 30.404

0.010654 0.037839 0.076991

0.051308 0.049804 0.045669

0.087615 0.11122 0.085639

941.12 501.11 241.12

−0.22613 0.46626 7.4245

300.00 400.00 500.00

20.000 20.000 20.000

21.859 17.379 9.7192

0.045748 0.057542 0.10289

2.7336 11.494 22.208

3.6486 12.644 24.266

0.0098108 0.035620 0.061366

0.051422 0.049297 0.049635

0.086075 0.096839 0.13354

987.39 608.17 292.48

−0.25095 0.15187 2.7261

300.00 400.00 500.00

35.000 35.000 35.000

22.232 18.377 13.607

0.044980 0.054417 0.073492

2.4136 10.681 19.322

3.9878 12.586 21.894

0.0086740 0.033379 0.054113

0.051645 0.049231 0.048429

0.084373 0.088711 0.097089

1048.5 717.32 462.83

−0.27869 −0.043488 0.67011

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., J. Chem. Eng. Data, 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density of the equation of state ranges from 0.1% at low temperatures in the liquid and vapor to 0.5% at the highest temperatures. The uncertainty in heat capacities is 2%, and the uncertainty in vapor pressure is 0.4% at temperatures above 270 K. The uncertainty in vapor pressure increases at lower temperatures due to the lack of experimental data. In the critical region, the uncertainties are higher for all properties except vapor pressure.

2-405

2-406 TABLE 2-301

Thermodynamic Properties of Sulfur Hexafluoride

Temperature K

Pressure MPa

222.38 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 318.73

0.22436 0.25012 0.30561 0.37011 0.44448 0.52962 0.62644 0.73591 0.85899 0.99671 1.1502 1.3204 1.5088 1.7164 1.9447 2.1952 2.4696 2.7698 3.0984 3.7539

222.38 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 290.00 295.00 300.00 305.00 310.00 318.73

0.22436 0.25012 0.30561 0.37011 0.44448 0.52962 0.62644 0.73591 0.85899 0.99671 1.1502 1.3204 1.5088 1.7164 1.9447 2.1952 2.4696 2.7698 3.0984 3.7539

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

0.078884 0.079415 0.080489 0.081642 0.082875 0.084192 0.085600 0.087108 0.088731 0.090487 0.092400 0.094507 0.096852 0.099503 0.10256 0.10617 0.11059 0.11626 0.12417 0.19636

22.560 22.842 23.405 23.996 24.610 25.244 25.894 26.559 27.238 27.932 28.640 29.363 30.105 30.867 31.653 32.470 33.327 34.240 35.238 38.575

22.578 22.862 23.430 24.026 24.647 25.288 25.948 26.623 27.315 28.022 28.746 29.488 30.251 31.037 31.852 32.703 33.600 34.562 35.623 39.312

7.6872 6.9355 5.7314 4.7696 3.9949 3.3655 2.8502 2.4248 2.0711 1.7747 1.5246 1.3120 1.1301 0.97310 0.83661 0.71677 0.61023 0.51377 0.42353 0.19636

36.790 36.943 37.235 37.528 37.820 38.110 38.397 38.681 38.960 39.232 39.496 39.749 39.988 40.209 40.407 40.573 40.695 40.751 40.693 38.575

38.515 38.678 38.987 39.293 39.595 39.892 40.183 40.465 40.739 41.001 41.249 41.481 41.693 41.879 42.034 42.147 42.202 42.174 42.006 39.312

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

0.11968 0.12094 0.12342 0.12596 0.12855 0.13117 0.13380 0.13643 0.13908 0.14172 0.14438 0.14704 0.14973 0.15244 0.15519 0.15801 0.16093 0.16399 0.16732 0.17871

0.063933 0.066189 0.069989 0.073230 0.076028 0.078482 0.080672 0.082667 0.084519 0.086274 0.087968 0.089631 0.091292 0.092978 0.094719 0.096558 0.098567 0.10090 0.10399

0.10620 0.11001 0.11631 0.12154 0.12596 0.12981 0.13330 0.13661 0.13994 0.14344 0.14730 0.15173 0.15702 0.16356 0.17202 0.18353 0.20035 0.22773 0.28254

578.22 565.05 539.86 514.71 489.70 464.88 440.25 415.80 391.49 367.25 343.00 318.65 294.10 269.23 243.92 218.04 191.41 163.77 134.40 0

−0.32279 −0.29395 −0.24573 −0.20297 −0.16256 −0.12208 −0.079442 −0.032633 0.020496 0.082465 0.15646 0.24678 0.35955 0.50392 0.69427 0.95460 1.3279 1.8996 2.8784 11.978

0.19135 0.19124 0.19106 0.19093 0.19083 0.19077 0.19074 0.19072 0.19071 0.19070 0.19069 0.19066 0.19059 0.19048 0.19030 0.19003 0.18960 0.18895 0.18791 0.17871

0.069340 0.070316 0.072196 0.074101 0.076031 0.077991 0.079983 0.082010 0.084077 0.086191 0.088360 0.090597 0.092917 0.095345 0.097913 0.10068 0.10372 0.10722 0.11155

0.080626 0.081915 0.084476 0.087187 0.090076 0.093176 0.096531 0.10020 0.10426 0.10882 0.11403 0.12013 0.12746 0.13657 0.14840 0.16473 0.18917 0.23074 0.31922

112.78 112.82 112.77 112.56 112.19 111.65 110.92 109.99 108.86 107.50 105.91 104.06 101.93 99.489 96.707 93.543 89.944 85.845 81.148 0

24.151 23.609 22.653 21.791 21.015 20.317 19.693 19.139 18.652 18.230 17.872 17.578 17.349 17.185 17.086 17.049 17.065 17.098 17.036 11.978

Saturated Properties 12.677 12.592 12.424 12.249 12.066 11.878 11.682 11.480 11.270 11.051 10.822 10.581 10.325 10.050 9.7505 9.4189 9.0428 8.6013 8.0537 5.0926 0.13009 0.14419 0.17448 0.20966 0.25032 0.29713 0.35086 0.41240 0.48285 0.56348 0.65592 0.76219 0.88492 1.0276 1.1953 1.3951 1.6387 1.9464 2.3611 5.0926

Single-Phase Properties 225.00 300.00 375.00 450.00 525.00

0.10000 0.10000 0.10000 0.10000 0.10000

225.00 265.11

1.0000 1.0000

265.11 300.00 375.00 450.00 525.00

1.0000 1.0000 1.0000 1.0000 1.0000

300.00 375.00 450.00 525.00

5.0000 5.0000 5.0000 5.0000

0.054962 0.040537 0.032243 0.026801 0.022943

37.124 43.122 50.424 58.668 67.581

38.943 45.589 53.525 62.399 71.940

0.19969 0.22504 0.24859 0.27014 0.28973

0.069072 0.089337 0.10425 0.11482 0.12235

0.078403 0.098060 0.11277 0.12326 0.13075

117.21 135.40 151.16 165.37 178.44

20.832 8.9939 4.9674 3.1241 2.1242

0.079249 0.090528

22.807 27.948

22.886 28.038

0.12079 0.14178

0.066092 0.086313

0.10946 0.14352

570.46 366.71

−0.30187 0.083989

0.56543 0.45567 0.33888 0.27473 0.23243

1.7686 2.1946 2.9509 3.6399 4.3024

39.238 42.504 50.051 58.408 67.387

41.007 44.699 53.002 62.048 71.689

0.19070 0.20379 0.22845 0.25042 0.27022

0.086239 0.091458 0.10508 0.11528 0.12264

0.10893 0.10595 0.11588 0.12496 0.13181

107.47 122.90 145.14 162.15 176.71

18.222 10.582 5.2070 3.1619 2.1142

9.6705 2.2583 1.5252 1.2155

0.10341 0.44281 0.65566 0.82269

32.693 47.826 57.112 66.464

33.210 50.040 60.390 70.578

0.15873 0.20902 0.23421 0.25515

0.096509 0.10970 0.11717 0.12378

0.16433 0.14796 0.13497 0.13716

257.24 119.18 151.89 173.02

0.56212 6.1481 3.1819 2.0030

12.618 11.046

18.194 24.669 31.015 37.312 43.585

300.00 375.00 450.00 525.00

10.000 10.000 10.000 10.000

10.303 5.9487 3.3384 2.4910

0.097057 0.16810 0.29955 0.40144

31.996 44.159 55.295 65.252

32.966 45.840 58.291 69.266

0.15625 0.19435 0.22473 0.24729

0.095358 0.11121 0.11863 0.12474

0.14728 0.18759 0.15056 0.14433

332.30 145.44 154.01 177.25

0.14584 2.6476 2.5753 1.6978

300.00 375.00 450.00 525.00

25.000 25.000 25.000 25.000

11.266 9.1060 7.0706 5.5874

0.088766 0.10982 0.14143 0.17897

30.853 41.023 51.697 62.312

33.072 43.768 55.233 66.786

0.15199 0.18375 0.21161 0.23536

0.095576 0.10890 0.11870 0.12549

0.13478 0.14958 0.15437 0.15348

455.74 314.62 248.92 234.29

−0.17165 0.14966 0.47538 0.57992

300.00 375.00 450.00 525.00

50.000 50.000 50.000 50.000

12.151 10.625 9.2093 8.0206

0.082298 0.094118 0.10859 0.12468

29.792 39.323 49.451 59.888

33.906 44.029 54.881 66.122

0.14767 0.17775 0.20410 0.22720

0.098484 0.10946 0.11846 0.12538

0.12950 0.14045 0.14799 0.15134

565.06 454.58 394.01 361.13

−0.30996 −0.19915 −0.10589 −0.058213

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is de Reuck, K. M., Craven, R. J. B., and Cole, W. A., “Report on the Development of an Equation of State for Sulphur Hexafluoride,” IUPAC Thermodynamic Tables Project Centre, London, 1991. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties of the equation of state are 0.1% in density, 2% in heat capacity, and 5% in the speed of sound, except in the critical region.

2-407

200

250

00 13

14

15

700.

600.

40

30

20

80.

2. 60.

por ted va

0.9

−20

0.8

satura

0.8

0.7

0.6

0.5

0.4 x=

0.3

0.2

iqu dl

0.1

ate tur sa

1. 30.

20.

5 1.8

0k

15.

0.4

s=

1.8

5 1.7

0 1.7

5 1.6

1.60

1.55

1.50

1.45

1.40

1.35

1.30

1.25

1.20

1.15

1.10

1.05

1.00

0.95

−30

0.90

0.6

J/(k

g·K

)

0.6 0.85

4.

40.

−10

id

freezing line

6.

100.

0

0.4

8.

240

230

210

T = 220 °C

200

190

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

T = 10 °C

30

20

2.

0.8

10.

150.

c.p.

30

1.

3

g/m

0. k ρ = 50

20

0

.

800

500 20.

200.

4.

Pressure (MPa)

.

900

450

300.

10

−10

1600

.

1700.

1800.

−30

T = −20 °C

−40

6.

.

00

10

400

400.

10. 8.

.

00

11

12

reference state: h = 200.0 kJ/kg, s = 1.00 kJ/(kg·K) for saturated liquid at 0 °C

350

.

00

.

00

.

.

Sulfur Hexafluoride

300

00

2-408

150 20.

−40

10. triple point = 50.77 °C

0.2 150

solid + vapor region

200

250

300

350

400

450

0.2 500

Enthalpy (kJ/kg) Pressure-enthalpy diagram for sulfur hexafluoride (SF6). Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the equation of state of de Reuck, K. M., Craven, R.J.B., and Cole, W. A., “Report on the Development of an Equation of State for Sulphur Hexafluoride,” IUPAC Thermodynamic Tables Project Centre, London, 1991. FIG. 2-30

THERMODYNAMIC PROPERTIES

2-409

Enthalpy-concentration diagram for aqueous sulfuric acid at 1 atm. Reference states: enthalpies of pure-liquid components at 32° F and vapor pressures are zero. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. The upper ends of the tie lines in this region are assumed to be pure water. (Hougen and Watson, Chemical Process Principles, part I, Wiley, New York, 1943.) FIG. 2-31

TABLE 2-302

Saturated SUVA AC 9000

DuPont bulletin T–AC–9000–SI, 1994 (16 pp.) gives tables and a chart to 100 bar, 235°C. With a stated composition of 23% wt CH2F2 (R23), 25% wt CHF2CF3 (R125), and 52% wt CH2FCH3 (R134a) this is apparently identical to KLEA 66, to which the reader is referred.

2-410 TABLE 2-303 Temperature K

Thermodynamic Properties of Toluene Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa −0.57170 −0.57178 −0.56718 −0.55455 −0.53604 −0.51329 −0.48747 −0.45928 −0.42904 −0.39664 −0.36162 −0.32305 −0.27947 −0.22860 −0.16688 −0.088509 0.016497 0.16681 0.40139 0.81616 1.7226 4.9610 19.734

Saturated Properties 178.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 580.00 591.75

3.9393E-08 5.5336E-08 1.0833E-06 1.1479E-05 7.7542E-05 0.00037312 0.0013829 0.0041774 0.010727 0.024170 0.048980 0.090988 0.15731 0.25622 0.39698 0.58974 0.84559 1.1766 1.5964 2.1207 2.7691 3.5688 4.1264

178.00 180.00 200.00 220.00 240.00 260.00 280.00 300.00 320.00 340.00 360.00 380.00 400.00 420.00 440.00 460.00 480.00 500.00 520.00 540.00 560.00 580.00 591.75

3.9393E-08 5.5336E-08 1.0833E-06 1.1479E-05 7.7542E-05 0.00037312 0.0013829 0.0041774 0.010727 0.024170 0.048980 0.090988 0.15731 0.25622 0.39698 0.58974 0.84559 1.1766 1.5964 2.1207 2.7691 3.5688 4.1264

10.580 10.559 10.349 10.144 9.9416 9.7408 9.5402 9.3385 9.1347 8.9275 8.7157 8.4979 8.2722 8.0367 7.7887 7.5246 7.2394 6.9258 6.5719 6.1562 5.6334 4.8513 3.1690 2.6618E-08 3.6974E-08 6.5146E-07 6.2760E-06 3.8866E-05 0.00017271 0.00059503 0.0016815 0.0040644 0.0086767 0.016770 0.029922 0.050060 0.079525 0.12122 0.17889 0.25767 0.36515 0.51366 0.72590 1.0524 1.6605 3.1690

0.094517 0.094708 0.096627 0.098582 0.10059 0.10266 0.10482 0.10708 0.10947 0.11201 0.11474 0.11768 0.12089 0.12443 0.12839 0.13290 0.13813 0.14439 0.15216 0.16244 0.17751 0.20613 0.31556 37,569,000. 27,046,000. 1,535,000. 159,340. 25,729. 5,790.1 1,680.6 594.71 246.04 115.25 59.631 33.420 19.976 12.575 8.2496 5.5901 3.8810 2.7386 1.9468 1.3776 0.95020 0.60222 0.31556

−31.779 −31.508 −28.792 −26.046 −23.242 −20.358 −17.378 −14.290 −11.084 −7.7538 −4.2939 −0.70017 3.0306 6.9017 10.917 15.082 19.404 23.896 28.580 33.494 38.729 44.585 50.827

−31.779 −31.508 −28.792 −26.046 −23.242 −20.358 −17.378 −14.289 −11.083 −7.7511 −4.2882 −0.68946 3.0496 6.9336 10.968 15.160 19.521 24.066 28.823 33.839 39.220 45.321 52.129

−0.11617 −0.11466 −0.10035 −0.087266 −0.075068 −0.063529 −0.052490 −0.041839 −0.031496 −0.021403 −0.011516 −0.0018023 0.0077653 0.017209 0.026551 0.035811 0.045015 0.054196 0.063400 0.072709 0.082291 0.092723 0.10409

0.094332 0.094377 0.095732 0.098418 0.10209 0.10651 0.11147 0.11682 0.12243 0.12819 0.13403 0.13988 0.14571 0.15148 0.15718 0.16280 0.16836 0.17388 0.17943 0.18518 0.19153 0.20008

0.13565 0.13561 0.13627 0.13855 0.14205 0.14648 0.15162 0.15729 0.16337 0.16976 0.17640 0.18325 0.19032 0.19764 0.20531 0.21350 0.22253 0.23298 0.24610 0.26495 0.30025 0.44025

1887.6 1876.5 1768.3 1664.8 1565.9 1471.5 1381.2 1294.5 1210.9 1130.0 1051.3 974.11 898.08 822.67 747.37 671.66 594.94 516.53 435.57 350.82 260.06 157.29 0

12.076 12.186 13.353 14.646 16.071 17.632 19.331 21.165 23.130 25.218 27.421 29.730 32.133 34.618 37.172 39.776 42.408 45.035 47.606 50.033 52.136 53.371 50.827

13.556 13.683 15.016 16.475 18.066 19.793 21.655 23.649 25.769 28.003 30.342 32.770 35.275 37.840 40.447 43.073 45.690 48.257 50.713 52.955 54.768 55.520 52.129

0.13852 0.13640 0.11869 0.10601 0.097048 0.090897 0.086914 0.084624 0.083666 0.083758 0.084678 0.086250 0.088329 0.090796 0.093548 0.096491 0.099533 0.10258 0.10550 0.10811 0.11005 0.11031 0.10409

0.054735 0.055322 0.061435 0.067930 0.074733 0.081779 0.089005 0.096358 0.10379 0.11127 0.11875 0.12622 0.13364 0.14100 0.14832 0.15558 0.16283 0.17013 0.17758 0.18540 0.19407 0.20499

0.063049 0.063636 0.069750 0.076246 0.083057 0.090124 0.097403 0.10486 0.11248 0.12026 0.12819 0.13630 0.14463 0.15326 0.16234 0.17214 0.18316 0.19639 0.21403 0.24190 0.30118 0.55687

136.02 136.69 143.14 149.27 155.12 160.70 166.00 170.96 175.47 179.42 182.67 185.07 186.44 186.62 185.42 182.60 177.87 170.89 161.17 148.11 130.92 108.57 0

821.69 783.08 496.50 328.33 225.22 159.61 116.50 87.391 67.283 53.119 42.979 35.624 30.237 26.277 23.381 21.313 19.928 19.156 19.007 19.581 21.067 23.353 19.734

Single-Phase Properties 200.00 325.00 383.28

0.10000 0.10000 0.10000

383.28 450.00 575.00 700.00

0.10000 0.10000 0.10000 0.10000

200.00 325.00 450.00 489.95

1.0000 1.0000 1.0000 1.0000

489.95 575.00 700.00

1.0000 1.0000 1.0000

200.00 325.00 450.00 575.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000

200.00 325.00 450.00 575.00 700.00

10.000 10.000 10.000 10.000 10.000

10.350 9.0842 8.4614 0.032692 0.027344 0.021125 0.017270 10.354 9.0933 7.6732 7.0876 0.30691 0.23372 0.18116

0.096623 0.11008 0.11818 30.589 36.572 47.338 57.904 0.096581 0.10997 0.13032 0.14109 3.2583 4.2786 5.5199

−28.794 −10.267 −0.097277

−28.784 −10.256 −0.085459

−0.10036 −0.028962 −0.00022242

0.095737 0.12386 0.14084

0.13627 0.16493 0.18440

1768.7 1191.0 961.57

−0.56720 −0.42130 −0.31629 34.622 18.656 8.1724 4.5991

30.118 39.369 60.145 84.697

33.177 43.026 64.879 90.487

0.086560 0.11021 0.15289 0.19311

0.12744 0.14828 0.18229 0.20915

0.13765 0.15755 0.19099 0.21767

185.37 202.98 230.86 255.08

−28.811 −10.303 12.928 21.616

−28.715 −10.193 13.058 21.757

−0.10045 −0.029074 0.031073 0.049581

0.095778 0.12389 0.16000 0.17110

0.13625 0.16482 0.20893 0.22749

1771.8 1196.3 715.87 556.21

43.717 59.065 84.035

46.976 63.344 89.555

0.10105 0.13184 0.17301

0.16645 0.18481 0.21011

0.18937 0.19848 0.22077

174.71 210.35 244.28

19.469 9.4517 4.8519

−0.56742 −0.42281 −0.13702 0.083851

10.373 9.1328 7.7765 5.6051 1.1698

0.096400 0.10950 0.12859 0.17841 0.85487

−28.887 −10.460 12.548 41.699 80.250

−28.405 −9.9129 13.191 42.591 84.524

−0.10083 −0.029562 0.030217 0.087531 0.15419

0.095961 0.12404 0.16002 0.19430 0.21547

0.13619 0.16434 0.20635 0.28257 0.24819

1785.8 1219.1 761.15 285.95 197.87

−0.56831 −0.42907 −0.18092 1.3449 5.8706

10.397 9.1800 7.8890 6.1856 3.2697

0.096178 0.10893 0.12676 0.16166 0.30584

−28.980 −10.647 12.131 40.014 73.814

−28.018 −9.5580 13.398 41.630 76.873

−0.10130 −0.030150 0.029259 0.084396 0.13965

0.096187 0.12422 0.16009 0.19242 0.21978

0.13612 0.16382 0.20400 0.24975 0.30359

1802.9 1246.4 810.93 425.25 189.44

−0.56932 −0.43597 −0.22080 0.38997 3.9797

−2.7551 19.397 45.521 74.845

−0.038333 0.019038 0.070122 0.11620

0.12727 0.16256 0.19323 0.21782

0.16019 0.19397 0.22291 0.24523

1609.1 1308.6 1104.9 967.84

−0.48598 −0.38806 −0.31962 −0.27316

−0.00069710 0.050390 0.096461

0.17022 0.20000 0.22392

0.19432 0.22274 0.24550

2238.9 2087.2 1977.5

−0.40813 −0.35992 −0.32978

325.00 450.00 575.00 700.00

100.00 100.00 100.00 100.00

9.7930 8.9423 8.1649 7.4598

0.10211 0.11183 0.12248 0.13405

−12.967 8.2140 33.274 61.439

450.00 575.00 700.00

500.00 500.00 500.00

10.545 10.097 9.7036

0.094835 0.099035 0.10305

3.6375 27.662 54.974

51.055 77.179 106.50

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. Validated equations for the viscosity and thermal conductivity are not currently available for this fluid. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in density in the equation of state are 0.05% in the liquid phase up to 540 K, 0.5% up to the critical temperature, 1% at higher temperatures, 0.5% at pressures from 100 to 500 MPa, and 0.2% in the vapor phase. The uncertainty for the saturated-liquid density (and densities near atmospheric pressure) approaches 0.01% around 300 K. The uncertainties in vapor pressure are 0.3% from 270 to 305 K, 0.05% from 305 to 425 K, 0.1% up to 555 K, and 0.15% up to the critical temperature. The uncertainty in heat capacities is 0.5% and rises to 3% in the critical region. The uncertainty in the speed of sound is 1% up to 500 K and 100 MPa and rises to 2% at higher pressures and higher temperatures.

2-411

2-412

PHYSICAL AND CHEMICAL DATA TABLE 2-304

Saturated Solid/Vapor Water* Volume, ft3/lb

Enthalpy, Btu/lb

Entropy, Btu/(lb)(°F)

Temp., °F

Pressure, lb/in2 abs.

Solid

Vapor

Solid

Vapor

Solid

Vapor

−160 −150 −140 −130 −120

4.949.−8 1.620.−7 4.928.−7 1.403.−6 3.757.−6

0.01722 0.01723 0.01724 0.01725 0.01726

3.607.+9 1.139.+9 3.864.+8 1.400.+8 5.386.+7

−222.05 −218.82 −215.49 −212.08 −208.58

990.38 994.80 999.21 1003.63 1008.05

−0.4907 −0.4801 −0.4695 −0.4590 −0.4485

3.5549 3.4387 3.3301 3.2284 3.1330

−110 −100 −90 −80 −70

9.517.−6 2.291.−5 5.260.−5 1.157.−4 2.443.−4

0.01728 0.01729 0.01730 0.01731 0.01732

2.189.+7 9.352.+6 4.186.+6 1.955.+6 9.501.+5

−204.98 −201.28 −197.49 −193.60 −189.61

1012.47 1016.89 1021.31 1025.73 1030.15

−0.4381 −0.4277 −0.4173 −0.4069 −0.3965

3.0434 2.9591 2.8796 2.8045 2.7336

−60 −50 −45 −40 −35

4.972.−4 9.776.−4 1.354.−3 1.861.−3 2.540.−3

0.01734 0.01735 0.01736 0.01737 0.01737

4.788.+5 2.496.+5 1.824.+5 1.343.+5 9.961.+4

−185.52 −181.34 −179.21 −177.06 −174.88

1034.58 1039.00 1041.21 1043.42 1045.63

−0.3862 −0.3758 −0.3707 −0.3655 −0.3604

2.6664 2.6028 2.5723 2.5425 2.5135

−30 −25 −20 −15 −10

3.440.−3 4.627.−3 6.181.−3 8.204.−3 1.082.−2

0.01738 0.01739 0.01739 0.01740 0.01741

7.441.+4 5.596.+4 4.237.+4 3.228.+4 2.475.+4

−172.68 −170.46 −168.21 −165.94 −163.65

1047.84 1050.05 1052.26 1054.47 1056.67

−0.3552 −0.3501 −0.3449 −0.3398 −0.3347

2.4853 2.4577 2.4308 2.4046 2.3791

−5 0 5 10 15

1.419.−2 1.849.−2 2.396.−2 3.087.−2 3.957.−2

0.01741 0.01742 0.01743 0.01744 0.01744

1.909.+4 1.481.+4 1.155.+4 9.060.+3 7.144.+3

−161.33 −158.98 −156.61 −154.22 −151.80

1058.88 1061.09 1063.29 1065.50 1067.70

−0.3295 −0.3244 −0.3193 −0.3142 −0.3090

2.3541 2.3297 2.3039 2.2827 2.2600

16 18 20 22 24

4.156.−2 4.581.−2 5.045.−2 5.552.−2 6.105.−2

0.01745 0.01745 0.01745 0.01746 0.01746

6.817.+3 6.210.+3 5.662.+3 5.166.+3 4.717.+3

−151.32 −150.34 −149.36 −148.38 −147.39

1068.14 1069.02 1069.90 1070.38 1071.66

−0.3080 −0.3060 −0.3039 −0.3019 −0.2998

2.2555 2.2466 2.2378 2.2291 2.2205

26 28 30 31 32

6.708.−2 7.365.−2 8.080.−2 8.461.−2 8.858.−2

0.01746 0.01746 0.01747 0.01747 0.01747

4.311.+3 3.943.+3 3.608.+3 3.453.+3 3.305.+3

−146.40 −145.40 −144.40 −143.90 −143.40

1072.53 1073.41 1074.29 1074.73 1075.16

−0.2978 −0.2957 −0.2937 −0.2927 −0.2916

2.2119 2.2034 2.1950 2.1908 2.1867

*Condensed from Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1967 and 1972. Reproduced by permission. The validity of many standard reference tables has been critically reviewed by Jancso, Pupezin, and van Hook, J. Phys. Chem., 74 (1970):2984. Current information on the properties of solid, vapor, and liquid water properties can be found at http://www.iapws.org. The notation 4.949.−8, 3.607.+9, etc., means 4.949 × 10−8, 3.607 × 109, etc.

2-412

TABLE 2-305 Temperature K

Thermodynamic Properties of Water Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0 0.51875 1.2742 2.0278 2.7808 3.5339 4.2873 5.0414 5.7964 6.5526 7.3104 8.0701 8.8320 9.5966 10.364 11.136 11.911 12.692 13.477 14.269 15.068 15.875 16.690 17.515 18.352 19.200 20.064 20.943 21.841 22.762 23.709 24.688 25.707 26.777 27.917 29.160 30.585 32.422 36.314

1.1E-05 0.51877 1.2742 2.0279 2.7810 3.5340 4.2876 5.0419 5.7972 6.5538 7.3121 8.0725 8.8354 9.6013 10.371 11.144 11.923 12.706 13.496 14.293 15.098 15.913 16.737 17.573 18.421 19.285 20.165 21.065 21.987 22.935 23.915 24.932 25.996 27.119 28.324 29.648 31.180 33.180 37.548

0 0.001876 0.004527 0.007082 0.009551 0.011941 0.014260 0.016511 0.018700 0.020830 0.022906 0.024932 0.026911 0.028847 0.030743 0.032602 0.034427 0.036222 0.037988 0.039729 0.041448 0.043147 0.044830 0.046498 0.048156 0.049807 0.051454 0.053102 0.054756 0.056422 0.058106 0.059821 0.061577 0.063396 0.065309 0.067371 0.069715 0.072737 0.079393

42.785 42.954 43.201 43.446 43.690 43.931 44.169 44.404 44.634 44.860 45.079 45.291 45.496 45.691 45.876 46.050 46.211 46.359

45.055 45.280 45.609 45.936 46.261 46.582 46.900 47.212 47.519 47.819 48.111 48.393 48.665 48.924 49.170 49.400 49.613 49.807

0.16494 0.16174 0.15741 0.15344 0.14981 0.14647 0.14339 0.14054 0.13791 0.13546 0.13317 0.13104 0.12904 0.12715 0.12537 0.12369 0.12208 0.12054

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.075978 0.075669 0.075095 0.074412 0.073645 0.072811 0.071927 0.071008 0.070070 0.069124 0.068180 0.067247 0.066331 0.065438 0.064570 0.063731 0.062920 0.062140 0.061390 0.060671 0.059984 0.059327 0.058702 0.058109 0.057548 0.057023 0.056536 0.056089 0.055690 0.055347 0.055071 0.054881 0.054808 0.054902 0.055258 0.056100 0.058152 0.064521

0.076023 0.075688 0.075429 0.075320 0.075294 0.075317 0.075373 0.075456 0.075567 0.075708 0.075883 0.076098 0.076357 0.076664 0.077026 0.077447 0.077934 0.078495 0.079136 0.079869 0.080706 0.081662 0.082757 0.084013 0.085464 0.087149 0.089124 0.091464 0.094275 0.097713 0.10201 0.10754 0.11491 0.12526 0.14100 0.16852 0.23108 0.46736

1402.3 1434.1 1472.1 1501.4 1523.2 1538.7 1548.7 1553.9 1554.8 1552.0 1545.8 1536.5 1524.3 1509.5 1492.2 1472.5 1450.6 1426.5 1400.4 1372.2 1342.0 1309.8 1275.7 1239.6 1201.5 1161.3 1119.1 1074.6 1027.9 978.54 926.44 871.23 812.49 749.57 681.27 604.73 513.19 400.66 0

−0.24142 −0.23515 −0.22720 −0.22024 −0.21393 −0.20804 −0.20241 −0.19690 −0.19140 −0.18581 −0.18005 −0.17404 −0.16769 −0.16092 −0.15366 −0.14581 −0.13728 −0.12794 −0.11767 −0.10631 −0.09369 −0.07959 −0.06372 −0.04578 −0.02534 −0.00189 0.025264 0.057002 0.094527 0.13949 0.19425 0.26220 0.34857 0.46172 0.61660 0.84473 1.2251 1.9542 3.7410

561.04 574.04 592.73 610.28 626.05 639.71 651.18 660.55 668.00 673.76 678.02 681.00 682.83 683.64 683.52 682.53 680.70 678.05 674.59 670.28 665.12 659.07 652.06 644.05 634.95 624.68 613.15 600.26 585.95 570.21 553.08 534.74 515.43 495.46 475.03 454.10 432.51 414.93

1791.2 1433.7 1084.0 853.84 693.54 577.02 489.49 421.97 368.77 326.10 291.36 262.69 238.77 218.60 201.43 186.68 173.91 162.77 152.98 144.31 136.58 129.64 123.37 117.66 112.42 107.57 103.05 98.792 94.746 90.857 87.074 83.342 79.600 75.773 71.759 67.382 62.244 55.247

0.025553 0.025657 0.025816 0.025982 0.026158 0.026350 0.026568 0.026821 0.027118 0.027469 0.027883 0.028372 0.028944 0.029608 0.030369 0.031230 0.032187 0.033234

0.033947 0.034073 0.034270 0.034483 0.034716 0.034980 0.035287 0.035653 0.036091 0.036617 0.037249 0.038004 0.038903 0.039963 0.041203 0.042634 0.044269 0.046114

409.00 413.92 420.99 427.89 434.63 441.18 447.54 453.68 459.58 465.22 470.57 475.61 480.32 484.67 488.65 492.22 495.39 498.12

Saturated Properties

2-413

273.16 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 647.1

0.000612 0.000992 0.001920 0.003537 0.006231 0.010546 0.017213 0.027188 0.041682 0.062194 0.090535 0.12885 0.17964 0.24577 0.33045 0.43730 0.57026 0.73367 0.9322 1.1709 1.4551 1.7905 2.1831 2.6392 3.1655 3.7690 4.4569 5.2369 6.1172 7.1062 8.2132 9.448 10.821 12.345 14.033 15.901 17.969 20.265 22.064

273.16 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440

0.000612 0.000992 0.001920 0.003537 0.006231 0.010546 0.017213 0.027188 0.041682 0.062194 0.090535 0.12885 0.17964 0.24577 0.33045 0.43730 0.57026 0.73367

55.497 55.501 55.440 55.315 55.139 54.919 54.662 54.371 54.049 53.698 53.321 52.918 52.490 52.038 51.563 51.064 50.541 49.994 49.421 48.824 48.199 47.545 46.861 46.145 45.393 44.603 43.770 42.889 41.954 40.956 39.885 38.725 37.456 36.048 34.451 32.577 30.210 26.729 17.874 0.000269 0.000426 0.000797 0.001420 0.002424 0.003978 0.006304 0.009681 0.014448 0.021014 0.029859 0.041537 0.056683 0.076014 0.10034 0.13055 0.16765 0.21276

0.018019 0.018018 0.018038 0.018078 0.018136 0.018209 0.018294 0.018392 0.018502 0.018623 0.018754 0.018897 0.019051 0.019217 0.019394 0.019583 0.019786 0.020003 0.020234 0.020482 0.020748 0.021033 0.021340 0.021671 0.022030 0.022420 0.022847 0.023316 0.023836 0.024417 0.025072 0.025823 0.026698 0.027741 0.029026 0.030697 0.033101 0.037413 0.055948 3711.0 2345.4 1254.3 704.01 412.60 251.39 158.62 103.30 69.213 47.586 33.491 24.075 17.642 13.156 9.9666 7.6601 5.9649 4.7002

592.65 477.26 351.65 264.35 203.74 161.25 130.92 108.77 92.178 79.440 69.427 61.373 54.749 49.181 44.405 40.237 36.550 33.259

17.071 17.442 18.031 18.673 19.369 20.117 20.922 21.784 22.707 23.695 24.750 25.875 27.074 28.347 29.699 31.128 32.638 34.230

9.2163 9.3815 9.6414 9.9195 10.213 10.518 10.833 11.157 11.487 11.823 12.162 12.504 12.848 13.192 13.538 13.883 14.228 14.573

2-414

TABLE 2-305

Thermodynamic Properties of Water (Continued)

Temperature K

Pressure MPa

Density mol/dm3

Volume dm3/mol

450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 647.1

0.93220 1.1709 1.4551 1.7905 2.1831 2.6392 3.1655 3.7690 4.4569 5.2369 6.1172 7.1062 8.2132 9.4480 10.821 12.345 14.033 15.901 17.969 20.265 22.064

0.26711 0.33209 0.40925 0.50035 0.60738 0.73265 0.87884 1.0491 1.2473 1.4780 1.7471 2.0620 2.4325 2.8720 3.3994 4.0434 4.8497 5.9009 7.3737 9.8331 17.874

3.7438 3.0113 2.4435 1.9986 1.6464 1.3649 1.1379 0.95318 0.80174 0.67659 0.57238 0.48497 0.41110 0.34819 0.29417 0.24732 0.20620 0.16946 0.13562 0.10170 0.055948

55.317 53.212

0.018078 0.018793

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.048177 0.050469 0.053005 0.055809 0.058919 0.062388 0.066289 0.070723 0.075827 0.081789 0.088873 0.097461 0.10813 0.12178 0.13994 0.16540 0.20384 0.26923 0.40819 0.94736

500.41 502.24 503.60 504.45 504.78 504.55 503.71 502.23 500.05 497.10 493.31 488.58 482.79 475.80 467.41 457.33 445.11 429.99 410.21 379.64 0

30.307 27.653 25.265 23.118 21.187 19.450 17.886 16.475 15.197 14.035 12.973 11.997 11.093 10.248 9.4499 8.6837 7.9329 7.1743 6.3669 5.3854 3.7410

35.904 37.663 39.512 41.455 43.502 45.666 47.969 50.442 53.130 56.102 59.456 63.341 67.981 73.721 81.108 91.052 105.17 126.66 163.44 250.01

14.917 15.261 15.606 15.952 16.300 16.653 17.011 17.377 17.755 18.149 18.563 19.007 19.489 20.024 20.634 21.350 22.229 23.374 25.018 27.938

−0.22024 −0.17843

610.32 678.97

67.038 47.254 19.298 10.567 6.6444 4.5167 3.2280 2.3885 1.8122 1.4006

25.053 27.008 35.861 46.367 57.964 70.385 83.466 97.085 111.15 125.58

−0.22022 −0.16113 −0.11435

610.73 684.10 673.37

29.473 19.741 10.615 6.6387 4.5077 3.2212 2.3837 1.8089 1.3982

36.427 38.799 47.636 58.735 70.983 84.000 97.573 111.57 125.89

−0.22012 −0.16222 −0.04945 0.047232

612.54 686.54 646.52 604.15

Saturated Properties 46.492 46.609 46.708 46.788 46.848 46.885 46.898 46.883 46.838 46.758 46.641 46.478 46.264 45.988 45.636 45.188 44.613 43.855 42.801 41.095 36.314

49.982 50.134 50.263 50.367 50.442 50.487 50.500 50.475 50.411 50.302 50.142 49.925 49.641 49.278 48.819 48.242 47.506 46.550 45.238 43.156 37.548

0.11907 0.11764 0.11627 0.11493 0.11362 0.11233 0.11105 0.10979 0.10852 0.10724 0.10595 0.10462 0.10324 0.10180 0.10026 0.098600 0.096755 0.094631 0.092029 0.088324 0.079393

0.034362 0.035561 0.036821 0.038137 0.039503 0.040920 0.042391 0.043920 0.045519 0.047197 0.048968 0.050848 0.052856 0.055017 0.057361 0.059939 0.062831 0.066197 0.070465 0.077576

Single-Phase Properties 300 372.76

0.1 0.1

372.76 400 500 600 700 800 900 1000 1100 1200

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

300 400 453.03

1 1 1

453.03 500 600 700 800 900 1000 1100 1200

1 1 1 1 1 1 1 1 1

300 400 500 537.09

5 5 5 5

537.09 600 700

5 5 5

0.032769 0.030397 0.024154 0.020086 0.017201 0.015044 0.013369 0.012030 0.010936 0.010024 55.340 52.060 49.243 0.28559 0.25158 0.20466 0.17377 0.15134 0.13418 0.12058 0.10951 0.10032 55.439 52.173 46.267 43.151 1.4072 1.1320 0.91269

30.517 32.898 41.401 49.786 58.136 66.471 74.799 83.123 91.444 99.763

2.0277 7.5196

2.0295 7.5214

0.007081 0.02347

0.074406 0.067921

0.075315 0.075938

45.138 45.900 48.619 51.387 54.256 57.240 60.347 63.581 66.941 70.426

48.190 49.189 52.759 56.365 60.069 63.887 67.827 71.893 76.085 80.402

0.13257 0.13516 0.14313 0.14970 0.15541 0.16050 0.16514 0.16943 0.17342 0.17718

0.02801 0.02717 0.02717 0.028103 0.029225 0.030431 0.031687 0.032963 0.034228 0.035458

0.037444 0.036170 0.035693 0.036513 0.037592 0.038778 0.040024 0.041293 0.042554 0.043781

0.018070 0.019209 0.020307

2.0263 9.5914 13.717

2.0444 9.6106 13.737

0.007077 0.028834 0.038518

0.074353 0.065422 0.061169

0.075270 0.076628 0.079348

3.5015 3.9749 4.8861 5.7547 6.6074 7.4524 8.2932 9.1313 9.9677

46.529 48.111 51.123 54.087 57.121 60.258 63.511 66.885 70.380

50.030 52.086 56.009 59.842 63.729 67.710 71.804 76.016 80.347

0.11863 0.12295 0.13011 0.13602 0.14121 0.14590 0.15021 0.15422 0.15799

0.034718 0.030084 0.029002 0.029629 0.030651 0.031821 0.033051 0.034290 0.035504

0.048846 0.041065 0.038358 0.038495 0.039301 0.040358 0.041522 0.042719 0.043905

0.018038 0.019167 0.021614 0.023175

2.0204 9.5643 17.474 20.685

2.1106 9.6601 17.582 20.801

0.007057 0.028766 0.046415 0.052622

0.074119 0.065337 0.058082 0.056215

0.075070 0.076438 0.083643 0.090740

0.71063 0.88340 1.0957

46.785 49.734 53.286

50.338 54.151 58.765

0.10762 0.11436 0.12148

0.046699 0.034611 0.031678

0.079952 0.051045 0.043318

1501.5 1543.5 471.99 490.31 548.31 598.61 643.92 685.47 724.03 760.17 794.33 826.85 1503.0 1511.3 1392.0 501.02 535.74 592.58 640.55 683.48 722.85 759.50 794.01 826.77 1509.8 1520.9 1250.0 1087.8 498.04 561.07 624.59

14.362 10.407 6.5536

55.203 54.653 62.680

853.83 282.91 12.256 13.285 17.270 21.407 25.564 29.669 33.685 37.592 41.382 45.054 853.67 218.80 150.24 15.021 17.051 21.329 25.550 29.687 33.718 37.630 41.420 45.088 853.00 219.84 118.27 100.01 18.032 21.062 25.547

800 900 1000 1100 1200

5 5 5 5 5

300 400 500 584.15

10 10 10 10

584.15 600 700 800 900 1000 1100 1200

10 10 10 10 10 10 10 10

0.77805 0.68224 0.60918 0.55109 0.50355 55.561 52.312 46.517 38.213 3.0787 2.7628 1.9625 1.6157 1.3945 1.2345 1.1111 1.0119

1.2853 1.4658 1.6416 1.8146 1.9859

56.576 59.855 63.197 66.632 70.172

63.002 67.183 71.405 75.705 80.101

0.12714 0.13207 0.13652 0.14061 0.14444

0.031683 0.032430 0.033447 0.034565 0.035704

0.041848 0.041922 0.042571 0.043465 0.044458

674.39 717.57 756.57 792.63 826.45

4.4532 3.1856 2.3599 1.7924 1.3865

73.950 86.626 99.971 113.64 127.51

29.806 33.891 37.821 41.606 45.257

0.017998 0.019116 0.021497 0.026169

2.0131 9.5311 17.389 25.105

2.1931 9.7222 17.604 25.367

0.007031 0.028682 0.046244 0.060543

0.073834 0.065233 0.058028 0.054835

0.074829 0.076208 0.082910 0.11032

1518.2 1532.7 1271.3 847.33

−0.21999 −0.16351 −0.05669 0.29540

614.81 689.57 651.64 526.83

852.28 221.13 119.55 81.795

0.32482 0.36195 0.50956 0.61893 0.71709 0.81002 0.90002 0.98820

45.852 47.183 52.145 55.851 59.334 62.798 66.314 69.910

49.100 50.802 57.241 62.040 66.505 70.898 75.314 79.792

0.10117 0.10405 0.11405 0.12046 0.12572 0.13035 0.13456 0.13846

0.055964 0.047271 0.034838 0.033089 0.033219 0.033947 0.034908 0.035954

0.128640 0.092535 0.051779 0.045603 0.044062 0.043952 0.044427 0.045164

472.51 503.34 602.20 662.61 710.98 753.03 791.02 826.16

9.9124 9.4382 6.3228 4.3529 3.1289 2.3241 1.7683 1.3695

76.543 71.110 69.301 78.476 90.516 103.50 116.73 130.00

20.267 21.036 25.704 30.054 34.176 38.111 41.882 45.506

300 400 500 600 700 800 900 1000 1100 1200

100 100 100 100 100 100 100 100 100 100

57.573 54.500 49.914 43.935 36.179 26.768 19.073 14.734 12.246 10.631

0.017369 0.018349 0.020034 0.022761 0.027640 0.037359 0.052429 0.067868 0.081656 0.094062

1.8921 9.0423 16.289 23.820 31.916 40.700 48.805 55.188 60.470 65.222

3.6290 10.877 18.292 26.097 34.680 44.435 54.048 61.975 68.635 74.628

0.006516 0.027360 0.043895 0.058109 0.071320 0.084331 0.095669 0.10404 0.11039 0.11561

0.069812 0.063582 0.057324 0.052776 0.049610 0.047143 0.043932 0.041345 0.040131 0.039810

0.071696 0.073086 0.075607 0.081104 0.091576 0.10108 0.088057 0.071678 0.062539 0.057826

1667.9 1717.3 1555.7 1300.4 1020.0 813.97 765.30 792.50 832.67 872.28

−0.21618 −0.17905 −0.12564 −0.02079 0.21155 0.65939 1.0399 1.0944 0.98401 0.83544

654.50 741.80 730.42 645.83 510.14 351.46 257.03 232.07 223.70 219.07

856.88 243.50 138.92 101.51 79.363 62.042 53.250 51.518 52.497 54.415

300 400 500 600 700 800 900 1000 1100 1200

500 500 500 500 500 500 500 500 500 500

63.750 60.862 57.695 54.316 50.847 47.385 44.018 40.814 37.834 35.124

0.015686 0.016431 0.017332 0.018411 0.019667 0.021104 0.022718 0.024501 0.026432 0.028470

1.5247 7.9635 14.264 20.481 26.606 32.615 38.492 44.233 49.839 55.312

9.3678 16.179 22.930 29.687 36.439 43.167 49.851 56.484 63.055 69.547

0.003746 0.023347 0.038412 0.050731 0.061141 0.070124 0.077998 0.084987 0.091251 0.096900

0.063403 0.059634 0.055769 0.052734 0.050315 0.048442 0.047068 0.046126 0.045537 0.045218

0.068296 0.067603 0.067522 0.067584 0.067436 0.067080 0.066596 0.066041 0.065356 0.064451

2228.6 2258.7 2200.7 2093.8 1970.5 1850.1 1743.4 1655.7 1589.3 1543.9

−0.19915 −0.19486 −0.18339 −0.16883 −0.15188 −0.13256 −0.11124 −0.08910 −0.06907 −0.05511

763.82 929.09 1096.6 1097.9 935.15 738.72 572.49 445.17 350.97 282.78

1089.4 320.18 189.08 141.83 118.47 104.70 95.388 88.418 83.021 78.952

400 500 600 700 800 900 1000 1100 1200

1000 1000 1000 1000 1000 1000 1000 1000 1000

65.942 63.253 60.572 57.937 55.384 52.937 50.611 48.415 46.349

0.015165 0.015810 0.016509 0.017260 0.018056 0.018890 0.019759 0.020655 0.021575

7.4792 13.357 19.141 24.836 30.435 35.938 41.354 46.695 51.976

22.644 29.167 35.650 42.096 48.491 54.828 61.113 67.350 73.551

0.019833 0.034391 0.046212 0.056150 0.064689 0.072155 0.078776 0.084722 0.090117

0.057934 0.055063 0.053055 0.051393 0.050059 0.049062 0.048373 0.047942 0.047713

0.065743 0.064967 0.064676 0.064219 0.063663 0.063101 0.062594 0.062176 0.061861

2718.6 2677.2 2602.3 2513.7 2423.7 2338.7 2261.5 2193.2 2133.6

−0.19303 −0.19158 −0.18789 −0.18439 −0.18105 −0.17779 −0.17459 −0.17139 −0.16808

1172.7 2199.5 3250.5 3202.2 2408.7 1610.7 1052.9 703.41 487.61

329.93 190.55 137.73 108.98 91.430 80.198 72.716 67.520 63.774

2-415

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Wagner, W., and Pruss, A., “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use,” J. Phys. Chem. Ref. Data 31(2):387–535, 2002. The source for viscosity is International Association for the Properties of Water and Steam, Revised Release on the IAPS Formulation 1985 for the Viscosity of Ordinary Water Substance, IAPWS, 1997. The source for thermal conductivity is the International Association for the Properties of Water and Steam, Revised Release on the IAPS Formulation 1985 for the Thermal Conductivity of Ordinary Water Substance, IAPWS, 1998. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainty in density of the equation of state is 0.0001% at 1 atm in the liquid phase, and 0.001% at other liquid states at pressures up to 10 MPa and temperatures to 423 K. In the vapor phase, the uncertainty is 0.05% or less. The uncertainties rise at higher temperatures and/or pressures, but are generally less than 0.1% in density except at extreme conditions. The uncertainty in pressure in the critical region is 0.1%. The uncertainty of the speed of sound is 0.15% in the vapor and 0.1% or less in the liquid, and increases near the critical region and at high temperatures and pressures. The uncertainty in isobaric heat capacity is 0.2% in the vapor and 0.1% in the liquid, with increasing values in the critical region and at high pressures. The uncertainties of saturation conditions are 0.025% in vapor pressure, 0.0025% in saturated-liquid density, and 0.1% in saturated-vapor density. The uncertainties in the saturated densities increase substantially as the critical region is approached. For the uncertainties in the viscosity and thermal conductivity, see the IAPWS Release.

2-416

PHYSICAL AND CHEMICAL DATA TABLE 2-306

Thermodynamic Properties of Water Substance along the Melting Line

P, bar

T, °C

103 v f , m3/kg

h f , kJ/kg

s f, kJ/kg⋅K

cpf, kJ/kg⋅K

cmelt, kJ/kg⋅K

106α f, K−1

106K f,T bar−1

6.117  10-3t 1.01325 50 100 150

0.0100 0.0026 −0.3618 −0.7410 −1.1249

1.00021 1.00016 0.99770 0.99523 0.99278

0 0.0719 3.5140 6.9794 10.3964

0 −0.0001 −0.0054 −0.0110 −0.0167

4.219 4.218 4.196 4.174 4.152

3.969 3.970 3.997 4.023 4.047

−67.42 −67.17 −54.92 −42.52 −30.24

50.90 50.88 50.30 49.73 49.17

200 250 300 400 500

−1.5166 −1.9151 −2.3206 −3.1532 −4.0156

0.99037 0.98798 0.98562 0.98098 0.97643

13.7648 17.0843 20.3547 26.7472 32.9403

−0.0225 −0.0285 −0.0347 −0.0474 −0.0607

4.132 4.112 4.092 4.056 4.022

4.070 4.092 4.113 4.150 4.184

−18.05 −5.93 6.12 30.09 53.97

48.63 48.11 47.59 46.61 45.68

600 800 1000

−4.909 −6.790 −8.803

0.97196 0.96326 0.95493

38.932 50.300 60.836

−0.0747 −0.1046 −0.1371

3.992 3.937 3.893

4.215 4.270 4.320

77.87 126.18 175.98

44.80 43.19 41.74

Condensed from U. Grigull, Private communication, January 18, 1995. Materials prepared at Technical University München, Germany by U. Grigull and S. Marek. For a table as a function of temperature, see Grigull, U. and S. Marek, Warme u. Stoff., 30 (1994): 1–8. t = the triple point (at 6.117 × 10−3 bar, 0.01 °C); vf = 0.0010021 m3/kg: α f = −67.42 × 10−6/K. Other equations for properties are given by Jones, F. E. and G. L. Harris, J. Res. N.I.S.T., 97, 3 (1992): 335–340, and by Wagner, W. and A. Pruss, J. Phys. Chem. Ref. Data, 22, 3 (1993): 783–787. Steam tables include Walker, W. A., U.S. Naval Ordn. Lab. rept. NOLTR NOLTR-66-217 = AD 651105 (0–1000 bar, 0–150°C), 1967 (72 pp.); Grigull, U., J. Straub, et al., Steam Tables in S.I. Units (0.01–1000 bar, 0–1000 °C), Springer-Verlag, Berlin, 1990 (133 pp.); Tseng, C. M., T. A. Hamp, et al., Atomic Energy of Canada rept. (30 props, sat liq & vap., 1–220 bar), AECL-5910 1977 (90 pp.). For dissociation, see e.g., Knonicek, V., Rozpr. Cesko Acad Ved., Rada techn ved (0.01–100 bar, 1000–5000 K). 77, 1 (1967). The proceedings of the 10th international conference on the properties of steam were edited by Sytchev, V. V. and A. A. Aleksandrov, Plenum, NY, 1984; and for the 11th conference by Pichal, M. and O. Sifner, Hemisphere, 1989 (550 pp.). Current information on the properties of solid, vapor, and liquid water properties can be found at http://www.iapws.org. For electrical conductivity, see e.g., Marshall, W. L., J. Chem. Eng. Data, 32 (1987): 221–226.

TABLE 2-307 Temperature K

Thermodynamic Properties of Xenon Pressure MPa

Density mol/dm3

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

0.044263 0.045137 0.045673 0.046232 0.046815 0.047426 0.048068 0.048743 0.049455 0.050210 0.051013 0.051869 0.052787 0.053776 0.054847 0.056017 0.057304 0.058735 0.060344 0.062183 0.064330 0.066910 0.070156 0.074566 0.081686 0.11905

−0.16663 0.21574 0.43915 0.66366 0.88953 1.1170 1.3464 1.5779 1.8119 2.0487 2.2887 2.5324 2.7802 3.0328 3.2907 3.5550 3.8265 4.1068 4.3976 4.7014 5.0218 5.3646 5.7397 6.1681 6.7103 8.2423

−0.16301 0.22176 0.44706 0.67390 0.90261 1.1335 1.3669 1.6033 1.8429 2.0864 2.3341 2.5866 2.8446 3.1089 3.3802 3.6598 3.9488 4.2491 4.5628 4.8930 5.2441 5.6230 6.0416 6.5247 7.1435 8.9377

−0.00099337 0.0013152 0.0026109 0.0038765 0.0051149 0.0063292 0.0075220 0.0086959 0.0098535 0.010997 0.012130 0.013254 0.014373 0.015488 0.016605 0.017726 0.018857 0.020002 0.021170 0.022371 0.023618 0.024935 0.026360 0.027976 0.030022 0.036074 0.077430 0.074250 0.072554 0.070955 0.069442 0.068004 0.066633 0.065319 0.064056 0.062836 0.061653 0.060498 0.059367 0.058252 0.057147 0.056042 0.054930 0.053801 0.052641 0.051434 0.050157 0.048773 0.047224 0.045388 0.042934 0.036074

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas

0.022064 0.021486 0.021176 0.020884 0.020609 0.020349 0.020104 0.019873 0.019656 0.019453 0.019264 0.019091 0.018933 0.018794 0.018675 0.018580 0.018515 0.018487 0.018507 0.018591 0.018767 0.019077 0.019604 0.020521 0.022329

0.044446 0.044730 0.044972 0.045274 0.045640 0.046075 0.046585 0.047179 0.047868 0.048666 0.049590 0.050663 0.051914 0.053385 0.055128 0.057218 0.059766 0.062932 0.066972 0.072312 0.079722 0.090763 0.10917 0.14668 0.26914

653.38 629.12 614.85 600.43 585.85 571.10 556.14 540.97 525.55 509.85 493.84 477.47 460.71 443.49 425.74 407.38 388.30 368.36 347.41 325.20 301.44 275.68 247.28 215.12 176.78 0

−0.63704 −0.60760 −0.58689 −0.56326 −0.53642 −0.50604 −0.47173 −0.43300 −0.38927 −0.33987 −0.28392 −0.22039 −0.14796 −0.064982 0.030670 0.14178 0.27205 0.42662 0.61272 0.84107 1.1284 1.5023 2.0135 2.7688 4.0593 8.3700

72.081 68.920 67.037 65.137 63.229 61.320 59.415 57.519 55.633 53.759 51.897 50.048 48.209 46.379 44.554 42.730 40.903 39.071 37.238 35.418 33.653 32.016 30.622 29.654 29.742

516.89 457.99 427.78 400.07 374.57 351.01 329.16 308.84 289.88 272.13 255.47 239.78 224.96 210.92 197.57 184.83 172.63 160.90 149.54 138.48 127.61 116.78 105.75 94.067 80.469

0.013321 0.013615 0.013809 0.014020 0.014247 0.014491 0.014750 0.015025 0.015314 0.015618 0.015938 0.016273 0.016625 0.016996 0.017389 0.017808 0.018258 0.018748 0.019288 0.019894 0.020590 0.021413 0.022428 0.023764 0.025741

0.022698 0.023442 0.023957 0.024539 0.025194 0.025929 0.026755 0.027681 0.028722 0.029898 0.031232 0.032757 0.034515 0.036563 0.038982 0.041885 0.045436 0.049883 0.055621 0.063311 0.074154 0.090577 0.11833 0.17512 0.35495

128.33 130.83 132.15 133.36 134.47 135.46 136.35 137.13 137.79 138.35 138.80 139.13 139.35 139.45 139.43 139.29 139.02 138.62 138.07 137.36 136.47 135.36 133.95 132.11 129.38 0

64.799 57.103 53.274 49.835 46.729 43.913 41.350 39.009 36.862 34.888 33.066 31.378 29.805 28.333 26.946 25.629 24.367 23.142 21.938 20.732 19.500 18.207 16.800 15.185 13.131 8.3700

3.1160 3.3040 3.4175 3.5348 3.6563 3.7826 3.9143 4.0523 4.1975 4.3511 4.5144 4.6894 4.8787 5.0855 5.3151 5.5745 5.8750 6.2336 6.6778 7.2524 8.0324 9.1495 10.852 13.688 19.481

13.386 14.014 14.392 14.781 15.180 15.592 16.018 16.460 16.920 17.399 17.901 18.429 18.987 19.579 20.212 20.892 21.630 22.437 23.333 24.341 25.501 26.874 28.575 30.845 34.410

0.021719 0.021011 0.020884

143.95 177.58 205.37

39.340 18.027 10.987

3.8225 5.5664 7.2216

16.051 23.114 30.061

Saturated Properties 161.40 170.00 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 289.73

0.081748 0.13343 0.17325 0.22153 0.27933 0.34774 0.42789 0.52091 0.62797 0.75025 0.88893 1.0452 1.2203 1.4155 1.6321 1.8712 2.1344 2.4229 2.7382 3.0820 3.4561 3.8623 4.3032 4.7818 5.3025 5.8420

22.592 22.155 21.895 21.630 21.361 21.085 20.804 20.516 20.220 19.916 19.603 19.279 18.944 18.596 18.232 17.852 17.451 17.026 16.572 16.082 15.545 14.945 14.254 13.411 12.242 8.4000

161.40 170.00 175.00 180.00 185.00 190.00 195.00 200.00 205.00 210.00 215.00 220.00 225.00 230.00 235.00 240.00 245.00 250.00 255.00 260.00 265.00 270.00 275.00 280.00 285.00 289.73

0.081748 0.13343 0.17325 0.22153 0.27933 0.34774 0.42789 0.52091 0.62797 0.75025 0.88893 1.0452 1.2203 1.4155 1.6321 1.8712 2.1344 2.4229 2.7382 3.0820 3.4561 3.8623 4.3032 4.7818 5.3025 5.8420

0.062613 0.098131 0.12476 0.15646 0.19385 0.23755 0.28826 0.34671 0.41374 0.49026 0.57731 0.67606 0.78791 0.91450 1.0578 1.2202 1.4049 1.6157 1.8581 2.1394 2.4705 2.8685 3.3629 4.0140 4.9911 8.4000

15.971 10.190 8.0157 6.3913 5.1587 4.2096 3.4691 2.8842 2.4170 2.0397 1.7322 1.4791 1.2692 1.0935 0.94537 0.81952 0.71181 0.61892 0.53818 0.46742 0.40477 0.34861 0.29736 0.24913 0.20036 0.11905

11.189 11.261 11.298 11.332 11.362 11.388 11.409 11.426 11.437 11.442 11.442 11.434 11.420 11.397 11.365 11.322 11.268 11.199 11.114 11.009 10.878 10.713 10.500 10.209 9.7610 8.2423

12.495 12.621 12.687 12.748 12.803 12.852 12.893 12.928 12.954 12.973 12.981 12.980 12.968 12.945 12.908 12.856 12.787 12.699 12.588 12.450 12.277 12.059 11.779 11.400 10.823 8.9377

200.00 300.00 400.00

0.10000 0.10000 0.10000

0.061174 0.040297 0.030132

16.347 24.816 33.188

11.693 12.971 14.227

13.328 15.453 17.546

Single-Phase Properties 2-417

0.080422 0.089046 0.095069

0.012825 0.012529 0.012489

2-418

TABLE 2-307

Thermodynamic Properties of Xenon (Concluded)

Temperature K

Pressure MPa

Density mol/dm3

500.00 600.00 700.00

0.10000 0.10000 0.10000

0.024077 0.020054 0.017184

200.00 218.61

1.0000 1.0000

218.61 300.00 400.00 500.00 600.00 700.00

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

200.00 282.15

5.0000 5.0000

282.15 300.00 400.00 500.00 600.00 700.00

5.0000 5.0000 5.0000 5.0000 5.0000 5.0000

Volume dm3/mol

Int. energy kJ/mol

Enthalpy kJ/mol

Entropy kJ/(molK)

Cv kJ/(molK)

Cp kJ/(molK)

Sound speed m/s

Joule-Thomson K/MPa

Therm. cond. mW/(mK)

Viscosity µPas 36.679 42.919 48.797

Single-Phase Properties (Concluded)

200.00 300.00 400.00 500.00 600.00 700.00

10.000 10.000 10.000 10.000 10.000 10.000

20.545 19.370 0.64740 0.42338 0.30719 0.24284 0.20128 0.17205 20.777 12.969 4.3744 3.0178 1.6846 1.2596 1.0212 0.86365

41.533 49.866 58.193 0.048673 0.051625 1.5446 2.3619 3.2553 4.1179 4.9682 5.8123 0.048130 0.077107

15.479 16.728 17.977 1.5644 2.4644 11.437 12.735 14.081 15.373 16.646 17.910 1.4588 6.3800

19.632 21.715 23.796 1.6131 2.5160 12.982 15.097 17.337 19.491 21.614 23.723 1.6995 6.7656

0.099724 0.10352 0.10673

0.012480 0.012477 0.012475

0.020841 0.020822 0.020811

229.73 251.71 271.90

7.4876 5.3775 3.9611

8.7865 10.259 11.644

0.0086285 0.012943

0.019883 0.019137

0.047017 0.050348

543.44 482.05

−0.43967 −0.23885

57.759 50.560

0.060816 0.069105 0.075560 0.080369 0.084241 0.087491

0.016178 0.013071 0.012653 0.012555 0.012521 0.012506

0.032313 0.023342 0.021812 0.021347 0.021143 0.021033

139.05 174.22 204.53 229.84 252.27 272.69

31.834 17.710 10.799 7.3466 5.2698 3.8786

4.6396 5.9066 7.4612 8.9749 10.414 11.777

18.280 23.825 30.617 37.129 43.291 49.110

0.0080923 0.028774

0.019975 0.021130

0.045835 0.17857

562.91 199.64

−0.48921 3.2255

59.693 29.468

324.93 88.606

310.61 244.05

0.22860 0.33137 0.59361 0.79390 0.97922 1.1579

10.045 11.189 13.369 14.889 16.278 17.615

11.188 12.846 16.337 18.859 21.175 23.404

0.044447 0.050182 0.060390 0.066031 0.070256 0.073694

0.024500 0.017212 0.013445 0.012902 0.012724 0.012645

0.22292 0.057127 0.027264 0.023853 0.022627 0.022028

131.10 155.40 202.15 231.38 255.44 276.67

14.382 14.736 9.6973 6.6609 4.7873 3.5219

15.582 9.7191 8.7332 9.8856 11.137 12.377

32.133 29.671 33.661 39.357 45.049 50.547

21.038 13.284 3.8169 2.6239 2.0706 1.7304

0.047533 0.075281 0.26199 0.38111 0.48294 0.57792

1.3406 6.6026 12.321 14.257 15.817 17.251

1.8159 7.3554 14.941 18.068 20.646 23.030

0.0074790 0.029493 0.052029 0.059048 0.063756 0.067433

0.020097 0.018200 0.014472 0.013339 0.012977 0.012819

0.044678 0.088497 0.037766 0.027456 0.024556 0.023262

584.79 258.88 204.59 235.98 260.98 282.62

−0.53901 2.0533 7.6552 5.6862 4.1777 3.0965

61.959 29.649 11.158 11.280 12.172 13.210

341.91 94.113 39.566 42.848 47.602 52.559

819.84 690.23 597.21 538.16 504.61 487.44

−0.80697 −0.77003 −0.68464 −0.60849 −0.56578 −0.55679

91.685 62.741 46.987 38.256 33.282 30.399

582.16 293.11 188.96 141.62 118.14 106.02

−0.96827 −1.0167 −1.0606 −1.1030

96.278 80.779 71.145 64.734

544.58 424.61 352.34 304.58

200.00 300.00 400.00 500.00 600.00 700.00

100.00 100.00 100.00 100.00 100.00 100.00

23.707 20.755 18.043 15.689 13.750 12.188

0.042181 0.048181 0.055422 0.063739 0.072728 0.082045

0.22766 3.3951 6.2728 8.8555 11.158 13.232

4.4458 8.2132 11.815 15.229 18.431 21.437

0.00066778 0.015969 0.026343 0.033970 0.039814 0.044451

0.022077 0.018781 0.017019 0.015937 0.015222 0.014724

0.038788 0.036811 0.035156 0.033084 0.030987 0.029184

400.00 500.00 600.00 700.00

500.00 500.00 500.00 500.00

25.007 23.787 22.695 21.712

0.039989 0.042040 0.044062 0.046058

4.1551 6.3322 8.3949 10.364

24.150 27.352 30.426 33.393

0.012220 0.019372 0.024980 0.029555

0.020738 0.019369 0.018396 0.017664

0.032768 0.031337 0.030178 0.029185

1164.9 1121.3 1086.3 1058.1

The values in these tables were generated from the NIST REFPROP software (Lemmon, E. W., McLinden, M. O., and Huber, M. L., NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties—REFPROP, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, Md., 2002, Version 7.1). The primary source for the thermodynamic properties is Lemmon, E. W., and Span, R., “Short Fundamental Equations of State for 20 Industrial Fluids,” J. Chem. Eng. Data 51(3):785–850, 2006. The source for viscosity and thermal conductivity is McCarty, R. D., Correlations for the Thermophysical Properties of Xenon, National Institute of Standards and Technology, Boulder, Colo., 1989. Properties at the triple point temperature and the critical point temperature are given in the first and last entries of the saturation tables, respectively. In the single-phase table, when the temperature range for a given isobar includes a vapor-liquid phase boundary, the temperature of phase equilibrium is noted, and properties for both the saturated liquid and saturated vapor are given (with liquid properties given in the upper line). Lines are omitted from the temperature-pressure grid of the single-phase table, when the system would be in the solid phase or if there are potential problems with the source property surface. The uncertainties in the equation of state are 0.2% in density up to 100 MPa, rising to 1% at higher pressures, 0.2% in vapor pressure, 1% in the speed of sound, and 2% in heat capacities. For viscosity, estimated uncertainty is less than 5%. For thermal conductivity, estimated uncertainty is less than 6%.

THERMODYNAMIC PROPERTIES

2-419

TABLE 2-308

Surface Tension (N/m) of Saturated Liquid Refrigerants*

R no.

−50

−25

0

25

50

75

100

125

150

0.0279 0.0188 0.0092 0.0197 0.0115

0.0244 0.0152 0.0056 0.0156 0.0065

0.0210 0.0118 0.0025 0.0117 0.0025

0.0178 0.0085 0.0002 0.0081

0.0146 0.0055 — 0.0047 —

0.0116 0.0029 — 0.0018 —

0.0087 0.0007 — — —

0.0060 — — — —

0.0036 — — — —

—

0.0231 0.0154

0.0069 0.0172 0.0109 0.0047 0.0082

0.0032 0.0144 0.0082 0.0022 0.0050

0.0002 0.0118 0.0056

0.0192

0.0201 0.0138 0.0075 0.0117

0.0021

— 0.0092 0.0033 — 0.0000

— 0.0067 0.0012 — —

— 0.0045 — — —

142b 152a 170 290 C318

0.0213 0.0201 0.0100

0.0178 0.0166 0.0051

—

0.0143

0.0145 0.0132 0.0032 0.0101 0.0113

0.0113 0.0100 0.0005 0.0082 0.0085

0.0083 0.0068 — 0.0041 0.0048

0.0055 0.0038 — 0.0016 0.0033

0.0029 0.0011 — — 0.0011

— — — —

— — — — —

502 503 600 600a 718

0.0159 0.0094

0.0121 0.0053 0.0180

—

—

0.0086 0.0018 0.0150 0.0132 0.0755

0.0054 — 0.0122 0.0101 0.0720

0.0026 — 0.0094 0.0073 0.0680

— 0.0068 0.0047 0.0636

— — 0.0043 0.0024 0.0590

— — 0.0020 0.0005 0.0540

— — 0.0001 — 0.0488

0.0096 0.0055 0.0136

0.0044 0.0013 0.0102

0.0005

—

—

—

—

—

0.0100 0.0171

0.0070

0.0041

0.0014

Temperature, °C 11 12 13 22 23 32 113 114 115 134a

744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data. Values and equations were given by Srinivasan, K., Can. J. Chem. Eng. (27 liquids), 68 (1990): 493; Lielmezs, J. and T. A. Herrick, Chem. Eng. J. (34 liquids), 32 (1986): 165–169. Somayajulu, G. R., Int. J. Thermophys. (64 liquids); 9, 4 (1988): 559–566; Ibrahim, N. and S. Murad, Chem. Eng. Commun. (29 polar liquids), 79 (1979): 165–174; Yaws, C. L.; Morachevsky, A. G. and I. B. Sladkov, Physico-Chemical Properties of Molecular Inorganic Compounds (200 compounds), Khimiya, Leningrad, 1987, Jasper, J., J. Phys. Chem. Ref. Data (2200 compounds), 1, (1972): 841–1009; and Vargaftik, N. B., B. N. Volkov, et al., J. Phys. Chem. Ref. Data (water), 12, 3 (1983): 817–820. See also Escobedo, J. and Mansoori, G. R., AIChE J., 42(5), May 1996: 1425–1433.

TABLE 2-309

Surface Tension ␴ (dyn/cm) of Various Liquids

Compound Acetic acid Acetone Aniline

Benzene

Benzonitrile Bromobenzene n-Butane Carbon disulfide Carbon tetrachloride

Chlorobenzene

T, K



293 333 298 308 318 293 313 333 353 293 313 333 353 293 323 363 293 323 373 203 233 293 293 313 288 308 328 348 368 293 323 373

27.59 23.62 24.02 22.34 21.22 42.67 40.5 38.33 36.15 28.88 26.25 23.67 21.2 39.37 35.89 31.26 35.82 32.34 26.54 23.31 19.69 12.46 32.32 29.35 27.65 25.21 22.76 20.31 17.86 33.59 30.01 24.06

Compound p-Cresol Cyclohexane Cyclopentane Diethyl ether 2,3-Dimethylbutane Ethyl acetate

Ethyl benzoate Ethyl bromide Ethyl mercaptan Formamide n-Heptane

T, K



313 373 293 313 333 293 313 288 303 293 313 293 313 333 353 373 293 313 333 283 303 288 303 298 338 373 293 313 333 353

34.88 29.32 25.24 22.87 20.49 22.61 19.68 17.56 16.2 17.38 15.38 23.97 21.65 19.32 17 14.68 35.04 32.92 30.81 25.36 23.04 23.87 22.68 57.02 53.66 50.71 20.14 18.18 16.22 14.26

Compound

T, K



Isobutyric acid

293 313 333 363 293 323 373 423 473 293 313 333 313 333 373 293 313 333 363 293 313 333 353 373 293 313 333

25.04 23.2 21.36 18.6 24.62 20.05 12.9 6.3 0.87 22.56 20.96 19.41 39.27 37.13 32.96 23.71 22.15 20.6 18.27 29.98 26.83 24.68 22.53 20.38 37.21 34.6 31.98

Methyl formate

Methyl alcohol Phenol n-Propyl alcohol

n-Propyl benzene

Pyridine

Methyl formate values from D. B. Macleod, Trans. Faradaay Soc. 19:38, 1923. All others from J. J. Jasper, J. Phys. Chem. Ref. Data 1:841, 1972.

2-420

PHYSICAL AND CHEMICAL DATA TABLE 2-310

Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure* Temperature, °C

−50

−25

0

25

50

75

100

125

150

14

158

166

173

180

187

194

200

206

212

170 290 600 600a 718

272 — — — —

286 227 — — —

299 238 200 201 —

311 249 210 211 —

323 258 220 221 —

334 268 228 229 —

344 277 237 237 473

355 286 245 246 490

364 294 252 253 505

290 —

248 305 235

258 318 246

269 330 257

279 341 267

288 352 277

297 363 286

307 373 295

316 384 303

R. no.

744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data. Values for the velocity of sound for all compounds listed in Table 2-184 are given in their respective tables earlier in this section.

TABLE 2-311

Velocity of Sound (m/s) in Saturated Liquid Refrigerants* Temperature, °C

R. no.

−50

−25

0

25

50

75

100

125

150

14

182

—

—

—

—

—

—

—

—

290 600 600a 718

1210 1290 1205 —

982 1163 1078 —

884 1031 947 1402

719 896 812 1495

551 759 661 1542

367 609 528 1554

— 477 378 1543

— 325 208 1514

— 142 — 1468

751 644 1022

525 372 859

272

—

—

—

—

—

874 1184

694

524

335

744 1150 1270

*Dashes indicate inaccessible states; blanks indicate no available data. Values for the velocity of sound for all compounds listed in Table 2-184 are given in their respective tables earlier in this section.

TRANSPORT PROPERTIES INTRODUCTION Extensive tables of the viscosity and thermal conductivity of air and of water or steam for various pressures and temperatures are given with the thermodynamic-property tables. The thermal conductivity and the viscosity for the saturated-liquid state are also tabulated for many fluids along with the thermodynamic-property tables earlier in this section. UNITS CONVERSIONS For this subsection the following units conversions are applicable: Diffusivity: to convert square centimeters per second to square feet per hour, multiply by 3.8750; to convert square meters per second to square feet per hour, multiply by 38,750. Pressure: to convert bars to pounds-force per square inch, multiply by 14.504. Temperature: °F = 9⁄5 °C + 32; °R = 9⁄5 K. Thermal conductivity: to convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779; and to convert British thermal unit–feet per hour–square foot–degree Fahrenheit to watts per meter-kelvin, multiply by 1.7307. Viscosity: to convert pascal-seconds to centipoises, multiply by 1000. ADDITIONAL REFERENCES An extensive coverage of the general pressure and temperature variation of thermal conductivity is given in the monograph by Vargaftik, Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978, now published in English translation by CRC Press, Miami, FL. For a similar work on viscosity, see Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Tables and

polynomial fits for refrigerants in both the gaseous and the liquid state are contained in ASHRAE Handbook—Fundamentals, SI ed., ASHRAE, Atlanta, 2005. Other sources for viscosity include Fischer & Porter Co. catalog 10-A-94, “Fluid densities and viscosities,” 1953 (200 industrial fluids in 48 pp.) and van Velzen, D., R. L. Cardozo et al., EURATOM Ispra, Italy rept. 4735 e, 1972 (160 pp.). Liquid viscosity, 314 cpds, is summarized in I&EC Fundtls., 11 (1972): 20–26. Five hundred forty-nine binary and ternary systems are discussed in Skubla, P., Coll. Czech. Chem. Commun., 46 (1981): 303–339. See also Duhne, C. R., Chem. Eng. (NY), 86, 15 (July 16, 1979): 83–91 (equations and 326 liquids); and Rao, K. V. K., Chem. Eng. (NY), 90, 11 (May 30, 1983): 90–91 (nomograph, 87 liquids). For rheology, non-Newtonian behavior, and the like, see, for instance, Barnes, H., The Chem. Engr. (UK), (June 24, 1993): 17–23; Hyman, W. A., I&EC Fundtls., 16 (1976): 215–218; and Ferguson, J. and Z. Kemblowski, Applied Fluid Rheology, Elsevier, 1991 (325 pp.). Other sources for thermal conductivity include Ho, C. Y., R. W. Powell et al., J. Phys. Chem. Ref. Data, 1 (1972) and 3, suppl. 1 (1974); Childs, Ericks et al., N.B.S. Monogr. 131, 1973; Jamieson, D. T., J. B. Irving et al., Liquid Thermal Conductivity, H.M.S.O., Edinburgh, Scotland, 1975 (220 pp.). Other references include Poling, Prausnitz, and O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2000; Vargaftik, Vinogradov, and Yargin, Handbook of Physical Properties of Liquids and Gases, Begell House, New York, 1996; Yaws, Chemical Properties Handbook: Physical, Thermodynamics, Environmental Transport, Safety & Health Related Properties for Organic & Inorganic Chemicals, McGraw-Hill, New York, 1998; and Riazi, Characterization and Properties of Petroleum Fractions, ASTM, West Conshohocken, Pa., 2005. Free web resources include the NIST Webbook at http://webbook.nist.gov and the KDB (Korea thermophysical properties) database at http://www.cheric.org/research/kdb/.

TABLE 2-312

Vapor Viscosity of Inorganic and Organic Substances (Pa
s)

2-421

Cmpd. no.

Name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O

CAS no.

Mol. wt.

C1

C2

C3

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138

1.222E-07 1.423E-07 2.7449E-08 1.20E-07 3.1005E-08 5.1964E-08 1.2025E-06 6.523E-07 1.7154E-07 4.302E-08 1.425E-06 4.1855E-08 1.7531E-07 9.2121E-07 2.5082E-08 3.134E-08 1.1184E-07 7.4266E-08 4.2137E-08 3.779E-07 6.9022E-08 1.56E-07 4.0138E-08 1.3874E-06 7.3534E-08 2.232E-07 5.29E-07 7.8796E-08 6.0259E-07 2.696E-07 3.4387E-08 7.5626E-08 7.0728E-08 1.4031E-06 1.2114E-07 6.9744E-07 4.2898E-08 1.05E-06 1.006E-07 3.4205E-07 5.4539E-08 3.1378E-08 2.7856E-06 8.0079E-08 2.128E-06 8.0223E-08 2.148E-06 5.8204E-08 1.1127E-06 3.137E-06 2.1709E-06 2.60E-07 1.065E-07 3.12E-07 1.696E-07 8.60E-08 3.638E-07 3.8802E-07 1.4427E-07

0.787 0.7574 1.0123 0.7915 0.9762 0.8857 0.4952 0.579 0.7418 0.9114 0.5039 0.9806 0.72 0.60529 0.96663 0.9676 0.8002 0.8289 0.92271 0.6005 0.84014 0.7181 0.90735 0.4434 0.93798 0.7146 0.632 0.90476 0.5309 0.6715 0.94604 0.83521 0.84383 0.4611 0.76972 0.5462 0.91349 0.4867 0.77881 0.59764 0.88896 0.96513 0.377 0.8178 0.4273 0.80561 0.46 0.9262 0.5338 0.3742 0.45853 0.7423 0.7942 0.6711 0.7693 0.8706 0.6417 0.6367 0.7438

77 272.14 7.4948 105.3 23.139 38.805 291.4 410.8 138.4 54.3 108.3 30.8 176.17 83.24

C4

7.9 152.43 91.197 45.387 409 74.746 180 34.714 678.22 184.9 226 199.64 134.7 71.798 64.391 537 92.661 305.25 358.7 95.108 234.21 43.687 663.14 65.855 886 75.207 290 44.581 94.7 491.5 208 98.3 94.7 139 96.6 35.8 208.3 205.08 166.15

−41,400

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

150.15 353.33 289.81 200.15 178.45 229.32 192.40 185.45 286.15 189.63 80.00 195.41 235.65 83.78 403.00 278.68 442.29 395.45 260.40 321.35 257.85 458.15 243.95 342.20 265.85 429.24 154.55 179.47 136.95 164.25 134.86 220.00 196.15 183.85 158.45 87.80 134.26 167.62 199.65 185.30 157.46 133.02 147.43 176.75 267.95 161.25 194.67 161.11 68.15 250.33 89.56 200.00 227.95 134.80 209.63 230.00 150.35 155.97 285.39

4.171E-06 6.842E-06 8.315E-06 5.213E-06 4.329E-06 5.476E-06 6.468E-06 4.174E-06 7.680E-06 3.985E-06 5.508E-06 6.378E-06 5.122E-06 6.742E-06 8.274E-06 7.077E-06 1.089E-05 8.578E-06 6.079E-06 5.324E-06 5.680E-06 9.122E-06 5.151E-06 6.186E-06 1.383E-05 1.187E-05 5.195E-06 8.626E-06 3.340E-06 4.553E-06 3.559E-06 5.157E-06 4.580E-06 3.961E-06 3.772E-06 1.795E-06 3.770E-06 4.044E-06 4.216E-06 3.425E-06 3.833E-06 3.520E-06 3.329E-06 4.017E-06 6.220E-06 3.284E-06 9.749E-06 5.048E-06 4.434E-06 8.361E-06 5.132E-06 8.900E-06 5.611E-06 4.127E-06 7.091E-06 8.468E-06 3.805E-06 4.175E-06 6.113E-06

1000 1000 1000 1000 1000 1000 600 1000 1000 1000 2000 1000 1000 3273.1 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 600 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 800 1000 1000 1000 1500 800 1250 1000 1000 1000 1000 1000 1000 700 1000 1000 1000

2.605E-05 2.094E-05 2.966E-05 2.572E-05 2.571E-05 2.271E-05 1.923E-05 2.523E-05 2.532E-05 2.213E-05 6.227E-05 3.551E-05 2.155E-05 1.001E-11 1.992E-05 2.486E-05 2.441E-05 2.087E-05 2.363E-05 1.698E-05 2.129E-05 1.886E-05 2.045E-05 1.768E-05 2.967E-05 2.623E-05 3.396E-05 4.081E-05 1.966E-05 2.457E-05 2.369E-05 2.260E-05 2.259E-05 2.207E-05 2.259E-05 2.325E-05 2.360E-05 2.229E-05 1.993E-05 1.720E-05 2.427E-05 2.466E-05 1.893E-05 2.134E-05 2.208E-05 1.948E-05 5.203E-05 2.693E-05 4.654E-05 2.789E-05 4.267E-05 3.992E-05 2.348E-05 2.824E-05 3.143E-05 2.454E-05 2.534E-05 2.618E-05 2.108E-05

2-422

TABLE 2-312

Vapor Viscosity of Inorganic and Organic Substances (Pa
s) (Continued)

Cmpd. no.

Name

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide

Formula C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS

CAS no.

Mol. wt.

C1

C2

C3

95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5

108.138 108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133

8.7371E-08 1.4305E-07 3.3699E-07 4.0854E-08 1.0881E-06 6.77E-08 7.9581E-08 5.2312E-08 1.3326E-06 2.3619E-07 3.026E-07 1.7578E-06 3.915E-08 1.2638E-07 2.64E-08 7.1748E-08 5.5065E-08 6.1192E-08 3.272E-08 5.6914E-07 2.4999E-07 1.4125E-07 1.1379E-07 2.9444E-07 7.7147E-08 2.334E-07 1.603E-07 1.5913E-07 2.0135E-07 1.4321E-07 7.6787E-07 1.4906E-07 1.1989E-07 3.3628E-08 4.3184E-07 1.948E-06 6.5492E-08 2.7228E-06 4.3934E-07 7.7484E-07 4.138E-07 1.691E-07 9.2797E-08 4.4172E-08 3.9833E-08 1.9377E-06 2.757E-07 6.8567E-07 7.822E-07 8.4576E-07 9.9104E-07 3.2282E-08 0.00000268 3.5538E-06 5.0372E-07 5.2195E-08 4.7238E-08 5.2854E-07 8.6101E-08

0.80775 0.7451 0.60751 0.97182 0.48359 0.8367 0.8376 0.89422 0.4537 0.67465 0.64991 0.4265 0.91427 0.7248 0.9487 0.7982 0.8341 0.82546 0.9302 0.50744 0.6878 0.8097 0.8502 0.728 0.79906 0.714 0.763 0.7639 0.73421 0.7785 0.5741 0.7617 0.79108 0.9426 0.6035 0.41 0.86232 0.39531 0.64867 0.57978 0.5999 0.7114 0.7819 0.91098 0.91566 0.4093 0.6841 0.52542 0.4994 0.487 0.4723 0.97742 0.3975 0.3766 0.54462 0.85584 0.90849 0.6112 0.8345

98.538 159.8 221.17 6.6762 330.86 36.7 104.97 58.008 445 139 167.14 370.34 22.264 176.88 71 109.38 79.56 77.434 39.13 273.3 0.5962 83.243 93.816 154.74 80.765 260 205 193.14 111.98 98.159 276.16 105.9 84.37 39.587 247 495.8 59.455 445.07 169.64 198.7 269.5 124 93.399 492.69 133.2 278.82 371.6 398 436.89 534 1176.1 227.44 69.036 302.85 167.86

C4

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

304.19 307.93 177.14 245.25 182.48 279.69 296.60 242.00 169.67 179.28 138.13 145.59 189.64 267.15 243.51 304.55 280.05 206.89 247.56 229.15 60.00 210.15 282.85 370.10 175.30 248.39 256.15 326.14 176.19 237.49 178.01 200.00 172.71 301.15 223.35 156.85 169.20 154.56 215.00 136.95 357.05 187.65 204.81 159.95 226.10 240.91 180.96 145.19 392.70 402.94 396.58 188.44 131.65 212.72 160.00 274.18 122.93 174.88 291.67

6.688E-06 6.731E-06 3.480E-06 8.353E-06 4.797E-06 6.671E-06 6.917E-06 5.714E-06 3.778E-06 4.409E-06 3.369E-06 4.150E-06 4.238E-06 4.365E-06 3.755E-06 5.070E-06 4.715E-06 3.632E-06 4.761E-06 4.091E-06 4.137E-06 7.685E-06 1.038E-05 1.538E-05 3.278E-06 5.850E-06 6.127E-06 8.313E-06 5.487E-06 7.164E-06 5.895E-06 5.515E-06 4.742E-06 6.450E-06 5.364E-06 3.720E-06 4.046E-06 5.148E-06 8.001E-06 5.478E-06 8.016E-06 4.218E-06 4.089E-06 4.497E-06 5.701E-06 6.006E-06 5.563E-06 3.211E-06 7.936E-06 7.900E-06 7.957E-06 5.405E-06 3.688E-06 4.097E-06 3.300E-06 5.089E-06 3.739E-06 4.544E-06 6.231E-06

1000 1000 1000 600 1000 900 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 480 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

2.108E-05 2.120E-05 1.834E-05 2.024E-05 2.308E-05 1.928E-05 2.346E-05 2.381E-05 2.118E-05 2.191E-05 2.309E-05 2.441E-05 2.118E-05 1.605E-05 1.730E-05 1.605E-05 1.622E-05 1.701E-05 1.944E-05 1.488E-05 1.744E-05 3.502E-05 3.696E-05 3.895E-05 1.781E-05 2.569E-05 2.588E-05 2.611E-05 2.887E-05 2.824E-05 3.175E-05 2.599E-05 2.611E-05 2.176E-05 2.239E-05 2.212E-05 2.388E-05 2.891E-05 3.317E-05 1.172E-12 2.055E-05 2.049E-05 1.881E-05 2.388E-05 2.225E-05 2.194E-05 2.744E-05 2.021E-05 1.796E-05 1.749E-05 1.801E-05 2.762E-05 2.722E-05 2.202E-05 1.766E-05 1.804E-05 2.511E-05 2.766E-05 2.350E-05

2-423

119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181

Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne

C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10

120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2

194.184 88.105 170.207 101.190 170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144

4.8739E-08 2.7334E-07 2.8451E-08 1.29E-07 6.344E-08 2.9236E-07 2.5906E-07 1.0613E-07 3.214E-06 4.934E-07 4.2231E-07 6.3441E-08 9.2371E-08 1.6175E-07 4.107E-07 2.1696E-06 2.0789E-06 1.3744E-07 8.6706E-08 2.8132E-07 4.3403E-08 6.761E-07 2.5704E-08 7.9129E-08 1.3974E-07 1.0498E-07 8.5992E-08 5.53E-07 5.1539E-07 3.2769E-08 6.36E-07 2.1174E-07 8.8742E-07 1.2269E-07 4.758E-07 6.829E-08 2.8608E-07 6.432E-07 3.253E-07 3.1338E-07 8.4633E-08 6.672E-08 7.2834E-08 2.572E-07 9.1526E-08 8.9656E-08 8.8629E-08 7.7509E-08 4.697E-08 5.9501E-07 1.2463E-07 9.6724E-08 1.7514E-07 8.4032E-08 1.5773E-07 1.0652E-07 9.782E-08 9.8882E-08 8.006E-08 5.2127E-07 4.3636E-08 2.9986E-07 5.5562E-07

0.8749 0.7393 0.93622 0.744 0.8287 0.62458 0.67988 0.8066 0.3572 0.5924 0.58154 0.8369 0.7908 0.7163 0.57143 0.3812 0.4163 0.7557 0.83923 0.6792 0.94806 0.5804 0.94738 0.79565 0.74266 0.76988 0.8427 0.6061 0.5726 0.9729 0.6638 0.7087 0.5404 0.82167 0.6405 0.8774 0.6958 0.5854 0.7162 0.6238 0.79185 0.82837 0.81279 0.6502 0.78346 0.78236 0.78376 0.81089 0.8932 0.52758 0.7322 0.78044 0.70737 0.80073 0.7189 0.77022 0.7772 0.7755 0.81293 0.5444 0.90747 0.62647 0.5337

51.885 129.93 117.03 219.5 702.84 98.902 52.7 667 239.17 239.21 73.63 102.32 142.27 230.06 577.77 352.7 122.8 75.512 238.46

3,590

354.9 83.193 98.58 100.41 58.148 273.66 288.76 61.6 157.42 251.82 161.7 54.864 184.25 325.3 −9.6 692.2 94.487 85.752 89.874 248.6 100.28 100.14 100.18 69.927 57.6 274.02 395 97.798 157.14 96.779 163.3 105.85 99.53 99.825 65.274 237.01 42.32 178.17 244.38

107

6,000

413.80 284.95 300.03 210.15 263.57 309.58 90.35 200.00 189.60 192.15 178.20 238.45 258.15 175.15 161.84 134.71 169.41 284.29 260.15 329.00 160.65 193.55 235.00 180.00 140.00 204.15 125.26 199.25 145.65 167.55 53.48 357.88 129.95 131.35 181.15 275.60 281.45 187.55 20.00 295.13 229.80 182.57 265.83 239.15 230.00 234.15 238.15 154.12 229.92 192.22 291.31 217.15 177.83 269.25 228.55 223.00 217.35 217.50 133.39 170.05 192.62 141.25 183.65

8.433E-06 1.226E-05 5.933E-06 4.429E-06 3.511E-06 3.214E-06 2.643E-06 6.029E-06 4.632E-06 4.953E-06 3.673E-06 4.733E-06 5.344E-06 3.392E-06 3.103E-06 2.659E-06 5.714E-06 6.863E-06 7.150E-06 8.359E-06 5.356E-06 5.069E-06 4.532E-06 3.371E-06 3.219E-06 4.224E-06 3.441E-06 5.768E-06 2.994E-06 4.779E-06 4.148E-06 9.491E-06 4.192E-06 6.752E-06 7.025E-06 7.882E-06 8.751E-06 5.037E-06 3.531E-06 3.254E-06 4.444E-06 3.391E-06 5.088E-06 4.440E-06 4.516E-06 4.485E-06 4.550E-06 3.169E-06 4.832E-06 3.932E-06 3.274E-06 4.444E-06 3.631E-06 5.457E-06 4.567E-06 4.650E-06 4.397E-06 4.403E-06 2.871E-06 3.567E-06 4.235E-06 2.947E-06 3.851E-06

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 2000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

1.953E-05 3.995E-05 1.831E-05 1.970E-05 1.593E-05 1.284E-05 2.583E-05 2.651E-05 2.274E-05 2.384E-05 1.893E-05 1.915E-05 1.975E-05 1.989E-05 1.729E-05 1.914E-05 2.726E-05 2.264E-05 2.655E-05 2.477E-05 3.032E-05 2.750E-05 1.787E-05 1.781E-05 2.150E-05 1.946E-05 2.742E-05 2.857E-05 2.088E-05 2.718E-05 5.874E-05 2.446E-05 2.963E-05 3.579E-05 3.419E-05 2.776E-05 2.954E-05 2.768E-05 7.561E-05 1.377E-05 1.836E-05 1.878E-05 1.834E-05 1.838E-05 1.864E-05 1.812E-05 1.809E-05 1.962E-05 2.124E-05 1.787E-05 1.399E-05 1.933E-05 2.005E-05 1.934E-05 1.945E-05 1.970E-05 1.909E-05 1.907E-05 2.064E-05 1.811E-05 2.209E-05 3.758E-13 1.782E-05

2-424

TABLE 2-312

Vapor Viscosity of Inorganic and Organic Substances (Pa
s) (Continued)

Cmpd. no.

Name

182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240

Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether

Formula H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O

CAS no.

Mol. wt.

C1

C2

C3

C4

302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5

32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122

2.3489E-07 1.797E-07 9.17E-08 4.924E-07 1.278E-08 4.5101E-14 3.9314E-08 1.1202E-07 5.2542E-08 8.4213E-08 9.113E-08 5.2546E-07 3.0663E-07 8.0599E-08 1.3226E-06 1.163E-06 1.648E-06 5.6409E-07 7.4106E-08 4.0824E-07 2.4344E-08 1.869E-07 8.8783E-08 5.0602E-07 8.5423E-07 5.6844E-07 3.9342E-08 4.995E-08 4.0748E-08 3.733E-07 4.8806E-08 6.5281E-07 8.5736E-08 2.40E-07 2.00E-07 9.0798E-07 3.7026E-08 3.9771E-08 1.977E-07 2.6098E-07 2.6552E-08 8.6219E-08 6.9755E-06 1.5035E-07 9.4257E-08 3.1573E-07 1.925E-07 1.0826E-07 8.6077E-08 1.637E-07 4.889E-07 7.2131E-08 1.1164E-06 1.0546E-07 1.0871E-07 9.605E-07 9.0981E-07 3.5642E-07 4.4941E-08

0.7151 0.685 0.9273 0.6702 1.0631 3.0005 1.0134 0.7822 0.88063 0.82573 0.8222 0.59006 0.69655 0.8392 0.4885 0.4787 0.4444 0.5863 0.82436 0.5923 0.97376 0.7096 0.80279 0.55258 0.47389 0.553 0.91086 0.89479 0.92709 0.6177 0.92549 0.5294 0.80277 0.68 0.704 0.495 0.92849 0.92242 0.7453 0.68276 0.98316 0.83591 0.3154 0.7338 0.7845 0.66404 0.7091 0.77382 0.81669 0.76706 0.6096 0.80319 0.4537 0.77106 0.78135 0.4856 0.49288 0.6327 0.90199

205.05 −0.59

140

157.7 340 −521.83

76,111

100.3 102.08 93.57 105.67 205 77.332 504.3 316 510.66 231.9 83.086 208.22 −91.597 192 77.075 199.82 239.34 227.18 44.662 256.5 310.59 100.77 210 187 355.89 131.22 133.4 72.564 1034.5 108.5 90.183 173.59 109 93.349 71.294 107.97 342.23 99.437 374.74 93.745 70.639 381 260.08 232.2

18,720

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

274.69 13.95 206.45 200.00 300.00 285.50 250.00 227.15 177.95 407.95 288.15 90.69 240.00 301.15 250.00 170.45 196.32 179.69 260.75 159.53 150.00 450.15 155.95 135.58 139.39 160.15 157.48 175.30 183.45 187.35 139.05 146.58 299.15 280.15 269.15 130.73 146.62 115.00 182.55 160.00 186.48 167.23 174.15 150.00 189.15 256.15 127.93 180.15 171.64 150.18 224.95 240.00 119.55 176.00 150.00 298.97 132.81 185.65 133.97

7.460E-06 6.517E-07 1.285E-05 9.594E-06 2.576E-06 9.931E-06 1.058E-05 5.415E-06 5.037E-06 9.639E-06 7.242E-06 3.468E-06 7.523E-06 7.714E-06 6.505E-06 4.769E-06 4.781E-06 5.167E-06 5.515E-06 3.572E-06 2.621E-06 1.000E-05 3.423E-06 3.083E-06 3.263E-06 3.893E-06 3.947E-06 4.052E-06 5.112E-06 3.993E-06 4.698E-06 2.934E-06 6.232E-06 6.331E-06 6.062E-06 2.722E-06 3.800E-06 3.165E-06 5.574E-06 4.552E-06 4.534E-06 4.341E-06 5.117E-06 3.448E-06 3.901E-06 7.481E-06 3.242E-06 3.968E-06 4.065E-06 4.450E-06 5.265E-06 4.162E-06 2.366E-06 3.707E-06 3.707E-06 6.727E-06 3.423E-06 4.316E-06 3.725E-06

1673.15 3000 800 1000 425 472.68 480 1000 1000 1000 1000 1000 1000 1000 800 800 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 600 1000 1000 1000

4.225E-05 4.330E-05 4.512E-05 4.358E-05 4.421E-06 2.019E-05 2.050E-05 2.261E-05 2.304E-05 2.293E-05 2.440E-05 2.800E-05 3.128E-05 2.464E-05 2.125E-05 2.045E-05 2.350E-05 2.628E-05 2.034E-05 2.022E-05 2.191E-05 2.109E-05 2.111E-05 1.918E-05 1.820E-05 2.112E-05 2.125E-05 2.312E-05 2.463E-05 2.118E-05 2.917E-05 1.930E-05 1.994E-05 2.175E-05 2.181E-05 2.046E-05 2.259E-05 2.327E-05 3.009E-05 2.573E-05 2.364E-05 2.588E-05 3.029E-05 2.157E-05 1.951E-05 2.642E-05 2.327E-05 2.076E-05 2.265E-05 2.956E-05 2.456E-05 1.685E-05 1.865E-05 1.983E-05 2.242E-05 1.312E-05 2.174E-05 2.288E-05 2.284E-05

2-425

241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303

Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine

C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N

3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8

90.187 46.144 118.176 88.148 58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132 59.110

5.8223E-08 3.8926E-07 7.1455E-07 1.571E-07 7.646E-07 6.4318E-07 7.19E-07 2.4391E-07 6.5592E-07 8.2005E-07 4.070E-07 2.115E-06 1.467E-06 3.0465E-07 7.1902E-08 1.0344E-07 7.0165E-08 1.20E-07 7.0111E-08 6.6329E-08 3.8673E-08 6.1447E-07 3.2095E-07 1.0321E-07 3.1191E-08 7.9611E-08 1.752E-07 8.1701E-08 8.0901E-08 6.1515E-11 5.0324E-05 3.3253E-08 5.7084E-07 1.1453E-07 1.101E-06 1.196E-07 4.0828E-08 2.27E-07 6.3412E-08 9.4314E-08 1.8903E-07 1.1749E-07 2.463E-07 1.164E-07 1.6378E-06 8.8646E-08 2.7467E-08 4.1022E-08 5.765E-07 4.3478E-07 1.0094E-07 8.536E-08 4.3511E-08 6.0758E-07 4.9054E-08 7.942E-07 1.2003E-06 5.4749E-07 1.7526E-07 1.61E-07 1.0111E-07 2.1372E-07 1.62E-07

0.88057 0.63159 0.49832 0.733 0.5476 0.5389 0.6659 0.702 0.6081 0.61423 0.6485 0.4642 0.5123 0.62218 0.8013 0.77301 0.8062 0.74 0.80701 0.82027 0.91142 0.50705 0.61839 0.7589 0.92925 0.7948 0.6941 0.79241 0.79062 1.8808 0.077611 0.9351 0.52446 0.7968 0.5634 0.84797 0.8766 0.6767 0.84758 0.7932 0.7031 0.7649 0.6653 0.7615 0.44337 0.81492 0.97555 0.90585 0.53498 0.5272 0.799 0.80872 0.908 0.53845 0.90125 0.5491 0.494 0.53893 0.72691 0.7457 0.7821 0.6894 0.7285

48.298 169.45 303.31 111.578 284 400.16 5.3 280 54.714 114.58 367.5 305.7 125.4 705.34 92.051 220.47 100.36 180 89.582 76.204 50.646 287.19 709.09 121.26 55.092 106.6 206.8 97.709 99.338 3604.6 32.426 271.76 126.34 96.3 212.68 191.74 41.718 98.279 175.9 103.78 208.7 107.94 636.11 85.198 235.2 238.27 103.1 88.273 102.73 173.45 415.8 479.78 283.52 119.93 159.3 89.5 178.57 117

−26,218

160.17 116.34 249.95 164.55 278.65 353.43 30.00 183.63 63.15 66.46 244.60 182.30 110.00 305.04 255.15 219.66 285.65 268.15 238.15 191.91 253.05 223.15 301.31 246.00 216.38 289.65 257.65 241.55 252.85 255.55 171.45 223.95 193.55 462.65 54.35 80.15 283.07 182.00 143.42 239.15 410.95 200.00 196.29 234.18 108.02 160.75 197.45 167.45 163.83 372.38 314.06 243.15 404.15 136.87 85.47 200.00 187.35 199.00 170.00 252.45 180.26 178.15 188.36

3.908E-06 3.196E-06 5.057E-06 3.944E-06 8.264E-06 7.125E-06 5.884E-06 3.752E-06 4.372E-06 3.964E-06 5.756E-06 8.854E-06 7.618E-06 3.231E-06 4.483E-06 3.335E-06 4.957E-06 4.499E-06 4.219E-06 3.542E-06 4.995E-06 4.170E-06 3.266E-06 4.510E-06 3.677E-06 5.267E-06 4.583E-06 4.498E-06 4.611E-06 2.075E-06 3.406E-06 4.579E-06 3.758E-06 1.196E-05 3.773E-06 4.922E-06 3.288E-06 3.740E-06 3.305E-06 5.150E-06 9.111E-06 4.452E-06 4.003E-06 5.079E-06 2.813E-06 3.638E-06 4.766E-06 4.242E-06 3.621E-06 6.010E-06 7.514E-06 5.324E-06 8.072E-06 3.788E-06 2.702E-06 4.732E-06 4.471E-06 3.914E-06 4.297E-06 6.105E-06 3.927E-06 3.802E-06 4.540E-06

1000 1000 1000 1000 1000 1000 3273.1 1000 1970 1000 1000 1000 1500 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1500 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

2.434E-05 5.762E-13 1.714E-05 2.235E-05 2.616E-05 1.900E-05 1.573E-04 2.432E-05 6.432E-05 1.975E-12 2.625E-05 4.000E-05 5.737E-05 1.314E-05 1.669E-05 1.767E-05 1.672E-05 1.688E-05 1.697E-05 1.781E-05 1.996E-05 1.585E-05 1.345E-05 1.741E-05 1.813E-05 1.743E-05 1.755E-05 1.774E-05 1.733E-05 2.700E-05 1.868E-05 2.057E-05 1.681E-05 2.498E-05 6.371E-05 4.184E-05 1.436E-05 2.042E-05 2.124E-05 2.058E-05 2.068E-05 2.098E-05 2.019E-05 2.023E-05 2.176E-05 2.275E-05 2.320E-05 2.141E-05 1.879E-05 1.340E-05 2.283E-05 2.093E-05 2.090E-05 2.135E-05 2.480E-05 2.490E-05 2.461E-05 1.765E-05 2.373E-05 2.397E-05 2.060E-05 2.122E-05 2.223E-05

2-426

TABLE 2-312

Vapor Viscosity of Inorganic and Organic Substances (Pa
s) (Concluded)

Cmpd. no.

Name

304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

Formula C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

CAS no.

Mol. wt.

C1

C2

C3

103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

3.0387E-07 7.3919E-07 6.0741E-07 3.5532E-08 7.9457E-08 4.543E-08 1.1085E-07 2.1671E-07 6.3863E-07 2.273E-07 6.863E-07 5.3986E-07 3.9067E-06 2.2452E-08 7.0859E-07 5.1567E-09 3.778E-07 5.0784E-07 8.5988E-08 8.1458E-07 1.03E-06 8.7268E-07 2.7081E-07 3.5585E-08 2.411E-07 1.2434E-06 7.8498E-07 6.8812E-07 1.107E-07 8.2418E-07 3.4066E-08 2.8471E-08 3.594E-08 5.9537E-08 1.388E-07 6.7484E-07 2.379E-07 3.6429E-08 1.7096E-08 6.8293E-07 8.3436E-07 9.3485E-07

0.61945 0.5423 0.5863 0.95654 0.84656 0.9173 0.8008 0.76757 0.5254 0.6845 0.6112 0.6349 0.3845 0.97631 0.51971 1.1561 0.6533 0.5614 0.82841 0.50257 0.5497 0.49397 0.6955 0.8987 0.6845 0.4832 0.49855 0.51063 0.746 0.4931 0.95252 0.96571 0.9052 0.81842 0.7599 0.5304 0.71517 0.95924 1.1146 0.52199 0.49713 0.47683

210.35 263.73 367.29 65.878 61 152.51 16.28 295.1 229.8 217 34.5 470.1 652.24 271.01 328.55 68.172 380.29 569.4 323.79 187.93 165.3 223 447.7 362.79 330.88 72.4 371.44 43.528 30.83 125 90.245 98 230.17 102.84 324.17 365.86 371.96

C4

19,000

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

173.55 87.89 180.25 142.61 159.95 213.15 388.85 250.00 242.54 460.65 197.67 205.15 297.93 700.15 329.35 279.01 164.65 237.38 176.99 373.96 234.94 178.18 236.50 267.76 158.45 156.08 247.79 229.33 165.78 387.91 398.40 354.00 247.57 288.45 180.35 173.15 119.36 178.35 273.16 225.30 247.98 286.41

3.351E-06 2.093E-06 4.203E-06 4.085E-06 4.132E-06 4.832E-06 9.439E-06 1.410E-05 5.158E-06 1.009E-05 8.280E-06 9.790E-06 1.355E-05 1.346E-05 4.837E-06 3.465E-06 4.006E-06 4.592E-06 4.520E-06 7.930E-06 6.049E-06 4.008E-06 6.756E-06 3.344E-06 3.210E-06 3.689E-06 4.975E-06 4.520E-06 3.488E-06 7.958E-06 9.208E-06 7.581E-06 3.506E-06 4.677E-06 4.660E-06 4.459E-06 3.907E-06 5.260E-06 8.882E-06 4.735E-06 5.225E-06 6.037E-06

1000 1000 1000 1000 1000 1000 1000 500 1000 1000 1000 5000 694.19 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1073.15 1000 1000 1000

1.812E-05 2.477E-05 2.550E-05 2.632E-05 2.583E-05 2.418E-05 2.429E-05 2.475E-05 1.858E-05 2.091E-05 3.844E-05 1.195E-04 2.883E-05 1.906E-05 1.554E-05 1.516E-05 2.710E-05 1.847E-05 2.461E-05 1.900E-05 2.926E-05 2.000E-05 2.782E-05 1.517E-05 2.230E-05 2.418E-05 1.803E-05 1.760E-05 1.786E-05 1.812E-05 2.352E-05 2.179E-05 1.660E-05 1.558E-05 2.407E-05 2.140E-05 1.263E-12 2.749E-05 4.082E-05 1.899E-05 1.894E-05 1.836E-05

The vapor viscosity is calculated by

C1T C2 µ 2 1  C3/T  C4/T where µ is the viscosity in Pas and T is the temperature in K. Viscosities are at either 1 atm or the vapor pressure, whichever is lower. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

TABLE 2-313

Viscosity of Inorganic and Organic Liquids (Pa⭈s)

2-427

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane m-Cresol

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6 108-39-4

44.053 59.067 60.052 102.089 58.079 41.052 26.037 56.063 72.063 53.063 28.960 17.031 108.138 39.948 121.137 78.112 110.177 122.121 103.121 182.218 108.138 136.191 124.203 154.208 159.808 157.008 108.965 94.939 54.090 54.090 58.122 90.121 90.121 74.122 74.122 56.106 56.106 56.106 116.158 134.218 90.187 90.187 54.090 72.106 88.105 69.105 44.010 76.141 28.010 153.823 88.004 70.906 112.557 64.514 119.378 50.488 78.541 78.541 108.138

−5.895 1.5525 −9.03 −14.164 −14.918 −10.906 6.224 −12.032 −28.12 2.019 −20.077 −6.743 −15.407 −8.8685 −12.632 7.5117 −8.4562 −12.947 −20.236 13.354 −14.152 −11.46 −11.459 −9.9265 16.775 −20.611 −10.015 −8.103 −10.143 17.844 −7.2471 −393.86 −390.03 0.87669 −16.323 −10.773 −10.346 −10.335 −17.488 −23.802 −10.807 −10.903 −3.4644 −10.057 −9.817 −10.136 18.775 −10.306 −4.9735 −8.0738 −9.9212 −9.5412 0.15772 −10.216 −14.109 −25.132 −13.994 −15.458 59.686

668.21 1376.4 1212.3 1350.3 1023.4 872.02 −151.8 867.34 2280.2 239.7 285.15 598.3 1518.7 204.29 2668.2 294.68 1024.4 2557.9 1737.4 −232.91 2652 1497 1334.4 1576.3 −314 1656.5 823.43 570.8 472.79 −310.2 534.82 19042 18609 1602.9 3141.7 591.61 522.3 521.39 1478.2 1887.2 966.74 932.82 334.5 903.73 1388 1006.4 −402.92 703.01 97.67 1121.1 300.5 456.62 540.5 702 1049.2 1381.9 949.4 1086 −3517.9

C3

C4

C5

−0.84323 −2.0126 −0.322 0.4492 0.5961 −2.6554 0.19534 2.3956 −1.8975 1.784 −0.7341 0.60172 −0.38305

−6.2382E-22 −3.69E-27

10 10

−1.2937E-22

10

−2.794 −0.30635 1.3531 −3.2685

1.7488E+20

−8.052

−0.043397 0.00049694 −0.21119 −3.9763 1.4415 −0.11122 −0.32958 −0.028241 −4.5058 −0.57469 59.978 60.014 −2.1475

−4.6625E-27 −0.049479 −0.055844 3.3866E+22

10 1 1 −9.9231

−0.011847 −0.013184 0.91828 1.8479 −0.014851 0.023034 −1.0811 −0.13186 −0.238 −0.1337 −4.6854

−6.9171E-26

10

−4.4999E-27 −6.163E-17

10 6

9.0312E+12

−5

−1.1088 −0.4726 −1.6075 −0.072 0.5377 2.0811 0.50223 0.654 −9.838

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

150.15 353.33 289.81 210.00 190.00 229.32 193.15 185.45 286.15 220.00 59.15 195.41 235.65 83.78 403.00 278.68 258.27 395.52 270.00 321.35 257.85 275.65 243.95 342.20 265.85 242.43 154.55 179.47 136.95 250.00 134.86 220.00 196.15 190.00 158.45 87.80 134.26 167.62 250.00 200.00 157.46 133.02 147.43 176.75 267.95 161.25 216.58 161.58 68.15 250.00 89.56 172.12 250.00 150.00 209.63 175.43 150.35 250.00 273.15

3.446E-03 1.728E-03 1.265E-03 4.834E-03 1.655E-03 8.220E-04 1.958E-04 1.773E-03 1.359E-03 8.040E-04 3.430E-04 5.240E-04 3.429E-03 2.950E-04 2.451E-03 7.761E-04 2.047E-03 1.530E-03 1.977E-03 6.100E-03 2.092E-02 1.887E-03 2.513E-03 1.427E-03 1.353E-03 2.842E-03 5.260E-03 1.316E-03 1.081E-03 2.547E-04 2.243E-03 2.020E+02 4.410E+04 3.237E-01 3.327E+01 1.770E-02 1.483E-03 6.810E-04 1.496E-03 1.030E-02 8.717E-03 2.288E-02 1.369E-03 3.602E-03 2.561E-03 1.031E-02 2.488E-04 2.590E-03 2.689E-04 2.032E-03 1.408E-03 1.020E-03 1.422E-03 2.748E-03 1.970E-03 1.501E-03 5.728E-03 5.515E-04 8.670E-02

294.00 494.30 391.05 412.70 329.44 436.40 273.15 353.22 460.00 350.50 130.00 393.15 426.73 150.00 563.15 545.00 442.29 600.80 450.00 664.00 478.60 458.15 472.03 723.15 350.00 429.24 311.50 276.71 284.00 400.00 420.00 544.00 540.80 391.90 372.90 335.60 276.87 274.03 399.26 456.46 373.15 358.13 373.15 348.05 436.42 390.75 303.15 441.60 131.37 455.00 145.10 333.72 540.00 373.15 353.20 403.15 423.15 308.85 564.68

2.216E-04 2.895E-04 3.890E-04 2.783E-04 2.351E-04 1.350E-04 9.819E-05 2.181E-04 2.086E-04 2.215E-04 4.276E-05 4.858E-05 2.736E-04 3.823E-05 3.730E-04 7.106E-05 3.333E-04 1.680E-04 3.009E-04 2.660E-04 1.821E-04 2.121E-04 1.788E-04 1.076E-04 6.021E-04 3.310E-04 3.321E-04 3.732E-04 1.773E-04 4.880E-05 3.566E-05 3.440E-04 2.890E-04 3.877E-04 3.715E-04 1.220E-04 1.982E-04 2.022E-04 2.521E-04 2.359E-04 2.475E-04 2.851E-04 1.271E-04 2.660E-04 3.087E-04 2.344E-04 5.652E-05 1.640E-04 6.515E-05 2.030E-04 3.897E-04 2.820E-04 1.291E-04 1.567E-04 3.410E-04 5.951E-05 1.154E-04 2.767E-04 1.630E-04

2-428

TABLE 2-313

Viscosity of Inorganic and Organic Liquids (Pa⭈s) (Continued)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide

C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si C2H6S C2H6OS

95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6 75-18-3 67-68-5

108.138 108.138 120.192 52.035 56.106 84.159 100.159 98.143 82.144 70.133 68.117 42.080 116.224 156.265 142.282 172.265 158.281 140.266 174.347 138.250 4.032 187.861 187.861 173.835 130.228 147.002 147.002 147.002 98.959 98.959 84.933 112.986 112.986 105.136 73.137 74.122 90.187 66.050 66.050 52.023 101.190 102.175 114.185 90.121 104.148 54.090 45.084 86.175 112.213 112.213 112.213 94.199 46.068 73.094 100.202 194.184 60.170 62.134 78.133

C1

C2

−0.033937 390.77 −1.6355 1052.9 −24.988 1807.9 −12.086 994.23 −3.4968 397.94 −33.763 2497.2 280.87 −31869 −44.877 3227.7 −11.641 1154.3 −3.2612 614.16 −4.1508 599.77 −3.524 342.54 −11.338 1304.1 −10.115 1111.9 −9.6489 1181.1 −12.305 2324.1 −69.985 5818.8 −15.868 1434.8 −11.464 1510.1 −2.3633 791.93 0.000001348 −10.457 1101.1 −17.582 1635.4 −10.013 921.31 10.027 206 −1.9265 387.67 −30.6 2153.4 31.63 −1080 −8.991 870.2 15.312 −41.12 −13.071 940.03 −10.872 1033.1 −11.269 1195.3 −375.21 17177 −17.57 1385.7 10.197 −63.8 −5.135 667.5 10.501 −52.181 −10.072 710.48 −17.723 850.2 −1.7366 599.8 −11.5 993 −15.097 1426.9 −10.968 885.49 −10.631 1086.4 0.10842 300.2 −10.93 699.5 7.2565 221.4 −10.716 1140.5 −11.796 1463.5 −11.344 1168.9 −10.577 1172.6 −10.62 448.99 −20.425 1515.5 −12.08 1112.2 16.961 −423.16 −17.641 −37.347

1067.5 2835

C3 −1.4547 −1.3891 2.0556

C4 5.0187E+12 3.6844E+17

−1.1087 3.2236 −38.837 3,994,500 4.887 0.066511 −1.156 −1.0308 −1.1599 0.000092396 −0.015659 −0.24367 9.0522E+34 −0.055494 8.0715 0.68071 −0.012754 −1.2272

C5 −5 −7

−2.002

−15

−0.0031354 0.9932 −3.1607 −1.1335 2.9371 −6.114 −0.2805 −3.919 0.3733 −0.00067435 0.012736 66.66 0.85647 −3.226 −0.8553 −3.3459 −0.14677 1.0601 −1.4237 0.022 0.51512

1.5037E+14

−6

−3.6367

0.5

−1.1719E-18

7

1.362E+15

−6

−1.6831 −2.7946 −0.047736 0.04513 −0.14244 0.000083967 1.4444 0.09654 −3.8178 1.0317 3.7937

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

293.15 273.15 200.00 245.25 182.48 279.69 296.60 242.00 200.00 225.00 138.13 145.59 189.64 267.15 240.05 304.55 285.00 206.89 247.56 229.15 20.35 210.15 282.85 220.60 175.30 248.39 256.15 326.14 176.19 237.49 208.38 200.00 172.71 293.15 223.35 200.00 225.00 154.56 215.00 137.00 250.00 187.65 204.81 159.95 226.10 240.91 200.00 220.00 239.66 223.16 184.99 188.44 131.65 240.00 160.00 274.18

9.600E-03 9.770E-02 6.363E-03 3.250E-04 8.345E-04 1.260E-03 6.330E-02 8.960E-03 4.018E-03 1.122E-03 7.531E-03 9.601E-04 1.155E-02 2.381E-03 2.780E-03 6.798E-03 1.937E-02 4.975E-03 4.364E-03 3.786E-03 1.348E-06 5.331E-03 2.042E-03 2.920E-03 5.931E-03 2.540E-03 2.727E-03 8.543E-04 4.076E-03 1.839E-03 1.407E-03 3.312E-03 1.381E-02 8.130E-01 1.191E-03 7.359E-04 1.113E-03 1.229E-03 5.231E-04 1.830E-03 7.479E-04 2.259E-03 4.569E-03 4.375E-03 2.950E-03 3.796E-04 5.917E-04 1.103E-03 1.992E-03 5.310E-03 8.315E-03 6.093E-03 7.398E-04 2.041E-03 9.669E-03 6.030E-02

558.04 563.72 400.00 320.12 367.94 443.04 520.08 428.58 373.15 325.00 405.60 318.40 431.95 488.15 494.16 543.15 503.00 443.75 512.35 505.60 20.35 381.15 404.51 488.80 414.15 547.16 453.57 447.21 330.45 400.00 373.93 361.25 369.52 589.28 329.10 373.15 365.25 343.15 283.65 343.15 357.05 341.45 397.55 337.45 366.15 371.00 308.15 331.13 392.70 484.92 396.58 382.90 248.31 425.15 362.93 612.80

2.160E-04 1.940E-04 2.881E-04 1.260E-04 1.278E-04 2.070E-04 1.650E-04 4.402E-04 2.877E-04 3.167E-04 1.416E-04 1.080E-04 2.440E-04 3.583E-04 1.550E-04 2.304E-04 2.727E-04 2.064E-04 1.848E-04 2.167E-04 1.348E-06 5.071E-04 5.120E-04 2.950E-04 1.989E-04 2.340E-04 3.761E-04 3.039E-04 3.407E-04 2.557E-04 2.374E-04 3.301E-04 3.495E-04 1.090E-04 2.260E-04 1.141E-04 2.354E-04 1.026E-04 2.257E-04 6.050E-05 2.193E-04 2.110E-04 2.194E-04 2.378E-04 4.695E-04 1.186E-04 1.734E-04 2.509E-04 3.045E-04 1.540E-04 2.956E-04 2.336E-04 1.490E-04 2.981E-04 2.147E-04 2.730E-04

225.00 6.696E-04 291.67 2.253E-03

310.48 464.00

2.528E-04 3.547E-04

2-429

119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181

Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan 1-Hexyne 2-Hexyne

C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S C6H10 C6H10

120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9 693-02-7 764-35-2

194.184 88.105 170.207 101.190 170.335 282.547 30.069 46.068 88.105 45.084 106.165 150.175 116.158 116.158 112.213 98.186 28.053 60.098 62.068 43.068 44.053 74.079 144.211 130.228 88.148 100.159 62.134 102.132 88.148 163.506 37.997 96.102 48.060 34.033 30.026 45.041 46.026 68.074 4.003 240.468 114.185 100.202 130.185 116.201 116.201 114.185 114.185 98.186 132.267 96.170 226.441 100.159 86.175 116.158 102.175 102.175 100.159 100.159 84.159 82.144 118.240 82.144 82.144

−11.488 −46.166 −12.373 −15.404 −7.8244 −18.315 −7.0046 7.875 14.354 19.822 −13.563 −40.706 −12.24 −15.485 −22.11 −6.894 1.8878 −53.908 −20.515 −11.012 −8.521 −9.8417 −13.037 −11.311 −11.331 −11.452 −9.7574 −8.9215 0.7109 7.8744 8.18 −10.064 −10.758 −10.501 −11.24 40.153 −48.529 −10.923 −9.6312 −19.991 −10.443 −9.4622 −40.543 −66.654 11.225 −9.3874 −13.929 −10.819 −11.812 −2.7947 −20.182 −10.745 −6.3276 −46.402 −39.324 −82.705 −11.445 −13.684 −10.36 −4.2684 −10.073 −4.7263 −3.7464

1922.6 3086.2 2017.5 1390 1191.9 2283.5 276.38 781.98 −154.6 −0.12598 1208.6 3035 1836.4 1325.6 1673 818.6 78.865 4030.8 2468.5 967.4 634.2 876.4 2346 1337.2 908.46 1172.7 729.43 950.8 386.51 −106.34 −75.6 1058.7 558.81 427.78 751.69 −912.39 3394.7 894.63 −3.841 2245.1 1063.2 877.07 3328.3 5325.8 25.319 1204.9 1321.9 841.33 1291.9 563.86 2203.5 1021.4 640 3448.6 3841 7404.9 1187.2 1283.4 775.85 647.6 1123.3 594.43 624.2

5.104 0.5564 −0.49963 0.95485 −0.6087 −3.0418 −3.7887 −4.9793 0.377 4.2655 0.021868 0.6432 1.641 −0.5941 −2.1554 5.9704 1.2435

3.9572E+23 −3.1108E-18

2.4998E+12

−0.3314 −0.1708 −0.02982 0.00042478 −0.00010095 −0.14912 −0.32687 −1.7754 −2.6884 −3.5148 −0.17162 −0.016459 0.0086309 −0.024579 −7.5664 5.3903 −0.00068418 −1.458 1.1982 −0.031488 −0.23445 4.1804 7.66 −3.2694 −0.32618 0.40382 0.076469 −1.1636 1.2289 −0.000055427 −0.694 5.0849 3.6933 6.4721 0.0029076 0.33755 −0.082348 −1.0087 −0.16515 −0.86247 −1.084

4.2757E+13

1.6873E+24 −1.065E-08 1.4022E+22 −2.2512E-28 1.0082E+11

5.6884E+21 −2.1166E-30 1.5016

413.80 284.95 293.15 260.00 −10 262.15 309.58 7 90.35 200.00 220.00 192.15 178.20 250.00 258.15 250.00 200.00 253.15 104.00 284.29 −5 260.15 250.00 160.65 245.00 235.00 180.00 140.00 204.15 125.26 250.00 200.00 −6 167.55 53.48 232.15 129.95 131.35 181.15 −10 273.15 281.45 200.00 10 2.20 295.13 229.80 −10 180.15 265.83 9.9041 239.15 −4.3444 230.00 234.15 250.00 154.12 229.92 192.22 291.31 217.15 −10 174.65 269.25 10.485 250.00 0.41014 223.00 217.35 217.50 133.39 170.05 192.62 141.25 183.65

1.068E-03 1.525E-03 4.120E-03 9.454E-04 3.020E-03 4.243E-03 1.247E-03 1.315E-02 1.132E-03 1.727E-03 8.012E-03 6.643E-03 6.705E-03 1.319E-03 6.406E-03 9.605E-04 6.334E-04 2.487E-03 1.340E-01 7.909E-04 1.918E-03 7.435E-04 4.717E-02 1.765E-02 7.908E-03 3.319E-03 9.520E-03 9.848E-04 1.156E-03 1.010E-02 7.317E-04 1.599E-03 1.448E-03 7.450E-04 7.331E-04 7.408E-03 2.319E-03 1.575E-03 3.628E-06 3.814E-03 2.510E-03 4.420E-03 9.242E-03 8.805E-02 4.036E-01 2.427E-03 1.642E-03 4.700E-03 3.097E-03 2.528E-03 3.536E-03 2.378E-03 2.400E-03 5.854E-03 2.822E-02 4.919E-01 2.561E-03 2.563E-03 7.108E-03 3.550E-03 6.035E-03 8.332E-03 2.483E-03

561.15 374.65 613.44 382.35 526.40 616.93 300.00 440.00 473.15 289.73 413.10 486.55 466.95 394.65 404.94 378.15 250.00 483.15 576.00 329.00 283.85 345.00 510.10 417.15 326.15 386.55 308.15 372.25 337.01 447.96 140.00 453.15 235.45 194.82 254.05 493.00 373.71 304.50 5.10 575.30 425.95 432.16 496.15 448.60 432.90 421.15 424.18 429.92 450.09 447.20 564.15 401.45 406.08 478.85 429.90 412.40 400.70 396.65 336.63 432.00 425.81 412.00 435.00

3.154E-04 4.610E-04 1.130E-04 2.118E-04 1.680E-04 2.078E-04 3.587E-05 1.416E-04 9.061E-05 2.236E-04 2.326E-04 3.109E-04 2.822E-04 2.533E-04 2.956E-04 2.599E-04 6.143E-05 1.723E-04 2.520E-04 3.123E-04 2.863E-04 2.486E-04 2.165E-04 2.522E-04 1.949E-04 2.207E-04 2.626E-04 2.480E-04 2.086E-04 1.550E-04 5.954E-05 1.542E-04 2.087E-04 2.587E-04 2.210E-04 1.821E-04 5.444E-04 3.392E-04 2.532E-06 2.088E-04 2.924E-04 1.430E-04 3.754E-04 3.190E-04 2.723E-04 2.040E-04 2.318E-04 1.420E-04 2.087E-04 1.777E-04 2.054E-04 2.744E-04 1.340E-04 4.019E-04 3.343E-04 3.274E-04 2.108E-04 2.185E-04 1.966E-04 1.377E-04 2.172E-04 2.083E-04 1.368E-04

2-430

TABLE 2-313

Viscosity of Inorganic and Organic Liquids (Pa⭈s) (Continued)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

C3

C4

182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240

Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether

H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10 C4H10O C4H8 C4H8O2 C4H10O

302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82-2 79-41-4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5 75-65-0 115-11-7 554-12-1 557-17-5

32.045 2.016 80.912 36.461 27.025 20.006 34.081 88.105 59.110 104.061 86.089 16.042 32.042 73.094 74.079 40.064 86.089 31.057 136.148 68.117 72.149 102.132 88.148 70.133 70.133 66.101 88.148 104.214 68.117 102.132 80.589 98.186 114.185 114.185 114.185 84.159 82.144 82.144 115.034 60.095 72.106 76.161 60.052 88.148 100.159 57.051 74.122 86.132 90.187 48.107 100.116 158.238 86.175 102.175 58.122 74.122 56.106 88.105 74.122

−75.781 −11.661 −11.633 −116.34 −21.927 353.99 −10.905 −11.497 −31.157 −19.834 −14.527 −6.1572 −25.317 −4.648 13.557 −2.8737 10.848 −17.044 −21.971 −10.481 −12.596 −1.035 −25.882 −10.755 −8.4453 −3.6585 −11.278 −10.97 −1.8842 −12.206 −12.002 −11.358 −6.1534 −6.6904 −6.6915 −1.8553 −4.8515 −6.7424 −10.517 −11.104 −1.0598 −10.842 −39.641 −11.27 −11.394

4175.4 24.7 316.38 3834.6 1266.5 13928 762.11 1365.7 1926 2784.5 1497.7 178.15 1789.2 1832 −187.3 301.35 75 1074 2267.4 648.37 889.11 1048.5 3359.4 705.48 639.21 441.1 949.12 1067.3 433.58 1141.7 1009.7 1213.1 3219 3150.5 3173.2 612.62 679.07 788.86 745.32 627.18 520.68 863.65 2113.3 888.42 1168.7

9.6508 −0.261 0.56191 16.864 1.5927 −41.717 −0.11863 0.036966 2.925 1.1161 0.51747 −0.95239 2.069 −1.2191 −3.6592 −1.2271 −3.297 0.84203 1.4173 −0.041947 0.20469 −1.5474 1.5787 −0.011113 −0.38409 −1.0547 −0.00012343 −0.017484 −1.3238 0.15014

−7.27E-09 −4.1E-16

−11.216 −11.272 −11.075 −10.628 −0.099 −12.579 −12.86 −11.391 −13.912 51.356 −10.385 −4.841 −10.705

737.75 1048.9 990.72 645 496 2224.2 946.91 1090.8 797.09 −1249.5 599.59 696.7 788.94

0.019308 0.00030493

−2.5875E-10 −2,962

−9.0606E-24

C5 3 10 4 −0.5

10

−1.4494 −1.392 −1.3046 −1.3774 −0.93238 −0.69862 0.036581 −1.4961 −0.00074603 4.308 0.024736 −0.007539

0.025885 −1.5939 0.26191 1.0752E-07 0.45308 −9.4593 −0.046088 −0.9194 −0.048383

3.694E+24

−9.8759

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

274.69 13.95 185.15 158.97 259.83 189.79 187.68 250.00 250.00 404.15 288.15 90.69 175.47 301.15 250.00 170.45 275.00 179.69 288.15 159.53 150.00 298.15 155.95 135.58 139.39 160.15 157.48 175.30 183.45 200.00 139.05 146.58 299.15 280.15 269.15 248.15 146.62 115.00 275.00 160.00 186.48 167.23 250.00 150.00 189.15

1.450E-03 2.546E-05 9.207E-04 1.000E-03 2.754E-04 1.550E-03 5.726E-04 2.938E-03 6.737E-04 1.939E-03 1.664E-03 2.063E-04 1.193E-02 3.995E-03 6.135E-04 6.045E-04 6.126E-04 1.236E-03 2.299E-03 1.321E-03 3.542E-03 1.774E-03 3.776E+01 3.675E-03 3.164E-03 1.915E-03 5.239E-03 6.930E-03 1.628E-03 3.339E-03 8.730E-03 4.590E-02 2.584E-02 3.729E-02 1.107E-01 9.288E-04 7.669E-03 4.086E-02 4.070E-04 9.133E-04 2.266E-03 3.409E-03 6.104E-04 5.390E-03 5.222E-03

522.52 33.00 206.45 318.15 298.85 368.92 350.00 450.00 453.15 580.00 434.15 188.00 337.85 478.15 425.00 373.15 400.00 273.15 472.65 314.00 310.00 450.15 404.15 304.30 311.70 390.15 343.31 396.58 364.00 375.90 353.60 457.68 548.80 491.20 493.60 353.15 433.60 420.80 314.70 280.50 535.50 339.80 304.90 331.70 389.15

2.190E-04 3.906E-06 8.206E-04 5.780E-05 1.821E-04 1.190E-04 8.089E-05 2.649E-04 1.214E-04 3.593E-04 3.582E-04 2.263E-05 3.442E-04 2.392E-04 1.198E-04 8.846E-05 1.636E-04 2.275E-04 2.149E-04 1.739E-04 1.928E-04 2.859E-04 3.051E-04 2.034E-04 1.841E-04 1.476E-04 2.006E-04 2.286E-04 2.035E-04 2.539E-04 1.070E-04 1.650E-04 8.026E-05 1.360E-04 2.356E-04 2.742E-04 1.301E-04 1.129E-04 2.891E-04 1.731E-04 7.577E-05 2.474E-04 3.134E-04 2.143E-04 2.170E-04

127.93 180.15 171.64 150.18 260.00 240.00 119.55 176.00 110.00 295.56 132.81 250.00 133.97

4.722E-03 4.305E-03 4.977E-03 2.023E-03 8.635E-04 3.646E-02 2.506E-02 5.554E-03 1.072E-02 5.440E-03 2.253E-03 8.002E-04 6.390E-03

303.92 367.55 553.10 279.11 400.00 518.15 333.41 372.00 310.95 451.21 266.25 352.60 312.20

1.703E-04 2.212E-04 9.292E-05 2.827E-04 2.229E-04 2.519E-04 2.038E-04 2.120E-04 1.588E-04 1.010E-04 2.270E-04 2.593E-04 2.127E-04

2-431

241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303

Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid Propionitrile Propyl acetate Propyl amine

C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2 C3H5N C5H10O2 C3H9N

3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4 107-12-0 109-60-4 107-10-8

90.187 46.144 118.176 88.148 58.079 128.171 20.180 75.067 28.013 71.002 61.040 44.013 30.006 268.521 142.239 128.255 158.238 144.255 144.255 126.239 160.320 124.223 254.494 128.212 114.229 144.211 130.228 130.228 128.212 128.212 112.213 146.294 110.197 90.035 31.999 47.998 212.415 86.132 72.149 102.132 88.148 88.148 86.132 86.132 70.133 104.214 104.214 68.117 68.117 178.229 94.111 119.121 148.116 40.064 44.096 60.095 60.095 122.207 58.079 74.079 55.079 102.132 59.110

−10.569

952.38

−0.063873

160.17

7.103E-03

368.69

2.333E-04

−11.632 −6.921 −10.34 −19.308 −17.945 −4.438 16.004

1251.6 790.773 519.61 1822.5 115.57 746.5 −181.61

0.071692 −0.654 −0.013899 1.218 1.428 −0.9385 −5.1551

249.95 164.55 151.15 353.43 25.09 200.00 63.15

1.972E-03 4.284E-03 9.377E-04 9.077E-04 1.602E-04 3.421E-03 2.633E-04

438.65 450.00 278.65 633.15 44.13 387.22 124.00

2.382E-04 1.052E-04 1.929E-04 1.892E-04 2.706E-05 3.027E-04 3.331E-05

−9.5556 19.329 −246.65 −16.403 −12.94 −68.54 −48.851 −39.863 −98.854 −21.921 −11.319 −2.3409 −22.688 −10.191 −7.556 −60.795 −19.907 16.792 −11.736 −20.804 −11.19 −11.498 −3.8552

981.64 −381.68 3150.3 2119.5 1257.6 3165.3 4095 4089 7183.8 1603.9 1428 715.52 2466 1072.4 881.09 4617.8 2791.7 1353.6 1415.2 1834.6 1057.4 1362.1 684.22

−0.19453 −4.8618 49.98 0.6881 0.37191 9.0919 5.294 3.7631 12.283 1.5971 −0.022545 −1.222 1.5703 −0.030553 −0.52502 7.028 0.94296 −4.6357 0.0003618 1.3403

244.60 210.00 109.50 305.04 255.15 218.15 285.55 280.00 238.15 191.91 253.05 223.15 301.31 246.00 211.15 289.65 280.00 241.55 252.85 255.55 171.45 223.95 193.55

1.344E-03 2.065E-04 3.858E-04 4.012E-03 2.606E-03 3.310E-03 1.030E-02 1.733E-02 2.310E-01 5.699E-03 3.026E-03 3.206E-03 3.926E-03 2.479E-03 2.660E-03 6.652E-03 1.569E-02 4.576E-01 2.161E-03 2.039E-03 6.590E-03 4.837E-03 3.614E-03

374.35 283.09 180.05 603.15 468.15 593.15 528.75 485.20 471.70 420.02 492.95 487.20 589.86 447.15 454.96 512.85 468.35 452.90 446.15 440.65 453.52 472.19 468.00

3.078E-04 7.730E-05 3.791E-05 2.068E-04 3.468E-04 5.000E-05 3.670E-04 2.852E-04 3.334E-04 2.127E-04 1.913E-04 2.172E-04 2.057E-04 3.425E-04 1.460E-04 3.576E-04 2.901E-04 1.899E-04 1.913E-04 2.075E-04 1.420E-04 2.000E-04 1.868E-04

−4.1476 −10.94 −19.299 −10.846 −53.509 −37.067 −36.561 −410.49 −11.055 −2.8695 −10.667 −6.9168 −11.677 −1.7273 −3.7241 −22.472 −43.335 −11.31 195.25 −6.3528 −17.156 23.467 −8.8918 −11.208 −9.9177 −23.931 −5.7136 17.797 −9.8074

94.04 415.96 2088.6 980.01 1836.6 2856.7 3542.2 18371 1005.3 596.32 659.56 818.76 1091.2 424.34 516.54 2566.9 3881.7 1280 −11072 240.85 646.25 116.07 2357.6 1079.8 839.53 1834.6 703.62 −252.43 1010.4

54.36 77.55 283.07 182.00 143.42 270.00 253.15 200.00 250.00 234.18 108.02 220.00 197.45 167.45 163.83 372.38 291.45 243.15 404.15 136.87 85.47 146.95 185.26 199.00 170.00 252.45 250.00 250.00 188.36

7.170E-04 3.790E-03 3.486E-03 4.129E-03 3.529E-03 3.773E-03 1.649E-02 3.823E+01 9.009E-04 1.024E-03 1.045E-02 1.643E-03 3.746E-03 2.323E-03 1.902E-03 1.920E-03 1.270E-02 2.370E-03 1.229E-03 5.772E-04 9.458E-03 2.069E+01 3.917E-01 3.080E-03 2.912E-03 2.275E-03 7.372E-04 1.002E-03 3.060E-03

150.00 208.80 543.84 376.15 465.15 458.65 410.90 392.20 375.46 375.14 303.22 385.15 399.79 378.00 415.20 610.03 555.40 522.40 557.65 298.15 360.00 370.35 355.30 508.80 321.15 414.32 370.50 473.15 321.00

6.990E-05 1.300E-04 2.091E-04 2.553E-04 4.796E-05 3.516E-04 3.844E-04 5.828E-04 2.354E-04 2.232E-04 2.051E-04 2.385E-04 2.463E-04 1.898E-04 9.980E-05 2.849E-04 1.940E-04 1.420E-04 1.986E-04 1.416E-04 4.275E-05 4.735E-04 4.892E-04 1.130E-04 2.562E-04 3.430E-04 2.171E-04 1.045E-04 2.908E-04

−2.14E-17

10

−0.22541

1

−0.000013519

2

4.6342E+22

−10

2.3041E+24 2.6663E+31

−10.09 −13.039

0.015575 −1.0071 −1.207 1.1091 −0.0054565 7.1409 3.7344 3.3364 61.985 0.0039301 −1.2025 −0.59628 0.10658 −1.342 −1.1167 1.5749 4.3983 −29.084 −0.58229 1.1101 −5.3372 −0.91376 −0.16735 1.9124 −0.78123 −4.291 −0.25697

−0.000019627 −8.0487E-37 −0.0095612

3.0548E+24

−7.3439E-11 2.8801E+09

2 12.84 1.2201

−10

4 −4.0267

2-432

TABLE 2-313

Viscosity of Inorganic and Organic Liquids (Pa⭈s) (Concluded)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

C3

304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H2O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

120.192 42.080 88.105 76.161 76.161 76.094 108.095 104.079 104.149 118.088 64.064 146.055 80.063 166.131 230.304 198.388 72.106 132.202 88.171 114.229 84.140 92.138 133.404 184.361 101.190 59.110 120.192 120.192 114.229 114.229 213.105 227.131 156.308 172.308 86.089 52.075 62.498 161.490 18.015 106.165 106.165 106.165

−18.282 −92.082 −73.735 −5.7244 −10.153 −804.54 −14.846

1549.7 1907.3 2668.2 638.2 840.71 30487 1829.4

1.0454 15.639 10.993 −0.76415 −0.093763 130.79 0.3729

−22.675 −13.422 46.223 3.8305 −88.793

1758 3431.8 −1378 41.21 6400.7

1.6701

−215.09 −14.493 −10.321 −11.167 −10.843 5.5351 −16.671 −226.08 0.388 −4.1103 −3.7067 10.142 −11.756 −9.6461 −12.928 −4.0309 −10.707 −11.504 52.176 −69.778 −22.407 −2.2333 0.26297 −10.37 −52.843 −11.91 −15.489 −7.381

11612 1710.8 900.92 1193.2 1165.2 632.38 1342.5 6805.7 736.5 1005.3 585.78 −130.41 1483.1 1281.2 1137.5 990.76 1818.5 3301 −4951.9 5905.2 1462.8 320.37 276.55 823.31 3703.6 1094.9 1393.5 911.7

C4

C5

−0.043098 −0.018364

1 1

−0.15449

1

−8.7475 −2.1342 10.709 31.849 0.4417 −0.069128 0.096226 −2.6576 0.8388 37.542 −1.7063 −1.0188 −1.0926 −3.2199 −0.040387 −0.29478 0.25725 −1.1771 −0.39102 −8.5676 8.0214 1.7006 −1.2915 −1.7282 5.866 0.13825 0.63711 −0.54152

−0.026882 3.0895E+28

1 −12

9.5986E+11

-5

−0.060853

1

1.0017E+19

−8

−3.6929E-28

10

570,980

−5.879E-29

−2

10

Tmin, K

Viscosity at Tmin

Tmax, K

Viscosity at Tmax

200.00 87.90 180.25 142.61 159.95 213.15 388.85

6.774E-03 1.550E-02 5.852E-03 6.477E-03 4.641E-03 9.500E+02 3.643E-04

432.39 333.15 353.97 325.71 340.87 500.80 454.00

2.357E-04 5.150E-05 2.810E-04 2.784E-04 2.656E-04 3.310E-04 1.965E-04

242.54 460.65 225.00 223.15 289.95

1.919E-03 2.550E-03 6.900E-04 5.388E-04 2.477E-03

418.31 644.80 400.00 318.69 318.15

2.268E-04 3.040E-04 6.557E-05 2.383E-04 9.456E-04

329.35 277.65 164.65 237.40 293.15 373.96 250.00 178.18 236.50 267.67 250.00 200.00 247.79 229.33 165.78 172.22 398.40 353.15 247.57 288.45 225.00 173.15 130.00 178.35 273.16 225.30 247.98 286.41

1.736E-02 3.350E-03 5.505E-03 1.300E-02 1.040E-03 1.999E-04 1.269E-03 1.569E-02 2.955E-03 3.450E-03 6.135E-04 5.156E-04 2.495E-03 3.477E-03 8.636E-03 1.305E-02 2.150E-03 1.167E-02 3.240E-03 2.089E-02 1.237E-03 8.764E-04 2.425E-03 3.170E-03 1.702E-03 1.834E-03 1.735E-03 7.021E-04

723.15 554.40 373.15 576.00 303.15 454.00 393.15 383.78 387.00 540.00 359.05 308.15 449.27 442.53 541.15 387.91 676.80 625.00 511.20 590.15 345.65 364.00 400.00 434.52 646.15 413.10 418.10 413.10

1.522E-04 1.810E-04 2.446E-04 2.100E-04 9.125E-04 8.859E-05 2.625E-04 2.428E-04 3.798E-04 1.740E-04 2.028E-04 1.612E-04 1.663E-04 1.942E-04 4.530E-05 2.049E-04 3.290E-04 1.601E-04 1.570E-04 1.856E-04 2.654E-04 1.273E-04 8.272E-05 2.090E-04 5.028E-05 2.189E-04 2.459E-04 2.169E-04

The liquid viscosity is calculated by

µ exp(C1  C2/T  C3 ln T  C4T c5) where µ is the viscosity in Pas and T is the temperature in K. Viscosities are at either 1 atm or the vapor pressure, whichever is higher. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source shold be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

TABLE 2-314

Vapor Thermal Conductivity of Inorganic and Organic Substances [W/(m⭈K)]

2-433

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane

C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6

44.05 59.07 60.05 102.09 58.08 41.05 26.04 56.06 72.06 53.06 28.96 17.03 108.14 39.95 121.14 78.11 110.18 122.12 103.12 182.22 108.14 136.19 124.20 154.21 159.81 157.01 108.97 94.94 54.09 54.09 58.12 90.12 90.12 74.12 74.12 56.11 56.11 56.11 116.16 134.22 90.19 90.19 54.09 72.11 88.11 69.11 44.01 76.14 28.01 153.82 88.00 70.91 112.56 64.51 119.38 50.49 78.54 78.54

3.7272E-07 0.00013195 0.000001691 0.00042004 −26.8 4.901E-08 0.000075782 0.024098 0.0009265 0.0013784 0.00031417 9.6608E-06 0.00059858 0.000633 0.025389 0.00001652 0.00047951 0.0001163 0.00015674 0.0001235 0.00023476 0.00096451 0.00015525 2.8646E-06 1.0404E-06 0.00027085 0.00019745 0.000038314 0.000088221 −20,890 0.051094 −295.44 −918.39 0.0011484 4.5894E-06 0.000096809 0.000067737 0.000078576 5.86E-09 0.1807 0.00097826 0.9719 0.000037269 1138.6 0.002751 5.2879E-07 3.69 0.0003467 0.00059882 0.00016599 0.000092004 0.0009993 0.0004783 −19.283 0.00043073 −22136 0.00004861 0.00009154

1.8129 0.97 1.6692 0.8066 0.9098 2.1091 1.0327 0.3285 0.7035 1.2093 0.7786 1.3799 0.7527 0.6221 0.28547 1.3117 0.7818 0.9705 0.95503 0.9495 0.8639 0.69225 0.9446 1.4098 1.4685 0.7932 0.8824 1.0484 1.0273 0.9593 0.45253 −0.21463 −0.21199 0.87647 1.4484 1.1153 1.0709 1.0565 2.376 0.0082225 0.78643 −0.111 1.1427 0.95596 0.2734 1.6715 −0.3838 0.7345 0.6863 0.94375 1.0164 0.5472 0.8994 0.20238 0.83878 0.7666 1.1407 1.0681

C3 728.3 658 439.37 −126500000 −36.227 1325.3 627.58 50594 −0.7116 354.04 70 1018.3 491 463.4 740 711.32 778.7 187.8 519.99 715.78 −391.35 278.33 647 287.38 75.316 −9.382E+10 5455.5 91602 334420 3253.7 781.82 −65.881 14.63 −401.32 −129.42 1531.5 1167.2 −43.844 7.5086E+09 −314.55 −381.9 964 479 57.13 1449.6 270.83 458.6 1845.5 −715050 1874.5 −4.8749E+10 593 746.6

C4

−95,400 142,620 31,432 577,830 112,460 2,121.7 241,830 1,228,600 189,410

193,840 278,930 156,820 165,880 99,063 1,979,800 −7.6032E+08 −2.8842E+09

129,390 105,920 69,280 1,691,500 67,115 3,163,200 79,421 95,650 1,860,000 501.92

163,000 −271,300,000

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

273.15 494.30 294.70 412.70 329.44 354.75 189.35 325.84 414.15 350.50 70.00 200.00 426.73 90.00 563.15 339.15 442.29 522.40 464.15 579.24 478.60 458.15 472.03 373.15 300.00 429.24 311.50 273.00 284.00 268.74 272.65 469.57 481.38 370.70 372.90 266.91 273.15 274.03 273.00 456.46 371.61 358.13 281.22 347.95 436.40 273.00 194.67 273.15 70.00 349.79 145.10 200.00 400.00 273.15 334.33 213.15 319.67 308.85

0.00973 0.02189 0.01049 0.01864 0.01363 0.01170 0.01011 0.01534 0.02027 0.01133 0.00603 0.01446 0.01809 0.00585 0.02317 0.01407 0.01861 0.02090 0.02180 0.02213 0.02167 0.01936 0.02071 0.01123 0.00452 0.01302 0.01018 0.00669 0.01172 0.01281 0.01357 0.02426 0.02110 0.02097 0.02435 0.01252 0.01105 0.01200 0.00783 0.02151 0.01832 0.01749 0.01268 0.01419 0.05192 0.00706 0.00887 0.00776 0.00576 0.00812 0.00505 0.00551 0.01579 0.00960 0.00854 0.00590 0.01225 0.01222

994 1000 686.88 1000 1000 994.75 1000 1000 1000 990.5 2000 900 1000 3273.1 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 500 1000 1000 1000 1000 1000 1000 1000 1000 712.94 1000 1000 1273.15 1257 800 1000 1000 1000 1000 1000 526.32 1000 1500 1000 1500 1000 1000 1000 1000 1000 1000 750 1000 1000

0.10124 0.06206 0.05236 0.06981 0.11362 0.10298 0.09545 0.08028 0.06867 0.11107 0.11675 0.11523 0.06796 0.09525 0.05618 0.09542 0.06427 0.05452 0.06713 0.04899 0.06636 0.06398 0.06171 0.06347 0.00956 0.04495 0.05321 0.04158 0.09071 0.16809 0.13799 0.10046 0.08332 0.06536 0.10161 0.12049 0.13926 0.13704 0.07634 0.07465 0.08610 0.08470 0.09644 0.11186 0.03792 0.07660 0.09025 0.03745 0.08724 0.04595 0.08108 0.03002 0.07935 0.07920 0.04920 0.05448 0.08065 0.08389

2-434

TABLE 2-314

Vapor Thermal Conductivity of Inorganic and Organic Substances [W/(m
K)] (Continued)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

m-Cresol o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate Dimethylsilane

C7H8O C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4 C2H8Si

108-39-4 95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3 1111-74-6

108.14 108.14 108.14 120.19 52.03 56.11 84.16 100.16 98.14 82.14 70.13 68.12 42.08 116.22 156.27 142.28 172.27 158.28 140.27 174.35 138.25 4.03 187.86 187.86 173.83 130.23 147.00 147.00 147.00 98.96 98.96 84.93 112.99 112.99 105.14 73.14 74.12 90.19 66.05 66.05 52.02 101.19 102.17 114.19 90.12 104.15 54.09 45.08 86.18 112.21 112.21 112.21 94.20 46.07 73.09 100.20 194.18 60.17

C1

C2

0.00019307 0.9248 0.00018648 0.9302 0.00019063 0.9282 1.6743E-07 1.8369 0.000026933 1.137 −449,910 0.27364 0.000000859 1.7709 0.0032207 0.5991 −1,095.5 −0.023408 0.0000901 1.0897 9.5461E-06 1.4641 0.0010949 0.71644 −91.383 0.89718 0.0000813 1.0674 −4.9825 0.04928 −668.4 0.9323 0.00017047 0.9313 −0.3072 0.489 0.000027232 1.257 0.00012058 1.0111 0.000016707 1.2128 0.00028527 0.9874 0.00021231 0.8052 0.00015878 0.8636 0.00021302 0.8719 0.0032694 0.58633 −1,067.8 0.754 −1,420 0.7614 −1,520.8 0.754 0.0001315 1.0113 0.00021054 0.9574 0.0014796 0.69531 0.000057603 1.1148 0.000062435 1.103 −11,633 0.4621 0.00001706 1.248 −0.0044894 0.6155 0.0018097 0.67406 0.000059249 1.0713 2.4194E-06 1.4456 0.000013015 1.1897 0.00051305 0.8076 0.00019879 0.9423 −8.5357 −0.0056423 0.00046265 0.81968 3.7962E-06 1.4462 0.00021761 0.9187 1.6085 −0.1103 0.000034741 1.1646 0.008856 0.4215 0.013298 0.3692 0.012144 0.3854 0.00022578 0.892 0.059975 0.2667 0.014449 0.3612 0.000022421 1.2137 0.00012822 0.9324 0.0011808 0.742

C3

C4

710 709.37 716.91 −449.46 112,760 28.119 −1.0001E+10 −9.8654E+12 243 608.69 509,290 498,780 −7.8355E+09 655 632.62 175.55 346,040 −283,310,000 697.6 −1107 −67,349,000 −4.071E+09 757.67 −67,500 −29,400,000 751.7 740 −206.08 153,850 −200.51 21,807 649.51 659.5 1,620 1259.9 300,890 −3.0361E+09 −4.5040E+09 −4.3328E+09 1,023.8 1,414 2,657.4 849.98 913.43 −3.7939E+09 −112.8 77,960 −3266.3 1179.7 174,850 101.84 45,974 360.19 306.8 1882.1 539.34 217 2160.3 −99.956 −50.645 0.1027 52.191 697 1,018.6 595.22 −146.91 752.5 1131

154,510 106,230 −65,622,000 104,530 132,070 2,989,300 130,820 764,580 852,540 803,590 1,098,800 728,130 131,830 6,400

Thermal cond. at Tmin, K Tmin

Tmax, K

Thermal cond. at Tmax

475.43 464.15 475.13 380.00 252.00 285.66 325.00 434.00 428.58 356.12 273.00 317.38 240.37 431.95 488.15 447.30 543.15 504.07 443.75 512.35 447.15 233.15 381.15 404.51 370.10 323.15 446.23 453.57 447.21 330.45 356.59 312.90 361.25 369.52 541.54 273.15 200.00 365.25 248.95 303.65 221.50 357.05 328.05 397.55 337.45 366.15 300.13 280.03 331.13 392.70 402.94 396.58 382.90 248.31 425.15 362.93 556.85 253.55

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1500 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 600 1000 1000 993.65 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1500 1000 1000 1000 1000

0.06716 0.06736 0.06762 0.08181 0.06749 0.14994 0.14198 0.09535 0.12704 0.10116 0.14429 0.10148 0.15854 0.07629 0.10382 0.10286 0.06034 0.09389 0.09175 0.07482 0.07667 0.44547 0.03351 0.03729 0.03356 0.07330 0.06430 0.06066 0.06417 0.07025 0.06498 0.04931 0.06881 0.06647 0.07463 0.09804 0.05181 0.08089 0.08447 0.05206 0.04826 0.08967 0.09444 0.13085 0.08099 0.08279 0.09199 0.12209 0.10506 0.09500 0.09196 0.09376 0.06310 0.19458 0.07539 0.09962 0.04587 0.09296

0.02316 0.02230 0.02319 0.01534 0.01302 0.01356 0.01380 0.02399 0.02291 0.01914 0.01061 0.01360 0.01061 0.02022 0.02381 0.02173 0.02508 0.02591 0.02149 0.02709 0.02092 0.11474 0.00940 0.01077 0.00687 0.01244 0.01561 0.01507 0.01564 0.01132 0.01177 0.00847 0.01220 0.01222 0.03044 0.01148 0.00764 0.01743 0.01016 0.00938 0.00803 0.01836 0.01598 0.02015 0.01554 0.01936 0.01288 0.01845 0.01581 0.01884 0.01948 0.01952 0.01613 0.01139 0.02001 0.01797 0.01981 0.01291

2-435

117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179

Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne Hexyl mercaptan

C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10 C6H14S

75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4 111-31-9

62.13 0.00023614 78.13 0.00064761 194.18 −25,190 88.11 6.4032E-07 170.21 0.00014629 101.19 0.0001123 170.33 0.000005719 282.55 −375.32 30.07 0.000073869 46.07 −0.010109 88.11 1.3575E-07 45.08 0.3935 106.17 0.000017537 150.17 0.00002012 116.16 0.00017727 116.16 829.29 112.21 0.0000748 98.19 0.0043244 28.05 8.6806E-06 60.10 0.1655 62.07 −8,145,800 43.07 0.00077079 44.05 −0.0003788 74.08 508 144.21 2.5804E-06 130.23 0.0052833 88.15 0.00021652 100.16 −152,400 62.13 0.0015251 102.13 1.0507E-07 88.15 5.8174E-08 163.51 2.9354E-06 38.00 0.00012144 96.10 0.000053432 48.06 0.0004104 34.03 0.003959 30.03 44.847 45.04 0.00025893 46.03 0.0003754 68.07 −644,950 4.00 0.00226 240.47 −114.41 114.19 1,556.7 100.20 −0.070028 130.19 0.00019376 116.20 -0.061993 116.20 0.00017569 114.19 1,348.6 114.19 2,049.3 98.19 0.00002133 132.27 0.0083145 96.17 0.000060732 226.44 0.000004438 100.16 −7,157,100 86.18 −650.5 116.16 0.00021014 102.17 −4,935,500 102.18 0.00018361 100.16 −1.2158 100.16 −0.33262 84.16 0.000064256 82.14 6.9682E-06 118.24 0.074318

0.9204 0.7716 0.3639 1.7194 0.9377 0.9958 1.4699 1.0708 1.1689 0.6475 1.9681 0.0131 1.3144 1.1513 0.9428 1.0156 1.1103 0.5429 1.4559 0.1798 −0.30502 0.7713 1.115 0.9023 1.4669 0.52982 0.94192 −0.049106 0.70243 1.9854 2.0116 1.4153 0.93831 1.1576 0.8333 0.4834 −0.7096 0.9083 0.8459 0.2862 0.7305 1.0566 1.0284 0.38068 0.92434 0.2792 0.97218 1.0313 1.0323 1.2885 0.51862 1.0586 1.4949 −0.05819 0.8053 0.91616 −0.1653 0.97199 0.026637 0.12054 1.1355 1.347 0.30035

638 1013.3 −6,869,000,000 745.89 183.2 579.4 −8,783,600,000 500.73 −7,332 1,380 560.65 −89.583 712.4 8,955,300,000 686 333.67 299.72 3,827.9 1,832,500,000 446.16 −5,641 2,170,000,000

82,563

98,000

−268,000 1,710,000 125,410

570,470 −29,403 1,600,000 −1.1842E+13 197,930

1,415.7 632.16 80,955,000 1,347.5

378,180

−372.68

57,690

−9.3122E+11 35,085

760.75 723 997.4 −3,493.5 5,353,200 723.6 674.4 −1.6794E+10 −1.7372E+13 −18.63 440 −2,211,400,000 17,049,000,000 −7,049.9 −2,400,500 739.56 −3,336 −1,642,000 686.56 14,832,000,000 22,983,000,000 487.8 2,253 532,590 −102.79 143,140 682 4,089,000,000 −4.5826E+13 −1,412,100,000 727.64 1,563,100,000 −1.5752E+13 677.05 −1,711.6 −13,176,000 −2,472.6 −5,493,400 445.15 64,810 −214.35 110,480 4,470.1 1,775,800

310.48 462.15 561.15 337.85 531.46 279.65 489.47 616.93 184.55 293.15 273.15 289.73 409.35 486.55 466.95 394.65 404.95 376.62 170.00 390.41 470.45 329.00 273.15 327.46 500.66 417.15 326.15 386.55 308.15 400.00 273.15 373.15 70.00 357.88 235.45 194.82 254.05 493.00 373.71 304.50 30.00 575.30 425.95 339.15 496.15 449.45 432.90 420.55 424.18 366.79 450.09 372.93 560.01 401.45 339.09 478.85 429.90 412.40 273.00 273.00 336.63 354.35 425.81

0.01520 0.02059 0.02060 0.01427 0.02188 0.01055 0.02354 0.02563 0.00886 0.01475 0.00847 0.01622 0.02007 0.01855 0.02306 0.01583 0.02180 0.01832 0.00879 0.02272 0.02513 0.01610 0.01004 0.01426 0.02353 0.01967 0.01717 0.01889 0.01487 0.01540 0.01133 0.01281 0.00654 0.01546 0.00955 0.00827 0.01256 0.02930 0.02008 0.01367 0.03124 0.02454 0.01967 0.01583 0.02413 0.02345 0.02484 0.01943 0.01951 0.01845 0.02289 0.01827 0.02568 0.01842 0.01704 0.02381 0.02220 0.02421 0.00775 0.00800 0.01644 0.01485 0.02151

1000 1000 1000 768.01 1000 1000 1000 1000 1000 1000 990.21 1000 1000 1000 1000 1000 1000 1000 590.92 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 550 573.15 700 600 1000 1000 994.05 1000 1000 1000 2000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

0.08319 0.06379 0.04529 0.05855 0.05449 0.08515 0.09301 0.06968 0.15807 0.13417 0.10682 0.10532 0.09859 0.05524 0.06973 0.10314 0.09505 0.09659 0.06613 0.08915 0.09896 0.09659 0.18063 0.11921 0.06492 0.07348 0.08882 0.12768 0.08195 0.09499 0.03690 0.02352 0.05675 0.03874 0.07531 0.05589 0.11532 0.07973 0.07733 0.13631 0.58820 0.07649 0.11110 0.11493 0.06605 0.10722 0.08596 0.11287 0.11145 0.10518 0.07899 0.08751 0.08055 0.11472 0.12003 0.06816 0.11104 0.09022 0.10523 0.10980 0.10850 0.08546 0.08167

2-436

TABLE 2-314

Vapor Thermal Conductivity of Inorganic and Organic Substances [W/(m
K)] (Continued)

Cmpd. no.

Name

180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237

1-Hexyne 2-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl Isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane 2-Methyl-2-propanol

Formula

CAS no.

C6H10 693-02-7 C6H10 764-35-2 H4N2 302-01-2 H2 1333-74-0 HBr 10035-10-6 HCl 7647-01-0 CHN 74-90-8 HF 7664-39-3 H2S 7783-06-4 C4H8O2 79-31-2 C3H9N 75-31-0 C3H4O4 141-82-2 C4H6O2 79-41-4 CH4 74-82-8 CH4O 67-56-1 C3H7NO 79-16-3 C3H6O2 79-20-9 C3H4 74-99-7 C4H6O2 96-33-3 CH5N 74-89-5 C8H8O2 93-58-3 C5H8 598-25-4 C5H12 78-78-4 C5H10O2 116-53-0 C5H12O 123-51-3 C5H10 563-46-2 C5H10 513-35-9 C5H6 78-80-8 C5H12O 628-28-4 C5H12S 628-29-5 C5H8 598-23-2 C5H10O2 623-42-7 CH5ClSi 993-00-0 C7H14 108-87-2 C7H14O 590-67-0 C7H14O 7443-70-1 C7H14O 7443-52-9 C6H12 96-37-7 C6H10 693-89-0 C6H10 1120-62-3 CH4Cl2Si 75-54-7 C3H8O 540-67-0 C4H8O 78-93-3 C3H8S 624-89-5 C2H4O2 107-31-3 C5H12O 625-44-5 C6H12O 108-10-1 C2H3NO 624-83-9 C4H10O 598-53-8 C5H10O 563-80-4 C4H10S 1551-21-9 CH4S 74-93-1 C5H8O2 80-62-6 C9H18O2 3004-93-1 C6H14 107-83-5 C6H14O 628-80-8 C4H10 75-28-5 C4H10O 75-65-0

Mol. wt.

C1

82.14 0.000058116 82.14 0.000011631 32.05 0.00043196 2.02 0.002653 80.91 0.00049725 36.46 0.001865 27.03 4.6496E-06 20.01 0.000034629 34.08 1.381E-07 88.11 0.000214 59.11 0.00028183 104.06 0.00033075 86.09 0.00019847 16.04 8.3983E-06 32.04 5.7992E-07 73.09 0.034177 74.08 −25,343 40.06 0.00026544 86.09 0.4734 31.06 −55.13 136.15 0.000023963 68.12 0.0002509 72.15 0.0008968 102.13 0.0001799 88.15 2,053.4 70.13 0.00019098 70.13 0.00021736 66.10 0.00015498 88.15 0.000023993 104.21 0.079414 68.12 0.000065855 102.13 1,333.1 80.59 0.00037057 98.19 0.0000719 114.19 0.00011359 114.19 0.069565 114.19 0.075448 84.16 0.0024385 82.14 0.0040082 82.14 0.0019845 115.03 0.00041077 60.10 0.00024036 72.11 −4,202,700 76.16 0.0034805 60.05 −800,040 88.15 0.00020053 100.16 −2,483,300 57.05 0.0026136 74.12 2.1191 86.13 −5,935,000 90.19 0.0071536 48.11 0.00002653 100.12 0.00072502 158.24 0.0001813 86.18 0.000061119 102.17 0.93312 58.12 0.089772 74.12 1.1776E-06

C2 1.0724 1.2753 0.86603 0.7452 0.63088 0.49755 1.3669 1.1224 1.8379 0.9248 0.92094 0.81895 0.9284 1.4268 1.7862 0.3312 −0.1934 0.8921 −0.1111 1.065 1.1308 0.899 0.7742 0.9457 0.90109 0.9341 0.9171 0.9364 1.1976 0.23442 1.072 0.9962 0.81367 1.1274 1.0311 0.1633 0.155 0.61774 0.54462 0.6393 0.75688 0.93177 −0.1524 0.61906 −0.2285 0.95381 −0.046517 0.62 −0.19015 −0.089497 0.53907 1.1631 0.7395 0.92912 1.0861 −0.1172 0.18501 1.6618

C3

C4

−77.165 −202.84 641.48 12 331.62 358 −210.76 18.744 −352.09 698 619.17 777.75 678.69 −49.654

123,900 122,990

2,070 11,164,000 222.19 533.57 −448,200,000 −67.272 253.4 456 704.6 8,755,500,000 84.07 112.3 15.366 58.59 2,671.9 −36.369 12,317,000,000 609.17 667 709.27 208.7 218.44 223.01 242.12 227.11 591.5 588.14 2,084,600,000 1,810.8 248,100,000 644.42 1,313,100,000 1,631.7 1,453.4 3,098,800,000 2,700.7 29.996 365.68 793.45 −59.592 1,154.3 639.23

1,195,600 −67,259,000,000 79,869 1,649,600

58,295 46,041

125,720 149,500 230,640 155,720 177,690 137,400 35,667 1,366,100 106,430

1,209,500 1,252,500 477,570 559,040 434,120 −1.4577E+13 166,290 −1.5034E+12 −1.5798E+13 126,720 3,575,500 −2.7994E+13 241,730 32,519 204,360 141,260 2,961,700 1,114,700

Thermal cond. at Tmin, K Tmin

Tmax, K

Thermal cond. at Tmax

344.48 357.67 386.65 22.00 206.45 190.00 273.15 350.00 212.80 427.85 304.92 580.00 434.15 111.63 273.00 478.15 330.09 249.94 353.35 266.82 472.65 314.00 273.15 450.15 404.15 304.30 311.71 305.40 273.15 396.58 302.15 375.90 281.85 374.08 441.15 438.15 440.15 344.96 348.64 338.05 314.70 273.00 352.79 339.80 300.00 331.70 389.65 312.00 303.92 367.55 171.64 273.15 373.45 518.15 333.41 372.00 261.43 333.82

1000 1000 1000 1600 600 700 673.15 450 600 1000 1000 1000 1000 600 684.37 1000 1000 1000 1000 650 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 766.87

0.09155 0.08466 0.10430 0.64299 0.01812 0.03213 0.04185 0.03160 0.03258 0.07497 0.10081 0.05327 0.07210 0.08425 0.06726 0.07895 0.11878 0.09675 0.06904 0.07917 0.05588 0.08902 0.11176 0.07253 0.11843 0.09771 0.09504 0.08664 0.08586 0.07960 0.10120 0.10543 0.06357 0.10399 0.08238 0.08888 0.08908 0.10227 0.09578 0.09888 0.04813 0.09447 0.11740 0.08415 0.13148 0.08863 0.12433 0.06864 0.09451 0.12847 0.07516 0.07704 0.07637 0.06195 0.10242 0.08117 0.11701 0.07325

0.01679 0.01506 0.02828 0.01718 0.00551 0.00880 0.00985 0.02356 0.00724 0.02206 0.01804 0.02590 0.02176 0.01263 0.01303 0.02498 0.01415 0.01154 0.01569 0.01259 0.01784 0.01326 0.01198 0.02266 0.02116 0.01348 0.01320 0.01304 0.01173 0.01966 0.01468 0.01495 0.01155 0.02056 0.02322 0.02415 0.02435 0.01592 0.01544 0.01501 0.01109 0.01419 0.01546 0.01653 0.01369 0.01729 0.01869 0.01221 0.01606 0.01760 0.00459 0.01171 0.01680 0.02383 0.01606 0.01828 0.01273 0.01839

2-437

238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane alpha-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde Propionic acid

C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O C3H6O2

115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6 79-09-4

56.11 −488.1 88.11 −200.9 74.12 0.011136 90.19 0.0023574 46.14 12.248 118.18 0.21276 88.15 0.0002084 58.08 0.00032359 128.17 0.000091828 20.18 0.0011385 75.07 0.0011282 28.01 0.00033143 71.00 2.1443 61.04 0.00003135 44.01 0.001096 30.01 0.0004096 268.52 0.000049571 142.24 −11.621 128.26 −0.065771 158.24 0.000178 144.25 −30.715 144.26 0.00016337 126.24 0.000021269 160.32 0.047041 124.22 0.000016681 254.49 −291.08 128.21 −110.89 114.23 −8,758 144.21 0.00018263 130.23 −0.0030238 130.23 0.00016915 128.21 −0.0020184 128.21 8.1833E-08 112.21 0.0000133 146.29 −3,965.5 110.20 0.000060734 90.03 0.05881 32.00 0.00044994 48.00 0.0043147 212.41 4.7796E-06 86.13 −4,918,700 72.15 −684.4 102.13 0.00024601 88.15 2,896 88.15 0.00019575 86.13 −0.01719 86.13 22.775 70.13 2.7081E-06 104.21 0.00022307 104.21 0.00011261 68.12 0.000052415 68.12 0.00025623 178.23 0.00010167 94.11 0.038846 119.12 0.00016675 148.12 0.0000593 40.06 0.000061629 44.10 −1.12 60.10 −613.84 60.10 7.3907E-07 122.21 0.00010242 58.08 1,165.1 74.08 0.00022286

0.8877 −0.1321 0.4831 0.67434 −0.5611 −0.022299 0.93034 0.8892 1.0345 0.6646 0.6895 0.7722 −0.30545 1.1119 0.667 0.7509 1.2652 0.025653 0.27198 0.9288 −0.1075 0.97256 1.2943 0.29733 1.218 1.0615 −0.000042988 0.8448 0.9283 0.8745 0.97238 1.0027 2.0418 1.3554 0.5213 1.0516 0.278 0.7456 0.47999 1.4851 −0.10297 0.764 0.8946 0.8985 0.9692 0.4832 1.0019 1.5493 0.93358 1.034 1.0948 1.0073 0.988 0.2392 0.91777 1.046 1.0731 0.10972 0.7927 1.7419 1.0486 0.90419 0.91704

−1,448,500,000 104,000 2,170.3 1,804.1 −1,067 −194.68 364.832 623.22 731.78 8.7 679.11 16.323 1,860.3 −91.6 540 45.6 3,332.3 2,248.3 −3,482.3 753 8,107 709.74 662.21 2,460.6 −199.41 −6,019,900,000 51,384 −2.7121E+10 741.3 −13,352 698.55 −20,406 504.59 −1,851,900,000 −124.91 14,815 56.699 700.09 643.13 2,691,100,000 −1,055,000,000 696.42 12,735,000,000 664.04 −3,798 191,000,000 41.075 794.16 693.05 −51.09 1,423.7 797 985.81 730.1 765.5 1.8579 −9,834.6 −1,157,400,000 701.56 5,472,900,000 678.21

266.25 350.00 312.20 368.69 216.25 438.65 273.00 278.65 491.14 30.00 238,800 387.22 373.72 63.15 1,216,700 144.09 128,000 374.35 182.30 121.38 603.05 −135,100,000 468.15 −1,580,300 423.97 528.75 −156,830,000 485.20 471.70 420.02 1,367,200 492.95 144,580 423.85 589.86 −1.0701E+09 447.15 339.00 513.05 468.35 452.90 446.15 440.65 394.41 472.19 158,300 399.35 569.00 80.00 161.85 543.84 −2.3179E+13 376.15 273.15 458.65 410.90 392.20 −1,235,000 273.00 273.00 8,301.3 303.22 385.15 399.79 101,160 313.33 329.27 610.03 937,170 454.99 439.43 557.65 70,128 238.65 −7,535,800 231.11 370.35 355.30 431.65 321.15 414.32 −846,000,000 281,220 155,660 2,715,200 1,708,700 73,041

0.01276 0.01402 0.01648 0.01802 0.01108 0.01969 0.01161 0.01493 0.02243 0.00846 0.01580 0.00602 0.00648 0.01365 0.00891 0.01094 0.02502 0.02228 0.02130 0.02484 0.02436 0.02599 0.02051 0.02559 0.01981 0.02491 0.02117 0.01503 0.02450 0.02380 0.02545 0.02046 0.02050 0.01926 0.02505 0.01967 0.01269 0.00691 0.00931 0.02529 0.01705 0.01288 0.02349 0.02084 0.02372 0.00877 0.00898 0.01546 0.01890 0.02019 0.01517 0.01653 0.02490 0.02183 0.01669 0.01864 0.00980 0.01114 0.02135 0.02049 0.02262 0.01263 0.02124

1000 1000 1000 1000 1000 1000 1000 1000 1000 3273.1 1000 2000 1000 1000 1000 750 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 2000 1000 1000 1000 1000 1000 990.95 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 720.25 1000 1000 1000 1000

0.15513 0.10886 0.09079 0.08398 0.09590 0.07255 0.08958 0.09273 0.06730 0.24616 0.06887 0.11638 0.06377 0.06553 0.07133 0.05567 0.07147 0.10522 0.10597 0.06209 0.09895 0.07905 0.09771 0.07598 0.07956 0.07395 0.10893 0.11053 0.06391 0.10288 0.08229 0.10597 0.10923 0.10295 0.07845 0.08394 0.02537 0.12655 0.06990 0.08299 0.11788 0.12707 0.07002 0.11087 0.09509 0.12002 0.12082 0.11472 0.07858 0.08412 0.09608 0.11119 0.05208 0.06936 0.05461 0.04615 0.09526 0.14599 0.07034 0.12428 0.08421 0.10983 0.07487

2-438

TABLE 2-314

Vapor Thermal Conductivity of Inorganic and Organic Substances [W/(m
K)] (Concluded)

Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

C3

301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

C3H5N C5H10O2 C3H9N C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

107-12-0 109-60-4 107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

55.08 102.13 59.11 120.19 42.08 88.11 76.16 76.16 76.09 108.09 104.08 104.15 118.09 64.06 146.06 80.06 166.13 230.30 198.39 72.11 132.20 88.17 114.23 84.14 92.14 133.40 184.36 101.19 59.11 120.19 120.19 114.23 114.23 213.10 227.13 156.31 172.31 86.09 52.07 62.50 161.49 18.02 106.17 106.17 106.17

0.001321 1,325.3 0.2833 0.16992 0.0000449 740.1 0.00018367 0.0087425 0.0001666 −5,678,600 0.0000955 0.010048 0.00032875 10.527 0.00048883 1.0702 0.00017531 0.000078652 −163.62 9.5521E-06 0.00007754 0.00085604 0.000015235 0.00013384 0.00002392 0.0000952 5.3701E-06 0.000106 0.00027648 0.000098408 0.00008498 0.00001758 0.000020248 0.00020544 0.00018189 0.038012 2,498.8 −3,279,500 0.000054197 −229.41 3510.8 6.2041E-06 3.0593E-09 4.9707E-06 9.9305E-08

1.2202 1 0.055046 0.021288 1.2018 0.9732 0.9627 0.51733 0.9765 −0.045252 0.928 0.4033 0.8172 −0.7732 0.6518 −0.2348 0.8901 0.95174 0.9193 1.4561 1.0778 0.7297 1.2816 0.98115 1.2694 1.0423 1.4751 1.0161 0.901 1.0452 1.061 1.3114 1.2284 0.87137 0.88744 0.68615 0.95209 −0.12941 1.0632 0.59582 0.225 1.3973 2.4182 1.3787 1.9229

51,822 12,235,000,000 1,325.9 −54.484 421 5,646,000,000 646.01 2,358.1 706 2,615,700,000 63.6 553.74 740.97 −1,333 −117.08 2,010.4 909.56 −282.82 −1.0876E+09 662.22 729 531.99 −111.88 645.95 537 1,243.3 599.09 91 167.68 720.49 708 392.9 −174.72 807.3 803.39 34,663 2.0167E+10 1.7104E+09 −70.589 −169,430,000 401,720,000 −569.28 −225.64 −469.93

C4

1,817,600 1,624,800

334,590 −3.5415E+13 685,570 1,506,400 78,863 1,277,000 289,490

213,840 124,120

132,900 132,200

147,800 8,721,900 −1.2727E+13 9,0617

121,060 66,786 113,460

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

370.50 374.65 321.00 432.39 225.45 353.97 325.71 340.87 460.75 454.00 333.55 418.31 591.00 250.00 273.15 317.90 832.00 373.15 526.73 339.12 480.77 394.27 379.44 357.31 383.78 387.00 508.62 273.15 273.15 449.27 442.53 355.15 387.91 629.60 625.00 469.08 520.30 345.65 278.25 259.25 363.85 273.16 320 320 320

0.01278 0.01520 0.01709 0.02022 0.01054 0.01403 0.01616 0.01654 0.02624 0.02593 0.01761 0.01837 0.02685 0.00745 0.01163 0.01386 0.03328 0.00950 0.02517 0.01564 0.02395 0.01801 0.01964 0.01525 0.01901 0.01125 0.02422 0.01018 0.01280 0.02238 0.02098 0.01846 0.02001 0.02474 0.02410 0.02259 0.02486 0.01515 0.01123 0.00963 0.01198 0.01574 0.00867 0.01492 0.01019

990.5 1000 1000 1000 1000 1000 1000 1000 1000 1000 702.45 1000 1000 900 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1073.15 1000 1000 1000

0.11209 0.10832 0.10000 0.07658 0.12737 0.10893 0.08624 0.08439 0.08302 0.12665 0.03837 0.07276 0.05342 0.03969 0.04587 0.04930 0.04297 0.05598 0.08615 0.13419 0.07676 0.07579 0.10528 0.07139 0.10007 0.05684 0.08942 0.09680 0.10734 0.07816 0.07583 0.10847 0.10079 0.04675 0.04635 0.09798 0.08899 0.12177 0.08222 0.08300 0.04135 0.10652 0.09965 0.08084 0.09060

The vapor thermal conductivity is calculated by

C1T C2 k  1  C3/T  C4/T 2 where k is the thermal conductivity in W/(m·K) and T is the temperature in K. Thermal conductivites are at either 1 atm or the vapor pressure, whichever is lower. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source shold be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007) . The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

TABLE 2-315 Thermal Conductivity of Inorganic and Organic Liquids [W/(m
K)]

2-439

Cmpd. no.

Name

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetylene Acrolein Acrylic acid Acrylonitrile Air Ammonia Anisole Argon Benzamide Benzene Benzenethiol Benzoic acid Benzonitrile Benzophenone Benzyl alcohol Benzyl ethyl ether Benzyl mercaptan Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene Butane 1,2-Butanediol 1,3-Butanediol 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene Butyl acetate Butylbenzene Butyl mercaptan sec-Butyl mercaptan 1-Butyne Butyraldehyde Butyric acid Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Carbon tetrafluoride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane

Formula C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C2H2 C3H4O C3H4O2 C3H3N Mixture H3N C7H8O Ar C7H7NO C6H6 C6H6S C7H6O2 C7H5N C13H10O C7H8O C9H12O C7H8S C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O2 C4H10O2 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C10H14 C4H10S C4H10S C4H6 C4H8O C4H8O2 C4H7N CO2 CS2 CO CCl4 CF4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl

CAS no.

Mol. wt.

C1

C2

75-07-0 60-35-5 64-19-7 108-24-7 67-64-1 75-05-8 74-86-2 107-02-8 79-10-7 107-13-1 132259-10-0 7664-41-7 100-66-3 7440-37-1 55-21-0 71-43-2 108-98-5 65-85-0 100-47-0 119-61-9 100-51-6 539-30-0 100-53-8 92-52-4 7726-95-6 108-86-1 74-96-4 74-83-9 590-19-2 106-99-0 106-97-8 584-03-2 107-88-0 71-36-3 78-92-2 106-98-9 590-18-1 624-64-6 123-86-4 104-51-8 109-79-5 513-53-1 107-00-6 123-72-8 107-92-6 109-74-0 124-38-9 75-15-0 630-08-0 56-23-5 75-73-0 7782-50-5 108-90-7 75-00-3 67-66-3 74-87-3 540-54-5 75-29-6

44.05 59.07 60.05 102.09 58.08 41.05 26.04 56.06 72.06 53.06 28.96 17.03 108.14 39.95 121.14 78.11 110.18 122.12 103.12 182.22 108.14 136.19 124.20 154.21 159.81 157.01 108.97 94.94 54.09 54.09 58.12 90.12 90.12 74.12 74.12 56.11 56.11 56.11 116.16 134.22 90.19 90.19 54.09 72.11 88.11 69.11 44.01 76.14 28.01 153.82 88.00 70.91 112.56 64.51 119.38 50.49 78.54 78.54

0.311 0.39363 0.214 0.23638 0.2878 0.33192 0.33363 0.2703 0.2441 0.28941 0.28472 1.169 0.23494 0.1819 0.28485 0.23444 0.20996 0.2391 0.21284 0.25867 0.17847 0.2029 0.20316 0.19053 −0.2185 0.16983 0.1799 0.1912 0.21966 0.22231 0.27349 0.064621 −0.0032865 0.2136 0.22787 0.22153 0.21378 0.21153 0.21721 0.18707 0.21143 0.2069 0.22334 0.21915 0.1967 0.2597 0.4406 0.2333 0.2855 0.1589 0.20771 0.2246 0.1841 0.2438 0.1778 0.41067 0.20143 0.21232

−0.000436 −0.00037053 −0.0001834 −0.00024263 −0.000427 −0.00043243 −0.00083655 −0.0003764 −0.0002904 −0.00041691 −0.0017393 −0.002314 −0.00026477 −0.0003176 −0.00025225 −0.00030572 −0.0002146 −0.0002325 −0.00021587 −0.00022516 −0.000065843 −0.0002226 −0.00019912 −0.00015145 0.0042143 −0.0001981 −0.000262 −0.000299 −0.0003436 −0.0003664 −0.00071267 0.00067625 0.0011463 −0.0002034 −0.00030727 −0.00035023 −0.00035445 −0.00035056 −0.00026563 −0.00020037 −0.000258 −0.0002568 −0.0003515 −0.00024846 −0.000168 −0.00031 −0.0012175 −0.000275 −0.001784 −0.0001987 −0.00078883 −0.000064 −0.0001917 −0.000419 −0.0002023 −0.0008478 −0.00028925 −0.0003149

C3

C4

C5

−0.00000411

−0.000017753

5.1555E-07 −1.0491E-06 −1.5525E-06

−0.000000788

3.1041E-08

−2.0108E-11

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

150.15 353.33 289.81 200.15 178.45 229.32 192.40 185.45 286.15 189.63 75.00 195.41 235.65 83.78 403.00 278.68 258.27 395.45 260.40 321.35 257.85 275.65 243.95 342.20 266.00 242.43 154.55 179.47 136.95 164.25 134.86 220.00 196.15 183.85 158.45 87.80 134.26 167.62 199.65 185.30 157.46 133.02 147.43 176.75 267.95 161.25 216.58 161.11 68.15 250.33 89.56 172.12 227.95 134.80 209.63 175.43 150.35 155.97

0.2455 0.2627 0.1609 0.1878 0.2116 0.2328 0.1727 0.2005 0.1610 0.2104 0.1543 0.7168 0.1726 0.1264 0.1832 0.1492 0.1545 0.1472 0.1566 0.1863 0.1615 0.1415 0.1546 0.1387 0.1299 0.1218 0.1394 0.1375 0.1726 0.1621 0.1868 0.1626 0.1618 0.1762 0.1792 0.1908 0.1662 0.1528 0.1642 0.1499 0.1708 0.1727 0.1715 0.1752 0.1517 0.2097 0.1769 0.1890 0.1639 0.1092 0.1371 0.1902 0.1404 0.1873 0.1354 0.2619 0.1579 0.1632

294.00 494.30 391.05 412.70 343.15 349.32 250.00 325.84 484.50 350.50 125.00 400.05 512.50 150.00 563.15 413.10 442.29 596.00 464.15 664.00 478.60 528.60 472.03 723.15 584.00 429.24 414.14 370.10 284.00 268.74 400.00 469.57 481.38 391.90 372.90 266.91 276.87 274.03 453.75 473.15 371.61 358.13 281.22 382.15 573.15 390.75 300.00 319.37 125.00 349.79 145.10 410.00 404.87 373.15 400.00 350.00 400.95 386.70

0.1828 0.2105 0.1423 0.1363 0.1413 0.1809 0.1245 0.1477 0.1034 0.1433 0.0673 0.2433 0.0992 0.0418 0.1428 0.1082 0.1150 0.1005 0.1126 0.1092 0.1470 0.0852 0.1092 0.0810 0.0316 0.0848 0.0714 0.0805 0.1221 0.1238 0.0709 0.1509 0.1888 0.1339 0.1133 0.1281 0.1156 0.1155 0.0967 0.0923 0.1156 0.1149 0.1245 0.1242 0.1004 0.1386 0.0754 0.1455 0.0625 0.0894 0.0933 0.0659 0.1065 0.0875 0.0969 0.1139 0.0855 0.0905

2-440

TABLE 2-315 Thermal Conductivity of Inorganic and Organic Liquids [W/(m
K)] (Continued) Cmpd. no. 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

Name m-Cresol o-Cresol p-Cresol Cumene Cyanogen Cyclobutane Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene Cyclopropane Cyclohexyl mercaptan Decanal Decane Decanoic acid 1-Decanol 1-Decene Decyl mercaptan 1-Decyne Deuterium 1,1-Dibromoethane 1,2-Dibromoethane Dibromomethane Dibutyl ether m-Dichlorobenzene o-Dichlorobenzene p-Dichlorobenzene 1,1-Dichloroethane 1,2-Dichloroethane Dichloromethane 1,1-Dichloropropane 1,2-Dichloropropane Diethanol amine Diethyl amine Diethyl ether Diethyl sulfide 1,1-Difluoroethane 1,2-Difluoroethane Difluoromethane Di-isopropyl amine Di-isopropyl ether Di-isopropyl ketone 1,1-Dimethoxyethane 1,2-Dimethoxypropane Dimethyl acetylene Dimethyl amine 2,3-Dimethylbutane 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane Dimethyl disulfide Dimethyl ether N,N-Dimethyl formamide 2,3-Dimethylpentane Dimethyl phthalate

Formula C7H8O C7H8O C7H8O C9H12 C2N2 C4H8 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C3H6 C6H12S C10H20O C10H22 C10H20O2 C10H22O C10H20 C10H22S C10H18 D2 C2H4Br2 C2H4Br2 CH2Br2 C8H18O C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4Cl2 C2H4Cl2 CH2Cl2 C3H6Cl2 C3H6Cl2 C4H11NO2 C4H11N C4H10O C4H10S C2H4F2 C2H4F2 CH2F2 C6H15N C6H14O C7H14O C4H10O2 C5H12O2 C4H6 C2H7N C6H14 C8H16 C8H16 C8H16 C2H6S2 C2H6O C3H7NO C7H16 C10H10O4

CAS no.

Mol. wt.

C1

C2

108-39-4 95-48-7 106-44-5 98-82-8 460-19-5 287-23-0 110-82-7 108-93-0 108-94-1 110-83-8 287-92-3 142-29-0 75-19-4 1569-69-3 112-31-2 124-18-5 334-48-5 112-30-1 872-05-9 143-10-2 764-93-2 7782-39-0 557-91-5 106-93-4 74-95-3 142-96-1 541-73-1 95-50-1 106-46-7 75-34-3 107-06-2 75-09-2 78-99-9 78-87-5 111-42-2 109-89-7 60-29-7 352-93-2 75-37-6 624-72-6 75-10-5 108-18-9 108-20-3 565-80-0 534-15-6 7778-85-0 503-17-3 124-40-3 79-29-8 590-66-9 2207-01-4 6876-23-9 624-92-0 115-10-6 68-12-2 565-59-3 131-11-3

108.14 108.14 108.14 120.19 52.03 56.11 84.16 100.16 98.14 82.14 70.13 68.12 42.08 116.22 156.27 142.28 172.27 158.28 140.27 174.35 138.25 4.03 187.86 187.86 173.83 130.23 147.00 147.00 147.00 98.96 98.96 84.93 112.99 112.99 105.14 73.14 74.12 90.19 66.05 66.05 52.02 101.19 102.17 114.19 90.12 104.15 54.09 45.08 86.18 112.21 112.21 112.21 94.20 46.07 73.09 100.20 194.18

0.18241 0.19186 0.17971 0.1855 0.4685 0.22262 0.19813 0.1715 0.17557 0.20926 0.2066 0.21776 0.24348 0.18374 0.20383 0.2063 0.206 0.228 0.20237 0.20134 0.20839 1.264 0.1426 0.13622 0.17558 0.19418 0.16694 0.16994 0.16977 0.18881 0.214 0.23847 0.18 0.19653 0.0218 0.2587 0.2495 0.21065 0.27019 0.23171 0.37296 0.1844 0.19162 0.22076 0.22078 0.22998 0.22773 0.2454 0.1774 0.1807 0.18092 0.17675 0.21373 0.31174 0.26 0.17964 0.13905

−0.00011109 −0.0001303 −0.00012037 −0.00020895 −0.00086594 −0.00034082 −0.0002505 −0.0001255 −0.00012392 −0.00026037 −0.0002696 −0.00027783 −0.00042568 −0.0001925 −0.0002 −0.00025 −0.0002 −0.000223 −0.00024187 −0.00020826 −0.00023622 −0.00016402 −0.0001179 −0.00022499 −0.00022246 −0.0001667 −0.0001637 −0.0001799 −0.00026083 −0.000266 −0.00033366 −0.00023144 −0.00025012 0.0010315 −0.00054343 −0.000407 −0.0002623 −0.000661 −0.00038503 −0.00088707 −0.000239 −0.0002762 −0.00027624 −0.00031271 −0.00030372 −0.00034804 −0.000338 −0.0002436 −0.0002177 −0.0002108 −0.0002077 −0.0002447 −0.0005638 −0.000255 −0.000246 0.0001509

C3

−0.000001355 4.2097E-07 3.443E-07 2.5762E-07

−3.978E-07

C4

C5

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

285.39 304.19 307.93 177.14 245.25 182.48 279.69 296.60 242.00 169.67 179.28 138.13 145.59 189.64 267.15 243.51 304.75 280.05 206.89 247.56 229.15 20.40 210.15 282.85 220.60 175.30 248.39 262.87 326.14 176.19 253.15 178.01 200.00 172.71 301.15 223.35 156.85 169.20 154.56 215.00 136.95 176.85 187.65 204.81 159.95 226.10 240.91 180.96 145.19 239.66 223.16 184.99 188.44 131.65 250.00 160.00 273.15

0.1507 0.1522 0.1426 0.1485 0.2561 0.1604 0.1281 0.1343 0.1456 0.1651 0.1583 0.1794 0.1815 0.1472 0.1504 0.1454 0.1451 0.1656 0.1523 0.1498 0.1543 1.2640 0.1081 0.1029 0.1260 0.1552 0.1255 0.1269 0.1111 0.1429 0.1467 0.1791 0.1337 0.1533 0.2096 0.1583 0.1857 0.1663 0.1763 0.1489 0.2563 0.1421 0.1398 0.1642 0.1708 0.1613 0.1439 0.1842 0.1420 0.1285 0.1339 0.1383 0.1676 0.2375 0.1963 0.1403 0.1506

475.43 464.15 475.13 413.15 252.00 285.66 353.87 563.15 428.58 356.12 322.40 333.15 240.37 431.95 488.15 447.30 543.15 503.00 443.75 512.35 447.15 20.40 498.40 404.51 370.10 523.15 446.23 351.71 548.00 416.90 356.59 325.00 438.00 457.60 673.15 453.15 433.15 365.25 363.15 372.80 302.56 357.05 400.10 460.00 337.45 366.15 300.13 403.15 331.15 392.70 402.94 596.15 382.90 320.03 425.15 362.93 556.85

0.1296 0.1314 0.1225 0.0992 0.2503 0.1253 0.1095 0.1008 0.1225 0.1165 0.1197 0.1252 0.1412 0.1006 0.1062 0.0945 0.0974 0.1158 0.0950 0.0946 0.1028 1.2640 0.0609 0.0885 0.0923 0.0778 0.0926 0.1124 0.0712 0.0801 0.1192 0.1300 0.0786 0.0821 0.1022 0.0989 0.0732 0.1148 0.0756 0.0882 0.1282 0.0991 0.0811 0.0937 0.1153 0.1188 0.1233 0.1091 0.0967 0.0952 0.0960 0.0529 0.1200 0.1313 0.1516 0.0904 0.0997

2-441

116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178

Dimethylsilane Dimethyl sulfide Dimethyl sulfoxide Dimethyl terephthalate 1,4-Dioxane Diphenyl ether Dipropyl amine Dodecane Eicosane Ethane Ethanol Ethyl acetate Ethyl amine Ethylbenzene Ethyl benzoate 2-Ethyl butanoic acid Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylenediamine Ethylene glycol Ethyleneimine Ethylene oxide Ethyl formate 2-Ethyl hexanoic acid Ethylhexyl ether Ethylisopropyl ether Ethylisopropyl ketone Ethyl mercaptan Ethyl propionate Ethylpropyl ether Ethyltrichlorosilane Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 Heptadecane Heptanal Heptane Heptanoic acid 1-Heptanol 2-Heptanol 3-Heptanone 2-Heptanone 1-Heptene Heptyl mercaptan 1-Heptyne Hexadecane Hexanal Hexane Hexanoic acid 1-Hexanol 2-Hexanol 2-Hexanone 3-Hexanone 1-Hexene 3-Hexyne

C2H8Si C2H6S C2H6OS C10H10O4 C4H8O2 C12H10O C6H15N C12H26 C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C6H12O2 C8H16 C7H14 C2H4 C2H8N2 C2H6O2 C2H5N C2H4O C3H6O2 C8H16O2 C8H18O C5H12O C6H12O C2H6S C5H10O2 C5H12O C2H5Cl3Si F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O He C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H16O C7H14O C7H14O C7H14 C7H16S C7H12 C16H34 C6H12O C6H14 C6H12O2 C6H14O C6H14O C6H12O C6H12O C6H12 C6H10

1111-74-6 75-18-3 67-68-5 120-61-6 123-91-1 101-84-8 142-84-7 112-40-3 112-95-8 74-84-0 64-17-5 141-78-6 75-04-7 100-41-4 93-89-0 88-09-5 105-54-4 1678-91-7 1640-89-7 74-85-1 107-15-3 107-21-1 151-56-4 75-21-8 109-94-4 149-57-5 5756-43-4 625-54-7 565-69-5 75-08-1 105-37-3 628-32-0 115-21-9 7782-41-4 462-06-6 353-36-6 593-53-3 50-00-0 75-12-7 64-18-6 110-00-9 7440-59-7 629-78-7 111-71-7 142-82-5 111-14-8 111-70-6 543-49-7 106-35-4 110-43-0 592-76-7 1639-09-4 628-71-7 544-76-3 66-25-1 110-54-3 142-62-1 111-27-3 626-93-7 591-78-6 589-38-8 592-41-6 928-49-4

60.17 62.13 78.13 194.18 88.11 170.21 101.19 170.33 282.55 30.07 46.07 88.11 45.08 106.17 150.17 116.16 116.16 112.21 98.19 28.05 60.10 62.07 43.07 44.05 74.08 144.21 130.23 88.15 100.16 62.13 102.13 88.15 163.51 38.00 96.10 48.06 34.03 30.03 45.04 46.03 68.07 4.00 240.47 114.19 100.20 130.19 116.20 116.20 114.19 114.19 98.19 132.27 96.17 226.44 100.16 86.18 116.16 102.17 102.18 100.16 100.16 84.16 82.14

0.25547 0.23942 0.3142 0.21593 0.3027 0.18686 0.2224 0.2047 0.2178 0.35758 0.2468 0.2501 0.30059 0.1999 0.20771 0.2175 0.21043 0.17662 0.18334 0.4194 0.36434 0.088067 0.3097 0.26957 0.2587 0.20954 0.19356 0.21928 0.22873 0.23392 0.2137 0.22717 0.19769 0.2758 0.20962 0.2595 0.445 0.37329 0.3847 0.302 0.2198 −0.013833 0.20926 0.21816 0.215 0.202 0.2239 0.21134 0.2026 0.2108 0.19664 0.2037 0.21098 0.20749 0.22196 0.22492 0.1855 0.2193 0.21391 0.21076 0.23493 0.19112 0.20996

−0.0004411 −0.0003311 −0.00030809 −0.00020805 −0.0004827 −0.00014953 −0.000314 −0.0002326 −0.0002233 −0.0011458 −0.000264 −0.0003563 −0.000581 −0.00023823 −0.00021265 −0.0002407 −0.00024903 −0.0002014 −0.0002228 −0.001591 −0.0004433 0.00094712 −0.0004023 −0.0003984 −0.00033 −0.00022251 −0.00024102 −0.00032568 −0.0002913 −0.0003206 −0.0002515 −0.0003298 −0.00017713 −0.0016297 −0.00028034 −0.0005008 −0.001023 −0.00065 −0.0001065 −0.000108 −0.00031405 0.022913 −0.0002215 −0.0003015 −0.000303 −0.0002 −0.000226 −0.00024776 −0.0002234 −0.000246 −0.00016623 −0.0002252 −0.00026652 −0.00021917 −0.00032053 −0.0003533 −0.000146 −0.00022 −0.00026042 −0.00024 −0.0002912 −0.000083519 −0.0002692

6.1866E-07 6.602E-07

0.000001306 −1.3114E-06

−1.5448E-07

−0.0054872 1.033E-07

−2.5241E-07

1.1554E-07

−5.1407E-07

0.0004585

122.93 174.88 291.67 413.80 284.95 300.03 210.15 263.57 309.58 90.35 159.05 189.60 192.15 178.20 238.45 258.15 175.15 161.84 134.71 104.00 284.29 260.15 195.20 160.65 193.55 235.00 180.00 140.00 204.15 125.26 199.25 145.65 167.55 53.48 238.15 129.95 131.35 204.00 275.70 281.45 187.55 2.20 295.13 229.80 182.57 265.83 239.15 230.00 234.15 238.15 154.12 229.92 192.22 291.31 217.15 177.83 269.25 228.55 223.00 217.35 217.50 133.39 170.05

0.2013 0.1815 0.2243 0.1298 0.1652 0.1420 0.1564 0.1434 0.1487 0.2591 0.2048 0.1826 0.2133 0.1575 0.1570 0.1554 0.1668 0.1440 0.1533 0.2681 0.2383 0.2457 0.2312 0.2056 0.1948 0.1573 0.1502 0.1737 0.1693 0.1938 0.1636 0.1791 0.1637 0.1886 0.1429 0.1944 0.3106 0.2407 0.3553 0.2716 0.1609 0.0149 0.1439 0.1543 0.1597 0.1488 0.1699 0.1544 0.1503 0.1522 0.1650 0.1519 0.1598 0.1436 0.1578 0.1621 0.1462 0.1690 0.1558 0.1586 0.1716 0.1708 0.1642

253.55 310.48 464.00 561.15 374.47 531.46 382.00 489.47 616.93 300.00 353.15 350.21 293.15 413.10 549.40 516.50 453.15 404.94 376.62 280.00 390.41 470.45 329.00 283.85 433.15 500.66 466.40 391.20 450.10 308.15 495.00 400.07 371.05 130.00 353.15 292.59 283.15 234.00 493.00 373.71 304.50 4.80 575.30 553.15 371.58 496.15 573.15 432.90 553.15 424.05 366.79 450.09 372.93 560.01 533.15 370.00 603.15 575.00 412.40 400.85 466.00 336.63 354.35

0.1436 0.1366 0.1713 0.0992 0.1219 0.1074 0.1025 0.0908 0.0800 0.0695 0.1536 0.1253 0.1870 0.1015 0.0909 0.0932 0.0976 0.0951 0.0994 0.0763 0.1913 0.2434 0.1773 0.1565 0.1158 0.0981 0.0811 0.0919 0.0976 0.1351 0.0892 0.0952 0.1107 0.0639 0.1106 0.1130 0.1553 0.2212 0.3322 0.2616 0.1242 0.0204 0.0818 0.0830 0.1024 0.1028 0.0944 0.1041 0.0790 0.1065 0.1017 0.1023 0.1116 0.0848 0.0839 0.0942 0.0974 0.0928 0.1065 0.1146 0.0992 0.1048 0.1146

2-442

TABLE 2-315 Thermal Conductivity of Inorganic and Organic Liquids [W/(m
K)] (Continued) Cmpd. no.

Name

Formula

CAS no.

Mol. wt.

C1

C2

179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236

Hexyl mercaptan 1-Hexyne 2-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide Isobutyric acid Isopropyl amine Malonic acid Methacrylic acid Methane Methanol N-Methyl acetamide Methyl acetate Methyl acetylene Methyl acrylate Methyl amine Methyl benzoate 3-Methyl-1,2-butadiene 2-Methylbutane 2-Methylbutanoic acid 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene 2-Methyl-1-butene-3-yne Methylbutyl ether Methylbutyl sulfide 3-Methyl-1-butyne Methyl butyrate Methylchlorosilane Methylcyclohexane 1-Methylcyclohexanol cis-2-Methylcyclohexanol trans-2-Methylcyclohexanol Methylcyclopentane 1-Methylcyclopentene 3-Methylcyclopentene Methyldichlorosilane Methylethyl ether Methylethyl ketone Methylethyl sulfide Methyl formate Methylisobutyl ether Methylisobutyl ketone Methyl isocyanate Methylisopropyl ether Methylisopropyl ketone Methylisopropyl sulfide Methyl mercaptan Methyl methacrylate 2-Methyloctanoic acid 2-Methylpentane Methyl pentyl ether 2-Methylpropane

C6H14S C6H10 C6H10 H4N2 H2 HBr HCl CHN HF H2S C4H8O2 C3H9N C3H4O4 C4H6O2 CH4 CH4O C3H7NO C3H6O2 C3H4 C4H6O2 CH5N C8H8O2 C5H8 C5H12 C5H10O2 C5H12O C5H10 C5H10 C5H6 C5H12O C5H12S C5H8 C5H10O2 CH5ClSi C7H14 C7H14O C7H14O C7H14O C6H12 C6H10 C6H10 CH4Cl2Si C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C2H3NO C4H10O C5H10O C4H10S CH4S C5H8O2 C9H18O2 C6H14 C6H14O C4H10

111-31-9 693-02-7 764-35-2 302-01-2 1333-74-0 10035-10-6 7647-01-0 74-90-8 7664-39-3 7783-06-4 79-31-2 75-31-0 141-82−2 79-41−4 74-82-8 67-56-1 79-16-3 79-20-9 74-99-7 96-33-3 74-89-5 93-58-3 598-25-4 78-78-4 116-53-0 123-51-3 563-46-2 513-35-9 78-80-8 628-28-4 628-29-5 598-23-2 623-42-7 993-00-0 108-87-2 590-67-0 7443-70-1 7443-52-9 96-37-7 693-89-0 1120-62-3 75-54-7 540-67-0 78-93-3 624-89-5 107-31-3 625-44-5 108-10-1 624-83-9 598-53-8 563-80-4 1551-21-9 74-93-1 80-62-6 3004-93-1 107-83-5 628-80-8 75-28-5

118.24 82.14 82.14 32.05 2.02 80.91 36.46 27.03 20.01 34.08 88.11 59.11 104.06 86.09 16.04 32.04 73.09 74.08 40.06 86.09 31.06 136.15 68.12 72.15 102.13 88.15 70.13 70.13 66.10 88.15 104.21 68.12 102.13 80.59 98.19 114.19 114.19 114.19 84.16 82.14 82.14 115.03 60.10 72.11 76.16 60.05 88.15 100.16 57.05 74.12 86.13 90.19 48.11 100.12 158.24 86.18 102.17 58.12

0.2058 0.21492 0.2119 1.3675 −0.0917 0.234 0.8045 0.43454 0.7516 0.4842 0.21668 0.237 0.28918 0.2306 0.41768 0.2837 0.23743 0.2777 0.23648 0.26082 0.33446 0.22142 0.1983 0.21246 0.22284 0.17471 0.19447 0.19636 0.20385 0.22235 0.20698 0.20348 0.21748 0.24683 0.1791 0.21558 0.21839 0.21828 0.1929 0.20023 0.1994 0.21956 0.27304 0.2197 0.22136 0.3246 0.222 0.2301 0.2822 0.24154 0.2332 0.20978 0.26119 0.2583 0.20911 0.19334 0.21698 0.20455

−0.0002324 −0.0002899 −0.00027048 −0.0015895 0.017678 −0.0004636 −0.002102 −0.0007008 −0.0010874 −0.001184 −0.0002556 −0.000332 −0.0002614 −0.00025201 −0.0024528 −0.000281 −0.0002362 −0.000417 −0.00041639 −0.0003506 −0.00067427 −0.00022759 −0.0002822 −0.00033581 −0.0002516 −0.0001256 −0.0002901 −0.000291 −0.0002874 −0.0003044 −0.00024439 −0.0003106 −0.00025913 −0.00038854 −0.0002291 −0.00022728 −0.00025776 −0.0002557 −0.0002492 −0.00025581 −0.00026149 −0.00032153 −0.0004518 −0.0002505 −0.00028938 −0.000468 −0.00032217 −0.00028899 −0.00042037 −0.0003774 −0.0003044 −0.00026468 −0.00038345 −0.000379 −0.00021852 −0.00028038 −0.00028998 −0.00036589

C3

−0.000382

3.5588E-06

8.033E-07

C4

−3.3324E-06

C5

1.0266E-07

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

192.62 141.25 183.65 274.69 13.95 185.15 273.15 259.83 189.79 193.15 227.15 177.95 407.95 288.15 90.69 175.47 301.15 175.15 170.45 196.32 179.69 260.75 159.53 113.25 357.15 155.95 135.58 139.39 160.15 157.48 175.30 183.45 187.35 139.05 273.15 299.15 280.15 269.15 130.73 146.62 115.00 182.55 160.00 186.48 167.23 174.15 150.00 189.15 256.15 127.93 180.15 171.64 150.18 290.15 208.20 119.55 176.00 113.54

0.1610 0.1740 0.1622 0.9309 0.0754 0.1482 0.2303 0.2525 0.5452 0.2555 0.1586 0.1779 0.1825 0.1580 0.2245 0.2344 0.1663 0.2047 0.1655 0.1920 0.2392 0.1621 0.1533 0.1744 0.1330 0.1551 0.1551 0.1558 0.1578 0.1744 0.1641 0.1465 0.1689 0.1928 0.1165 0.1476 0.1462 0.1495 0.1603 0.1627 0.1693 0.1609 0.2008 0.1730 0.1730 0.2431 0.1737 0.1754 0.1745 0.1933 0.1784 0.1644 0.2036 0.1483 0.1636 0.1598 0.1659 0.1630

425.81 344.48 357.67 623.15 31.00 290.62 323.15 298.85 394.45 292.42 482.75 305.55 644.00 530.00 180.00 337.85 478.15 386.15 249.94 421.00 283.15 547.90 314.00 368.13 480.90 404.15 304.30 311.70 305.40 463.15 396.58 302.15 493.15 281.85 374.08 548.80 484.20 484.80 344.95 348.64 338.05 314.70 341.34 352.79 339.80 373.15 390.00 451.42 312.00 370.00 435.90 357.91 279.11 363.45 555.20 389.25 432.30 400.00

0.1068 0.1151 0.1152 0.3770 0.0847 0.0993 0.1252 0.2251 0.3227 0.1380 0.0933 0.1356 0.1208 0.0970 0.0915 0.1888 0.1245 0.1167 0.1324 0.1132 0.2079 0.0967 0.1097 0.0888 0.1019 0.1239 0.1062 0.1057 0.1161 0.0814 0.1101 0.1096 0.0897 0.1373 0.0934 0.0908 0.0936 0.0943 0.1069 0.1110 0.1110 0.1184 0.1188 0.1313 0.1230 0.1500 0.0964 0.0996 0.1510 0.1019 0.1005 0.1151 0.1542 0.1206 0.0878 0.0842 0.0916 0.0582

2-443

237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299

2-Methyl-2-propanol 2-Methyl propene Methyl propionate Methylpropyl ether Methylpropyl sulfide Methylsilane α-Methyl styrene Methyl tert-butyl ether Methyl vinyl ether Naphthalene Neon Nitroethane Nitrogen Nitrogen trifluoride Nitromethane Nitrous oxide Nitric oxide Nonadecane Nonanal Nonane Nonanoic acid 1-Nonanol 2-Nonanol 1-Nonene Nonyl mercaptan 1-Nonyne Octadecane Octanal Octane Octanoic acid 1-Octanol 2-Octanol 2-Octanone 3-Octanone 1-Octene Octyl mercaptan 1-Octyne Oxalic acid Oxygen Ozone Pentadecane Pentanal Pentane Pentanoic acid 1-Pentanol 2-Pentanol 2-Pentanone 3-Pentanone 1-Pentene 2-Pentyl mercaptan Pentyl mercaptan 1-Pentyne 2-Pentyne Phenanthrene Phenol Phenyl isocyanate Phthalic anhydride Propadiene Propane 1-Propanol 2-Propanol Propenylcyclohexene Propionaldehyde

C4H10O C4H8 C4H8O2 C4H10O C4H10S CH6Si C9H10 C5H12O C3H6O C10H8 Ne C2H5NO2 N2 F3N CH3NO2 N2O NO C19H40 C9H18O C9H20 C9H18O2 C9H20O C9H20O C9H18 C9H20S C9H16 C18H38 C8H16O C8H18 C8H16O2 C8H18O C8H18O C8H16O C8H16O C8H16 C8H18S C8H14 C2H2O4 O2 O3 C15H32 C5H10O C5H12 C5H10O2 C5H12O C5H12O C5H10O C5H10O C5H10 C5H12S C5H12S C5H8 C5H8 C14H10 C6H6O C7H5NO C8H4O3 C3H4 C3H8 C3H8O C3H8O C9H14 C3H6O

75-65-0 115-11-7 554-12-1 557-17-5 3877-15-4 992-94-9 98-83-9 1634-04-4 107-25-5 91-20-3 7440-01-9 79-24-3 7727-37-9 7783-54-2 75-52-5 10024-97-2 10102-43-9 629-92-5 124-19-6 111-84-2 112-05-0 143-08-8 628-99-9 124-11-8 1455-21-6 3452-09-3 593-45-3 124-13-0 111-65-9 124-07-2 111-87-5 123-96-6 111-13-7 106-68-3 111-66-0 111-88-6 629-05-0 144-62-7 7782-44-7 10028-15-6 629-62-9 110-62-3 109-66-0 109-52-4 71-41-0 6032-29-7 107-87-9 96-22-0 109-67-1 2084-19-7 110-66-7 627-19-0 627-21-4 85-01-8 108-95-2 103-71-9 85-44-9 463-49-0 74-98-6 71-23-8 67-63-0 13511-13-2 123-38-6

74.12 56.11 88.11 74.12 90.19 46.14 118.18 88.15 58.08 128.17 20.18 75.07 28.01 71.00 61.04 44.01 30.01 268.52 142.24 128.26 158.24 144.25 144.26 126.24 160.32 124.22 254.49 128.21 114.23 144.21 130.23 130.23 128.21 128.21 112.21 146.29 110.20 90.03 32.00 48.00 212.41 86.13 72.15 102.13 88.15 88.15 86.13 86.13 70.13 104.21 104.21 68.12 68.12 178.23 94.11 119.12 148.12 40.06 44.10 60.10 60.10 122.21 58.08

0.21258 0.2802 0.22534 0.24817 0.21103 0.2774 0.19657 0.2253 0.28035 0.17096 0.2971 0.247 0.2654

−0.00029864 −0.000786 −0.0002683 −0.0003774 −0.00025985 −0.00054608 −0.0002118 −0.00037273 −0.0004646 −0.00010059 −0.017356 −0.0002814 −0.001677

0.3276 0.10112 0.1878 0.21229 0.21523 0.209 0.204 0.2292 0.20829 0.20468 0.20244 0.20954 0.2137 0.20143 0.2156 0.203 0.2316 0.20955 0.2132 0.21732 0.20467 0.2012 0.2095 0.3074 0.2741 0.17483 0.20649 0.22697 0.2537 0.1848 0.2006 0.21875 0.2161 0.21569 0.21361 0.20597 0.2086 0.22102 0.21282 0.13753 0.18831 0.16326 0.22946 0.23081 0.26755 0.2203 0.20161 0.1831 0.2498

−0.000405 0.0010293 −0.00022 −0.0002799 −0.000264 −0.0002 −0.00023 −0.00022922 −0.00025738 −0.00021343 −0.00024588 −0.0002252 −0.00021102 −0.00029483 −0.0002 −0.0002407 −0.00023733 −0.0002494 −0.00024969 −0.0002675 −0.0002142 −0.00025334 −0.00028101 −0.00138 0.00075288 −0.00021911 −0.00033227 −0.000576 −0.0001434 −0.0001603 −0.00027849 −0.00024866 −0.00024081 −0.00030777 −0.00024518 −0.00024536 −0.000322 −0.0002856 −0.000025247 −0.0001 −0.00017777 −0.00021345 −0.0004078 −0.00066457 −0.0002155 −0.00021529 −0.00020275 −0.00030075

6.516E-07

1.1728E-07 0.0005911

−0.00000943 9.572E-08

−2.5228E-06 1.177E-07 0.000000344

2.774E-07

−0.000007421

298.97 132.81 185.65 133.97 160.17 116.34 249.95 164.55 151.15 353.43 25.00 183.63 63.15

0.1233 0.1873 0.1755 0.1976 0.1694 0.2139 0.1436 0.1671 0.2101 0.1354 0.1167 0.1953 0.1595

404.96 395.20 475.00 373.00 368.69 216.25 438.65 328.18 341.10 646.97 44.00 387.22 124.00

0.0916 0.0713 0.0979 0.1074 0.1152 0.1593 0.1037 0.1156 0.1219 0.1059 0.0457 0.1380 0.0575

244.60 277.59 110.00 305.04 255.15 219.66 285.55 268.15 238.15 191.91 253.05 223.15 301.31 246.00 216.38 289.65 257.65 241.55 252.85 255.55 171.45 223.95 193.55 462.65 60.00 77.35 283.07 182.00 143.42 239.15 273.15 200.00 196.29 234.18 108.02 160.75 197.45 167.45 163.83 372.38 314.06 243.15 404.15 136.87 85.47 200.00 185.26 199.00 170.00

0.2285 0.1011 0.1869 0.1452 0.1501 0.1510 0.1469 0.1675 0.1537 0.1553 0.1484 0.1547 0.1458 0.1495 0.1518 0.1451 0.1696 0.1522 0.1501 0.1535 0.1588 0.1532 0.1605 0.1774 0.1913 0.2180 0.1445 0.1704 0.1782 0.1505 0.1568 0.1631 0.1673 0.1593 0.1804 0.1666 0.1602 0.1671 0.1660 0.1281 0.1569 0.1200 0.1432 0.1750 0.2128 0.1772 0.1617 0.1428 0.1987

374.35 277.59 176.40 603.05 593.15 423.97 528.75 578.65 471.70 420.02 492.95 423.85 589.86 573.15 398.83 512.85 570.15 452.90 499.00 440.65 394.41 472.19 399.35 643.20 150.00 161.85 543.84 513.15 445.00 458.65 353.15 392.20 375.46 375.14 303.22 385.15 399.79 313.33 329.27 610.03 454.99 439.43 557.65 238.65 350.00 370.35 425.00 431.65 453.15

0.1760 0.1011 0.0759 0.0796 0.0829 0.0971 0.0983 0.0961 0.1002 0.0966 0.0972 0.1053 0.0809 0.0805 0.0980 0.1004 0.0944 0.1021 0.0887 0.1073 0.0992 0.1001 0.1083 0.1267 0.0671 0.2306 0.0873 0.0875 0.0655 0.1190 0.1440 0.1095 0.1227 0.1254 0.1203 0.1115 0.1105 0.1201 0.1188 0.1221 0.1428 0.0851 0.1104 0.1335 0.0689 0.1405 0.1101 0.0956 0.1135

2-444

TABLE 2-315 Thermal Conductivity of Inorganic and Organic Liquids [W/(m
K)] (Concluded) Cmpd. no.

Name

300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345

Propionic acid Propionitrile Propyl acetate Propyl amine Propylbenzene Propylene Propyl formate 2-Propyl mercaptan Propyl mercaptan 1,2-Propylene glycol Quinone Silicon tetrafluoride Styrene Succinic acid Sulfur dioxide Sulfur hexafluoride Sulfur trioxide Terephthalic acid o-Terphenyl Tetradecane Tetrahydrofuran 1,2,3,4-Tetrahydronaphthalene Tetrahydrothiophene 2,2,3,3-Tetramethylbutane Thiophene Toluene 1,1,2-Trichloroethane Tridecane Triethyl amine Trimethyl amine 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 1,3,5-Trinitrobenzene 2,4,6-Trinitrotoluene Undecane 1-Undecanol Vinyl acetate Vinyl acetylene Vinyl chloride Vinyl trichlorosilane Water m-Xylene o-Xylene p-Xylene

Formula C3H6O2 C3H5N C5H10O2 C3H9N C9H12 C3H6 C4H8O2 C3H8S C3H8S C3H8O2 C6H4O2 F4Si C8H8 C4H6O4 O2S F6S O3S C8H6O4 C18H14 C14H30 C4H8O C10H12 C4H8S C8H18 C4H4S C7H8 C2H3Cl3 C13H28 C6H15N C3H9N C9H12 C9H12 C8H18 C8H18 C6H3N3O6 C7H5N3O6 C11H24 C11H24O C4H6O2 C4H4 C2H3Cl C2H3Cl3Si H2O C8H10 C8H10 C8H10

CAS no.

Mol. wt.

C1

C2

79-09-4 107-12-0 109-60-4 107-10-8 103-65-1 115-07-1 110-74-7 75-33-2 107-03-9 57-55-6 106-51-4 7783-61-1 100-42-5 110-15-6 7446-09-5 2551-62-4 7446-11-9 100-21-0 84-15-1 629-59-4 109-99-9 119-64-2 110-01-0 594-82-1 110-02-1 108-88-3 79-00-5 629-50-5 121-44-8 75-50-3 526-73-8 95-63-6 540-84-1 560-21-4 99-35-4 118-96-7 1120-21-4 112-42-5 108-05-4 689-97-4 75-01-4 75-94-5 7732-18-5 108-38-3 95-47-6 106-42-3

74.08 55.08 102.13 59.11 120.19 42.08 88.11 76.16 76.16 76.09 108.09 104.08 104.15 118.09 64.06 146.06 80.06 166.13 230.30 198.39 72.11 132.20 88.17 114.23 84.14 92.14 133.40 184.36 101.19 59.11 120.19 120.19 114.23 114.23 213.10 227.13 156.31 172.31 86.09 52.07 62.50 161.49 18.02 106.17 106.17 106.17

0.1954 0.26626 0.2332 0.2632 0.18707 0.24719 0.2247 0.21706 0.2202 0.2152 0.26524

−0.000164 −0.0003307 −0.0003096 −0.0004278 −0.00019846 −0.00048824 −0.000264 −0.00028952 −0.00028535 −0.0000497 −0.00028676

0.20215 0.28215 0.38218 0.2544 0.92882

−0.0002201 −0.0002585 −0.0006254 −0.0006595 −0.0030803

0.16853 0.20293 0.19428 0.14563 0.20414 0.17835 0.20571 0.20463 0.20731 0.20447 0.1918 0.23813 0.18854 0.19216 0.1659 0.16815 0.18421 −1.5128 0.20515 0.21211 0.256 0.22838 0.2333 0.21831 −0.432 0.20044 0.19989 0.20003

−0.00010817 −0.00021798 −0.000249 −0.0000536 −0.00021217 −0.00023704 −0.00020028 −0.00024252 −0.00024997 −0.00022612 −0.0002453 −0.00038397 −0.0001963 −0.0002105 −0.00022686 −0.00020535 −0.00016097 0.0079553 −0.00023933 −0.00021815 −0.0003542 −0.00035173 −0.00039223 −0.00029122 0.0057255 −0.00023544 −0.0002299 −0.00023573

C3

C4

0.000000412

0.00000266

−0.000010066

−0.000008078

1.861E-09

C5

Tmin, K

Thermal cond. at Tmin

Tmax, K

Thermal cond. at Tmax

252.45 180.26 178.15 188.36 173.55 87.89 180.25 142.61 159.95 213.15 388.85

0.1540 0.2067 0.1780 0.1972 0.1526 0.2043 0.1771 0.1758 0.1746 0.2046 0.1537

543.15 370.50 434.82 333.15 583.15 340.49 483.15 325.71 340.87 460.75 545.00

0.1063 0.1437 0.0986 0.1664 0.0713 0.0809 0.0971 0.1228 0.1229 0.1923 0.1090

242.54 460.65 197.67 223.15 289.95

0.1488 0.1631 0.2586 0.1072 0.2593

418.31 642.00 400.00 318.69 481.47

0.1101 0.1162 0.1320 0.0442 0.0624

329.35 279.01 164.65 237.38 176.98 373.96 234.94 178.18 236.50 267.76 158.45 156.08 247.79 229.33 165.78 172.22 398.40 357.20 247.57 288.45 180.35 173.15 119.36 178.35 273.16 225.30 247.98 286.41

0.1329 0.1421 0.1533 0.1329 0.1666 0.0897 0.1587 0.1614 0.1482 0.1439 0.1529 0.1782 0.1399 0.1439 0.1283 0.1328 0.1201 0.0445 0.1459 0.1492 0.1921 0.1675 0.1865 0.1664 0.5672 0.1474 0.1429 0.1325

723.15 526.73 339.12 480.77 394.27 426.00 357.31 474.85 482.00 508.62 483.15 276.02 449.27 442.53 372.39 387.91 629.60 393.20 469.08 561.20 410.00 278.25 345.60 434.52 633.15 413.10 417.58 413.10

0.0903 0.0881 0.1098 0.1199 0.1205 0.0774 0.1342 0.0895 0.0868 0.0895 0.0733 0.1322 0.1004 0.0990 0.0814 0.0885 0.0829 0.0590 0.0929 0.0897 0.1108 0.1305 0.0977 0.0918 0.4272 0.1032 0.1039 0.1027

The liquid thermal conductivity is calculated by k C1  C2T  C3T 2  C4T 3  C5T 4. where k is the thermal conductivity in W/(mK) and T is the temperature in K. Thermal conductivites are at either 1 atm or the vapor pressure, whichever is higher. All substances are listed by chemical family in Table 2-6 and by formula in Table 2-7. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AICHE and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). The number of digits provided for values at Tmin and Tmax was chosen for uniformity of appearance and formatting; these do not represent the uncertainties of the physical quantities, but are the result of calculations from the standard thermophysical property formulations within a fixed format.

TRANSPORT PROPERTIES TABLE 2-316

2-445

Transport Properties of Selected Gases at Atmospheric Pressure* Viscosity, 10−4 Pa⋅s Temperature, K

Thermal conductivity, W/(m⋅K) Temperature, K Substance

250

300

400

500

Acetone Acetylene Benzene

0.0080 0.0162 0.0077

0.0115 0.0213 0.0104

0.0201 0.0332 0.0195

0.0310 0.0452 0.0335

Bromine CCl4 Chlorine

0.0038 0.0053 0.0071

0.0048 0.0067 0.0089

0.0067 0.0099 0.0124

Deuterium Propylene R 11 R 12 R 13

0.122 0.0114 0.0072 0.0091

0.141 0.0168 0.0078 0.0097 0.0121

0.176 0.0226 0.0119 0.0151 0.0185

0.0208 0.0248

R 21 R 22 SO2

0.0080 0.0078

0.0088 0.0109 0.0096

0.0135 0.0170 0.0143

0.0181 0.0230 0.0200

600

300

400

500

600

0.101 0.135 0.101

0.128 0.164 0.127

0.156

0.0561 0.0524

0.077 0.104 0.076

0.0126 0.0156

0.0190

0.101 0.136

0.203 0.131 0.178

0.260 0.162 0.218

0.291 0.191 0.259

0.0580

0.111 0.073 0.094 0.108 0.123

0.126 0.087 0.110 0.126 0.145

0.153 0.115 0.144 0.162 0.190

0.178 0.141

0.201

0.0430

0.100 0.109

0.115 0.129 0.129

0.154 0.168 0.175

0.0290 0.0256

250

Prandtl number, dimensionless Temperature, K

0.217

250

300

400

500

0.860 0.827 0.796

0.797 0.814 0.781 0.766

0.762 0.761 0.745 0.759

0.708 0.757

0.820

0.779 0.771

0.773 0.760

0.154

0.256

*An approximate interpolation scheme is to plot the logarithm of the viscosity or the thermal conductivity versus the logarithm of the absolute temperature. At 250 K the viscosity of gaseous argon is to be read as 1.95 × 10−5 Pa⋅s = 0.0000195 Ns/m2.

TABLE 2-317

Lower and Upper Flammability Limits, Flash Point, and Autoignition Temperature for Selected Hydrocarbons

2-446

CAS no.

Formula

LFL

LFL rating

UFL

UFL rating

Flash point ºC

Flash point rating

74828 74840 74986 106978 75285 109660 78784 463821 1100543 142825 565593 111659 540841 111842 124185

CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C5H12 C6H14 C7H16 C7H16 C8H18 C8H18 C9H20 C10H22

5.00 2.90 2.00 1.50 1.80 1.30 1.30 1.40 1.05 1.00 1.12 0.80 0.95 0.70 0.70

12 12 12 12 12 14 12 11 13 14 4 12 11 12 12

15.00 13.00 9.50 9.00 8.40 8.00 8.00 7.50 7.68 7.00 6.75 6.50 6.00 5.60 5.40

12 12 12 12 12 14 12 11 13 14 4 12 11 12 12

−187.15 −130.15 −104.00 −69.00 −82.59 −40.00 −56.00 −72.15 −21.65 −4.15 −6.00 12.85 −12.22 30.85 45.85

4 4 4 6 5 11 11 4 11 11 4 11 11 11 11

536.85 471.85 449.85 287.85 460.00 242.85 420.00 450.00 224.85 203.85 335.00 205.85 411.00 204.85 200.85

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

Olefins Ethy1ene Propy1ene 1- Butene 2- Butene 1-Pentene

74851 115071 106989 107017 109671

C2H4 C3H6 C4H8 C4H8 C5H10

2.70 2.00 1.60 1.70 1.50

11 11 11 8 11

36.00 11.00 9.30 9.70 8.70

11 11 11 8 11

−146.15 −108.15 −79.81 <−34.00 −18.15

4 11 5 1 11

450.00 455.00 383.85 325.00 272.85

11 11 11 8 11

Acetylenes Acetylene Vinylacetylene Methylacetylene

74862 689974 613878

C2H2 C4H4 C3H4

2.50 1.80 1.70

12 6 11

80.00 100.00 16.80

12 4 11

−18.15 −66.15 −86.15

11 4 4

305.00 310.00 340.00

11 6 11

Aromatics Benzene Toluene o-Xylene Ethylbenzene Isopropylbenzene (Cumene) Anthracene

71432 108883 95476 100414 98828 120127

C6H6 C7H8 C8H10 C8H10 C9H12 C14H10

1.40 1.20 1.00 1.00 0.88 0.65

11 11 11 11 12 8

7.10 7.10 6.00 6.70 6.50 5.46

11 11 11 11 12 4

−11.15 4.85 16.85 15.00 31.00 120.85

11 11 11 11 12 4

561.85 535.85 463.85 431.85 423.85 540.00

11 11 11 11 11 8

Cyclic hydrocarbons Cyclopropane Furan Cyclopentadiene Cyclohexane Methylcyclohexane Phenol Dicyclopentadiene

75194 110009 542927 110827 108872 108952 77736

C3H6 C4H4O C5H6 C6H12 C7H14 C6H6O C10H12

2.40 2.30 1.62 1.30 1.10 1.52 1.00

12 11 4 8 11 6 4

10.40 14.30 7.70 7.80 6.70 8.76 6.76

12 11 4 8 11 6 4

92.00 −36.15 −28.15 −18.15 −6.00 79.85 26.00

4 11 4 4 11 11 4

497.85 510.00 429.85 245.00 250.00 714.85 505.00

11 6 4 11 11 11 5

Alcohols Methyl alcohol Ethyl alcohol 2-propen-1-ol n-Propyl alcohol Isopropyl alcohol n-Butyl alcohol 2-Butanol 2-Methyl-l-propanol 2-Methyl-2-propanol Cyclohexanol

67561 64175 107186 71238 67630 71363 78922 78831 75650 108930

CH4O C2H6O C3H6O C3H8O C3H8O C4H10O C4H10O C4H10O C4H10O C6H12O

7.30 3.30 2.50 2.10 2.20 1.40 1.68 1.70 1.90 1.20

11 8 11 8 8 8 4 11 4 11

36.00 19.00 18.00 13.50 12.00 11.20 9.80 10.90 9.00 9.30

11 8 11 8 8 8 4 11 4 10

10.85 12.85 20.85 24.63 21.85 37.19 23.85 27.85 10.85 67.85

11 4 11 5 4 5 4 11 4 11

463.85 356.00 377.85 440.00 399.00 359.00 405.00 408.00 480.00 300.00

11 8 11 8 5 5 8 11 8 11

50000 75070 107028 123386 123739 4170303 78842 123728 98011

CH2O C2H4O C3H4O C3H6O C4H6O C4H6O C4H8O C4H8O C5H4O2

7.00 4.00 2.80 2.60 2.10 2.10 1.60 2.50 2.10

11 11 11 11 11 8 4 8 8

73.00 60.00 31.00 16.10 15.50 15.50 10.60 12.50 19.30

11 11 11 11 11 8 4 4 4

−52.96 −38.15 −26.00 −30.00 8.00 13.00 −18.15 − 7.15 72.85

3 11 11 11 11 6 4 4 4

430.00 175.00 233.85 206.85 231.85 232.00 204.85 215.76 316.00

11 8 11 11 11 5 11 5 5

115106 107255 60297 101848

C2H6O C3H6O C4H10O C12H10O

3.30 2.60 1.90 0.80

11 11 11 11

27.30 39.00 48.00 1.50

11 11 11 11

−41.15 −56.00 −45.00 95.85

11 11 11 11

350.00 287.00 160.00 618.00

11 11 11 11

Compound

Paraffin hydrocarbons Methane Ethane Propane Butane Isobutane Pentane Isopentane 2,2-Dimethylpropane Hexane Heptane 2,3-Dimethylpentane Octane Isooctane Nonane Decane

Aldehydes Formaldehyde Acetaldehyde Acrolein Propionaldehyde Crotonic aldehyde Crotonaldehyde Isobutyraldehyde Butyraldehyde 2-Furancarboxaldehyde Ethers Dimethyl ether Viny1 methyl ether Diethyl ether Diphenyl ether

Autoignition T ºC

Autoignition rating

Ketones Acetone Methylethyl ketone Methy1 phenyl ketone

67641 78933 98862

C3H6O C4H8O C8H8O

2.60 1.80 1.10

8 11 8

13.00 10.00 6.89

8 11 4

−18.15 −6.15 82.00

4 11 8

465.00 515.85 571.11

8 11 11

Acids Acetic acid Hydrocyanic acid Formic acid

64197 74908 64186

C2H4O2 HCN CH2O2

4.05 6.00 15.70

8 11 10

16.00 41.00 38.00

4 11 11

42.85 −18.00 48.00

11 6 12

426.85 537.85 480.00

11 11 11

107313 109944

C2H4O2 C3H6O2

5.90 2.80

11 8

20.00 16.00

11 8

−19.15 −19.50

11 6

455.85 455.00

11 8

Esters Methyl formate Ethyl formate

Methyl acetate Vinyl acetate Ethyl acetate Propyl acetate Isopropyl acetate Butyl acetate Isobutyl acetate Amyl acetate Inorganic Hydrogen Ammonia Cyanogen

79209 108054 141786 109604 108214 123864 110190 628637 1333740 7664417 460195

3.20 2.60 2.20 2.05 2.00 1.70 1.30 1.10

8 11 11 5 5 8 11 11

16.00 13.40 11.40 8.00 7.80 7.60 10.50 7.50

8 11 11 4 4 8 11 11

−15.50 −8.15 −4.15 − 4.50 21.85 17.85 23.00

6 11 11 − 6 4 11 11

476.85 426.85 426.85 450.00 460.00 425.00 422.85 360.00

14 11 11 4 4 8 11 11

H2 NH3 C2N2

4.00 16.00 6.00

11 11 11

75.00 25.00 32.00

11 11 11

−260.15 Gas −62.15

4 3 4

400.00 650.85 661.85

11 11 4

C3H6O2 C4H6O2 C4H8O2 C5H10O2 C5H10O2 C6H12O2 C6H12O2 C7H14O2

Oxides Carbon monoxide Ethylene oxide Propylene oxide Dioxan Mesityl oxide Styrene oxide

630380 75218 75569 123911 141797 100425

CO C2H4O C3H6O C4H8O2 C6H10O C8H8O

12.50 3.00 2.10 2.00 1.30 1.10

11 11 11 11 4 11

74.00 100.00 21.50 22.00 7.20 −

11 8 11 11 4 −

−206.15 −19.81 −37.15 11.85 30.00 31.85

4 5 11 11 4 11

608.85 428.85 465.00 180.00 344.85 463.00

11 11 11 11 4 11

Peroxides Di-tert-butyl peroxide

110054

C8H18O2

1.00

6

7.05

4

1.00

−

79.85

4

CS2 H2S COS C2H6S

1.30 4.30 12.00 2.20

11 14 11 8

50.00 45.50 29.00 19.70

11 14 11 8

−30.00 −107.15 −88.15 −36.00

− − − −

90.00 260.00 250.00 205.00

11 11 6 11

CH3Cl C2H5Cl C3H7Cl C2H2Cl2 C3H6Cl2 CH2Cl2 C2H5ClO C2HCl3 C3Cl6 C2H3Cl C6H5Cl C7H7Cl

8.10 3.80 2.80 6.20 3.40 14.00 4.90 8.00 2.90 3.60 1.30 1.10

11 11 11 11 11 6 11 11 1 11 11 11

17.20 15.40 10.70 16.00 14.50 22.00 15.90 40.00 15.70 33.00 7.10 8.00

11 11 11 6 11 4 11 5 1 11 11 4

−70.15 −50.00 −35.00 12.85 13.00 − 9.15 40.55 32.22 121.85 −78.15 31.85 60.00

4 11 11 11 11 4 12 11 10 11 11 11

631.85 518.85 592.85 412.85 556.85 615.00 425.00 410.00 610.00 471.85 637.85 585.00

11 11 11 11 11 11 11 11 11 11 11 11

CH3Br

10.00

11

15.00

11

−48.15

4

537.22

11

Sulfur containing Carbon disulfide Hydrogen sulfide Carbon oxysulfide Dimethyl sulfide Chlorine containing Methyl chloride Ethyl chloride Isopropyl chloride Ethylene dichloride Propylene dichloride Dichloromethane 2-Chloroethanol Trichloroethylene Hexachlorobutadiene Vinyl chloride Chlorobenzene Benzyl chloride Bromides Methyl bromide

75150 7783064 463581 75183 74873 75003 75296 107062 78875 75092 107073 79016 87683 75014 108907 100447 74839

Glycols Ethylene glycol Diethylene glycol Triethylene glycol

107211 111466 112276

C2H6O2 C4H10O3 C6H14O4

3.20 2.00 0.90

11 10 11

33.00 17.10 9.20

10 10 11

110.85 123.85 156.00

11 11 11

395.85 223.85 370.85

11 11 11

Amines Methyl amine Ethyl amine Dimethyl amine Isopropylamine Trimethylamine Isopropylamine Allylamine Diethyl amine tert-Butylamine Triethyl amine Cyclohexylamine Monoethanolamine Diethanolamine Dimethylethanolamine

74895 75047 124403 75310 75503 75310 107119 109897 75349 121448 108918 141435 111422 108010

CH5N C2H7N C2H7N C3H7N C3H9N C3H9N C3H7N C4H11N C4H11N C6H15N C6H13N C2H7NO C4H11NO2 C4H11NO

4.90 3.50 2.80 2.00 2.00 2.00 2.20 1.70 1.70 1.20 0.66 3.02 1.69 1.67

11 8 11 11 11 11 11 12 11 11 11 4 4 4

20.70 14.00 14.40 10.40 11.60 10.40 22.00 10.10 8.90 8.00 8.42 24.06 16.25 11.36

11 8 11 11 11 11 11 12 11 11 4 4 4 4

−62.00 −39.00 −57.00 −37.00 − 7.15 −37.00 −28.89 −39.00 − 9.00 −12.15 31.00 85.00 151.85 41.00

7 6 7 11 11 11 11 11 11 12 13 11 11 11

430.00 383.85 400.00 400.00 190.00 400.00 373.89 310.00 380.00 231.85 293.00 410.00 661.85 295.00

11 7 11 12 11 12 11 12 5 10 11 5 11 11

C3H3N C6H15N B2H6 C5H8O2 − C8H8 C12H10 C4H6O2 C8H4O3

2.42 1.30 0.90 2.10 0.80 1.10 0.60 2.80 1.70

11 11 11 11 8 11 11 4 8

17.34 11.00 98.00 12.50 5.00 6.10 5.80 25.00 10.50

13 11 11 11 8 11 11 4 4

0.00 70.00 −148.15 11.00 −6.00 31.85 113.00 5.85 151.85

11 11 4 11 4 11 11 4 6

480.85 616.85 51.85 435.85 290.00 490.00 540.00 468.00 583.85

11 11 11 11 4 11 11 5 11

Miscellaneous Acrylonitrile Aniline Diborane Methyl methacrylate Naptha Styrene Biphenyl Methyl acrylate Phthalic anhydride

107131 62533 19287457 80626 8030306 100425 92524 96333 85449

2-447

LFL and UFL are lower and upper flammability limits respectively in volume percent fuel in air. Values in this table were taken from the Design Institute for Physical Properties (DIPPR) of the American Institute of Chemical Engineers (AIChE), copyright 2007 AIChE and reproduced with permission of AIChE, the DIPPR Environmental Safety Property Data and Estimations Steering Committee and of the DIPPR Evaluated Process Design Data Project Steering Committee. Their source should be cited as T. N. Rogers, D. A. Zei, R. L. Rowley, W. V. Wilding, J. L. Oscarson, Y. Yang, N. A. Zundel, T. E. Daubert, R. P. Danner, DIPPR® Data Compilation of Pure Chemical Properties, Design Institute for Physical Properties, AIChE, New York (2007). Flammability limits are from B. Lewis and G. Von Elbe, Combustion, Flames and Explosions of Gases, New York: Harcourt Brace Jovanovich (1987). Flash point data are from N. I. Sax, Dangerous Properties of Industrial Materials, 6th ed., New York: Van Nostrand Reinhold (1984). Autoignition data are from I. Glassman, Combustion, 3d ed., New York: Academic Press (1996). The ratings reflect DIPPR® ESP’s effort to provide a critical evaluation and quality assessment of each data point with 15 being the highest score possible. The rating is not directly correlated with the estimated experimental uncertainty.

2-448

PHYSICAL AND CHEMICAL DATA

TABLE 2-318

Viscosities of Liquids: Coordinates for Use with Fig. 2-32

Liquid

X

Y

Liquid

X

Y

Acetaldehyde Acetic acid, 100% Acetic acid, 70% Acetic anhydride Acetone, 100% Acetone, 35% Acetonitrile Acrylic acid Allyl alcohol Allyl bromide Allyl iodide Ammonia, 100% Ammonia, 26% Amyl acetate Amyl alcohol Aniline Anisole Arsenic trichloride Benzene Brine, CaCl2, 25% Brine, NaCl, 25% Bromine Bromotoluene Butyl acetate Butyl acrylate Butyl alcohol Butyric acid Carbon dioxide Carbon disulfide Carbon tetrachloride Chlorobenzene Chloroform Chlorosulfonic acid Chlorotoluene, ortho Chlorotoluene, meta Chlorotoluene, para Cresol, meta Cyclohexanol Cyclohexane Dibromomethane Dichloroethane Dichloromethane Diethyl ketone Diethyl oxalate Diethylene glycol Diphenyl Dipropyl ether Dipropyl oxalate Ethyl acetate Ethyl acrylate Ethyl alcohol, 100% Ethyl alcohol, 95% Ethyl alcohol, 40% Ethyl benzene Ethyl bromide 2-Ethyl butyl acrylate Ethyl chloride Ethyl ether Ethyl formate 2-Ethyl hexyl acrylate Ethyl iodide Ethyl propionate Ethyl propyl ether Ethyl sulfide Ethylene bromide Ethylene chloride Ethylene glycol Ethylidene chloride Fluorobenzene Formic acid Freon-11 Freon-12 Freon-21 Freon-22

15.2 12.1 9.5 12.7 14.5 7.9 14.4 12.3 10.2 14.4 14.0 12.6 10.1 11.8 7.5 8.1 12.3 13.9 12.5 6.6 10.2 14.2 20.0 12.3 11.5 8.6 12.1 11.6 16.1 12.7 12.3 14.4 11.2 13.0 13.3 13.3 2.5 2.9 9.8 12.7 13.2 14.6 13.5 11.0 5.0 12.0 13.2 10.3 13.7 12.7 10.5 9.8 6.5 13.2 14.5 11.2 14.8 14.5 14.2 9.0 14.7 13.2 14.0 13.8 11.9 12.7 6.0 14.1 13.7 10.7 14.4 16.8 15.7 17.2

4.8 14.2 17.0 12.8 7.2 15.0 7.4 13.9 14.3 9.6 11.7 2.0 13.9 12.5 18.4 18.7 13.5 14.5 10.9 15.9 16.6 13.2 15.9 11.0 12.6 17.2 15.3 0.3 7.5 13.1 12.4 10.2 18.1 13.3 12.5 12.5 20.8 24.3 12.9 15.8 12.2 8.9 9.2 16.4 24.7 18.3 8.6 17.7 9.1 10.4 13.8 14.3 16.6 11.5 8.1 14.0 6.0 5.3 8.4 15.0 10.3 9.9 7.0 8.9 15.7 12.2 23.6 8.7 10.4 15.8 9.0 5.6 7.5 4.7

Freon-113 Glycerol, 100% Glycerol, 50% Heptane Hexane Hydrochloric acid, 31.5% Iodobenzene Isobutyl alcohol Isobutyric acid Isopropyl alcohol Isopropyl bromide Isopropyl chloride Isopropyl iodide Kerosene Linseed oil, raw Mercury Methanol, 100% Methanol, 90% Methanol, 40% Methyl acetate Methyl acrylate Methyl i-butyrate Methyl n-butyrate Methyl chloride Methyl ethyl ketone Methyl formate Methyl iodide Methyl propionate Methyl propyl ketone Methyl sulfide Napthalene Nitric acid, 95% Nitric acid, 60% Nitrobenzene Nitrogen dioxide Nitrotoluene Octane Octyl alcohol Pentachloroethane Pentane Phenol Phosphorus tribromide Phosphorus trichloride Propionic acid Propyl acetate Propyl alcohol Propyl bromide Propyl chloride Propyl formate Propyl iodide Sodium Sodium hydroxide, 50% Stannic chloride Succinonitrile Sulfur dioxide Sulfuric acid, 110% Sulfuric acid, 100% Sulfuric acid, 98% Sulfuric acid, 60% Sulfuryl chloride Tetrachloroethane Thiophene Titanium tetrachloride Toluene Trichloroethylene Triethylene glycol Turpentine Vinyl acetate Vinyl toluene Water Xylene, ortho Xylene, meta Xylene, para

12.5 2.0 6.9 14.1 14.7 13.0 12.8 7.1 12.2 8.2 14.1 13.9 13.7 10.2 7.5 18.4 12.4 12.3 7.8 14.2 13.0 12.3 13.2 15.0 13.9 14.2 14.3 13.5 14.3 15.3 7.9 12.8 10.8 10.6 12.9 11.0 13.7 6.6 10.9 14.9 6.9 13.8 16.2 12.8 13.1 9.1 14.5 14.4 13.1 14.1 16.4 3.2 13.5 10.1 15.2 7.2 8.0 7.0 10.2 15.2 11.9 13.2 14.4 13.7 14.8 4.7 11.5 14.0 13.4 10.2 13.5 13.9 13.9

11.4 30.0 19.6 8.4 7.0 16.6 15.9 18.0 14.4 16.0 9.2 7.1 11.2 16.9 27.2 16.4 10.5 11.8 15.5 8.2 9.5 9.7 10.3 3.8 8.6 7.5 9.3 9.0 9.5 6.4 18.1 13.8 17.0 16.2 8.6 17.0 10.0 21.1 17.3 5.2 20.8 16.7 10.9 13.8 10.3 16.5 9.6 7.5 9.7 11.6 13.9 25.8 12.8 20.8 7.1 27.4 25.1 24.8 21.3 12.4 15.7 11.0 12.3 10.4 10.5 24.8 14.9 8.8 12.0 13.0 12.1 10.6 10.9

TRANSPORT PROPERTIES

Nomograph for viscosities of liquids at 1 atm. For coordinates see Table 2-318. To convert centipoises to pascalseconds, multiply by 0.001.

FIG. 2-32

2-449

2-450

PHYSICAL AND CHEMICAL DATA

TABLE 2-320

TABLE 2-319

Viscosity of Sucrose Solutions* Viscosity in centipoises

Percentage sucrose by weight

Temp., °C

20

40

0 5 10 15 20 25 30 35 40 45

3.818 3.166 2.662 2.275 1.967 1.710 1.510 1.336 1.197 1.074

14.82 11.60 9.830 7.496 6.223 5.206 4.398 3.776 3.261 2.858

60

Temp., °C

113.9 74.9 56.7 44.02 34.01 26.62 21.30 17.24

50 55 60 65 70 75 80 85 90 95

Percentage sucrose by weight 20

40

60

0.974 0.887 0.811 0.745 0.688 0.637 0.592 0.552

2.506 2.227 1.989 1.785 1.614 1.467 1.339 1.226 1.127 1.041

14.06 11.71 9.87 8.37 7.18 6.22 5.42 4.75 4.17 3.73

*International Critical Tables, vol. 5, p. 23. Bingham and Jackson, Bur. Standards Bull. 14 (1919): 59.

FIG. 2-33 and TABLE 2-320

Nomograph (right) for thermal conductivity of organic liquids. (From Mallu and Rao, Hydroc. Proc. 78, 1988.)

TRANSPORT PROPERTIES TABLE 2-321

2-451

Prandtl Number of Air* Pressure, bar

Temperature, K

1

5

10

20

30

40

50

60

70

80

90

100

80 90 100 120 140

mix 0.796 0.786 0.773 0.763

2.31 1.76 0.872 0.813 0.782

2.32 1.77 1.54 0.89 0.82

2.35 1.78 1.53 1.44 0.94

2.37 1.79 1.53 1.65 1.20

2.40 1.81 1.53 1.54 1.59

2.42 1.82 1.53 1.48 2.14

2.45 1.83 1.53 1.43 2.43

2.48 1.85 1.53 1.40 2.07

2.51 1.87 1.54 1.38 1.78

2.54 1.89 1.54 1.36 1.62

2.57 1.91 1.55 1.34 1.52

160 180 200 240 280

0.754 0.745 0.738 0.724 0.710

0.765 0.754 0.743 0.727 0.711

0.78 0.763 0.749 0.729 0.713

0.84 0.792 0.766 0.737 0.717

0.92 0.830 0.788 0.746 0.721

1.03 0.876 0.812 0.756 0.726

1.13 0.932 0.841 0.767 0.731

1.25 1.00 0.87 0.78 0.737

1.37 1.07 0.90 0.80 0.742

1.65 1.14 0.95 0.81 0.75

1.83 1.20 0.97 0.81 0.75

1.72 1.25 1.00 0.82 0.76

300 350 400 450 500

0.705 0.699 0.694 0.691 0.689

0.707 0.699 0.694 0.691 0.689

0.708 0.699 0.694 0.691 0.689

0.712 0.701 0.695 0.691 0.689

0.715 0.703 0.696 0.692 0.689

0.717 0.705 0.697 0.692 0.690

0.721 0.707 0.698 0.693 0.690

0.725 0.709 0.699 0.693 0.690

0.728 0.711 0.700 0.694 0.690

0.732 0.712 0.701 0.695 0.691

0.737 0.714 0.703 0.695 0.691

0.742 0.716 0.704 0.696 0.691

600 700 800 900 1000

0.690 0.696 0.705 0.709 0.711

0.690 0.696 0.704 0.709 0.711

0.690 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.704 0.708 0.711

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.689 0.695 0.703 0.708 0.710

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

0.690 0.695 0.702 0.708 0.709

*Compiled by P. E. Liley from tables of specific heat at constant pressure, thermal conductivity, and viscosity given in SI units for integral kelvin temperatures and pressures in bars by Vasserman. Thermophysical Properties of Air and Its Components and Thermophysical Properties of Liquid Air and Its Components. Nauka, Moscow, and in translated form by the National Bureau of Standards, Washington. The number of significant figures given above reflects the similar numbers appearing for the constituent properties in the source references. While reasonable agreement occurs for atmospheric pressure with some other works, the fragmentary data available for the saturated, etc., states show large deviations.

TABLE 2-322

Prandtl Number of Liquid Refrigerants* Temperature, K Refrigerant

Trichlorofluoromethane Dichlorodifluoromethane Chlorotrifluoromethane Bromotrifluoromethane Dichlorofluoromethane Chlorodifluoromethane Methyl chloride Trichlorotrifluoroethane Dichlorotetrafluoroethane Chloropentafluoroethane Ethane Propane Octafluorocyclobutane Dichlorodifluoromethane/difluoroethane Chlorodifluoromethane/chloropentafluoroethane Trifluoromethane/chlorotrifluoromethane Methylene fluoride/chloropentafluoroethane Butane Isobutane (2-methyl propane) Ammonia Water Ethylene Propylene

No. 11 12 13 13B1 21 22 40 113 114 115 170 290 C318 500 502 503 504 600 600a 717 718 1150 1270

180

200

220

240

260

280

300

11.9 5.25 2.96 3.75

8.64 4.27 2.67 3.27 5.72

6.73 3.65 2.69 2.94 4.50

5.33 3.27 3.05 2.83 3.87

4.74 3.08 3.57 3.03 3.48

4.18 3.04 3.61 3.25

2.93 2.42

2.79 2.40

2.77 2.45

2.87 2.60

— 15.13 7.85

3.23 2.53 — 11.18 6.16

8.59 5.21

6.94 4.67

5.77 4.40

5.06 4.46

2.55 5.28 —

2.29 4.46 — 5.78 5.73

2.22 3.88 — 4.23 4.71

2.40 3.44 11.2 3.40 4.13

2.70 3.16 8.74 3.13 3.81

3.02 7.35 3.01

3.16 6.37 3.13

2.10

2.09 4.90 6.19 8.26

2.24 3.60 5.20 6.36

2.43 3.04 4.44 5.18 1.97

2.89 2.79 3.83 4.49 1.76

— 1.74 2.24

— 1.78 1.88

— 2.07 1.71

— 2.70 1.71

7.00 4.80 4.68 — 25.7 —

8.35

3.76

— — 1.85 3.80

320

340

360

380

3.19 — 4.52 3.16

3.44 —

4.00 — —

— —

3.17

3.18 2.85 7.04 4.78 4.90

3.54

—

—

5.87 3.35

5.96 3.72

2.69 3.44 3.93 1.54

— 2.85 3.22 3.66 1.40

— 3.30 3.07 3.53 1.29

— — 3.02 3.53 1.24

10.3 4.4 1.88

5.69 — 2.24

3.65 — 3.91

2.60 — 4.73

6.23 4.82

— 5.61

5.18

—

—

—

—

—

— — —

— —

— —

3.77 1.25

4.68 1.34

1.99 —

1.59 — —

*Dashes indicate inaccessible states. Average uncertainty is about 20 percent. Values derived from formulations for thermal conductivity, specific heat at constant pressure, and viscosity contained in Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. For further details see M. W. Johnson, M.S.M.E. thesis, Purdue University, West Lafayette, Ind., 1976.

2-452

PHYSICAL AND CHEMICAL DATA

TABLE 2-323

Thermophysical Properties of Miscellaneous Saturated Liquids Temperature, °C

Substance

Property

Acetaldehyde ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr Acetic acid

−50

−40

−30

−20

−10

0

863 2.05 460 0.211 4.47

852 2.08 404 0.206 4.08

840 2.11 358 0.200 3.78

828 2.14 321 0.195 3.52

816 2.17 290 0.189 3.33

804 2.20 263 0.184 3.14

10

20

794 2.24 241 0.182 2.97

783 2.28 222 0.180 2.81

ρ (kg/m3) cp (kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

30

40

50

60

1049 2.031 1210 0.173 14.2

1039

1028

1018

1006

1102 0.170

1010 0.168

795 0.167

600 0.165

70

80

90

100

995

984

972

960

0.163

0.161

Aniline

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

1039 2.024 10200 0.186 111

1030 2.047 6500 0.184 72

1022 2.071 4400 0.182 50

1013 2.093 3160 0.180 36.7

1005 2.113 2370 0.177 28.3

996 2.132 1850 0.174 22.7

987 2.17 1510 0.171 19.2

978 2.20 1270 0.169 16.5

969 2.23 1090 0.168 14.5

960 2.27 935 0.167 12.7

951 2.32 825 0.167 11.5

Butanol

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

845 1.947 34700 0.175 3860

841 1.996 22400 0.174 2570

837 2.046 14700 0.173 1740

833 2.100 10300 0.172 1260

829 2.153 7400 0.171 930

825 2.202 5190 0.170 670

817 2.262 3870 0.168 120

810 2.345 2950 0.167 41

803 2.437 2300 0.166 33.8

797 2.524 1780 0.165 27.2

791 2.621 1410 0.164 22.5

784

776

768

760

753

1140 0.163

930 0.162

760 630 535 0.161 0.160 0.159

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1362 0.988 630 0.194 3.21

1348 0.989 580 0.190 3.02

1334 0.990 535 0.186 2.85

1320 0.991 496 0.182 2.70

1306 0.993 463 0.178 2.58

1292 0.996 435 0.174 2.49

1278 1.004 405 0.170 2.39

1263 1.017 375 0.166 2.30

350 0.161

330 0.158

0.156

0.154

0.152

0.150

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

789 2.068 1175 0.122 19.9

779 2.081 980 0.120 17.0

769 2.094 820 0.119 14.4

759 2.106 710 0.118 12.7

750 2.119 605 0.117 11.0

740

731

721

540 0.116

0.114

0.112

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

2.01 6400 0.188 68.4

2.04 4790 0.186 52.5

2.08 3650 0.184 41.3

2.13 2825 0.181 33.2

2.19 2220 0.179 27.2

806 2.27 1770 0.177 22.7

798 2.35 1470 0.175 19.7

789 2.43 1200 0.173 16.9

781 2.52 1000 0.171 14.7

776 2.62 835 0.168 13.0

763 2.73 700 0.165 11.6

754 2.83 590 0.162 10.3

745 2.93 500 0.159 9.2

735 3.03 435 0.156 8.4

725 3.19 370 0.153 7.7

716 3.30 314 0.151 6.9

947

935

924

912

888

876

863

851

838

825

811

797

580

510

901 2.01 455 0.145 6.3

400 0.142

370 0.139

345 0.136

310 0.133

280 0.130

250 230 220 0.127 0.123 0.119

Carbon disulfide

Cyclohexane

Ethanol

Ethyl acetate

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) 1090 k (W/m⋅K) Pr

Ethylamine

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

761 2.95 580 0.204 8.39

750 2.97 500 0.201 7.39

739 2.98 435 0.199 6.51

729 3.00 390 0.196 5.97

718 3.01 350 0.194 5.43

707 3.03 320 0.191 5.08

695

683

671

658

646

633

620

607

Ethyl ether

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

790 2.135 550 0.159 7.39

780 2.156 470 0.155 6.54

769 2.179 410 0.151 5.92

758 2.205 365 0.147 5.48

747 2.233 330 0.144 5.12

736 2.265 290 0.140 4.69

725 2.299 265 0.139 4.38

714 2.332 233 0.134 4.05

702 2.36 214 0.129 3.92

689 2.39 197 0.125 3.77

676 2.43 181 0.120 3.67

666 2.47 166 0.116 3.54

653 2.51 153 0.112 3.43

640

625

611

140

129

118

Ethyl iodide

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

0.656 0.663 0.670 0.677 730 0.092 5.37

0.684 655 0.090 4.98

0.691 590 0.088 4.63

0.698 539 0.086 4.30

0.705 495 0.085 4.11

0.712 455 0.083 3.90

0.718 420 0.081 3.72

0.724 390 0.080 3.53

Ethylene glycol

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1127 2.272 57000 0.254 510

1120 2.327 33300 0.255 305

1113 2.381 20200 0.256 190

1106 2.431 13400 0.258 126

1099 2.484 9100 0.259 87.3

1092 2.536 7070 0.260 69.0

1085 2.586 4000

1077 2.636 3450

1070 1063 1056 2.685 2.734 2.779 3000 2440 2000

Formic acid

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1241

1231

1220

1209

1196

1184

1170

1156

1140

0.265

2260 0.261

1800 0.257

1470 0.257

1220 0.253

1030 0.250

890 0.246

780 0.243

680 615 550 0.240 0.236 0.232

1124

1108

TRANSPORT PROPERTIES TABLE 2-323

2-453

Thermophysical Properties of Miscellaneous Saturated Liquids (Concluded) Temperature, °C

Substance Gasoline

Glycerol

Kerosene

Property

−50

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) 1710 k (W/m⋅K) 0.131 Pr ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

—

−40

1400 0.128 —

−30

−20

784 1.88 1170 990 0.125 0.123 15.1 —

—

−10

0

775 1.92 850 0.121 13.5 —

10

20

30

40

50

60

70

80

90

100

767 1.97 735 0.120 12.1

759 2.02 645 0.118 11.0

751 2.06 530 0.116 9.41

743 2.11 464 0.114 8.59

735 2.15 410 0.112 7.87

721 2.20 367 0.110 7.34

717 2.25 330 0.108 6.88

708 2.30 298 0.106 6.47

699 2.35 270 0.104 6.10

690 2.41 246 0.102 5.81

681 2.46 225 0.100 5.54

1276

1270

1260 2.393 1.5.+6 0.284 12650

1254 2.406

1248 2.457

1242 2.504

2.548

2.588

2.625 2.657 2.686

0.285

0.287

0.288

0.289

0.291

0.293 0.294 0.295

2.28 73

2.32 66

2.35 60

2.38 55

1.2.+7 4.0.+6

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) 1150 k (W/m⋅K) Pr

725

500

360

275

781 1.91 215 0.140 2.93

774 1.96 173 0.139 2.44

767 2.02 149 0.139 2.17

760 2.07 126 0.138 1.89

754 2.13 108 0.138 1.67

748 2.18 95 0.137 1.51

742 2.23 83 0.137 1.35

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k(W/m⋅K) Pr

2.30 2305 0.225 23.6

2.32 1800 0.222 18.8

2.35 1410 0.219 15.1

2.37 1170 0.216 12.9

2.40 975 0.212 11.0

2.42 820 0.209 9.53

2.45 692 0.206 8.23

2.47 590 0.203 7.18

783 2.49 510 0.199 6.38

774 2.52 455 0.195 5.88

766 2.55 400 0.192 5.31

756 2.65 355 0.189 4.98

746 2.78 315 0.187 4.68

736 2.94 271 0.184 4.34

725 3.13 240 0.182 4.13

711 3.30 218 0.180 3.99

Methyl formate

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

1069 1.84 830 0.217 7.04

1056 1.86 711 0.213 6.21

1043 1.88 618 0.209 5.56

1030 1.90 544 0.205 5.04

1017 1.92 481 0.200 4.62

1003 1.95 430 0.195 4.30

989 1.99 380 0.191 3.96

975 2.03 345 0.186 3.77

960 2.08 315 0.180 3.64

944

929

913

897

880

863

845

Oil, castor

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Methanol

Oil, olive

Pentane

Propanol

Sulfuric acid

2,420,000 986,000 451,000 231,000 125,000 74,000 43,000 0.182 0.181 0.180 0.179 0.178 0.177 0.176 0.175 0.174 0.17

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

138,000 0.170

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

693 2.060 489 0.142 7.14

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

849 1.955 20,200 13,500 9500 6900 0.167 0.166 0.165 236

684 2.084 428 0.139 6.42

674 2.110 379 0.136 5.88

665 2.137 339 0.132 5.49

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Turpentine

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

923 1.535 1670 0.149 17.8

913 1.556 1345 0.147 14.2

904 1.579 1100 0.144 12.1

36,300 0.167

24,500 0.166

17,000 12,400 0.166 0.165 0.165 0.164 0.164

646 2.206 279 0.125 4.92

636 2.239 254 0.122 4.66

626 2.273 234 0.119 4.47

616

606

596

585

574

562

209 0.115

190 0.112

175 0.108

161 0.105

148 0.101

137 124 113 0.098 0.095 0.091

811

814

796

788

779

770

761

752

5110

819 2.219 3900

2900

2245

1720 0.171

1400 0.169

1130 0.168

921 0.167

760 0.165

630 508 447 0.164 0.163 0.162

1834 1.382 25,400

15,700

11,500

8820

7220

6090

5190

829 1.80 380 0.124 5.5

820 1.83 355 0.122 5.3

810 1.87 325 0.119 5.1

820

730

675

48,400 35,200 0.314 932 1.514 2120 0.152 21.1

52,000 0.168

656 2.167 307 0.128 5.20

ρ (kg/m3) cp(kJ/kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr

Toluene

914 1.633 84,000 0.169 810

895 1.602 915 0.142 10.3

886 1.633 770 0.139 9.0

876 1.652 670 0.137 8.1

867 1.675 590 0.134 7.4

858 1.701 520 0.132 6.7

848 1.73 470 0.129 6.3

839 1.76 420 0.126 5.9

1.72 2250 0.130 29.8

1.76 1780 0.129 24.3

1.80 1490 0.128 20.9

1270 0.127 18.4

1070 0.126 16.1

1.93 925 0.125 14.3

550

747

800 1.92 295 0.117 4.8

538

743

790 1.97 270 0.114 4.7

2-454

PHYSICAL AND CHEMICAL DATA

TABLE 2-324

Diffusivities of Pairs of Gases and Vapors (1 atm) Dv in cm2/s

Substance Acetic acid Acetone n-Amyl alcohol sec-Amyl alcohol Amyl butyrate Amyl formate i-Amyl formate Amyl isobutyrate Amyl propionate Aniline Anthracene Argon Benzene Benzidine Benzyl chloride n-Butyl acetate i-Butyl acetate n-Butyl alcohol i-Butyl alcohol Butyl amine i-Butyl amine i-Butyl butyrate i-Butyl formate i-Butyl isobutyrate i-Butyl proprionate i-Butyl valerate Butyric acid i-Butyric acid Cadmium Caproic acid i-Caproic acid Carbon dioxide

Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorobenzene Chloroform Chloropicrin m-Chlorotoluene o-Chlorotoluene p-Chlorotoluene Cyanogen chloride Cyclohexane n-Decane Diethylamine 2,3-Dimethyl butane Diphenyl n-Dodecane Ethane Ethanol Ether (diethyl) Ethyl acetate Ethyl alcohol Ethyl benzene Ethyl n-butyrate Ethyl i-butyrate Ethylene Ethyl formate Ethyl propionate Ethyl valerate Eugenol Formic acid Helium n-Heptane n-Hexane Hexyl alcohol Hydrogen

Temp., °C 0 0 0 30 0 0 0 0 0 0 30 0 20 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 25 500‡ 0 0 450‡ 0 30 0 25 0 0 0 0 15 45 90 0 15 0 126 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 20 38 15 0 0 25 500

Air

A

H2

0.1064 .109 .0589 .072 .040 .0543 .058 .0419 .046 .0610 .075 .0421

0.416 .361 .235

.077 .0298 .066 .058 .0612 .0703 .088 .0727 .0821 .0853 .0468 .0705 .0457 .0529 .0424 .067 .0679

.306

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

0.0716 .0422

.171 .1914

.0347

0.194 0.0797

.0528

.2364 .2716

.0425 .0476

.2771

.0483

.185

.0327

.191 .203 .173 .264 .271

.0364 .0366 .0308 .0476 .0471 .17

.050 .0513 .138

.550

.139

0.096

0.153

.163 .0996*

.00215†

.9 .0892

.369 .651 .293

.185 1.0 0.0636

.319

.0744

.063 .137

0.116

.075 .091 .088 .054 .059 .051 .111 0.0719

.0760

.086 .306

.0841

.0884 .0657

.301

.0753

.0751

.0610 .308 .459 .377 .298 .273

.0778 .0715 .089 .102 .0658 .0579 .0591 .0840 .068 .0512 .0377 .1308

.0813 .0686 .0546 .0487

.375

.0685

.224 .229 .486 .337 .236 .205

.0407 .0413 .0573 .0450 .0367

.510

Ref. 8 6, 16 8 5 8 8 8 8 8 8 5 8 18 8, 15 8 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 13 8 8 8 19 1, 9 18 8 8 18 16, 17 5 6 10 8 8 8 10 3 6 3 8 3 8 3 8 20 7, 8 8 5 8 8 8 8 8 8 4, 8 8 8 8 8 19

.0874

.641 .705 .066§ .0663 .0499 .611

.290 .200

.0753 .697 4.2

.0757 .674

.0351 .550 .646

.535

.625

0.459 .537

0.486 .726

0.272

0.277

3 8 8 2 18

TRANSPORT PROPERTIES TABLE 2-324

2-455

Diffusivities of Pairs of Gases and Vapors (1 atm) (Concluded) Dv in cm2/s

Substance

Temp., °C

Air

Hydrogen cyanide Hydrogen peroxide Iodine Mercury Mesitylene Methane Methyl acetate Methyl alcohol Methyl butyrate Methyl i-butyrate Methyl cyclopentane Methyl formate Methyl propionate Methyl valerate Naphthalene Nitrogen

0 60 0 0 0 500 0 0 0 0 15 0 0 0 0 0 25 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 30 0

0.173 .188 .07 .112 .056

Nitrous oxide n-Octane Oxygen Phosgene Propionic acid Propyl acetate n-Propyl alcohol i-Propyl alcohol n-Propyl benzene i-Propyl benzene n-Propyl bromide i-Propyl bromide Propyl butyrate Propyl formate n-Propyl iodide i-Propyl iodide n-Propyl isobutyrate i-Propyl isobutyrate Propyl propionate Propyl valerate Safrol i-Safrol Sulfur hexafluoride Toluene Trimethyl carbinol 2,2,4-Trimethyl pentane 2,2,3-Trimethyl heptane n-Valeric acid i-Valeric acid Water

H2

O2

N2

CO2

N2O

CH4

C2H6

C2H4

n-C4H10

i-C4H10

0.070 .13

0.53 1.1

.084 .132 .0633 .0639 0.0731 .0872 .0735 0.0569 .0513

.333 .506 .242 .257 .318

0.0567 .0879 .0446 .0451 0.0742

0.0758

.295

.0528 0.181 0.165 .096

0.535 .0505 0.0642 .178 .095 .0829 .067 .085 .0818 .101 .0481 .0489 .085 .0902 .0530 .0712 .079 .0802 .0549 .059 .057 .0466 .0434 .0455

.271 .697

0.0705

0.0710 0.181

.139

.330

.0588

.315

.0577

.206 .281

.0364 .0490

.212

.0388

.212 .189

.0395 .0341

.418 .076 .088 .087

30 90 0 0 0 450

A

0.071

0.0618

.288

0.0688

.270 0.050 0.0544 0.220

.212 .75

0.148

0.163

0.0960

0.0908

Ref. 10 11 8, 12, 14 8, 12, 13 8 18 8 8 8 8 3 8 8 8 8 8 2 8 8 3 8 10 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 4, 8 5 8

0.0705

3

0.0684

3 8 8 8, 20 18

.0376 .138 1.3

*320 mmHg. † 40 atm. ‡ Also at other temperatures. § Strong function of concentration. References 1 Amdur, Irvine, Mason, and Ross, J. Chem. Phys., 20, 436 (1952). 2 Boyd, Stein, Steingrimsson, and Rumpel, J. Chem. Phys., 19, 548 (1951). 3 Cummings and Ubbelohde, J. Chem. Soc. (London), 1953, p. 3751. 4 Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 5 Gilliland, Ind. Eng. Chem., 26, 681 (1934). 6 Gorynnova and Kuvskinskii, Zhur. Tekh. Fiz., 18, 1421 (1948). 7 Hansen, Dissertation, Jena, 1907. 8 International Critical Tables, vol. 5, p. 62. 9 Jeffries and Drickamer, J. Chem. Phys., 22, 436 (1954). 10 Klotz and Miller, J. Am. Chem. Soc., 69, 2557 (1947). 11 McMurtrie and Keyes, J. Am. Chem. Soc., 70, 3755 (1948). 12 Mullaly and Jacques, Phil. Mag., 48, 6, 1105 (1924). 13 Spier, Physica, 6 (1939): 453; 7, 381 (1940). 14 Topley and Whytlaw-Gray, Phil. Mag., 4, 873 (1927). 15 Trautz and Ludwig, Ann. Physik, 5, 5, 887 (1930). 16 Trautz and Muller, Ann. Physik, 22, 353 (1935). 17 Trautz and Ries, Ann. Physik, 8, 163 (1931). 18 Walker and Westenberg, J. Chem. Phys., 32, 136 (1960). 19 Westenberg and Walker, J. Chem. Phys., 26, 1753 (1957). 20 Winkelmann, Wied. Ann., 22, 152 (1884); 23, 203 (1884); 26, 105 (1885); 33, 445 (1888); 36, 92 (1889).

2-456

PHYSICAL AND CHEMICAL DATA

In this table are a representative selection of diffusion coefficients. The subsection “Prediction and Correlation of Physical Properties” should be consulted for estimation techniques. As general references, the works by Hirschfelder, Curtiss, and Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964; Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge, New York, 1970; Reid and Sherwood, The Properties of Gases and Liquids, TABLE 2-325

McGraw-Hill, New York, 1964; and Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971, may be found useful. The most exhaustive recent compilation for gases is by Mason and Marrero, J. Phys. Chem. Ref. Data, 1 (1972). Unfortunately, the Mason and Marrero work cites only equations and equation constants and not direct tabulations. For these, the LandoltBörnstein series is suggested.

Diffusivities in Liquids (25 °C)

Dilute solutions and 1 atm unless otherwise noted; use DLµ/T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute

Solvent

DL × 105, sq cm/sec

Acetal* Acetamide* Acetamide* Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid* Acetonitrile Acetylene Allyl alcohol* Allyl alcohol Ammonia* i-Amyl alcohol* i-Amyl alcohol Benzene Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzoic acid Benzoic acid Benzoic acid Benzoic acid Benzoic acid Bromine Bromine Bromine Bromobenzene Bromoform* Bromoform Bromoform Bromoform* Bromoform Bromoform n-Butanol Caffeine Carbon dioxide Carbon dioxide Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 218 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 100 atm.) Carbon disulfide (50 mole %, 50 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %) Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Chloral* Chloral hydrate

Ethanol Ethanol Water Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water Water Ethanol Water Water Ethanol Water Carbon tetrachloride n-Decane 2,4-Dimethyl pentane n-Dodecane n-Heptane n-Hexadecane n-Octadecane Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Benzene Carbon disulfide Water Benzene Acetone i-Amyl alcohol Ethanol Ethyl ether Methanol n-Propanol Water Water Ethanol Water n-Butanol i-Butanol Chlorobenzene 2,4-Dimethyl pentane n-Heptane Methyl cyclohexane n-Octane Toluene Benzene Cyclohexane Decalin Dioxane Ethanol n-Heptane Kerosene Methanol i-Octane Tetralin Ethanol Water

1.25 0.68 1.19 3.31 2.11 1.49 0.13 2.26 1.24 1.66 1.78, 2.11 1.06 1.19 1.7, 2.0, 2.3 0.87 1.0 1.53 1.72 2.49 1.40 2.47 0.96 0.86 2.62 1.38 0.91 0.043 1.49 2.7 4.1 1.3 2.30 2.90 0.53 1.08 3.62 2.20 0.94 0.96 0.63 4.0 1.96 3.57 2.42 3.00 3.63 3.0 3.5 3.10 2.06 2.04 1.49 0.776 1.02 1.50 3.17 0.961 2.30 2.57 0.735 0.68 0.77

Estimated possible, error,  %1 5 5 3

3 5 5 6 5 8

5

5 6 6 1

3 2 2 2 2 2 2 2 2 2 5 7

Ref. 11 11 11 4 1, 4 4 4 4 11 11 1, 24 11 11 1, 11 11 11, 25 7 26 26 26 26 26 26 4 4 4 4 4 11 11 11 25 11 11 11 11 23 11 1, 11, 18, 25 11 11 1, 3, 5, 20, 24, 28 14 14 14 14 14 14 14 14 7, 9 9, 10* 9 9 9, 10* 9 9 9 9 9 11 11

TRANSPORT PROPERTIES TABLE 2-325

Diffusivities in Liquids (25 °C) (Continued)

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Chlorine Chlorobenzene Chloroform Chloroform Cinnamic acid Cinnamic acid Cinnamic acid Cinnamic acid 1,1′-Dichloropropanol Dicyanodiamide* Diethyl ether Diethyl ether 2,4-Dimethyl pentane (50 mole %) 2,4-Dimethyl pentane (50 mole %) Ethanol* Ethyl acetate Ethylene dichloride Formic acid Formic acid Formic acid Formic acid Formic acid Formic acid Glucose Glycerol Glycerol Glycerol* n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) Hexamethylene tetramine Hydrogen chloride* Hydrogen Hydrogen sulfide Hydroquinone* Hydroquinone* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodobenzene Lactose* Maltose* Mannitol* Methanol Nicotine* Nitric acid* Nitrobenzene Nitrogen Nitrous oxide Oxalic acid*

Solvent

DL × 105, sq cm/sec

Water Benzene Benzene Ethanol Acetone Benzene Carbon tetrachloride Toluene Water Water Benzene Water n-Dodecane n-Hexadecane Water Ethyl benzoate Benzene Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water i-Amyl alcohol Ethanol Water n-Dodecane n-Hexadecane n-Octadecane n-Tetradecane Water Water Water Water Ethanol Water Acetic acid Anisole Benzene Bromobenzene Carbon disulfide Carbon tetrachloride Chloroform Cyclohexane Dioxane Ethanol Ethyl acetate Ethyl ether Ethylene bromide n-Heptane n-Hexane Mesitylene Methanol Methyl cyclohexane n-Octane Tetrabromoethane n-Tetradecane Toluene m-Xylene Ethanol Water Water Water Water Water Water Carbon tetrachloride Water Water Water

1.44 2.66 2.50 1.38 2.41 1.12 0.76 2.41 1.0 1.18 2.73 0.85 1.44 0.88 1.28 0.94 2.8 3.77 2.28 1.89 0.094 2.65 1.37 0.69 0.12 0.56 0.94 1.58 1.00 0.92 1.29 0.67 3.10 5.85 (4.4) 1.61 0.53 0.88, 1.12 1.13 1.25 1.98 1.25 3.2 1.45 2.30 1.80 1.07 1.30 2.2 3.61 0.93 3.4, 2.5 4.15 1.49 1.74 2.1 2.76 2.0 0.96 2.1 1.82 1.09 0.49 0.48 0.65 1.6 0.60 2.98 1.00 1.9 1.8 1.61

Estimated possible, error,  %1 4 6 3

6 4

4

10 6 6

3 5

10 8 3

3 5 5 5 8 2

2

Ref. 1, 28 25 1, 25 11 4 4 4 4 11 11 25 2 26 26 1, 7, 9,* 11,* 22 6 1, 25 4 4 4 4 11 11 11 11 1, 11* 26 26 26 26 11 4, 11,* 12* 1, 11, 24(?) 1 11 2, 11* 11 11 9, 19, 23 4, 11, 19 11, 19, 23 9, 11, 19 11, 23 4 9 4, 11* 11, 19 11 11 9, 11, 19 4, 9 9 19 4 4 11 4 11 9, 11 11 11 11 11 1, 7, 11 11 11 7 1, 24 1, 11 11

2-457

2-458

PHYSICAL AND CHEMICAL DATA TABLE 2-325

Diffusivities in Liquids (25 °C) (Concluded)

Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.

Solute Oxygen Oxygen Oxygen Pentaerythritol* Phenol Phenol Phenol Phenol Phenol Phenol n-Propanol Pyridine* Pyridine Pyrogallol Raffinose* Resorcinol* Resorcinol* Saccharose* Stearic acid* Succinic acid* Sucrose Sulfur dioxide Sulfuric acid* Tartaric acid* 1,1,2,2-Tetrabromoethane Toluene Toluene Toluene Toluene Toluene Urea Urea Urethane Water

Solvent Glycerol*-water (106 poise) Sucrose*-water (125 poise) Water Water i-Amyl alcohol Benzene Carbon disulfide Chloroform Ethanol Ethyl ether Water Ethanol Water Water Water Ethanol Water Water Ethanol Water Water Water Water Water 1,1,2,2-Tetrachloroethane n-Decane n-Dodecane n-Heptane n-Hexane n-Tetradecane Ethanol Water Water Glycerol

DL × 105, sq cm/sec

Estimated possible, error,  %1

Ref.

0.24

13

0.25

13

2.5 0.77 0.2 1.68 3.7 2.0 0.89 3.9 1.1 1.24 0.76 0.74 0.41 0.46 0.87 0.49 0.65 0.94 0.56 1.7 1.97 0.80 0.61 2.09 1.38 3.72 4.21 1.02 0.73 1.37 1.06 0.021

References 1 Arnold, J. Am. Chem. Soc., 52, 3937 (1930). 2 Calvet, J. Chim. Phys., 44, 47 (1947). 3 Carlson, J. Am. Chem. Soc., 33, 1027 (1911). 4 Chang and Wilke, J. Phys. Chem., 59, 592 (1955). 5 Davidson and Cullen, Trans. Inst. Chem. Eng., 35, 51 (1957). 6 Dummer, Z. Anorg. Chem., 109, 31 (1949). 7 Gerlach, Ann. Phys. (Leipzig), 10, 437 (1931). 8 Gosting and Akeley, J. Am. Chem. Soc., 74, 2058 (1952). 9 Hammond and Stokes, Trans. Faraday Soc., 49, 890 (1953); 49, 886 (1953). 10 Hammond and Stokes, Trans. Faraday Soc., 52, 781 (1956). 11 International Critical Tables, vol. 5, p. 63. 12 James, Hollingshead, and Gordon, J. Chem. Phys., 7, 89 (1939); 7, 836 (1939). 13 Jordon, Ackermann, and Berger, J. Am. Chem. Soc., 78, 2979 (1956). 14 Koeller and Drickamer, J. Chem. Phys., 21, 575 (1953). 15 Kolthoff and Miller, J. Am. Chem. Soc., 63, 1013 (1941).

20 4

3 7 7 4 5 4 4 5 6 3 10 4

2

1, 3, 15, 21, 24 11 11 1 11 11 11 11 1, 7, 11 11 11 11 11 11 11 11 11 11 2, 27 15, 17 11 11 11 4 4 4 4 4 11 8, 11 11, 25 16

TRANSPORT PROPERTIES TABLE 2-326

2-459

Thermal Conductivities of Some Building and Insulating Materials* k = Btu/(h⋅ft2)(°F/ft)

Material

Apparent density ρ, lb/ft3 at room temperature

Aerogel, silica, opacified

8.5

Asbestos-cement boards Asbestos sheets Asbestos slate

120 55.5 112 112 29.3 29.3 36 36 36 36 43.5 43.5 0.2

Asbestos

Aluminum foil (7 air spaces per 2.5 in.) Ashes, wood Asphalt Boiler scale (Note 1) Bricks: Alumina (92–99% Al2O3 by wt.) fused Alumina (64–65% Al2O3 by wt.) (See also Bricks, fire clay) Building brick work Carbon Chrome brick (32% Cr2O3 by wt.) Diatomaceous earth, natural, across strata (Note 2) Diatomaceous, natural, parallel to strata (Note 2) Diatomaceous earth, molded and fired (Note 2) Diatomaceous earth and clay, molded and fired (Note 2) Diatomaceous earth, high burn, large pores (Note 3)

132

Kaolin insulating firebrick (Note 4) Magnesite (86.8% MgO, 6.3% Fe2O3, 3% CaO, 2.6% SiO2 by wt.)

Silicon carbide brick, recrystallized (Note 3)

Calcium carbonate, natural White marble Chalk Calcium sulfate (4H2O), artificial plaster (artificial) (building) Cambric (varnished) Carbon, gas Carbon stock Cardboard, corrugated

k

120 290 20 51 0 60 −200 0 0 100 200 400 −200 0 38 177 0–100 20

0.013 .026 .43 .096 .087 .114 .043 .090 .087 .111 .120 .129 .090 .135 .025 .038 .041 .43

427 1315 800 1100 20

96.7 200 200 200

200 650 1315

1.8 2.7 0.62 .63 .4 3.0 .67 .85 1.0

27.7 27.7

204 871

0.051 .077

27.7 27.7 38 38

204 871 204 871

.081 .106 .14 .18

42.3 42.3

204 871

.14 .19

27 27 19 19

200 1000 200 600 1000 1400 500 1150 200 760

.13 .34 .58 .85 .95 1.02 0.15 .26 .050 .113

158 158 158

204 650 1200

2.2 1.6 1.1

129 129 129 129 129 162

600 800 1000 1200 1400 30

10.7 9.2 8.0 7.0 6.3 1.3 1.7 0.4 .22 .43 .25 .091 2.0 0.55 3.6 0.037

115 115

37 37

Fire clay (Missouri)

Kaolin insulating brick (Note 3)

t, °C

96 84.6 132 77.9 94

40 75 25 38 0–100 −184 0

Material Cotton wool Cork board Cork (regranulated) (ground) Diatomaceous earth powder, coarse (Note 2) fine (Note 2) molded pipe covering (Note 2) 4 vol. calcined earth and 1 vol. cement, poured and fired (Note 2) Dolomite Ebonite Enamel, silicate Felt, wool Fiber insulating board Fiber, red (with binder, baked) Gas carbon Glass Borosilicate type Window glass Soda glass Granite Graphite, longitudinal powdered, through 100 mesh Gypsum (molded and dry) Hair felt (perpendicular to fibers) Ice Infusorial earth, see diatomaceous earth Kapok Lampblack Lava Leather, sole Limestone (15.3 vol. % H2O) Linen Magnesia (powdered) Magnesia (light carbonate) Magnesium oxide (compressed) Marble Mica (perpendicular to planes) Mill shavings Mineral wool

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

5 10 8.1 9.4 20.0 20.0 17.2 17.2 26.0 26.0

30 30 30 30 38 871 204 871 204 871

0.024 .025 .026 .025 .036 .082 .040 .074 .051 .088

61.8 61.8 167

204 871 50

38 20.6 14.8 80.5

139

30 78 17 57.5 0.88 10 62.4 103 49.7 13 49.9

20 40

0 21 21 20 40 21

0.020 .038 .49 .092 .54 .05 .35 0.034 .32 1.2–1.7 0.25 0.033–0.05 0.0225 .024 .075 .14 3.4 2.9 0.88 .17 .14 .075 .087 .109 0.075–0.092 0.19 1.06 0.03

38 24–127 30 94 0 100 21 21 30 30

.026 .096 .064 .022 .86 .27 0.09–0.097 0.16 .028 .04 .034

24 30 47 21 20 50

9.4 19.7

Paper Paraffin wax Petroleum coke Porcelain Portland cement, see concrete Pumice stone Pyroxylin plastics Rubber (hard) (para) (soft) Sand (dry) Sandstone Sawdust Scale (Note 1) Silk varnished Slag, blast furnace Slag wool Slate Snow Sulfur (monoclinic) (rhombic) Wall board, insulating type Wall board, stiff paste board Wood shavings

.16 .23 1.0 0.10 0.5–0.75 30 0.03 21 .028 20 .27 20–97 .097 0–100 2.0 0.2–0.73 30–75 0.63 0.3–0.61 0.3–0.44 1.0–2.3 20 95 40 0.104 20 .25 30 .021 0 1.3

30 30 0 100 500 200 90 21–66

74.8 94.6 140 12 6.3 12 34.7 14.8 43 8.8

2-460

PHYSICAL AND CHEMICAL DATA

TABLE 2-326

Thermal Conductivities of Some Building and Insulating Materials* (Concluded) k = Btu/(h⋅ft2)(°F/ft) Apparent density ρ, lb/ft3 at room temperature

Material Celluloid Charcoal flakes

87.3 11.9 15

Clinker (granular) Coke, petroleum Coke, petroleum (20–100 mesh) Coke (powdered) Concrete (cinder) (stone) (1:4 dry)

62

t, °C

k

30 80 80 0–700 100 500 400 0–100

.12 .043 .051 .27 3.4 2.9 0.55 .11 .20 .54 .44

Material Wood (across grain): Balsa Oak Maple Pine, white Teak White fir Wood (parallel to grain): Pine Wool, animal

Apparent density ρ, lb/ft3 at room temperature

t, °C

k

7–8 51.5 44.7 34.0 40.0 28.1

30 15 50 15 15 60

0.025–0.03 0.12 .11 .087 .10 .062

34.4 6.9

21 30

.20 .021

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941. International Critical Tables, McGraw-Hill, 1929, and other sources. Note 1: B. Kamp [Z. tech. Physik, 12, 30 (1931)] shows the effect of increased porosity in decreasing thermal conductivity of boiler scale. Partridge [University of Michigan, Eng. Research Bull., 15, 1930] has published a 170-page treatise on Formation and Properties of Boiler Scale. Note 2: Townshend and Williams, Chem. & Met., 39, 219 (1932). Note 3: Norton, Refractories, 2d ed., McGraw-Hill, New York, 1942. Note 4: Norton, private communication. TABLE 2-327

Thermal-Conductivity-Temperature Table for Metals* Thermal conductivities tabulated in watts per meter-kelvin Temperature, K

Substance

10

20

40

60

80

100

200

300

400

500

7 38,000 470 47 240

32 13,500 230 196 100

121 2,300 110 810 45

174 850 80 1,400 31

160 380 60 1,650 24

125 300 48 1,490 22

55 237 32 480 18

36 273 26 272 16

26 240 22 196 14

20 237 20 146 12

165 900 400 250 4

305 250 570 450 9

400 150 450 380 16

327 120 250 250 18

230 110 180 190 19

170 110 158 160 20

45 105 111 120 23

25 104 90 100 25

15 101 87 85 27

Copper Gallium Gold Graphite† Graphite‡

19,000 2,200 2,800 27 81

10,700 640 1,500 108 420

2,100 250 520 135 1,630

850 200 380 81 2,980

570 170 350 54 4,290

483 140 345 39 4,980

413 100 327 15 3,250

398 85 315 10 2,000

Hastelloy Inconel Iridium Iron Lead

1 2 1,300 710 175

3 4 1,900 1,000 57

4 8 750 560 43

5 10 360 270 42

6 11 230 170 41

7 11 172 132 40

9 14 147 94 37

10 15 145 80 35

11

13

143 69 34

140 61 33

Magnesium Magnesium oxide Manganese Manganin Mercury

1,200 1,100 2 2 54

1,300 3,100 2 4 40

620 2,200 4 9 35

290 950 5 11 33

190 460 5 13 33

169 260 6 13 32

159 75 7 17 32

156 48 8 22 8

153 36 9 28 10

Molybdenum Nickel Nylon Palladium Platinum

150 2,600 0.04 1,200 1,200

280 1,700 0.10 610 490

350 570 0.17 160 130

250 290 0.20 100 92

210 200 0.23 88 82

179 158 0.25 80 79

143 106 0.28 78 75

138 91 0.30 78 73

PTFE§ Pyrex Quartz Rhodium Rubber

0.94 0.12 1,200 2,900

1.43 0.20 480 3,900

1.94 0.33 82 1,000 0.13

2.1 0.42 40 370 0.15

2.15 0.51 30 250 0.16

2.16 0.57

2.20 0.88

2.25 1.1

190 0.17

160 0.20

Alumina Aluminum Antimony Beryllium oxide Bismuth Boron Cadmium Chromium Cobalt Constantan

Selenium (axis) Silica Silver Tantalum Tellurium Tin Titanium Tungsten Uranium Zinc Zirconium

1200

1400

16 232

10 220

8 93

7 99

6 105

111

70

47

33

25

12 99 85 70 30

81

71

65

62

61

392

388

383

371

357

342

312 7 1,460

309 5 1,140

304 4 930

292 3 680

278 3 530

262 2 440

2 370

55 31

43 19

33 22

28 24

31 26

151 27 9 34 11

149 21

146 13

84 10

98 8

112 7

40 12

13

14

134 80

130 72

126 66

118 67

112 72

105 76

100 80

78 72

80 72

72

73

78

78

81

358 61

62

2.3 1.6

2.5 2.1

150 0.22

145 0.24

140 0.25

140

57

25

15

10

8

6

16,500 108 300

5,200 146 93

1,100 88 29

630 68 17

500 62 13

430 59 11

425 58 6

4 1.34 424 57 4

3 1.52 420 58 3

2 1.70 413 58 3

14

320 28

130 39 880

100

110

59

101 37 330 20 150 42

90 33 310 22 135 38

84 31 280 23 130 34

72 26 190 26 123 25

67 21 180 28 120 23

62 20 170 30 116 22

60 20 150 32 110 21

600

1.87 405 59

800

2.22 389 59

1000

2.60 374 60

19 140 110 21

*Especially at low temperatures, the thermal conductivity can often be markedly reduced by even small traces of impurities. This table, for the highest-purity specimens available, should thus be used with caution in applications with commercial materials. From Perry, Engineering Manual, 3d ed., McGraw-Hill, New York, 1976. A more detailed table appears as Section 5.5.6 in the Heat Exchanger Design Handbook, Hemisphere Pub. Corp., Washington, DC, 1983. † Parallel to basal plane. ‡ Perpendicular to basal plane. § Also known as Teflon, etc.

TRANSPORT PROPERTIES TABLE 2-328

Thermal Conductivity of Chromium Alloys* k = Btu/(h⋅ft2)(°F/ft)

American Iron and Steel Institute Type No.

k at 212 °F

k at 932 °F

9.4 8.8 8.0 9.3 14.4 15.1 12.5 21.2

12.4 12.5 10.8 12.8 16.6 15.2 14.2 19.5

301, 302, 302B, 303, 304, 316† 308 309, 310 321, 347 403, 406, 410, 414, 416† 430, 430F† 442 501, 502†

*Table 3-322 is based on information from manufacturers. † Shelton and Swanger (National Bureau of Standards), Trans. Am. Soc. Steel Treat., 21, 1061–1078 (1933).

TABLE 2-329 Thermal Conductivity of Some Alloys at High Temperature*

TABLE 2-330 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation* k = Btu/(h⋅ft2)(°F/ft) at approximately room temperature

Material Soft flexible materials in sheet form: Chemically treated wood fiber Eel grass between paper Felted cattle hair Flax fibers between paper Hair and asbestos fibers, felted Insulating hair, and jute Jute and asbestos fibers, felted Loose materials: Cork, regranulated, fine particles Charcoal, 6 mesh Diatomaceous earth, powdered Glass wool, curled Gypsum in powdered form Mineral wool, fibrous

Thermal conductivity, Btu/(ft)(hr)(°R) °R

Kovar

Advance

Monel

Hastelloy A

Inconel

Nichrome V

5.6 6.2 6.8 7.3 7.8

6.0 6.5 7.0 7.6 8.1

5.5 6.1 6.7 7.3 7.8

Sawdust Wood shavings, from planer Semiflexible materials in sheet form: Flax fiber Semirigid materials in board form: Corkboard Mineral wool, block, with binder Stiff fibrous materials in sheet form: Wood pulp Sugar-cane fiber Cellular gypsum

500 600 700 800 900

7.8 8.3 8.6 8.7 8.7

11.4 12.6 13.9 15.1

9.0 10.2 11.2 12.3 13.4

1000 1100 1200 1300 1400

8.9 9.2 9.5 9.8 10.2

16.4 17.6 18.8 20.0 21.2

14.4 15.4 16.5 17.6 18.7

8.4 9.0 9.5 10.1 10.7

8.6 9.1 9.7 10.2 10.8

8.4 9.0 9.5 10.1 10.7

1500 1600 1700 1800 1900

10.5 10.8 11.1 11.3 11.5

22.5 23.8 25.0 26.2 27.4

19.8 20.8 21.9 23.0 24.0

11.3 11.8 12.3 12.9 13.4

11.3 11.8 12.4 13.0 13.6

11.3 11.9 12.4 13.0 13.5

2000 2100 2200

11.8 12.1 12.3

28.7 30.0

25.1 26.1 27.2

14.0 14.6 15.1

14.0 14.5 15.0

14.1 14.7 15.3

2-461

Apparent density, lb/cu ft room temp.

k

2.2 3.4–4.6 11–13 4.9 7.8 6.1–6.3 10.0

0.023 0.021–0.022 0.022 .023 .023 0.022–0.023 0.031

8–9 15.2 10.6 4–10 26–34 6 10 14 18 12 8.8

.025 .031 .026 .024 0.043–0.05 0.0217 .0225 .0233 .0242 .034 .034

13.0

.026

7.0 10.6 16.7

.0225 .025 .031

16.2–16.9 13.2–14.8 8 12 18 24 30

.028 .028 .029 .037 .049 .064 .083

*Abstracted from U.S. Bur. Standards Letter Circ. 227, Apr. 19, 1927.

*Silverman, J. Metals, 5, 631 (1953). Copyright American Institute of Mining, Metallurgical and Petroleum Engineers, Inc.

TABLE 2-331

Thermal Conductivities of Insulating Materials at High Temperatures* k = Btu/(h⋅ft2)(°F/ft)

Material Laminated asbestos felt (approx. 40 laminations per in) Laminated asbestos felt (approx. 20 laminations per in) Corrugated asbestos (4 plies per in) 85% magnesia (density, 13 lb/ft3) Diatomaceous earth, asbestos and bonding material Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth powder (density, 18 lb/ft3) Rock wool

For temperatures, °F up to 700 500 300 600 1600 1600 2000 2500

Mean temperatures, °F 100

200

300

400

500

0.033 .045 .050 .034 .045 .054 .127 .128 .039 .030

0.037 .050 .058 .036 .047 .056 .130 .131 .042 .034

0.040 .055 .069 .038 .049 .058 .133 .135 .044 .039

0.044 .060

0.048 .065

.040 .050 .060 .137 .139 .048 .044

.053 .063 .140 .143 .051 .050

Asbestos cement, 1.2; 85% magnesia cement, 0.05; asbestos and rock wool cement, 0.075 approx. *Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

600

800

1000

0.055 .065 .143 .148 .054 .057

0.060 .069 .150 .155 .061

0.065 .073 .158 .163 .068

1500

2000

0.176 .183

0.203

2-462

PHYSICAL AND CHEMICAL DATA

TABLE 2-332 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt)* k = Btu/(h⋅ft2)(°F/ft) Weight, lb/cu ft

Material Asbestos Burned infusorial earth for pipe coverings Insulating composition (loose) Cotton Silk hair Silk Wool Pulverized cork Infusorial earth (loose)

TABLE 2-333 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber)* k = Btu/(h⋅ft2)(°F/ft)

Temperatures, °F 32

100

200

300

400

600

800

36.0

0.087 0.097 0.110 0.117 0.121 0.125 0.130

12.5

.043 .046 .052 .057 .062 .073 .085

25.0 5.0 9.1 6.3 8.5 10.0 22.0

.040 .032 .026 .025 .022 .021 .035

.046 .035 .030 .028 .027 .026 .039

.050 .053 .055 .039 .034 .034 .033 .032 .045 .047 .050 .053

Temperatures, °F

Material

Weight, lb/cu ft

32

−50

−100

−200

−300

Asbestos Asbestos Cotton Silk

44.0 29.0 5.0 6.3

0.135 .0894 .0325 .0290

0.132 .0860 .0302 .0256

0.130 .0820 .0276 .0235

0.125 .0720 .0235 .0196

0.100 .0545 .0198 .0155

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.

TABLE 2-334

Thermal Diffusivity (m2/s) of Selected Elements* Temperature, K

Element

20

40

60

80

100

Aluminum Beryllium Chromium Copper Gold

0.50

200

400

600

800

1000

0.012

0.0014

4.4. − 4

0.038 0.16 0.005

0.0037 0.0040 4.5. − 4

5.9. − 4 6.9. − 4 2.3. − 4

2.0. − 4 3.1. − 4 1.8. − 4

2.3. − 4 0.0036 1.2. − 4 2.2. − 4 1.5. − 4

1.1. − 4 1.5. − 4 4.1. − 5 1.3. − 4 1.3. − 4

9.4. − 5 4.0. − 5 2.6. − 5 1.1. − 4 1.2. − 4

8.4. − 5 2.6. − 5 2.0. − 5 1.0. − 4 1.2. − 4

7.4. − 5 2.1. − 5 1.7. − 5 9.0. − 5 1.1. − 4

6.6. − 5 1.7. − 5 1.4. − 5 9.0. − 5 9.8. − 5

Iridium Iron Lead Molybdenum Nickel

0.046 0.043 9.3. − 5 0.0095 0.033

3.2. − 3 3.9. − 5 0.0014 0.0017

4.9. − 4 3.3. − 5 4.0. − 4 3.1. − 4

1.6. − 4 3.1. − 5 2.0. − 4 1.3. − 4

8.4. − 5 8.2. − 5 2.9. − 5 1.3. − 4 8.0. − 5

5.6. − 5 3.1. − 5 2.6. − 5 6.3. − 5 3.1. − 5

4.8. − 5 1.8. − 5 2.3. − 5 5.1. − 5 1.9. − 5

4.4. − 5 1.3. − 5 2.0. − 5 4.5. − 5 1.3. − 5

4.1. − 5 1.1. − 5 1.3. − 5 4.2. − 5 1.4. − 5

3.5. − 5 1.0. − 5 1.5. − 5 3.8. − 5 1.5. − 5

Platinum Silver Zinc

0.0029 0.031 0.0046

1.6. − 4 0.0013 3.1. − 4

6.3. − 5 4.5. − 4 1.0. − 4

4.3. − 5 2.8. − 4 7.0. − 5

3.6. − 5 2.3. − 4 5.5. − 5

2.7. − 5 1.8. − 4 4.7. − 5

2.5. − 5 1.7. − 4 3.9. − 5

2.5. − 5 1.6. − 4 3.4. − 5

2.5. − 5 1.5. − 4 1.8. − 5

2.5. − 5 1.4. − 4 2.2. − 5

*Tables for up to 24 temperatures for 47 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. The notation 3.2. − 4 signifies 2.3 × 10−4.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-335

2-463

Thermophysical Properties of Selected Nonmetallic Solid Substances Density, kg/m3

Material Alumina Asphalt Bakelite Beryllia Brick

3975 2110 1300 3000 1925

Brick, fireclay Carbon, amorphous Clay Coal Cotton

2640 1950 1460 1350 80

Diamond Granite Hardboard Magnesite Magnesia

3500 2630 1000 3025 3635

Emissivity

Specific heat, kJ/(kg⋅K)

Thermal conductivity, W/(m⋅K)

0.82 0.93

0.765 0.920 1.465 1.030 0.835

36 0.06 1.4 270 0.72

11.9 0.03 0.74 88 0.45

0.960 0.724 0.880 1.26 1.30

1.0 1.6 1.3 0.26 0.06

0.39 1.13 1.01 0.15 0.58

0.509 0.775 1.38 1.13 0.943

2300 2.79 0.15 4.0 48

1290 1.37 0.11 1.2 14

0.93 0.86 0.91 0.80

0.38 0.72

Oak Paper Pine Plaster board Plywood

770 930 525 800 540

0.90 0.83 0.84 0.91

Pyrex Rubber Rubber, foam Salt Sandstone

2250 1150 70 2150

0.92 0.92 0.90 0.34 0.59

Silica Sapphire Silicon carbide Soil

3975 3160 2050

Teflon Thoria Urethane foam Vermiculite

2200 4160 70 120

2.38 1.34 2.75

Thermal diffusivity, m2/s × 106

0.18 0.011 0.12 0.17 0.12

0.10 0.01 0.54

0.74 0.09

0.854 0.745

1.4 0.2 0.03 7.1 2.9

0.79 0.48 0.86 0.38

0.743 0.765 0.675 1.84

1.3 46 110 0.52

15 230 0.14

0.92 0.28

0.35 0.71 1.05 0.84

0.26 14 0.03 0.06

0.34 4.7 0.36 0.60

1.22 0.835 2.00

0.18

1.8

NOTE: Difficulties of accurately characterizing many of the specimens mean that many of the values presented here must be regarded as being of order of magnitude only. For some materials, actual measurement may be the only way to obtain data of the required accuracy. To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428; to convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Fahrenheit, multiply by 0.23885.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES* INTRODUCTION Physical property values, sufficiently accurate for many engineering applications, can be estimated in the absence of reliable experimental data. The purpose of this subsection is to provide a set of recommended prediction methods for general engineering use; it is not a comprehensive review, and many alternative methods are available in the literature. Methods recommended were selected on the basis of accuracy, generality, and, in most cases, simplicity. They generally correspond to the methods tested and given priority in the DIPPR® 801 database project.† Properties included in this subsection are divided into 10 categories: (1) physical constants: critical properties, normal melting and boiling points, acentric factor, radius of gyration, and dipole moment; (2) vapor

*Some material in this subsection has been retained from the corresponding subsection in the Seventh Edition, which was coauthored by Thomas E. Daubert and Evan Buck. †The Design Institute for Physical Properties (DIPPR®) is an industrial consortium under the auspices of AIChE; Project 801, Evaluated Process Design Data, is a pure-component database of industrially important compounds.

pressure: liquid and solid; (3) thermal properties: enthalpy and Gibbs’ energy of formation and ideal gas entropy; (4) latent enthalpy: vaporization, fusion, and sublimation; (5) heat capacity: ideal gas, gas, liquid, and solid; (6) density: gas, liquid, and solid; (7) viscosity: gas and liquid; (8) thermal conductivity: gas and liquid; (9) surface tension; and (10) flammability properties: flash point, flammability limits, and autoignition temperature. Each of the 10 subsections provides a definition of the relevant properties and a description of one or more recommended prediction methods. Each description lists the type of method, its uncertainty, its limitations, and the expected uncertainty of the predicted value. A numerical example is also given to illustrate use of the method. For brevity, symbols used for physical properties and for variables and constants in the equations are defined under Nomenclature and are not generally defined after their use except where doing so clarifies usage. A list of equation and table numbers in which variables appear is included in the Nomenclature section for quick cross-referencing. Although emphasis is on pure-component properties, some mixture estimation techniques have been included for physical constants, density, viscosity, thermal conductivity, surface tension, and flammability. Correlation and estimation of properties that are inherently multicomponent (e.g., diffusion coefficients, mixture excess properties, activity coefficients, etc.) are treated elsewhere in the Handbook.

2-464

PHYSICAL AND CHEMICAL DATA

UNITS The International Metric System (SI) of units has been used throughout this subsection. Where possible, the estimation equations are set up in dimensionless groups. This makes transparent any conversion factors that should be applied to obtain the property in a desired set of units and eliminates the requirement of specific units for variables. For example, rather than use Pc as a variable with defined units, the dimensionless group Pc / Pa is used. When a value for Pc expressed in

any units (say, Pc = 6.53 MPa) is inserted into this group, the result is dimensionless with an explicit indication of conversion factors that must be included: PC 6.53 MPa 6.53 MPa 106 Pa  =  =   = 6.53  106 Pa Pa Pa MPa





Section 1 of this handbook should be used for appropriate unit conversion factors.

Nomenclature Physical constants h k NA R Properties A, B, C B, B(T) Bm CP C op Cv Hi k kb LFL M P P Pc Pr P* P*meas Pr* Pt* Rg So Ss Sr Svib T Tb Tbr Tc Tm Tmeas Tr Tt UFL V Vm Vr wi xi yi Z Zc Zi Gfo Gfs Hfo Hfs Hfus Hrxn Hsub Hv Sfs Sfo Sfus Zv  c



Definition Planck’s constant Boltzmann’s constant Avogadro’s number Gas constant Definition Molecular principal moments of inertia Second virial coefficient Second virial coefficient for a mixture Constant-pressure molar heat capacity Ideal gas constant-pressure molar heat capacity Constant-volume molar heat capacity Enthalpy of compound i Thermal conductivity Thermal conductivity at Tb Lower flammability limit Molecular weight Pressure Parachor Critical pressure Reduced pressure; Pr = P/Pc Vapor pressure Measured vapor pressure value Reduced vapor pressure; Pr* = P*/Pc Vapor pressure at triple point Radius of gyration Ideal gas entropy Standard state entropy Rotational contribution to entropy Vibrational contribution to entropy Temperature Normal boiling point temperature Reduced temperature at Tb; Tbr = Tb/Tc Critical temperature Melting temperature T at which a dependent property was measured Reduced temperature; Tr = T/Tc Triple point temperature Upper flammability limit Molar volume Mixture molar volume Reduced volume; Vr = ZTr/Pr Mass fraction of component i Mole fraction of component i Mole fraction of component i in vapor phase Compressibility factor; Z = PV/RT Critical compressibility factor; Zc = PcVc/RTc Compressibility factor of reference fluid i Ideal gas standard Gibbs energy of formation Standard state Gibbs energy of formation Ideal gas standard enthalpy of formation Standard state enthalpy of formation Enthalpy of fusion Enthalpy change per mole of reaction as written Enthalpy of sublimation Enthalpy of vaporization Standard state entropy of formation Ideal gas entropy of formation Latent entropy of fusion Change in compressibility factor upon vaporization Molar density;  = V−1 Critical molar density; c = Vc−1

Value 6.626  10−34 Js 1.3806  10−23 J/(moleculeK) 6.022  1026 molecule/kmol 8.3143 Pam3/(kmolK) Typical units kgm2 m3/kmol m3/kmol J/(kmolK) J/(kmolK) J/(kmolK) J/kmol W/(mK) W/(mK) % kg/kmol Pa unitless Pa unitless Pa Pa unitless Pa m J/(kmolK) J/(kmolK) J/(kmolK) J/(kmolK) K K unitless K K K unitless K % m3/kmol m3/kmol unitless unitless unitless unitless unitless unitless unitless J/kmol J/kmol J/kmol J/kmol J/kmol J/kmol J/kmol J/kmol J/(kmolK) J/(kmolK) J/(kmolK) unitless kmol/m3 kmol/m3

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Nomenclature (Continued) Properties r   o m  b  r Eq. variables

Definition Reduced molar density; r = /c Acentric factor Viscosity Viscosity at low pressure Surface tension Surface tension of mixture Complementary reduced temperature (= 1 − Tr) Complementary reduced normal boiling temperature (= 1 − Tbr) Dipole moment Reduced dipole moment [defined in Eq. (2-62)] Definition

a a, b, c, . . .

EoS constant GC values for Cp and 

a, b, c ai a, b a, b ai, bi, di – a–  A, B, C, . . .

Correlation coefficients GC values Terms in second virial correlation Chickos correlation parameters GC values for liquid Cp EoS constant for mixture Correlation constants/parameters

A Ai b bi, ci, . . . b –i b B(i)

Factor in liquid k correlation Constants in Cop correlation EoS constant Reference EoS constants GC value for AIT EoS constant for mixture Second virial expansion terms

C

Number of components in mixture

Ci Ci,int Cij (C op)i Csj Ctj fi f(i) F Gij gEc gEr h K LFLi n nA nE ni

GC values for Tb or  Sum of intramolecular group-group interactions Group-group intramolecular interaction pair GC values for ideal gas heat capacity Chickos: GC value for C—H group Chickos: GC value for functional group Halogen correction for ∆Hsub correlation Vapor pressure deviation function Factor in surface tension equation Adjustable mixture viscosity parameter UNIFAC combinatorial excess Gibbs energy UNIFAC residual excess Gibbs energy Parameter in Riedel vapor pressure equation Parameter in Riedel vapor pressure equation GC contribution Number of nonhydrogen atoms Number of atoms in molecule Number of occurrnces of element E in compound Number of occurrences of group i

nf ns

Chickos: no. of different functional groups Chickos: no. of different nonring or aromatic C—H groups bonded to functional groups Total no. of halogen and H atoms attached to C and Si atoms for Hsub correlation Total number of groups in molecule

nx N

NC Nfi Ngi NH

Number of C atoms Chickos: no. of functional groups of type i Chickos: no. of C—H groups of type i bonded to other C atoms Number of H atoms

Typical units unitless unitless Pas Pas mN/m mN/m unitless unitless D unitless (Equations), [Tables] (2-66), [2-354] (2-51), (2-52), (2-53), (2-97), [2-356] (2-21), (2-23) (2-43), (2-97), [2-346, 2-356] (2-61) (2-39), (2-40), (2-41) (2-51), [2-348] (2-76) (2-2), (2-20), (2-22), (2-34), (2-36), (2-50), (2-65), (2-68), (2-83), (2-85), (2-87), (2-88), (2-95), (2-96), (2-101), above (2-114) (2-108), [2-358] (2-45), (2-46) (2-66), [2-354] (2-65), [2-353] (2-124), [2-363, 2-364] (2-75) (2-58), (2-59), (2-60), (2-61) (2-55), (2-72), (2-73), (2-74), (2-75), (2-76), (2-77), (2-80), (2-81), (2-82), (2-98), (2-113), (2-119) (2-15), (2-87), [2-341, 2-355] (2-16) (2-16), [2-342] (2-49), [2-347] (2-41), [2-344] (2-41), [2-345] (2-43), [2-346] (2-25) (2-116), (2-117) (2-98) (2-99) (2-99) following (2-24) following (2-24) (2-122), [2-361, 2-362] (2-15) (2-1), (2-30), (2-31), (2-48) (2-54) (2-27), (2-43), (2-49), (2-51), (2-53), (2-87), (2-109), (2-115), (2-122), (2-123), (2-124) (2-41) (2-41) (2-43) (2-12), (2-15) (2-16), (2-43), (2-49), (2-51), (2-53), (2-54), (2-87), (2-97), (2-109), (2-115) (2-121) (2-41) (2-41) (2-121)

2-465

2-466

PHYSICAL AND CHEMICAL DATA Nomenclature (Continued) Eq. variables NCR NO NR NS Nsi NX – Pc Pc,ij q qi Qk ri r* Rk (So)i t tm1,i tm2,i – Tc Tc,ij xP U* UFLi Vc,ij Z(0) Z(1) Zc,ij ZRA ⎯ ZRA , , , . . . (Tr) c mn   i   E P T V (Hfo)i ki Pi si    i i j   A, B, C j r*   mn ξ –  ij

Definition Chickos: no. of CH2 groups in nonaromatic ring to form cyclic paraffin of same ring size Chickos: Number of O atoms Chickos: no. of nonaromatic rings Chickos: Number of S atoms Chickos: no. of C—H groups of type i bonded to at least one functional group Number of halogen atoms Pseudocritical pressure for mixture Cross term in mixing rule Rackett equation power for Zc UNIFAC molecular surface area UNIFAC group surface area UNIFAC molecular volume Dimensionless separation distance UNIFAC group volume GC value for entropy Chickos: total no. of functional groups First-order GC contribution for Tm Second-order GC contribution for Tm Pseudocritical temperature for mixture Cross term in mixing rule Term in Pailhes method = log(1 atm/P) Dimensionless intermolecular potential GC contribution Cross term in mixing rule Compressibility factor of simple fluid Acentric deviation term for Z Cross term in mixing rule Modified Rackett correlation parameter Modified Rackett parameter for mixture Correlation parameters for k EoS temperature-dependent function Parameter in Riedel vapor pressure equation Viscosity group-group interactions Reference EoS constant Stoichiometric coefficient in FP correlation Nonlinear correction term in correlation Reference EoS constant = 0 for nonlinear molecules; = 1 for linear EoS parameter Contribution of element E to heat capacity GC contribution to Pc GC contribution to Tc GC contribution to Vc GC value for enthalpy of formation GC for thermal conductivity at Tb GC for parachor Chickos: GC value for group i Lennard-Jones well depth parameter EoS parameter UNIFAC molecular volume fraction Volume fraction of component i Stoichiometric coefficient (+ for product and − for reactant) for compound i in reaction Frequency of vibrational mode j UNIFAC molecular surface fraction UNIFAC group surface fraction Characteristic rotational T of molecule Characteristic vibrational T of mode j Lennard-Jones size parameter Rotational external symmetry number Modified reduced dipole moment Parameter in Riedel vapor pressure equation Parameter in correlation of k for gases UNIFAC interaction factor Viscosity dedimensionalizing factor Pseudoacentric factor for mixture Cross term in mixing rule

Acronyms and abbreviations CC CS DIPPR® EoS

(Equations), [Tables] (2-40) (2-121) (2-40) (2-121) (2-41) (2-121) (2-73) (2-79) (2-69), (2-70), (2-81) following (2-100) following (2-100) following (2-100) (2-4) following (2-100) (2-27), [2-343] (2-41) (2-13), [2-339] (2-13), [2-340] (2-72), (2-73), (2-80) (2-78), (2-80) (2-14) (2-4) (2-123), [2-361, 2-362] (2-78), (2-79) (2-63), (2-64), [2-351] (2-63), (2-64), [2-352] (2-78), (2-79) (2-70) (2-82) (2-106), (2-107), (2-108), (2-109), [2-358] (2-66), [2-354] following (2-24) (2-100), [2-357] (2-65), [2-353] (2-120), (2-121) (2-43), (2-53), [2-346, 2-349] (2-65), [2-353] (2-1), (2-48) (2-66), [2-354] (2-54), [2-350] (2-7), (2-10), [2-336, 2-337] (2-6), (2-9), [2-336, 2-337] (2-8), (2-11), [2-336, 2-337] (2-27), [2-343] (2-109), [2-359] (2-115), [2-360] (2-41), [2-344, 2-345] following (2-4) (2-66), [2-354] following (2-100) (2-80), following (2-100), (2-113) (2-28), (2-29), (2-30) (2-47) following (2-100) following (2-100) before and following (2-31) (2-1), (2-31), (2-48) (2-4) following (2-31) (2-85), (2-86) following (2-24) (2-106) (2-100) (2-89), (2-90), (2-91), (2-92), (2-93), (2-94) (2-74) (2-79) Definition Computational chemistry Corresponding states Design Institute for Physical Properties Equation of state

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

2-467

Nomenclature (Concluded) Acronyms and abbreviations

Definition

GC LJ MC MD QSPR TRC

Group contributions Lennard-Jones Monte Carlo Molecular dynamics Quantitative structure-property relationships Thermodynamics Research Center

GENERAL REFERENCES

(a) Prediction methods:

[PGL4] Reid, R. C., J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987. [PGL5] Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001.

(b) Property databases:

[DIPPR] Rowley, R. L., et al., DIPPR® Data Compilation of Pure Chemicals Properties, Design Institute for Physical Properties, AIChE, New York, 2007. [TRC] TRC Thermodynamic Tables—Non-Hydrocarbons, Thermodynamics Research Center, The Texas A&M University System, College Station, Tex., extant 2004; TRC Thermodynamic Tables—Hydrocarbons, Thermodynamics Research Center, The Texas A&M University System, College Station, Tex., extant 2004. [JANAF] Chase, M. W., Jr., et al., “JANAF Thermochemical Tables,” J. Phys. Chem. Ref. Data, 14, suppl. 1, 1985. [SWS] Stull, D. R., F. F. Westrum, Jr., and G. C. Sinke, The Chemical Thermodynamics of Organic Compounds, John Wiley & Sons, New York, 1969.

CLASSIFICATION OF ESTIMATION METHODS Physical property estimation methods may be classified into six general areas: (1) theory and empirical extension of theory, (2) corresponding states, (3) group contributions, (4) computational chemistry, (5) empirical and quantitative structure property relations (QSPR) correlations, and (6) molecular simulation. A quick overview of each class is given below to provide context for the methods and to define the general assumptions, accuracies, and limitations inherent in each. Theory and Empirical Extension of Theory Methods based on theory generally provide better extrapolation capability than empirical fits of experimental data. Assumptions required to simplify the theory to a manageable equation suggest accuracy limitations and possible improvements, if necessary. For example, the ideal gas isobaric heat capacity, rigorously obtained from statistical mechanics under the assumption of independent harmonic vibrational modes, is [Rowley, R. L., Statistical Mechanics for Thermophysical Property Calculations, Prentice-Hall, Englewood Cliffs, N.J., 1994] 8  δ 3n 6δ CPo  +

2 R j =1 A

δ

{ 10

Θj  T



2

eΘ  T  ΘT (e  1)2

nonlinear molecules linear molecules

j

j

(2-1)

where Θj is the characteristic temperature for the jth vibrational frequency in a molecule of nA atoms. The temperature dependence of this equation is exact to the extent that the frequencies are harmonic. Corrections for anharmonicity can be applied (albeit with difficulty) where warranted. Extension of theory often requires introduction of empirical models and parameters in lieu of terms that cannot be rigorously calculated. Good accuracy is expected in the region where the model parameters were fitted to experimental data, but only limited accuracy when an empirical model is extrapolated to other conditions. For example, a simplified theory suggests that vapor pressure should have the form B ln P* A   (2-2) T where the empirical parameter B is given by ∆Hv B  R ∆Zv

(2-3)

and ∆Hv and ∆Zv are differences between the vapor and liquid enthalpies and compressibility factors, respectively. Vapor pressures over a narrow temperature range can be effectively correlated using Eq. (2-2), but this equation should not be used to extrapolate vapor pressures over a wide range of temperatures. Corresponding States (CS) The principle of CS applies to conformal fluids [Leland, T. L., Jr., and P. S. Chappelear, Ind. Eng. Chem., 60 (1968): 15]. Two fluids are conformal if their intermolecular interactions are equivalent when scaled in dimensionless form. For example, the Lennard-Jones (LJ) intermolecular pair potential energy U can be written in dimensionless form as U* 4[(r*12  r*6)]

(2-4)

where r* rσ, U* Uε, σ is the LJ size parameter, and ε is the LJ attractive well depth parameter. At equivalent scaled temperatures kTε (k is Boltzmann’s constant) and pressures Pσ3ε, all LJ fluids will have identical dimensionless properties because the molecules interact through the identical scaled intermolecular potential given by Eq. (2-4). Generalization of this scaling principle is commonly done using critical temperature Tc and critical pressure Pc as scaling factors. At the same reduced coordinates (Tr = T/Tc and Pr = P/Pc) all conformal fluids will have the same dimensionless properties; for example, Z = Z(Tr, Pr) where the compressibility factor is defined as Z = PV/RT. A correlation of experimental data for one fluid can be used as the reference for the properties of all conformal fluids. Nonconformality is the main accuracy limitation. For instance, interactions between nonspherical or polar molecules may not be adequately represented by Eq. (2-4), and so the scaled properties of these fluids will not conform to those of a fluid with interactions well represented by Eq. (2-4). A correction for nonconformality is usually made by the addition of one or more reference fluids whose deviations from the first reference fluid are used to characterize the effect of nonconformality. For example, in the Lee-Kesler method [Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510] n-octane is used as a second, nonspherical reference fluid, and deviations of n-octane scaled properties from those of the spherical reference fluid at equivalent reduced conditions are assumed to be a linear function of the acentric factor. An extended Lee-Kesler method [Wilding, W. V., and R. L. Rowley, Int. J. Thermophys., 7 (1986): 525] uses three reference fluids: n-octane to correct for size-shape nonconformality relative to methane and water to correct for polar effects. Some CS methods [e.g., Teja, A., S. I. Sandler, and N. C. Patel, Chem. Eng. J., 21 (1981): 21] utilize different reference fluids for different classes of fluids to maintain close conformality between the fluid whose properties are to be estimated and the reference fluids. Group Contributions (GC) Chemical and physical properties generally correlate well with molecular structure. GC methods assume a summative behavior of the structural groups of the constituent molecules. For example, ethanol (CH3—CH2—OH) properties would be obtained as the sum of contributions from the —CH3,—CH2, and —OH groups. The contribution of each group is obtained by regression of experimental data that include as many different compounds containing that group as possible. Structural groups must be used exactly as defined in the original correlation. A general priniciple in deciding how to make the groupings is to give the more specific group priority. For example, although the structural piece —COOCH3 is ambiguous in a methyl ester, a more specific group value available for an ester —COO— would take precedence over a combination the two smaller groups —(C苷O)— and

2-468

PHYSICAL AND CHEMICAL DATA

—O— whose values were most likely regressed only from ketone and ether data, respectively. Excellent accuracy can usually be expected from GC methods when group values were regressed from large quantities of experimental data. However, if the ratio of number of groups to regressed experimental data is large, significant errors can result when the method is applied to new compounds (extrapolation). Such excessive specificity in the group definitions leads to poor extrapolation capabilities even though the fit of the regressed data may be excellent. Computational Chemistry (CC) Commercial software is available that solves the Schrödinger equation for approximate forms of the wave function. Various levels of sophistication (termed model chemistry) for the wave function can be chosen at the expense of computational time. Results include structural information (bond lengths, bond angles, dihedral angles, etc.), electron/charge distribution information, internal vibrational modes (for ideal gas properties), and energy of the molecule, valid for the chosen model chemistry. Because calculations are performed on individual molecules, they are primarily suited for ideal gas properties. Relative energies for the same model chemistry are more accurately obtained than absolute energies, so enthalpies and entropies of reaction are common industrial uses of CC predictions in addition to individual structural and ideal gas properties. Empirical QSPR Correlations In quantitative structure property relationship (QSPR) methods, physical properties are correlated with molecular descriptors that characterize the molecular and electronic structure of the molecule. Large amounts of experimental data are used to statistically determine the most significant descriptors to be used in the correlation and their contributions. The resultant correlations are simple to apply if the descriptors are available. Descriptors must generally be generated by the user with computational chemistry software, although the DIPPR® 801 database now contains a table of molecular descriptors for most of the compounds in it. QSPR methods are often very accurate for specific families of compounds for which the correlation was developed, but extrapolation problems are even more of an issue than with GC methods. Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton’s equations of motion for a small number (on the order of 103) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time.

Recommended Methods Organic molecules: The Ambrose method is recommended for all three critical properties of hydrocarbons and n-alcohols. The Joback method is recommended for Tc and Pc of all other organic molecules. Fedors’ method is recommended for Vc of these compounds, but the Joback method can also be used. Inorganic molecules: The simple correlation Tc = 1.64Tb is recommended if the normal boiling point is known. Critical pressure Pc is best obtained by extrapolating vapor pressure data to Tc, and Vc is best obtained from a correlation of liquid density extrapolated to Tc. Other recent methods including the Wilson-Jasperson (Wilson, G. M., and L.V. Jasperson, Critical Constants Tc, Pc, Estimation Based on Zero, First and Second Order Methods, AIChE Spring Meeting, New Orleans, La., 1996) and Marrero-Pardillo [Marrero-Morejon, J., and E. PardilloFontdevila, AIChE J., 45 (1999): 615] methods have proved to be as good as or better for some classes of compounds than the methods presented here; however, their application is more difficult. Method: Ambrose method. Reference: Ambrose, D., Correlation and Estimation of Vapour-Liquid Critical Properties. I. Critical Temperatures of Organic Compounds, Natl. Phys. Lab. Report Chem. 92 (1978); Correlation and Estimation of Vapour-Liquid Critical Properties. II. Critical Pressures and Volumes of Organic Compounds, Natl. Phys. Lab Report Chem. 98 (1979). Classification: Group contributions. Expected uncertainty: ~6 K for Tc (about 1 percent), ~2 bar for Pc (about 5 percent), ~8 cm3/mol for Vc (about 3 percent). Applicability: Organic compounds. Input data: Tb, M, group contributions ∆T, ∆P, and ∆V from Table 2-336. Description: A GC method with first-order contributions and corrections (delta Platt number) for branched alkanes. Variables Tc, Pc, and Vc are given by the following relations: Tc = Tb 1 + 1.242 + ∆T

 

(2-5)

(2-7)

P

Vc = 40 + ∆V  cm3/mol

(2-8)

Example Use the Ambrose method to estimate the critical constants of 2,2,4-trimethylpentane. Required data: From the DIPPR® 801 database, Tb = 372.39 K and M = 114.229 kg/kmol. Structure: H3C

CH CH3 3

H 3C

PcVc Zc =  RTc

(2-6)

−2

c

PHYSICAL CONSTANTS Critical Properties The critical temperature Tc, pressure Pc, and volume Vc of a compound are important, widely used constants. They are important in determining the phase boundaries of a compound and (particularly Tc and Pc) are required input parameters for most thermal and volumetric property calculations of the equilibrium phases using CS or analytical equations of state. Most estimation methods employ weighted group, atom, or bond contributions. The critical temperature of a compound is the temperature above which a liquid phase cannot be formed, no matter the pressure of the system. The critical pressure is the vapor pressure of the compound at the critical temperature. The critical volume is the volume occupied by a set amount of a compound (typically 1 mol) at its critical temperature and pressure. The critical compressibility factor Zc is determined from the experimental or predicted values of the critical properties by the definition

−1

 M P  =  0.339 + ∆ kg/kmol bar

CH3

Group contributions from Table 2-336: Group

ni

T

P

Alkyl carbons >CH— (correction) >C< (correction) Delta Platt no.

8 1 1 0

0.138 −0.043 −0.120 −0.023

0.226 −0.006 −0.030 −0.026

V 55.1 −8 −17 —

Calculations using Eqs. (2-6), (2-7), and (2-8):

∆

T

= (8)(0.138) + (1)(−0.043) + (1)(−0.120) = 0.941

Tc = Tb(1.4581) = (372.39 K)(1.4581) = 543.0 K

∆

P

= (8)(0.226) + (1)(−0.006) + (1)(−0.030) = 1.772

M Pc  =  0.339 + kg/kmol bar Pc = 25.63 bar



∆

−2

P

114.229 = 2 = 25.63 (0.339 + 1.772)

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-336

Ambrose Groupa Contributions for Critical Constants

Group Carbon atoms in alkyl groups Corrections >CH (each) >C< (each) Double bonds (nonaromatic) Triple bonds Delta Platt number,b multiply by Aliphatic functional groups: O >CO CHO COOH COOOC COO NO2 NH2 NH >N CN S SH SiH3 OSi(CH3)2 F Cl Br I Halogen correction in aliphatic compounds: F is present F is absent, but Cl, Br, I are present Aliphatic alcoholsc Ring compound increments (listed only when different from aliphatic values): CH2, >CH, >C< >CH in fused ring Double bond O NH S Aromatic compounds: Benzene Pyridine C4H4 (fused as in naphthalene) F Cl Br I OH Corrections for nonhalogenated substitutions: First Each subsequent Ortho pairs containing OH Ortho pairs with no OH Highly fluorinated aliphatic compounds: CF3, CF2, >CF CF2, >CF (ring) >CF (in fused ring) H (monosubstitution) Double bond (nonring) Double bond (ring) (other increments as in nonfluorinated compounds)

T

P

V

0.138

0.226

55.1

−0.043 −0.120 −0.050 −0.200 −0.023

−0.006 −0.030 −0.065 −0.170 −0.026

−8 −17 −20 −40 —

0.138 0.220 0.220 0.578 1.156 0.330 0.370 0.208 0.208 0.088 0.423 0.105 0.090 0.200 0.496 0.055 0.055 0.055 0.055

0.160 0.282 0.220 0.450 0.900 0.470 0.420 0.095 0.135 0.170 0.360 0.270 0.270 0.460 — 0.223 0.318 0.500 —

20 60 55 80 160 80 78 30 30 30 80 55 55 119 — 14 45 67 90

0.125 0.055 d

e

0.090 0.030 −0.030 0.090 0.090 0.090

0.182 0.182 — — — —

0.448 0.448 0.220 0.080 0.080 0.080 0.080 0.198

0.924 0.850 0.515 0.183 0.318 0.600 0.850 −0.025

0.010 0.030 −0.080 −0.040

0 0.020 −0.050 −0.050

0.200 0.140 0.030 −0.050 −0.150 −0.030

0.550 0.420 — −0.350 −0.500 —

15 44.5 44.5 −15 10 — 30 f

a Ambrose, D., Correlation and Estimation of Vapour-Liquid Critical Properties. I. Critical Temperatures of Organic Compounds, Natl. Phys. Lab Report Chem. 92 (1978); Correlation and Estimation of Vapour-Liquid Critical Properties. II. Critical Pressures and Volumes of Organic Compounds, Natl. Phys. Lab Report Chem. 98 (1979). b The delta Platt number is defined as the Platt number of the isomer minus the Platt number of the corresponding alkane. (For n-alkanes the Platt number is n − 3.) The Platt number is the total number of groups of four carbon atoms three bonds apart [Platt, J. R., J. Chem. Phys., 15(1947): 419; 56(1952): 328]. This correction is used only for branched alkanes. c Includes naphthenic alcohols and glycols but not aromatic alcohols such as xylenol. d First determine the hydrocarbon homomorph, i.e., substitute CH3 for each OH and calculate ∆T for this compound. Subtract 0.138 from ∆T for each OH substituted. Next, add 0.87 − 0.11n + 0.003n2 where n = [Tb/K (alcohol) − 314]/19.2. Exceptions include methanol ( ∆T = 0), ethanol ( ∆T = 0.939), and any alcohol whose value of n exceeds 10. e Determine the hydrocarbon homomorph as in footnote d. Calculate ∆p and subtract 0.226 for each OH substituted. Add 0.100 − 0.013n, where n is computed as in footnote d. f When estimating the critical volumes of aromatic substances, use ring compound values, if available, and correct for double bonds.

2-469

2-470

PHYSICAL AND CHEMICAL DATA

∆

V

TABLE 2-337 Joback * Group Contributions for Critical Constants

= (8)(55.1) + (1)(−8) + (1)(−17) = 415.8

Group

Vc = (40 + 415.8) cm3mol = 455.8 cm3mol

Results: Property

DIPPR® recommended value

Ambrose estimation

% Difference

543.8 25.70 468.0

543.0 25.63 455.8

−0.15 0.27 −2.6

Tc /K Pc /bar Vc /(cm3/mol)

Method: Joback method. Reference: Joback, K. G., M.S. Thesis in Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., June 1984. Classification: Group contributions. Expected uncertainty: 7 K (~1 percent) for Tc; 2 bar (~5 percent) for Pc. Applicability: Organic compounds. Input data: Tb, group contributions ∆T, ∆P, ∆V from Table 2-337, and the number of atoms in the molecule nA. Description: A GC method with first-order contributions. Variables Tc, Pc, and Vc are given by the following relations: Tc = Tb 0.584 + 0.965 ∆T −



2 −1

 ∆  T

Pc  = 0.113 + 0.0032nA − ∆ P bar



−2



Vc = 17.5 + ∆V  cm3/mol

(2-9) (2-10) (2-11)

where ∆T, ∆P, ∆V are group contributions from Table 2-337 and nA is the number of atoms in the molecule. Example Estimate the critical constants of o-xylene by using the Joback method. Structure:

CH3

Required input data: From the DIPPR® 801 database, Tb = 417.58 K. From Table 2-337:

苷CH (ring) 苷C < (ring) CH3

T

P

V

0.0141 0.0189 0.0164 0.0067 0.0113 0.0129 0.0117 0.0026 0.0027 0.0020

−0.0012 0 0.0020 0.0043 −0.0028 −0.0006 0.0011 0.0028 −0.0008 0.0016

65 56 41 27 56 46 38 36 46 37

0.0100 0.0122 0.0042 0.0082 0.0143

0.0025 0.0004 0.0061 0.0011 0.0008

48 38 27 41 32

0.0111 0.0105 0.0133 0.0068

−0.0057 −0.0049 0.0057 −0.0034

27 58 71 97

0.0741 0.0240 0.0168 0.0098 0.0380 0.0284 0.0379 0.0791 0.0481 0.0143

0.0112 0.0184 0.0015 0.0048 0.0031 0.0028 0.0030 0.0077 0.0005 0.0101

28 −25 18 13 62 55 82 89 82 36

0.0243 0.0295 0.0130 0.0169 0.0255 0.0085 0.0496 0.0437

0.0109 0.0077 0.0114 0.0074 −0.0099 0.0076 −0.0101 0.0064

38 35 29 9 — 34 91 91

0.0031 0.0119 0.0019

0.0084 0.0049 0.0051

63 54 38

Results:

ni

T

P

V

Property

DIPPR® 801 recommendation

Joback estimation

% Difference

4 2 2

0.0082 0.0143 0.0141

0.0011 0.0008 −0.0012

41 32 65

Tc /K Pc /bar Vc /(cm3/mol)

630.3 37.32 370

630.37 35.86 375.5

0.00 −3.92 1.49

From Eqs. (2-9), (2-10), and (2-11):

∆

T

*Joback, K. G., M.S. thesis in chemical engineering, Massachusetts Institute of Technology, Cambridge, Mass., June 1984.

CH3

Group

Nonring increments: CH3 >CH2 >CH >C< 苷CH2 苷CH 苷C< 苷C苷 CH C Ring increments: CH2  >CH >C< 苷CH 苷C< Halogen increments: F Cl Br I Oxygen increments: OH (alcohol) OH (phenol) O (nonring) O (ring) >C苷O (nonring) >C苷O (ring) CH苷O (aldehyde) COOH (acid) COO (ester) 苷O (except as above) Nitrogen increments: NH2 >NH (nonring) >NH (ring) >N (nonring) N苷 (nonring) N苷 (ring) CN NO2 Sulfur increments: SH S (nonring) S (ring)

= (4)(0.0082) + (2)(0.0143) + (2)(0.0141) = 0.0896

Example Estimate the critical constants of sec-butanol by using the Joback method. Required input data: From DIPPR® 801 database, Tb = 372.7 K, M = 74.1216 kg/kmol. Structure: H 3C

Tc = Tb[0.584 + 0.965 (0.896) − (0.896)2]−1

OH

CH3

Tc = Tb(1.5096) = (417.58 K)(1.5096) = 630.37 K nA = 18

∆ = (4)(0.0011) + (2)(0.0008) + (2)(−0.0012) = 0.00360 P



Pc  = 0.113 + 0.0032nA − bar

∆

−2

P

= [0.113 + (0.0032)(18) − 0.0036]−2 = 35.86

Pc = 35.86 bar

∆

V = (4)(41) + (2)(32) + (2)(65) = 358

Vc = (17.5 + 358) cm3mol = 375.5 cm3mol

Group contributions from Table 2-337: Group

ni

T

CH3 >CH2 >CH OH (alcohol)

2 1 1 1

0.0141 0.0189 0.0164 0.0741

From Eqs. (2-9), (2-10), and (2-11):

P −0.0012 0 0.0020 0.0112

V 65 56 41 28

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-338 Fedors*Method Atomic and Structural Contributions Atomic increments Atom C H O O (alcohols) N N (amines) F Cl Br I S

Calculation using Eq. (2-12):

∆

V

Structural increments V

Feature

V

34.426 9.172 20.291 18.000 48.855 47.422 22.242 52.801 71.774 96.402 50.866

Three-member ring Four-member ring Five-member ring Six-member ring Double bond Triple bond Ring attached to another

−15.824 −17.247 −39.126 −39.508 5.028 0.797 35.524

*Fedors, R. F., AIChE J., 25 (1979): 202.

∆

T

= (2)(0.0141) + (1)(0.0189) + (1)(0.0164) + (1)(0.0741) = 0.1376

Tc = Tb(1.4330) = (372.7 K)(1.4330) = 534.1 K

∆

P

= (2)(−0.0012) + (1)(0.0020) + (1)(0.0112) = 0.0108



1 Pc  = (0.1022 + 0.0032nA)−2 =  bar 0.1022 + (0.0032)(15)

nA = 15

 = 44.33 2

Pc = 44.33 bar

∆

V

= (2)(65) + (1)(56) + (1)(41) + (1)(28) = 255

Vc = (17.5 + 255) cm3mol = 272.5 cm3mol

= (4)(34.426) + (10)(9.172) + (1)(18.000) = 247.4

Vc = (26.6 + 247.4) cm3mol = 274.0 cm3mol

This value differs from the DIPPR® 801 recommended value of 269 cm3/mol by 1.9 percent. Mixtures Application of CS methods and analytical equations of state (EoS) to mixtures typically requires pseudocritical temperatures and pressures. These values are obtained from the pure-component critical properties by applying mixing rules specific to the method under consideration. While pseudocritical values should be used for mixture CS and EoS applications, the values are usually quite different from the mixture’s true critical properties. A variety of methods are available for estimating true critical properties for mixtures (see PGL4), but only modest accuracy can be expected. Normal Melting Point The normal melting point is defined as the temperature at which melting occurs at atmospheric pressure. Methods to estimate the melting point have not been particularly effective because the melting point depends strongly on solid crystal structure and that structure is not effectively correlated with standard GC or CS methods. If the triple point temperature is known, then the melting point is best estimated as being equal to the triple point temperature. However, rarely is the triple point temperature available if the melting point has not also been determined. Recommended Method The method of Constantinou and Gani is recommended with caution. Reference: Constantinou, L., and R. Gani, AIChE J., 40 (1994): 1697. Classification: Group contributions. Expected uncertainty: 25 percent. Applicability: Organic compounds. Input data: First-order and second-order group contributions from molecular structure. Description: A group contribution method given by Tm = (102.425 K) ⋅ ln

Results:

 N t

i m1,i

i

Property Tc /K Pc /bar Vc /(cm3/mol)

DIPPR® 801 recommendation

Joback estimation

% Difference

536.2 42.02 269

534.1 44.33 272.5

−0.39 5.50 1.30

Method: Fedor method. Reference: Fedors, R. F., AIChE J., 25 (1979): 202. Classification: Atom/structure contributions. Expected uncertainty: ~8 cm3/mol (3 percent). Applicability: Organic compounds. Input data: Molecular structure, atom and structural increments, ∆V in Table 2-238. Description: An atom contribution method with structural increments for Vc, given by N Vc = 26.6 + ni∆V  i=1 cm3/mol

(2-12)

Example Use Fedors’ method to estimate the critical volume of secbutanol. Group contributions from Table 2-238: OH CH3 Group

ni

C H Ο (alcohol)

4 10 1



+ Nj tm2, j j

(2-13)

where Ni, Nj = number of first- and second-order groups, respectively tm1,i = first-order group contributions from Table 2-339 tm2,i = second-order group contributions from Table 2-340 Example Estimate the melting point of 2,6-dimethylpyridine. Structure and group contributions:

H3C

N

CH3

Group

Ni

tm1,i

−CH3 −C5H3(N)− Six-member ring

2 1 1

0.4640 12.6275

tm2,i

1.5656

Calculation using Eq. (2-13): Tm = (102.425 K) ln [(2)(0.4640) + 12.6275 + 1.5656] = 278 K

where ∆V are structural contributions from Table 2-238.

H3C

2-471

V 34.426 9.172 18.000

The predicted value is 4 percent higher than the recommended experimental value of 267 K in the DIPPR® 801 database. Normal Boiling Point The normal boiling temperature Tb is the temperature at which the vapor pressure of the liquid equals 101.325 kPa (1.0 atm). In some (usually older) literature sources, Tb values were reported at ambient pressures rather than at 101.325 kPa. If two or more such values are available, they can be be used to obtain Tb by using Eq. (2-2) to linearly interpolate ln P* vs. 1/T values. If there are sufficient vapor pressure data available, then Tb may be found from a regression of the data using an appropriate vapor pressure equation [e.g., Eqs. (2-21)–(2-26)]. If one or a few vapor pressure data at low pressure are available, a common occurrence, then the method of Pailhes can be used to estimate Tb. The most accurate method for prediction of normal boiling temperatures without experimental data is the Nannoolal method.

2-472

PHYSICAL AND CHEMICAL DATA TABLE 2-339 First-Order Groups and Their Contributions for Melting Point* Group CH3 >CH2 >CH >C< CH苷CH2 CH苷CH >C苷CH2 >C苷CH >C苷C< CH苷C苷CH2 >ACH >AC >ACCH3 >ACCH2 >ACCH< OH >ACOH COCH3 COCH2 CHO COOCH3

tm1,i

Group

tm1,i

Group

tm1,i

0.4640 0.9246 0.3557 1.6479 1.6472 1.6322 1.7899 2.0018 5.1175 3.3439 1.4669 0.2098 1.8635 0.4177 −1.7567 3.5979 13.7349 4.8776 5.6622 4.2927 4.0823

COOCH2 OOCH OCH3 OCH2 OCH< OCH2F CH2NH2 >CHNH2 NHCH3 CH2NH >CHNH >NCH3 NCH2 >ACNH2 C5H3(N) CH2CN COOH CH2Cl >CHCl >CCl CHCl2

3.5572 4.2250 2.9248 2.0695 4.0352 4.5047 6.7684 4.1187 4.5341 6.0609 3.4100 4.0580 0.9544 10.1031 12.6275 4.1859 11.5630 3.3376 2.9933 9.8409 5.1638

CCl3 >ACCl CH2NO2 >CHNO2 >ACNO2 CH2SH I Br CCH CC >C苷CCl >ACF CF3 COO CCl2F CClF2 F (other) CONH2 CON(CH3)2 CH3S >CH2S

10.2337 2.7336 5.5424 4.9738 8.4724 3.0044 4.6089 3.7442 3.9106 9.5793 1.5598 2.5015 3.2411 3.4448 7.4756 2.7523 1.9623 31.2786 11.3770 5.0506 3.1468

*

Constantinou, L., and R. Gani, AIChE J., 40 (1994): 1697.

Recommended Method Pailhes method. Classification: Group contributions. Expected uncertainty: 3 K (~1 to 2 percent). Applicability: Organic compounds. Reference: Pailhes, F., Fluid Phase Equilib., 41 (1988): 97. Input data: Molecular structure and one measured vapor pres* sure value Pmeas , (often at a low pressure). The method requires estimation of Tc and Pc using a group contribution method. The original Pailhes method used the Lyderson method [Lydersen, A. L., AIChE J., 21 (1975): 510], but the Joback method is illustrated here for consistency with the previous examples on critical constant estimation.

TABLE 2-340

Description: A simple group contribution method given by log(Pc /bar) + (1 − Tbr)xP 2 Tb = Tmeas  − 3xP − 1.49xP log (Pc /bar)

(2-14)

where Tb = estimate of normal boiling point Pc = critical pressure estimated from group contributions Tbr = reduced normal boiling point estimated from Eq. (2-9) xP = log(1 atm/P*meas) Tmeas = temperature at which experimental vapor pressure is known

Second-Order Groups and Their Contributions for Melting Point*

Group

tm21,i

Group

tm21,i

CH(CH3)2 C(CH3)3 CH(CH3)CH(CH3) CH(CH3)C(CH3)2 C(CH3)2C(CH3)2 Three-member ring Five-member ring Six-member ring Seven-member ring CHn苷CHmCHp苷CHk [k, n, m, p = 0, 1, 2] CH3CHm苷CHn [m, n = 0, 1, 2]

0.0381 −0.2355 0.4401 −0.4923 6.0650 1.3772 0.6824 1.5656 6.9709 1.9913

CHCOOH; CCOOH ACCOOH CH3COOCH; CH3COOC COCH2COO or COCHCOO or COCCOO COOCO ACCOO CHOH COH CHm(OH)CHn(OH) [m, n = 0, 1, 2] CHm cyclicOH [m = 0, 1]

−3.1034 28.4324 0.4838 0.0127 −2.3598 −2.0198 −0.5480 0.3189 0.9124 9.5209

CH2CHm苷CHn [m, n = 0, 1, 2]

−0.5870

CHCHm苷CHn or CCHm苷CHn [m, n = 0, 1, 2] Alicyclic side chain: CcyclicCm [m > 1] CH3CH3

−0.2361

CHCHO; CCHO

0.2476

−2.8298 1.4880 2.0547

CH3COCH2

−0.2951

CH3COCH; CH3COC

−0.2986

Ccyclic(苷O) ACCHO

0.7143 −0.6697

*Constantinou, L., and R. Gani, AIChE J., 40 (1994): 1697.

CHm(OH)CHn(NHp) [m, n, p = 0, 1, 2, 3] CHm(NH2)CHn(NH2) [m, n = 0, 1, 2] CHm cyclicNHpCHn cyclic [m, n, p = 0, 1, 2] CHmOCHn苷CHp [m, n, p = 0, 1, 2] ACOCHm [m = 0, 1, 2, 3] CHm cyclicSCHn cyclic [m, n = 0, 1, 2] CHm苷CHnF [m, n = 0, 1, 2] CHm苷CHnBr [m, n = 0, 1, 2] ACBr ACl

2.7826 2.5114 1.0729 0.2476 0.1175 −0.2914 −0.0514 −1.6425 2.5832 −1.5511

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Example The vapor pressure of n-decylacetate at 348.65 K is 106.66 Pa. Estimate the normal boiling point of this compound. Structure and group contributions from Table 2-337:

C3H

O

Group

ni

T

CH3 >CH2 COO (ester)

2 9 1

0.0141 0.0189 0.0481

CH3 P −0.0012 0 0.0005

Group contribution calculations using Eqs. (2-9) and (2-10):

∆

T

= (2)(0.0141) + (9)(0.0189) + (1)(0.0481) = 0.2464

P

nA = 38

= (2)(−0.0012) + (1)(0.0005) = −0.0019

Pc = (0.113 + 0.0032nA + 0.0019)−2 bar = 17.88 bar = 17.65 atm

Calculation of auxiliary quantities: 1 atm 101325 Pa xP = log  = log  = 2.9777 P*meas 106.66 Pa Calculation of normal boiling point using Eq. (2-14): log17.88 + (1 − 0.76016)(2.9777) Tb  = 348.65  − 3(2.9777) − 1.49(2.9777)2 log 17.88 K Tb = 525.3 K

The estimated value is 1.6 percent higher than the DIPPR® 801 recommended value of 517.15 K. Recommended Method Nannoolal method. Reference: Nannoolal, Y., et al., Fluid Phase Equilib., 226 (2004): 45. Classification: Group contributions. Expected uncertainty: 7 K (on the order of 2 percent). Applicability: Organic compounds for which group values are available. Input data: GC values Ci in Table 2-341; intramolecular groupgroup interactions Cij in Table 2-342. Description: A GC method that includes second-order corrections for steric effects and intramolecular interactions. Variable Tb is found from N

ni ⋅ Ci

Tb i=1  =  + 84.3395 0.6583 n + 1.6868 K

(2-15)

where n = number of nonhydrogen atoms ni = number of occurrences of group i N = number of groups Ci = group contribution from Table 2-341 or Eq. (2-16) Corrections for intramolecular group-group interactions Ci,int are made by dividing the sum of all unique group pairs within the molecule by n. This can be written as 1 N N Ci,int =  Cij (2-16) n i=1 j>i or thought of as the sum of terms in the upper half (above and to the right of the diagonal) of the square matrix formed by listing each group in the molecule along the row and column. The values for the interactions are shown in this format in Table 2-342. Example Estimate the normal boiling point of di-isopropanolamine by using the Nannoolal method. Structure:

Corrections OH::OH OH::NH

ni

Ci

2 4 2 1

177.3066 266.8769 390.2446 223.0992

1/9 2/9

291.7985 286.9698

Group total 354.6132 1067.508 780.4892 223.0992 32.42206 63.77107

Total

2521.902

Note that there are three interacting groups (OH, OH, NH) in the molecule which gives two OH::NH interactions and one OH::OH interaction which are then divided by n (= 9) as in Eq. (2-16) to give the frequency.

Tb 2521.902 + 84.3395 = 509.3  =  K 90.6583 + 1.6868

Tb = 509.3 K

The calculated value differs by −2.4 percent from the DIPPR® 801 recommended value of 521.9 K.

Characterizing and Correlating Constants Acentric Factor The acentric factor of a compound ω is defined in terms of the reduced vapor pressure evaluated at a reduced temperature of 0.7 as ω = −log P*r − 1.0000 (2-17)



Tr = 0.7

It is primarily used as a third parameter (beyond Tc and Pc) in CS predictions as a measure of deviations from nonspherical molecular shape, hence the name, suggesting molecular interactions that are not between centers of molecules. However, as defined in Eq. (2-17), ω also contains polarity information, and ω increases slightly with increasing polarity for molecules of similar size and shape. The value of ω is close to zero for small, spherically shaped, nonpolar molecules (argon, methane, etc.). It increases in value with larger deviations of molecular shape from spherical (longer chain lengths, less chain branching, etc.) and with increasing molecular polarity. When possible, ω should be obtained from experimental vapor pressure correlations by using Eq. (2-17), but an accurate estimation of ω can be made by using the critical constants and a single vapor pressure point by application of CS vapor pressure equations. Recommended Method Ambrose-Walton modification of LeeKesler vapor pressure equations. References: Ambrose, D., and J. Walton, Pure & Appl. Chem., 61 (1989): 1395; Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510. Classification: Corresponding states. Expected uncertainty: Generally within 5 percent. Applicability: Most organic compounds. Input data: Tc, Pc, and a single vapor pressure point (e.g., the normal boiling point). Description: See Eq. (2-25) for the equations used in this method. The vapor pressure equation is inverted to obtain the acentric factor from a single experimental vapor pressure point. Example Calculate the acentric factor of chlorobenzene with a known value for Tb. Input information: From the DIPPR® 801 database, Tb = 404.87 K, Tc = 632.35 K, and Pc = 45.1911 bar. Calculation of auxiliary quantities: 404.87 T Tbr = b =  = 0.64 632.35 Tc

τ = 1 − 0.64 = 0.36

(−5.97616)(0.36) + (1.29874)(0.36)1.5 − (0.60394)(0.36)2.5 − (1.06841)(0.36)5 f (0) =  0.64 = − 3.0034

OH H3C

Group

Calculation using Eq. (2-15):

Tbr = 0.584 + 0.965(0.2464) − (0.2464)2 = 0.76106

∆

Group contributions and values:

CH3 >C(c)<(e) ΟΗ sec NH

O

2-473

CH3 NH

OH

(−5.03365)(0.36) + (1.11505)(0.36)1.5 − (5.41217)(0.36)2.5 − (7.46628)(0.36)5 f (1) =  0.64 = − 3.1788

2-474

PHYSICAL AND CHEMICAL DATA

TABLE 2-341

Group Contributions for the Nannoolal* Method for Normal Boiling Point

Table-specific nomenclature: (e) = connected to N, O, F, Cl; (ne) = not connected to N, O, F, Cl; (r) = in a ring; (c) = in a chain; (a) = aromatic, not necessarily carbon; (Ca) = aromatic carbon; b = any nonhydrogen atom ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

Group CH3  (ne) CH3  (e) CH3(a)  C(c)H2  >C(c)H >C(c)< >C(c)<(e) >C(c)<(Ca)  C(r)H2  >C(r)H >C(r)< >C(r)<(e, c) >C(r)<(e, r) >C(r)<(Ca) 苷 C(a)H 苷 C(a)<(ne) 苷 C(a)<(e) (a)苷C(a)<2(a) F (C, Si)  CF苷C< F (C,Si)(F)(2b) F (C,Si)([F, Cl])(b) F (C,Si)([F, Cl]2) F (Ca) Cl (C,Si) Cl (C,Si)([F, Cl]) Cl (C, Si)([F, Cl]2) Cl (Ca)  CCl苷C< Br (C,Si) Br (Ca) I (C, Si)  OH tert HO (C,Si) sec HO (C,Si) long HO (C,Si) short  OH (Ca) (C,Si) O (C,Si) >(OC2)< NH2  (C, Si) NH2  (Ca) (C,Si) NH (C,Si) (C,Si)2>N (C,Si) COOH (C) (C) COO (C) HCOO (C)  C(c)OO  CON<  CONH  CONH2 O苷C<(Can)2 CHO (Can) SH (C) (C) S (C) (C) S S (C) S(a) (C)CN >C(c)苷C(c)< >C(c)苷 C(c)<(Ca) (e)C(c)苷 C(c)< H2C(c)苷 C< >C(r)苷 C(r)< CC HCC (Ca) O(a) (Ca) 苷N(a) (r5) 苷N(a) (r6) NO2  (C) NO2  (Ca) >Si< >Si<(O) NO3  PO4  O苷N O (C)

Description CH3  not connected to N, O, F, or Cl CH3  connected to N, O, F, or Cl CH3  connected to an aromatic atom (not necessarily C) CH2  in a chain >CH in a chain >C< in a chain >C< in a chain connected to at least one F, Cl, N, or O >C< in a chain connected to at least one aromatic carbon  CH2 in a ring >CH in a ring >C< in a ring >C< in a ring; connected to at least one N, O, Cl, or F which are not part of the ring >C< in a ring connected to at least one N or O which are part of the ring >C< in a ring connected to at least one aromatic carbon aromatic 苷CH aromatic 苷C< not connected to O, N, Cl, or F aromatic 苷C< connected to O, N, Cl, or F aromatic 苷C< with three aromatic neighbors F connected to C or Si F  on a C苷C (vinylfluoride) F connected to C or Si already substituted with at least one F and two other atoms F connected to a C or Si already substituted with one F or Cl and one other atom F connected to C or Si already substituted with two F or Cl atoms F  connected to an aromatic carbon Cl connected to C or Si not already substituted with F or Cl Cl connected to C or Si already substituted with one F or Cl Cl connected to C or Si already substituted with at least two F or Cl Cl connected to aromatic C Cl on a C苷C (vinylchloride) Br connected to a nonaromatic C or Si Br connected to an aromatic C I connected to C or Si  OH connected to tertiary carbon  OH connected to secondary C or Si  OH connected to primary C or Si; chain > 4 C or Si  OH connected to primary C or Si; chain < 5 C or Si  OH connected to an aromatic C (phenols) ether  O connected to two C or Si >(OC2)< (epoxide) NH2  connected to either C or Si NH2  connected to an aromatic C  NH connected to two C or Si (secondary amine) >N connected to three C or Si (tertiary amine)  COOH connected to C  COO connected to two C (ester) HCOO connected to C (formic acid ester)  COO in ring, C is connected to C (lactone)  CON< disubstituted amide  CONH (monosubstituted amide)  CONH2 (amide)  CO connected to two nonaromatic C (ketones) CHO connected to nonaromatic C (aldehydes)  SH connected to C (thioles)  S connected to two C  SS (disulfide) connected to two C  S in an aromatic ring  CN (cyanide) connected to C >C苷C< (both C have at least one non-H neighbor) noncyclic >C苷C< connected to at least one aromatic C noncyclic >C苷C< with at least one F, Cl, N, or O H2C苷C< (1-ene) cyclic >C苷C< CC HCC (1-yne)  O in an aromatic ring with aromatic C neighbors aromatic  N in a five-member ring, free electron pair aromatic 苷N in a six-member ring NO2  connected to aliphatic C NO2  connected to aromatic C >Si< >Si< connected to at least one O nitrate (esters of nitric acid) phosphates nitrites (esters of nitrous acid)

Value 177.3066 251.8338 157.9527 239.4531 240.6785 249.5809 266.8769 201.0115 239.4957 222.1163 209.9749 250.9584 291.2291 244.3581 235.3462 315.4128 348.2779 367.9649 106.5492 49.2701 53.1871 78.7578 103.5672 −19.5575 330.9117 287.1863 267.4170 205.7363 292.5816 419.4959 377.6775 556.3944 349.9409 390.2446 443.8712 488.0819 361.4775 146.4836 820.7118 321.1759 441.4388 223.0992 126.2952 1080.3139 636.2020 642.0427 1142.6119 1052.6072 1364.5333 1487.4109 618.9782 553.8090 434.0811 461.5784 864.5074 304.3321 719.2462 475.7958 586.1413 500.2434 412.6276 475.9623 512.2893 422.2307 37.1936 453.3397 306.7139 866.5843 821.4141 282.0181 207.9312 920.3617 1153.1344 494.2668

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-341

2-475

Group Contributions for the Nannoolal* Method for Normal Boiling Point (Concluded)

Table-specific nomenclature: (e) = connected to N, O, F, Cl; (ne) = not connected to N, O, F, Cl; (r) = in a ring; (c) = in a chain; (a) = aromatic, not necessarily carbon; (Ca) = aromatic carbon; b = any nonhydrogen atom ID

Group

Description

Value

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 94 95 96 97 99 100 101 102 103 104 105 106 107 108 109 111 113 115 116 117

ONC  C苷O O C苷O COCl >Si<(F,Cl) O苷C( O)2 OCN SCN (C) (C) SO2  (C) (C)2>Sn<(C)2 AsCl2  GeCl3  (C)2>Ge<(C)2 >C苷C苷C< >C苷C C苷C<(r) >C苷C C苷C<(c) CHO (Ca) (C,Si)苷N (O苷C< (C)2)a  O O C C CC ( C苷O O C苷O)r (C,Si)a NH (Ca,Si)a  OCON< >N (C苷O)N< (C,Si)2>N< (C,Si)2 F(C,Si)(Cl)(b)2  OCOO >SO4  SO2N< . . .苷CNC苷NC苷. . . >S苷O (S) CN >N(C苷O) (N) CN >P<  ON苷(C,Si) >Se< >Al<

ONC (oxime) anhydride connected to two C COCl connected to C (acid chloride) >Si< connected to at least one F or Cl noncyclic carbonate OCN connected to C or Si (cyanate) SCN (thiocyanate) connected to C noncyclic sulfone connected to two C (sulfones) >Sn< connected to four carbons AsCl2 connected to C GeCl3  connected to carbons >Ge< connected to four carbons cumulated double bond conjugated double bond in a ring conjugated double bond in a chain CHO connected to aromatic C (aldehydes) double-bonded amine connected to at least one C or Si  CO connected to two C with at least one aromatic C (ketones) peroxide conjugated triple bond cyclic anhydride connected to two C  NH connected to two C or Si with at least one aromatic (secondary amines)  CO connected to O and N (carbamate)  CO connected to two N (urea) Quaternary amine connected to four C or Si F connected to C or Si already substituted with at least one Cl and two other atoms  CO connected to two O (carbonates) S(苷O)2 connected to two O (sulfates)  S(苷O)2 connected to N imadizole sulfoxide  CN (cyanide) connected to S  CO connected to N  CN (cyanide) connected to N phosphorus connected to at least 1 C or S (phosphine)  ON苷connected to C or Si (isoazole) >Se< connected to at least one C or Si >Al< connected to at least one C or Si

1041.0851 1251.2675 778.9151 540.0895 879.7062 660.4645 1018.4865 1559.9840 510.4223 1149.9670 1209.2972 347.7717 664.0903 957.6388 928.9954 560.1024 229.2288 606.1797 273.1755 1218.1878 2082.3288 201.3224 886.7613 1045.0343 −109.6269 111.0590 1573.3769 1483.1289 1506.8136 484.6371 1379.4485 659.7336 492.0707 971.0365 428.8911 612.9506 562.1791 761.6006

118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133

C苷C C苷O (C苷O)C([F,Cl]2,3) (C苷O)C([F,Cl]2,3)2 C[F,Cl]3 (C)2C[F,Cl]2 No hydrogen One-hydrogen (3,4) ring 5-ring Ortho pair(s) Meta pair(s) Para pair(s) ((C苷)(C)CCC3) C2CCC2 C3CCC2 C3CCC3

 C苷O connected to sp2 carbon carbonyl connected to C with two or more halogens carbonyl connected to two C, each with at least two halogens carbon with three halogens secondary carbon with two halogens component has no hydrogen component has one hydrogen a three- or four-member nonaromatic ring a five-member nonaromatic ring Ortho- position counted only once and only if there are no meta or para pairs Meta- position counted only once and only if there are no para or ortho pairs Para- position counted only once and only if there are no meta or ortho pairs carbon with four carbon neighbors and one double-bonded carbon neighbor carbon with four carbon neighbors, two on each side carbon with five carbon neighbors carbon with six carbon neighbors

Corrections 40.4205 −82.2328 −247.8893 −20.3996 15.4720 −172.4201 −99.8035 −62.3740 −40.0058 −27.2705 −3.5075 16.1061 25.8348 35.8330 51.9098 111.8372

*

Nannoolal, Y., et al., Fluid Phase Equilib. 226 (2004): 45.

(−0.64771)(0.36) + (2.41539)(0.36)1.5 − (4.26979)(0.36)2.5 + (3.25259)(0.36)5 f (2) =  0.64 = − 0.037 Calculation using Eq. (2-25) at the normal boiling point: 1.01325 ln  = − 3.798 = f (0) + ωf (1) + ωf (2) = 3.0034 + 3.1788ω + 0.037ω2 45.1911 ω = 0.249 The value obtained from the Ambrose-Walton-Lee-Kesler method compares favorably with the value of 0.2499 recommended in the DIPPR® 801 database (obtained from the vapor pressure correlation).

Radius of Gyration The radius of gyration Rg is a measure of the mass distribution about the center of mass of a molecule. Radius Rg increases with molecular size. It is useful in CS applications to separate molecular size and shape effects from polar effects. It is defined in terms of the principal moments of inertia of a molecule (A, B, and C) as Rg =

(AB) N  M 1/2

A

(2-18)

for planar molecules and as Rg =

2π(ABC) N 

M 1/3

A

(2-19)

2-476

PHYSICAL AND CHEMICAL DATA

TABLE 2-342 OH OH(a) COOH O >(OC2)< COOC CO CHO

OH OH(a) COOH O >(OC2)< COOC CO CHO O(a) S(na) S(a) SH NH2 >NH OCN CN

Intermolecular Interaction Corrections for the Nannoolal et al.* Method for Normal Boiling Point OH

OH(a)

COOH

O

>(OC2)<

COOC

CO

CHO

291.7985

0 288.6155

146.7286 −1477.9671 117.2044

135.3991 130.3742 612.8821 91.4997

226.4980 0 0 178.7845 1006.388

211.6814 −1184.9784 −183.2986 322.5671 0 431.0990

46.3754 0 −55.9871 15.6980 22.5208 22.5208 −303.9653

0 43.9722 0 17.0400 163.5475 0 −391.3690 582.1763

O(a)

S(na)

S(a)

435.0923 0 0 329.0050 0 707.9404 176.5481 674.6858 0

−74.0193 0 0 394.5505 0 0 0 0 0 −11.9406

Nitrate OH OH(a) COOH O >(OC2)< COOC CO CHO O(a) S(na) S(a) SH NH2 >NH OCN CN Nitrate 苷N(a)−(r5) 苷N(a)−(r6)

0 −1048.124 0 963.6518 0 −205.6165 −3628.903 140.9644 0 0 0 0 663.8009 0 −263.0807 0 65.1432

苷N(a)(r5) 0 0 0 0 0 0 0 0 −888.612 0 −348.740 0 0 0 0 0 0 0

0 0 0 0 0 0 381.0107 397.575 0 0 0

SH

NH2

>NH

OCN

CN

38.6974 0 0 0 0 0 0 0 0 0 0 217.6360

314.6126 797.4327 0 124.3549 0 182.6291 0 0 395.4093 −562.306 0 0 174.0258

286.9698 0 0 101.8475 0 317.0200 −215.3532 0 0 0 0 0 510.3473 239.8076

0 0 0 0 0 0 0 0 0 0 0 0 0 0 −356.5017

306.3979 0 0 293.5974 0 517.0677 −574.2230 0 0 0 −101.232 0 0 0 0 0

苷N(a)(r6) 1334.6747 −614.3624 0 0 0 0 124.1943 0 0 0 0 0 27.2735 758.9855 0 −370.9729 0 0 −271.9449

*Nannoolal, Y., et al., Fluid Phase Equilib., 226 (2004): 45.

for nonplanar molecules. Radii of gyration can be calculated from these defining equations and principal moments of inertia obtained from spectral data or from computational chemistry software. Recommended Method Principal moments of inertia. Classification: Computational chemistry. Expected uncertainty: Less than 5 percent. Applicability: All molecules. Input data: M and molecular structure. Description: Computational chemistry software is used to optimize the geometry of the molecule and obtain the principal moments of inertia to be used in Eqs. (2-18) and (2-19). Example Calculate the radius of gyration for hydrazine. Input information: From the DIPPR® 801 database, M = 32.0452 kg/kmol. The structure of hydrazine is H2N—NH2 Calculation of the principal moments of inertia: Optimizing hydrazine with HF/6-31G model chemistry gives the following principal moments of inertia:

A = 12.24050 amu⋅Bohr2

B = 72.41081 amu⋅Bohr2

C = 79.16893 amu⋅Bohr

2

Conversion from atomic units to SI gives 5.29177 × 10−11 m A = (12.24050 amu⋅Bohr2)  Bohr



= 5.692 × 10−47 kg⋅m2

1.66054 × 10−27 kg  amu

 2



4.65010−48 kg⋅ m2 B = (72.41081 amu⋅Bohr2)  = 3.367 × 10−46 kg⋅m2 amu⋅Bohr2





4.65010−48 kg ⋅ m2 C = (79.16893 amu⋅Bohr2)  = 3.681 × 10−46 kg⋅m2 amu ⋅ Bohr2





Calculation using Eq. (2-19): (ABC)1/3 = [(5.692 × 10−47)(3.367 × 10−46)(3.681 × 10−46)]1/3 kg⋅m2 = 1.918 × 10−46 kg⋅m2

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Rg =

2π(1.918 × 10 kg⋅m )(6.022⋅10 kmol )  = 1.505 × 10

32.0452 kg/kmol −46

2

−1

26

−10

m

This is 3.8 percent below the DIPPR® 801 database value of 1.564 × 10−10 m which was obtained from spectral principal moments of inertia.

Dipole Moment The dipole moment of a molecule is the first moment of the electric charge density expansion. All normal paraffins have a value of zero. Charge separation within the molecule due to electronegativity differences between bonded atoms increases the dipole moment. Computational chemistry software uses the electron density distribution of the optimized molecule to calculate dipole moments. Recommended Method Electron density distribution. Classification: Computational chemistry. Expected uncertainty: Uncertainty varies depending upon the model chemistry chosen, but it can be as large as 60 percent. Applicability: All molecules. Input data: Molecular structure.

2-477

be estimated by the methods shown earlier if experimental values are unavailable. Recommended Method 1 Riedel method. Reference: Riedel, L., Chem. Ing. Tech., 26 (1954): 679 Classification: Empirical extension of theory and corresponding states. Expected uncertainty: Varies strongly depending upon relative T, but 1 percent or less above Tb is typical with uncertainties of 5 to 30 percent near the triple point. Applicability: Most organic compounds. Input data: Tb, Tc, Pc. Description: Equation (2-22) in reduced form B ln Pr = A +  + C ln Tr + DT6r Tr

(2-24)

Constants for this equation are determined from the following set of relationships:

Example Calculate the dipole moment for methanol. Draw structure and optimize molecule using computational chemistry software: The dipole moment obtained from a geometry optimized with the HF/631G model chemistry for methanol is 2.288 D. This value is 35 percent larger than the experimental gas-phase value of 1.700 D in the DIPPR® 801 database.

36 6 ψ = − 35 +  + 42 ln Tbr − T br Tbr

3.758Kψ + ln (Pc /1.01325 bar) αc =  Kψ − ln Tbr

ln (Pc /1.01325 bar) h = Tbr  1 − Tbr

D = K(αc − 3.758)

VAPOR PRESSURE

C = αc − 42D

A = 35D

Liquids Vapor pressure is the most important of the basic thermodynamic properties of fluids. It is the pressure of equilibrium, coexisting liquid and vapor phases at a specified temperature. The vapor pressure curve is a monotonic function of temperature from its minimum value (the triple point pressure) at the triple point temperature Tt to its maximum value (the critical pressure) at Tc. Liquid vapor pressure data over a limited temperature range can be correlated with the Antoine [Antoine, C, C.R., 107 (1888): 681, 836] equation B P∗ ln  = A −  Τ/Κ + C Pa

(2-20)

Data from the triple point to the critical point can be correlated with either a modified form of the Wagner equation [Wagner, W., “A New Correlation Method for Thermodynamic Data Applied to the VaporPressure Curve of Argon, Nitrogen, and Water,” J.T.R. Watson (trans. and ed.), IUPAC Thermodynamic Tables Project Centre, London, 1977; Ambrose, D., J. Chem. Thermodyn., 18 (1986): 45; Ambrose, D., and N. B. Ghiassee, J. Chem. Thermodyn., 19 (1987): 903, 911] aτ + bτ1.5 + cτ2.5 + dτ5 ln Pr* =  1−τ

where τ  1 − Tr

(2-21)

Values of the constant K [Vetere, A., Ind. Eng. Chem Res., 30 (1991): 2487]: Class

Value

P∗ B T T ln  = A +  + C ln  + D  Pa T/K K K



E

(2-22)

Generally, E in Eq. (2-22) is assigned a value of 6, but values of 2 or 1 have also been used, particularly when correlating low-temperature data. While the Wagner equation can be used to correlate most fluids over the whole liquid range, a fifth term is often required for alcohols [Poling, B. E., Fluid Phase Equil., 116 (1996): 102]: (for alcohols) (2-23)

Correlation of experimental data within a few tenths of a percent over the entire fluid range can usually be obtained with either the Wagner or Riedel equations. Two prediction methods are recommended for liquid vapor pressure. The first method is based on the Riedel equation; the second is a CS method. Both methods require Tc and Pc as input, but these can

K = −0.120 + 0.025h K = 0.373 − 0.030h K = 0.0838

Acids Alcohols All other organic compounds

Example Estimate the vapor pressure of chlorobenzene at 50 K intervals from 300 to 600 K. Input information: From the DIPPR® 801 database, Tb = 404.87 K, Tc = 632.35 K, and Pc = 45.1911 bar. Auxiliary Quantities: K = 0.0838

Tbr = 404.87/632.35 = 0.640

36 ψ = − 35 +  + 42 ln 0.640 − (0.640)6 = 2.431 0.640 (3.758)(0.0838)(2.431) + ln(45.191/1.01325) αc =  = 7.0248 (0.0838)(2.431) − ln(0.640) D = (0.0838)(7.0248 − 3.758) = 0.2738 B = − (36)(0.2738) = − 9.8552

or the Riedel [Riedel, L., Chem. Ing. Tech., 26 (1954): 679] equation

aτ + bτ1.5 + cτ2.5 + dτ5 + eτ6 ln Pr* =  1−τ

B = − 36D

C = 7.0248 − (42)(0.2738) = − 4.4729 A = − (35)(0.2738) = 9.5814

Calculation using Eq. (2-24) at each T (detailed calculation shown for T = 500 K): Tr = 500/632.35 = 0.7907 9.8552 ln Pr = 9.5814 −  − 4.4729 ln 0.7907 + (0.2738)(0.7907)6 = − 1.7651 0.7907 Pr = exp(−1.7651) = 0.1712

P = PrPc = (0.1712)(45.1911 bar) = 7.74 bar

T/K

Tr

ln Pr

P/bar

PDIPPR/bar

% Error

300 350 400 450 500 550 600

0.4744 0.5535 0.6326 0.7116 0.7907 0.8698 0.9488

−7.8532 −5.5704 −3.9323 −2.7101 −1.7651 −1.0067 −0.3705

0.0176 0.172 0.886 3.01 7.74 16.51 31.20

0.0175 0.172 0.880 2.98 7.67 16.39 31.11

0.3 0.1 0.6 0.9 0.9 0.8 0.3

Recommended Method 2 Ambrose-Walton method. References: Ambrose, D., and J. Walton, Pure & Appl. Chem., 61 (1989): 1395; Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510.

2-478

PHYSICAL AND CHEMICAL DATA

Classification: Corresponding states. Expected uncertainty: Varies strongly with relative T, but less than 1 percent is typical above Tb if the acentric factor is known. Applicability: Most organic compounds. Input data: Tb, Tc, Pc, and ω. Description: The acentric factor is used to linearly interpolate within the simple-fluid and deviation terms for the ln P* values of the reference fluids, which themselves have been correlated with the Wagner vapor pressure equation. ln Pr* = f (0) + ωf (1) + ω2f (2)

−5.03365τ + 1.11505τ − 5.41217τ − 7.46628τ f (1) =  1−τ 2.5

5

(2-25)

−0.64771τ + 2.41539τ1.5 − 4.26979τ2.5 + 3.25259τ5 f (2) =  1−τ where τ = 1 − Tr. Example Repeat the calculation of the liquid vapor pressure of chlorobenzene at 50 K intervals from 300 to 600 K. Input information: From the DIPPR® 801 database, Tc = 632.35 K, Pc = 45.1911 bar, and ω = 0.249857. Auxiliary quantities: Tr = 500/632.35 = 0.7907

τ = 1 − 0.7907 = 0.2093

Simple-fluid and deviation vapor pressure terms at each T (shown for T = 500 K): (−5.97616)(0.2093) + (1.29874)(0.2093)1.5 − (0.60394)(0.2093)2.5 − (1.06841)(0.2093)5 f (0) =  = − 1.4405 0.7907 (−5.03365)(0.2093) + (1.11505)(0.2093)1.5 − (5.41217)(0.2093)2.5 − (7.46628)(0.2093)5 f (1) =  = − 1.3383 0.7907 (−0.64771)(0.2093) + (2.41539)(0.2093)1.5 − (4.26979)(0.2093)2.5 + (3.25259)(0.2093)5 f (2) =  = 0.0145 0.7907

Calculation using Eq. (2-25): ln Pr* = − 1.4405 + (0.249857)(−1.3383) + (0.249857)2 (0.0145) = − 1.774 P* = (45.1911 bar)[exp(−1.774)] = 7.667 bar

T

τ

f (0)

f (1)

f (2)

300 350 400 450 500 550 600

0.5256 0.4465 0.3674 0.2884 0.2093 0.1302 0.0512

−5.9228 −4.3006 −3.1036 −2.1800 −1.4405 −0.8289 −0.3068

−7.5966 −5.0017 −3.3106 −2.1576 −1.3383 −0.7318 −0.2612

−0.3050 −0.1439 −0.0437 0.0043 0.0145 0.0036 −0.0081

ln P*r

% P*/bar P*DIPPR/bar Error

−7.840 0.0178 −5.559 0.174 −3.933 0.885 −2.719 2.98 −1.774 7.67 −1.012 16.43 −0.373 31.14

0.0175 0.172 0.880 2.98 7.67 16.39 31.11

The liquid and solid vapor pressures are identical at the triple point. A good vapor pressure correlation that is valid at the triple point may be used to obtain the triple point pressure. Estimating solid vapor pressures by using Eq. (2-26) generally requires an estimation of ∆Hsub, and so the illustrative example is combined with the example on enthalpy of sublimation in the section on latent enthalpy. THERMAL PROPERTIES

−5.97616τ + 1.29874τ1.5 − 0.60394τ2.5 − 1.06841τ5 f (0) =  1−τ 1.5

where Tt = triple point temperature P*t = triple point pressure ∆Hsub = enthalpy of sublimation

1.4 1.5 0.5 0.0 0.0 0.3 0.1

Solids Below the triple point, the pressure at which the solid and vapor phases of a pure component are in equilibrium at any given temperature is the vapor pressure of the solid. It is a monotonic function of temperature with a maximum at the triple point. Solid vapor pressures can be correlated with the same equations used for liquids. Estimation of solid vapor pressure can be made from the integrated form of the Clausius-Clapeyron equation

Enthalpy of Formation The standard enthalpy (heat) of formation is the enthalpy change upon formation of 1 mol of the compound in its standard state from its constituent elements in their standard states. Two different standard enthlapies of formation are commonly defined based on the chosen standard state. The standard state enthalpy of formation ∆Hfs uses the naturally occurring phase at 298.15 K and 1 bar as the standard state; the ideal gas standard enthalpy (heat) of formation ∆Hfo uses the compound in the ideal gas state at 298.15 K and 1 bar as the standard state. In both cases, the standard state for the elements is their naturally occurring state of aggregation at 298.15 K and 1 atm. Sources for data include DIPPR®, TRC, SWS, JANAF, and Daubert, T. E., and R. P. Danner, Technical Data Book—Petroleum Refining, 5th ed., American Petroleum Institute, Washington, extant 1994. The Domalski-Hearing method is the most accurate general method for estimating either ∆Hfs or ∆Hfo if the appropriate GC values are available, but a CC method is also as accurate for estimating ∆Hfo if an isodesmic reaction can be formulated and used. The Domalski-Hearing method also applies to entropies, and the entropy predictive equations are listed in this section for convenience because they are equivalent in form to the enthalpy equations. However, discussion and illustration of the estimation methods for entropy are delayed to the next subsection. Recommended Method Domalski-Hearing method. Reference: Domalski, E. S., and E. D. Hearing, J. Phys. Chem. Ref. Data, 22 (1993): 805. Classification: Group contributions. Expected uncertainty: 3 percent. Applicability: Organic compounds for which group contributions have been regressed. Input data: Molecular structure. Description: GC values from Table 2-343 are directly additive for both enthalpy of formation and absolute third-law entropies: N H!f ni (∆H!f )i  kJ/mol =

i=1

N S! ni (S!)i  J⋅mol−1 K−1 =

i=1

where (∆Hof )i = enthalpy of formation GC value from Table 2-343 and (So)i = entropy GC value from Table 2-343. Group values in Table 2-343 are defined by the central, nonhydrogen group and the atoms bonded to that group. Thus, C—(2H)(2C) represents a C atom to which 2 H and 2 C atoms are bonded. For example, propane (CH3—CH2—CH3) is composed of three groups: two C—(3H)(C) and one C—(2H)(2C). Example Estimate the standard and ideal gas enthalpies of formation of o-toluidine. Input information: The melting point (256.8 K) and boiling point (473.49 K) given for o-toluidine in the DIPPR® 801 database bracket 298.15 K, and so the standard state phase at 298.15 K and 1 bar must be liquid. Structure: CH3

P∗ ∆Hsub T ln * =  1 − t Pt RTt T





(2-26)

(2-27)

NH2

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-343

2-479

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. Group

∆Hfo

So

∆Hfs liq.

Ss liq.

∆Hfs solid

Ss solid 56.69 23.01 −16.89 0.00 −33.19 0.00 0.00 0.00

21.75

CH Groups C(3H)(C) C(2H)(2C) C(H)(3C) CH3 corr (tertiary) C(4C) CH3 corr (quaternary) CH3 corr (tert/quat) CH3 corr (quat/quat) Cd(2H) Cd(H)(C) Cd(2C) Cd(H)(Cd) Cd(C)(Cd) Cd(Cd)(Cb) Cd(H)(Cb) Cd(C)(Cb) Cd(H)(Ct) C(4H), Methane Cd(2Cb) C(2H)(C)(Cd) C(H)(2C)(Cd) CH3 corr (tertiary) C(3C)(Cd) CH3 corr (quaternary) C(H)(C)(2Cd) C(2H)(2Cd) C(2H)(Cd)(Cb) C(H)(C)(Cd)(Cb) cis (unsat) corr tertButyl cis corr Ct(H) Ct(C) Ct(Cd) Ct(Cb) Ct(Ct) C(2H)(C)(Ct) C(H)(2C)(Ct) CH3 corr (tertiary) C(3C)(Ct) CH3 corr (quaternary) C(2H)(2Ct) C(2C)(2Ct) Ca Cb(H)(2Cb) Cb(C)(2Cb) Cb(Cd)(2Cb) Cb(Ct)(2Cb) Cb(3Cb) C(2C)(2Cb) C(2H)(C)(Cb) C(H)(2C)(Cb) C(Cb)(3C) C(2H)(2Cb) C(H)(C)(2Cb) C(H)(3Cb) C(3Cb)(C) C(4Cb) Cbf(Cbf)(2Cb) Cbf(Cb)(2Cbf) Cbf(3Cbf) Cb(2Cb)(Cbf) Cb(Cb)(2Cbf) ortho corr, hydrocarbons meta corr, hydrocarbons Cyclopropane rsc (unsub) Cyclobutane rsc Cyclopentane rsc (unsub) Cyclohexane rsc (unsub) Cycloheptane rsc Cyclooctane rsc Cyclononane rsc Cyclodecane rsc

−42.26 −20.63 −1.17 −2.26 19.20 −4.56 −1.80 −0.64 −26.32 36.32 44.14 28.28 36.78

127.32 39.16 −53.60 0.00 −149.49 0.00 0.00 0.00 115.52 33.05 −50.84 27.74 −61.33

−47.61 −25.73 −4.77 −2.18 17.99 −4.39 −1.77 −0.64 21.75 31.05 39.16 22.18 30.42

83.30 32.38 −23.89 0.00 −98.65 0.00 0.00 0.00 86.19 28.58 −29.83 13.30 −41.92

28.28 37.95 28.28 −74.48 32.88 −20.88 −1.63 −2.26 22.13 −4.56 −1.17 −18.92

27.74 −51.97 27.74 206.92

22.18 38.58 22.18

13.30

−46.74 −29.41 −5.98 −2.34 12.47 −4.35 −2.70 −2.24 22.43 25.48 32.97 17.53 27.91 56.07 17.53

13.30

17.53

38.20 −50.38 0.00 −150.23 0.00 −53.60 42.08

31.67 −28.07 0.00 −108.20 0.00 −23.89 19.32

49.91 −24.35 −6.49 −2.34 12.51 −4.35 −5.98 −21.60

4.85 17.24 113.50 115.10 121.42 120.76 120.76 −19.70 −3.16 −2.26

5.06 0.00 101.96 26.32 39.92 17.77 25.94 42.80 −45.69 0.00

0.00 0.00 67.57 14.25

5.73 17.57 110.34 101.66

32.36

103.28 103.28 −29.41

−4.56 −41.14

0.00

142.67 13.81 23.64 24.17 24.17 21.66

26.28 48.31 −35.61 −33.85 −33.85 −36.57

−21.34 −4.52 18.28 −46.43

42.59 −48.00 −147.19

22.46 1.26 −0.63 115.15 110.89 26.75 0.68 26.34 40.65 52.91 51.99

−2.18 22.83 −4.39 −39.08 20.67 134.68 8.16 19.16 19.12 19.12 17.21

0.00 0.00 14.39 28.87 −19.50 −9.04 −9.04

−24.81 −5.82 18.70 −26.50 −21.47

47.40 −13.90 −96.10 51.97 28.12

0.00

15.83 11.50 −0.90

−5.54

−2.50 0.00 134.86 126.04 116.22 78.18 73.97 70.78

3.26 0.00 111.58 106.64 22.84 −1.77 23.50 38.10 50.40 50.61

0.00 0.00

−6.86 27.04 20.10 16.00 3.59

30.83 −25.73 −5.02 −2.18 20.79 −4.39 −4.77 −24.43 −24.73 −6.90 5.27 17.48 104.47 107.15 114.77 119.00 104.80 −22.13

51.48 42.24 10.07 15.89 2.96

−2.34 26.38 −4.35 131.08 6.53 13.90 20.27 20.07 17.03 52.81 −22.10 −3.50 21.57 −21.44 16.40 34.48 116.25 64.89 14.10 12.00 1.94 −8.77 47.93 5.00 2.00 114.43 34.00 10.94

21.75 21.75

0.00 0.00 −16.89

0.00 0.00

0.00 0.00

22.75 −5.50 −10.00 −10.00 −6.00 26.90 22.85 −12.62 −6.00 2.00 7.00 0.00 0.00

2-480

PHYSICAL AND CHEMICAL DATA

TABLE 2-343

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties (Continued )

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. Group

∆Hfo

So

∆Hfs liq.

Ss liq.

∆Hfs solid

Ss solid

CHO Groups CO(2H), formaldehyde CO(C)(CO) CO(H)(CO) CO(CO)(Cb) CO(O)(CO) CO(Cd)(O) CO(C)(O) CO(H)(O) CO(2O) CO(H)(Cd) CO(2Cb) CO(C)(Cb) CO(H)(Cb) CO(O)(Cb) CO(2C) CO(H)(C) CO(C)(Cd) O(2CO), aliphatic O(2CO), aromatic O(Cd)(CO) O(C)(CO) O(H)(CO) O(Cb)(CO) O(C)(O) O(H)(O) O(2Cd) O(H)(Cd) O(C)(Cd) O(2Cb) O(C)(Cb) O(H)(Cb) O(2C) O(H)(C) Cd(H)(CO) Cd(C)(CO) Cd(O)(Cd) Cd(O)(C) Cd(O)(H) Ct(CO) Cb(CO)(2Cb) Cb(O)(2Cb) C(2H)(2CO) C(CO)(3C) C(H)(CO)(2C) C(2H)(CO)(C) C(3H)(CO) C(2H)(CO)(Cd) C(2H)(CO)(Ct) C(2H)(CO)(Cb) C(H)(CO)(C)(Cb) C(H)(O)(CO)(C) C(4O) C(H)(3O) C(3O)(C) C(2O)(2C) C(H)(2O)(C) C(2H)(2O) C(2H)(O)(Cb) C(2H)(O)(Cd) C(H)(CO)(C)(Cb) C(H)(CO)(2Cb) C(O)(3Cb) C(O)(3C) (ethers, esters) C(H)(O)(2C) (ethers, esters) C(O)(3C) (alcohols, peroxides) C(H)(O)(2C) (alcohols, peroxides) C(2H)(O)(C) C(3H)(O) O(CO)(O) C(2C)(O)(Cb) C(H)(C)(2O)

−108.60 −121.29 −105.98 −112.30 −123.75 −136.73 −137.24 −124.39 −111.88 −126.96 −110.00 −148.82 −121.35 −125.00 −132.67 −124.39

224.54

−214.50 −238.30 −198.03 −188.87 −254.30 −167.00 −20.75 −72.26 −139.29

34.16

62.59 62.59 147.03

64.31 147.03

36.03 101.71

−135.04 −123.30 −155.56 −149.37 −142.42 −122.00 −153.05 −119.00 −145.22 −138.12 −140.00 −152.76 −142.42 −230.50 −220.90 −201.42 −196.02 −285.64 −165.50 −23.50 −101.75 −137.32

−129.33 −77.66 −92.55 −160.30 −101.42 −159.33 32.30

121.50 29.33 121.50 35.19

−133.72 −85.27 −104.85 −191.75 −110.83 −191.50 26.61

36.78 44.14 36.32

−61.34 −50.84 33.05

30.42 39.08 31.05

15.50 −4.75 −30.74 23.93 −0.25 −21.84 −42.26 −16.95 −25.48 −16.20 126.63 −152.46 −113.97 −114.39 −53.56 −57.78 −62.22 −33.76 −27.49

9.50 −19.46 −13.50 −26.10 −32.90 −42.26 −88.00 15.30

−43.72

39.58 127.32

37.49

−141.92 −52.80 −144.60 −43.05 43.43 127.32

−140.75

32.72 94.68

33.81 93.55

−117.75 −120.81 −134.10 −153.60

32.90 32.13

−123.00

−42.92

−116.00 −143.70 −160.18 −145.00 −157.95

23.72 32.13

−235.00 −207.00 38.28 38.28

−210.60 −282.15 −170.00 −30.20 −105.30

12.09 21.78 45.32

23.31

−96.20 −122.87 −199.25 −119.00 −199.66 7.82

3.14

43.89 26.78 43.89 −41.92 −29.83 28.58

28.62 28.62 27.53

−85.98 −24.52 39.87 83.30

27.91 32.97 25.48 144.52 8.15 1.00 −19.10 24.02 −9.83 −27.90 −46.74

24.73 56.69

123.43 −133.34 −107.74 −99.54 −41.30 −51.42 −62.89 −29.17 −28.62

−46.71

14.81 −14.39

8.08

0.79 −21.00 −11.13 −27.60 −35.80 −47.61 −90.00 25.80

−94.68 −25.31 −122.48 −29.83 32.59 83.30

10.50 −5.61 −23.06 26.15 −3.89 −24.14 −47.61 −19.62 −26.61 −11.67

−10.59

0.08 1.59

23.85 −14.39 3.72 60.46 −0.50 −20.08 −12.25 −29.08 −33.00 −46.74 −80.50 29.30 −52.50

−14.77 6.95 24.73 56.69

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-343

2-481

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties (Continued)

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. Group

∆Hfo

So

∆Hfs liq.

Ss liq.

∆Hfs solid −46.74 −34.00 −13.90 −2.34 1.00 −4.35 −26.00 −33.31 −6.30 −46.00 47.80 101.00 18.97

Ss solid

CHN and CHNO Groups C(3H)(N) C(2H)(C)(N) C(H)(2C)(N) CH3 corr (tertiary) C(3C)(N) CH3 corr (quaternary) C(2H)(2N) C(2H)(Cb)(N) N(2H)(C) (first, amino acids) N(2H)(C) (second, amino acids) N(H)(2C) N(3C) N(2H)(N) N(H)(C)(N) N(2C)(N) N(2Cb)(N) N(H)(Cb)(N) N(2CO)(N) N(H)(2Cd) N(C)(2Cd) N(2H)(Cb) N(H)(C)(Cb) N(2C)(Cb) N(C)(2Cb) N(H)(2Cb) N(3Cb) NI(C) NI(Cb) NA(C) NA(Cb) NA(oxide)(C) C(2H)(C)(NA) C(H)(2C)(NA) C(3C)(NA) Cd(H)(N) Cd(C)(N) Cb(N)(2Cb) Cb(NO)(2Cb) Cb(NO2)(2Cb) Cb(CNO)(2Cb) Cb(CN)(2Cb) Cb(NA)(2Cb) Cb(H)(2NI) CO(H)(N) CO(C)(N) CO(Cb)(N) (amides) CO(Cb)(N) (amino acids) CO(Cd)(N) CO(2N) N(2H)(CO) (amides, ureas) N(2H)(CO) (amino acids) N(H)(C)(CO) (amides, ureas) N(H)(C)(CO) (amino acids) N(2C)(CO) N(H)(Cb)(CO) N(H)(2CO) N(C)(2CO) N(Cb)(2CO) N−(2Cb)(CO) N(C)(Cb)(CO) C(3H)(CN), acetonitrile C(2H)(C)(CN) C(H)(2C)(CN) C(3C)(CN) C(2C)(2CN) C(2H)(Cd)(CN) Cd(H)(CN) Ct(CN) C(3H)(NO2), nitromethane C(2H)(2NO2), dinitromethane C(H)(3NO2), trinitromethane C(4NO2), tetranitromethane C(2H)(C)(NO2)

−42.26 −28.30 −16.70 −2.26 0.29 −4.56 −30.00 −24.14 19.25 19.25 67.55 116.50 47.70 89.16 120.71

127.32 42.26 −63.55 0.00 −152.59 0.00

−47.61 −30.80 −14.65 −2.18 5.10 −4.39

83.30 32.38 −20.00 0.00 −87.99 0.00

124.40 126.90 33.96 −61.71 122.18

−26.09 0.33 0.33 51.50 112.00 25.30 75.00 119.00

71.71 71.71 32.09 −38.62 60.58 22.05 −26.94

87.50

73.40

83.55 120.64 19.25 59.00 126.40 120.44 83.55 123.15 81.46 69.00 109.50 109.50 40.80 −20.70 −2.66 11.50 −16.00 −5.74 −1.30 21.50 −1.45 −177.63 151.00 22.55 6.30 −124.39 −133.26

50.50 97.38 −11.00 26.25 109.40 97.38 50.50 121.80 73.68 54.50 104.85 104.85 22.65 −25.70 −5.42 15.50 −15.50 −5.62 1.50

−24.43

−171.80 −111.00 −63.00 −63.00 −16.28 −16.28 45.00 −20.84 −91.00 −11.64 9.12

126.90

47.01

−43.53

71.71

36.40

137.35 66.90 73.62 45.40 88.92 −21.60 36.55 96.50 89.30 45.40 107.50

56.69 23.01 0.00 0.00 39.00 48.75

70.00

57.00 103.00 103.00 −29.41

−28.30

79.95

85.25

122.38 20.08

64.75

147.03 56.70

−188.00 −185.00

93.55

96.00 88.25

−190.50 −63.90 −63.90 −17.10 −17.10 62.00 56.20

10.50 −13.00 −3.95 9.75 23.00 −32.50 155.69 121.20 18.65 0.25

−37.57 110.46 50.45

−194.60 −177.75 −177.75

40.00

−203.10 −65.25 −59.75 −9.80 5.50 55.00 −3.50 −30.80 64.00

69.00 18.00 33.03

60.85 72.00 74.04 94.52 113.50 137.96 95.31 146.65 264.60 −74.86 −58.90 −0.30 82.30 −60.50

252.60 167.25 67.86

158.41 284.14

203.60

40.56 66.07 81.50 116.20 66.40 117.28 250.20 −112.60 −104.90 −32.80 38.30 −93.50

149.62 106.02 −17.91

69.85 69.00 102.07

96.15 74.57

92.72 171.75 −48.00 −99.00

2-482

PHYSICAL AND CHEMICAL DATA

TABLE 2-343

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties (Continued)

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. Group

∆Hfo

∆Hfs liq.

So

Ss liq.

∆Hfs solid

Ss solid

CHN and CHNO Groups C(H)(2C)(NO2) C(3C)(NO2) C(2H)(Cb)(NO2) C(H)(C)(2NO2) C(2C)(2NO2) C(H)(C)(CO)(N) C(2H)(CO)(N) C(H)(Cb)(CO)(N) O(C)(NO) O(C)(NO2) N(H)(C)(NO2) N(H)(Cb)(NO2) N(H)(CO)(NO2) N(C)(2NO2) N(C)(Cb)(NO2) N(2C)(NO) N(2C)(NO2) C(2H)(C)(N3) C(H)(2C)(N3) C(2H)(Cb)(N3) C(3Cb)(N3) Cb(N3)(2Cb)

−53.00 −36.65 −62.00 −36.80 −28.50 −18.70 −3.10

115.32

−24.23 −79.71

166.11 191.92

100.30 183.00 90.00 88.00

−82.50 −61.20 −82.76 −88.80 −77.20

−46.50 −108.96

−89.00 −76.55 −81.00 −91.50 −90.30 −11.65 −30.95 127.50

−124.00 16.50 −14.00

53.50 167.00 59.00 50.00 321.70 255.00 327.40

274.00 347.00 328.60 320.00

−4.00 24.00

150.50 55.00 40.00

346.50 303.50 CHS and CHSO Groups

C(3H)(S) C(2H)(C)(S) C(H)(2C)(S) CH3 corr (tertiary) C(3C)(S) CH3 corr (quaternary) CH3 corr (tert/quat) CH3 corr (quat/quat) C(2H)(Cb)(S) C(2H)(Cd)(S) C(2H)(2S) Cb(S)(2Cb) Cd(H)(S) Cd(C)(S) S(C)(H) S(Cb)(H) S(2C) S(H)(Cd) S(C)(Cd) S(2Cd) S(Cb)(C) S(C)(S) S(Cb)(S) S(2S) S(2Cb) S(H)(S) S(H)(CO) CO(C)(S) C(3H)(SO) C(2H)(C)(SO) C(H)(2C)(SO) CH3 corr (tertiary) C(3C)(SO) CH3 corr (quaternary) C(2H)(Cd)(SO) cis correction Cb(SO)(2Cb) O(SO)(H) O(C)(SO) SO(2C) SO(2Cb) SO(2O) SO(C)(Cb) C(3H)(SO2) C(2H)(C)(SO2) C(H)(2C)(SO2) CH3 corr (tertiary) C(3C)(SO2) CH3 corr (quaternary)

−42.26 −23.17 −5.88 −2.26 13.52 −4.56 −1.80 −0.64 −18.53 −25.93 −25.10 −4.75 36.32 45.73 18.64 48.10 46.99 25.52 54.39 102.60 76.21 27.62 57.45 12.59 102.60 7.95 −5.90 −132.67 −42.26 −29.16

127.32 41.87 −47.36 0.00 −145.38 0.00 0.00 0.00

−47.61 −26.77 −6.07 −2.18 16.69 −4.39 −1.77 −0.64 −23.82 −32.44

83.30 41.09 −16.61 0.00 −86.86 0.00 0.00 0.00

−46.74

56.69

−2.34

0.00

−4.35 −2.70 −2.24

0.00 0.00 0.00

43.72 33.05 −51.92 137.67 57.34 55.19

−5.61 31.05

−10.59 28.58

1.00 25.48

1.59

0.06 28.51 29.82

85.95 89.04 29.80

58.20 14.36

35.44 30.84

−2.26 4.56 −4.56 −27.56 4.11 15.48 −158.60 −92.60 −66.78 −62.26 −213.00 −72.00 −42.26 −27.03 −14.00 −2.26 1.52 −4.56

0.00

68.59 50.50

42.00 40.60

56.07 68.59 130.54 64.31 127.32

93.02 −152.76 −47.61 −36.88

33.81 83.30

−46.74

56.69

−2.18 0.97 −4.39 −32.63 5.27 25.44

0.00

−2.34

0.00

0.00

−4.35

0.00

0.00

5.73 7.55

0.00 0.08

75.73

−108.98

22.18

127.32

−47.61 −33.76

83.30

−46.74 −35.96

56.69

0.00

−2.18 2.00 −4.39

0.00

−2.34 3.78 −4.35

0.00

0.00 5.06

0.00

0.00

0.00

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-343

2-483

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties (Continued)

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. Group

∆Hfo

So

∆Hfs liq.

Ss liq.

∆Hfs solid

Ss solid

CHN and CHNO Groups CH3 corr (quat/quat) C(2H)(Cd)(SO2) C(H)(C)(Cd)(SO2) C(2H)(Cb)(SO2) C(2H)(Ct)(SO2) Cb(SO2)(2Cb) Cd(H)(SO2) Cd(C)(SO2) Ct(SO2) SO2(Cd)(Cb) SO2(2Cd) SO2(2C) SO2(C)(Cb) SO2(2Cb) SO2(SO2)(Cb) SO2(2O) SO2(C)(Cd) SO2(Ct)(Cb) O(SO2)(H) O(C)(SO2)

−0.64 −29.49 −71.99 −29.80 16.36 15.48 51.58 64.01 177.10 −291.55 −306.70 −288.58 −289.10 −287.76 −325.18 −417.30 −316.80 −296.30 −158.60 −91.40

C(3H)(F), methyl fluoride C(3H)(Cl), methyl chloride C(3H)(Br), methyl bromide C(3H)(I), methyl iodide C(C)(3F) C(2H)(C)(F) C(H)(2C)(F) C(3C)(F) C(H)(C)(2F) C(2C)(2F) C(C)(Cl)(2F) C(H)(C)(Cl)(F) C(C)(3Cl) C(H)(C)(2Cl) C(2H)(C)(Cl) C(2C)(2Cl) C(H)(2C)(Cl) C(3C)(Cl) C(C)(3Br) C(H)(C)(2Br) C(2H)(C)(Br) C(2C)(2Br) C(H)(2C)(Br) C(3C)(Br) C(C)(3I) C(H)(C)(2I) C(2H)(C)(I) C(2C)(2I) C(H)(2C)(I) C(3C)(I) C(H)(C)(Br)(Cl) N(C)(2F) C(H)(C)(Cl)(O) C(2H)(I)(O) C(C)(2Cl)(F) C(C)(Br)(2F) C(C)(2Br)(F) C(Br)(Cl)(F) Cd(H)(F) Cd(H)(Cl) Cd(H)(Br) Cd(H)(I) Cd(C)(Cl) Cd(2F) Cd(2Cl) Cd(2Br) Cd(2I) Cd(Cl)(F) Cd(Br)(F) Cd(Cl)(Br) Ct(F)

−247.00 −81.90 −37.66 14.30 −673.81 −221.12 −204.46 −202.92 −454.74 −411.39 −462.70 −271.14 −81.98 −79.10 −69.45 −79.56 −55.61 −43.70

87.37

−0.64 −49.05

−2.24

25.44

7.55

0.08

−341.14

−356.62

32.10

−305.40 −361.75

CHX and CHXO Groups 231.93 243.60 254.94 263.14 178.22 146.80 55.76

−61.10 −11.70 −709.07

135.56

164.32 74.48 169.45

−487.23 −400.37 −466.00

138.31

202.14 183.28 159.24 95.41 71.34 −24.26 233.05

−112.93 −102.60 −86.90 −101.80 −71.17 −56.78

145.91 128.45 104.27

−21.78

173.31

−42.65

113.00

−10.75 7.26

84.69 −13.46

−27.31 −7.40

108.78 33.54

228.45 177.78

48.74 68.46 −18.45 −32.64 −90.37 15.90 −322.54 −394.55

88.10 −3.21 191.21

−165.12 4.37 50.94 102.36 −5.06 −329.90 −11.51

137.24 147.85 159.91 169.45 62.76 155.63 175.41 199.16

−235.10

175.61 177.82 188.70

66.53 170.29

4.14

141.71 149.70

−12.67 −2.23 −32.08

−85.65

3.65

24.78 48.60

−343.87

−428.77

115.35

2-484

PHYSICAL AND CHEMICAL DATA

TABLE 2-343

Domalski-Hearing* Group Contribution Values for Standard State Thermal Properties (Concluded)

This table is a partial listing of GC values available from the original Domalski-Hearing tables. Table-specific nomenclature: Cd = carbon with double bond; Ct = carbon with triple bond; Cb = carbon in benzene ring; Ca = allenic carbon; corr = correction term; Cbf = fused benzene ring; NA = azo nitrogen; NI = imino nitrogen. ∆Hfo

Group

∆Hfs liq.

So

Ss liq.

∆Hfs solid

Ss solid

−191.20 −32.20 19.90 73.70 0.00 −58.41 −55.11 −419.59 −696.66 −44.06 −7.24 −92.56 −225.29 −216.67 −175.49 −117.09 −35.46

54.19 55.47 74.85 61.08 0.00

−194.00 −32.00 13.50 70.40 0.00 −74.75

39.79 43.37 54.45

6.96 25.00 14.00 6.30 0.00 0.00 18.50 40.60 83.55 0.00 0.00 0.00 112.00 6.00 −6.00 8.00 8.00 6.00 10.00 8.50 0.00 34.43 23.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

CHX and CHXO Groups Ct(Cl) Ct(Br) Ct(I) Cb(F)(2Cb) Cb(Cl)(2Cb) Cb(Br)(2Cb) Cb(I)(2Cb) cis corr(I)(I) C(2H)(CO)(Cl) C(H)(CO)(2Cl) CO(C)(F) C(Cb)(3F) C(2H)(Cb)(Br) C(2H)(Cb)(I) C(2H)(Cb)(Cl) CO(C)(Cl) CO(Cb)(Cl) CO(C)(Br) CO(C)(I) C(H)(C)(CO)(Cl) C(C)(CO)(2Cl) ortho corr(I)(I) ortho corr(F)(F) ortho corr(Cl)(Cl) ortho corr(alkyl)(X) cis corr(Cl)(Cl) cis corr(CH3)(Br) ortho corr(F)(Cl) ortho corr(F)(Br) ortho corr(F)(I) meta corr(I)(I) meta corr(COCl)(COCl) ortho corr(COCl)(COCl) ortho corr(F)(CF3) meta corr(F)(CF3) ortho corr(F)(CH3) ortho corr(F)(F’) ortho corr(Cl)(Cl’) meta corr(F)(F) meta corr(Cl)(Cl) ortho corr(Cl)(CHO) ortho corr(F)(COOH) ortho corr(Cl)(COCl) ortho corr(F)(OH) ortho corr(Cl)(COOH) ortho corr(Br)(COOH) ortho corr(I)(COOH) ortho corr(NH2)(NH2) meta corr(NH2)(NH2) ortho corr(OH)(Cl) cis corr(CH3)(I)

140.00 151.30 35.53 −181.26 −17.03 36.35 94.50 3.00 −44.26 −40.40 −379.84 −691.79 −29.49 7.31 −73.79 −200.54

67.52 77.08 88.60 98.26 0.00

179.08

176.66

−148.54 −83.94 −39.88 7.56 20.90 9.50 2.51 −4.00 −4.00 13.50 37.25 85.40 0.00 0.00 0.00 111.00 2.00 −3.30 8.00 8.00 0.00 −5.00 −6.75 20.00 0.00 25.50 0.00 0.00 0.00 −10.00 0.00 7.50 −4.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00

−212.99

5.50 25.50 8.50 0.00 0.00 0.00 19.50 42.50 85.20 20.08 16.06 0.00 0.00 0.00 8.00 8.00 8.50 4.00 0.00 20.00 0.00 20.00 20.00 20.00 20.00 0.00 14.00 11.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

*Domalski, E. S, and E. D. Hearing, J. Phys. Chem. Ref. Data, 22 (1993): 805

Group contributions: Group Cb(H)(2Cb) Cb(C)(2Cb) Cb(N)(2Cb) C(3H)(C) N(2H)(Cb)

ni

Hf! gas

Hf! liq.

S! gas

Ss liq.

4 1 1 1 1

13.81 23.64 −1.30 −42.26 19.25

8.16 19.16 1.50 −47.61 −11.00

48.31 −35.61 −43.53 127.32 126.90

28.87 −19.50 −24.43 83.30 71.71

Total

54.57

−5.31

368.32

226.56

Calculation from Eq. (2-27): ∆Hof  = 54.57 kJ/mol

∆Hsf  = −5.31 kJ/mol

So  = 368.32 J/(mol⋅ K)

So  = 226.56 J/(mol⋅ K)

The recommended DIPPR® 801 standard enthalpies of formation are ∆Hof = 53.20 kJmol and ∆Hsf = − 4.72 kJmol; the estimated values are higher than the recommended values by 2.6 and 12.5 percent, respectively. The recommended DIPPR® 801 standard entropies are So = 355.8 J(mol⋅K) and Ss = 231.2 J(mol⋅K). The estimated values differ from these by 3.5 and − 2.0 percent, respectively.

Recommended Method Isodesmic reaction. Reference: Foresman, J. B., and A. Frisch, Exploring Chemistry with Electronic Structure Methods, 2d ed., Gaussian Inc., Pittsburgh, Pa., 1996. Classification: Computational chemistry.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Expected uncertainty: 5 to 10 percent depending upon the level of theory and basis set size used. Applicability: Compounds for which an isodesmic reaction can be formulated. Input data: Experimental ∆Hof values for all other participants in the isodesmic reaction. Description: While ab initio calculations of absolute enthalpies are not currently as accurate as GC methods, relative enthalpies of molecules calculated with the same level of theory and basis set can be very accurate, as in the case of isodesmic reactions. An isodesmic reaction is one in which the number and type of bonds are preserved during the reaction. For example, the reaction of acetaldehyde with ethane to form acetone and methane H

O HC

 H 3C

O

CH3 H3C

CH3

CH3

 H

C

H

H

is isodesmic with 12 single bonds and 1 double bond in both reactants and products. To use this method, one devises an isodesmic reaction involving the compound for which ∆Hfo is to be determined and other compounds for which experimental ∆Hfo values are available. Ab initio calculations are performed on all the participating compounds, all at the same level of theory and basis set size, to obtain the enthalpy for each at 298.15 K. The enthalpy of reaction is then calculated from ∆Hrxn = νiHi

(2-28)

where νi = stoichiometric coefficient of i (+ for products, − for reactants). The enthalpy of reaction is also related to ∆Hof by ∆Hrxn = νi(∆Hof )i

Example Estimate the standard ideal gas enthalpy of formation of acetaldehyde. Input information: The isodesmic reaction shown above will be used. The recommended ∆Hfo values from DIPPR® 801 for the other three compounds are Acetone

Methane

Ethane

−215.70 kJ/mol

−74.52 kJ/mol

−83.82 kJ/mol

Ab initio calculations of enthalpy: With structures optimized using HF/631G(d) model chemistry and energies calculated with B3LYP/6-311 + G(3df,2p), the following enthalpies are obtained (including the zero point energy): Acetone

ideal gas So at 298.15 K and 1 bar can be found in various literature sources (DIPPR, JANAF, TRC, SWS, and Daubert, T. E., and R. P. Danner, Technical Data Book-Petroleum Refining, 5th ed., American Petroleum Institute, Washington, extant 1994). Very good estimates for Ss or So can be obtained by using the Domalski-Hearing method. Excellent So values can also be obtained from statistical mechanics by using experimental vibrational frequencies or values of the frequencies generated from computational chemistry. The standard state ∆Sfs and standard ideal gas ∆Sfo entropies of formation at 298.15 K and 1 bar are related to the standard entropies by nA

nA

∆Ssf = Sscompound − νiSselement,i

∆Sof = Socompound − νiSselement,i (2-30)

i=1

i=1

where Sselement,i is the absolute entropy of element i in its standard state at 298.15 K and 1 bar. Recommended Method Domalski-Hearing method. Reference: Domalski, E. S., and E. D. Hearing, J. Phys. Chem. Ref. Data, 22 (1993): 805. Classification: Group contributions. Expected uncertainty: 3 percent. Applicability: Organic compounds for which group contributions have been regressed. Input data: Molecular structure. Description: See description given under Enthalpy of Formation above. Example Estimate the standard and ideal gas entropies of formation of o-toluidine. Standard state entropies: Estimation of Ss and So using the Domalski-Hearing method was illustrated above in the Enthalpy of Formation section. The standard entropies of formation can be obtained from the values determined in that example. Formula: C7H9N. The standard state entropies of the elements from the DIPPR® 801 database are as follows: Compound: νi: Ssi/[J(kmol⋅K)]:

N2 1/2 1.9151  105

H2 9/2 1.3057  105

C, graphite 7 5740

(2-29)

With experimental values available for all ∆Hof except the desired compound, its value can be back-calculated from Eq. (2-29).

−5.071 × 105 kJ/mol

2-485

Methane

Ethane

Acetaldehyde

−1.063 × 105 kJ/mol

−2.095 × 105 kJ/mol

−4.039 × 105 kJ/mol

Calculation using Eq. (2-28): ∆Hrxn = (−1.063 − 5.071 + 2.095 + 4.039) × 105 kJmol = − 41.67 kJmol Calculation using Eq. (2-29): ∆Hof,acetaldehyde = ∆Hof,acetone + ∆Hof,methane − ∆Hof,ethane − ∆Hrxn kJ kJ ∆Hof,acetaldehyde = (−215.70 − 74.52 + 83.82 + 41.67)  = −164.73  mol mol ®

The estimated value is 1.0 percent above the DIPPR 801 recommended value of −166.40 kJmol.

Entropy Absolute or third-law entropies (relative to a perfectly ordered crystal at 0 K) of a compound in its standard state Ss or of an

Entropies of formation can be calculated from these values by using Eq. (2-30): 105 J 9 1 ∆Ssf = 0.22656 −  (1.9151) −  (1.3057) − (7)(0.0574)  2 2 kmol⋅K









J = − 7.008⋅105  kmol⋅K 105 J 1 9 ∆Sof = 0.36832 −  (1.9151) −  (1.3057) − (7)(0.0574)  2 2 kmol⋅K









J = − 6.867⋅105  kmol⋅K

Recommended Method Statistical mechanics. Classification: Theory and computational chemistry. Expected uncertainty: 0.2 percent if vibrational frequencies (or their characteristic temperatures) are experimentally available; uncertainty depends upon model chemistry if frequencies are determined from computational chemistry, but generally within about 5 percent. Applicability: Ideal gases. Input data: M; σ (external symmetry number); characteristic rotational temperature(s) (ΘA for linear molecules; ΘA, ΘB, and ΘC for nonlinear molecules); and 3nA − 6 + δ characteristic vibrational temperatures Θj. Description: For harmonic frequencies, the rigorous temperature dependence of So is given by S So 3 M  =  ln 6175  + r + R 2 kg/kmol R



Θj



 T (e

ΘjT

3nA − 6 + δ



j=1



− 1)−1 − ln (1 − e−Θ T) j

(2-31)

2-486

PHYSICAL AND CHEMICAL DATA

0 where δ = e 1

vapor and saturated liquid at a temperature between the triple point and critical point (at the corresponding vapor pressure). Variable ∆Hv is related to the vapor pressure P* by the thermodynamically exact Clapeyron equation

nonlinear linear

S r = R

{

1 πT 3 e 3 ln    σ ΘAΘ B ΘC

 

ln

 12

σΘ  12

Te

d ln P∗ d ln P∗ ∆Hv = −R ∆Z v  = RT 2 ∆Z v  d(1T) dT

nonlinear

where ∆Zv = ZG − ZL ZG = Z of saturated vapor ZL = Z of saturated liquid

linear

A

T = 298.15 K

Experimental heats of vaporization can be effectively correlated with

Example Calculate S for ammonia. o

Θj/K

Θj/T

1207.91 1850.16 1850.16 3688.19 3821.36 3821.36

4.051 6.205 6.205 12.370 12.817 12.817

∆Hv = A(1 − Tr)B + CT + DT + ET r

Structure: NH3. Input data: M = 17 kg/kmol. McQuarrie [McQuarrie, D. A., Statistical Mechanics, Harper & Row, New York, 1976] gives the following 3m − 6 + δ = 12 − 6 + 0 = 6 characteristic vibrational temperatures (K): 1360, 2330, 2330, 4800, 4880, 4880. The characteristic rotational temperatures given by McQuarrie are ΘA = 13.6 K, ΘB = 13.6 K, and ΘC = 8.92 K. For NH3, σ = 3. Vibrational contribution: The table below shows a spreadsheet calculation of the vibrational terms inside the summation sign in Eq. (2-31).

(298.15 K)3πc3 S 1 r = ln e  ⋅  R 3 (13.6 K)(13.6 K)(8.92 K)





1 − Tr ∆Hv = ∆Hv,ref  1 − Tr,ref



f = 5.81593

12

Calculation using Eq. (2-31): So298 3  =  ln (6175⋅17) + 5.81593 + 0.1186 = 23.277 ∆Hrxn = R 2

ν (∆H ) i

o f i

J So298 = 1.935 × 105  kmol⋅K

∆G = ∆H − T∆S o f

o f

and

∆G = ∆H − T∆S s f

s f

s f

(2-32)

and predicted values of ∆G and ∆G are obtained from Eq. (2-32) by estimating the enthalpies and entropies of formation as shown above. s f

o f

3 r



0.38

∆Hv = R∆Zv (−B + CT + DET E+1)

(2-35)

(2-36)

The ZG and ZL values can be evaluated by using the methods given in the subsection on densities below. Example Calculate ∆Hv for anisole at 452 K. Input data: The vapor pressure coefficients in the DIPPR® 801 database, based on Eq. (2-22), are A = 128.06

B = −9307.7

C = −16.693

D = 0.014919

E=1

The vapor pressure at 452 K is P∗ 9307.7 ln  = 128.06 −  − 16.693 ln 452 + 0.014919(452)1 = 12.155 Pa 452 P∗ = exp (12.155)⋅Pa = 1.901 × 105Pa

Determine ∆Z: Required data from the DIPPR® 801 database for this calculation are Tc = 645.6 K, Pc = 4.25 MPa, and ω = 0.35017. These values are used to determine the reduced conditions and the values of ZG and ZL from the Lee-Kesler corresponding states method as discussed in the subsection on density.

LATENT ENTHALPY Enthalpy of Vaporization The enthalpy (heat) of vaporization ∆Hv is the difference between the molar enthalpies of the saturated

(2-34)

If an accurate correlation for P* and accurate values for ZG and ZL are available, Eq. (2-33) is the preferred method for obtaining enthalpies of vaporization. Otherwise, the CS methods shown below should be used. Recommended Method 1 Vapor pressure correlation. Classification: Extension of theory. Expected uncertainty: Varies significantly with temperature and with the quality and temperature range of the vapor pressure data used in the correlation. Applicability: Organic compounds for which group contributions have been regressed. Input data: Correlations for P*, ZG, and ZL. If Tr < 0.8, then ZG can be set to 1. Description: An expression for ∆Hv can be obtained from Eq. (2-33) by using an appropriate vapor pressure correlation. If one differentiates the Riedel vapor pressure correlation, Eq. (2-22), in accordance with Eq. (2-33), one obtains the heat of vaporization as

The calculated value differs from the DIPPR® 801 recommended value of 1.9266 × 105 J(kmol⋅K) by 0.5 percent.

Gibbs’ Energy of Formation The standard Gibbs energy of formation is the Gibbs energy change upon formation of 1 mol of the compound in its standard state from its constituent elements in their standard states. The standard state Gibbs energy of formation ∆Gsf uses the naturally occurring phase at 298.15 K and 1 bar as the standard state, while the ideal gas Gibbs energy of formation ∆Gof uses the compound in the ideal gas state at 298.15 K and 1 bar as the standard state. In both cases, the standard state for the elements is their naturally occurring state of aggregation at 298.15 K and 1 atm. Sources for data include DIPPR, TRC, JANAF, and Daubert and Danner, (Daubert, T. E., and R. P. Danner Technical Data Book—Petroleum Refining, 5th ed., American Petroleum Institute, Washington, extant 1994). The Gibbs energies of formation are related to the corresponding enthalpies and entropies of formation by

2 r

A simple method for obtaining ∆Hv at one temperature from a known value at a reference temperature, say, at the normal boiling point, is to truncate Eq. (2-34) after the B term, set B = 0.38, and take a ratio of the ∆Hv values at the two conditions to give the Watson [Thek, R. E., and L. I. Stiel, AIChE J., 12 (1966): 599; 13(1967): 626] correlation

Svib 0.08929 0.01457 0.01457 0.00006 0.00004 0.00004 Sum 0.1186

Rotational contribution:

o f

(2-33)

452 = 0.7 Tr =  645.6

0.1901 Pr =  = 0.045 4.25

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Interpolation of the Pr values in Tables 2-351 and 2-352 at a Tr of 0.7 gives 0.050 − 0.010

0.045 − 0.010 Z(1) G = −0.0064 +  (−0.0507 + 0.0064) = − 0.0452 0.050 − 0.010

(1) ZG = Z(0) G + ωZG = 0.9554 + (0.35017)(−0.0452) = 0.94

∆ZV = ZG – ZL = 0.94

Calculation using Eq. (2-36):



J ∆Hv = 8.314  (0.94)[9307.7 − (16.693)(452) + (0.014919)(1)(452)2] mol⋅K kJ = 37.59  mol⋅K This value is 0.2 percent higher than the value of 37.51 kJ/(mol⋅K) obtained from the DIPPR® 801 database.

Recommended Method 2 Corresponding states correlation. Reference: PGL5, p. 7.18. Classification: Corresponding states. Expected uncertainty: Less than about 6 percent. Applicability: Organic compounds. Input data: Tc, Pc, and ω. Description: The following correlation is used: where τ = 1 − Tr

(2-37)

Example Repeat the above calculation for anisole’s ∆Hv at 452 K. Input data: Tc = 645.6 K, Pc = 4.25 MPa, and ω = 0.35017. Auxiliary quantities: From the previous example, the reduced temperature is Tr = 0.7





J kJ ∆Hv = (10.785) 8.314  (294 K) = 26.36  mol⋅K mol⋅K This value is 2.4 percent above the DIPPR® 801 recommended value of 25.73 kJ(mol⋅K).

Enthalpy of Fusion The enthalpy (heat) of fusion ∆Hfus is the difference between the molar enthalpies of the equilibrium liquid and solid at the melting temperature and 1.0 atm pressure. There is no generally applicable, high-accuracy estimation method for ∆Hfus, but the GC method of Chickos can be used to obtain approximate results if the melting temperature is known. Recommended Method Chickos method. Reference: Chickos, J. S., et al., J. Org. Chem., 56 (1991): 927. Classification: QSPR and group contributions. Expected uncertainty: Considerable variation but generally less than 50 percent. Applicability: Only valid at the melting temperature. The method is based on the ∆Sfus between a solid at 0 K and the liquid at the Tm so no solid-solid transitions are taken into account. Values of ∆Hfus will be overestimated if there are solid-solid transitions for the actual material. Input data: Tm and molecular structure. Description: ∆Hfus ∆Sfus Tm  =    = (Tm K)(a + b) J/mol J/(mol⋅K) K



ng

no nonaromatic rings nonaromatic rings

ns

nf

j=1

k=1

b = Ngi ∆si + Nsj Csj ∆sj + NfkCtk ∆sk i=1



This value is 2.2 percent below the DIPPR® 801 recommended value of 37.51 kJ(mol⋅K).

Recommended Method 3 Vetere method for ∆Hv at Tb. Reference: Vetere, A., Fluid Phase Equilib., 106 (1995): 1. Classification: Corresponding states. Expected uncertainty: About 4 percent. Applicability: Valid only at the normal boiling point. Input data: Tc, Pc, and Tb. Description: The following correlation is used:

where τb = 1 − Tbr

F=1



{

J kJ ∆Hv = (6.838) 8.314  (645.6 K) = 36.70  mol⋅K mol⋅K

0.5066 ⋅ bar τ b0.38 ln(Pc /bar) − 0.513 +  PcT 2b r  0.38 τb + F(1 − τ b ) ln Tbr





0 a = 35.19N + 4.289(N − 3N ) R CR R

∆Hv  = 7.08(0.3)0.354 + 10.95(0.35017)(0.3)0.456 = 6.838 RTc

∆Hv  = RTb

τb = 1 − Tbr = 0.369

(0.369)0.38 ln 55.5 − 0.513 + 0.5066/(55.5)(0.631)2 ∆Hv  =  = 10.785 RTb (0.369) + [1 − (0.369)0.38] ln 0.631

τ = 1 − 0.7 = 0.3

Calculation using Eq. (2-37):



294 Tbr =  = 0.631 466 Calculation using Eq. (2-38):

At this low pressure, ZL is very small compared to ZG and may be neglected; so

∆Hv  = 7.08τ 0.354 + 10.95ωτ 0.456 RTc

For most compounds, F = 1; compounds that dimerize (e.g., SO2, NO, NO2) and alcohols with more than two C atoms are assigned F = 1.05. Example Calculate ∆Hv at the normal boiling point for acetaldehyde. Input data: Recommended values from the DIPPR® 801 database are Tc = 466.0 K, Pc = 5.55 MPa, and Tb = 294.0 K. Auxiliary quantities: From the previous example, the reduced temperature is

0.045 − 0.010 Z(0) G = 0.9904 +  (0.9504 − 0.9904) = 0.9554



2-487

 (2-38)

(2-39) (2-40)

(2-41)

where Ngi = number of C—H groups of type i bonded to other carbon atoms ng = number of different nonring or aromatic C—H groups bonded to other carbon atoms Nsj = number of C—H groups of type j bonded to at least one functional group or atom ns = number of different nonring or aromatic C—H groups bonded to at least one functional group or atom Nfk = number of functional groups of type k nf = number of different functional groups or atoms t = total number of functional groups or atoms with the exception that F atoms count as one regardless of number of occurences Csj = value from Table 2-344 for C—H group j bonded to at least one functional group or atom Ctk = value from Table 2-345 for functional group k NR = number of nonaromatic rings NCR = number of —CH2— groups in nonaromatic ring(s) required to form cyclic paraffin of same ring size(s) ∆si = contribution from Table 2-344 for group i ∆sk = contribution from Table 2-345 for group k

2-488

PHYSICAL AND CHEMICAL DATA

TABLE 2-344

Cs (CᎏH) Group Values for Chickos Estimation* of ⌬Hfus

Group

Description

Cs

∆s

Group

Description

Cs

∆s

CH3 >CH2 >CH >C < CH2= CH= >C= CH C

methyl methylene secondary C tertiary C terminal alkene alkene subst. alkene term. alkyne alkyne

1.0 1.0 0.69 0.67 1.0 3.23 1.0 1.0 1.0

18.33 9.41 −16.91 −38.70 14.56 4.85 −11.38 10.88 2.18

CHAr CAr CAr CAr >CrH >Cr < CrH= >Cr= Cr or =Cr=

aromatic C ar. C bonded to paraffinic C ar. C bonded to olefinic C or non-C group ar. C bonded to acetylinic C ring structure ring structure ring structure ring structure ring structure

1.0 1.0 1.0 1.0 0.76 1.0 0.62 0.86 1.0

6.44 −10.33 −4.27 −2.51 −15.98 −32.97 −4.35 −11.72 −5.36

*Chickos, J. S., et al., J. Org. Chem., 56 (1991): 927. Tm = 304.5 K

Note that nonaromatic ring CH2 groups are accounted for in the a term and are not included in the b term. Example Calculate ∆Hfus at the melting point for (a) benzothiophene, (b)

furfuryl alcohol, and (c) cis-crotonaldehyde. Structures:

O

(a)

(b) NCR = 5

NR = 1

O

(c)

Group

a = 35.19 + (5 − 3)(4.289) = 43.77

Group

Description

N

C

苷CH 苷C 苷C 苷CH 苷CH S

aromatic (Ng type) ring (Ng type) ring (Ns type) ring (Ng type) ring (Ns type) ring

4 1 1 1 1 1

1 1 0.86 1 0.62 1

∆s

Total

6.44 −11.72 −11.72 −4.35 −4.35 2.18 Total

25.76 −11.72 −10.08 −4.35 −2.70 2.18 −0.91

TABLE 2-345 Ct (Functional) Group Values for Chickos Estimation* of ⌬H fus Description

C1

C2

C3

C4

s

OH OH O O >C=O >C=O CHO COOH COO NH2 NH2 >NH >NH >N >N =N =N CN NO2 CONH2 CONH SH S S SO2 F F F Cl Br I

alcohol phenol nonring ether ring ether nonring ketone ring ketone aldehyde acid ester aliphatic aromatic nonring ring nonring ring ring aromatic nitrile nitro primary amide secondary amide

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

12.6 1.0 1.0 1.0 1.0 1.0 1.0 1.83 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0

18.9 1.0 1.0 1.0

26.4 1.0 1.0 1.0

1.13 16.57 1.09 1.34 3.14 −1.88 19.66 14.90 3.68 16.23 15.48 −2.18 1.84 −15.90 −17.07 1.67 7.32 9.62 17.36 26.19 −0.42 17.99 7.20 2.18 3.26 14.73 13.01 15.90 8.37 17.95 16.95

1.88 1.0

1.72 1.0

1.0 1.0

0.36 1.0 1.0 1.0 2.0 1.0

*Chickos, J. S., et al., J. Org. Chem., 56 (1991): 927.

1.0 1.0 1.0 1.93 0.82

NCR = 5

a = 35.19 + (5 − 3)(4.289) = 43.77

Description

苷CH 苷CH 苷C< 苷O CH2 OH

ring (Ng type) ring (Ns type) ring (Ns type) ring ether Ns type alcohol

Tm = 258.52 K

N

C

2 1 1 1 1 1

1 0.62 0.86 1 1 12.6

∆s

Total

−4.35 −8.70 −4.35 −2.70 −11.72 −10.08 1.34 1.34 9.41 9.41 1.13 14.24 Total 3.51

from DIPPR® 801 database

∆Hfus = (Tm K)(a + b) Jmol = (258.52)(43.77 + 3.51) Jmol = 12.22 kJmol This value is 7 percent lower than the DIPPR® 801 recommended value of 13.13 kJ/mol. (c)

Group

nonring ring nonring on aliph. C on olefinic C on ring C

This value is 10 percent higher than the DIPPR® 801 recommended value of 11.83 kJ/mol.

CH3

(a) t = 1 (1 total “functional group”), so the C1 column in Table 2-345 is used. NR = 1

∆Hfus = (Tm K)(a + b) Jmol = (304.5)(43.77 − 0.91) Jmol = 13.05 kJmol

(b) t = 2 (2 total “functional groups”), so the C2 column in Table 2-345 is used.

OH

S

from DIPPR® 801 database

t=1

Group CH3 苷CH– 苷CH– CHO

NR = 0

a=0

Description

N

C

∆s

Total

nonring (Ng type) nonring (Ng type) nonring (Ns type) aldehyde

1 1 1 1

1 1 3.23 1

18.33 4.85 4.85 19.66

18.33 4.85 15.67 19.66

Total Tm = 158.38 K

58.51

from DIPPR® 801 database

∆Hfus = (TmK)(a + b) Jmol = (158.38)(0 + 58.51) Jmol = 9.27 kJmol This value is 5 percent higher than the DIPPR® 801 recommended value of 8.86 kJ/mol.

Enthalpy of Sublimation The enthalpy (heat) of sublimation ∆Hfus is the difference between the molar enthalpies of the equilibrium vapor and solid along the sublimation curve below the triple point. The effects of pressure on ∆Hfus and melting temperature are very small so that Tt and the normal melting point are nearly equal and ∆Hsub(Tt) = ∆Hv(Tt) + ∆Hfus(Tt)

(2-42)

Equation (2-42) can be used to estimate ∆Hsub at the triple point if ∆Hv is accurately known at Tt. Because ∆Hv is usually obtained from Eq. (2-33), ∆Hv(T) correlations may be less accurate near Tt where P*(Tt) is very small and difficult to measure. In this case, it is better to estimate ∆Hsub directly by using the following recommended

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES method. Although Eqs. (2-33) and (2-34) apply to Tt, ∆Hsub is only a weak function of temperature and can generally be taken as a constant from the triple point temperature down to the first solid-solid phase transition. Recommended Method Goodman method. Reference: Goodman, B., et al., Int. J. Thermophys. 25 (2004): 337. Classification: QSPR and group contributions. Expected uncertainty: 6 percent. Applicability: Organic compounds for which group contributions have been regressed. Input data: Molecular structure and radius of gyration RG. Description:

 + n a + n β N

∆Hsub(Tt) 12 RG  = 698.04 + 3.83798 × 10  m RK

N

i i

i=1

2 i i

i=1

(2-43)

N

ni f +  i

2-489

Input data: The value of RG from the DIPPR® 801 database is 4.455 × 10−10 m. Calculation using Eq. (2-43): ∆Hsub(Tt)  = 698.04 + (3.83798 × 1012)(4.455 × 10−10) + 6657.049 RK



3 + (32)(−2.21614) +  (−1543.66) 6





kJ kJ ∆Hsub(Tt) = (8273.134 K) 0.008314  = 68.78  mol⋅K mol The estimated value is 5.6 percent above the DIPPR® 801 recommended value of 65.11 kJ/mol. Estimate the solid vapor pressure from Eq. (2-26): The solid vapor pressure at 301.15 K can be calculated from Eq. (2-26) by using the estimated ∆Hsub. Recommended values for Tt and P*t from the DIPPR® 801 database are 325.65 K and 182.957 Pa, respectively.

nx

i=1

where ai = GC values from Table 2-346 βi = nonlinear corrections for >CH2 and Ar—CH = groups fi = halogen corrections nx = total number of all halogen and hydrogen atoms attached to C and Si atoms ∆Hsub and the solid vapor pressure for 1,2,3trichlorobenzene at 301.15 K. Structure:

Example Calculate



P∗ = (182.957 Pa)[exp(−2.067)] = 23.16 Pa The estimated value is 0.3 percent above the DIPPR® 801 recommended value of 23.09 Pa.

HEAT CAPACITY

Cl

∂V   dP ∂T

CP = CoP − T

Group contributions: Linear groups

Nonlinear and correction terms

Group

ni

ai

Group

ni

βi

ArCH苷 Ar >C苷 Cl

3 3 3

626.7621 348.8092 1243.445

Ar CH苷 Cl nx

3 3 6

−2.21614

fi −1543.66

ni ai 6657.049 i

TABLE 2-346

Description

CH3 >CH2 >CH >C< CH2= CH= >C= Ar CH= Ar >C= Ar O Ar N= Ar S O OH COH

methyl methylene secondary C tertiary C terminal alkene alkene substituted alkene aromatic C subst. aromatic C furan O pyridine N thiophene S ether alcohol aldehyde

Nonlinear terms >CH2 ArCH=

P

0

2

2

The second term, giving the deviation of the real fluid heat capacity from the ideal gas value, can be neglected at low to moderate pressures, or it can be calculated directly from an appropriate EoS. Ideal gas heat capacities are available from several sources (DIPPR, JANAF, TRC, and SWS). Two common correlating equations for CPo are the Aly-Lee [Aly, F. A., and L. L. Lee, Fluid Phase Equilib., 6

methylene aromatic C

ai 736.5889 561.3543 111.0344 −800.517 572.6245 541.2918 117.9504 626.7621 348.8092 763.284 1317.056 911.2903 970.4474 3278.446 2402.093

Group >C=O COO COOH NH2 NH >N NO2 SH S SS F Cl Br >Si< >Si(O)

i 9.5553 −2.21614

*Goodman, B., et al., Int. J. Thermophys., 25 (2004): 337.

(2-44)

P

Group Contributions and Corrections* for Hsub

Group



Heat capacity CP is defined as the energy required to change the temperature of a unit mass (specific heat) or mole (molar heat capacity) of the material by one degree. Typical units are J/(kg⋅K). Gases The heat capacity of a gas is related rigorously to the ideal gas heat capacity CPo by

Cl Cl

P* 68.78kJ /mol 325.65 ln  =  1 −  = − 2.067 182.957 Pa [0.008314 kJ(mol⋅K)](325.65 K) 301.15

Description

ai

ketone ester acid primary amine sec. amine tertiary amine nitro thiol/mercaptan sulfide disulfide fluoride chloride bromide silane siloxane

1816.093 2674.525 5006.188 2219.148 1561.222 325.9442 3661.233 1921.097 1930.84 2782.054 626.4494 1243.445 669.9302 −83.7034 −16.0597

Halogen correction terms F Cl Br

F fraction Cl fraction Br fraction

fi −1397.4 −1543.66 5812.49

2-490

PHYSICAL AND CHEMICAL DATA

(1981): 169] equation A2T CoP = Ao + A1  sinh(A2T)





2

A4T + A3  cosh(A4T)





2

(2-45)

From experimental frequencies: J J 8 C0P =  + 0.2769 R = (4.2769) 8.3143  = 35.56  2 mol⋅K mol⋅K





J J 8 C0P =  + 0.197 R = (4.197) 8.3143  = 34.90  2 mol⋅K mol⋅K



4

C = AiT



From computational chemistry frequencies:

and a polynomial form (generally fourth-order) o P



i

(2-46)

i=0

Ideal gas heat capacities may also be estimated from several techniques, of which two of the most accurate and commonly used are recommended here. Recommended Method 1 Statistical mechanics. Reference: Rowley, R. L., Statistical Mechanics for Thermophysical Property Calculations, Prentice-Hall, Englewood Cliffs, N.J., 1994. Classification: Theory and computational chemistry. Expected uncertainty: 0.2 percent if vibrational frequencies (or their characteristic temperatures) are experimentally available; accuracy depends upon model chemistry if frequencies are determined from computational chemistry, but generally within 3 percent. Applicability: Ideal gases. Input data: 3nA − 6 + δ vibrational frequencies νj, or the corresponding characteristic vibrational temperatures Θj. The two are related by







The value calculated from experimental frequencies is 0.1 percent lower than the DIPPR® 801 recommended value of 35.61 J/(mol⋅K); the value calculated from frequencies generated from computational chemistry software is 2.0 percent lower than the DIPPR® 801 value.

Recommended Method 2 Benson method as implemented in CHETAH program. References: Benson, S. W., et al., Chem. Rev., 69 (1969): 279; CHETAH Version 8.0: The ASTM Computer Program for Chemical Thermodynamic and Energy Release Evaluation (NIST Special Database 16). Classification: Group contributions. Expected uncertainty: 4 percent. Applicability: Ideal gases of organic compounds. Input data: Table 2-347 group values at the seven specified temperatures. Description: Groups are summed at each individual temperature: N

Θj = hνj k

8−δ Co P =  + 2 R



j=1

Θj

eΘ  T

− 1)   T  (e 2

j

Θj  T

2

δ=e

0 1

nonlinear linear

where ni = number of occurrences of group i and (Cpo)i = individual group contribution. Either Eq. (2-45) or Eq. (2-46) can be used to interpolate between the discrete temperatures. Example Calculate the ideal gas heat capacity of isoprene (2-methyl-1,3butadiene) at 400 K. Structure: CH3

(2-48) Example Calculate the ideal gas heat capacity of ammonia at 300 K. Structure:

H2C

苷CH2 苷C(2C) CH3-(苷C) 苷CH-(C)

H N

CH

CH2

No.

Value, J/(mol⋅K)

2 1 1 1

26.62 19.3 32.82 21.05

Contribution, J/(mol⋅K)

Total

H Input data: McQuarrie (McQuarrie, D. A., Statistical Mechanics, Harper & Row, New York, 1976) gives the following 3m − 6 + δ = 12 − 6 + 0 = 6 characteristic vibrational temperatures (K): 1360, 2330, 2330, 4800, 4880, and 4880. Alternatively, a computational chemistry package gives the following scaled frequencies for HF/6-31G+ model chemistry (1013 Hz): 3.24, 4.97, 4.97, 9.90, 10.26, and 10.26. Calculation: The table on the left uses the experimental Θ values to determine the individual terms in the summation of Eq. (2-48); the table on the right uses the scaled frequencies from computational chemistry software and Eq. (2-47) to obtain Θ values and the individual terms in Eq. (2-48).

C

Group identification and values: Group

H

(2-49)

i=1

where nA = number of atoms in molecule and δ = 1 for linear molecules and 0 for nonlinear molecules. Description: For harmonic frequencies, the rigorous temperature dependence of CPo is given by 3nA −6 + δ

CoP = ni ⋅(Cop)i

(2-47)

53.24 19.3 32.82 21.05 126.41

The value of 126.4 J(mol⋅K) is 3.1 percent below the DIPPR® 801 recommended value of 130.4 J(mol⋅K).

Liquids Liquid heat capacity increases with increasing temperature, although a minimum occurs near the triple point for many compounds. Usually liquid heat capacity is correlated as a function of temperature with a polynomial equation. A cubic equation is usually adequate. Estimation of liquid heat capacity can be done by using a number of methods [Ruzicka, V., and E. S. Domalski, J. Phys. Chem. Ref. Data, 22 (1993): 597, 619; Chueh, C. F., and A. C. Swanson, Chem. Eng.

HF/6-31G + scaled frequencies

Experimental frequencies Θ/K

Θ/(300 K)

Term

νscaled/Hz

Θ/K

Θ/(300 K)

Term

1360 2330 2330 4800 4880 4880

4.533 7.767 7.767 16.000 16.267 16.267

0.2256 0.0256 0.0256 0.0000 0.0000 0.0000

3.24 4.97 4.97 9.90 10.26 10.26

1555.0 2385.3 2385.3 4751.4 4924.2 4924.2

5.183 7.951 7.951 15.838 16.414 16.414

0.1524 0.0223 0.0223 0.0000 0.0000 0.0000

Sum

0.2769

Sum

0.1970

NOTE: Empirical scaling factors have been developed for each model chemistry to help correct theoretical frequencies for anharmonic effects [Scott, A. P., and L. Radom, J. Phys. Chem., 100 (1996): 16502].

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-347

2-491

Benson* and CHETAH† Group Contributions for Ideal Gas Heat Capacity

Table-specific nomenclature: Cb = carbon in benzene ring; Ct = carbon with a triple bond, (苷C) = carbon with a double bond; Cp = carbon in fused ring; Naz = azide; Nim = imino. Group

298 K

400 K

500 K

600 K

800 K

1000 K

45.17 45.17 45.17 45.17 45.21 45.17 45.17 45.17 45.17 45.17 45.17 45.17 45.17 45.17

54.5 54.5 54.5 54.5 54.42 54.54 54.54 54.54 54.54 54.5 54.5 54.5 54.5 54.5

61.83 61.83 61.83 61.83 61.95 61.83 61.83 61.83 61.83 61.83 61.83 61.83 61.83 61.83

20.59 19.42 17.12 20.59 38.51 37.67 35.79 28.76 38.93 53.16

22.35 20.93 19.25 22.35 39.77 39.35 38.3 31.27 40.18 56.93

23.02 21.89 20.59 23.02 40.6 40.18 39.85 33.32 41.02 59.86

37.8 39.14 38.17 40.18 40.98 40.98 63.71 40.18 41.9 39.77 38.01 39.35 39.47 36.84 38.34 41.73 39.72 41.36 40.1 38.72 40.85

45.46 46.34 43.2 47.3 49.35 49.77 72.58 47.3 48.1 46.46 45.46 46.46 46.5 44.58 45.84 51.32 46.97 48.3 47.17 45.92 50.98

51.74 51.65 47.26 52.74 55.25 55.25 78.82 52.74 52.49 51.07 51.03 51.49 51.61 49.94 51.15 59.23 52.24 53.29 52.49 51.28 59.48

36.63 35.12 32.57 35.16 36.54 35.5 36.38 34.28 33.7 31.77

40.73 41.11 38.09 40.18 41.06 40.35 41.44 39.6 38.97 35.41

42.9 43.99 41.44 42.7 43.53 43.11 44.24 42.65 42.07 38.97

25.53 36.75 27.17 19.97 33.07 32.23 27.17 27.63 34.11 34.58 33.99

27.63 38.47 30.43 25.2 35.58 34.32 30.43 31.56 36.5 37.34 36.71

28.46 37.51 31.69 26.71 35.58 34.49 31.23 33.32 33.91 37.51 36.67

1500 K

CH3 Groups CH3(Cb) CH3(CO) CH3(Ct) CH3(C) CH3(N) CH3(O) CH3(PO) CH3(P) CH3(P苷N) CH3(Si) CH3(SO2) CH3(SO) CH3(S) CH3(苷C)

25.91 25.91 25.91 25.91 25.95 25.91 25.91 25.91 25.91 25.91 25.91 25.91 25.91 25.91

32.82 32.82 32.82 32.82 32.65 32.82 32.82 32.82 32.82 32.82 32.82 32.82 32.82 32.82

39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35 39.35

73.59 73.59 73.59 73.59 73.67 73.59 73.59 73.59 73.59 73.59

73.59

Ct Groups Ct(Cb) Ct(Ct) Ct(C) Ct(苷C) CtBr CtCl CtF CtH CtI Ct(CN)

10.76 14.82 13.1 10.76 34.74 33.07 28.55 22.06 35.16 43.11

14.82 16.99 14.57 14.82 36.42 35.16 31.65 25.07 36.84 47.3

14.65 18.42 15.95 14.65 37.67 36.42 33.99 27.17 38.09 50.65

24.28 23.32 26.58 24.28 41.77 37.04 64.04

CH2 Groups CH2(2CO) CH2(2C) CH2(2O) CH2(2苷C) CH2(Cb,O) CH2(Cb,SO2) CH2(Cb,S) CH2(Cb,苷C) CH2(C,Cb) CH2(C,CO) CH2(C,Ct) CH2(C,N) CH2(C,O) CH2(C,SO2) CH2(C,SO) CH2(C,S) CH2(C,苷C) CH2(苷C,O) CH2(苷C,SO2) CH2(苷C,SO) CH2(苷C,S)

16.03 23.02 11.85 19.67 15.53 15.53 38.09 19.67 24.45 25.95 20.72 21.77 20.89 17.12 19.05 22.52 21.43 19.51 20.34 18.42 22.23

26.66 29.09 21.18 28.46 26.26 27.5 49.02 28.46 31.85 32.23 27.46 28.88 28.67 24.99 26.87 29.64 28.71 29.18 28.51 26.62 28.59

CH(2C,Cb) CH(2C,CO) CH(2C,Ct) CH(2C,N) CH(2C,O) CH(2C,SO2) CH(2C,S) CH(2C,苷C) CH(3C) CH(C,2O)

20.43 18.96 16.7 19.67 20.09 18.5 20.3 17.41 19 22.02

27.88 25.87 23.48 26.37 27.79 26.16 27.25 24.74 25.12 23.06

C(2C,2O) C(3C,Cb) C(3C,CO) C(3C,Ct) C(3C,N) C(3C,O) C(3C,SO2) C(3C,SO) C(3C,S) C(3C,苷C) C(4C)

19.25 19.72 9.71 0.33 18.42 18.12 9.71 12.81 19.13 16.7 18.29

19.25 28.42 18.33 7.33 25.95 25.91 18.33 19.17 26.25 25.28 25.66

32.15 34.53 31.48 35.16 34.66 34.66 57.43 35.16 37.59 36.42 33.19 34.74 34.74 31.56 33.28 36 34.83 36.21 34.95 29.05 34.45

59.65 60.28

60.28 57.6 59.44 61.11

60.11

CH Groups 33.07 30.89 28.67 31.81 33.91 31.65 32.57 30.72 30.01 27.67

44.7 46.55

47.22 46.76

C Groups 23.02 33.86 23.86 14.36 30.56 30.35 23.86 20.26 31.18 31.1 30.81

31.94

34.45 33.99

2-492

PHYSICAL AND CHEMICAL DATA

TABLE 2-347

Benson* and CHETAH† Group Contributions for Ideal Gas Heat Capacity (Continued)

Table-specific nomenclature: Cb = carbon in benzene ring; Ct = carbon with a triple bond, (苷C) = carbon with a double bond; Cp = carbon in fused ring; Naz = azide; Nim = imino. Group

298 K

400 K

500 K

600 K

800 K

1000 K

1500 K 25.32

Aromatic (Cb and Cp Groups) Cb(Cb) Cb(CO) Cb(Ct) Cb(C) Cb(N) Cb(O) Cb(Si) Cb(SO2) Cb(SO) Cb(S) Cb(苷C) Cb(苷Nim) CbBr CbCl CbF CbH CbI Cb(CHN2) Cb(CN) Cb(N3) Cb(NCO) Cb(NCS) Cb(NO2) Cb(SO2OH) Cp(2Cb,Cp) Cp(3Cp) Cp(Cb,2Cp)

13.94 11.18 15.03 11.18 16.53 16.32 11.18 11.18 11.18 16.32 15.03 16.53 32.65 30.98 26.37 13.56 33.49 47.3 41.86 34.74 55.25 32.23 38.93 65.42 12.56 8.37 12.56

17.66 13.14 16.62 13.14 21.81 22.19 13.14 13.14 13.14 22.19 16.62 21.81 36.42 35.16 31.81 18.59 37.25

20.47 15.4 18.33 15.4 24.86 25.95 15.4 15.4 15.4 25.95 18.33 24.86 39.35 38.51 35.58 22.85 40.18

22.06 17.37 19.76 17.37 26.45 27.63 17.37 17.37 17.37 27.63 19.76 26.45 41.44 40.6 38.09 26.37 41.44

24.11 20.76 22.1 20.76 27.33 28.88 20.76 20.76 20.76 28.88 22.1 27.33 43.11 42.7 41.02 31.56 43.11

24.91 22.77 23.48 22.77 27.46 28.88 22.77 22.77 22.77 28.88 23.48 27.46 43.95 43.53 42.7 35.2 43.95

48.14

52.74

55.67

59.86

62.79

64.04

70.32

74.51

79.95

82.88

50.23 79.49 15.49 12.14 15.49

59.44 84.51 17.58 14.65 17.58

66.56 97.61 19.25 16.74 19.25

76.18 109.25 21.77 19.67 21.77

80.37 113.31 23.02 21.35 23.02

苷C(2C) 苷C(CO,O) 苷C(C,Cb) 苷C(C,CO) 苷C(C,O) 苷C(C,SO2) 苷C(C,S) 苷C(C,苷C) 苷CC(苷C,O) 苷CH(Cb) 苷CH(CO) 苷CH(Ct) 苷CH(C) 苷CH(O) 苷CH(SO2) 苷CH(S) 苷CH(苷C) 苷CH2 苷C苷

17.16 23.4 18.42 22.94 17.16 15.49 14.65 18.42 18.42 18.67 31.73 18.67 17.41 17.41 12.72 17.41 18.67 21.35 16.32

19.3 29.3 22.48 29.22 19.3 26.04 14.94 22.48 22.9 24.24 37.04 24.24 21.05 21.05 19.55 21.05 24.24 26.62 18.42

22.02 32.44 25.87 31.98 22.02 38.51 17.12 25.87 26.29 31.06 40.31 31.06 27.21 27.21 28.63 27.21 31.06 35.58 20.93

24.28 33.57 27.21 33.53 24.28 44.62 18.46 27.21 27.21 34.95 43.45 34.95 32.02 32.02 32.94 32.02 34.95 42.15 22.19

25.45 34.03 27.71 34.32 25.45 47.47 20.93 27.71 27.71 37.63 46.21 37.63 35.37 35.37 36.29 35.37 37.63 47.17 23.02

O(2C) O(2O) O(2苷C) O(Cb,CO) O(CO,O) O(C,Cb) O(C,CO) O(C,O) O(C,苷C) O(苷C,CO) OH(Cb) OH(CO) OH(C) OH(O) O(CN)(Cb) O(CN)(C) O(CN)(苷C) O(NO2)(C) O(NO)(C) (CO)Cl(C) (CO)H(Cb) (CO)H(CO)

14.23 15.49 14.02 8.62 1.51 2.6 11.64 15.49 12.72 6.03 18 15.95 18.12 21.64 34.74 41.86 54.42 39.93 38.09 42.28 33.53 28.13

15.49 15.49 16.32 11.3 6.28 3.01 15.86 15.49 13.9 12.47 18.84 20.85 18.63 24.24

15.49 15.49 17.58 13.02 9.63 4.94 18.33 15.49 14.65 16.66 20.09 24.28 20.18 26.29

15.91 15.49 18.84 14.32 11.89 7.45 19.8 15.49 15.49 18.79 21.77 26.54 21.89 27.88

18.42 17.58 21.35 16.24 15.28 11.89 20.55 17.58 17.54 20.8 25.12 30.01 25.2 29.93

19.25 17.58 22.6 17.5 17.33 14.99 21.05 17.58 18.96 21.77 27.63 32.44 27.67 31.44

48.3 43.11 46.04 44.2 32.78

55.5 46.88 49.39 48.77 37.25

65.3 50.23 51.9 59.48 41.4

68.61 55.67 55.67 68.56 47.84

72.75 58.18 57.76 74.01 50.73

24.07 25.03 25.03

24.07

40.73

85.81

苷C苷, 苷C, 苷CH Groups 20.89 31.31 24.82 31.02 20.89 33.32 16.03 24.82 24.82 28.25 38.8 28.25 24.32 24.32 24.82 24.32 28.25 31.44 19.67

26.62 28.13

28.13 41.77 41.77 40.27 40.27 41.77 55.21 23.86

Oxygen Groups 20.09

20.09

37.34 33.65 34.2

60.69

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-347

2-493

Benson* and CHETAH† Group Contributions for Ideal Gas Heat Capacity (Continued)

Table-specific nomenclature: Cb = carbon in benzene ring; Ct = carbon with a triple bond, (苷C) = carbon with a double bond; Cp = carbon in fused ring; Naz = azide; Nim = imino. Group

298 K

400 K

500 K

600 K

800 K

1000 K

40.52 40.52 40.52 48.77 21.05

46.71 46.71 46.71 63.12 26.32

51.07 51.07 51.07 74.68 29.54

55.25 85.64 49.52 68.98 82.88 77.86 46.71 57.85 77.44 74.93 68.52 57.3 57.35 59.86 56.09 54.42 59.9 58.18 56.72 53.75 68.23 53.41 46.88 69.07 48.39 63.21 54.42 74.17 56.3 59.86 59.02 56.51 58.18 55.67 53.16 47.72 46.88 43.95 48.56

56.93 88.66 52.07 70.99 86.23 82.88 52.03 63.46 84.14 80.79 76.06 65.26 64.88 67.81 64.04 63.62 68.15 66.14 64.25 58.81 74.93 59.82

56.09 89.66 53.12 71.24 87.9 85.39 53.24 65.84 87.9 83.72 79.99 69.95 70.32 73.67 69.9 69.49 73.8 72 69.36 61.62 79.53 64.38

74.93 54.83 69.9 59.31 79.7 57.72 62.37 61.53 59.86 61.11 59.44 57.76 51.9 51.49 49.39 52.74

78.28 58.64 74.51 61.95 81.58 56.93 63.62 62.79 61.53 62.79 61.53 60.69 55.25 54.83 53.16 55.67

17.29 26.16 34.45 27.33 29.05 24.99 20.3 20.93 17.66 45.63 21.35 30.93 28.59 13.94 26.29 14.57 29.3 33.78 34.7 33.78 38.93 22.35 28.34

21.89 28.42 37.8 28.59 30.93 27.46 22.1 22.94 20.05 50.9 28.3 33.28 33.07 16.91 30.1 17.75 32.65 39.39 41.69 39.39 43.95 23.82 28.71

23.4 28.76 38.47 34.91 38.68 27.92 22.14 27.08 21.43 53.54 32.98 34.28 36.21 18.21 32.36 18.96 34.74 43.83 46.97 43.83 48.14 23.9 29.51

1500 K

Oxygen Groups (CO)H(C) (CO)H(N) (CO)H(O) (CO)H(苷C) CO(Cb)(O)

29.43 29.43 29.43 24.32 9.12

32.94 32.94 32.94 30.22 11.51

CBr(3C) CBr3(C) CCl(3C) CCl2(2C) CCl3(C) CClF2(C) CF(3C) CF2(2C) CF3(Cb) CF3(C) CF3(S) CH2Br(Cb) CH2Br(C) CH2Br(苷C) CH2Cl(C) CH2F(C) CH2I(Cb) CH2I(C) CH2I(O) CHBr(2C) CHBrCl(C) CHCl(2C) CHCl(C,O) CHCl2(C) CHF(2C) CHF2(C) CHI(2C) CHI2(C) CI(3C) 苷CBr2 苷CBrCl 苷CBrF 苷CCl2 苷CClF 苷CF2 苷CHBr 苷CHCl 苷CHF 苷CHI

39.35 72.12 36.96 51.07 68.23 57.35 28.46 39.01 52.32 53.16 41.36 30.51 38.09 40.6 37.25 33.91 33.91 38.51 34.41 37.38 51.9 35.45 37.67 50.65 30.56 41.44 38.64 56.93 41.15 51.49 50.65 45.21 47.72 43.11 40.6 33.91 33.07 28.46 36.84

47.72 78.65 43.87 62.29 75.35 67.39 37.09 46.97 64.04 62.79 54.46 46.46 46.04 47.72 44.79 41.86 45.17 46.04 43.91 44.62 58.6 42.7 41.44 58.6 37.84 50.23 45.67 63.42 49.18 55.25 53.16 50.23 52.32 48.97 46.04 39.77 38.51 35.16 41.86

36.92 36.92 36.92 39.77 16.65 Halide Groups 52.74 82.92 47.72 66.76 79.95 73.25 42.7 53.24 72 68.65 62.08 52.2 52.74 54.42 51.49 50.23 53.7 54 51.19 50.06 63.3 48.89 43.95 64.46 43.83 57.35 50.9 69.61 54.08 58.18 56.51 53.58 55.67 52.74 50.23 44.37 43.11 39.77 45.63 Nitrogen Groups

CH2(N3)(C) 苷CH(N3) N(2C,Cb) N(2C,CO) N(2C,SO2) N(2C,SO) N(2C,S) N(3C) N(Cb,2CO) N(C,2CO) Nb pyridN NF2(C) NH(2Cb) NH(2CO) NH(2C) NH(Cb,CO) NH(C,Cb) NH(C,CO) NH(C,N) NH2(Cb) NH2(CO) NH2(C) NH2N 苷Naz(C) 苷Naz(N)

64.46 54.42 2.6 13.02 25.2 17.58 15.99 14.57 4.1 4.48 10.88 26.5 9.04 15.03 17.58 2.39 15.99 2.76 20.09 23.94 17.04 23.94 25.53 11.3 8.87

8.46 19.17 26.58 24.61 21.64 19.09 12.81 12.99 13.48 34.58 13.06 23.19 21.81 6.32 20.47 6.49 24.28 27.25 24.03 27.25 30.98 17.16 17.5

13.69 23.52 31.56 25.62 25.99 22.73 17.71 18.04 15.95 40.9 17.29 28.05 25.66 9.96 23.9 10.3 27.21 30.64 29.85 30.64 35.16 20.59 23.06

27.21

39.97

37.67 51.4 51.4 55.25

2-494

PHYSICAL AND CHEMICAL DATA

TABLE 2-347

Benson* and CHETAH† Group Contributions for Ideal Gas Heat Capacity (Continued)

Table-specific nomenclature: Cb = carbon in benzene ring; Ct = carbon with a triple bond, (苷C) = carbon with a double bond; Cp = carbon in fused ring; Naz = azide; Nim = imino. Group

298 K

400 K

500 K

600 K

800 K

1000 K

1500 K

Nitrogen Groups 苷NazH 苷Nim(Cb) 苷Nim(C) 苷NimH

18.33 12.56 10.38 12.35

S(2Cb) S(2C) S(2S) S(2苷C) S(Cb,S) S(C,Cb) S(C,S) S(C,苷C) SH(Cb) SH(CO) SH(C) SO(2Cb) SO(2C) SO2(2Cb) SO2(2C) SO2(2苷C) SO2(Cb,SO2) SO2(Cb,苷C) SO2(C,Cb) S(CN)(Cb) S(CN)(C) S(CN)(苷C)

8.37 20.89 19.67 20.05 12.1 12.64 21.89 17.66 21.43 31.94 24.53 23.94 37.17 34.99 48.22 48.22 41.06 41.4 41.61 39.77 46.88 59.44

8.41 20.76 20.93 23.36 14.19 14.19 22.69 21.26 22.02 33.86 25.95 38.05 41.98 46.17 50.1 50.1 48.14 48.14 48.14

113.23 −39.64

134.95

47.72 61.95 52.7 45.21 50.19 80.79 36.21 61.62 41.4 72.42 43.11 51.90 51.49 56.93

56.93

64.04

66.22 54 63.67 101.3 46.71 74.47 55.84

20.47

22.77

24.86

28.34

31.06

13.98 19.17

16.53 27

17.96 32.27

19.21 38.22

19.25 41.52

11.47 21.22 21.77 26.33 17.37 16.91 23.06 24.15 25.24 34.2 28.38 47.93 45.17 62.54 59.77 59.77 61.66 61.16 60.74

15.91 22.65 22.19 33.24 20.01 19.34 22.52 24.57 29.26 35.58 30.56 47.97 45.96 66.39 64.38 64.38 65.76 65.8 65.38

19.72 23.98 22.6 40.73 21.35 20.93 21.43 24.57 32.82 34.49 32.27 47.09 46.76 66.81 66.47 66.47 67.1 66.64 66.64

198.62

219.72

70.74

80.79

85.81

77.52 60.69 74.17 117.2 53.96 83.72 66.39

86.48 66.14 82.08 129.76 58.81 90.46 73.75

99.58 72 92.84 146.09 64.92 99.54 82.92

108.41 79.11 99.2 156.13 67.77 104.48 87.32

50.23

56.09

61.11

68.65

73.67

63.21 69.28

72.83 78.19

80.37 84.76

90.41 93.51

97.11 98.74

−11.05 −7.03 −7.95 −10.88 −12.64 −16.37 −14.56

−7.87 −6.2 −7.41 −9.63 18.09 −19.25 −10.88

−5.78 −5.57 −6.78 −8.63 24.35 −23.86 0.84

−10.97 −5.44 −5.99 −6.91 −15.91 −17.33 −12.56 −3.77 −16.74 −15.91 −2.89 −15.32 −15.32

−6.4 4.6 −1.21 −5.36 −11.72 −12.26 −10.88 9.21 −12.01 −11.3 3.6 −18.46 −18.46

−1.8 9.21 0.33 −4.35 −8.08 −9.46 −10.05 17.58 −9.08 −7.53 5.4 −23.32 −23.32

35.33

Sulfur Groups 9.38 21.01 21.35 23.15 15.57 15.53 23.06 23.27 23.32 33.99 27.25 40.6 43.95 56.72 55.88 55.88 56.59 55.88 56.3

Boron and Silicon Groups Si(4C) SiH3(C)

154.5

171.2

252.91

Monovalent Ligands CH2(CN)(C) CH2(NCS)(C) CH2(NO2)(C) CH(CN)(2C) CH(NO2)(2C) CH(NO2)2(C) C(CN)(3C) C(CN)2(2C) C(NO2)(3C) 苷CH(CHN2) 苷CH(CN) 苷CH(NCS) 苷CH(NO2) 苷C(CN)2

105.9

3,4 Member Ring Corrections cyclobutane ring cyclobutene ring cyclopropane ring ethylene oxide ring ethylene sulfide ring thietane ring trimethylene oxide ring

−19.3 −10.59 −12.77 −8.37 −11.93 −19.21 −19.25

−16.28 −9.17 −10.59 −11.72 −10.84 −17.5 −20.93

1,4 dioxane ring cyclohexane ring cyclohexene ring cyclopentadiene ring cyclopentane ring cylopentene ring furan ring piperidine ring pyrrolidine ring tetrahydrofuran ring thiacyclohexane ring thiolane ring thiophene ring

−19.21 −24.28 −17.92 −14.44 −27.21 −25.03 −20.51 −24.7 −25.83 −25.12 −26.04 −20.51 −20.51

−20.8 −17.16 −12.72 −11.85 −23.02 −22.39 −18 −19.67 −23.36 −24.28 −17.83 −19.55 −19.55

−13.14 −7.91 −8.79 −12.56 −11.13 −16.37 −17.58

−2.8 −5.11 −6.36

5,6 Member Ring Corrections −15.91 −12.14 −8.29 −8.96 −18.84 −20.47 −15.07 −12.14 −20.09 −20.09 −9.38 −15.4 −15.4

13.81 3.39 −1.55 −4.52

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

2-495

Benson* and CHETAH† Group Contributions for Ideal Gas Heat Capacity (Concluded)

TABLE 2-347

Table-specific nomenclature: Cb = carbon in benzene ring; Ct = carbon with a triple bond, (苷C) = carbon with a double bond; Cp = carbon in fused ring; Naz = azide; Nim = imino. Group

298 K

400 K

500 K

600 K

800 K

1000 K

−1.63 −1.63 −1.63 −1.63 −1.63 −1.26 2.93 3.68

−1.09 −1.09 −1.09 −1.09 −1.09

1500 K

7 and 8 Member Ring Corrections −38.01 −44.16

cycloheptane ring cyclooctane ring

Gauche and 1,5 Repulsion Corrections −5.61 −5.61 −5.61 −5.61 −5.61 −2.09

but-2-ene structure CC苷CC but-3-ene structure CCC苷C cis- between 2 t-butyl groups cis- involving 1 t-butyl group cis-(not with t-butyl group) ortho- between Cl atoms ortho- between F atoms other ortho- (nonpolar-nonpolar)

−4.56 −4.56 −4.56 −4.56 −4.56 5.02 −0.84 5.65

4.69

−3.39 −3.39 −3.39 −3.39 −3.39 2.09 −0.42 5.44

−2.55 −2.55 −2.55 −2.55 −2.55 −2.51 1.26 4.9

−0.21

2.76

*Benson, S. W., et al., Chem. Rev., 69 (1969): 279. †CHETAH Version 8.0: The ASTM Computer Program for Chemical Thermodynamic and Energy Release Evaluation (NIST Special Database 16).

Prog., 69, 7 (1973): 83; Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510; Tarakad, R. R., and R. P. Danner, AIChE J., 23 (1977): 944] and thermodynamic differentiation. The most accurate and generally applicable method is that by Ruzicka and Domalski. Recommended Method Ruzicka-Domalski. References: Ruzicka, V., and E. S. Domalski, J. Phys. Chem. Ref. Data, 22 (1993): 597, 619. Classification: Group contributions. Expected Uncertainty: 4 percent. Applicability: Organic compounds for which group values are available. Input data: Molecular structure and Table 2-348 values. Description: Groups are summed to find the temperature coefficients for a cubic polynomial correlation:







C T T P = A + B  + D  R 100 K 100 K N

N

A = niai

B = nibi i =1

i=1



2

(2-50)

N

D = nidi

(2-51)

ference in the heat capacity of the two equilibrium solid phases that exist on either side of the transition temperature. The heat capacity generally rises steeply with increasing temperature near the triple point. For a quick estimation of solid heat capacity specifically at 298.15 K, the very simple modification of Kopp’s rule [Kopp, H., Ann. Chem. Pharm. (Liebig), 126 (1863): 362] by Hurst and Harrison [Hurst, J. E., and B. K. Harrison, Chem. Eng. Comm., 112 (1992): 21] can be used. At other temperatures and to obtain the temperature dependence of the solid heat capacity, the method given below by Goodman et al. should be used. Recommended Method 1 Goodman method. Reference: Goodman, B. T., et al., J. Chem. Eng. Data, 49 (2004): 24. Classification: Group contributions. Expected uncertainty: 10 percent. Applicability: Organic compounds for which group values are available. Input data: Molecular structure and Table 2-349 group values. Description:

i=1



CP A T  =   J(mol⋅K) 1000 K

where ni = number of occurrences of group i and ai, bi, di = individual group contributions.



i=1

i=1



(2-53)

Example Estimate the solid heat capacity for p-cresol at 307.93 K.

Group contributions: Group

ni

ai

bi

di

C(3C,O) (alcohol) O(H)(C) C(3H)(C)

1 1 3

−44.690 12.952 3.8452

31.769 −10.145 −0.33997

−4.8791 2.6261 0.19489

−20.202

20.604

Sum



N

where ni = number of occurences of group i ai = individual group contribution βi = nonlinear correction terms for chain and aromatic carbons

CH3 H3C CH3 OH

J CP = 8.3143  mol⋅K

N

(2-52)

A = exp 6.7796 + ni ai + n2i βi

Example Estimate the liquid heat capacity for 2-methyl-2-propanol at 340 K. Structure:

0.79267

H3C

OH

−1.668

− 1.668    −20.202 + 20.604   100 100 340

Structure:

340

2

J = 254.16  mol⋅K This value is 0.7 percent higher than the DIPPR® 801 recommended value of 252.40 J(mol⋅K).

Solids Solid heat capacity increases with increasing temperature and is proportional to T 3 near absolute zero. The heat capacity at a solidsolid phase transition becomes large, and there can be a substantial dif-

Group contributions: Group

ni

ai

βi

CH3 ArCH苷 Ar >C苷 OH

1 4 2 1

0.20184 0.082478 0.012958 0.10341

0 −0.00033 0 0

From Eq. (2-53): A = exp[6.7796 + 0.20184 + (4)(0.082478) + (2)(0.012958) + 0.10341 + (4)2 (−0.00033)] = 1694.9

2-496

PHYSICAL AND CHEMICAL DATA

TABLE 2-348 Liquid Heat Capacity Group Parameters for Ruzicka-Domalski Method* Table-specific nomenclature: Ct refers to a carbon atom with a triple bond; Cb refers to a carbon atom in benzene ring; 苷C refers to a carbon atom with a double bond; Cp refers to a carbon atom in a fused benzene ring; 苷C苷 refers to an allenic carbon atom. Group Definition

a

b

d

T range (K)

Group Definition

Hydrocarbon Groups C(3H,C) C(2H,2C) C(H,3C) C(4C) 苷C(2H) 苷C(H,C) 苷C(2C) 苷C(H,苷C) 苷C(C,苷C) C(3H,苷C) C(2H,C,苷C) C(H,2C,苷C) C(3C,苷C) C(2H,2苷C) Ct(H) Ct(C) 苷C苷 Ct(Cb) Cb(H) Cb(C) Cb(苷C) Cb(Cb) C(2H,C,Ct) C(3H,Ct) C(3H,Cb) C(2H,C,Cb) C(H,2C,Cb) C(3C,Cb) C(2H,2Cb) C(H,3Cb) 苷C(H,Cb) 苷C(C,Cb) Cp(Cp,2Cb) Cp(2Cp,Cb) Cp(3Cp)

3.8452 2.7972 −0.42867 −2.9353 4.1763 4.0749 1.9570 3.6968 1.0679 3.8452 2.0268 −0.87558 −4.8006 1.4973 9.1633 1.4822 3.0880 12.377 2.2609 1.5070 −5.7020 5.8685 2.0268 3.8452 3.8452 1.4142 −0.10495 1.2367 −18.583 −46.611 3.6968 1.0679 −3.5572 −11.635 26.164

−0.33997 −0.054967 0.93805 1.4255 −0.47392 −1.0735 −0.31938 −1.6037 −0.50952 −0.33997 −0.20137 0.82109 2.6004 −0.46017 −4.6695 1.0770 −0.62917 −7.5742 −0.2500 −0.13366 5.8271 −0.86054 −0.20137 −0.33997 −0.33997 0.56919 1.0141 −1.3997 11.344 24.987 −1.6037 −0.50952 2.8308 6.4068 −11.353

0.19489 0.10679 0.0029498 −0.085271 0.099928 0.21413 0.11911 0.55022 0.33607 0.19489 0.11624 0.18415 −0.040688 0.52861 1.1400 −0.19489 0.25779 1.3760 0.12592 0.011799 −1.2013 −0.063611 0.11624 0.19489 0.19489 0.0053465 −0.071918 0.41385 −1.4108 −3.0249 0.55022 0.33607 −0.39125 −0.78182 1.2756

80–490 80–490 85–385 145–395 90–355 90–355 140–315 130–305 130–305 80–490 90–355 110–300 165–295 130–300 150–275 150–285 140–315 230–550 180–670 180–670 230–550 295–670 90–355 80–490 80–490 180–470 180–670 220–295 300–420 375–595 130–305 130–305 250–510 370–510 385–480

2.8647 −1.9986 −0.42564 0.08488 0.41360 0.41360 −0.82721 0.19403 0.33288 −0.92054 0.80145 0.15892 0.24596 0.27199 0.82003 0.44241 0.19165 −0.0055745 −0.0097873 −0.40942 −0.62960 0.39346 1.2520

125–345 125–345 245–310 180–355 140–360 140–360 275–360 168–360 190–420 245–340 240–420 180–420 165–415 120–300 120–240 155–300 120–240 210–365 230–460 245–370 250–320 210–365 245–345

Halogen Groups C(C,3F) C(2C,2F) C(C,3Cl) C(H,C,2Cl) C(2H,C,Cl) C(2H,苷C,Cl) C(H,2C,Cl) C(2H,C,Br) C(H,2C,Br) C(2H,C,I) C(C,2Cl,F) C(C,Cl,2F) C(C,Br,2F) 苷C(H,Cl) 苷C(2F) 苷C(2Cl) 苷C(Cl,F) Cb(F) Cb(Cl) Cb(Br) Cb(I) C(Cb,3F) C(2H,Cb,Cl)

15.423 −8.9527 8.5430 10.880 9.6663 9.6663 −2.0600 6.3944 10.784 0.037620 13.532 7.2295 8.7956 7.1564 7.6646 9.3249 7.8204 3.0794 4.5479 2.2857 2.9033 7.4477 16.752

−9.2464 10.550 2.6966 −0.35391 −1.8601 −1.8601 5.3281 −0.10298 −2.4754 5.6204 −3.2794 0.41759 −0.19165 −0.84442 −2.0750 −1.2478 −0.69005 0.46959 0.22250 2.2573 2.9763 −0.92230 −6.7938

Nitrogen Groups C(3H,N) C−(2H,C,N) C(2H,Cb,N) C(H,2C,N) C(3C,N) N(2H,C) N(2H,Cb)

3.8452 2.4555 2.4555 2.6322 1.9630 8.2758 8.2758

−0.33997 1.0431 1.0431 −2.0135 −1.7235 −0.18365 −0.18365

a

b

d

T range (K)

Nitrogen Groups

0.19489 −0.24054 −0.24054 0.45109 0.31086 0.035272 0.035272

80−490 190–375 190–375 240–370 255–375 185–455 185–455

N(H,2C) N(3C) N(H,C,Cb) N(2C,Cb) N(C,2Cb) Cb(N) N(2H,N) N(H,C,N) N(2C,N) N(H,Cb,N) C(2H,C,CN) C(3C,CN) 苷C(H,CN) Cb(CN) C(2H,C,NO2) O(C,NO2) Cb(NO2) N(H,2Cb) (pyrrole) Nb(2Cb)

−0.10987 4.5942 0.49631 −0.23640 4.5942 −0.78169 6.8050 1.1411 −1.0570 −0.74531 11.976 2.5774 9.0789 1.9389 18.520 −2.0181 15.277 −7.3662 0.84237

0.73024 −2.2134 3.4617 16.260 −2.2134 1.5059 −0.72563 3.5981 4.0038 3.6258 −2.4886 3.5218 −0.86929 3.0269 −5.4568 10.505 −4.4049 6.3622 1.25560

0.89325 0.55316 −0.57161 −2.5258 0.55316 −0.25287 0.15634 −0.69350 −0.71494 −0.53306 0.52358 −0.58466 0.32986 −0.47276 1.05080 −1.83980 0.71161 −0.68137 −0.20336

170–400 160–360 240–380 285–390 160–360 240–455 215–465 205–300 205–300 295–385 185–345 295–345 195–345 265–480 190–300 180–350 280–415 255–450 210–395

2.6261 0.54075 0.54075 −0.87263 0.19489 −0.27140 −4.9593 −4.9593 0.69508 −0.016124 −4.8791 −0.44354 0.37860 −1.44210 0.31655 −0.31693 −1.53670 −0.79259 0.19489 0.47121 0.49646 −0.25674 −1.27110 −0.15377 −0.15377 −0.18312 6.0326 −2.82740 −5.12730 −2.89620 0.53631 −3.24270 3.05310 2.74830 −3.04360 −3.05670 −0.12758 −2.68490

155–505 195–475 195–475 285–400 80–490 135–505 260–460 260–460 185–460 130–170 200–355 170–310 130–350 320–350 300–535 170–310 275–335 285–530 80–490 180–465 185–375 225–360 180–430 220–430 220–430 185–380 275–355 300–465 280–340 180–445 195–350 320–345 175–440 230–500 195–355 195–430 175–500 175–500

0.19489 −0.08349 −0.31234 −0.72356 −0.75674 0.47368 0.47368 0.45625 0.45625 0.17938 0.45625 −0.06131

80–490 130–390 150–390 190–365 260–375 130–380 130–380 165–390 165–390 170–350 165–390 205–345

Oxygen Groups O(H,C) O(H,C) (diol) O(H,Cb) (diol) O(H,Cb) C–(3H,O) C–(2H,C,O) C–(2H,Cb,O) C–(2H,苷C,O) C(H,2C,O) (alcohol) C(H,2C,O) (ether, ester) C(3C,O) (alcohol) C(3C,O) (ether, ester) O(2C) O(C,Cb) O(2Cb) C(2H,2O) C(2C,2O) Cb(O) C(3H,CO) C(2H,C,CO) C(H,2C,CO) C(3C,CO) CO(H,C) CO(H,苷C) CO(H,Cb) CO(2C) CO(C,苷C) CO(C,Cb) CO(H,O) CO(C,O) CO(苷C,O) CO(O,CO) O(C,CO) O(H,CO) 苷C(H,CO) 苷C(C,CO) Cb(CO) CO(Cb,O)

12.952 −10.145 5.2302 −1.5124 5.2302 −1.5124 −7.9768 8.10450 3.8452 −0.33997 1.4596 1.4657 −35.127 28.409 −35.127 28.409 2.2209 −1.4350 0.98790 0.39403 −44.690 31.769 −3.3182 2.6317 5.0312 −1.5718 −22.5240 13.1150 −4.5788 0.94150 1.0852 1.5402 −12.955 9.10270 −1.0686 3.52210 3.8452 −0.33997 6.6782 −2.44730 3.92380 −2.12100 −2.2681 1.75580 −3.82680 7.67190 −8.00240 3.63790 −8.00240 3.63790 5.4375 0.72091 41.507 −32.632 −47.21100 24.36800 13.11800 16.12000 29.24600 3.42610 41.61500 −12.78900 23.99000 6.25730 −21.43400 −4.01640 −27.58700 −0.16485 −9.01080 15.14800 −12.81800 15.99700 12.15100 −1.67050 16.58600 5.44910 Sulfur Groups

C(3H,S) C(2H,C,S) C(H,2C,S) C(3C,S) Cb(S) S(H,C) S(H,Cb) S(2C) S(2Cb) S(C,S) S(Cb,S) S(2Cb) (thiophene)

3.84520 1.54560 −1.64300 −5.38250 −4.45070 10.99400 10.99400 9.23060 9.23060 6.65900 9.23060 3.84610

−0.33997 0.88228 2.30700 4.50230 4.43240 −3.21130 −3.21130 −3.00870 −3.00870 −1.35570 −3.00870 0.36718

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

2-497

TABLE 2-348 Liquid Heat Capacity Group Parameters for Ruzicka-Domalski Method* (Concluded) Table-specific nomenclature: Ct refers to a carbon atom with a triple bond; Cb refers to a carbon atom in benzene ring; 苷C refers to a carbon atom with a double bond; Cp refers to a carbon atom in a fused benzene ring; 苷C苷 refers to an allenic carbon atom. Group Definition

a

b

d

T range (K)

Ring Strain Contributions Hydrocarbons (ring strain) cyclopropane cyclobutane cyclopentane (unsub) cyclopentane (sub) cyclohexane cycloheptane cyclooctane spiropentane cyclopentene cyclohexene cycloheptene cyclooctene cyclohexadiene cyclooctadiene cycloheptatriene cyclooctatetraene indan 1H-indene tetrahydronaphthalene

4.4297 1.2313 −0.33642 0.21983 −2.0097 −11.460 −4.1696 5.9700 0.21433 −1.2086 −5.6817 −14.885 −8.9683 −7.2890 −8.7885 −12.914 −6.1414 −3.6501 −6.3861

−4.3392 −2.8988 −2.8663 −1.5118 −0.72656 4.9507 0.52991 −3.7965 −2.5214 −1.5041 1.5073 7.4878 6.4959 3.1119 8.2530 13.583 3.5709 2.4707 2.6257

1.0222 0.75099 0.70123 0.28172 0.14758 −0.74754 −0.018423 0.74612 0.63136 0.42863 −0.19810 −1.0879 −1.5272 −0.43040 −2.4573 −4.0230 −0.48620 −0.60531 −0.19578

155–240 140–300 180–300 135–365 145–485 270–300 295–320 175–310 140–300 160–320 220–300 260–330 170–300 205–320 200–310 275–330 170–395 280–375 250–320

Group Definition

a

b

decahydronaphthalene −6.8984 0.66846 hexahydroindan −3.9271 −0.29239 dodecahydrofluorene −19.687 8.8265 tetradecahydrophenanthrene −0.67632 −1.4753 hexadecahydropyrene 61.213 −30.927 Nitrogen compounds ethyleneimine 15.281 −2.3360 pyrrolidine 12.703 1.3109 piperidine 25.681 −7.0966 Oxygen compounds ethylene oxide 6.8459 −5.8759 trimethylene oxide −7.0148 7.3764 1, 3-dioxolane −2.3985 −0.48585 furan 9.6704 −2.8138 tetrahydrofuran 3.2842 −5.8260 tetrahydropyran −13.017 3.7416 Sulfur compounds thiacyclobutane −0.73127 −1.3426 thiacyclopentane −3.2899 0.38399 thiacyclohexane −12.766 5.2886

d

T range (K)

−0.070012 0.048561 −1.4031 −0.13087 3.2269

235–485 210–425 315–485 315–485 310–485

−0.13720 −1.18130 0.14304

195–330 170–400 265–370

1.2408 −2.1901 0.10253 0.11376 1.2681 −0.15622

135–325 185–300 175–300 190–305 160–320 295–325

0.40114 0.089358 −0.59558

200–320 170–390 295–340

*Ruzicka, V., and E. S. Domalski, J. Phys. Chem. Ref. Data, 22 (1993): 597, 619.

From Eq. (2-52):

This neglects the excess heat capacity, which if available can be added to the mole fraction average to improve the estimated value.

J J 1694.9 CP =  (307.93)0.79267  = 159.1  mol⋅K mol⋅K 1000 This value is 2.5 percent higher than the DIPPR® 801 recommended value of 155.2 J(mol⋅K).

Recommended Method 2 Modified Kopp’s rule.

Reference: Kopp, H., Ann. Chem. Pharm. (Liebig), 126 (1863): 362; Hurst, J. E., and B. K. Harrison, Chem. Eng. Comm., 112 (1992): 21.

Classification: Group contributions. Expected uncertainty: 10 percent. Applicability: 298.15 K; organic compounds that are solids at 298.15 K. Input data: Compound chemical formula and element contributions of Table 2-350. Description: N CP = nE ∆E  (2-54) J/(mol⋅K) E =1 where N = number of different elements in compound nE = number of occurrences of element E in compound ∆E = contribution of element E from Table 2-350 Example Estimate the solid heat capacity at 298.15 K for dibenzothiophene. Structure: C12H8S. Group values from Table 2-350: ∆C = 10.89

∆H = 7.56

∆S = 12.36

Calculation using Eq. (2-54):

This value is 2.5 percent higher than the DIPPR® 801 recommended value of 198.45 J(mol⋅K).

Mixtures The heat capacity of liquid and vapor mixtures can be estimated as mole fraction averages of the pure-component values C

i=1

Density is defined as the mass of a substance per unit volume. Density is given in kg/m3 in SI units, but lbm/ft3 and g/cm3 are common AES and cgs units, respectively. Other commonly used forms of density include molar density (density divided by molecular weight) in kmol/m3, relative density (density relative to water at 15°C), and the older term specific gravity (density relative to water at 60°F). Often the inverse of density, specific volume, and the inverse of molar density, molar volume, are correlated and used to convey equivalent information. Gases Gases/vapors are compressible and their densities are strong functions of both temperature and pressure. Equations of state (EoS) are commonly used to correlate molar densities or molar volumes. The most accurate EoS are those developed for specific fluids with parameters regressed from all available data for that fluid. Super EoS are available for some of the most industrially important gases and may contain 50 or more constants specific to that chemical. Different predictive methods may be used for gas densities depending upon the conditions: 1. At very low densities (high temperatures, generally above the critical, and very low pressures, generally below a few bar), the ideal gas EoS PV Z   =1 RT

(2-56)

may be applied. 2. At moderate densities (below 40 percent of the critical density), the virial equation truncated after the second virial coefficient

Cp = (12)(10.89) + (8)(7.56) + (1)(12.36) = 203.52 J(mol⋅K)

Cp,m = xiCp,i

DENSITY

(2-55)

B(T) Z=1+  V

(2-57)

may be used. Second virial coefficients B(T) are available in the DIPPR® 801 database for many chemicals and can be estimated for others by using the Tsonopoulos method.

2-498

PHYSICAL AND CHEMICAL DATA TABLE 2-349 Group Values and Nonlinear Correction Terms for Estimation of Solid Heat Capacity with the Goodman et al.* Method Group

Description

ai

Group

Description

CH3 >CH2 >CH >C< CH2苷 CH苷 >C苷 苷C苷 #CH #C Ar CH苷 Ar >C苷 Ar O Ar N苷 Ar >N Ar NH Ar S O OH COH >C苷O COO COOH COOCO

methyl methylene secondary C tertiary C terminal alkene alkene subst. alkene allene terminal alkyne alkyne arom. C subst. arom. C furan O pyridine N subst. pyrrole N pyrrole N thiophene S ether alcohol aldehyde ketone ester acid anhydride

0.20184 0.11644 0.030492 −0.04064 0.18511 0.11224 0.028794 0.053464 −0.02914 0.13298 0.082478 0.012958 0.066027 0.056641 0.008938 −0.05246 0.090926 0.064068 0.10341 0.15699 0.12939 0.13686 0.21019 0.33091

CO3 NH2 >NH >N 苷NH #N N苷N NO2 N苷C苷O SH S SS 苷S >S苷O F Cl Br I >Si< >Si(O) cyc >Si(O) P(苷O)(O)3 >P >P(苷O)

carbonate primary amine secondary amine tertiary amine double-bond NH nitrile diazide nitro isocyanate thiol/mercaptan sulfide disulfide sulfur double bond sulfoxide fluoride chloride bromide iodide silane linear siloxane cyclic siloxane phosphate phosphine phospine oxide

ai 0.2517 0.056138 −0.00717 −0.01661 0.17689 0.015355 0.3687 0.23327 0.2698 0.21123 0.14232 0.31457 0.13753 0.040002 0.15511 0.16995 0.19112 0.11318 0.12213 0.10125 0.063438 0.15016 0.069602 0.21875

Nonlinear Terms i

Usage

−0.00188 −0.00033

Methylene Aromatic carbon

Groups >CH2 Ar苷CH

*Goodman, B. T., W. V. Wilding, J. L. Oscarson, and R. L. Rowley, J. Chem. Eng. Data, 49 (2004): 24

Recommended Method Tsonopoulos method. Reference: Tsonopoulos, C., AIChE J., 20 (1974): 263; 21 (1975): 827; 24 (1978): 1112. Classification: Corresponding states. Expected uncertainty: 8 percent for B(T). Applicability: Nonpolar organic compounds and some classes of polar compounds. Input data: Class of fluid, ω, Pc, Tc, and µ. Description: BP c = B(0) + ωB(1) + B(2) RTc

(2-58)

where ω = acentric factor Pc = critical pressure Tc = critical temperature

0.331 0.423 0.008 B(1) = 0.0637 +  −  −  T2r T3r T8r b a B(2) = 6 − 8 Tr Tr µ µr =  D

   bar  K

TABLE 2-350 Element Contributions to Solid Heat Capacity for the Modified Kopp’s Rule*† Element

E

Element

E

Element

E

C H O N S F Cl Br I Al B

10.89 7.56 13.42 18.74 12.36 26.16 24.69 25.36 25.29 18.07 10.10

Ba Be Ca Co Cu Fe Hg K Li Mg Mn

32.37 12.47 28.25 25.71 26.92 29.08 27.87 28.78 23.25 22.69 28.06

Mo Na Ni Pb Si Sr Ti V W Zr All others

29.44 26.19 25.46 31.60 17.00 28.41 27.24 29.36 30.87 26.82 26.63

*Kopp, H., Ann. Chem. Pharm. (Liebig), 126 (1863): 362. †Hurst, J. E., and B. K. Harrison, Chem. Eng. Comm., 112 (1992): 21.

2

Pc

Tc

(2-61) −2

(2-62)

where µ = dipole moment. The values of a and b used in Eq. (2-61) depend upon the class of fluid, as given in the table below: Class

0.330 0.1385 0.0121 0.000607 B(0) = 0.1445 −  −  −  − (2-59) Tr T2r T3r T8r

(2-60)

a

Nonpolar fluids 0 Ketones, aldehydes, −21.4µr − 4.308 × 1019µ8r nitriles, ethers, esters, NH3, H2S, HCN Monoalkylhalides, −2.188 × 1016µr4 − 7.831 × 1019µ8r mercaptans, sulfides 1-Alcohols except 0.0878 methanol Methanol 0.0878

b 0 0 0 0.00908 + 69.57µr 0.0525

Example Estimate the molar volume of ammonia at 430 K and 2.82 MPa. Input properties: Recommended values from the DIPPR® 801 database are Tc = 405.65 K, Pc = 11.28 MPa, µ = 1.469 D, and ω = 0.252608. Reduced conditions: Tr = (430 K)(405.65 K) = 1.06 Pr = (2.82 MPa)(11.28 MPa) = 0.25 µr = (1.469)2(112.8)(405.65)2 = 0.0014793 Second virial coefficient from Eqs. (2-59) to (2-61): B(0) = 0.1445 – 0.3301.06 – 0.1385(1.06)2 − 0.0121(1.06)3

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES − 0.000607(1.06)8 = − 0.301 B(1) = 0.0637 + 0.331(1.06)2 − 0.423(1.06)3 − 0.008(1.06)8 = − 0.00189 a = (−21.4)(0.0014793) − (4.308 × 1019)(0.0014793)8 = − 0.033

2-499

Example Estimate the molar volume of saturated decane vapor at 540.5 K. Input properties: Recommended values from the DIPPR® 801 database are Tc = 617.7 K, Pc= 2.11 MPa, P*(540.5 K) = 0.6799 MPa (vapor pressure), and ω = 0.492328. Reduced conditions: Tr = (540.5 K)/(617.7 K) = 0.875

b=0

Pr = (0.6799 MPa)/(2.11 MPa) = 0.322

B(2) = (−0.033)(1.06)6 = − 0.023 From Eq. (2-58):

LK compressiblity factor: Since vapor phase values are needed, the appropriate values from Tables 2-351 and 2-352 that can be used to double-interpolate are

BPc (RTc) = − 0.301 − (0.252608)(0.00189) − 0.023 = − 0.324 B = (− 0.324)[0.008314 m3 ⋅MPa(kmol⋅K)](405.65 K)(11.28 MPa) = − 0.097 m3kmol Molar volume from Eq. (2-56): m ⋅MPa −0.097    (430 K) 0.0083143 mkmol⋅K km ol 1 +  b a V 2.82 MPa

RT B V=  1+  = P V = 1.162 m kmol 3

Note that the ideal gas value, 1.268 m3/kmol, deviates by 9.1 percent from this more accurate value. The truncated virial EoS should be valid for this density since ρ = V−1 = 0.86 kmolm3 is much less than 40 percent of the critical density (the DIPPR® 801 recommended value for the critical density is 13.8 kmol/m3).

3. For higher gas densities, the Lee-Kesler method described below provides excellent predictions for nonpolar and slightly polar fluids. Extended four-parameter corresponding-states methods are available for polar and slightly associating compounds. Recommended Method Lee-Kesler method. Reference: Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510. Classification: Corresponding states. Expected uncertainty: 1 percent except near the critical point where errors can be up to 30 percent. Applicability: Nonpolar and moderately polar compounds. An extended Lee-Kesler method, not described here, may be used for polar and slightly associating compounds [Wilding, W. V., and R. L. Rowley, Int. J. Thermophys., 8 (1986): 525]. Input data: Tc, Pc, ω, Z(0),Z(1). Description: Z = Z(0) + ωZ(1) (2-63) where Z = compressibility factor Z(0) = compressibility factor of simple fluid obtained from Table 2-351. Z(1) = deviation from simple fluid obtained from Table 2-352. Analytical expressions for Z(0) and Z(1) can also be generated by using Z(0) = Z0

Tr\Pr

0.2

0.85 0.90

0.8810 0.9015

0.4 (0.7222) 0.7800

3

3



Z(0)

Z1 − Z0 Z(1) =  0.3978

(2-64)

where Z0 and Z1 are determined from γ −γ PrVr B C D c4 Zi =  = 1 +  + 2 + 5 +  β + 2 exp 2 Vr Vr Tr Vr Vr Vr Tr3Vr2





(2-65)

b2 b3 b4 B = b1 −  − 2 − 3 Tr Tr Tr c2 c3 C = c1 −  + 2 Tr Tr d2 D = d1 +  Tr

as applied to the simple reference fluid and to the acentric reference fluid (n-octane), respectively. The constants for Eq. (2-65) for the two reference fluids are given in Table 2-353.

Z(1) Tr\Pr

0.2

0.4

0.85 0.90

−0.0715 −0.0442

(−0.1503) −0.1118

Double linear interpolation within these values gives Z(0) = 0.8058 and Z(1) = −0.1025. From Eq. (2-63): Z = 0.8058 + (0.492328)(−0.1025) = 0.7553 Note: If the analytical form available in Eq. (2-65) is used, the following more accurate values are obtained: Z(0) = 0.8131, Z(1) = − 0.1067, and Z = 0.7606. Molar volume:

ZRT V=  = P

m3⋅MPa (0.7553) 0.0083143  kmol⋅K (540.5 K)





0.6799 MPa

m3 = 4.992  kmol

4. Cubic EoS can be used to obtain both vapor and liquid densities as an alternative method to those mentioned above.

Recommended Method Cubic EoS. Classification: Empirical extension of theory. Expected uncertainty: Varies depending upon compound and conditions, but a general expectation is perhaps 10 to 20 percent. Applicability: Nonpolar and moderately polar compounds. Input data: Tc, Pc, ω. Description: The more common cubic EoS can be written in the form V V a α(Tr) Z =  −   V−b V2 + δV + ε RT

(2-66)

where a, b, δ, and ε are constants that depend upon the model EoS chosen, as does the temperature dependence of the function α(Tr). Definitions of these constants and α(Tr) for some of the more commonly used EoS models are shown in Table 2-354. The corresponding relations for many other EoS models in this same form are available [Soave, G., Chem. Eng. Sci., 27 (1972): 1197]. The independent parameters a and b in these models can be regressed from experimental data to correlate densities or obtained from known critical constants to predict density data. In the latter case, the relationships between a and b and the critical constants shown in Table 2-354 were obtained from the critical point requirements ∂P

∂2P = 0 = 2 ∂V T T=T

`  ∂V

C



`

T T=TC

(2-67)

2-500

TABLE 2-351

Simple Fluid Compressibility Factors Z (0)

Values in parentheses are for the opposite phase and may be used to interpolate to or near the phase boundary [PGL4; Wilding, W. V., J. K. Johnson, and R. L. Rowley, Int. J. Thermophys., 8(1987):717]. Tr\Pr 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

0.010

0.050

0.100

0.200

0.400

0.600

0.800

1.000

1.200

1.500

2.000

3.000

5.000

7.000

0.0029 0.0026 0.0024 0.0022

0.0145 0.0130 0.0119

0.0290 0.0261 0.0239

0.0579 0.0522 0.0477

0.1158 0.1043 0.0953

0.1737 0.1564 0.1429

0.2315 0.2084 0.1904

0.2892 0.2604 0.2379

0.3470 0.3123 0.2853

0.4335 0.3901 0.3563

0.5775 0.5195 0.4744

0.8648 0.7775 0.7095

1.4366 1.2902 1.1758

2.0048 1.7987 1.6373

2.8507 2.5539 2.3211

0.0110

0.0221

0.0442

0.0882

0.1322

0.1762

0.2200

0.2638

0.3294

0.4384

0.6551

1.0841

1.5077

2.1338

(0.9648)

0.0021

0.0103

(0.9741)

(0.8699)

0.9804

0.0098

0.0195

(0.0020)

(0.9000)

(0.7995)

0.9849

0.0093

0.0186

(0.0019)

(0.9211)

(0.8405)

0.9881

0.9377

0.0178

0.0356

(0.0018

(0.0089)

(0.8707)

(0.7367)

0.9904

0.9504

0.8958

0.0344

(0.0086)

(0.0172)

(0.7805)

0.9598

0.9165

0.0336

0.0670

(0.0085)

(0.0169)

(0.8181)

(0.6122)

0.9319

0.8539

0.0661

0.0985

(0.0168)

(0.0332)

(0.6659)

(0.4746)

0.8810

0.0661

0.0983

(0.0336)

(0.7222)

(0.5346)

0.9015

0.7800

0.1006

0.1321

(0.0364)

(0.0685)

(0.6040)

(0.4034)

0.8059

0.6635

0.1359

(0.7350)

(0.1047)

(0.4499)

0.8206

0.6967

0.1410

(0.0822)

(0.1116)

0.4853)

0.7240

0.5580

(0.1312)

(0.1532)

0.9922 0.9935 0.9946

0.9669 0.9725

0.0207

0.9436

0.90

0.9954

0.9768

0.9528

0.93

0.9959

0.9790

0.9573

0.95 0.97

0.9961 0.9963

0.9803 0.9815

0.9600 0.9625

10.000

0.0413

0.0825

0.1236

0.1647

0.2056

0.2465

0.3077

0.4092

0.6110

1.0094

1.4017

1.9801

0.0390

0.0778

0.1166

0.1553

0.1939

0.2323

0.2899

0.3853

0.5747

0.9475

1.3137

1.8520

0.0371

0.0741

0.1109

0.1476

0.1842

0.2207

0.2753

0.3657

0.5446

0.8959

1.2398

1.7440

0.0710

0.1063

0.1415

0.1765

0.2113

0.2634

0.3495

0.5197

0.8526

1.1773

1.6519

0.0687

0.1027

0.1366

0.1703

0.2038

0.2538

0.3364

0.4991

0.8161

1.1241

1.5729

0.1001

0.1330

0.1656

0.1981

0.2464

0.3260

0.4823

0.7854

1.0787

1.5047

0.1307

0.1626

0.1942

0.2411

0.3182

0.4690

0.7598

1.0400

1.4456

0.1301

0.1614

0.1924

0.2382

0.3132

0.4591

0.7388

1.0071

1.3943

0.1630

0.1935

0.2383

0.3114

0.4527

0.7220

0.9793

1.3496

0.1664

0.1963

0.2405

0.3122

0.4507

0.7138

0.9648

1.3257

0.1705

0.1998

0.2432

0.3138

0.4501

0.7092

0.9561

1.3108

0.1779

0.2055

0.2474

0.3164

0.4504

0.7052

0.9480

1.2968

0.1844

0.2097

0.2503

0.3182

0.4508

0.7035

0.9442

1.2901

0.1959

0.2154

0.2538

0.3204

0.4514

0.7018

0.9406

1.2835

0.2901 0.4648 0.5146 0.6026 0.6880 0.7443 0.7858 0.8438 0.8827 0.9103 0.9308 0.9463 0.9583 0.9678 0.9754 0.9865 0.9941 0.9993 1.0031 1.0057 1.0097 1.0115

0.2237 0.2370 0.2629 0.4437 0.5984 0.6803 0.7363 0.8111 0.8595 0.8933 0.9180 0.9367 0.9511 0.9624 0.9715 0.9847 0.9936 0.9998 1.0042 1.0074 1.0120 1.0140

0.2583 0.2640 0.2715 0.3131 0.4580 0.5798 0.6605 0.7624 0.8256 0.8689 0.9000 0.9234 0.9413 0.9552 0.9664 0.9826 0.9935 1.0010 1.0063 1.0101 1.0156 1.0179

0.3229 0.3260 0.3297 0.3452 0.3953 0.4760 0.5605 0.6908 0.7753 0.8328 0.8738 0.9043 0.9275 0.9456 0.9599 0.9806 0.9945 1.0040 1.0106 1.0153 1.0221 1.0249

0.4522 0.4533 0.4547 0.4604 0.4770 0.5042 0.5425 0.6344 0.7202 0.7887 0.8410 0.8809 0.9118 0.9359 0.9550 0.9827 1.0011 1.0137 1.0223 1.0284 1.0368 1.0401

0.7004 0.6991 0.6980 0.6956 0.6950 0.6987 0.7069 0.7358 0.7761 0.8200 0.8617 0.8984 0.9297 0.9557 0.9772 1.0094 1.0313 1.0463 1.0565 1.0635 1.0723 1.0741

0.9372 0.9339 0.9307 0.9222 0.9110 0.9033 0.8990 0.8998 0.9112 0.9297 0.9518 0.9745 0.9961 1.0157 1.0328 1.0600 1.0793 1.0926 1.1016 1.1075 1.1138 1.1136

1.2772 1.2710 1.2650 1.2481 1.2232 1.2021 1.1844 1.1580 1.1419 1.1339 1.1320 1.1343 1.1391 1.1452 1.1516 1.1635 1.1728 1.1792 1.1830 1.1848 1.1834 1.1773

0.9115 0.9174 0.9227

0.8338

0.98

0.9965

0.9821

0.9637

0.9253

0.8398

0.7360

0.99

0.9966

0.9826

0.9648

0.9277

0.8455

0.7471

1.00 1.01 1.02 1.05 1.10 1.15 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 3.50 4.00

0.9967 0.9968 0.9969 0.9971 0.9975 0.9978 0.9981 0.9985 0.9988 0.9991 0.9993 0.9994 0.9995 0.9996 0.9997 0.9998 0.9999 1.0000 1.0000 1.0000 1.0001 1.0001

0.9832 0.9837 0.9842 0.9855 0.9874 0.9891 0.9904 0.9926 0.9942 0.9954 0.9964 0.9971 0.9977 0.9982 0.9986 0.9992 0.9996 0.9998 1.0000 1.0002 1.0004 1.0005

0.9659 0.9669 0.9679 0.9707 0.9747 0.9780 0.9808 0.9852 0.9884 0.9909 0.9928 0.9943 0.9955 0.9964 0.9972 0.9983 0.9991 0.9997 1.0001 1.0004 1.0008 1.0010

0.9300 0.9322 0.9343 0.9401 0.9485 0.9554 0.9611 0.9702 0.9768 0.9818 0.9856 0.9886 0.9910 0.9929 0.9944 0.9967 0.9983 0.9994 1.0002 1.0008 1.0017 1.0021

0.8509 0.8561 0.8610 0.8743 0.8930 0.9081 0.9205 0.9396 0.9534 0.9636 0.9714 0.9775 0.9823 0.9861 0.9892 0.9937 0.9969 0.9991 1.0007 1.0018 1.0035 1.0043

0.7574 0.7671 0.7761 0.8002 0.8323 0.8576 0.8779 0.9083 0.9298 0.9456 0.9575 0.9667 0.9739 0.9796 0.9842 0.9910 0.9957 0.9990 1.0013 1.0030 1.0055 1.0066

0.5887 (0.1703)

0.6138 (0.2324)

0.6353 0.6542 0.6710 0.7130 0.7649 0.8032 0.8330 0.8764 0.9062 0.9278 0.9439 0.9563 0.9659 0.9735 0.9796 0.9886 0.9948 0.9990 1.0021 1.0043 1.0075 1.0090

Table 2-352

Acentric Deviations Z(1) from the Simple Fluid Compressibility Factor

Values in parentheses are for the opposite phase and may be used to interpolate to or near the phase boundary [PGL4; Wilding, W. V., J. K. Johnson, and R. L. Rowley, Int. J. Thermophys., 8(1987):717]. Tr\Pr 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

0.010

0.050

0.100

0.200

0.400

0.600

0.800

1.000

1.200

1.500

2.000

3.000

5.000

7.000

10.000

−0.0008 −0.0009 −0.0010 −0.0009

−0.0040 −0.0046 −0.0048

−0.0081 −0.0093 −0.0095

−0.0161 −0.0185 −0.0190

−0.0323 −0.0370 −0.0380

−0.0484 −0.0554 −0.0570

−0.0645 −0.0738 −0.0758

−0.0806 −0.0921 −0.0946

−0.0966 −0.1105 −0.1134

−0.1207 −0.1379 −0.1414

−0.1608 −0.1834 −0.1879

−0.2407 −0.2738 −0.2799

−0.3996 −0.4523 −0.4603

−0.5572 −0.6279 −0.6365

−0.7915 −0.8863 −0.8936

−0.0047

−0.0094

−0.0187

−0.0374

−0.0560

−0.0745

−0.0929

−0.1113

−0.1387

−0.1840

−0.2734

−0.4475

−0.6162

−0.8606

−0.0090

−0.0181

−0.0360

−0.0539

−0.0716

−0.0893

−0.1069

−0.1330

−0.1762

−0.2611

−0.4253

−0.5831

−0.8099

−0.0172

−0.0343

−0.0513

−0.0682

−0.0849

−0.1015

−0.1263

−0.1669

−0.2465

−0.3991

−0.5446

−0.7521

−0.0164

−0.0326

−0.0487

−0.0646

−0.0803

−0.0960

−0.1192

−0.1572

−0.2312

−0.3718

−0.5047

−0.6928

−0.0309

−0.0461

−0.0611

−0.0759

−0.0906

−0.1122

−0.1476

−0.2160

−0.3447

−0.4653

−0.6346

−0.0294

−0.0438

−0.0579

−0.0718

−0.0855

−0.1057

−0.1385

−0.2013

−0.3184

−0.4270

−0.5785

−0.0417

−0.0550

−0.0681

−0.0808

−0.0996

−0.1298

−0.1872

−0.2929

−0.3901

−0.5250

−0.0526

−0.0648

−0.0767

−0.0940

−0.1217

−0.1736

−0.2682

−0.3545

−0.4740

−0.0509

−0.0622

−0.0731

−0.0888

−0.1138

−0.1602

−0.2439

−0.3201

−0.4254

−0.0604

−0.0701

−0.0840

−0.1059

−0.1463

−0.2195

−0.2862

−0.3788

−0.0602

−0.0687

−0.0810

−0.1007

−0.1374

−0.2045

−0.2661

−0.3516

−0.0607

−0.0678

−0.0788

−0.0967

−0.1310

−0.1943

−0.2526

−0.3339

−0.0623

−0.0669

−0.0759

−0.0921

−0.1240

−0.1837

−0.2391

−0.3163

−0.0641

−0.0661

−0.0740

−0.0893

−0.1202

−0.1783

−0.2322

−0.3075

−0.0680

−0.0646

−0.0715

−0.0861

−0.1162

−0.1728

−0.2254

−0.2989

−0.0879 −0.0223 −0.0062 0.0220 0.0476 0.0625 0.0719 0.0819 0.0857 0.0864 0.0855 0.0838 0.0816 0.0792 0.0767 0.0719 0.0675 0.0634 0.0598 0.0565 0.0497 0.0443

−0.0609 −0.0473 0.0227 0.1059 0.0897 0.0943 0.0991 0.1048 0.1063 0.1055 0.1035 0.1008 0.0978 0.0947 0.0916 0.0857 0.0803 0.0754 0.0711 0.0672 0.0591 0.0527

−0.0678 −0.0621 −0.0524 0.0451 0.1630 0.1548 0.1477 0.1420 0.1383 0.1345 0.1303 0.1259 0.1216 0.1173 0.1133 0.1057 0.0989 0.0929 0.0876 0.0828 0.0728 0.0651

−0.0824 −0.0778 −0.0722 −0.0432 0.0698 0.1667 0.1990 0.1991 0.1894 0.1806 0.1729 0.1658 0.1593 0.1532 0.1476 0.1374 0.1285 0.1207 0.1138 0.1076 0.0949 0.0849

−0.1118 −0.1072 −0.1021 −0.0838 −0.0373 0.0332 0.1095 0.2079 0.2397 0.2433 0.2381 0.2305 0.2224 0.2144 0.2069 0.1932 0.1812 0.1706 0.1613 0.1529 0.1356 0.1219

−0.1672 −0.1615 −0.1556 −0.1370 −0.1021 −0.0611 −0.0141 0.0875 0.1737 0.2309 0.2631 0.2788 0.2846 0.2848 0.2819 0.2720 0.2602 0.2484 0.2372 0.2268 0.2042 0.1857

−0.2185 −0.2116 −0.2047 −0.1835 −0.1469 −0.1084 −0.0678 0.0176 0.1008 0.1717 0.2255 0.2628 0.2871 0.3017 0.3097 0.3135 0.3089 0.3009 0.2915 0.2817 0.2584 0.2378

−0.2902 −0.2816 −0.2731 −0.2476 −0.2056 −0.1642 −0.1231 −0.0423 0.0350 0.1058 0.1673 0.2179 0.2576 0.2876 0.3096 0.3355 0.3459 0.3475 0.3443 0.3385 0.3194 0.2994

(−0.0740)

−0.0009 (−0.0457)

−0.0314 (−0.0009) −0.0205 (0.0008)

−0.0137 (−0.0008)

−0.0093

0.75

−0.0064

0.80

−0.0044

0.85 0.90

−0.0029 −0.0019

−0.0045 (−0.2270)

−0.0043 (−0.1438)

−0.0041 (0.0949)

−0.0772 (0.0039)

−0.0507 (−0.0038)

−0.0339 (−0.0037)

−0.0228 −0.0152 −0.0099

−0.0086 (−0.2864)

−0.0082 (−0.1857)

−0.0078 (−0.1262)

−0.1161 (−0.0075)

−0.0744 (−0.0072)

−0.0487 (−0.0073)

−0.0319 −0.0205

−0.0156 (−0.2424)

−0.0148 (−0.1685)

−0.0143 (−0.1298)

−0.1160 (−0.0139)

−0.0715 (−0.0144)

−0.0442 (−0.0179)

0.93

−0.0015

−0.0075

−0.0154

−0.0326

0.95

−0.0012

−0.0062

−0.0126

−0.0262

0.97 0.98

−0.0010 −0.0009

−0.0050 −0.0044

−0.0101 −0.0090

−0.0208 −0.0184

−0.0282 (−0.2203)

−0.0272 (−0.1682)

−0.0268 (−0.1503)

−0.1118 (−0.0286)

−0.0763 (−0.0340)

−0.0589 (−0.0444)

−0.0450 −0.0390

−0.0401 (−0.2185)

−0.0391 (−0.1692)

−0.0396 (−0.1580)

−0.1662 (−0.0424)

−0.1110 (−0.0490)

−0.0770 (−0.0714)

−0.0641

0.99

−0.0008

−0.0039

−0.0079

−0.0161

−0.0335

−0.0531

1.00 1.01 1.02 1.05 1.10 1.15 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 3.50 4.00

−0.0007 −0.0006 −0.0005 −0.0003 0.0000 0.0002 0.0004 0.0006 0.0007 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 0.0007 0.0006 0.0006 0.0005 0.0005

−0.0034 −0.0030 −0.0026 −0.0015 0.0000 0.0011 0.0019 0.0030 0.0036 0.0039 0.0040 0.0040 0.0040 0.0040 0.0039 0.0037 0.0035 0.0033 0.0031 0.0029 0.0026 0.0023

−0.0069 −0.0060 −0.0051 −0.0029 0.0001 0.0023 0.0039 0.0061 0.0072 0.0078 0.0080 0.0081 0.0081 0.0079 0.0078 0.0074 0.0070 0.0066 0.0062 0.0059 0.0052 0.0046

−0.0140 −0.0120 −0.0102 −0.0054 0.0007 0.0052 0.0084 0.0125 0.0147 0.0158 0.0162 0.0163 0.0162 0.0159 0.0155 0.0147 0.0139 0.0131 0.0124 0.0117 0.0103 0.0091

−0.0285 −0.0240 −0.0198 −0.0092 0.0038 0.0127 0.0190 0.0267 0.0306 0.0323 0.0330 0.0329 0.0325 0.0318 0.0310 0.0293 0.0276 0.0260 0.0245 0.0232 0.0204 0.0182

−0.0435 −0.0351 −0.0277 −0.0097 0.0106 0.0237 0.0326 0.0429 0.0477 0.0497 0.0501 0.0497 0.0488 0.0477 0.0464 0.0437 0.0411 0.0387 0.0365 0.0345 0.0303 0.0270

−0.0503 (−0.1464)

−0.0514 (−0.1418)

−0.0540 (−0.1532)

−0.1647 (−0.0643)

−0.1100 (−0.0828)

−0.0796 (−0.1621)

−0.0588 −0.0429 −0.0303 −0.0032 0.0236 0.0396 0.0499 0.0612 0.0661 0.0677 0.0677 0.0667 0.0652 0.0635 0.0617 0.0579 0.0544 0.0512 0.0483 0.0456 0.0401 0.0357

2-501

2-502

PHYSICAL AND CHEMICAL DATA

TABLE 2-353 Constants for the Two Reference Fluids Used in Lee-Kesler Method* Constant

Simple reference fluid

Acentric reference fluid

b1 b2 b3 b4 c1 c2 c3 c4 d1  104 d2  104 β γ

0.1181193 0.265728 0.154790 0.030323 0.0236744 0.0186984 0.0 0.042724 0.155488 0.623689 0.65392 0.060167

0.2026579 0.331511 0.027655 0.203488 0.0313385 0.0503618 0.016901 0.041577 0.48736 0.0740336 1.226 0.03754

Tr = (353.15 K)(405.65 K) = 0.871 α ={1 + [0.48 + (1.574)(0.252608) − (0.176)(0.252608)2][1 − (0.871)0.5]}2 = 1.119 Rearrange and solve Eq. (2-66) for V: RT aα P=  −  V−b V(V + b)



V 41.352  m3/mol

PV3 − RTV2 + (aα − bRT − Pb2)V − abα = 0

or m3

 + 4.037 × 10 − 0.029  mol  m /mol  3

V

2

−6

3



m6  mol2





V ×  − 1.25 × 10−10 = 0 m3/mol

*Lee, B. I., and M. G. Kesler, AIChE J., 21 (1975): 510.

Vapor root (initial guess of V = 7.1 × 10−7 m3/mol from ideal gas equation): Vvap = 5.395 × 10−4 m3mol

Of the cubic EoS given in Table 2-354, the Soave and PengRobinson are the most accurate, but there is no general rule for which EoS produces the best estimated volumes for specific fluids or conditions. The Peng-Robinson equation has been better tuned to liquid densities, while the Soave equation has been better tuned to vapor-liquid equilibrium and vapor densities. In solving the cubic equation for volume, a convenient initial guess to find the vapor root is the ideal gas value, while an initial value of 1.05b is convenient to locate the liquid root.

Liquid root (initial guess of V = 2.72 × 10−5 m3mol from 1.05b): Vliq = 4.441 × 10−5 m3mol

Pc = 112.8 bar

a = 4.611 × 106 cm6 ⋅barmol2

b = 23.262 cm3mol

α = 1.103 aα RT P =  −  V−b V 2 + 2bV − b2

ω = 0.252608

P∗ (353.15 K) = 41.352 bar (vapor pressure at 353.15 K)

ρ liq = 1Vliq = 22.516 kmolm3

and

The corresponding values and equation for the Peng-Robinson EoS are

Example Estimate the molar density of liquid and vapor saturated ammonia at 353.15 K, using the Soave and Peng-Robinson EoS. Required properties: Recommended values in the DIPPR® 801 database are Tc = 405.65 K

ρvap = 1Vvap = 1.854 kmolm3

and

or

EoS parameters (shown for Soave EoS): PV 3 + (bP − RT)V2 + (aα − 2bRT − 3Pb2)V + (bP3 + RTb2 − abα) = 0 3

0.42748(RTc)2 a =  = Pc

0.42748

2

83.145  mol⋅K (405.65 K) bar⋅cm

112.8 bar



V 41.352  m3/mol

m3

 + 3.651 × 10 − 0.0284  mol  m /mol  3

V

cm6⋅bar = 4.311 × 106  mol2

0.08664(RTc) b =  = Pc

2

3



−6

m6  mol2





V ×  − 1.018 × 10−10 = 0 m3/mol Vvap = 5.286 × 10−4 m3mol and ρvap = 1.892 kmolm3



bar⋅cm3  0.08664 83.145  mol ⋅ K (405.65 K)



Vliq = 3.914 × 10−5 m3mol and ρliq = 25.55 kmolm3

112.8 bar

The liquid density calculated from the Soave EoS is 24.2 percent below the DIPPR® 801 recommended value of 29.69 kmol/m3, while that calculated from the Peng-Robinson EoS is 13.9 percent below the recommended value.

cm3 = 25.906  mol

TABLE 2-354

Relationships for Eq. (2-66) for Common Cubic EoS

EoS

δ

ε

α(Tr)

aPc/(RTc)2

bPc/(RTc)

van der Waals* Relich-Kwong† Soave‡ Peng-Robinson§

0 0 b 2b

0 0 0 −b2

1

0.42188 0.42748 0.42748 0.45724

0.125 0.08664 0.08664 0.0778

Tr−0.5 [1 + (0.48 + 1.574ω − 0.176ω2)(1 − Tr0.5)]2 [1 + (0.37464 + 1.54226ω − 0.2699ω2)(1 − Tr0.5)]2

*van der Waal, J. H., Z. Phys. Chem., 5 (1890): 133. †Redlich, O., and J. N. S. Kwong, Chem. Rev., 44 (1949): 233. ‡Soave, G., Chem. Eng. Sci., 27 (1972): 1197. §Peng, D. Y., and D. B. Robinson, Ind. Eng. Chem. Fundam., 15 (1976): 59.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Liquids For most liquids, the saturated molar liquid density ρ can be effectively correlated with A ρ=  B[1+(1−T/C) ]

(2-68)

D

adapted from the Rackett prediction equation [Rackett, H. G., J. Chem. Eng. Data, 15 (1970): 514]. The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The liquid density decreases approximately linearly from the triple point to the normal boiling point and then nonlinearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. The recommended method for estimation of saturated liquid density for pure organic compounds is the Rackett prediction method. Recommended Method Rackett method. Reference: Rackett, H. G., J. Chem. Eng. Data, 15 (1970): 514. Classification: Corresponding states. Expected uncertainty: 8 percent as purely predictive equation; 2 percent if ZRA (see Description below) or some liquid density data are available. Applicability: Saturated liquid densities of organic compounds. Input data: Tc, Pc, and Zc (or, equivalently, Vc). Description: A predictive form of the equation is given by





1 RTc  = V =  Z qc ρ Pc

where q = 1.0 + (1.0 − Tr)27

(2-69)

A modification of the Rackett method by Spencer and Danner [Spencer, C. F., and R. P. Danner, J. Chem. Eng. Data, 17 (1972): 236] replaces Zc with an adjustable parameter ZRA





1 RTc q  = V =  Z RA ρ Pc

(2-70)

to provide better estimations of liquid density away from the critical point [Eq. (2-70) gives the correct critical density only when ZRA = Zc]. An alternative to this modification when several liquid density data points are available is to replace the 2/7 power in q of Eq. (2-69) with an adjustable parameter. This generally provides good agreement with the experimental values and permits accurate extrapolation of the densities all the way to the critical point. Example Estimate the saturated liquid density of acetonitrile at 376.69 K.

Required properties: The recommended values from the DIPPR® 801 database are Tc = 545.5 K

Pc = 4.83 MPa

Zc = 0.184

Calculate supporting quantities: Tr = (376.69 K)(545.5 K) = 0.691 q = 1 + (1 − 0.691)27 = 1.715 Calculate saturated liquid density from Eq. (2-69):

ρ=

c

4.83 × 106 Pa  Pa⋅m3 8.314  (545.5 K) mol⋅K



d (0.184)

−1.715

kmol = 19.42  m3

This estimated value is 16.1 percent above the DIPPR® 801 recommended value of 16.726 kmol/m3.

2-503

Calculate ρsat from Eq. (2-70): Kratzke [Kratzke, H., and S. Muller, J. Chem. Thermo., 17 (1985): 151] reported an experimental density of 18.919 kmol/m3 at 298.08 K. Use of this experimental value in Eq. (2-70) to calculate ZRA gives Tr = (298.08 K)(545.5 K) = 0.546

ZRA =

c

ρ=

q = 1 + (1 – 0.546)27 = 1.798

d

4.83 × 106 Pa  Pa⋅m3 kmol 8314   (545.5 K) 18.919  k mol⋅K m3





c

4.83 × 106 Pa  Pa⋅m3 8.314  (545.5 K) mol⋅K



d (0.202)

−1.715

1/1.798

= 0.202

kmol = 16.577  m3

The value obtained by the modified Rackett method is 0.9 percent below the DIPPR® 801 recommended value. Note, however, that with ZRA = 0.202, Eq. (270) gives ρc = 5.28 kmolm3 as opposed to the DIPPR® 801 recommended value of 5.79 kmol/m3. If the power is regressed from the Kratzke density, one obtains q1 = 0.452 and ρ = 15.68 kmolm3 (4 percent below the experimental value), while still retaining ρc = 5.79 kmol/m3.

Solids Solid density data are sparse and usually available only within a narrow temperature range. For most solids, density decreases approximately linearly with increasing temperature. Prediction of solid densities is an inexact science, but reasonable correlation has been found between the density of the liquid phase at the triple point and the solid that is stable at the triple point conditions. Recommended Method Goodman method. Reference: Goodman, B. T., et al., J. Chem. Eng. Data, 49 (2004): 1512. Classification: Empirical correlation. Expected uncertainty: 6 percent. Applicability: Organic compounds; applicable to the stable solid phase at the triple point temperature Tt, to either the next solid-phase transition temperature or to approximately 0.3Tt. Input data: Liquid density at the triple point. Description: The density for the solid phase that is stable at the triple point has been correlated as a function of temperature and the liquid density at Tt:



T ρs = 1.28 − 0.16  Tt

ρ (T ) L

(2-71)

t

Example Estimate the density of solid naphthalene at 281.46 K.

Required properties: The recommended values from the DIPPR® 801 database for Tt and the liquid density at Tt are Tt = 353.43 K

ρL(Tt) = 7.6326 kmolm3

From Eq. (2-71):



281.46 K ρs = 1.28 − 0.16  353.43 K

= 8.797  7.6326  m m kmol 3

kmol 3

The estimated value is 4.3 percent lower than the DIPPR® 801 recommended value of 9.1905 kmol/m3.

Mixtures Both liquid and vapor densities can be estimated using pure-component CS and EoS methods by treating the fluid as a pseudo-pure component with effective parameters calculated from the pure-component parameters and using ad hoc mixing rules. To apply the Lee-Kesler CS method to mixtures, pseudo-pure fluid constants are required. One of the simplest set of mixing rules for these quantities is [Prausnitz, J. M., and R. D. Gunn, AIChE J., 4 (1958): 430, 494; Joffe, J., Ind. Eng. Chem. Fundam., 10 (1971): 532]: ⎯ C Tc = xiTc,i i=1

(2-72)

2-504

PHYSICAL AND CHEMICAL DATA where

C

xZ

i=1

i c,i

⎯ = ⎯ Pc  RTc C

xiVc,i

C ⎯ ZRA = 0.29056 − 0.08775 xiωi

(2-73)

and

i=1

T Tr =  ⎯ Tc

(2-82)

i=1

C

⎯ = xω ω

i i

(2-74)

i=1

The procedures are identical to those for pure components with the replacement of Tc, Pc, and ω with the effective mixture values calculated by using these equations. To use a cubic EoS for a mixture, mixing rules are used to calculate effective mixture parameters in terms of the pure-component values. Although there are more complex mixing rules available that may improve prediction accuracy, the simplest forms are recommended here for their simplicity and reasonable accuracy without adjustable parameters: ⎯ C (2-75) b = xi bi i=1

⎯= a⎯α

C

 x (a α )  i

i

i

12

2

(2-76)

i=1

Mixture calculations are then identical to the pure-component calculations using these effective mixture parameters for the pure-component aα and b values. The actual mixture second virial coefficient Bm is related to the pure-component values by C

Example Estimate the saturated liquid density of a liquid mixture of 50 mol % ethane(1) and 50 mol % n-decane(2) at 377.6 K. Required properties: The recommended values from the DIPPR® 801 database for the required properties are as follows:

Ethane Decane

Bii = Bi

where

13 V13 c,i + Vc, j Vc,ij =  2



ωi + ωj ω ij =  2



3

Zc,i + Zc,j Zc,ij =  (2-78) 2

Zc,ijRTc,ij Pc,ij =  Vc,ij

(2-79)

These interaction parameters are used in place of the corresponding pure-component parameters to determine the Bij values. The modified Rackett method has also been extended to liquid mixtures [Spencer, C. F., and R. P. Danner, J. Chem. Eng. Data, 17 (1972): 236] using the following combining and mixing rules as modified by Li [Li, C. C., Can. J. Chem. Eng., 19 (1971): 709]: xiVc,i φi =  C xV

j=1 j

⎯ C Tc =

C

φ φ T i =1 j=1 i

j c,ij

(2-80)

c,j

Recommended Method Spencer-Danner-Li mixing rules with Rackett equation. References: Spencer, C. F., and R. P. Danner, J. Chem. Eng. Data, 17 (1972): 236; Li, C. C., Can. J. Chem. Eng., 19 (1971): 709. Classification: Corresponding states. Expected uncertainty: About 7 percent on average; higher near the Tc of any of the components. Applicability: Saturated (at the bubble point) liquid mixtures. Input data: Tc, Vc, and xi. Description: The predictive form of the equation is given by 1 = V = R ρ

xiTc,i ⎯q

Z   P i=1

ω

0.1455 0.617

48.72 21.1

0.0995 0.4923

(0.5)(0.1455) φ1 =  = 0.191; (0.5)(0.1455) + (0.5)(0.617)

φ2 = 0.809

 2 K)(6 17.7 K) = 434.3 K Tc,12 = (305.3 ⎯⎯ ⎯ T c = φ 2T + 2φ φ T + φ 2T  1 c,1 2 c,2 1 2 c,12 K = (0.191)2(305.32) + (2)(0.191)(0.809)(434.3) + (0.809)2(617.7) ⎯ Tc = 549.68 K Calculations from Eqs. (2-81) and (2-82):

(2-77)

This requires calculation of all possible binary pair interaction virials (Bij, i ≠ j) for the mixture. Again the pure-component methods can be used to provide estimates of these values by using the following combining rules:

C

Pc / bar

305.32 617.7

Auxiliary quantities from Eq. (2-80):

q = 1 + (1 − 0.687)2/7 = 1.718 ⎯ ZRA = 0.29056 − 0.08775[(0.5)(0.0995) + (0.5)(0.4923)] = 0.2646

i=1 j=1

Tc,ij = T c,iTc, j

Vc /(m3kmol−1)

Tr = (377.6 K)/(549.63 K) = 0.687

C

Bm = xi xjB ij

Tc,ij = T  c,iTc,j

Tc /K

c,i

RA

q = 1.0 + (1.0 − Tr)27

(2-81)

m3⋅bar V = 0.08314  K⋅kmol



+   (0.2646)   48.72 bar 21.1 bar (0.5)(305.32 K)

(0.5)(617.7 K)

1.718

m3 = 0.151  kmol

The experimental value [Reamer, H. H., and B. H. Sage, J. Chem. Eng. Data, 7 (1962): 161] is 0.149 m3/kmol, and the error in the estimated value is 1.3 percent. VISCOSITY Viscosity is defined as the shear stress per unit area at any point in a confined fluid, divided by the velocity gradient in the direction perpendicular to the direction of flow. The absolute viscosity η is the shear stress at a point, divided by the velocity gradient at that point. The SI unit of viscosity is Pa⋅s [1 kg/(m⋅s)], but the cgs unit of poise (P) [1 g/(cm⋅s)] is also commonly used. Because many common fluids have viscosities on the order of 0.01 P, the unit of centipoise (cP) is also frequently used (1 cP =1 mPa⋅s). The kinematic viscosity ν is defined as the ratio of the absolute viscosity to density at the same temperature and pressure. The SI unit for ν is m2/s, but again cgs units are very common and ν is often given in stokes (St) (1 cm2/s) or centistokes (cSt) (0.01 cm2/s). Gases Experimental data for gases and vapors at low density are often correlated with AT B η o = 2 1 + C/T+D/T

(2-83)

Over smaller temperature ranges, parameters C and D may not be necessary as ln η is often reasonably linear with ln T. Care should be taken in extrapolating using Eq. (2-83) as there can be unintended mathematical poles where the denominator approaches zero.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Numerous methods have been developed for estimation of vapor viscosity. For nonpolar vapors, the Yoon-Thodos CS method works well, but for polar fluids the Reichenberg method is preferred. Both methods are illustrated below. Recommended Method Yoon-Thodos method. Reference: Yoon, P., and G. Thodos, AIChE J., 16 (1970): 300. Classification: Corresponding states. Expected uncertainty: 5 percent. Applicability: Nonpolar and slightly polar organic vapors. Input data: Tc, Pc, and M. Description: The correlation for viscosity as a function of reduced temperature is ηo 46.1Tr0.618 − 20.4 exp(−0.449Tr) + 19.4 exp(−4.058Tr) + 1  =  Pa⋅s 2.173424 × 1011(Tc /K)1/6(M/gmol)−1(Pc /Pa)−2/3

TABLE 2-355

2-505

Reichenberg* Group Contribution Values

Group

Ci

Group

Ci

CH3 >CH2 >CH >C< 苷CH2 苷CH >C苷 CH C >CH2 ring >CH ring >C< ring 苷CH ring >C苷 ring

9.04 6.47 2.67 −1.53 7.68 5.53 1.78 7.41 5.24 6.91 1.16 0.23 5.90 3.59

F Cl Br OH alcohol >O >C苷O CHO COOH COO or HCOO NH2 >NH 苷N ring CN >S ring

4.46 10.06 12.83 7.96 3.59 12.02 14.02 18.65 13.41 9.71 3.68 4.97 18.13 8.86

*Reichenberg, D., AIChE J., 21 (1975): 181.

(2-84) Example Estimate the low-pressure vapor viscosity of propane at 353 K.

Required constants: The DIPPR® 801 database recommends the following values: Tc = 369.83 K Pc = 4.248 MPa M = 44.0956 g/mol

Example Estimate the low-pressure vapor viscosity of ethyl acetate at 401.25 K.

Required constants: The DIPPR® 801 database recommends the following values: M = 88.1051 gmol

Tc = 523.3 K

Pc = 3.88 MPa

O

Calculation using Eq. (2-84):

ηo  = Pa ⋅ s

H3C Group

(46.1)(0.9545)0.618 − 20.4 exp[−0.449)(0.9545)]+19.4 exp[−4.058(0.9545)]+1  (2.173424 × 1011)(369.83 )−12(44.0956)−12(4.248× 106)−23

−CH3 >CH2 —COO—

ni 2 1 1

O

CH3 Ci

Contribution

9.04 6.47 13.41

18.08 6.47 13.41 Total

= 9.84 × 10−6 The estimated value is 1.5 percent higher than the DIPPR® 801 recommended value of 9.70 × 10−6 Pa⋅s.

Recommended Method Reichenberg method. Reference: Reichenberg, D., AIChE J., 21 (1975): 181. Classification: Group contributions and corresponding states. Expected uncertainty: 5 percent. Applicability: Nonpolar and polar organic and inorganic vapors. Input data: Tc, Pc, M, µ, and molecular structure. Description: The temperature dependence of the viscosity is given by ATr2 ηo  =  [1 + 0.36Tr(Tr − 1)]16 Pa⋅s



1 + 270(µ*r)4  Tr + 270(µ*r)4



µr* = 52.46µr

(2-86)

and Eq. (2.62).

For organic compounds, A is found from the group values Ci, listed in Table 2-355, using

(kg  kmol) M

A = 10

12

(Tc / K)

 N

(2-87)

nC

i=1 i

i

For inorganic gases, A is obtained from A = 1.6104 × 10−10

M

−16

   g/mol  Pa  K  12

Pc

23

Tc

37.96

Tr = (401.25 K)(523.3 K) = 0.767 From Eqs. (2-62) and (2-86): (1.78)2(38.8) = 0.024 µ*r = 52.46  (523.3)2 From Eq. (2-87): (88.1051)12(523.3) A = 10−7  = 1.294 × 10−5 37.96 Calculation using Eq. (2-84):

(2-85)

where the parameter A is determined from group contributions and the modified reduced dipole µ*r is found from

−7

µ = 1.78 D

Supporting quantities: Structural groups:

Reduced temperature: Tr = (353 K)/(369.83 K) = 0.9545

(2-88)

(1.294 × 10−5)(0.767)2 1 + (270)(0.024)4 ηo  = 1.003 × 10−5  =  [1 + (0.36)(0.767)(0.767 − 1)]16 0.767 + (270)(0.024)4 Pa⋅s The estimated value is 1.5 percent lower than the DIPPR® 801 recommended value of 1.018 × 10−5 Pa⋅s.

The dependence of viscosity upon pressure is principally a density effect. Estimation of vapor viscosity at elevated pressures is commonly done by correlating density deviations from the low-pressure values, which are in turn estimated by using the procedures mentioned above. Several methods are available, but the method developed by Jossi et al. and extended to polar fluids by Stiel and Thodos is relatively accurate and easy to apply. Recommended Method Jossi-Stiel-Thodos Method. References: Stiel, L. I., and G. Thodos, AIChE J., 10 (1964): 26; Jossi, J. A., L. I. Stiel, and G. Thodos, AIChE J., 8 (1962): 59. Classification: Empirical correlation and corresponding states. Expected uncertainty: 9 percent—often less for nonpolar gases, larger for polar gases. Applicability: Nonassociating gases; ρr < 2.6. Input data: M, Tc, Pc, Zc, µ,ηo (low-pressure viscosity at same T may be estimated by using methods given above), and ρ (may be calculated from T and P by using density methods given above).

2-506

PHYSICAL AND CHEMICAL DATA

Description: Deviation of η from the low-pressure value ηo is given by one of the following correlations depending upon its polarity and reduced density range: For nonpolar gases, 0.1 < ρr < 3.0: η − ηo

ξ + 1   mPa⋅s

14

(2-89)

ξ = 1.656ρ  mPa⋅s

1.111 r

(2-90)

For polar gases, 0.1 < ρr ≤ 0.9: η − ηo

ξ = 0.0607(9.045ρ + 0.63)  mPa⋅s

1.739

r

(2-91)





η − ηo  ξ mPa⋅s

= 0.6439 − 0.1005ρr



which is analogous to the Riedel [Riedel, L., Chem. Ing. Tech., 26 (1954): 83] vapor pressure equation. Currently the most accurate method for predicting pure liquid viscosity is the following GC method. Recommended Method Hsu method. Reference: Hsu, H.-C., Y.-W. Sheu, and C.-H. Tu, Chem. Eng. J., 88 (2002): 27. Classification: Group contributions. Expected uncertainty: 20 percent. Applicability: Organic liquids; Tr < 0.75. Input data: Pc and molecular structure. Description: The temperature dependence of the liquid viscosity is given by η ln  mPa⋅s



(2-92)

N

c

i=1

N

N

+  d ln   (2-97) = a + T b +  bar T i

i=1

i

i

i=1

Pc

i

2

i=1

where Pc is critical pressure and the coefficients a, b, c, and d are the sum of the group contributions obtained from Table 2-356.

For polar gases, 2.2 < ρr ≤ 2.6: log 4 − log

(2-96)

N

For polar gases, 0.9 < ρr  2.2:



or an extension of it. For example, the DIPPR® 801 database uses the equation B ln η = A +  + C ln T + DTE T

For polar gases, ρr ≤ 0.1: η − ηo

(2-95)

T

= 1.0230 + 0.23364ρr

+ 0.58533ρ2r − 0.40758ρ3r + 0.093324ρ4r

log 4 − log

B ln η = A + 

ξ = 0.6439 − 0.1005ρ   mPa⋅s 

Example Estimate the liquid viscosity of benzotrifluoride at 303.15 K.

η − ηo

Structural information:

r

F

− 0.000475(ρ3r − 10.65)2

(2-93)

F

where ρc = Pc (ZcRTc) and

F −12

−23

   kg/kmol  MPa

T ξ = 2173.4 c K

16

M

Pc

(2-94)

Example Estimate the vapor viscosity of CO2 at 350 K and 20 MPa if ηo = 0.0174 mPa⋅s and Z = 0.4983 (estimated from Lee-Kesler method, see section on density). Required properties: From the DIPPR® 801 database, M = 44.01 kgkmol Zc = 0.274

Pc = 7.383 MPa Tc = 304.21 K µ = 0 D (nonpolar)

Group >C< (苷CH)A (苷C<)A (F)3

Number 1 5 1 1

a

1.0031 −0.8570 0.7896 1.5394 Total −0.9529

ξ = 2173.4(304.21) (44.01)

−12

−23

(7.383)

= 224.1

0.0001c −6.0316 2.4376 −0.9222 17.8121 23.0463

d 1.1972 0.1311 0.1928 −2.9915 −0.9460

Supporting values: Pc = 32.1 MPa

Auxiliary quantities: 16

100b −0.3677 −0.0098 −0.0231 0.8465 0.4067

Calculation using Eq. (2-97):

7.383 MPa

kmol ρc =  = 10.654  m3 0.274[8.314 m3Pa/(K⋅mol)](304.21 K) ρ P 20 MPa ρr =  =  =  ρc ZRTρc 0.4983[8.314 m3⋅Pa/(K⋅mol)](350 K)(10.654 m3⋅kmol) = 1.295 Calculation using Eq. (2-89) for nonpolar fluids: η − ηo

+ 1 224.1  mPa⋅s

14

= 1.0230 + 0.23364(1.295) + 0.58533(1.295)2

− 0.40758(1.295)3 + 0.093324(1.295)4 = 1.684 1.6844 − 1 η =  mPa⋅s + 0.0174 mPa⋅s = 0.0489 mPa⋅s 224.1 This differs from the experimental value of 0.0473 mPa⋅s by 3.4 percent.

Liquids Liquid viscosity can be correlated as a function of temperature for low pressures. Usually the correlation is based on the Andrade equation [Andrade, E. N. da C., Nature, 125 (1930): 309]

η 230463  = exp −0.9529 + (0.004067)(303.15) + 2 mPa⋅s (303.15)



= 0.610



− 0.9460 ln 32.1

The estimated value is 20 percent higher than the DIPPR® 801 recommended value of 0.509 mPa⋅s.

Liquid Mixtures Almost all methods for estimating liquid mixture viscosity interpolate between the pure-component values at the same temperature. The Grunberg-Nissan [Grunberg, L., and A. H. Nissan, Nature, 164 (1949): 799] equation C

1 ln η = xi ln ηi +  i

2

C

C



xi xj Gij i=1 j=1

(2-98)

is commonly used for nonaqueous mixtures. The parameter Gij generally must be regressed from an experimental mixture viscosity. However, in the case of hydrocarbon mixtures Gij can be set to zero with expected errors in the mixture viscosity of about 15 percent.

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-356

Group Contributions for the Hsu et al. Method*

Table-specific nomenclature: R = in nonaromatic ring, A = in aromatic ring, RC = attached to nonaromatic ring, AC = attached to aromatic ring, X = halogen, (−X)n = n X atoms attached to same C atom Group

a

100b

0.0001c

d

−1.0563 −0.2382 0.0060 0.4028 −0.3677 −0.2656 0.1612 0.4305 −0.1106 −0.0111 −0.1778 0.7637 −0.0120 1.7694 −0.0098 −0.0231 0.0444 0.0683 1.2165 −0.8665

0.8928 0.7556 1.4157 4.5094 −6.0316 0.9860 1.9408 3.1287 4.4245 0.3265 0.8437 7.2433 2.0143 19.0452 2.4376 −0.9222 8.1690 8.8426 34.2857 −14.7474

−0.0019 −0.1765 0.0751 0.6679 1.1972 −0.4417 0.2507 1.0465 −24.1836 25.0542 −1.5184 8.5951 −0.3677 13.3885 0.1311 0.1928 −0.4351 −0.1685 −11.6500 −2.7574

−0.5310 −1.0010 −0.3645 2.4275 0.8509 −0.3634 0.6362 −0.4377 0.0985 −0.2847 0.0872 −0.2612 0.1142 −0.2405 0.0413 −0.1841 −0.1693 −0.0113 −0.2162 0.0006 1.5834 0.0111 0.2848 1.0614 0.1801 0.1322 0.0427 0.0779 −0.0965 −0.3285 1.2621 0.5248 −1.2592 0.6232

9.5499 13.8366 29.8404 78.5417 77.1759 23.2329 50.0840 17.2243 2.9405 −4.3145 6.4296 3.7241 6.7008 3.8828 27.4079 12.6878 20.0309 9.4694 1.9325 5.4231 34.5474 7.2831 9.3746 49.1049 12.9392 15.8672 12.1837 4.4123 6.0066 1.9387 23.1473 14.2694 −23.9353 27.5184

−1.0300 0.3418 0.4246 0.9650 −6.9285 −0.0172 −1.0539 0.7139 0.1149 8.3131 0.5389 0.2386 0.7348 −12.4994 0.0002 1.1139 0.0279 0.6071 0.4686 0.8717 −0.4244 3.6587 2.1486 2.8583 2.8987 −0.0701 −0.0948 0.9549 0.3464 0.1148 1.3950 3.7646 0.8329 7.7525

−0.2126 −0.1723 0.4842 −0.0180 0.2208 0.5975 −0.2774 0.0432 26.4615 0.1316 1.5664 −4.5188 0.6712 0.1910

7.0544 5.7804 6.1893 23.2752 14.9707 14.0415 31.8007 20.9135 394.1670 45.5193 60.8742 −62.0987 37.9465 12.0578

0.1336 1.6467 0.4718 1.0653 0.1171 −0.0031 0.0001 1.8795 0.3530 1.2172 −4.6399 1.2353 1.9199 −0.0276

C, H Groups CH4 CH3 CH2 >CH >C< 苷CH2 苷CH 苷C< CH C (CH2)R (>CH)R (苷CH)R cycloalkene (>C<)R spirocyclane (苷CH)A (苷C<)A cycloalkene (苷C<)A bi/terphenyl (苷C<)A naphthalene (苷C<)A turpentine (苷C<)A tetralin

−1.7296 0.0570 −0.1497 −2.2942 1.0031 0.9256 1.3365 −3.5020 87.6040 −91.6154 6.0416 −33.8745 1.2028 −56.2158 −0.8570 0.7896 2.0973 0.4392 27.3350 14.2586

OH primary for C<3 OH primary for C>2 OH secondary OH tertiary (OH)RC OH polyhydric (OH)AC OH alkoxyalcohol O (O)R (O)AC CHO >CO (>CO)R HCOOH COOH for C<7 COOH for C>6 HCOO COO for C<8 COO for C>7 >CHO (CO)O(CO) anhydride O(CO)O carbonate (>NO)R NO2 苷CHNO2 (NO2)AC S SH primary SH secondary SH tertiary CSO for C<13 CSO for C>12 >SO

5.7852 1.4351 −2.6895 −18.5630 16.7808 −0.0125 −2.0856 −2.6991 −0.7185 −29.8045 −2.3454 −0.8288 −2.6622 45.9143 −2.7291 −4.0451 −0.6721 −3.3731 −0.0635 −2.5390 −5.4872 −11.8236 −8.0314 −16.9531 −13.0333 −1.9653 −1.2954 −3.2767 −2.1030 −0.2481 −12.3498 −15.2678 3.7475 −32.8607

O, S Groups

N, X Groups NH2 NH N< (NH2)AC (NH)AC (N<)AC HCONH2 HCONH HCON< CONH2 CONH COONH2 COONH (>NH)R

−1.1345 −6.9489 −2.1403 −6.3646 −1.7592 −1.2982 −1.5435 −8.1097 −122.3280 −6.7363 8.9977 17.8400 −10.1316 −0.1589

2-507

2-508

PHYSICAL AND CHEMICAL DATA TABLE 2-356

Group Contributions for the Hsu et al. Method* (Concluded)

Table-specific nomenclature: R = in nonaromatic ring, A = in aromatic ring, RC = attached to nonaromatic ring, AC = attached to aromatic ring, X = halogen, (−X)n = n X atoms attached to same C atom Group

a

100b

0.0001c

d

0.1120 −0.1324 −0.0086 −0.3851 −0.1934 −1.0770 −0.3220 −0.4130 −0.0623 0.2607 −1.1189 0.8465 −0.2352 −0.3682 −0.5629 0.0109 0.1403 −0.6623 −0.3420 −0.1635 −0.2787 −0.0245 −0.0470

6.98437 7.7955 8.6310 3.0118 3.7798 0.1882 8.8683 13.3194 4.1382 11.3406 1.3134 17.8121 −0.1505 4.6451 3.6831 5.9474 10.3743 −2.4228 1.4253 3.0150 4.3362 7.2061 8.2815

0.9719 0.6293 −0.6443 0.5524 −0.4748 1.2223 0.1702 −1.1972 −0.2644 1.8461 2.6681 −2.9915 −0.2893 −0.0751 0.3613 −14.5771 −1.1972 0.7385 73.6293 0.0621 0.5635 −18.9106 0.4485

N, X Groups −4.7601 −2.7194 0.9435 −1.7997 1.5851 −3.0561 −1.3357 4.2070 −0.3083 −9.4982 −10.3980 1.5394 0.4079 −0.8565 −3.4552 54.2824 −2.1710 −0.7586 −279.0030 −8.1919 −1.4672 70.9918 −2.3300

(苷N)R CN (CN)AC Cl primary 苷CHCl (Cl)2 (Cl)3 (Cl)4 (Cl)AC F primary (F)2 (F)3 (F)AC (F)(Cl) (F)(Cl)2 (F)2(Cl) (F)2(Cl)2 Br primary Br secondary (Br)AC I primary (I)AC (CO)Cl

*Hsu, H.-C., Y.-W. Sheu, and C.-H. Tu, Chem. Eng. J., 88 (2002): 27

Estimation of liquid mixture viscosity without any mixture data is difficult because the viscosity is strongly affected by large molecular size differences and strong cross interactions between the different types of molecules. Viscosity-composition plots for aqueous mixtures can have maxima or minima, and viscosities for these mixtures are particularly difficult to estimate. The UNIFAC-VISCO method described below can be used to predict liquid viscosity of organic mixtures without any mixture data. It is relatively successful even for large differences in molecular size, but it is currently limited in scope by the small number of group contributions available. Recommended Method UNIFAC-VISCO method. Reference: Chevalier, J. L., P. Petrino, and Y. Gaston-Bonhomme, Chem. Eng. Sci., 43 (1988): 1303; Gaston-Bonhomme, Y., P. Petrino, and J. L. Chevalier, Chem. Eng. Sci., 49 (1994): 1799. Classification: Group contributions. Expected uncertainty: 20 percent. Applicability: Organic liquids. Input data: Molecular structure; pure-component molar volumes and viscosities at the mixture temperature. Description: The equations for this method are η ln  = mPa⋅s





ηi

comp

E c

g

Groups, area fractions, volume fractions: Group CH3 CH2 CHar OH Group

g

where Vm is the mixture molar volume and Vi is the pure-component molar volume of component i. The combinatorial and residual excess Gibbs energies are calculated exactly as in the UNIFAC method (see Sec. 13) with the exception that the group interactions ψmm are calculated by using



N1

N2

1 1 0 1

0 0 6 0

0.9011 0.6744 0 1 Total 2.5755

q1

r2

q2

0.848 0.54 0 1.2 2.588

0 0 3.1878 0 3.1878

0 0 2.4 0 2.4

where in the above table qi = Ni,kQk k=1

m



Q 0.8480 0.5400 0.4000 1.2000

r1

CH3 CH2 CHar OH

(2-99)

i

αmn Ψmn = exp −  298.15

R 0.9011 0.6744 0.5313 1.0000

N

E r

x ln  ⋅  +  − 

mPa⋅s V RT RT i=1 Vi

Required input: Values from the DIPPR® 801 database for the pure components at 298.15 K are η1 = 1.0774 mPa⋅s, η2 = 0.5997 mPa⋅s, V1 = 0.05862 m3kmol, and V2 = 0.08948 m3kmol.

(2-100)

The interaction parameters αmn are obtained from Table 2-357. Example Estimate the viscosity of a mixture of 51.13 mol % ethanol(1) and 48.87 mol % benzene(2) at 298.15 K.

N

and

ri = Ni,k Rk k=1

x1q1 (0.5113)(2.588) θ1 =  =  = 0.53 4 (0.5113)(2.588) + (0.4887)(2.4) x q

ii

θ2 = 0.47

i=1

(0.5113)(2.5755) x1r1 =  = 0.458 φ1 =  4 (0.5113)(2.5755) + (0.4887)(3.1878) x r

ii i=1

UNIFAC combinatorial term: 2 2 gCE φi θi  = xi ln  + 5 xiqi ln  = 0.124 xi φi RT i=1 i=1

φ2 = 0.542

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-357

2-509

UNIFAC-VISCO* Group Interaction Parameters ␣mn

m/n

CH2

CH3

CH2cy

CHar

Cl

CO

COO

OH

CH3OH

CH2 CH3 CH2cy CHar Cl CO COO OH CH3OH

0 −709.5 −538.1 −623.7 −710.3 586.2 541.6 −634.5 −526.1

66.53 0 187.3 237.2 375.3 −21.56 −44.25 1209.0 653.1

224.9 −130.7 0 50.89 −163.3 740.6 416.2 −138 751.3

406.7 −119.5 8.958 0 −139.8 −117.9 −36.17 197.7 51.31

60.30 82.41 251.4 177.2 0 −4.145 240.5 195.7 −140.9

859.5 11.86 −125.4 128.4 −404.3 0 22.92 664.1 −22.59

1172.0 −172.4 −165.7 −49.85 −525.4 29.20 0 68.35 −286.2

498.6 594.4 694.4 419.3 960.2 221.5 186.8 0 −23.91

−219.7 −228.7 −381.53 −88.81 −165.4 55.52 69.62 416.4 0

*Chevalier, J. L., P. Petrino, and Y. Gaston-Bonhomme, Chem. Eng. Sci., 43 (1988): 1303; Gaston-Bonhomme, Y., P. Petrino, and J. L. Chevalier, Chem. Eng. Sci., 49 (1994): 1799.

Group interactions:

Benzene Group

αmn

m/n group

CH3

CH2

CHar

OH

CH3 CH2 CHar OH

0 66.53 237.2 1209

−709.5 0 −623.7 −634.5

−119.5 406.7 0 197.7

594.4 498.6 419.3 0

Ψmn

m/n group

CH3

CH2

CHar

OH

CH3 CH2 CHar OH

1.000 0.800 0.451 0.017

10.801 1.000 8.100 8.399

1.493 0.256 1.000 0.515

0.136 0.188 0.245 1.000

The α mn values were obtained from Table 2-357, and Ψmn values were calculated from Eq. (2-100).

CH3 CH2 CHar OH

N

X

XQ

Θ

ln γ

0 0 6 0 Sum 6

0 0 1 0

0 0 0.4 0 0.4

0 0 1 0

0.257 −0.728 0.000 2.270

Here Θ and ln γ are defined by the same equations shown above for the mixture groups. UNIFAC residual term: 2 gEr  = xi RT i=1

4

 N

m,i



(lnγm − lnγm,i) = 0.3425

m=1

where Nm and ln γm refer to the mixture and Nm,i and ln γm,i refer to the purecomponent values. Mixture volume:

Group fractions in the mixture: Group CH3 CH2 CHar OH

X

XQ

Θ

lnγ

0.1145 0.1145 0.6565 0.1145

0.097083 0.061822 0.262618 0.137382 0.558905

0.17370 0.11061 0.46988 0.24581

0.293 −0.873 0.066 1.077

N 0.5113 0.5113 2.9322 0.5113 Sum 4.4661

2 m3 m3 Vm = xiVi = 0.5113 0.05862  + 0.4887 0.08948  kmol kmol i=1





i=1



 Θ Ψ −   4

i

0.05862

 + 0.124 − 0.3425 = −0.4523

ΘiΨm,i

4

i,m

i=1



+ 0.4887 = 0.5113 ln 1.0774   0.07370 

0.08948 ln 0.5997  0.07370

and





m3 = 0.07370  kmol

η ln  mPa⋅s

ln γm = Qm 1 − ln



Using Eq. (2-99):

XmQm 4 Here Θm = 

XiQi

4



η = exp (− 0.4523) mPa⋅s = 0.636 mPa⋅s

ΘjΨ j,i

i=1

j=1

The estimated value is 6.6 percent below the reported experimental value of 0.681 mPa⋅s [Kouris, S., and C. Panayiotou, J. Chem. Eng. Data, 34 (1989): 200].

Group fractions in pure components: Ethanol Group CH3 CH2 CHar OH

N 1 1 0 1 Sum 3

X

XQ

Θ

0.3333 0.3333 0.0000 0.3333

0.283 0.180 0.000 0.400 0.863

0.3277 0.2087 0.0000 0.4637

ln γ 0.5306 −0.9405 0.2095 0.6179

THERMAL CONDUCTIVITY Thermal conductivity k is a measure of the rate at which heat conducts through the material and is defined as the proportionality constant in Fourier’s law of heat conduction that relates the gradient of temperature to the heat flux or flow per unit area. In SI units, it has the units of W/(m⋅K). The conduction mechanism in low-density media, gases,

2-510

PHYSICAL AND CHEMICAL DATA

is primarily via molecular collisions, and k increases with increasing temperature (increasing molecular velocity). In dense media such as liquids, the energy transfer by conduction occurs more efficiently through the attractive intermolecular potential wells between the molecules than through collisions. As a result, liquid thermal conductivity generally decreases with increasing temperature (except for water, aqueous solutions, and a few multihydroxy and multiamine compounds), corresponding to the decrease in density with increased temperature. Below or near the normal boiling point, the temperature dependence of liquid thermal conductivity is nearly linear for modest temperature ranges and can be represented by k = A − BT

monatomic

 1.7614 −  T

kM R  = 1.30 +  ηCv Cv

0.3523

(2-102)

linear molecules (2-103)

r



kM R  = 1.15 + 2.033  ηCv Cv

nonlinear molecules

(2-104)

Example Estimate the low-pressure thermal conductivity of toluene vapor at 500 K. Required properties from the DIPPR® 801 database: M = 92.138 gmol

η (500 K) = 1.1408 × 10 Pa⋅s −5

Cv = Cp − R = (170.78 − 8.314) J(mol⋅K) = 162.47 J(mol⋅K) Auxiliary quantities: Tr = 500591.75 = 0.845

RCv = (8.314)(162.47) = 0.0512

From Eq. (2-104):



(2-105)

0.215 + 0.28288α − 1.061β + 0.26665γ Ψ = 1 + α  0.6366 + βγ + 1.061αβ



C α = v − 1.5 R

(2-106)

β = 0.7862 − 0.7109ω + 1.3168ω2 (2-107)

γ = 2.0 + 10.5T2r

Example Estimate the low-pressue thermal conductivity of naphthalene vapor at 500 K. Required properties from the DIPPR® 801 database: Tc = 748.4 K

M = 128.17 gmol

ω = 0.30203

−5

η(500 K) = 1.0173 × 10 Pa⋅s Cv = Cp − R = (219.82 − 8.314) J(mol⋅K) = 211.51 J(mol⋅K) Auxiliary quantities [Eqs. (2-106) and(2-107)]: Tr = 500748.4 = 0.6681

RCv = (8.314)(211.51) = 0.0393

γ = 2.0 + (10.5)(0.6681) = 6.6866 2

α = (0.0393)−1 − 1.5 = 23.9388

β = 0.7862 − (0.7109)(0.30203) + (1.3168)(0.30203)2 = 0.6916 Ψ = 1 + 23.9388 ×

0.215 + 0.28288(23.9388) − 1.061(0.6916) + 0.26665(6.6866)

   0.6366 + 0.6916(6.6866) + 1.061(23.9388)(0.6916)

= 9.4273 From Eq. (2-105):

where η = viscosity at same conditions as desired for k. Note: Because this method is only applicable at low pressures, Cv may be calculated as Cpo − R, where Cpo is the ideal gas isobaric heat capacity.

Tc = 591.75 K



kM R  = 3.75 Ψ  ηCv Cv

(2-101)

where B is generally in the range of 1 × 10−4 to 3 × 10−4 W(m⋅K2). Gases Methods for estimating low-pressure gas thermal conductivities are based on kinetic theory and generally correlate the dimensionless group kMηCv (M = molecular weight, η = viscosity, Cv = isochoric heat capacity) known as the Eucken factor. The method of Stiel and Thodos is recommended for pure nonpolar compounds, and the method of Chung is recommended for pure polar compounds. Recommended Method Stiel-Thodos method. Reference: Stiel, L. I., and G. Thodos, AIChE J., 10 (1964): 26. Classification: Empirical extension of theory. Expected uncertainty: 15 percent. Applicability: Pure nonpolar gases at low pressure. Input data: M, Tc, η, and Cv. Description: The following equations may be used depending upon the molecular shape: kM  = 2.5 ηCv

Applicability: Pure organic gases at low pressure. Input data: Cv, ω, Tc, M, and η. Description: The following equations apply:







J (1.1408 × 10−5Pa⋅s) 162.47  mol⋅K k = [1.15 + (2.033)(0.0512)]  g 92.138  mol mW = 25.2  m⋅K

The estimated value is 18 percent below the DIPPR® 801 recommended value of 30.76 mW/(m⋅K).

Recommended Method Chung-Lee-Starling method. Reference: Chung, T.-H., L. L. Lee, and K. E. Starling, Ind. Eng. Chem. Fundam., 23 (1984): 8. Classification: Corresponding states. Expected uncertainty: 15 percent.

k = (3.75)(9.4273)







J (1.1408 × 10−5Pa⋅s) 8.314  mol⋅K  g 128.17  mol



mW = 23.33  m⋅K

The estimated value is 1.0 percent above the DIPPR® 801 recommended value of 23.09 mW/(m⋅K).

Liquids The Baroncini method provides accurate liquid thermal conductivity estimates if the compound clearly belongs to one of the specific families designated by the correlation. The Sastri-Rao method is recommended for other specific families as listed in the discussion of the method given below. As a general method for estimating thermal conductivity of pure liquids at ambient pressure for all other compounds, the Missenard method is recommended. Recommended Method Baroncini method. Reference: Baroncini, C., et al., Int. J. Thermophys., 2 (1981): 21. Classification: Empirical correlation. Expected uncertainty: 10 percent. Applicability: Particularly accurate for the following families: acetates, aliphatic ethers, halogenated compounds, dicarboxylic acids, ketones, aliphatic alcohols, aliphatic acids, propionates and butyrates, and unsaturated aliphatic esters. Input data: M, Tb (normal boiling point), Tc. Description: α

−β

−γ

   g/mol  K

k Tb  = A  W(m⋅K) K

M

Tc

(1 − Tr)0.38  T16 r

where A, α, β, and γ are obtained from Table 2-358.

(2-108)

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES From Eq. (2-108):

TABLE 2-358 Correlation Parameters for Baroncini et al. Method* for Estimation of Thermal Conductivity Family

A







Saturated hydrocarbons Olefins Cycloparaffins Aromatics Alcohols Organic acids Ketones Esters Ethers Refrigerants R20, R21, R22, R23 Others

0.00350 0.0361 0.0310 0.0346 0.00339 0.00319 0.00383 0.0415 0.0385

1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

0.5 1 1 1 0.5 0.5 0.5 1 1

0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167

0.562 0.494

0 0

0.5 0.5

−0.167 −0.167

*

Baroncini, C., et al., Int. J. Thermophys., 2 (1981): 21.

Example Estimate the thermal conductivity of liquid p-cresol at 400 K. Required properties from DIPPR® 801 database: M = 108.1378 gmol

Tc = 704.65 K

Tb = 475.133 K

Auxiliary properties:

k  = (0.00339)(475.13)1.2(108.1378)−12 (704.65)−0.167 W(m⋅K) (1 − 0.5677)0.38  = 0.142 0.567716 The estimated value is 7.6 percent higher than the DIPPR® 801 recommended value of 0.132 W/(m⋅K).

Recommended Method Sastri-Rao method. Reference: Sastri, S. R. S., and K. K. Rao, Chem. Eng. J., 74 (1999): 161. Classification: Group contributions and corresponding states. Expected uncertainty: 10 percent. Applicability: Particularly accurate for the following families: alkanes, alkenes, aromatic alcohols, cycloalkanes, epoxides, aliphatic acids, condensed rings. Input data: Tc , Tb (normal boiling point), and molecular structure. Description: The thermal conductivity at the normal boiling point kb is calculated from the sum of the group contributions given in Table 2-359. This value is then scaled to the desired reduced temperature by using a factor αβ in which the power β is related to the fractional reduced temperature distance between the normal boiling point and the critical point as shown in the following equations: N

Tr = TTc = (400 K)(704.65 K) = 0.5677

kb = ni ∆ki i=1

From Table 2-358 for alcohols: A = 0.00339

α = 1.2

1 β=  2

1 − Tr β=1−  1 − Tbr



Group

ki (W/mK) 0.0545 −0.0008 −0.0600 −0.1230 0.0545 0.0020 −0.0630 0.1200 0.1130 0.0100 0.0830 0.0680 0.0175 0.0730 0.0070

Group

ki (W/mK)

COOH NH2 NH NH ring >N >N ring CN NO2 S F perfluoro F other Cl Br I Hc 3-ring other ringd

0.0650 0.0880 0.0065 0.0450 −0.0605 0.0135 0.0645 0.0700 0.0100 0.0568 0.0510 0.0550 0.0415 0.0245 0.0675 0.1500 0.1100

Corrections for multigroup interactions Hydrocarbons with 4 or fewer carbon atoms Single CH3 group + nonhydrocarbon groups other than COOH, Br, I f Two hydrocarbon + nonhydrocarbon groups other than COOH, Br, I f Unsaturated aliphatic compounds with three hydrocarbon groups Special groups Cl(CH2)nCl More than one nonhydrocarbon group with hydrocarbon groups Nonhydrocarbon groups but no hydrocarbon groups a

0.0150(5 − nC) 0.0600 0.0285 0.0285 0.0350 0.0095 0.1165

e

All rings are treated as separate rings in polycyclic compounds. Used only for aliphatic primary alcohols and phenols having no branch chains. Used in methane, formic acid, formates, etc. d Used for all rings in polycyclic compounds with at least one nonhydrocarbon ring. e The number of carbon atoms is nC. f Aliphatic nonhydrocarbon liquids such as methylformate, aceticanhydride, and ethylformate, having more than one type of nonhydrocarbon group and (1) one or two methyl groups or (2) one ethyl group only, require two correction factors. One is due to the hydrocarbon groups, and one is due to the presence of more than one type of nonhydrocarbon group *SOURCE: Sastri, S. R. S., and K. K. Rao, Chem. Eng. J., 74 (1999): 161. b c

γ



k = kbαβ

(2-109)

where γ = 1.23 for alcohols and 0.2 for all other compounds and α = 0.856 for alcohols and 0.16 for all other compounds.

γ = 0.167

TABLE 2-359 Sastri-Rao* Group Contributions for Liquid Thermal Conductivity at the Normal Boiling Point Hydrocarbons CH3 CH2 >CH >C< 苷CH2 苷CH 苷C< 苷C苷 ringa Nonhydrocarbons O OH primaryb OH other >CO >CHO COO

2-511

2-512

PHYSICAL AND CHEMICAL DATA

Example Estimate the thermal conductivity of liquid methyl formate at 323 K. Molecular structure: O

O



ki

ni

ni ki

CH3 1 0.0545 COO 1 0.0070 H 1 0.0675 Corrections for multigroup interactions Single CH3 group + nonhydrocarbon groups More than one nonhydrocarbon with hydrocarbon groups

0.0545 0.0070 0.0675 0.0600 0.0095 Total

0.1985

Required input properties from DIPPR® 801 recommended values: Tc = 487.2 K

Tb = 304.9 K

Tr = TTc = 323487.2 = 0.6630

Tbr = 304.9487.2 = 0.6258

From Eq. (2-109): α = 0.16

γ = 0.2

1 − 0.633 β = 1 −  1 − 0.6257





0.2

(nonalcohol values)

= 0.0207

kb = 0.1985 W(m⋅K)

k = [0.1985 W(m⋅K)] (0.16)0.0207 = 0.1911 W(m⋅K) The estimated value is 10.2 percent above the DIPPR® 801 recommended value of 0.1734 W/(m⋅K).

Recommended Method Missenard method. Reference: Missenard, A., Comptes Rendus, 260 (1965): 5521. Classification: Corresponding states. Expected uncertainty: 20 percent. Applicability: Organic compounds; nonassociating. Input data: Tc, m (number of atoms in molecule), ρ273 (liquid density at 273.15 K), Tb, M, Cp,273 (liquid heat capacity at 273.15 K). Description: 8.4 k273 T ⋅ b  =  m14 mW(m⋅K) K

ρ273

−12

   g/m  g/mol



12

12

M

3

Cp,273 ×  J(mol⋅K)



mW 141.3  [3 + 20(1 − 0.5673)23] m⋅K k273[3 + 20(1 − Tr)23] k =  =  3 + 20 (1 − 0.4425)23 3 + 20(1 − Tr,273)23

CH3

Group contributions from Table 2-359: Group

From Eq. (2-111):

mW = 123.3  m⋅K The estimated value is 4.5 percent above the DIPPR® 801 recommended value of 118.0 mW/(m⋅K).

Liquid Mixtures The thermal conductivity of liquid mixtures generally shows a modest negative deviation from a linear mass-fraction-weighted average of the pure-component values. Although more complex methods with some improved accuracy are available, two simple methods are recommended here that require very little additional information. The first method applies only to binary mixtures while the second can be used for multiple components. Recommended Method Filippov correlation. References: Filippov, L. P., Vest. Mosk. Univ., Ser. Fiz. Mat. Estestv. Nauk, 10 (1955): 67; Filippov, L. P., and N. S. Novoselova, Sugden, Vest. Mosk. Univ., Ser. Fiz. Mat. Estestv. Nauk, 10 (1955): 37. Classification: Empirical correlation. Expected uncertainty: 4 to 8 percent. Applicability: Binary liquid mixtures. Input data: Pure-component thermal conductivities ki at mixture conditions; wi. Description: The mixture thermal conductivity is calculated from the pure-component values using k = w1k1 + w2k2 − 0.72w1w2 k2 − k1

where wi is the mass fraction of pure fluid i. Recommended Method Li correlation. References: Li, C. C., AIChE J., 22 (1976): 927. Classification: Empirical correlation. Expected uncertainty: 4 to 8 percent. Applicability: Liquid mixtures. Input data: Pure-component thermal conductivities ki at mixture conditions; ρL,i Description: The mixture thermal conductivity is correlated as a function of the mixture volume fractions φi: C

k=

(2-110)



i =1

k273[3 + 20 (1 − Tr)23] k =  3 + 20 (1 − Tr,273)23

(2-112)

C

2ki kj

φφ 

k +k j =1

(2-113)

i j

i

j

−1 xiρL,i where φi =  C

xjρ−1L,j

(2-111)

j=1

where Tr,273 = (273 K)/Tc. Example Estimate the thermal conductivity of m-xylene at 350 K. ®

Required properties from DIPPR 801 database: Tc = 617 K

m = 18

ρ273 = 7.6812 kmolm3

Tb = 412.27 K

M = 106.165 kgkmol

Cp,273 = 200.64 kJ/(kmol⋅K)

Auxiliary properties: Tr = 350617 = 0.5673

Tbr = 412.27617 = 0.6682

Tr,273 = 273617 = 0.4425

Example Estimate the thermal conductivity of a mixture containing 30.2 mol % diethyl ether(1) and 69.8 mol % methanol(2) at 273.15 K and 0.1 MPa, using the Filippov and Li correlations. Auxiliary data: The pure-component thermal conductivities and molar densities at 273.15 K recommended in the DIPPR® 801 database are k1 = 0.1383 W(m⋅K)

ρ1 = 9.9335 kmolm3

M1 = 74.1216 kgkmol

k2 = 0.2069 W(m⋅K)

ρ2 = 25.371 kmolm

M2 = 32.0419 kgkmol

3

The mass fractions corresponding to the mole fractions given above are w1 = 0.5

From Eq. (2-110):

w2 = 0.5

The volume fractions are  = (8.4)(412.27)  mW/(m⋅K) k273

(0.007681)12(106.165)−12(200.64)(18)−0.25

12

= 141.3

(0.302)(9.9335)−1 = 0.525 φ1 =  (0.302)(9.9335)−1 + (0.698)(25.371)−1

φ2 = 0.475

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Calculation using Eq. (2-112): W k = [(0.5)(0.1383) + (0.5)(0.2069) − (0.72)(0.5)(0.5)0.2069 − 0.1383]  m⋅K = 0.160 W(m⋅K)

Input Data: ρL, molecular structure, and Table 2-360. Description: Equation (2-114) is used with P calculated from N

P = ni ∆Pi

(2-115)

i =1

Calculation using Eq. (2-113):

2-513

Group values for the parachor are given in Table 2-360.



(0.525)(0.475)(2)(0.1383)(0.2069) k = (0.525)2(0.1383) + 2⋅  0.1383 + 0.2069

Example Estimate the surface tension of ethylacetylene at 237.45 K. Structure:



W + (0.475)2(0.2069)  m⋅K

HC

CH3

C

= 0.167 W(m⋅K) The Filippov value is 7.5 percent lower than the experimental value of 0.173 W/(m⋅K) [Jamieson, D. T., and B. K. Hastings, Thermal Conductivity, Proceedings of the Eighth Conference, C. Y. Ho and R. E. Taylor, eds., Plenum Press, New York, 1969]; the Li value is 3.5 percent lower than the experimental value.

Group

ni

∆Pi

ni ∆Pi

 CH  C >CH2 (n = 1–11) CH3

1 1 1 1

43.64 28.64 39.92 55.25

43.64 28.64 39.92 55.25

Solids There is no reliable method for estimating solid thermal conductivity at this time. SURFACE TENSION The surface layer at a vapor-liquid interface is in tension and will contract to minimize the surface area. Qualitatively, the surface tension is due to the larger attractive forces that molecules at the interface experience from molecules in the dense liquid phase than from those in the low-density gas phase. Quantitatively, surface tension is defined as the force in the surface plane per unit length. Jasper [Jasper, J. J., J. Phys. Chem. Ref. Data, 1 (1972): 841] has made a critical evaluation of experimental surface tension data for approximately 2200 pure chemicals and correlated surface tension σ (mN/m = dyn/cm) with temperature as σ = A − BT Jasper’s evaluation also includes values of A and B for most of the tabulated chemicals. Surface tension decreases with increasing temperature and increasing pressure. Pure Liquids An approach suggested by Macleod [Macleod, D. B., Trans. Faraday Soc., 19 (1923): 38] and modified by Sugden [Sugden, S. J., Chem. Soc., 125 (1924): 32] relates σ to the liquid and vapor molar densities and a temperature-independent parameter called the parachor P σ ρL − ρV  = P⋅  mN/m 103 kmol/m3

 



4

(2-114)

where ρL and ρV are the saturated molar liquid and vapor densities, respectively. At low temperatures, where ρL >> ρV, the vapor density can be neglected, but at higher temperatures the density of both phases must be calculated. At the critical point the surface tension is zero as ρL = ρV. Quayle [Quayle, O. R., Chem. Rev., 53 (1953): 439] proposed a group contribution method for estimating P that has been improved in recent years by Knotts et al. [ Knotts, T. A., et. al., J. Chem. Eng. Data, 46 (2001): 1007]. This method using P is recommended when groups are available; otherwise, the Brock-Bird [Brock, J. R., and R. B. Bird, AIChE J., 1 (1955): 174] corresponding-states method as modified by Miller [Miller, D. G., Ind. Eng. Chem. Fundam., 2 (1963): 78] may be used to estimate surface tension for compounds that are not strongly polar or associating. Recommended Method Parachor method. References: Macleod, D. B., Trans. Faraday Soc., 19 (1923): 38; Sugden, S. J., Chem. Soc., 125 (1924): 32; Knotts, T. A., et al., J. Chem. Eng. Data, 46 (2001): 1007. Classification: Group contributions and QSPR. Expected uncertainty: 4 percent. Applicability: Organic compounds for which group values are available.

Total

167.45

Required properties: The DIPPR® 801 database gives ρL = 13.2573 kmol/m3 at 237.45 K. Calculation using Eq. (2-114):





13.2573 σ = (167.45)  1000



4

mN N  = 0.02429  m m

The estimated value is 0.9 percent above the DIPPR® 801 recommended value of 0.02407 N/m.

Recommended Method Brock-Bird method. Reference: Brock, J. R., and R. B. Bird, AIChE J., 1 (1955): 174; Miller, D. G., Ind. Eng. Chem. Fundam., 2 (1963): 78. Classification: Corresponding states. Expected uncertainty: 5 percent. Applicability: Nonpolar and moderately polar organic compounds. Input data: Tc, Pc, and Tb. Description: σ Pc  = (5.553 × 10−5)  mN/m Pa

  K 2/3

Tc

1/3

F(1 − Tr)11/9

(2-116)

where Tbr [ln(Pc /Pa) − 11.5261] F =  − 1.3281 1 − Tbr

(2-117)

Example Estimate the surface tension for ethyl mercaptan at 303.15 K. Required properties from DIPPR® 801: Tc = 499.15 K

Pc = 5.49 × 106 Pa

Tb = 308.15 K

Supporting quantities: Tr = (303.15 K)/(499.15 K) = 0.6073 Tbr = (308.15 K)/(499.15 K) = 0.6173 F = {0.6173 [ln (5.49 × 106) − 11.5261]/(1 − 0.6173)} − 1.3281 = 5.113 [from Eq. (2-117)] From Eq. (2-116): σ = (5.553 × 10−5)(5.49 × 106)2/3(499.15)1/3(5.113)(1 − 0.6073)11/9 mN/m = 22.36 mN/m The estimated value is 1.4 percent lower than the DIPPR® 801 recommended value of 22.68 mN/m.

2-514

PHYSICAL AND CHEMICAL DATA TABLE 2-360

Knotts* Group Contributions for the Parachor in Estimating Surface Tension

Group (a) Nonring C CH3 >CH2 (n = 1–11) >CH2 (n = 12–20) >CH2 (n > 20) >CH >C< 苷CH2 苷CH 苷C< 苷C苷 CH C Branch corrections Per branch sec-sec adjacency sec-tert adjacency tert-tert adjacency (b) Nonaromatic ring C CH2 >CH >C< 苷CH 苷C< >CH (fused ring) Ring corrections Three-member ring Four-member ring Five-member ring Six-member ring Seven-member ring (c) Aromatic ring C >CH >C C (fused arom/arom) C (fused arom/aliph) Arom ring corr ortho para meta subst. naphthalene corr (d) Oxygen groups OH (alc, primary) OH (alc, sec) OH (alc, tertiary) OH (phenol) O (nonring) O (ring) O (aromatic) >C苷O (nonring) >C苷O (ring) O苷CH (aldehyde) CHOOH (formic) COOH (acid) OCHO (formate) COO (ester) COOCO (acid anhyd) OC(苷O)O (ring)

Pi 55.25 39.92 40.11 40.51 28.90 15.76 49.76 34.57 24.50 24.76 43.64 28.64 −6.02 −2.73 −3.61 −6.10 39.21 23.94 7.19 34.07 18.85 22.05 12.67 15.76 7.04 5.19 3.00 34.36 16.07 19.73 14.41 −0.60 3.40 2.24 −7.07 31.42 22.68 20.66 30.32 20.61 21.67 23.54 47.02 50.04 66.06 94.01 74.57 82.29 64.97 115.07 84.05

Pi

Group (e) Nitrogen groups RNH2 (primary R) RNH2 (sec R) RNH2 (tert R) ANH2 (attached to arom ring) >NH (nonring) >NH (ring) >NH (in arom ring) >N- (nonring) >N- (ring) N苷 (nonring) >N (aromatic) HCN (hyd cyanide) CN CN (aromatic) (f) Nitrogen and oxygen groups C苷ONH2 (amides) C苷ONH- (amides) C苷ON< (amides) NHCHO >NCHO N苷O NO2 NO2 (aromatic) (g) Sulfur groups R-SH (primary R) R-SH (sec R) R-SH (tert R) SH (aromatic) S (nonring) S (ring) S (aromatic) >S苷O (nonring) >SO2 (nonring) >SO2 (ring) (h) Halogen groups F Cl Br I F (aromatic) Cl (aromatic) Br (aromatic) I (aromatic) (i) Si groups SiH4 >SiH >Si< >Si< (ring) (j) Other inorganic groups PO4 >P >B >Al ClO3

44.98 44.63 46.44 46.53 29.04 31.97 33.92 10.77 15.71 23.24 26.49 80.94 65.23 67.54 93.43 73.64 57.05 91.69 77.12 64.32 73.86 75.05 66.89 63.34 65.33 68.30 51.37 51.75 51.47 72.21 93.20 90.13 21.81 26.24 51.16 54.56 66.30 70.39 90.84 92.04 105.11 54.50 44.93 28.64 115.59 48.84 22.65 25.06 106.03

*Knotts, T. A., et al., J. Chem. Eng. Data, 46 (2001): 1007.

Liquid Mixtures Compositions at the liquid-vapor interface are not the same as in the bulk liquid, and so simple (bulk) compositionweighted averages of the pure-fluid values do not provide quantitative estimates of the surface tension at the vapor-liquid interface of a mixture. The behavior of aqueous mixtures is more difficult to correlate and estimate than that of nonpolar mixtures because small amounts of organic material can have a pronounced effect upon the surface concentrations and the resultant surface tension. These effects are usually modeled with thermodynamic methods that account for the activity coefficients. For example, a UNIFAC method [Suarez, J. T. C. TorresMarchal, and P. Rasmussen, Chem. Eng. Sci., 44 (1989): 782] is recommended and illustrated in PGL5. For nonaqueous systems the extension of the parachor method, used above for pure fluids, is a simple and reasonably effective method for estimating σ for mixtures.

Recommended Method Parachor correlation. Reference: Hugill, J. A., and A. J. van Welsenes, Fluid Phase Equil., 29 (1986): 383; Macleod, D. B., Trans. Faraday Soc., 19 (1923): 38; Sugden, S. J., Chem. Soc., 125 (1924). Classification: Corresponding states. Expected uncertainty: 3 to 10 percent. Applicability: Nonaqueous mixtures. Input data: Liquid and vapor ρ at mixture T; parachors of pure components; xi. Description: σm ρL,m ρV,m − PV,m   = PL,m  mN/m 103 kmol/m3 103 kmol/m3





4

(2-118)

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES where σm = surface tension of the mixture PL,m = parachor of liquid mixture PV,m = parachor of vapor mixture ρL,m = liquid mixture molar density ρV,m = vapor mixture molar density The following definitions are used for the liquid and vapor mixture parachors: 1 PL,m =  2

C

C

i j

i

C

1 PV,m =  2



x x (P + P ) i =1 j =1 j

C



y y (P + P )(2-119) i=1 j =1 i j

i

j

where xi is the mole fraction of component i in the liquid and yi is the mole fraction of component i in the vapor. Note that ρV is generally very small compared to ρL at temperatures substantially lower than Tc and can often be neglected. Example Estimate the surface tension for a 16.06 mol % n-pentane(1) + 83.94 mol % dichloromethane(2) mixture at 298.15 K.

The autoignition temperature is the minimum temperature for a substance to initiate self-combustion in air in the absence of an ignition source. Recommended Methods Flash point: Thornton method. Reference: As described by N. Y. Shebeko, A. V. Ivanov, and E. N. Alekhina, “Calculation of Flash Points and Ignition Temperatures of Organic Compounds,” Soviet Chem. Ind. 16(1984):1371. Classification: Atomic contributions. Expected uncertainty: 5 K. Applicability: Organic compounds. Input data: Number of carbon, hydrogen, sulfur, halogen, and oxygen atoms; vapor pressure correlation. Description: A simple atom contribution method is given by Psys P* −  =0 1 + 4.76(2β − 1)

P

ρL/(kmol·m ) at 298.15 K

231.1 146.6

8.6173 15.5211

NH − NX NO β = NC + NS +  −  4 2

−3

Mixture parachor from Eq. (2-119) and mixture density: PL,m = (0.1606)2(231.1) + (0.1606)(0.8394)(231.1 + 146.6) + (0.8394)2(146.6) = 160.17 ρL,m =

C

−1

−1

+   ρ =   8.6173 15.5211 i=1

xi

0.1606

i

0.8394

kmol kmol = 13.752   m3 m3

Calculation using Eq. (2-118): Because the temperature is low, the density of the vapor can be neglected, and σm mN  = [(160.17)(0.013752)]4 = 23.54  m mN/m

(2-120)

where P* = vapor pressure at flash point temperature Psys = total system pressure, typically 1.01325 × 105 Pa β = stoichiometric coefficient, defined by

Required properties from DIPPR® 801:

n-Pentane Dichloromethane

2-515

(2-121)

where NC = number of carbon atoms in compound NS = number of sulfur atoms in compound NH = number of hydrogen atoms in compound NX = number of halogen atoms in compound NO = number of oxygen atoms in compound Procedure: Step 1. Determine the number of carbon, sulfur, hydrogen, halogen, and oxygen atoms in the compound. Step 2. Calculate β from Eq. (2-121). Step 3. Substitute a temperature-dependent expression for vapor pressure in Eq. (2-120) for P*. Step 4. Solve Eq. (2-120) for temperature. This temperature is the flash point estimate. Example Estimate the flash point of phenol. Structure:

OH

The estimated value is 2.9 percent below the experimental value of 24.24 mN/m reported by De Soria et al. [De Soria, M. L. G., et al., J. Colloid Interface Sci., 103 (1985): 354].

FLAMMABILITY PROPERTIES Flash points, lower and upper flammability limits, and autoignition temperature are important properties for determining safe operating limits when processing organic compounds. As with any property, experimental values are preferable to predicted values, and prediction techniques for these properties are only modestly accurate. The flash point is the lowest temperature at which a liquid gives off sufficient vapor to form an ignitable mixture with air near the surface of the liquid or within the vessel used. ASTM test methods include procedures using a closed-cup apparatus (ASTM D 56, ASTM D 93, and ASTM D 3828), which is preferred, and an open-cup apparatus (ASTM D 92 and ASTM D 1310). Closed-cup values are typically lower than open-cup values. When several values are available, the lowest reasonable temperature is usually accepted in order to ensure safe operations. The lower and upper flammability limits are the boundary-line equilibrium mixtures of vapor or gas with air, which if ignited will just propagate a flame away from the ignition source. Each of these limits has a temperature at which the flammability limits are reached. The lower flammability limit temperature corresponds approximately to the flash point, but since the flash point is determined with downward flame propagation and nonuniform mixtures and the lower flammability temperature is determined with upward flame propagation and unifrom vapor mixtures, the measured lower flammability temperature is often somewhat lower than the flash point.

Atomic contributions: Atom type

Number

C H O

6 6 1 1 6 β=6+  −  =7 4 2

The DIPPR® 801 correlation for the vapor pressure of phenol is 10,113 P* = exp(95.444 −  − 10.09 ln T + 6.7603 × 10−18 T 6) T When this expression is used in Eq. (2-120) and solved for temperature, one obtains TFP = 348.7 K, which is 1.2 percent below the DIPRR recommended value of 353 K.

Recommended Method Flammability limits: Pintar method. Reference: Pintar, A. J., Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. Classification: Group contributions. Expected uncertainty: 25 percent.

2-516

PHYSICAL AND CHEMICAL DATA TABLE 2-361 Group Contributions for Pintar* Flammability Limits Method for Organic Compounds LFLi

UFLi

Group

LFLi

UFLi

17.2750 13.7022 10.0000 5.5291 2.7250 2.1797 −3.0156 −6.0312 4.6752 10.3801 4.8890 1.2955 4.6740 73.8338 57.4447 57.4447 57.4447 45.0633 4.2821 17.5470 −2.9697 −5.9764 −8.0982 −1.2615 −2.1224 −5.1300 −8.0405 −16.0809 −21.9000 −22.0000 −44.0000 −60.0000 3.7078

3.8461 1.4959 0.2183 −0.8422 2.8206 0.5856 −2.2427 −4.4854 0.6009 −1.2148 1.6121 −1.1840 2.4751 9.6661 7.6126 7.2450 7.9291 −5.9925 2.0269 0.7842 1.4008 3.1943 4.2024 0.3984 0.6847 1.1952 4.0018 8.0036 12.0054 11.4300 22.8600 34.2900 1.8802

>NH N苷 CN 苷C苷N 苷NNH2 >NNH2 NO2 SH S SO SO2 SO3 SO4 CO3 OPO2 P苷 PO PO4苷 SiC† SiO† SiH† SiCl† SiN† SiSi Al B Cr Na cis trans Nonarom⋅ring Add’l⋅ring

3.2709 −7.2149 8.0990 2.5963 −3.5071 0.5861 −3.1507 7.9424 11.0079 3.9115 5.5400 2.8600 0.1800 2.4103 7.1419 47.6909 7.1515 −11.5096 −2.2855 2.5034 8.3130 4.1010 15.8960 — — 47.3806 — — −6.8350 0.5821 2.9082 14.2712

−1.9112 −2.3309 3.6918 −1.1463 −0.29897 2.4811 0.8011 0.5344 −1.9832 −4.1834 — — — −3.4894 — — — −6.0260 −3.0576 1.4282 −24.4160 9.7131 1.6577 — — — — — 1.8040 0.9183 3.7760 3.1127

Group CH3 >CH2 >CH >C< H OH O OO 苷C苷O CHO COOH COO COOCO C6H5 m-C6H4 o-C6H4 p-C6H4 Arom. ring 苷  Cl Cl2 Cl3 F F2 F3 Br Br2 Br3 I I2 I3 NH2

*Pintar, A. J., Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. †Does not include contribution of atoms attached to silicon. TABLE 2-362 Group Contributions for Pintar* Flammability Limits Method for Inorganic Compounds Group B Br C Cl F Fe H N Na Ni

LFLi

UFLi

Group

LFLi

UFLi

24.5190 — 32.0745 14.2658 — — 10.3452 −24.5487 — —

−1.3818 — −0.6259 −1.4231 — — 0.6500 1.8453 — —

>NNH2 O P S SiC† SiO† SiH† SiCl† SiN† SiSi

0.5861 −24.3242 24.8302 26.6776 −2.2855 2.5034 8.3130 4.1010 15.8960 —

−0.2990 2.2990 — 1.3171 −3.0576 1.4282 −24.4160 9.7131 1.6577 —

*Pintar, A. J., Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. †Does not include contribution of atoms attached to silicon.

Applicability: Organic and inorganic compounds. Input data: Group contributions from Tables 2-361 and 2-362. Description: A simple GC method with first-order contributions with lower flammability limit (LFL) and upper flammability limit (UFL) in volume % given by

Structure:

CH3

Group contributions:

100% LFL = 

niLFLi

(2-122)

100% UFL = 

niUFLi

(2-123)

where ni = number of groups of type i in molecule LFLi = contribution of group i to LFL UFLi = contribution of group i to UFL Example Estimate the lower and upper flammability limits of toluene.

Group

LFL

UFL

CH3 C6H5

17.2750 73.8338

3.8461 9.6661

Calculations using Eqs. (2-122) and (2-123): 100% LFL =  = 1.10% 17.2750 + 73.8338 100% UFL =  = 7.40% 3.8461 + 9.6661

PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES

2-517

TABLE 2-363 Group Contributions for Pintar*Autoignition Temperature Method for Organic Compounds Group

bi

CH3 >CH2 >CH >C< H OH O OO 苷C苷O CHO COOH COO COOCO C6H5 mC6H4 oC6H4 pC6H4 Aromatic ring 苷  Cl Cl2

301.91 −10.86 −275.17 −570.43 391.48 324.10 −18.60 −397.61 57.65 195.20 370.75 43.90 46.11 380.27 153.15 77.48 99.87 −1339.65 578.72 1116.50 347.39 726.03

Group Cl3 F F2 F3 Br Br2 Br3 I I2 I3 NH2 >NH N苷 CN 苷C苷N 苷NNH2 >NNH2 NO2 SH S SO SO2

bi

Group

bi

1073.47 360.60 755.54 1082.00 420.96 607.69 1260.00 310.53 — — 354.11 9.88 −249.91 469.67 −273.70 378.27 −215.02 292.57 273.84 −60.75 −91.10 —

SO3 SO4 CO3 P苷 PO OPO2 PO4苷 SiC† SiO† SiH† SiCl† SiN† SiSi Al B Cr Na cis trans Nonarom.ring Add’l.ring Zn

— −31.71 442.26 −334.91 −549.59 — −329.45 −147.69 −136.99 −310.52 −200.88 — — — — — 534.29 −29.19 −38.31 605.97 565.11 349.02

*Pintar, A. J., Estimation of Autoignition Temperature, Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. †Does not include contribution of atoms attached to silicon.

TABLE 2-364 Group Contributions for Pintar*Autoignition Temperature Method for Inorganic Compounds Group

bi

Group

bi

Group

bi

B Br C Cl F Fe H

−457.14 — 489.19 395.42 — −2050.90 204.55

N Na Ni >NNH2 O P S

0.71 — −1595.10 −215.02 −13.39 108.45 −3.57

SiC† SiO† SiH† SiCl† SiN† SiSi

−147.69 −136.99 −310.52 −200.88 — —

*Pintar, A. J., Estimation of Autoignition Temperature, Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. †Does not include contribution of atoms attached to silicon.

The values recommended in the DIPPR® 801 database are 1.2 and 7.1 percent, respectively. Flammability temperatures are found by determining the temperature at which the vapor pressure equals the partial pressure corresponding to the LFL or UFL.

Recommended Methods Autoignition temperature: Pintar method. Reference: Pintar, A. J., Estimation of Autoignition Temperature, Technical Support Document DIPPR Project 912, Michigan Technological University, Houghton, 1996. Classification: Group contributions. Expected uncertainty: 25 percent. Applicability: Organic and inorganic compounds. Input data: Group contributions from Tables 2-363 and 2-364. Description: A simple GC method with first-order contributions is given by AIT = ni bi

(2-124)

where ni is the number of groups of type i in the molecule and bi is the contribution of group i to the autoignition temperature. A more accurate but somewhat more complicated logarithmic GC method was also developed by Pintar in the same reference cited here.

Example Estimate the autoignition temperature of 2,3-dimethylpentane. Structure and group information:

H3C

CH3

H3C

CH3

Group

ni

CH3 >CH2 >CH

4 1 2

bi 301.91 −10.86 −275.17

Calculation using Eq. (2-124): AIT = 4(301.91) − 10.86 + 2(−275.17) = 646.4 K The estimated value is 6.3 percent above the DIPPR® 801 recommended value of 608.15 K.

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Section 3

Mathematics

Bruce A. Finlayson, Ph.D. Rehnberg Professor, Department of Chemical Engineering, University of Washington; Member, National Academy of Engineering (Section Editor, numerical methods and all general material) Lorenz T. Biegler, Ph.D. Bayer Professor of Chemical Engineering, Carnegie Mellon University (Optimization)

MATHEMATICS General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Mathematical Constants. . . . . . . . . . . . . . . . . . . . . . . . . . The Real-Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-3 3-4 3-4 3-5

INFINITE SERIES Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations with Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests for Convergence and Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . Series Summation and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-25 3-25 3-26 3-26

MENSURATION FORMULAS Plane Geometric Figures with Straight Boundaries . . . . . . . . . . . . . . . . Plane Geometric Figures with Curved Boundaries . . . . . . . . . . . . . . . . Solid Geometric Figures with Plane Boundaries . . . . . . . . . . . . . . . . . . Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irregular Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-6 3-6 3-7 3-7 3-8 3-8

COMPLEX VARIABLES Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Functions (Analytic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-27 3-27 3-27 3-27 3-27 3-28

ELEMENTARY ALGEBRA Operations on Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations, Combinations, and Probability. . . . . . . . . . . . . . . . . . . . . Theory of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-8 3-9 3-9 3-10 3-10

DIFFERENTIAL EQUATIONS Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations of the First Order . . . . . . . . . . . . . . . . Ordinary Differential Equations of Higher Order . . . . . . . . . . . . . . . . . Special Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-29 3-30 3-30 3-31 3-32

ANALYTIC GEOMETRY Plane Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Analytic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-11 3-13

DIFFERENCE EQUATIONS Elements of the Calculus of Finite Differences . . . . . . . . . . . . . . . . . . . Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-34 3-34

PLANE TRIGONOMETRY Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Circular Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations between Angles and Sides of Triangles . . . . . . . . . . . . . . . . . . Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations for Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .

3-16 3-16 3-17 3-17 3-18 3-18

INTEGRAL EQUATIONS Classification of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-36 3-36 3-37

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-37 3-39 3-39 3-39 3-39

DIFFERENTIAL AND INTEGRAL CALCULUS Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Calculus Applied to Thermodynamics . . . . . . . . . . . . . . . Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-18 3-21 3-22

3-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

3-2

MATHEMATICS

MATRIX ALGEBRA AND MATRIX COMPUTATIONS Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-40 3-41

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS Approximation Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-43

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Nonlinear Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods for Multiple Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . Interpolation and Finite Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Integration (Quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Ordinary Differential Equations as Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations-Boundary Value Problems . . . . . . . . . Numerical Solution of Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Partial Differential Equations. . . . . . . . . . . . . . . Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-43 3-44 3-44 3-44 3-45 3-47 3-47 3-48 3-51 3-54 3-54 3-54 3-59

OPTIMIZATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gradient-Based Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . Optimization Methods without Derivatives . . . . . . . . . . . . . . . . . . . . . . Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixed Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Optimization Models . . . . . . . . . . . . . . . . . . . . . . . . . .

3-60 3-60 3-65 3-66 3-67 3-70

STATISTICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enumeration Data and Probability Distributions . . . . . . . . . . . . . . . . . . Measurement Data and Sampling Densities. . . . . . . . . . . . . . . . . . . . . . Tests of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factorial Design of Experiments and Analysis of Variance . . . . . . . . . .

3-70 3-72 3-73 3-78 3-84 3-86 3-86

DIMENSIONAL ANALYSIS PROCESS SIMULATION Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Modules or Blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commercial Packages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-89 3-89 3-89 3-90 3-90

GENERAL REFERENCES: Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C. (1972); Finlayson, B.A., Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York (1980), Ravenna Park, Seattle (2003); Jeffrey, A., Mathematics for Engineers and Scientists, Chapman & Hall/CRC, New York (2004); Jeffrey, A., Essentials of Engineering Mathematics, 2d ed., Chapman &

Hall/CRC, New York (2004); Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, 2d ed., CRC Press, New York (2002); Wrede, R. C., and Murray R. Spiegel, Schaum's Outline of Theory and Problems of Advanced Calculus, 2d ed., McGraw-Hill, New York (2006); Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 1st ed., CRC Press, New York (2002); http:// eqworld.ipmnet.ru/.

MATHEMATICS GENERAL The basic problems of the sciences and engineering fall broadly into three categories: 1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems and stability problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. The mathematical treatment of engineering problems involves four basic steps: 1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process. 2. Solution. Appropriate mathematical and numerical operations are accomplished so that logical deductions may be drawn from the mathematical model. 3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world. 4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed. The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form Input of x – output of x  production of x = accumulation of x or Rate of input of x  rate of output of x  rate of production of x = rate of accumulation of x

FIG. 3-1

Boundary conditions.

satisfy the differential equation inside the region and the prescribed conditions on the boundary. In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions. The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”). Prototypes are: 1. Elliptic. Laplace’s equation ∂2u ∂2u 2 + 2 = 0 ∂x ∂y Poisson’s equation ∂2u ∂2u 2 + 2 = g(x,y) ∂x ∂y These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of g(x, y). Steady heattransfer and mass-transfer problems are elliptic. 2. Parabolic. The heat equation ∂u ∂2u ∂2u  = 2 + 2 ∂t ∂x ∂y describes unsteady or propagation states of diffusion as well as heat transfer. 3. Hyperbolic. The wave equation ∂2u ∂2u ∂2u = 2 + 2  ∂t2 ∂x ∂y describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear.

where x mass, energy, etc. These statements may be abbreviated by the statement Input − output + production = accumulation Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. For steady state or boundary-value problems (Fig. 3-1) the solution must

FIG. 3-2

Propagation problem. 3-3

3-4

MATHEMATICS

The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available. Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem. The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking and statistical methods are reused. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas: 1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable. Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed. 2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms of probability statements. Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers. The mathematical notation is given in Table 3-1. MISCELLANEOUS MATHEMATICAL CONSTANTS Numerical values of the constants that follow are approximate to the number of significant digits given. π = 3.1415926536 e = 2.7182818285 γ = 0.5772156649 ln π = 1.1447298858 log π = 0.4971498727 Radian = 57.2957795131° Degree = 0.0174532925 rad Minute = 0.0002908882 rad Second = 0.0000048481 rad

 m − ln n = 0.577215 n

γ = lim

n→∞

Pi Napierian (natural) logarithm base Euler’s constant Napierian (natural) logarithm of pi, base e Briggsian (common logarithm of pi, base 10

1

TABLE 3-1

Mathematical Signs, Symbols, and Abbreviations

 (") : ⬋ < ⬏ > ⬐  ∼ ⬑ ≠ ⬟ ∝ ∞ ∴  3   n  ⬔ ⊥  |x| log or log10 loge or ln e a° a′ a a″ a sin cos tan ctn or cot sec csc vers covers exsec sin−1 sinh cosh tanh sinh−1 f(x) or φ(x) ∆x

dx dy/dx or y′ d2y/dx2 or y″ dny/dxn ∂y/∂x ∂ny/∂xn ∂nz  ∂x∂y





The natural numbers, or counting numbers, are the positive integers: 1, 2, 3, 4, 5, . . . . The negative integers are −1, −2, −3, . . . . A number in the form a/b, where a and b are integers, b ≠ 0, is a rational number. A real number that cannot be written as the quotient of two integers is called an irrational number, e.g., 2, 3, 5, π, 3 e, 2.

nth partial derivative with respect to x and y integral of

b

integral between the limits a and b

a

y˙ y¨ ∆ or ∇2

first derivative of y with respect to time second derivative of y with respect to time the “Laplacian” ∂2 ∂2 ∂2 2 + 2 + 2 ∂x ∂y ∂z sign of a variation sign for integration around a closed path



m=1

THE REAL-NUMBER SYSTEM

plus or minus (minus or plus) divided by, ratio sign proportional sign less than not less than greater than not greater than approximately equals, congruent similar to equivalent to not equal to approaches, is approximately equal to varies as infinity therefore square root cube root nth root angle perpendicular to parallel to numerical value of x common logarithm or Briggsian logarithm natural logarithm or hyperbolic logarithm or Naperian logarithm base (2.718) of natural system of logarithms an angle a degrees prime, an angle a minutes double prime, an angle a seconds, a second sine cosine tangent cotangent secant cosecant versed sine coversed sine exsecant anti sine or angle whose sine is hyperbolic sine hyperbolic cosine hyperbolic tangent anti hyperbolic sine or angle whose hyperbolic sine is function of x increment of x summation of differential of x derivative of y with respect to x second derivative of y with respect to x nth derivative of y with respect to x partial derivative of y with respect to x nth partial derivative of y with respect to x

δ





MATHEMATICS There is a one-to-one correspondence between the set of real numbers and the set of points on an infinite line (coordinate line). Order among Real Numbers; Inequalities a > b means that a − b is a positive real number. If a < b and b < c, then a < c. If a < b, then a  c < b  c for any real number c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c < d, then a + c < b + d. If 0 < a < b and 0 < c < d, then ac < bd. If a < b and ab > 0, then 1/a > 1/b. If a < b and ab < 0, then 1/a < 1/b. Absolute Value For any real number x, |x| = x −x



a < 0, and n odd, it is the unique negative root, and (3) if a < 0 and n even, it is any of the complex roots. In cases (1) and (2), the root can be found on a calculator by taking y = ln a/n and then x = e y. In case (3), see the section on complex variables. ALGEBRAIC INEQUALITIES Arithmetic-Geometric Inequality Let An and Gn denote respectively the arithmetic and the geometric means of a set of positive numbers a1, a2, . . . , an. The An ≥ Gn, i.e., a1 + a2 + ⋅ ⋅ ⋅ + an  n

if x ≥ 0 if x < 0

Properties If |x| = a, where a > 0, then x = a or x = −a. |x| = |−x|; −|x| ≤ x ≤ |x|; |xy| = |x| |y|. If |x| < c, then −c < x < c, where c > 0. ||x| − |y|| ≤ |x + y| ≤ |x| + |y|. x2 = |x|. a = c , then a + b = c + d , a − b = c − d , Proportions If       b d d b d b a−b c−d  = . a+b c+d

Form

Example

(∞)(0) 00 ∞0 1∞

xe−x xx (tan x)cos x (1 + x)1/x

x→∞ x → 0+ − x→aπ x → 0+

∞−∞ 0  0 ∞  ∞

1 − x− 1 x+ sin x  x ex  x

x→∞ x→0 x→∞

n

(a a

1 2

⋅ ⋅ ⋅ ar)1/r ≤ neAn

r=1

where e is the best possible constant in this inequality. Cauchy-Schwarz Inequality Let a = (a1, a2, . . . , an), b = (b1, b2, . . . , bn), where the ai’s and bi’s are real or complex numbers. Then

 a b  ≤  |a |  |b | 2

n

n

a−n = 1/an

a≠0

(ab)n = anbn (an)m = anm, n

anam = an + m

 a=a if a > 0 mn m n a = a, a > 0  1/n

n

am a > 0 am/n = (am)1/n = , a0 = 1 (a ≠ 0) 0a = 0 (a ≠ 0) Logarithms log ab = log a + log b, a > 0, b > 0 log an = n log a log (a/b) = log a − log b n log  a = (1/n) log a The common logarithm (base 10) is denoted log a or log10 a. The natural logarithm (base e) is denoted ln a (or in some texts log e a). If the text is ambiguous (perhaps using log x for ln x), test the formula by evaluating it. Roots If a is a real number, n is a positive integer, then x is called the nth root of a if xn = a. The number of nth roots is n, but not all of them are necessarily real. The principal nth root means the following: (1) if a > 0 the principal nth root is the unique positive root, (2) if

n

k k

k

2

k

k=1

2

k=1

The equality holds if, and only if, the vectors a, b are linearly dependent (i.e., one vector is scalar times the other vector). Minkowski’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers. Then for any real number p > 1,

 |a + b | ≤  |a | +  |b | 1/p

n

k

k

1/p

n

p

k

k=1

1/p

n

p

k

k=1

p

k=1

Hölder’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers, and let p and q be positive numbers with 1/p + 1/q = 1. Then

 a b  ≤  |a |  |b | n

1/p

n

k k

Integral Exponents (Powers and Roots) If m and n are positive integers and a, b are numbers or functions, then the following properties hold:

≥ (a1a2 ⋅ ⋅ ⋅ an)1/n

The equality holds only if all of the numbers ai are equal. Carleman’s Inequality The arithmetic and geometric means just defined satisfy the inequality

k=1

Indeterminants

3-5

k

k=1

k=1

1/q

n

p

q

k

k=1

The equality holds if, and only if, the sequences |a1|p, |a2|p, . . . , |an|p and |b1|q, |b2|q, . . . , |bn|q are proportional and the argument (angle) of the complex numbers akb k is independent of k. This last condition is of course automatically satisfied if a1, . . . , an and b1, . . . , bn are positive numbers. Lagrange’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Then

 a b =  a  b − 2

n

n

k=1

n

2 k

k k

k=1

2 k

k=1



(akbj − aj bk)2

1≤k≤j≤n

Example Two chemical engineers, John and Mary, purchase stock in the same company at times t1, t2, . . . , tn, when the price per share is respectively p1, p2, . . . , pn. Their methods of investment are different, however: John purchases x shares each time, whereas Mary invests P dollars each time (fractional shares can be purchased). Who is doing better? While one can argue intuitively that the average cost per share for Mary does not exceed that for John, we illustrate a mathematical proof using inequalities. The average cost per share for John is equal to n

x pi

1 n Total money invested i=1  =  =  pi nx Number of shares purchased n i=1 The average cost per share for Mary is nP n  = n n P 1 

 i

i = 1 pi = 1 pi

3-6

MATHEMATICS

Thus the average cost per share for John is the arithmetic mean of p1, p2, . . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only if p1 = p2 = ⋅⋅⋅ = pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors a = (p1−1/2, p2−1/2, . . . , pn−1/2)

Then a ⋅ b = 1 + ⋅⋅⋅ + 1 = n, and so by the Cauchy-Schwarz inequality n 1 (a ⋅ b)2 = n2 ≤  i = 1 pi

n

p

i

i=1

with the equality holding only if p1 = p2 = ⋅⋅⋅ = pn. Therefore n

p

i

b = (p11/2, p21/2, . . . , pn1/2)

i=1 n   n 1 ≤ n

 i = 1 pi

MENSURATION FORMULAS REFERENCES: Liu, J., Mathematical Handbook of Formulas and Tables, McGraw-Hill, New York (1999); http://mathworld.wolfram.com/SphericalSector. html, etc.

Area of Regular Polygon of n Sides Inscribed in a Circle of Radius r A = (nr 2/2) sin (360°/n)

Let A denote areas and V volumes in the following.

Perimeter of Inscribed Regular Polygon

PLANE GEOMETRIC FIGURES WITH STRAIGHT BOUNDARIES

P = 2nr sin (180°/n)

Triangles (see also “Plane Trigonometry”) A = a bh where b = base, h = altitude. Rectangle A = ab where a and b are the lengths of the sides. Parallelogram (opposite sides parallel) A = ah = ab sin α where a, b are the lengths of the sides, h the height, and α the angle between the sides. See Fig. 3-3. Rhombus (equilateral parallelogram) A = aab where a, b are the lengths of the diagonals. Trapezoid (four sides, two parallel) A = a(a + b)h where the lengths of the parallel sides are a and b, and h = height. Quadrilateral (four-sided) A = aab sin θ where a, b are the lengths of the diagonals and the acute angle between them is θ. Regular Polygon of n Sides See Fig. 3-4. 180° 1 A =  nl 2 cot  where l = length of each side n 4 180° l R =  csc  where R is the radius of the circumscribed circle n 2 180° l r =  cot  where r is the radius of the inscribed circle n 2 Radius r of Circle Inscribed in Triangle with Sides a, b, c r=



s (s − a)(s − b)(s − c)

where s = a(a + b + c)

Radius R of Circumscribed Circle abc R =  4 s( s −a )( s − b )( s −c)

FIG. 3-3

Parallelogram.

FIG. 3-4

Regular polygon.

Area of Regular Polygon Circumscribed about a Circle of Radius r A = nr 2 tan (180°/n) Perimeter of Circumscribed Regular Polygon 180° P = 2nr tan  n PLANE GEOMETRIC FIGURES WITH CURVED BOUNDARIES Circle (Fig. 3-5) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ l = chord length subtended by θ H = maximum rise of arc above chord, r − H = d θ = central angle (rad) subtended by arc S C = 2πr = πD (π = 3.14159 . . .) S = rθ = aDθ l = 2 r2 − d2 = 2r sin (θ/2) = 2d tan (θ/2) θ 1 1 d =  4 r2 −l2 =  l cot  2 2 2 S d l θ =  = 2 cos−1  = 2 sin−1  r r D

FIG. 3-5

Circle.

MENSURATION FORMULAS

3-7

Frustum of Pyramid (formed from the pyramid by cutting off the top with a plane

1⋅A 2)h V = s (A1 + A2 + A where h = altitude and A1, A2 are the areas of the base; lateral area of a regular figure = a (sum of the perimeters of base) × (slant height). FIG. 3-6

Ellipse.

Volume and Surface Area of Regular Polyhedra with Edge l FIG. 3-7

Parabola.

A (circle) = πr = dπD A (sector) = arS = ar 2θ A (segment) = A (sector) − A (triangle) = ar 2(θ − sin θ) 2

2

Ring (area between two circles of radii r1 and r2 ) The circles need not be concentric, but one of the circles must enclose the other. A = π(r1 + r2)(r1 − r2) Ellipse (Fig. 3-6)

r1 > r2

Let the semiaxes of the ellipse be a and b A = πab C = 4aE(e)

where e2 = 1 − b2/a2 and E(e) is the complete elliptic integral of the second kind, π 1 2 E(e) =  1 −  e2 + ⋅ ⋅ ⋅ 2 2

 



2+ b2)/2 ]. [an approximation for the circumference C = 2π (a Parabola (Fig. 3-7) 2 2x + 4 x +y2 y2 Length of arc EFG = 4 x2 +y2 +  ln  y 2x 4 Area of section EFG =  xy 3 Catenary (the curve formed by a cord of uniform weight suspended freely between two points A, B; Fig. 3-8) y = a cosh (x/a) Length of arc between points A and B is equal to 2a sinh (L/a). Sag of the cord is D = a cosh (L/a) − a. SOLID GEOMETRIC FIGURES WITH PLANE BOUNDARIES Cube Volume = a3; total surface area = 6a2; diagonal = a3, where a = length of one side of the cube. Rectangular Parallelepiped Volume = abc; surface area = 2  2(ab + ac + bc); diagonal =  a2 + b +c2, where a, b, c are the lengths of the sides. Prism Volume = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Pyramid Volume = s (area of base) × (altitude); lateral area of regular pyramid = a (perimeter of base) × (slant height) = a (number of sides) (length of one side) (slant height).

FIG. 3-8

Catenary.

Type of surface

Name

Volume

Surface area

4 equilateral triangles 6 squares 8 equilateral triangles 12 pentagons 20 equilateral triangles

Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron

0.1179 l3 1.0000 l3 0.4714 l3 7.6631 l3 2.1817 l3

1.7321 l2 6.0000 l2 3.4641 l2 20.6458 l2 8.6603 l2

SOLIDS BOUNDED BY CURVED SURFACES Cylinders (Fig. 3-9) V = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Right Circular Cylinder V = π (radius)2 × (altitude); lateral surface area = 2π (radius) × (altitude). Truncated Right Circular Cylinder V = πr 2h; lateral area = 2πrh h = a (h1 + h2) Hollow Cylinders Volume = πh(R2 − r 2), where r and R are the internal and external radii and h is the height of the cylinder. Sphere (Fig. 3-10) V (sphere) = 4⁄ 3πR3, jπD3 V (spherical sector) = wπR2hi = 2 (open spherical sector), i 1 (spherical cone) V (spherical segment of one base) = jπh1(3r 22 + h12) V (spherical segment of two bases) = jπh 2(3r 12 + 3r 22 + h 22 ) A (sphere) = 4πR2 = πD2 A (zone) = 2πRh = πDh A (lune on the surface included between two great circles, the inclination of which is θ radians) = 2R2θ. Cone V = s (area of base) × (altitude). Right Circular Cone V = (π/3) r 2h, where h is the altitude and r is the radius of the base; curved surface area = πr  r2 + h2, curved sur2 face of the frustum of a right cone = π(r1 + r2)  h2 +( r1 −r 2), where r1, r2 are the radii of the base and top, respectively, and h is the altitude; volume of the frustum of a right cone = π(h/3)(r 21 + r1r2 + r 22) = h/3(A1 + A2 +  A 1A2), where A1 = area of base and A2 = area of top. Ellipsoid V = (4 ⁄3) πabc, where a, b, c are the lengths of the semiaxes. Torus (obtained by rotating a circle of radius r about a line whose distance is R > r from the center of the circle) V = 2π2Rr 2

FIG. 3-9

Cylinder.

Surface area = 4π2Rr

FIG. 3-10

Sphere.

3-8

MATHEMATICS

Prolate Spheroid (formed by rotating an ellipse about its major axis [2a]) Surface area = 2πb + 2π(ab/e) sin e

43

1+e b2 Surface area = 2πa2 + π  ln  e 1−e

b

2

where a, b are the major and minor axes and e = eccentricity (e < 1). Oblate Spheroid (formed by the rotation of an ellipse about its minor axis [2b]) Data as given previously.

 y ds

S = 2π

V = ⁄ πab

−1

2

Area of a Surface of Revolution a

y/d x) 2 dx and y = f(x) is the equation of the plane where ds = 1 +(d curve rotated about the x axis to generate the surface. Area Bounded by f(x), the x Axis, and the Lines x = a, x = b

 f(x) dx b

A=

V = 4 ⁄3πa2b

For process vessels, the formulas reduce to the following: Hemisphere # # V =  D3, A =  D2 12 2 For a hemisphere (concave up) partially filled to a depth h1, use the formulas for spherical segment with one base, which simplify to

[ f(x) ≥ 0]

a

Length of Arc of a Plane Curve If y = f(x), Length of arc s =

dy  1

+

 dx dx

Length of arc s =

dx  1

+

 dy dy

b

2

a

If x = g(y), 2

d

c

V = #h12(Rh1/3) = #h12(D/2 − h1/3)

If x = f(t), y = g(t),

A = 2#Rh1 #Dh1 For a hemisphere (concave down) partially filled from the bottom, use the formulas for a spherical segment of two bases, one of which is a plane through the center, where h = distance from the center plane to the surface of the partially filled hemisphere.

Length of arc s =

dy dx  

+  dt  dt

dt t1

2

2

t0

In general, (ds)2 = (dx)2 + (dy)2.

V = #h(R2h2/3) = #h(D2/4 − h2/3)

IRREGULAR AREAS AND VOLUMES

A = 2#Rh = #Dh

Irregular Areas Let y0, y1, . . . , yn be the lengths of a series of equally spaced parallel chords and h be their distance apart (Fig. 3-11). The area of the figure is given approximately by any of the following:

Cone For a cone partially filled, use the same formulas as for right circular cones, but use r and h for the region filled. Ellipsoid If the base of a vessel is one-half of an oblate spheroid (the cross section fitting to a cylinder is a circle with radius of D/2 and the minor axis is smaller), then use the formulas for one-half of an oblate spheroid. V 0.1745D3, S 1.236D2, minor axis D/3 V 0.1309D , S 1.084D , minor axis D/4 3

2

AT = (h/2)[(y0 + yn) + 2(y1 + y2 + ⋅ ⋅ ⋅ + yn − 1)]

(trapezoidal rule)

As = (h/3)[(y0 + yn) + 4(y1 + y3 + y5 + ⋅ ⋅ ⋅ + yn − 1) + 2(y2 + y4 + ⋅ ⋅ ⋅ + yn − 2)]

(n even, Simpson’s rule)

The greater the value of n, the greater the accuracy of approximation. Irregular Volumes To find the volume, replace the y’s by crosssectional areas Aj and use the results in the preceding equations.

MISCELLANEOUS FORMULAS See also “Differential and Integral Calculus.” Volume of a Solid Revolution (the solid generated by rotating a plane area about the x axis)

 [ f(x)] dx b

V=π

2

a

where y = f(x) is the equation of the plane curve and a ≤ x ≤ b.

FIG. 3-11

Irregular area.

ELEMENTARY ALGEBRA REFERENCES: Stillwell, J. C., Elements of Algebra, CRC Press, New York (1994); Rich, B., and P. Schmidt, Schaum's Outline of Elementary Algebra, McGraw-Hill, New York (2004).

OPERATIONS ON ALGEBRAIC EXPRESSIONS An algebraic expression will here be denoted as a combination of letters and numbers such as 3ax − 3xy + 7x2 + 7x 3/ 2 − 2.8xy Addition and Subtraction Only like terms can be added or subtracted in two algebraic expressions.

Example (3x + 4xy − x2) + (3x2 + 2x − 8xy) = 5x − 4xy + 2x2. Example (2x + 3xy − 4x1/2) + (3x + 6x − 8xy) = 2x + 3x + 6x − 5xy − 4x1/2. Multiplication Multiplication of algebraic expressions is term by term, and corresponding terms are combined. Example (2x + 3y − 2xy)(3 + 3y) = 6x + 9y + 9y2 − 6xy2. Division This operation is analogous to that in arithmetic. Example Divide 3e2x + ex + 1 by ex + 1.

ELEMENTARY ALGEBRA PROGRESSIONS

Dividend Divisor ex + 1 | 3e2x + ex + 1 3ex − 2 quotient 3e2x + 3ex

An arithmetic progression is a succession of terms such that each term, except the first, is derivable from the preceding by the addition of a quantity d called the common difference. All arithmetic progressions have the form a, a + d, a + 2d, a + 3d, . . . . With a = first term, l = last term, d = common difference, n = number of terms, and s = sum of the terms, the following relations hold:

−2e + 1 −2ex − 2 x

+ 3 (remainder) Therefore, 3e + e + 1 = (e + 1)(3e − 2) + 3. 2x

x

x

x

s (n − 1) l = a + (n − 1)d =  +  d 2 n

Operations with Zero All numerical computations (except division) can be done with zero: a + 0 = 0 + a = a; a − 0 = a; 0 − a = −a; (a)(0) = 0; a0 = 1 if a ≠ 0; 0/a = 0, a ≠ 0. a/0 and 0/0 have no meaning. Fractional Operations −x x −x x −x x ax x −  = −  =  =  ;  =  ;  =  , if a ≠ 0. y y −y −y y −y y ay

n n n s =  [2a + (n − 1)d] =  (a + l) =  [2l − (n − 1)d] 2 2 2



y t = yt ;

x z xz    = ; y y y

x

z

xz

s (n − 1)d 2s a = l − (n − 1)d =  −  =  − l 2 n n

 

xt x/y x t =   = z/t y z yz

l−a 2(s − an) 2(nl − s) d=== n−1 n(n − 1) n(n − 1)

Factoring That process of analysis consisting of reducing a given expression into the product of two or more simpler expressions called factors. Some of the more common expressions are factored here: (2) x + 2xy + y = (x + y) 2

l−a 2s n=+1= d l+a The arithmetic mean or average of two numbers a, b is (a + b)/2; of n numbers a1, . . . , an is (a1 + a2 + ⋅ ⋅ ⋅ + an)/n. A geometric progression is a succession of terms such that each term, except the first, is derivable from the preceding by the multiplication of a quantity r called the common ratio. All such progressions have the form a, ar, ar 2, . . . , ar n − 1. With a = first term, l = last term, r = ratio, n = number of terms, s = sum of the terms, the following relations hold: [a + (r − 1)s] (r − 1)sr n − 1 l = ar n − 1 =  =  r rn − 1

(1) (x2 − y2) = (x − y)(x + y) 2

2

(3) x3 − y3 = (x − y)(x2 + xy + y2) (4) (x3 + y3) = (x + y)(x2 − xy + y2) (5) (x4 − y4) = (x − y)(x + y)(x2 + y2) (6) x5 + y5 = (x + y)(x4 − x3y + x2y2 − xy3 + y4) (7) xn − yn = (x − y)(xn − 1 + xn − 2y + xn − 3y2 + ⋅ ⋅ ⋅ + yn − 1) Laws of Exponents (an)m = anm; an + m = an ⋅ am; an/m = (an)1/m; an − m = an/am; a1/m = ma; a1/2 = a;  x2 = |x| (absolute value of x). For x > 0, y > 0, xy  = x n n n  y; for x > 0  xm = xm/n;  1 /x = 1/x

a(r n − 1) a(1 − r n) rl − a lr n − l s==== r−1 1−r r − 1 rn − rn − 1 log l − log a l (r − 1)s s−a a= = , r =  , log r =  rn − l rn − 1 s−l n−1 log l − log a log[a + (r − 1)s] − log a n =  + 1 =  log r log r

THE BINOMIAL THEOREM If n is a positive integer,

; of n The geometric mean of two nonnegative numbers a, b is ab numbers is (a1a2 . . . an)1/n. The geometric mean of a set of positive numbers is less than or equal to the arithmetic mean.

n(n − 1) (a + b)n = an + nan − 1b +  an − 2 b2 2! n n(n − 1)(n − 2) n n−j j +  an − 3b3 + ⋅ ⋅ ⋅ + bn =

a b j 3! j=0





n n! where =  = number of combinations of n things taken j at j j!(n − j)! a time. n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ n, 0! = 1. Example Find the sixth term of (x + 2y)12. The sixth term is obtained by setting j = 5. It is

5 x 12

n

(2y)5 = 792x7(2y)5

Example Find the sum of 1 + a + d + ⋅ ⋅ ⋅ + 1⁄64. Here a = 1, r = a, n = 7. Thus

a(1⁄64) − 1 s =  = 127/64 a−1 a ar n s = a + ar + ar 2 + ⋅ ⋅ ⋅ + ar n − 1 =  −  1−r 1−r a If |r| < 1, then lim s =  n→∞ 1−r which is called the sum of the infinite geometric progression.

Example The present worth (PW) of a series of cash flows Ck at the end of year k is

 j = (1 + 1) 14

Example

12 − 5

3-9

14

= 214.

j=0

If n is not a positive integer, the sum formula no longer applies and an infinite series results for (a + b)n. The coefficients are obtained from the first formulas in this case. Example (1 + x)1/2 = 1 + ax − a ⋅ dx2 + a ⋅ d ⋅ 3⁄ 6 x3 ⋅ ⋅ ⋅ (convergent for

x2 < 1).

Additional discussion is under “Infinite Series.”

n Ck PW = k k = 1 (1 + i) where i is an assumed interest rate. (Thus the present worth always requires specification of an interest rate.) If all the payments are the same, Ck = R, the present worth is n 1 PW = R k k = 1 (1 + i)

This can be rewritten as R PW =  1+i

n



k=1

R 1 =  (1 + i)k − 1 1 + i

n−1



j=0

1 j (1 + i)

3-10

MATHEMATICS

This is a geometric series with r = 1/(1 + i) and a = R/(1 + i). The formulas above give R (1 + i)n − 1 PW (=s) =   (1 + i)n i The same formula applies to the value of an annuity (PW) now, to provide for equal payments R at the end of each of n years, with interest rate i.

A progression of the form a, (a + d )r, (a + 2d)r 2, (a + 3d)r 3, etc., is a combined arithmetic and geometric progression. The sum of n such terms is a − [a + (n − 1)d]r n rd(1 − r n − 1) s =  +  1−r (1 − r)2 a If |r| < 1, lim s =  + rd/(1 − r)2. n→∞ 1−r The non-zero numbers a, b, c, etc., form a harmonic progression if their reciprocals 1/a, 1/b, 1/c, etc., form an arithmetic progression. Example The progression 1, s, 1⁄5, 1⁄7, . . . , 1⁄31 is harmonic since 1, 3, 5, 7, . . . , 31 form an arithmetic progression.

Cubic Equations A cubic equation, in one variable, has the form x3 + bx2 + cx + d = 0. Every cubic equation having complex coefficients has three complex roots. If the coefficients are real numbers, then at least one of the roots must be real. The cubic equation x3 + bx2 + cx + d = 0 may be reduced by the substitution x = y − (b/3) to the form y3 + py + q = 0, where p = s(3c − b2), q = 1⁄27(27d − 9bc + 2b3). This equation has the solutions y1 = A + B, y2 = −a(A + B) + (i3/2)(A − B), 3 y3 = −a(A + B) − (i3/2)(A − B), where i2 = −1, A =  − q /2 + R,  3 3 2 B =  − q /2 − R, and R = (p/3) + (q/2) . If b, c, d are all real and if  R > 0, there are one real root and two conjugate complex roots; if R = 0, there are three real roots, of which at least two are equal; if R < 0, there are three real unequal roots. If R < 0, these formulas are impractical. In this case, the roots are given by yk = " 2  − p /3 cos [(φ/3) + 120k], k = 0, 1, 2 where φ = cos−1

ability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.

THEORY OF EQUATIONS Linear Equations A linear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or x = −B/A. Linear equations in n variables have the form a11 x1 + a12 x2 + ⋅ ⋅ ⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅ ⋅ ⋅ + a2n xn = b2 ⯗ am1 x1 + am2 x2 + ⋅ ⋅ ⋅ + amn xn = bm The solution of the system may then be found by elimination or matrix methods if a solution exists (see “Matrix Algebra and Matrix Computations”). Quadratic Equations Every quadratic equation in one variable is expressible in the form ax 2 + bx + c = 0. a ≠ 0. This equation has two solutions, say, x1, x2, given by

4ac  x1 b2− −b   =  x2 2a




If a, b, c are real, the discriminant b2 − 4ac gives the character of the roots. If b2 − 4ac > 0, the roots are real and unequal. If b2 − 4ac < 0, the roots are complex conjugates. If b2 − 4ac = 0 the roots are real and equal. Two quadratic equations in two variables can in general be solved only by numerical methods (see “Numerical Analysis and Approximate Methods”).

3

Example y3 − 7y + 7 = 0. p = −7, q = 7, R < 0. Hence 28 φ  cos   + 120k 3 3

yk = −

PERMUTATIONS, COMBINATIONS, AND PROBABILITY

Example Two dice may be thrown in 36 separate ways. What is the prob-

2

and the upper sign applies if q > 0, the lower if q < 0.

The harmonic mean of two numbers a, b is 2ab/(a + b).

Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written n! P(n, r) =  = n(n − 1)(n − 2) ⋅⋅⋅ (n − r + 1) (n − r)! Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/ [r!(n − r)!]. An important relation is r! C(n, r) = P(n, r). If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).

q /4  −p /27

where

27 φ  ,  = 3.6311315 rad = 3°37′52″ 28 3

φ = cos−1

The roots are approximately −3.048917, 1.692021, and 1.356896.

Example Many equations of state involve solving cubic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving Z 3 − Z 2 + cZ + d = 0,

d<0

where c and d depend on critical constants of the chemical species. In this case, only positive solutions, Z > 0, are desired.

Quartic Equations See Abramowitz and Stegun (1972, p. 17). General Polynomials of the nth Degree Denote the general polynomial equation of degree n by P(x) = a0 x n + a1 x n − 1 + ⋅ ⋅ ⋅ + an − 1 x + an = 0 If n > 4, there is no formula which gives the roots of the general equation. For fourth and higher order (even third order), the roots can be found numerically (see “Numerical Analysis and Approximate Methods”). However, there are some general theorems that may prove useful. Remainder Theorems When P(x) is a polynomial and P(x) is divided by x − a until a remainder independent of x is obtained, this remainder is equal to P(a). Example P(x) = 2x4 − 3x2 + 7x − 2 when divided by x + 1 (here a = −1) results in P(x) = (x + 1)(2x3 − 2x2 − x + 8) − 10 where −10 is the remainder. It is easy to see that P(−1) = −10. Factor Theorem If P(a) is zero, the polynomial P(x) has the factor x − a. In other words, if a is a root of P(x) = 0, then x − a is a factor of P(x). If a number a is found to be a root of P(x) = 0, the division of P(x) by (x − a) leaves a polynomial of degree one less than that of the original equation, i.e., P(x) = Q(x)(x − a). Roots of Q(x) = 0 are clearly roots of P(x) = 0. Example P(x) = x3 − 6x2 + 11x − 6 = 0 has the root + 3. Then P(x) =

(x − 3)(x2 − 3x + 2). The roots of x2 − 3x + 2 = 0 are 1 and 2. The roots of P(x) are therefore 1, 2, 3.

Fundamental Theorem of Algebra Every polynomial of degree n has exactly n real or complex roots, counting multiplicities. Every polynomial equation a0 x n + a1 x n − 1 + ⋅⋅⋅ + an = 0 with rational coefficients may be rewritten as a polynomial, of the same degree, with integral coefficients by multiplying each coefficient by the least common multiple of the denominators of the coefficients. Example The coefficients of 3⁄2 x4 + 7⁄3 x3 − 5⁄6 x2 + 2x − j = 0 are rational numbers. The least common multiple of the denominators is 2 × 3 = 6. Therefore, the equation is equivalent to 9x4 + 14x3 − 5x2 + 12x − 1 = 0.

ANALYTIC GEOMETRY Determinants Consider the system of two linear equations a11x1 + a12x2 = b1 If the first equation is multiplied by a22 and the second by −a12 and the results added, we obtain (a11a22 − a21a12)x1 = b1a22 − b2a12 The expression a11a22 − a21a12 may be represented by the symbol a11 a12 = a11a22 − a21a12 a21 a22 This symbol is called a determinant of second order. The value of the square array of n2 quantities aij, where i = 1, . . . , n is the row index, j = 1, . . . , n the column index, written in the form |A| =







= a31



a11 a12 a21 a22 ⯗ an1 an2

a13 ⋅ ⋅ ⋅ a1n ⋅ ⋅ ⋅ ⋅ ⋅ a2n an3 ⋅ ⋅ ⋅ ann





a11 a12 a13 a a a21 a22 a23 The minor of a23 is M23 = 11 12 a31 a32 a31 a32 a33





The cofactor Aij of the element aij is the signed minor of aij determined by the rule Aij = (−1) i + jMij. The value of |A| is obtained by forming any of the n n equivalent expressions j = 1 aij Aij , i = 1 aij Aij, where the elements aij must be taken from a single row or a single column of A.

aa









a13 a a a a − a32 11 13 + a33 11 12 a23 a21 a23 a21 a22

12 22



In general, Aij will be determinants of order n − 1, but they may in turn be expanded by the rule. Also,



is called a determinant. The n2 quantities aij are called the elements of the determinant. In the determinant |A| let the ith row and jth column be deleted and a new determinant be formed having n − 1 rows and columns. This new determinant is called the minor of aij denoted Mij. Example

Example a11 a12 a13 a21 a22 a23 = a31A31 + a32A32 + a33A33 a31 a32 a33

a21x1 + a22x2 = b2



3-11

n

a

j=1

n

ji



A jk = a ij A jk = |A| 0 j=1

i=k i≠k

Fundamental Properties of Determinants 1. The value of a determinant |A| is not changed if the rows and columns are interchanged. 2. If the elements of one row (or one column) of a determinant are all zero, the value of |A| is zero. 3. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 6. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving unchanged the remaining elements. 7. The value of a determinant is not changed if one row (or column) is multiplied by a constant and added to another row (or column).

ANALYTIC GEOMETRY REFERENCES: Fuller, G., Analytic Geometry, 7th ed., Addison Wesley Longman (1994); Larson, R., R. P. Hostetler, and B. H. Edwards, Calculus with Analytic Geometry, 7th ed., Houghton Mifflin (2001); Riddle, D. F., Analytic Geometry, 6th ed., Thompson Learning (1996); Spiegel, M. R., and J. Liu, Mathematical Handbook of Formulas and Tables, 2d ed., McGraw-Hill (1999); Thomas, G. B., Jr., and R. L. Finney, Calculus and Analytic Geometry, 9th ed., Addison-Wesley (1996).

Analytic geometry uses algebraic equations and methods to study geometric problems. It also permits one to visualize algebraic equations in terms of geometric curves, which frequently clarifies abstract concepts. PLANE ANALYTIC GEOMETRY Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis,

FIG. 3-12

Rectangular coordinates.

positive if to the right, negative to the left. y (ordinate) is the distance of the point from the x axis, measured parallel to the y axis, positive if above, negative if below the x axis. The quadrants are labeled 1, 2, 3, 4 in the drawing, the coordinates of points in the various quadrants having the depicted signs. Another common coordinate system is the polar coordinate system (Fig. 3-13). In this system the position of a point is designated by the pair (r, θ), r =  x2 +y2 being the distance to the origin 0(0,0) and θ being the angle the line r makes with the positive x axis (polar axis). To change from polar to rectangular coordinates, use x = r cos θ and y = r sin θ. To change from rectangular to x2 +y2 and θ = tan−1 (y/x) if x ≠ 0; θ = π/2 polar coordinates, use r =  if x = 0. The distance between two points (x1, y1), (x2, y2) is defined 2 2 by d =  (x1 −x +(y1 − y 2) 2) in rectangular coordinates or by d = 2 2 r 1 +r −2 r1 r2cos (θ1 − θ  2 2) in polar coordinates. Other coordinate systems are sometimes used. For example, on the surface of a sphere latitude and longitude prove useful. The Straight Line (Fig. 3-14) The slope m of a straight line is the tangent of the inclination angle θ made with the positive x axis. If

FIG. 3-13

Polar coordinates.

FIG. 3-14

Straight line.

3-12

MATHEMATICS

(x1, y1) and (x2, y2) are any two points on the line, slope = m = (y2 − y1)/(x2 − x1). The slope of a line parallel to the x axis is zero; parallel to the y axis, it is undefined. Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is −1 (the exception being that case when the lines are parallel to the coordinate axes). Every equation of the type Ax + By + C = 0 represents a straight line, and every straight line has an equation of this form. A straight line is determined by a variety of conditions: Given conditions

Geometric Properties of a Curve When the Equation Is Given The analysis of the properties of an equation is facilitated by the investigation of the equation by using the following techniques: 1. Points of maximum, minimum, and inflection. These may be investigated by means of the calculus. 2. Symmetry. Let F(x, y) = 0 be the equation of the curve.

Equation of line

(1) (2) (3) (4) (5)

Parallel to x axis Parallel y axis Point (x1, y1) and slope m Intercept on y axis (0, b), m Intercept on x axis (a, 0), m

(6)

Two points (x1, y1), (x2, y2)

(7)

Two intercepts (a, 0), (0, b)

y = constant x = constant y − y1 = m(x − x1) y = mx + b y = m(x − a) y2 − y1 y − y1 =  (x − x1) x2 − x1 x/a + y/b = 1

The angle β a line with slope m1 makes with a line having slope m2 is given by tan β = (m2 − m1)/(m1m2 + 1). A line is determined if the length and direction of the perpendicular to it (the normal) from the origin are given (see Fig. 3-15). Let p = length of the perpendicular and α the angle that the perpendicular makes with the positive x axis. The equation of the line is x cos  + y sin  = p. The equation of a line perpendicular to a given line of slope m and passing through a point (x1, y1) is y − y1 = −(1/m) (x − x1). The distance from a point (x1, y1) to a line with equation Ax + By + C = 0 is |Ax1 + By1 + C| d =  A2 + B2  Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. Example Consider y = bxn. B = log b. Taking logarithms log y = n log x + log b. Let Y = log y, X = log x, B = log b. The equation then has the form Y = nX + B, which is a linear equation. Consider k = k0 exp (−E/RT), taking logarithms loge k = loge k0 − E/(RT). Let Y = loge k, B = loge k0, and m = −E/R, X = 1/T, and the result is Y = mX + B. Next consider y = a + bxn. If the substitution t = x n is made, then the graph of y is a straight line versus t. Asymptotes The limiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that n n b f(x) = ak x k + kk k=0 k=1 x

Condition on F(x, y)

Symmetry

F(x, y) = F(−x, y) F(x, y) = F(x, −y) F(x, y) = F(−x, −y) F(x, y) = F(y, x)

With respect to y axis With respect to x axis With respect to origin With respect to the line y = x

3. Extent. Only real values of x and y are considered in obtaining the points (x, y) whose coordinates satisfy the equation. The extent of them may be limited by the condition that negative numbers do not have real square roots. 4. Intercepts. Find those points where the curves of the function cross the coordinate axes. 5. Asymptotes. See preceding discussion. 6. Direction at a point. This may be found from the derivative of the function at a point. This concept is useful for distinguishing among a family of similar curves. Example y2 = (x2 + 1)/(x2 − 1) is symmetric with respect to the x and y axis, the origin, and the line y = x. It has the vertical asymptotes x = 1. When x = 0, y2 = −1; so there are no y intercepts. If y = 0, (x2 + 1)/(x2 − 1) = 0; so there are no x intercepts. If |x| < 1, y2 is negative; so |x| > 1. From x2 = (y2 + 1)/(y2 − 1), y = 1 are horizontal asymptotes and |y| > 1. As x → 1+, y → + ∞; as x → + ∞, y → + 1. The graph is given in Fig. 3-16. Conic Sections The curves included in this group are obtained from plane sections of the cone. They include the circle, ellipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixed line, called the directrix. This ratio is the eccentricity e. If e = 0, the conic is a circle; if 0 < e < 1, the conic is an ellipse; if e = 1, the conic is a parabola; if e > 1, the conic is a hyperbola. Every conic section is representable by an equation of second degree. Conversely, every equation of second degree in two variables represents a conic. The general equation of the second degree is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Let  be defined as the determinant 2A B D = B 2C E D E 2F The table characterizes the curve represented by the equation.



B2 − 4AC < 0

n

and

lim f(x) = akxk x→∞

k=0

n

then the equation of the asymptote is y = k = 0 ak x k. If n = 1, then the asymptote is (in general oblique) a line. In this case, the equation of the asymptote may be written as y = mx + b

m = lim f′(x) x→∞

b = lim [f(x) − xf′(x)] x→∞

FIG. 3-15

Determination of line.

≠0

A < 0 A ≠ C, an ellipse A < 0 A = C, a circle A > 0, no locus

=0

FIG. 3-16

Point



B2 − 4AC = 0

Parabola 2 parallel lines if Q = D2 + E2 − 4(A + C)F > 0 1 straight line if Q = 0, no locus if Q < 0

Graph of y2 = (x2 + 1)/(x2 − 1).

B2 − 4AC > 0

Hyperbola

2 intersecting straight lines

ANALYTIC GEOMETRY

3-13

Some common equations in parametric form are given below. (1) (x − h)2 + (y − k)2 = a2 (x − h) (y − k) (2)  + =1 a2 b2 2

2

(3) x2 + y2 = a2

Circle (Fig. 3-23) Parameter is angle θ.

x = h + a cos θ y = k + a sin θ x = h + a cos φ y = k + a sin φ −at x= t2 + 1 



Ellipse (Fig. 3-20) Parameter is angle φ. dy Circle Parameter is t =  = slope of tangent at (x, y). dx

a y= t2 +1 

(4) x2 = y + k

Parabola (Fig. 3-22)

x2 y2 (5) 2  2 = 1 a b

Hyperbola with the origin at the center (Fig. 3-21)

x (6) y = a cosh  a

 

(7) Cycloid

s x = a sinh−1  a

Catenary (such as hanging cable under gravity) Parameter s = arc length from (0, a) to (x, y).

y2 = a2 + s2 x = a(φ − sin φ) y = a(1 − cos φ)

Fig. 3.24

Similarly, x = a cos φ, y = b sin φ are the parametric equations of the ellipse x2/a2 + y2/b2 = 1 with parameter φ.

Example 3x2 + 4xy − 2y2 + 3x − 2y + 7 = 0.



6 = 4 3

4 −4 −2



3 −2 = −596 ≠ 0, B2 − 4AC = 40 > 0 14

SOLID ANALYTIC GEOMETRY

The curve is therefore a hyperbola.

The following tabulation gives the form of the more common equations. Polar equation

Type of curve

(1) r = a (2) r = 2a cos θ (3) r = 2a sin θ (4) r2 − 2br cos (θ − β) + b2 − a2 = 0

Circle, Fig. 3-17 Circle, Fig. 3-18 Circle, Fig. 3-19 Circle at (b, β), radius a

ke (5) r =  1 − e cos θ

e = 1 parabola, Fig. 3-22 0 < e < 1 ellipse, Fig. 3-20 e > 1 hyperbola, Fig. 3-21

Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos φ, y = a sin φ where φ is the parameter.

Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems [see Morse, P. M., and H. Feshbach, Methods of Theoretical Physics, vols. I and II, McGraw-Hill, New York (1953)]. The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers (x, y, z) is used to represent each point. The cylindrical coordinate system (r, θ, z; Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (r, θ) coupled with the z coordinate. As before, x = r cos θ, y = r sin θ, z = z and r2 = x2 + y2, y/x = tan θ. If r is held constant and θ and z are allowed to vary, the locus of (r, θ, z) is a right circular cylinder of radius r along the z axis. The locus of r = C is a circle, and θ = constant is a plane containing the z axis and making an angle θ with the xz plane. Cylindrical coordinates are convenient to use when the problem has an axis of symmetry. The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (ρ, φ, θ) illustrated in Fig. 3-27. The relations are x =

FIG. 3-17

Circle center (0,0) r = a.

FIG. 3-18

Circle center (a,0) r = 2a cos θ.

FIG. 3-19

Circle center (0,a) r = 2a sin θ.

FIG. 3-20

Ellipse, 0 < e < 1.

FIG. 3-21

Hyperbola, e > 1.

FIG. 3-22

Parabola, e = 1.

3-14

MATHEMATICS

FIG. 3-23

Circle.

FIG. 3-24

Cycloid.

ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, and r = ρ sin φ. θ = constant is a plane containing the z axis and making an angle θ with the xz plane. φ = constant is a cone with vertex at 0. ρ = constant is the surface of a sphere of radius ρ, center at the origin 0. Every point in the space may be given spherical coordinates restricted to the ranges 0 ≤ φ ≤ π, ρ ≥ 0, 0 ≤ θ < 2π. Lines and Planes The distance between two points (x1, y1, z1), 2 2 2 (x2, y2, z2) is d =  (x −x +(y −y +( z1 −z 1 2) 1 2) 2). There is nothing in the geometry of three dimensions quite analogous to the slope of a line in the plane case. Instead of specifying the direction of a line by a trigonometric function evaluated for one angle, a trigonometric function evaluated for three angles is used. The angles α, β, γ that a line segment makes with the positive x, y, and z axes, respectively, are called the direction angles of the line, and cos α, cos β, cos γ are called the direction cosines. Let (x1, y1, z1), (x2, y2, z2) be on the line. Then cos α = (x2 − x1)/d, cos β = (y2 − y1)/d, cos γ = (z2 − z1)/d, where d = the distance between the two points. Clearly cos2 α + cos2 β + cos2 γ = 1. If two lines are specified by the direction cosines (cos α1, cos β1, cos γ1), (cos α2, cos β2, cos γ2), then the angle θ between the lines is cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2. Thus the lines are perpendicular if and only if θ = 90° or cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = 0. The equation of a line with direction cosines (cos α, cos β, cos γ) passing through (x1, y1, z1) is (x − x1)/cos α = (y − y1)/cos β = (z − z1)/cos γ. The equation of every plane is of the form Ax + By + Cz + D = 0. The numbers A B C  ,  ,  A2 + B2 + C2  A2 + B2 + C2  A2 + B2 + C2  are direction cosines of the normal lines to the plane. The plane through the point (x1, y1, z1) whose normals have these as direction cosines is A(x − x1) + B(y − y1) + C(z − z1) = 0. Example Find the equation of the plane through (1, 5, −2) perpendicular to the line (x + 9)/7 = (y − 3)/−1 = z/8. The numbers (7, −1, 8) are called direction numbers. They are a constant multiple of the direction cosines. cos α = 7/114, cos β = −1/114, cos γ = 8/114. The plane has the equation 7(x − 1) − 1(y − 5) + 8(z + 2) = 0 or 7x − y + 8z + 14 = 0.

FIG. 3-25

Cartesian coordinates.

The distance from the point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is |Ax + By + Cz + D|

1 1 1 d =  2 2 2

+   B+ C A

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x = f(t), y = g(t), z = h(t) in the parameter t. Example The equation of a straight line in space is (x − x1)/a = (y − y1)/b = (z − z1)/c. Since all these quantities must be equal (say, to t), we may write x = x1 + at, y = y1 + bt, z = z1 + ct, which represent the parametric equations of the line. FIG. 3-26

Cylindrical coordinates.

Example The equations z = a cos βt, y = a sin βt, z = bt, a, β, b positive constants, represent a circular helix.

FIG. 3-27

Spherical coordinates.

FIG. 3-28

Parabolic cylinder.

ANALYTIC GEOMETRY

3-15

Surfaces The locus of points (x, y, z) satisfying f(x, y, z) = 0, broadly speaking, may be interpreted as a surface. The simplest surface is the plane. The next simplest is a cylinder, which is a surface generated by a straight line moving parallel to a given line and passing through a given curve. Example The parabolic cylinder y = x2 (Fig. 3-28) is generated by a straight line parallel to the z axis passing through y = x2 in the plane z = 0. A surface whose equation is a quadratic in the variables x, y, and z is called a quadric surface. Some of the more common such surfaces are tabulated and pictured in Figs. 3-28 to 3-36.

FIG. 3-29

FIG. 3-30

FIG. 3-31

Elliptic paraboloid.

FIG. 3-33

y2 z2 x2 FIG. 3-32 Cone. 2 + 2 + 2 = 0 b c a

y2 x2 2 + 2 + cz = 0 b a

y2 z2 x2 Ellipsoid. 2 + 2 + 2 = 1 (sphere if a = b = c) b c a FIG. 3-34

y2 x2 Hyperbolic paraboloid. 2 − 2 + cz = 0 b a

FIG. 3-35

y2 x2 Elliptic cylinder. 2 + 2 = 1 b a

FIG. 3-36

Hyperbolic cylinder.

y2 z2 x2 Hyperboloid of one sheet. 2 + 2 − 2 = 1 b c a

y2 z2 x2 Hyperboloid of two sheets. 2 + 2 − 2 = −1 b c a

y2 x2 2 − 2 = 1 b a

3-16

MATHEMATICS

PLANE TRIGONOMETRY REFERENCES: Gelfand, I. M., and M. Saul, Trigonometry, Birkhäuser, Boston (2001); Heineman, E. Richard, and J. Dalton Tarwater, Plane Trigonometry, 7th ed., McGraw-Hill (1993).

ANGLES An angle is generated by the rotation of a line about a fixed center from some initial position to some terminal position. If the rotation is clockwise, the angle is negative; if it is counterclockwise, the angle is positive. Angle size is unlimited. If α, β are two angles such that α + β = 90°, they are complementary; they are supplementary if α + β = 180°. Angles are most commonly measured in the sexagesimal system or by radian measure. In the first system there are 360 degrees in one complete revolution; one degree = 1⁄90 of a right angle. The degree is subdivided into 60 minutes; the minute is subdivided into 60 seconds. In the radian system one radian is the angle at the center of a circle subtended by an arc whose length is equal to the radius of the circle. Thus 2# rad = 360°; 1 rad = 57.29578°; 1° = 0.01745 rad; 1 min = 0.00029089 rad. The advantage of radian measure is that it is dimensionless. The quadrants are conventionally labeled as Fig. 3-37 shows.

II

I

III

IV

FIG. 3-37

Quadrants.

FIG. 3-38

Triangles.

FIG. 3-39

Graph of y = sin x.

FIG. 3-40

Graph of y = cos x.

FIG. 3-41

Graph of y = tan x.

FUNCTIONS OF CIRCULAR TRIGONOMETRY The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-38 for the various quadrants. Clearly r =  x2 +y2 ≥ 0. The fundamental functions (see Figs. 3-39, 3-40, 3-41) are Plane Trigonometry Sine of θ = sin θ = y/r Cosine of θ = cos θ = x/r Tangent of θ = tan θ = y/x

Secant of θ = sec θ = r/x Cosecant of θ = csc θ = r/y Cotangent of θ = cot θ = x/y

Values of the Trigonometric Functions for Common Angles θ°

θ, rad

sin θ

cos θ

tan θ

0 30 45 60 90

0 π/6 π/4 π/3 π/2

0 1/2 2/2 3/2 1

1 3/2 2/2 1/2 0

0 3/3 1 3 +∞

If 90° ≤ θ ≤ 180°, sin θ = sin (180° − θ); cos θ = −cos (180° − θ); tan θ = −tan (180° − θ). If 180° ≤ θ ≤ 270°, sin θ = −sin (270° − θ); cos θ = −cos (270° − θ); tan θ = tan (270° − θ). If 270° ≤ θ ≤ 360°, sin θ = −sin (360° − θ); cos θ = cos (360° − θ); tan θ = −tan (360° − θ). The reciprocal properties may be used to find the values of the other functions. If it is desired to find the angle when a function of it is given, the procedure is as follows: There will in general be two angles between 0° and 360° corresponding to the given value of the function. Given (a > 0)

Find an acute angle θ0 such that

Required angles are

sin θ = +a cos θ = +a tan θ = +a sin θ = −a cos θ = −a tan θ = −a

sin θ0 = a cos θ0 = a tan θ0 = a sin θ0 = a cos θ0 = a tan θ0 = a

θ0 and (180° − θ0) θ0 and (360° − θ0) θ0 and (180° + θ0) 180° + θ0 and 360° − θ0 180° − θ0 and 180° + θ0 180° − θ0 and 360° − θ0

Relations between Functions of a Single Angle sec θ = 1/ cos θ; csc θ = 1/sin θ, tan θ = sin θ/cos θ = sec θ/csc θ = 1/cot θ; sin2 θ + cos2 θ = 1; 1 + tan2 θ = sec2 θ; 1 + cot2 θ = csc2 θ. For 0 ≤ θ ≤ 90° the following results hold: θ θ sin θ = 2 sin  cos  2 2

 

PLANE TRIGONOMETRY

3-17

θ θ cos θ = cos2  − sin2  2 2 The cofunction property is very important. cos θ = sin (90° − θ), sin θ = cos (90° − θ), tan θ = cot (90° − θ), cot θ = tan (90° − θ), etc. Functions of Negative Angles sin (−θ) = −sin θ, cos (−θ) = cos θ, tan (−θ) = −tan θ, sec (−θ) = sec θ, csc (−θ) = −csc θ, cot (−θ) = −cot θ.



and



FIG. 3-42

Identities Sum and Difference Formulas Let x, y be two angles. sin (x  y) = sin x cos y  cos x sin y; cos (x  y) = cos x cos y " sin x sin y; tan (x  y) = (tan x  tan y)/(1 " tan x tan y); sin x  sin y = 2 sin a(x  y) cos a(x " y); cos x + cos y = 2 cos a(x + y) cos a(x − y); cos x − cos y = −2 sin a(x + y) sin a(x − y); tan x  tan y = [sin (x  y)]/(cos x cos y); sin2 x − sin2 y = cos2 y − cos2 x = sin (x + y) sin (x − y); cos2 x − sin2 y = cos2 y − sin2 x = cos (x + y) cos (x − y); sin (45° + x) = cos (45° − x); sin (45° − x) = cos (45° + x); tan (45°  x) = cot (45° " x). Multiple and Half Angle Identities Let x = angle, sin 2x = 2 sin x cos x; sin x = 2 sin ax cos ax; cos 2x = cos2 x − sin2x = 1 − 2 sin2x = 2 cos2x − 1. tan 2x = (2 tan x)/(1 − tan2 x); sin 3x = 3 sin x − 4 sin3x; cos 3x = 4 cos3 x − 3 cos x. tan 3x = (3 tan x − tan3 x)/(1 − 3 tan2 x); sin 4x = 4 sin x cos x − 8 sin3 x cos x; cos 4x = 8 cos4 x − 8 cos2 x + 1. x sin  =  a (1 −co s x) 2

 

x (1   cos  = a +o csx) 2 1 − cos x sin x ==  sin x 1 + cos x 1 + cos x 1 − cos x

x tan  = 2

Triangle.

b + c, A = area, r = radius of the inscribed circle, R = radius of the circumscribed circle, and h = altitude. In any triangle α + β + γ = 180°. Law of Sines sin α/a = sin β/b = sin γ/c 1/(2R). Law of Tangents a + b tan a(α + β) b + c tan a(β + γ) a + c tan a(α + γ)  =  ;  =  ;  =  a − b tan a(α − β) b − c tan a(β − γ) a − c tan a(α − γ) Law of Cosines a2 = b2 + c2 − 2bc cos α; b2 = a2 + c2 − 2ac cos β; c2 = a2 + b2 − 2ab cos γ. Other Relations In this subsection, where appropriate, two more formulas can be generated by replacing a by b, b by c, c by a, α by β, β by γ, and γ by α. cos α = (b2 + c2 − a2)/2bc; a = b cos γ + c cos β; sin α = (2/bc)  s( s −a )( s − b )( s −c) ; ;  ; cos   =  2 bc bc

α sin  = 2

Example y = sin−1 a, y is 30°.

The complete solution of the equation x = sin y is y = (−1)n sin−1 x + n(180°), −π/2 ≤ sin−1 x ≤ π/2 where sin−1 x is the principal value of the angle whose sine is x. The range of principal values of the cos−1 x is 0 ≤ cos−1 x ≤ π and −π/2 ≤ tan−1 x ≤ π/2. If these restrictions are allowed to hold, the following formulas result:

x −x2 1 sin x = cos 1 −x = tan 2 = cot−1  x −x 1 −1

−1

2

−1

1 1 π = sec−1 2 = csc−1  =  − cos−1 x x 2 −x 1

α

s(s − a)

1 1 a2 sin β sin γ A =  bh =  ab sin γ =  = s( s−  a)( s−  b)( s− c) = rs 2 2 2 sin α

INVERSE TRIGONOMETRIC FUNCTIONS y = sin −1 x = arcsin x is the angle y whose sine is x.

(s − b)(s − c)

where r =

(s − a)(s − b)(s − c) 

s

R = a/(2 sin α) = abc/4A; h = c sin a = a sin γ = 2rs/b. Right Triangle (Fig. 3-43) Given one side and any acute angle α or any two sides, the remaining parts can be obtained from the following formulas:  + b)( c−  b) = c sin α = b tan α a = (c  +a)( c−  a) = c cos α = a cot α b = (c a b a 2 2 c = a, + b sin α =  , cos α =  , tan α =  , β = 90° − α c c b

x 1 = cot−1 2 = sec−1  x −x 1

1 a2 b2 tan α c2 sin 2α A =  ab =  =  =  2 2 tan α 2 4 Oblique Triangles (Fig. 3-44) There are four possible cases. 1. Given b, c and the included angles α, b−c 1 1 1 1 tan  (β + γ)  (β + γ) = 90° −  α; tan  (β − γ) =  b+c 2 2 2 2

π 1 = csc−1 2 =  − sin−1 x −x 2 1

1 1 1 1 b sin α β =  (β + γ) +  (β − γ); γ =  (β + γ) −  (β − γ); a =  2 2 2 2 sin β

−x 1 cos−1 x = sin−1 1 −x2 = tan−1  x 2

x 1 tan−1 x = sin−1 2 = cos−1 2 −x +x 1 1 +x2 1 1 = cot−1  = sec−1 1 +x2 = csc−1  x x RELATIONS BETWEEN ANGLES AND SIDES OF TRIANGLES Solutions of Triangles (Fig. 3-42) Let a, b, c denote the sides and α, β, γ the angles opposite the sides in the triangle. Let 2s = a +

FIG. 3-43

Right triangle.

FIG. 3-44

Oblique triangle.

3-18

MATHEMATICS

2. Given the three sides a, b, c, s = a (a + b + c); r=



(s − a)(s − b)(s − c)  s

1 r 1 r 1 r tan  α =  ; tan  β =  ; tan  γ =  2 s−a 2 s−b 2 s−c 3. Given any two sides a, c and an angle opposite one of them α, sin γ = (c sin α)/a; β = 180° − a − γ; b = (a sin β)/(sin α). There may be two solutions here. γ may have two values γ1, γ2; γ1 < 90°, γ2 = 180° − γ1 > 90°. If α + γ2 > 180°, use only γ1. This case may be impossible if sin γ > 1. 4. Given any side c and two angles α and β, γ = 180° − α − β; a = (c sin α)/(sin γ); b = (c sin β)/(sin γ).

cosh y  cosh x sinh y; cosh (x  y) = cosh x cosh y  sinh x sinh y; 2 sinh2 x/2 = cosh x − 1; 2 cosh2 x/2 = cosh x + 1; sinh (−x) = −sinh x; cosh (−x) = cosh x; tanh (−x) = −tanh x. When u = a cosh x, v = a sinh x, then u2 − v2 = a2; which is the equation for a hyperbola. In other words, the hyperbolic functions in the parametric equations u = a cosh x, v = a sinh x have the same relation to the hyperbola u2 − v2 = a2 that the equations u = a cos θ, v = a sin θ have to the circle u2 + v2 = a2. Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh−1 x or arcsinh x. sinh−1 x = loge (x +  x2 +1) 1 1+x −1 2  1 tanh −1 x =  loge  ; cosh x = loge (x  x); 2 1−x 1 x+1 1 + 1 −x2 coth−1 x =  loge  ; sech−1 x = loge  ; 2 x−1 x



HYPERBOLIC TRIGONOMETRY The hyperbolic functions are certain combinations of exponentials ex and e−x. ex + e−x ex − e−x sinh x ex − e−x cosh x =  ; sinh x =  ; tanh x =  =  2 2 cosh x ex + e−x 1 ex + e−x 1 cosh x 2 coth x =  =  =  ; sech x =  =  ; ex − e−x tanh x sinh x cosh x ex + e−x 1 2 csch x =  =  sinh x ex − e−x Fundamental Relationships sinh x + cosh x = ex; cosh x − sinh x = e−x; cosh2 x − sinh2 x = 1; sech2 x + tanh2 x = 1; coth2 x − csch2 x = 1; sinh 2x = 2 sinh x cosh x; cosh 2x = cosh2 x + sinh2 x = 1 + 2 sinh2 x = 2 cosh2 x − 1. tanh 2x = (2 tanh x)/(1 + tanh2 x); sinh (x  y) = sinh x



+x2 1 + 1 csch−1 = loge  x





Magnitude of the Hyperbolic Functions cosh x ≥ 1 with equality only for x = 0; −∞ < sinh x < ∞; −1 < tanh x < 1. cosh x ∼ ex/2 as x → ∞; sinh x → ex/2 as x → ∞. APPROXIMATIONS FOR TRIGONOMETRIC FUNCTIONS For small values of θ (θ measured in radians) sin θ ≈ θ, tan θ ≈ θ; cos θ ≈ 1 − (θ2/2). The behavior ratio of the functions as θ → 0 is given by the following: lim sin θ/θ = 1; sin θ/tan θ = 1. θ→0

DIFFERENTIAL AND INTEGRAL CALCULUS REFERENCES: Char, B. W., et al., Maple V Language Reference Manual, Springer-Verlag, New York (1991); Wolfram, S., The Mathematics Book, 5th ed., Wolfram Media (2003).

The dimensions are then x = 40 ft, y = 40 ft, h = 16,000/(40 × 40) = 10 ft. Symbolically, the original cost relationship is written

DIFFERENTIAL CALCULUS

and the volume relation

An Example of Functional Notation Suppose that a storage warehouse of 16,000 ft3 is required. The construction costs per square foot are $10, $3, and $2 for walls, roof, and floor respectively. What are the minimum cost dimensions? Thus, with h = height, x = width, and y = length, the respective costs are Walls = 2 × 10hy + 2 × 10hx = 20h(y + x) Roof = 3xy Floor = 2xy Total cost = 2xy + 3xy + 20h(x + y) = 5xy + 20h(x + y)

(3-1)

and the restriction Total volume = xyh

(3-2)

Solving for h from Eq. (3-2), h = volume/xy = 16,000/xy 320,000 Cost = 5xy +  (y + x) = 5xy + 320,000 xy In this form it can be shown that the minimum x = y; therefore Cost = 5x2 + 640,000 (1/x)

(3-3) 1 1 (3-4) + x y cost will occur for





Cost = f(x, y, h) = 5xy + 20h(y + x) Volume = g(x, y, h) = xyh = 16,000 In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriction of the problem, relative to volume, the function g(x, y, z) = xyh becomes a constraint function. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables (x, y, h) and no degrees of freedom, that is, freedom of independent selection. Limits The limit of function f(x) as x approaches a (a is finite or else x is said to increase without bound) is the number N. lim f(x) = N x→a

This states that f(x) can be calculated as close to N as desirable by making x sufficiently close to a. This does not put any restriction on f(x) when x = a. Alternatively, for any given positive number ε, a number δ can be found such that 0 < |a − x| < δ implies that |N − f(x)| < ε. The following operations with limits (when they exist) are valid: lim bf(x) = b lim f(x)

By evaluation, the smallest cost will occur when x = 40. Cost = 5(1600) + 640,000/40 = $24,000

x→a

x→a

lim [f(x) + g(x)] = lim f(x) + lim g(x) x→a

x→a

x→a

DIFFERENTIAL AND INTEGRAL CALCULUS lim [f(x)g(x)] = lim f(x) ⋅ lim g(x) x→a

x→a

dy 1 = dx dx/dy

x→a

lim f(x) x→a f(x) lim  =  lim g(x) x→a g(x)

d df  f n = nf n − 1  dx dx

if lim g(x) ≠ 0

x→a

x→a

See “Indeterminant Forms” below when g(a) 0. Continuity A function f(x) is continuous at the point x = a if



df df dv =× dx dv dx

h→0

Rigorously, it is stated f(x) is continuous at x = a if for any positive ε there exists a δ > 0 such that |f(a + h) − f(a)| < ε for all x with |x − a| < δ. For example, the function (sin x)/x is not continuous at x = 0 and therefore is said to be discontinuous. Discontinuities are classified into three types: y = sin x/x at x = 0 y = 1/x at x = 0 y = 10/(1 + e1/x) at x = 0+ y = 0+ x=0 y=0 x = 0− y = 10

Here

exists. This implies continuity at x = a. Conversely, a function may be continuous but not have a derivative. The derivative function is f(x + h) − f(x) df f′(x) =  = lim  h dx h→0 Differentiation Define ∆y = f(x + ∆x) − f(x). Then dividing by ∆x ∆y f(x + ∆x) − f(x)  =  ∆x ∆x

then

∆y dy lim  =  ∆x→0 ∆x dx

Example Find the derivative of y = sin x. dy sin (x + ∆x) − sin(x)  = lim  dx ∆x→0 ∆x sin x cos ∆x + sin ∆x cos x − sin x = lim  ∆x→0 ∆x

Differential Operations The following differential operations are valid: f, g, . . . are differentiable functions of x, c and n are constants; e is the base of the natural logarithms.

d dg df  (f × g) = f  + g  dx dx dx

dax  = (ln a) ax dx

(3-14)

d d d d d  x2 +  y3 =  x +  xy +  A dx dx dx dx dx dy dy 2x + 3y2  = 1 + y + x  + 0 dx dx

by rules (3-10), (3-10), (3-6), (3-8), and (3-5) respectively. dy 2x − 1 − y  =  dx x − 3y2

Thus

Differentials dex = ex dx

(3-15a)

d(ax) = ax log a dx

(3-15b)

d ln x = (1/x) dx

(3-16)

d log x = (log e/x)dx

(3-17)

d sin x = cos x dx

(3-18)

d cos x = −sin x dx

(3-19)

d tan x = sec x dx

(3-20)

d cot x = −csc2 x dx

(3-21)

d sec x = tan x sec x dx

(3-22)

d csc x = −cot x csc x dx

(3-23)

d sin−1 x = (1 − x2)−1/2 dx

(3-24)

d cos−1x = −(1 − x2)−1/2 dx

(3-25)

d tan−1 x = (1 + x2)−1 dx

(3-26)

d cot−1 x = −(1 + x2)−1 dx

(3-27)

d sec x = x (x − 1)

sin ∆x = cos x since lim  = 1 ∆x→0 ∆x

d df dg  (f + g) =  +  dx dx dx

(3-13)

−1

sin x(cos ∆x − 1) sin ∆x cos x = lim  + lim  ∆x→0 ∆x→0 ∆x ∆x

dx =1 dx

(3-12)

dg df g df  = g f g − 1  + f g ln f  dx dx dx

2

dy f(x + ∆x) − f(x) lim   = ∆x→0 dx ∆x

dc =0 dx

(chain rule)

(3-11)

Example Derive dy/dx for x2 + y3 = x + xy + A.

Derivative The function f(x) has a derivative at x = a, which can be denoted as f ′(a), if f(a + h) − f(a) lim  h→0 h

Call

(3-9) (3-10)

g(df/dx) − f(dg/dx) d f   =  g2 dx g

lim [f(a + h) − f(a)] = 0

1. Removable 2. Infinite 3. Jump

dx ≠0 dy

if

3-19

(3-5) (3-6)

(3-8)

−1/2

2

dx

(3-28)

d csc−1 x = −x−1(x2 − 1)−1/2 dx

(3-29)

d sinh x = cosh x dx

(3-30)

d cosh x = sinh x dx

(3-31)

d tanh x = sech2 x dx

(3-32)

d coth x = −csch2 x dx

(3-33)

d sech x = −sech x tanh x dx

(3-34)

d csch x = −csch x coth x dx

(3-35)

d sinh−1 x = (x2 + 1)−1/2 dx

(3-36)

d cosh−1 = (x2 − 1)−1/2 dx

(3-37)

−1

(3-7)

−1

2 −1

d tanh x = (1 − x ) dx

(3-38)

d coth−1 x = −(x2 − 1)−1 dx

(3-39)

d sech−1 x = −(1/x)(1 − x2)−1/2 dx

(3-40)

d csch−1 x = −x−1(x2 + 1)−1/2 dx

(3-41)

3-20

MATHEMATICS

Example Find dy/dx for y = x cos (1 − x2).

Using

d dy d  = x  cos (1 − x2) + cos (1 − x2)  x dx dx dx

(3-8)

d d  cos (1 − x2) = −sin (1 − x2)  (1 − x2) dx dx

(3-19)

= −sin (1 − x2)(0 − 2x)

(3-5), (3-10)

dx 1  =  x−1/2 dx 2

(3-10)

Indeterminate Forms: L’Hospital’s Theorem Forms of the type 0/0, ∞/∞, 0 × ∞, etc., are called indeterminates. To find the limiting values that the corresponding functions approach, L’Hospital’s theorem is useful: If two functions f(x) and g(x) both become zero at x = a, then the limit of their quotient is equal to the limit of the quotient of their separate derivatives, if the limit exists, or is +∞ or −∞. sin x x sin x d sin x cos x lim  = lim  = lim  = 1 x→0 x→0 x→0 x dx 1

Example Find lim . n→0 Here

1 dy  = 2x3/2 sin (1 − x2) +  x−1/2 cos (1 − x2) dx 2

(1.1)x x

Example Find lim .  1000 x→∞

Example Find the derivative of tan x with respect to sin x.

(1.1)x d(1.1)x (ln 1.1)(1.1)x = lim  = lim  lim  x→∞ x1000 x→∞ dx1000 x→∞ 1000x999

v = sin x y = tan x

Using

d tan x dy dy dx == d sin x dv dx dv 1 d tan x = d sin x dx  dx = sec2 x/cos x

(3-12) (3-9) (3-18), (3-20)

Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be applied numerically.

1.1x Obviously lim  = ∞ since repeated application of the rule will reduce the x→∞ x1000 denominator to a finite number 1000! while the numerator remains infinitely large.

Example Find lim x3 e−x. x→∞

6 x3 lim x3 e−x = lim x = lim x = 0 x→∞ x→∞ e x→∞ e

Example Find lim (1 − x)1/x. x→0

y = (1 − x)1/x

Let

ln y = (1/x) ln (1 − x) ln(1 − x) lim (ln y) = lim  = −1 x→0 x→0 x

Example Given the following table of values for differentiable functions f and g; evaluate the following quantities: f(x)

f′(x)

g(x)

g′(x)

1 3 4

3 0 −2

1 2 10

4 4 3

−4 7 6

d  [f(x) + g(x)]| x = 4 = f′(4) + g′(4) = 10 + 6 = 16 dx f′(1)g(1) − f(1)g′(1)

′

1 ⋅ 4 − 3(−4) 16 =  =  = 1 gf (1) =  [g(1)] (4) 16 2

lim y = e−1

Therefore,

x

2

x→0

Partial Derivative The abbreviation z = f(x, y) means that z is a function of the two variables x and y. The derivative of z with respect to x, treating y as a constant, is called the partial derivative with respect to x and is usually denoted as ∂z/∂x or ∂f(x, y)/∂x or simply fx. Partial differentiation, like full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations. Example Find ∂z/∂x and ∂z/∂y for z = ye x + xey. 2

∂e ∂x ∂z  = y  + ey  ∂x ∂x ∂x x2

Higher Differentials The first derivative of f(x) with respect to x is denoted by f′ or df/dx. The derivative of the first derivative is called the second derivative of f(x) with respect to x and is denoted by f″, f (2), or d 2 f/dx 2; and similarly for the higher-order derivatives. Example Given f(x) = 3x3 + 2x + 1, calculate all derivative values at x = 3. df(x)  = 9x2 + 2 dx 2

x = 3, f ′(3) = 9(9) + 2 = 83

d f(x) = 18x  dx2

x = 3, f″(3) = 18(3) = 54

d3f(x) = 18  dx3

x = 3, f″(3) = 18

n

d f(x) =0  dxn

for n ≥ 4

If f ′(x) > 0 on (a, b), then f is increasing on (a, b). If f ′(x) < 0 on (a, b), then f is decreasing on (a, b). The graph of a function y = f(x) is concave up if f ′ is increasing on (a, b); it is concave down if f ′ is decreasing on (a, b). If f ″(x) exists on (a, b) and if f ″(x) > 0, then f is concave up on (a, b). If f ″(x) < 0, then f is concave down on (a, b). An inflection point is a point at which a function changes the direction of its concavity.

∂z ∂ey 2 ∂y  = ex  + x  ∂y ∂y ∂y

= 2xyex + e y

= ex + xey

2

2

Order of Differentiation It is generally true that the order of differentiation is immaterial for any number of differentiations or variables provided the function and the appropriate derivatives are continuous. For z = f(x, y) it follows: ∂3f ∂3f ∂3f =  = 2  2 ∂y ∂x ∂y ∂x ∂y ∂x ∂y General Form for Partial Differentiation 1. Given f(x, y) = 0 and x = g(t), y = h(t). df ∂f dx ∂f dy Then  =   +   dt ∂x dt ∂y dt d 2f ∂2f dx 2 = 2  dt ∂x dt

∂2f

∂2f

∂f

d +  +  +2 ∂x ∂y dt dt ∂y  dt ∂x dt

∂f d 2y + ∂y dt2

2

dx dy

2

dy

2

2

x 2

DIFFERENTIAL AND INTEGRAL CALCULUS Example Find df/dt for f = xy, x = ρ sin t, y = ρ cos t. ∂(xy) d ρ cos t df ∂(xy) d ρ sin t =  +  dt dt dt ∂x ∂y = y(ρ cos t) + x(−ρ sin t) = ρ2 cos2 t − ρ2 sin2 t







The total differential can be written as ∂z ∂z dz =  dx +  dy ∂x y ∂y x and the following condition guarantees path independence. ∂ ∂z ∂ ∂z   =  ∂y ∂x y ∂x ∂y x







2. Given f(x, y) = 0 and x = g(t, s), y = h(t, s). ∂f ∂f ∂x ∂f ∂y Then =+ ∂t ∂x ∂t ∂y ∂t

or





y

(∂y/∂x)z

y

2



 

H=U+pV A = U − TS G = H − TS = U + pV − TS = A + pV S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T and p) are specified, for example. Likewise, V is specified once T and p are specified; it is therefore a state function. All applications are for closed systems with constant mass. If a process is reversible and only p-V work is done, the first law and differentials can be expressed as follows.

∂u  + f′  ∂x 2

2

∂2f ∂u ∂u ∂2u  = f″   + f′  ∂x ∂y ∂x ∂y ∂x ∂y ∂u  + f′  ∂y 2

dU = T dS − p dV dH = T dS + V dp dA = −S dT − p dV dG = −S dT + V dp

2

MULTIVARIABLE CALCULUS APPLIED TO THERMODYNAMICS Many of the functional relationships needed in thermodynamics are direct applications of the rules of multivariable calculus. This section reviews those rules in the context of the needs of themodynamics. These ideas were expounded in one of the classic books on chemical engineering thermodynamics [see Hougen, O. A., et al., Part II, “Thermodynamics,” in Chemical Process Principles, 2d ed., Wiley, New York (1959)]. State Functions State functions depend only on the state of the system, not on past history or how one got there. If z is a function of two variables, x and y, then z(x,y) is a state function, since z is known once x and y are specified. The differential of z is dz = M dx + N dy (M dx + N dy)

Alternatively, if the internal energy is considered a function of S and V, then the differential is: ∂U ∂U dU =  dS +  dV ∂S V ∂V S This is the equivalent of Eq. (3-43) and gives the following definitions.





∂U ∂U T=  , p=−  ∂S V ∂V S Since the internal energy is a state function, then Eq. (3-44) must be satisfied. ∂2U ∂2U = ∂V ∂S ∂S ∂V



This is



∂p

∂T

= −   ∂V ∂S S

is independent of the path in x-y space if and only if ∂M ∂N = ∂y ∂x

(3-45)

x

Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U; enthalpy, H; and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows:

Rule 3. Given f(u) = 0 where u = g(x,y), then ∂f ∂u ∂f ∂u  = f′(u)  +  = f′(u)  ∂x ∂x ∂y ∂y

C

x

   

d 2u

− u2 1 = −2  sin u cos u u



z

Alternatively, divide Eq. (3-43) by dy when holding some other variable w constant to obtain ∂z ∂z ∂x ∂z (3-46)  =   +  ∂y w ∂x y ∂y w ∂y x Also divide both numerator and denominator of a partial derivative by dw while holding a variable y constant to get (∂z/∂w)y ∂z ∂z ∂w (3-47)  ==   ∂x y (∂x/∂w)y ∂w y ∂x y



The line integral

z

∂y ∂z  =− ∂x∂z = − (∂y/∂z) ∂x  ∂y

1 = 2 sin u cos u  (−2x)(1 − x2)−1/2 2

2

x

Rearrangement gives

df d sin2 u d1 −x2  =   dx du dx

∂2f ∂u 2 = f″  ∂y ∂y

∂z

∂z

Example Find df/dx for f = sin2 u and u = 1−x2

2

(3-44)

dy 0 = ∂x dx +  ∂y

 + f′(u) dx

∂2f ∂u 2 = f″  ∂x ∂x

2

Example Suppose z is constant and apply Eq. (3-43).

Differentiation of Composite Function dy ∂f/∂x ∂f Rule 1. Given f(x, y) = 0, then  = −   ≠ 0 . dx ∂f/∂y ∂y Rule 2. Given f(u) = 0 where u = g(x), then du df  = f′(u)  dx dx 2

(3-43)



∂z ∂z = ∂y ∂x ∂x ∂y 2

∂f ∂f ∂x ∂f ∂y =+ ∂s ∂x ∂s ∂y ∂s

du d 2f 2 = f ″(u)  dx dx



3-21

(3-42)

V

This is one of the Maxwell relations, and the other Maxwell relations can be derived in a similar fashion by applying Eq. (3-44). See Sec. 4, Thermodynamics, “Constant-Composition Systems.”

3-22

MATHEMATICS

Partial Derivatives of All Thermodynamic Functions The various partial derivatives of the thermodynamic functions can be classified into six groups. In the general formulas below, the variables U, H, A, G or S are denoted by Greek letters, while the variables V, T, or p are denoted by Latin letters. Type I (3 possibilities plus reciprocals) ∂a ∂p General:  ; Specific:  ∂b c ∂T V Eq. (3-45) gives (∂V/∂T)p ∂p ∂V ∂p  =−   =− (∂V/∂p)T V p T ∂T ∂T ∂V





  

Type II (30 possibilities plus reciprocals) ∂α ∂G General:  ; Specific:  ∂b c ∂T V The differential for G gives ∂G ∂p  = −S + V  ∂T V ∂T V Using the other equations for U, H, A, or S gives the other possibilities. Type III (15 possibilities plus reciprocals) ∂a ∂V General:  ; Specific:  ∂b α ∂T S First expand the derivative using Eq. (3-45). (∂S/∂T)V ∂V ∂S ∂V  =−   =− (∂S/∂V)T ∂T S ∂T V ∂S T Then evaluate the numerator and denominator as type II derivatives. CV ∂V   T ∂V CV ∂p T  =  = −  ∂ p ∂ V S ∂T T ∂V −    ∂T p ∂V T ∂T p













  



 

 

These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Thomson coefficient for constant H. ∂T  ∂p

1 ∂V =  −V + T  Cp ∂T



H

 

Type IV (30 possibilities plus reciprocals)





p

Use Eq. (3-47) to introduce a new variable. ∂G  ∂A

∂G =  p ∂T

(∂G/∂T)p ∂T  = ∂A p (∂A/∂T)p

   p

This operation has created two type II derivatives; by substitution we obtain ∂G S  =  ∂A p S + p (∂V/∂T)p



Type V (60 possibilities plus reciprocals) ∂α ∂G General:  ; Specific:  ∂b β ∂p



∂G



∂T = −S  A ∂p



A

A

+V

The derivative is type III and can be evaluated by using Eq. (3-45). ∂G  ∂p



(∂A/∂p)T = S + V (∂A/∂T)p A

A

Sp (∂V/∂p)T =  + V S + p (∂V/∂T)p

These derivatives are also of interest for free expansions or isentropic changes. Type VI (30 possibilities plus reciprocals) ∂α ∂G General:  ; Specific:  ∂β γ ∂A H We use Eq. (3-47) to obtain two type V derivatives.



∂G  ∂A

H



(∂G/∂T)H = (∂A/∂T)H

These can then be evaluated using the procedures for Type V derivatives. INTEGRAL CALCULUS Indefinite Integral If f ′(x) is the derivative of f(x), an antiderivative of f ′(x) is f(x). Symbolically, the indefinite integral of f ′(x) is

 f ′(x) dx = f(x) + c

where c is an arbitrary constant to be determined by the problem. By virtue of the known formulas for differentiation the following relationships hold (a is a constant):

 (du + dv + dw) =  du +  dv +  dw  a dv = a  dv v  v dv =  + c (n ≠ −1) n+1

(3-49)

dv  = ln |v| + c v

(3-51)

(3-48)

n+1

n

a  a dv =  +c ln a  e dv = e + c  sin v dv = −cos v + c  cos v dv = sin v + c  sec v dv = tan v + c  csc v dv = −cot v + c  sec v tan v dv = sec v + c  csc v cot v dv = −csc v + c dv 1 v  =  tan  + c v +a a a

(3-50)

v

v

(3-52)

v

(3-53) (3-54) (3-55)

2

(3-56)

2

(3-57) (3-58) (3-59)

−1

2

Start from the differential for dG. Then we get

 ∂p

∂G  ∂p

v

p

∂G ∂α General:  ; Specific:  ∂β c ∂A

The two type II derivatives are then evaluated.

(3-60)

2

dv  = sin a −v 

−1

2

2

v +c a

(3-61)

dv 1 v−a  =  ln    + c v −a 2a v+a 2

(3-62)

2

dv   = ln |v +  v a| + c v a 

(3-63)

 sec v dv = ln (sec v + tan v) + c

(3-64)

2

2

2

2

DIFFERENTIAL AND INTEGRAL CALCULUS

 csc v dv = ln (csc v − cot v) + c

(3-65)

=3

Example Find



2

2

7 =− 6

2

 (3 +x dx4x)

Example Find  . Let 3 + 4x = y4; then 4dx = 4y3 dy and 1/4

2

7 ln |v| + c = − 7 ln |2 − 3x2| + c =− 

x dx   = (3 + 4x)

6

1/4

Example—Constant of Integration By definition the derivative of x3 is 3x2, and x3 is therefore the integral of 3x2. However, if f = x3 + 10, it follows that f′ = 3x2, and x3 + 10 is therefore also the integral of 3x2. For this reason the constant c in ∫ 3x2 dx = x3 + c must be determined by the problem conditions, i.e., the value of f for a specified x. Methods of Integration In practice it is rare when generally encountered functions can be directly integrated. For example, the integrand in ∫ si n x dx which appears quite simple has no elementary function whose derivative is  si nx. In general, there is no explicit way of determining whether a particular function can be integrated into an elementary form. As a whole, integration is a trial-and-error proposition which depends on the effort and ingenuity of the practitioner. The following are general procedures which can be used to find the elementary forms of the integral when they exist. When they do not exist or cannot be found either from tabled integration formulas or directly, the only recourse is series expansion as illustrated later. Indefinite integrals cannot be solved numerically unless they are redefined as definite integrals (see “Definite Integral”), i.e., F(x) = ∫ f(x) dx, x indefinite, whereas F(x) = ∫ a f(t) dt, definite. Direct Formula Many integrals can be solved by transformation in the integrand to one of the forms given previously. Example Find ∫ x2 3 x3  + 10 dx. Let v = 3x3 + 10 for which dv = 9x2 dx.

Thus

 x 3x +01 dx =  (3x + 10) 2

3

3

1/2



9

v

1/2

 e dx− 1

y−1 e −1 dy dx (1/y) dy  =   =   = ln  = ln  y e e −1 y−1 y −y x

x

x2 −a2 Let x = a sec θ  x2 +a2 Let x = a tan θ 

x2(A + B + C) + x(−3A + B) + (2A − 2B − 4C) =  (x + 2)(x − 2)(x − 1) Equate coefficients and solve for A, B, and C. A+B+C=0 −3A + B = 0 2A − 2B − 4C = 1 A = 1⁄12, B = d, C = −s 1 1 1 1 =+−  x3 − x2 − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1) Hence

dx dx dx dx   =+− x − x − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)

a2 −x2 Let x = a sin θ 



3

θ 2 −sin 2/31  cos θ dθ   (2/3) sin θ  3 2

2

2

2

Parts An extremely useful formula for integration is the relation

49x 2 2 2  dx. Let x =  sin θ; then dx =  cos θ dθ. x2 3 3

2 (2/3)  − x2 dx = 3  2 x

x

A(x − 2)(x − 1) + B(x + 2)(x − 1) + C(x + 2)(x − 2) =  (x + 2)(x − 2)(x − 1)

Trigonometric Substitution This technique is particularly well adapted to integrands in the form of radicals. For these the function is transformed into a trigonometric form. In the latter form they may be more easily recognizable relative to the identity formulas. These functions and their transformations are

3

2

Partial Fractions Rational functions are of the type f(x)/g(x) where f(x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of f is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types: Type 1 Reducible denominator to linear unequal factors. For example, 1 1 =   x3 − x2 − 4x + 4 (x + 2)(x − 2)(x − 1)

27



4

Example Find  . Let ex = y; then ex dx = dy or dx = 1/y dy. x

dv

2 (3x3 + 10)3/2 + c =

Example Find

2

General The number of possible transformations one might use are unlimited. No specific overall rules can be given. Success in handling integration problems depends primarily upon experience and ingenuity. The following example illustrates the extent to which alternative approaches are possible.

[by Eq. (3-50)]

⁄

32

 y (y − 3) dy

C A B =++ x+2 x−2 x−1 9

1 v3/2 + c = 

y4 − 3  y3 dy 4 1  =  y 4

1 y7 3 y3 1 1 =   −   + c =  (3 + 4x)7/4 −  (3 + 4x)3/4 + c 4 7 4 3 28 4

(x2 dx)

1 (3x3 + 10)1/2(9x2 dx) = 1 =  9

2

−9 x2 4 3 = −  − 3 sin−1  x + c in terms of x x 2

 dvv , with v 2  3x and dv 6x dx

6

2

2

Algebraic Substitution Functions containing elements of the type (a + bx)1/n are best handled by the algebraic transformation yn = a + bx.

7x dx x dx 7 −6x dx  =7=− 2 − 3x 2 − 3x 6 2 − 3x

Thus

cos θ  dθ = 3  cot θ dθ sin θ

= −3 cot θ − 3θ + c by trigonometric transform

Example Find ∫ (3x2 + ex − 10) dx using Eq. (3-48). ∫ (3x2 + ex − 10) dx = 3 ∫ x2 dx + ∫ ex dx − 10 ∫ dx = x3 + ex − 10x + c (by Eqs. 3-50, 3-53). 7x dx 2 . Let v = 2 − 3x2; dv = −6x dx 2 − 3x

3-23

d(uv) = u dv + v du

and or

 u dv +  v du  u dv = uv −  v du uv =

3-24

MATHEMATICS

No general rule for breaking an integrand can be given. Experience alone limits the use of this technique. It is particularly useful for trigonometric and exponential functions. Example Find  xex dx. Let u=x



dv = ex dx

and

du = dx

v = ex

 xe dx = xe −  e dx x

Therefore

x

these restrictions gives rise to so-called improper integrals and requires special handling. These occur when ∞ 1. The limits of integration are not both finite, i.e., ∫ 0 e−x dx. 2. The function becomes infinite within the interval of integration, i.e., 1 1  dx  x 0 Techniques for determining when integration is valid under these conditions are available in the references. Properties The fundamental theorem of calculus states

x

= xex − ex + c

 f(x) dx = F(b) − F(a) b

Example Find  ex sin x dx. Let u = ex

a

dv = sin x dx v = −cos x

du = e dx x

 e sin x dx = −e cos x +  e cos x dx x

x

Other properties of the definite integral are

x

u=e

 c[ f(x) dx] = c  f(x) dx b

dv = cos x dx v = sin x

x

Again

dF(x)/dx = f(x)

where

du = ex dx

a

x

x

b

x

 f(x) dx = 

i

i+1

x4  dx − 2!



for all x

+ ⋅ ⋅ ⋅ + f(ξn − 1)(b − xn − 1)

 f(x) dx = f(b)

∂  ∂a

 f(x) dx = −f(a)

b

a

b

a

∂f(x, α)  dx if a and b are constant ∂α b

dF(α) = dα

a

 dx  f(x, α) dα =  dα  f(x, α) dx b



i+1

d

c

b

c

f(x, y) dy, the Leibniz rule gives

a(x)

 f(x) dx indefinite integral where dF/dx = f(x) F(a, b) =  f(x) dx definite integral F(α) =  f(x, α) dx F(x) =

b

a b

a(x)

∂f  dy ∂x

2

the incorrect value dx 1  = −   (x − 1) x−1 2

0

2

2

= −2

0

It should be noted that f(x) = 1/(x − 1)2 becomes unbounded as x → 1 and by Rule 2 the integral diverges and hence is said not to exist. Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. Change of Variable This substitution is basically the same as previously indicated for indefinite integrals. However, for definite integrals, the limits of integration must also be changed: i.e., for x = φ(t),

 f(x) dx =  b

a

There are certain restrictions of the integration definition, “The function f(x) must be continuous in the finite interval (a, b) with at most a finite number of finite discontinuities,” which must be observed before integration formulas can be generally applied. Relaxing two of

b(x)

2

i=1

where the points x1, x2, . . . , xn are equally spaced. Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the function but not on the dummy variable of integration x. Symbolically



dx  . Direct application of the formula would yield (x − 1) 0

− xi)

(3-66)

a

b(x)

When F(x) =

xi − 1 ≤ ξi − 1 ≤ xi

n

i

d

a

Example Find

 f(x) dx = lim f(ξ )(x

or

∂  ∂b

dF db da  =  f [x, b(x)] −  f [x, a(x)] + dx dx dx

The definite integral of f(x) is defined as n→∞

c

a

i=1

a

a

 f(x) dx b

f(x) dx +

 f(x) dx = (b − a)f(ξ) for some ξ in (a, b) x6  dx + ⋅ ⋅ ⋅ 3!

− xi) = f(ξ1)(x2 − a) + f(ξ2)(x3 − x2)

b

c

b

Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation ∫ as it turns out treats the integrand in the same way for both. Consider the function f(x) = 10 − 10e−2x. Define x1 = a and xn = b, and suppose it is desirable to compute the area between the curve and the coordinate axis y = 0 and bounded by x1 = a, xn = b. Obviously, by a sufficiently large number of rectangles this area could be approximated as closely as desired by the formula

f(ξ )(x

b

b

a

x3 x5 x7 =x−+−+⋅⋅⋅ 3 5.2! 7.3!

n−1

a

a

a

x4 x6 2 e− x = 1 − x2 +  −  + ⋅ ⋅ ⋅ 2! 3!

 dx −  x dx + 

2

a

 f(x) dx = − f(x) dx

2

2

b

1

b

Example Find  e− x dx. Since

dx =

2

a

Series Expansion When an explicit function cannot be found, the integration can sometimes be carried out by a series expansion.

e

b

1

c = (ex/2)(sin x − cos x) +  2

−x2

a

 [ f (x) + f (x)] dx =  f (x) dx +  f (x) dx

 e sin x dx = −e cos x + e sin x −  e sin x dx + c x

b

a

where

t = t0 when x = a t = t1 when x = b

t1

t0

f [φ(t)]φ′(t) dt

INFINITE SERIES

 16−x dx. Let

x

4

Example Find

 16−x dx = 16 

π/2

2

0

0

0



1

0

1 dx =  (α > −1) α+1

Then multiplying both sides by dα and integrating between a and b,

 dα  x b

1

α

cos2 θ dθ = 16[aθ + d sin 2θ]0π/2 = 4π

a

Integration It is sometimes useful to generate a double integral to solve a problem. By this approach, the fundamental theorem indicated by Eq. (3-66) can be used. Example Find

α

Consider

x = 4 sin θ (x = 0, θ = 0) dx = 4 cos θ dθ (x = 4, θ = π/2)

4

Then

1

2

0

3-25

xb − xa  dx ln x

0



b

dx =

a

b+1 dα  = ln  α+1 a+1





But also

 dα  x b

1

dx =

0



1

Therefore

 dx  x 1

α

a

0

b

α

0

a



1

dα =

0

xb − xa b+1  dx = ln  ln x a+1



xb − xa  dx ln x



INFINITE SERIES REFERENCES: de Brujin, N. G., Asymptotic Methods in Analysis, Dover, New York (1981); Folland, G. B., Advanced Calculus, Prentice-Hall, Saddle River, N.J. (2002); Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic, New York (2000); Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003).

DEFINITIONS A succession of numbers or terms that are formed according to some definite rule is called a sequence. The indicated sum of the terms of a sequence is called a series. A series of the form a0 + a1(x − c) + a2(x − c)2 + ⋅ ⋅ ⋅ + an(x − c)n + ⋅ ⋅ ⋅ is called a power series. Consider the sum of a finite number of terms in the geometric series (a special case of a power series). Sn = a + ar + ar 2 + ar 3 + ⋅ ⋅ ⋅ + ar n − 1

(3-67)

For any number of terms n, the sum equals 1 − rn Sn = a  1−r In this form, the geometric series is assumed finite. In the form of Eq. (3-67), it can further be defined that the terms in the series be nonending and therefore an infinite series. S = a + ar + ar 2 + ⋅ ⋅ ⋅ + ar n + ⋅ ⋅ ⋅

(3-68)

However, the defined sum of the terms [Eq. (3-67)] 1 − rn r≠1 Sn = a  1−r while valid for any finite value of r and n now takes on a different interpretation. In this sense it is necessary to consider the limit of Sn as n increases indefinitely: S = lim Sn n→∞

1 − rn = a lim  n→∞ 1 − r For this, it is stated the infinite series converges if the limit of Sn approaches a fixed finite value as n approaches infinity. Otherwise, the series is divergent. On this basis an analysis of 1 − rn S = a lim  n→∞ 1 − r shows that if r is less than 1 but greater than −1, the infinite series is convergent. For values outside of the range −1 < r < 1, the series is divergent because the sum is not defined. The range −1 < r < 1 is called the region of convergence. (We assume a ≠ 0.)

There are also two types of convergent series. Consider the new series 1 1 1 1 S = 1 −  +  −  + ⋅ ⋅ ⋅ + (−1)n + 1  + ⋅ ⋅ ⋅ 2 3 4 n

(3-69)

It can be shown that the series (3-69) does converge to the value S = log 2. However, if each term is replaced by its absolute value, the series becomes unbounded and therefore divergent (unbounded divergent): 1 1 1 1 S=1+++++⋅⋅⋅ 2 3 4 5

(3-70)

In this case the series (3-69) is defined as a conditionally convergent series. If the replacement series of absolute values also converges, the series is defined to converge absolutely. Series (3-69) is further defined as an alternating series, while series (3-70) is referred to as a positive series. OPERATIONS WITH INFINITE SERIES 1. The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection “Tests for Convergence and Divergence.” 2. If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series; it can in fact be made divergent or oscillatory (Riemann’s theorem). 3. A series of positive terms, if convergent, has a sum independent of the order of its terms; but if divergent, it remains divergent however its terms are rearranged. 4. An oscillatory series can always be made to converge by grouping terms. 5. A power series can be inverted, provided the first-degree term is not zero. Given y = b1 x + b2 x2 + b3 x3 + b4 x4 + b5 x5 + b6 x6 + b7 x7 + ⋅ ⋅ ⋅ then

x = B1 y + B2 y2 + B3 y3 + B4 y4 + B5 y5 + B6 y6 + B7 y7 + ⋅ ⋅ ⋅

where B1 = 1/b1 B2 = −b2 /b13 B3 = (1/b15 ) (2b22 − b1b3 ) B4 = (1/b17 )(5b1b2 b3 − b12 b4 − 5b23 ) Additional coefficients are available in the references. 6. Two series may be added or subtracted term by term provided each is a convergent series. The joint sum is equal to the sum (or difference) of the individuals.

3-26

MATHEMATICS

7. The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. 8. A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. If f(x) = a0 + a1x + a2x2 + ⋅ ⋅ ⋅ , then



x2

x1

f(x) dx =



x2

x1

a0 dx +



x2

x1

a1x dx +



x2

x1

a2 x2 dx + ⋅ ⋅ ⋅

9. A power series may be differentiated term by term and represents the function df(x)/dx within the same region of convergence as f(x).

Harmonic Progression n

In general, the problem of determining whether a given series will converge or not can require a great deal of ingenuity and resourcefulness. There is no all-inclusive test which can be applied to all series. As the only alternative, it is necessary to apply one or more of the developed theorems in an attempt to ascertain the convergence or divergence of the series under study. The following defined tests are given in relative order of effectiveness. For examples, see references on advanced calculus. 1. Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms. 2. nth-Term Test. A series is divergent if the nth term of the series does not approach zero as n becomes increasingly large. 3. Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 4. Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 5. Cauchy’s Root Test. If the nth root of the absolute value of the nth term, as n becomes unbounded, approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 6. Maclaurin’s Integral Test. Suppose an is a series of positive terms and f is a continuous decreasing function such that f(x) ≥ 0 for 1∞ ≤ x < ∞ and f(n) = an. Then the series and the improper integral ∫ 1 f(x) dx either both converge or both diverge.

n



j=1 n



j=1

Example Find a series expansion for f(x) = ln (1 + x) about x0 = 0. f′(x) = (1 + x)−1, thus

f″(0) = −1, 3

f′′′(1) = 2, etc.

4

which converges for −1 < x ≤ 1.

Maclaurin’s Series x2 x3 f(x) = f(0) + xf′(0) +  f″(0) +  f′′′(0) + ⋅ ⋅ ⋅ 2! 3! This is simply a special case of Taylor’s series when h is set to zero. Exponential Series xn x2 x3 ex = 1 + x +  +  + ⋅ ⋅ ⋅ +  + ⋅ ⋅ ⋅ − ∞ < x < ∞ 2! 3! n! Logarithmic Series x−1 1 x−1 ln x =  +   x x 2



x−1

1 x−1

+⋅⋅⋅ + 3  x 2

x−1

+  +⋅⋅⋅  x + 1 3 x + 1 1

x2 x4 x6 cos x = 1 −  +  −  + ⋅ ⋅ ⋅ 2! 4! 6!

3

3

(x > a)

(x > 0)

−∞ < x < ∞ −∞ < x < ∞

x3 1 3 x5 1 3 5 x7 sin−1 x = x +  +  ⋅  ⋅  +  ⋅  ⋅  ⋅  + ⋅ ⋅ ⋅ 6 2 4 5 2 4 6 7

[a + (k − 1)d] = a + (a + d) + (a + 2d)

+ (a + 3d) + ⋅ ⋅ ⋅ + [a + (n − 1)]d 1 = na +  n(n − 1)d 2

Geometric Progression j=1

f′(0) = 1,

f′′′(x) = 2(1 + x)−3, etc.

xn x x x ln (x + 1) = x −  +  −  + ⋅ ⋅ ⋅ + (−1)n + 1  + ⋅ ⋅ ⋅ 2 3 4 n

n

1 − rn = a + ar + ar 2 + ar 3 + ⋅ ⋅ ⋅ + ar n − 1 = a  1−r

f(0) = 0,

f″(x) = −(1 + x)−2,

2

Arithmetic Progression

j−1

1

f″(x0) f′′′(x0) or f(x) = f(x0) + f ′(x0) (x − x0) +  (x − x0)2 +  (x − x0)3 + ⋅ ⋅ ⋅ 2! 3!

n2(n + 1)2 j 3 =  = 13 + 23 + 33 + ⋅ ⋅ ⋅ + n3 4

n

1

x2 x3 f(h + x) = f(h) + xf′(h) +  f ″(h) +  f ′′′(h) + ⋅ ⋅ ⋅ 2! 3!

Trigonometric Series* x3 x5 x7 sin x = x −  +  −  + ⋅ ⋅ ⋅ 3! 5! 7!

n(n + 1)(2n + 1) j 2 =  = 12 + 22 + 32 + 42 + ⋅ ⋅ ⋅ + n2 6

ar

1

Taylor’s Series

ln x = 2

n(n + 1) j==1+2+3+4+⋅⋅⋅+n 2

k=1

1

Binomial Series (see also Elementary Algebra)

Sums for the First n Numbers to Integer Powers j=1

1

The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series.

SERIES SUMMATION AND IDENTITIES



1

k=0

n(n − 1)(n − 2) n(n − 1) (1  x)n = 1  nx +  x2   x3 + ⋅ ⋅ ⋅ (x2 < 1) 2! 3!

TESTS FOR CONVERGENCE AND DIVERGENCE

n

1

=+++++⋅⋅⋅+

 a + kd a a + d a + 2d a + 3d a + 4d a + nd

r≠1

1 1 1 tan−1 x = x −  x3 +  x5 −  x7 + ⋅ ⋅ ⋅ 3 5 7

(x2 < 1)

(x2 < 1)

Taylor Series The Taylor series for a function of two variables, expanded about the point (x0, y0), is * tan x series has awkward coefficients and should be computed as sin x (sign)  . 2 1 − sin  x





COMPLEX VARIABLES ∂f f(x, y) = f(x0, y0) +  ∂x 1 ∂2f +  2 2! ∂x

 



x 0, y0

∂f (x − x0) +  ∂y

∂2f (x − x0)2 + 2  x 0, y0 ∂x∂y





x 0, y0

x 0, y0

(y − y0)

(x − x0)(y − y0)

∂2f + 2 ∂y



x 0, y0



(y − y0)2 + ⋅ ⋅ ⋅

3-27

Partial Sums of Infinite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent series either converges rapidly, in which case almost any test (among those presented in the preceding subsections) will do; or it converges slowly, in which case it is not going to be of much use unless there is some way to get at its sum without adding up an unreasonable number of terms. To find out, as accurately as possible, how fast a convergent series converges and how fast a divergent series diverges, see Boas, R. P., Jr., Am. Math. Mon. 84: 237–258 (1977).

COMPLEX VARIABLES REFERENCES: Ablowitz, M. J., and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge University Press, New York (2003); Asmar, N., and G. C. Jones, Applied Complex Analysis with Partial Differential Equations, Prentice-Hall, Upper Saddle River, N.J. (2002); Brown, J. W., and R. V. Churchill, Complex Variables and Applications, 7th ed., McGraw-Hill, New York (2003); Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003); Kwok, Y. K., Applied Complex Variables for Scientists and Engineers, Cambridge University Press, New York (2002); McGehee, O. C., An Introduction to Complex Analysis, Wiley, New York (2000); Priestley, H. A., Introduction to Complex Analysis, Oxford University Press, New York (2003).

Numbers of the form z = x + iy, where x and y are real, i = −1, are called complex numbers. The numbers z = x + iy are representable in the plane as shown in Fig. 3-45. The following definitions and terminology are used: 1. Distance OP = r = modulus of z written |z|. |z| =  x2 +y2. 2. x is the real part of z. 3. y is the imaginary part of z. 4. The angle θ, 0 ≤ θ < 2π, measured counterclockwise from the positive x axis to OP is the argument of z. θ = arctan y/x = arcsin y/r = arccos x/r if x ≠ 0, θ = π/2 if x = 0 and y > 0. 5. The numbers r, θ are the polar coordinates of z. 6. z⎯ = x − iy is the complex conjugate of z. 2

ALGEBRA Let z1 = x1 + iy1, z2 = x2 + iy2. Equality z1 = z2 if and only if x1 = x2 and y1 = y2. Addition z1 + z2 = (x1 + x2) + i(y1 + y2). Subtraction z1 − z2 = (x1 − x2) + i(y1 − y2). Multiplication z1 ⋅ z2 = (x1x2 − y1y2) + i(x1y2 + x2y1). x2y1 − x1y2 x1x2 + y1y2 Division z1 /z2 =  + i  , z2 ≠ 0. x 22 + y22 x 22 + y22 SPECIAL OPERATIONS zz = x2 + y2 = |z|2; z1 z2 = z1  z2; z1 = z1; z1z2 = z1z2; |z1 ⋅ z2| = |z1| ⋅ |z2|; arg (z1 ⋅ z2) = arg z1 + arg z2; arg (z1 /z2) = arg z1 − arg z2; i4n = 1 for n any 2n integer; i = −1 where n is any odd integer; z + z = 2x; z − ⎯z = 2iy. Every complex quantity can be expressed in the form x + iy. TRIGONOMETRIC REPRESENTATION By referring to Fig. 3-45, there results x = r cos θ, y = r sin θ so that z = x + iy = r (cos θ + i sin θ), which is called the polar form of the

complex number. cos θ + i sin θ = e iθ. Hence z = x + iy = re iθ. z = x − iy = re−iθ. Two important results from this are cos θ = (eiθ + e−iθ)/2 and sin θ = (eiθ − e−iθ)/2i. Let z1 = r1e iθ1, z2 = r2e iθ2. This form is convenient for multiplication for z1z2 = r1 r2 e i(θ1 + θ2) and for division for z1 /z2 = (r1 /r2)ei(θ1 − θ2), z2 ≠ 0. POWERS AND ROOTS If n is a positive integer, zn = (reiθ)n = r neinθ = r n(cos nθ + i sin nθ). If n is a positive integer, θ + 2kπ θ + 2kπ z1/ n = r 1/ nei[(θ + 2kπ)/n] = r 1/n cos  + i sin  n n

 







and selecting values of k = 0, 1, 2, 3, . . . , n − 1 give the n distinct values of z1/n. The n roots of a complex quantity are uniformly spaced around a circle, with radius r 1/n, in the complex plane in a symmetric fashion. Example Find the three cube roots of −8. Here r = 8, θ = π. The roots are z0 = 2(cos π/3 + i sin π/3) = 1 + i 3, z1 = 2(cos π + i sin π) = −2, z2 = 2(cos 5π/3 + i sin 5π/3) = 1 − i 3. ELEMENTARY COMPLEX FUNCTIONS Polynomials A polynomial in z, anzn + an − 1zn − 1 + ⋅ ⋅ ⋅ + a0, where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been defined. Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m is counted m times. Exponential Functions The exponential function ez is defined by the equation ez = ex + iy = ex ⋅ eiy = ex(cos y + i sin y). Properties: e0 = 1; ez1 ⋅ ez2 = ez1 + z2; ez1/ez2 = ez1 − z2; ez + 2kπi = ez. Trigonometric Functions sin z = (eiz − e−iz)/2i; cos z = (eiz + e−iz)/2; tan z = sin z/cos z; cot z = cos z/sin z; sec z = 1/cos z; csc z = 1/sin z. Fundamental identities for these functions are the same as their real counterparts. Thus cos2 z + sin2 z = 1, cos (z1  z2) = cos z1 cos z2 " sin z1 sin z2, sin (z1  z2) = sin z1 cos z2  cos z1 sin z2. The sine and cosine of z are periodic functions of period 2π; thus sin (z + 2π) = sin z. For computation purposes sin z = sin (x + iy) = sin x cosh y + i cos x sinh y, where sin x, cosh y, etc., are the real trigonometric and hyperbolic functions. Similarly, cos z = cos x cosh y − i sin x sinh y. If x = 0 in the results given, cos iy = cosh y, sin iy = i sinh y. Example Find all solutions of sin z = 3. From previous data sin z = sin x cosh y + i cos x sinh y = 3. Equating real and imaginary parts sin x cosh y = 3, cos x sinh y = 0. The second equation can hold for y = 0 or for x = π/2, 3π/2, . . . . If y = 0, cosh 0 = 1 and sin x = 3 is impossible for real x. Therefore, x =  π/2, 3π/2, . . . (2n + 1)π/2, n = 0, 1, 2, . . . . However, sin 3π/2 = −1 and cosh y ≥ 1. Hence x = π/2, 5π/2, . . . . The solution is z = [(4n + 1)π]/2 + i cosh−13, n = 0, 1, 2, 3, . . . . Example Find all solutions of ez = −i. ez = ex(cos y + i sin y) = −i. Equating real and imaginary parts gives ex cos y = 0, ex sin y = −1. From the first y = π/2, 3π/2, . . . . But ex > 0. Therefore, y = 3π/2, 7π/2, −π/2, . . . . Then x = 0. The solution is z = i[(4n + 3)π]/2.

FIG. 3-45

Complex plane.

Two important facets of these functions should be recognized. First, the sin z is unbounded; and, second, ez takes all complex values except 0.

3-28

MATHEMATICS

Hyperbolic Functions sinh z = (ez − e−z)/2; cosh z = (ez + e−z)/2; tanh z = sinh z/cosh z; coth z = cosh z/sinh z; csch z = 1/sinh z; sech z = 1/cosh z. Identities are: cosh2 z − sinh2 z = 1; sinh (z1 + z2) = sinh z1 cosh z2 + cosh z1 sinh z2; cosh (z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2; cosh z + sinh z = ez; cosh z − sinh z = e−z. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 2πi. That is, sinh (z + 2πi) = sinh z. Logarithms The logarithm of z, log z = log |z| + i(θ + 2nπ), where log |z| is taken to the base e and θ is the principal argument of z, that is, the particular argument lying in the interval 0 ≤ θ < 2π. The logarithm of z is infinitely many valued. If n = 0, the resulting logarithm is called the principal value. The familiar laws log z1 z2 = log z1 + log z2, log z1 /z2 = log z1 − log z2, log zn = n log z hold for the principal value. General powers of z are defined by zα = eα log z. Since log z is infinitely many valued, so too is zα unless α is a rational number. DeMoivre’s formula can be derived from properties of ez. zn = rn (cos θ + i sin θ)n = rn (cos nθ + i sin nθ) (cos θ + i sin θ) = cos nθ + i sin nθ

Cauchy-Riemann equations are satisfied and ∂u ∂u ∂v ∂v u, v,  ,  ,  ,  ∂x ∂y ∂x ∂y are continuous in a region of the complex plane, then f(z) is analytic in that region. Example w = zz = x2 + y2. Here u = x2 + y2, v = 0. ∂u/∂x = 2x, ∂u/∂y = 2y, ∂v/∂x = ∂v/∂y = 0. These are continuous everywhere, but the Cauchy-Riemann equations hold only at the origin. Therefore, w is nowhere analytic, but it is differentiable at z = 0 only. Example w = ez = ex cos y + iex sin y. u = ex cos y, v = ex sin y. ∂u/∂x = ex cos y, ∂u/∂y = −ex sin y, ∂v/∂x = ex sin y, ∂v/∂y = ex cos y. The continuity and CauchyRiemann requirements are satisfied for all finite z. Hence ez is analytic (except at ∞) and dw/∂z = ∂u/∂x + i(∂v/∂x) = ez. 1 z

Thus

COMPLEX FUNCTIONS (ANALYTIC) In the real-number system a greater than b(a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow: |z| ≥ x; |z| ≥ y; |z1  z2| ≤ |z1| + |z2|; |z1 − z2| ≥ ||z1| − |z2||; |z| ≥ (|x| + |y|)/2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities.  Example |z − 3| ≤ 5. This is equivalent to (x −3 )2 +y2 ≤ 5, which is the

set of all points within and on the circle, centered at x = 3, y = 0 of radius 5.

Example |z − 1| ≤ x represents the set of all points inside and on the

parabola 2x = y2 + 1 or, equivalently, 2x ≥ y2 + 1.

Functions of a Complex Variable If z = x + iy, w = u + iv and if for each value of z in some region of the complex plane one or more values of w are defined, then w is said to be a function of z, w = f(z). Some of these functions have already been discussed, e.g., sin z, log z. All functions are reducible to the form w = u(x, y) + iv(x, y), where u, v are real functions of the real variables x and y. Example z3 = (x + iy)3 = x3 + 3x2(iy) + 3x(iy)2 + (iy)3 = (x3 − 3xy2) +

i(3x2y − y3).

Differentiation The derivative of w = f(z) is f(z + ∆z) − f(z) dw lim   = ∆z→0 dz ∆z and for the derivative to exist the limit must be the same no matter how ∆z approaches zero. If w1, w2 are differentiable functions of z, the following rules apply: dw2 d(w1  w2) dw1  =    dz dz dz

d(w1w2) dw1 dw2  = w2  + w1  dz dz dz

x − iy x +y

y x +y

It is easy to see that dw/dz exists except at z = 0. Thus 1/z is analytic except at z = 0.

Singular Points If f(z) is analytic in a region except at certain points, those points are called singular points. Example 1/z has a singular point at zero. Example tan z has singular points at z = (2n + 1)(π/2), n = 0, 1, 2, . . . . The derivatives of the common functions, given earlier, are the same as their real counterparts. Example (d/dz)(log z) = 1/z, (d/dz)(sin z) = cos z. Harmonic Functions Both the real and the imaginary parts of any analytic function f = u + iv satisfy Laplace’s equation ∂2φ/∂x2 + ∂2φ/∂y2 = 0. A function which possesses continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function. Example ez = ex cos y + iex sin y. u = ex cos y, ∂u/∂x = ex cos y, ∂2u/∂x2 = ex cos y, ∂u/∂y = −ex sin y, ∂2u/∂y2 = −ex cos y. Clearly ∂2u/∂x2 + ∂2u/∂y2 = 0. Similarly, v = ex sin y is also harmonic. If w = u + iv is analytic, the curves u(x, y) = c and v(x, y) = k intersect at right angles, if wi(z) ≠ 0. Integration In much of the work with complex variables a simple extension of integration called line or curvilinear integration is of fundamental importance. Since any complex line integral can be expressed in terms of real line integrals, we define only real line integrals. Let F(x,y) be a real, continuous function of x and y and c be any continuous curve of finite length joining the points A and B (Fig. 3-46). F(x,y) is not related to the curve c. Divide c up into n segments, ∆si, whose projection on the x axis is ∆xi and on the y axis is ∆yi. Let (εi, ηi) be the coordinates of an arbitrary point on ∆si. The limits of the sums

F(ε , η ) ∆s =  F(x, y) ds n

lim

∆si→0 i=1

d(w1/w2) w2(dw1/dz) − w1(dw2/dz)  =  w22 dz dw1n dw1  = nw1n − 1  dz dz For w = f(z) to be differentiable, it is necessary that ∂u/∂x = ∂v/∂y and ∂v/∂x = −∂u/∂y. The last two equations are called the CauchyRiemann equations. The derivative ∂u dw ∂u ∂v ∂v =+i=−i dz ∂x ∂x ∂y ∂y and

If f(z) possesses a derivative at zo and at every point in some neighborhood of z0, then f(z) is said to be analytic or homomorphic at z0. If the

x x +y

Example w =  =  = −i 2 2 2 2 2 2

n

FIG. 3-46

Line integral.

i

i

i

c

DIFFERENTIAL EQUATIONS lim

F(ε , η ) ∆x =  F(x, y) dx

lim

F(ε , η ) ∆y =  F(x, y) dy

3-29

n

i

∆si→0 i=1

i

i

c

n

i

∆si→0 i=1

i

i

c

are known as line integrals. Much of the initial strangeness of these integrals will vanish if it be observed that the ordinary definite integral b ∫a f(x) dx is just a line integral in which the curve c is a line segment on the x axis and F(x, y) is a function of x alone. The evaluation of line integrals can be reduced to evaluation of ordinary integrals. Example ∫c y(1 + x) dy, where c: y = 1 − x21 from (−1, 0) to (1, 0). Clearly y = 1 − x2, dy = −2x dx. Thus ∫c y(1 + x) dy = −2 ∫ −1 (1 − x2)(1 + x)x dx = −8⁄15. Example ∫c x2y ds, c is the square whose vertices are (0, 0), (1, 0), (1, 1),  x2 + dy2. When dx = 0, ds = dy. From (0, 0) to (1, 0), y = 0, dy = (0, 1). ds = d 0. Similar arguments for the other sides give

 x y ds =  0.x dx +  y dy +  x dx +  0.y dy = a − s = ⁄ 1

2

1

0

2

2

0

c

0

0

16

1

1

Let f(z) be any function of z, analytic or not, and c any curve as above. The complex integral is calculated as ∫c f(z) dz = ∫c (u dx − v dy) + i ∫c (v dx + u dy), where f(z) = u(x, y) + iv(x, y). Properties of line integrals are the same as those for ordinary integrals. That is, ∫c [ f(z)  g(z)] dz = ∫c f(z) dz  ∫c g(z) dz; ∫c kf(z) dz = k ∫c f(z) dz for any constant k, etc.

 (x + iy) dz along c: y = x, 0 to 1 + i. This becomes  (x + iy) dz =  (x dx − y dy) + i  (y dx + x dy) =  x dx −  x dx + i  x dx + i  x dx = − ⁄

Example

2

c

2

2

c

c

1

2

c

1

1

1

2

0

2

0

0

0

16

+ 5i/6

FIG. 3-47

Conformal transformation.

Conformal Mapping Every function of a complex variable w = f(z) = u(x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude and sense. Every analytic function, except at those points where f ′(z) = 0, is a conformal transformation. See Fig. 3-47. Example w = z2. u + iv = (x2 − y2) + 2ixy or u = x2 − y2, v = 2xy. These are the transformation equations between the (x, y) and (u, v) planes. Lines parallel to the x axis, y = c1 map into curves in the u, v plane with parametric equations u = x2 − c12, v = 2c1x. Eliminating x, u = (v2/4c12) − c12, which represents a family of parabolas with the origin of the w plane as focus, the line v = 0 as axis and opening to the right. Similar arguments apply to x = c2. The principles of complex variables are useful in the solution of a variety of applied problems, including Laplace transforms and process control (Sec. 8).

DIFFERENTIAL EQUATIONS REFERENCES: Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Academic Press, New York (1965); Aris, R., and N. R. Amundson, Mathematical Methods in Chemical Engineering, vol. 2, First-Order Partial Differential Equations with Applications, Prentice-Hall, Englewood Cliffs, N.J. (1973); Asmar, N., and G. C. Jones, Applied Complex Analysis with Partial Differential Equations, Prentice-Hall, Upper Saddle River, N.J. (2002); Boyce, W. E., and R. C. Di Prima, Elementary Differential Equations and Boundary Value Problems, 7th ed., Wiley, New York (2004); Braun, M., Differential Equations and Their Applications: An Introduction to Applied Mathematics, 4th ed., Springer-Verlag, New York (1993); Bronson, R., and G. Costa, Schaum’s Outline of Differential Equations, 3d ed., McGraw-Hill, New York (2007); Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000); Courant, R., and D. Hilbert, Methods of Mathematical Physics, vols. I and II, Interscience, New York (1953, 1962); Duffy, D., Green’s Functions with Applications, Chapman and Hall/CRC (2001); Kreyszig, E., Advanced Engineering Mathematics, 8th ed., Wiley, New York (1999); Morse, P. M., and H. Feshbach, Methods of Theoretical Physics, vols. I and II, McGraw-Hill, New York (1953); Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman and Hall/CRC (2002); Polyanin, A. D., and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2d ed., Chapman and Hall/CRC (2002); Ramkrishna, D., and N. R. Amundson, Linear Operator Methods in Chemical Engineering with Applications to Transport and Chemical Reaction Systems, Prentice-Hall, Englewood Cliffs, N.J. (1985).

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. k

Example The rate of the homogeneous bimolecular reaction A + B → C is characterized by the differential equation dx/dt = k(a − x)(b − x), where a = initial

concentration of A, b = initial concentration of B, and x = x(t) = concentration of C as a function of time t.

Example The differential equation of heat conduction in a moving fluid with velocity components vx, vy is ∂u ∂u ∂u K ∂2u ∂2u  + vx  + vy  =  2 + 2 ∂t ∂x ∂y ρcp ∂x ∂y where u = u(x, y, t) = temperature, K = thermal conductivity, ρ = density, and cp = specific heat at constant pressure.





ORDINARY DIFFERENTIAL EQUATIONS When the function involved in the equation depends upon only one variable, its derivatives are ordinary derivatives and the differential equation is called an ordinary differential equation. When the function depends upon several independent variables, then the equation is called a partial differential equation. The theories of ordinary and partial differential equations are quite different. In almost every respect the latter is more difficult. Whichever the type, a differential equation is said to be of nth order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinary differential equation which includes the maximum possible number of “arbitrary” constants is called the general solution. The maximum number of “arbitrary” constants is exactly equal to the order of the

3-30

MATHEMATICS

differential equation. If any set of specific values of the constants is chosen, the result is called a particular solution. Example The general solution of (d2x/dt2) + k2x = 0 is x = A cos kt + B sin kt, where A, B are arbitrary constants. A particular solution is x = a cos kt + 3 sin kt.

Distinct Real Roots If the roots of the characteristic equation are distinct real roots, r1 and r2, say, the solution is y = Aer1x + Ber2x, where A and B are arbitrary constants. Example y″ + 4y′ + 3 = 0. The characteristic equation is m2 + 4m + 3 = 0. The roots are −3 and −1, and the general solution is y = Ae−3x + Be−x.

In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution.

Multiple Real Roots If r1 = r2, the solution of the differential equation is y = e r1x(A + Bx).

Example y = x(dy/dx) − d(dy/dx)2 has the general solution y = cx − dc2, where c is an arbitrary constant; y = x2 is a singular solution, as is easily verified.

Example y″ + 4y + 4 = 0. The characteristic equation is m2 + 4m + 4 = 0 with roots −2 and −2. The solution is y = e−2x(A + Bx). Complex Roots If the characteristic roots are p  iq, then the solution is y = epx(A cos qx + B sin qx).

ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M(x, y) dx + N(x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = ∫ f(x) dx + c, which may or may not be expressible in simpler form.

Example The differential equation My″ + Ay′ + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2  kM, the roots of the characteristic equation A Mm2 + Am + k = 0 are complex −   i 2M and the solution is

Exact Equations The equation M(x, y) dx + N(x, y) dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x. In this case there exists a function w = f(x, y) such that ∂f/∂x = M, ∂f/∂y = N, and f(x, y) = C is the required solution. f(x, y) is found as follows: treat y as though it were constant and evaluate ∫ M(x, y) dx. Then treat x as though it were constant and evaluate ∫ N(x, y) dy. The sum of all unlike terms in these two integrals (including no repetitions) is f(x, y). Example (2xy − cos x) dx + (x2 − 1) dy = 0 is exact for ∂M/∂y = 2x, ∂N/∂x = 2x. ∫ M dx = ∫ (2xy − cos x) dx = x2y − sin x, ∫ N dy = ∫ (x2 − 1) dy = x2y − y. The solution is x2y − sin x − y = C, as may easily be verified. Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P(x)y = Q(x). Its general solution is y = e−∫ P dx

 Qe

∫ P dx

dx + C



Example A tank initially holds 200 gal of a salt solution in which 100 lb is dissolved. Six gallons of brine containing 4 lb of salt run into the tank per minute. If mixing is perfect and the output rate is 4 gal/min, what is the amount A of salt in the tank at time t? The differential equation of A is dA/dt = [2/(100 + t)]A = 4. Its general solution is A = (4/3) (100 + t) + C/(100 + t)2. At t = 0, ′A = 100; so the particular solution is A = (4/3) (100 + t) − (1/3) 106/(100 + t)2. ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. Equation y(n) = f(x)* Such a differential equation can be solved by n integrations. The solution will contain n arbitrary constants. Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y″ + ay′ + by = 0 depends upon the nature of the roots of the characteristic equation m2 + am + b = 0 obtained by substituting the trial solution y = emx in the equation. *The superscript (n) means n derivatives.

2

y = e−(At/2M)

Example Two liquids A and B are boiling together in a vessel. Experimentally it is found that the ratio of the rates at which A and B are evaporating at any time is proportional to the ratio of the amount of A (say, x) to the amount of B (say, y) still in the liquid state. This physical law is expressible as (dy/dt)/(dx/dt) = ky/x or dy/dx = ky/x, where k is a proportionality constant. This equation may be written dy/y = k(dx/x), in which the variables are separated. The solution is ln y = k ln x + ln c or y = cxk.

A k −   M

2M

−  t + ic sin  −  t   c cos  M

2M M

2M
k

A

2

1

k

A

2

2

This solution is oscillatory, representing undercritical damping.

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. Second-Order Equations: Dependent Variable Missing Such an equation is of the form dy d 2y F x,  , 2 = 0 dx dx It can be reduced to a first-order equation by substituting p = dy/dx and dp/dx = d 2y/dx2. Second-Order Equations: Independent Variable Missing Such an equation is of the form









dy d 2y F y,  , 2 = 0 dx dx dy d 2y dp  = p, 2 = p  dx dx dy The result is a first-order equation in p, Set





dp F y, p, p  = 0 dy Example The capillary curve for one vertical plate is given by

  

d 2y 4y dy 1+  2 =  dx c2 dx

2 3/2

Its solution by this technique is



c c −1 c 2 x +  c2 −y2 −  c2 − h  − cosh−1  0 =  cosh 2 y h0 where c, h0 are physical constants.



Example The equation governing chemical reaction in a porous catalyst in plane geometry of thickness L is d 2c D 2 = k f(c), dx

dc  (0) = 0, dx

c(L) = cv

where D is a diffusion coefficient, k is a reaction rate parameter, c is the concentration, k f(c) is the rate of reaction, and c0 is the concentration at the

DIFFERENTIAL EQUATIONS dc boundary. Making the substitution p =  gives (Finlayson, 1980, p. 92) ds k dp p  =  f(c) dc D



p2 k c f(c) dc = 2 D c(0) If the reaction is very fast, c(0) ≈ 0 and the average reaction rate is related to p(L). This variable is given by 1/2 2k c0 f(c) dc p(L) =  D 0 Thus, the average reaction rate can be calculated without solving the complete problem. Integrating gives

 



Linear Nonhomogeneous Differential Equations Linear Differential Equations Right-Hand Member f(x) ≠ 0 Again the specific remarks for y″ + ay′ + by = f(x) apply to differential equations of similar type but higher order. We shall discuss two general methods. Method of Undetermined Coefficients Use of this method is limited to equations exhibiting both constant coefficients and particular forms of the function f(x). In most cases f(x) will be a sum or product of functions of the type constant, xn (n a positive integer), emx, cos kx, sin kx. When this is the case, the solution of the equation is y = H(x) + P(x), where H(x) is a solution of the homogeneous equations found by the method of the preceding subsection and P(x) is a particular integral found by using the following table subject to these conditions: (1) When f(x) consists of the sum of several terms, the appropriate form of P(x) is the sum of the particular integrals corresponding to these terms individually. (2) When a term in any of the trial integrals listed is already a part of the homogeneous solution, the indicated form of the particular integral is multiplied by x. Form of Particular Integral

If f(x) is

Then P(x) is

a (constant) axn aerx c cos kx d sin kx

A (constant) An xn + An − 1 xn − 1 + ⋅⋅⋅A1 x + A0 Berx




n rx

gx e cos kx hxnerx sin kx

A cos kx + B sin kx




(Anxn + ⋅⋅⋅ + A0)erx cos kx + (Bn xn + ⋅⋅⋅ + B0)erx sin kx

f1 dv v′ =  =  R(x) dx f1 f 2′ − f2 f 1′ and since f1, f2, and R are known u, v may be found by direct integration. Perturbation Methods If the ordinary differential equation has a parameter that is small and is not multiplying the highest derivative, perturbation methods can give solutions for small values of the parameter. Example Consider the differential equation for reaction and diffusion in a catalyst; the reaction is second order: c″ = ac2, c′(0) = 0, c(1) = 1. The solution is expanded in the following Taylor series in a. c(x, a) = c0(x) + ac1(x) + a2c2(x) + . . . The goal is to find equations governing the functions {ci(x)} and solve them. Substitution into the equations gives the following equations: c0″(x) + a c″1(x) + a2c″2(x) + . . . = a[c0(x) + ac1(x) + a2c2(x) + . . . ]2 c′0(0) + ac′1(0) + a2c′2(0) + . . . = 0 c0(1) + ac1(1) + a2c2(1) + . . . = 1 Like terms in powers of a are collected to form the individual problems. c″0 = 0,

(m + 1)2 = 0 so that the homogeneous solution is y = (c1 + c2x)e−x. To find a particular solution we use the trial solution from the table, y = a1e2x + a2 cos x + a3 sin x + a4x3 + a5x2 + a6x + a7. Substituting this in the differential equation collecting and equating like terms, there results a1 = s, a2 = 0, a3 = −a, a4 = 1, a5 = −6, a6 = 18, and a7 = −24. The solution is y = (c1 + c2x)e−x + se2x − a sin x + x3 − 6x2 + 18x − 24.

Method of Variation of Parameters This method is applicable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y″ + a(x)y′ + b(x)y = R(x) and let the solution of the homogeneous equation, found by some method, be y = c1 f1(x) + c2 f2(x). It is now assumed that a particular integral of the differential equation is of the form P(x) = uf1 + vf2 where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf1 + vf2 satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be u′f1 + v′f2 = 0 Then

and

u′f′1 + v′f′2 = R(x)

f2 du u′ =  = −  R(x) f1 f 2′ − f2 f 1′ dx

c′0(0) = 0,

c″1 = c 02,

c0(1) = 1

c′1(0) = 0,

c″2 = 2c0 c1,

c1(1) = 0

c′2(0) = 0,

c2(1) = 0

The solution proceeds in turn. (x2 − 1) c1(x) =  , 2

c0(x) = 1,

5 − 6x2 + x4 c2(x) =  12

SPECIAL DIFFERENTIAL EQUATIONS [SEE ABRAMOWITZ AND STEGUN (1972)] Euler’s Equation The linear equation x ny(n) + a1x n − 1y(n − 1) + ⋅ ⋅ ⋅ + an − 1xy′ + any = R(x) can be reduced to a linear equation with constant coefficients by the change of variable x = et. To solve the homogeneous equation substitute y = x r into it, cancel the powers of x, which are the same for all terms, and solve the resulting polynomial for r. In case of multiple or complex roots there results the form y = x r(log x)r and y = x α[cos (β log x) + i sin (β log x)]. Bessel’s Equation The linear equation x2(d 2y/dx2) + (1 − 2α) x(dy/dx) + [β 2 γ 2 x 2γ + (α2 − p2γ 2)]y = 0 is the general Bessel equation. By series methods, not to be discussed here, this equation can be shown to have the solution y = AxαJp(βxγ) + BxαJ−p(βxγ)

Since the form of the particular integral is known, the constants may be evaluated by substitution in the differential equation. Example y″ + 2y′ + y = 3e2x − cos x + x3. The characteristic equation is

3-31

α

α

p not an integer or zero

y = Ax Jp(βx ) + Bx Yp(βx ) where

γ

γ

p an integer

(−1)   k!Γ(p + k + 1)

x Jp(x) =  2

k

2k

(x/2)

k=0

 Γ(n) =  x

x J−p(x) =  2

∞

p

–p

∞

(−1)k(x/2)2k

 k!Γ(k + 1 − p)

p not an integer

k=0

∞

n − 1 −x

0

e dx

n>0

is the gamma function. For p an integer

  k!(p + k)!

x Jp(x) =  2

p ∞

(−1)k(x/2)2k

k=0

(Bessel function of the first kind of order p) [ Jp(x) cos (pπ) − J−p(x)] Yp(x) =  sin (pπ) (replace right-hand side by limiting value if p is an integer or zero). The series converge for all x. Much of the importance of Bessel’s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them.

3-32

MATHEMATICS

Legendre’s Equation The Legendre equation (1 − x2)y″ − 2xy′ + n(n + 1)y = 0, n ≥ 0, has the solution Pn for n an integer. The polynomials Pn are the so-called Legendre polynomials, P0(x) = 1, P1(x) = x, P2(x) = a(3x2 − 1), P3(x) = a(5x3 − 3x), . . . . For n positive and not an integer, see Abramowitz and Stegun (1972). Laguerre’s Equation The Laguerre equation x(d 2y/dx2) + (c − x) (dy/dx) − ay = 0 is satisfied by the confluent hypergeometric function. See Abramowitz and Stegun (1972) and Kreszig (1999). Hermite’s Equation The Hermite equation y″ − 2xy′ + 2ny = 0 is satisfied by the Hermite polynomial of degree n, y = AHn(x) if n is a positive integer or zero. H0(x) = 1, H1(x) = 2x, H2(x) = 4x2 − 2, H3(x) = 8x3 − 12x, H4(x) = 16x4 − 48x2 + 12, Hr + 1(x) = 2xHr(x) − 2rHr − 1(x). Chebyshev’s Equation The equation (1 − x2)y″ − xy′ + n2y = 0 for n a positive integer or zero is satisfied by the nth Chebyshev polynomial y = ATn(x). T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1; Tr + 1(x) = 2xTr(x) − Tr − 1(x). PARTIAL DIFFERENTIAL EQUATIONS The analysis of situations involving two or more independent variables frequently results in a partial differential equation. Example The equation ∂T/∂t = K(∂2T/∂x2) represents the unsteady onedimensional conduction of heat. Example The equation for the unsteady transverse motion of a uniform beam clamped at the ends is ρ ∂2y ∂4y =0 4 +   EI ∂t2 ∂x

Example The expansion of a gas behind a piston is characterized by the simultaneous equations ∂u ∂u c2 ∂ρ +u+=0 ∂t ∂x ρ ∂x

and

∂ρ ∂ρ ∂u +u+ρ=0 ∂t ∂x ∂x

The partial differential equation ∂2f/∂x ∂y = 0 can be solved by two integrations yielding the solution f = g(x) + h(y), where g(x) and h(y) are arbitrary differentiable functions. This result is an example of the fact that the general solution of partial differential equations involves arbitrary functions in contrast to the solution of ordinary differential equations, which involve only arbitrary constants. A number of methods are available for finding the general solution of a partial differential equation. In most applications of partial differential equations the general solution is of limited use. In such applications the solution of a partial differential equation must satisfy both the equation and certain auxiliary conditions called initial and/or boundary conditions, which are dictated by the problem. Examples of these include those in which the wall temperature is a fixed constant T(x0) = T0, there is no diffusion across a nonpermeable wall, and the like. In ordinary differential equations these auxiliary conditions allow definite numbers to be assigned to the constants of integration. Partial Differential Equations of Second and Higher Order Many of the applications to scientific problems fall naturally into partial differential equations of second order, although there are important exceptions in elasticity, vibration theory, and elsewhere. A second-order differential equation can be written as ∂2u ∂2u ∂2u a 2 + b  + c 2 = f ∂x ∂x∂y ∂y where a, b, c, and f depend upon x, y, u, ∂u/∂x, and ∂u/∂y. This equation is hyperbolic, parabolic, or elliptic, depending on whether the discriminant b2 − 4ac is >0, =0, or <0, respectively. Since a, b, c, and f depend on the solution, the type of equation can be different at different x and y locations. If the equation is hyperbolic, discontinuities can be propagated. See Courant and Hilbert (1953, 1962) and LeVeque, R. J., Numerical Methods for Conservation Laws, Birkhäuser, Basel (1992). Phenomena of propagation such as vibrations are characterized by equations of “hyperbolic” type which are essentially different in their

properties from other classes such as those which describe equilibrium (elliptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows: Elliptic Laplace’s equation ∂2u/∂x2 + ∂2u/∂y2 = 0 and Poisson’s equation ∂2u/∂x2 + ∂2u/∂y2 = g(x, y) do not contain the variable time explicitly and consequently represent equilibrium configurations. Laplace’s equation is satisfied by static electric or magnetic potential at points free from electric charges or magnetic poles. Other important functions satisfying Laplace’s equation are the velocity potential of the irrotational motion of an incompressible fluid, used in hydrodynamics; the steady temperature at points in a homogeneous solid, and the steady state of diffusion through a homogeneous body. Parabolic The heat equation ∂T/∂t = ∂ 2T/∂x2 + ∂2T/∂y2 represents nonequilibrium or unsteady states of heat conduction and diffusion. Hyperbolic The wave equation ∂2u/∂t2 = c2(∂2u/∂x2 + ∂2u/∂y2) represents wave propagation of many varied types. Quasilinear first-order differential equations are like ∂u ∂u a+b=f ∂x ∂y where a, b, and f depend on x, y, and u, with a2 + b2 ≠ 0. This equation can be solved using the method of characteristics, which writes the solution in terms of a parameter s, which defines a path for the characteristic. dx dy du  = a,  = b,  = f ds ds ds These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and f all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Courant and Hilbert (1953, 1962); Rhee, H. K., R. Aris, and N. R. Amundson, First-Order Partial Differential Equations, vol. I, Theory and Applications of Single Equations, Prentice-Hall, Englewood Cliffs, N.J. (1986); and LeVeque (1992), ibid. An example of a linear hyperbolic equation is the advection equation for flow of contaminants when the x and y velocity components are u and v, respectively. ∂c ∂c ∂c +u+v=0 ∂t ∂x ∂y The equations for flow and adsorption in a packed bed or chromatography column give a quasilinear equation. ∂c ∂c df ∂c φ  + φ u  + (1 − φ)   = 0 ∂t ∂x dc ∂t Here n = f(c) is the relation between concentration on the adsorbent and fluid concentration. The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundary conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. Similarity Variables The physical meaning of the term “similarity” relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, “separation of variables” (not the classical concept) and the use of “continuous transformation groups.” The basic theory is available in Ames (1965).

DIFFERENTIAL EQUATIONS Example The equation ∂θ/∂x = (A/y)(∂2θ/∂y2) with the boundary conditions θ = 0 at x = 0, y > 0; θ = 0 at y = ∞, x > 0; θ = 1 at y = 0, x > 0 represents the nondimensional temperature θ of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form θ = f(y/xn) = f(u), where θ = 0 at u = ∞ and θ = 1 at u = 0. The purpose here is to replace the independent variables x and y by the single variable u when it is hoped that a value of n exists which will allow x and y to be completely eliminated in the equation. In this case since u = y/xn, there results after some calculation ∂θ/∂x = −(nu/x)(dθ/du), ∂2θ/∂y2 = (1/x2n)(d2θ/du2), and when these are substituted in the equation, −(1/x)nu(dθ/du) = (1/x3n)(A/u)(d2θ/du2). For this to be a function of u only, choose n = s. There results (d2θ/du2) + (u2/3A)(dθ/du) = 0. Two integrations and use of the boundary conditions for this ordinary differential equation give the solution



∞

θ=

u



exp (−u3/9A) du 

∞

exp (−u3/9A) du

0

Group Method The type of transformation can be deduced using group theory. For a complete exposition, see Ames (1965) and Hill, J. M., Differential Equations and Group Methods for Scientists and Engineers, CRC Press, New York (1992); a shortened version is in Finlayson (1980, 2003). Basically, a similarity transformation should be considered when one of the independent variables has no physical scale (perhaps it goes to infinity). The boundary conditions must also simplify (and combine) since each transformation leads to a differential equation with one fewer independent variable. Example A similarity variable is found for the problem ∂c ∂ D(c) ∂c  =    , c(0,t) = 1, c(∞,t) = 0, c(x,0) = 0 ∂t ∂x D ∂x Note that the length dimension goes to infinity, so that there is no length scale in the problem statement; this is a clue to try a similarity transformation. The transformation examined here is





t = a t,

x = a x, c = a c With this substitution, the equation becomes α

β

γ

∂c ∂c ∂ aα − γ  = a2β − γ  D(a−γ c)  ∂t ∂x ∂x Group theory says a system is conformally invariant if it has the same form in the new variables; here, that is γ = 0,





α − γ = 2β − γ,

or α = 2β

separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say λ. Thus the problem has again been reduced to the solution of ordinary differential equations. Example Laplace’s equation ∂2V/∂x2 + ∂2V/∂y2 = 0 plus the boundary conditions V(0, y) = 0, V(l, y) = 0, V(x, ∞) = 0, V(x, 0) = f(x) represents the steadystate potential in a thin plate (in z direction) of infinite extent in the y direction and of width l in the x direction. A potential f(x) is impressed (at y = 0) from x = 0 to x = 1, and the sides are grounded. To obtain a solution of this boundaryvalue problem assume V(x, y) = f(x)g(y). Substitution in the differential equation yields f″(x)g(y) + f(x)g″(y) = 0, or g″(y)/g(y) = −f″(x)/f(x) = λ2 (say). This system becomes g″(y) − λ2g(y) = 0 and f″(x) + λ2f(x) = 0. The solutions of these ordinary differential equations are respectively g(y) = Aeλy + Be−λy, f(x) = C sin λx + D cos λx. Then f(x)g(y) = (Aeλy + Be−λy) (C sin λx + D cos λx). Now V(0, y) = 0 so that f(0)g(y) = (Aeλy + Be−λy) D  0 for all y. Hence D = 0. The solution then has the form sin λx (Aeλy + Be−λy) where the multiplicative constant C has been eliminated. Since V(l, y) = 0, sin λl(Aeλy + Be−λy)  0. Clearly the bracketed function of y is not zero, for the solution would then be the identically zero solution. Hence sin λl = 0 or λn = nπ/l, n = 1, 2, . . . where λn = nth eigenvalue. The solution now has the form sin (nπx/l)(Aenπy/l + Be−nπy/l). Since V(x, ∞) = 0, A must be taken to be zero because ey becomes arbitrarily large as y → ∞. The solution then reads Bn sin (nπx/l)e−nπy/l, where Bn is the multiplicative constant. The differential equation is linear and homogeneous so that n∞= 1 Bne−nπy/l sin (nπx/l) is also a solution. Satisfaction of the last boundary condition is ensured by taking 2 Bn =  l

 f(x) sin (nπx/l) dx = Fourier sine coefficients of f(x) l

0

Further, convergence and differentiability of this series are established quite easily. Thus the solution is ∞ nπx V(x, y) = Bne−nπy/l sin  l n=1

Example The diffusion problem in a slab of thickness L ∂c ∂2c  = D 2 , c(0, t) = 1, c(L, t) = 0, c(x, 0) = 0 ∂t ∂x can be solved by separation of variables. First transform the problem so that the boundary conditions are homogeneous (having zeros on the right-hand side). Let x c(x, t) = 1 −  + u(x, t) L Then u(x, t) satisfies ∂2u ∂u  = D 2 , ∂t ∂x

The invariants are β δ= α

x η = δ , t

x u(x, 0) =  − 1, L

c(x, t) = f(η)tγ/α

Since both sides are constant, this gives the following ordinary differential equations to solve. 1 dT 1 d 2X = −λ   = −λ,   DT dt X dx2 The solution of these is T = A e−λDt,

The use of the 4 and D0 makes the analysis below simpler. The result is



f(0) = 1,

f(∞) = 0

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffusivity is constant, the solution is the error function, a tabulated function.

erf η =



0

e



dξ 

∞

−ξ

e

 x + E sin λ  x) e−λDt u = A (B cos λ Apply the boundary condition that u(0,t) = 0 to give B = 0. Then the solution is

 x)e−λDt u = A (sin λ where the multiplicative constant E has been eliminated. Apply the boundary condition at x = L.

c(x,t) = 1 − erf η = erfc η 2

 x + E sin λ x X = B cos λ

The combined solution for u(x,t) is

d D(c) df df    + 2η  = 0, dη D0 dη dη

−ξ

u(L, t) = 0

1 dT 1 d 2X = DT dt X dx2

We can take γ = 0 and δ = β/α = a. Note that the boundary conditions combine because the point x = ∞ and t = 0 give the same value of η and the conditions on c at x = ∞ and t = 0 are the same. We thus make the transformation x η =  , c(x, t) = f(η) 4 D  0t

η

u(0, t) = 0,

Assume a solution of the form u(x, t) = X(x) T(t), which gives

and the solution is



3-33

2

 L)e−λDt 0 = A (sin λ

dξ

0

Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U = f(x)g(y). If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be

This can be satisfied by choosing A = 0, which gives no solution. However, it can also be satisfied by choosing λ such that

 L = 0, sin λ

L=nπ λ

nπ λ= L2 2 2

Thus

3-34

MATHEMATICS

The combined solution can now be written as





Since the initial condition must be satisfied, we use an infinite series of these functions. u=

e

A  L ∞

sin nπx

n

−n 2π 2Dt/L 2

n=1

At t = 0, we satisfy the initial condition.



∞ sin nπx x  − 1 = An  L L n=1





sin mπx  L and integrating over x: 0 → L. (This is the same as minimizing the mean-square error of the initial condition.) This gives



L

0

 

Example The equation ∂c/∂t = D(∂2c/∂x2) represents the diffusion in a semi-infinite medium, x ≥ 0. Under the boundary conditions c(0, t) = c0, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t, ∞ ∂2c 1 ∞ −st ∂c e−st 2 dt =  e  dt 0 ∂x D 0 ∂t

This is done by multiplying the equation by

AmL = 2

∂y dLt[y(x, t)] Lt  =  ∂x dx since L[y(x, t)] is “really” only a function of x. Otherwise the results are similar. These facts coupled with the linearity of the transform, i.e., L[af(t) + bg(t)] = aL[ f(t)] + bL[g(t)], make it a useful device in solving some linear differential equations. Its use reduces the solution of ordinary differential equations to the solution of algebraic equations for L[y]. The inverse transform must be obtained either from tables or by use of complex inversion methods. whereas

sin nπx 2 2 2 u = A  e−n π Dt/L L

x mπx ( − 1) sin  dx L L



2

dF sF = (1/D)sF − c(x, 0) =   dx2 D where F(x, s) = Lt[c(x, t)]. Hence s d 2F −  F=0  dx2 D The other boundary condition transforms into F(0, s) = c0 /s. Finally the solution of the ordinary differential equation for F subject to F(0, s) = c0 /s and F remains x. Reference to a table shows that the funcfinite as x → ∞ is F(x, s) = (c0 /s)e−s/D tion having this as its Laplace transform is or



which completes the solution.

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see “Integral Transforms (Operational Methods)”]. The one-sided Laplace trans∞ form indicated by L[ f(t)] is defined by the equation L[ f(t)] = ∫0 f(t)e− st dt. It has numerous important properties. The ones of interest here are L[ f ′(t)] = sL[f(t)] − f(0); L[ f″(t)] = s2L[f(t)] − sf(0) − f′(0); L[ f (n)(t)] = snL[ f(t)] − sn − 1f(0) − sn − 2f′(0) − ⋅ ⋅ ⋅ − f (n − 1)(0) for ordinary derivatives. For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if ∂y y = y(x, t), Lt  = sL[y(x, t)] − y(x, 0) ∂t

 



2 c(x, t) = c0 1 −  π 



t x/2D

0

x 2 e−u du C0 erfc  4Dt 







Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981); and Lagerstrom, P. A., Matched Asymptotic Expansions: Ideas and Techniques, Springer-Verlag, New York (1988).

DIFFERENCE EQUATIONS REFERENCES: Elaydi, Saber, and S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York (1999); Fulford, G., P. Forrester, and A. Jones, Modelling with Differential and Difference Equations, Cambridge University Press, New York (1997); Goldberg, S., Introduction to Difference Equations, Dover (1986); Kelley, W. G., and A. C. Peterson, Difference Equations : An Introduction with Applications, 2d ed., Harcourt/Academic Press, San Diego (2001).

Certain situations are such that the independent variable does not vary continuously but has meaning only for discrete values. Typical illustrations occur in the stagewise processes found in chemical engineering such as distillation, staged extraction systems, and absorption columns. In each of these the operation is characterized by a finite betweenstage change of the dependent variable in which the independent variable is the integral number of the stage. The importance of difference equations is twofold: (1) to analyze problems of the type described and (2) to obtain approximate solutions of problems which lead, in their formulation, to differential equations. In this subsection only problems of analysis are considered; the application to approximate solutions is considered under “Numerical Analysis and Approximate Methods.” ELEMENTS OF THE CALCULUS OF FINITE DIFFERENCES Let y = f(x) be defined for discrete equidistant values of x, which will be denoted by xn. The corresponding value of y will be written yn = f(xn). The first forward difference of f(x) denoted by ∆f(x) = f(x + h) − f(x) where h = xn − xn − 1 = interval length. Example Let f(x) = x2. Then ∆f(x) = (x + h)2 − x2 = 2hx + h2.

The second forward difference is obtained by taking the difference of the first; thus ∆∆f(x) = ∆2f(x) = ∆f(x + h) − ∆f(x) = f(x + 2h) − 2f(x + h) + f(x). Example f(x) = x2, ∆2f(x) = ∆[∆f(x)] = ∆2hx + ∆h2 = 2h(x + h) − 2hx + h2 −

h2 = 2h2.

Similarly the nth forward difference is defined by the relation ∆nf(x) = ∆[∆n − 1f(x)]. Other difference relations are also quite useful. Some of these are ∇f(x) = f(x) − f(x − h), which is called the backward difference, and δf(x) = f [x + (h/2)] − f [x − (h/2)], called the central difference. Some properties of the operator ∆ are quite important. If C is any constant, ∆C = 0; if f(x) is any function of period h, ∆f(x) = 0 (in fact, periodic functions of period h play the same role here as constants do in the differential calculus); ∆[f(x) + g(x)] = ∆f(x) + ∆g(x); ∆m[∆nf(x)] = ∆m + nf(x); ∆[f(x)g(x)] = f(x) ∆g(x) + g(x + h) ∆f(x) g(x) ∆f(x) − f(x) ∆g(x) f(x) ∆  =  g(x) g(x)g(x + h)

 

Example ∆(x sin x) = x∆ sin x + sin (x + h) ∆x = 2x sin (h/2) cos [x + (h/2)] +

h sin (x + h).

DIFFERENCE EQUATIONS A difference equation is a relation between the differences and the independent variable, φ(∆ny, ∆n − 1y, . . . , ∆y, y, x) = 0, where φ is some

DIFFERENCE EQUATIONS given function. The general case in which the interval between the successive points is any real number h, instead of 1, can be reduced to that with interval size 1 by the substitution x = hx′. Hence all further difference-equation work will assume the interval size between successive points is 1.

3-35

Clearly y is a function of the stage number x. α is a combination of system constants. By using the trial solution yx = cβx, there results β2 − (α + 1)β + α = 0, so that β1 = 1, β2 = α. The general solution is yx = c1 + c2αx. By using the side conditions, c1 = c0 − c2, c2 = (ym + 1 − c0)/(αm + 1 − 1). The desired solution is (yx − c0)/(ym + 1 − c0) = (αx − 1)/(αm + 1 − 1).

Example f(x + 1) − (α + 1)f(x) + αf(x − 1) = 0. Common notation usually is

Example yx + 3 − 3yx + 2 + 4yx = 0. By setting yx = cβ x, there results β3 − 3β2 + 4 = 0 or β1 = −1, β2 = 2, β3 = 2. The general solution is yx = c1(−1)x + 2x(c2 + c3x).

Example yx + 2 + 2yx yx + 1 + yx = x2.

Example yx + 1 − 2yx + 2yx − 1 = 0. β1 = 1 + i, β2 = 1 − i. p = 1 + 1 = 2, θ = π/4. The solution is yx = 2 x/2 [c1 cos (xπ/4) + ic2 sin (xπ/4)].

yx = f(x). This equation is then written yx + 1 − (α + 1)yx + αyx − 1 = 0.

Example yx + 1 − yx = 2x. The order of the difference equation is the difference between the largest and smallest arguments when written in the form of the second example. The first and second examples are both of order 2, while the third example is of order 1. A linear difference equation involves no products or other nonlinear functions of the dependent variable and its differences. The first and third examples are linear, while the second example is nonlinear. A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitrary constants. The techniques for solving difference equations resemble techniques used for differential equations. Equation Dny = a The solution of ∆ny = a, where a is a constant, is a polynomial of degree n plus an arbitrary periodic function of period 1. That is, y = (axn/n!) + c1xn − 1 + c2xn − 2 + ⋅ ⋅ ⋅ + cn + f(x), where f(x + 1) = f(x). Example ∆3y = 6. The solution is y = x3 + c1x2 + c2x + c3 + f(x); c1, c2, c3 are arbitrary constants, and f(x) is an arbitrary periodic function of period 1. Equation yx + 1 - yx = φ(x) This equation states that the first difference of the unknown function is equal to the given function φ(x). The solution by analogy with solving the differential equation dy/dx = φ(x) by integration is obtained by “finite integration” or summation. When there are only a finite number of data points, this is easily accomplished by writing yx = y0 + tx = 1 φ(t − 1), where the data points are numbered from 1 to x. This is the only situation considered here. Examples If φ(x) = 1, yx = x. If φ(x) = x, yx = [x(x − 1)]/2. If φ(x) = ax, a ≠ 0,

yx = a /(a − 1). In all cases y0 = 0.

Constant Coefficients and Q(x) ≠ 0 (Nonhomogeneous) In this case the general solution is found by first obtaining the homogeneous solution, say, yHx and adding to it any particular solution with Q(x) ≠ 0, say, y xP. There are several means of obtaining the particular solution. Method of Undetermined Coefficients If Q(x) is a product or linear combination of products of the functions ebx, a x, x p (p a positive integer or zero) cos cx and sin cx, this method may be used. The “families” [ax], [ebx], [sin cx, cos cx] and [x p, x p − 1, . . . , x, 1] are defined for each of the above functions in the following way: The family of a term fx is the set of all functions of which fx and all operations of the form a x + y, cos c(x + y), sin c(x + y), (x + y)p on fx and their linear combinations result in. The technique involves the following steps: (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume y Px is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. Example yx + 1 − 3yx + 2yx − 1 = 1 + ax. a ≠ 0. The homogeneous solution is yxH = c1 + c2 2x. The family of 1 is 1 and of ax is ax. However, 1 is a solution of the homogeneous system. Therefore, try yxP = Ax + Bax. Substituting in the equation there results a yx = c1 + c2 2x − x +  ax, a ≠ 1, a ≠ 2 (a − 1)(a − 2) If a = 1, yx = c1 + c22x − 2x. If a = 2, yx = c1 + c22x − x + x2x.

x

Other examples may be evaluated by using summation, that is, y2 = y1 + φ(1), y3 = y2 + φ(2) = y1 + φ(1) + φ(2), y4 = y3 + φ(3) = y1 + φ(1) + φ(2) + φ(3), . . . , yx = y1 + tx =− 11 φ(t). Example yx + 1 − ryx = 1, r constant, x > 0 and y0 = 1. y1 = 1 + r, y2 = 1 + r + r2, . . . , yx = 1 + r + ⋅⋅⋅ + rx = (1 − rx + 1)/(1 − r) for r ≠ 1 and yx = 1 + x for r = 1. Linear Difference Equations The linear difference equation of order n has the form Pnyx + n + Pn − 1yx + n − 1 + ⋅ ⋅ ⋅ + P1yx + 1 + P0yx = Q(x) with Pn ≠ 0 and P0 ≠ 0 and Pj ; j = 0, . . . , n are functions of x. Constant Coefficient and Q(x) = 0 (Homogeneous) The solution is obtained by trying a solution of the form yx = cβ x. When this trial solution is substituted in the difference equation, a polynomial of degree n results for β. If the solutions of this polynomial are denoted by β1, β2, . . . , βn then the following cases result: (1) if all the βj’s are n real and unequal, the solution is yx = j = 1 cj β jx, where the c1, . . . , cn are arbitrary constants; (2) if the roots are real and repeated, say, βj has multiplicity m, then the partial solution corresponding to βj is β jx(c1 + c2 x + ⋅ ⋅ ⋅ + cm xm − 1); (3) if the roots are complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ, the partial solution corresponding to this pair is px(c1 cos θx + ic2 sin θx); and (4) if the roots are multiple complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ are m-fold, then the partial solution corresponding to these is px[(c1 + c2 x + ⋅ ⋅ ⋅ + cm x m − 1) cos θx + i(d1 + d2 x + ⋅ ⋅ ⋅ + dm x m − 1) sin θx]. Example The equation yx + 1 − (α + 1)yx + αyx − 1 = 0, y0 = c0 and ym + 1 =

xm + 1/k represents the steady-state composition of transferable material in the raffinate stream of a staged countercurrent liquid-liquid extraction system.

Example The family of x23x is [x23x, x3x, 3x]. Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system yx + 2 + Ayx + 1 + Byx = φ(x). Assume that the homogeneous solution has been found by some technique and write yxH = c1ux + c2vx. Assume that a particular solution yxP = Dxux + Exvx. Ex and Dx can be found by solving the equations: ux + 1φ(x) Ex + 1 − Ex =  ux + 1vx + 2 − ux + 2vx + 1 vx + 1φ(x) Dx + 1 − Dx =  vx + 1ux + 2 − vx + 2ux + 1 by summation. The general solution is then yx = yxP + yxH. Variable Coefficients The method of variation of parameters applies equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. Equation yx + 1 - axyx = 0 By assuming that this equation is valid for x ≥ 0 and y0 = c, the solution is yx = c  xn = 1 an − 1. x+2 x+1

Example yx + 1 +  yx = 0. The solution is x n+1 x+1 2 3 yx = c  −  = c(−1)x  ⋅  ⋅ ⋅ ⋅  = (−1)xc(x + 1) n x 1 2 n=1





Example yx + 1 − xyx = 0. The solution is yx = c(x − 1)!

3-36

MATHEMATICS

Reduction of Order If one homogeneous solution, say, ux, can be found by inspection or otherwise, an equation of lower order can be obtained by the substitution vx = yx /ux. The resultant equation must be satisfied by vx = constant or ∆vx = 0. Thus the equation will be of reduced order if the new variable Ux = ∆(yx /ux) is introduced. Example (x + 2)yx + 2 − (x + 3)yx + 1 + yx = 0. By observation ux = 1 is a solution. Set Ux = ∆yx = yx + 1 − yx. There results (x + 2)Ux + 1 − Ux = 0, which is of degree one lower than the original equation. The complete solution for yx is finally x 1 yx = c0  + c1 n = 0 n! Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx + 2 + Axyx + 1 + Bxyx = φ(x). If there exists ax, bx such that ax + bx = −Ax and axbx = Bx, then the difference equation may be written yx + 2 − (ax + bx) yx + 1 + axbxyx = φ(x). First solve Ux + 1 − bxUx = φ(x) and then yx + 1 − axyx = Ux.

The substitution ux = yx /fx reduces afx fx + 1yx + z + bfx fx + 2yx + 1 + cfx + 1 fx + 2yx = φ(x) to an equation with constant coefficients. Example x(x + 1)yx + 2 + 3x(x + 2)yx + 1 − 4(x + 1)(x + 2)yx = x. Set ux = yx /fx = yx /x. Then yx = xux, yx + 1 = (x + 1)ux + 1 and yx + 2 = (x + 2)ux + 2. Substitution in the equation yields x(x + 1)(x + 2)ux + 2 + 3x(x + 2)(x + 1)uu + 1 − 4x(x + 1)(x + 2) ux = x or ux + 2 + 3ux + 1 − 4ux = 1/(x + 1)(x + 2), which is a linear equation with constant coefficients. Nonlinear Difference Equations: Riccati Difference Equation The Riccati equation yx + 1yx + ayx + 1 + byx + c = 0 is a nonlinear difference equation which can be solved by reduction to linear form. Set y = z + h. The equation becomes zx + 1zx + (h + a)zx + 1 + (h + b)zx + h2 + (a + b)h + c = 0. If h is selected as a root of h2 + (a + b)h + c = 0 and the equation is divided by zx + 1zx there results [(h + b)/zx + 1] + [(h + a)/zx] + 1 = 0. This is a linear equation with constant coefficients. The solution is

Example yx + 2 − (2x + 1)yx + 1 + (x2 + x)yx = 0. Set ax = x, bx = x + 1. Solve ux + 1 − (x + 1)ux = 0 and then yx + 1 − xyx = ux. Substitution If it is possible to rearrange a difference equation so that it takes the form afx + 2yx + 2 + bfx + 1yx + 1 + cfxyx = φ(x) with a, b, c constants, then the substitution ux = fxyx reduces the equation to one with constant coefficients. Example (x + 2) yx + 2 − 3(x + 1) yx + 1 + 2x yx = 0. Set ux = x yx. The equation becomes ux + 2 − 3ux + 1 + 2ux = 0, which is linear and easily solved by previous methods. 2

2

2

2

1 yx = h +  a+h x 1 c −  −  b+h (a + h) + (b + h)





Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation.

INTEGRAL EQUATIONS REFERENCES: Courant, R., and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York (1953); Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985); Porter, D., and D. S. G. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications, Cambridge University Press (1990); Statgold, I., Green’s Functions and Boundary Value Problems, 2d ed., Interscience, New York (1997).

The limits of integration are fixed, and these problems are analogous to boundary value problems. An eigenvalue problem is a homogeneous equation of the second kind, and solutions exist only for certain λ.

An integral equation is any equation in which the unknown function appears under the sign of integration and possibly outside the sign of integration. If derivatives of the dependent variable appear elsewhere in the equation, the equation is said to be integrodifferential.

See Linz (1985) for further information and existence proofs. If the unknown function u appears in the equation in any way except to the first power, the integral equation is said to be nonlinear. b The equation u(x) = f(x) + ∫a K(x, t)[u(t)]3/2 dt is nonlinear. The differential equation du/dx = g(x, u) is equivalent to the nonlinear integral x equation u(x) = c + ∫a g[t, u(t)] dt. An integral equation is said to be singular when either one or both of the limits of integration become infinite or if K(x, t) becomes infinite for one or more points of the interval under discussion.

CLASSIFICATION OF INTEGRAL EQUATIONS Volterra integral equations have an integral with a variable limit. The Volterra equation of the second kind is

 K(x, t)u(t) dt x

u(x) = f(x) + λ

a

whereas a Volterra equation of the first kind is

 K(x, t)u(t) dt

 K(x, t)u(t) dt b

u(x) = λ



a

∞



x

u(t)  dt are both x−t singular. The kernel of the first equation is cos (xt), and that of the second is (x − t)−1.

Example u(x) = x +

0

cos (xt)u(t) dt and f(x) =

0

x

u(x) = λ

a

Equations of the first kind are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. A Fredholm equation of the second kind is

 K(x, t)u(t) dt b

u(x) = f(x) + λ

RELATION TO DIFFERENTIAL EQUATIONS The Leibniz rule (see “Integral Calculus”) can be used to show the equivalence of the initial-value problem consisting of the secondorder differential equation d 2y/dx2 + A(x)(dy/dx) + B(x)y = f(x) together with the prescribed initial conditions y(a) = y0, y′(a) = y′0 to the integral equation.

a

 K(x, t)u(t) dt

 K(x, t)y(t) dt + F(x) x

y(x) =

whereas a Fredholm equation of the first kind is

a

b

u(x) =

a

where

K(x, t) = (t − x)[B(t) − A′(t)] − A(t)

INTEGRAL TRANSFORMS (OPERATIONAL METHODS)

 (x − t)f(t) dt + [A(a)y + y′](x − a) + y x

F(x) =

and

0

0

3-37

Equations of the type considered here occur quite frequently in practice in what can be called “cause-and-effect” systems.

0

a

This integral equation is a Volterra equation of the second kind. Thus the initial-value problem is equivalent to a Volterra integral equation of the second kind. Example d 2y/dx2 + x2(dy/dx) + xy = x, y(0) = 1, y′(0) x= 0. Here A(x) = x2, B(x) = x, f(x) = x. The equivalent integral equation is y(x) = ∫ 0 K(x, t)y(t) dt + F(x) x where K(x, t) = t(x − t) − t2 and F(x) = ∫ 0 (x − t)t dt + 1 = x3/6 + 1. Combining these x 3 y(x) = ∫ 0 t[x − 2t]y(t) dt + x /6 + 1. Eigenvalue problems can also be related. For example, the problem (d 2y/dx2) + λy = 0 with y(0) = 0, y(a) = 0 is equivalent to the integral a equation y(x) = λ ∫ 0 K(x, t)y(t) dt, where K(x, t) = (t/a)(a − x) when t < x and K(x, t) = (x/a)(a − t) when t > x. The differential equation may be recovered from the integral equation by differentiating the integral equation by using the Leibniz rule. METHODS OF SOLUTION

Example In a certain linear system, the effect E(t) due to a cause C = λE at time τ is a function only of the elapsed time t − τ. If the system has the activity level 1 at time t < 0, thet cause λE and effect (E) relation is given by the integralt equation E(t) = 1 + λ ∫ 0 K(t − τ)E(τ) dτ. Let K(t − τ) = t − τ. Then E(t) = 1 + λ ∫ 0 (t − τ)E(τ) dτ. By using the transform method













1/p L[1] p t E(t) = L−1  = L−1 2 = L−1  = cosh λ 1 − λL[K(t)] 1 − λ/p p2 − λ

Method of Successive Approximations Consider the equation b y(x) = f(x) + λ ∫ a K(x, t)y(t) dt. In this method a unique solution is obtained in sequence form as follows: Substitute in the right-hand member of the equation y0(t) for y(t). Upon integration there results b y1(t) = f(x) + λ ∫ a K(x, t)y0(t) dt. Continue in like manner by replacing y0 by y1, y1 by y2, etc. A series of functions y0(x), y1(x), y2(x), . . . are obtained which satisfy the equations

 K(x, t)y b

In general, the solution of integral equations is not easy, and a few exact and approximate methods are given here. Often numerical methods must be employed, as discussed in “Numerical Solution of Integral Equations.” Equations of Convolution Type The equation u(x) = f(x) + x λ ∫ 0 K(x − t)u(t) dt is a special case of the linear integral equation of the second kind of Volterra type. The integral part is the convolution integral discussed under “Integral Transforms (Operational Methods)”; so the solution can be accomplished by Laplace transforms; L[u(x)] = L[f(x)] + λL[u(x)]L[K(x)] or L[f(x)] L[u(x)] =  , 1 − λL[K(x)]



L[f(x)] u(x) = L−1  1 − λL[K(x)]

yn(x) = f(x) + λ

n−1

a

2 b ∫a

(t) dt b

Then yn(x) = f(x) + λ ∫ ab K(x, t)f(t) dt + λ K(x, t) ∫ a K(t, t1)f(t1) dt1 dt + b λ3 ∫ ab K(x, t) ∫ a K(t, t1) ∫ ab K(t1, t2)f(t2) dt2 dt1 dt + ⋅ ⋅ ⋅ + Rn, where Rn is the remainder, and





max. y0 |Rn| ≤ |λn| a ≤ x ≤ b Mn(b − a)n where M = maximum value of |K| in the rectangle a ≤ t ≤ b, a ≤ x ≤ b. If |λ|M(b − a) < 1, lim Rn = 0. Then yn(x) → y(x), which is the unique n→∞ solution.



INTEGRAL TRANSFORMS (OPERATIONAL METHODS) REFERENCES: Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000); Churchill, R. V., Operational Mathematics, 3d ed., McGraw-Hill, New York (1972); Davies, B., Integral Transforms and Their Applications, 3d ed., Springer (2002); Duffy, D. G., Transform Methods for Solving Partial Differential Equations, Chapman & Hall/CRC, New York (2004); Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997).

The term “operational method” implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a very powerful tool in the applications of mathematics, but it is limited to linear problems. Most integral transforms are special cases of the equation g(s) = b ∫ a f(t)K(s, t) dt in which g(s) is said to be the transform of f(t) and K(s, t) is called the kernel of the transform. A tabulation of the more important kernels and the interval (a, b) of applicability follows. The first three transforms are considered here. Name of transform

(a, b)

Laplace

(0, ∞)

Fourier

(−∞, ∞)

Fourier cosine

(0, ∞)

K(s, t) e−st 1  e−ist  2π

π2 cos st π2 sin st

Fourier sine

(0, ∞)

Mellin

(0, ∞)

ts − 1

Hankel

(0, ∞)

tJν (st), ν ≥ −a

LAPLACE TRANSFORM The Laplace transform of a function f(t) is defined by F(s) = L{f(t)} = ∞ ∫ 0 e−stf(t) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions f for which this improper integral converges. The Laplace transform is used in process control (see Sec. 8). The function L[f(t)] = g(s) is called the direct transform, and L−1[g(s)] = f(t) is called the inverse transform. Both the direct and the inverse transforms are tabulated for many often-occurring functions. In general, 1  + i∞ st L−1[g(s)] =  e g(s) ds 2πi  − i∞ and to evaluate this integral requires a knowledge of complex variables, the theory of residues, and contour integration. A function is said to be piecewise continuous on an interval if it has only a finite number of finite (or jump) discontinuities. A function f on 0 < t < ∞ is said to be of exponential growth at infinity if there exist constants M and α such that |f(t)| ≤ Meαt for sufficiently large t. Sufficient Conditions for the Existence of Laplace Transform Suppose f is a function which is (1) piecewise continuous on every finite interval 0 < t < T, (2) of exponential growth at infinity, and (3) ∫ 0δ |f(t)| dt exist (finite) for every finite δ > 0. Then the Laplace transform of f exists for all complex numbers s with sufficiently large real part. Note that condition 3 is automatically satisfied if f is assumed to be piecewise continuous on every finite interval 0 ≤ t < T. The function f(t) = t−1/2 is not piecewise continuous on 0 ≤ t ≤ T but satisfies conditions 1 to 3. Let Λ denote the class of all functions on 0 < t < ∞ which satisfy conditions 1 to 3.



3-38

MATHEMATICS

Example Let f(t) be the Heaviside step function at t = t0; i.e., f(t) = 0 for t ≤ t0, and f(t) = 1 for t > t0. Then ∞ T e−st0 1 L{ f(t)} = e−st dt = lim e−st dt = lim  (e−st0 − e−sT) =  T→∞ t T→∞ s s t0 0





TABLE 3-2

Laplace Transforms

f(t)

g(s)

Example Let f(t) = eat, t ≥ 0, where a is a real number. Then L{eat} =

∫0 e

−(s − a)

1

1/s

e (1 − at)

tn, (n a + integer)

n!  sn + 1

t sin at  2a

s  (s2 + a2)2

tn, n ≠ + integer

Γ(n + 1)  sn + 1

1 2 sin at sinh at 2a

s  s4 + 4a4

cos at

s  s2 + a2

cos at cosh at

s2  s4 + 4a4

sin at

a  s2 + a2

s2 1  (sinh at + sin at)  s4 − a4 2a

cosh at

s  s2 − a2

s3 a (cosh at + cos at)  4 s − a4

sinh at

a  s2 − a2

sin at  t

a tan−1  s

e−at

1  s+a

J0(at)

1  s2 +a2 

e−bt cos at

s+b  (s + b)2 + a2

Jn(at) nan  t

( s2 + a2 − s)n (n > 0)

e−bt sin at

a  (s + b)2 + a2

) J0 (2 at

1  e−a/s s

k erfc  2t

1   e−k s s

dt = 1/(s − a), provided Re s > a.

Properties of the Laplace Transform 1. The Laplace transform is a linear operator: L{af(t) + bg(t)} = aL{ f(t)} + bL{g(t)} for any constants a, b and any two functions f and g whose Laplace transforms exist. 2. The Laplace transform of a real-valued function is real for real s. If f(t) is a complex-valued function, f(t) = u(t) + iv(t), where u and v are real, then L{ f(t)} = L{u(t)} + iL{v(t)}. Thus L{u(t)} is the real part of L{ f(t)}, and L{v(t)} is the imaginary part of L{ f(t)}. 3. The Laplace transform of a function in the class Λ has derivatives of all orders, and L{tkf(t)} = (−1)kd kF(s)/ds k, k = 1, 2, 3, . . . .



∞

2as a Example e−st sin at dt =  , s > 0. By property 3,  = 0 (s2 + a2)2 s2 + a2 ∞ −st e t sin at dt = L{t sin at}.



0

Example By applying property 3 with f(t) = 1 and using the preceding

results, we obtain



dk 1 k! L{tk} = (−1)k k  =  ds s sk + 1 provided Re s > 0; k = 1, 2, . . . . Similarly, we obtain





g(s) s 2 (s + a)

provided s > 0. ∞

f(t) −at

dk 1 k! L{tkeat} = (−1)k k  =  ds s − a (s − a)k + 1

1 Example L{te } = 2 , s > 0. (s + a) −at

5. Time-shift property. Let u(t − a) be the unit step function at t = a. Then L{f(t − a)u(t − a)} = e−asF(s). 6. Transform of a derivative. Let f be a differentiable function such that both f and f ′ belong to the class Λ. Then L{ f ′(t)} = sF(s) − f(0). 7. Transform of a higher-order derivative. Let f be a function which has continuous derivatives up to order n on (0, ∞), and suppose that f and its derivatives up to order n belong to the class Λ. Then L{ f ( j)(t)} = s jF(s) − s j − 1f(0) − s j − 2 f ′(0) − ⋅ ⋅ ⋅ − sf ( j − 2)(0) − f ( j − 1)(0) for j = 1, 2, . . . , k. Example L{ f″(t)} = s2L{ f(t)} − sf(0) − f′(0) L{ f ″′(t)} = s3L{ f(t)} − s2f(0) − sf′(0) − f ″(0)

Example Solve y″ + y = 2et, y(0) = y′(0) = 2. L[y″] = −y′(0) − sy(0) + s2L[y] =

−2 − 2s + s2L[y]. Thus

2 −2 − 2s + s2L[y] + L[y] = 2L[et] =  s−1 1 1 s 2s2 L[y] =  =++ (s − 1)(s2 + 1) s − 1 s2 + 1 s2 + 1 Hence y = et + cos t + sin t.

A short table (Table 3-2) of very common Laplace transforms and inverse transforms follows. The references include more detailed ∞ tables. NOTE: Γ(n + 1) = ∫ 0 xne−x dx (gamma function); Jn(t) = Bessel function of the first kind of order n. t 1 0 1 8. L f(t) dt =  L[ f(t)] +  f(t) dt a s s a







1 s

 s −1 a 

Example Find f(t) if L[ f(t)] = 2  . 2 2 Therefore f(t) =





1 1 L  sinh at =  . s2 − a2 a

sinh at   1a sinh at dt dt = a1  − t. a t

t

0

0

  

f(t) 9. L  = t

2

   ⋅⋅⋅

∞

∞

f(t) L  = tk

g(s) ds

s

∞

s

g(s)(ds)k

s



4. Frequency-shift property (or, equivalently, the transform of an exponentially modulated function). If F(s) is the Laplace transform of a function f(t) in the class Λ, then for any constant a, L{eat f(t)} = F(s − a).

k integrals



sin at t

 

∞

Example L  =



∞

L[sin at] ds =

s

s

a ds s  = cot−1  s2 + a2 a

10. The unit step function u(t − a) =

 10

ta

L[u(t − a)] = e−as/s

11. The unit impulse function is δ(a) = u′(t − a) =

∞0

at t = a elsewhere

L[u′(t − a)] = e−as

12. L−1[e−asg(s)] = f(t − a)u(t − a) (second shift theorem). 13. If f(t) is periodic of period b, i.e., f(t + b) = f(t), then b 1 L[f(t)] =  e−stf(t) dt −bs 0 1−e





Example The partial differential equations relating gas composition to position and time in a gas chromatograph are ∂y/∂n + ∂x/∂θ = 0, ∂y/∂n = x − y, where x = mx′, n = (kGaP/Gm)h, θ = (mkGaP/ρB)t and GM = molar velocity, y = mole fraction of the component in the gas phase, ρB = bulk density, h = distance from the entrance, P = pressure, kG = mass-transfer coefficient, and m = slope of the equilibrium line. These equations are equivalent to ∂2y/∂n ∂θ + ∂y/∂n + ∂y/∂θ = 0, where the boundary conditions considered here are y(0, θ) = 0 and x(n, 0) = y(n, 0) + (∂y/∂n) (n, 0) = δ(0) (see property 11). The problem is conveniently

INTEGRAL TRANSFORMS (OPERATIONAL METHODS) solved by using the Laplace transform of y with respect to n; write ∞ g(s, θ) = ∫ 0 e−nsy(n, θ) dn. Operating on the partial differential equation gives s(dg/dθ) − (∂y/∂θ) (0, θ) + sg − y(0, θ) + dg/dθ = 0 or (s + 1) (dg/dθ) + sg = (∂y/∂θ) (0, θ) + y(0, θ) = 0. The second boundary condition gives g(s, 0) + sg(s, 0) − y(0, 0) = 1 or g(s, 0) + sg(s, 0) = 1 (L[δ(0)] = 1). A solution of the ordinary differential equation for g consistent with this second condition is

3-39

TABLE 3-3 z-Transforms f(k)

1 g(s, θ) =  e−sθ /(s + 1) s+1

θ) where Inversion of this transform gives the solution y(n, θ) = e−(n + θ) I0(2 n I0 = zero-order Bessel function of an imaginary argument. For large u, In(u) ∼ u e /2 π u. Hence for large n, θ −  n)2] exp [−( y(n, θ) ∼  2π1/2(nθ)1/4

g*(z)

1(k)

1  1 − z−1

k ∆t

∆t z−1  (1 − z−1)2

(k ∆t)n − 1

∂n − 1 1 lim (−1)n − 1   a→0 ∂an − 1 1 − e−a∆tz−1





−1

sin a k ∆t

z sin a ∆t  (1 − 2 z−1 cos a ∆t + z−2)

or for sufficiently large n, the peak concentration occurs near θ = n.

cos a k ∆t

1 − z−1 cos a ∆t  (1 − 2 z−1 cos a ∆t + z−2)

Other applications of Laplace transforms are given under “Differential Equations.”

e−ak∆t

1  1 − e−a∆tz−1

CONVOLUTION INTEGRAL

e−bk∆t cos a k ∆t

1 − z−1 e−b∆t cos a ∆t  1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t

The convolution integral (faltung) of two functions f(t), r(t) is x(t) = f(t)°r(t) = ∫ 0t f(τ)r(t − τ) dτ.

1  e−bk∆t sin a k ∆t b

z−1 e−b∆t sin a ∆t 1   b 1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t

 τ sin (t − τ) dτ = t − sin t. t

Example t° sin t =

0

L[ f(t)]L[h(t)] = L[ f(t)°h(t)]

The Fourier transform is given by

Z-TRANSFORM

See Ogunnaike, Babatunde A., and W. Harmon Ray, Process Dynamics, Modeling, and Control, Oxford University Press (1994); Seborg, D., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York (2003). The z-transform is useful when data is available at only discrete points. Let

1 F[ f(t)] =  2  π

tk = k ∆t,

k = 0, 1, 2, . . .

Then the function f *(t) is k=0

∞

∞

k=0

k=0

g*(s) = L[ f*(t)] = f(tk) e−stk = f(tk) e−s∆tk s∆t

For convenience, replace e by z and call g*(z) the z-transform of f*(t). ∞

g*(z) = f(tk) z−k k=0

The z-transform is used in process control when the signals are at intervals of ∆t. A brief table (Table 3-3) is provided here. The z-transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. Example The difference equation for y(k) is y(k) + a1 y(k − 1) + a2 y(k − 2) = b1u(k) Take the z-transform (1 + a1z−1 + a2z−2) y*(z) = b1u*(z) Then

u*(z) y*(z) = −1 1 + a1z + a2z−2

The inverse transform must be found, usually from a table of inverse transforms.

FOURIER TRANSFORM REFERENCES: Bateman, H., Tables of Integral Transforms, vol. I, McGraw-Hill, New York (1954); Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997).

f(t)e−ist dt = g(s)



∞

−∞

g(s)eist dt = f(t)

In brief, the condition for the Fourier transform to exist is that ∞ ∫ -∞ |f(t)| dt < ∞, although certain functions may have a Fourier transform even if this is violated. Example The function f(t) =

e a

ist

Take the Laplace transform of this.

−∞

1 F−1[g(s)] =  π 2

∞

f*(t) = f(tk) δ(t − tk)

∞

and its inverse by

f*(t) = f(tk) be the value of f at the sample points



0

e a

dt +

−ist

0

 0 elsewhere has F[ f(t)] = 1−a≤t≤a

a

−a

e−ist dt =

2 sin sa  cos st dt =  s a

dt = 2

0

Properties of the Fourier Transform Let F[f (t)] = g(s); F−1[g(s)] = f(t). 1. F[ f (n)(t)] = (is)nF[ f(t)]. 2. F[af(t) + bh(t)] = aF[ f(t)] + bF[h(t)]. 3. F[ f(−t)] = g(−s). 1 s 4. F[ f(at)] =  g  , a > 0. a a 5. F[e−iwt f(t)] = g(s + w). 6. F[ f(t + t1)] = eist1g(s). 7. F[ f(t)] = G(is) + G(−is) if f(t) = f(−t) ( f even) F[ f(t)] = G(is) − G(−is) if f(t) = −f(−t) ( f odd) where G(s) = L[f(t)]. This result allows the use of the Laplacetransform tables to obtain the Fourier transforms.



Example Find F[e−a|t|] by property 7. e−a|t| is even. So L[e−at] = 1/(s + a). Therefore, F[e−a|t|] = 1/(is + a) + 1/(−is + a) = 2a/(s2 + a2). FOURIER COSINE TRANSFORM The Fourier cosine transform is given by

π2 

∞

Fc[f(t)] = g(s) =

0

f(t) cos st dt

3-40

MATHEMATICS

and its inverse by

π2 

The Fourier sine transform Fs is obtainable by replacing the cosine by the sine in these integrals. They can be used to solve linear differential equations; see the transform references.

∞

F c−1[g(s)] = f(t) =

g(s) cos st ds

0

MATRIX ALGEBRA AND MATRIX COMPUTATIONS REFERENCES: Anton, H., and C. Rorres, Elementary Linear Algebra with Applications, 9th ed., Wiley (2004); Bernstein, D. S., Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, N.J. (2005); Kolman, B., and D. R. Hill, Introductory Linear Algebra: An Applied First Course, 8th ed., Prentice-Hall, Englewood Cliffs, N.J. (2004); Lay, D. C., Linear Algebra and Its Applications, 3d ed., Addison Wesley (2002); Lipschutz, S., and M. Lipson, Schaum’s Outline of Linear Algebra, McGraw-Hill, New York (2000); Noble, B., and J. W. Daniel, Applied Linear Algebra, 3d ed., Prentice-Hall, Englewood Cliffs, N.J. (1987); Press, W. H., et al., Numerical Recipes, Cambridge University Press, Cambridge (1986).

MATRIX ALGEBRA Matrices A rectangular array of rows and n columns a11 a A = (aij) = 21 ⯗ am1



mn quantities, arranged in m … …

a1n a2n

…

amn



is called a matrix. The elements aij may be real or complex. The notation aij means the element in the ith row and jth column, i is called the row index, j the column index. If m = n the matrix is said to be square and of order n. A matrix, even if it is square, does not have a numerical value, as a determinant does. However, if the matrix A is square, a determinant can be formed which has the same elements as the matrix A. This is called the determinant of the matrix and is written det (A) or |A|. If A is square and det (A) ≠ 0, A is said to be nonsingular; if det (A) = 0, A is said to be singular. A matrix A has rank r if and only if it has a nonvanishing determinant of order r and no nonvanishing determinant of order > r. Equality of Matrices Let A = (aij), B = (bij). Two matrices A and B are equal (=) if and only if they are identical; that is, they have the same number of rows and the same number of columns and equal corresponding elements (aij = bij for all i and j). Addition and Subtraction The operations of addition (+) and subtraction (−) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A  B = (aij  bij); i.e., addition and subtraction are of corresponding elements. Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A′ or AT. Example



1 A= 2

3 1

 

1 A = 3 4



4 , 6

T

2 1 6

T T

Note that (A ) = A.

Multiplication Let A = (aij), i = 1, . . . , m1; j = 1, . . . , m2. B = (bij), i = 1, . . . , n1, j = 1, . . . , n2. The product AB is defined if and only if the number of columns of A (m2) equals the number of rows of B(n1), i.e., n1 = m2. For two such matrices the product P = AB is defined by summing the element by element products of a row of A by a column of B. This is the row by column rule. Thus n1

pij = aikbkj k=1

The resulting matrix has m1 rows and n2 columns. Example

  3 1 5

2 1 4

0 −2

1 0

5 1



6 = 3



−4 −2 −8

3 1 5

17 6 29

24 9 42



It is helpful to remember that the element pij is formed from the ith row of the first matrix and the jth column of the second matrix. The matrix product is not commutative. That is, AB ≠ BA in general.

Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = I, where I is the identity matrix of order n. 1 0.... 0 0 1.. ⯗ 1 0 0 ....0 1





The inverse B is a square matrix of the order of A, designated by A−1. Thus AA−1 = A−1A = I. A square matrix A has an inverse if and only if A is nonsingular. Certain relations are important: (1)

(AB)−1 = B−1A−1

(2)

(AB)T = BTAT

(3)

(A−1)T = (AT )−1

(4)

(ABC)−1 = C−1B−1A−1

Scalar Multiplication Let c be any real or complex number. Then cA = (caij). Adjugate Matrix of a Matrix Let Aij denote the cofactor of the element aij in the determinant of the matrix A. The matrix BT where B = (Aij) is called the adjugate matrix of A written adj A = BT. The elements bij are calculated by taking the matrix A, deleting the ith row and jth column, and calculating the determinant of the remaining matrix times (−1)i + j. Then A−1 = adj A/|A|. This definition may be used to calculate A−1. However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under “Numerical Analysis and Approximate Methods.” Linear Equations in Matrix Form Every set of n nonhomogeneous linear equations in n unknowns a11 x1 + a12 x2 + ⋅ ⋅ ⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅ ⋅ ⋅ + a2n xn = b2   an1 x1 + an2 x2 + ⋅ ⋅ ⋅ + ann xn = bn can be written in matrix form as AX = B, where A = (aij), X T = [x1 ⋅ ⋅ ⋅ xn], and BT = [b1 ⋅ ⋅ ⋅ bn]. The solution for the unknowns is X = A−1B. Special Square Matrices 1. A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements a11, . . . , ann) are zero. If A is triangular, det (A) = a11. a22 . . . ann. 2. A diagonal matrix is one such that all elements both above and below the main diagonal are zero (i.e., aij = 0 for all i ≠ j). If all diagonal elements are equal, the matrix is called scalar. If A is diagonal, A = (aij), A−1 = (1/aij). 3. If aij = aji for all i and j (i.e., A = AT ), the matrix is symmetric. 4. If aij = −aji for i ≠ j but the aij are not all zero, the matrix is skew. 5. If aij = −aji for all i and j (i.e., aii = 0), the matrix is skew symmetric. 6. If AT = A−1, the matrix A is orthogonal. 7. If the matrix A* = (aij)T, aij = complex conjugate of aij, A* is the hermitian transpose of A. 8. If A = A−1, A is involutory. 9. If A = A*, A is hermitian. 10. If A = −A*, A is skew hermitian. 11. If A−1 = A*, A is unitary. If A is any matrix, then AAT and ATA are square symmetric matrices, usually of different order.

MATRIX ALGEBRA AND MATRIX COMPUTATION



5

1 4 −2

Example Let A = 3 2



35 AAT = 22 8

2 −2 0 1

38 13 18 17

13 21 7 18

18 7 10 5





22 51 3



3 4 1 5

5 0 1 5 , AT = 3 1 0

3 1 0

8 3 , 9

ATA =





MATRIX COMPUTATIONS



17 18 5 26

Using a program such as MATLAB, these are easily calculated.

Matrix Calculus Differentiation Let the elements of A = [aij(t)] be differentiable dA daij(t) functions of t. Then  =  . dt dt













−sin t . cos t

cos t sin t cos t dA , = Example A = −cos sin t t sin t  dt

Integration The integral ∫ A dt = [∫ aij(t) dt].









2 t 2t Example A = t2 e2t , ∫ A dt = tt3/2 . /3 et

The matrix B = A − λI is called the characteristic (eigen) matrix of A. Here A is square of order n, λ is a scalar parameter, and I is the n × n identity. det B = det (A − λI) = 0 is the characteristic (eigen) equation for A. The characteristic equation is always of the same degree as the order of A. The roots of the characteristic equation are called the eigenvalues of A.

 

  

 

Example A = 13 28 , B = 13 28 − λ0 λ0 = 1−λ3



2 8−λ

is the characteristic matrix and f(λ) = det (B) = det (A − λI) = (1 − λ)(8 − λ) − 6 = 2 − 9λ + λ2 = 0 is the characteristic equation. The eigenvalues of A are the roots of λ2 − 9λ + 2 = 0, which are (9  73)/2.

A nonzero matrix Xi, which has one column and n rows, called a column vector satisfying the equation (A − λI)Xi = 0 and associated with the ith characteristic root λi is called an eigenvector. Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vectors in Rn and for matrices that are analogous to the notion of length of a geometric vector. Let Rn denote the set of all vectors with n components, x = (x1, . . . , xn). In dealing with matrices it is convenient to treat vectors in Rn as columns, and so x = (x1, . . . , xn)T; however, we shall here write them simply as row vectors. A norm on Rn is a real-valued function f defined on Rn with the following properties: 1. f(x) ≥ 0 for all x  Rn. 2. f(x) = 0 if and only if x = (0, 0, . . . , 0). 3. f(ax) = |a|f(x) for all real numbers a and x  Rn. 4. f(x + y) $ f(x) + f(y) for all x, y  Rn. The usual notation for a norm is f(x) = x. The norm of a matrix is κ(A)  A  A−1 n  A x A = supx ≠ 0  = maxk |ajk|  x j=1 The norm is useful when doing numerical calculations. If the computer’s floating-point precision is 10−6, then κ = 106 indicates an illconditioned matrix. If the floating-point precision is 10−12 (double precision), then a matrix with κ = 1012 may be ill-conditioned. Two other measures are useful and are more easily calculated:

where

(k)

3-41

maxk |akk | |det A| Ratio =  V =  , αi = (a 2i1 + a 2i2 + . . . a 2in)1/2 (k) , mink |akk | α1 α2 . . . αn where akk(k) are the diagonal elements of the LU decomposition.

The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. LU Factorization of a Matrix To every m × n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and an m × n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessary in carrying out the Gauss elimination to interchange two rows of A to produce a (nonzero) pivot, such as if we start with 0 2 A= 1 6 If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normalized (as assumed previously), so that it has unit elements on the main diagonal. Solution of Ax = b by Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = b implies that Lz = b. Thus to solve Ax = b we first solve Lz = b for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = b. Note that the systems Lz = b and Ux = z are both in triangular forms and thus can be easily solved. The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency. The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiplication count for an n × n matrix and m right-hand sides is

 

1 1 operation count =  n3 −  n + mn2 3 3 If an inverse is desired, it can be calculated by solving for the LU decomposition and then solving n problems with right-hand sides consisting of all zeros except one entry. Thus 4n2/3 − n/3 multiplications are required for the inverse. The determinant is given by n

Det A =  aii(i) i=1

where aii(i) are the diagonal elements obtained in the LU decomposition. A tridiagonal matrix is one in which the only nonzero entries lie on the main diagonal and the diagonal just above and just below the main diagonal. The set of equations can be written as ai xi − 1 + bi xi + ci xi + 1 = di The LU decomposition is b1 = b1 for k=2,n do ak a′k = , b′k − 1

ak b′k = bk −  ck − 1 b′k − 1

enddo d′1 = d1 for k=2,n do d′k = dk − a′k d′k − 1 enddo xn = d′n /b′n for k=n−1,1 do d′k − ck xk + 1 xk =  b′k enddo

3-42

MATHEMATICS

The operation count for an n × n matrix with m right-hand sides is

>> a1 = a’*a

2(n − 1) + m(3n − 2) If

|bi| > |ai| + |ci |

no pivoting is necessary, and this is true for many boundary-value problems and partial-differential equations. Sparse matrices are ones in which the majority of the elements are zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Duff, I. S., J. K. Reid, and A. M. Erisman (eds.), Direct Methods for Sparse Matrices, Clarendon Press, Oxford (1986); Saad, Y., Iterative Methods for Sparse Linear Systems, 2d ed., Society for Industrial and Applied Mathematics, Philadelphia (2003). The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n × n matrix. Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions. See Amundson, N. R., Mathematical Methods in Chemical Engineering, Prentice-Hall, Englewood Cliffs, N.J. (1966, p. 50). QR Factorization of a Matrix If A is an m × n matrix with m ≥ n, there exists an m × m unitary matrix Q = [q1, q2,…,qm] and an m × n right triangular matrix R such that A = QR. The QR factorization is frequently used in the actual computations when the other transformations are unstable. Singular-Value Decomposition If A is an m × n matrix with m ≥ n and rank k ≤ n, consider the two following matrices. AA* and

A*A

An m × m unitary matrix U is formed from the eigenvectors ui of the first matrix.

V [v1,v2, . . .,vn] Then matrix A can be decomposed into A UV* where  is a k × k diagonal matrix with diagonal elements dii = i > 0 for 1 ≤ i ≤ k. The eigenvalues of * are  2i. The vectors ui for k + 1 ≤ i ≤ m and vi for k + 1 ≤ i ≤ n are eigenvectors associated with the eigenvalue zero; the eigenvalues for 1 ≤ i ≤ k are 2i. The values of i are called the singular values of matrix A. If A is real, then U and V are real and hence orthogonal matrices. The value of the singular-value decomposition comes when a process is represented by a linear transformation and the elements of A, aij, are the contribution to an output i for a particular variable as input variable j. The input may be the size of a disturbance, and the output is the gain [Seborg, D. E., T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York (2004)]. If the rank is less than n, not all the variables are independent and they cannot all be controlled. Furthermore, if the singular values are widely separated, the process is sensitive to small changes in the elements of the matrix, and the process will be difficult to control. Example Consider the following example from Noble and Daniel [Applied Linear Algebra, Prentice-Hall (1987)] with the MATLAB commands to do the analysis. Define the following real matrix with m = 3 and n = 2 (whose rank k = 1). >> a = [ 1 1 22 22]

9

9

9

%(n × n or 2 × 2)

>> a2 = a*a´ a2 = 2

4

4

4

8

8

4

8

8

%(m × m or 3 × 3)

>> [v,d1]=eig(a1) v = −0.7071 0.7071

%(n × n or 2 × 2)

0.7071 0.7071

d1 = 0 0 0 18 >> [u,d2]=eig(a2) u = 0.8944 0.2981

0.3333

−0.4472 0.5963

0.6667

−0.7454

0.6667

0

%(m × m or 3 × 3)

d2 = 0 0 0 0 0 0 0 0 18 Thus, 18 and the eigenfunctions are the rows of v and u. The second column of v is associated with the eigenvalue 21 18, and the third column of u is associated with the eigenvalue 21 18. If A is square and nonsingular, the vector x that minimizes 2 1

||Ax  b||

(3-71)

is obtained by solving the linear equations x A−1b When A is not square, then the solution to Ax = b

U [u1,u2, . . ., um] An n × n unitary matrix V is formed from the eigenvectors vi of the second matrix.

a1 = 9

is x Vy where yi b′i / i for i 1, . . ., k, b¢ = UTb, and yk1, yk2, . . . , ym are arbitrary. The matrices U and V are those obtained in the singular-value decomposition. The solution which minimizes the norm, Eq. (3-71), is x with yk1, yk2, . . . , ym zero. These techniques can be used to monitor process variables. See Montgomery, D. C., Introduction to Statistical Quality Control, 4th ed., Wiley, New York (2001); Piovos, M. J., and K. A. Hoo, “Multivariate Statistics for Process Control,” IEEE Control Systems 22(5):8 (2002).

Principal Component Analysis (PCA) PCA is used to recognize patterns in data and reduce the dimensionality of the problem. Let the matrix A now represent data with the columns of A representing different samples and the rows representing different variables. The covariance matrix is defined as ATA cov(A)  m 1 This is just the same matrix discussed with singular value decomposition. For data analysis, though, it is necessary to adjust the columns to have zero mean by subtracting from each entry in the column the average of the column entries. Once this is done, the loadings are the vi and satisfy cov(A) vi 2i vi and the score vector ui is given by Avi iui In process analysis, the columns of A represent different measurement techniques (temperatures, pressures, etc.) and the rows represent the measurement output at different times. In that case

NUMERICAL ANALYSIS AND APPROXIMATE METHODS the columns of A are adjusted to have a zero mean and a variance of 1.0 (by dividing each entry in the column by the variance of the column). The goal is to represent the essential variation of the process with as few variables as possible. The ui, vi pairs are arranged in descending order according to the associated i. The i can be thought of as the variance, and the ui, vi pair captures the greatest amount of variation in the data. Instead of having to deal with n variables, one can capture most of the variation of the

3-43

data by using only the first few pairs. An excellent example of this is given by Wise, B. M., and B. R. Kowalski, “Process Chemometrics,” Chap. 8 in Process Analytical Chemistry, F. McLennan and B. Kowalski (eds.), Blackie Academic & Professional, London (1995). When modeling a slurry-fed ceramic melter, they were able to capture 97 percent of the variation by using only four eigenvalues and eigenvectors, even though there were 16 variables (columns) measured.

NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS APPROXIMATION IDENTITIES

Approximation

For the following relationships the sign  means approximately equal to, when X is small. These equations are derived by using a Taylor’s series (see “Series Summation and Identities”). Approximation 1 1"X 1X

Approximation X  X1 1 2

Approximation

(1  X)n  1 ± nX

(1  X)−n  1 " nX

(a  X)2  a2 ± 2aX

ex  1 + X

sin X  X(X rad) 2Y + X (Y  + X)   Y 2

tan X  X





X2 X 2 +  X2  Y +   small Y 2Y Y

NUMERICAL ANALYSIS AND APPROXIMATE METHODS REFERENCES: Buchanan, G. R., Schaum’s Outline of Finite Element Analysis, McGraw-Hill, New York (1995); Burden, R. L., J. D. Faires, and A. C. Reynolds, Numerical Analysis, 8th ed., Brookes Cole (2004); Chapra, S. C., and R. P. Canal, Numerical Methods for Engineers, 5th ed., McGraw-Hill, New York (2006); Finlayson, B. A., Nonlinear Analysis in Chemical Engineering, McGraw-Hill (1980), Ravenna Park (2003); Finlayson, B. A., and L. T. Biegler, “Mathematics in Chemical Engineering,” Ullmann’s Encyclopedia of Industrial Chemistry, vol. 20, VCH, Weinheim (2006); Gunzburger, M. D., Finite Element Methods for Viscous Incompressible Flows, Academic Press (1989); Kardestuncer, H., and D. H. Norrie (eds.), Finite Element Handbook, McGraw-Hill, New York (1987); Lau, H. T., A Numerical Library in C for Scientists and Engineers, CRC Press (1994); Lau, H. T., A Numerical Library in Java for Scientists and Engineers, CRC Press (2004); Morton, K. W., and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press (1994); Press, W. H., et al., Numerical Recipes, Cambridge University Press, Cambridge (1986); Quarteroni, A., and A. Valli, Numerical Approximation of Partial Differential Equations, Springer (1997); Reddy, J. N., and D. K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, 2d ed., CRC Press (2000); Scheid, F., Schaum’s Outline of Numerical Analysis, 2d ed., McGraw-Hill, New York (1989); Schiesser, W. E., The Numerical Method of Lines, Academic Press (1991); Shampine, L., Numerical Solution of Ordinary Differential Equations, Chapman & Hall (1994); Zienkiewicz, O. C., R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, vol. 1, 6th ed., Elsevier Butterworth-Heinemann (2005); Zienkiewicz, O. C., and R. L. Taylor, The Finite Element Method: Solid Mechanics, vol. 2, 5th ed., Butterworth-Heinemann (2000); Zienkiewicz, O. C., and R. L. Taylor, The Finite Element Method: Fluid Mechanics, vol. 2, 5th ed., Butterworth-Heinemann (2000).

INTRODUCTION The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as True value = approximate value + error

The four sources of error are as follows: 1. Gross errors. These result from unpredictable human, mechanical, or electrical mistakes. 2. Round-off errors. These are the consequence of using a number specified by m correct digits to approximate a number which requires more than m digits for its exact specification. For example, approximate the irrational number 2 by 1.414. Such errors are often present in experimental data, in which case they may be called inherent errors, due either to empiricism or to the fact that the computer dictates the number of digits. Such errors may be especially damaging in areas such as matrix inversion or the numerical solution of partial differential equations when the number of algebraic operations is extremely large. 3. Truncation errors. These errors arise from the substitution of a finite number of steps for an infinite sequence of steps which would yield the exact result. To illustrate this error consider the infinite series for e−x: e−x = 1 − x + x2/2 − x3/6 + ET(x), where ET is the truncation error, ET = (1/24)e−εx4, 0 < ε < x. If x is positive, ε is also positive. Hence e−ε < 1. The approximation e−x ≈ 1 − x + x2/2 − x3/6 is in error by a positive amount smaller than (1/24)x4. 4. Inherited errors. These arise as a result of errors occurring in the previous steps of the computational algorithm. The study of errors in a computation is related to the theory of probability. In what follows a relation for the error will be given in certain instances. A variety of general-purpose computer programs are available commercially. Mathematica (http://www.wolfram.com/), Maple (http:// www.maplesoft.com/), and Mathcad (http://www.mathcad.com/) all have the capability of doing symbolic manipulation so that algebraic solutions can be obtained. For example, Mathematica can solve some ordinary and partial differential equations analytically; Maple can make simple graphs and do linear algebra and simple computations; and Mathcad can do simple calculations. In this section, examples are given for the use of Matlab (http://www.mathworks.com/), which is a package of numerical analysis tools, some of which are accessed by simple commands and others of which are accessed by writing programs in C. Spreadsheets can also be used to solve certain problems, and these are described below too. A popular program used in chemical engineering education is Polymath (http://www.polymath-software. com/), which can numerically solve sets of linear or nonlinear equations, ordinary differential equations as initial-value problems, and perform data analysis and regression.

3-44

MATHEMATICS

NUMERICAL SOLUTION OF LINEAR EQUATIONS See the section entitled “Matrix Algebra and Matrix Computation.” NUMERICAL SOLUTION OF NONLINEAR EQUATIONS IN ONE VARIABLE Special Methods for Polynomials Consider a polynomial equation of degree n: P(x) = a0 x + a1x n

n−1

+ a2 x

n−2

+ ⋅⋅⋅ + an − 1x + an = 0

(3-72)

with real coefficients. P(x) has exactly n roots, which may be real or complex. If the roots are complex, they occur in pairs with their complex conjugates. One can obtain an upper and lower bound for the real roots by the following device: If a0 > 0 in Eq. (3-72) and if in Eq. (3-72) the first negative coefficient is preceded by k coefficients which are positive or zero, and if G is the greatest of the absolute values of the negative k coefficients, then each real root is less than 1 +  G/a 0. Example P(x) = x5 + 3x4 − 7x2 − 40x + 2 = 0. Here a03 = 1, G = 40, and k = 3 since we must supply 0 as the coefficient for x3. Thus 1 + 40 ≈ 4.42 is an upper bound for the real roots. The largest real root is 2.19. A lower bound to the real roots may be found by applying the criterion to the equation P(−x). Example P(−x) = −x5 + 3x4 − 7x2 + 40x + 2 = 0, which is equivalent to x5 − 3x4 + 7x2 − 40x − 2 = 0 since a0 must be +. Then a0 = 1, G = 40, and k = 1. Hence −(1 + 40) = −41 is a lower bound. The smallest real root is −3.41. Thus all real roots are between −41 and 4.42. One last result is helpful in getting an estimate of how many positive and negative real roots there are. Descartes Rule The number of positive real roots of a polynomial with real coefficients is either equal to the number of changes in sign v or is less than v by a positive even integer. The number of negative roots of P(x) is either equal to the number of variations of sign of P(−x) or is less than this by a positive even integer.

f(x) by the tangent line at x = xk in each successive step. If f′(x) and f″(x) have the same sign throughout an interval a ≤ x ≤ b containing the solution, with f(a), f(b) of opposite signs, then the process converges starting from any x0 in the interval a ≤ x ≤ b. The process is second order. (0.5)x − 0.5 0.3

Example f(x) = x − 1 +  f ′(x) = 1 − 2.3105[0.5]x An approximate root (obtained graphically) is 2. Step k

xk

f(xk)

f′(xk)

0 1 2

2 1.6054 1.4632

0.1667 0.0342 0.0055

0.4224 0.2407 0.1620

Method of False Position This variant is commenced by finding x0 and x1 such that f(x0), f(x1) are of opposite signs. Then α1 = slope of secant line joining [x0, f(x0)] and [x1, f(x1)] so that x1 − x0 x2 = x1 −  f(x1) f(x1) − f(x0) In each of the following steps αk is the slope of the line joining [xk, f(xk)] to the most recently determined point where f(xj) has the opposite sign from that of f(xk). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method. Method of Wegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure xk + 1 = F(xk) is revised by setting ˆxk + 1 = F(xk) and then taking xk + 1 = qxk + (1 − q)ˆxk + 1, where q is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q’s are: Behavior of successive substitution process

Range of optimum q

Oscillatory convergence Oscillatory divergence Monotonic convergence Monotonic divergence

0
Example f(x) = x4 − 13x2 + 4x − 2 = 0 has three changes in sign; therefore,

without Wegstein without Wegstein

there are either three or one positive root. f(−x) = x − 13x − 4x − 2 has one change in sign. Therefore, there is one negative root. Using MATLAB, one defines the vector

At each step q may be calculated to give a locally optimum value by setting

C(i) = ai−1

xˆ k + 1 − xˆ k q =  xˆ k + 1 − 2 xˆ k + xˆ k− 1

4

2

and uses the command roots(c) to find all the roots.

Methods for Nonlinear Equations in One Variable Successive Substitutions Let f(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F(x), then an iterative scheme can be set up in the form xk + 1 = F(xk). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writing x = F(x), of which there will usually be several forms. However, if a is a root of f(x) = 0, and if |F′(a)| < 1, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xk + 1 is proportional to the first power of the error in xk for large k. One way of writing the equation is xk + 1 = xk + β f(xk). The choice of β is made such that |1 + β df/dx(a)| < 1. Convergence is guaranteed by the theorem given for simultaneous equations.

Methods of Perturbation Let f(x) = 0 be the equation. In general, the iterative relation is xk + 1 = xk − [ f(xk)/αk] where the iteration begins with x0 as an initial approximation and αk as some functional, derived below. Newton-Raphson Procedure This variant chooses αk = f′(xk) where f′ = df/dx and geometrically consists of replacing the graph of

The Wegstein method is a secant method applied to g(x)  x − F(x). In Microsoft Excel, roots are found by using Goal Seek or Solver. Assign one cell to be x, put the equation for f(x) in another cell, and let Goal Seek or Solver find the value of x that makes the equation cell zero. In MATLAB, the process is similar except that a function (m-file) is defined and the command fzero(‘f’,x0) provides the solution x, starting from the initial guess x0. METHODS FOR MULTIPLE NONLINEAR EQUATIONS Method of Successive Substitutions Write a system of equations as αi = fi(α),

or α = f(α)

The following theorem guarantees convergence. Let α be the solution to αi = fi(α). Assume that given h > 0, there exists a number 0 < µ < 1 such that n ∂fi  ≤ µ for |xi − αi| < h, i = 1, . . . , n

∂xj j=1

 

x ik + 1 = fi(x ik) Then

x ik → αi

as k increases [see Finlayson (2003)].

NUMERICAL ANALYSIS AND APPROXIMATE METHODS Newton-Raphson Method To solve the set of equations Fi(x1, x2, . . . , xn) = 0,

or Fi({xj}) = 0,

Divided Differences of Higher Order and Higher-Order Interpolation The first-order divided difference f[x0, x1] was defined previously. Divided differences of second and higher order are defined iteratively by f [x1, x2] − f[x0, x1] f[x0, x1, x2] =  x2 − x0

or Fi(x) = 0

one uses a truncated Taylor series to give n ∂Fi 0 = Fi({x k}) +  j = 1 ∂xj



xk

(x jk + 1 − x jk)

⯗

Thus one solves iteratively from one point to another.

f[x1, . . . , xk] − f[x0, x1, . . . , xk − 1] f[x0, x1, . . . , xk] =  xk − x0

n

A

k ij

(x jk + 1 − x jk) = −Fi({xk})

j=1

∂Fi A kij =  ∂xj

where



xk

This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Finlayson (2003). Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable effort. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations fi(x1, x2, . . . , xn) = 0

i = 1, . . . , n

3-45

(3-73)

and a convenient form for computational purposes is k′ f(xj) f[x0, x1, . . . , xk] =  j = 0 (xj − x0)(xj − x1) ⋅⋅⋅ (xj − xk) for any k ≥ 0, where the ′ means that the term (xj − xj) is omitted in the denominator. For example, f(x0) f(x1) f(x2) f [x0, x1, x2] =  +  +  (x0 − x1)(x0 − x2) (x1 − x0)(x1 − x2) (x2 − x0)(x2 − x1) If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that f(x) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton’s fundamental interpolation formula with divided differences f(x) ≈ f(x0) + (x − x0) f[x0, x1] + (x − x0)(x − x1)f[x0, x1, x2]

and replacing them by

+ ⋅⋅⋅ + (x − x0)(x − x1) ⋅⋅⋅ (x − xn − 1) f [x0, x1, . . . , xn] + En(x)

fi(x1, x2, . . . , xn, λ) = 0

i = 1, . . . , n

(3-74)

where λ is introduced in such a way that the functions (3-74) depend in a simple way upon λ and reduce to an easily solvable system for λ = 0 and to the original equations (3-73) for λ = 1. A system of ordinary differential equations, with independent variable λ, is then constructed by differentiating Eqs. (3-74) with respect to λ. There results n

∂fi dxj

∂fi

+=0

 ∂x dλ ∂λ j=1

(3-75)

j

where x1, . . . , xn are considered as functions of λ. Equations (3-75) are integrated, with initial conditions obtained from Eqs. (3-74) with λ = 0, from λ = 0 to λ = 1. If the solution can be continued to λ = 1, the values of x1, . . . , xn for λ = 1 will be a solution of the original equations. If the integration becomes infinite, the parameter λ must be introduced in a different fashion. Integration of the differential equations (which are usually nonlinear in λ) may be accomplished by using techniques described under “Numerical Solution of Ordinary Differential Equations.” Other Methods Other methods can be found in the literature. See Chan, T. F. C., and H. B. Keller, SIAM J. Sci. Stat. Comput. 3:173–194 (1982); Seader, J. D., “Computer Modeling of Chemical Processes,” AIChE Monograph Series 81(15) (1986). INTERPOLATION AND FINITE DIFFERENCES Linear Interpolation If a function f(x) is approximately linear in a certain range, then the ratio

1 En(x) = error =  f n + 1(ε)π(x) (n + 1)! where minimum (x0, . . . , x) < ε < maximum (x0, x1, . . . , xn, x) and π(x) = (x − x0)(x − x1) ⋅⋅⋅ (x − xn). In order to use the previous equation most effectively one may first form a divided-difference table. For example, for third-order interpolation the difference table is where

x0

f(x0)

x1

f(x1)

x2

f(x2)

x3

f(x3)

f(x) ≈ f(x0) + (x − x0)f [x0, x1] x − x0 ≈ f(x0) +  [ f(x1) − f(x0)] x1 − x0 1 ≈  [(x1 − x)f(x0) − (x0 − x)f(x1)] x1 − x0

f [x0, x1, x2]

f [x1, x2]

f [x0, x1, x2, x3]

f [x1, x2, x3]

f [x2, x3]

where each entry is given by taking the difference between diagonally adjacent entries to the left, divided by the abscissas corresponding to the ordinates intercepted by the diagonals passing through the calculated entry. Equally Spaced Forward Differences If the ordinates are equally spaced, i.e., xj − xj − 1 = ∆x for all j, then the first differences are denoted by ∆f(x0) = f(x1) − f(x0) or ∆y0 = y1 − y0, where y = f(x). The differences of these first differences, called second differences, are denoted by ∆2y0, ∆2y1, . . . , ∆2yn. Thus ∆2y0 = ∆y1 − ∆y0 = y2 − y1 − y1 + y0 = y2 − 2y1 + y0 And in general j

∆ jy0 = (−1)n

f(x1) − f(x0)  = f [x0, x1] x1 − x0 is approximately independent of x0, x1 in the range. The linear approximation to the function f(x), x0 < x < x1 then leads to the interpolation formula

f [x0, x1]

n=0

where

n y j

j−n

= binomial coefficients. n =  n!( j − n)! j

j!

If the ordinates are equally spaced, xn + 1 − xn = ∆x yn = y(xn) then the first and second differences are denoted by ∆yn = yn + 1 − yn ∆2yn = ∆yn + 1 − ∆yn = yn + 2 − 2yn + 1 + yn

3-46

MATHEMATICS

A new variable is defined

1.0 at that point:

xα − x0 α= ∆x and the finite interpolation formula through the points y0, y1, . . . , yn is written as follows: α (α − 1) α (α − 1) ⋅⋅⋅ (α − n + 1) yα = y0 + α ∆y0 +  ∆2y0 + ⋅⋅⋅ +  ∆ny0 2! n! (3-76) Keeping only the first two terms gives a straight line through (x0, y0)–(x1, y1); keeping the first three terms gives a quadratic function of position going through those points plus (x2, y2). The value α = 0 gives x = x0; α = 1 gives x = x1, and so on. Equally Spaced Backward Differences Backward differences are defined by ∇yn = yn − yn − 1 ∇ 2yn = ∇yn − ∇yn − 1 = yn − 2 yn − 1 + yn − 2 The interpolation polynomial of order n through the points y0, y−1, . . . , y−n is α (α + 1) α (α + 1) ⋅⋅⋅ (α + n − 1) yα = y0 + α ∇y0 +  ∇2y0 + ⋅⋅⋅ +  ∇ny0 2! n! The value of α = 0 gives x = x0; α = −1 gives x = x−1, and so on. Alternatively, the interpolation polynomial of order n through the points y1, y0, y−1, . . . , y−n is α (α − 1) yα = y1 + (α − 1) ∇y1 +  ∇ 2y1 (3-77) 2! (α − 1) α (α + 1) ⋅⋅⋅ (α + n − 2) + ⋅⋅⋅ +  ∇ny1 n! Now α = 1 gives x = x1; α = 0 gives x = x0. Central Differences The central difference denoted by



 h h δ f(x) = δf x +  − δf x −  = f(x + h) − 2f(x) + f(x − h) 2 2 h h δf(x) = f x +  − f x −  2 2 2







h h δnf(x) = δn − 1 f x +  − δn − 1 f x −  2 2



is useful for calculating at the interior points of tabulated data. Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space m

Pm(x) = cj x j j=0

This polynomial is of degree m (highest power is x m) and order m + 1 (m + 1 parameters {cj}). If we are given a set of m + 1 points y1 = f(x1), y2 = f(x2), . . . , ym + 1 = f(xm + 1) then Lagrange’s formula gives a polynomial of degree m that goes through the m + 1 points: (x − x2) (x − x3) ⋅⋅⋅ (x − xm + 1) Pm(x) =  y1 (x1 − x2) (x1 − x3) ⋅⋅⋅ (x1 − xm + 1) (x − x1) (x − x3) ⋅⋅⋅ (x − xm + 1) +  y2 + ⋅⋅⋅ (x2 − x1) (x2 − x3) ⋅⋅⋅ (x2 − xm + 1) (x − x1) (x − x2) ⋅⋅⋅ (x − xm + 1) +  ym + 1 (xm + 1 − x1) (xm + 1 − x2) ⋅⋅⋅ (xm + 1 − xm) Note that each coefficient of yj is a polynomial of degree m that vanishes at the points {xj} (except for one value of j) and takes the value of

Pm(xj) = yj,

j = 1, 2, . . . , m + 1

If the function f(x) is known, the error in the approximation is, per Anderson, E., et al., LAPACK Users’ Guide, SIAM(1992), |xm + 1 − x1|m + 1 |error(x)| ≤  maxx1 ≤ x ≤ xm + 1 | f (m + 2)(x)| (m + 2)! The evaluation of Pm(x) at a point other than at the defining points can be made with Neville’s algorithm [Press et al. (1986)]. Let P1 be the value at x of the unique function passing through the point (x1, y1); or P1 = y1. Let P12 be the value at x of the unique polynomial passing through the points x1 and x2. Likewise, Pijk . . . r is the unique polynomial passing through the points xi, xj, xk, . . . xr. Then use the table x1

y1 = P1

x2

y2 = P2

x3

y3 = P3

x4

y4 = P4

P12 P23 P34

P123

P1234

P234

These entries are defined using (x − xi + m) Pi(i + 1) . . . (i + m − 1) + (xi − x) P(i + 1)(i + 2) . . . (i + m) Pi(i + 1) . . . (i + m) =  xi − xi + m For example, consider P1234. The terms on the right-hand side involve P123 and P234. The “parents,” P123 and P234, already agree at points 2 and 3. Here i = 1, m = 3; thus, the parents agree at xi + 1, . . . , xi + m − 1 already. The formula makes Pi(i + 1) . . . (i + m) agree with the function at the additional points xi + m and xi. Thus, Pi(i + 1) . . . (i + m) agrees with the function at all the points {xi, xi + 1, . . . xi + m}. Spline Functions Splines are functions that match given values at the points x1, . . . , xNT and have continuous derivatives up to some order at the knots, or the points x2, . . . , xNT − 1. Cubic splines are most common; see de Boor, C., A Practical Guide to Splines, Springer (1978). The function is represented by a cubic polynomial within each interval (xi, xi + 1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero; this gives the natural cubic splines. Take yi = y(xi) at each of the points xi, and let ∆xi = xi + 1 − xi. Then, in the interval (xi, xi + 1), the function is represented as a cubic polynomial. Ci(x) = a0i + a1i x + a2i x2 + a3i x3 The interpolating function takes on specified values at the knots and has continuous first and second derivatives at the knots. Within the ith interval, the function is (x − xi) Ci(x) = Ci(xi) + C′i (xi)(x − xi) + C″i (xi)  2∆ xi (x − xi)3 + [C″i (xi + 1) − C″i (xi)]  6∆xi where Ci(xi) = yi. The second derivative C″i (xi) = y″i is found by solving the following tridiagonal system of equations: yi − yi − 1 yi + 1 − yi y″i − 1∆xi − 1 + y″i 2(∆xi − 1 + ∆xi) + y″i + 1∆xi = − 6  −  ∆xi ∆xi − 1





Since the continuity conditions apply only for i = 2, . . . , NT − 1, we have only NT − 2 conditions for the NT values of y″i . Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y″1 , y″NT. If these values are zero, we get the natural cubic splines; they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT − 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are

NUMERICAL ANALYSIS AND APPROXIMATE METHODS known at each of the knots, the first derivatives are given by ∆xi yi + 1 − yi ∆xi y′i =  − y″i  − y″i + 1  6 ∆xi 3 The function itself is then known within each element.

in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate f given x. One can estimate the value of the first derivative at x0 using df  dx

NUMERICAL DIFFERENTIATION Numerical differentiation should be avoided whenever possible, particularly when data are empirical and subject to appreciable observation errors. Errors in data can affect numerical derivatives quite strongly; i.e., differentiation is a roughening process. When such a calculation must be made, it is usually desirable first to smooth the data to a certain extent. Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula [Eq. (3-76) or (3-77)] may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange’s formulas must be used. By differentiating three-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points: Three-Point Formulas Let x0, x1, x2 be the three points. 1 h2 f′(x0) =  [−3f(x0) + 4f(x1) − f(x2)] +  f′″(ε) 2h 3 1 h2 f′(x1) =  [−f(x0) + f(x2)] −  f′″(ε) 2h 6 h2 1 f ′(x2) =  [ f(x0) − 4f(x1) + 3f(x2)] +  f′″(ε) 2h 3 where the last term is an error term min. xj < ε < max. xj. j

y0 = j[5f0 + 2 f1 − f2] y1 = s[ f0 + f1 + f2]



x0

f[x0(1 + ε)] − f[x0] ≈  εx0

(a first-order formula) or df  dx



x0

f[x0(1 + ε)] − f[x0(1 − ε)] ≈  2εx0

(a second-order formula). The value of ε is important; a value of 10−6 is typical, but smaller or larger values may be necessary depending on the computer precision and the application. One must also be sure that the value of x0 is not zero and use a different increment in that case. NUMERICAL INTEGRATION (QUADRATURE) A multitude of formulas have been developed to accomplish numerical integration, which consists of computing the value of a definite integral from a set of numerical values of the integrand. Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral b ∫ a f(x) dx is to be evaluated. Trapezoidal Rule This formula consists of subdividing the interval a ≤ x ≤ b into n subintervals a to a + h, a + h to a + 2h, . . . and replacing the graph of f(x) by the result of joining the ends of adjacent ordinates by line segments. If fj = f(xj) = f(a + jh), f0 = f(a), fn = f(b), the integration formula is

 f(x) dx = h2 [ f + 2f + 2f + ⋅ ⋅ ⋅ + 2f b

j

Smoothing Techniques These techniques involve the approximation of the tabular data by a least-squares fit of the data by using some known functional form, usually a polynomial (for the concept of least squares see “Statistics”). In place of approximating f(x) by a single least-squares polynomial of degree n over the entire range of the tabulation, it is often desirable to replace each tabulated value by the value taken on by a least-squares polynomial of degree n relevant to a subrange of 2M + 1 points centered, when possible, at the point for which the entry is to be modified. Thus each smoothed value replaces a tabulated value. Let fj = f(xj) be the tabular points and yj = smoothed values. First-Degree Least Squares with Three Points

3-47

0

1

n−1

2

a

where

nh3 (b − a)3 |En| =  |f″(ε)| =  |f ″(ε)| 12 12n2

+ fn] + En

a<ε
This procedure is not of high accuracy. However, if f ″(x) is continuous in a < x < b, the error goes to zero as 1/n2, n → ∞. Parabolic Rule (Simpson’s Rule) This procedure consists of subdividing the interval a < x < b into n/2 subintervals, each of length 2h, where n is an even integer. By using the notation as above the integration formula is

 f(x) dx = h3 [ f + 4 f + 2 f + 4 f + ⋅ ⋅ ⋅ b

0

1

2

3

a

+ 4 fn − 3 + 2fn − 2 + 4 fn − 1 + fn] + En

y2 = j[−f0 + 2 f1 + 5f2] The derivatives at all the points are 1 f′0 = f 1′ = f 2′ =  h [y2 − y1] 2 Second-Degree Least Squares with Five Points For five evenly spaced points x−2, x−1, x0, x1, and x2 (separated by distance h) and their ordinates f−2, f−1, f0, f1, and f2, assume a parabola is fit by least squares. Then the derivative at the center point is f′0 = 1/10h [−2f−2 − f−1 + f1 + 2f2] The derivatives at the other points are f′−2 = 1/70h [−54 f−2 + 13 f−1 + 40 f0 + 27 f1 − 26 f2] f′−1 = 1/70h [−34 f−2 + 3 f−1 + 20 f0 + 17 f1 − 6 f2] f′1 = 1/70h [6 f−2 − 17 f−1 − 20 f0 − 3 f1 + 34 f2] f ′2 = 1/70h [26 f−2 − 27 f−1 − 40 f0 − 13 f1 + 54 f2] Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer,

where

(b − a)5 (IV) nh5 |En| =  | f (IV)(ε)| =  | f (ε)| 180 180n4

a<ε
This method approximates f(x) by a parabola on each subinterval. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. Gaussian Quadrature Gaussian quadrature provides a highly accurate formula based on irregularly spaced points, but the integral needs to be transformed onto the interval 0 to 1. x = a + (b − a)u,

dx = (b − a)du

 f(x) dx = (b − a)  f(u) du b

1

a

0

 f(u) du = W f(u ) 1

m

i

0

i

i=1

The quadrature is exact when f is a polynomial of degree 2m − 1 in x. Because there are m weights and m Gauss points, we have 2m parameters that are chosen to exactly represent a polynomial of degree

3-48

MATHEMATICS

2m − 1, which has 2m parameters. The Gauss points and weights are given in the table. Gaussian Quadrature Points and Weights m

ui

Wi

2

0.21132 48654 0.78867 51346 0.11270 16654 0.50000 00000 0.88729 83346 0.06943 18442 0.33000 94783 0.66999 05218 0.93056 81558 0.04691 00771 0.23076 53450 0.50000 00000 0.76923 46551 0.95308 99230

0.50000 00000 0.50000 00000 0.27777 77778 0.44444 44445 0.27777 77778 0.17392 74226 0.32607 25774 0.32607 25774 0.17392 74226 0.11846 34425 0.23931 43353 0.28444 44444 0.23931 43353 0.11846 34425

3

4

5

I1 = 0.24491148225216

 f(x) dx = 

1

I=

−x

sin x dx

b

(3-78)

0

Using the Gaussian quadrature formulas gives the following values for various values of m. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results. m

I

2 3 4 5

0.24609643062368 0.24583487736505 0.24583700444293 0.24583700700700

Romberg’s Method Romberg’s method uses extrapolation techniques to improve the answer [Press et al. (1986)]. If we let I1 be the value of the integral obtained using interval size h = ∆x, and I2 be the value of I obtained when using interval size h/2, and I0 the true value of I, then the error in a method is approximately hm, or I1 ≈ I0 + chm h I2 ≈ I0 + c  2

I2 = 0.24560560017077 I3 = 0.24577915369183 I4 = 0.24582254357310 J1 = 0.24583697281030 J2 = 0.24583700486552 J3 = 0.24583700686685 K1 = 0.24583701555059 K2 = 0.24583700753396 L1 = 0.24583700486175

Singularities When the integrand has singularities, a variety of techniques can be tried. The integral may be divided into one part that can be integrated analytically near the singularity and another part that is integrated numerically. Sometimes a change of argument allows analytical integration. Series expansion might be helpful, too. When the domain is infinite, it is possible to use Gauss-Legendre or Gauss-Hermite quadrature. Also a transformation can be made. For example, let u = 1/x and then

Example Calculate the value of the following integral.

e

Computer Methods These methods are easily programmed in a spreadsheet program such as Microsoft Excel. In MATLAB, the trapezoid rule can be calculated by using the command trapz(x,y), where x is a vector of x values xi and y is a vector of values y(xi). Alternatively, use the commands F = @(x) exp(-x).*sin(x) Q = quad(F,0,1)

a

1/b



1 1 2 f  du u u

ab > 0

Two-Dimensional Formula Two-dimensional integrals can be calculated by breaking down the integral into one-dimensional integrals.

 b

a

g2(x)

g1(x)

 G(x) dx b

f(x, y) dx dy = G(x) =

a



g2(x)

f(x, y) dy

g1(x)

Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)

  f(x, y) dx dy = W 1

1

mx

xi

0



1/a

0

i=1

my

W

yj

f(xi, yj)

j=1

m

Replacing the ≈ by an equality (an approximation) and solving for c and I0 gives 2mI2 − I1 I0 =  2m − 1 This process can also be used to obtain I1, I2, . . . , by halving h each time, and then calculating new estimates from each pair, calling them J1, J2, . . . ; that is, in the formula above, replace I0 with J1. The formulas are reapplied for each pair of J to obtain K1, K2, . . . The process continues until the required tolerance is obtained. I1 I2 I3 I4 J1 J2 J3 K1 K2 L1

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equations become two-point boundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. A higher-order differential equation y(n) + F(y(n − 1), y(n − 2), . . . , y′, y) = 0

Romberg’s method is most useful for a low-order method (small m) because significant improvement is then possible.

with initial conditions

Example Evaluate the same intergral (3-78) by using the trapezoid rule and then apply the Romberg method. Use 11, 21, 41, and 81 points with m = 2. To achieve six-digit accuracy, any result from J2 through L1 is suitable, even though the base results (I1 through I4) are not that accurate.

can be converted into a set of first-order equations using dyi − 1 d d(i − 1)y =  y(i − 2) =  yi  y(i − 1) =  dt dt(i − 1) dt The higher-order equation can be written as a set of first-order equations. dy1  = y2 dt

I1 = 0.24491148225216

I2 = 0.24560560017077 J1 = 0.24583697281030

I3 = 0.24577915369183 I4 = 0.24582254357310 J2 = 0.24583700486552 J3 = 0.24583700686685 K1 = 0.24583701555059 K2 = 0.24583700753396 L1 = 0.24583700486175

Gi(y(n − 1)(0), y(n − 2)(0), . . . , y(0), y(0)) = 0,

i = 1, . . . , n

NUMERICAL ANALYSIS AND APPROXIMATE METHODS dy2  = y3 dt dy3  = y4 dt ... dyn  = −F(yn − 1, yn − 2, . . . , y2, y1) dt

25 1408 2197 1 yn + 1 = yn +  k1 +  k3 +  k4 −  k5 216 2565 4104 5 16 6656 28561 9 2 zn + 1 = yn +  k1 +  k3 +  k4 −  k5 +  k6 135 12825 56430 50 55

The initial conditions would have to be specified for variables y1(0), . . . , yn(0), or equivalently y(0), . . . , y(n − 1)(0). The set of equations is then written as dy  = f(y, t) dt All the methods in this section are described for a single equation; the methods apply to multiple equations. Euler’s method is first-order. yn + 1 = yn + ∆t f(yn) and errors are proportional to ∆t. The second-order Adams-Bashforth method is ∆t yn + 1 = yn +  [3 f(yn) − f(yn − 1)] 2 Errors are proportional to ∆t2, and high-order methods are available. Notice that the higher-order explicit methods require knowing the solution (or the right-hand side) evaluated at times in the past. Since these were calculated to get to the current time, this presents no problem except for starting the problem. Then it may be necessary to use Euler’s method with a very small step size for several steps in order to generate starting values at a succession of time points. The error terms, order of the method, function evaluations per step, and stability limitations are listed in Finlayson (2003). The advantage of the high-order Adams-Bashforth method is that it uses only one function evaluation per step yet achieves high-order accuracy. The disadvantage is the necessity of using another method to start. Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for f(t, y). The first-order Runge-Kutta method is Euler’s method. A second-order Runge-Kutta method is ∆t yn + 1 = yn +  [ f n + f(tn + ∆t, yn + ∆t f n)] 2 while the midpoint scheme is also a second-order Runge-Kutta method. ∆t ∆t yn + 1 = yn + ∆t f tn +  , yn +  f n 2 2





A popular fourth-order Runge-Kutta method is the Runge-KuttaFeldberg formulas, which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. k1 = ∆t f(tn, yn) ∆t k1 k2 = ∆t f tn +  , yn +  4 4

 



The value of yn + 1 − zn + 1 is an estimate of the error in yn + 1 and can be used in step-size control schemes. Usually one would use a high-order method to achieve high accuracy. The Runge-Kutta-Feldberg method is popular because it is high order and does not require a starting method (as does an AdamsBashforth method). However, it does require four function evaluations per time step, or four times as many as a fourth-order Adams-Bashforth method. For problems in which the function evaluations are a significant portion of the calculation time, this might be important. Given the speed and availability of desktop computers, the efficiency of the methods is most important only for very large problems that are going to be solved many times. For other problems, the most important criterion for choosing a method is probably the time the user spends setting up the problem. The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations δfi dyi n dt = i=1 Aij yj, Aij =  δyjy For linear problems, the eigenvalues do not change, so that the stability and oscillation limits must be satisfied for every eigenvalue of the matrix A. When solving nonlinear problems, the equations are linearized about the solution at the local time, and the analysis applies for small changes in time, after which a new analysis about the new solution must be made. Thus, for nonlinear problems, the eigenvalues keep changing, and the largest stable time step changes, too. The stability limits are: Euler method, λ ∆t ≤ 2 Runge-Kutta, 2nd order, λ ∆t < 2 Runge-Kutta-Feldberg, λ ∆t < 3.0 Richardson extrapolation can be used to improve the accuracy of a method. Suppose we step forward one step ∆t with a pth-order method. Then redo the problem, this time stepping forward from the same initial point, but in two steps of length ∆t/2, thus ending at the same point. Call the solution of the one-step calculation y1 and the solution of the two-step calculation y2. Then an improved solution at the new time is given by 2 py2 − y1 y= 2p − 1 This gives a good estimate provided ∆t is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg’s method was used for quadrature. The error term in the various methods can be used to deduce a step size that will give a user-specified accuracy. Most packages today are based on a user-specified tolerance; the step-size is changed during the calculation to achieve that accuracy. The accuracy itself is not guaranteed, but it improves as the tolerance is decreased. Implicit Methods By using different interpolation formulas involving yn + 1, it is possible to derive implicit integration methods. Implicit methods result in a nonlinear equation to be solved for yn + 1 so that iterative methods must be used. The backward Euler method is a first-order method. yn + 1 = yn + ∆t f(yn + 1)



3 3 9 k3 = ∆t f tn +  ∆t, yn +  k1 +  k2 8 32 32









12 1932 7200 7296 k4 = ∆t f tn +  ∆t, yn +  k1 −  k2 +  k3 13 2197 2197 2197

439 3680 845 k5 = ∆t f tn + ∆t, yn +  k1 − 8k2 +  k3 −  k4 216 513 4104

∆t 11 8 3544 1859 k6 = ∆t f tn +  , yn −  k1 + 2k2 −  k3 +  k4 −  k5 2 27 2565 4104 40



3-49



Errors are proportional to ∆t for small ∆t. The trapezoid rule is a second-order method. ∆t yn + 1 = yn +  [ f(yn) + f(yn + 1)] 2 Errors are proportional to ∆ t2 for small ∆ t. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method

3-50

MATHEMATICS

or Newton-Raphson method. See Bogacki, M. B, K. Alejski, and J. Szymanewski, Comp. Chem. Eng. 13: 1081–1085 (1989) for an application to dynamic distillation problems. The best packages for stiff equations (see below) use Gear’s backward difference formulas. The formulas of various orders are [Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971)] (1) yn + 1 = yn + ∆t f(yn + 1) 4 1 2 (2) yn + 1 =  yn −  yn − 1 +  ∆t f(yn + 1) 3 3 3 18 9 2 6 (3) yn + 1 =  yn −  yn − 1 +  yn − 2 +  ∆t f(yn + 1) 11 11 11 11 48 36 16 3 12 (4) yn + 1 =  yn −  yn − 1 +  yn − 2 −  yn − 3 +  ∆t f(yn + 1) 25 25 25 25 25 300 300 200 75 12 (5) yn + 1 =  yn −  yn − 1 +  yn − 2 −  yn − 3 +  yn − 4 137 137 137 137 137 60 +  ∆t f(yn + 1) 137 Stiffness The concept of stiffness is described for a system of linear equations. dy  = Ay dt Let λi be the eigenvalues of the matrix A. The stiffness ratio is defined as maxi |Re (λi)| SR =  (3-79) mini |Re (λi)| 3 6 SR = 20 is not stiff, SR = 10 is stiff, and SR = 10 is very stiff. If the problem is nonlinear, then the solution is expanded about the current state. n ∂fi dyi  = fi [y(tn)] +  [yj − yj(tn)] dt j = 1 ∂yj The question of stiffness then depends on the solution at the current time. Consequently nonlinear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step size used. Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor. The time constants for different phenomena are: 1. Time for device flow-through L φAL tflow =  =  Q u where Q is the volumetric flow rate, A is the cross sectional area, L is the length of the packed bed, and φ is the void fraction; 2. Time for reaction 1 tr × n =  k where k is a rate constant (time−1); 3. Time for diffusion inside the catalyst εR2 tinternal diffusion =  De where ε is the porosity of the catalyst, R is the catalyst radius, and De is the effective diffusion coefficient inside the catalyst; 4. Time for heat transfer is R2 ρsCsR2 tinternal heat transfer =  =  α ke where ρs is the catalyst density, Cs is the catalyst heat capacity per unit mass, ke is the effective thermal conductivity of the catalyst, and

α is the thermal diffusivity. For example, in the model of a catalytic converter for an automobile [Ferguson, N. B., and B. A. Finlayson, AIChE J. 20:539–550 (1974)], the time constants for internal diffusion was 0.3 seconds; internal heat transfer, 21 seconds; and device flow-through, 0.003 seconds. The device flow-through is so fast that it might as well be instantaneous. The stiffness is approximately 7000. Implicit methods must be used to integrate the equations. Alternatively, a quasistate model can be developed [Ramirez, W. F., Computational Methods for Process Simulations, 2d ed., ButterworthHeinemann, Boston (1997)]. Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are dxn M  = Vn + 1yn + 1 − Lnxn − V nyn + Ln − 1xn − 1 dt xn − 1 − xn = En(xn − 1 − x*,n) N

x =1 i

i=1

where x and y are the mole fraction in the liquid and vapor, respectively; L and V are liquid and vapor flow rates, respectively; M is the holdup; and the superscript is the stage number. The efficiency is E, and the concentration in equilibrium with the vapor is x*. The first equation is an ordinary differential equation for the mass of one component on the stage, while the third equation represents a constraint that the mass fractions add to one. This is a differential-algebraic system of equations. Differential-algebraic equations can be written in the general notation dy F t, y,  = 0 dt





To solve the general problem using the backward Euler method, replace the nonlinear differential equation with the nonlinear algebraic equation for one step. yn + 1 − yn F t, yn + 1,  = 0 ∆t This equation must be solved for yn + 1. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998); and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland: Elsevier (1989)]. Differential-algebraic systems are more complicated than differential systems because the solution may not always be defined. Pontelides et al. [Comp. Chem. Eng. 12: 449–454 (1988)] introduced the term index to identify the possible problems. The index is defined as the minimum number of times the equations need to be differentiated with respect to time to convert the system to a set of ordinary differential equations. These higher derivatives may not exist, and the process places limits on which variables can be given initial values. Sometimes the initial values must be constrained by the algebraic equations. For a differential-algebraic system modeling a distillation tower, Pontelides et al. show that the index depends on the specification of pressure for the column. Byrne and Ponzi [Comp. Chem. Eng. 12: 377–382 (1988); also Chan, T. F. C., and H. B. Keller, SIAM J. Sci. Stat. Comput. 3: 173–194 (1982)] also list several chemical engineering examples of differential-algebraic systems and solve one involving two-phase flow. Computer Software Efficient computer packages are available for solving ordinary differential equations as initial value problems. The packages are widely available and good enough that most chemical engineers use them and do not write their own. Here we discuss three of them: RKF45, LSODE, and EPISODE. In each of the packages, the user specifies the differential equation to be solved and a





NUMERICAL ANALYSIS AND APPROXIMATE METHODS desired error criterion. The package then integrates in time and adjusts the step size to achieve the error criterion within the limitations imposed by stability. A popular explicit, Runge-Kutta package is RKF45. Notice there that an estimate of the truncation error at each step is available. Then the step size can be reduced until this estimate is below the userspecified tolerance. The method is thus automatic, and the user is assured of the results. Note, however, that the tolerance is set on the local truncation error, namely from one step to another, whereas the user is usually interested in the global truncation error, or the error after several steps. The global error is generally made smaller by making the tolerance smaller, but the absolute accuracy is not the same as the tolerance. If the problem is stiff, then very small step sizes are used; the computation becomes very lengthy. The RKF45 code discovers this and returns control to the user with a message indicating the problem is too hard to solve with RKF45. A popular implicit package is LSODE, a version of Gear’s method (Gear, ibid.) written by Alan Hindmarsh at Lawrence Livermore Laboratory. In this package, the user specifies the differential equation to be solved and the tolerance desired. Now the method is implicit and therefore stable for any step size. The accuracy may not be acceptable, however, and sets of nonlinear equations must be solved. Thus, in practice the step size is limited but not nearly so much as in the RungeKutta methods. In these packages, both the step size and order of the method are adjusted by the package. Suppose we are calculating with a kth order method. The truncation error is determined by the (k + 1)th order derivative. This is estimated using difference formulas and the values of the right-hand sides at previous times. An estimate is also made for the kth and (k + 2)th derivative. Then it is possible to estimate the error in a (k − 1)th order method, a kth order method, and a (k + 1)th order method. Furthermore, the step size needed to satisfy the tolerance with each of these methods can be determined. Then we can choose the method and step size for the next step that achieves the biggest step, with appropriate adjustments due to the different work required for each order. The package generally starts with a very small step size and a first-order method, the backward Euler method. Then it integrates along, adjusting the order up (and later down) depending on the error estimates. The user is thus assured that the local truncation error meets the tolerance. There is a further difficulty, since the set of nonlinear equations must be solved. Usually a good guess of the solution is available, since the solution is evolving in time and past history can be extrapolated. Thus, the Newton-Raphson method will usually converge. The package protects itself, though, by only doing a few iterations. If convergence is not reached within this many iterations, then the step size is reduced and the calculation is redone for that time step. The convergence theorem for the Newton-Raphson method (p. 3-50) indicates that the method will converge if the step size is small enough. Thus the method is guaranteed to work. Further economies are possible. The Jacobian needed in the Newton-Raphson method can be fixed over several time steps. Then, if the iteration does not converge, the Jacobian can be reevaluated at the current time-step. If the iteration still does not converge, then the step-size is reduced and a new Jacobian is evaluated. Also the successive substitution method can be used, which is even faster, except that it may not converge. However, it, too, will converge if the time step is small enough. The Runge-Kutta methods give extremely good accuracy, especially when the step size is kept small for stability reasons. If the problem is stiff, though, backward difference implicit methods must be used. Many chemical reactor problems are stiff, necessitating the use of implicit methods. In the MATLAB suite of ODE solvers, the ode45 uses a revision of the RKF45 program, while the ode15s program uses an improved backward difference method. Shampine and Reichelt [SIAM J. Sci. Comp. 18:1–22 (1997)] give details of the programs in MATLAB. Fortunately, many packages are available. On the NIST web page http://gams.nist.gov/, choose “problem decision tree” and then “differential and integral equations” to find packages that can be downloaded. On the Netlib web site http://www.netlib.org/, choose “ode” to find packages that can be downloaded. Using Microsoft Excel to solve ordinary differential equations is cumbersome, except for the simplest problems. Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations; other aspects involve

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the integration of ordinary differential equations; applications include chaos and fractals as well as unusual operation of some chemical engineering equipment. Kubicek, M., and M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures, SpringerVerlag, Berlin (1983), give an excellent introduction to the subject and the details needed to apply the methods. Chan, T. F. C., and H. B. Keller, SIAM J. Sci. Stat. Comput. 3:173–194 (1982), give more details of the algorithms. A concise survey with some chemical engineering examples is given in Doherty, M. F., and J. M. Ottino, Chem. Eng. Sci. 43:139–183 (1988). Bifurcation results are closely connected with stability of the steady states, which is essentially a transient phenomenon. Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Finlayson, et al. (2006). Molecular Dynamics Special integration methods have been developed for molecular dynamics calculations due to the structure of the equations. A very large number of equations are to be integrated, with the following form based on molecular interactions between molecules. d2ri = Fi ({r}) Fi({r}) = −∇V mi  dt2 The symbol mi is the mass of the ith particle, ri is the position of the ith particle, Fi is the force acting on the ith particle, and V is the potential energy that depends upon the location of all the particles (but not their velocities). Since the major part of the calculation lies in the evaluation of the forces, or potentials, a method must be used that minimizes the number of times the forces are calculated to move from one time to another time. Rewrite this equation in the form of an acceleration as d2ri 1 =  Fi({r}) ai  dt2 mi In the Verlet method, this equation is written by using central finite differences (see “Interpolation and Finite Differences”). Note that the accelerations do not depend upon the velocities. ri(t + ∆t) = 2ri (t) − ri (t − ∆t) + ai(t)∆t2 The calculations are straightforward, and no explicit velocity is needed. The storage requirement is modest, and the precision is modest (it is a second-order method). Note that one must start the calculation with values of {r} at times t and t − ∆t. In the Verlet velocity method, an equation is written for the velocity, too. dVi  = ai dt The trapezoid rule [see “Numerical Integration (Quadrature)”] is applied to obtain 1 vi(t + ∆t) = vi (t) +  [ai (t) + ai(t + ∆t)] ∆t 2 The position of the particles is expanded in a Taylor series. 1 ri(t + ∆t) = ri (t) + vi ∆t +  ai (t) ∆t2 2 Beginning with values of {r} and {v} at time 0, one calculates the new positions and then the new velocities. This method is second-order in ∆t, too. For additional details, see Allen, M. P., and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford (1989); Frenkel, D., and B. Smit, Understanding Molecular Simulation, Academic Press (2002); Haile, J. M., Molecular Dynamics Simulation, Wiley (1992); Leach, A. R., Molecular Modelling: Principles and Applications, Prentice-Hall (2001); Schlick, T., Molecular Modeling and Simulations, Springer, New York (2002). ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial

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MATHEMATICS

locations: at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods: finite difference, orthogonal collocation, and Galerkin finite element methods. Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-48 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point xi is assigned the symbol ci = c(xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are: ci + 1 − ci d 2c ∆x ci − ci − 1 d 2c ∆x dc dc  =  − 2  + ⋅⋅⋅ ,  =  + 2  + ⋅ ⋅ ⋅ dx i ∆x dx i 2 dx i ∆x dx i 2









ci + 1 − ci − 1 d 3c ∆x2 dc  =  − 3  + ⋅ ⋅ ⋅ dx i 2∆x dx i 3! The truncation error in the first two expressions is proportional to ∆x, and the methods are said to be first-order. The truncation error in the third expression is proportional to ∆x2, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is: d 2c ci + 1 − 2ci + ci − 1 d4c 2∆x2 − 4  + ⋅ ⋅ ⋅ 2 =  dx i ∆x2 dx i 4! The truncation error is proportional to ∆x2. To solve a differential equation, it is evaluated at a point i and then these expressions are inserted for the derivatives.









Example Consider the equation for convection, diffusion, and reaction in a tubular reactor. 1 d 2c dc  2 −  = Da R(c) Pe dx dx The finite difference representation is 1 ci + 1 − 2ci + ci − 1 ci + 1 − ci − 1 −  = Da R(ci)   Pe ∆x2 2∆x This equation is written for i = 2 to n − 1, or the internal points. The equations would then be coupled but would also involve the values of c1 and cn, as well. These are determined from the boundary conditions. If the boundary condition involves a derivative, it is important that the derivatives be evaluated using points that exist. Three possibilities exist: c2 − c1 dc  = dx 1 ∆x



−3c1 + 4c2 − c3 dc  =  dx 1 2∆x The third alternative is to add a false point, outside the domain, as c0 = c(x = −∆x). c2 − c0 dc  = dx 1 2∆x Since this equation introduces a new variable, c0, another equation is needed and is obtained by writing the finite difference equation for i = 1, too.





FIG. 3-48

Finite difference mesh; ∆x uniform.

The sets of equations can be solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form. Frequently, the transport coefficients, such as diffusion coefficient or thermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look like d dc  D(c)  = 0 dx dx This could be written as two equations. dJ dc −=0 J = −D(c)  dx dx Because the coefficient depends on c, the equations are more complicated. A finite difference method can be written in terms of the fluxes at the midpoints, i + 1/2. ci + 1 − ci Ji + 1/2 − Ji − 1/2 −  = 0 Ji +1/2 = −D(ci + 1/2)  ∆x ∆x These are combined to give the complete equation. D(ci + 1/2) (ci + 1 − ci) − D(ci − 1/2) (ci − ci − 1) =0  ∆x2 This represents a set of nonlinear algebraic equations that can be solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve D(cki + 1/2) (cik++11 − c ik + 1) − D(cik− 1/2) (cik + 1 − c ik−+11) =0  ∆x2 The advantage of this approach is that it is easier to program than a full NewtonRaphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it may be necessary to use the full Newton-Raphson method. There are two common ways to evaluate the transport coefficient at the midpoint: Use the average value of the solution on each side to evaluate the diffusivity, or use the average value of the diffusivity on each side. Both methods have truncation error ∆x2 (Finlayson, 2003). The spacing of the grid points need not be uniform; see Finlayson (2003) and Finlayson et al. (2006) for the formulas in that case.





Example A reaction diffusion problem is solved with the finite difference method. d 2c 2 = φ2c, dx

dc  (0) = 0, dx

c(1) = 1

The solution is derived for φ = 2. It is solved several times, first with two intervals and three points (at x = 0, 0.5, 1), then with four intervals, then with eight intervals. The reason is that when an exact solution is not known, one must use several ∆x and see that the solution converges as ∆x approaches zero. With two intervals, the equations are as follows. The points are x1 = 0, x2 = 0.5, and x3 = 1.0; and the solution at those points are c1, c2, and c3, respectively. A false boundary is used at x0 = −0.5. c0 − c2  = 0, 2∆x

c0 − 2c1 + c2 − φ2c1 = 0,  ∆x2

c1 − 2c2 + c3 − φ2c2 = 0,  ∆x2

c3 = 1

The solution is c1 = 0.2857, c2 = 0.4286, and c3 = 1.0. The problem is solved again with four and then eight intervals. The value of concentration at x = 0 takes the following values for different ∆x. These values are extrapolated using the Richardson extrapolation technique to give c(0) = 0.265718. Using this value as the best estimate of the exact solution, the errors in the solution are tabulated versus ∆x. Clearly the errors go as ∆x2 (decreasing by a factor of 4 when ∆x decreases by a factor of 2), thus validating the solution. The exact solution is 0.265802. n−1

∆x

c(0)

2 4 8

0.5 0.25 0.125

0.285714 0.271043 0.267131

n−1

∆x

Error in c(0)

2 4 8

0.5 0.25 0.125

0.02000 0.00532 0.00141

Finite Difference Methods Solved with Spreadsheets A convenient way to solve the finite difference equations for simple problems is to use a computer spreadsheet. The equations for the problem solved in the example can be cast into the following form 2c2 c1 =  2 + φ2∆x2

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-53

the residual.





NT NT d 2bi Residual = ai  − φ2R aibi(x) 2 dx i=1 i=1 The residual is then made orthogonal to the set of basis functions.

db  b (x)  a  − φ R a b (x)
dx = 0 dx 1

i

0

FIG. 3-49

Finite difference method using spreadsheets.

ci + 1 + ci − 1 ci =  2 + φ2∆x2 cn + 1 = 1 Let us solve the problem using 6 nodes, or 5 intervals. Then the connection between the cell in the spreadsheet and the nodal value is shown in Fig. 3-49. The following equations are placed into the various cells. A1: = 2*B1/(2.+(phi*dx)**2) B1: = (A1 + C1)/(2.+(phi*dx)**2) F1: = 1.

The equation in cell B1 is copied into cells C1 though E1. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See “Interpolation and Finite Differences: Lagrange Interpolation Formulas.” N

c(x) = amPm (x) m= 0

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C., J. Math. Phys. 17:123–199 (1938); and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22:1483–1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 − N)]N − 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). Galerkin Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced: the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. d 2c 2 = φ2R(c) dx dc  (0) = 0, c(1) = 1 dx The unknown solution is expanded in a series of known functions {bi(x)} with unknown coefficients {ai}. NT

c(x) = ai bi(x) i=1

The trial solution is substituted into the differential equation to obtain

2

NT

j

i=1

NT

i

2

i i

2

j = 1, . . . , NT

i=1

This is the process that makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, the problem is solved. This equation is integrated by parts to give the following equation NT

−

i=1



1

0

dbj dbi   dxai = φ2 dx dx

 b (x)R a b (x) dx 1

NT

j

i i

0

i=1

j = 1, . . . , NT − 1

(3-80)

This equation defines the Galerkin method and a solution that satisfies this equation (for all j = 1, . . . , ∞) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1, . . . , NT − 1, and the boundary condition is applied. NT

a b (1) = c i

i

B

i=1

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points xi and xi+1. x− x i c(x) = ci(1 − u) + ci+1 u u   xi+1− xi A finite element method based on these functions would have an error proportional to ∆x2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from xi-1 and xi+1 as one element. Then the interpolation would be c(x) ci1N1(u)ciN2(u)ci1N3(u)



1 N1(u) 2(u1) u  2



N2(u) 4u(1u)



1 N3 (u) 2u u −  2 A finite element method based on these functions would have an error proportional to ∆x3. Thus, it would converge faster than one based on linear interpolation. A variety of other finite element functions can be used as well, including B splines (see “Interpolation and Finite Differences: Spline Functions”). Adaptive Meshes In many two-point boundary value problems, the difficulty in the problem is the formation of a boundary layer region, or a region in which the solution changes very dramatically. In such cases, it is prudent to use small mesh spacing there, either with the finite difference method or the finite element method. If the region is known a priori, small mesh spacings can be assumed at the boundary layer. If the region is not known, though, other techniques must be used. These techniques are known as adaptive mesh techniques. The mesh size is made small where some property of the solution is large. For example, if the truncation error of the method is nth order, then the nth-order derivative of the solution is evaluated and a small mesh is used where it is large. Alternatively, the residual (the differential equation with the numerical

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MATHEMATICS

solution substituted into it) can be used as a criterion. It is also possible to define the error that is expected from a method one order higher and one order lower. Then a decision about whether to increase or decrease the order of the method can be made, taking into account the relative work of the different orders. This provides a method of adjusting both the mesh spacing (∆x, or sometimes called h) and the degree of polynomial (p). Such methods are called h-p methods. Many finite element programs have the capability to do this mesh refinement automatically. Singular Problems and Infinite Domains If the solution being sought has a singularity, it may be difficult to find a good numerical solution. Sometimes even the location of the singularity may not be known. One method of solving such problems is to refine the mesh near the singularity, relying on the better approximation due to a smaller ∆x. Another approach is to incorporate the singular trial function into the approximation. Thus, if the solution approaches f(x) as x goes to zero and f(x) becomes infinite, one may define a new variable u(x) = y(x) − f(x) and derive an equation for u. The differential equation is more complicated, but the solution is better near the singularity. Press et al. (1986). Sometimes the domain is semi-infinite, as in boundary layer flow. The domain can be transformed from the x domain (0 − ∞) to the η domain (1 − 0) using the transformation η = exp (−x). Another approach is to use a variable mesh, perhaps with the same transformation. For example, use η = exp (−βx) and a constant mesh size in η; the value of β is found experimentally. Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution. A location that is far enough away must be found by trial and error. Packages to solve boundary value problems are available on the Internet. On the NIST web page http://gams.nist.gov/, choose “problem decision tree” and then “differential and integral equations” and then “ordinary differential equations” and “multipoint boundary value problems.” On the Netlib web site http://www.netlib.org/, search on “boundary value problem.” Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. NUMERICAL SOLUTION OF INTEGRAL EQUATIONS In this subsection is considered a method of solving numerically the Fredholm integral equation of the second kind:

 k(x, t)u(t) dt b

u(x) = f(x) + λ

for u(x)

(3-81)

a

The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-81) can be replaced by an integration formula, and Eq. (3-81) may be written

 c k(x, t )u(t ) n

u(x) = f(x) + λ(b − a)

i

i

i

(3-82)

i=1

where t1, . . . , tn are points of subdivision of the t axis, a ≤ t ≤ b, and the c’s are coefficients whose values depend upon the type of numerical integration formula used. Now Eq. (3-82) must hold for all values of x, a ≤ x ≤ b; so it must hold for x = t1, x = t2, . . . , x = tn. Substituting for x successively t1, t2, . . . , tn and setting u(ti) = ui, f(ti) = fi, we get n linear algebraic equations for the n unknowns u1, . . . , un. That is, ui = fi + λ(b − a)[c1k(ti, t1)u1 + c2 k(ti, t2)u2 + ⋅⋅⋅ + cnk(ti, tn)un] i = 1, 2, . . . , n These uj may be solved for by the methods under “Numerical Solution of Linear Equations and Associated Problems” and substituted into Eq. (3-82) to yield an approximate solution for Eq. (3-81). Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of u’s be computed. Thus the gaussian integration formulas are useful because of the economy they offer. Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia (1985). There are methods that are analogous to the usual methods for

integrating differential equations (Runge-Kutta, predictor-corrector, Adams methods, etc.). Explicit methods are fast and efficient until the time step is very small to meet the stability requirements. Then implicit methods are used, even though sets of simultaneous algebraic equations must be solved. The major part of the calculation is the evaluation of integrals, however, so that the added time to solve the algebraic equations is not excessive. Thus, implicit methods tend to be preferred. Volterra equations of the first kind are not well posed, and small errors in the solution can have disastrous consequences. The boundary element method uses Green’s functions and integral equations to solve differential equations. See Brebbia, C. A., and J. Dominguez, Boundary Elements—An Introductory Course, 2d ed., Computational Mechanics Publications, Southhampton (1992); and Mackerle, J., and C. A. Brebbia (eds.), Boundary Element Reference Book, Springer-Verlag (1988). MONTE CARLO SIMULATIONS Some physical problems, such as those involving interaction of molecules, are usually formulated as integral equations. Monte Carlo methods are especially well-suited to their solution. This section cannot give a comprehensive treatment of such methods, but their use in calculating the value of an integral will be illustrated. Suppose we wish to calculate the integral G=



Ω0

g(x)f(x) dx

where the distribution function f(x) satisfies: f(x) ≥ 0,



Ω0

f(x) dx = 1

The distribution function f(x) can be taken as constant; for example, 1/Ω0. We choose variables x1, x2, . . . , xN randomly from f(x) and form the arithmetic mean 1 GN =  g(xi) N i The quantity GN is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of GN is G, if G exists [Kalos, M. H., and P. A. Whitlock, Monte Carlo Methods, vol. 1, Wiley, New York (1986)]. The error in the calculation is given by σ1 ε= N1/2 where σ 21 is calculated from σ 21 =



Ω0

g2(x) f(x) dx − G2

Thus the number of terms needed to achieve a specified accuracy % can be calculated once an estimate of σ 21 is known. σ12 N= ε Various methods, such as influence sampling, can be used to reduce the number of calculations needed. See also Lapeyre, B., Introduction to Monte-Carlo Methods for Transport and Diffusion Equations, Oxford University Press (2003), and Liu, J. S., Monte Carlo Strategies in Scientific Computing, Springer (2001). Some computer programs are available that perform simple Monte Carlo calculations using Microsoft Excel. NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS The numerical methods for partial differential equations can be classified according to the type of equation (see “Partial Differential Equations”): parabolic, elliptic, and hyperbolic. This section uses the finite difference method to illustrate the ideas, and these results can be programmed for simple problems. For more complicated problems, though, it is common to rely on computer packages. Thus, some discussion is given to the issues that arise when using computer packages. Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

3-55

is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerkin finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. The finite volume method is described under Hyperbolic Equations. Example Consider the diffusion equation, with boundary and initial conditions.

∂c ∂2c  = D 2 ∂t ∂x c(x, 0) = 0 c(0, t) = 1, c(1, t) = 0

We denote by ci the value of c(xi, t) at any time. Thus, ci is a function of time, and differential equations in ci are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2, . . . , n (see Fig. 3-50). dci ci + 1 − 2ci + ci − 1  = D  dt ∆x2 This can be written in the general form of a set of ordinary differential equations by defining the matrix AA. dc  = AAc dt This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. When Euler’s method is used to integrate in time, the equations become c in + 1 − cin cni+ 1 − 2c ni + cni− 1  = D  ∆t ∆x2 where c in = c(xi, tn). Notice that if the solution is known at every point at one time n, then it is a straightforward calculation to find the solution at every point at the new time n+1. If Euler’s method is used for integration, the time step is limited by 2 ∆t ≤  |λ|max whereas if the Runge-Kutta-Feldberg method is used, the 2 in the numerator is replaced by 3.0. The largest eigenvalue of AA is bounded by Gerschgorin’s Theorem. n 4D |λ|max ≤ max 2 < j < n |AA ji | = 2 ∆x i=2 This gives the well-known stability limit D 1 ∆t 2 ≤  ∆x 2 The smallest eigenvalue is independent of ∆x (it is Dπ 2/L2) so that the ratio of largest to smallest eigenvalue is proportional to 1/∆x2. Thus, the problem becomes stiff as ∆x approaches zero. See Eq. (3-79). The effect of the increased stiffness is that a smaller and smaller time step (∆t) must be taken as the mesh is refined (∆x2 —> 0). At the same time, the number of points is increasing, so that the computation becomes very lengthy. Implicit methods are used to overcome this problem. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. c in + 1 − cin cni+ 1 − 2cni + cni− 1 cin++11 − 2c in + 1 + cni −+11 + Dθ   = D(1 − θ)  ∆t ∆x2 ∆x2 Now the equations are of the form D∆tθ n + 1 D∆tθ n + 1 D∆tθ n + 1 − ci + 1 + 1 + 2  ci −  ci − 1 ∆x2 ∆x2 ∆x2





D∆t(1 − θ) n = c in +  (ci + 1 − 2c in + cni− 1) ∆x2 and require solving a set of simultaneous equations, which have a tridiagonal structure. Using θ = 0 gives the Euler method (as above),

FIG. 3-50

Computational molecules. h = ∆x = ∆y.

θ = 0.5 gives the Crank-Nicolson method, and θ = 1 gives the backward Euler method. The Crank-Nicolson method is also the same as applying the trapezoid rule to do the integration. The stability limit is given by 0.5 D∆t ≤  ∆x2 1 − 2θ

3-56

MATHEMATICS

The price of using implicit methods is that one now has a system of equations to solve at each time step, and the solution methods are more complicated (particularly for nonlinear problems) than the straightforward explicit methods. Phenomena that happen quickly can also be obliterated or smoothed over by using a large time step, so implicit methods are not suitable in all cases. The engineer must decide if he or she wants to track those fast phenomena, and choose an appropriate method that handles the time scales that are important in the problem. Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see “Ordinary Differential Equations—Boundary Value Problems”) with the methods for initialvalue problems (see “Numerical Solution of Ordinary Differential Equations as Initial Value Problems”). Fast Fourier transforms can also be used on regular grids (see “Fast Fourier Transform”). Elliptic Equations Elliptic equations can be solved with both finite difference and finite element methods. One-dimensional elliptic problems are two-point boundary value problems. Two- and threedimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider: how the equations are discretized to form sets of algebraic equations and how the algebraic equations are then solved. The prototype elliptic problem is steady-state heat conduction or diffusion, ∂2T ∂2T k  + =Q ∂x2 ∂y2





possibly with a heat generation term per unit volume, Q. The boundary conditions taken here are T = f(x, y) on the boundary (S) with f a known function. Illustrations are given for constant thermal conductivity k while Q is a known function of position. The finite difference formulation is given using the following nomenclature: Ti, j = T(i∆x, j∆y) The finite difference formulation is then (see Fig. 3-50) Ti + 1, j − 2Ti, j + Ti − 1, j Ti, j + 1 − 2Ti, j + Ti, j − 1 +  = Qi, j  ∆x2 ∆y2

(3-83)

Ti, j = f(xi, yj) on S If the boundary is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. Equation (3-83) provides a set of linear equations that must be solved. These equations and their boundary conditions may be written in matrix form as At f where t is the set of temperatures at all the points, f is the set of heat generation terms at all points, and A is formed from the coefficients of Tij in Eq. (3-83). The solution can be obtained simply by solving the set of linear equations. For three-dimensional problems, the matrix A is sparse, and iterative methods are used. These include Gauss-Seidel, alternating direction, overrelaxation methods, conjugate gradient, and multigrid methods. In Gauss-Seidel methods, one writes the equation for Tij in terms of the other temperatures and cycles through all the points over and over. In the alternating direction method, one solves along one line (that is, x = constant), keeping the side values fixed, and then repeats this for all lines, and then repeats the process. Multigrid methods solve the problem on successively refined grids, which has advantages for both convergence and error estimation. Conjugate gradient methods frequently use a preconditioned matrix. The equation is multiplied by another matrix, which is chosen so that the resulting problem is easier to solve than the original one. Finding such matrices

is an art, but it can speed convergence. The generalized minimal residual method is described in http://mathworld.wolfram.com/ GeneralizedMinimalResidualMethod.html. Additional resources can be found at http://www.netlib.org/linalg/html_templates/Templates. html. When the problem is nonlinear, the iterative methods may not converge, or the mesh may have to be refined before they converge, so some experimentation is sometimes necessary. Spreadsheets can be used to solve two-dimensional problems on rectangular grids. The equation for Tij is obtained by rearranging Eq. (3-83). Qi,j x2 x2 2 1+ 2 Ti, j = Ti+ 1, j + Ti1, j +  (Ti, j+ 1 + T1, j−1) − x2  k y y2 This equation is inserted into a cell and copied throughout the space represented by all the cells; when the iteration feature is turned on, the solution is obtained. The Galerkin finite element method (FEM) is useful for solving elliptic problems and is particularly effective when the domain or geometry is irregular. As an example, cover the domain with triangles and define a trial function on each triangle. The trial function takes the value 1.0 at one corner and 0.0 at the other corners and is linear in between. See Fig. 3-51. These trial functions on each triangle are pieced together to give a trial function on the whole domain. General treatments of the finite element method are available (see references). The steps in the solution method are similar to those described for boundary value problems, except now the problems are much bigger so that the numerical analysis must be done very carefully to be efficient. Most engineers, though, just use a finite element program without generating it. There are three major caveats that must be addressed. The first one is that the solution is dependent on the mesh laid down, and the only way to assess the accuracy of the solution is to solve the problem with a more refined mesh. The second concern is that the solution obeys the shape of the trial function inside the element. Thus, if linear functions are used on triangles, a three-dimensional view of the solution, plotting the solution versus x and y, consists of a series of triangular planes joined together at the edges, as in a geodesic dome. The third caveat is that the Galerkin finite element method is applied to both the differential equations and the boundary conditions. Computer programs are usually quite general and may allow the user to specify boundary conditions that are not realistic. Also, natural boundary conditions are satisfied if no other boundary condition (ones involving derivatives) is set at a node. Thus, the user of finite element codes must be very clear what boundary conditions and differential equations are built into the computer code. When the problem is nonlinear, the Newton-Raphson method is used to iterate from an initial guess. Nonlinear problems lead to complicated integrals to evaluate, and they are usually evaluated using Gaussian quadrature. One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Chang, M. W., and B. A. Finlayson, Int. J. Num. Methods Eng. 15, 935–942 (1980), and Finlason, B. A. (1992). The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. The trial functions in the finite element method are not limited to linear ones. Quadratic functions and even higher-order functions are frequently used. The same considerations hold as for boundary value problems: The higher-order trial functions converge faster, but require more work. It is possible to refine both the mesh h and the power of polynomial in the trial function p in an hp method. Some problems have constraints on some of the variables. For flow problems, the pressure must usually be approximated by using a trial function that is one order lower than the polynomial used to approximate the velocity. Hyperbolic Equations The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation





∂c ∂c ∂2c  + u  = D 2 ∂t ∂x ∂x and the chromatography equation.

NUMERICAL ANALYSIS AND APPROXIMATE METHODS

1

3-57

1 0.9 0.8 0.7 0.6 0.5

0.5

0.4 0.3 0.2 0.1 0 0.7

0 0.7 0.6

1

0.5

0.6

0.1

0.6

0.3

0.4

0.2

0.8

0.4

0.6

0.3

1

0.5

0.8

0.4

0.4

0.2 0.1

0.2 0

0.2 0

0

0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7

0.6

1

0.5

0.8

0.4

0.6

0.3 0.4

0.2 0.1

0.2 0

FIG. 3-51

0

Trial functions for Galerkin finite element method: linear polynomial on triangle.

(See “Partial Differential Equations.”) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. Equations for several methods are given here, as taken from the book by Finlayson [Finlayson, B. A. (1992)]. If the convective term is treated with a centered difference expression, the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give dci ci − ci − 1 ci + 1 − 2ci + ci − 1  + u  = D  dt ∆x ∆x2 The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation u∆x ci + 1 − 2ci + ci − 1 dci ci + 1 − ci − 1  + u  = D +   2 dt 2∆x ∆x2





Thus the diffusion coefficient has been changed from u∆x D to D +  2 Alternatively, the diffusion coefficient has been multiplied by the factor ux Pecell D& = D 1 +  = D 1+  2D 2







u x uL x x where Pecell =  =   = Pe  is called the cell Peclet number. D D L L When the diffusion coefficient is very small (or diffusion is slow compared with convection), the Peclet number will be large. In that case, extraneous diffusion will be included in the solution unless the mesh size (denoted by ∆x) is small compared with the characteristic length of the problem. To avoid this problem (by keeping the factor small), very fine meshes must be used, and the smaller the diffusion coefficient, the smaller the required mesh size. A variety of other methods are used to obtain a good solution without using extremely fine meshes. The flux correction methods keep track of the flux of material into and out of a cell (from one node to another) and put limits on the flux to make sure that no more material

3-58

MATHEMATICS

i –1st cell

ith cell

i−1 i−1/2 i

i+1/2

∆x FIG. 3-52

Nomenclature for finite volume method.

leaves the cell than is there originally plus the input amount. See Finlayson, ibid., for many examples. All the methods have a limit to the time step that is set by the convection term. Essentially, the time step should not be so big as to take the material farther than it can go at its velocity. This is usually expressed as a Courant number limitation. ut Co  ≤ 1 x Some methods require a smaller limit, depending upon the amount of diffusion present (see Finlayson, ibid., Appendix). In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to ∆x so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. Finite Volume Methods Finite volume methods are utilized extensively in computational fluid dynamics. In this method, a mass balance is made over a cell, accounting for the change in what is in the cell, and the flow in and out. Figure 3-52 illustrates the geometry of the ith cell. A mass balance made on this cell (with area A perpendicular to the paper) is Ax (cni 1  cni) tA(Jj1/2  Ji1/2) where J is the flux due to convection and diffusion, positive in the +x direction. 'c J uc  D , 'x

cici1/2 Ji1/2 ui1/2 ci1/2  D   ∆x

The concentration at the edge of the cell is taken as 1 ci1/2 (ci  ci1/2) 2 Rearrangement for the case when the velocity u is the same for all nodes gives u(ci1  c i1) D cni1  cni    2 (ci1  2ci  ci1) t 2x x This is the same equation obtained by using the finite difference method. This isn’t always true, and the finite volume equations are easy to derive. In two and three dimensions, the mesh need not be rectangular, as long as it is possible to compute the velocity normal to an edge of the cell. The finite volume method is useful for applications involving filling, such as injection molding, when only part of the cell is filled with fluid. Such applications do involve some approximations, since the interface is not tracked precisely, but they are useful engineering approximations. Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation ∂T ∂2T ∂2T + −Q ρCp  = k  ∂x2 ∂y2 ∂t





Most engineers use computer packages to solve such problems. If there is both convection and diffusion in the problem, the same considerations apply: A fine mesh is needed when the Peclet number is large. The upstream weighting and Petrov-Galerkin methods can be used, but it is important to apply the smoothing only in the direction of flow, since smoothing in the direction transverse to the flow direction would be incorrect. Some transverse smoothing is unavoidable, but the engineer needs to be sure that the smoothing is just enough to allow a good solution without creating large errors. Computer Software When you are choosing computer software to solve your problem, there are a number of important considerations. The first decision is whether to use an approximate, engineering flow model, developed from correlations, or to solve the partial differential equations that govern the problem. Correlations are quick and easy to apply, but they may not be appropriate to your problem or give the needed detail. When you are using a computer package to solve partial differential equations, the first task is always to generate a mesh covering the problem domain. This is not a trivial task, and special methods have been developed to permit importation of a geometry from a computer-aided design (CAD) program. Then the mesh must be created automatically. If the boundary is irregular, the finite element method is especially well suited, although special embedding techniques can be used in finite difference methods (which are designed to be solved on rectangular meshes). Another capability to consider is the ability to track free surfaces that move during the computation. This phenomenon introduces the same complexity that occurs in problems with a large Peclet number, with the added difficulty that the free surface moves between mesh points and improper representation can lead to unphysical oscillations. The method used to solve the equations is important, and both explicit and implicit methods (as described above) can be used. Implicit methods may introduce unacceptable extra diffusion, so the engineer needs to examine the solution carefully. The methods used to smooth unphysical oscillations from node to node are also important, and the engineer needs to verify that the added diffusion or smoothing does not give inaccurate solutions. Since current-day problems are mostly nonlinear, convergence is always an issue since the problems are solved iteratively. Robust programs provide several methods for convergence, each of which is best in some circumstance or other. It is wise to have a program that includes many iterative methods. If the iterative solver is not very robust, the only recourse to solving a steadystate problem may be to integrate the time-dependent problem to steady state. The solution time may be long, and the final result may be further from convergence than would be the case if a robust iterative solver were used. A variety of computer programs are available on the Internet, some free. First consider general-purpose programs. On the NIST web page http://gams.nist.gov/, choose “problem decision tree” then “differential and integral equations” and then “partial differential equations.” The programs are organized by type of problem (elliptic, parabolic, and hyperbolic) and by the number of spatial dimensions (one or more than one). On the Netlib web site http://www.netlib.org/, search on “partial differential equation.” Lau (1994, 2004) provides many programs in C++ (also see http://www.nr.com/). The multiphysics program Comsol Multiphysics (formerly FEMLAB) also solves many standard equations arising in mathematical physics. Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http://www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http://www.comsol.com/), and ANSYS (http://www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing

NUMERICAL ANALYSIS AND APPROXIMATE METHODS convergence), and has extensive graphical output capabilities. Other packages are also available (see http://cfd-online.com/), and these may contain features and correlations specific to the engineer’s industry. One important point to note is that for turbulent flow, all the programs contain approximations, using the k-epsilon models of turbulence, or large eddy simulations; the direct numerical simulation of turbulence is too slow to apply to very big problems, although it does give insight (independent of any approximations) that is useful for interpreting turbulent phenomena. Thus, the method used to include those turbulent correlations is important, and the method also may affect convergence or accuracy.

The Fourier transform and inverse transform are



∞

−∞

y(t)eiωt dt

1 y(t) =  2π



∞

−∞

Y(ω)e d

Suppose a signal y(t) is sampled at equal intervals yn = y(n∆),

of {yj} and is of length N/2. Similarly, Yko is the kth component of the Fourier transform of the odd components of {yj} and is of length N/2. W is a constant, which is taken to the kth power. W = e2πi/N Since Yk has N components, while Y ek and Yko have N/2 components, Y ek and Yko are repeated once to give N components in the calculation of Yk. This decomposition can be used recursively. Thus, Y ek is split into even and odd terms of length N/4. Yke = Ykee + W k Ykeo Yko = Ykoe + W k Ykoo

FAST FOURIER TRANSFORM

Y(ω) =

n = . . . , −2, −1, 0, 1, 2, . . .

∆ = sampling interval (e.g., time between samples)

This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector {yj } is filled with zeroes, if need be, to make N = 2 p for some p. The standard Fourier transform takes N 2 operations to calculate, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant; at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100. The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations [Gottlieb, D., and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia (1977); Trefethen, L. N., Spectral Methods in Matlab, SIAM, Philadelphia (2000)]. Suppose we have a grid of equidistant points

The Nyquist critical frequency or critical angular frequency is 1 fc =  , 2∆

π ωc =  ∆

If a function y(t) is bandwidth-limited to frequencies smaller than fc, such as Y(ω) = 0

for ω > ωc

then the function is completely determined by its samples yn. Thus, the entire information content of a signal can be recorded by sampling at a rate ∆−1 = 2fc. If the function is not bandwidth-limited, then aliasing occurs. Once a sample rate ∆ is chosen, information corresponding to frequencies greater than fc is simply aliased into that range. The way to detect this in a Fourier transform is to see if the transform approaches zero at fc; if not, aliasing has occurred, and a higher sampling rate is needed. Next, suppose we have N samples, where N is even yk = y(tk)

tk = k∆

k = 0,1,2, . . . , N − 1

and the sampling interval is ∆. With only N values {yk}, it is not possible to determine the complete Fourier transform Y(ω). We calculate the value Y(ωn) at the discrete points 2πn ωn =  , N∆ Yn =

N N n = −  , . . . , 0, . . . ,  2 2

N−1

xn = n∆x,

n = 0, 1, 2, . . . , 2N − 1,

1 Yk =  2N

2N − 1



y(xn)e−2ikπx n /L , k = −N, − N + 1, . . . , 0, . . . , N − 1, N

n=0

The inverse transformation is 1 N y(x) =  Yke2ikπx/L L k = −N Differentiate this to get 2πik dy 1 N  =  Yk  e2ikπx/L L dx L k = −N Thus at the grid points



N 2πik dy  = Yk  e2ikπx n /L L dx n k = −N

The process works as follows. From the solution at all grid points the Fourier transform is obtained using FFT, {Yk}. Then this is multiplied by 2πik/L to obtain the Fourier transform of the derivative. 2πik Yk = Yk  L

2πikn/N

k=0

Y(ωn) = ∆Yn The discrete inverse Fourier transform is 1 yk =  N

L ∆x =  2N

The solution is known at each of these grid points {y(xn)}. First the discrete Fourier transform is taken:

ye k

3-59

N−1

Ye n

−2πikn/N

n=0

The fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. Yk = Yke + W k Yko Here Yk is the kth component of the Fourier transform of y, and Yke is the kth component of the Fourier transform of the even components

Then the inverse Fourier transform is taken using FFT, giving the value of the derivative at each of the grid points.



dy 1 N  =  Yk e2ikπx n /L dx n L k = −N The spectral method is used for direct numerical simulation (DNS) of turbulence. The Fourier transform is taken of the differential equation, and the resulting equation is solved. Then the inverse transformation gives the solution. When there are nonlinear terms, they are calculated at each node in physical space, and the Fourier transform is taken of the result. This technique is especially suited to time-dependent problems, and the major computational effort is in the fast Fourier transform.

3-60

MATHEMATICS

OPTIMIZATION REFERENCES: General references include the following textbooks. For nonlinear programming, Fletcher, R., Practical Methods of Optimization, Wiley (1987); Nocedal, J., and S. J. Wright, Numerical Optimization, Springer, New York (1999); Conn, A. R., N. Gould, and P. Toint, Trust Region Methods, SIAM, Philadelphia (2000); Edgar, T. F., D. M. Himmelblau, and L. S. Lasdon, Optimization of Chemical Processes, McGraw-Hill (2002). For linear programming, Dantzig, G. B., Linear Programming and Extensions, Princeton, N.J.: Princeton University Press, (1963); Hillier, F., and G. J. Lieberman, Introduction to Operations Research, Holden-Day, San Francisco (1974). For mixed integer programming, Biegler, L. T., I. E. Grossmann, and A. W. Westerberg, Systematic Methods for Chemical Process Design, Prentice-Hall, Englewood Cliffs, N.J. (1997); Nemhauser, G. L., and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience, New York (1988). For global optimization, Floudas, C. A., Deterministic Global Optimization: Theory, Algorithms and Applications, Kluwer Academic Publishers (2000); Horst, R., and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin (1993); Tawarmalani, M., and N. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer Academic Publishers (2002). Many useful resources including descriptions, trial software, and examples can be found on the NEOS server (http://www-neos.mcs.anl.gov) maintained at Argonne National Laboratory. Background material for this section includes the two previous sections on matrix algebra and numerical analysis.

INTRODUCTION Optimization is a key enabling tool for decision making in chemical engineering. It has evolved from a methodology of academic interest into a technology that continues to have a significant impact on engineering research and practice. Optimization algorithms form the core tools for (1) experimental design, parameter estimation, model development, and statistical analysis; (2) process synthesis analysis, design, and retrofit; (3) model predictive control and real-time optimization; and (4) planning, scheduling, and the integration of process operations into the supply chain. As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. Mixed integer problems also include discrete variables. These can be written as mixed integer nonlinear programs (MINLP), or as mixed integer linear programs (MILP), if all variables appear linearly in the constraint and objective functions. For the latter an important case occurs when all the variables are integer; this gives rise to an integer programming (IP) problem. IP problems can be further classified into many special problems (e.g., assignment, traveling salesperson, etc.), which are not shown in Fig. 3-53. Similarly, the MINLP problem also gives rise to special problem classes, although here the main distinction is whether its relaxation is convex or nonconvex. Optimization

Mixed Integer (Discrete)

Nondifferentiable

Differentiable

MILP Convex

Nonconvex

IP LP FIG. 3-53

QP

Local

Min f(x, y) subject to h(x, y) = 0 g(x, y) ≤ 0 x ∈ ᑬn, y ∈ {0, 1}

(3-84)

where f(x, y) is the objective function (e.g., cost, energy consumption, etc.), h(x, y) = 0 are the equations that describe the performance of the system (e.g., material balances, production rates), and the inequality constraints g(x, y) ≤ 0 can define process specifications or constraints for feasible plans and schedules. Note that the operator max f(x) is equivalent to min [−f(x)]. We define the real n vector x to represent the continuous variables while the t vector y represents the discrete variables, which, without loss of generality, are often restricted to take values of 0 or 1 to define logical or discrete decisions, such as assignment of equipment and sequencing of tasks. (These variables can also be formulated to take on other integer values as well.) Problem (3-84) corresponds to a mixed integer nonlinear program when any of the functions involved are nonlinear. If all functions are linear, it corresponds to a mixed integer linear program. If there are no 0–1 variables, then problem (3-84) reduces to a nonlinear program (3-85) or linear program (397) depending on whether the functions are linear. We start with continuous variable optimization and consider in the next section the solution of NLP problems with differentiable objective and constraint functions. If only local solutions are required for the NLP problem, then very efficient large-scale methods can be considered. This is followed by methods that are not based on local optimality criteria; we consider direct search optimization methods that do not require derivatives as well as deterministic global optimization methods. Following this, we consider the solution of mixed integer problems and outline the main characteristics of algorithms for their solution. Finally, we conclude with a discussion of optimization modeling software and its implementation on engineering models. GRADIENT-BASED NONLINEAR PROGRAMMING For continuous variable optimization we consider (3-84) without discrete variable y. The general NLP problem (3-85) is presented here: Min f(x) subject to h(x) = 0

(3-85)

g(x) ≤ 0 and we assume that the functions f(x), h(x), and g(x) have continuous first and second derivatives. A key characteristic of (3-85) is whether the problem is convex or not, i.e., whether it has a convex objective function and a convex feasible region. A function φ(x) of x in some domain X is convex if and only if for all points x1, x2 ∈ X φ[α x1 + (1 − α)x2] ≤ αφ[x1 + (1 − α)x2] + (1 − α)φ(x2) (3-86)

NLP (Continuous)

MINLP

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continuous optimization problems can be written as the following mixed integer optimization problem:

Global

Classes of optimization problems and algorithms.

Direct Search

holds for all  ∈ (0, 1). [Strict convexity requires that the inequality (3-86) be strict.] Convex feasible regions require g(x) to be a convex function and h(x) to be linear. If (3-85) is a convex problem, then any local solution is guaranteed to be a global solution to (3-85). Moreover, if the objective function is strictly convex, then this solution x* is unique. On the other hand, nonconvex problems may have multiple local solutions, i.e., feasible solutions that minimize the objective function within some neighborhood about the solution. We consider first methods that find only local solutions to nonconvex problems, as more difficult (and expensive) search procedures are required to find a global solution. Local methods are currently very

OPTIMIZATION

∇x f(x*) = 0

p T∇xx f(x*)p ≥ 0

for all p ≠ 0

(3-87)

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows: ∇xx f(x*) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. Now consider the imposition of inequality [g(x) ≤ 0] and equality constraints [h(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) ≤ 0 act as “fences” in the valley, and equality constraints h(x) = 0 act as “rails.” Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [− ∇g(x*)] and rails [− ∇h(x*)] on the ball are balanced by the force of gravity [− ∇f(x*)]. This condition can be stated by the following Karush-KuhnTucker (KKT) necessary conditions for constrained optimality: Balance of Forces It is convenient to define the L function L(x,λ,ν) = f(x) + g(x)Tλ + h(x)Tν, along with “weights” or multipliers λ and ν for the constraints. The stationarity condition (balance of forces acting on the ball) is then given by ∇L(x, λ, ν) = ∇f(x) + ∇h(x)λ + ∇g(x)ν = 0

(3-88)

Feasibility Both inequality and equality constraints must be satisfied (ball must lie on the rail and within the fences): h(x) = 0,

g(x) ≤ 0

(3-89)

Complementarity Inequality constraints are either strictly satisfied (active) or inactive, in which case they are irrelevant to the solu-

10

9

h(x*)=0

8 7 6

f (x*)

5 4

100

3

x*

50

2

−2

g(x*)

0

h(x* )

1

1

2

10 2

g(x)≤0 3

4

5

6

7

8

9

10

x1 FIG. 3-55

Constrained minimum.

tion. In the latter case the corresponding KKT multiplier must be zero. This is written as ν Tg(x) = 0,

ν ≥0

(3-90)

Constraint Qualification For a local optimum to satisfy the KKT conditions, an additional regularity condition is required on the constraints. This can be defined in several ways. A typical condition is that the active constraints at x* be linearly independent; i.e., the matrix: [∇h(x*)|∇g A(x*)] is full column rank, where g A is the vector of inequality constraints with elements that satisfy g A,i(x*) = 0. With this constraint qualification, the KKT multipliers (λ, ν) are guaranteed to be unique at the optimal solution. Second-Order Conditions As with unconstrained optimization, nonnegative (positive) curvature is necessary (sufficient) in all the allowable (i.e., constrained) nonzero directions p. The necessary second-order conditions can be stated as pT ∇xx L(x*)p ≥ 0 for all p ≠ 0, ∇h(x*)Tp = 0, ∇g(x*)Tp ≥ 0,∇g A(x*)Tp = 0

(3-91)

and the corresponding sufficient conditions require first the inequality in (3-91) to be strict. Note that for the example in Fig. 3-54, the allowable directions p span the entire space for x while in Fig. 3-55 there are no allowable directions p. Example To illustrate the KKT conditions, consider the following unconstrained NLP problem:

9 8

Min (x1)2 − 4x1 + 3/2 (x2)2 − 7x2 + x1x2 + 9 − ln x1 − ln x2

7

corresponding to the contour plot in Fig. 3-54. The optimal solution can be found by solving for the first-order conditions (3-87): 2x − 4 + x − 1/x 1.3475 ∇f(x) = 3x1 − 7 + x2 − 1/x1 = 0 ⇒ x* = 2.0470 2 1 2



6

x2

10

x2

efficient and have been developed to deal with very large NLP problems. Moreover, by considering the structure of convex NLP problems (including LP and QP problems), even more powerful methods can be applied. To study these methods, we first consider conditions for local optimality. Local Optimality Conditions: A Kinematic Interpretation Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function f(x), given in Fig. 3-54, as a smooth valley in space of the variables x1 and x2. For the contour plot of this unconstrained problem Min f(x), consider a ball rolling in this valley to the lowest point of f(x), denoted by x*. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector ∇f(x) and second-derivative (Hessian) matrix ∇xxf(x) to state the necessary first- and second-order conditions for unconstrained optimality:

3-61

5







(3-92)

(3-93)

and f(x*) = − 2.8742. Checking the second-order conditions leads to 4



* 2 1 ∇xx f(x*) = 2 + 1/(x1) 1 3 + 1/(x2*)2

100

3 50 2

−2

0

x*

1



2.5507 1 ⇒ ∇xx f(x*) = 1 3.2387

10 2

(3-94)



(positive definite)

Now consider the constrained NLP problems 1

2

3

4

5

x1 FIG. 3-54



Unconstrained minimum.

6

7

8

9

10

Min (x *1)2 − 4x1 + 3/2(x 2*)2 − 7x2 + x1x2 + 9 − ln x1 − ln x2 subject to 4 − x1x2 ≤ 0 2x1 − x2 = 0

(3-95)

that correspond to the plot in Fig. 3-54. The optimal solution can be found by

3-62

MATHEMATICS

applying the first-order KKT conditions (3-88) to (3-90): (L(x, λ ,ν) = (f(x) + (h(x)λ + (g(x)ν =

2x1 − 4 + x2 − 1x1

 3x − 7 + x − 1x  2

1

2

−x 2 + λ + − x2 ν = 0 −1 1

   

g(x) = 4 − x1x2 ≤ 0,

h(x) = 2x1 − x2 = 0

g(x) ν = (4 − x1x2) ν,

(3-96)

ν≥0

⇓ x* =

2.8284, 1.4142

λ* = 1.036,

ν* = 1.068

and f(x*) = −1.8421. Checking the second-order conditions (3-91) leads to (xxL(x*,λ*,ν*) = (xx[f(x*) + h(x*)λ* + g(x*)ν*] = =

0.068

[(h(x*) | (gA(x*)]Tp =

2.5

−1 2

0.068 3.125

2 + 1(x1) 1−ν



2

1−ν 3 + 1(x2)2





− 2.8284 T p = 0, − 1.4142



p≠0

However, note that because [(h(x*) | (gA(x*)] is nonsingular, there are no nonzero vectors p that satisfy the allowable directions. Hence, the sufficient second-order conditions [pT (xxL(x*,λ*,ν*) p > 0, for all allowable p] are vacuously satisfied for this problem.

Convex Cases of NLP Problems Linear programs and quadratic programs are special cases of (3-85) that allow for more efficient solution, based on application of KKT conditions (3-88) through (3-91). Because these are convex problems, any locally optimal solution is a global solution. In particular, if the objective and constraint functions in (3-85) are linear, then the following linear program (LP) Min cTx subject to Ax = b Cx ≤ d

(3-97)

can be solved in a finite number of steps, and the optimal solution lies at a vertex of the polyhedron described by the linear constraints. This is shown in Fig. 3-56, and in so-called primal degenerate cases, multiple vertices can be alternate optimal solutions, with the same values of the objective function. The standard method to solve (3-97) is the simplex method, developed in the late 1940s (see Dantzig, 1963) although, starting from Karmarkar’s discovery in 1984, interior point methods have become quite advanced and competitive for highly constrained problems [Wright, S. J., Primal-Dual Interior Point Methods,

SIAM, Philadelphia (1996)]. The simplex method proceeds by moving successively from vertex to vertex with improved objective function values. Methods to solve (3-97) are well implemented and widely used, especially in planning and logistical applications. They also form the basis for MILP methods discussed later. Currently, state-of-theart LP solvers can handle millions of variables and constraints, and the application of further decomposition methods leads to the solution of problems that are two or three orders of magnitude larger than this. See the general references of Hillier and Lieberman (1974) and Edgar et al. (2002) for more details. Also, the interior point method is described below from the perspective of more general NLP problems. Quadratic programs (QPs) represent a slight modification of (3-97) and can be stated as Min cTx + 1/2 xTQx subject to Ax = b Cx ≤ d

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. Solving the General NLP Problem Solution techniques for (3-85) deal with satisfaction of the KKT conditions (3-88) through (391). Many NLP solvers are based on successive quadratic programming (SQP) as it allows the construction of a number of NLP algorithms based on the Newton-Raphson method for equation solving (see “Numerical Analysis” section). SQP solvers have been shown to require the fewest function evaluations to solve NLP problems [Schittkowski, K., Lecture Notes in Economics and Mathematical Systems, no. 282, Springer-Verlag, Berlin (1987)], and they can be tailored to a broad range of process engineering problems with different structure. The SQP strategy applies the equivalent of a Newton step to the KKT conditions of the nonlinear programming problem, and this leads to a fast rate of convergence. By adding slack variables s, the first-order KKT conditions can be rewritten as ∇f(x) + ∇h(x) λ + ∇g(x) ν = 0 h(x) = 0

Min Min

Linear Program

FIG. 3-56

Contour plots of linear programs.

(3-98)

Linear Program (Alternate Optima)

(3-99a) (3-99b)

OPTIMIZATION

3-63

Min

Min Min

Convex Objective Functions Linear Constraints FIG. 3-57

Contour plots of convex quadratic programs.

g(x) + s = 0

(3-99c)

SVe = 0

(3-99d)

(s, ν) ≥ 0

(3-99e)

where e = [1, 1, . . . , 1]T, S = diag{s}, and V = diag{ν}. SQP methods find solutions that satisfy (3-99) by generating Newton-like search directions at iteration k. However, Eqs. (3-99d) and active bounds (3-99e) are dependent at the solution and serve to make the KKT system illconditioned near the solution. SQP algorithms treat these conditions in two ways. In the active set strategy, discrete decisions are made regarding the active constraint set i ∈ I = {i|gi(x*) = 0} and (3-99d) is replaced by si = 0, i ∈ I, and νi = 0, i ∉ I. Determining the active set is a combinatorial problem, and a straightforward way to determine an estimate of the active set [and to satisfy (3-99e)] is to formulate and solve, at a point xk, the following QP at iteration k: Min ∇ f(xk)Tp + 1/2 pT ∇xx L(xk, λk, νk)p subject to h(xk) + ∇h(xk)Tp = 0 g(xk) + ∇g(xk)Tp + s = 0, s ≥ 0

(3-100)

The KKT conditions of (3-100) are given by ∇f(xk) + ∇2L(xk, λk, ν k)p + ∇h(xk)λ + ∇g(xk)ν = 0 (3-101a) h(xk) + ∇h(xk) T p = 0

(3-101b)

g(xk) + ∇g(xk) T p + s = 0

(3-101c)

SVe = 0

(3-101d)

(s, ν) ≥ 0

(3-101e)

where the Hessian of the Lagrange function ∇xx L(x, λ, ν) = ∇xx[f(x) + h(x)T λ + g(x)T ν] is calculated directly or through a quasi-Newton approximation (created by differences of gradient vectors). It is easy to show that (3-101a) through (3-101c) correspond to a Newton-Raphson step for (3-99a) through (3-99c) applied at iteration k. Also, selection of the active set is now handled at the QP level by satisfying the conditions (3-101d) and (3-101e). To evaluate and change candidate active sets, QP algorithms apply inexpensive matrix updating strategies to the KKT matrix associated with (3-100). Details of this approach can be found in Nocedal and Wright (1999) and Fletcher (1987). As alternatives that avoid the combinatorial problem of selecting the active set, interior point (or barrier) methods modify the NLP problem (3-85) to form Min f(xk) − µΣi lnsi subject to h(xk) = 0 g(xk) + s = 0

(3-102)

(3-103)

and for µ > 0, s > 0, and ν > 0, Newton steps generated to solve (3-103) are well behaved and analogous to (3-101), with a modification on the right-hand side of (3-101d). A detailed description of this algorithm, called IPOPT, can be found in Wächter and Biegler [Math. Prog. 106(1), 25–57 (2006)]. Both active set and interior point methods possess clear tradeoffs. Interior point methods may require more iterations to solve (3-102) for various values of µ, while active set methods require the solution of the more expensive QP subproblem (3-100). Thus, if there are few inequality constraints or an active set is known (say from a good starting guess, or a known QP solution from a previous iteration), then solving (3-100) is not expensive and the active set method is favored. On the other hand, for problems with many inequality constraints, interior point methods are often faster, as they avoid the combinatorial problem of selecting the active set. This is especially true for large-scale problems and when a large number of bounds are active. Examples that demonstrate the performance of these approaches include the solution of model predictive control (MPC) problems [Rao et al., J. Optim. Theory Appl. 99:723 (1998); Albuquerque et al., Comp. Chem. Eng. 23:283 (1997)] and the solution of large optimal control problems using barrier NLP solvers. For instance, IPOPT allows the solution of problems with more than 1,000,000 variables and up to 50,000 degrees of freedom [see Biegler et al., Chem. Eng. Sci. 57(4):575–593 (2002); Laird et al., ASCE J. Water Resource Management and Planning 131(2):125 (2005)]. Other Gradient-Based NLP Solvers In addition to SQP methods, a number of NLP solvers have been developed and adapted for large-scale problems. Generally these methods require more function evaluations than of SQP methods, but they perform very well when interfaced to optimization modeling platforms, where function evaluations are cheap. All these can be derived from the perspective of applying Newton steps to portions of the KKT conditions. LANCELOT (Conn et al., 2000) is based on the solution of boundconstrained subproblems. Here an augmented Lagrangian is formed from (3-85) and the following subproblem is solved: Min f(x) + λTh(x) + νT[g(x) + s] + 1/2 ρ||h(x), g(x) + s||2 subject to s ≥ 0

where the solution to (3-103) has s > 0 for the penalty parameter µ > 0, and decreasing µ to 0 leads to solution of problem (3-85). The KKT conditions for this problem can be written as ∇f(x*) + ∇h(x*)λ + ∇g(x*)ν = 0

h(x*) = 0 g(x*) + s = 0 SVe = µe

(3-104)

The above subproblem can be solved very efficiently for fixed values of the multipliers λ and ν and penalty parameter ρ. Here a gradient projection trust region method is applied. Once subproblem (3-104) is solved, the multipliers and penalty parameter are updated in an outer loop and the cycle repeats until the KKT conditions for (3-85) are satisfied. LANCELOT works best when exact second derivatives are available. This promotes a fast convergence rate in solving each

3-64

MATHEMATICS

subproblem and allows a bound-constrained trust region method to exploit directions of negative curvature in the Hessian matrix. Reduced gradient methods are active set strategies that rely on partitioning the variables and solving (3-99) in a nested manner. Without loss of generality, problem (3-85) can be rewritten as Min f(z) subject to c(z) = 0, a ≤ z ≤ b. Variables are partitioned as nonbasic variables (those fixed to their bounds), basic variables (those that can be solved from the equality constraints), and superbasic variables (those remaining variables between bounds that serve to drive the optimization); this leads to zT = [zNT, zBT, zST]. This partition is derived from local information and may change over the course of the optimization iterations. The corresponding KKT conditions can be written as ∇N f(z) + ∇N c(z)γ = βa − βb

(3-105a)

∇B f(z) + ∇B c(z)γ = 0

(3-105b)

∇S f(z) + ∇S c(z)γ = 0

(3-105c)

c(z) = 0

(3-105d)

zN,j = aj or bj,

βa,j ≥ 0,

βb,j = 0 or βb,j ≥ 0,

βa,j = 0 (3-105e)

where γ and β are the KKT multipliers for the equality and bound constraints, respectively, and (3-105e) replaces the complementarity conditions (3-90). Reduced gradient methods work by nesting equations (3-105b,d) within (3-105a,c). At iteration k, for fixed values of zNk and zSk, we can solve for zB by using (3-105d) and for γ by using (3-105b). Moreover, linearization of these equations leads to sensitivity information (i.e., constrained derivatives or reduced gradients) that indicates how zB changes with respect to zS and zN. The algorithm then proceeds by updating zS by using reduced gradients in a Newton-type iteration to solve Eq. (3-105c). Following this, bound multipliers β are calculated from (3-105a). Over the course of the iterations, if the variables zB or zS exceed their bounds or if some bound multipliers β become negative, then the variable partition needs to be changed and Eqs. (3-105) are reconstructed. These reduced gradient methods are embodied in the popular GRG2, CONOPT, and SOLVER codes (Edgar et al., 2002). The SOLVER code has been incorporated into Microsoft Excel. CONOPT [Drud, A., ORSA J. Computing 6: 207–216 (1994)] is an efficient and widely used code in several optimization modelling environments. MINOS (Murtagh and Saunders, Technical Report SOL 83-20R, Stanford University, 1987) is a well-implemented package that offers a variation on reduced gradient strategies. At iteration k, Eq. (3-105d) is replaced by its linearization c(z Nk , z Bk , z Sk) + ∇B c(zk)T (zB − z Bk ) + ∇Sc(zk)T (zS − z Sk) = 0 (3-106) and Eqs. (3-105a–c, e) are solved with (3-106) as a subproblem by using concepts from the reduced gradient method. At the solution of this subproblem, constraints (3-105d) are relinearized and the cycle repeats until the KKT conditions of (3-105) are satisfied. The augmented lagrangian function from (3-104) is used to penalize movement away from the feasible region. For problems with few degrees of freedom, the resulting approach leads to an extremely efficient method even for very large problems. MINOS has been interfaced to a number of modelling systems and enjoys widespread use. It performs especially well on large problems with few nonlinear constraints. However, on highly nonlinear problems it is usually less reliable than other reduced gradient methods. Algorithmic Details for NLP Methods All the above NLP methods incorporate concepts from the Newton-Raphson method for equation solving. Essential features of these methods are that they provide (1) accurate derivative information to solve for the KKT conditions, (2) stabilization strategies to promote convergence of the Newton-like method from poor starting points, and (3) regularization of the Jacobian matrix in Newton’s method (the so-called KKT matrix) if it becomes singular or ill-conditioned. 1. NLP methods provide first and second derivatives. The KKT conditions require first derivatives to define stationary points, so accurate first derivatives are essential to determine locally optimal solutions for differentiable NLPs. Moreover, Newton-Raphson methods that are applied to the KKT conditions, as well as the task of checking second-order KKT conditions, necessarily require second-

derivative information. (Note that second-order conditions are not checked by methods that do not use second derivatives.) With the recent development of automatic differentiation tools, many modeling and simulation platforms can provide exact first and second derivatives for optimization. When second derivatives are available for the objective or constraint functions, they can be used directly in LANCELOT as well as SQP and reduced gradient methods. Otherwise, on problems with few superbasic variables, both reduced gradient methods and SQP methods [with reduced gradient methods applied to the QP subproblem (3-100)] can benefit from positive definite quasi-Newton approximations [Nocedal and Wright (1999)] applied to reduced second-derivative quantities (the so-called reduced Hessian). Finally, for problems with least squares functions (see “Statistics” subsection), as in data reconciliation, parameter estimation, and model predictive control, one can often assume that the values of the objective function and its gradient at the solution are vanishingly small. Under these conditions, one can show that the multipliers (λ, ν) also vanish and ∇xxL(x, λ, ν) can be substituted by ∇xx f(x*). This Gauss-Newton approximation has been shown to be very efficient for the solution of least squares problems [see Nocedal and Wright (1999)]. 2. Line search and trust region methods promote convergence from poor starting points. These are commonly used with the search directions calculated from NLP subproblems such as (3-100). In a trust region approach, the constraint ||p|| ≤ ∆ is added and the iteration step is taken if there is sufficient reduction of some merit function (e.g., the objective function weighted with some measure of the constraint violations). The size of the trust region ∆ is adjusted based on the agreement of the reduction of the actual merit function compared to its predicted reduction from the subproblem (see Conn et al., 2000, for details). Such methods have strong global convergence properties and are especially appropriate for ill-conditioned NLPs. This approach has been applied in the KNITRO code [Byrd, Hribar, and Nocedal, SIAM J. Optimization 9(4):877 (1999)]. Line search methods can be more efficient on problems with reasonably good starting points and well-conditioned subproblems, as in real-time optimization. Typically, once a search direction is calculated from (3-100), or other related subproblem, a step size α ∈ (0, 1) is chosen so that xk + αp leads to a sufficient decrease of a merit function. As a recent alternative, a novel filter stabilization strategy (for both line search and trust region approaches) has been developed based on a bicriterion minimization, with the objective function and constraint infeasibility as competing objectives [Fletcher et al., SIAM J. Optimization 13(3):635 (2002)]. This method often leads to better performance than that based on merit functions. 3. Regularization of the KKT matrix for the NLP subproblem is essential for good performance of general-purpose algorithms. For instance, to obtain a unique solution to (3-100), active constraint gradients must be full rank and the Hessian matrix, when projected into the null space of the active constraint gradients, must be positive definite. These properties may not hold far from the solution, and corrections to the Hessian in SQP may be necessary (see Fletcher, 1987). Regularization methods ensure that subproblems such as (3-100) remain well-conditioned; they include addition of positive constants to the diagonal of the Hessian matrix to ensure its positive definiteness, judicious selection of active constraint gradients to ensure that they are linearly independent, and scaling the subproblem to reduce the propagation of numerical errors. Often these strategies are heuristics built into particular NLP codes. While quite effective, most of these heuristics do not provide convergence guarantees for general NLPs. From the conceptual descriptions as well as algorithmic details given above, it is clear that NLP solvers are complex algorithms that have required considerable research and development to turn them into reliable and efficient software tools. Practitioners who are confronted with engineering optimization problems should therefore leverage these efforts, rather than write their own codes. Table 3-4 presents a sampling of available NLP codes that represent the above classifications. Much more information on these and other codes can be found on the NEOS server (www-neos.mcs.anl.gov) and the NEOS Software Guide: http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide.

OPTIMIZATION TABLE 3-4

Representative NLP Solvers

Method CONOPT (Drud, 1994) GRG2 (Edgar et al., 2002) IPOPT KNITRO (Byrd et al., 1997) LANCELOT LOQO MINOS NPSOL SNOPT SOCS SOLVER SRQP

Algorithm type

Stabilization

Reduced gradient

Line search

Reduced gradient

Line search

SQP, barrier SQP, barrier

Line search Trust region

Augmented lagrangian, bound constrained SQP, barrier Reduced gradient, augmented lagrangian SQP, active set Reduced space SQP, active set SQP, active set Reduced gradient Reduced space SQP, active set

Trust region

Second-order information Exact and quasi-Newton Quasi-Newton

Line search Line search

Exact Exact and quasi-Newton Exact and quasi-Newton Exact Quasi-Newton

Line search Line search

Quasi-Newton Quasi-Newton

Line search Line search Line search

Exact Quasi-Newton Quasi-Newton

OPTIMIZATION METHODS WITHOUT DERIVATIVES A broad class of optimization strategies does not require derivative information. These methods have the advantage of easy implementation and little prior knowledge of the optimization problem. In particular, such methods are well suited for “quick and dirty” optimization studies that explore the scope of optimization for new problems, prior to investing effort for more sophisticated modeling and solution strategies. Most of these methods are derived from heuristics that naturally spawn numerous variations. As a result, a very broad literature describes these methods. Here we discuss only a few important trends in this area. Classical Direct Search Methods Developed in the 1960s and 1970s, these methods include one-at-a-time search and methods based on experimental designs (EVOP). At that time, these direct search methods were the most popular optimization methods in chemical engineering. Methods that fall into this class include the pattern search of Hooke and Jeeves [J. ACM 8:212 (1961)], the conjugate direction method of Powell (1964), simplex and complex searches, in particular Nelder-Mead [Comput. J. 7: 308 (1965)], and the adaptive random search methods of Luus-Jaakola [AIChE J. 19: 760 (1973)], Goulcher and Cesares Long [Comp. Chem. Engr. 2: 23 (1978)] and

Banga et al. [in State of the Art in Global Optimization, C. Floudas and P. Pardalos (eds.), Kluwer, Dordrecht, p. 563 (1996)]. All these methods require only objective function values for unconstrained minimization. Associated with these methods are numerous studies on a wide range of process problems. Moreover, many of these methods include heuristics that prevent premature termination (e.g., directional flexibility in the complex search as well as random restarts and direction generation). To illustrate these methods, Fig. 3-58 illustrates the performance of a pattern search method as well as a random search method on an unconstrained problem. Simulated Annealing This strategy is related to random search methods and derives from a class of heuristics with analogies to the motion of molecules in the cooling and solidification of metals [Laarhoven and Aarts, Simulated Annealing: Theory and Applications, Reidel Publishing, Dordrecht (1987)]. Here a temperature parameter  can be raised or lowered to influence the probability of accepting points that do not improve the objective function. The method starts with a base point x and objective value f(x). The next point x& is chosen at random from a distribution. If f(x&) < f(x), the move is accepted with x& as the new point. Otherwise, x& is accepted with probability p(, x&, x). Options include the Metropolis distribution p(, x, x&) = exp{−[f(x&)− f(x)]/} and the Glauber distribution, p(, x, x&) = exp{−[f(x&)−f(x)]/}/(1 + exp{−[f(x&)−f(x)]/}). The  parameter is then reduced, and the method continues until no further progress is made. Genetic Algorithms This approach, first proposed in Holland, J. H., Adaptations in Natural and Artificial Systems [University of Michigan Press, Ann Arbor (1975)], is based on the analogy of improving a population of solutions through modifying their gene pool. It also has similar performance characteristics as random search methods and simulated annealing. Two forms of genetic modification, crossover or mutation, are used and the elements of the optimization vector x are represented as binary strings. Crossover deals with random swapping of vector elements (among parents with highest objective function values or other rankings of population) or any linear combinations of two parents. Mutation deals with the addition of a random variable to elements of the vector. Genetic algorithms (GAs) have seen widespread use in process engineering, and a number of codes are available. Edgar et al. (2002) describe a related GA that is available in MS Excel. Derivative-free Optimization (DFO) In the past decade, the availability of parallel computers and faster computing hardware and the need to incorporate complex simulation models within optimization studies have led a number of optimization researchers to reconsider classical direct search approaches. In particular, Dennis and Torczon [SIAM J. Optim. 1: 448 (1991)] developed a multidimensional search algorithm that extends the simplex approach of Nelder

(b)

(a)

Examples of optimization methods without derivatives. (a) Pattern search method. (b) Random search method. , first phase; ∆, second phase; , third phase.

FIG. 3-58

*

3-65

3-66

MATHEMATICS

and Mead (1965). They note that the Nelder-Mead algorithm fails as the number of variables increases, even for very simple problems. To overcome this, their multidimensional pattern search approach combines reflection, expansion, and contraction steps that act as line search algorithms for a number of linear independent search directions. This approach is easily adapted to parallel computation, and the method can be tailored to the number of processors available. Moreover, this approach converges to locally optimal solutions for unconstrained problems and observes an unexpected performance synergy when multiple processors are used. The work of Dennis and Torczon (1991) has spawned considerable research on the analysis and code development for DFO methods. For instance, Conn et al. [Math. Programming, Series B, 79(3): 397 (1997)] constructed a multivariable DFO algorithm that uses a surrogate model for the objective function within a trust region method. Here points are sampled to obtain a well-defined quadratic interpolation model, and descent conditions from trust region methods enforce convergence properties. A number of trust region methods that rely on this approach are reviewed in Conn et al. (1997). Moreover, a number of DFO codes have been developed that lead to black box optimization implementations for large, complex simulation models [see Audet and Dennis, SIAM J. Optim.13: 889 (2003); Kolda et al., SIAM Rev. 45(3): 385 (2003)]. These include the DAKOTA package at Sandia National Lab (Eldred, 2002; http://endo.sandia.gov/DAKOTA/software.html) and FOCUS developed at Boeing Corporation (Booker et al., CRPC Technical Report 98739, Rice University, February 1998). Direct search methods are easy to apply to a wide variety of problem types and optimization models. Moreover, because their termination criteria are not based on gradient information and stationary points, they are more likely to favor the search for globally optimal rather than locally optimal solutions. These methods can also be adapted easily to include integer variables. However, no rigorous convergence properties to globally optimal solutions have yet been discovered. Also, these methods are best suited for unconstrained problems or for problems with simple bounds. Otherwise, they may have difficulties with constraints, as the only options open for handling constraints are equality constraint elimination and addition of penalty functions for inequality constraints. Both approaches can be unreliable and may lead to failure of the optimization algorithm. Finally, the performance of direct search methods scales poorly (and often exponentially) with the number of decision variables. While performance can be improved with the use of parallel computing, these methods are rarely applied to problems with more than a few dozen decision variables. GLOBAL OPTIMIZATION Deterministic optimization methods are available for nonconvex nonlinear programming problems of the form (3-85) that guarantee convergence to the global optimum. More specifically, one can show under mild conditions that they converge to an % distance to the global optimum in a finite number of steps. These methods are generally more expensive than local NLP methods, and they require the exploitation of the structure of the nonlinear program. Because there are no optimality conditions like the KKT conditions for global optimization, these methods work by first partitioning the problem domain (i.e., containing the feasible region) into subregions. Upper bounds on the objective function are computed over all subregions of the problem. In addition, lower bounds can be derived from convex relaxations of the objective function and constraints for each subregion. The algorithm then proceeds to eliminate all subregions that have infeasible constraint relaxations or lower bounds that are greater than the least upper bound. After this, the remaining regions are further partitioned to create new subregions, and the cycle continues until the upper and lower bounds converge. This basic concept leads to a wide variety of global algorithms, with the following features that can exploit different problem classes. Bounding strategies relate to the calculation of upper and lower bounds. For the former, any feasible point or, preferably, a locally optimal point in the subregion can be used. For the lower bound, convex relaxations of the objective and constraint functions are derived.

The refining step deals with the construction of partitions in the domain and further partitioning them during the search process. Finally, the selection step decides on the order of exploring the open subregions. For simplicity, consider the problem Min f(x) subject to g(x) ≤ 0 where each function can be defined by additive terms. Convex relaxations for f(x) and g(x) can be derived in the following ways: • Convex additive terms remain unmodified in these functions. • Concave additive unary terms are replaced by a linear underestimating function that matches the term at the bounds of the subregion. • Nonconvex polynomial terms can be replaced by a set of binary terms, with new variables introduced to define the higher-order polynomials. • Binary terms can be relaxed by using the McCormick underestimator; e.g., the binary term xz is replaced by a new variable w and linear inequality constraints w ≥ xlz + zlx − xlzl w ≥ xuz + zux − xuzu w ≤ xuz + zlx − xuzl

(3-107)

w ≤ xlz + zux − xlzu where the subregions are defined by xl ≤ x ≤ xu and zl ≤ z ≤ zu. Thus the feasible region and the objective function are replaced by convex envelopes to form relaxed problems. Solving these convex relaxed problems leads to global solutions that are lower bounds to the NLP in the particular subregion. Finally, we see that gradient-based NLP solvers play an important role in global optimization algorithms, as they often yield the lower and upper bounds for the subregions. The spatial branch and bound global optimization algorithm can therefore be given by the following steps: 0. Initialize algorithm: calculate upper and lower bounds over the entire (relaxed) feasible region. For iteration k with a set of partitions Mkj and bounds in each subregion fLj and fUj: 1. Bound. Define best upper bound: fU = minj fUj and delete (fathom) all subregions j with lower bounds fLj ≥ fU . If the remaining subregions satisfy fLj ≥ fU − ε, stop. 2. Refine. Divide the remaining active subregions into partitions Mk,j1 and Mk,j2. (Many branching rules are available for this step.) 3. Select. Solve the convex relaxed NLP in the new partitions to obtain fLj1 and fLj2. Delete the partition if there is no feasible solution. 4. Update. Obtain upper bounds fUj1 and fUj2 to new partitions, if present. Set k = k+1, update partition sets, and go to step 1. Example To illustrate the spatial branch and bound algorithm, consider the global solution of 5 Min f(x) =  x4  20x3 + 55x2  57x, 2

subject to

0.5 ≤ x ≤ 2.5 (3-108)

As seen in Fig. 3-59, this problem has local solutions at x* = 2.5 and at x* = 0.8749. The latter is also the global solution with f(x*) = −19.7. To find the global solution, we note that all but the −20x3 term in (3-108) are convex, so we replace this term by a new variable and a linear underestimator within a particular subregion, i.e., 5 Min fL(x) =  x4 − 20w + 55x2−57x 2 subject to x l ≤ x ≤ xu xu − x x −x w = xl3   + xu3 l xu − xl xu − xl

(3-109)

In Fig. 3-59 we also propose subregions that are created by simple bisection partitioning rules, and we use a “loose” bounding tolerance of ε = 0.2. In each partition the lower bound fL is determined by (3-109) and the upper bound fU is determined by the local solution of the original problem in the subregion. Figure 3-60 shows the progress of the spatial branch and bound algorithm as the partitions are refined and the bounds are updated. In Fig. 3-60, note the definitions of the partitions for the nodes, and the sequence numbers in each node that show the order in which the partitions are

OPTIMIZATION MIXED INTEGER PROGRAMMING

−13

Mixed integer programming deals with both discrete and continuous decision variables. For the purpose of illustration we consider discrete decisions as binary variables, that is, yi = 0 or 1, and we consider the mixed integer problem (3-84). Unlike in local optimization methods, there are no optimality conditions, such as the KKT conditions, that can be applied directly. Instead, as in the global optimization methods, a systematic search of the solution space, coupled with upper and lower bounding information, is applied. As with global optimization problems, large mixed integer programs can be expensive to solve, and some care is needed in the problem formulation. Mixed Integer Linear Programming If the objective and constraint functions are all linear, then (3-84) becomes a mixed integer linear programming problem given by

−14 −15

f(x)

−16 −17 −18

Min Z = aTx + cTy subject to Ax + By ≤ b x ≥ 0, y ∈ {0, 1}t

−19 −20 0.5

FIG. 3-59

3-67

1

1.5 x

2.5

2

Global optimization example with partitions.

processed. The grayed partitions correspond to the deleted subregions, and at termination of the algorithm we see that fLj ≥ fU − ε (that is, −19.85 ≥ −19.7 − 0.2), with the gray subregions in Fig. 3-59 still active. Further partitioning in these subregions will allow the lower and upper bounds to converge to a tighter tolerance. Note that a number of improvements can be made to the bounding, refinement, and selection strategies in the algorithm that accelerate the convergence of this method. A comprehensive discussion of all these options can be found in Floudas (2000), Tawarlamani and Sahinidis (2002), and Horst and Tuy (1993). Also, a number of efficient global optimization codes have recently been developed, including αBB, BARON, LGO, and OQNLP. An interesting numerical comparison of these and other codes can be found in Neumaier et al., Math. Prog. B 103(2): 335 (2005).

Note that if we relax the t binary variables by the inequalities 0 ≤ y ≤ 1, then (3-110) becomes a linear program with a (global) solution that is a lower bound to the MILP (3-110). There are specific MILP classes where the LP relaxation of (3-110) has the same solution as the MILP. Among these problems is the well-known assignment problem. Other MILPs that can be solved with efficient special-purpose methods are the knapsack problem, the set covering and set partitioning problems, and the traveling salesperson problem. See Nemhauser and Wolsey (1988) for a detailed treatment of these problems. More generally, MILPs are solved with branch and bound algorithms, similar to the spatial branch and bound method of the previous section, that explore the search space. As seen in Fig. 3-61, binary variables are used to define the search tree, and a number of bounding properties can be noted from the structure of (3-110). Upper bounds on the objective function can be found from any feasible solution to (3-110), with y set to integer values. These can be found at the bottom or “leaf” nodes of a branch and bound tree (and sometimes at intermediate nodes as well). The top, or root, node in

-1-107.6 -19.7 x ≤1.5≤x

-2-34.5 -19.7

-3-44.1 -13.9 x≤2 ≤x -14-17-21.66 -21.99 -13.55 -15.1

x≤1≤x -4-22.38 -19.7

-6-19.6

x≤0.75≤x -7-20.5 -19.7

x≤0.875≤x -8-19.85 -19.7 FIG. 3-60

-9-19.87 -19.7

-5-23.04 -19.5 x≤1.25≤x -13-10-107.6 -20.12 -19.7 -19.5

x≤1.75≤x -15-17.4

x≤1.125≤x -11-19.58

(3-110)

-12-19.03

Spatial branch and bound sequence for global optimization example.

-16-16.25

-node#fL fU ( x *)

x≤2.25≤x -18-15.7

-19-15.76

3-68

MATHEMATICS -1Z=5 (0.5, 1, 0) y1

-5Z=6.33 (0,1,0.67) y3 -7Inf. (0,1,0) FIG. 3-61

-6Z=7.625 (0, 0.875, 1)

-node#Z (y1, y2, y3)

-2Z=6.25 (1, 0.75, 0) y2 -4Inf. (1,0,1)

-3Z=7 (1,1,0)

Branch and bound sequence for MILP example.

the tree is the solution to the linear programming relaxation of (3-110); this is a lower bound to (3-110). On the other hand, as one proceeds down the tree with a partial assignment of the binary variables, a lower bound for any leaf node in that branch can be found from solution of the linear program at this intermediate node with the remaining binary variables relaxed. This leads to the following properties: • Any intermediate node with an infeasible LP relaxation has infeasible leaf nodes and can be fathomed (i.e., all remaining children of this node can be eliminated). • If the LP solution at an intermediate node is not less than an existing integer solution, then the node can be fathomed. These properties lead to pruning of the search tree. Branching then continues in the tree until the upper and lower bounds converge. This basic concept leads to a wide variety of MILP algorithms with the following features. LP solutions at intermediate nodes are relatively easy to calculate with the simplex method. If the solution of the parent node is known, multiplier information from this solution can be used to calculate (via a very efficient pivoting operation) the LP solution at the child node. Branching strategies to navigate the tree take a number of forms. More common depth-first strategies expand the most recent node to a leaf node or infeasible node and then backtrack to other branches in the tree. These strategies are simple to program and require little storage of past nodes. On the other hand, breadth-first strategies expand all the nodes at each level of the tree, select the node with the lowest objective function, and then proceed until the leaf nodes are reached. Here, more storage is required but generally fewer nodes are evaluated than in depth-first search. In addition, selection of binary variable for branching is based on a number of criteria, including choosing the variable with the relaxed value closest to 0 or 1, or the one leading to the largest change in the objective. Additional description of these strategies can be found in Biegler et al. (1997) and Nemhauser and Wolsey (1988). Example To illustrate the branch and bound approach, we consider the MILP: Min Z = x + y1 +2y2 + 3y3 subject to −x + 3y1 + y2 + 2y3 ≤ 0

(3-111)

− 4y1 − 8y2 − 3y3 ≤ −10 x ≥ 0, y1, y2, y3 = {0, 1} The solution to (3-111) is given by x = 4, y1 = 1, y2 = 1, y3 = 0, and Z = 7. Here we use a depth-first strategy and branch on the variables closest to 0 or 1. Fig. 3-61 shows the progress of the branch and bound algorithm as the binary variables are selected and the bounds are updated. The sequence numbers for each node in Fig. 3-61 show the order in which they are processed. The grayed partitions correspond to the deleted nodes, and at termination of the algorithm we see that Z = 7 and an integer solution is obtained at an intermediate node where coincidentally y3 = 0. A number of improvements that can be made to the branching rules will accelerate the convergence of this method. A comprehensive discussion of all these options can be found in Nemhauser and Wolsey (1988). Also, a number of efficient MILP codes have recently been developed, including CPLEX, OSL, XPRESS, and ZOOM. All these serve as excellent large-scale optimization codes as well. A detailed description and availability of these and other MILP solvers

can be found in the NEOS Software Guide: http://www-fp.mcs.anl.gov/otc/ Guide/SoftwareGuide.

Mixed Integer Nonlinear Programming Without loss of generality, we can rewrite the MINLP in (3-84) as: Min Z = f(x) + cTy subject to g(x) + By ≤ b

(3-112)

x ≥ 0, y ∈ {0, 1}t where the binary variables are kept as separate linear terms. We develop several MINLP solution strategies by drawing from the material in the preceding sections. MINLP strategies can be classified into two types. The first deals with nonlinear extensions of the branch and bound method discussed above for MILPs. The second deals with outer approximation decomposition strategies that provide lower and upper bounding information for convergence. Nonlinear Branch and Bound The MINLP (3-112) can be solved in a similar manner to (3-110). If the functions f(x) and g(x) in (3-112) are convex, then direct extensions to the branch and bound method can be made. A relaxed NLP can be solved at the root node, upper bounds to the solution of (3-112) can be found at the leaf nodes, and the bounding properties due to NLP solutions at intermediate nodes still hold. However, this approach is more expensive than the corresponding MILP method. First, NLPs are more expensive than LPs to solve. Second, unlike with relaxed LP solutions, NLP solutions at child nodes cannot be updated directly from solutions at parent nodes. Instead, the NLP needs to be solved again (but one hopes with a better starting guess). The NLP branch and bound method is used in the SBB code interfaced to GAMS. In addition, Leyffer, [Comput. Optim. Appl. 18: 295 (2001)] proposed a hybrid MINLP strategy nested within an SQP algorithm. At each iteration, a mixed integer quadratic program is formed, and a branch and bound algorithm is executed to solve it. If f(x) and g(x) are nonconvex, additional difficulties can occur. In this case, nonunique, local solutions can be obtained at intermediate nodes, and consequently lower bounding properties would be lost. In addition, the nonconvexity in g(x) can lead to locally infeasible problems at intermediate nodes, even if feasible solutions can be found in the corresponding leaf node. To overcome problems with nonconvexities, global solutions to relaxed NLPs can be solved at the intermediate nodes. This preserves the lower bounding information and allows nonlinear branch and bound to inherit the convergence properties from the linear case. However, as noted above, this leads to much more expensive solution strategies. Outer Approximation Decomposition Methods Again, we consider the MINLP (3-112) with convex f(x) and g(x). Note that the NLP with binary variables fixed at y– Min Z = f(x) + c Ty– subject to g(x) + By– ≤ b

(3-113)

x≥0 if feasible, leads to a solution that is an upper bound on the MINLP

OPTIMIZATION solution. In addition, linearizations of convex functions lead to underestimation of the function itself: (x) ≥ (xk) + (xk)T(x − xk)

(3-114)

Consequently, linearization of (3-112) at a point xk, to form the problem subject to g(x k) + (g(x k)T(x−x k) + By ≤ b

(3-115)

leads to overapproximation of the feasible region and underapproximation of the objective function in (3-112). Consequently, solution of (3-115) is a lower bound to the solution of (3-112). Adding more linearizations from other points does not change the bounding property, so for a set of points xl, l = 1, . . ., k, the problem Min Z = 




l = 1, . . . , k

(3-116)

where  is a scalar variable, still has a solution that is a lower bound to (3-112). The outer approximation strategy is depicted in Fig. 3-62. The outer approximation algorithm [Duran, M., and I. E. Grossmann, Math. Programming 36: 307 (1986)] begins by initializing the problem, either with a predetermined starting guess or by solving a relaxed NLP based on (3-112). An upper bound to the solution is then generated by fixing the binary variables to their current values yk and solving the NLP (3-113). This solution determines the continuous variable values xk for the MILP (3-116). [If (3-113) is an infeasible problem, any point may be chosen for xk, or the linearizations could be omitted.] Note that this MILP also contains linearizations from previous solutions of (3-113). Finally, the integer cut )|yi  yki | ≥ 1 is added i to (3-116) to avoid revisiting previously encountered values of binary variables. (In convex problems the integer cut is not needed, but it helps to accelerate solution of the MILP.) Solution of (3-116) yields new values of y and (without the integer cut) must lead to a lower bound to the solution of (3-112). Consequently, if the objective function of the lower bounding MILP is greater than the least upper bound determined in solutions of (3-113), then the algorithm terminates. Otherwise, the new values of y are used to solve the next NLP (3-113). Compared to nonlinear branch and bound, the outer approximation algorithm usually requires very few solutions of the MILP and NLP subproblems. This is especially advantageous on problems where the NLPs are large and expensive to solve. Moreover, there are two variations of outer approximation that may be suitable for particular problem types: In Generalized Benders Decomposition (GBD) [see Sahinidis and Grossmann, Computers and Chem. Eng. 15: 481 (1991)], the lower

Initialize x 0, y 0 Upper bound withy fixed NLP (3-113) Update y Lower bound MILP (3-116) + integer cuts LB ≥ UB FIG. 3-62

Outer approximation MINLP algorithm.

Min  T




subject to  ≥ f(xl)  cTy  ν l [g(xl)  By] l 1, . . . , k (3-117) )|yi  yli| ≥ 1 where ν is the vector of KKT multipliers from the solution of (3-113) at iteration l. This MILP can be derived through a reformulation of the MILP used in Fig. 3-62 with the inactive constraints from (3-113) dropped. Solution of (3-117) leads to a weaker lower bound than (3116), and consequently, more solutions of the NLP and MILP subproblems are needed to converge to the solution. However, (3-117) contains only a single continuous variable and far fewer inequality constraints and is much less expensive to solve than (3-116). Thus, GBD is favored over outer approximation if (3-113) is relatively inexpensive to solve or solution of (3-116) is too expensive. The extended cutting plane (ECP) algorithm [Westerlund and Pettersson, Computers and Chem. Engng. 19: S131 (1995)] is complementary to GBD. While the lower bounding problem in Fig. 3-62 remains essentially the same, the continuous variables xk are chosen from the MILP solution and the NLP (3-113) is replaced by a simple evaluation of the objective and constraint functions. As a result, only MILP problems [(3-116) plus integer cuts] need be solved. Consequently, the ECP approach has weaker upper bounds than outer approximation and requires more MILP solutions. It has advantages over outer approximation when the NLP (3-113) is expensive to solve. Additional difficulties arise for the outer approximation algorithm and its GBD and ECP extensions when neither f(x) nor g(x) is convex. Under these circumstances, the lower bounding properties resulting from the linearization and formulation of the MILP subproblem are lost, and the MILP solution may actually exclude the solution of (3-112). Hence, these algorithms need to be applied with care to nonconvex problems. To deal with nonconvexities, one can relax the linearizations in (3-116) through the introduction of additional deviation variables that can be penalized in the objective function. Alternately, the linearizations in (3-116) can be replaced by valid underestimating functions, such as those derived for global optimization [e.g., (3-107)]. However, this requires knowledge of structural information for (3-112) and may lead to weak lower bounds in the resulting MILP. Finally, the performance of both MILP and MINLP algorithms is strongly dependent on the problem formulations (3-110) and (3-112). In particular, the efficiency of the approach is impacted by the lower bounds produced by the relaxation of the binary variables and subsequent solution of the linear program in the branch and bound tree. A number of approaches have been proposed to improve the quality of the lower bounds, including these: • Logic-based methods such as generalized disjunctive programming (GDP) can be used to formulate MINLPs with fewer discrete variables that have tighter relaxations. Moreover, the imposition of logic-based constraints prevents the generation of unsuitable alternatives, leading to a less expensive search. In addition, constrained logic programming (CLP) methods offer an efficient search alternative to MILP solvers for highly combinatorial problems. See Raman and Grossmann, Computers and Chem. Engng. 18(7): 563 (1994) for more details. • Convex hull formulations of MILPs and MINLPs lead to relaxed problems that have much tighter lower bounds. This leads to the examination of far fewer nodes in the branch and bound tree. See Grossmann and Lee, Comput. Optim. Applic. 26: 83 (2003) for more details. • Reformulation and preprocessing strategies including bound tightening of the variables, coefficient reduction, lifting facets, and special ordered set constraints frequently lead to improved lower bounds and significant performance improvements in mixed integer programming algorithms. A number of efficient codes are available for the solution of MINLPs. These include the AlphaECP, BARON, DICOPT, MINLP, and SBB solvers; descriptions of these can be found on the http://www.gamsworld.org/minlp/solvers.htm web site. l

x ≥ 0, y ∈ {0, 1}t

subject to  ≥ f(x l) + (f(x l)T(x − x l) + cTy g(x l) + (g(x l)T(x − x l) + By ≤ b x ≥ 0, y ∈ {0,1}t

bounding problem in Fig. 3-62 is replaced by the following MILP:

i

Min Z = f(x k) + (f(x k)T(x − x k) + cTy

3-69

3-70

MATHEMATICS

DEVELOPMENT OF OPTIMIZATION MODELS The most important aspect to a successful optimization study is the formulation of the optimization model. These models must reflect the realworld problem so that meaningful optimization results are obtained; they also must satisfy the properties of the problem class. For instance, NLPs addressed by gradient-based methods need to have functions that are defined in the variable domain and have bounded and continuous first and second derivatives. In mixed integer problems, proper formulations are also needed to yield good lower bounds for efficient search. With increased understanding of optimization methods and the development of efficient and reliable optimization codes, optimization practitioners now focus on the formulation of optimization models that are realistic, well posed, and inexpensive to solve. Finally, convergence properties of NLP, MILP, and MINLP solvers require accurate first (and often second) derivatives from the optimization model. If these contain numerical errors (say, through finite difference approximations), then the performance of these solvers can deteriorate considerably. As a result of these characteristics, modeling platforms are essential for the formulation task. These are classified into two broad areas: optimization modeling platforms and simulation platforms with optimization. Optimization modeling platforms provide general-purpose interfaces for optimization algorithms and remove the need for the user to interface to the solver directly. These platforms allow the general formulation for all problem classes discussed above with direct interfaces to state-of-the-art optimization codes. Three representative platforms are the GAMS (General Algebraic Modeling Systems), AMPL (A Mathematical Programming Language), and AIMMS (Advanced Integrated Multidimensional Modeling Software). All three require problem model input via a declarative modeling language and provide exact gradient and Hessian information through automatic differentiation strategies. Although it is possible, these platforms were not designed to handle externally added procedural models. As a result, these platforms are best applied on optimization models that can be developed entirely within their modeling framework. Nevertheless, these platforms are widely used for large-scale research and industrial applications. In addition, the MATLAB platform allows the flexible formulation of optimization models as well, although it currently has only limited capabilities for automatic differentiation and limited optimization solvers. More information on these and other modeling platforms can be found at http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide. Simulation platforms with optimization are often dedicated, application-specific modeling tools to which optimization solvers have been interfaced. These lead to very useful optimization studies, but because they were not originally designed for optimization models, they need to be used with some caution. In particular, most of these platforms do not provide exact derivatives to the optimization solver; often they are approximated through finite differences. In addition, the models themselves are constructed and calculated through numerical procedures, instead of through an open declarative language. Examples of these include widely used process simulators such as Aspen/Plus, PRO/II, and Hysys. Also note that more recent platforms such as Aspen Custom Modeler and gPROMS include declarative models and exact derivatives.

For optimization tools linked to procedural models, note that reliable and efficient automatic differentiation tools are available that link to models written, say, in FORTRAN and C, and calculate exact first (and often second) derivatives. Examples of these include ADIFOR, ADOL-C, GRESS, Odyssee, and PADRE. When used with care, these can be applied to existing procedural models and, when linked to modern NLP and MINLP algorithms, can lead to powerful optimization capabilities. More information on these and other automatic differentiation tools can be found at http://www-unix.mcs.anl.gov/ autodiff/AD_Tools/. For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes: • Simultaneous formulations are based on converting the differential equation model to algebraic constraints via discretization (say, with a Runge-Kutta method) and solving a large-scale NLP or MINLP. This approach requires efficient large-scale optimization solvers but also leads to very flexible and efficient problem formulations, particularly for optimal control and state-constrained problems. It also allows the treatment of unstable dynamics. A review of these strategies can be found in Biegler et al. [Chem. Engng. Sci. 57(4): 575 (2002)] and Betts [Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, Philadelphia (2001)]. • Sequential formulations require the linkage of NLP or MINLP solvers to embedded ODE or PDE solvers. This leads to smaller optimization problems only in the decision (or control) variables. However, an important aspect to this formulation is the need to recover accurate gradient information from the differential equation solver. Here, methods based on the solution of direct and adjoint sensitivity equations have been derived and automated [see Cao et al., SIAM J. Sci. Comp. 24(3): 1076 (2003) for a description of these sensitivity methods]. For the former, commercial tools such as Aspen Custom Modeler and gPROMS have been developed. Finally, the availability of automatic differentiation and related sensitivity tools for differential equation models allows for considerable flexibility in the formulation of optimization models. In van Bloemen Waanders et al. (Sandia Technical Report SAND2002-3198, October 2002), a seven-level modeling hierarchy is proposed that matches optimization algorithms with models that range from completely open (fully declarative model) to fully closed (entirely procedural without sensitivities). At the lowest, fully procedural level, only derivative-free optimization methods are applied, while the highest, declarative level allows the application of an efficient large-scale solver that uses first and second derivatives. This report notes that, based on the modeling level, optimization solver performance can vary by several orders of magnitude.

STATISTICS REFERENCES: Baird, D. C., Experimentation: An Introduction to Measurement Theory and Experiment Design, 3d ed., Prentice-Hall, Englewood Cliffs, N.J. (1995); Box, G. P., J. S. Hunter, and W. G. Hunter, Statistics for Experimenters: Design, Innovation, and Discovery, 2d ed., Wiley, New York (2005); Cropley, J. B., “Heuristic Approach to Complex Kinetics,” pp. 292–302 in Chemical Reaction Engineering—Houston, ACS Symposium Series 65, American Chemical Society, Washington (1978); Lipschutz, S., and J. J. Schiller, Jr., Schaum’s Outline of Theory and Problems of Introduction to Probability and Statistics, McGraw-Hill, New York (1988); Moore, D. S., and G. P. McCabe, Introduction to the Practice of Statistics, 4th ed., Freeman, San Francisco (2003); Montgomery, D. C., and G. C. Runger, Applied Statistics and Probability for Engineers, 3d ed., Wiley, New York

(2002); and Montgomery, D. C., G. C. Runger, and N. F. Hubele, Engineering Statistics, 3d ed., Wiley, New York (2004).

INTRODUCTION Statistics represents a body of knowledge which enables one to deal with quantitative data reflecting any degree of uncertainty. There are six basic aspects of applied statistics. These are: 1. Type of data 2. Random variables

STATISTICS 3. Models 4. Parameters 5. Sample statistics 6. Characterization of chance occurrences From these can be developed strategies and procedures for dealing with (1) estimation and (2) inferential statistics. The following has been directed more toward inferential statistics because of its broader utility. Detailed illustrations and examples are used throughout to develop basic statistical methodology for dealing with a broad area of applications. However, in addition to this material, there are many specialized topics as well as some very subtle areas which have not been discussed. The references should be used for more detailed information. Section 8 discusses the use of statistics in statistical process control (SPC). Type of Data In general, statistics deals with two types of data: counts and measurements. Counts represent the number of discrete outcomes, such as the number of defective parts in a shipment, the number of lost-time accidents, and so forth. Measurement data are treated as a continuum. For example, the tensile strength of a synthetic yarn theoretically could be measured to any degree of precision. A subtle aspect associated with count and measurement data is that some types of count data can be dealt with through the application of techniques which have been developed for measurement data alone. This ability is due to the fact that some simplified measurement statistics serve as an excellent approximation for the more tedious count statistics. Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes would tend to be different, with heads and tails occurring randomly over a period of time. Given a long strand of synthetic fiber, the tensile strength of successive samples would tend to vary significantly from sample to sample. Counts and measurements are characterized as random variables, that is, observations which are susceptible to chance. Virtually all quantitative data are susceptible to chance in one way or another. Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probability, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value 1 for a head and 0 for a tail. Given a fair coin, the probability of observing a head on a toss would be a .5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as x

P(x)

0 1

.5 .5

where P(x) stands for what is called a probability function. This term is reserved for count data, in that probabilities can be defined for particular outcomes. The probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. For measurement data which are considered continuous, the term probability density is used. For example, consider a spinner wheel which conceptually can be thought of as being marked off on the circumference infinitely precisely from 0 up to, but not including, 1. In spinning the wheel, the probability of the wheel’s stopping at a specified marking point at any particular x value, where 0 ≤ x < 1, is zero, for example, stopping at the value x =  .5. For the spinning wheel, the probability density function would be defined by f(x) = 1 for 0 ≤ x < 1. Graphically, this is shown in Fig. 3-63. The relative-probability concept refers to the fact that density reflects the relative likelihood of occurrence; in this case, each number between 0 and 1 is equally likely. For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. Example For the density of Fig. 3-63 the P[0 ≤ x ≤ .4] = .4

3-71

Density function.

FIG. 3-63

P[.2 ≤ x ≤ .9] = .7 P[.6 ≤ x < 1] = .4 and so forth. Since the probability associated with any particular point value is zero, it makes no difference whether the limit point is defined by a closed interval (≤ or ≥) or an open interval (< or >).

Many different types of models are used as the foundation for statistical analysis. These models are also referred to as populations. Parameters As a way of characterizing probability functions and densities, certain types of quantities called parameters can be defined. For example, the center of gravity of the distribution is defined to be the population mean, which is designated as µ. For the coin toss µ = .5, which corresponds to the average value of x; i.e., for half of the time x will take on a value 0 and for the other half a value 1. The average would be .5. For the spinning wheel, the average value would also be .5. Another parameter is called the standard deviation, which is designated as σ. The square of the standard deviation is used frequently and is called the popular variance, σ2. Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean µ. If the spread is broad, then the standard deviation will be larger than if it were more constrained. For specified probability and density functions, the respective means and variances are defined by the following: Probability functions

Probability density functions

 x f(x) dx

E(x) = µ = x P(x)

E(x) = µ =

Var(x) = σ2 = (x − µ)2 P(x)

Var(x) = σ2 =

x

x

x

 (x − µ) f(x) dx 2

x

where E(x) is defined to be the expected or average value of x. Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as x, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like µ and σ are not random variables; they are fixed constants. Example In an experiment, six random numbers (rounded to four decimal places) were observed from the uniform distribution f(x) = 1 for 0 ≤ x < 1: .1009 .3754 .0842 .9901 .1280 .6606 The sample mean corresponds to the arithmetic average of the observations, which will be designated as x1 through x6, where 1 n x =  xi with n = 6, n i=1

x⎯ = 0.3899

The sample standard deviation s is defined by the computation s=

(xi − x)2  n−1



3-72

MATHEMATICS =





n x 2i − ( xi)2  n(n − 1)

(3-118)

In effect, this represents the root of a statistical average of the squares. The divisor quantity (n − 1) will be referred to as the degrees of freedom. The sample value of the standard deviation for the data given is .3686. The value of n  1 is used in the denominator because the deviations from the sample average must total zero, or

)(xi  x– ) 0 Thus knowing n − 1 values of xi  x– permits calculation of the nth value of xi  x– . The sample mean and sample standard deviation are obtained by using Microsoft Excel with the commands AVERAGE(B2:B7) and STDEV(B2:B7) when the observations are in cells B2 to B7.

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of “average” of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s2 have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. For the corresponding population, the parameter values are µ = .50 and σ = .2887, which are obtained by calculating the integrals defined above with f(x) = 1 and 0 ≤ x ≤ 1. If, instead of using individual observations only, averages of six were reported, then the corresponding population parameter values would be µ = .50 and σx = σ/6 = .1179.  The corresponding variance for an average will be written occasionally as Var (x) = var (x)/n. In effect, the variance of an average is inversely proportional to the sample size n, which reflects the fact that sample averages will tend to cluster about µ much more closely than individual observations. This is illustrated in greater detail under “Measurement Data and Sampling Densities.” Characterization of Chance Occurrences To deal with a broad area of statistical applications, it is necessary to characterize the way in which random variables will vary by chance alone. The basic foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x − µ)/σ, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x − µ)/(σ/ n). A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. Also, in many applications involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines. The normal probability distribution function can be obtained in Microsoft Excel by using the NORMDIST function and supplying the desired mean and standard deviation. The cumulative value can also be determined. In MATLAB, the corresponding command is randn. ENUMERATION DATA AND PROBABILITY DISTRIBUTIONS Introduction Many types of statistical applications are characterized by enumeration data in the form of counts. Examples are the number of lost-time accidents in a plant, the number of defective items in a sample, and the number of items in a sample that fall within several specified categories. The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately

interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections. Binomial Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials x = number of successes in n trials p = probability of observing a success on any one trial pˆ = x/n, the proportion of successes in n trials Probability Law x n x P(x) = P  = p (1 − p)n − x x = 0, 1, 2, . . . , n x n

 

where

nx =  x!(n − x)! n!

Properties E(x) = np E( p ˆ)=p

Var(x) = np(1 − p) Var( p ˆ ) = p(1 − p)/n

Geometric Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing the first success on the xth trial is defined by the geometric distribution. Nomenclature p = probability of observing a success on any one trial x = the number of trials to obtain the first success Probability Law Properties

P(x) = p(1 − p)x − 1 E(x) = 1/p

x = 1, 2, 3, . . .

Var (x) = (1 − p)/p2

Poisson Probability Distribution Nature In monitoring a moving threadline, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadline by a broken-filament detector mounted adjacent to the threadline. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include: 1. The number of cars (arrivals) that pass a point on a high-speed highway between 10:00 and 11:00 A.M. on Wednesdays 2. The number of customers arriving at a bank between 10:00 and 10:10 A.M. 3. The number of telephone calls received through a switchboard between 9:00 and 10:00 A.M. 4. The number of insurance claims that are filed each week 5. The number of spinning machines that break down during 1 day at a large plant. Nomenclature x = total number of arrivals in a total length L or total period T a = average rate of arrivals for a unit length or unit time λ = aL = expected or average number of arrivals for the total length L λ = aT = expected or average number of arrivals for the total time T Probability Law Given that a is constant for the total length L or period T, the probability of observing x arrivals in some period L or T

STATISTICS is given by λx P(x) =  e−λ x! Properties E(x) = λ

Example A bin contains 300 items, of which 240 are good and 60 are defective. In a sample of 6 what is the probability of selecting 4 good and 2 defective items by chance?

x = 0, 1, 2, . . .

240 60  4  2 P(x) = = .2478  3006

Var (x) = λ

Example The number of broken filaments in a threadline has been averaging .015 per yard. What is the probability of observing exactly two broken filaments in the next 100 yd? In this example, a = .015/yd and L = 100 yd; therefore λ = (.015)(100) = 1.5: (1.5)2 P(x = 2) =  e−1.5 = .2510 2!

Example A commercial item is sold in a retail outlet as a unit product. In the past, sales have averaged 10 units per month with no seasonal variation. The retail outlet must order replacement items 2 months in advance. If the outlet starts the next 2-month period with 25 items on hand, what is the probability that it will stock out before the end of the second month? Given a = 10/month, then λ = 10 × 2 = 20 for the total period of 2 months: ∞

25

26

0

P(x ≥ 26) = P(x) = 1 − P(x)



20x −20 −20 2025 20 202 e =e 1 +  +  + ⋅⋅⋅ + 

0  x! 1 2! 25! 25



= .887815 Therefore P(x ≥ 26) = .112185 or roughly an 11 percent chance of a stockout.

Hypergeometric Probability Distribution Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probability of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. Nomenclature N = total group size n = sample group size X = number of items in the total group with a specified attribute A N − X = number of items in the total group with the other attribute B x = number of items in the sample with a specified attribute A n − x = number of items in the sample with the other attribute B

Category A Category B Total

3-73

Population

Sample

X N−X N

x n−x n

Multinomial Distribution Nature For an experiment in which successive outcomes can be classified into two or more categories and the probabilities associated with the respective outcomes remain constant, then the experiment can be characterized through the multinomial distribution. Nomenclature n = total number of trials k = total number of distinct categories pj = probability of observing category j on any one trial, j = 1, 2, . . . , k xj = total number of occurrences in category j in n trials Probability Law n! P(x1, x2, . . . , xk) =  p1x1 p2x2 ⋅ ⋅ ⋅ pkx k x1!x2! . . . xk! Example In tossing a die 12 times, what is the probability that each face value will occur exactly twice? 12! 1 2 1 2 1 2 1 2 1 2 1 2 p(2, 2, 2, 2, 2, 2) =        = .003438 2!2!2!2!2!2! 6 6 6 6 6 6

     

To compute these probabilities in Microsoft Excel, put the value of x in cell B2, say, and use the functions Binomial distribution: = BINOMDIST(B2, n,p,0) Poisson distribution: = POISSON(B2,*,0) Hypergeometric distribution: = HYPGEOMDIST(B2,n,X,N) To generate a table of values with these probability distributions, in Microsoft Excel use the following functions: Bernoulli random values: = CRITBINOM(1,p,RAND()) Binomial random values: = CRITBINOM(n,p,RAND()) The factorial function is FACT(n) in Microsoft Excel and factorial(n) in MATLAB. Be sure that x is an integer.

MEASUREMENT DATA AND SAMPLING DENSITIES Introduction The following example data are used throughout this subsection to illustrate concepts. Consider, for the purpose of illustration, that five synthetic-yarn samples have been selected randomly from a production line and tested for tensile strength on each of 20 production days. For this, assume that each group of five corresponds to a day, Monday through Friday, for a period of 4 weeks:

Probability Law P(x) =

 n − x  x  n N−X

X

N

nX E(x) =  N N−n var (x) = nP(1 − P)  N−1 Example What is the probability that an appointed special committee of 4 has no female members when the members are randomly selected from a candidate group of 10 males and 7 females?

 4  0 = .0882 P(x = 0) = 10

7

4 17

Monday 1

Tuesday 2

Wednesday 3

Thursday 4

Friday 5

Groups of 25 pooled

36.48 35.33 35.92 32.28 31.61 x = 34.32 s = 2.22

38.06 31.86 33.81 30.30 35.27 33.86 3.01

35.28 36.58 38.81 33.31 33.88 35.57 2.22

36.34 36.25 30.46 37.37 37.52 35.59 2.92

36.73 37.17 33.07 34.27 36.94 35.64 1.85

35.00 2.40

6

7

8

9

10

38.67 32.08 33.79 32.85 35.22 x = 34.52 s = 2.60

36.62 33.05 35.43 36.63 31.46 34.64 2.30

35.03 36.22 32.71 32.52 27.23 32.74 3.46

35.80 33.16 35.19 32.91 35.44 34.50 1.36

36.82 36.49 32.83 32.43 34.16 34.54 2.03

34.19 2.35

3-74

MATHEMATICS

11

12

13

14

15

39.63 34.38 36.51 30.00 39.64 x = 36.03 s = 4.04

34.52 37.39 34.16 35.76 37.63 35.89 1.59

36.05 35.36 35.00 33.61 36.98 35.40 1.25

36.64 31.18 36.13 37.51 39.05 36.10 2.96

31.57 36.21 33.84 35.01 34.95 34.32 1.75

35.55 2.42

Monday 16

Tuesday 17

Wednesday 18

Thursday 19

Friday 20

Groups of 25 pooled

37.68 36.38 38.43 39.07 33.06 x = 36.92 s = 2.38

35.97 35.92 36.51 33.89 36.01 35.66 1.02

33.71 32.34 33.29 32.81 37.13 33.86 1.90

35.61 37.13 31.37 35.89 36.33 35.27 2.25

36.65 37.91 42.18 39.25 33.32 37.86 3.27

35.91 2.52

Pooled sample of 100: x = 35.16

s = 2.47

Even if the process were at steady state, tensile strength, a key property would still reflect some variation. Steady state, or stable operation of any process, has associated with it a characteristic variation. Superimposed on this is the testing method, which is itself a process with its own characteristic variation. The observed variation is a composite of these two variations. Assume that the table represents “typical” production-line performance. The numbers themselves have been generated on a computer and represent random observations from a population with µ = 35 and a population standard deviation σ = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supervisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. An objective of statistical analysis is to serve as a guide in decision making in the context of normal variation. In the case of the production supervisor, it is to make a decision, with a high probability of being correct, that something has in fact changed the operation. Suppose that an engineering change has been made in the process and five new tensile samples have been tested with the results: 36.81 38.34 34.87 39.58 36.12

x = 37.14 s = 1.85

In this situation, management would inquire whether the product has been improved by increased tensile strength. To answer this question, in addition to a variety of analogous questions, it is necessary to have some type of scientific basis upon which to draw a conclusion. A scientific basis for the evaluation and interpretation of data is contained in the accompanying table descriptions. These tables characterize the way in which sample values will vary by chance alone in the context of individual observations, averages, and variances. Table number

Designated symbol

Variable

Example What proportion of tensile values will fall between 34 and 36? z1 = (34 − 35)/2.45 = −.41

3-5

z

3-5

z

x − µ  σ/  n

Averages

3-6

t

x − µ  s/ n

Averages when σ is unknown*

3-7

χ2

(s2/σ2)(df)

3-8

F

s12 /s 22

Observations*

z2 = (36 − 35)/2.45 = .41

P[−.41 ≤ z ≤ .41] = .3182, or roughly 32 percent The value 0.3182 is interpolated from Table 3-4 using z = 0.40, A = 0.3108, and z = 0.45, A = 0.3473.

Example What midrange of tensile values will include 95 percent of the sample values? Since P[−1.96 ≤ z ≤ 1.96] = .95, the corresponding values of x are x1 = 35 − 1.96(2.45) = 30.2 x2 = 35 + 1.96(2.45) = 39.8 or

P[30.2 ≤ x ≤ 39.8] = .95

Normal Distribution of Averages An examination of the tensilestrength data previously tabulated would show that the range (largest minus the smallest) of tensile strength within days averages 5.12. The average range in x values within each week is 2.37, while the range in the four weekly averages is 1.72. This reflects the fact that averages tend to be less variable in a predictable way. Given that the variance of x is var (x) = σ2, then the variance of x based on n observations is var (x) = σ2/n. For averages of n observations, the corresponding relationship for the Z-scale relationship is σ z = (x − µ) /σ/ n or z x = µ +   n The Microsoft Excel function NORMDIST(X, , , 1) gives the probability that x ≤ µ. The command CONFIDENCE(, , n) gives the confidence interval about the mean for a sample size n. To obtain 95 percent confidence limits, use  = .025; 2 = 1 – A.

Sampling distribution of

x−µ  σ

*When sampling from a gaussian distribution.

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve; this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very special properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. Basically, the tabled values represent area (proportions or probability) associated with a scaling variable designated by Z in Fig. 3-64. The normal curve is centered at 0, and for particular values of Z, designated as z, the tabulated numbers represent the corresponding area under the curve between 0 and z. For example, between 0 and 1 the area is .3413. (Get this number from Table 3-5. The value of A includes the area on both sides of zero. Thus we want A/2. For z = 1, A = 0.6827, A/2 = 0.3413. For z = 2, A/2 = 0.4772.) The area between 0 and 2 is .4772; therefore, the area between 1 and 2 is .4772 − .3413 = .1359. Also, since the normal curve is symmetric, areas to the left can be determined in exactly the same way. For example, the area between −2 and +1 would include the area between −2 and 0, .4772 (the same as 0 to 2), plus the area between 0 and 1, .3413, or a total area of .8185. Any types of observation which are applicable to the normal curve can be transformed to Z values by the relationship z = (x − µ)/σ and, conversely, Z values to x values by x = µ + σz, as shown in Fig. 3-64. For example, for tensile strength, with µ = 35 and σ = 2.45, this would dictate z = (x − 35)/2.45 and x = 35 + 2.45z.

Variances* Ratio of two independent sample variances* FIG. 3-64

Transformation of z values.

STATISTICS

3-75

TABLE 3-5 Ordinates and Areas between Abscissa Values -z and +z of the Normal Distribution Curve A = area between µ − σz and µ + σz z

X

Y

A

1−A

z

X

0  .10  .20

µ µ  .10σ µ  .20σ

0.399 .397 .391

0.0000 .0797 .1585

1.0000 .9203 .8415

1.50 1.60 1.70

µ  1.50σ µ  1.60σ µ  1.70σ

0.1295 .1109 .0940

0.8664 .8904 .9109

0.1336 .1096 .0891

 .30  .40

µ  .30σ µ  .40σ

.381 .368

.2358 .3108

.7642 .6892

1.80 1.90

µ  1.80σ µ  1.90σ

.0790 .0656

.9281 .9446

.0719 .0574

 .50  .60  .70

µ  .50σ µ  .60σ µ  .70σ

.352 .333 .312

.3829 .4515 .5161

.6171 .5485 .4839

2.00 2.10 2.20

µ  2.00σ µ  2.10σ µ  2.20σ

.0540 .0440 .0335

.9545 .9643 .9722

.0455 .0357 .0278

 .80  .90

µ  .80σ µ  .90σ

.290 .266

.5763 .6319

.4237 .3681

2.30 2.40

µ  2.30σ µ  2.40σ

.0283 .0224

.9786 .9836

.0214 .0164

1.00 1.10 1.20

µ  1.00σ µ  1.10σ µ  1.20σ

.242 .218 .194

.6827 .7287 .7699

.3173 .2713 .2301

2.50 2.60 2.70

µ  2.50σ µ  2.60σ µ  2.70σ

.0175 .0136 .0104

.9876 .9907 .9931

.0124 .0093 .0069

1.30 1.40

µ  1.30σ µ  1.40σ

.171 .150

.8064 .8385

.1936 .1615

2.80 2.90

µ  2.80σ µ  2.90σ

.0079 .0060

.9949 .9963

.0051 .0037

1.50

µ  1.50σ

.130

.8664

.1336

3.00 4.00 5.00

µ  3.00σ µ  4.00σ µ  5.00s

.0044 .0001 .000001

.9973 .99994 .9999994

.0027 .00006 .0000006

0.000  .126  .253  .385  .524  .674  .842

µ µ  0.126σ µ  .253σ µ  .385σ µ  .524σ µ  .674σ µ  .842σ

0.3989 .3958 .3863 .3704 .3477 .3178 .2800

.0000 .1000 .2000 .3000 .4000 .5000 .6000

1.0000 0.9000 .8000 .7000 .6000 .5000 .4000

1.036 1.282 1.645 1.960 2.576 3.291 3.891

µ  1.036σ µ  1.282σ µ  1.645σ µ  1.960σ µ  2.576σ µ  3.291σ µ  3.891σ

Y

0.2331 .1755 .1031 .0584 .0145 .0018 .0002

A

0.7000 .8000 .9000 .9500 .9900 .9990 .9999

1−A

0.3000 .2000 .1000 .0500 .0100 .0010 .0001

This table is obtained in Microsoft Excel with the function Y NORMDIST(X, , , 0). If X > , A = 2 NORMDIST(X, , , 1) – 1. If X < , A = 1 – 2 NORMDIST(X, , , 1)

Example What proportion of daily tensile averages will fall between 34 and 36? Using Table 3-5 z1 = (34 − 35)/(2.45/5) = −.91

z2 = (36 − 35)/(2.45/5) = .91

P[−.91 ≤ z ≤ .91] = .637, or roughly 64 percent Using Microsoft Excel, the more precise answer is P[34 ≤ x ≤ 36] 2 NORMDIST(36, 35, 2.455 , 1)  1 ,1) .6386

1  2 NORMDIST(34, 35, 2.455

Example What midrange of daily tensile averages will include 95 percent of the sample values? Using Table 3-5 x1 = 35 − 1.96(2.45/5) = 32.85 x2 = 35 + 1.96(2.45/5) = 37.15 P[32.85 ≤ x ≤ 37.15] = .95 or using Microsoft Excel, 35 ± CONFIDENCE(.05, 2.45, 5) = 35 ± 2.15 t Distribution of Averages

The normal curve relies on a knowledge of σ, or in special cases, when it is unknown, s can be used with the normal curve as an approximation when n > 30. For example, with n > 30 the intervals x ± s and x ± 2s will include roughly 68 and 95 percent of the sample values respectively when the distribution is normal. In applications sample sizes are usually small and σ unknown. In these cases, the t distribution can be used where t = (x − µ)/(s/n )

or

x = µ + ts/n 

The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual observations x are not normally distributed, sample averages of x have distributions which tend toward normality as n gets large. Even for small n of 5

through 10, the approximation is usually relatively accurate. It is sometimes called the Student’s t distribution. In reference to the tensile-strength table, consider the summary statistics x and s by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution of t values summarized in Table 3-6. Since the t distribution relies on the sample standard deviation s, the resultant distribution will differ according to the sample size n. To designate this difference, the respective distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus 1. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). The Microsoft Excel function TDIST(X, df,1) gives the right-tail probability, and TDIST(X, df, 2) gives twice that. The probability that t ≤ X is 1 – TDIST(X, df, 1) when X ≥ 0 and TDIST(abs(X),df,1) when X < 0. The probability that –X ≤ t ≤ + X is 1–TDIST(X, df, 2). Example For a sample size n = 5, what values of t define a midarea of 90 percent? For 4 df the tabled value of t corresponding to a midarea of 90 percent is 2.132; i.e., P[−2.132 ≤ t ≤ 2.132] = .90. Using Microsoft Excel, TINV(.1, 4) = 2.132. Example For a sample size n = 25, what values of t define a midarea of 95 percent? For 24 df the tabled value of t corresponding to a midarea of 95 percent is 2.064; i.e., P[−2.064 ≤ t ≤ 2.064] = .95. Example Also, 1 – TDIST(2.064, 24, 2) = .9500. Using Microsoft Excel, TINV(.05, 24) = 2.064.

3-76

MATHEMATICS

TABLE 3-6

Values of t

t.40

t.30

t.20

t.10

t.05

t.025

t.01

t.005

1 2 3 4 5

0.325 .289 .277 .271 .267

0.727 .617 .584 .569 .559

1.376 1.061 0.978 .941 .920

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

6 7 8 9 10

.265 .263 .262 .261 .260

.553 .549 .546 .543 .542

.906 .896 .889 .883 .879

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

11 12 13 14 15

.260 .259 .259 .258 .258

.540 .539 .538 .537 .536

.876 .873 .870 .868 .866

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

16 17 18 19 20

.258 .257 .257 .257 .257

.535 .534 .534 .533 .533

.865 .863 .862 .861 .860

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

21 22 23 24 25

.257 .256 .256 .256 .256

.532 .532 .532 .531 .531

.859 .858 .858 .857 .856

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

26 27 28 29 30

.256 .256 .256 .256 .256

.531 .531 .530 .530 .530

.856 .855 .855 .854 .854

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

40 60 120 ∞

.255 .254 .254 .253

.529 .527 .526 .524

.851 .848 .845 .842

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

df

Above values refer to a single tail outside the indicated limit of t. For example, for 95 percent of the area to be between −t and +t in a two-tailed t distribution, use the values for t0.025 or 2.5 percent for each tail. This table is obtained in Mircrosoft Excel using the function TINV(,df), where α is .05.

(a + b)2 df =  a2/(n1 − 1) + b2/(n2 − 1)

and

Chi-Square Distribution For some industrial applications, product uniformity is of primary importance. The sample standard deviation s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where χ 2 = (s2/σ2) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. The probability distribution function is −(df )2 p(y) = y0ydf − 2 exp  2





where y0 is chosen such that the integral of p(y) over all y is one. In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures could be computed. The corresponding df would be 4, 24, and 99 respectively. These numbers would represent sample values from the respective distributions which are summarized in Table 3-7. In a manner similar to the use of the t distribution, chi square can be interpreted in a direct probabilistic sense corresponding to a midarea of (1 − α): P[χ 21 ≤ (s2/σ2)(df) ≤ χ 22 ] = 1 − α where χ corresponds to a lower-tail area of α/2 and χ 22 an upper-tail area of α/2. The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance σ2. When the population is not normal but skewed, chi-square probabilities could be substantially in error. 2 1

Example On the basis of a sample size n = 5, what midrange of values will include the sample ratio s/σ with a probability of 95 percent? Use Table 3-7 for 4 df and read χ 21 = 0.484 for a lower tail area of 0.05/2, 2.5 percent, and read χ 22 = 11.1 for an upper tail area of 97.5 percent. P[.484 ≤ (s2/σ2)(4) ≤ 11.1] = .95 or

P[.35 ≤ s/σ ≤ 1.66] = .95

The Microsoft Excel functions CHIINV(.025, 4) and CHIINV(.975, 4) give the same values.

Example On the basis of a sample size n = 25, what midrange of values will include the sample ratio s/σ with a probability of 95 percent? Example What is the sample value of t for the first day of tensile data? Sample t = (34.32 − 35)/(2.22/5) = −.68 Note that on the average 90 percent of all such sample values would be expected to fall within the interval 2.132.

t Distribution for the Difference in Two Sample Means with Equal Variances The t distribution can be readily extended to the difference in two sample means when the respective populations have the same variance σ: (x1 − x2) − (µ1 − µ2) t =  sp1 /n +1 /n2 1 

(3-119)

where s 2p is a pooled variance defined by (n1 − 1)s12 + (n2 − 1)s22 s 2p =  (n1 − 1) + (n2 − 1)

(3-120)

In this application, the t distribution has (n1 + n2 − 2) df. t Distribution for the Difference in Two Sample Means with Unequal Variances When population variances are unequal, an approximate t quantity can be used: (x1 − x2) − (µ1 − µ2) t =  a + b  with

a = s12 /n1

b = s22 /n2

P[12.4 ≤ (s2/σ2)(24) ≤ 39.4] = .95 or

P[.72 ≤ s/σ ≤ 1.28] = .95

This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 95 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor.

The chi-square distribution can be applied to other types of application which are of an entirely different nature. These include applications which are discussed under “Goodness-of-Fit Test” and “Two-Way Test for Independence of Count Data.” In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity F(γ1, γ 2) = s12 /s22 where γ1 and γ2 correspond to the respective df’s for the sample variances. In statistical applications, it turns out that the primary area of interest is found when the ratios are greater than 1. For this reason, most tabled values are defined for an upper-tail area. However,

TABLE 3-7

STATISTICS

3-77

99

99.5

Percentiles of the c2 Distribution Percent

df

10

90

95

97.5

0.000039 .0100 .0717 .207 .412

0.00016 .0201 .115 .297 .554

0.00098 .0506 .216 .484 .831

0.0039 .1026 .352 .711 1.15

0.0158 .2107 .584 1.064 1.61

2.71 4.61 6.25 7.78 9.24

3.84 5.99 7.81 9.49 11.07

5.02 7.38 9.35 11.14 12.83

6.63 9.21 11.34 13.28 15.09

7.88 10.60 12.84 14.86 16.75

6 7 8 9 10

.676 .989 1.34 1.73 2.16

.872 1.24 1.65 2.09 2.56

1.24 1.69 2.18 2.70 3.25

1.64 2.17 2.73 3.33 3.94

2.20 2.83 3.49 4.17 4.87

10.64 12.02 13.36 14.68 15.99

12.59 14.07 15.51 16.92 18.31

14.45 16.01 17.53 19.02 20.48

16.81 18.48 20.09 21.67 23.21

18.55 20.28 21.96 23.59 25.19

11 12 13 14 15

2.60 3.07 3.57 4.07 4.60

3.05 3.57 4.11 4.66 5.23

3.82 4.40 5.01 5.63 6.26

4.57 5.23 5.89 6.57 7.26

5.58 6.30 7.04 7.79 8.55

17.28 18.55 19.81 21.06 22.31

19.68 21.03 22.36 23.68 25.00

21.92 23.34 24.74 26.12 27.49

24.73 26.22 27.69 29.14 30.58

26.76 28.30 29.82 31.32 32.80

16 18 20 24 30

5.14 6.26 7.43 9.89 13.79

5.81 7.01 8.26 10.86 14.95

6.91 8.23 9.59 12.40 16.79

7.96 9.39 10.85 13.85 18.49

9.31 10.86 12.44 15.66 20.60

23.54 25.99 28.41 33.20 40.26

26.30 28.87 31.41 36.42 43.77

28.85 31.53 34.17 39.36 46.98

32.00 34.81 37.57 42.98 50.89

34.27 37.16 40.00 45.56 53.67

40 60 120

20.71 35.53 83.85

22.16 37.48 86.92

24.43 40.48 91.58

26.51 43.19 95.70

29.05 46.46 100.62

51.81 74.40 140.23

55.76 79.08 146.57

59.34 83.30 152.21

63.69 88.38 158.95

66.77 91.95 163.64

1 2 3 4 5

0.5

1

2.5

5

For large values of degrees of freedom the approximate formula



2 χ 2a = n 1 −  + za 9n

2  9n

3

where za is the normal deviate and n is the number of degrees of freedom, may be used. For example, χ.299 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 60(1.1379)3 = 88.4 for the 99th percentile for 60 degrees of freedom. The Microsoft Excel function CHIDIST(X, df), where X is the table value, gives 1 – Percent. The function CHIINV (1– Percent, df) gives the table value.

defining F2 to be that value corresponding to an upper-tail area of α/2, then F1 for a lower-tail area of α/2 can be determined through the identity F1(γ1, γ 2) = 1/F2(γ 2, γ1) The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. The Microsoft Excel function FDIST(X, df1, df2) gives the upper percent points of Table 3-8, where X is the tabular value. The function FINV(Percent, df1, df2) gives the table value.

µ. The magnitude of µ can be only large enough to retain the t quantity above −2.132 and small enough to retain the t quantity below +2.132. This can be found by rearranging the quantities within the bracket; i.e., P[35.38 ≤ µ ≤ 38.90] = .90. This states that we are 90 percent sure that the interval from 35.38 to 38.90 includes the unknown parameter µ. In general, s s P x − t  ≤ µ ≤ x + t  = 1 − α  n  n

Example For two sample variances with 4 df each, what limits will bracket their ratio with a midarea probability of 90 percent? Use Table 3-8 with 4 df in the numerator and denominator and upper 5 percent points (to get both sides totaling 10 percent). The entry is 6.39. Thus:

where t is defined for an upper-tail area of α/2 with (n − 1) df. In this application, the interval limits (x + t s/n ) are random variables which will cover the unknown parameter µ with probability (1 − α). The converse, that we are 100(1 − α) percent sure that the parameter value is within the interval, is not correct. This statement defines a probability for the parameter rather than the probability for the interval.

P[1/6.39 ≤ s 12 /s 22 ≤ 6.39] = .90 or

P[.40 ≤ s1 /s2 ≤ 2.53] = .90

The Microsoft Excel function FINV(.05, 4, 4) gives 6.39, too.

Confidence Interval for a Mean For the daily sample tensilestrength data with 4 df it is known that P[−2.132 ≤ t ≤ 2.132] = .90. This states that 90 percent of all samples will have sample t values which fall within the specified limits. In fact, for the 20 daily samples exactly 16 do fall within the specified limits (note that the binomial with n = 20 and p = .90 would describe the likelihood of exactly none through 20 falling within the prescribed limits—the sample of 20 is only a sample). Consider the new daily sample (with n = 5, x = 37.14, and s = 1.85) which was observed after a process change. In this case, the same probability holds. However, in this instance the sample value of t cannot be computed, since the new µ, under the process change, is not known. Therefore P[−2.132 ≤ (37.14 − µ)/(1.85/5) ≤ 2.132] = .90. In effect, this identity limits the magnitude of possible values for





Example What values of t define the midarea of 95 percent for weekly samples of size 25, and what is the sample value of t for the second week? P[−2.064 ≤ t ≤ 2.064] = .95 (34.19 − 35)/(2.35/25) = 1.72.

and

Example For the composite sample of 100 tensile strengths, what is the 90 percent confidence interval for µ? Use Table 3-6 for t.05 with df ≈ ∞.





2.47 2.47 P 35.16 − 1.645  < µ < 35.16 + 1.645  = .90  100  100 or

P[34.75 ≤ µ ≤ 35.57] = .90

Confidence Interval for the Difference in Two Population Means The confidence interval for a mean can be extended to

3-78

MATHEMATICS

TABLE 3-8 F Distribution Upper 5% Points (F.95)

Degrees of freedom for denominator

Degrees of freedom for numerator 1

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

60

120

∞

1 2 3 4 5

161 18.5 10.1 7.71 6.61

200 19.0 9.55 6.94 5.79

216 19.2 9.28 6.59 5.41

225 19.2 9.12 6.39 5.19

230 19.3 9.01 6.26 5.05

234 19.3 8.94 6.16 4.95

237 19.4 8.89 6.09 4.88

239 19.4 8.85 6.04 4.82

241 19.4 8.81 6.00 4.77

242 19.4 8.79 5.96 4.74

244 19.4 8.74 5.91 4.68

246 19.4 8.70 5.86 4.62

248 19.4 8.66 5.80 4.56

249 19.5 8.64 5.77 4.53

250 19.5 8.62 5.75 4.50

251 19.5 8.59 5.72 4.46

252 19.5 8.57 5.69 4.43

253 19.5 8.55 5.66 4.40

254 19.5 8.53 5.63 4.37

6 7 8 9 10

5.99 5.59 5.32 5.12 4.96

5.14 4.74 4.46 4.26 4.10

4.76 4.35 4.07 3.86 3.71

4.53 4.12 3.84 3.63 3.48

4.39 3.97 3.69 3.48 3.33

4.28 3.87 3.58 3.37 3.22

4.21 3.79 3.50 3.29 3.14

4.15 3.73 3.44 3.23 3.07

4.10 3.68 3.39 3.18 3.02

4.06 3.64 3.35 3.14 2.98

4.00 3.57 3.28 3.07 2.91

3.94 3.51 3.22 3.01 2.85

3.87 3.44 3.15 2.94 2.77

3.84 3.41 3.12 2.90 2.74

3.81 3.38 3.08 2.86 2.70

3.77 3.34 3.04 2.83 2.66

3.74 3.30 3.01 2.79 2.62

3.70 3.27 2.97 2.75 2.58

3.67 3.23 2.93 2.71 2.54

11 12 13 14 15

4.84 4.75 4.67 4.60 4.54

3.98 3.89 3.81 3.74 3.68

3.59 3.49 3.41 3.34 3.29

3.36 3.26 3.18 3.11 3.06

3.20 3.11 3.03 2.96 2.90

3.09 3.00 2.92 2.85 2.79

3.01 2.91 2.83 2.76 2.71

2.95 2.85 2.77 2.70 2.64

2.90 2.80 2.71 2.65 2.59

2.85 2.75 2.67 2.60 2.54

2.79 2.69 2.60 2.53 2.48

2.72 2.62 2.53 2.46 2.40

2.65 2.54 2.46 2.39 2.33

2.61 2.51 2.42 2.35 2.29

2.57 2.47 2.38 2.31 2.25

2.53 2.43 2.34 2.27 2.20

2.49 2.38 2.30 2.22 2.16

2.45 2.34 2.25 2.18 2.11

2.40 2.30 2.21 2.13 2.07

16 17 18 19 20

4.49 4.45 4.41 4.38 4.35

3.63 3.59 3.55 3.52 3.49

3.24 3.20 3.16 3.13 3.10

3.01 2.96 2.93 2.90 2.87

2.85 2.81 2.77 2.74 2.71

2.74 2.70 2.66 2.63 2.60

2.66 2.61 2.58 2.54 2.51

2.59 2.55 2.51 2.48 2.45

2.54 2.49 2.46 2.42 2.39

2.49 2.45 2.41 2.38 2.35

2.42 2.38 2.34 2.31 2.28

2.35 2.31 2.27 2.23 2.20

2.28 2.23 2.19 2.16 2.12

2.24 2.19 2.15 2.11 2.08

2.19 2.15 2.11 2.07 2.04

2.15 2.10 2.06 2.03 1.99

2.11 2.06 2.02 1.98 1.95

2.06 2.01 1.97 1.93 1.90

2.01 1.96 1.92 1.88 1.84

21 22 23 24 25

4.32 4.30 4.28 4.26 4.24

3.47 3.44 3.42 3.40 3.39

3.07 3.05 3.03 3.01 2.99

2.84 2.82 2.80 2.78 2.76

2.68 2.66 2.64 2.62 2.60

2.57 2.55 2.53 2.51 2.49

2.49 2.46 2.44 2.42 2.40

2.42 2.40 2.37 2.36 2.34

2.37 2.34 2.32 2.30 2.28

2.32 2.30 2.27 2.25 2.24

2.25 2.23 2.20 2.18 2.16

2.18 2.15 2.13 2.11 2.09

2.10 2.07 2.05 2.03 2.01

2.05 2.03 2.01 1.98 1.96

2.01 1.98 1.96 1.94 1.92

1.96 1.94 1.91 1.89 1.87

1.92 1.89 1.86 1.84 1.82

1.87 1.84 1.81 1.79 1.77

1.81 1.78 1.76 1.73 1.71

30 40 60 120 ∞

4.17 4.08 4.00 3.92 3.84

3.32 3.23 3.15 3.07 3.00

2.92 2.84 2.76 2.68 2.60

2.69 2.61 2.53 2.45 2.37

2.53 2.45 2.37 2.29 2.21

2.42 2.34 2.25 2.18 2.10

2.33 2.25 2.17 2.09 2.01

2.27 2.18 2.10 2.02 1.94

2.21 2.12 2.04 1.96 1.88

2.16 2.08 1.99 1.91 1.83

2.09 2.00 1.92 1.83 1.75

2.01 1.92 1.84 1.75 1.67

1.93 1.84 1.75 1.66 1.57

1.89 1.79 1.70 1.61 1.52

1.84 1.74 1.65 1.55 1.46

1.79 1.69 1.59 1.50 1.39

1.74 1.64 1.53 1.43 1.32

1.68 1.58 1.47 1.35 1.22

1.62 1.51 1.39 1.25 1.00

Interpolation should be performed using reciprocals of the degrees of freedom.

include the difference between two population means. This interval is based on the assumption that the respective populations have the same variance σ2:

Example For the first week of tensile-strength samples compute the 90 percent confidence interval for σ2 (df = 24, corresponding to n = 25, using 5 percent and 95 percent in Table 3-7):

1+ 1/n 2 ≤ µ1 − µ2 ≤ (x1 − x2) + tsp1/n 1+  2 (x1 − x2) − tsp1/n 1/n

24(2.40)2 24(2.40)2  ≤ σ2 ≤  36.4 13.8

Example Compute the 95 percent confidence interval based on the original 100-point sample and the subsequent 5-point sample: 99(2.47)2 + 4(1.85)2 s p2 =  = 5.997 103

3.80 ≤ σ2 ≤ 10.02

or

sp = 2.45

With 103 df and α = .05, t = 1.96 using t.025 in Table 3-6. Therefore /100 + 1 /5 = −1.98  2.20 (35.16 − 37.14)  1.96(2.45) 1 or

−4.18 ≤ (µ1 − µ2) ≤ .22

Note that if the respective samples had been based on 52 observations each rather than 100 and 5, the uncertainty factor would have been .94 rather than the observed 2.20. The interval width tends to be minimum when n1 = n2.

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance σ2 when the parent population is normally distributed. For a 100(1 − α) percent confidence interval (df)s2 (df)s2 ≤ σ2 ≤   2 χ2 χ12 where χ 12 corresponds to a lower-tail area of α/2 and χ 22 to an upper-tail area of α/2.

or

1.95 ≤ σ ≤ 3.17

TESTS OF HYPOTHESIS General Nature of Tests The general nature of tests can be illustrated with a simple example. In a court of law, when a defendant is charged with a crime, the judge instructs the jury initially to presume that the defendant is innocent of the crime. The jurors are then presented with evidence and counterargument as to the defendant’s guilt or innocence. If the evidence suggests beyond a reasonable doubt that the defendant did, in fact, commit the crime, they have been instructed to find the defendant guilty; otherwise, not guilty. The burden of proof is on the prosecution. Jury trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. Suppose that a company has been using a certain supplier of raw materials in one of its chemical processes. A new supplier approaches the company and states that its material, at the same cost, will increase the process yield. If the new supplier has a good reputation, the company might be willing to run a limited test. On the basis of the test results it would then make a decision to change suppliers or not. Good management would dictate that an improvement must be demonstrated

STATISTICS (beyond a reasonable doubt) for the new material. That is, the burden of proof is tied to the new material. In setting up a test of hypothesis for this application, the initial assumption would be defined as a null hypothesis and symbolized as H0. The null hypothesis would state that yield for the new material is no greater than for the conventional material. The symbol µ0 would be used to designate the known current level of yield for the standard material and µ for the unknown population yield for the new material. Thus, the null hypothesis can be symbolized as H0: µ ≤ µ0. The alternative to H0 is called the alternative hypothesis and is symbolized as H1: µ > µ0. Given a series of tests with the new material, the average yield x would be compared with µ0. If x < µ0, the new supplier would be dismissed. If x > µ0, the question would be: Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt? If the confidence interval for µ included µ0, the answer would be no, but if it did not include µ0, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases. Test of Hypothesis for a Mean Procedure Nomenclature µ = mean of the population from which the sample has been drawn σ = standard deviation of the population from which the sample has been drawn µ0 = base or reference level H0 = null hypothesis H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 t = tabled t value corresponding to the significance level α. For a two-tailed test, each corresponding tail would have an area of α/2, and for a one-tailed test, one tail area would be equal to α. If σ2 is known, then z would be used rather than the t. t = (x − µ0)/(s/ n) = sample value of the test statistic. Assumptions 1. The n observations x1, x2, . . . , xn have been selected randomly. 2. The population from which the observations were obtained is normally distributed with an unknown mean µ and standard deviation σ. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference designated by µ0. This can take one of three forms: Form 1

Form 2

Form 3

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value of t does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 3. The rule for accepting H0 is specified by selection of the α level as indicated in Fig. 3-65. For forms 2 and 3 the α area is defined to be in the upper or the lower tail respectively. The parameter α is the probability of rejecting the null hypothesis when it is actually true. 4. The decision rules for each of the three forms are defined as follows: If the sample t falls within the acceptance region, accept H0 for lack of contrary evidence. If the sample t falls in the critical region, reject H0 at a significance level of 100α percent.

3-79

FIG. 3-65 Acceptance region for two-tailed test. For a one-tailed test, area =  on one side only.

Example Application. In the past, the yield for a chemical process has been established at 89.6 percent with a standard deviation of 3.4 percent. A new supplier of raw materials will be used and tested for 7 days. Procedure 1. The standard of reference is µ0 = 89.6 with a known σ = 3.4. 2. It is of interest to demonstrate whether an increase in yield is achieved with the new material; H0 says it has not; therefore, H0: µ ≤ 89.6 H1: µ > 89.6 3. Select α = .05, and since σ is known (the new material would not affect the day-to-day variability in yield), the test statistic would be z with a corresponding critical value cv(z) = 1.645 (Table 3-6, df = ∞). 4. The decision rule: Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 5. A 7-day test was carried out, and daily yields averaged 91.6 percent with a sample standard deviation s = 3.6 (this is not needed for the test of hypothesis). 6. For the data sample z = (91.6 − 89.6)/(3.4/7) = 1.56. 7. Since the sample z < cv(z), accept the null hypothesis for lack of contrary evidence; i.e., an improvement has not been demonstrated beyond a reasonable doubt.

Example Application. In the past, the break strength of a synthetic yarn has averaged 34.6 lb. The first-stage draw ratio of the spinning machines has been increased. Production management wants to determine whether the break strength has changed under the new condition. Procedure 1. The standard of reference is µ0 = 34.6. 2. It is of interest to demonstrate whether a change has occurred; therefore, H0: µ = 34.6 H1: µ ≠ 34.6 3. Select α = .05, and since with the change in draw ratio the uniformity might change, the sample standard deviation would be used, and therefore t would be the appropriate test statistic. 4. A sample of 21 ends was selected randomly and tested on an Instron with the results x = 35.55 and s = 2.041. 5. For 20 df and a two-tailed α level of 5 percent, the critical values of t.025 (two tailed) are given by ±2.086 with a decision rule (Table 3-6, t.025, df = 20): Accept H0 if −2.086 < sample t < 2.086 Reject H0 if sample t < −2.086 or > 2.086 6. For the data sample t = (35.55 − 34.6)/(2.041/2 1) = 2.133. 7. Since 2.133 > 2.086, reject H0 and accept H1. It has been demonstrated that an improvement in break strength has been achieved.

Two-Population Test of Hypothesis for Means Nature Two samples were selected from different locations in a plastic-film sheet and measured for thickness. The thickness of the respective samples was measured at 10 close but equally spaced points in each of the samples. It was of interest to compare the average thickness of the respective samples to detect whether they were significantly different. That is, was there a significant variation in thickness between locations? From a modeling standpoint statisticians would define this problem as a two-population test of hypothesis. They would define the respective sample sheets as two populations from which 10 sample thickness determinations were measured for each. In order to compare populations based on their respective samples, it is necessary to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effect, the t statistic characterizes

3-80

MATHEMATICS

the way in which two sample means from two separate populations will tend to vary by chance alone when the population means and variances are equal. Consider the following: Population 1

Population 2

Normal

Sample 1

Normal

Sample 2

µ1

n1 x1 s12

µ2

n2 x2 s22

σ

2 1

σ

2 2

Consider the hypothesis µ1 = µ2. If, in fact, the hypothesis is correct, i.e., µ1 = µ2 (under the condition σ 12 = σ 22), then the sampling distribution of (x1 − x2) is predictable through the t distribution. The observed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflected through the α level), that is, x1 and x2 are not “too different” from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other “too much” and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected. Example Application. Two samples were selected from different locations in a plasticfilm sheet. The thickness of the respective samples was measured at 10 close but equally spaced points. Procedure 1. Demonstrate whether the thicknesses of the respective sample locations are significantly different from each other; therefore, H0: µ1 = µ2

H1: µ1 ≠ µ2

2. Select α = .05. 3. Summarize the statistics for the respective samples: Sample 1

Sample 2

1.473 1.484 1.484 1.425 1.448

1.367 1.276 1.485 1.462 1.439

1.474 1.501 1.485 1.435 1.348

1.417 1.448 1.469 1.474 1.452

x1 = 1.434

s1 = .0664

x2 = 1.450

s2 = .0435

4. As a first step, the assumption for the standard t test, that σ 12 = σ 22, can be tested through the F distribution. For this hypothesis, H0: σ 12 = σ 22 would be tested against H1: σ 12 ≠ σ 22. Since this is a two-tailed test and conventionally only the upper tail for F is published, the procedure is to use the largest ratio and the corresponding ordered degrees of freedom. This achieves the same end result through one table. However, since the largest ratio is arbitrary, it is necessary to define the true α level as twice the value of the tabled value. Therefore, by using Table 3-8 with α = .05 the corresponding critical value for F(9,9) = 3.18 would be for a true α = .10. For the sample, Sample F = (.0664/.0435)2 = 2.33 Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact σ 12 = σ 22. There is not sufficient evidence to reject the null hypothesis that σ 12 = σ 22. 5. For 18 df and a two-tailed α level of 5 percent the critical values of t are given by 2.101 (Table 3-6, t0.025, df = 18). 6. The decision rule: Accept H0 if −2.101 ≤ sample t ≤ 2.101 Reject H0 otherwise 7. For the sample the pooled variance estimate is given by Eq. (3-120).

or

9(.0664)2 + 9(.0435)2 = (.0664)2 + (.0435)2 = .00315 s p2 =   2 9+9 sp = .056

8. The sample statistic value of t is given by Eq. (3-119). 1.434 − 1.450 Sample t =  = −.64 0 0 .0561/1 +1/1 9. Since the sample value of t falls within the acceptance region, accept H0 for lack of contrary evidence; i.e., there is insufficient evidence to demonstrate that thickness differs between the two selected locations.

Test of Hypothesis for Paired Observations Nature In some types of applications, associated pairs of observations are defined. For example, (1) pairs of samples from two populations are treated in the same way, or (2) two types of measurements are made on the same unit. For applications of this type, it is not only more effective but necessary to define the random variable as the difference between the pairs of observations. The difference numbers can then be tested by the standard t distribution. Examples of the two types of applications are as follows: 1. Sample treatment a. Two types of metal specimens buried in the ground together in a variety of soil types to determine corrosion resistance b. Wear-rate test with two different types of tractor tires mounted in pairs on n tractors for a defined period of time 2. Same unit a. Blood-pressure measurements made on the same individual before and after the administration of a stimulus b. Smoothness determinations on the same film samples at two different testing laboratories Test of Hypothesis for Matched Pairs: Procedure Nomenclature di = sample difference between the ith pair of observations s = sample standard deviation of differences µ = population mean of differences σ = population standard deviation of differences µ0 = base or reference level of comparison H0 = null hypothesis H1 = alternative hypothesis α = significance level t = tabled value with (n − 1) df t = (d n), the sample value of t  − µ0)/(s/ Assumptions 1. The n pairs of samples have been selected and assigned for testing in a random way. 2. The population of differences is normally distributed with a mean µ and variance σ2. As in the previous application of the t distribution, this is a robust procedure, i.e., not sensitive to the normality assumption if the sample size is 10 or greater in most situations. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference level designated by µ0. For most applications of this type, the value of µ0 is defined to be zero; that is, it is of interest generally to demonstrate a difference not equal to zero. The hypothesis can take one of three forms: Form 1

Form 2

H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test

H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test

Form 3 H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test

2. If the null hypothesis is assumed to be true, say, in the case of a lower-tailed test, form 3, then the distribution of the test statistic t is known under the null hypothesis that limits µ = µ0. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero by chance alone when µ = µ0. If the sample value of t is too small, as in the case of a negative value, then this would be defined as sufficient evidence to reject the null hypothesis. 3. Select α. 4. The critical values or value of t would be defined by the tabled value of t with (n − 1) df corresponding to a tail area of α. For a twotailed test, each tail area would be α/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of α corresponding to forms 2 and 3 respectively. 5. The decision rule for each of the three forms would be to reject the null hypothesis if the sample value of t fell in that area of the t distribution defined by α, which is called the critical region.

STATISTICS Otherwise, the alternative hypothesis would be accepted for lack of contrary evidence. Example Application. Pairs of pipes have been buried in 11 different locations to determine corrosion on nonbituminous pipe coatings for underground use. One type includes a lead-coated steel pipe and the other a bare steel pipe. Procedure 1. The standard of reference is taken as µ0 = 0, corresponding to no difference in the two types. 2. It is of interest to demonstrate whether either type of pipe has a greater corrosion resistance than the other. Therefore, H0: µ = 0

H1: µ ≠ 0

3. Select α = .05. Therefore, with n = 11 the critical values of t with 10 df are defined by t = 2.228 (Table 3.5, t.025). 4. The decision rule: Accept H0 if −2.228 ≤ sample t ≤ 2.228 Reject H0 otherwise 5. The sample of 11 pairs of corrosion determinations and their differences are as follows:

Soil type

Lead-coated steel pipe

Bare steel pipe

d = difference

A B C D E F

27.3 18.4 11.9 11.3 14.8 20.8

41.4 18.9 21.7 16.8 9.0 19.3

−14.1 −0.5 −9.8 −5.5 5.8 1.5

G H I J K

17.9 7.8 14.7 19.0 65.3

32.1 7.4 20.7 34.4 76.2

−14.2 0.4 −6.0 −15.4 −10.9

6. The sample statistics, Eq. (3-118) 11 d 2 − ( d)2 s2 =  = 52.59 11 × 10

d = −6.245

s = 7.25

or

Sample t = (−6.245 − 0)/(7.25/11) = −2.86 7. Since the sample t of −2.86 < tabled t of −2.228, reject H0 and accept H1; that is, it has been demonstrated that, on the basis of the evidence, lead-coated steel pipe has a greater corrosion resistance than bare steel pipe.

Example Application. A stimulus was tested for its effect on blood pressure. Ten men were selected randomly, and their blood pressure was measured before and after the stimulus was administered. It was of interest to determine whether the stimulus had caused a significant increase in the blood pressure. Procedure 1. The standard of reference was taken as µ0 ≤ 0, corresponding to no increase. 2. It was of interest to demonstrate an increase in blood pressure if in fact an increase did occur. Therefore, H0: µ0 ≤ 0

H1: µ0 > 0

3. Select α = .05. Therefore, with n = 10 the critical value of t with 9 df is defined by t = 1.833 (Table 3-6, t.05, one-sided). 4. The decision rule: Accept H0 if sample t < 1.833 Reject H0 if sample t > 1.833 5. The sample of 10 pairs of blood pressure and their differences were as follows: Individual

Before

After

d = difference

1 2 3 4

138 116 124 128

146 118 120 136

8 2 −4 8

3-81

5

155

174

19

6 7 8 9 10

129 130 148 143 159

133 129 155 148 155

4 −1 7 5 −4

6. The sample statistics: d = 4.4 s = 6.85 Sample t = (4.4 − 0)/(6.85/10) = 2.03 7. Since the sample t = 2.03 > critical t = 1.833, reject the null hypothesis. It has been demonstrated that the population of men from which the sample was drawn tend, as a whole, to have an increase in blood pressure after the stimulus has been given. The distribution of differences d seems to indicate that the degree of response varies by individuals.

Test of Hypothesis for a Proportion Nature Some types of statistical applications deal with counts and proportions rather than measurements. Examples are (1) the proportion of workers in a plant who are out sick, (2) lost-time worker accidents per month, (3) defective items in a shipment lot, and (4) preference in consumer surveys. The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity p0, and since the distribution is a binomial proportion, the associated variance is [p0(1 − p0)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(1 − p) > 5. Test of Hypothesis for a Proportion: Procedure Nomenclature p = mean proportion of the population from which the sample has been drawn p0 = base or reference proportion [p0(1 − p0)]/n = base or reference variance p ˆ = x/n = sample proportion, where x refers to the number of observations out of n which have the specified attribute H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 z = tabled Z value corresponding to the significance level α. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-6. z = (ˆp − p0)/ p 1 − p /n, the sample value of the test 0( 0) statistic Assumptions 1. The n observations have been selected randomly. 2. The sample size n is sufficiently large to meet the requirement for the Z approximation. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population with a proportion p0 of items having the specified attribute. For example, in tossing a coin the population could be thought of as having an unbounded number of potential tosses. If it is assumed that the coin is fair, this would dictate p0 = 1/2 for the proportional number of heads in the population. The null hypothesis can take one of three forms: Form 1

Form 2

Form 3

H0: p = p0 H1: p ≠ p0 Two-tailed test

H0: p ≤ p0 H1: p > p0 Upper-tailed test

H0: p ≥ p0 H1: p < p0 Lower-tailed test

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MATHEMATICS

2. If the null hypothesis is assumed to be true, then the sampling distribution of the test statistic Z is known. Given a random sample, it is possible to predict how far the sample proportion x/n might deviate from its assumed population proportion p0 through the Z distribution. When the sample proportion deviates too far, as defined by the significance level α, this serves as the justification for rejecting the assumption, that is, rejecting the null hypothesis. 3. The decision rule is given by Form 1: Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise Form 2: Accept H0 if sample z < upper critical z Reject H0 otherwise Form 3: Accept H0 if lower critical z < sample z Reject H0 otherwise Example Application. A company has received a very large shipment of rivets. One product specification required that no more than 2 percent of the rivets have diameters greater than 14.28 mm. Any rivet with a diameter greater than this would be classified as defective. A random sample of 600 was selected and tested with a go–no-go gauge. Of these, 16 rivets were found to be defective. Is this sufficient evidence to conclude that the shipment contains more than 2 percent defective rivets? Procedure 1. The quality goal is p ≤ .02. It would be assumed initially that the shipment meets this standard; i.e., H0: p ≤ .02. 2. The assumption in step 1 would first be tested by obtaining a random sample. Under the assumption that p ≤ .02, the distribution for a sample proportion would be defined by the z distribution. This distribution would define an upper bound corresponding to the upper critical value for the sample proportion. It would be unlikely that the sample proportion would rise above that value if, in fact, p ≤ .02. If the observed sample proportion exceeds that limit, corresponding to what would be a very unlikely chance outcome, this would lead one to question the assumption that p ≤ .02. That is, one would conclude that the null hypothesis is false. To test, set H0: p ≤ .02

H1: p > .02

3. Select α = .05. 4. With α = .05, the upper critical value of Z = 1.645 (Table 3-6, t.05, df = ∞, one-sided). 5. The decision rule: Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 6. The sample z is given by (16/600) − .02 Sample z =  02)( .9 8)/ 600 (. = 1.17 7. Since the sample z < 1.645, accept H0 for lack of contrary evidence; there is not sufficient evidence to demonstrate that the defect proportion in the shipment is greater than 2 percent.

Test of Hypothesis for Two Proportions Nature In some types of engineering and management-science problems, we may be concerned with a random variable which represents a proportion, for example, the proportional number of defective items per day. The method described previously relates to a single proportion. In this subsection two proportions will be considered. A certain change in a manufacturing procedure for producing component parts is being considered. Samples are taken by using both the existing and the new procedures in order to determine whether the new procedure results in an improvement. In this application, it is of interest to demonstrate statistically whether the population proportion p2 for the new procedure is less than the population proportion p1 for the old procedure on the basis of a sample of data. Test of Hypothesis for Two Proportions: Procedure Nomenclature p1 = population 1 proportion p2 = population 2 proportion n1 = sample size from population 1 n2 = sample size from population 2

x1 = number of observations out of n1 that have the designated attribute x2 = number of observations out of n2 that have the designated attribute p ˆ 1 = x1/n1, the sample proportion from population 1 p ˆ 2 = x2/n2, the sample proportion from population 2 α = significance level H0 = null hypothesis H1 = alternative hypothesis z = tabled Z value corresponding to the stated significance level α p ˆ1−p ˆ2 z =  , the sample value of Z pˆ 1(1 − pˆ 1)/n1 + pˆ 2(1 − pˆ 2)/n2 Assumptions 1. The respective two samples of n1 and n2 observations have been selected randomly. 2. The sample sizes n1 and n2 are sufficiently large to meet the requirement for the Z approximation; i.e., x1 > 5, x2 > 5. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions p1 = p2. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated: Form 1

Form 2

Form 3

H0: p1 = p2 H1: p1 ≠ p2 Two-tailed test

H0: p1 ≤ p2 H1: p1 > p2 Upper-tailed test

H0: p1 ≥ p2 H1: p1 < p2 Lower-tailed test

2. The decision rule for form 1 is given by Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise Example Application. A change was made in a manufacturing procedure for component parts. Samples were taken during the last week of operations with the old procedure and during the first week of operations with the new procedure. Determine whether the proportional numbers of defects for the respective populations differ on the basis of the sample information. Procedure 1. The hypotheses are H0: p1 = p2

H1: p1 ≠ p2

2. Select α = .05. Therefore, the critical values of z are 1.96 (Table 3-5, A = 0.9500). 3. For the samples, 75 out of 1720 parts from the previous procedure and 80 out of 2780 parts under the new procedure were found to be defective; therefore, p ˆ 1 = 75/1720 = .0436

p ˆ 2 = 80/2780 = .0288

4. The decision rule: Accept H0 if −1.96 ≤ sample z ≤ 1.96 Reject H0 otherwise 5. The sample statistic: .0436 − .0288 Sample z =  0436)( .9 564)/ 1720 288)( .9 712)/ 2780 +(.0 (. = 2.53 6. Since the sample z of 2.53 > tabled z of 1.96, reject H0 and conclude that the new procedure has resulted in a reduced defect rate.

Goodness-of-Fit Test Nature A standard die has six sides numbered from 1 to 6. If one were really interested in determining whether a particular die was well balanced, one would have to carry out an experiment. To do this, it might be decided to count the frequencies of outcomes, 1 through 6, in tossing the die N times. On the assumption that the die is perfectly balanced,

STATISTICS one would expect to observe N/6 occurrences each for 1, 2, 3, 4, 5, and 6. However, chance dictates that exactly N/6 occurrences each will not be observed. For example, given a perfectly balanced die, the probability is only 1 chance in 65 that one will observe 1 outcome each, for 1 through 6, in tossing the die 6 times. Therefore, an outcome different from 1 occurrence each can be expected. Conversely, an outcome of six 3s would seem to be too unusual to have occurred by chance alone. Some industrial applications involve the concept outlined here. The basic idea is to test whether or not a group of observations follows a preconceived distribution. In the case cited, the distribution is uniform; i.e., each face value should tend to occur with the same frequency. Goodness-of-Fit Test: Procedure Nomenclature Each experimental observation can be classified into one of r possible categories or cells. r = total number of cells Oj = number of observations occurring in cell j Ej = expected number of observations for cell j based on the preconceived distribution N = total number of observations f = degrees of freedom for the test. In general, this will be equal to (r − 1) minus the number of statistical quantities on which the Ej’s are based (see the examples which follow for details). Assumptions 1. The observations represent a sample selected randomly from a population which has been specified. 2. The number of expectation counts Ej within each category should be roughly 5 or more. If an Ej count is significantly less than 5, that cell should be pooled with an adjacent cell. Computation for Ej On the basis of the specified population, the probability of observing a count in cell j is defined by pj. For a sample of size N, corresponding to N total counts, the expected frequency is given by Ej = Npj. Test Statistics: Chi Square r (Oj − Ej)2 χ2 =  with f df Ej j=1 Test of Hypothesis 1. H0: The sample came from the specified theoretical distribution H1: The sample did not come from the specified theoretical distribution 2. For a stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 if sample χ2 < tabled χ2 Example Application A production-line product is rejected if one of its characteristics does not fall within specified limits. The standard goal is that no more than 2 percent of the production should be rejected. Computation 1. Of 950 units produced during the day, 28 units were rejected. 2. The hypotheses: H0: the process is in control H1: the process is not in control 3. Assume that α = .05; therefore, the critical value of χ2(1) = 3.84 (Table 3-7, 95 percent, df = 1). One degree of freedom is defined since (r − 1) = 1, and no statistical quantities have been computed for the data. 4. The decision rule: Reject H0 if sample χ2 > 3.84 Accept H0 otherwise 5. Since it is assumed that p = .02, this would dictate that in a sample of 950 there would be on the average (.02)(950) = 19 defective items and 931 acceptable items: Category

Observed Oj

Expectation Ej = 950pj

Acceptable Not acceptable Total

922 28 950

931 19 950

3-83

(922 − 931)2 (28 − 19)2 Sample χ2 =  +  931 19 = 4.35 with critical χ2 = 3.84 6. Conclusion. Since the sample value exceeds the critical value, it would be concluded that the process is not in control.

Example Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation? Computation 1. The hypotheses: H0: there are no significant differences, in number of defective units, between workers H1: there are significant differences 2. Assume that α = .05. 3. Test statistic: No. of defective units 0 1 2 3 4 5 6 7 8 9 ≥10 Sum

Oj 3 7 9 12 9 6 3 2 0 1 0 52

Ej


10

2.06 6.64 10.73 11.55 9.33 6.03


8.70 pool




3.24 1.50 .60 .22 .10 52




6

5.66 pool

The expectation numbers Ej were computed as follows: For the Poisson distribution, λ = E(x); therefore, an estimate of λ is the average number of defective units per worker, i.e., λ = (1/52)(0 × 3 + 1 × 7 + ⋅ ⋅ ⋅ + 9 × 1) = 3.23. Given this approximation, the probability of no defective units for a worker would be (3.23)0/0!)e−3.23 = .0396. For the 52 workers, the number of workers producing no defective units would have an expectation E = 52(0.0396) = 2.06, and so forth. The sample chi-square value is computed from (10 − 8.70)2 (9 − 10.73)2 (6 − 5.66)2 χ2 =  +  + ⋅ ⋅ ⋅ +  8.70 10.73 5.66 = .522 4. The critical value of χ2 would be based on four degrees of freedom. This corresponds to (r − 1) − 1, since one statistical quantity λ was computed from the sample and used to derive the expectation numbers. 5. The critical value of χ2(4) = 9.49 (Table 3-7) with α = .05; therefore, accept H0.

Two-Way Test for Independence for Count Data Nature When individuals or items are observed and classified according to two different criteria, the resultant counts can be statistically analyzed. For example, a market survey may examine whether a new product is preferred and if it is preferred due to a particular characteristic. Count data, based on a random selection of individuals or items which are classified according to two different criteria, can be statistically analyzed through the χ2 distribution. The purpose of this analysis is to determine whether the respective criteria are dependent. That is, is the product preferred because of a particular characteristic? Two-Way Test for Independence for Count Data: Procedure Nomenclature 1. Each observation is classified according to two categories: a. The first one into 2, 3, . . . , or r categories b. The second one into 2, 3, . . . , or c categories 2. Oij = number of observations (observed counts) in cell (i, j) with i = 1, 2, . . . , r j = 1, 2, . . . , c

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MATHEMATICS

3. N = total number of observations 4. Eij = computed number for cell (i,j) which is an expectation based on the assumption that the two characteristics are independent 5. Ri = subtotal of counts in row i 6. Cj = subtotal of counts in column j 7. α = significance level 8. H0 = null hypothesis 9. H1 = alternative hypothesis 10. χ2 = critical value of χ2 corresponding to the significance level α and (r − 1)(c − 1) df c,r (Oij − Eij)2 11. Sample χ2 =  Eij i, j Assumptions 1. The observations represent a sample selected randomly from a large total population. 2. The number of expectation counts Eij within each cell should be approximately 2 or more for arrays 3 × 3 or larger. If any cell contains a number smaller than 2, appropriate rows or columns should be combined to increase the magnitude of the expectation count. For arrays 2 × 2, approximately 4 or more are required. If the number is less than 4, the exact Fisher test should be used. Test of Hypothesis Under the null hypothesis, the classification criteria are assumed to be independent, i.e., H0: the criteria are independent H1: the criteria are not independent For the stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 otherwise Computation for Eij Compute Eij across rows or down columns by using either of the following identities: R Eij = Cj i across rows N





C Eij = Ri j down columns N Sample c Value 2

(Oij − Eij)2 χ2 =  Eij i, j In the special case of r = 2 and c = 2, a more accurate and simplified formula which does not require the direct computation of Eij can be used: [|O11O22 − O12O21| − aN]2N χ2 =  R1R2C1C2

3. Select α = .05; therefore, with (r − 1)(c − 1) = 1 df, the critical value of χ2 is 3.84 (Table 3-7, 95 percent). 4. The decision rule: Accept H0 if sample χ2 < 3.84 Reject H0 otherwise 5. The sample value of χ2 by using the special formula is [|114 × 18 − 13 × 55| − 100]2200 Sample χ2 =  (169)(31)(127)(73) = 6.30 6. Since the sample χ2 of 6.30 > tabled χ2 of 3.84, reject H0 and accept H1. The relative proportionality of E11 = 169(127/200) = 107.3 to the observed 114 compared with E22 = 31(73/200) = 11.3 to the observed 18 suggests that when the consumer likes the feel, the consumer tends to like the product, and conversely for not liking the feel. The proportions 169/200 = 84.5 percent and 127/200 = 63.5 percent suggest further that there are other attributes of the product which tend to nullify the beneficial feel of the product.

LEAST SQUARES When experimental data is to be fit with a mathematical model, it is necessary to allow for the fact that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a linear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just linear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. See Press et al. (1986); for a description of maximum likelihood as it applies to both linear and nonlinear least squares. In a least squares parameter estimation, it is desired to find parameters that minimize the sum of squares of the deviation between the experimental data and the theoretical equation. i

1

2

M

)]2

where yi is the ith experimental data point for the value xi, y(xi; a1, a2,. . ., aM) is the theoretical equation at xi, and the parameters {a1, a2,. . ., aM} are to be determined to minimize χ2. This will also minimize the variance of the curve fit N [yi  y(xi; a1, a2,. . . , aM)]2 2  N i =1 Linear Least Squares When the model is a straight line, one is minimizing 2

N

(y  a  bx )

i=1 i

i

2

The linear correlation coefficient r is defined by

Application A market research study was carried out to relate the subjective “feel” of a consumer product to consumer preference. In other words, is the consumer’s preference for the product associated with the feel of the product, or is the preference independent of the product feel? Procedure 1. It was of interest to demonstrate whether an association exists between feel and preference; therefore, assume

N

(x  –x )(y  y–)

i =1 i

r=

i

   N N

(x  –x) (y  y–) i

i =1

2

i

2

i =1

and

H0: feel and preference are independent H1: they are not independent

N

2. A sample of 200 people was asked to classify the product according to two criteria: a. Liking for this product b. Liking for the feel of the product Like feel Yes No Cj

i

χ

Example

Like product

N

[y  y(x ; a , a , . . . , a

i=1

χ2

Yes

No

Ri

114 55 169

13 18 31

= 127 = 73 200

+2 (1  r2) (yi  y)2 i=1

where y is the average of the yi values. Values of r near 1 indicate a positive correlation; r near –1 means a negative correlation, and r near 0 means no correlation. These parameters are easily found by using standard programs, such as Microsoft Excel. Example Application. Brenner (Magnetic Method for Measuring the Thickness of Non-magnetic Coatings on Iron and Steel, National Bureau of Standards, RP1081, March 1938) suggests an alternative way of measuring the thickness of nonmagnetic coatings of galvanized zinc on iron and steel. This procedure is based on a nondestructive magnetic method as a substitute for the standard

STATISTICS destructive stripping method. A random sample of 11 pieces was selected and measured by both methods. Nomenclature. The calibration between the magnetic and the stripping methods can be determined through the model

study of stream velocity as a function of relative depth of the stream. Sample data Depth*

y = a + bx + ε where x = strip-method determination y = magnetic-method determination Sample data Thickness, 10−5 in Magnetic method, y

104 114 116 129 132 139

85 115 105 127 120 121

174 312 338 465 720

155 250 310 443 630

SLOPE(B1:B11, A1:A11), INTERCEPT(B1:B11, A1:A11), RSQ(B1:B11, A1:A11)

3.1950 3.2299 3.2532 3.2611 3.2516

.5 .6 .7 .8 .9

3.2282 3.1807 3.1266 3.0594 2.9759

The model is taken as a quadratic function of position: Velocity a  bx  cx 2 The parameters are easily determined by using computer software. In Microsoft Excel, the data are put into columns A and B and the graph is created as for a linear curve fit. This time, though, when adding the trendline, choose the polynomial icon and use 2 (which gives powers up to and including x2). The result is Velocity 3.195  0.4425x  0.7653x2 The value of r2 is .9993. Multiple Regression In multiple regression, any set of functions can be used, not just polynomials, such as M

give the slope b, the intercept a, and the value of r2. Here they are 3.20, 0.884, and 0.9928, respectively, so that r = 0.9964. By choosing Insert/Chart and Scatter Plot, the data are plotted. Once that is done, place the cursor on a data point and right-click; choose Format trendline with options selected to display the equation and r2, and you get Fig. 3-66. On the Macintosh, use CTRL-click. Polynomial Regression In polynomial regression, one expands the function in a polynomial in x. M

y(x) aj x j1 j=1

Example Application. Merriman (“The Method of Least Squares Applied to a Hydraulic Problem,” J. Franklin Inst., 233–241, October 1877) reported on a

700

.1 .2 .3 .4

*As a fraction of total depth.

This example is solved by using Microsoft Excel. Put the data into columns A and B as shown, using rows 1 through 11. Then the commands

y(x) a j fj(x) j=1

where the set of functions {fj(x)} is known and specified. Note that the unknown parameters {aj} enter the equation linearly. In this case, the spreadsheet can be expanded to have a column for x and then successive columns for fj(x). In Microsoft Excel, you choose Regression under Tools/Data Analysis, and complete the form. In addition to the actual correlation, you get the expected variance of the unknowns, which allows you to assess how accurately they were determined. In the example above, by creating a column for x and x2, one obtains an intercept of 3.195 with a standard error of .0039, b = .4416 with a standard error of .018, and c = −.7645 with a standard error of .018. Nonlinear Least Squares There are no analytic methods for determining the most appropriate model for a particular set of data.

y = 0.8844x + 3.1996 R2 = 0.9928

600

Magnetic method

Velocity, y, ft/s

0

Stripping method, x

3-85

500 R2 = 0.9928 400 300 200 100 0 0

100

200

300

400 Stripping method

FIG. 3-66

Plot of data and correlating line.

500

600

700

800

3-86

MATHEMATICS

In many cases, however, the engineer has some basis for a model. If the parameters occur in a nonlinear fashion, then the analysis becomes more difficult. For example, in relating the temperature to the elapsed time of a fluid cooling in the atmosphere, a model that has an asymptotic property would be the appropriate model (temp = a + b exp(−c time), where a represents the asymptotic temperature corresponding to t → ∞. In this case, the parameter c appears nonlinearly. The usual practice is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be applied only to problems in which there is a well-defined, clear association between the two variables; therefore, a test of hypothesis on the significance of the fit would be somewhat ludicrous. In addition, the generalization of the theory for the associate confidence intervals for nonlinear coefficients is not well developed. Example Application. Data were collected on the cooling of water in the atmosphere as a function of time. Sample data

If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E(dY), is zero. The variances are given by the following equation (Baird, 1995; Box et al., 2005): N ∂Y 2 σ2(dY) =  σ i2 ∂y i=1 i Thus, the variance of the desired quantity Y can be found. This gives an independent estimate of the errors in measuring the quantity Y from the errors in measuring each variable it depends upon.



Example Suppose one wants to measure the thermal conductivity of a solid (k). To do this, one needs to measure the heat flux (q), the thickness of the sample (d), and the temperature difference across the sample (∆T). Each measurement has some error. The heat flux (q) may be the rate of electrical heat ˙ divided by the area (A), and both quantities are measured to some input (Q) tolerance. The thickness of the sample is measured with some accuracy, and the temperatures are probably measured with a thermocouple to some accuracy. These measurements are combined, however, to obtain the thermal conductivity, and it is desired to know the error in the thermal conductivity. The formula is d ˙ k=Q A∆T

Time x

Temperature y

0 1 2 3 5

92.0 85.5 79.5 74.5 67.0

The variance in the thermal conductivity is then k 2 2 k 2 k k 2 σ k2 =  σ d2 +  σ Q˙ +  σ A2 +  ˙ d A ∆T Q

7 10 15 20

60.5 53.5 45.0 39.5

FACTORIAL DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCE

Model. MATLAB can be used to find the best fit of the data to the formula y = a + becx: a = 33.54, b = 57.89, c = 0.11. The value of χ2 is 1.83. Using an alternative form, y = a + b/(c + x), gives a = 9.872, b = 925.7, c = 11.27, and χ = 0.19. Since this model had a smaller value of χ2 it might be the chosen one, but it is only a fit of the specified data and may not be generalized beyond that. Both forms give equivalent plots.

ERROR ANALYSIS OF EXPERIMENTS Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E(y), the true mean, with variance σ2/n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day’s measurement, n. Of course, other factors (weather, weekends) may make the observations on different days not distributed identically. Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables {yi}, Y = α1y1 + α2y2 + . . . . Let the random variables y1, y2, . . . have means E(y1), E(y2), . . . and variances σ2(y1), σ2(y2), . . . . The variable Y has mean E(Y) = α1E(y1) + α2 E(y2) + . . . and variance (Cropley, 1978) n

n

σ2(Y) = α2i σ2(yi) + 2

i=1

n



αi αj Cov (yi, yj)

i=1 j=i+1

If the variables are uncorrelated and have the same variance, then

 α σ n

σ2(Y) =

2 i

2

i=1

Next suppose the model relating Y to {yi} is nonlinear, but the errors are small and independent of one another. Then a change in Y is related to changes in yi by ∂Y ∂Y dY =  dy1 +  dy2 + … ∂y2 ∂y1







 σ 2

2 ∆T

Statistically designed experiments consider, of course, the effect of primary variables, but they also consider the effect of extraneous variables and the interactions between variables, and they include a measure of the random error. Primary variables are those whose effect you wish to determine. These variables can be quantitative or qualitative. The quantitative variables are ones you may fit to a model in order to determine the model parameters (see the section “Least Squares”). Qualitative variables are ones you wish to know the effect of, but you do not try to quantify that effect other than to assign possible errors or magnitudes. Qualitative variables can be further subdivided into Type I variables, whose effect you wish to determine directly, and Type II variables, which contribute to the performance variability and whose effect you wish to average out. For example, if you are studying the effect of several catalysts on yield in a chemical reactor, each different type of catalyst would be a Type I variable because you would like to know the effect of each. However, each time the catalyst is prepared, the results are slightly different due to random variations; thus, you may have several batches of what purports to be the same catalyst. The variability between batches is a Type II variable. Since the ultimate use will require using different batches, you would like to know the overall effect including that variation, since knowing precisely the results from one batch of one catalyst might not be representative of the results obtained from all batches of the same catalyst. A randomized block design, incomplete block design, or Latin square design (Box et al., ibid.), for example, all keep the effect of experimental error in the blocked variables from influencing the effect of the primary variables. Other uncontrolled variables are accounted for by introducing randomization in parts of the experimental design. To study all variables and their interaction requires a factorial design, involving all possible combinations of each variable, or a fractional factorial design, involving only a selected set. Statistical techniques are then used to determine which are the important variables, what are the important interactions, and what the error is in estimating these effects. The discussion here is only a brief overview of the excellent book by Box et al. (2005). Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section “Two-Population Test of Hypothesis for Means” can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged

STATISTICS as shown in the table: several measurements for each treatment. The goal is to see if the treatments differ significantly from each other; that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. Estimating the Effect of Four Treatments Treatment

Treatment average Grand average

1

2

3

4

— — —

— — — —

— — — — —

— — — — — — —

—

— —

—

The data for k = 4 treatments is arranged in the table. For each treatment, there are nt experiments and the outcome of the ith experiment with treatment t is called yti. Compute the treatment average nt

y

Block Design with Four Treatments and Five Blocks Treatment

1

2

3

4

Block average

Block 1 Block 2 Block 3 Block 4 Block 5

— — — — —

— — — — —

— — — — —

— — — — —

— — — — —

The following quantities are needed for the analysis of variance table. Name

1

blocks

2 SB = k i = 1 (y i − y)

n−1

treatments

ST = n t = 1 (yt − y)2

residuals

SR = t = 1 i = 1 (yti − yi − yt + y)

(n − 1)(k − 1)

total

S = t = 1 i = 1 y

N = nk

n

k

k

k

sT2 ,  sR2

k

N = nt t=1

Next compute the sum of squares of deviations from the average within the tth treatment nt

St = (yti − yt)2 i=1

Since each treatment has nt experiments, the number of degrees of freedom is nt − 1. Then the sample variances are St s t2 =  nt − 1 The within-treatment sum of squares is k

SR = St

Now, if there is no difference between treatments, a second estimate of σ2 could be obtained by calculating the variation of the treatment averages about the grand average. Thus compute the betweentreatment mean square k ST 2 sT2 =  , ST = nt(y t − y) k−1 t=1 Basically the test for whether the hypothesis is true or not hinges on a comparison of the within-treatment estimate sR2 (with νR = N − k degrees of freedom) with the between-treatment estimate s2T (with νT = k − 1 degrees of freedom). The test is made based on the F distribution for νR and νT degrees of freedom (Table 3-8). Next consider the case that uses randomized blocking to eliminate the effect of some variable whose effect is of no interest, such as the batch-to-batch variation of the catalysts in the chemical reactor example. Suppose there are k treatments and n experiments in each treatment. The results from nk experiments can be arranged as shown in the block design table; within each block, the various treatments are applied in a random order. Compute the block average, the treatment average, as well as the grand average as before.

n

ST sT2 =  , k−1

2

2 ti

SR s R2 =  (n − 1)(k − 1)

and the F distribution for νR and νT degrees of freedom (Table 3-8). The assumption behind the analysis is that the variations are linear. See Box et al. (2005). There are ways to test this assumption as well as transformations to make if it is not true. Box et al. also give an excellent example of how the observations are broken down into a grand average, a block deviation, a treatment deviation, and a residual. For two-way factorial design in which the second variable is a real one rather than one you would like to block out, see Box et al. To measure the effects of variables on a single outcome a factorial design is appropriate. In a two-level factorial design, each variable is considered at two levels only, a high and low value, often designated as a + and −. The two-level factorial design is useful for indicating trends, showing interactions, and it is also the basis for a fractional factorial design. As an example, consider a 23 factorial design with 3 variables and 2 levels for each. The experiments are indicated in the factorial design table. Two-Level Factorial Design with Three Variables Variable

t=1

and the within-treatment sample variance is SR sR2 =  N−k

k−1

n

The key test is again a statistical one, based on the value of

i=1

k

dof

2 SA = nky 

yt =  nt

nt yt t=1 y =  , N

Formula

average

ti

Also compute the grand average

3-87

Run

1

2

3

1 2 3 4 5 6 7 8

− + − + − + − +

− − + + − − + +

− − − − + + + +

The main effects are calculated by calculating the difference between results from all high values of a variable and all low values of a variable; the result is divided by the number of experiments at each level. For example, for the first variable: [(y2 + y4 + y6 + y8) − (y1 + y3 + y5 + y7)] Effect of variable 1 =  4 Note that all observations are being used to supply information on each of the main effects and each effect is determined with the precision of a fourfold replicated difference. The advantage of a one-at-atime experiment is the gain in precision if the variables are additive and the measure of nonadditivity if it occurs (Box et al., 2005). Interaction effects between variables 1 and 2 are obtained by calculating the difference between the results obtained with the high and low value of 1 at the low value of 2 compared with the results obtained

3-88

MATHEMATICS

with the high and low value 1 at the high value of 2. The 12-interaction is [(y4 − y3 + y8 − y7) − (y2 − y1 + y6 − y5)] 12-interaction =  2 The key step is to determine the errors associated with the effect of each variable and each interaction so that the significance can be determined. Thus, standard errors need to be assigned. This can be done by repeating the experiments, but it can also be done by using higher-order interactions (such as 123 interactions in a 24 factorial design). These are assumed negligible in their effect on the mean but can be used to estimate the standard error. Then, calculated

effects that are large compared with the standard error are considered important, while those that are small compared with the standard error are considered to be due to random variations and are unimportant. In a fractional factorial design one does only part of the possible experiments. When there are k variables, a factorial design requires 2k experiments. When k is large, the number of experiments can be large; for k = 5, 25 = 32. For a k this large, Box et al. (2005) do a fractional factorial design. In the fractional factorial design with k = 5, only 16 experiments are done. Cropley (1978) gives an example of how to combine heuristics and statistical arguments in application to kinetics mechanisms in chemical engineering.

DIMENSIONAL ANALYSIS Dimensional analysis allows the engineer to reduce the number of variables that must be considered to model experiments or correlate data. Consider a simple example in which two variables F1 and F2 have the units of force and two additional variables L1 and L2 have the units of length. Rather than having to deduce the relation of one variable on the other three, F1 = fn (F2, L1, L2), dimensional analysis can be used to show that the relation must be of the form F1 /F2 = fn (L1 /L2). Thus considerable experimentation is saved. Historically, dimensional analysis can be done using the Rayleigh method or the Buckingham pi method. This brief discussion is equivalent to the Buckingham pi method but uses concepts from linear algebra; see Amundson, N. R., Mathematical Methods in Chemical Engineering, Prentice-Hall, Englewood Cliffs, N.J. (1966), p. 54, for further information. The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let {Qi} represent a set of fundamental units, like length, time, force, and so on. Let [Pi] represent the dimensions of a physical quantity Pi; there are n physical quantities. Then form the matrix αij

Q1 Q2 … Qm

[P1]

[P2]

…

[Pn]

α11 α21

α12 α22

… …

α1n α2n

αm1

αm2

…

αmn

in which the entries are the number of times each fundamental unit appears in the dimensions [Pi]. The dimensions can then be expressed as follows. [Pi] = Q1α1i Q2α2i⋅⋅⋅Qmαmi Let m be the rank of the α matrix. Then p = n − m is the number of dimensionless groups that can be formed. One can choose m variables {Pi} to be the basis and express the other p variables in terms of them, giving p dimensionless quantities. Example: Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below: Film coefficient = h = (F/LθT) Conduit internal diameter = D = (L) Fluid linear velocity = V = (L/θ) Fluid density = ρ = (M/L3) Fluid absolute viscosity = µ = (M/Lθ) Fluid thermal conductivity = k = (F/θT) Fluid specific heat = cp = (FL/MT) Dimensional constant = gc = (ML/Fθ2) The matrix α in this case is as follows.

[Pi]

Qj

F M L θ T

h

D

V

ρ

µ

k

Cp

gc

1 0 −1 −1 −1

0 0 1 0 0

0 0 1 −1 0

0 1 −3 0 0

0 1 −1 −1 0

1 0 0 −1 −1

1 −1 1 0 −1

−1 1 1 −2 0

Here m ≤ 5, n = 8, p ≥ 3. Choose D, V, µ, k, and gc as the primary variables. By examining the 5 × 5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5; thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, ρ, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. hD h [h] = D−1k+1; thus  =  is a dimensionless group D−1 k k ρVD ρ =  is a dimensionless group [ρ] = µ1V −1D −1; thus  µ1V−1D−1 µ Cpµ Cp =  is a dimensionless group [Cp] = k+1µ−1; thus  k k+1 µ−1 Thus, the dimensionless groups are [Pi] hD ρVD Cp µ : , ,   k Q1α1i Q2α2i⋅⋅⋅Qmαmi k µ The dimensionless group hD/k is called the Nusselt number, NNu, and the group Cp µ/k is the Prandtl number, NPr. The group DVρ/µ is the familiar Reynolds number, NRe , encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as or as

φ(NNu, NPr, NRe) = 0 NNu = φ1(NPr, NRe)

(3-121)

TABLE 3-9 Dimensionless Groups in the Engineering System of Dimensions Biot number Condensation number Number used in condensation of vapors Euler number Fourier number Froude number Graetz number Grashof number Mach number Nusselt number Peclet number Prandtl number Reynolds number Schmidt number Stanton number Weber number

NBi NCo NCv NEu NFo NFr NGz NGr NMa NNu NPe NPr NRe NSc NSt NWe

hL/k (h/k)(µ2/ρ2g)1/3 L3ρ2gλ/kµ∆t gc(−dp)/ρV2 kθ/ρcL2 V2/Lg wc/kL L3ρ2βg∆t/µ2 V/Va hD/k DVρc/k cµ/k DVρ/µ µ/ρDυ h/cVρ LV2ρ/σgc

PROCESS SIMULATION It has been found that these dimensionless groups may be correlated well by an equation of the type hD/k = K(cpµ/k)a(DVρ/µ)b in which K, a, and b are experimentally determined dimensionless constants. However, any other type of algebraic expression or perhaps simply a graphical relation among these three groups that accurately fits the experimental data would be an equally valid manner of expressing Eq. (3-121).

Naturally, other dimensionless groups might have been obtained in the example by employing a different set of five repeating quantities

3-89

that would not form a dimensionless group among themselves. Some of these groups may be found among those presented in Table 3-9. Such a complete set of three dimensionless groups might consist of Stanton, Reynolds, and Prandtl numbers or of Stanton, Peclet, and Prandtl numbers. Also, such a complete set different from that obtained in the preceding example will result from a multiplication of appropriate powers of the Nusselt, Prandtl, and Reynolds numbers. For such a set to be complete, however, it must satisfy the condition that each of the three dimensionless groups be independent of the other two.

PROCESS SIMULATION REFERENCES: Dimian, A., Chem. Eng. Prog. 90: 58–66 (Sept. 1994); Kister, H. Z., “Can We Believe the Simulation Results?” Chem. Eng. Prog., pp. 52–58 (Oct. 2002); Krieger, J. H., Chem. Eng. News 73: 50–61 (Mar. 27, 1995); Mah, R. S. H., Chemical Process Structure and Information Flows, Butterworths (1990); Seader, J. D., Computer Modeling of Chemical Processes, AIChE Monograph Series no. 15 (1985); Seider, W. D., J. D. Seader, and D. R. Lewin, Product and Process Design Principles: Synthesis, Analysis, and Evaluation, 2d ed., Wiley, New York (2004).

CLASSIFICATION Process simulation refers to the activity in which mathematical systems of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this section. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing the start-up of plants and are especially useful for the operation of chemical plants. THERMODYNAMICS The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, “Thermodynamics.” It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vaporliquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. PROCESS MODULES OR BLOCKS At the first level of detail, it is not necessary to know the internal parameters for all the units, since what is desired is just the overall performance. For example, in a heat exchanger design, it suffices to know the heat duty, the total area, and the temperatures of the output streams; the details such as the percentage baffle cut, tube layout, or baffle spacing can be specified later when the details of the proposed plant are better defined. It is important to realize the level of detail modeled by a commercial computer program. For example, a chemical reactor could be modeled as an equilibrium reactor, in which the input stream is brought to a new temperature and pressure and the

output stream is in chemical equilibrium at those new conditions. Or, it may suffice to simply specify the conversion, and the computer program will calculate the outlet compositions. In these cases, the model equations are algebraic ones, and you do not learn the volume of the reactor. A more complicated reactor might be a stirred tank reactor, and then you would have to specify kinetic information so that the simulation can be made, and one output would be either the volume of the reactor or the conversion possible in a volume you specify. Such models are also composed of sets of algebraic equations. A plug flow reactor is modeled as a set of ordinary differential equations as initialvalue problems, and the computer program must use numerical methods to integrate them. See “Numerical Solution of Ordinary Differential Equations as Initial Value Problems.” Kinetic information must be specified, and one learns the conversion possible in a given reactor volume, or, in some cases, the volume reactor that will achieve a given conversion. The simulation engineer determines what a reactor of a given volume will do for the specified kinetics and reactor volume. The design engineer, though, wants to achieve a certain result and wants to know the volume necessary. Simulation packages are best suited for the simulation engineer, and the design engineer must vary specifications to achieve the desired output. Distillation simulations can be based on shortcut methods, using correlations based on experience, but more rigorous methods involve solving for the vapor-liquid equilibrium on each tray. The shortcut method uses relatively simple equations, and the rigorous method requires solution of huge sets of nonlinear equations. The computation time of the latter is significant, but the rigorous method may be necessary when the chemicals you wish to distill are not well represented in the correlations. Then the designer must specify the number of trays and determine the separation that is possible. This, of course, is not what he or she wants: the number of trays needed to achieve a specified objective. Thus, again, some adjustment of parameters is necessary in a design situation. Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see “Numerical Solution of Ordinary Differential Equations as Initial-Value Problems”). If one wants to model a process unit that has significant flow variation, and possibly some concentration distributions as well, one can consider using computational fluid dynamics (CFD) to do so. These calculations are very time-consuming, though, so that they are often left until the mechanical design of the unit. The exception would occur when the flow variation and concentration distribution had a significant effect on the output of the unit so that mass and energy balances couldn’t be made without it. The process units are described in greater detail in other sections of the Handbook. In each case, parameters of the unit are specified (size, temperature, pressure, area, and so forth). In addition, in a computer simulation, the computer program must be able to take any input to the unit and calculate the output for those parameters. Since the entire calculation is done iteratively, there is no assurance that the

3-90

MATHEMATICS

6 6

6

2

1 Mixer

4 Reactor

5 Separator

3 FIG. 3-67

Prototype flowsheet.

input stream is a “reasonable” one, so that the computer codes must be written to give some sort of output even when the input stream is unreasonable. This difficulty makes the iterative process even more complicated. PROCESS TOPOLOGY A chemical process usually consists of a series of units, such as distillation towers, reactors, and so forth (see Fig. 3-67). If the feed to the process is known and the operating parameters of the units are specified by the user, then one can begin with the first unit, take the process input, calculate the unit output, carry that output to the input of the next unit, and continue the process. However, if the process involves a recycle stream, as nearly all chemical processes do, then when the calculation is begun, it is discovered that the recycle stream is unknown. This situation leads to an iterative process: the flow rates, temperature, and pressure of the unknown recycle stream are guessed, and the calculations proceed as before. When one reaches the end of the process, where the recycle stream is formed to return to the first unit, it is necessary to check to see if the recycle stream is the same as assumed. If not, an iterative procedure must be used to cause convergence. Possible techniques are described in “Numerical Solutions of Nonlinear Equations in One Variable” and “Numerical Solution of Simultaneous Equations.” The direct method (or successive substitution method) just involves calculating around the process over and over. The Wegstein method accelerates convergence for a single variable, and Broyden’s method does the same for multiple variables. The Newton method can be used provided there is some way to calculate the derivatives (possibly by using a numerical derivative). Optimization methods can also be used (see “Optimization” in this section). In the description given here, the recycle stream is called the tear stream: this is the stream that must be guessed to begin the calculation. When there are multiple recycle streams, convergence is even more difficult, since more guesses are necessary, and what happens in one recycle stream may cause difficulties for the guesses in other recycle streams. See Seader (1985) and Mah (1990). It is sometimes desired to control some stream by varying an operating parameter. For example, in a reaction/separation system, if there is an impurity that must be purged, a common objective is to set the purge fraction so that the impurity concentration into the reactor is kept at some moderate value. Commercial packages contain procedures for

doing this using what are often called control blocks. However, this can also make the solution more difficult to find. An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or to give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (tens of thousands) of nonlinear algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment, provided efficient software is available. Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in “Numerical Solution of Ordinary Differential Equations as Initial-Value Problems.” It must be realized, though, that a dynamic calculation can also be time-consuming, and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. COMMERCIAL PACKAGES Computer programs are provided by many companies, and the models range from empirical models to deterministic models. For example, if one wanted to know the pressure drop in a piping network, one would normally use a correlation for friction factor as a function of Reynolds number to calculate the pressure drop in each segment. A sophisticated turbulence model of fluid flow is not needed in that case. As computers become faster, however, more and more models are deterministic. Since the commercial codes have been used by many customers, the data in them have been verified, but possibly not for the case you want to solve. Thus, you must test the thermodynamics correlations carefully. In 2005, there were a number of computer codes, but the company names change constantly. Here are a few of them for process simulation: Aspen Tech (Aspen Plus), Chemstations (CHEMCAD), Honeywell (UniSim Design), ProSim (ProSimPlus), and SimSci-Esseor (Pro II). The CAPE-OPEN project is working to make details as transferable as possible.

Section 4

Thermodynamics

Hendrick C. Van Ness, D.Eng. Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society (Section Coeditor) Michael M. Abbott, Ph.D. Deceased; Professor Emeritus, Howard P. Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute (Section Coeditor)*

INTRODUCTION Postulate 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 2 (First Law of Thermodynamics) . . . . . . . . . . . . . . . . . . . . . . Postulate 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 4 (Second Law of Thermodynamics) . . . . . . . . . . . . . . . . . . . . Postulate 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-4 4-4 4-5 4-5 4-5

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Constant-Composition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U, H, and S as Functions of T and P or T and V . . . . . . . . . . . . . . . . . The Ideal Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-6 4-6 4-7 4-7

Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Throttling Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbines (Expanders) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: LNG Vaporization and Compression . . . . . . . . . . . . . . . .

4-15 4-15 4-16 4-16 4-16 4-17

4-17 4-18 4-18 4-19 4-19 4-19 4-19 4-20 4-20 4-21 4-21 4-21 4-21 4-22 4-23 4-26

4-26 4-27 4-27 4-27 4-28 4-28 4-28 4-29

OTHER PROPERTY FORMULATIONS Liquid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Vapor Phase Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-13 4-13

SYSTEMS OF VARIABLE COMPOSITION Partial Molar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs-Duhem Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Equation-of-State Parameters . . . . . . . . . . . . . . . . . . . . Partial Molar Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Gas Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fugacity and Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Fugacity Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Solution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Property Changes of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Property Relations Based on the Gibbs Energy. . . . . . . . Fundamental Residual-Property Relation. . . . . . . . . . . . . . . . . . . . . . Fundamental Excess-Property Relation . . . . . . . . . . . . . . . . . . . . . . . Models for the Excess Gibbs Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Binary Liquid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .

THERMODYNAMICS OF FLOW PROCESSES Mass, Energy, and Entropy Balances for Open Systems . . . . . . . . . . . . Mass Balance for Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Energy Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Balances for Steady-State Flow Processes . . . . . . . . . . . . . . . Entropy Balance for Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Equations of Balance for Open Systems . . . . . . . . . . . . Applications to Flow Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duct Flow of Compressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-14 4-14 4-14 4-14 4-14 4-15 4-15 4-15

EQUILIBRIUM Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Application of the Phase Rule . . . . . . . . . . . . . . . . . . . . . Duhem’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor/Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma/Phi Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Raoult’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Dew and Bubble Point Calculations . . . . . . . . . . . . . . . .

PROPERTY CALCULATIONS FOR GASES AND VAPORS Evaluation of Enthalpy and Entropy in the Ideal Gas State . . . . . . . . . 4-8 Residual Enthalpy and Entropy from PVT Correlations . . . . . . . . . . . . 4-9 Virial Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Cubic Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 Pitzer’s Generalized Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

*Dr. Abbott died on May 31, 2006. This, his final contribution to the literature of chemical engineering, is deeply appreciated, as are his earlier contributions to the handbook. 4-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

4-2

THERMODYNAMICS

Data Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solute/Solvent Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K Values, VLE, and Flash Calculations . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Flash Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation-of-State Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extrapolation of Data with Temperature. . . . . . . . . . . . . . . . . . . . . . . Example 5: VLE at Several Temperatures . . . . . . . . . . . . . . . . . . . . . Liquid/Liquid and Vapor/Liquid/Liquid Equilibria . . . . . . . . . . . . . . . . Chemical Reaction Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Property Changes of Reaction . . . . . . . . . . . . . . . . . . . . . . .

4-30 4-31 4-31 4-32 4-32 4-34 4-34 4-35 4-35 4-35 4-35

Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Single-Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . Complex Chemical Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . .

4-36 4-37 4-38

THERMODYNAMIC ANALYSIS OF PROCESSES Calculation of Ideal Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lost Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Steady-State Steady-Flow Proceses. . . . . . . . . . . . . . . . . . . . Example 7: Lost-Work Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-38 4-39 4-39 4-40

THERMODYNAMICS

4-3

Nomenclature and Units Correlation- and application-specific symbols are not shown. Symbol A A âi ⎯a i B ⎯ Bi Bˆ C Cˆ D B′ C′ D′ Bij Cijk CP CV fi fˆi G g g H Ki Kj k1 M M Mi ⎯ Mi MR ME ⎯E Mi ∆M ∆M°j m ⋅ m n n⋅ ni P

Definition Molar (or unit-mass) Helmholtz energy Cross-sectional area in flow Activity of species i in solution Partial parameter, cubic equation of state 2d virial coefficient, density expansion Partial molar second virial coefficient Reduced second virial coefficient 3d virial coefficient, density expansion Reduced third virial coefficient 4th virial coefficient, density expansion 2d virial coefficient, pressure expansion 3d virial coefficient, pressure expansion 4th virial coefficient, pressure expansion Interaction 2d virial coefficient Interaction 3d virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Fugacity of pure species i Fugacity of species i in solution Molar (or unit-mass) Gibbs energy Acceleration of gravity ≡ GE/RT Molar (or unit-mass) enthalpy

SI units J/mol [J/kg] m Dimensionless

Btu/lb mol [Btu/lbm] ft2 Dimensionless

cm3/mol

cm3/mol

2

3

cm /mol

cm3/mol

cm6/mol2

cm6/mol2

cm9/mol3

cm9/mol3

kPa−1

kPa−1

kPa−2

kPa−2

kPa−3

kPa−3

cm3/mol

cm3/mol

cm6/mol2

cm6/mol2

J/(mol·K)

Btu/(lb·mol·R)

J/(mol·K)

Btu/(lb·mol·R)

kPa kPa J/mol [J/kg]

psi psi Btu/(lb·mol) [Btu/lbm] ft/s2 Dimensionless Btu/(lb·mol) [Btu/lbm] Dimensionless Dimensionless

m/s2 Dimensionless J/mol [J/kg]

Equilibrium K value, yi /xi Dimensionless Equilibrium constant for Dimensionless chemical reaction j Henry’s constant for kPa solute species 1 Molar or unit-mass solution property (A, G, H, S, U, V) Mach number Dimensionless Molar or unit-mass pure-species property (Ai, Gi, Hi, Si, Ui, Vi) Partial property of species i in⎯ solution ⎯ ⎯ ⎯ ⎯ ⎯ (Ai, Gi, Hi, Si, Ui, Vi) Residual thermodynamic property (AR, GR, HR, SR, UR, VR) Excess thermodynamic property (AE, GE, HE, SE, UE, VE) Partial molar excess thermodynamic property Property change of mixing (∆ A, ∆G, ∆H, ∆S, ∆U, ∆V) Standard property change of reaction j (∆Gj°, ∆Hj°, ∆CP°) Mass kg Mass flow rate kg/s Number of moles Molar flow rate Number of moles of species i Absolute pressure kPa j

U.S. Customary System units

psi

Symbol

Definition

Pisat

Saturation or vapor pressure of species i Heat Volumetric flow rate Rate of heat transfer Universal gas constant Molar (or unit-mass) entropy

Q q ⋅ Q R S ⋅ SG T Tc U u V W Ws ⋅ Ws xi xi

psi

J m3/s J/s J/(mol·K) J/(mol·K) [J/(kg·K)] J/(K·s)

Btu ft3/s Btu/s Btu/(lb·mol·R) Btu/(lb·mol·R) [Btu/(lbm·R)] Btu/(R·s)

K K J/mol [J/kg]

J J J/s

R R Btu/(lb·mol) [Btu/lbm] ft/s ft3/(lb·mol) [ft3/lbm] Btu Btu Btu/s

Dimensionless m

Dimensionless ft

m/s m3/mol [m3/kg]

Z z E id ig l lv R t v ∞

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase Denotes phase transition, liquid to vapor Denotes residual thermodynamic property Denotes total value of property Denotes vapor phase Denotes value at infinite dilution

c cv fs n r rev

Denotes value for the critical state Denotes the control volume Denotes flowing streams Denotes the normal boiling point Denotes a reduced value Denotes a reversible process

α, β β εj

As superscripts, identify phases Volume expansivity Reaction coordinate for reaction j Defined by Eq. (4-196) Heat capacity ratio CP /CV Activity coefficient of species i in solution Isothermal compressibility Chemical potential of species i Stoichiometric number of species i in reaction j Molar density As subscript, denotes a heat reservoir Defined by Eq. (4-304) Fugacity coefficient of pure species i Fugacity coefficient of species i in solution Acentric factor

yi

Superscripts

Subscripts

Greek Letters

Γi(T) γ γi κ µi νi,j ρ σ Φi φi φˆ i psi

Rate of entropy generation, Eq. (4-151) Absolute temperature Critical temperature Molar (or unit-mass) internal energy Fluid velocity Molar (or unit-mass) volume

U.S. Customary System units

kPa

Work Shaft work for flow process Shaft power for flow process Mole fraction in general Mole fraction of species i in liquid phase Mole fraction of species i in vapor phase Compressibility factor Elevation above a datum level

Dimensionless

lbm lbm/s

SI units

ω

K−1 mol

°R−1 lb·mol

J/mol Dimensionless Dimensionless

Btu/(lb·mol) Dimensionless Dimensionless

kPa−1 J/mol Dimensionless

psi−1 Btu/(lb·mol) Dimensionless

mol/m3

lb·mol/ft3

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless

Dimensionless

GENERAL REFERENCES: Abbott, M. M., and H. C. Van Ness, Schaum’s Outline of Theory and Problems of Thermodynamics, 2d ed., McGraw-Hill, New York, 1989. Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001. Prausnitz, J. M., R. N. Lichtenthaler, and E. G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 3d ed., Prentice-Hall PTR, Upper Saddle River, N.J., 1999. Sandler, S. I., Chemical and Engineering Thermodynamics, 3d ed.,

Wiley, New York, 1999. Smith, J. M., H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGrawHill, New York, 2005. Tester, J. W., and M. Modell, Thermodynamics and Its Applications, 3d ed., Prentice-Hall PTR, Upper Saddle River, N.J., 1997. Van Ness, H. C., and M. M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, McGraw-Hill, New York, 1982.

INTRODUCTION Thermodynamics is the branch of science that lends substance to the principles of energy transformation in macroscopic systems. The general restrictions shown by experience to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive; they cannot be derived from anything more basic. The first law of thermodynamics states that energy is conserved, that although it can be altered in form and transferred from one place to another, the total quantity remains constant. Thus the first law of thermodynamics depends on the concept of energy, but conversely energy is an essential thermodynamic function because it allows the first law to be formulated. This coupling is characteristic of the primitive concepts of thermodynamics. The words system and surroundings are similarly coupled. A system can be an object, a quantity of matter, or a region of space, selected for study and set apart (mentally) from everything else, which is called the surroundings. An envelope, imagined to enclose the system and to separate it from its surroundings, is called the boundary of the system. Attributed to this boundary are special properties which may serve either to isolate the system from its surroundings or to provide for interaction in specific ways between the system and surroundings. An isolated system exchanges neither matter nor energy with its surroundings. If a system is not isolated, its boundaries may permit exchange of matter or energy or both with its surroundings. If the exchange of matter is allowed, the system is said to be open; if only energy and not matter may be exchanged, the system is closed (but not isolated), and its mass is constant. When a system is isolated, it cannot be affected by its surroundings. Nevertheless, changes may occur within the system that are detectable with measuring instruments such as thermometers and pressure gauges. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. For a closed system which interacts with its surroundings, a final static condition may likewise be reached such that the system is not only internally at equilibrium but also in external equilibrium with its surroundings. The concept of equilibrium is central in thermodynamics, for associated with the condition of internal equilibrium is the concept of state. A system has an identifiable, reproducible state when all its properties, such as temperature T, pressure P, and molar volume V, are fixed. The concepts of state and property are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see postulates 1 and 3 following) is recognized much more indirectly. The number of properties for which values must be specified in order to fix the state of a system depends on the nature of the system, and is ultimately determined from experience. When a system is displaced from an equilibrium state, it undergoes a process, a change of state, which continues until its properties attain new equilibrium values. During such a process, the system may be caused to interact with its surroundings so as to interchange energy in the forms of heat and work and so to produce in the system changes considered desirable for one reason or another. A process that proceeds so that the system is never displaced more than differentially from an equilibrium state is said to be reversible, because such a process can be reversed at any point by an infinitesimal change in external conditions, causing it to retrace the initial path in the opposite direction. 4-4

Thermodynamics finds its origin in experience and experiment, from which are formulated a few postulates that form the foundation of the subject. The first two deal with energy. POSTULATE 1 There exists a form of energy, known as internal energy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to the measurable coordinates that characterize the system. POSTULATE 2 (FIRST LAW OF THERMODYNAMICS) The total energy of any system and its surroundings is conserved. Internal energy is quite distinct from such external forms as the kinetic and potential energies of macroscopic bodies. Although it is a macroscopic property, characterized by the macroscopic coordinates T and P, internal energy finds its origin in the kinetic and potential energies of molecules and submolecular particles. In applications of the first law of thermodynamics, all forms of energy must be considered, including the internal energy. It is therefore clear that postulate 2 depends on postulate 1. For an isolated system the first law requires that its energy be constant. For a closed (but not isolated) system, the first law requires that energy changes of the system be exactly compensated by energy changes in the surroundings. For such systems energy is exchanged between a system and its surroundings in two forms: heat and work. Heat is energy crossing the system boundary under the influence of a temperature difference or gradient. A quantity of heat Q represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. The convention with respect to sign makes numerical values of Q positive when heat is added to the system and negative when heat leaves the system. Work is again energy in transit between a system and its surroundings, but resulting from the displacement of an external force acting on the system. Like heat, a quantity of work W represents an amount of energy, and is not a property of the system. The sign convention, analogous to that for heat, makes numerical values of W positive when work is done on the system by the surroundings and negative when work is done on the surroundings by the system. When applied to closed (constant-mass) systems in which only internal-energy changes occur, the first law of thermodynamics is expressed mathematically as dUt = dQ + dW

(4-1)

t

where U is the total internal energy of the system. Note that dQ and dW, differential quantities representing energy exchanges between the system and its surroundings, serve to account for the energy change of the surroundings. On the other hand, dUt is directly the differential change in internal energy of the system. Integration of Eq. (4-1) gives for a finite process ∆Ut = Q + W

(4-2)

where ∆U is the finite change given by the difference between the final and initial values of Ut. The heat Q and work W are finite quantities of heat and work; they are not properties of the system or functions of the thermodynamic coordinates that characterize the system. t

VARIABLES, DEFINITIONS, AND RELATIONSHIPS POSTULATE 3 There exists a property called entropy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to the measurable coordinates that characterize the system. For reversible processes, changes in this property may be calculated by the equation dQrev dSt =  (4-3) T t where S is the total entropy of the system and T is the absolute temperature of the system.

POSTULATE 4 (SECOND LAW OF THERMODYNAMICS) The entropy change of any system and its surroundings, considered together, resulting from any real process is positive, approaching zero when the process approaches reversibility. In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. The second law requires that the entropy of an isolated system either increase or, in the limit where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part, or that in the limit where the process is reversible, the total entropy of the system plus its surroundings be constant. The fundamental thermodynamic properties that arise in connection with the first and second laws of thermodynamics are internal energy and entropy. These properties together with the two laws for which they are essential apply to all types of systems. However, different types of systems are characterized by different sets of measurable coordinates or variables. The type of system most commonly encountered in chemical technology is one for which the primary characteristic variables are temperature T, pressure P, molar volume V, and composition, not all of which are necessarily independent. Such systems are usually made up of fluids (liquid or gas) and are called PVT systems.

4-5

For closed systems of this kind the work of a reversible process may always be calculated from dWrev = −PdV t

(4-4)

where P is the absolute pressure and Vt is the total volume of the system. This equation follows directly from the definition of mechanical work. POSTULATE 5 The macroscopic properties of homogeneous PVT systems at internal equilibrium can be expressed as functions of temperature, pressure, and composition only. This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system serves as a satisfactory model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., electric, magnetic, or gravitational) are negligible and that surface and viscous shear effects are unimportant. Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, and the functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume V, which can be measured, and leads immediately to the conclusion that there exists an equation of state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primary tool in applications of thermodynamics. Postulate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. These molar or unit-mass properties, represented by the plain symbols V, U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V t, U t, St) do depend on system size and are extensive. For a system containing n mol of fluid, Mt = nM, where M is a molar property. Applications of the thermodynamic postulates necessarily involve the abstract quantities of internal energy and entropy. The solution of any problem in applied thermodynamics is therefore found through these quantities.

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider a single-phase closed system in which there are no chemical reactions. Under these restrictions the composition is fixed. If such a system undergoes a differential, reversible process, then by Eq. (4-1)

where subscript n indicates that all mole numbers ni (and hence n) are held constant. Comparison with Eq. (4-5) shows that ∂(nU)

  ∂(nS) 

dUt = dQrev + dWrev Substitution for dQrev and dWrev by Eqs. (4-3) and (4-4) gives dUt = T dSt − P dVt Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It is equally well written as d(nU) = T d(nS) − P d(nV)

(4-5)

where n is the number of moles of fluid in the system and is constant for the special case of a closed, nonreacting system. Note that n  n1 + n2 + n3 + … = ni i

where i is an index identifying the chemical species present. When U, S, and V represent specific (unit-mass) properties, n is replaced by m. Equation (4-5) shows that for a single-phase, nonreacting, closed system, nU = u(nS, nV). ∂(nU) ∂(nU) Then d(nU) =  d(nS) +  d(nV) ∂(nS) nV,n ∂(nV) nS,n









∂(nU)

  ∂(nV) 

= T and

nV,n

= −P

nS,n

For an open single-phase system, we assume that nU = U (nS, nV, n1, n2, n3, . . .). In consequence, ∂(nU) d(nU) =  ∂(nS)





nV,n

∂(nU) d(nS) +  ∂(nV)





nS,n

∂(nU) d(nV) +  ∂ni i





dni

nS,nV,nj

where the summation is over all species present in the system and subscript nj indicates that all mole numbers are held constant except the ith. Define ∂(nU) µi   ∂ni nS,nV,nj





The expressions for T and −P of the preceding paragraph and the definition of µi allow replacement of the partial differential coefficients in the preceding equation by T, −P, and µi. The result is Eq. (4-6) of Table 4-1, where important equations of this section are collected. Equation (4-6) is the fundamental property relation for single-phase PVT systems, from which all other equations connecting properties of

4-6

THERMODYNAMICS

TABLE 4-1

Mathematical Structure of Thermodynamic Property Relations

Primary thermodynamic functions U = TS − PV + xiµi

(4-7)

H  U + PV

(4-8)

For homogeneous systems of constant composition

Fundamental property relations d(nU) = T d(nS) − P d(nV) + µi dni

i

dU = T dS − P dV

(4-6)

Maxwell equations

 = − ∂S

(4-14)

i

d(nH) = T d(nS) + nV dP + µi dni

(4-11)

d(nA) = − nS dT − P d(nV) + µi dni

(4-12)

d(nG) = − nS dT + nV dP + µi dni

(4-13)

dH = T dS + V dP

(4-9) (4-10)

∂P

∂H dH =  ∂T

∂H dT +  ∂P P



∂S dS =  ∂T

 ∂S

∂U dU =  ∂T



∂S dS =  ∂T

V

∂U dT +  ∂V



∂S dT +  ∂V

dV

(4-24)

T



dV

∂P

T

∂S = T  ∂P

T

∂T

∂U

V

∂S = T  ∂T

V

∂U

∂S = T  ∂V

T

∂H

(4-23)

T

V



dP

P



(4-22)

T

 dT + ∂P P

dP

∂S = T  ∂T P

∂V

∂V

(4-25)

T

T

(4-28)

∂V + V = V − T  ∂T



∂P − P = T  ∂T



V

−P

∂V dH = CP dT + V − T  ∂T



  dP

(4-32)

P

(4-29)

C ∂V dS = P dT −  dP T ∂T P

(4-33)

(4-30)

∂P dU = CV dT + T  ∂T

(4-34)

P

= CV



(4-21)

T

Total derivatives

P



∂S

P

 =C 

(4-20)

T

∂T = − ∂P

(4-17)

Partial derivatives ∂H  ∂T

∂S

V

dG = −S dT + V dP

(4-19)

P

∂T = ∂V

(4-16)

i

U, H, and S as functions of T and P or T and V

∂V

S

dA = −S dT − P dV

(4-18)

V

∂T

i

G  H − TS

S

∂P = ∂S

(4-15)

i

A  U − TS

∂P

∂T  ∂V

(4-31)



  − P dV V

CV ∂P dS =  dT +  dV T ∂T V



(4-35)

U  Internal energy; H  enthalpy; A  Helmoholtz energy; G  Gibbs energy.

such systems are derived. The quantity µ i is called the chemical potential of species i, and it plays a vital role in the thermodynamics of phase and chemical equilibria. Additional property relations follow directly from Eq. (4-6). Because ni = xin, where xi is the mole fraction of species i, this equation may be rewritten as d(nU) − T d(nS) + P d(nV) − µi d(xin) = 0 i

Expansion of the differentials and collection of like terms yield

dU − T dS + P dV − µ dx n + U − TS + PV − x µ dn = 0 i

i

i i

i

i

Because n and dn are independent and arbitrary, the terms in brackets must separately be zero. This provides two useful equations: dU = T dS − P dV + µi dxi i

U = TS − PV + xiµi i

The first is similar to Eq. (4-6). However, Eq. (4-6) applies to a system of n mol where n may vary. Here, however, n is unity and invariant. It is therefore subject to the constraints i xi = 1 and i dxi = 0. Mole fractions are not independent of one another, whereas the mole numbers in Eq. (4-6) are. The second of the preceding equations dictates the possible combinations of terms that may be defined as additional primary functions. Those in common use are shown in Table 4-1 as Eqs. (4-7) through (4-10). Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication of Eq. (4-8) of Table 4-1 by n and differentiation yield the general expression d(nH) = d(nU) + P d(nV) + nV dP Substitution for d(nU) by Eq. (4-6) reduces this result to Eq. (4-11). The total differentials of nA and nG are obtained similarly and are expressed by Eqs. (4-12) and (4-13). These equations and Eq. (4-6) are equivalent forms of the fundamental property relation, and appear under that heading in Table 4-1. Each expresses a total property—nU, nH, nA, and nG—as a function of a particular set of independent

variables, called the canonical variables for the property. The choice of which equation to use in a particular application is dictated by convenience. However, the Gibbs energy G is special, because of its relation to the canonical variables T, P, and {ni}, the variables of primary interest in chemical processing. Another set of equations results from the substitutions n = 1 and ni = xi. The resulting equations are of course less general than their parents. Moreover, because the mole fractions are not independent, mathematical operations requiring their independence are invalid. CONSTANT-COMPOSITION SYSTEMS For 1 mol of a homogeneous fluid of constant composition, Eqs. (4-6) and (4-11) through (4-13) simplify to Eqs. (4-14) through (4-17) of Table 4-1. Because these equations are exact differential expressions, application of the reciprocity relation for such expressions produces the common Maxwell relations as described in the subsection “Multivariable Calculus Applied to Thermodynamics” in Sec. 3. These are Eqs. (4-18) through (4-21) of Table 4-1, in which the partial derivatives are taken with composition held constant. U, H, and S as Functions of T and P or T and V At constant composition, molar thermodynamic properties can be considered functions of T and P (postulate 5). Alternatively, because V is related to T and P through an equation of state, V can serve rather than P as the second independent variable. The useful equations for the total differentials of U, H, and S that result are given in Table 4-1 by Eqs. (4-22) through (4-25). The obvious next step is substitution for the partial differential coefficients in favor of measurable quantities. This purpose is served by definition of two heat capacities, one at constant pressure and the other at constant volume: ∂H C P   ∂T

P

∂U CV   ∂T

V

 

(4-26) (4-27)

Both are properties of the material and functions of temperature, pressure, and composition.

VARIABLES, DEFINITIONS, AND RELATIONSHIPS Equation (4-15) of Table 4-1 may be divided by dT and restricted to constant P, yielding (∂H/∂T)P as given by the first equality of Eq. (4-28). Division of Eq. (4-15) by dP and restriction to constant T yield (∂H/∂P)T as given by the first equality of Eq. (4-29). Equation (4-28) is completed by Eq. (4-26), and Eq. (4-29) is completed by Eq. (4-21). Similarly, equations for (∂U/∂T)V and (∂U/∂V)T derive from Eq. (4-14), and these with Eqs. (4-27) and (4-20) yield Eqs. (4-30) and (4-31) of Table 4-1. Equations (4-22), (4-26), and (4-29) combine to yield Eq. (4-32); Eqs. (4-23), (4-28), and (4-21) to yield Eq. (4-33); Eqs. (4-24), (4-27), and (4-31) to yield Eq. (4-34); and Eqs. (4-25), (4-30), and (4-20) to yield Eq. (4-35). Equations (4-32) and (4-33) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. Equations (4-34) and (4-35) are general expressions for the internal energy and entropy of homogeneous fluids at constant composition as functions of temperature and molar volume. The coefficients of dT, dP, and dV are all composed of measurable quantities. The Ideal Gas Model An ideal gas is a model gas comprising imaginary molecules of zero volume that do not interact. Its PVT behavior is represented by the simplest of equations of state PVig = RT, where R is a universal constant, values of which are given in Table 1-9. The following partial derivatives, all taken at constant composition, are obtained from this equation: ∂P

∂Vig

 ∂T

Vig

=  =   ∂T P T

R P =  =  Vig T V

R

P

∂P

 ∂V

T

∂Uig

∂Hig

T

∂Sig

 ∂P

=0

T

R = − P T

∂Sig

 ∂V

T

R =  Vig

Moreover, Eqs. (4-32) through (4-35) become dHig = CPig dT

CPig R dSig =  dT −  dP T P

dU ig = CigV dT

CVig R dSig =  dT +  dV T Vig

(4-36)

The ideal gas state properties of mixtures are directly related to the ideal gas state properties of the constituent pure species. For those ig ig properties that are independent of P—Uig, Hig, CV , and CP —the mixture property is the sum of the properties of the pure constituent species, each weighted by its mole fraction: M = yiM ig

ig i

Z = 1.02

1

Pr

Z = 0.98

0.1

0.01

0.001 0

1

2 Tr

3

4

Region where Z lies between 0.98 and 1.02, and the ideal-gas equation is a reasonable approximation. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 104, McGraw-Hill, New York (2005).] FIG. 4-1

For the Gibbs energy, Gig = Hig − TSig; whence by Eqs. (4-37) and (4-38):

In these equations Vig, Uig, CVig, Hig, CPig, and Sig are ideal gas state values—the values that a PVT system would have were the ideal gas equation the true equation of state. They apply equally to pure species and to constant-composition mixtures, and they show that Uig, CVig, Hig, and CPig, are functions of temperature only, independent of P and V. The entropy, however, is a function of both T and P or of both T and V. Regardless of composition, the ideal gas volume is given by Vig = RT/P, and it provides the basis for comparison with true molar volumes through the compressibility factor Z. By definition, V V PV Z =  =  Vig RTP RT

10

P = − Vig

The first two of these relations when substituted appropriately into Eqs. (4-29) and (4-31) of Table 4-1 lead to very simple expressions for ideal gases: =   ∂V  ∂P

4-7

(4-37)

Gig = yiGigi + RT yi ln yi i

(4-39)

i

The ideal gas model may serve as a reasonable approximation to reality under conditions indicated by Fig. 4-1. Residual Properties The differences between true and ideal gas state properties are defined as residual properties MR: MR  M − Mig

(4-40)

where M is the molar value of an extensive thermodynamic property of a fluid in its actual state and Mig is its corresponding ideal gas state value at the same T, P, and composition. Residual properties depend on interactions between molecules and not on characteristics of individual molecules. Because the ideal gas state presumes the absence of molecular interactions, residual properties reflect deviations from ideality. The most commonly used residual properties are as follows: Residual volume VR  V − Vig Residual enthalpy HR  H − Hig R ig Residual entropy S  S − S Residual Gibbs energy GR  G − Gig

i

ig

where M can represent any of the properties listed. For the entropy, which is a function of both T and P, an additional term is required to account for the difference in partial pressure of a species between its pure state and its state in a mixture: Sig = yiSigi − R yi ln yi i

i

(4-38)

Useful relations connecting these residual properties derive from Eq. (4-17), an alternative form of which follows from the mathematical identity:

 RT

G  1 dG − G dT d   2 RT

RT

4-8

THERMODYNAMICS

Substitution for dG by Eq. (4-17) and for G by Eq. (4-10) gives, after algebraic reduction,

 RTG

V RT

H RT

d  =  dP − 2 dT

(4-41)

This equation may be written for the special case of an ideal gas and subtracted from Eq. (4-41) itself, yielding R

R



(4-42)

VR ∂(GR/RT)  =  RT ∂P



(4-43)

∂(GRRT) HR  = −T  ∂T RT



(4-44)





and

P

ZRT RT RT VR  V − Vig =  −  =  (Z − 1) P P P



(constant T)

Integration from P = 0 to arbitrary pressure P gives dP  (Z − 1)  P P

(constant T)

0

0

P

(4-45)

dP  P

(constant T)

(4-46)

SR HR GR  =  −  R RT RT

(4-47)

Equations (4-45) through (4-47) provide the basis for calculation of residual properties from PVT correlations. They may be put into generalized form by substitution of the relationships T = TcTr dT = Tc dTr

P = Pc Pr dP = Pc dPr The resulting equations are

This equation in combination with a rearrangement of Eq. (4-43) yields GR VR dP d  =  dP = (Z − 1)  RT RT P

P

Because G = H − TS and Gig = Hig − TSig, then by difference, GR = HR − TSR, and

T

Equation (4-43) provides a direct link to PVT correlations through the compressibility factor Z as given by Eq. (4-36). Thus, with V = ZRT/P,

GR  = RT

∂Z   ∂T

HR  = −T RT

R

G V H d  =  dP − 2 dT RT RT RT As a consequence,

Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 210–211, McGraw-Hill, New York (2005)] show that it is permissible here to set the lower limit of integration (GR/RT)P=0 equal to zero. Note also that the integrand (Z − 1)/P remains finite as P → 0. Differentiation of Eq. (4-45) with respect to T in accord with Eq. (4-44) gives

GR  = RT



Pr

0

dP (Z − 1) r Pr



(4-48)

Pr ∂Z HR dP  r (4-49)  = −Tr2 ∂Tr P Pr 0 RTc The terms on the right sides of these equations depend only on the upper limit Pr of the integrals and on the reduced temperature at which they are evaluated. Thus, values of GR/RT and HR/RTc may be determined once and for all at any reduced temperature and pressure from generalized compressibility factor data. r

PROPERTY CALCULATIONS FOR GASES AND VAPORS The most satisfactory calculation procedure for the thermodynamic properties of gases and vapors is based on ideal gas state heat capacities and residual properties. Of primary interest are the enthalpy and entropy; these are given by rearrangement of the residual property definitions: H = Hig + HR and S = Sig + SR These are simple sums of the ideal gas and residual properties, evaluated separately. EVALUATION OF ENTHALPY AND ENTROPY IN THE IDEAL GAS STATE For the ideal gas state at constant composition: dHig = CigP dT

dT dP and dSig = CigP  − R  T P

Integration from an initial ideal gas reference state at conditions T0 and P0 to the ideal gas state at T and P gives Hig = Hig0 + Sig = Sig0 +

C T

T0

ig P

C T

T0

ig P

dT

H=H +

C T

T0

ig P

dT + H

R

T

T0

ig P

i

Cig (4-52) P = A + BT + CT 2 + DT −2 R where A, B, C, and D are constants characteristic of the particular gas, and either C or D is zero. The ratio CPig /R is dimensionless; thus the units of CPig are those of R. Data for ideal gas state heat capacities are given for many substances in Table 2-155. Evaluation of the integrals ∫ CPig dT and ∫ (CPig /T) dT is accomplished by substitution for CPig, followed by integration. For temperature limits of T0 and T and with τ  T/T0, the equations that follow from Eq. (4-52) are

 CR dT = AT (τ − 1) + B2 T (τ ig P

T

dT P  − R ln  T P0

2

C D τ−1 − 1) +  T 03 (τ 3 − 1) +   3 T0 τ (4-53)



τ+1 D  RCT dT = A ln τ + BT + CT +   (τ − 1) τ T  2  T

(4-50)

2 0

0

T0

Substitution into the equations for H and S yields ig 0

C

dT P (4-51)  − R ln  + SR T P0 The reference state at T0 and P0 is arbitrarily selected, and the values assigned to Hig0 and Sig0 are also arbitrary. In practice, only changes in H and S are of interest, and fixed reference state values ultimately cancel in their calculation. The ideal gas state heat capacity CPig is a function of T but not of P. For a mixture the heat capacity is simply the molar average iyiCigP . ig Empirical equations relating CP to T are available for many pure gases; a common form is S = Sig0 +

T0

ig P

0

2 0

2

2 0



(4-54)

PROPERTY CALCULATIONS FOR GASES AND VAPORS Equations (4-50) and (4-51) may sometimes be advantageously expressed in alternative form through use of mean heat capacities: H = Hig0 + 〈CigP 〉H(T − T0) + HR

(4-55)

T P S = Sig0 + 〈CigP 〉 S ln  − R ln  + SR T0 P0

(4-56)

where 〈CigP 〉H and 〈CigP 〉S are mean heat capacities specific, respectively, for enthalpy and entropy calculations. They are given by the following equations: 〈CigP 〉H B C D = A +  T0(τ + 1) +  T 20(τ 2 + τ + 1) + 2  R 2 3 τT 0 〈CigP 〉S D = A + BT0 + CT 20 +   R τ 2T 20





τ+1

τ−1

   2  ln τ

(4-57) (4-58)

B =

yi yj Bij

(4-60)

C =

yi yj yk Cijk

(4-61)

j

(4-65) (4-66)

Values can often be found for B, but not so often for C. Generalized correlations for both B and C are given by Meng, Duan, and Li [Fluid Phase Equilibria 226: 109–120 (2004)]. For pressures up to several bars, the two-term expansion in pressure, with B′ given by Eq. (4-65), is usually preferred: Z = 1 + B′P = 1 + BPRT

(4-67)

For supercritical temperatures, it is satisfactory to ever higher pressures as the temperature increases. For pressures above the range where Eq. (4-67) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable: (4-68)

GR BP  =  RT RT

j

k

where yi, yj, and yk are mole fractions for a gas mixture and i, j, and k identify species. The coefficient Bij characterizes a bimolecular interaction between molecules i and j, and therefore Bij = Bji. Two kinds of second virial coefficient arise: Bii and Bjj, wherein the subscripts are the same (i = j), and Bij, wherein they are different (i ≠ j ). The first is a virial coefficient for a pure species; the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients: Ciii, Cjjj, and Ckkk are for the pure species, and Ciij = Ciji = Cjii, . . . are cross coefficients. Although the virial equation itself is easily rationalized on empirical grounds, the mixing rules of Eqs. (4-60) and (4-61) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by

(4-69)

Differentiation of Eq. (4-67) yields ∂Z

=  −    ∂T  dT T RT dB

B

P

P

By Eq. (4-46), HR P B dB  =   −  RT R T dT



(4-59)

The density series virial coefficients B, C, D, . . . depend on temperature and composition only. In practice, truncation is to two or three terms. The composition dependencies of B and C are given by the exact mixing rules

i

B′ = BRT C′ = (C − B2)(RT)2

Equations for residual enthalpy and entropy may be developed from each of these expressions. Consider first Eq. (4-67), which is explicit in volume. Equations (4-45) and (4-46) are therefore applicable. Direct substitution for Z in Eq. (4-45) gives

The residual properties of gases and vapors depend on their PVT behavior. This is often expressed through correlations for the compressibility factor Z, defined by Eq. (4-36). Analytical expressions for Z as functions of T and P or T and V are known as equations of state. They may also be reformulated to give P as a function of T and V or V as a function of T and P. Virial Equations of State The virial equation in density is an infinite series expansion of the compressibility factor Z in powers of molar density ρ (or reciprocal molar volume V−1) about the real gas state at zero density (zero pressure):

i

or three terms, with B′ and C′ depending on temperature and composition only. Moreover, the two sets of coefficients are related:

Z = 1 + Bρ + Cρ2

RESIDUAL ENTHALPY AND ENTROPY FROM PVT CORRELATIONS

Z = 1 + Bρ + Cρ2 + Dρ3 + · · ·

4-9



(4-70)

and by Eq. (4-47), SR P dB  =−   R R dT

(4-71)

An extensive set of three-parameter corresponding-states correlations has been developed by Pitzer and coworkers [Pitzer, Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York (1995)]. Particularly useful is the one for the second virial coefficient. The basic equation is BPc (4-72)  = B0 + ωB1 RTc with the acentric factor defined by Eq. (2-17). For pure chemical species B0 and B1 are functions of reduced temperture only. Substitution for B in Eq. (4-67) by this expression gives P Z = 1 + (B0 + ωB1)r Tr

(4-73)

By differentiation, dB1dTr ∂Z dB0dTr B0 B1  = Pr  − 2 + ωPr  − 2 Tr ∂Tr P Tr Tr Tr



r









dB dBij  =

yi yj  i j dT dT

(4-62)

Upon substitution of these equations into Eqs. (4-48) and (4-49), integration yields

dC dCijk  =

yi yj yk  dT dT i j k

(4-63)

GR Pr  = (B0 + ωB1)  RT Tr

An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real gas state at zero pressure (zero density): (4-64) Z = 1 + B′P + C′P2 + D′P3 + . . . Equation (4-64) is the virial equation in pressure, and B′, C′, D′, . . . are the pressure series virial coefficients. Again, truncation is to two

(4-74)

dB1 HR dB0  = Pr B0 − Tr  + ω B1 − Tr  RTc dTr dTr







(4-75)

The residual entropy follows from Eq. (4-47): dB1 SR dB0  = − Pr  + ω  R dTr dTr





(4-76)

4-10

THERMODYNAMICS

In these equations, B0 and B1 and their derivatives are well represented by Abbott’s correlations [Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics, 3d ed., p. 87, McGraw-Hill, New York (1975)]: 0.422 B0 = 0.083 −  Tr1.6

(4-77)

0.172 B1 = 0.139 −  Tr4.2

(4-78)

dB0 0.675  =  dTr T r2.6

(4-79)

dB1 0.722  =  dTr T r5.2

(4-80)

Although limited to pressures where the two-term virial equation in pressure has approximate validity, these correlations are applicable for most chemical processing conditions. As with all generalized correlations, they are least accurate for polar and associating molecules. Although developed for pure materials, these correlations can be extended to gas or vapor mixtures. Basic to this extension are the mixing rules for the second virial coefficient and its temperature derivative as given by Eqs. (4-60) and (4-62). Values for the cross coefficients Bij, with i ≠ j, and their derivatives are provided by Eq. (4-72) written in extended form: RTcij 0 Bij =  (B + ωij B1) Pcij

(4-81)

where B0, B1, dB0 /dTr, and dB1/dTr are the same functions of Tr as given by Eqs. (4-77) through (4-80). Differentiation produces RTcij dB0 dBij dB1  =   + ωij  dT Pcij dT dT



dBij R dB0 dB1  =   + ωij  dT Pcij dTrij dTrij







(4-82)

where Trij = T/Tcij. The following combining rules for ωij, Tcij, and Pcij are given by Prausnitz, Lichtenthaler, and de Azevedo [Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., pp. 132 and 162, Prentice-Hall, Englewood Cliffs, N.J. (1986)]:

with and

ωi + ωj ωij =  2

(4-83)

Tcij = (TciTcj)12(1 − kij)

(4-84)

ZcijRTcij Pcij =  Vcij

(4-85)

Zci + Zcj Zcij =  2

(4-86)

13 V13 ci + Vcj Vcij =  2





3

(4-87)

In Eq. (4-84), kij is an empirical interaction parameter specific to an i − j molecular pair. When i = j and for chemically similar species, kij = 0. Otherwise, it is a small (usually) positive number evaluated from minimal PVT data or, absence data, set equal to zero. When i = j, all equations reduce to the appropriate values for a pure species. When i ≠ j, these equations define a set of interaction parameters without physical significance. For a mixture, values of Bij and dBij /dT from Eqs. (4-81) and (4-82) are substituted into Eqs. (4-60) and (4-62) to provide values of the mixture second virial coefficient

B and its temperature derivative. Values of HR and SR are then given by Eqs. (4-70) and (4-71). A primary virtue of Abbott’s correlations for second virial coefficients is simplicity. More complex correlations of somewhat wider applicability include those by Tsonopoulos [AIChE J. 20: 263–272 (1974); ibid., 21: 827–829 (1975); ibid., 24: 1112–1115 (1978); Adv. in Chemistry Series 182, pp. 143–162 (1979)] and Hayden and O’Connell [Ind. Eng. Chem. Proc. Des. Dev. 14: 209–216 (1975)]. For aqueous systems see Bishop and O’Connell [Ind. Eng. Chem. Res., 44: 630–633 (2005)]. Because Eq. (4-68) is explicit in P, it is incompatible with Eqs. (4-45) and (4-46), and they must be transformed to make V (or molar density ρ) the variable of integration. The resulting equations are given by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 216–217, McGraw-Hill, New York (2005)]: GR  = Z − 1 − ln Z + RT

 (Z − 1) dρρ ρ

(4-88)

0

  dρρ

ρ ∂Z HR  = Z − 1 − T  0 ∂T RT

(4-89)

ρ

By differentiation of Eq. (4-68), ∂Z

= ρ + ρ  ∂T dT dT dB

dC

2

ρ

Substituting in Eqs. (4-88) and (4-89) for Z by Eq. (4-68) and in Eq. (4-89) for the derivative yields, upon integration and reduction, GR 3  = 2Bρ +  Cρ2 − ln Z RT 2

(4-90)

HR dB T dC  = B − T  ρ + C −   ρ2 RT dT 2 dT







(4-91)

The residual entropy is given by Eq. (4-47). In a process calculation, T and P, rather than T and ρ (or T and V), are usually the favored independent variables. Applications of Eqs. (4-90) and (4-91) therefore require prior solution of Eq. (4-68) for Z or ρ. With Z = P/ρRT, Eq. (4-68) may be written in two equivalent forms: BP CP2 Z 3 − Z 2 −  Z − 2 = 0 RT (RT)



(4-92)

 

B 1 P ρ3 +  ρ2 +  ρ −  = 0 (4-93) C C CRT In the event that three real roots obtain for these equations, only the largest Z (smallest ρ), appropriate for the vapor phase, has physical significance, because the virial equations are suitable only for vapors and gases. Data for third virial coefficients are often lacking, but generalized correlations are available. Equation (4-68) may be rewritten in reduced form as



Pr Pr Z = 1 + Bˆ + Cˆ  Tr Z Tr Z



2

(4-94)

where Bˆ is the reduced second virial coefficient given by Eq. (4-72). Thus by definition, BPc (4-95) Bˆ   = B0 + ωB1 RTc The reduced third virial coefficient Cˆ is defined as CP2c Cˆ   R2Tc2 A Pitzer-type correlation for Cˆ is then written as Cˆ = C0 + ωC1

(4-96)

(4-97)

PROPERTY CALCULATIONS FOR GASES AND VAPORS Correlations for C0 and C1 with reduced temperature are 0.02432 0.00313 C0 = 0.01407 +  −  Tr Tr10.5

(4-98)

0.05539 0.00242 C1 = − 0.02676 +  −  Tr2.7 Tr10.5

(4-99)

The first is given by, and the second is inspired by, Orbey and Vera [AIChE J. 29: 107–113 (1983)]. Equation (4-94) is cubic in Z; with Tr and Pr specified, solution for Z is by iteration. An initial guess of Z = 1 on the right side usually leads to rapid convergence. Another class of equations, known as extended virial equations, was introduced by Benedict, Webb, and Rubin [J. Chem. Phys. 8: 334–345 (1940); 10: 747–758 (1942)]. This equation contains eight parameters, all functions of composition. It and its modifications, despite their complexity, find application in the petroleum and natural gas industries for light hydrocarbons and a few other commonly encountered gases [see Lee and Kesler, AIChE J., 21: 510–527 (1975)]. Cubic Equations of State The modern development of cubic equations of state started in 1949 with publication of the RedlichKwong (RK) equation [Chem. Rev., 44: 233–244 (1949)], and many others have since been proposed. An extensive review is given by Valderrama [Ind. Eng. Chem. Res. 42: 1603–1618 (2003)]. Of the equations published more recently, the two most popular are the Soave-Redlich-Kwong (SRK) equation, a modification of the RK equation [Chem. Eng. Sci. 27: 1197–1203 (1972)] and the PengRobinson (PR) equation [Ind. Eng. Chem. Fundam. 15: 59–64 (1976)]. All are encompased by a generic cubic equation of state, written as RT a(T) P =   −  V − b (V + %b)(V + σb)

(4-100)

For a specific form of this equation, % and σ are pure numbers, the same for all substances, whereas parameters a(T) and b are substancedependent. Suitable estimates of the parameters in cubic equations of state are usually found from values for the critical constants Tc and Pc. The procedure is discussed by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 93–94, McGraw-Hill, New York (2005)], and for Eq. (4-100) the appropriate equations are given as α(Tr)R2Tc2 a(T) = ψ  Pc

(4-101)

RT b = Ω c (4-102) Pc Function α(Tr) is an empirical expression, specific to a particular form of the equation of state. In these equations ψ and Ω are pure numbers, independent of substance and determined for a particular equation of state from the values assigned to % and σ. As an equation cubic in V, Eq. (4-100) has three volume roots, of which two may be complex. Physically meaningful values of V are always real, positive, and greater than parameter b. When T > Tc, solution for V at any positive value of P yields only one real positive root. When T = Tc, this is also true, except at the critical pressure, where three roots exist, all equal to Vc. For T < Tc, only one real positive (liquidlike) root exists at high pressures, but for a range of lower pressures there are three. Here, the middle root is of no significance; the smallest root is a liquid or liquidlike volume, and the largest root is a vapor or vaporlike volume. Equation (4-100) may be rearranged to facilitate its solution either for a vapor or vaporlike volume or for a liquid or liquidlike volume. Vapor: Liquid:

a(T) RT V−b V =  + b −   P (V + %b)(V + σb) P RT − bP − VP V = b + (V + %b)(V + σb)  a(T)



(4-103a)



(4-103b)

4-11

Solution for V is most convenient with the solve routine of a software package. An initial estimate for V in Eq. (4-103a) is the ideal gas value RT/P; for Eq. (4-103b) it is V = b. In either case, iteration is initiated by substituting the estimate on the right side. The resulting value of V on the left is returned to the right side, and the process continues until the change in V is suitably small. Equations for Z equivalent to Eqs. (4-103) are obtained by substituting V = ZRT/P. Vapor: Liquid:

Z−β Z = 1 + β − qβ  (Z + %β)(Z + σβ) 1+β−Z Z = β + (Z + %b)(Z + σb)  qβ



(4-104a)



(4-104b)

where by definition

bP β  RT

(4-105)

and

a(T) q  bRT

(4-106)

These dimensionless quantities provide simplification, and when combined with Eqs. (4-101) and (4-102), they yield Pr β=Ω  Tr

(4-107)

Ψα(Tr) q=  ΩTr

(4-108)

In Eq. (4-104a) the initial estimate is Z = 1; in Eq. (4-104b) it is Z = β. Iteration follows the same pattern as for Eqs. (4-103). The final value of Z yields the volume root through V = ZRT/P. Equations of state, such as the Redlich-Kwong (RK) equation, which expresses Z as a function of Tr and Pr only, yield two-parameter corresponding-states correlations. The SRK equation and the PR equation, in which the acentric factor ω enters through function α(Tr; ω) as an additional parameter, yield three-parameter corresponding-states correlations. The numerical assignments for parameters %, σ, Ω, and Ψ are given in Table 4-2. Expressions are also given for α(Tr; ω) for the SRK and PR equations. As shown by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 218–219, McGrawHill, New York (2005)], Eqs. (4-104) in conjunction with Eqs. (4-88), (4-89), and (4-47) lead to GR  = Z − 1 − ln(Z − β) − qI RT

(4-109)

HR d ln α(Tr)  = Z − 1 +  − 1 qI RT d ln Tr

(4-110)



TABLE 4-2 of State*



Parameter Assignments for Cubic Equations

For use with Eqs. (4-104) through (4-106) Eq. of state

α(Tr)

σ

%

Ω

Ψ

RK (1949) SRK (1972) PR (1976)

Tr−1/2 αSRK(Tr; ω)† αPR(Tr; ω)‡

1 1 1 + 2

0 0 1 − 2

0.08664 0.08664 0.07780

0.42748 0.42748 0.45724

*Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 98, McGraw-Hill, New York (2005). † α SRK(Tr ; ω) = [1 + (0.480 + 1.574ω − 0.176ω 2) (1 − Tr1/2)]2 ‡ α PR(Tr; ω) = [1 + (0.37464 + 1.54226ω − 0.26992ω 2) (1 − Tr1/2)]2

4-12

THERMODYNAMICS SR d ln α(Tr)  = ln (Z − β) +  qI R d ln Tr Z + σβ 1 I =  ln  σ−% Z + %β



where



(4-111)

HR (HR)0 (HR)1  =  + ω RTc RTc RTc

(4-118)

(4-112)

SR (SR)0 (SR)1  =  + ω R R R

(4-119)

Preliminary to application of these equations Z is found by solution of either Eq. (4-104a) or (4-104b). Cubic equations of state may be applied to mixtures through expressions that give the parameters as functions of composition. No established theory prescribes the form of this dependence, and empirical mixing rules are often used to relate mixture parameters to purespecies parameters. The simplest realistic expressions are a linear mixing rule for parameter b and a quadratic mixing rule for parameter a b = xi bi

(4-113)

a =

xi xj aij

(4-114)

i

Pitzer’s original correlations for Z and the derived quantities were determined graphically and presented in tabular form. Since then, analytical refinements to the tables have been developed, with extended range and accuracy. The most popular Pitzer-type correlation is that of Lee and Kesler [AIChE J. 21: 510–527 (1975); see also Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 5th, 6th, and 7th eds., App. E, McGraw-Hill, New York (1996, 2001, 2005)]. These tables cover both the liquid and gas phases and span the ranges 0.3 ≤ Tr ≤ 4.0 and 0.01 ≤ Pr ≤ 10.0. They list values of Z0, Z1, (HR)0/RTc, (HR)1/RTc, (SR)0/R, and (SR)1/R. Lee and Kesler also included a Pitzer-type correlation for vapor pressures: 0 1 ln P sat (4-120) r (Tr) = ln P r (Tr) + ω ln P r (Tr)

i j

with aij = aji. The aij are of two types: pure-species parameters (like subscripts) and interaction parameters (unlike subscripts). Parameter bi is for pure species i. The interaction parameter aij is often evaluated from pure-species parameters by a geometric mean combining rule aij = (aiaj)1/2

(4-115)

These traditional equations yield mixture parameters solely from parameters for the pure constituent species. They are most likely to be satisfactory for mixtures comprised of simple and chemically similar molecules. Pitzer’s Generalized Correlations In addition to the corresponding-states coorelation for the second virial coefficient, Pitzer and coworkers [Thermodynamics, 3d ed., App. 3, McGrawHill, New York (1995)] developed a full set of generalized correlations. They have as their basis an equation for the compressibility factor, as given by Eq. (2-63): Z = Z0 + ωZ1

(2-63)

where Z0 and Z1 are each functions of reduced temperature Tr and reduced pressure Pr. Acentric factor ω is defined by Eq. (2-17). Correlations for Z appear in Sec. 2. Generalized correlations are developed here for the residual enthalpy and residual entropy from Eqs. (4-48) and (4-49). Substitution for Z by Eq. (2-63) puts Eq. (4-48) into generalized form:



GR  = RT

Pr

dP (Z0 − 1) r + ω Pr

0



Pr

dP Z1 r Pr

0

(4-116)

Differentiation of Eq. (2-63) yields ∂Z

∂Z0

=   ∂T ∂T r

Pr

r

Pr

∂Z1 + ω  ∂Tr





Pr

Substitution for (∂Z∂Tr)P in Eq. (4-49) gives r



HR  = − T 2r RTc

Pr

0

∂Z0  ∂Tr



Pr

dPr  − ωT 2r Pr

∂Z   ∂T Pr

1

0

Pr

r

dPr  Pr

(4-117)

GR SR 1 HR  =   −  R RT Tr RTc



By Eq. (4-47),



Combination of Eqs. (4-116) and (4-117) leads to SR  = − R

∂Z  T   ∂T Pr

0

r

0

r

Pr



dP + Z 0 − 1 r − ω Pr

∂Z  T   ∂T Pr

1

r

0

r

Pr



dP + Z1 r Pr

If the first terms on the right sides of Eq. (4-117) and of this equation (including the minus signs) are represented by (HR)0/RTc and (SR)0/R and if the second terms, excluding ω but including the minus signs, are represented by (HR)1/RTc and (SR)1/R, then

where

6.09648 ln P0r (Tr) = 5.92714 −  − 1.28862 ln Tr + 0.169347T 6r Tr (4-121)

15.6875 and ln P1r (Tr) = 15.2518 −  − 13.4721 ln Tr + 0.43577T 6r Tr (4-122) The value of ω to be used with Eq. (4-120) is found from the correlation by requiring that it reproduce the normal boiling point; that is, ω for a particular substance is determined from ln Prsat − ln Pr0(Tr ) ω= (4-123)  ln Plr(Tr ) where Tr is the reduced normal boiling point and Prsat is the reduced vapor pressure corresponding to 1 standard atmosphere (1.01325 bar). Although the tables representing the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters Tc, Pc, and ω for a mixture to the pure-species values and to composition. One such set is given by Eqs. (2-80) through (2-82) in the Seventh Edition of Perry’s Chemical Engineers’ Handbook (1997). These equations define pseudoparameters, so called because the defined values of Tpc, Ppc, and ω have no physical significance for the mixture. The Lee-Kesler correlations provide reliable data for nonpolar and slightly polar gases; errors of less than 3 percent are likely. Larger errors can be expected in applications to highly polar and associating gases. The quantum gases (e.g., hydrogen, helium, and neon) do not conform to the same corresponding-states behavior as do normal fluids. Prausnitz, Lichtenthaler, and de Azevedo [Molecular Thermodynamics of Fluid-Phase Equilibria, 3d ed., pp. 172–173, Prentice-Hall PTR, Upper Saddle River, N.J. (1999)] propose the use of temperaturedependent effective critical parameters. For hydrogen, the quantum gas most commonly found in chemical processing, the recommended equations are n

n

n

n

n

43.6 T c =  K 1 + 21.8/2.016T

(for H2)

(4-124)

20.5 Pc  =  bar 1 + 44.2/2.016T

(for H2)

(4-125)

51.5 Vc  =  cm3mol−1 1 − 9.91/2.016T

(for H2)

(4-126)

where T is absolute temperature in kelvins. Use of these effective critical parameters for hydrogen requires the further specification that ω = 0.

OTHER PROPERTY FORMULATIONS

4-13

OTHER PROPERTY FORMULATIONS LIQUID PHASE Although residual properties have formal reality for liquids as well as for gases, their advantageous use as small corrections to ideal gas state properties is lost. Calculation of property changes for the liquid state are usually based on alternative forms of Eqs. (4-32) through (4-35), shown in Table 4-1. Useful here are the definitons of two liquid-phase properties—the volume expansivity β and the isothermal compressibility κ: 1 ∂V β   V ∂T



(4-127)



∂V dV =  ∂T

∆Mlv  Mv − Ml l

(4-128)

T

∂V

dP  dT +   ∂P P

T

∆Hlv = T ∆Slv

If V is constant,

β =  κ V

∂P  ∂T



dPsat ∆Hlv = T ∆V lv  dT

(4-129)

(4-130)

Because liquid-phase isotherms of P versus V are very steep and closely spaced, both β and κ are small. Moreover (outside the critical region), they are weak functions of T and P and are often assumed constant at average values. Integration of Eq. (4-129) then gives V ln 2 = β(T2 − T1) − κ(P2 − P1) (4-131) V1 Substitution for the partial derivatives in Eqs. (4-32) through (4-35) by Eqs. (4-127) and (4-130) yields

(4-139)

This equation follows from Eq. (4-15), because vaporization at the vapor pressure Psat occurs at constant T. As shown by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 221, McGraw-Hill, New York (2005)] the heat of vaporization is directly related to the slope of the vapor-pressure curve.

This equation in combination with Eqs. (4-127) and (4-128) becomes dV  = β dT − κ dP V

(4-138)

v

where M and M are molar properties for states of saturated liquid and saturated vapor. Some experimental values of the enthalpy change of vaporization ∆Hlv, usually called the latent heat of vaporization, are listed in Table 2-150. The enthalpy change and entropy change of vaporization are directly related:

P

1 ∂V κ−   V ∂P For V = f (T, P),

treatment of this transition is facilitated by definition of property changes of vaporization ∆Mlv:

(4-140)

Known as the Clapeyron equation, this exact thermodynamic relation provides the connection between the properties of the liquid and vapor phases. In application an empirical vapor pressure versus temperature relation is required. The simplest such equation is B ln P sat = A −  T

(4-141)

where A and B are constants for a given chemical species. This equation approximates Psat over its entire temperature range from triple point to critical point. It is also a sound basis for interpolation between reasonably spaced values of T. More satisfactory for general use is the Antoine equation

dH = CP dT + (1 − βT)V dP

(4-132)

dT dS = CP  − βV dP T

B ln P sat = A −  T+C

(4-133)

β dU = CV dT +  T − P dV κ



(4-134)

The Wagner equation is useful for accurate representation of vapor pressure data over a wide temperature range. It expresses the reduced vapor pressure as a function of reduced temperature

β CV dS =  dT +  dV T κ

(4-135)



Integration of these equations is most common from the saturatedliquid state to the state of compressed liquid at constant T. For example, Eqs. (4-132) and (4-133) in integral form become H = Hsat +



P

P

(1 − βT)V dP

(4-136)



(4-137)

sat

S = Ssat −

P

P

sat

β V dP

Again, β and V are weak functions of pressure for liquids, and are often assumed constant at the values for the saturated liquid at temperature T. An alternative treatment of V comes from Eq, (4-131), which for this application can be written V = V exp[−κ(P − P )] sat

sat

LIQUID/VAPOR PHASE TRANSITION The isothermal vaporization of a pure liquid results in a phase change from saturated liquid to saturated vapor at vapor pressure Psat. The

Aτ + Bτ1.5 + Cτ 3 + Dτ 6 ln P sat r =  1−τ where

(4-142)

(4-143)

τ  1 − Tr

and A, B, C, and D are constants. Values of the constants for either the Wagner equation or the Antoine equation are given for many species by Poling, Prausnitz, and O’Connell [The Properties of Gases and Liquids, 5th ed., App. A, McGraw-Hill, New York (2001)]. Latent heats of vaporization are functions of temperature, and experimental values at a particular temperture are often not available. Recourse is then made to approximate methods. Trouton’s rule of 1884 provides a simple check on whether values calculated by other methods are reasonable: ∆H lvn  ∼ 10 RTn Here, Tn is the absolute temperature of the normal boiling point, and ∆Hnlv is the latent heat at this temperature. The units of ∆Hnlv, R, and Tn must be chosen so that ∆Hnlv/RTn is dimensionless. A much more accurate equation is that of Riedel [Chem. Ing. Tech. 26: 679–683 (1954)]: 1.092(ln Pc − 1.013) ∆H lvn  =  0.930 − Tr RTn n

(4-144)

4-14

THERMODYNAMICS

where Pc is the critical pressure in bars and Tr is the reduced temperature at Tn. This equation provides reasonable approximations; errors rarely exceed 5 percent. Estimates of the latent heat of vaporization of a pure liquid at any temperature from the known value at a single temperature may be based on an experimental value or on a value estimated by Eq. (4-144). n

Watson’s equation [Ind. Eng. Chem. 35: 398–406 (1943)] has found wide acceptance: 1 − Tr 0.38 ∆H lv2 =  (4-145)  1 − Tr ∆H lv1 This equation is simple and fairly accurate.



2

1



THERMODYNAMICS OF FLOW PROCESSES The thermodynamics of flow encompasses mass, energy, and entropy balances for open systems, i.e., for systems whose boundaries allow the inflow and outflow of fluids. The common measures of flow are as follows: Mass flow rate m⋅ molar flow rate n⋅ volumetric flow rate q velocity u Also

m˙ = Mn˙

and

q = uA

where M is molar mass. Mass flow rate is related to velocity by m˙ = uAρ

(4-146)

where A is the cross-sectional area of a conduit and ρ is mass density. If ρ is molar density, then this equation yields molar flow rate. Flow rates m⋅, n⋅, and q measure quantity per unit of time. Although velocity u does not represent quantity of flow, it is an important design parameter. MASS, ENERGY, AND ENTROPY BALANCES FOR OPEN SYSTEMS Mass and energy balances for an open system are written with respect to a region of space known as a control volume, bounded by an imaginary control surface that separates it from the surroundings. This surface may follow fixed walls or be arbitrarily placed; it may be rigid or flexible. Mass Balance for Open Systems Because mass is conserved, the time rate of change of mass within the control volume equals the net rate of flow of mass into the control volume. The flow is positive when directed into the control volume and negative when directed out. The mass balance is expressed mathematically by dmcv  + ∆(m˙)fs = 0 dt

(4-147)

The operator ∆ signifies the difference between exit and entrance flows, and the subscript fs indicates that the term encompasses all flowing streams. When the mass flow rate m⋅ is given by Eq. (4-146), dmcv  + ∆(ρuA)fs = 0 dt

(4-148)

This form of the mass balance equation is often called the continuity equation. For the special case of steady-state flow, the control volume contains a constant mass of fluid, and the first term of Eq. (4-148) is zero. General Energy Balance Because energy, like mass, is conserved, the time rate of change of energy within the control volume equals the net rate of energy transfer into the control volume. Streams flowing into and out of the control volume have associated with them energy in its internal, potential, and kinetic forms, and all contribute to the energy change of the system. Energy may also flow across the control surface as heat and work. Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 47–48, McGraw-Hill, New York (2005)] show that the general energy balance for flow processes is





d(mU)cv 1 ˙ (4-149)  + ∆ H +  u2 + zg m˙ = Q˙ + W fs dt 2 ⋅ The work rate ⋅ W may be of several forms. Most commonly there is shaft work Ws. Work may be associated with expansion or contraction of the control volume, and there may be stirring work. The velocity u in the kinetic energy term is the bulk mean velocity as defined by the

equation u = m˙ ρA; z is elevation above a datum level, and g is the local acceleration of gravity. Energy Balances for Steady-State Flow Processes Flow processes for which the first term of Eq. (4-149) is zero are said to occur at steady state. As discussed with respect to the mass balance, this means that the mass of the system within the control volume is constant; it also means that no changes occur with time in the properties of the fluid within the control volume or at its entrances and exits. No expansion of the control volume is possible under these circumstances. The only work of the process is shaft work, and the general energy balance, Eq. (4-149), becomes 1 ⋅ ⋅ (4-150) ∆ H +  u2 + zg m⋅ = Q + Ws fs 2 Entropy Balance for Open Systems An entropy balance differs from an energy balance in a very important way—entropy is not conserved. According to the second law, the entropy changes in the system and surroundings as the result of any process must be positive, with a limiting value of zero for a reversible process. Thus, the entropy changes resulting from the process sum not to zero, but to a positive quantity called the entropy generation term. The statement of balance, expressed as rates, is therefore


















Net rate of change in entropy of flowing streams

Time rate of change of entropy in control volume

+

+

Time rate of change of entropy in surroundings




Total rate = of entropy generation

The equivalent equation of entropy balance is d(mS)cv dStsurr ⋅ +  = SG ≥ 0 (4-151) ∆(Sm⋅ )fs +  dt dt ⋅ where SG is the entropy generation term. In accord with the second law, it must be positive, with zero as a limiting value. This equation is the general rate form of the entropy balance, applicable at any instant. The three terms on the left are the net rate of gain in entropy of the flowing streams, the time rate of change of the entropy of the fluid contained within the control volume, and the time rate of change of the entropy of the surroundings. The entropy change of the surroundings results from heat transfer between system and surroundings. Let Q⋅ j represent the heat-transfer rate at a particular location on the control surface associated with a surroundings temperature Tσ, j. In accord with Eq. (4-3), the rate of ⋅entropy change in the surroundings as a result of this transfer is ⋅ , defined with respect to the sys− Qj Tσ,j. The minus sign converts Q j tem, to a heat rate with respect to the surroundings. The third term in Eq. (4-151) is therefore the sum of all such quantities, and Eq. (4-151) can be written Q⋅ j d(mS)cv ⋅ ∆(Sm⋅ )fs +  (4-152) −  = SG ≥ 0 dt j Tσ, j For any process, the two kinds of irreversibility are (1) those internal to the control volume and (2) those resulting from heat transfer across finite temperature differences that may exist between the

THERMODYNAMICS OF FLOW PROCESSES TABLE 4-3

4-15

Equations of Balance General equations of balance

dmcv  + ∆(m˙)fs = 0 dt





d(mU)cv 1  + ∆ H +  u2 + zg m˙ dt 2

˙ = Q˙ + W

(4-147)

∆(m˙)fs = 0

(4-149)

(4-152)

fs

d(mS)cv Q˙ j  + ∆(Sm˙ )fs −  = S˙ G ≥ 0 dt j Tσ , j

(4-153)

m˙ 1 = m˙ 2 = m˙

(4-154)

1 ˙s ∆ H +  u2 + zg m⋅ = Q˙ + W 2 fs

(4-150)

∆u2 ∆H +  + g ∆z = Q + Ws 2

(4-155)

Q˙ j ∆(Sm˙ )fs −  = S˙ G ≥ 0 j Tσ , j

(4-156)

Qj ∆S −  = SG ≥ 0 j Tσ, j

(4-157)





⋅ system and surroundings. In the limiting case where SG = 0, the process is completely reversible, implying that • The process is internally reversible within the control volume. • Heat transfer between the control volume and its surroundings is reversible. Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restricted case is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m⋅ is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. APPLICATIONS TO FLOW PROCESSES Duct Flow of Compressible Fluids Thermodynamics provides equations interrelating pressure changes, velocity, duct cross-sectional area, enthalpy, entropy, and specific volume within a flowing stream. Considered here is the adiabatic, steady-state, one-dimensional flow of a compressible fluid in the absence of shaft work and changes in potential energy. The appropriate energy balance is Eq. (4-155). With Q, Ws, and ∆z all set equal to zero, ∆u2 ∆H +  = 0 2 In differential form,

dH = − u du

(4-158)

The continuity equation given by Eq. (4-148) here becomes d(ρuA) = d(uA/V) = 0, whence dV du dA  −  −  =0 V u A

dP βu2 dS u2 dA V(1 − M2)  + T 1 +   −   = 0 CP dx A dx dx



βu2/ CP + M2 dS du 1 u2 dA u − T    + 2   = 0 dx 1 − M2 dx 1−M A dx









the second law, the irreversibilities of fluid friction in adiabatic flow cause an entropy increase in the fluid in the direction of flow. In the limit as the flow approaches reversibility, this increase approaches zero. In general, then, dS/dx ≥ 0. Pipe Flow For a pipe of constant cross-sectional area, dA/dx = 0, and Eqs. (4-160) and (4-161) reduce to

(4-160)





Mach number M is the ratio of the speed of fluid in the duct to the speed of sound in the fluid. The derivatives in these equations are rates of change with length as the fluid passes through a duct. Equation (4-160) relates the pressure derivative, and Eq. (4-161), the velocity derivative, to the entropy and area derivatives. According to





When flow is subsonic, M2 < 1; all terms on the right in these equations are then positive, and dP/dx < 0 and du/dx > 0. Pressure therefore decreases and velocity increases in the direction of flow. The velocity increase is, however, limited, because these inequalities would reverse if the velocity were to become supersonic. This is not possible in a pipe of constant cross-sectional area, and the maximum fluid velocity obtainable is the speed of sound, reached at the exit of the pipe. Here, dS/dx reaches its limiting value of zero. For a discharge pressure low enough, the flow becomes sonic and lengthening the pipe does not alter this result; the mass rate of flow decreases so that the sonic velocity is still obtained at the outlet of the lengthened pipe. According to the equations for supersonic pipe flow, pressure increases and velocity decreases in the direction of flow. However, this flow regime is unstable, and a supersonic stream entering a pipe of constant cross section undergoes a compression shock, the result of which is an abrupt and finite increase in pressure and decrease in velocity to a subsonic value. Nozzles Nozzle flow is quite different from pipe flow. In a properly designed nozzle, its cross-sectional area changes with length in such a way as to make the flow nearly frictionless. The limit is reversible flow, for which the rate of entropy increase is zero. In this event dS/dx = 0, and Eqs. (4-160) and (4-161) become dP u 2 dA 1  =  2  dx VA 1 − M dx









du dA u 1  = −  2  dx A 1 − M dx

The characteristics of flow depend on whether the flow is subsonic (M < 1) or supersonic (M > 1). The possibilities are summarized in Table 4-4. Thus, for subsonic flow in a converging nozzle, the velocity increases and the pressure decreases as the cross-sectional area

TABLE 4-4

Nozzle Characteristics Subsonic: M < 1

(4-161)

βu2/CP + M2 ds du u = T    dx dx 1− M2

dp T 1 + βu2/CP ds  = −    1 − M2 dx V dx

(4-159)

Smith, Abbott, and Van Ness [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 255–258, McGraw-Hill, New York (2005)] show that these basic equations in combination with Eq. (4-15) and other property relations yield two very general equations



Balance equations for single-stream steady-flow processes

Balance equations for steady-flow processes

dA  dx dP  dx du  dx

Supersonic: M > 1

Converging

Diverging

Converging

Diverging

−

+

−

+

−

+

+

−

+

−

−

+

4-16

THERMODYNAMICS

diminishes. The maximum possible fluid velocity is the speed of sound, reached at the exit. A converging subsonic nozzle can therefore deliver a constant flow rate into a region of variable pressure. Supersonic velocities characterize the diverging section of a properly designed converging/diverging nozzle. Sonic velocity is reached at the throat, where dA/dx = 0, and a further increase in velocity and decrease in pressure require a diverging cross-sectional area to accommodate the increasing volume of flow. The pressure at the throat must be low enough for the velocity to become sonic. If this is not the case, the diverging section acts as a diffuser—the pressure rises and the velocity decreases in the conventional behavior of subsonic flow in a diverging section. An analytical expression relating velocity to pressure in an isentropic nozzle is readily derived for an ideal gas with constant heat capacities. Combination of Eqs. (4-15) and (4-159) for isentropic flow gives

1

H

H

P1

(H)S 2 2

P2

S

u du = − V dP Integration, with nozzle entrance and exit conditions denoted by 1 and 2, yields u22 − u21 = − 2

P 2γP V  V dP =  1 −   γ−1  P P2

1

1

P1

2

(γ −1)/γ

The rate form of this equation is ˙ s = m˙ ∆H = m˙ (H2 − H1) W

1

(4-163) (4-164)

When inlet conditions T1 and P1 and discharge pressure P2 are known, the value of H1 is fixed. In Eq. (4-163) both H2 and Ws are unknown, and the energy balance alone does not allow their calculation. However, if the fluid expands reversibly and adiabatically, i.e., isentropically, in the turbine, then S2 = S1. This second equation establishes the final state of the fluid and allows calculation of H2. Equation (4-164) then gives the isentropic work: Ws(isentropic) = (∆H)S

Adiabatic expansion process in a turbine or expander. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 269, McGraw-Hill, New York (2005).] FIG. 4-2

(4-162)

where the final term is obtained upon elimination of V by PV γ = const, an equation valid for ideal gases with constant heat capacities. Here, γ  CP/CV. Throttling Process Fluid flowing through a restriction, such as an orifice, without appreciable change in kinetic or potential energy undergoes a finite pressure drop. This throttling process produces no shaft work, and in the absence of heat transfer, Eq. (4-155) reduces to ∆H = 0 or H2 = H1. The process therefore occurs at constant enthalpy. The temperature of an ideal gas is not changed by a throttling process, because its enthalpy depends on temperature only. For most real gases at moderate conditions of T and P, a reduction in pressure at constant enthalpy results in a decrease in temperature, although the effect is usually small. Throttling of a wet vapor to a sufficiently low pressure causes the liquid to evaporate and the vapor to become superheated. This results in a considerable temperature drop because of the evaporation of liquid. If a saturated liquid is throttled to a lower pressure, some of the liquid vaporizes or flashes, producing a mixture of saturated liquid and saturated vapor at the lower pressure. Again, the large temperature drop results from evaporation of liquid. Turbines (Expanders) High-velocity streams from nozzles impinging on blades attached to a rotating shaft form a turbine (or expander) through which vapor or gas flows in a steady-state expansion process which converts internal energy of a high-pressure stream into shaft work. The motive force may be provided by steam (turbine) or by a high-pressure gas (expander). In any properly designed turbine, heat transfer and changes in potential and kinetic eneregy are negligible. Equation (4-155) therefore reduces to Ws = ∆H = H2 − H1

S

(4-165)

The absolute value |Ws |(isentropic) is the maximum work that can be produced by an adiabatic turbine with given inlet conditions and given

discharge pressure. Because the actual expansion process is irreversible, turbine efficiency is defined as Ws η   Ws(isentropic) where Ws is the actual shaft work. By Eqs. (4-163) and (4-165), ∆H η=  (∆H)S

(4-166)

Values of η usually range from 0.7 to 0.8. The HS diagram of Fig. 4-2 compares the path of an actual expansion in a turbine with that of an isentropic expansion for the same intake conditions and the same discharge pressure. The isentropic path is the dashed vertical line from point 1 at intake pressure P1 to point 2′ at P2. The irreversible path (solid line) starts at point 1 and terminates at point 2 on the isobar for P2. The process is adiabatic, and irreversibilities cause the path to be directed toward increasing entropy. The greater the irreversiblity, the farther point 2 lies to the right on the P2 isobar, and the lower the value of η. Compression Processes Compressors, pumps, fans, blowers, and vacuum pumps are all devices designed to bring about pressure increases. Their energy requirements for steady-state operation are of interest here. Compression of gases may be accomplished in rotating equipment (high-volume flow) or for high pressures in cylinders with reciprocating pistons. The energy equations are the same; indeed, based on the same assumptions, they are the same as for turbines or expanders. Thus, Eqs. (4-159) through (4-161) apply to adiabatic compression. The isentropic work of compression, as given by Eq. (4-165), is the minimum shaft work required for compression of a gas from a given initial state to a given discharge pressure. A compressor efficiency is defined as Ws(isentropic) η   Ws In view of Eqs. (4-163) and (4-165), this becomes (∆H)S η  ∆H

(4-167)

Compressor efficiencies are usually in the range of 0.7 to 0.8. The compression process is shown on an HS diagram in Fig. 4-3. The vertical dashed line rising from point 1 to point 2′ represents the reversible adiabatic (isentropic) compression process from P1 to P2.

SYSTEMS OF VARIABLE COMPOSITION

Example 1: LNG Vaporization and Compression A port facility for unloading liquefied natural gas (LNG) is under consideration. The LNG arrives by ship, stored as saturated liquid at 115 K and 1.325 bar, and is unloaded at the rate of 450 kg s-1. It is proposed to vaporize the LNG with heat discarded from a heat engine operating between 300 K, the temperature of atmospheric air, and 115 K, the temperature of the vaporizing LNG. The saturated-vapor LNG so produced is compressed adiabatically to 20 bar, using the work produced by the heat engine to supply part of the compression work. Estimate the work to be supplied from an external source. For estimation purposes we need not be concerned with the design of the heat engine, but assume that a suitable engine can be built to deliver 30 percent of the work of a Carnot engine operating between the temperatures of 300 and 115 K. The equations that apply to Carnot engines can be found in any thermodynamics text.

2 2 H

H

P2

(H)S

W = QH − QC

By the first law:

1 P1

S

QH TH  =  QC TC

By the second law:



S Adiabatic compression process. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 274, McGraw-Hill, New York (2005).] FIG. 4-3

The actual irreversible compression process follows the solid line from point 1 upward and to the right in the direction of increasing entropy, terminating at point 2. The more irreversible the process, the farther this point lies to the right on the P2 isobar, and the lower the efficiency η of the process. Liquids are moved by pumps, usually by rotating equipment. The same equations apply to adiabatic pumps as to adiabatic compressors. Thus, Eqs. (4-163) through (4-165) and Eq. (4-167) are valid. However, application of Eq. (4-163) requires values of the enthalpy of compressed (subcooled) liquids, and these are seldom available. The fundamental property relation, Eq. (4-15), provides an alternative. For an isentropic process, dH = V dP

(constant S)

Ws(isentropic) = (∆H)S =



TH W = QC  −1 TC

In combination:

Combining this with Eq. (4-165) yields

4-17

Here, |W| is the work produced by the Carnot engine; |QC| is the heat transferred at the cold temperature, i.e., to vaporize the LNG; TH and TC are the hot and cold temperatures of the heat reservoirs between which the heat engine operates, or 300 and 115 K, respectively. LNG is essentially pure methane, and enthalpy values from Table 2-281 of the Seventh Edition of Perry’s Chemical Engineers’ Handbook provide its heat of vaporization: ∆Hlv = Hv − Hl = 802.5 − 297.7 = 504.8 kJ kg−1 For a flow rate of 450 kg s−1, QC = (450)(504.8) = 227,160 kJ s−1 The equation for work gives 300 W = (227,160)  − 1 = 3.654 × 105 kJ s−1 = 3.654 × 105 kW 115 This is the reversible work of a Carnot engine. The assumption is that the actual power produced is 30 percent of this, or 1.096 × 105 kW. The enthalpy and entropy of saturated vapor at 115 K and 1.325 bar are given in Table 2-281 of the Seventh Edition of Perry’s as Hv = 802.5 kJ kg−1 and Sv = 9.436 kJ kg−1 K−1





Isentropic compression of saturated vapor at 1.325 to 20 bar produces superheated vapor with an entropy of 9.436 kJ kg−1 K−1. Interpolation in Table 2-282 at 20 bar yields an enthalpy of H = 1026.2 kJ kg−1 at 234.65 K. The enthalpy change of isentropic compression is then ∆HS = 1026.2 − 802.5 = 223.7 kJ kg−1

 V dP P2

P1

The usual assumption for liquids (at conditions well removed from the critical point) is that V is independent of P. Integration then gives

For a compressor efficiency of 75 percent, the actual enthalpy change of compression is

Ws(isentropic) = (∆H)S = V(P2 − P1)

223.7 ∆H ∆H = S =  = 298.3 kJ kg−1 η 0.75 The actual enthalpy of superheated LNG at 20 bar is then H = 802.5 + 298.3 = 1100.8 kJ kg−1 Interpolation in Table 2-282 of the Seventh Edition of Perry’s indicates an actual temperature of 265.9 K for the compressed LNG, which is quite suitable for its entry into the distribution system. The work of compression is W = m ∆H = (450 kg s−1)(298.3 kJ kg−1) = 1.342 × 105 kJ s−1 = 1.342 × 105 kW

(4-168)

Also useful are Eqs. (4-132) and (4-133). Because temperature changes in the pumped fluid are very small and because the properties of liquids are insensitive to pressure (again at conditions not close to the critical point), these equations are usually integrated on the assumption that CP, V, and β are constant, usually at initial values. Thus, to a good approximation ∆H = CP ∆T + V(1 − βT) ∆P

(4-169)

T ∆S = CP ln 2 − βV∆P T1

(4-170)

The estimated power which must be supplied from an external source is ˙ = 1.342 × 105 − 1.096 × 105 = 24,600 kW W

SYSTEMS OF VARIABLE COMPOSITION The composition of a system may vary because the system is open or because of chemical reactions even in a closed system. The equations developed here apply regardless of the cause of composition changes. PARTIAL MOLAR PROPERTIES For a homogeneous PVT system comprised of any number of chemical species, let symbol M represent the molar (or unit-mass) value of an

extensive thermodynamic property, say, U, H, S, A, or G. A total-system property is then nM, where n = Σ i ni and i is the index identifying chemical species. One might expect the solution property M to be related solely to the properties Mi of the pure chemical species which comprise the solution. However, no such generally valid relation is known, and the connection must be established experimentally for every specific system. Although the chemical species which make up a solution do not have their own individual properties, a solution property may be arbitrarily

4-18

THERMODYNAMICS

apportioned among the individual species. Once an apportioning recipe is adopted, the assigned property values are quite logically treated as though they were indeed properties of the individual species in solution, and reasoning on this basis leads to valid conclusions. For a homogeneous PVT system, postulate 5 requires that



P,n

∂(nM) dT +  ∂P





T,n

∂(nM) dP +  ∂ni i





dni

T,P,nj

where subscript n indicates that all mole numbers ni are held constant, and subscript nj signifies that all mole numbers are held constant except the ith. This equation may also be written ∂M d(nM) = n  ∂T

∂M dT + n  ∂P P,x



∂(nM) dP +  ∂ni T,x i







dni

T,P,nj





(4-171)

T,P,nj

The basis for calculation of partial properties from solution properties is provided by this equation. Moreover, ∂M d(nM) = n  ∂T

∂M dT + n  ∂P P,x





T,x

⎯ dP + Mi dni (4-172) i

This equation, valid for any equilibrium phase, either closed or open, attributes changes in total property nM to changes in T and P and to mole-number changes resulting from mass transfer or chemical reaction. The following are mathematical identities: dni = d(xi n) = xi dn + n dxi

d(nM) = n dM + M dn

Combining these expressions with Eq. (4-172) and collecting like terms give ∂M dM −  ∂T



∂M dT −  ∂P P,x





T,x

⎯ ⎯ dP − Mi dxi n + M − Mi xi dn = 0

 

i

i



Because n and dn are independent and arbitrary, the terms in brackets must separately be zero, whence ∂M dM =  ∂T

∂M dT +  ∂P P,x



and



T,x

⎯ M = xi Mi

⎯ dP + Mi dxi

∂M

 ∂T

P,x

∂M dT +  ∂P



T,x

⎯ dP − xi dMi = 0

(4-175)

i

This general result, the Gibbs-Duhem equation, imposes a constraint on how the partial properties of any phase may vary with temperature, pressure, and composition. For the special case where T and P are constant, ⎯ (4-176)

xi dMi = 0 (constant T, P) i

where subscript x indicates that all mole fractions are held constant. The derivatives in the summation are ⎯ called partial molar properties. They are given the generic symbol Mi and are defined by ∂(nM) ⎯ Mi   ∂ni

i

Because this equation and Eq. (4-173) are both valid in general, their right sides can be equated, yielding

The total differential of nM is therefore



⎯ ⎯ dM = xi dMi + Mi dxi i

nM = M(T, P, n1, n2, n3, …) ∂(nM) d(nM) =  ∂T

solution properties in the parent equation are related linearly (in the algebraic sense). Gibbs-Duhem Equation Differentiation of Eq. (4-174) yields

(4-173)

i

(4-174)

i

The first of these equations is merely a special case of Eq. (4-172); however, Eq. (4-174) is a vital new relation. Known as the summability equation, it provides for the calculation of solution properties from partial properties, a purpose opposite to that of Eq. (4-171). Thus a solution property apportioned according to the recipe of Eq. (4-171) may be recovered simply by adding the properties attributed to the individual species, each weighted by its mole fraction in solution. The equations for partial molar properties are valid also for partial specific properties, in which case m replaces n and {xi} are mass fractions. Equation (4-171) applied to the definitions of Eqs. (4-8) through (4-10) yields the partial-property relations ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ Hi = Ui + PVi Ai = Ui − TSi Gi = Hi − TSi These equations illustrate the parallelism that exists between the equations for a constant-composition solution and those for the corresponding partial properties. This parallelism exists whenever the

Symbol M may represent the molar value of any extensive thermodynamic property, say, V, U, H, S, or G. When M  H, the derivatives (∂H/∂H)P and (∂H/∂P)T are given by Eqs. (4-28) and (4-29), and Eqs. (4-173), (4-174), and (4-175) specialize to ∂V dH = CP dT + V − T  ∂T



⎯

  dP + H dx i

i

(4-177)

i

P,x

⎯ H = xiHi

(4-178)

i

∂V CP dT + V − T  ∂T



⎯

  dP − x dH = 0 i

i

(4-179)

i

P,x

Similar equations are readily derived when M takes on other identities. Equation (4-171), which defines a partial molar property, provides a general means by which partial-property values may be determined. However, for a binary solution an alternative method is useful. Equation (4-174) for a binary solution is ⎯ ⎯ M = x1M1 + x2M2 Moreover, the Gibbs-Duhem equation for a solution at given T and P, Eq. (4-176), becomes ⎯ ⎯ x1 dM1 + x2 dM2 = 0 These two equations combine to yield ⎯ dM M1 = M + x2  dx1

(4-180a)

⎯ dM M2 = M − x1  dx1

(4-180b)

Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calculations are complex, and direct use of Eq. (4-171) is appropriate. Partial Molar Equation-of-State Parameters The parameters in equations of state as applied to mixtures are related to composition by mixing rules. For the second virial coefficient B =

yiyjBij

(4-60)

i j

The partial molar second virial coefficient is by definition ∂(nB) ⎯ Bi   ∂ni





(4-181)

T,nj

Because B is independent of P, this is in accord with Eq. (4-171). These ⎯ two equations lead through derivation to useful expressions for Bi, as shown in detail by Van Ness and Abbott [Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase

SYSTEMS OF VARIABLE COMPOSITION Equilibria, pp. 137–140, McGraw-Hill, New York (1982)]. The simplest result is ⎯ Bi = 2 ykBki − B (4-182) k

An analogous expression follows from Eq. (4-114) for parameter a in the generic cubic equation of state given by Eqs. (4-100), (4-103), and (4-104): a⎯ = 2 y a − a (4-183)

k

i

k ki

This expression is independent of the combining rule [e.g., Eq. (4-114)] used for aki. For the linear ⎯ mixing rule of Eq. (4-113) for b, the result of derivation is simply bi = bi. Partial Molar Gibbs Energy Implicit in Eq. (4-13) is the relation ∂(nG) µi =  ∂ni T,P,n





j

Comparison with Eq. (4-171) indicates the following identity: ⎯ (4-184) µi = Gi The reciprocity relation for an exact differential applied to Eq. (4-13) produces not only the Maxwell relation, Eq. (4-21), but also two other useful equations: ∂(nV) ⎯ ∂µi =  = Vi (4-185)  ∂ni T,P,n ∂P T,n

 

∂µi  ∂T

∂(nS) = −  ∂ni P,n









j

⎯ = − Si

(4-186)

T,P,nj

⎯ ∂µ ∂µ dµi  dGi = i dT + i ∂T P,n ∂P ⎯ ⎯ ⎯ dGi = − Si dT + Vi dP



and

⎯ ig RT V i = V igi =  P



T,n

⎯ ⎯ ⎯ dUi = T dSi − P dVi ⎯ ⎯ ⎯ dHi = T dSi + Vi dP ⎯ ⎯ ⎯ dAi = − Si dT − P dVi

(4-187)

Because the enthalpy is independent of pressure, ⎯ ig H i = H igi

(4-193)

ig i

where S is evaluated at the mixture T and P. The entropy of an ideal gas does depend on pressure, and here ⎯ig (4-194) Si = Sigi − R ln yi where Sigi is evaluated at the mixture T and P. ⎯ ⎯ ⎯ From the definition of the Gibbs energy, Gigi = H igi − TSiig. In combination with Eqs. (4-193) and (4-194), this becomes ⎯ Gigi = H igi − TSigi + RT ln yi ⎯ or µigi  Gigi = Gigi + RT ln yi (4-195) Elimination of Gigi from this equation is accomplished through Eq. (4-17), written for pure species i as an ideal gas: RT dGigi = V igi dP =  dP = RT d ln P P

(constant T)

G igi = Γi(T) + RT ln P

By Eq. (4-172)

Gid = yiΓi(T) + RT ln (yiP) i

(4-188) (4-189) (4-190)

These equations again illustrate the fact that for every equation providing a linear relation among the thermodynamic properties of a constant-composition solution there exists a parallel relationship for the partial properties of the species in solution. The following equation is a mathematical identity:



nG 1 nG d    d(nG) − 2 dT RT RT RT ⎯ Substitution for d(nG) by Eq. (4-13), with µi = Gi , and for G by Eq. (4-10) gives, after algebraic reduction, ⎯ Gi nG nV nH d  =  dP − 2 dT +  dni RT RT RT i RT

(4-192)

(4-196)

where integration constant Γi(T) is a function of temperature only. Equation (4-195) now becomes ⎯ (4-197) µ igi = Gigi = Γi(T) + RT ln (yiP)

dP

Similarly, in view of Eqs. (4-14), (4-15), and (4-16),



limit of zero pressure, and provides a conceptual basis upon which to build the structure of solution thermodynamics. Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 391–394, McGraw-Hill, New York (2005)] develop the following property relations for the ideal gas model.

Integration gives

Because µ = f (T, P),

4-19

(4-191)

This result is a useful alternative to the fundamental property relation given by Eq. (4-13). All terms in this equation have units of moles; moreover, the enthalpy rather than the entropy appears on the right side. SOLUTION THERMODYNAMICS Ideal Gas Mixture Model The ideal gas mixture model is useful because it is molecularly based, is analytically simple, is realistic in the

(4-198)

i

A dimensional ambiguity is apparent with Eqs. (4-196) through (4-198) in that P has units, whereas ln P must be dimensionless. In practice this is of no consequence, because only differences in Gibbs energy appear, along with ratios of the quantities with units of pressure in the arguments of the logarithm. Consistency in the units of pressure is, of course, required. Fugacity and Fugacity Coefficient The chemical potential µi plays a vital role in both phase and chemical reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics that discourage its use in the solution of practical problems. The Gibbs energy, and hence µi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, µi approaches negative infinity when either P or yi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of µi but that does not exhibit its less desirable characteristics. The origin of the fugacity concept resides in Eq. (4-196), an equation valid only for pure species i in the ideal gas state. For a real fluid, an analogous equation is written as Gi  Γi(T) + RT ln fi

(4-199)

in which a new property fi replaces the pressure P. This equation serves as a partial definition of the fugacity fi. Subtraction of Eq. (4-196) from Eq. (4-199), both written for the same temperature and pressure, gives fi Gi − Gigi = RT ln  P

4-20

THERMODYNAMICS

According to the definition of Eq. (4-40), Gi − Gigi is the residual Gibbs energy GRi. The dimensionless ratio fi /P is another new property called the fugacity coefficient φi. Thus, GRi = RT ln φi

(4-200)

fi φi   P

(4-201)

where

The definition of fugacity is completed by setting the ideal gas state fugacity of pure species i equal to its pressure, f iig = P. Thus for the special case of an ideal gas, GRi = 0, φi = 1, and Eq. (4-196) is recovered from Eq. (4-199). The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-197), is written for species i in a fluid mixture (4-202) µi  Γi(T) + RT ln fˆi where the partial pressure yi P is replaced by fˆi, the fugacity of species i in solution. Because it is not a partial property, it is identified by a circumflex rather than an overbar. Subtracting Eq. (4-197) from Eq. (4-202), both written for the same temperature, pressure, and composition, yields fˆi µi − µigi = RT ln  yiP The residual Gibbs energy of a mixture is defined by GR  G − Gig, and definition of a partial molar residual Gibbs energy is ⎯R the ⎯ analogous ⎯ Gi  Gi − Gigi = µi − µigi . Therefore ⎯R (4-203) Gi = RT ln φˆ i (4-204)

The dimensionless ratio φˆ i is called the fugacity coefficient of species i in solution. Equation (4-203) is the ⎯ analog of Eq. (4-200), which relates φi to GRi. For an ideal gas, GRi is necessarily zero; therefore φˆ iig = 1 and ˆfiig = yiP. Thus the fugacity of species i in an ideal gas mixture is equal to its partial pressure. Evaluation of Fugacity Coefficients Combining Eq. (4-200) with Eq. (4-45) gives GR ln φ =  = RT

dP  (Z − 1)  P P

(4-205)

0

Subscript i is omitted, with the understanding that φ here is for a pure species. Clearly, all correlations for GR/RT are also correlations for ln φ. Equation (4-200) with Eqs. (4-48) and (4-73) yields ln φ =

dP P  (Z − 1)  = (B + ωB )  P T Pr

r

0

0

r

1

r

(4-206)

r

This equation, used in conjunction with Eqs. (4-77) and (4-78), provides a useful generalized correlation for the fugacity coefficients of pure species. A more comprehensive generalized correlation results from Eqs. (4-200) and (4-116): ln φ = An alternative form is where ln φ0  By Eq. (4-207),

 (Z Pr

0

0

dP dP  (Z − 1)  +ω Z  P P Pr

Pr

r

0

0

0

r

r

1

r

ln φ = ln φ0 + ω ln φ1 dP − 1) r Pr

and

φ = (φ )(φ ) 0

1 ω

This equation takes on new meaning when Gigi (T, P) is replaced by Gi (T, P), the Gibbs energy of pure species i in its real physical state of gas, liquid, or solid at the mixture T and P. The ideal solution is therefore defined as one for which ⎯ (4-209) µidi = Gidi  Gi(T, P) + RT ln xi where superscript id denotes an ideal solution property and xi represents the mole fraction because application is usually to liquids. This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-185) ⎯ and (4-186), applied to an ideal solution with µi replaced by Gi, are written as ⎯ ⎯ ∂Gidi ∂Gidi ⎯ ⎯ Vidi =  and Sidi = −  ∂P T, x ∂T P,x







Differentiation of Eq. (4-209) yields ⎯ ⎯ ∂Gidi ∂Gidi ∂Gi   =  and ∂P T,x ∂T ∂P T









ln φ1 

(4-207) dP Z P Pr

0

r

1

r

(4-208)

Correlations may therefore be presented for φ0 and φ1, as was done by Lee and Kesler [AIChE J. 21: 510–527 (1975)].



∂Gi

+ R ln x =  ∂T

i

P,x

P

Equation (4-17) implies that ∂Gi

 ∂P

= Vi

∂Gi

= −S  ∂T

and

i

T

fˆi φˆ i   yiP

where by definition

Ideal Solution Model The ideal gas model is useful as a standard of comparison for real gas behavior. This is formalized through residual properties. The ideal solution is similarly useful as a standard to which real solution behavior may be compared. The partial molar Gibbs energy or chemical potential of species i in an ideal gas mixture is given by Eq. (4-195), written as ⎯ µigi = Gigi = Gigi (T, P) + RT ln yi

P

In combination these sets of equations provide ⎯ id (4-210) Vi = Vi ⎯id and Si = Si − R ln xi (4-211) ⎯id ⎯id ⎯id Because Hi = Gi + TSi , substitutions by Eqs. (4-209) and (4-211) yield ⎯id Hi = Hi (4-212) The summability relation, Eq. (4-174), written for the special case of an ideal solution, may be applied to Eqs. (4-209) through (4-212): Gid = xiGi + RT xi ln xi

(4-213)

V id = xiVi

(4-214)

i

i

i

Sid = xiSi − R xi ln xi

(4-215)

Hid = xiHi

(4-216)

i

i

i

A simple equation for the fugacity of a species in an ideal solution follows from Eq. (4-209). For the special case of species i in an ideal solution, Eq. (4-202) becomes ⎯ µidi = Gidi = Γi(T) + RT ln fˆ idi When this equation and Eq. (4-199) are combined with Eq. (4-209), Γi (T) is eliminated, and the resulting expression reduces to (4-217) fˆiid = xi fi This equation, known as the Lewis-Randall rule, shows that the fugacity of each species in an ideal solution is proportional to its mole fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the same T and P. Division of both sides of Eq. (4-217) by xi P and substitution of φˆ iid for fˆidi xiP [Eq. (4-204)] and of φi for fi/P [Eq. (4-201)] give the alternative form (4-218) φˆ iid = φi

SYSTEMS OF VARIABLE COMPOSITION

∆S, and ∆H are the Gibbs energy change of mixing, the volume change of mixing, the entropy change of mixing, and the enthalpy change of mixing. For an ideal solution, each excess property is zero, and for this special case the equations reduce to those shown in the third column of Table 4-5. Property changes of mixing and excess properties are easily calculated one from the other. The most common property changes of mixing are the volume change of mixing ∆V and the enthalpy change of mixing ∆H, commonly called the heat of mixing. These properties are identical to the corresponding excess properties. Moreover, they are directly measurable, providing an experimental entry into the network of equations of solution thermodynamics.

Thus the fugacity coefficient of species i in an ideal solution equals the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules not too different in size and of the same chemical nature. Thus, a mixture of isomers conforms very closely to ideal solution behavior. So do mixtures of adjacent members of a homologous series. Excess Properties An excess property ME is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same T, P, and composition. Thus, ME  M − Mid

(4-219)

FUNDAMENTAL PROPERTY RELATIONS BASED ON THE GIBBS ENERGY

where M represents the molar (or unit-mass) value of any extensive thermodynamic property (say, V, U, H, S, G). This definition is analogous to the definition of a residual property as given by Eq. (4-40). However, excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mix⎯E are defined analogously: tures. Partial molar excess properties M i ⎯E ⎯ ⎯id Mi = Mi − Mi (4-220)

Of the four fundamental property relations shown in the second column of Table 4-1, only Eq. (4-13) has as its special or canonical variables T, P, and {ni}. It is therefore the basis for extension to several useful supplementary thermodynamic properties. Indeed an alternative form has been developed as Eq. (4-191). These equations are the first two entries in the upper left quadrant of Table 4-6, which is now to be filled out with important derived relationships. Fundamental Residual-Property Relation Equation (4-191) is general and may be written for the special case of an ideal gas ⎯ig nGig nVig Gi nHig d  =  dP − 2 dT +  dni RT RT RT i RT

Of particular interest is the partial molar excess Gibbs energy. Equation (4-202) may be written as ⎯ Gi = Γi(T) + RT ln fˆi



In accord with Eq. (4-217) for an ideal solution, this becomes ⎯id Gi = Γi(T) + RT ln xi fi



⎯ The left side is the partial excess Gibbs energy GEi ; the dimensionless ˆ ratio fixi fi on the right is the activity coefficient of species i in solution, given the symbol γi, and by definiton, fˆi γi   (4-221) xifi ⎯E Thus, Gi = RT ln γi (4-222) ⎯E Comparison with Eq. (4-203) shows⎯that Eq. (4-222) relates γ⎯ i to Gi exactly as Eq. (4-203) relates φˆ i to GRi . For an ideal solution, GEi = 0, and therefore γi = 1. Property Changes of Mixing A property change of mixing is defined by ∆M  M − xiMi



∂(GRRT) VR  =  ∂P RT





(4-223)





(4-225)

V = ∆V

SE = S − xiSi + R xi ln xi

(4-226)

HE = H − xiHi

(4-227)

V = V − xiVi

j

GR  = xi ln φˆ i RT i

G = ∆G − RT xi ln xi

i

(4-239)

P,x

∂(nGRRT) ln φˆ i =  (4-240) ∂ni T,P,n This equation demonstrates that ln φˆ i is a partial property with respect to GR/RT. The summability relation therefore applies, and

(4-224)

i

E



Also implicit in Eq. (4-237) is the relation

Relations Connecting Property Changes of Mixing and Excess Properties ME in relation to M ME in relation to ∆M

G = G − xiGi − RT xi ln xi

(4-238)

T, x

∂(GRRT) HR  = −T  ∂T RT

where M represents a molar thermodynamic property of a homogeneous solution and Mi is the molar property of pure species i at the T and P of the solution and in the same physical state. Applications are usually to liquids. Each of Eqs. (4-213) through (4-216) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property [Eq. (4-219)], yielding the equations of the first column of Table 4-5. In view of Eq. (4-223) these may be written as shown in the second column of Table 4-5, where ∆G, ∆V,

E



Similarly, the result of division by dT and restriction to constant P and composition is

i

TABLE 4-5



Subtraction of this equation from Eq. (4-191) gives ⎯ nGR nVR GRi nHR d  =  dP − 2 dT +  dni (4-236) RT RT RT i RT ⎯ ⎯ ⎯ where the definitions GR  G − Gig and GRi  Gi − Gigi have been imposed. Equation (4-236) is the fundamental residual-property relation. An alternative form follows by introduction of the fugacity coefficient given by Eq. (4-203). The result is listed as Eq. (4-237) in the upper left quadrant of Table 4-6. Limited forms of this equation are particularly useful. Division by dP and restriction to constant T and composition lead to

⎯ ⎯id fˆi Gi − Gi = RT ln  xifi

By difference

4-21

(4-241)

Expressions for ∆Mid

(4-228)

∆G = RT xi ln xi

(4-232)

(4-229)

∆V = 0

(4-233)

SE = ∆S + R xi ln xi

(4-230)

∆Sid = − R xi ln xi

(4-234)

HE = ∆H

(4-231)

∆H id = 0

(4-235)

E

i

E

id

i

id

i

i

i

i

i

i

4-22

THERMODYNAMICS

TABLE 4-6

Fundamental Property Relations for the Gibbs Energy and Related Properties General equations for an open system

Equations for 1 mol (constant composition)

d(nG) = nV dP − nS dT + µi dni

(4-13)

dG = V dP − S dT

(4-17)

i

⎯ Gi nH nV nG d  =  dP − 2 dT +  dni RT RT RT RT i

(4-191)

H V G d  =  dP − 2 dT RT RT RT

(4-253)

nHR nV R nGR d  =  dP − 2 dT + ln φˆ i dni RT RT RT i

(4-237)

HR VR GR d  =  dP − 2 dT RT RT RT

(4-254)

nG E nVE nHE d  =  dP − 2 dT + ln γi dni RT RT RT i

(4-248)

HE GE VE d  =  dP − 2 dT RT RT RT

(4-255)

  



  

Equations for partial molar properties (constant composition) ⎯ ⎯ ⎯ dGi = dµi = Vi dP − Si dT

Gibbs-Duhem equations (4-256)

V dP − S dT = xi dµi

(4-260)

i

 

(4-257)

⎯ Gi H V  dP − 2 dT = xi d  RT RT RT i



(4-258)

HR VR  dP − 2 dT = xi d ln φˆ i RT RT i

(4-262)



(4-259)

HE VE  dP − 2 dT = xi d ln γi RT RT i

(4-263)

⎯ ⎯ ⎯ Hi G µi V d i = d  = i dP −  dT RT 2 RT RT RT ⎯ ⎯ ⎯R GRi Vi dP − HiR dT = d ln φˆ i =  d  2 RT RT RT ⎯ ⎯E ⎯E Vi GEi Hi d  = d ln γi =  dP − 2 dT RT RT RT

Application of Eq. (4-240) to an expression giving GR as a function of composition yields an equation for ln φˆ i. In the simplest case of a gas mixture for which the virial equation [Eq. (4-67)] is appropriate, Eq. (4-69) provides the relation

Differentiation in accord with Eqs. (4-240) and (4-181) yields P ⎯ ln φˆ i =  Bi RT

(4-242)

P ln φˆ 1 =  (B11 + y22δ12) RT

(4-243a)

P ln φˆ 2 =  (B22 + y21δ12) RT

(4-243b)

⎯ where Bi is given by Eq. (4-182). For a binary system these equations reduce to

δ12  2B12 − B11 − B22

For the special case of pure species i, these equations reduce to P ln φi =  Bii RT

(4-244)

For the generic cubic equation of state [Eqs. (4-104)], GR/RT is given by Eq. (4-109), which in view of Eq. (4-200) for a pure species becomes ln φi = Zi − 1 − ln(Zi − βi) − qiIi

(4-245)

For species i in solution Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 562–563, McGraw-Hill, New York (2005)] show that ⎯ bi qiI (4-246) ln φˆ i =  (Z − 1) − ln(Z − β) − ⎯ b Symbols without subscripts represent mixture properties, and I is given by Eq. (4-112).

(4-261)

Fundamental Excess-Property Relation Equations for excess properties are developed in much the same way as those for residual properties. For the special case of an ideal solution, Eq. (4-191) becomes ⎯id nHid Gi nGid nVid d  =  dP −  dT +  dni 2 RT RT RT i RT



nGR P  =  (nB) RT RT

where





Subtraction of this equation from Eq. (4-191) yields ⎯ GEi nHE nGE nVE d  =  dP − 2 dT +  dni (4-247) RT RT RT i RT ⎯ ⎯ ⎯ where the definitions GE  G − Gid and GEi  Gi − Gidi have been imposed. Equation (4-247) is the fundamental excess-property relation. An alternative form follows by introduction of the activity coefficient as given by Eq. (4-222). This result is listed as Eq. (4-248) in the upper left quadrant of Table 4-6. The following equations are in complete analogy to those for residual properties.





∂(GERT) VE  =  ∂P RT





∂(GERT) HE  = −T  ∂T RT



∂(nGERT) ln γi =  ∂ni



(4-249)

T, x





(4-250)

P, x

(4-251)

T, P,nj

This last equation demonstrates that ln γi is a partial property with respect to GE/RT, implying also the validity of the summability relation GE  = xi ln γi RT i

(4-252)

The equations of the upper left quadrant of Table 4-6 reduce to those of the upper right quadrant for n = 1 and dni = 0. Each equation in the upper left quadrant has a partial-property analog, as shown in the lower left quadrant. Each equation of the upper left quadrant is a special case of Eq. (4-172) and therefore has associated with it a Gibbs-Duhem equation of the form of Eq. (4-173). These are shown in the lower right quadrant. The equations of Table 4-6 store an enormous amount of information, but they are so general that their direct

SYSTEMS OF VARIABLE COMPOSITION application is seldom appropriate. However, by inspection one can write a vast array of relations valid for particular applications. For example, one sees immediately from Eqs. (4-258) and (4-259) that ⎯R Vi ∂ ln φˆ i (4-264)  =  ∂P T,x RT ⎯R Hi ∂ ln φˆ i (4-265)  = − 2 RT ∂T P,x ⎯E Vi ∂ ln γi (4-266)  =  ∂P T,x RT ⎯E Hi ∂ ln γi (4-267)  = − 2 RT ∂T P,x

















Excess properties find application in the treatment of liquid solutions. Of primary importance for engineering calculations is the excess Gibbs energy GE, because its canonical variables are T, P, and composition, the variables usually specified or sought in design calculations. Knowing GE as a function of T, P, and composition, one can in principle compute from it all other excess properties. The excess volume for liquid mixtures is usually small, and in accord with Eq. (4-249) the pressure dependence of GE is usually ignored. Thus, engineering efforts to model GE center on representing its composition and temperature dependence. For binary systems at constant T, GE becomes a function of just x1, and the quantity most conveniently represented by an equation is GE/x1x2RT. The simplest procedure is to express this quantity as a power series in x1: (constant T)

An equivalent power series with certain advantages is the RedlichKister expansion [Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48: 49–61 (1952)]: GE  = A + B(x1 − x2) + C(x1 − x2)2 + · · · x1x2RT In application, different truncations of this expansion are appropriate, and for each truncation specific expressions for ln γ1 and ln γ2 result from application of Eq. (4-251). When all parameters are zero, GE/RT = 0, and the solution is ideal. If B = C = . . . = 0, then GE  =A x1x2RT where A is a constant for a given temperature. The corresponding equations for ln γ1 and ln γ2 are ln γ1 = Ax22

(4-268a)

ln γ 2 = Ax21

(4-268b)

The symmetric nature of these relations is evident. The infinite dilution values of the activity coefficients are ln γ 1∞ = ln γ 2∞ = A. If C = · · · = 0, then E

G  = A + B(x1 − x2) = A + B(2x1 − 1) x1x2RT and in this case GE/x1x2RT is linear in x1. The substitutions A + B = A21 and A−B = A12 transform this expression to the Margules equation: E

G  = A21x1 + A12x2 x1x2RT

Application of Eq. (4-251) yields ln γ1 = x22 [A12 + 2(A21 − A12) x1]

(4-270a)

ln γ2 = x [A21 + 2(A12 − A21) x2]

(4-270b)

2 1

When x1 = 0, ln γ 1∞ = A12; when x2 = 0, ln γ 2∞ = A21. An alternative equation is obtained when the reciprocal quantity x1x2RT/GE is expressed as a linear function of x1: x1x2  = A′ + B′(x1 − x2) = A′ + B′(2x1 − 1) GE/RT This may also be written as x1x2  = A′(x1 + x2) + B′(x1 − x2) = (A′ + B′)x1 + (A′ − B′) x2 GE/RT The substitutions A′ + B′ = 1/A′21 and A′ − B′ = 1/A′12 ultimately produce

MODELS FOR THE EXCESS GIBBS ENERGY

GE  = a + bx1 + cx21 + · · · x1x2RT

4-23

(4-269)

A′12 A′21 GE  =  A′12x1 + A′21x2 x1x2RT

(4-271)

The activity coefficients implied by this equation are given by −2

A′12 x1 ln γ1 = A′12 1 +  A′21 x2





A′21x2 ln γ2 = A′21 1 +  A′12 x1





(4-272a)

−2

(4-272b)

These are the van Laar equations. When x1 = 0, ln γ 1∞= A′12; when x2 = 0, ln γ 2∞ = A′21. The Redlich-Kister expansion, the Margules equations, and the van Laar equations are all special cases of a very general treatment based on rational functions, i.e., on equations for GE given by ratios of polynomials [Van Ness and Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, Sec. 5-7, McGraw-Hill, New York (1982)]. Although providing great flexibility in the fitting of VLE data for binary systems, they are without theoretical foundation, with no basis in theory for their extension to multicomponent systems. Nor do they incorporate an explicit temperature dependence for the parameters. Theoretical developments in the molecular thermodynamics of liquid solution behavior are often based on the concept of local composition, presumed to account for the short-range order and nonrandom molecular orientations resulting from differences in molecular size and intermolecular forces. Introduced by G. M. Wilson [J. Am. Chem. Soc. 86: 127−130 (1964)] with the publication of a model for GE, this concept prompted the development of alternative local composition models, most notably the NRTL (Non-Random Two-Liquid) equation of Renon and Prausnitz [AIChE J. 14: 135−144 (1968)] and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz [AIChE J. 21: 116−128 (1975)]. The Wilson equation, like the Margules and van Laar equations, contains just two parameters for a binary system (Λ12 and Λ21) and is written as GE  = − x1 ln(x1 + x2Λ12) − x2 ln(x2 + x1Λ21) RT

(4−273)

Λ12 Λ21 ln γ1 = − ln (x1 + x2Λ12) + x2  −  x1 + x2Λ12 x2 + x1Λ21



(4-274a)

Λ21 Λ12 ln γ2 = − ln (x2 + x1Λ21) − x1  −  x1 + x2Λ12 x2 + x1Λ21



(4-274b)

 

At infinite dilution, ln γ 1∞ = −ln Λ12 + 1 − Λ21

ln γ 2∞ = −ln Λ21 + 1 − Λ12

Both Λ12 and Λ21 must be positive numbers.

THERMODYNAMICS

The NRTL equation contains three parameters for a binary system and is written as

 

+  (x + x G ) 

b12 τ12 =  RT

2

1

12

2

G21τ21

2

1

2

21

2

(4-276a) (4-276b)

G21 = exp(−ατ21) b21 τ21 =  RT

where α, b12, and b21, parameters specific to a particular pair of species, are independent of composition and temperature. The infinite dilution values of the activity coefficients are ln γ ∞1 = τ21 + τ12 exp (−ατ12)

ln γ ∞2 = τ12 + τ21 exp (−ατ21)

 x Λ

2000

T S

j

(4-277)

ij

H

0

0

T S

H

0

1

x1

G

0

x1

(a)

G

x1 (d)

1000

H G

0

1

1

x1 (c)

1000 H

0

1000

1000 0

T S

T S

J mol1

J mol1

0

(4-280)

2000

1000

H

1

x1 (e)

TS

0 H G

1000

G

0

1

0

x1

1

(f)

Property changes of mixing at 50!C for six binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); (f ) ethanol(1)/water(2). [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 455, McGraw-Hill, New York (2005).]

FIG. 4-4

(4-278)

kj

0

(b) TS

1000

j

1000

1000

1000

G

k

g = gC + gR

2000

1000

1000

ij

where Λij = 1 for i = j, etc. All indices in these equations refer to the same species, and all summations are over all species. For each ij pair there are two parameters, because Λij ≠ Λji. For example, in a ternary system the three possible ij pairs are associated with the parameters Λ12, Λ21; Λ13, Λ31; and Λ23, Λ32. The temperature dependence of the parameters is given by −aij Vj Λij =  exp  i≠j (4-279) RT Vi where Vj and Vi are the molar volumes of pure liquids j and i and aij is a constant independent of composition and temperature. Molar volumes Vj and Vi, themselves weak functions of temperature, form ratios that in practice may be taken as independent of T, and are usually evaluated at or near 25°C. The Wilson parameters Λij and NRTL parameters Gij inherit a Boltzmann-type T dependence from the origins of the expressions for E G , but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/liquid solubility) are in general only qualitatively correct. However, all parameters are found from data for binary (in contrast to multicomponent) systems, and this makes parameter determination for the local composition models a task of manageable proportions. The UNIQUAC equation treats g  GERT as made up of two additive parts, a combinatorial term gC, accounting for molecular size and shape differences, and a residual term gR (not a residual property), accounting for molecular interactions:

j

J mol1

J mol1

The local composition models have limited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constitutent binary systems. For example, the Wilson equation for multicomponent systems is written as GE  = − xi ln RT i

j

j

J mol1

G12 = exp(−ατ12)

and

G12τ12

+  (x + x G ) 

G12 ln γ2 = x21 τ12  x2 + x1G12 Here

(4-275)

 

2

xkΛki

 x Λ − 

x Λ j

G21τ21 G12τ12 GE  =  +  x1 + x2G21 x2 + x1G12 x1x2RT G21 ln γ1 = x22 τ21  x1 + x2G21

ln γi = 1 − ln

and

J mol1

4-24

SYSTEMS OF VARIABLE COMPOSITION

4-25

Function gC contains pure-species parameters only, whereas function gR incorporates two binary parameters for each pair of molecules. For a multicomponent system,

An expression for ln γ i is found by application of Eq. (4-251) to the UNIQUAC equation for g [Eqs. (4-280) through (4-282)]. The result is given by the following equations:

Φi θi g C = xi ln  + 5 qi xi ln  xi Φi i i

ln γ i = ln γ iC + ln γ Ri

g R = − qi xi ln i

(4-281)

 j θ τ j ji

xiri Φi  

xjrj

where

(4-282)

Ji Ji ln γ iC = 1 − Ji + ln Ji − 5qi 1 −  + ln  Li Li

(4-283)

τij ln γ iR = qi 1 − ln si − θj  sj j

j









(4-287)

(4-288)

where in addition to Eqs. (4-284) and (4-285),

xi qi θi  

xj qj

and

(4-286)

(4-284)

j

ri Ji = 

rj xj

(4-289)

qi Li = 

qj xj

(4-290)

si = θ l τli

(4-291)

j

Subscript i identifies species, and j is a dummy index; all summations are over all species. Note that τji ≠ τij; however, when i = j, then τii = τjj = 1. In these equations ri (a relative molecular volume) and qi (a relative molecular surface area) are pure-species parameters. The influence of temperature on g enters through the interaction parameters τji of Eq. (4-282), which are temperature-dependent:

j

l

− (uji − uii) τji = exp  RT

(4-285)

Parameters for the UNIQUAC equation are therefore values of uji − uii.

Again subscript i identifies species, and j and l are dummy indices. Values for the parameters of the commonly used models for the excess Gibbs energy are given by Gmehling, Onken, and Arlt [VaporLiquid Equilibrium Data Collection, Chemistry Data Series, vol. 1, parts 1–8, DECHEMA, Frankfurt/Main (1974–1990)].

0

TS E

GE

J mol1

500

H

J mol1

J mol1

HE

E

GE

500

TS E

GE

1000 TS E HE

0

0

1

0

1 x1

x1

(a)

(b)

(c)

GE HE

GE

J mol1

500 0 TS

500

GE

500

0

E

H

E

0 500

1

J mol1

J mol1

1000

1

x1

HE

TS E

1000

TS E

0

0

1

0

1

x1

x1

x1

(d )

(e)

(f )

Excess properties at 50!C for six binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); (f ) ethanol(1)/water(2). [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 420, McGraw-Hill, New York (2005).]

FIG. 4-5

4-26

THERMODYNAMICS 0 0.6

0.6

0.4

0.4

In 1

0.4

In 2

In 1

0.2

0.2 In 2

In 2 In 1

0.6

0.2

0.8 0

1

x1

0

(a)

In 1

1.0

In 2

1.5 In 1

1.0

In 1

In 2

In 2

1

0.5

0.5

x1

1

x1 (c)

1.5

0

0

(b)

3 2

1

x1

1

0

1

x1

(d )

0

1

x1

(e)

(f)

Activity coefficients at 50!C for six binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); (f) ethanol(1)/water(2). [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 445, McGraw-Hill, New York (2005).]

FIG. 4-6

Behavior of Binary Liquid Solutions Property changes of mixing and excess properties find greatest application in the description of liquid mixtures at low reduced temperatures, i.e., at temperatures well below the critical temperature of each constituent species. The properties of interest to the chemical engineer are VE ( ∆V), HE ( ∆H), SE, ∆S, GE, and ∆G. The activity coefficient is also of special importance because of its application in phase equilibrium calculations. The volume change of mixing (VE = ∆V), the heat of mixing (HE = ∆H), and the excess Gibbs energy GE are experimentally accessible, ∆V and ∆H by direct measurement and GE (or ln γ i ) indirectly by reduction of vapor/liquid equilibrium data. Knowledge of HE and

GE allows calculation of SE by Eq. (4-10), written for excess properties as HE − GE SE =  (4-292) T with ∆S then given by Eq. (4-230). Figure 4-4 displays plots of ∆H, ∆S, and ∆G as functions of composition for six binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-5; the activity coefficients, derived from Eq. (4-251), appear in Fig. 4-6. The properties shown here are insensitive to pressure and for practical purposes represent solution properties at 50°C and low pressure (P ≈ 1 bar).

EQUILIBRIUM dUt − dW − TdSt ≤ 0

CRITERIA

Combination gives

The equations developed in preceding sections are for PVT systems in states of internal equilibrium. The criteria for internal thermal and mechanical equilibrium simply require uniformity of temperature and pressure throughout the system. The criteria for phase and chemical reaction equilibria are less obvious. If a closed PVT system of uniform T and P, either homogeneous or heterogeneous, is in thermal and mechanical equilibrium with its surroundings, but is not at internal equilibrium with respect to mass transfer or chemical reaction, then changes in the system are irreversible and necessarily bring the system closer to an equilibrium state. The first and second laws written for the entire system are dQ dUt = dQ + dW dSt ≥  T

Because mechanical equilibrium is assumed, dW = −PdVt, whence dUt + PdVt − TdSt ≤ 0 The inequality applies to all incremental changes toward the equilibrium state, whereas the equality holds at the equilibrium state where change is reversible. Constraints put on this expression produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, dUSt ,V ≤ 0. Particularly important is fixing T and P; this produces t

t

d(Ut + PVt − TSt)T,P ≤ 0

or dGtT,P ≤ 0

EQUILIBRIUM This expression shows that all irreversible processes occurring at constant T and P proceed in a direction such that the total Gibbs energy of the system decreases. Thus the equilibrium state of a closed system is the state with the minimum total Gibbs energy attainable at the given T and P. At the equilibrium state, differential variations may occur in the system at constant T and P without producing a change in Gt. This is the meaning of the equilibrium criterion dGtT,P = 0

(4-293)

This equation may be applied to a closed, nonreactive, two-phase system. Each phase taken separately is an open system, capable of exchanging mass with the other; Eq. (4-13) is written for each phase: d(nG)′ = −(nS)′ dT + (nV)′ dP + µ′i dni′

phase. The number of these variables is 2 + (N − 1)π. The masses of the phases are not phase rule variables, because they have nothing to do with the intensive state of the system. The equilibrium equations that may be written express chemical potentials or fugacities as functions of T, P, and the phase compositions, the phase rule variables: 1. Equation (4-295) for each species, giving (π − 1)N phase equilibrium equations 2. Equation (4-296) for each independent chemical reaction, giving r equations The total number of independent equations is therefore (π − 1)N + r. Because the degrees of freedom of the system F is the difference between the number of variables and the number of equations, F = 2 + (N − 1)π − (π − 1)N − r

i

d(nG)″ = −(nS)″ dT + (nV)″ dP + µ″i dn″i

or

i

where the primes and double primes denote the two phases; the presumption is that T and P are uniform throughout the two phases. The change in the Gibbs energy of the two-phase system is the sum of these equations. When each total-system property is expressed by an equation of the form nM = (nM)′ + (nM)″, this sum is given by d(nG) = (nV)dP − (nS)dT + µ′i dn′i + µ″i dni″ i

i

If the two-phase system is at equilibrium, then application of Eq. (4-293) yields dGtT,P  d(nG)T,P = µ′i dni′ + µi″dn″i = 0 i

i

The system is closed and without chemical reaction; material balances therefore require that dn″i = −dn′i, reducing the preceding equation to

i (µ′ − µ″)dn′ = 0 i

i

Because the dn′i are independent and arbitrary, it follows that µ′i = µ″i. This is the criterion of two-phase equilibrium. It is readily generalized to multiple phases by successive application to pairs of phases. The general result is (4-294)

Substitution for each µi by Eq. (4-202) produces the equivalent result: (4-295) fˆi ′ = fˆi″ = fˆi″′ = · · · These are the criteria of phase equilibrium applied in the solution of practical problems. For the case of equilibrium with respect to chemical reaction within a single-phase closed system, combination of Eqs. (4-13) and (4-293) leads immediately to

i µ dn = 0 i

i

F=2−π+N−r

(4-297)

The number of independent chemical reactions r can be determined as follows: 1. Write formation reactions from the elements for each chemical compound present in the system. 2. Combine these reaction equations so as to eliminate from the set all elements not present as elements in the system. A systematic procedure is to select one equation and combine it with each of the other equations of the set so as to eliminate a particular element. This usually reduces the set by one equation for each element eliminated, although two or more elements may be simultaneously eliminated. The resulting set of r equations is a complete set of independent reactions. More than one such set is often possible, but all sets number r and are equivalent. Example 2: Application of the Phase Rule a. For a system of two miscible nonreacting species in vapor/liquid equilibrium, F=2−π+N−r=2−2+2−0=2

i

µi′ = µi″ = µi″′ = · · ·

4-27

(4-296)

For a system in which both phase and chemical reaction equilibrium prevail, the criteria of Eqs. (4-295) and (4-296) are superimposed. PHASE RULE The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states not all these variables are independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and it is called the number of degrees of freedom of the system. It is the number of variables that may be arbitrarily specified and that must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. For a system containing N chemical species distributed at equilibrium among π phases, the phase rule variables are T and P, presumed uniform throughout the system, and N − 1 mole fractions in each

The 2 degrees of freedom for this system may be satisfied by setting T and P, or T and y1, or P and x1, or x1 and y1, etc., at fixed values. Thus for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the 2 degrees of freedom are used up, no further specification is possible that would restrict the phase rule variables. For example, one cannot in addition require that the system form an azeotrope (assuming this is possible), for this requires x1 = y1, an equation not taken into account in the derivation of the phase rule. Thus the requirement that the system form an azeotrope imposes a special constraint, making F = 1. b. For a gaseous system consisting of CO, CO2, H2, H2O, and CH4 in chemical reaction equilibrium, F=2−π+N−r=2−1+5−2=4 The value of r = 2 is found from the formation reactions: C + 12O2 → CO

C + O2 → CO2

H2 + 12O2 → H2O

C + 2H2 → CH4

Systematic elimination of C and O2 from this set of chemical equations reduces the set to two. Three possible pairs of equations may result, depending on how the combination of equations is effected. Any pair of the following three equations represents a complete set of independent reactions, and all pairs are equivalent. CH4 + H2O → CO + 3H2 CO + H2O → CO2 + H2 CH4 + 2H2O → CO2 + 4H2 The result, F = 4, means that one is free to specify, for example, T, P, and two mole fractions in an equilibrium mixture of these five chemical species, provided nothing else is arbitrarily set. Thus it cannot simultaneously be required that the system be prepared from specified amounts of particular constituent species.

Duhem’s Theorem Because the phase rule treats only the intensive state of a system, it applies to both closed and open systems. Duhem’s theorem, on the other hand, is a rule relating to closed systems only: For any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely

4-28

THERMODYNAMICS

determined by any two properties of the system, provided only that the two properties are independently variable at the equilibrium state. The meaning of completely determined is that both the intensive and extensive states of the system are fixed; not only are T, P, and the phase compositions established, but so also are the masses of the phases. VAPOR/LIQUID EQUILIBRIUM Vapor/liquid equilibrium (VLE) relationships (as well as other interphase equilibrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but measurements are seldom easy, even for binary systems, and they become ever more difficult as the number of species increases. This is the incentive for application of thermodynamics to the calculation of phase equilibrium relationships. The general VLE problem treats a multicomponent system of N constituent species for which the independent variables are T, P, N − 1 liquid-phase mole fractions, and N − 1 vapor-phase mole fractions. (Note that Σixi = 1 and Σiyi = 1, where xi and yi represent liquid and vapor mole fractions, respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations (4-298) fˆi l = fˆiv i = 1, 2, . . . , N where superscripts l and v denote the liquid and vapor phases, respectively. In practice, either T or P and either the liquid-phase or vapor-phase composition are specified, thus fixing 1 + (N − 1) = N independent variables. The remaining N variables are then subject to calculation, provided that sufficient information is available to allow determination of all necessary thermodynamic properties. Gamma/Phi Approach For many VLE systems of interest, the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactory for the vapor phase. Liquidphase behavior, on the other hand, is described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is given by Eq. (4-221), and the vapor-phase fugacity is given by Eq. (4-204). These are here written as fˆil = γi xi fi and fˆiv = φˆ vi yiP By Eq. (4-298),

γi xi fi = φˆ iyi P

i = 1, 2, . . . , N

(4-299)

Identifying superscripts l and v are omitted here with the understanding that γi and fi are liquid-phase properties, whereas φˆ i is a vaporphase property. Applications of Eq. (4-299) represent what is known as the gamma/phi approach to VLE calculations. Evaluation of φˆ i is usually by Eq. (4-243), based on the two-term virial equation of state. The activity coefficient γi is ultimately based on Eq. (4-251) applied to an expression for GE/RT, as described in the section “Models for the Excess Gibbs Energy.” The fugacity fi of pure compressed liquid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor φvi = φisat by an integrated form of Eq. (4-205), most commonly by Eq. (4-242) evaluated for pure species i at temperature T and the corresponding vapor pressure P = Pisat. Equation (4-298) written for pure species i becomes fiv = fil = fisat

(4-300)

where fisat indicates the value both for saturated liquid and for saturated vapor. Division by Pisat yields corresponding fugacity coefficients: fisat fiv fil =  =   sat sat Pi Pi P sat i or

φvi = φ li = φ sat i

(4-301)

The second step is the evaluation of the change in fugacity of the liquid with a change in pressure to a value above or below Pisat. For this isothermal change of state from saturated liquid at Pisat to liquid at pressure P, Eq. (4-17) is integrated to give Gi − Gisat =



P

Vi dP

sat Pi

Equation (4-199) is then written twice: for Gi and for Gsat i . Subtraction provides another expression for Gi − Gisat: fi Gi − Gisat = RT ln  fisat Equating the two expressions for Gi − Gisat yields 1 fi ln  =  fisat RT



P

Vi dP

sat Pi

Because Vi, the liquid-phase molar volume, is a very weak function of P at temperatures well below Tc, an excellent approximation is usually obtained when evaluation of the integral is based on the assumption that Vi is constant at the value for saturated liquid Vil: fi Vil(P − Psat i ) ln  =  sat fi RT sat sat Substituting f sat i = φ i P i and solving for fi give

Vil(P − P sat i ) sat fi = φ sat i P i exp  RT

(4-302)

The exponential is known as the Poynting factor. Equation (4-299) may now be written as

where

yiPΦi = xi γi Psat i = 1, 2, . . . , N i −Vil(P − P sat φˆ i i ) Φi = sat exp  RT φi

(4-303) (4-304)

If evaluation of φ isat and φˆ i is by Eqs. (4-244) and (4-243), this reduces to ⎯ l sat PBi − Psat i Bii − Vi (P − Pi ) Φi = exp  RT

(4-305)

⎯ where Bi is given by Eq. (4-182). The N equations represented by Eq. (4-303) in conjunction with Eq. (4-305) may be solved for N unknown phase equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. When Eq. (4-303) is applied to VLE for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a very simple expression. For ideal gases, fugacity coefficients φˆ i and φsat i are unity, and the right side of Eq. (4-304) reduces to the Poynting factor. For the systems of interest here, this factor is always very close to unity, and for practical purposes Φi = 1. For ideal solutions, the activity coefficients γi are also unity, and Eq. (4-303) reduces to yiP = xiP sat i

i = 1, 2, . . . , N

(4-306)

an equation which expresses Raoult’s law. It is the simplest possible equation for VLE and as such fails to provide a realistic representation of real behavior for most systems. Nevertheless, it is useful as a standard of comparison. Modified Raoult’s Law Of the qualifications that lead to Raoult’s law, the one least often reasonable is the supposition of solution ideality for the liquid phase. Real solution behavior is reflected by values of activity coefficients that differ from unity. When γi of Eq. (4-303) is retained in the equilibrium equation, the result is the modified Raoult’s law: yiP = xi γi P sat i

i = 1, 2, . . . , N

(4-307)

EQUILIBRIUM This equation is often adequate when applied to systems at low to moderate pressures and is therefore widely used. Bubble point and dew point calculations are only a bit more complex than the same calculations with Raoult’s law. Activity coefficients are functions of temperature and liquid-phase composition and are correlated through equations for the excess Gibbs energy. When an appropriate correlating equation for GE is not available, suitable estimates of activity coefficients may often be obtained from a group contribution correlation. This is the “solution of groups” approach, wherein activity coefficients are found as sums of contributions from the structural groups that make up the molecules of a solution. The most widely applied such correlations are based on the UNIQUAC equation, and they have their origin in the UNIFAC method (UNIQUAC Functional-group Activity Coefficients), proposed by Fredenslund, Jones, and Prausnitz [AIChE J. 21: 1086–1099 (1975)], and given detailed treatment by Fredenslund, Gmehling, and Rasmussen [Vapor-Liquid Equilibrium Using UNIFAC, Elsevier, Amsterdam (1977)]. Subsequent development has led to a variety of applications, including liquid/liquid equilibria [Magnussen, Rasmussen, and Fredenslund, Ind. Eng. Chem. Process Des. Dev. 20: 331–339 (1981)], solid/liquid equilibria [Anderson and Prausnitz, Ind. Eng. Chem. Fundam. 17: 269–273 (1978)], solvent activities in polymer solutions [Oishi and Prausnitz, Ind. Eng. Chem. Process Des. Dev. 17: 333–339 (1978)], vapor pressures of pure species [Jensen, Fredenslund, and Rasmussen, Ind. Eng. Chem. Fundam. 20: 239–246 (1981)], gas solubilities [Sander, Skjold-Jørgensen, and Rasmussen, Fluid Phase Equilib. 11: 105–126 (1983)], and excess enthalpies [Dang and Tassios, Ind. Eng. Chem. Process Des. Dev. 25: 22–31 (1986)]. The range of applicability of the original UNIFAC model has been greatly extended and its reliability enhanced. Its most recent revision and extension is treated by Wittig, Lohmann, and Gmehling [Ind. Eng. Chem. Res. 42: 183–188 (2003)], wherein are cited earlier pertinent papers. Because it is based on temperature-independent parameters, its application is largely restricted to 0 to 150°C. Two modified versions of the UNIFAC model, based on temperaturedependent parameters, have come into use. Not only do they provide a wide temperature range of applicability, but also they allow correlation of various kinds of property data, including phase equilibria, infinite dilution activity coefficients, and excess properties. The most recent revision and extension of the modified UNIFAC (Dortmund) model is provided by Gmehling et al. [Ind. Eng. Chem. Res. 41: 1678–1688 (2002)]. An extended UNIFAC model called KT-UNIFAC is described in detail by Kang et al. [Ind. Eng. Chem. Res. 41: 3260–3273 (2003)]. Both papers contain extensive literature citations. The UNIFAC model has also been combined with the predictive Soave-Redlich-Kwong (PSRK) equation of state. The procedure is most completely described (with background literature citations) by Horstmann et al. [Fluid Phase Equilibria 227: 157–164 (2005)]. Because Σ iyi = 1, Eq. (4-307) may be summed over all species to yield P = xi γi P sat i

(4-308)

4-29

with parameters i 1 2

Ai 14.3145 13.8193

Bi 2756.22 2696.04

Ci −45.090 −48.833

Activity coefficients are given by Eq. (4-274), the Wilson equation: ln γ1 = −ln(x1 + x2Λ12) + x2λ

(B)

ln γ2 = −ln(x2 + x1Λ21) − x1λ

(C)

Λ12 Λ21 λ   −  x1 + x2Λ12 x2 + x1Λ21

where

− aij Vj Λij =  exp  RT Vi

By Eq. (4-279)

i≠j

with parameters [Gmehling et al., Vapor-Liquid Data Collection, Chemistry Data Series, vol. 1, part 3, DECHEMA, Frankfurt/Main (1983)] a12 cal mol−1 985.05

a21 cal mol−1 453.57

V1 cm3 mol−1 74.05

V2 cm3 mol−1 131.61

When T and x1 are given, the calculation is direct, with final values for vapor pressures and activity coefficients given immediately by Eqs. (A), (B), and (C). In all other cases either T or x1 or both are initially unknown, and calculations require trial or iteration. a. BUBL P calculation: Find y1 and P, given x1 and T. Calculation here is direct. For x1 = 0.40 and T = 325.15 K (52!C), Eqs. (A), (B), and (C) yield the values listed in the table on the following page. Equations (4-308) and (4-307) then become sat P = x1γ1P sat 1 + x2 γ2 P 2 = (0.40)(1.8053)(87.616) + (0.60)(1.2869)(58.105) = 108.134 kPa

(0.40)(1.8053)(87.616) x1γ1Psat 1 y1 =  =  = 0.5851 P 108.134 b. DEW P calculation: Find x1 and P, given y1 and T. With x1 an unknown, the activity coefficients cannot be immediately calculated. However, an iteration scheme based on Eqs. (4-309) and (4-307) is readily devised, and is part of any solve routine of a software package. Starting values result from setting each γi = 1. For y1 = 0.4 and T = 325.15 K (52!C), results are listed in the accompanying table. c. BUBL T calculation: Find y1 and T, given x1 and P. With T unknown, neither the vapor pressures nor the activity coefficients can be initally calculated. An iteration scheme or a solve routine with starting values for the unknowns is required. Results for x1 = 0.32 and P = 80 kPa are listed in the accompanying table. d. DEW T calculation: Find x1 and T, given y1 and P. Again, an iteration scheme or a solve routine with starting values for the unknowns is required. For y1 = 0.60 and P = 101.33 kPa, results are listed in the accompanying table. e. Azeotrope calculations: As noted in Example 1a, only a single degree of freedom exists for this special case. The most sensitive quantity for identifying the azeotropic state is the relative volatility, defined as

i

Alternatively, Eq. (4-307) may be solved for xi, in which case summing over all species yields

y1/x1 α12   y2/x2

1 P = 

yi /γiP sati

Because yi = xi for the azeotropic state, α12 = 1. Substitution for the two ratios by Eq. (4-307) provides an equation for calculation of α12 from the thermodynamic functions:

(4-309)

i

Example 3: Dew and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x1, and y1, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoult’s law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. The necessary vapor pressures and activity coefficients are supplied by data correlations. For the system acetone(1)/n-hexane(2), vapor pressures are given by Eq. (4-142), the Antoine equation: Bi ln P sat i = 1, 2 (A) i /kPa = Ai −  TK + Ci

γ 1Psat 1 α12 =  γ 2P sat 2 Because α12 is a monotonic function of x1, the test of whether an azeotrope exists at a given T or P is provided by values of α12 in the limits of x1 = 0 and x1 = 1. If both values are either > 1 or < 1, no azeotrope exists. But if one value is < 1 and the other > 1, an azeotrope necessarily exists at the given T or P. Given T, the azeotropic composition and pressure is found by seeking the value of P that makes x1 = y1 or that makes α12 = 1. Similarly, given P, one finds the azeotropic composition and temperature. Shown in the accompanying table are calculated azeotropic states for a temperature of 46!C and for a pressure of 101.33 kPa. At 46°C, the limiting values of α12 are 8.289 at x1 = 0 and 0.223 at x1 = 1.

4-30 T/K a. b. c. d. e. f.

325.15 325.15 317.24 322.98 319.15 322.58

THERMODYNAMICS P1sat/ kPa P2sat/ kPa 87.616 87.616 65.830 81.125 70.634 79.986

58.105 58.105 43.591 53.779 46.790 53.021

γ1

γ2

1.8053 3.5535 2.1286 1.6473 1.2700 1.2669

1.2869 1.0237 1.1861 1.3828 1.9172 1.9111

x1

y1

0.4000 0.5851 0.1130 0.4000 0.3200 0.5605 0.4550 0.6000 0.6445 = 0.6445 0.6454 = 0.6454

P/kPa 108.134 87.939 80.000 101.330 89.707 101.330

Given values are italic; calculated results are boldface.

Data Reduction Correlations for GE and the activity coefficients are based on VLE data taken at low to moderate pressures. Groupcontribution methods, such as UNIFAC, depend for validity on parameters evaluated from a large base of such data. The process of finding a suitable analytic relation for g ( GERT) as a function of its independent variables T and x1, thus producing a correlation of VLE data, is known as data reduction. Although g is in principle also a function of P, the dependence is so weak as to be universally and properly neglected. Given here is a brief description of the treatment of data taken for binary systems under isothermal conditions. A more comprehensive development is given by Van Ness [J. Chem. Thermodyn. 27: 113–134 (1995); Pure & Appl. Chem. 67: 859–872 (1995)]. Presumed in all that follows is the existence of an equation inherently capable of correlating values of GE for the liquid phase as a function of x1: g  GE/RT = G(x1; α, β, . . .)

(4-310)

where α, β, etc., represent adjustable parameters. The measured variables of binary VLE are x1, y1, T, and P. Experimental values of the activity coefficient of species i in the liquid are related to these variables by Eq. (4-303), written as yi*P* γ i* =  Φi xiPsat i

i = 1, 2

(4-311)

where Φi is given by Eq. (4-305) and the asterisks denote experimental values. A simple summability relation analogous to Eq. (4-252) defines an experimental value of g*: g*  x1 ln γ *1 + x2 ln γ*2

Subtraction of Eq. (4-316) from Eq. (4-315) gives γ *1 γ1 dg* dg d ln γ * d ln γ *  −  = ln  − ln  − x1 1 + x2 2 γ *2 γ2 dx1 dx1 dx1 dx1



The differences between like terms represent residuals between derived and experimental values. Defining these residuals as δg  g − g*

γ1 dδg d ln γ * d ln γ *  = δ ln  − x1 1 + x2 2 γ2 dx1 dx1 dx1



(4-314a)

dg ln γ2 = g − x1  dx1

(4-314b)

These two equations combine to yield γ1 dg  = ln  γ2 dx1

(4-315)

This equation is valid for derived property values. The corresponding experimental values are given by differentiation of Eq. (4-312): dg* d ln γ * d ln γ *  = x1 1 + ln γ *1 + x2 2 − ln γ *2 dx1 dx1 dx1 or

dg* γ* d ln γ * d ln γ *  = ln 1 + x1 1 + x2 2 dx1 γ *2 dx1 dx1

(4-316)



If a data set is reduced so as to yield parameters—α, β, etc.—that make the δ g residuals scatter about zero, then the derivative on the left is effectively zero, and the preceding equation becomes γ d ln γ * d ln γ * δ ln 1 = x1 1 + x2 2 γ2 dx1 dx1

(4-317)

The right side of this equation is the quantity required by Eq. (4-313), the Gibbs-Duhem equation, to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs-Duhem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. The data reduction procedure just described provides parameters in the correlating equation for g that make the δg residuals scatter about zero. This is usually accomplished by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y1. Equation (4-303) is solved for yi P and written for species 1 and for species 2. Adding the two equations gives

Moreover, Eq. (4-263), the Gibbs-Duhem equation, may be written for experimental values in a binary system at constant T and P as

dg ln γ1 = g + x2  dx1

γ γ γ* δ ln 1  ln 1 − ln 1 γ2 γ2 γ *2

and

puts this equation into the form

(4-312)

d ln γ* d ln γ* (4-313) x1 1 + x2 2 = 0 dx1 dx1 Because experimental measurements are subject to systematic error, sets of values of ln γ *1 and ln γ *2 may not satisfy, i.e., may not be consistent with, the Gibbs-Duhem equation. Thus Eq. (4-313) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. Values of g provided by the equation used to correlate the data, as represented by Eq. (4-310), are called derived values, and produce derived values of the activity coefficients by Eqs. (4-180) with M  g:



x1γ1Psat x2γ2Psat 1 2 P=  +  Φ1 Φ2

(4-318)

x1γ1Psat 1 y1 =  Φ1P

(4-319)

whence by Eq. (4-303),

These equations allow calculation of the primary residuals: δP  P − P*

and

δy1  y1 − y*1

If the experimental values P* and y*1 are closely reproduced by the correlating equation for g, then these residuals, evaluated at the experimental values of x1, scatter about zero. This is the result obtained when the data approach thermodynamic consistency. When they do not, these residuals fail to scatter about zero and the correlation for g does not properly reproduce the experimental values P* and y*1. Such a correlation is unnecessarily divergent. An alternative is to base data reduction on just the P-x1 data subset; this is possible because the full P-x1-y1 data set includes redundant information. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters α, β, etc., that yield pressures by Eq. (4-318) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals δP. Known as Barker’s method [Austral. J. Chem. 6: 207−210 (1953)], it provides the best possible fit of the experimental pressures. When experimental y*1 values are not consistent with the P*-x1 data, Barker’s method cannot lead to calculated y1 values that closely match the experimental y*1 values. With experimental error usually concentrated in the y*1 values, the calculated y1 values are likely to be more nearly correct. Because Barker’s method requires only the P*-x1 data subset, the measurement of y*1 values is not usually worth the extra effort, and the correlating parameters α, β, etc., are usually best determined without them. Hence, many P*-x1 data subsets appear in the literature; they are of course not subject to a test for consistency by the Gibbs-Duhem equation.

EQUILIBRIUM

4-31

For the solvent, species 2, the analog of Eq. (4-319) is Henry’s law

x2γ2Psat 2 y2 =  Φ2P

f^1

x1(γ1γ1∞)k1 x2γ2P sat 2 Because y1 + y2 = 1, P =  +  φˆ 1 Φ2

f^1

0

1

Plot of solute fugacity fˆ1 versus solute mole fraction. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 555, McGraw-Hill, New York (2005).] FIG. 4-7

The world’s store of VLE data has been compiled by Gmehling et al. [Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. 1, parts 1–8, DECHEMA, Frankfurt am Main (1979–1990)]. Solute/Solvent Systems The gamma/phi approach to VLE calculations presumes knowledge of the vapor pressure of each species at the temperature of interest. For certain binary systems species 1, designated the solute, is either unstable at the system temperature or is supercritical (T > Tc). Its vapor pressure cannot be measured, and its fugacity as a pure liquid at the system temperature f1 cannot be calculated by Eq. (4-302). Equations (4-303) and (4-304) are applicable to species 2, designated the solvent, but not to the solute, for which an alternative approach is required. Figure 4-7 shows a typical plot of the liquidphase fugacity of the solute fˆ1 versus its mole fraction x1 at constant temperature. Since the curve representing fˆ1 does not extend all the way to x1 = 1, the location of f1, the liquid-phase fugacity of pure species 1, is not established. The tangent line at the origin, representing Henry’s law, provides alternative information. The slope of the tangent line is Henry’s constant, defined as (4-320)

This is the definition of k1 for temperature T and for a pressure equal to the vapor pressure of the pure solvent P2sat. The activity coefficient of the solute at infinite dilution is fˆ1 1 fˆ lim γ1 = lim  =  lim 1 → x1→0 x1 f1 x 0 f1 1 x1

x1→0

In view of Eq. (4-320), this becomes γ 1∞ = k1 f1, or k1 f1 =  γ 1∞

(4-321)

where γ 1∞ represents the infinite dilution value of the activity coefficient of the solute. Because both k1 and γ 1∞ are evaluated at P2sat, this pressure also applies to f1. However, the effect of P on a liquid-phase fugacity, given by a Poynting factor, is very small and for practical purposes may usually be neglected. The activity coefficient of the solute then becomes fˆ1 y1Pφˆ 1 y1Pφˆ 1γ 1∞ γ1   =  =  x1 f1 x1 f1 x1k1 For the solute, this equation takes the place of Eqs. (4-303) and (4-304). Solution for y1 gives x1(γ1γ 1∞)k1 y1 =   φˆ 1P

(4-324)

The same correlation that provides for the evaluation of γ1 also allows evaluation of γ 1∞. There remains the problem of finding Henry’s constant from the available VLE data. For equilibrium fˆ1  fˆ1l = fˆ1v = y1Pφˆ 1

x1

fˆ k1  lim 1 x1→0 x1

(4-323)

(4-322)

Division by x1 gives

fˆ1 y1  = Pφˆ 1  x1 x1

Henry’s constant is defined as the limit as x1 →0 of the ratio on the left; therefore y1 ˆ∞ k1 = Psat 2 φ1 lim  x1 → 0 x1 The limiting value of y1/x1 can be found by plotting y1/x1 versus x1 and extrapolating to zero. K Values, VLE, and Flash Calculations A measure of the distribution of a chemical species between liquid and vapor phases is the K value, defined as the equilibrium ratio: y Ki  i xi

(4-325)

It has no thermodynamic content, but may make for computational convenience through elimination of one set of mole fractions in favor of the other. It does characterize “lightness” of a constituent species. A “light” species, with K > 1, tends to concentrate in the vapor phase whereas a “heavy” species, with K < 1, tends to concentrate in the liquid phase. The rigorous evaluation of a K value follows from Eq. (4-299): y γi fi Ki  i =  xi φˆ iP

(4-326)

When Raoult’s law applies, Eq. (4-326) reduces to Ki = PisatP. For modified Raoult’s law, Ki = γiPisatP. With Ki = yi xi, these are alternative expressions of Raoult’s law and modified Raoult’s law. Were Raoult’s valid, K values could be correlated as functions of just T and P. However, Eq. (4-326) shows that they are in general functions of T, P, {xi}, and {yi}, making convenient and accurate correlation impossible. Those correlations that do exist are approximate and severely limited in application. The nomographs for K values of light hydrocarbons as functions of T and P, prepared by DePriester [Chem. Eng. Progr. Symp. Ser. No. 7, 49: 1–43 (1953)], do allow for an average effect of composition, but their essential basis is Raoult’s law. The defining equation for K can be rearranged as yi = Ki xi. The sum Σiyi = 1 then yields

i K x = 1 i i

(4-327)

With the alternative rearrangement xi = yi/Ki, the sum Σi xi = 1 yields yi

=1

i  K

(4-328)

i

Thus for bubble point calculations, where the xi are known, the problem is to find the set of K values that satisfies Eq. (4-327), whereas for dew point calculations, where the yi are known, the problem is to find the set of K values that satisfies Eq. (4-328). The flash calculation is a very common application of VLE. Considered here is the P, T flash, in which are calculated the quantities and compositions of the vapor and liquid phases in equilibrium at known T, P, and overall composition. This problem is determinate on the basis of Duhem’s theorem: For any closed system formed initally from

4-32

THERMODYNAMICS

given masses of prescribed chemical species, the equilibrium state is completely determined when any two independent variables are fixed. The independent variables are here T and P, and systems are formed from given masses of nonreacting chemical species. For 1 mol of a system with overall composition represented by the set of mole fractions {zi}, let L represent the molar fraction of the system that is liquid (mole fractions {xi}) and let V represent the molar fraction that is vapor (mole fractions {yi}). The material balance equations are L +V=1

and

zi = xiL + yiV

r

t

P

W

M

i = 1, 2, . . . , N

Combining these equations to eliminate L gives zi = xi(1 − V ) + yiV

P

i = 1, 2, . . . , N

u

0

V

(4-329)

Substitute xi = yi /Ki and solve for yi: ziKi yi =  1 + V(Ki − 1)

i = 1,2, . . . , N s

Because Σiyi = 1, this equation, summed over all species, yields ziKi

=1

i  1 + V (K − 1)

(4-330)

A subcritical isotherm on a PV diagram for a pure fluid. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 557, McGraw-Hill, New York (2005).] FIG. 4-8

i

The initial step in solving a P, T flash problem is to find the value of V which satisfies this equation. Note that V = 1 is always a trivial solution. Example 4: Flash Calculation The system of Example 3 has the overall composition z1 = 0.4000 at T = 325.15 K and P = 101.33 kPa. Determine V, x1, and y1. The BUBL P and DEW P calculations at T = 325.15 K of Example 3a and 3b show that for x1 = z1, Pbubl = 108.134 kPa, and for y1 = z1, Pdew = 87.939 kPa. Because P here lies between these values, the system is in two-phase equilibrium, and a flash calculation is appropriate. The modified Raoult’s law K values are given by (γ1)(Psat 1 ) K1 =  P

(γ2)(Psat 2 ) and K2 =  P

Equation (4-329) may be solved for V :

z1 − x1 V=  y1 − x1

Equation (4-330) here becomes (z2)(K2) (z1)(K1)  +  = 1 1 + V(K1 − 1) 1 + V(K2 − 1) A trial calculation illustrates the nature of the solution. Vapor pressures are taken from Example 3a or 3b; a trial value of x1 then allows calculation of γ1 and γ2 by Eqs. (B) and (C) of Example 3. The values of K1, K2, and V that result are substituted into the summation equation. In the unlikely event that the sum is indeed unity, the chosen value of x1 is correct. If not, then successive trials easily lead to this value. Note that the trivial solution giving V = 1 must be avoided. More elegant solution procedures can of course be employed. The answers are x1 = 0.2373 y1 = 0.5190 V = 0.5775 with

γ1 = 2.5297

γ2 = 1.0997

K1 = 2.1873

K2 = 0.6306

Equation-of-State Approach Although the gamma/phi approach to VLE is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactory for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for GE is included only for the local composition equations, and it is at best approximate. A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-204), may be applied to each phase: Liquid: fˆi l = φˆ li xiP Vapor: fˆiv = φˆ vi yi P By Eq. (4-298),

xiφˆ il = yiφˆ vi i = 1, 2, . . . , N

(4-331)

This introduces compositions xi and yi into the equilibrium equations, but neither is explicit, because the φˆ i are functions, not only of T and

P, but of composition. Thus, Eq. (4-331) represents N complex relationships connecting T, P, {xi}, and {yi}. Two widely used cubic equations of state appropriate for VLE calculations, both special cases of Eq. (4-100) [with Eqs. (4-101) and (4-102)], are the Soave-Redlich-Kwong (SRK) equation and the Peng-Robinson (PR) equation. The present treatment is applicable to both. The pure numbers ε, σ, Ψ, and Ω and expressions for α(Tri) specific to these equations are listed in Table 4-2. The associated expression for φˆ i is given by Eq. (4-246). The simplest application of equations of state in vapor/liquid equilibrium is to the calculation of vapor pressures Pisat of pure liquids. Vapor pressures can of course be measured, but values are also implicit in cubic equations of state. A subcritical PV isotherm, generated by a cubic equation of state, is shown in Fig. 4-8. Three segments are evident. The very steep one on the left (rs) is characteristic of liquids. Note that as P → ∞,V → b, where b is a constant in the cubic equation. The gently sloping segment on the right (tu) is characteristic of vapors; here P → 0 as V → ∞. The middle segment (st), with both a minimum (note P < 0) and a maximum, provides a transition from liquid to vapor, but has no physical meaning. The actual transition occurs along a horizontal line, such as connects points M and W. For pure species i, Eq. (4-331) reduces to φvi = φli, which may be written as ln φvi = ln φli

(4-332)

For given T, line MW lies at the vapor pressure Pisat if and only if the fugacity coefficients for points M and W satisfy Eq. (4-332). These points then represent saturated liquid and vapor phases in equilibrium at temperature T. The fugacity coefficients in Eq. (4-332) are given by Eq. (4-245): p

p

p

ln φip = Zi − 1 − ln(Zi − βi) − qi Ii

p = l, v

(4-333)

Expressions for Z iv and Zil come from Eqs. (4-104): Zvi − βi Ziv = 1 + βi − qiβi  v (Zi + %βi)(Ziv + σβi) 1 + βi − Zli Zli = βi + (Z li + %βi)(Zli + σβi)  qiβi



(4-334)



(4-335)

and Iip comes from Eq. (4-112): 1 Z pi + σβi Iip =  ln  σ− % Zip + %βi

p = l, v

(4-336)

EQUILIBRIUM The equation-of-state parameters are independent of phase. As defined by Eq. (4-105), βi is a function of P and here becomes biPsat i βi   RT

(4-337)

The remaining equation-of-state parameters, given by Eqs. (4-101), (4-102), and (4-106), are functions of T only and are written here as α(Tr )R2Tc2 ai(T) = ψ  Pc

(4-338)

RT bi = Ω c Pc

(4-339)

ai(T) qi   biRT

(4-340)

i

i

i

i

i

The eight equations (4-332) through (4-337) may be solved for the eight unknowns Pisat, βi, Zil, Ziv, Iil, Iiv, ln φil, and ln φiv. Perhaps more useful is the reverse calculation whereby an equationof-state parameter is evaluated from a known vapor pressure. Thus, Eqs. (4-332) and (4-333) may be combined and solved for qi, yielding Zvi − Zli + ln [(Zli − βi)/(Ziv − βi)] qi =  Iiv − Iil

(4-341)

Expressions for Zil, Ziv, Iil, Iiv, and βi are given by Eqs. (4-334) through (4-337). Because Zil and Ziv depend on qi, an iterative procedure is indicated, with a starting value for qi from a generalized correlation as given by Eqs. (4-338), (4-339), and (4-340). For mixtures the presumption is that the equation of state has exactly the same form as when written for pure species. Equations (4-104) are therefore applicable, with parameters β and q given by Eqs. (4-105) and (4-106). Here, these parameters, and therefore b and a(T), are functions of composition. Liquid and vapor mixtures in equilibrium in general have different compositions. The PV isotherms generated by an equation of state for these different compositions are represented in Fig. 4-9 by two similar lines: the solid line for the liquid-phase composition and the dashed line for the vapor-phase composition. They are displaced from each other because the equationof-state parameters are different for the two compositions.

Each line includes three segments as described for the isotherm of Fig. 4-8: the leftmost segment representing a liquid phase and the rightmost segment, a vapor phase, both with the same composition. Each left segment contains a bubble point (saturated liquid), and each right segment contains a dew point (saturated vapor). Because these points for a given line are for the same composition, they do not represent phases in equilibrium and do not lie at the same pressure. Shown in Fig. 4-9 is a bubble point B on the solid line and a dew point D on the dashed line. Because they lie at the same P, they represent phases in equilibrium, and the lines are characterized by the liquid and vapor compositions. For a BUBL P calculation, the temperature and the liquid composition are known, and this fixes the location of the PV isotherm for the composition of the liquid phase (solid line). The problem then is to locate a second (dashed) line for a vapor composition such that the line contains a dew point D on its vapor segment that lies at the pressure of the bubble point B on the liquid segment of the solid line. This pressure is the phase equilibrium pressure, and the composition for the dashed line is that of the equilibrium vapor. This equilibrium condition is shown by Fig. 4-9. In the absence of a theory to prescribe the composition dependence of parameters for cubic equations of state, empirical mixing rules are used to relate mixture parameters to pure-species parameters. The simplest realistic expressions are a linear mixing rule for parameter b and a quadratic mixing rule for parameter a, as shown by Eqs. (4-113) and (4-114). A common combining rule is given by Eq. (4-115). The general mole fraction variable xi is used here because application is to both liquid and vapor mixtures. These equations, known as van der Waals prescriptions, provide for the evaluation of mixture parameters solely from parameters for the pure constituent species. They find application primarily for mixtures comprised of simple and chemically similar molecules. Useful in the application of cubic equations of state to mixtures are partial equation-of-state parameters. For the parameters of the generic cubic, represented by Eqs. (4-104), (4-105), and (4-106), the definitions are ∂(na) a⎯i   ∂ni



∂(nb) ⎯ bi   ∂ni



(4-343)

∂(nq) ⎯ qi   ∂ni



(4-344)



 

P

T,nj

T,nj

n(na) nq =  RT(nb)

D whence V

Two PV isotherms at the same T for mixtures. The solid line is for a liquid-phase composition; the dashed line is for a vapor-phase composition. Point B represents a bubble point with the liquid-phase composition; point D represents a dew point with the vapor-phase composition. When these points lie at the same P (as shown), they represent phases in equilibrium. [Smith, Van Ness, and Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 560, McGraw-Hill, New York (2005).] FIG. 4-9

(4-342)

T,nj

These are general equations, valid regardless of the particular mixing or combining rules adopted for the composition dependence of mixture parameters. Parameter q is defined in relation to parameters a and b by Eq. (4-106). Thus,

B 0

4-33

∂(nq) ⎯ qi   ∂ni





T,nj

⎯ bi a⎯i =q 1+  −  a b





(4-345)

Any two of the three partial parameters form an independent pair, and any one of them can be found from ⎯ the other two. Because q, a, and b ⎯ ≠ a⎯ b are not linearly related, q i i iRT. l v Values of φˆ i and φˆ i as given by Eq. (4-246) are implicit in an equation of state and with Eq. (4-331) allow calculation of mixture VLE. Although more complex, the same basic principle applies as for pure-species VLE. With φˆ li a function of T, P, and {xi}, and φˆ Vi a function of T, P, and {yi}, Eq. (4-331) represents N relations among the 2N variables: T, P, (N −1) xi s, and (N−1) yi s. Thus, specification of N of these variables, usually either T or P and either the liquid- or vaporphase composition, allows solution for the remaining N variables by BUBL P, DEW P, BUBL T, and DEW T calculations.

4-34

THERMODYNAMICS

Because of limitations inherent in empirical mixing and combining rules, such as those given by Eqs. (4-113) through (4-115), the equationof-state approach has found primary application to systems exhibiting modest deviations from ideal solution behavior in the liquid phase, e.g., to systems containing hydrocarbons and cryogenic fluids. However, since 1990, extensive research has been devoted to developing mixing rules that incorporate the excess Gibbs energy or activity coefficient data available for many systems. The extensive literature on this subject is reviewed by Valderrama [Ind. Eng. Chem. Res. 42: 1603–1618 (2003)] and by Twu, Sim, and Tassone [Chem. Eng. Progress 98:(11): 58–65 (Nov. 2002)]. The idea here is to exploit the connection between fugacity coefficients and activity coefficients provided by their definitions: fˆi/xiP φˆ fˆi γi   =  = i fi/P xifi φi ln γi = ln φˆ i − ln φi

Therefore,

(4-346)

Because γi is a liquid-phase property, this equation is written for the liquid phase. Substituting for ln φˆ i and ln φi by Eqs. (4-245) and Eq. (4-246) gives Z−β b ⎯I + qI ln γi = i (Z − 1) − Zi + 1 − ln  − q i i i Zi − βi b Symbols without subscripts are mixture properties. Solution for ⎯ qi yields Z−β 1 b ⎯ =  q 1 − Zi + i (Z − 1) − ln  + qiIi − ln γi i Zi − βi I b





GE GE  =  RT RT

(4-348)

T0

E



I

where

Integration of the first equation from T0 to T gives H dT  −  RT T

E

T0

(4-349)

2

T0

Integration of the second equation from T1 to T yields

In addition,

Integrate from T2 to T:

 C dT T

E P



(4-350)

T1

∂CPE dCEP =  ∂T





1 ∂C     RT ∂T T

T

2

T0

T

E P

T1 T2

(4-351)

dT dT dT

P, x

This general equation employs excess Gibbs energy data at temperature T0, excess enthalpy (heat-of-mixing) data at T1, and excess heat capacity data at T2. Integral I depends on the temperature dependence of CPE. Excess heat capacity data are uncommon, and the T dependence is rarely known. Assuming CPE independent of T makes the integral zero, and the closer T0 and T1 are to T, the less the influence of this assumption. When no information is available for CPE and excess enthalpy data are available at only a single temperature, CPE must be assumed zero. In this case only the first two terms on the right side of Eq. (4-351) are retained, and it more rapidly becomes imprecise as T increases. For application of Eq. (4-351) to binary systems at infinite dilution of one of the constituent species, it is divided by the product x1x2. GE GE  =  x1x2RT x1x2RT



HE

 − 1  −  x x RT  T T T0

T

T1

1 2

T1

0

CEP T T T −  ln  −  − 1 1 x1x2R T0 T0 T







The assumption here is that C is independent of T, making I = 0. As shown by Smith, Van Ness and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., p. 437, McGraw-Hill, New York (2005)], E P

GE

 x x RT

xi = 0

1 2

 ln γ ∞i

The preceding equation may therefore be written as HE ln γ ∞i = (ln γ ∞i )T −  x1x2RT 0



CEP −  x1x2R



− 1   T T T

T1, xi = 0

T1

0

−   − 1   ln  T T T xi = 0

T

T

0

0

T1

(4-352)

Example 5: VLE at Several Temperatures For the methanol(1)/ acetone(2) system at a base temperature of T0 = 323.15 K (50!C), both VLE data [Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978: 67 (1978)] and excess enthalpy data [Morris et al., J. Chem. Eng. Data 20: 403–405 (1975)] are available. The VLE data are well correlated by the Margules equations. As noted in connection with Eq. (4-270), parameters A12 and A21 relate directly to infinite dilution values of the activity coefficients. Thus, we have from the VLE data at 323.15 K:

dHE = CPE dT constant P, x

HE = H1E +

T1

constant P, x

and the excess-property analog of Eq. (4-26):

GE GE  =  RT RT

T1

0



2

E

G H d  = − 2 dT RT RT

T

CPE T T T −  ln  −  − 1 1 − I R T0 T0 T

i

Application of this equation in the solution of VLE problems is illustrated by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 7th ed., pp. 569–572, McGraw-Hill, New York (2005)]. Extrapolation of Data with Temperature Liquid-phase excessproperty data for binary systems at near-ambient temperatures appear in the literature. They provide for the extrapolation of GE correlations with temperature. The key relations are Eq. (4-250), written as

HE −  RT

− 1     T T

(4-347)

⎯ is a partial property, the summability equation provides an Because q i exact mixing rule: q = xi ⎯ qi

Combining this equation with Eqs. (4-349) and (4-350) leads to



dT

P, x

CEP = CPE + 2

∂C   ∂T T

T2

E P

P, x

dT

A12 = ln γ ∞1 = 0.6281

and A21 = ln γ ∞2 = 0.6557

These values allow calculation of equilibrium pressures through Eqs. (4-270) and (4-308) for comparison with the measured pressures of the data set. Values of Pisat required in Eq. (4-308) are the measured values reported with the data set. The root-mean-square (rms) value of the pressure differences is given in Table 4-7 as 0.08 kPa, thus confirming the suitability of the Margules equation for this system. Vapor-phase mole fractions were not reported; hence no value can be given for rms δy1. Experimental VLE data at 372.8 and 397.7 K are given by Wilsak et al. [Fluid Phase Equilib. 28: 13–37 (1986)]. Values of ln γ ∞i and hence of the Margules parameters for these higher temperatures are found from Eq. (4-352) with CPE = 0. The required excess enthalpy values at T0 are HE

 x x RT 1 2

T0, x1=0

= 1.3636

and

HE

 x x RT 1 2

T0, x2 = 0

= 1.0362

EQUILIBRIUM TABLE 4-7 T, K 323.15 372.8 397.7

VLE Results for Methanol(1)/Acetone(2)

A12 = ln γ ∞1

A21 = ln γ ∞2

RMS δP/kPa

RMS % δP

0.6281 0.4465 (0.4607) 0.3725 (0.3764)

0.6557 0.5177 (0.5271) 0.4615 (0.4640)

0.08 0.85 (0.83) 2.46 (1.39)

0.12 0.22 0.32

RMS δy1 0.004 (0.006) 0.014 (0.013)

4-35

the νi are stoichiometric coefficients and the Ai stand for chemical formulas. The νi themselves are called stoichiometric numbers, and associated with them is a sign convention such that the value is positive for a product and negative for a reactant. More generally, for a system containing N chemical species, any or all of which can participate in r chemical reactions, the reactions are represented by the equations 0 = νi,j Ai j = I, II, . . . , r

(4-357)

i

Results of calculations with the Margules equations are displayed as the primary entries at each temperature in Table 4-7. The values in parentheses are from the gamma/phi approach as reported in the papers cited. Results for the higher temperatures indicate the quality of predictions based only on vapor-pressure data for the pure species and on mixture data at 323.15 K. Extrapolations based on the same data to still higher temperatures can be expected to become progressively less accurate.

Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for a binary system.

Equation (4-295) is the basis for both liquid/liquid equilibria (LLE) and vapor/liquid/liquid equilibria (VLLE). Thus for LLE with superscripts α and β denoting the two phases, Eq. (4-295) is written as i = 1, 2, . . . , N

(4-353)

Eliminating fugacities in favor of activity coefficients gives xαi γ αi = xβi γ βi

i = 1, 2, . . . , N

(4-354)

For most LLE applications, the effect of pressure on the γi can be ignored, and Eq. (4-354) then constitutes a set of N equations relating equilibrium compositions to one another and to temperature. For a given temperature, solution of these equations requires a single expression for the composition dependence of GE suitable for both liquid phases. Not all expressions for GE suffice, even in principle, because some cannot represent liquid/liquid phase splitting. The UNIQUAC equation is suitable, and therefore prediction is possible by UNIFAC models. A special table of parameters for LLE calculations is given by Magnussen et al. [Ind. Eng. Chem. Process Des. Dev. 20: 331–339 (1981)]. A comprehensive treatment of LLE is given by Sorensen et al. [Fluid Phase Equilib.2: 297–309 (1979); 3: 47–82 (1979); 4: 151–163 (1980)]. Data for LLE are collected in a three-part set compiled by Sorensen and Arlt [Liquid-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. 5, parts 1–3, DECHEMA, Frankfurt am Main (1979–1980)]. For vapor/liquid/liquid equilibria, Eq. (4-295) becomes i = 1, 2 , . . . , N (4-355) fˆαi = fˆβi = fˆvi where α and β designate the two liquid phases. With activity coefficients applied to the liquid phases and fugacity coefficients to the vapor phase, the 2N equilibrium equations for subcritical VLLE are xαi γ αi f αi = yiφˆ iP xβi γ βi f βi = yiφˆ iP

− for a reactant species

+

sign (νi, j) =

for a product species

If species i does not participate in reaction j, then vi,j = 0. The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reaction j ∆n1,j ∆n2,j ∆nN,j  =  =…=  ν1,j ν2,j νN,j

∆ni, j = νi,j ∆ε j

i = 1, 2, . . . , N j = I, II, . . . , r

all i

(4-356)

As for LLE, an expression for GE capable of representing liquid/liquid phase splitting is required; as for VLE, a vapor-phase equation of state for computing the φˆ i is also needed.

∆ni = ∆ni,j = νi,j ∆ε j j

For a phase in which a chemical reaction occurs according to the equation ν1A1 + ν2A2 + · · · → ν3A3 + ν4A4 + · · ·

i = 1, 2, . . . , N

(4-360)

j

If the initial number of moles of species i is ni and if the convention is adopted that εj = 0 for each reaction in this initial state, then 0

ni = ni + νi,j ε j

i = 1, 2, . . . , N

0

(4-361)

j

Equation (4-361) is the basic expression of material balance for a closed system in which r chemical reactions occur. It shows for a reacting system that at most r mole-number-related quantities ε j are capable of independent variation. It is not an equilibrium relation, but merely an accounting scheme, valid for tracking the progress of the reactions to arbitrary levels of conversion. The reaction coordinate has units of moles. A change in ε j of 1 mol signifies a mole of reaction, meaning that reaction j has proceeded to such an extent that the change in mole number of each reactant and product is equal to its stoichiometric number. CHEMICAL REACTION EQUILIBRIA The general criterion of chemical reaction equilibria is given by Eq. (4-296). For a system in which just a single reaction occurrs, Eq. (4-361) becomes ni = ni + νiε

whence dni = νi dε

Substitution for dni in Eq. (4-296) leads to

i ν µ = 0 i i

(4-362)

Generalization of this result to multiple reactions produces

i ν

µ =0

i,j i

CHEMICAL REACTION STOICHIOMETRY

(4-359)

Because the total change in mole number ∆ni is just the sum of the changes ∆ ni,j resulting from the various reactions,

0

}

(4-358)

All these terms are equal, and they can be equated to the change in a single quantity ε j, called the reaction coordinate for reaction j, thereby giving

LIQUID/LIQUID AND VAPOR/LIQUID/ LIQUID EQUILIBRIA

fˆαi = fˆβi

where

j = I, II, . . . , r

(4-363)

Standard Property Changes of Reaction For the reaction aA + bB → lL + mM a standard property change is defined as the property change resulting when a mol of A and b mol of B in their standard states at temperature

4-36

THERMODYNAMICS

T react to form l mol of L and m mol of M in their standard states also at temperature T. A standard state of species i is its real or hypothetical state as a pure species at temperature T and at a standard state pressure P°. The standard property change of reaction j is given the symbol ∆M°j, and its general mathematical definition is ∆M°j  νi, j M°i

For species present as gases in the actual reactive system, the standard state is the pure ideal gas at pressure P°. For liquids and solids, it is usually the state of pure real liquid or solid at P°. The standard state pressure P° is fixed at 100 kPa. Note that the standard states may represent different physical states for different species; any or all the species may be gases, liquids, or solids. The most commonly used standard property changes of reaction are ∆G°j  νi,jG°i = νi,jµ°i

(4-365)

∆H°j  νi,jH°i

(4-366)

∆C°P  νi,jC°P

(4-367)

i

i

j

i

i

The standard Gibbs energy change of reaction ∆G°j is used in the calculation of equilibrium compositions. The standard heat of reaction ∆H°j is used in the calculation of the heat effects of chemical reaction, and the standard heat capacity change of reaction is used for extrapolating ∆H°j and ∆G°j with T. Numerical values for ∆H°j and ∆G°j are computed from tabulated formation data, and ∆C°P is determined from empirical expressions for the T dependence of the C°P [see, e.g., Eq. (4-52)]. Equilibrium Constants For practical application, Eq. (4-363) must be reformulated. The initial step is elimination of the µi in favor of fugacities. Equation (4-199) for species i in its standard state is subtracted from Eq. (4-202) for species i in the equilibrium mixture, giving j

i

µi = G°i + RT ln aˆi (4-368) ˆ where by definition aˆi  fi/f°i and is called an activity. Substitution of this equation into Eq. (4-364) yields, upon rearrangement,

i [ν

(G°i + RT ln aˆi)] = 0

i ν

G°i + RT ln aˆνi = 0

i, j

or

or

i, j

However, the standard state for gases is the ideal gas state at the standard state pressure, for which f i° = P°. Therefore, y φˆ iP aˆi = i  P° and Eq. (4-369) becomes

i (y φˆ ) i i

νi,j

νj

 P° P

= Kj

all j

(4-371)

where νj  Σiνi,j and P° is the standard state pressure of 100 kPa, expressed in the same units used for P. The yi’s may be eliminated in favor of equilibrium values of the reaction coordinates ε j (see Example 6). Then, for fixed temperature Eqs. (4-371) relate the ε j to P. In principle, specification of the pressure allows solution for the ε j. However, the problem may be complicated by the dependence of the φˆ i on composition, i.e., on the ε j. If the equilibrium mixture is assumed an ideal solution, then [Eq. (4-218)] each φˆ i becomes φi, the fugacity coefficient of pure species i at the mixture T and P. This quantity does not depend on composition and may be determined from experimental data, from a generalized correlation, or from an equation of state. An important special case of Eq. (4-371) results for gas-phase reactions when the phase is assumed an ideal gas. In this event φˆ i = 1, and

i (y ) i

νi, j

νj

 P° P

= Kj

all j

(4-372)

In the general case the evaluation of the φˆ i requires an iterative process. An initial step is to set each φˆ i equal to unity and to solve the problem by Eq. (4-372). This provides a set of yi values, allowing evaluation of the φˆ i by, for example, Eq. (4-243) or (4-246). Equation (4-371) can then be solved for a new set of yi values, with the process continued to convergence. For liquid-phase reactions, Eq. (4-369) is modified by introduction of the activity coefficient γi = fˆi xifi, where xi is the liquid-phase mole fraction. The activity is then

i,j

fˆ f aˆi  i = γ i xi i fi° fi°



i

− νi,jG°i i ln aˆνi =  RT i i, j

The right side of this equation is a function of temperature only for given reactions and given standard states. Convenience suggests setting it equal to ln Kj, whence

i aˆ

νi,j i

where, by definition,

fˆ y φˆ iP aˆi  i = i  f i° f i°

(4-364)

i

i

The application of Eq. (4-369) requires explicit introduction of composition variables. For gas-phase reactions this is accomplished through the fugacity coefficient

= Kj

all j

−∆G°j Kj  exp  RT





(4-369) (4-370)

Quantity Kj is the chemical reaction equilibrium constant for reaction j, and ∆G°j is the corresponding standard Gibbs energy change of reaction [see Eq. (4-365)]. Although called a “constant,” Kj is a function of T, but only of T. The activities in Eq. (4-369) provide the connection between the equilibrium states of interest and the standard states of the constituent species, for which data are presumed available. The standard states are always at the equilibrium temperature. Although the standard state need not be the same for all species, for a particular species it must be the state represented by both G°i and the f°i upon which activity âi is based.

Both fi and fi° represent fugacity of pure liquid i at temperature T, but at pressures P and P°, respectively. Except in the critical region, pressure has little effect on the properties of liquids, and the ratio fi fi° is often taken as unity. When this is not acceptable, this ratio is evaluated by the equation V (P − P°)  V dP   RT P

fi 1 ln  =  f°i RT

i

i

P°

When the ratio fi fi° is taken as unity, aˆi = γ i xi, and Eq. (4-369) becomes

i (γ x )

νi,j

i i

= Kj

all j

(4-373)

Here the difficulty is to determine the γ i’s, which depend on the xi’s. This problem has not been solved for the general case. Two courses are open: the first is experiment; the second, assumption of solution ideality. In the latter case, γ i = 1, and Eq. (4-373) reduces to

i (x ) i

νi,j

= Kj

all j

(4-374)

the “law of mass action.” The significant feature of Eqs. (4-372) and (4-374), the simplest expressions for gas- and liquid-phase reaction

EQUILIBRIUM equilibrium, is that the temperature-, pressure-, and compositiondependent terms are distinct and separate. The effect of temperature on the equilibrium constant follows from Eq. (4-41) written for pure species j in its standard state (wherein the pressure Po is fixed): d(Gj°/RT) − Hj°  =  dT RT 2 With Eqs. (4-365) and (4-366) this equation easily extends to relate standard property changes of reaction: d(∆Gj°RT) − ∆Hj°  =  dT RT 2

(4-375)

C6H6 + 3H2 → C6H12

d ln Kj ∆Hj°  =  dT RT 2

(4-376)

For an endothermic reaction, ∆Hj° is positive and K j increases with increasing T; for an exothermic reaction, it is negative and Kj decreases with increasing T. Because the standard state pressure is constant, Eq. (4-28) may be extended to relate standard properties of reaction, yielding dT d∆Sj° = ∆C°Pj  T

and

Integration of these equations from reference temperature T0 (usually 298.15 K) to temperature T gives ∆H° = ∆H°0 + R

P

T0



 ∆RC° dT + ∆RS° +  ∆RC° dTT T

T

P

0

T0

P

T0

Substituting ∆S°0 = (∆G°0 − ∆H°0)/T0, rearranging, and defining τ  TT0 give finally







∆CP°  dT + T0 R T



∆C°P dT   T0 R T T

(4-379)

When heat capacity equations have the form of Eq. (4-52), the integrals are evaluated by equations of exactly the form of Eqs. (4-53) and (4-54), but with parameters A, B, C, and D replaced by ∆A, ∆B, ∆C, and ∆D, in accord with Eq. (4-364). Thus for the ideal gas standard state ∆B  ∆RC° dT = ∆A T (τ − 1) +  T (τ 2 T

P

2 0

0

2

T0

∆D τ − 1 +   T0 τ



0

i

Each mole fraction is therefore given by yi = ni (4 − 3ε). Assume first that the equilibrium mixture is an ideal gas, and apply Eq. (4-372), written for a single reaction, with subscript j omitted and ν = − 3: ε  4 − 3ε 15 −3 P ν νi = 3  = K = 0.02874 i yi  1 − ε 3 − 3ε P° 1   4 − 3ε 4 − 3ε



whence









4 − 3ε 3 ε   (15)−3 = 0.02874 1 − ε 3 − 3ε





and

ε = 0.815

Thus, the assumption of ideal gases leads to a calculated conversion of 81.5 percent. An alternative assumption is that the equilibrium mixture is an ideal solution. This requires application of Eq. (4-371). However, in the case of an ideal solution Eq. (4-218) indicates that φˆ idi = φi, in which case Eq. (4-371) for a single reaction becomes

i (y φ )   P° i i

νi

P

ν

=K

P φi = exp(Bi0 + ωBi1) r Tr i

i

T

P

cyclohexane

(4-380)

τ+1  ∆RC° dTT = ∆A lnτ + ∆BT + ∆CT + τ∆DT   (τ − 1) (4-381) 2  T0

hydrogen

nC = ε

For purposes of illustration we evaluate the pure-species fugacity coefficients by Eq. (4-206), written here as

∆C − 1) +  T 30(τ 3 − 1) 3



benzene

nH = 3 − 3ε

i n = 4 − 3ε

(4-378)

where for simplicity subscript j has been supressed. The definition of G leads directly to ∆G° = ∆H° − T ∆S°. Combining this equation with Eqs. (4-370), (4-377), and (4-378) yields

1 −∆G°0 ∆H°0 τ − 1 ln K =  +   −  T RT0 RT0 τ

0

(4-377)

T ∆CP° dT ∆S° = ∆S°0 + R   T0 R T

1 −∆G° −∆H°0 ln K =  =  −  RT RT T

is carried out over a catalyst formulated to repress side reactions. Operating conditions cover a pressure range from 10 to 35 bar and a temperature range from 450 to 670 K. Reaction rate increases with increasing T, but because the reaction is exothermic the equilibrium conversion decreases with increasing T. A comprehensive study of the effect of operating variables on the chemical equilibrium of this reaction has been published by J. Carrero-Mantilla and M. LlanoRestrepo, Fluid Phase Equilib. 219: 181–193 (2004). Presented here are calculations for a single set of operating conditions, namely, T = 600 K, P = 15 bar, and a molar feed ratio H2/C6H6 = 3, the stoichiometric value. For these conditions we determine the fractional conversion of benzene to cyclohexane. Carrero-Mantilla and Llano-Restrepo express ln K as a function of T by an equation which for 600 K yields the value K = 0.02874. A feed stream containing 3 mol H2 for each 1 mol C6H6 is the basis of calculation, and for this single reaction, Eq. (4-361) becomes ni = ni + νiε, yielding nB = 1 − ε

∆C °  dT R T

In the more extensive compilations of data, values of ∆G° and ∆H° for formation reactions are given for a wide range of temperatures, rather than just at the reference temperature T0 = 298.15 K. [See in particular TRC Thermodynamic Tables—Hydrocarbons and TRC Thermodynamic Tables—Non-hydrocarbons, serial publications of the Thermodynamics Research Center, Texas A & M Univ. System, College Station, Tex.; “The NBS Tables of Chemical Thermodynamic Properties,” J. Phys. Chem. Ref. Data 11, supp. 2 (1982).] Where data are lacking, methods of estimation are available; these are reviewed by Poling, Prausnitz, and O’Connell, The Properties of Gases and Liquids, 5th ed., chap. 6, McGraw-Hill, New York, 2000. For an estimation procedure based on molecular structure, see Constantinou and Gani, Fluid Phase Equilib. 103: 11–22 (1995). See also Sec. 2. Example 6: Single-Reaction Equilibrium The hydrogenation of benzene to produce cyclohexane by the reaction

In view of Eq. (4-370) this may also be written as

d∆Hj° = ∆CP°j dT

4-37

2 0

2

The following table shows values for the various quantities in this equation. Note that Tc and Pc for hydrogen are effective values as calculated by Eqs. (4-124) and (4-125) and used with ω = 0.

2 0

Equations (4-379) through (4-381) together allow an equation to be written for lnK as a function of T for any reaction for which appropriate data are available.

C6H6 H2 C6H12

Tc

Tr

Pc

Pr

ω

B0

B1

φ

562.2 42.8 553.6

1.067 14.009 1.084

48.98 19.78 40.73

0.306 0.758 0.368

0.21 0.00 0.21

− 0.2972 0.0768 − 0.2880

0.008 0.139 0.016

0.919 1.004 0.908

4-38

THERMODYNAMICS

The equilibrium equation now becomes: ε  0.919 4 − 3ε P ν 15 ν = 3  i (yiφi)  3 − 3ε 1−ε P° 1  0.908 1.004 4 − 3ε 4 − 3ε



i







The first term on the right is the definition of the chemical potential; therefore, −3



ε = 0.816

Solution yields

This result is hardly different from that based on the ideal gas assumption. The fugacity coefficients in the equilibrium equation clearly cancel one another. This is not uncommon in reaction equilibrium calculations, as there are always products and reactions, making the ideal gas assumption far more useful than might be expected. Carrero-Mantilla and Llano-Restrep present results for a wide range of conditions, both for the ideal gas assumption and for calculations wherein φˆ i values are determined from the Soave-Redlich-Kwong equation of state. In no case are these calculated conversions significantly divergent.

Complex Chemical Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy Gt in accord with Eq. (4-293). The treatment here is limited to gas-phase reactions for which the problem is to find the equilibrium composition for given T and P and for a given initial feed. 1. Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ak as the total number of atomic masses of the kth element in the feed. Further, let aik be the number of atoms of the kth element present in each molecule of chemical species i. The material balance for element k is then

i n a

i ik

i n a

or

i ik

= Ak

k = 1, 2 , . . . , w

− Ak = 0

(4-382)

k = 1, 2, . . . , w

2. Multiply each element balance by λk, a Lagrange multiplier:

 n a

λk

i ik



− Ak = 0

i

k = 1, 2, . . . , w

Summed over k, these equations give

k λ  i n a k

i ik



− Ak = 0

3. Form a function F by addition of this sum to Gt: F = Gt + λk k

 n a

i ik

− Ak

i



t

Function F is identical with G , because the summation term is zero. However, the partial derivatives of F and Gt with respect to ni are different, because function F incorporates the constraints of the material balances. 4. The minimum value of both F and Gt is found when the partial derivatives of F with respect to ni are set equal to zero: ∂F

∂Gt =  ∂ni T,P,n

 ∂n i

j





T,P,nj

µi + λkaik = 0

= K = 0.02874

+ λkaik = 0 k

i = 1, 2, . . . , N

(4-383)

k

However, the chemical potential is given by Eq. (4-368); for gas-phase reactions and standard states as the pure ideal gases at Po, this equation becomes fˆi µi = G°i + RT ln  P° If G°i is arbitrarily set equal to zero for all elements in their standard states, then for compounds G°i = ∆G°f i, the standard Gibbs energy change of formation of species i. In addition, the fugacity is eliminated—in favor of the fugacity coefficient by Eq. (4-204), fˆi = yiφˆ iP. With these substitutions, the equation for µi becomes y φˆ iP µi = ∆G°f + RT ln i  P° i

Combination with Eq. (4-383) gives y φˆ iP ∆G°f + RT ln i  + λkaik = 0 P° k i

i = 1, 2, . . . , N (4-384)

If species i is an element, ∆G°f is zero. There are N equilibrium equations [Eqs. (4-384)], one for each chemical species, and there are w material balance equations [Eqs. (4-382)], one for each element—a total of N + w equations. The unknowns in these equations are the ni’s (note that yi = ni /Σi ni), of which there are N, and the λk’s, of which there are w—a total of N + w unknowns. Thus the number of equations is sufficient for the determination of all unknowns. Equation (4-384) is derived on the presumption that the set {φˆ i} is known. If the phase is an ideal gas, then each φˆ i is unity. If the phase is an ideal solution, each φˆ i becomes φi and can at least be estimated. For real gases, each φˆ i is a function of the set {yi}, the quantities being calculated. Thus an iterative procedure is indicated, initiated with each φˆ i set equal to unity. Solution of the equations then provides a preliminary set {yi}. For low pressures or high temperatures this result is usually adequate. Where it is not satisfactory, an equation of state with the preliminary set {yi} gives a new and more nearly correct set {φˆ i} for use in Eq. (4-384). Then a new set {yi} is determined. The process is repeated to convergence. All calculations are well suited to computer solution. In this procedure, the question of what chemical reactions are involved never enters directly into any of the equations. However, the choice of a set of species is entirely equivalent to the choice of a set of independent reactions among the species. In any event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results. A detailed example of a complex gas-phase equilibrium calculation is given by Smith, Van Ness, and Abbott [Introduction to Chemical Engineering Thermodynamics, 5th ed., Example 15.13, pp. 602–604; 6th ed., Example 13.14, pp. 511–513; 7th ed., Example 13.14, pp. 527–528, McGraw-Hill, New York (1996, 2001, 2005)]. General application of the method to multicomponent, multiphase systems is treated by Iglesias-Silva et al. [Fluid Phase Equilib. 210: 229–245 (2003)] and by Sotyan, Ghajar, and Gasem [Ind. Eng. Chem. Res. 42: 3786–3801 (2003)]. i

THERMODYNAMIC ANALYSIS OF PROCESSES Real irreversible processes can be subjected to thermodynamic analysis. The goal is to calculate the efficiency of energy use or production and to show how wasted energy is apportioned among the steps of a process. The treatment here is limited to steady-state steady-flow processes, because of their predominance in chemical technology.

CALCULATION OF IDEAL WORK In any steady-state steady-flow process requiring work, a minimum amount must be expended to bring about a specific change of state in the flowing fluid. In a process producing work, a maximum amount is

THERMODYNAMIC ANALYSIS OF PROCESSES attainable for a specific change of state in the flowing fluid. In either case, the limiting value obtains when the specific change of state is accomplished completely reversibly. The implications of this requirement are that 1. The process is internally reversible within the control volume. 2. Heat transfer external to the control volume is reversible. The second item means that heat exchange between system and surroundings must occur at the temperature of the surroundings, presumed to constitute a heat reservoir at a constant and uniform temperature Tσ . This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperatures of the flowing fluid to that of the surroundings. Because Carnot engines and heat pumps are cyclic, they undergo no net change of state. These conditions are implicit in the entropy balance of Eq. (4-156) when S˙G = 0. If in addition there is but a single surroundings temperature Tσ, this equation becomes Q˙ ∆(Sm˙)fs −  = 0 (4-385) Tσ The energy balance for a steady-state steady-flow process as given by Eq. (4-150) is





1 ∆ H +  u2 + zg m˙ 2

fs

˙s = Q˙ + W

(4-150)

Combining this equation with Eq. (4-385) to eliminate Q˙ yields





1 ∆ H +  u2 + zg m˙ 2

fs

˙ s(rev) = Tσ ∆(Sm˙)fs + W

LOST WORK Work that is wasted as the result of irreversibilities in a process is ˙ lost, and it is defined as the difference between the called lost work W actual work of a process and the ideal work for the process. Thus by definition,

The rate form is





fs

− Tσ ∆(Sm˙)fs

(4-386)

In most applications to chemical processes, the kinetic and potential energy terms are negligible compared with the others; in this event Eq. (4-386) is written as ˙ ideal = ∆(Hm˙)fs − Tσ ∆(Sm˙)fs (4-387) W For the special case of a single stream flowing through the system, Eq. (4-387) becomes ˙ ideal = m˙ (∆H − Tσ ∆S) W

(4-388)

Division by m˙ puts this equation on a unit-mass basis: Wideal = ∆H − Tσ ∆S

(4-389)

A completely reversible process is hypothetical, devised solely to find the ideal work associated with a given change of state. Its only connection with an actual process is that it brings about the same change of state as the actual process, allowing comparison of the actual work of a process with the work of the hypothetical reversible process. Equations (4-386) through (4-389) give the work of a completely reversible process associated with given property changes in the flowing streams. When the same property changes occur in an actual ˙ s (or Ws) is given by an energy balance, and process, the actual work W comparison can be made of the actual work with the ideal work. When ˙ ideal (or Wideal) is positive, it is the minimum work required to bring W about a given change in the properties of the flowing streams, and it is ˙ s. In this case a thermodynamic efficiency ηt is defined smaller than W as the ratio of the ideal work to the actual work: ˙ eal W ηt(work required) = id (4-390) ˙s W ˙ ideal (or Wideal) is negative, W ˙ ideal is the maximum work When W obtainable from a given change in the properties of the flowing ˙ s. In this case, the thermodynamic effistreams, and it is larger than W ciency is defined as the ratio of the actual work to the ideal work: ˙s W ηt(work produced) =  (4-391) ˙ ideal W

Wlost  Ws − Wideal

(4-392)

˙ lost  W ˙s−W ˙ ideal W

(4-393)

The actual work rate comes from Eq. (4-150):





1 2 ˙ s= ∆ H +  u + zg m˙ W 2

− Q˙

fs

Subtracting the ideal work rate as given by Eq. (4-386) yields ˙ lost = Tσ ∆(Sm˙ )fs − Q˙ (4-394) W For the special case of a single stream flowing through the control volume, ˙ lost = m˙ Tσ ∆S − Q˙ (4-395) W Division of this equation by m˙ gives Wlost = Tσ ∆S − Q

(4-396)

where the basis is now a unit amount of fluid flowing through the control volume. The total rate of entropy generation (in both system and surroundings) as a result of a process is

˙ s(rev) indicates that the shaft work is for a completely where W ˙ ideal. Thus reversible process. This work is called the ideal work W 1 2 ˙ ideal = ∆ H +  W u + zg m˙ 2

4-39

Q˙ S˙ G = ∆(Sm˙ )fs −  Tσ

(4-397)

Division by m˙ provides an equation based on a unit amount of fluid flowing through the control volume: Q SG = ∆S −  Tσ

(4-398)

Equations (4-397) and (4-398) are special cases of Eqs. (4-156) and (4-157). Multiplication of Eq. (4-397) by Tσ gives Tσ S˙ G = Tσ ∆(Sm˙ )fs − Q˙ Because the right sides of this equation and of Eq. (4-394) are identical, it follows that ˙ lost = Tσ S˙ G W (4-399) For flow on the basis of a unit amount of fluid, this becomes Wlost = Tσ SG

(4-400)

Because the second law of thermodynamics requires

therefore

S˙ G ≥ 0

and

SG ≥ 0

˙ lost ≥ 0 W

and

Wlost ≥ 0

When a process is completely reversible, the equality holds and the lost work is zero. For irreversible processes the inequality holds, and the lost work, i.e., the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear: The greater the irreversibility of a process, the greater the rate of entropy generation and the greater the amount of energy that becomes unavailable for work. Thus every irreversibility carries with it a price. ANALYSIS OF STEADY-STATE STEADY-FLOW PROCESSES Many processes consist of a number of steps, and lost-work calculations are then made for each step separately. Writing Eq. (4-399) for each step of the process and summing give

W˙

lost

= Tσ S˙ G

4-40

THERMODYNAMICS

Dividing Eq. (4-399) by this result yields ˙ lost W S˙ G =   ˙ W

lost S˙ G

TABLE 4-8 States and Values of Properties for the Process of Fig. 4-10* Point P, bar

Thus an analysis of the lost work, made by calculation of the fraction that each individual lost-work term represents of the total lost work, is the same as an analysis of the rate of entropy generation, made by expressing each individual entropy generation term as a fraction of the sum of all entropy generation terms. An alternative to the lost-work or entropy generation analysis is a work analysis. This is based on Eq. (4-393), written as

W˙

lost

˙s−W ˙ ideal =W

(4-402)

A work analysis then gives each of the individual work terms in the ˙ s. summation on the right as a fraction of W ˙ s and W ˙ ideal are negative, and For a work-producing process, W ˙ ideal > W ˙ s. Equation (4-401) in this case is best written as W ˙ ideal = W ˙ s + W ˙ lost W

55.22 1.01 1.01 55.22 1.01 1.01 1.01

T, K

Composition

300 295 295 147.2 79.4 90 300

Air Pure O2 91.48% N2 Air 91.48% N2 Pure O2 Air

State

H, J/mol S, J/(mol·K)

Superheated Superheated Superheated Superheated Saturated vapor Saturated vapor Superheated

12,046 13,460 12,074 5,850 5,773 7,485 12,407

82.98 118.48 114.34 52.08 75.82 83.69 117.35

*Properties on the basis of Miller and Sullivan, U.S. Bur. Mines Tech. Pap. 424 (1928).

(4-401)

For a work-requiring process, all these work quantities are positive ˙s>W ˙ ideal. The preceding equation is then expressed as and W ˙ s= W ˙ ideal + W ˙ lost W

1 2 3 4 5 6 7

Thus, by Eq. (4-387), ˙ ideal = −144 − (300)(−2.4453) = 589.6 J W Calculation of actual work of compression: For simplicity, the work of compression is calculated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercooling and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics: nγ RT1 ˙s= W  0.8(γ − 1)

(4-403)

A work analysis here expresses each of the individual work terms on ˙ ideal. A work analysis cannot be carried out the right as a fraction of W ˙ ideal is negative, indiin the case where a process is so inefficient that W ˙ s is positive, indicating that the process should produce work; but W cating that the process in fact requires work. A lost-work or entropy generation analysis is always possible. Example 7: Lost-Work Analysis A work analysis follows for a simple Linde system for the separation of air into gaseous oxygen and nitrogen, as depicted in Fig. 4-10. Table 4-8 lists a set of operating conditions for the numbered points of the diagram. Heat leaks into the column of 147 J/mol of entering air and into the exchanger of 70 J/mol of entering air have been assumed. Take Tσ = 300 K. The basis for analysis is 1 mol of entering air, assumed to contain 79 mol % N2 and 21 mol % O2. By a material balance on the nitrogen, 0.79 = 0.9148 x, whence x = 0.8636 mol of nitrogen product 1 − x = 0.1364 mol of oxygen product

P2 1

−1



where n = number of stages, here taken as 3 γ = ratio of heat capacities, here taken as 1.4 T1 = initial absolute temperature, equal to 300 K P2/P1 = overall pressure ratio, equal to 54.5 R = universal gas constant, equal to 8.314 J/(mol·K) The efficiency factor of 0.8 is already included in the equation. Substitution of the remaining values gives





(3)(1.4)(8.314)(300) ˙ s =  W (54.5)0.4(3)(1.4) − 1 = 15,171 J (0.8)(0.4) The heat transferred to the surroundings during compression as a result of intercooling and aftercooling to 300 K is found from the first law: ˙ s = (12,046 − 12,407) −15,171 = −15,532 J Q˙ = m˙ (∆H) − W Calculation of lost work: Equation (4-394) may be applied to each of the major units of the process. For the compressor/cooler,

Calculation of ideal work: If changes in kinetic and potential energies are neglected, Eq. (4-387) is applicable. From the tabulated data, ∆(Hm˙ )fs = (13,460)(0.1364) + (12,074)(0.8636) − (12,407)(1) = −144 J ∆(Sm˙ )fs = (118.48)(0.1364) + (114.34)(0.8636) − (117.35)(1)= − 2.4453 JK

(γ − 1)nγ

P

˙ lost = (300)[(82.98)(1) − (117.35)(1)] − (−15,532) W = 5221.0 J For the exchanger, ˙ lost = (300)[(118.48)(0.1364) + (114.34)(0.8636) + (52.08)(1) W − (75.82)(0.8636) − (83.69)(0.1364) − (82.98)(1)] − 70 = 2063.4 J Finally, for the rectifier, ˙ lost = (300)[(75.82)(0.8636) + (83.69)(0.1364) − (52.08)(1)] − 147 = 7297.0 J W Work analysis: Because the process requires work, Eq. (4-402) is appropriate for a work analysis. The various terms of this equation appear as entries in the following table and are on the basis of 1 mol of entering air. ˙s % of W ˙ ideal W ˙ lost: W ˙ Wlost: ˙ lost: W ˙s W

FIG. 4-10

Diagram of simple Linde system for air separation.

Compressor/cooler Exchanger Rectifier

589.6 J 5,221.0 J 2,063.4 J 7,297.0 J 15,171.0 J

3.9 34.4 13.6 48.1 100.0

The thermodynamic efficiency of this process as given by Eq. (4-390) is only 3.9 percent. Significant inefficiencies reside with each of the primary units of the process.

Section 5

Heat and Mass Transfer*

Hoyt C. Hottel, S.M. Deceased; Professor Emeritus of Chemical Engineering, Massachusetts Institute of Technology; Member, National Academy of Sciences, National Academy of Arts and Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Radiation)† James J. Noble, Ph.D., P.E., CE [UK] Research Affiliate, Department of Chemical Engineering, Massachusetts Institute of Technology; Fellow, American Institute of Chemical Engineers; Member, New York Academy of Sciences (Radiation Section Coeditor) Adel F. Sarofim, Sc.D. Presidential Professor of Chemical Engineering, Combustion, and Reactors, University of Utah; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Radiation Section Coeditor) Geoffrey D. Silcox, Ph.D. Professor of Chemical Engineering, Combustion, and Reactors, University of Utah; Member, American Institute of Chemical Engineers, American Chemical Society, American Society for Engineering Education (Conduction, Convection, Heat Transfer with Phase Change, Section Coeditor) Phillip C. Wankat, Ph.D. Clifton L. Lovell Distinguished Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society (Mass Transfer Section Coeditor) Kent S. Knaebel, Ph.D. President, Adsorption Research, Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society; Professional Engineer (Ohio) (Mass Transfer Section Coeditor)

HEAT TRANSFER Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-3

HEAT TRANSFER BY CONDUCTION Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction with Resistances in Series . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Conduction with Resistances in Series and Parallel . . . . Conduction with Heat Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two- and Three-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . .

5-3 5-3 5-3 5-3 5-5 5-5 5-5 5-5

Unsteady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction: Lumped and Distributed Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Correlation of First Eigenvalues by Eq. (5-22) . . . . . . . . Example 3: One-Dimensional, Unsteady Conduction Calculation . . Example 4: Rule of Thumb for Time Required to Diffuse a Distance R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction: Semi-infinite Plate . . . . . . . . . . . . . .

5-6

5-6 5-7

HEAT TRANSFER BY CONVECTION Convective Heat-Transfer Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Heat-Transfer Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . .

5-7 5-7

5-6 5-6 5-6

*The contribution of James G. Knudsen, Ph.D., coeditor of this section in the seventh edition, is acknowledged. † Professor H. C. Hottel was the principal author of the radiation section in this Handbook, from the first edition in 1934 through the seventh edition in 1997. His classic zone method remains the basis for the current revision. 5-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

5-2

HEAT AND MASS TRANSFER

Overall Heat-Transfer Coefficient and Heat Exchangers. . . . . . . . . . Representation of Heat-Transfer Coefficients . . . . . . . . . . . . . . . . . . Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Natural Flow for Various Geometries. . . . . . . . . . . . . . . . . . Simultaneous Heat Transfer by Radiation and Convection . . . . . . . . Mixed Forced and Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . Enclosed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Comparison of the Relative Importance of Natural Convection and Radiation at Room Temperature. . . . . . . . . . . . . . . Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow in Round Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow in Noncircular Ducts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Turbulent Internal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . Coiled Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow-through Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonnewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-7 5-7 5-8 5-8 5-8 5-8 5-8 5-8 5-9 5-9 5-9 5-10 5-10 5-10 5-10 5-12 5-12

HEAT TRANSFER WITH CHANGE OF PHASE Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling (Vaporization) of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-12 5-12 5-12 5-14 5-14 5-15

HEAT TRANSFER BY RADIATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Radiation Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Radiation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Displacement Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Properties of Opaque Surfaces . . . . . . . . . . . . . . . . . . . . . . . . Emittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . View Factors and Direct Exchange Areas . . . . . . . . . . . . . . . . . . . . . . . . Example 7: The Crossed-Strings Method . . . . . . . . . . . . . . . . . . . . . . Example 8: Illustration of Exchange Area Algebra . . . . . . . . . . . . . . . Radiative Exchange in Enclosures—The Zone Method. . . . . . . . . . . . . Total Exchange Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit Matrix Solution for Total Exchange Areas . . . . . . . . . . . . . . . Zone Methodology and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . The Limiting Case of a Transparent Medium . . . . . . . . . . . . . . . . . . . The Two-Zone Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multizone Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Examples from Furnace Design . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Radiation Pyrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: Furnace Simulation via Zoning. . . . . . . . . . . . . . . . . . . . Allowance for Specular Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . An Exact Solution to the Integral Equations—The Hohlraum . . . . . Radiation from Gases and Suspended Particulate Matter . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emissivities of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Calculations of Gas Emissivity and Absorptivity . . . . . . Flames and Particle Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange with Participating Media. . . . . . . . . . . . . . . . . . . . . Energy Balances for Volume Zones—The Radiation Source Term . .

5-16 5-16 5-16 5-16 5-18 5-19 5-19 5-20 5-23 5-24 5-24 5-24 5-24 5-25 5-25 5-26 5-26 5-27 5-28 5-28 5-29 5-30 5-30 5-30 5-30 5-31 5-32 5-34 5-35 5-35

Weighted Sum of Gray Gas (WSGG) Spectral Model . . . . . . . . . . . . The Zone Method and Directed Exchange Areas. . . . . . . . . . . . . . . . Algebraic Formulas for a Single Gas Zone . . . . . . . . . . . . . . . . . . . . . Engineering Approximations for Directed Exchange Areas. . . . . . . . Example 12: WSGG Clear plus Gray Gas Emissivity Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Engineering Models for Fuel-Fired Furnaces . . . . . . . . . . . . . . . . . . . . Input/Output Performance Parameters for Furnace Operation . . . . The Long Plug Flow Furnace (LPFF) Model. . . . . . . . . . . . . . . . . . . The Well-Stirred Combustion Chamber (WSCC) Model . . . . . . . . . Example 13: WSCC Furnace Model Calculations . . . . . . . . . . . . . . . WSCC Model Utility and More Complex Zoning Models . . . . . . . . .

5-35 5-36 5-37 5-38 5-38 5-39 5-39 5-39 5-40 5-41 5-43

MASS TRANSFER Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fick’s First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient . . . . Self-Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuity and Flux Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux Expressions: Simple Integrated Forms of Fick’s First Law . . . . Stefan-Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Mixtures—Low Pressure—Nonpolar Components . . . . . . . . Binary Mixtures—Low Pressure—Polar Components . . . . . . . . . . . . Binary Mixtures—High Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Pressure/Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stokes-Einstein and Free-Volume Theories . . . . . . . . . . . . . . . . . . . . Dilute Binary Nonelectrolytes: General Mixtures . . . . . . . . . . . . . . . Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids . Dilute Binary Mixtures of a Nonelectrolyte in Water . . . . . . . . . . . . . Dilute Binary Hydrocarbon Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute Dilute Dispersions of Macromolecules in Nonelectrolytes . . . . . . . . Concentrated, Binary Mixtures of Nonelectrolytes . . . . . . . . . . . . . . Binary Electrolyte Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion of Fluids in Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Dilute Systems . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Concentrated Systems . . . . . . . . . . . . . . . . HTU (Height Equivalent to One Transfer Unit) . . . . . . . . . . . . . . . . NTU (Number of Transfer Units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Mass-Transfer Coefficients ^ k G and ^ kL . . . . . . . . . . . . . Simplified Mass-Transfer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Total Pressure on ^ k G and ^ kL. . . . . . . . . . . . . . . . . . . . . . . . . Effects of Temperature on ^ k G and ^ kL. . . . . . . . . . . . . . . . . . . . . . . . . . Effects of System Physical Properties on ^ kG and ^ kL . . . . . . . . . . . . . . . . Effects of High Solute Concentrations on ^ k G and ^ kL . . . . . . . . . . . . . Influence of Chemical Reactions on ^ k G and ^ kL . . . . . . . . . . . . . . . . . . Effective Interfacial Mass-Transfer Area a . . . . . . . . . . . . . . . . . . . . . Volumetric Mass-Transfer Coefficients ^ k Ga and ^ k La . . . . . . . . . . . . . . Chilton-Colburn Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-45 5-45 5-45 5-45 5-45 5-45 5-45 5-49 5-49 5-49 5-50 5-50 5-50 5-52 5-52 5-52 5-52 5-53 5-53 5-53 5-54 5-55 5-55 5-55 5-55 5-55 5-55 5-57 5-57 5-58 5-59 5-59 5-60 5-61 5-61 5-61 5-61 5-62 5-68 5-68 5-74 5-74 5-74 5-83 5-83 5-83

HEAT TRANSFER GENERAL REFERENCES: Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966. Arpaci, Convection Heat Transfer, Prentice-Hall, 1984. Arpaci, Introduction to Heat Transfer, Prentice-Hall, 1999. Baehr and Stephan, Heat and Mass Transfer, Springer, Berlin, 1998. Bejan, Convection Heat Transfer, Wiley, 1995. Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. Edwards, Radiation Heat Transfer Notes, Hemisphere Publishing, 1981. Hottel and Sarofim, Radiative Transfer, McGraw-Hill, 1967. Incropera and DeWitt, Fundamentals of Heat and Mass Transfer, 5th ed., Wiley, 2002. Kays and Crawford, Convective Heat and Mass Transfer, 3d ed., McGraw-Hill, 1993. Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999. Modest, Radiative Heat Transfer, McGraw-Hill, 1993. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor and Francis, London, 1980. Pletcher, Anderson, and Tannehill, Computational Fluid Mechanics and Heat Transfer, 2d ed., Taylor and Francis, London, 1997. Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998. Siegel and Howell, Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis, London, 2001.

MODES OF HEAT TRANSFER Heat is energy transferred due to a difference in temperature. There are three modes of heat transfer: conduction, convection, and radiation. All three may act at the same time. Conduction is the transfer of energy between adjacent particles of matter. It is a local phenomenon and can only occur through matter. Radiation is the transfer of energy from a point of higher temperature to a point of lower energy by electromagnetic radiation. Radiation can act at a distance through transparent media and vacuum. Convection is the transfer of energy by conduction and radiation in moving, fluid media. The motion of the fluid is an essential part of convective heat transfer.

HEAT TRANSFER BY CONDUCTION FOURIER’S LAW

THERMAL CONDUCTIVITY

The heat flux due to conduction in the x direction is given by Fourier’s law

The thermal conductivity k is a transport property whose value for a variety of gases, liquids, and solids is tabulated in Sec. 2. Section 2 also provides methods for predicting and correlating vapor and liquid thermal conductivities. The thermal conductivity is a function of temperature, but the use of constant or averaged values is frequently sufficient. Room temperature values for air, water, concrete, and copper are 0.026, 0.61, 1.4, and 400 W(m ⋅ K). Methods for estimating contact resistances and the thermal conductivities of composites and insulation are summarized by Gebhart, Heat Conduction and Mass Diffusion, McGraw-Hill, 1993, p. 399.

. dT Q = −kA  dx

(5-1)

. T1 − T2 Q = kA  ∆x

(5-2)

. where Q is the rate of heat transfer (W), k is the thermal conductivity [W(m⋅K)], A is the area perpendicular to the x direction, and T is temperature (K). For the homogeneous, one-dimensional plane shown in Fig. 5-1a, with constant k, the integrated form of (5-1) is

where ∆x is the thickness of the plane. Using the thermal circuit shown in Fig. 5-1b, Eq. (5-2) can be written in the form . T1 − T2 T1 − T2 Q=  =  (5-3) ∆xkA R where R is the thermal resistance (K/W).

STEADY-STATE CONDUCTION One-Dimensional Conduction In the absence of energy source . terms, Q is constant with distance, as shown in Fig. 5-1a. For steady conduction, the integrated form of (5-1) for a planar system with constant k and A is Eq. (5-2) or (5-3). For the general case of variables k (k is a function of temperature) and A (cylindrical and spherical systems with radial coordinate r, as sketched in Fig. 5-2), the average heattransfer area and thermal conductivity are defined such that . ⎯⎯ T1 − T2 T1 − T2 Q = kA  =  (5-4) ∆x R For a thermal conductivity that depends linearly on T, k = k0 (1 + γT)

T1

˙ Q

˙ Q T1

∆x

T2

T2 ∆x kA

x

(a)

(5-5)

r1

r

T1

r2

(b)

Steady, one-dimensional conduction in a homogeneous planar wall with constant k. The thermal circuit is shown in (b) with thermal resistance ∆x(kA).

T2

FIG. 5-1

FIG. 5-2

The hollow sphere or cylinder. 5-3

5-4

HEAT AND MASS TRANSFER

Nomenclature and Units—Heat Transfer by Conduction, by Convection, and with Phase Change Symbol A Ac Af Ai Ao Aof AT Auf A1 ax b bf B1 Bi c cp D Di Do f Fo gc g G Gmax Gz h ⎯ h hf hf hfi hi ho ham hlm ⎯k k L m m. NuD ⎯⎯ NuD Nulm n′ p pf p′ P Pr q Q. Q Q/Qi r R

Definition

SI units

Area for heat transfer m2 Cross-sectional area m2 Area for heat transfer for finned portion of tube m2 Inside area of tube External area of bare, unfinned tube m2 External area of tube before tubes are attached = Ao m2 Total external area of finned tube m2 Area for heat transfer for unfinned portion of finned tube m2 First Fourier coefficient Cross-sectional area of fin m2 Geometry: b = 1, plane; b = 2, cylinder; b = 3, sphere Height of fin m First Fourier coefficient Biot number, hR/k Specific heat J(kg⋅K) Specific heat, constant pressure J(kg⋅K) Diameter m Inner diameter m Outer diameter m Fanning friction factor Dimensionless time or Fourier number, αtR2 Conversion factor 1.0 kg⋅m(N⋅s2) Acceleration of gravity, 9.81 m2/s m2/s Mass velocity, m. Ac; Gv for vapor mass velocity kg(m2⋅s) Mass velocity through minimum free area between rows of tubes normal to the fluid stream kg(m2⋅s) Graetz number = Re Pr Heat-transfer coefficient W(m2⋅K) Average heat-transfer coefficient W(m2⋅K) Heat-transfer coefficient for finned-tube exchangers based on total external surface W(m2⋅K) Outside heat-transfer coefficient calculated for a bare tube for use with Eq. (5-73) W(m2⋅K) Effective outside heat-transfer coefficient based on inside area of a finned tube W(m2⋅K) Heat-transfer coefficient at inside tube surface W(m2⋅K) Heat-transfer coefficient at outside tube surface W(m2⋅K) Heat-transfer coefficient for use with ∆Tam, see Eq. (5-33) W(m2⋅K) Heat-transfer coefficient for use with ∆TIm; see Eq. (5-32) W(m2⋅K) Thermal conductivity W(m⋅K) Average thermal conductivity W(m⋅K) Length of cylinder or length of flat plate in direction of flow or downstream distance. Length of heat-transfer surface m Fin parameter defined by Eq. (5-75). Mass flow rate kg/s Nusselt number based on diameter D, hD/k ⎯ Average Nusselt number based on diameter D, hDk Nusselt number based on hlm Flow behavior index for nonnewtonian fluids Perimeter m Fin perimeter m Center-to-center spacing of tubes in a bundle m Absolute pressure; Pc for critical pressure kPa Prandtl number, να Rate of heat transfer W Amount of heat transfer J Rate of heat transfer W Heat loss fraction, Q[ρcV(Ti − T∞)] Distance from center in plate, cylinder, or sphere m Thermal resistance or radius K/W or m

Symbol Rax ReD S S S1 t tsv ts T Tb ⎯ Tb TC Tf TH Ti Te Ts T∞ U V VF V∞ WF x x zp

Definition Rayleigh number, β ∆T gx3να Reynolds number, GDµ Volumetric source term Cross-sectional area Fourier spatial function Time Saturated-vapor temperature Surface temperature Temperature Bulk or mean temperature at a given cross section Bulk mean temperature, (Tb,in + Tb,out)/2 Temperature of cold surface in enclosure Film temperature, (Ts + Te)/2 Temperature of hot surface in enclosure Initial temperature Temperature of free stream Temperature of surface Temperature of fluid in contact with a solid surface Overall heat-transfer coefficient Volume Velocity of fluid approaching a bank of finned tubes Velocity upstream of tube bank Total rate of vapor condensation on one tube Cartesian coordinate direction, characteristic dimension of a surface, or distance from entrance Vapor quality, xi for inlet and xo for outlet Distance (perimeter) traveled by fluid across fin

SI units

W/m3 m2 s K K K or °C K K K K K K K K K W(m2⋅K) m3 m/s m/s kg/s m

m

Greek Symbols α β β′ Γ ∆P ∆t ∆T ∆Tam ∆TIm ∆x δ1 δ1,0 δ1,∞ δS ε ζ θθi λ µ ν ρ σ σ τ Ω

Thermal diffusivity, k(ρc) Volumetric coefficient of expansion Contact angle between a bubble and a surface Mass flow rate per unit length perpendicular to flow Pressure drop Temperature difference Temperature difference Arithmetic mean temperature difference, see Eq. (5-32) Logarithmic mean temperature difference, see Eq. (5-33) Thickness of plane wall for conduction First dimensionless eigenvalue First dimensionless eigenvalue as Bi approaches 0 First dimensionless eigenvalue as Bi approaches ∞ Correction factor, ratio of nonnewtonian to newtonian shear rates Emissivity of a surface Dimensionless distance, r/R Dimensionless temperature, (T − T∞)(Ti − T∞) Latent heat (enthalpy) of vaporization (condensation) Viscosity; µl, µL viscosity of liquid; µG, µg, µv viscosity of gas or vapor Kinematic viscosity, µρ Density; ρL, ρl for density of liquid; ρG, ρv for density of vapor Stefan-Boltzmann constant, 5.67 × 10−8 Surface tension between and liquid and its vapor Time constant, time scale Efficiency of fin

m2/s K−1 ° kg(m⋅s) Pa K K K K m

J/kg kg(m⋅s) m2/s kg/m3 W(m2⋅K4) N/m s

HEAT TRANSFER BY CONDUCTION and the average heat thermal conductivity is ⎯ ⎯ k = k0 (1 + γT ) (5-6) ⎯ where T = 0.5(T1 + T2). For cylinders and spheres, A is a function of radial position (see Fig. 5-2): 2πrL and 4πr2, where L is the length of the cylinder. For constant k, Eq. (5-4) becomes . T1 − T2 Q =  cylinder (5-7) [ln(r2r1)](2πkL) and . T1 − T2 Q =  sphere (5-8) (r2 − r1)(4πkr1r2) Conduction with Resistances in Series A steady-state temperature profile in a planar composite wall, with three constant thermal conductivities and no source terms, is shown in Fig. 5-3a. The corresponding thermal circuit is given in Fig. 5-3b. The rate of heat transfer through each of the layers is the same. The total resistance is the sum of the individual resistances shown in Fig. 5-3b: T1 − T2 T1 − T2 Q. =  =  ∆XA ∆XB ∆XC RA + RB + RC  +  +  kAA kBA kCA

(5-9)

Additional resistances in the series may occur at the surfaces of the solid if they are in contact with a fluid. The rate of convective heat transfer, between a surface of area A and a fluid, is represented by Newton’s law of cooling as . Tsurface − Tfluid Q = hA(Tsurface − Tfluid) =  (5-10) 1(hA) where 1/(hA) is the resistance due to convection (K/W) and the heattransfer coefficient is h[W(m2⋅K)]. For the cylindrical geometry shown in Fig. 5-2, with convection to inner and outer fluids at temperatures Ti and To, with heat-transfer coefficients hi and ho, the steady-state rate of heat transfer is . Q=

Ti − To Ti − To =  ln(r2r1) Ri + R1 + Ro 1 1  +  +  2πkL 2πr1Lhi 2πr2Lho

(5-11)

Example 1: Conduction with Resistances in Series and Parallel Figure 5-4 shows the thermal circuit for a furnace wall. The outside surface has a known temperature T2 = 625 K. The temperature of the surroundings

B

T1

T2

(a)

Ti1

Ti2

T2

∆ xA

∆x B

∆ xC

kA A

kBA

kC A

(b)

Steady-state temperature profile in a composite wall with constant thermal conductivities kA, kB, and kC and no energy sources in the wall. The thermal circuit is shown in (b). The total resistance is the sum of the three resistances shown. FIG. 5-3

T2

∆ xD

∆x B

∆ xS

kD

kB

kS

Tsur

1 hR

FIG. 5-4 Thermal circuit for Example 1. Steady-state conduction in a furnace wall with heat losses from the outside surface by convection (hC) and radiation (hR) to the surroundings at temperature Tsur. The thermal conductivities kD, kB, and kS are constant, and there are no sources in the wall. The heat flux q has units of W/m2.

Tsur is 290 K. We want to estimate the temperature of the inside wall T1. The wall consists of three layers: deposit [kD = 1.6 W(m⋅K), ∆xD = 0.080 m], brick [kB = 1.7 W(m⋅K), ∆xB = 0.15 m], and steel [kS = 45 W(m⋅K), ∆xS = 0.00254 m]. The outside surface loses heat by two parallel mechanisms—convection and radiation. The convective heat-transfer coefficient hC = 5.0 W(m2⋅K). The radiative heat-transfer coefficient hR = 16.3 W(m2⋅K). The latter is calculated from hR = ε2σ(T22 + T2sur)(T2 + Tsur)

(5-12)

where the emissivity of surface 2 is ε2 = 0.76 and the Stefan-Boltzmann constant σ = 5.67 × 10−8 W(m2⋅K4). Referring to Fig. 5-4, the steady-state heat flux q (W/m2) through the wall is . T1  T2 Q q =  =  = (hC + hR)(T2 − Tsur) ∆XD ∆XB ∆XS A   +  +  kD kB kS Solving for T1 gives ∆xD ∆xB ∆xS T1 = T2 +  +  +  (hC + hR)(T2 − Tsur) kD kB kS









0.080 0.15 0.00254 T1 = 625 +  +  +  (5.0 + 16.3)(625 − 290) = 1610 K 1.6 1.7 45

Conduction with Heat Source Application of the law of conservation of energy to a one-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equations for steady-state conduction in a flat plate of thickness 2R (b = 1), a cylinder of diameter 2R (b = 2), and a sphere of diameter 2R (b = 3). The parameter b is a measure of the curvature. The thermal conductivity is constant, and there is convection at the surface, with heat-transfer coefficient h and fluid temperature T∞.



Q T1

T1

hc



d dT S  rb−1  +  rb−1 = 0 dr dr k

.

C

1

.

=Q/A

and

where resistances Ri and Ro are the convective resistances at the inner and outer surfaces. The total resistance is again the sum of the resistances in series.

A

q

5-5

dT(0)  =0 dr

(symmetry condition)

(5-13)

dT −k  = h[T(R) − T∞] dr The solutions to (5-13), for uniform S, are T(r)  T∞ 1 r 

 1   SR2k 2b R

    bBi 2

1



b 1, plate, thickness 2R b 2, cylinder, diameter 2R b 3, sphere, diameter 2R

(5-14) where Bi = hR/k is the Biot number. For Bi << 1, the temperature in the solid is uniform. For Bi >> 1, the surface temperature T(R) T∞. Two- and Three-Dimensional Conduction Application of the law of conservation of energy to a three-dimensional solid, with the

5-6

HEAT AND MASS TRANSFER

heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equation for steady-state conduction in rectangular coordinates. ∂ ∂T ∂ ∂T ∂ ∂T  k +  k +  k + S = 0 ∂x ∂x ∂y ∂y ∂z ∂z













(5-15)

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., PrenticeHall, 1999, p. 164). UNSTEADY-STATE CONDUCTION Application of the law of conservation of energy to a three-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equation for unsteady-state conduction in rectangular coordinates. ∂T ∂ ∂T ∂ ∂T ∂ ∂T ρc  =  k  +  k  +  k  + S ∂t ∂x ∂x ∂y ∂y ∂z ∂z













(5-16)

The energy storage term is on the left-hand side, and ρ and c are the density (kg/m3) and specific heat [J(kg  K)]. Solutions to (5-16) are generally obtained numerically (see General References and Sec. 3). The one-dimensional form of (5-16), with constant k and no source term, is ∂T ∂2T  = α ∂t ∂x2

(5-17)

where α k(ρc) is the thermal diffusivity (m2/s). One-Dimensional Conduction: Lumped and Distributed Analysis The one-dimensional transient conduction equations in rectangular (b = 1), cylindrical (b = 2), and spherical (b = 3) coordinates, with constant k, initial uniform temperature Ti, S = 0, and convection at the surface with heat-transfer coefficient h and fluid temperature T∞, are α ∂ b1 ∂T ∂T    r  rb1 ∂r ∂t ∂r



for t , 0, at r 0, at r R,





b 1, plate, thickness 2R b 2, cylinder, diameter 2R b 3, sphere, diameter 2R



and



Plate

Cylinder

Sphere

A1

B1

S1

2sinδ1  δ1 + sinδ1cosδ1

2Bi2  2 2 δ1(Bi + Bi + δ21)

cos(δ1ζ)

2J1(δ1)  δ1[J20(δ1) + J21(δ1)]

4Bi2  2 2 δ1(δ1 + Bi2)

J0(δ1ζ)

2Bi[δ21 + (Bi − 1)2]12  δ21 + Bi2 − Bi

6Bi2  δ21(δ21 + Bi2 − Bi)

sinδ1ζ  δ1ζ

The time scale is the time required for most of the change in θθi or Q/Qi to occur. When t = τ, θθi = exp(−1) = 0.368 and roughly twothirds of the possible change has occurred. When a lumped analysis is not valid (Bi > 0.2), the single-term solutions to (5-18) are convenient: θ Q  = A1 exp (− δ21Fo)S1(δ1ζ) and  = 1 − B1 exp (−δ21Fo) (5-21) θi Qi where the first Fourier coefficients A1 and B1 and the spatial functions S1 are given in Table 5-1. The first eigenvalue δ1 is given by (5-22) in conjunction with Table 5-2. The one-term solutions are accurate to within 2 percent when Fo > Foc. The values of the critical Fourier number Foc are given in Table 5-2. The first eigenvalue is accurately correlated by (Yovanovich, Chap. 3 of Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998, p. 3.25) δ1,∞ δ1  (5-22) [1  (δ1,∞δ1,0)n]1n Equation (5-22) gives values of δ1 that differ from the exact values by less than 0.4 percent, and it is valid for all values of Bi. The values of δ1,∞, δ1,0, n, and Foc are given in Table 5-2. Example 2: Correlation of First Eigenvalues by Eq. (5-22) As an example of the use of Eq. (5-22), suppose that we want δ1 for the flat plate  5, and n = 2.139. Equawith Bi = 5. From Table 5-2, δ1,∞ π2, δ1,0 Bi tion (5-22) gives π2 δ1 

1.312 [1  (π2/5 )2.139]12.139

Example 3: One-Dimensional, Unsteady Conduction Calcula-

(5-18)

The solutions to (5-18) can be compactly expressed by using dimensionless variables: (1) temperature θθi = [T(r,t) − T∞](Ti − T∞); (2) heat loss fraction QQi = Q[ρcV(Ti − T∞)], where V is volume; (3) distance from center ζ = rR; (4) time Fo = αtR2; and (5) Biot number Bi = hR/k. The temperature and heat loss are functions of ζ, Fo, and Bi. When the Biot number is small, Bi < 0.2, the temperature of the solid is nearly uniform and a lumped analysis is acceptable. The solution to the lumped analysis of (5-18) is



Geometry

The tabulated value is 1.3138.

T Ti (initial temperature) ∂T (symmetry condition)  0 ∂r ∂T  k  h(T  T∞) ∂r

θ hA  = exp −  t θi ρcV

TABLE 5-1 Fourier Coefficients and Spatial Functions for Use in Eqs. (5-21)



Q hA  = 1 − exp −  t Qi ρcV

(5-19)

where A is the active surface area and V is the volume. The time scale for the lumped problem is ρcV τ=  (5-20) hA

tion As an example of the use of Eq. (5-21), Table 5-1, and Table 5-2, consider the cooking time required to raise the center of a spherical, 8-cm-diameter dumpling from 20 to 80°C. The initial temperature is uniform. The dumpling is heated with saturated steam at 95°C. The heat capacity, density, and thermal conductivity are estimated to be c = 3500 J(kgK), ρ = 1000 kgm3, and k = 0.5 W(mK), respectively. Because the heat-transfer coefficient for condensing steam is of order 104, the Bi → ∞ limit in Table 5-2 is a good choice and δ1 = π. Because we know the desired temperature at the center, we can calculate θθi and then solve (5-21) for the time. 80 − 95 θ T(0,t) − T∞  =  =  = 0.200 20 − 95 θi Ti − T∞ For Bi → ∞, A1 in Table 5-1 is 2 and for ζ = 0, S1 in Table 5-1 is 1. Equation (5-21) becomes αt θ  = 2 exp (−π2Fo) = 2 exp −π2 2 R θi





TABLE 5-2 First Eigenvalues for Bi Æ 0 and Bi Æ • and Correlation Parameter n The single-term approximations apply only if Fo ≥ Foc. Geometry

Bi → 0

Bi → ∞

n

Foc

Plate Cylinder Sphere

δ1 → Bi  δ1 → 2Bi  δ1 → 3Bi 

δ1 → π2 δ1 → 2.4048255 δ1 → π

2.139 2.238 2.314

0.24 0.21 0.18

HEAT TRANSFER BY CONVECTION

where erf(z) is the error function. The depth to which the heat penetrates in time t is approximately (12αt)12. If the heat-transfer coefficient is finite,

Solving for t gives the desired cooking time. θ R (0.04 m) 0.2 t = − 2 ln  = −  ln  = 43.5 min 2θi απ 1.43 × 10−7(m2s)π2 2 2

2

Example 4: Rule of Thumb for Time Required to Diffuse a Distance R A general rule of thumb for estimating the time required to diffuse a distance R is obtained from the one-term approximations. Consider the equation for the temperature of a flat plate of thickness 2R in the limit as Bi → ∞. From Table 5-2, the first eigenvalue is δ1 = π2, and from Table 5-1, θ π  = A1 exp −  θi 2

αt 2 cosδ1ζ R

  2

When t R2α, the temperature ratio at the center of the plate (ζ 0) has decayed to exp(π24), or 8 percent of its initial value. We conclude that diffusion through a distance R takes roughly R2α units of time, or alternatively, the distance diffused in time t is about (αt)12.

One-Dimensional Conduction: Semi-infinite Plate Consider a semi-infinite plate with an initial uniform temperature Ti. Suppose that the temperature of the surface is suddenly raised to T∞; that is, the heat-transfer coefficient is infinite. The unsteady temperature of the plate is T(x,t) − T∞ x  = erf  Ti − T∞ 2 αt





5-7

(5-23)

T(x,t)T∞  Ti  T∞ x x hαt  hx h2αt = erfc  −exp  +  erfc  +  k k2 2 αt k 2αt 











(5-24)

where erfc(z) is the complementary error function. Equations (5-23) and (5-24) are both applicable to finite plates provided that their halfthickness is greater than (12αt)12. Two- and Three-Dimensional Conduction The one-dimensional solutions discussed above can be used to construct solutions to multidimensional problems. The unsteady temperature of a rectangular, solid box of height, length, and width 2H, 2L, and 2W, respectively, with governing equations in each direction as in (5-18), is θ

 θ i

2H2L2W

θ =  θi

θ

θ

  θ  θ 2H

i

2L

i

(5-25)

2W

Similar products apply for solids with other geometries, e.g., semiinfinite, cylindrical rods.

HEAT TRANSFER BY CONVECTION CONVECTIVE HEAT-TRANSFER COEFFICIENT Convection is the transfer of energy by conduction and radiation in moving, fluid media. The motion of the fluid is an essential part of convective heat transfer. A key step in calculating the rate of heat transfer by convection is the calculation of the heat-transfer coefficient. This section focuses on the estimation of heat-transfer coefficients for natural and forced convection. The conservation equations for mass, momentum, and energy, as presented in Sec. 6, can be used to calculate the rate of convective heat transfer. Our approach in this section is to rely on correlations. In many cases of industrial importance, heat is transferred from one fluid, through a solid wall, to another fluid. The transfer occurs in a heat exchanger. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. Section 3 introduces dimensional analysis and the dimensionless groups associated with the heat-transfer coefficient. Individual Heat-Transfer Coefficient The local rate of convective heat transfer between a surface and a fluid is given by Newton’s law of cooling q h(Tsurface  Tfluid)

(5-26)

where h [W(m2K)] is the local heat-transfer coefficient and q is the energy flux (W/m2). The definition of h is arbitrary, depending on whether the bulk fluid, centerline, free stream, or some other temperature is used for Tfluid. The heat-transfer coefficient may be defined on an average basis as noted below. Consider a fluid with bulk temperature T, flowing in a cylindrical tube of diameter D, with constant wall temperature Ts. An energy balance on a short section of the tube yields . dT cpm  πDh(Ts  T) dx

(5-27)

. where cp is the specific heat at constant pressure [J(kgK)], m is the mass flow rate (kg/s), and x is the distance from the inlet. If the temperature of the fluid at the inlet is Tin, the temperature of the fluid at a downstream distance L is ⎯ T(L)  Ts hπDL (5-28)  exp   . Tin  Ts m cp





⎯ The average heat-transfer coefficient h is defined by ⎯ 1 L (5-29) h =  h dx L 0 Overall Heat-Transfer Coefficient and Heat Exchangers A local, overall heat-transfer coefficient U for the cylindrical geometry shown in Fig. 5-2 is defined by using Eq. (5-11) as . Q Ti − To  =  = 2πr1U(Ti − To) (5-30) 1 + ln(r2r1) + 1 ∆x    2πr1hi 2πk 2πr2ho where ∆x is a short length of tube in the axial direction. Equation (5-30) defines U by using the inside perimeter 2πr1. The outer perimeter can also be used. Equation (5-30) applies to clean tubes. Additional resistances are present in the denominator for dirty tubes (see Sec. 11). For counterflow and parallel flow heat exchanges, with high- and low-temperature fluids (TH and TC) and flow directions as defined in Fig. 5-5, the total heat transfer for the exchanger is given by . Q = UA ∆Tlm (5-31) where A is the area for heat exchange and the log mean temperature difference ∆Tlm is defined as (TH − TC)L − (TH − TL)0 ∆Tlm =  (5-32) ln[(TH − TC)L − (TH − TL)0] Equation (5-32) applies to both counterflow and parallel flow exchangers with the nomenclature defined in Fig. 5-5. Correction factors to ∆Tlm for various heat exchanger configurations are given in Sec. 11. In certain applications, the log mean temperature difference is replaced with an arithmetic mean difference: (TH − TC)L + (TH − TL)0 ∆Tam =  (5-33) 2 Average heat-transfer coefficients are occasionally reported based on Eqs. (5-32) and (5-33) and are written as hlm and ham. Representation of Heat-Transfer Coefficients Heat-transfer coefficients are usually expressed in two ways: (1) dimensionless relations and (2) dimensional equations. Both approaches are used below. The dimensionless form of the heat-transfer coefficient is the Nusselt



5-8

HEAT AND MASS TRANSFER For horizontal flat surfaces, the characteristic dimension for the correlations is [Goldstein, Sparrow, and Jones, Int. J. Heat Mass Transfer, 16, 1025–1035 (1973)] A L  (5-37) p

TH

TC

x=0

x=L

where A is the area of the surface and p is the perimeter. With hot surfaces facing upward, or cold surfaces facing downward [Lloyd and Moran, ASME Paper 74-WA/HT-66 (1974)],

(a)

⎯⎯ NuL TH

TC

x=0

104 , RaL , 107

(5-38)

0.15Ra13 L

107 , RaL , 1010

(5-39)

and for hot surfaces facing downward, or cold surfaces facing upward, ⎯⎯ NuL 0.27Ra14 105 , RaL , 1010 (5-40) L x=L

(b) Nomenclature for (a) counterflow and (b) parallel flow heat exchangers for use with Eq. (5-32).

FIG. 5-5

number. For example, with a cylinder of diameter D in cross flow, the local Nusselt number is defined as NuD = hD/k, where k is the thermal conductivity of the fluid. The subscript D is important because different characteristic lengths can be used to define Nu. The average Nus⎯⎯ ⎯ selt number is written NuD hDk. NATURAL CONVECTION Natural convection occurs when a fluid is in contact with a solid surface of different temperature. Temperature differences create the density gradients that drive natural or free convection. In addition to the Nusselt number mentioned above, the key dimensionless parameters for natural convection include the Rayleigh number Rax β ∆T gx3 να and the Prandtl number Pr να. The properties appearing in Ra and Pr include the volumetric coefficient of expansion β (K1); the difference ∆T between the surface (Ts) and free stream (Te) temperatures (K or °C); the acceleration of gravity g(m/s2); a characteristic dimension x of the surface (m); the kinematic viscosity ν(m2s); and the thermal diffusivity α(m2s). The volumetric coefficient of expansion for an ideal gas is β = 1T, where T is absolute temperature. For a given geometry, ⎯⎯ (5-34) Nux f(Rax, Pr) External Natural Flow for Various Geometries For vertical walls, Churchill and Chu [Int. J. Heat Mass Transfer, 18, 1323 (1975)] recommend, for laminar and turbulent flow on isothermal, vertical walls with height L, 0.387Ra16 ⎯⎯ L NuL 0.825   [1  (0.492Pr)916]827






2

(5-35)

⎯⎯ ⎯ where the fluid properties for Eq. (5-35) and NuL hLk are evaluated at the film temperature Tf = (Ts + Te)/2. This correlation is valid for all Pr and RaL. For vertical cylinders with boundary layer thickness much less than their diameter, Eq. (5-35) is applicable. An expression for uniform heating is available from the same reference. For laminar and turbulent flow on isothermal, horizontal cylinders of diameter D, Churchill and Chu [Int. J. Heat Mass Transfer, 18, 1049 (1975)] recommend ⎯⎯ 0.387Ra16 D NuL 0.60   [1  (0.559Pr)916]827





0.54Ra14 L




2

(5-36)

Fluid properties for (5-36) should be evaluated at the film temperature Tf = (Ts + Te)/2. This correlation is valid for all Pr and RaD.

Fluid properties for Eqs. (5-38) to (5-40) should be evaluated at the film temperature Tf = (Ts + Te)/2. Simultaneous Heat Transfer by Radiation and Convection Simultaneous heat transfer by radiation and convection is treated per the procedure outlined in Examples 1 and 5. A radiative heat-transfer coefficient hR is defined by (5-12). Mixed Forced and Natural Convection Natural convection is commonly assisted or opposed by forced flow. These situations are discussed, e.g., by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 340) and Raithby and Hollands (Chap. 4 of Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998, p. 4.73). Enclosed Spaces The rate of heat transfer across an enclosed space is described in terms of a heat-transfer coefficient based on the temperature difference between two surfaces: . ⎯ QA h  (5-41) TH  TC For rectangular cavities, the plate spacing between the two surfaces L is the characteristic dimension that defines the Nusselt and Rayleigh numbers. The temperature difference in the Rayleigh number, RaL β ∆T gL3να is ∆T TH  TC. For a horizontal rectangular cavity heated from below, the onset of advection requires RaL > 1708. Globe and Dropkin [J. Heat Transfer, 81, 24–28 (1959)] propose the correlation ⎯⎯ 0.074 NuL 0.069Ra13 3 × 105 < RaL < 7 × 109 (5-42) L Pr All properties in (5-42) are calculated at the average temperature (TH + TC)/2. For vertical rectangular cavities of height H and spacing L, with Pr ≈ 0.7 (gases) and 40 < H/L < 110, the equation of Shewen et al. [J. Heat Transfer, 118, 993–995 (1996)] is recommended:



⎯⎯ 0.0665Ra13 L NuL 1   1  (9000RaL)1.4




2

12

RaL < 106

(5-43)

All properties in (5-43) are calculated at the average temperature (TH + TC)/2. Example 5: Comparison of the Relative Importance of Natural Convection and Radiation at Room Temperature Estimate the heat losses by natural convection and radiation for an undraped person standing in still air. The temperatures of the air, surrounding surfaces, and skin are 19, 15, and 35°C, respectively. The height and surface area of the person are 1.8 m and 1.8 m2. The emissivity of the skin is 0.95. We can estimate the Nusselt number by using (5-35) for a vertical, flat plate of height L = 1.8 m. The film temperature is (19 + 35)2 = 27°C. The Rayleigh number, evaluated at the film temperature, is (1300)(35 − 19)9.81(1.8)3 β ∆T gL3 RaL =  =  = 8.53 × 109 1.589 × 10−5(2.25 × 10−5) να From (5-35) with Pr = 0.707, the Nusselt number is 240 and the average heattransfer coefficient due to natural convection is W ⎯ k ⎯⎯ 0.0263 h =  NuL =  (240) = 3.50  m2K L 1.8

HEAT TRANSFER BY CONVECTION

TABLE 5-3 Effect of Entrance Configuration on Values of C and n in Eq. (5-53) for Pr ª 1 (Gases and Other Fluids with Pr about 1)

The radiative heat-transfer coefficient is given by (5-12): hR = εskinσ(T2skin + T2sur)(Tskin + Tsur)

W = 0.95(5.67 × 10−8)(3082 + 2882)(308 + 288) = 5.71  m2⋅K The total rate of heat loss is . ⎯ ⎯ Q = hA(Tskin − Tair) + hRA(Tskin − Tsur)

Entrance configuration

= 3.50(1.8)(35 − 19) + 5.71(1.8)(35 − 15) = 306 W At these conditions, radiation is nearly twice as important as natural convection.

FORCED CONVECTION Forced convection heat transfer is probably the most common mode in the process industries. Forced flows may be internal or external. This subsection briefly introduces correlations for estimating heattransfer coefficients for flows in tubes and ducts; flows across plates, cylinders, and spheres; flows through tube banks and packed beds; heat transfer to nonevaporating falling films; and rotating surfaces. Section 11 introduces several types of heat exchangers, design procedures, overall heat-transfer coefficients, and mean temperature differences. Flow in Round Tubes In addition to the Nusselt (NuD = hD/k) and Prandtl (Pr = να) numbers introduced above, the key dimensionless parameter for forced convection in round tubes of diameter D is the. Reynolds number Re = GDµ, where G is the mass velocity G = m Ac and Ac is the cross-sectional area Ac = πD24. For internal flow in a tube or duct, the heat-transfer coefficient is defined as q = h(Ts − Tb)

(5-44)

where Tb is the bulk or mean temperature at a given cross section and Ts is the corresponding surface temperature. For laminar flow (ReD < 2100) that is fully developed, both hydrodynamically and thermally, the Nusselt number has a constant value. For a uniform wall temperature, NuD = 3.66. For a uniform heat flux through the tube wall, NuD = 4.36. In both cases, the thermal conductivity of the fluid in NuD is evaluated at Tb. The distance x required for a fully developed laminar velocity profile is given by [(xD)ReD] ≈ 0.05. The distance x required for fully developed velocity and thermal profiles is obtained from [(x/D)(ReD Pr)] ≈ 0.05. For a constant wall temperature, a fully developed laminar velocity profile, and a developing thermal profile, the average Nusselt number is estimated by [Hausen, Allg. Waermetech., 9, 75 (1959)] ⎯⎯ 0.0668(DL) ReD Pr NuD = 3.66 +  (5-45) 1 + 0.04[(DL) ReD Pr]23 For large values of L, Eq. (5-45) approaches NuD = 3.66. Equation (545) also applies to developing velocity and thermal profiles conditions if Pr >>1. The properties in (5-45) are evaluated at the bulk mean temperature ⎯ Tb = (Tb,in + Tb,out)2 (5-46) For a constant wall temperature with developing laminar velocity and thermal profiles, the average Nusselt number is approximated by [Sieder and Tate, Ind. Eng. Chem., 28, 1429 (1936)] 13 µ 0.14 ⎯⎯ D b NuD = 1.86  ReD Pr (5-47)  µs L The properties, except for µs, are evaluated at the bulk mean temperature per (5-46) and 0.48 < Pr < 16,700 and 0.0044 < µb µs < 9.75. For fully developed flow in the transition region between laminar and turbulent flow, and for fully developed turbulent flow, Gnielinski’s [Int. Chem. Eng., 16, 359 (1976)] equation is recommended: (f2)(ReD − 1000)(Pr) NuD =  K (5-48) 1 + 12.7(f2)12 (Pr23 − 1)





where 0.5 < Pr < 105, 2300 < ReD < 106, K = (Prb/Prs)0.11 for liquids (0.05 < Prb/Prs < 20), and K = (Tb/Ts)0.45 for gases (0.5 < Tb/Ts < 1.5). The factor K corrects for variable property effects. For smooth tubes, the Fanning friction factor f is given by f = 0.25(0.790 ln ReD − 1.64)−2

2300 < ReD < 106

5-9

(5-49)

Long calming section Open end, 90° edge 180° return bend 90° round bend 90° elbow

C

n

0.9756 2.4254 0.9759 1.0517 2.0152

0.760 0.676 0.700 0.629 0.614

For rough pipes, approximate values of NuD are obtained if f is estimated by the Moody diagram of Sec. 6. Equation (5-48) is corrected for entrance effects per (5-53) and Table 5-3. Sieder and Tate [Ind. Eng. Chem., 28, 1429 (1936)] recommend a simpler but less accurate equation for fully developed turbulent flow µb 13 NuD = 0.027 Re45  D Pr µs



0.14

(5-50)

where 0.7 < Pr < 16,700, ReD < 10,000, and L/D > 10. Equations (548) and (5-50) apply to both constant temperature and uniform heat flux along the tube. The properties are evaluated at the bulk temperature Tb, except for µs, which is at the temperature of the tube. For L/D⎯greater than about 10, Eqs. (5-48) and (5-50) provide an estimate ⎯ of NuD. In this case, the properties are evaluated at the bulk mean temperature per (5-46). More complicated and comprehensive predictions of fully developed turbulent convection are available in Churchill and Zajic [AIChE J., 48, 927 (2002)] and Yu, Ozoe, and Churchill [Chem. Eng. Science, 56, 1781 (2001)]. For fully developed turbulent flow of liquid metals, the Nusselt number depends on the wall boundary condition. For a constant wall temperature [Notter and Sleicher, Chem. Eng. Science, 27, 2073 (1972)], 0.93 NuD = 4.8 + 0.0156 Re0.85 D Pr

(5-51)

while for a uniform wall heat flux, 0.93 NuD = 6.3 + 0.0167 Re0.85 D Pr

(5-52)

In both cases the properties are evaluated at Tb and 0.004 < Pr < 0.01 and 104 < ReD < 106. Entrance effects for turbulent flow with simultaneously developing velocity and thermal profiles can be significant when L/D < 10. Shah and Bhatti correlated entrance effects for gases (Pr ≈ 1) to give an equation for the average Nusselt number in the entrance region (in Kaka, Shah, and Aung, eds., Handbook of Single-Phase Convective Heat Transfer, Chap. 3, Wiley-Interscience, 1987). ⎯⎯ NuD C (5-53)  = 1 + n NuD (xD) where NuD is the fully developed Nusselt number and the constants C and n are given in Table 5-3 (Ebadian and Dong, Chap. 5 of Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998, p. 5.31). The tube entrance configuration determines the values of C and n as shown in Table 5-3. Flow in Noncircular Ducts The length scale in the Nusselt and Reynolds numbers for noncircular ducts is the hydraulic diameter, Dh = 4Ac/p, where Ac is the cross-sectional area for flow and p is the wetted perimeter. Nusselt numbers for fully developed laminar flow in a variety of noncircular ducts are given by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 307). For turbulent flows, correlations for round tubes can be used with D replaced by Dh. For annular ducts, the accuracy of the Nusselt number given by (5-48) is improved by the following multiplicative factors [Petukhov and Roizen, High Temp., 2, 65 (1964)]. Di −0.16 Inner tube heated 0.86  Do





Di 0.6 1 − 0.14  Do where Di and Do are the inner and outer diameters, respectively. Outer tube heated

5-10

HEAT AND MASS TRANSFER

Example 6: Turbulent Internal Flow Air at 300 K, 1 bar, and 0.05 kg/s enters a channel of a plate-type heat exchanger (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999) that measures 1 cm wide, 0.5 m high, and 0.8 m long. The walls are at 600 K, and the mass flow rate is 0.05 kg/s. The entrance has a 90° edge. We want to estimate the exit temperature of the air. Our approach will use (5-48) to estimate the average heat-transfer coefficient, followed by application of (5-28) to calculate the exit temperature. We assume ideal gas behavior and an exit temperature of 500 K. The estimated bulk mean temperature of the air is, by (5-46), 400 K. At this temperature, the properties of the air are Pr = 0.690, µ = 2.301 × 10−5 kg(m⋅s), k = 0.0338 W(m⋅K), and cp = 1014 J(kg⋅K). We start by calculating the hydraulic diameter Dh = 4Ac/p. The cross-sectional area for flow Ac is 0.005 m2, and the wetted perimeter p is 1.02 m. The hydraulic diameter Dh = 0.01961 m. The Reynolds number is . m Dh 0.05(0.01961) ReD =  =  = 8521 Acµ 0.005(2.301 × 10−5)

External Flows For a single cylinder in cross flow, Churchill and Bernstein recommend [J. Heat Transfer, 99, 300 (1977)] 13 58 45 0.62 Re12 ⎯⎯ ReD D Pr NuD = 0.3 +  1+  (5-56) 23 14 [1 + (0.4Pr) ] 282,000 ⎯⎯ ⎯ where NuD = hDk. Equation (5-56) is for all values of ReD and Pr, provided that ReD Pr > 0.4. The fluid properties are evaluated at the film temperature (Te + Ts)/2, where Te is the free-stream temperature and Ts is the surface temperature. Equation (5-56) also applies to the uni⎯ form heat flux boundary condition provided h is based on the perimeteraveraged temperature difference between Ts and Te. For an isothermal spherical surface, Whitaker recommends [AIChE, 18, 361 (1972)] 14 ⎯⎯ 0.4 µe 23 NuD = 2 + (0.4Re12 (5-57)  D + 0.06ReD )Pr µs This equation is based on data for 0.7 < Pr < 380, 3.5 < ReD < 8 × 104, and 1 < (µeµs) < 3.2. The properties are evaluated at the free-stream temperature Te, with the exception of µs, which is evaluated at the surface temperature Ts. The average Nusselt number for laminar flow over an isothermal flat plate of length x is estimated from [Churchill and Ozoe, J. Heat Transfer, 95, 416 (1973)] ⎯⎯ 1.128 Pr12 Re12 x Nux =  (5-58) [1 + (0.0468Pr)23]14

 



h

The flow is in the transition region, and Eqs. (5-49) and (5-48) apply: f = 0.25(0.790 ln ReD − 1.64) = 0.25(0.790 ln 8521 − 1.64) = 0.008235 −2

−2

h

(f2)(ReD − 1000)(Pr) NuD =  K 1 + 12.7(f2)12(Pr23 − 1) (0.0082352)(8521 − 1000)(0.690) 400  =  1 + 12.7(0.0082352)12 (0.69023 − 1) 600





0.45

= 21.68

Entrance effects are included by using (5-53) for an open end, 90° edge: ⎯⎯ C 2.4254 (21.68) = 25.96 NuD = 1 + n NuD = 1 +  (xD) (0.80.01961)0.676 The average heat-transfer coefficient becomes W ⎯ 0.0338 k ⎯⎯ h =  NuD =  (25.96) = 44.75  m2⋅K Dh 0.01961 The exit temperature is calculated from (5-28): ⎯ hpL T(L) = Ts − (Ts − Tin)exp −  . mcP











This equation is valid for all values of Pr as long as Rex Pr > 100 and Rex < 5 × 105. The fluid properties are evaluated at the film temperature (Te + Ts)/2, where Te is the free-stream temperature and Ts is the surface temperature. For a uniformly heated flat plate, the local Nusselt number is given by [Churchill and Ozoe, J. Heat Transfer, 95, 78 (1973)] 0.886 Pr12 Re12 x Nux =  (5-59) [1 + (0.0207Pr)23]14







44.75(1.02)0.8 = 600 − (600 − 300)exp −  = 450 K 0.05(1014)

where again the properties are evaluated at the film temperature. The average Nusselt number for turbulent flow over a smooth, isothermal flat plate of length x is given by (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 315) ⎯⎯ Recr 0.8 13 0.43 Nux = 0.664 Re12 + 0.036 Re0.8 1−  (5-60) cr Pr x Pr Rex The critical Reynolds number Recr is typically taken as 5 × 105, Recr < Rex < 3 × 107, and 0.7 < Pr < 400. The fluid properties are evaluated at the film temperature (Te + Ts)/2, where Te is the free-stream temperature and Ts is the surface temperature. Equation (5-60) ⎯ also applies to the uniform heat flux boundary condition provided h is based on the average temperature difference between Ts and Te. Flow-through Tube Banks Aligned and staggered tube banks are sketched in Fig. 5-6. The tube diameter is D, and the transverse and longitudinal pitches are ST and SL, respectively. The fluid velocity upstream

We conclude that our estimated exit temperature of 500 K is too high. We could repeat the calculations, using fluid properties evaluated at a revised bulk mean temperature of 375 K.

 

Coiled Tubes For turbulent flow inside helical coils, with tube inside radius a and coil radius R, the Nusselt number for a straight tube Nus is related to that for a coiled tube Nuc by (Rohsenow, Hartnett, and Cho, Handbook of Heat Transfer, 3d ed., McGraw-Hill, 1998, p. 5.90) Nuc a a 0.8 (5-54)  = 1.0 + 3.6 1 −   Nus R R where 2 × 104 < ReD < 1.5 × 105 and 5 < R/a < 84. For lower Reynolds numbers (1.5 × 103 < ReD < 2 × 104), the same source recommends Nuc a (5-55)  = 1.0 + 3.4  Nus R





D V∞

ST

D ST

SL SL (a)

(b)

(a) Aligned and (b) staggered tube bank configurations. The fluid velocity upstream of the tubes is V∞.

FIG. 5-6





HEAT TRANSFER BY CONVECTION of the tubes is V∞. To estimate the overall heat-transfer coefficient for the tube bank, Mills proceeds as follows (Heat Transfer, 2d ed., PrenticeHall, 1999, p. 348). The Reynolds number for use in (5-56) is recalculated with an effective average velocity in the space between adjacent tubes: ⎯ V ST  =  (5-61) V∞ ST − (π4)D The heat-transfer coefficient increases from row 1 to about row 5 of the tube bank. The average Nusselt number for a tube bank with 10 or more rows is ⎯⎯10+ ⎯⎯ NuD = ΦNu1D (5-62) ⎯⎯ where Φ is an arrangement factor and Nu1D is the Nusselt number for the first row, calculated by using the velocity in (5-61). The arrangement factor is calculated as follows. Define dimensionless pitches as PT = ST/D and PL/D and calculate a factor ψ as follows. π 1−  if PL ≥ 1 4PT ψ= (5-63) π 1−  if PL < 1 4PTPL



The arrangement factors are 0.7 SLST − 0.3  Φaligned = 1 +  ψ1.5 (SLST + 0.7)2

(5-64)

2 Φstaggered = 1 +  3PL

(5-65)

⎯⎯ 1 + (N − 1)Φ ⎯⎯ NuD =  Nu1D N

(5-66)

where N is the number of rows. The fluid properties for gases are evaluated at the average mean film temperature [(Tin + Tout)/2 + Ts]/2. For liquids, properties are evaluated at the bulk mean temperature (Tin + Tout)/2, with a Prandtl number correction (Prb/Prs)0.11 for cooling and (Prb/Prs)0.25 for heating. Falling Films When a liquid is distributed uniformly around the periphery at the top of a vertical tube (either inside or outside) and allowed to fall down the tube wall by the influence of gravity, the fluid does not fill the tube but rather flows as a thin layer. Similarly, when a liquid is applied uniformly to the outside and top of a horizontal tube, it flows in layer form around the periphery and falls off the bottom. In both these cases the mechanism is called gravity flow of liquid layers or falling films. For the turbulent flow of water in layer form down the walls of vertical tubes the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)] is recommended: hlm = bΓ1/3

(5-67)

where b = 9150 (SI) or 120 (U.S. Customary) and is based on values of . Γ = WF = M/πD ranging from 0.25 to 6.2 kg/(ms) [600 to 15,000 lb/ (hft)] of wetted perimeter. This type of water flow is used in vertical vapor-in-shell ammonia condensers, acid coolers, cycle water coolers, and other process-fluid coolers. The following dimensional equations may be used for any liquid flowing in layer form down vertical surfaces: For

4Γ k3ρ2g  > 2100 hlm = 0.01  µ µ2

For

4Γ k2ρ4/3cg2/3  < 2100 ham = 0.50  µ Lµ1/3





k µ 1/3

cµ

1/3

µ

4Γ

1/3

(5-68a)

4Γ   µ  µ 1/3

necessarily decrease. Within the finite limits of 0.12 to 1.8 m (0.4 to 6 ft), this equation should give results of the proper order of magnitude. For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting πD/2 for L. For water flowing over a horizontal tube, data for several sizes of pipe are roughly correlated by the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)]. ham = b(Γ/D0)1/3

1/4

1/9

(5-68b)

w

Equation (5-68b) is based on the work of Bays and McAdams [Ind. Eng. Chem., 29, 1240 (1937)]. The significance of the term L is not clear. When L = 0, the coefficient is definitely not infinite. When L is large and the fluid temperature has not yet closely approached the wall temperature, it does not appear that the coefficient should

(5-69)

where b = 3360 (SI) or 65.6 (U.S. Customary) and Γ ranges from 0.94 to 4 kg/(m⋅s) [100 to 1000 lb/(h⋅ft)]. Falling films are also used for evaporation in which the film is both entirely or partially evaporated (juice concentration). This principle is also used in crystallization (freezing). The advantage of high coefficient in falling-film exchangers is partially offset by the difficulties involved in distribution of the film, maintaining complete wettability of the tube, and pumping costs required to lift the liquid to the top of the exchanger. Finned Tubes (Extended Surface) When the heat-transfer coefficient on the outside of a metal tube is much lower than that on the inside, as when steam condensing in a pipe is being used to heat air, externally finned (or extended) heating surfaces are of value in increasing substantially the rate of heat transfer per unit length of tube. The data on extended heating surfaces, for the case of air flowing outside and at right angles to the axes of a bank of finned pipes, can be represented approximately by the dimensional equation derived from 0.6 p′ VF0.6  hf = b  (5-70) p′ − D0 D0.4 0 −3 where b = 5.29 (SI) or (5.39)(10 ) (U.S. Customary); hf is the coefficient of heat transfer on the air side; VF is the face velocity of the air; p′ is the center-to-center spacing, m, of the tubes in a row; and D0 is the outside diameter, m, of the bare tube (diameter at the root of the fins). In atmospheric air-cooled finned tube exchangers, the air-film coefficient from Eq. (5-70) is sometimes converted to a value based on outside bare surface as follows: Af + Auf A hfo = hf  = hf T (5-71) Aof Ao in which hfo is the air-film coefficient based on external bare surface; hf is the air-film coefficient based on total external surface; AT is total external surface, and Ao is external bare surface of the unfinned tube; Af is the area of the fins; Auf is the external area of the unfinned portion of the tube; and Aof is area of tube before fins are attached. Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Trans. Am. Soc. Mech. Eng., 67, 621 (1945)]. Heat-transfer coefficients for finned tubes of various types are given in a series of papers [Trans. Am. Soc. Mech. Eng., 67, 601 (1945)]. For flow of air normal to fins in the form of short strips or pins, Norris and Spofford [Trans. Am. Soc. Mech. Eng., 64, 489 (1942)] correlate their results for air by the dimensionless equation of Pohlhausen: cpµ 2/3 zpGmax −0.5 hm = 1.0  (5-72)   cpGmax k µ for values of zpGmax/µ ranging from 2700 to 10,000. For the general case, the treatment suggested by Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recommended. Because of the wide variations in fin-tube construction, it is convenient to convert all coefficients to values based on the inside bare surface of the tube. Thus to convert the coefficient based on outside area (finned side) to a value based on inside area Kern gives the following relationship:



If there are fewer than 10 rows,

5-11







hfi = (ΩAf + Ao)(hf /Ai)



(5-73)

5-12

HEAT AND MASS TRANSFER

in which hfi is the effective outside coefficient based on the inside area, hf is the outside coefficient calculated from the applicable equation for bare tubes, Af is the surface area of the fins, Ao is the surface area on the outside of the tube which is not finned, Ai is the inside area of the tube, and Ω is the fin efficiency defined as Ω = (tanh mbf)/mbf

(5-74)

m = (hf pf /kax)1/2 m−1 (ft−1)

(5-75)

in which and bf = height of fin. The other symbols are defined as follows: pf is the perimeter of the fin, ax is the cross-sectional area of the fin, and k is the thermal conductivity of the material from which the fin is made. Fin efficiencies and fin dimensions are available from manufacturers. Ratios of finned to inside surface are usually available so that the terms A f, Ao, and Ai may be obtained from these ratios rather than from the total surface areas of the heat exchangers. JACKETS AND COILS OF AGITATED VESSELS See Secs. 11 and 18. NONNEWTONIAN FLUIDS A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior

and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generalized relationships exist to predict the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (NonNewtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as d ln (D ∆P/4L) n′ =  (5-76) d ln (8V/D) as determined from a plot of shear stress versus velocity gradient. For circular tubes, Gz > 100, n′ > 0.1, and laminar flow 1/3 Nulm = 1.75 δ1/3 s Gz

(5-77)

where δs = (3n′ + 1)/4n′. When natural convection effects are considered, Metzer and Gluck [Chem. Eng. Sci., 12, 185 (1960)] obtained the following for horizontal tubes:





PrGrD Nulm = 1.75 δ 1/3 Gz + 12.6  s L

γb

 γ 0.4 1/3

0.14

(5-78)

w

where properties are evaluated at the wall temperature, i.e., γ = gc K′8n′ −1 and τw = K′(8V/D)n′. Metzner and Friend [Ind. Eng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through laminar boundary layers are available in Skelland’s book (op. cit.).

HEAT TRANSFER WITH CHANGE OF PHASE In any operation in which a material undergoes a change of phase, provision must be made for the addition or removal of heat to provide for the latent heat of the change of phase plus any other sensible heating or cooling that occurs in the process. Heat may be transferred by any one or a combination of the three modes—conduction, convection, and radiation. The process involving change of phase involves mass transfer simultaneous with heat transfer. CONDENSATION Condensation Mechanisms Condensation occurs when a saturated vapor comes in contact with a surface whose temperature is below the saturation temperature. Normally a film of condensate is formed on the surface, and the thickness of this film, per unit of breadth, increases with increase in extent of the surface. This is called film-type condensation. Another type of condensation, called dropwise, occurs when the wall is not uniformly wetted by the condensate, with the result that the condensate appears in many small droplets at various points on the surface. There is a growth of individual droplets, a coalescence of adjacent droplets, and finally a formation of a rivulet. Adhesional force is overcome by gravitational force, and the rivulet flows quickly to the bottom of the surface, capturing and absorbing all droplets in its path and leaving dry surface in its wake. Film-type condensation is more common and more dependable. Dropwise condensation normally needs to be promoted by introducing an impurity into the vapor stream. Substantially higher (6 to 18 times) coefficients are obtained for dropwise condensation of steam, but design methods are not available. Therefore, the development of equations for condensation will be for the film type only. The physical properties of the liquid, rather than those of the vapor, are used for determining the coefficient for condensation. Nusselt [Z. Ver. Dtsch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation.

The Reynolds number of the condensate film (falling film) is 4Γ/µ, where Γ is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(sm) [lb/(hft)]. The thickness of the condensate film for Reynolds number less than 2100 is (3µΓ/ρ2g)1/3. Condensation Coefficients Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /πD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled length and ∆t is tsv − ts): Nusselt type: hL L3ρ2gλ  = 0.943  kµ ∆t k





1/4

L3ρ2g = 0.925  µΓ





1/3

(5-79)*

Dimensional: h = b(k3ρ2D/µbWF)1/3

(5-80)*

where b = 127 (SI) or 756 (U.S. Customary). For steam at atmospheric pressure, k = 0.682 J/(msK) [0.394 Btu/(hft°F)], ρ = 960 kg/m3 (60 lb/ft3), µb = (0.28)(10−3) Pas (0.28 cP), h = b(D/WF)1/3

(5-81)

where b = 2954 (SI) or 6978 (U.S. Customary). For organic vapors at normal boiling point, k = 0.138 J/(msK) [0.08 Btu/(hft°F)], ρ = 720 kg/m3 (45 lb/ft3), µb = (0.35)(10−3) Pas (0.35 cP), h = b(D/WF)1/3

(5-82)

where b = 457 (SI) or 1080 (U.S. Customary). Horizontal Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /2L. * If the vapor density is significant, replace ρ2 with ρl(ρl − ρv).

HEAT TRANSFER WITH CHANGE OF PHASE

5-13

FIG. 5-7 Chart for determining heat-transfer coefficient hm for film-type condensation of pure vapor, based on Eqs. (5-79) 4 ρ 2k3/µ  is in U.S. Customary units; and (5-83). For vertical tubes multiply hm by 1.2. If 4Γ/µf exceeds 2100, use Fig. 5-8. λ to convert feet to meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert British thermal units per hour–square foot–degrees Fahrenheit to watts per square meter–kelvins, multiply by 5.6780.

Nusselt type: hD D3ρ2gλ 1/4 D3ρ2g 1/3 = 0.76  (5-83)*  = 0.73  k kµ ∆t µΓ Dimensional: h = b(k3ρ2L/µbWF)1/3 (5-84)* where b = 205.4 (SI) or 534 (U.S. Customary). For steam at atmospheric pressure









* If the vapor density is significant, replace ρ with ρl(ρl − ρv). 2

h = b(L/WF)1/3

(5-85)

where b = 2080 (SI) or 4920 (U.S. Customary). For organic vapors at normal boiling point h = b(L/WF)1/3

(5-86)

where b = 324 (SI) or 766 (U.S. Customary). Figure 5-7 is a nomograph for determining coefficients of heat transfer for condensation of pure vapors.

5-14

HEAT AND MASS TRANSFER

Banks of Horizontal Tubes (Re < 2100) In the idealized case of N tubes in a vertical row where the total condensate flows smoothly from one tube to the one beneath it, without splashing, and still in laminar flow on the tube, the mean condensing coefficient hN for the entire row of N tubes is related to the condensing coefficient for the top tube h1 by hN = h1N−1/4

(5-87)

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theory. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950; 2138, 1952; 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundary. According to the Dukler theory, three fixed factors must be known to establish the value of the average film coefficient: the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group Nd defined as follows: 2/3 2 0.553 0.78 0.16 Nd = (0.250µ1.173 ρG ) L µG )/(g D ρL

(5-88)

Graphical relationships of these variables are available in Document 6058, ADI Auxiliary Publications Project, Library of Congress, Washington. If rigorous values for condensing-film coefficients are desired, especially if the value of Nd in Eq. (5-88) exceeds (1)(10−5), it is suggested that these graphs be used. For the case in which interfacial shear is zero, Fig. 5-8 may be used. It is interesting to note that, according to the Dukler development, there is no definite transition Reynolds number; deviation from Nusselt theory is less at low Reynolds numbers; and when the Prandtl number of a fluid is less than 0.4 (at Reynolds number above 1000), the predicted values for film coefficient are lower than those predicted by the Nusselt theory. The Dukler theory is applicable for condensate films on horizontal tubes and also for falling films, in general, i.e., those not associated with condensation or vaporization processes. Vapor Shear Controlling For vertical in-tube condensation with vapor and liquid flowing concurrently downward, if gravity controls, Figs. 5-7 and 5-8 may be used. If vapor shear controls, the Carpenter-Colburn correlation (General Discussion on Heat Transfer, London, 1951, ASME, New York, p. 20) is applicable: 1/2 hµl /kl ρl1/2 = 0.065(Pr)1/2 l Fvc

Fvc = fG2vm /2ρv 2 1/2 Gvi2 + GviGvo + Gvo Gvm =  3 and f is the Fanning friction factor evaluated at

where



(Re)vm = DiGvm /µv



(5-89a) (5-89b) (5-89c)

(5-89d)

Dukler plot showing average condensing-film coefficient as a function of physical properties of the condensate film and the terminal Reynolds number. (Dotted line indicates Nusselt theory for Reynolds number < 2100.) [Reproduced by permission from Chem. Eng. Prog., 55, 64 (1959).] FIG. 5-8

and the subscripts vi and vo refer to the vapor inlet and outlet, respectively. An alternative formulation, directly in terms of the friction factor, is h = 0.065 (cρkf/2µρv)1/2Gvm

(5-89e)

expressed in consistent units. Another correlation for vapor-shear-controlled condensation is the Boyko-Kruzhilin correlation [Int. J. Heat Mass Transfer, 10, 361 (1967)], which gives the mean condensing coefficient for a stream between inlet quality xi and outlet quality xo: hDi DiGT 0.8 (ρ /ρ (ρ /ρ  m)i +  m)o (5-90a)  = 0.024  (Pr)l0.43  kl µl 2 where GT = total mass velocity in consistent units ρ ρl − ρv  = 1 +  xi (5-90b) ρm i ρv





 ρ

ρl − ρv

=1+x  ρ ρ

and

(5-90c)

o

m

o

v

For horizontal in-tube condensation at low flow rates Kern’s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is valid: k 3l ρl (ρl − ρv)gλ 1/4 Lk 3l ρl(ρl − ρv)g 1/3 hm = 0.761  = 0.815  (5-91) WF µl πµl Di ∆t where WF is the total vapor condensed in one tube and ∆t is tsv − ts . A more rigorous correlation has been proposed by Chaddock [Refrig. Eng., 65(4), 36 (1957)]. Use consistent units. At high condensing loads, with vapor shear dominating, tube orientation has no effect, and Eq. (5-90a) may also be used for horizontal tubes. Condensation of pure vapors under laminar conditions in the presence of noncondensable gases, interfacial resistance, superheating, variable properties, and diffusion has been analyzed by Minkowycz and Sparrow [Int. J. Heat Mass Transfer, 9, 1125 (1966)].









BOILING (VAPORIZATION) OF LIQUIDS Boiling Mechanisms Vaporization of liquids may result from various mechanisms of heat transfer, singly or combinations thereof. For example, vaporization may occur as a result of heat absorbed, by radiation and convection, at the surface of a pool of liquid; or as a result of heat absorbed by natural convection from a hot wall beneath the disengaging surface, in which case the vaporization takes place when the superheated liquid reaches the pool surface. Vaporization also occurs from falling films (the reverse of condensation) or from the flashing of liquids superheated by forced convection under pressure. Pool boiling refers to the type of boiling experienced when the heating surface is surrounded by a relatively large body of fluid which is not flowing at any appreciable velocity and is agitated only by the motion of the bubbles and by natural-convection currents. Two types of pool boiling are possible: subcooled pool boiling, in which the bulk fluid temperature is below the saturation temperature, resulting in collapse of the bubbles before they reach the surface, and saturated pool boiling, with bulk temperature equal to saturation temperature, resulting in net vapor generation. The general shape of the curve relating the heat-transfer coefficient to ∆tb, the temperature driving force (difference between the wall temperature and the bulk fluid temperature) is one of the few parametric relations that are reasonably well understood. The familiar boiling curve was originally demonstrated experimentally by Nukiyama [J. Soc. Mech. Eng. ( Japan), 37, 367 (1934)]. This curve points out one of the great dilemmas for boiling-equipment designers. They are faced with at least six heat-transfer regimes in pool boiling: natural convection (+), incipient nucleate boiling (+), nucleate boiling (+), transition to film boiling (−), stable film boiling (+), and film boiling with increasing radiation (+). The signs indicate the sign of the derivative d(q/A)/d ∆tb. In the transition to film boiling, heat-transfer rate decreases with driving force. The regimes of greatest commercial interest are the nucleate-boiling and stable-film-boiling regimes. Heat transfer by nucleate boiling is an important mechanism in the vaporization of liquids. It occurs in the vaporization of liquids in

HEAT TRANSFER BY RADIATION kettle-type and natural-circulation reboilers commonly used in the process industries. High rates of heat transfer per unit of area (heat flux) are obtained as a result of bubble formation at the liquid-solid interface rather than from mechanical devices external to the heat exchanger. There are available several expressions from which reasonable values of the film coefficients may be obtained. The boiling curve, particularly in the nucleate-boiling region, is significantly affected by the temperature driving force, the total system pressure, the nature of the boiling surface, the geometry of the system, and the properties of the boiling material. In the nucleate-boiling regime, heat flux is approximately proportional to the cube of the temperature driving force. Designers in addition must know the minimum ∆t (the point at which nucleate boiling begins), the critical ∆t (the ∆t above which transition boiling begins), and the maximum heat flux (the heat flux corresponding to the critical ∆t). For designers who do not have experimental data available, the following equations may be used. Boiling Coefficients For the nucleate-boiling coefficient the Mostinski equation [Teplenergetika, 4, 66 (1963)] may be used: q 0.7 P 0.17 P 1.2 P 10 h = bPc0.69  1.8  + 4  + 10  (5-92) A Pc Pc Pc where b = (3.75)(10−5)(SI) or (2.13)(10−4) (U.S. Customary), Pc is the critical pressure and P the system pressure, q/A is the heat flux, and h is the nucleate-boiling coefficient. The McNelly equation [J. Imp. Coll. Chem. Eng. Soc., 7(18), (1953)] may also be used: 0.33 qc 0.69 Pkl 0.31 ρl −1 h = 0.225 l (5-93)  ρ Aλ σ v where cl is the liquid heat capacity, λ is the latent heat, P is the system pressure, kl is the thermal conductivity of the liquid, and σ is the surface tension. An equation of the Nusselt type has been suggested by Rohsenow [Trans. Am. Soc. Mech. Eng., 74, 969 (1952)]. hD/k = Cr(DG/µ)2/3(cµ/k)−0.7 (5-94a) in which the variables assume the following form: 1/2 1/2 gcσ gcσ hβ′ β′ W 2/3 cµ −0.7  = Cr   (5-94b)    k k g(ρL − ρv) µ g(ρL − ρv) A

  



  





 



 



 

The coefficient Cr is not truly constant but varies from 0.006 to 0.015.* It is possible that the nature of the surface is partly responsible for the variation in the constant. The only factor in Eq. (5-94b) not readily available is the value of the contact angle β′. Another Nusselt-type equation has been proposed by Forster and Zuber:† Nu = 0.0015 Re0.62 Pr1/3 (5-95) which takes the following form:

α  W 2σ cρLπ   kρv A ∆p

  1/2

ρL  ∆pgc



where α = k/ρc (all liquid properties) ∆p = pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Equations (5-94b) and (5-96) have been arranged in dimensional form by Westwater. The numerical constant may be adjusted to suit any particular set of data if one desires to use a certain criterion. However, surface conditions vary so greatly that deviations may be as large as 25 percent from results obtained. The maximum heat flux may be predicted by the KutateladseZuber [Trans. Am. Soc. Mech. Eng., 80, 711 (1958)] relationship, using consistent units:

Aq

(ρl − ρv)σg = 0.18gc1/4ρv λ  ρ2v max

 

  2 0.62

cµ  k

1/ 2

(5-96)





1/4

(5-97)

Alternatively, Mostinski presented an equation which approximately represents the Cichelli-Bonilla [Trans. Am. Inst. Chem. Eng., 41, 755 (1945)] correlation: (q/A)max P 0.35 P 0.9 1− (5-98) =b  Pc Pc Pc

 



where b = 0.368(SI) or 5.58 (U.S. Customary); Pc is the critical pressure, Pa absolute; P is the system pressure; and (q/A)max is the maximum heat flux. The lower limit of applicability of the nucleate-boiling equations is from 0.1 to 0.2 of the maximum limit and depends upon the magnitude of natural-convection heat transfer for the liquid. The best method of determining the lower limit is to plot two curves: one of h versus ∆t for natural convection, the other of h versus ∆t for nucleate boiling. The intersection of these two curves may be considered the lower limit of applicability of the equations. These equations apply to single tubes or to flat surfaces in a large pool. In tube bundles the equations are only approximate, and designers must rely upon experiment. Palen and Small [Hydrocarbon Process., 43(11), 199 (1964)] have shown the effect of tube-bundle size on maximum heat flux.

Aq

max

p gσ(ρl − ρv) = b  ρv λ  Do NT ρ2v





1/4

(5-99)

where b = 0.43 (SC) or 61.6 (U.S. Customary), p is the tube pitch, Do is the tube outside diameter, and NT is the number of tubes (twice the number of complete tubes for U-tube bundles). For film boiling, Bromley’s [Chem. Eng. Prog., 46, 221 (1950)] correlation may be used: kv3(ρl − ρv)ρv g h = b  µv Do ∆tb



1/4

α  ρL cρL ∆T π = 0.0015   µ λρv

5-15



1/4

(5-100)

where b = 4.306 (SI) or 0.620 (U.S. Customary). Katz, Myers, and Balekjian [Pet. Refiner, 34(2), 113 (1955)] report boiling heat-transfer coefficients on finned tubes.

HEAT TRANSFER BY RADIATION GENERAL REFERENCES: Baukal, C. E., ed., The John Zink Combustion Handbook, CRC Press, Boca Raton, Fla., 2001. Blokh, A. G., Heat Transfer in Steam Boiler Furnaces, 3d ed., Taylor & Francis, New York, 1987. Brewster, M. Quinn, Thermal Radiation Heat Transfer and Properties, Wiley, New York, 1992. Goody, R. M., and Y. L. Yung, Atmospheric Radiation—Theoretical Basis, 2d ed., Oxford University Press, 1995. Hottel, H. C., and A. F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. Modest, Michael F., Radiative Heat Transfer, 2d ed., Academic Press, New York, 2003. Noble, James J., “The Zone

Method: Explicit Matrix Relations for Total Exchange Areas,” Int. J. Heat Mass Transfer, 18, 261–269 (1975). Rhine, J. M., and R. J. Tucker, Modeling of GasFired Furnaces and Boilers, British Gas Association with McGraw-Hill, 1991. Siegel, Robert, and John R. Howell, Thermal Radiative Heat Transfer, 4th ed., Taylor & Francis, New York, 2001. Sparrow, E. M., and R. D. Cess, Radiation Heat Transfer, 3d ed., Taylor & Francis, New York, 1988. Stultz, S. C., and J. B. Kitto, Steam: Its Generation and Use, 40th ed., Babcock and Wilcox, Barkerton, Ohio, 1992.

* Reported by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956, p. 15. † Forster, J. Appl. Phys., 25, 1067 (1954); Forster and Zuber, J. Appl. Phys., 25, 474 (1954); Forster and Zuber, Conference on Nuclear Engineering, University of California, Los Angeles, 1955; excellent treatise on boiling of liquids by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956.

5-16

HEAT AND MASS TRANSFER

INTRODUCTION Heat transfer by thermal radiation involves the transport of electromagnetic (EM) energy from a source to a sink. In contrast to other modes of heat transfer, radiation does not require the presence of an intervening medium, e.g., as in the irradiation of the earth by the sun. Most industrially important applications of radiative heat transfer occur in the near infrared portion of the EM spectrum (0.7 through 25 µm) and may extend into the far infrared region (25 to 1000 µm). For very high temperature sources, such as solar radiation, relevant wavelengths encompass the entire visible region (0.4 to 0.7 µm) and may extend down to 0.2 µm in the ultraviolet (0.01- to 0.4-µm) portion of the EM spectrum. Radiative transfer can also exhibit unique action-at-a-distance phenomena which do not occur in other modes of heat transfer. Radiation differs from conduction and convection not only with regard to mathematical characterization but also with regard to its fourth power dependence on temperature. Thus it is usually dominant in high-temperature combustion applications. The temperature at which radiative transfer accounts for roughly one-half of the total heat loss from a surface in air depends on such factors as surface emissivity and the convection coefficient. For pipes in free convection, radiation is important at ambient temperatures. For fine wires of low emissivity it becomes important at temperatures associated with bright red heat (1300 K). Combustion gases at furnace temperatures typically lose more than 90 percent of their energy by radiative emission from constituent carbon dioxide, water vapor, and particulate matter. Radiative transfer methodologies are important in myriad engineering applications. These include semiconductor processing, illumination theory, and gas turbines and rocket nozzles, as well as furnace design. THERMAL RADIATION FUNDAMENTALS In a vacuum, the wavelength λ, frequency, ν and wavenumber η for electromagnetic radiation are interrelated by λ = cν = 1η, where c is the speed of light. Frequency is independent of the index of refraction of a medium n, but both the speed of light and the wavelength in the medium vary according to cm = c/n and λm = λn. When a radiation beam passes into a medium of different refractive index, not only does its wavelength change but so does its direction (Snell’s law) as well as the magnitude of its intensity. In most engineering heat-transfer calculations, wavelength is usually employed to characterize radiation while wave number is often used in gas spectroscopy. For a vacuum, air at ambient conditions, and most gases, n ≈ 1.0. For this reason this presentation sometimes does not distinguish between λ and λm. Dielectric materials exhibit 1.4 < n < 4, and the speed of light decreases considerably in such media. Æ In radiation heat transfer, the monochromatic intensity Iλ ≡ Iλ (r , Æ W, λ), is a fundamental (scalar) field variable which characterizes EM energy transport. Intensity defines the radiant energy flux passing through an infinitesimal area dA, oriented normal to a radiation beam Æ of arbitrary direction W. At steady state, the monochromatic intensity Æ Æ is a function of position r , direction W, and wavelength and has units 2 of W(m ⋅sr⋅µm). In the general case of an absorbing-emitting and scattering medium, characterizedÆ by some absorption coefficient K(m−1), intensity in the direction W will be modified by attenuation and by scattering of radiation into and out of the beam. For the special case of a nonabsorbing (transparent), nonscattering, medium of constant refractive index, the radiation intensity is constant and independent of Æ position in a given direction W. This circumstance arises in illumination theory where the light intensity in a room is constant in a given direction but may vary with respect to all other directions. The basic conservation law for radiation intensity is termed the equation of transfer or radiative transfer equation. The equation of transfer is a directional energy balance and mathematically is an integrodifferential equation. The relevance of the transport equation to radiation heat transfer is discussed in many sources; see, e.g., Modest, M. F., Radiative Heat Transfer, 2d ed., Academic Press, 2003, or Siegel, R., and J. R. Howell, Thermal Radiative Heat Transfer, 4th ed., Taylor & Francis, New York, 2001. Introduction to Radiation Geometry Consider a homogeneous medium of constant refractive index n. A pencil of radiation

originates at differential area element dAi and is incident on differenÆ Æ tial area element dAj. Designate n i and n j as the unit vectors normal Æ to dAi and dAj, and let r, with unit direction vector W, define the distance of separation between the area elements. Moreover, φi and φj Æ Æ Æ denote the confined angles between W and n i and n j, respectively [i.e., Æ Æ Æ Æ cosφi ≡ cos(W, r i) and cosφj ≡ cos(W, r j)]. As the beam travels toward dAj, it will diverge and subtend a solid angle cosφj dΩ j =  dAj sr r2 Æ

at dAi. Moreover, the projected area of dAi in the direction of W is Æ Æ givenÆby cos(W, r i) dAi = cosφi dAi. Multiplication of the intensity Iλ ≡ Æ Iλ(r , W, λ) by dΩj and the apparent area of dAi then yields an expression for the (differential) net monochromatic radiant energy flux dQi,j originating at dAi and intercepted by dAj. Æ

dQi,j ≡ Iλ(W, λ) cosφi cosφj dAi dAjr2

(5-101)

The hemispherical emissive power* E is defined as the radiant flux density (W/m2) associated with emission from an element of surface area dA into a surrounding unit hemisphere whose base is coplaÆ nar with dA. If the monochromatic intensity Iλ(W, λ) of emission from Æ the surface is isotropic (independent of the angle of emission, W), Eq. (5-101) may be integrated over the 2π sr of the surrounding unit hemisphere to yield the simple relation Eλ = πIλ, where Eλ ≡ Eλ(λ) is defined as the monochromatic or spectral hemispherical emissive power. Blackbody Radiation Engineering calculations involving thermal radiation normally employ the hemispherical blackbody emissive power as the thermal driving force analogous to temperature in the cases of conduction and convection. A blackbody is a theoretical idealization for a perfect theoretical radiator; i.e., it absorbs all incident radiation without reflection and emits isotropically. In practice, soot-covered surfaces sometimes approximate blackbody behavior. Let Eb,λ = Eb,λ (T,λ) denote the monochromatic blackbody hemispherical emissive power frequency function defined such that Eb,λ (T, λ)dλ represents the fraction of blackbody energy lying in the wavelength region from λ to λ + dλ. The function Eb,λ = Eb,λ (T,λ) is given by Planck’s law c1(λT)−5 Eb,λ (T,λ) =   2 5 ec λT − 1 nT 2

(5-102)

where c1 = 2πhc2 and c2 = hc/k are defined as Planck’s first and second constants, respectively. Integration of Eq. (5-102) over all wavelengths yields the StefanBoltzman law for the hemispherical blackbody emissive power Eb(T) =



∞

λ=0

Eb,λ (T, λ) dλ = n2σT4

(5-103)

where σ = c1(πc2) 15 is the Stephan-Boltzman constant. Since a blackbody is an isotropic emitter, it follows that the intensity of blackbody emission is given by the simple formula Ib = Ebπ = n2σT4π. The intensity of radiation emitted over all wavelengths by a blackbody is thus uniquely determined by its temperature. In this presentation, all references to hemispherical emissive power shall be to the blackbody emissive power, and the subscript b may be suppressed for expediency. For short wavelengths λT → 0, the asymptotic form of Eq. (5-102) is known as the Wien equation Eb,λ(T, λ) ≅ c1(λT)−5e−c λT (5-104)  n2T5 The error introduced by use of the Wien equation is less than 1 percent when λT < 3000 µm⋅K. The Wien equation has significant practical value in optical pyrometry for T < 4600 K when a red filter (λ = 0.65 µm) is employed. The long-wavelength asymptotic approximation for Eq. (5-102) is known as the Rayleigh-Jeans formula, which is accurate to within 1 percent for λT > 778,000 µm⋅K. The RaleighJeans formula is of limited engineering utility since a blackbody emits over 99.9 percent of its total energy below the value of λT = 53,000 µm⋅K. 4

2

*In the literature the emissive power is variously called the emittance, total hemispherical intensity, or radiant flux density.

Nomenclature and Units—Radiative Transfer a,ag,ag,1 ⎯ ⎯ P,Prod ⎯C⎯ p, C ij = ⎯s⎯⎯s i j

A, Ai c c1, c2 dp, rp Eb,λ = Eb,λ(T,λ) En(x) E Eb = n2σT4 fv Fb(λT) Fi,j ⎯⎯ F i,j F i,j h hi Hi H. HF Æ Æ Iλ ≡ Iλ(r , W, λ) k kλ,p K L. M, LM0 m n M, N pk P Qi Qi,j T U V W

WSGG spectral model clear plus gray weighting constants Heat capacity per unit mass, J⋅kg−1⋅K−1 Shorthand notation for direct exchange area Area of enclosure or zone i, m2 Speed of light in vacuum, m/s Planck’s first and second constants, W⋅m2 and m⋅K Particle diameter and radius, µm Monochromatic, blackbody emissive power, W(m2⋅µm) Exponential integral of order n, where n = 1, 2, 3,. . . Hemispherical emissive power, W/m2 Hemispherical blackbody emissive power, W/m2 Volumetric fraction of soot Blackbody fractional energy distribution Direct view factor from surface zone i to surface zone j Refractory augmented black view factor; F-bar Total view factor from surface zone i to surface zone j Planck’s constant, J⋅s Heat-transfer coefficient, W(m2⋅K) Incident flux density for surface zone i, W/m2 Enthalpy rate, W Enthalpy feed rate, W Monochromatic radiation intensity, W(m2⋅µm⋅sr) Boltzmann’s constant, J/K Monochromatic line absorption coefficient, (atm⋅m)−1 Gas absorption coefficient, m−1 Average and optically thin mean beam lengths, m Mass flow rate, kgh−1 Index of refraction Number of surface and volume zones in enclosure Partial pressure of species k, atm Number of WSGG gray gas spectral windows Total radiative flux originating at surface zone i, W Net radiative flux between zone i and zone j, W Temperature, K Overall heat-transfer coefficient in WSCC model Enclosure volume, m3 Leaving flux density (radiosity), W/m2 Greek Characters

α, α1,2 αg,1, εg, τg,1 β ∆Tge ≡ Tg − Te ε εg(T, r) ελ(T, Ω, λ) η = 1λ λ = cν ν ρ=1−ε σ Σ τg = 1 − εg Ω Φ Ψ(3)(x) ω

Surface absorptivity or absorptance; subscript 1 refers to the surface temperature while subscript 2 refers to the radiation source Gas absorptivity, emissivity, and transmissivity Dimensionless constant in mean beam length equation, LM = β⋅LM0 Adjustable temperature fitting parameter for WSCC model, K Gray diffuse surface emissivity Gas emissivity with path length r Monochromatic, unidirectional, surface emissivity Wave number in vacuum, cm−1 Wavelength in vacuum, µm Frequency, Hz Diffuse reflectivity Stefan-Boltzmann constant, W(m2⋅K4) Number of unique direct surface-to-surface direct exchange areas Gas transmissivity Solid angle, sr (steradians) Equivalence ratio of fuel and oxidant Pentagamma function of x Albedo for single scatter Dimensionless Quantities

NFD Deff =  (S 1G RA1) + NCR h NCR =  ⎯ 4σT3g,1 . 4 NFD = Hf σT Ref ⋅A1 ηg η′g = ηg(1 − Θ0) Θi = TiTRef

Effective firing density Convection-radiation number Dimensionless firing density Gas-side furnace efficiency Reduced furnace efficiency Dimensionless temperature Vector Notation

Æ

n i and Æ nj

Æ rÆ

W

Unit vectors normal to differential area elements dAi and dAj Position vector Arbitrary unit direction vector

Matrix Notation Column vector; all of whose elements are unity. [M × 1] Identity matrix, where δi,j is the Kronecker delta; i.e., δi,j = 1 for i = j and δi,j = 0 for i ≠ j. aI Diagonal matrix of WSGG gray gas surface zone a-weighting factors [M × M] agI Diagonal matrix of gray gas WSGG volume zone a-weighting factors [N × N] A = [Ai,j] Arbitrary nonsingular square matrix T A = [Aj,i] Transpose of A A−1 = [Ai,j]−1 Inverse of A DI = [Di⋅δi,j] Arbitrary diagonal matrix DI−1 = [δi,jDi] Inverse of diagonal matrix CDI CI⋅DI = [Ci⋅Di ⋅δi,j], product of two diagonal matrices AI = [Ai⋅δi,j] Diagonal matrix of surface zone areas, m2 [M × M] εI = [εi⋅δi,j] Diagonal matrix of diffuse zone emissivities [M × M] ρI = [ρi⋅δi,j] Diagonal matrix of diffuse zone reflectivities [M × M] 4 E = [Ei] = [σTi ] Column vector of surface blackbody hemispherical emissive powers, W/m2 [M × 1] EI = [Ei⋅δi,j] = [σT 4i ⋅δi,j] Diagonal matrix of surface blackbody emissive powers, W/m2 [M × M] Eg = [Eg,i] = [σT 4g,i] Column vector of gas blackbody hemispherical emissive powers, W/m2 [N × 1] EgI = [Ei⋅δi,j] = [σT 4i ⋅δi,j] Diagonal matrix of gas blackbody emissive powers, W/m2 [N × N] H = [Hi] Column vector of surface zone incident flux densities, W/m2 [M × 1] W = [Wi] Column vector of surface zone leaving flux densities, W/m2 [M × 1] Q = [Qi] ⎯⎯ Column vector of surface zone fluxes, W [M × 1] −1 R = [AI − ss⋅ρI] Inverse multiple-reflection matrix, m−2 [M × M] Diagonal matrix of WSGG Kp,i values for the ith KIp = [δi, j⋅Kp,i] zone and pth gray gas component, m−1 [N × N] q KI Diagonal matrix of WSGG-weighted gray gas absorption coefficients, m−1 [N × N] S′ Column vector for net volume absorption, W [N × 1] s⎯s⎯ = [s⎯⎯i ⎯s⎯j ] Array of direct surface-to-surface exchange areas, m2 [M × M] ⎯sg ⎯ = [s⎯⎯⎯g ] = g⎯s⎯ T Array of direct gas-to-surface exchange areas, m2 i j [M × N] g⎯g⎯ = [g⎯⎯i ⎯g⎯j ] Array of direct gas-to-gas exchange areas, m2 [N × N] ⎯⎯ ⎯⎯⎯ SS = [Si Sj] Array of total surface-to-surface exchange areas, m2 [M × M] ⎯ ⎯ ⎯⎯⎯ SG = [Si Gj] Array of total gas-to-surface exchange areas, m2 [M × N] ⎯⎯ ⎯⎯T GS = GS Array of total surface-to-gas exchange areas, m2 [N × M] ⎯ ⎯ ⎯ ⎯⎯ GG = [Gi Gj] Array of total gas-to-gas exchange areas, m2 [N × N] q q Array of directed surface-to-surface exchange SS = [SiSi] areas, m2 [M × M] q q Array of directed gas-to-surface exchange areas, m2 SG = [SiGi] [M × N] T q q GS ≠ SG Array of directed surface-to-gas exchange areas, m2 [N × M] q q GG = [GiGi] Array of directed gas-to-gas exchange areas, m2 [N × N] VI = [Vi⋅δi,j] Diagonal matrix of zone volumes, m3 [N × N] 1M I = [δi,j]

Subscripts b f h i, j n p r s λ Ref

Blackbody or denotes a black surface zone Denotes flux surface zone Denotes hemispherical surface emissivity Zone number indices Denotes normal component of surface emissivity Index for pth gray gas window Denotes refractory surface zone Denotes source-sink surface zone Denotes monochromatic variable Denotes reference quantity

CFD DO, FV EM RTE LPFF SSR WSCC WSGG

Computational fluid dynamics Discrete ordinate and finite volume methods Electromagnetic Radiative transfer equation; equation of transfer Long plug flow furnace model Source-sink refractory model Well-stirred combustion chamber model Weighted sum of gray gases spectral model

Abbreviations

HEAT AND MASS TRANSFER

The blackbody fractional energy distribution function is defined by  Eb,λ(T, λ) dλ Fb(λT) =  (5-105)  Eb,λ(T, λ) dλ λ

λ= 0 ∞

λ=0

The function Fb(λT) defines the fraction of total energy in the blackbody spectrum which lies below λT and is a unique function of λT. For purposes of digital computation, the following series expansion for Fb(λT) proves especially useful. 15 ∞ e−kξ 3 3ξ2 6ξ 6 Fb(λT) =  +  + 3

 ξ +  π4 k=1 k k2 k k



c2

where ξ =  λT

(5-106)

Equation (5-106) converges rapidly and is due to Lowan [1941] as referenced in Chang and Rhee [Int. Comm. Heat Mass Transfer, 11, 451–455 (1984)]. Numerically, in the preceding, h = 6.6260693 × 10−34 J⋅s is the Planck constant; c = 2.99792458 × 108 ms is the velocity of light in vacuum; and k = 1.3806505 × 10−23 JK is the Boltzmann constant. These data lead to the following values of Planck’s first and second constants: c1 = 3.741771 × 10−16 W⋅m2 and c2 = 1.438775 × 10−2 m⋅K, respectively. Numerical values of the Stephan-Boltzmann constant σ in several systems of units are as follows: 5.67040 × 10−8 W(m2⋅K4); 1.3544 × 10−12 cal(cm2⋅s⋅K4); 4.8757 × 10−8 kcal(m2⋅h⋅K4); 9.9862 × 10−9 CHU(ft2⋅h⋅K4); and 0.17123 × 10−8 Btu(ft2⋅h⋅°R4) (CHU = centigrade heat unit; 1.0 CHU = 1.8 Btu.)

Blackbody Displacement Laws The blackbody energy spectrum W Eb,λ(λT) is plotted logarithmically in Fig. 5-9 as  × 1013  2 5 2 5 m ⋅µm⋅K n T versus λT µm⋅K. For comparison a companion inset is provided in Cartesian coordinates. The upper abscissa of Fig. 5-9 also shows the blackbody energy distribution function Fb(λT). Figure 5-9 indicates that the wavelength-temperature product for which the maximum intensity occurs is λmaxT = 2898 µm⋅K. This relationship is known as Wien’s displacement law, which indicates that the wavelength for maximum intensity is inversely proportional to the absolute temperature. Blackbody displacement laws are useful in engineering practice to estimate wavelength intervals appropriate to relevant system temperatures. The Wien displacement law can be misleading, however, because the wavelength for maximum intensity depends on whether the intensity is defined in terms of frequency or wavelength interval. Two additional useful displacement laws are defined in terms of either the value of λT corresponding to the maximum energy per unit fractional change in wavelength or frequency, that is, λT = 3670 µm⋅K, or to the value of λT corresponding to one-half the blackbody energy, that is, λT = 4107 µm⋅K. Approximately one-half of the blackbody energy lies within the twofold λT range geometrically centered on λT = 3670 µm⋅K, that is, 36702 < λT < 36702 µm⋅K. Some 95 percent of the blackbody energy lies in the interval 1662.6 < λT < 16,295 µm⋅K. It thus follows that for the temperature range between ambient (300 K) and flame temperatures (2000 K or

5

W [ m ·µm·K ]

1013 ×

Eb,λ

Percentage of total blackbody energy found below λT, Fb (λT)

n2T 5

5-18

n2T 5

1013 ×

Eb,λ (λT)

2

λT [ µm . K]

Wavelength-temperature product λT [ µm . K] FIG. 5-9

Spectral dependence of monochromatic blackbody hemispherical emissive power.

HEAT TRANSFER BY RADIATION 3140°F), wavelengths of engineering heat-transfer importance are bounded between 0.83 and 54.3 µm. RADIATIVE PROPERTIES OF OPAQUE SURFACES Emittance and Absorptance The ratio of the total radiating power of any surface to that of a black surface at the same temperature is called the emittance or emissivity, ε of the surface.* In general, the monochromatic emissivity is a function of temperature, Æ direction, and wavelength, that is, ελ = ελ(T, W, λ). The subscripts n and h are sometimes used to denote the normal and hemispherical values, respectively, of the emittance or emissivity. If radiation is incident on a surface, the fraction absorbed is called the absorptance (absorptivity). Two subscripts are usually appended to the absorptance α1,2 to distinguish between the temperature of the absorbing surface T1 and the spectral energy distribution of the emitting surface T2. According to Kirchhoff’s law, the emissivity and absorptivity of a surface exposed to surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperatures of the surface and its surroundings differ, the total emissivity and absorptivity of the surface are often found to be unequal; but because the absorptivity is substantially independent of irradiation density, the monochromatic emissivity and absorptivity of surfaces are equal for all practical purposes. The difference between total emissivity and absorptivity depends on the variation of ελ with wavelength and on the difference between the temperature of the surface and the effective temperature of the surroundings. Consider radiative exchange between a real surface of area A1 at temperature T1 with black surroundings at temperature T2. The net radiant interchange is given by Q1,2 = A1



∞

λ= 0

or where and since

[ελ(T1, λ)⋅Eb,λ(T1,λ) − αλ(T1,λ)⋅Eb,λ(T2,λ]) dλ Q1,2 = A1(ε1σT41 − α1,2σT42) ε1(T1) =



∞

Eb,λ(T1,λ) ελ(T1,λ)⋅  dλ Eb(T1)

λ= 0

5-19

structure of the surface layer is quite complex. However, a number of generalizations concerning the radiative properties of opaque surfaces are possible. These are summarized in the following discussion. Polished Metals 1. In the infrared region, the magnitude of the monochromatic emissivity ελ is small and is dependent on free-electron contributions. Emissivity is also a function of the ratio of resistivity to wavelength rλ, as depicted in Fig. 5-11. At shorter wavelengths, bound-electron contributions become significant, ελ is larger in magnitude, and it sometimes exhibits a maximum value. In the visible spectrum, common values for ελ are 0.4 to 0.8 and ελ decreases slightly as temperature increases. For 0.7 < λ < 1.5 µm, ελ is approximately independent of temperature. For λ > 8 µm, ελ is approximately proportional to the square root of temperature since ελ-r and r - T. Here the Drude or Hagen-Rubens relation applies, that is, ελ,n ≈ 0.0365rλ , where r has units of ohm-meters and λ is measured in micrometers. 2. Total emittance is substantially proportional to absolute temperature, and at moderate temperatures εn = 0.058TrT , where T is measured in kelvins. 3. The total absorptance of a metal at temperature T1 with respect to radiation from a black or gray source at temperature T2 is equal to the emissivity evaluated at the geometric mean of T1 and T2. Figure 511 gives values of ελ and ελ,n, and their ratio, as a function of the product rT (solid lines). Although Fig. 5-11 is based on free-electron

(5-107a) (5-107b) (5-108)

αλ (T,λ) = ελ(T,λ), α1,2(T1,T2) =



∞

Eb,λ(T2,λ) ελ (T1,λ)⋅  dλ Eb(T2)

λ= 0

(5-109)

For a gray surface ε1 = α1,2 = ελ. A selective surface is one for which ελ(T,λ) exhibits a strong dependence on wavelength. If the wavelength dependence is monotonic, it follows from Eqs. (5-107) to (5109) that ε1 and α1,2 can differ markedly when T1 and T2 are widely separated. For example, in solar energy applications, the nominal temperature of the earth is T1 = 294 K, and the sun may be represented as a blackbody with radiation temperature T2 = 5800 K. For these temperature conditions, a white paint can exhibit ε1 = 0.9 and α1,2 = 0.1 to 0.2. In contrast, a thin layer of copper oxide on bright aluminum can exhibit ε1 as low as 0.12 and α1,2 greater than 0.9. The effect of radiation source temperature on low-temperature absorptivity for a number of representative materials is shown in Fig. 5-10. Polished aluminum (curve 15) and anodized (surface-oxidized) aluminum (curve 13) are representative of metals and nonmetals, respectively. Figure 5-10 thus demonstrates the generalization that metals and nonmetals respond in opposite directions with regard to changes in the radiation source temperature. Since the effective solar temperature is 5800 K (10,440°R), the extreme right-hand side of Fig. 5-10 provides surface absorptivity data relevant to solar energy applications. The dependence of emittance and absorptance on the real and imaginary components of the refractive index and on the geometric *In the literature, emittance and emissivity are often used interchangeably. NIST (the National Institute of Standards and Technology) recommends use of the suffix -ivity for pure materials with optically smooth surfaces, and -ance for rough and contaminated surfaces. Most real engineering materials fall into the latter category.

Variation of absorptivity with temperature of radiation source. (1) Slate composition roofing. (2) Linoleum, red brown. (3) Asbestos slate. (4) Soft rubber, gray. (5) Concrete. (6) Porcelain. (7) Vitreous enamel, white. (8) Red brick. (9) Cork. (10) White dutch tile. (11) White chamotte. (12) MgO, evaporated. (13) Anodized aluminum. (14) Aluminum paint. (15) Polished aluminum. (16) Graphite. The two dashed lines bound the limits of data on gray paving brick, asbestos paper, wood, various cloths, plaster of paris, lithopone, and paper. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1).

FIG. 5-10

5-20

HEAT AND MASS TRANSFER

Hemispherical emittance εh and the ratio of hemispherical to normal emittance εh/εn for a semi-infinite absorbing-scattering medium.

FIG. 5-12

Hemispherical and normal emissivities of metals and their ratio. Dashed lines: monochromatic (spectral) values versus r/λ. Solid lines: total values versus rT. To convert ohm-centimeter-kelvins to ohm-meter-kelvins, multiply by 10−2. FIG. 5-11

contributions to emissivity in the far infrared, the relations for total emissivity are remarkably good even at high temperatures. Unless extraordinary efforts are taken to prevent oxidation, a metallic surface may exhibit an emittance or absorptance which may be several times that of a polished specimen. For example, the emittance of iron and steel depends strongly on the degree of oxidation and roughness. Clean iron and steel surfaces have an emittance from 0.05 to 0.45 at ambient temperatures and 0.4 to 0.7 at high temperatures. Oxidized and/or roughened iron and steel surfaces have values of emittance ranging from 0.6 to 0.95 at low temperatures to 0.9 to 0.95 at high temperatures. Refractory Materials For refractory materials, the dependence of emittance and absorptance on grain size and impurity concentrations is quite important. 1. Most refractory materials are characterized by 0.8 < ελ < 1.0 for the wavelength region 2 < λ < 4 µm. The monochromatic emissivity ελ decreases rapidly toward shorter wavelengths for materials that are white in the visible range but demonstrates high values for black materials such as FeO and Cr2O3. Small concentrations of FeO and Cr2O3, or other colored oxides, can cause marked increases in the emittance of materials that are normally white. The sensitivity of the emittance of refractory oxides to small additions of absorbing materials is demonstrated by the results of calculations presented in Fig. 5-12. Figure 5-12 shows the emittance of a semi-infinite absorbing-scattering medium as a function of its albedo ω ≡ KS(Ka + KS), where Ka and KS are the scatter and absorption coefficients, respectively. These results are relevant to the radiative properties of fibrous materials, paints, oxide coatings, refractory materials, and other particulate media. They demonstrate that over the relatively small range 1 − ω = 0.005 to 0.1, the hemispherical emittance εh increases from approximately 0.15 to 1.0. For refractory materials, ελ varies little with temperature, with the exception of some white oxides which at high temperatures become good emitters in the visible spectrum as a consequence of the induced electronic transitions. 2. For refractory materials at ambient temperatures, the total emittance ε is generally high (0.7 to 1.0). Total refractory emittance decreases with increasing temperature, such that a temperature increase from 1000 to 1570°C may result in a 20 to 30 percent reduction in ε. 3. Emittance and absorptance increase with increase in grain size over a grain size range of 1 to 200 µm. 4. The ratio εhεn of hemispherical to normal emissivity of polished surfaces varies with refractive index n; e.g., the ratio decreases from a value of 1.0 when n = 1.0 to a value of 0.93 when n = 1.5 (common glass) and increases back to 0.96 at n = 3.0. 5. As shown in Fig. 5-12, for a surface composed of particulate

matter which scatters isotropically, the ratio εhεn varies from 1.0 when ω < 0.1 to about 0.8 when ω = 0.999. 6. The total absorptance exhibits a decrease with an increase in temperature of the radiation source similar to the decrease in emittance with an increase in the emitter temperature. Figure 5-10 shows a regular variation of α1,2 with T2. When T2 is not very different from T1, α1,2 = ε1(T2T1)m. It may be shown that Eq. (5-107b) is then approximated by Q1,2 = (1 + m4)εav A1 σ(T41 − T42)

(5-110)

where εav is evaluated at the arithmetic mean of T1 and T2. For metals m ≈ 0.5 while for nonmetals m is small and negative. Table 5-4 illustrates values of emittance for materials encountered in engineering practice. It is based on a critical evaluation of early emissivity data. Table 5-4 demonstrates the wide variation possible in the emissivity of a particular material due to variations in surface roughness and thermal pretreatment. With few exceptions the data in Table 5-4 refer to emittances εn normal to the surface. The hemispherical emittance εh is usually slightly smaller, as demonstrated by the ratio εhεn depicted in Fig. 5-12. More recent data support the range of emittance values given in Table 5-4 and their dependence on surface conditions. An extensive compilation is provided by Goldsmith, Waterman, and Hirschorn (Thermophysical Properties of Matter, Purdue University, Touloukian, ed., Plenum, 1970–1979). For opaque materials the reflectance ρ is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials are specular reflectors. In contrast, the intensity of the radiation reflected from a perfectly diffuse or Lambert surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractory materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculation purposes, is achieved by assuming that the total reflectance is the sum of diffuse and specular components ρD and ρS, as discussed in a subsequent section. VIEW FACTORS AND DIRECT EXCHANGE AREAS Consider radiative interchange between two finite black surface area elements A1 and A2 separated by a transparent medium. Since they are black, the surfaces emit isotropically and totally absorb all incident radiant energy. It is desired to compute the fraction of radiant energy, per unit emissive power E1, leaving A1 in all directions which is intercepted and absorbed by A2. The required quantity is defined as the direct view factor and is assigned the notation F1,2. Since the net radiant energy interchange Q1,2 ≡ A1F1,2E1 − A2F2,1E2 between surfaces A1 and A2 must be zero when their temperatures are equal, it follows

HEAT TRANSFER BY RADIATION TABLE 5-4

5-21

Normal Total Emissivity of Various Surfaces A. Metals and Their Oxides Surface

Aluminum Highly polished plate, 98.3% pure Polished plate Rough plate Oxidized at 1110°F Aluminum-surfaced roofing Calorized surfaces, heated at 1110°F. Copper Steel Brass Highly polished: 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Hard rolled, polished: But direction of polishing visible But somewhat attacked But traces of stearin from polish left on Polished Rolled plate, natural surface Rubbed with coarse emery Dull plate Oxidized by heating at 1110°F Chromium; see Nickel Alloys for Ni-Cr steels Copper Carefully polished electrolytic copper Commercial, emeried, polished, but pits remaining Commercial, scraped shiny but not mirrorlike Polished Plate, heated long time, covered with thick oxide layer Plate heated at 1110°F Cuprous oxide Molten copper Gold Pure, highly polished Iron and steel Metallic surfaces (or very thin oxide layer): Electrolytic iron, highly polished Polished iron Iron freshly emeried Cast iron, polished Wrought iron, highly polished Cast iron, newly turned Polished steel casting Ground sheet steel Smooth sheet iron Cast iron, turned on lathe Oxidized surfaces: Iron plate, pickled, then rusted red Completely rusted Rolled sheet steel Oxidized iron Cast iron, oxidized at 1100°F Steel, oxidized at 1100°F Smooth oxidized electrolytic iron Iron oxide Rough ingot iron

t, °F*

Emissivity*

440–1070 73 78 390–1110 100

0.039–0.057 0.040 0.055 0.11–0.19 0.216

390–1110 390–1110

0.18–0.19 0.52–0.57

476–674 494–710 530

0.028–0.031 0.033–0.037 0.030

70 73 75 100–600 72 72 120–660 390–1110 100–1000

0.038 0.043 0.053 0.096 0.06 0.20 0.22 0.61–0.59 0.08–0.26

176

0.018

66

0.030

72 242

0.072 0.023

77 390–1110 1470–2010 1970–2330

0.78 0.57 0.66–0.54 0.16–0.13

440–1160

0.018–0.035

350–440 800–1880 68 392 100–480 72 1420–1900 1720–2010 1650–1900 1620–1810

0.052–0.064 0.144–0.377 0.242 0.21 0.28 0.435 0.52–0.56 0.55–0.61 0.55–0.60 0.60–0.70

68 67 70 212 390–1110 390–1110 260–980 930–2190 1700–2040

0.612 0.685 0.657 0.736 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95

Surface Sheet steel, strong rough oxide layer Dense shiny oxide layer Cast plate: Smooth Rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough High temperature alloy steels (see Nickel Alloys). Molten metal Cast iron Mild steel Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 390°F. Mercury Molybdenum filament Monel metal, oxidized at 1110°F Nickel Electroplated on polished iron, then polished Technically pure (98.9% Ni, + Mn), polished Electroplated on pickled iron, not polished Wire Plate, oxidized by heating at 1110°F Nickel oxide Nickel alloys Chromnickel Nickelin (18–32 Ni; 55–68 Cu; 20 Zn), gray oxidized KA-2S alloy steel (8% Ni; 18% Cr), light silvery, rough, brown, after heating After 42 hr. heating at 980°F. NCT-3 alloy (20% Ni; 25% Cr.), brown, splotched, oxidized from service NCT-6 alloy (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Steel, see Iron. Tantalum filament Tin—bright tinned iron sheet Tungsten Filament, aged Filament Zinc Commercial, 99.1% pure, polished Oxidized by heating at 750°F. Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized

t, °F* 75 75 73 73 100–480 70–680 100–700

2370–2550 2910–3270 260–440 75 390 32–212 1340–4700 390–1110 74

Emissivity* 0.80 0.82 0.80 0.82 0.95 0.94 0.94–0.97

0.29 0.28 0.057–0.075 0.281 0.63 0.09–0.12 0.096–0.292 0.41–0.46 0.045

440–710

0.07–0.087

68 368–1844 390–1110 1200–2290

0.11 0.096–0.186 0.37–0.48 0.59–0.86

125–1894

0.64–0.76

70

0.262

420–914 420–980

0.44–0.36 0.62–0.73

420–980

0.90–0.97

520–1045

0.89–0.82

440–1160 1700–2960 80–2240 440–2510

0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182

440–1160 100–700

0.0198–0.0324 0.0221–0.0312

2420–5430 76

0.194–0.31 0.043 and 0.064

80–6000 6000

0.032–0.35 0.39

440–620 750 82 75

0.045–0.053 0.11 0.228 0.276

260–1160

0.81–0.79

1900–2560 206–520 209–362

0.526 0.952 0.959–0.947

B. Refractories, Building Materials, Paints, and Miscellaneous Asbestos Board Paper Brick Red, rough, but no gross irregularities Silica, unglazed, rough Silica, glazed, rough Grog brick, glazed See Refractory Materials below.

74 100–700 70 1832 2012 2012

0.96 0.93–0.945 0.93 0.80 0.85 0.75

Carbon T-carbon (Gebr. Siemens) 0.9% ash (this started with emissivity at 260°F. of 0.72, but on heating changed to values given) Carbon filament Candle soot Lampblack-waterglass coating

5-22

HEAT AND MASS TRANSFER

TABLE 5-4

Normal Total Emissivity of Various Surfaces (Concluded) B. Refractories, Building Materials, Paints, and Miscellaneous Surface

Same Thin layer on iron plate Thick coat Lampblack, 0.003 in. or thicker Enamel, white fused, on iron Glass, smooth Gypsum, 0.02 in. thick on smooth or blackened plate Marble, light gray, polished Oak, planed Oil layers on polished nickel (lube oil) Polished surface, alone +0.001-in. oil +0.002-in. oil +0.005-in. oil Infinitely thick oil layer Oil layers on aluminum foil (linseed oil) Al foil +1 coat oil +2 coats oil Paints, lacquers, varnishes Snowhite enamel varnish or rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black lacquer Flat black lacquer White lacquer

t, °F*

Emissivity*

260–440 69 68 100–700 66 72

0.957–0.952 0.927 0.967 0.945 0.897 0.937

70 72 70 68

0.903 0.931 0.895 0.045 0.27 0.46 0.72 0.82

212 212 212

0.087† 0.561 0.574

Surface Oil paints, sixteen different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface 26% Al, 27% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating to 620°F. Paper, thin Pasted on tinned iron plate On rough iron plate On black lacquered plate Plaster, rough lime Porcelain, glazed Quartz, rough, fused Refractory materials, 40 different poor radiators

t, °F*

Emissivity*

212

0.92–0.96

212

0.52

212

0.3

212 70 300–600

0.27–0.67 0.39 0.35

66 66 66 50–190 72 70 1110–1830

0.924 0.929 0.944 0.91 0.924 0.932



good radiators 73 76 70 170–295 100–200 100–200 100–200

0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.80–0.95

Roofing paper Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Serpentine, polished Water

69 74 76 74 32–212

0.65 – 0.75 0.70 0.80 – 0.85 0.85 – 0.90 0.91

} }{



0.945 0.859 0.900 0.95–0.963

*When two temperatures and two emissivities are given, they correspond, first to first and second to second, and linear interpolation is permissible. °C = (°F − 32)/1.8. †Although this value is probably high, it is given for comparison with the data by the same investigator to show the effect of oil layers. See Aluminum, Part A of this table.

thermodynamically that A1F1,2 = A2F2,1. The product of area and view factor ⎯s⎯1⎯s2 ≡ A1F1,2, which has the dimensions of area, is termed the direct surface-to-surface exchange area for finite black surfaces. Clearly, direct exchange areas are symmetric with respect to their subscripts, that is, ⎯s⎯i ⎯sj = ⎯s⎯j ⎯si, but view factors are not symmetric unless the associated surface areas are equal. This property is referred to as the symmetry or reciprocity⎯⎯relation for direct exchange areas. The ⎯⎯ shorthand notation ⎯s⎯1⎯s2 ≡ 12 = 21 for direct exchange areas is often found useful in mathematical developments. Equation (5-101) may also be restated as cosφi cosφj ∂2 ⎯s⎯i ⎯sj  =  (5-111) πr2 ∂Ai ∂Aj which leads directly to the required definition of the direct exchange area as a double surface integral —— cosφi cos φj s⎯⎯i ⎯sj =  dAj dAi (5-112) πr2 A A j

i

All terms in Eq. (5-112) have been previously defined. Suppose now that Eq. (5-112) is integrated over the entire confining surface of an enclosure which has been subdivided into M finite area elements. Each of the M surface zones must then satisfy certain conservation relations involving all the direct exchange areas in the enclosure M

s⎯⎯s⎯ = A

j=1 i j

i

for 1 ≤ i ≤ M

(5-113a)

for 1 ≤ i ≤ M

(5-113b)

or in terms of view factors M

F

j=1

i,j

=1

Contour integration is commonly used to simplify the evaluation of Eq. (5-112) for specific geometries; see Modest (op. cit., Chap. 4)

or Siegel and Howell (op. cit., Chap. 5). The formulas for two particularly useful view factors involving perpendicular rectangles of area xz and yz with common edge z and equal parallel rectangles of area xy and distance of separation z are given for perpendicular rectangles with common dimension z 1  X +Y

1 1 −1 (πX)FX,Y = X tan−1  + Y tan−1  −  X2 + Y2 tan X Y 1 4



(1 + X2)(1 + Y2) 1+X +Y



X2(1 + X2 + Y2) (1 + X )(X + Y )

+  ln   2 2 2 2 2

 2

X

2

2

Y2(1 + X2 + Y2)  (1 + Y2)(X2 + Y2)


Y2

(5-114a) and for parallel rectangles, separated by distance z,





(1 + X2)(1 + Y2) πXY  FX,Y = ln  1 + X2 + Y2 2





12

X + X1  + Y2 tan−1 2 1  +Y

Y −1 −1 + Y1  + X2 tan−1 2 − X tan X − Y tan Y 1  +X (5-114b) In Eqs. (5-114) X and Y are normalized whereby X = x/z and Y = y/z and the corresponding dimensional direct surface areas are given by ⎯s⎯⎯s = xzF and s⎯⎯⎯s = xyF , respectively. x y X,Y x y X,Y The exchange area between any two area elements of a sphere is independent of their relative shape and position and is simply the product of the areas, divided by the area of the entire sphere; i.e., any spot on a sphere has equal views of all other spots. Figure 5-13, curves 1 through 4, shows view factors for selected parallel opposed disks, squares, and 2:1 rectangles and parallel rectangles with one infinite dimension as a function of the ratio of the

HEAT TRANSFER BY RADIATION

FIG. 5-13

5-23

Radiation between parallel planes, directly opposed.

smaller diameter or side to the distance of separation. Curves 2 through 4 of Fig. 5-13, for opposed rectangles, can be computed with Eq. (5-114b). The view factors for two finite coaxial coextensive cylinders of radii r ≤ R and height L are shown in Fig. 5-14. The direct view factors for an infinite plane parallel to a system of rows of parallel tubes (see Fig. 5-16) are given as curves 1 and 3 of Fig. 5-15. The view factors for this two-dimensional geometry can be readily calculated by using the crossed-strings method. The crossed-strings method, due to Hottel (Radiative Transfer, McGraw-Hill, New York, 1967), is stated as follows: “The exchange area for two-dimensional surfaces, A1 and A2, per unit length (in the infinite dimension) is given by the sum of the lengths of crossed strings from the ends of A1 to the ends of A2 less the sum of the uncrossed strings from and to the same points all divided by 2.” The strings must be drawn so that all the flux from one surface to the other must cross each of a pair of crossed strings and neither of the pair of uncrossed strings. If one surface can see the other around both sides of an obstruction, two more pairs of strings are involved. The calculation procedure is demonstrated by evaluation of the tube-to-tube view factor for one row of a tube bank, as illustrated in Example 7. Example 7: The Crossed-Strings Method Figure 5-16 depicts the transverse cross section of two infinitely long, parallel circular tubes of diameter D and center-to-center distance of separation C. Use the crossed-strings method to formulate the tube-to-tube direct exchange area and view factor s⎯⎯t s⎯t and Ft,t, respectively. Solution: The circumferential area of each tube is At = πD per unit length in the infinite dimension for this two-dimensional geometry. Application of the crossed-strings procedure then yields simply

(a)

Distribution of radiation to rows of tubes irradiated from one side. Dashed lines: direct view factor F from plane to tubes. Solid lines: total view factor  F for black tubes backed by a refractory surface.

FIG. 5-15

2(EFGH − HJ) 2 ⎯s⎯s⎯ =  = D[sin−1(1R) + R  − 1 − R] t t 2 and

2 Ft,t = ⎯s⎯t ⎯stAt = [sin−1(1R) + R  − 1 − R]π

where EFGH and HJ = C are the indicated line segments and R ≡ CD ≥ 1. Curve 1 of Fig. 5-15, denoted by Fp,t, is a function of Ft,t, that is, Fp,t = (π/R)(21 − Ft,t).

The Yamauti principle [Yamauti, Res. Electrotech Lab. (Tokyo), 148 (1924); 194 (1927); 250 (1929)] is stated as follows; The exchange areas between two pairs of surfaces are equal when there is a one-to-one correspondence for all sets of symmetrically positioned pairs of differential elements in the two surface combinations. Figure 5-17 illustrates the Yamauti principle applied to surfaces in perpendicular planes having a common edge. With reference to Fig. 5-17, the Yamauti principle states ⎯⎯⎯⎯⎯⎯ that the diagonally opposed exchange areas are equal, that is, (1)(4) = ⎯⎯⎯⎯⎯⎯ (2)(3). Figure 5-17 also shows a more complex geometric construction for displaced cylinders for which the Yamauti principle also applies. Collectively the three terms reciprocity or symmetry principle, conservation

(b)

View factors for a system of two concentric coaxial cylinders of equal length. (a) Inner surface of outer cylinder to inner cylinder. (b) Inner surface of outer cylinder to itself.

FIG. 5-14

5-24

HEAT AND MASS TRANSFER it can be shown that the net radiative flux Qi,j between all such surface zone pairs Ai and Aj, making full allowance for all multiple reflections, may be computed from Qi,j = σ(AiF i,jT4j − AjF j,iT4i )

FIG. 5-16

Direct exchange between parallel circular tubes.

principle, and Yamauti principle are referred to as view factor or exchange area algebra. Example 8: Illustration of Exchange Area Algebra Figure 5-17 shows a graphical construction depicting four perpendicular opposed rectangles with a common edge. Numerically evaluate the direct exchange areas and⎯⎯view ⎯⎯⎯⎯ factors for the diagonally opposed (shaded) rectangles A1 and A4, that is, (1)(4), ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ as well as (1)(3 + 4). The dimensions of the rectangular construction are shown in Fig. 5-17 as x = 3, y = 2, and z = 1. Solution: Using shorthand notation for direct exchange areas, the conservation principle yields ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ (1 + 2)(3 + 4) = (1 + 2)(3) + (1 + 2)(4) = (1)(3) + (2)(3) + (1)(4) + (2)(4) ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ Now by the Yamauti principle we have (1)(4) ≡ (2)(3). Combination of these ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ two relations yields the first result (1)(4) = [(1 + 2)(3 + 4) − (1)(3) − (2)(4)]2. ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ For (1)(3 + 4), again conservation yields (1)(3 + 4) = (1)(3) + (1)(4), and substi ⎯⎯⎯⎯⎯⎯ tution of the expression for (1)(4) just obtained yields the second result, that is, ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯⎯⎯⎯ (1)(3 + 4) = [(1 + 2)(3 + 4) + (1)(3) − (2)(4)]2.0. All three required direct exchange areas in these two relations are readily evaluated from Eq. (5-114a). Moreover, these equations apply to opposed parallel rectangles as well as rectangles with a common edge oriented at any angle. Numerically it follows from ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ Eq. (5-114a) that for X = 13, Y = 23, and z = 3 that (1 + 2)(3 + 4) = 0.95990; for X = ⎯⎯⎯⎯⎯⎯ 1, Y = 2, and z = 1 that (1)(3) = 0.23285; and for X = 1⁄2, Y = 1, and z = 2 that ⎯⎯⎯⎯⎯⎯ ⎯⎯⎯ (2)(4) = 0.585747. Since A1 = 1.0, this leads to s1s4 = F1,4 = (0.95990 − 0.23285 − ⎯⎯⎯ 0.584747)2.0 = 0.07115 and s1s3+4 = F1,3+4 = (0.95990 + 0.23285 − 0.584747) 2.0 = 0.30400. Many literature sources document closed-form algebraic expressions for view factors. Particularly comprehensive references include the compendia by Modest (op. cit., App. D) and Siegel and Howell (op. cit., App. C). The appendices for both of these textbooks also provide a wealth of resource information for radiative transfer. Appendix F of Modest, e.g., references an extensive listing of Fortan computer codes for a variety of radiation calculations which include view factors. These codes are archived in the dedicated Internet web site maintained by the publisher. The textbook by Siegel and Howell also includes an extensive database of view factors archived on a CD-ROM and includes a reference to an author-maintained Internet web site. Other historical sources for view factors include Hottel and Sarofim (op. cit., Chap. 2) and Hamilton and Morgan (NACA-TN 2836, December 1952). RADIATIVE EXCHANGE IN ENCLOSURES—THE ZONE METHOD Total Exchange Areas When an enclosure contains reflective surface zones, allowance must be made for not only the radiant energy transferred directly between any two zones but also the additional transfer attendant to however many multiple reflections which occur among the intervening reflective surfaces. Under such circumstances,

FIG. 5-17

Illustration of the Yamauti principle.

(5-115)

Here, F i,j is defined as the ⎯⎯⎯total surface-to-surface view factor from Ai to Aj, and the quantity Si Sj ≡ AiF i,j is defined as the corresponding total surface-to-surface exchange area. In analogy with the direct exchange areas, the total surface-to-surface exchange areas ⎯⎯⎯ are⎯⎯also ⎯ symmetric and thus obey reciprocity, that is, AiF i,j = AjF j,i or Si Sj = Sj Si. When applied to an enclosure, total exchange areas and view factors also must satisfy appropriate conservation relations. Total exchange areas are functions of the geometry and radiative properties of the entire enclosure. They are also independent of temperature if all surfaces and any radiatively participating media are gray. The following subsection presents a general matrix method for the explicit evaluation of total exchange areas from direct exchange areas and other enclosure parameters. In what follows, conventional matrix notation is strictly employed as in A = [ai,j] wherein the scalar subscripts always denote the row and column indices, respectively, and all matrix entities defined here are denoted by boldface notation. Section 3 of this handbook, “Mathematics,” provides an especially convenient reference for introductory matrix algebra and matrix computations. General Matrix Formulation The zone method is perhaps the simplest numerical quadrature of the governing integral equations for radiative transfer. It may be derived from first principles by starting with the equation of transfer for radiation intensity. The zone method always conserves radiant energy since the spatial discretization utilizes macroscopic energy balances involving spatially averaged radiative flux quantities. Because large sets of linear algebraic equations can arise in this process, matrix algebra provides the most compact notation and the most expeditious methods of solution. The mathematical approach presented here is a matrix generalization of the original (scalar) development of the zone method due to Hottel and Sarofim (op. cit.). The present matrix development is abstracted from that introduced by Noble [Noble, J. J., Int. J. Heat Mass Transfer, 18, 261–269 (1975)]. Consider an arbitrary three-dimensional enclosure of total volume V and surface area A which confines an absorbing-emitting medium (gas). Let the enclosure be subdivided (zoned) into M finite surface area and N finite volume elements, each small enough that all such zones are substantially isothermal. The mathematical development in this section is restricted by the following conditions and/or assumptions: 1. The gas temperatures are given a priori. 2. Allowance is made for gas-to-surface radiative transfer. 3. Radiative transfer with respect to the confined gas is either monochromatic or gray. The gray gas absorption coefficient is denoted here by K(m−1). In subsequent sections the monochromatic absorption coefficient is denoted by Kλ(λ). 4. All surface emissivities are assumed to be gray and thus independent of temperature. 5. Surface emission and reflection are isotropic or diffuse. 6. The gas does not scatter. Noble (op. cit.) has extended the present matrix methodology to the case where the gaseous absorbing-emitting medium also scatters isotropically. In matrix notation the blackbody emissive powers for all surface and volume zones comprising the zoned enclosure are designated as 4 E = [Ei] = [σT4i ], an M × 1 vector, and Eg = [Eg,i] = [σTg,i ], an N × 1 vector, respectively. Moreover, all surface zones are characterized by three M × M diagonal matrices for zone area AI = [Ai⋅δi,j], diffuse emissivity εI = [εi⋅δi,j], and diffuse reflectivity, ρI = [(1 − εi)⋅δi,j], respectively . Here δi,j is the Kronecker delta (that is, δi,j = 1 for i = j and δi,j = 0 for i ≠ j). Two arrays of direct exchange areas are now defined; i.e., the matrix s⎯s⎯ = [s⎯⎯i ⎯sj] is the M × M array of direct surface-to-surface exchange ⎯ = [s⎯⎯⎯g ] is the M × N array of direct gas-toareas, and the matrix s⎯g i j ⎯ are surface exchange areas. Here the scalar elements of s⎯s⎯ and s⎯g computed from the integrals — — −Kr e 2 cos φi cos φj dAj dAi s⎯⎯i ⎯sj = (5-116a) πr A A i

j

HEAT TRANSFER BY RADIATION ⎯s⎯⎯g = i j

— Ai

™ Vj

e−Kr K 2 cos φi dVj dAi πr

(5-116b)

Equation (5-116a) is a generalization of Eq. (5-112) for the case K ≠ 0 while s⎯⎯i ⎯gj is a new quantity, which arises only for the case K ≠ 0. Matrix characterization of the radiative energy balance at each surface zone is facilitated via definition of three M × 1 vectors; the radiative surface fluxes Q = [Qi], with units of watts; and the vectors H = [Hi] and W = [Wi] both having units of W/m2. The arrays H and W define the incident and leaving flux densities, respectively, at each surface zone. The variable W is also referred to in the literature as the radiosity or exitance. Since W ∫ eI◊E + rI◊H, the radiative flux at each surface zone is also defined in terms of E, H, and W by three equivalent matrix relations, namely, Q = AI◊[W - H] = eAI◊[E - H] = rI-1◊eAI◊[E - W]

(5-117)

where the third form is valid if and only if the matrix inverse ρI exists. Two other ancillary matrix expressions are eAI◊E = rI◊Q + eAI◊W and AI◊H = sæsæ◊W + sægæ◊E (5-117a,b) -1

g

which lead to eI◊E = [I - rI◊AI−1◊sæsæ ]◊W - rI◊AI-1 sægæ◊Eg.

(5-117c)

The latter relation is especially useful in radiation pyrometry where true wall temperatures must be computed from wall radiosities. Explicit Matrix Solution for Total Exchange Areas For gray or monochromatic transfer, the primary working relation for zoning calculations via the matrix method is ææ ææ Q = eI◊AI◊E - SS◊E - SG◊Eg [M × 1] (5-118) Equation (5-118) makes full allowance for multiple reflections in an enclosure of any degree of complexity. To apply Eq. (5-118) for design or simulation purposes, the gas temperatures must be known and surface boundary conditions must be specified for each and every surface zone in the form of either Ei or Qi. In application of Eq. (5-118), physically impossible values of Ei may well result if physically unrealistic values of Qi are specified. ææ ææ In Eq. (5-118), SS and SG are defined as the required arrays of total surface-to-surface exchange areas and total gas-to-surface exchange areas, respectively. The matrices for total exchange areas are calculated explicitly from the corresponding arrays of direct exchange areas and the other enclosure parameters by the following matrix formulas: ææ Surface-to-surface exchange SS = eI◊AI◊R◊sæsæ◊eI [M × M] (5-118a) Gas-to-surface exchange

ææ SG = eI◊AI◊R◊sægæ

[M × N]

(5-118b)

where in Eqs. (5-118), R is the explicit inverse reflectivity matrix, defined as R = [AI - s⎯s⎯ ρI]−1 [M × M] (5-118c) While the R matrix is generally not symmetric, the matrix product ρI◊R is always symmetric. This fact proves useful for error checking. The most computationally significant aspect of the matrix method is that the inverse reflectivity matrix R always exists for any physically meaningful enclosure problem. More precisely, R always exists provided that K ≠ 0. For a transparent medium, R exists provided that there formally exists at least one surface zone Ai such that εi ≠ 0. An important computational corollary of this statement for transparent media is that the matrix [AI − sæsæ ] is always singular and demonstrates matrix rank M − 1 (Noble, op. cit.). ⎯⎯ ⎯⎯ Finally, the four matrix arrays s⎯s⎯, g⎯s⎯, SS, and SG of direct and total exchange areas must satisfy matrix conservation relations, i.e., Direct exchange areas AI◊1M = s⎯s⎯◊1M + s⎯g⎯⋅1N (5-119a) ⎯⎯ ⎯⎯ Total exchange areas eI◊AI◊1M = SS◊1M + SG◊1N (5-119b) Here 1M is an M × 1 column vector all of whose elements are unity. If eI = I or equivalently, ρI = 0, Eq. (5-118c) reduces to R = AI−1 with

5-25

⎯⎯ the result degenerate to simply SS = ⎯ ⎯that⎯ ⎯Eqs. (5-118a) and (5-118b)⎯⎯ ⎯s s⎯ and S ⎯ ⎯G = sg. Further, while the array SS is always symmetric, the array SG is generally not square. For purposes of digital computation, it is good practice to enter all data for direct exchange surface-to-surface areas s⎯s⎯ with a precision of at least five significant figures. This need arises because all the scalar elements of s⎯g⎯ can be calculated arithmetically from appropriate direct surface-to-surface exchange areas by using view factor algebra rather than via the definition of the defining integral, Eq. (5-116b). This process often involves small arithmetic differences between two numbers of nearly equal magnitude, and numerical significance is easily lost. Computer implementation of the matrix method proves straightforward, given the availability of modern software applications. In particular, several especially user-friendly GUI mathematical utilities are available that perform matrix computations using essentially algebraic notation. Many simple zoning problems may be solved with spreadsheets. For large M and N, the matrix method can involve management of a large amount of data. Error checks based on symmetry and conservation by calculation of the row sums of the four arrays of direct and total exchange areas then prove indispensable. Zone Methodology and Conventions For a transparent medium, no more than Σ = M(M − 1)2 of the M2 elements of the sæsæ array are unique. Further, surface zones are characterized into two generic types. Source-sink zones are defined as those for which temperature is specified and whose radiative flux Qi is to be determined. For flux zones, conversely, these conditions are reversed. When both types of zone are present in an enclosure, Eq. (5-118) may be partitioned to produce a more efficient computational algorithm. Let M = Ms + Mf represent the total number of surface zones where Ms is the number of source-sink zones and Mf is the number of flux zones. The flux zones are the last to be numbered. Equation (5-118) is then partitioned as follows: ææ ææ ææ Q1 E1 SG1 SS1,1 SS1,2 E1 εAI1,1 0 ææ  =  − ææ − æ æ ⋅Eg SG2 SS2,1 SS2,2 E2 0 εAI2,2 Q2 E2

 

  

   

(5-120) ⎯ Here the⎯ ⎯ dimensions of the submatrices εAI1,1 and SS1,1 are both Ms × Ms and SG1 has dimensions Ms × N. Partition algebra then yields the following two matrix equations for Q1, the Ms × 1 vector of unknown source-sink fluxes and E2, the Mf × 1 vector of unknown emissive powers for the flux zones, i.e., ⎯⎯ ⎯⎯ ⎯⎯ E2 = [εAI2,2 − SS2,2]−1⋅[Q2 + SS2,1◊E1 + SG2◊Eg] (5-120a) ⎯⎯ ⎯⎯ ⎯⎯ Q1 = εAI1,1◊E1 − SS1,1◊E1 − SS1,2◊E2 − SG1◊Eg (5-120b) The inverse matrix in Eq. (5-120a) formally does not exist if there is at least one flux zone such that εi = 0. However, well-behaved results are usually obtained with Eq. (5-120a) by utilizing a notional zero, say, εi ≈ 10−5, to simulate εi = 0. Computationally, E2 is first obtained from Eq. (5-120a) and then substituted into either Eq. (5-120b) or Eq. (5-118). Surface zones need not be contiguous. For example, in a symmetric enclosure, zones on opposite sides of the plane of symmetry may be “lumped” into a single zone for computational purposes. Lumping nonsymmetrical zones is also possible as long as the zone temperatures and emissivities are equal. An adiabatic refractory surface of area Ar and emissivity εr, for which Qr = 0, proves quite important in practice. A nearly radiatively adiabatic refractory surface occurs when differences between internal conduction and convection and external heat losses through the refractory wall are small compared with the magnitude of the incident and leaving radiation fluxes. For any surface zone, the radiant flux is given by Q = A(W − H) = εA(E − H) and Q = εAρ(E − W) (if ρ ≠ 0). These equations then lead to the result that if Qr = 0, Er = Hr = Wr for all 0 ≤ εr ≤ 1. Sufficient conditions for modeling an adiabatic refractory zone are thus either ⎯⎯⎯ to put εr = 0 or to specify directly that Qr = 0 with εr ≠ 0. If εr = 0, SrSj = 0 for all 1 ≤ j ≤ M which leads directly by definition to Qr = 0. For εr = 0, the refractory emissive power Er never enters the zoning calculations. For the special case of K ⎯⎯⎯= 0 and Mr = 1, a single (lumped) refractory, with Qr = 0 and εr ≠ 0, SrSj ≠ 0 and the refractory emissive power may be calculated from Eq. (5-120a) as a weighted sum of all other known blackbody emissive powers which

5-26

HEAT AND MASS TRANSFER

characterize the enclosure, i.e., Ms

where the following matrix conservation relations must also be satisfied,

⎯⎯⎯

S S ⋅E r j

j

Er =  ⎯⎯⎯

SrSj j = 1 Ms

(5-121)

with j ≠ r

j = 1

Equation (5-121) specifically includes those zones which may not have a direct view of the refractory. When Qr = 0, the refractory surface is said to be in radiative equilibrium with the entire enclosure. Equation (5-121) is indeterminate if εr = 0. If εr = 0, Er does indeed exist and may be evaluated with use of the statement Er = Hr = Wr. It transpires, however, that Er is independent of εr for all 0 ≤ εr ≤ 1. Moreover, since Wr = Hr when Qr = 0, for all 0 ≤ εr ≤ 1, the value specified for εr is irrelevant to radiative transfer in the entire enclosure. In particular it follows that if Qr = 0, then the vectors W, H, and Q for the entire enclosure are also independent of all 0 ≤ εr ≤ 1.0. A surface zone for which εi = 0 is termed a perfect diffuse mirror. A perfect diffuse mirror is thus also an adiabatic surface zone. The matrix method automatically deals with all options for flux and adiabatic refractory surfaces. The Limiting Case of a Transparent Medium For the special case of a transparent medium, K = 0, many practical engineering applications can be modeled with the zone method. These include combustion-fired muffle furnaces and electrical resistance furnaces. ææ When K → 0, sægæ → 0 and SG → 0. Equations (5-118) through (5-119) then reduce to three simple matrix relations ææ Q = εI◊AI◊E − SS◊E (5-122a) ⎯⎯ æ æ SS = εI◊AI◊R◊ss◊εI (5-122b) with again

R ≡ [AI − sæsæ◊ρI]−1

(5-122c)

The radiant surface flux vector Q, as computed from Eq. (5122a), always satisfies the (scalar) conservation condition 1MT ⋅Q = 0 or M

Qi = 0, which is a statement of the overall radiant energy balance.

i=1

The matrix conservation relations also simplify to AI◊1 = sæsæ◊1 M

(5-123a)

M

ææ εI◊AI◊1M = SS◊1M

and

ææ F = AI−1◊SS

G





and in particular

⎯⎯⎯ A1 S1S2 =  1ε1 + (A1A2)(ρ2ε2)

ε1ε2A1 ææ (ε1 + ε2 − 2ε1ε2)A1 /[ε1 + ε2 − ε1ε2] SS = ε1ε2A1 (ε1+ ε2 − 2ε1ε2)A1





and in particular ⎯⎯⎯ A1 S1S2 =  1ε1 + 1ε2 − 1

Case 3

Case 4

A1 A2

A2

A2 A2

F=

0

1

A1 / A2 1 – A1 / A2

Two infinite parallel plates where A1 = A2.

F=

0

1

1

0

(5-127a)

In the limiting case, where A1 has no negative curvature and is completely surrounded by a very much larger surface A2 such that A1 << ⎯⎯⎯ A2, Eq. (5-127a) leads to the even simpler result that S1S2 = ε1⋅A1. 2. Two parallel plates of equal area which are large compared to their distance of separation (infinite parallel plates). Case 2 is a limiting form of case 1 with A1 = A2. Algebraic manipulation then results in

A1

A planar surface A1 completely surrounded by a second surface A2 > A1.



ε1ε2A1A2 ε1ρ2 A21 + ε2ρ1A1A2 ææ /[ε1 ρ2A1 + ε2A2] SS = ε1ε2A1A2 ε2A22 + ε1(ρ2 − ε2)A1A2

A1 A1



Equation (5-127) is of general utility for any two-zone system for which εi ≠ 0. The total exchange areas for the four geometries shown in Fig. 5-18 follow directly from Eqs. (5-126) and (5-127). 1. A planar surface A1 completely surrounded by a second surface A2 > A1. Here F1,1 = 0, F1,2 = 1, and ⎯s⎯1⎯s2 = A1, resulting in

G A2

(5-125b)



Case 2

A2

F ◊1M = εI◊1M



(5-124b)

Case 1

(5-125a)

The Two-Zone Enclosure Figure 5-18 depicts four simple enclosure geometries which are particularly useful for engineering calculations characterized by only two surface zones. For M = 2, the reflectivity matrix R is readily evaluated in closed form since an explicit algebraic inversion formula is available for a 2 × 2 matrix. In this case knowledge of only Σ = 1 direct exchange area is required. Direct evaluation of Eqs. (5-122) then leads to ⎯ ⎯⎯ ⎯⎯⎯ S1S2 ææ ε1A1 − S1S2 ⎯ ⎯ ⎯ ⎯⎯⎯ SS = (5-126) S1S2 ε2A2 − S1S2 where 1 ⎯⎯⎯ S1S2 =  (5-127) ρ1 ρ2 1  +   ⎯s⎯ ⎯s⎯ +  ε1A1 ε2A2 1 2

(5-123b)

And the M × M arrays for all the direct and total view factors can be readily computed from F = AI−1◊sæsæ (5-124a)

F◊1M = 1M and

G

Concentric spheres or infinite cylinders where A1 < A2. Identical to Case 1. F=

0

1

A1 / A2 1 – A1 / A2

A1 G

A1

A2

A speckled enclosure with two surface zones.

F=

A 1 A2 1 (A1 + A2 ) A A 1 2

FIG. 5-18 Four enclosure geometries characterized by two surface zones and one volume zone. (Marks’ Standard Handbook for Mechanical Engineers, McGraw-Hill, New York, 1999, p. 4-73, Table 4.3.5.)

(5-127b)

HEAT TRANSFER BY RADIATION 3. Concentric spheres or cylinders where A2 > A1. Case 3 is mathematically identical to case 1. 4. A speckled enclosure with two surface zones. Here A21 A1A2 A1 A2 1 1 F=  such that sæsæ =  and Eqs. A + A A2 1 2 A1A2 A1 + A2 A1 A2









(5-126) and (5-127) then produce



⎯ ⎯ ⎯⎯⎯ 1 A1F1,2 = A2F2,1 = s1s2 +  1s⎯⎯s⎯ + 1s⎯⎯s⎯ 1 r

⎯⎯⎯ 1 S1S2 =  1(ε1A1) + 1(ε2A2)

(5-127c)

Physically, a two-zone speckled enclosure is characterized by the fact that the view factor from any point on the enclosure surface to the sink zone is identical to that from any other point on the bounding surface. This is only possible when the two zones are “intimately mixed.” The seemingly simplistic concept of a speckled enclosure provides a surprisingly useful default option in engineering calculations when the actual enclosure geometries are quite complex. Multizone Enclosures [M ≥ 3] Again assume K = 0. The major numerical effort involved in implementation of the zone method is the evaluation of the inverse reflection matrix R. For M = 3, explicit closedform algebraic formulas do indeed exist for the nine scalar elements of the inverse of any arbitrary nonsingular matrix. These formulas are so algebraically complex, however, that it proves impractical to present universal closed-form expressions for the total exchange areas, as has been done for the case M = 2. Fortunately, many practical furnace configurations can be idealized with zoning such that only relatively simple hand calculation procedures are required. Here the enclosure is modeled with only M = 3 surface zones, e.g., a single source, a single sink, and a lumped adiabatic refractory zone. This approach is sometimes termed the SSR model. The SSR model assumes that all adiabatic refractory surfaces are perfect diffuse mirrors. To implement the SSR procedure, it is necessary to develop⎯specialized algebraic formulas and to define a third black view factor F i,j with an overbar as follows. ⎯ Refractory Augmented Black View Factors Fi,j Let M = Mr + Mb, where Mb is the number of black surface zones and Mr is the number of adiabatic refractory zones. Assume εr = 0 or ρr = 1 or, equivalently, that all adiabatic refractory surfaces are perfect diffuse mirrors. ⎯ The view factor Fi,j is then defined as the refractory augmented black view factor, i.e., the direct view factor between any two black source-sink zones, Ai and Aj, with full allowance for⎯ reflections from all intervening refractory surfaces. The quantity Fi,j shall be referred to as F-bar, for expediency. Consider the special situation where Mb = 2, with any number of refractory zones Mr ≥ 1. By use of appropriate row and column reduction of the reflectivity matrix R, an especially useful relation can be derived that allows computation of the conventional total exchange area ⎯ ⎯⎯ ⎯ Si Sj from the corresponding refractory augmented black view factor Fi,j

or

1 ⎯⎯⎯ S1S2 =  ρ1 ρ2 1  +  + ⎯ ε1A1 ε2A2 A1F1,2



1 F 1,2 =  ρ1 A1 ρ2 1  +   +  ⎯ ε1 A2 ε2 F1,2





(5-130)

2 r

and if ⎯s⎯1⎯s1 = ⎯s⎯2⎯s2 = 0, Eq. (5-130) further reduces to

with the particular result



zone (Mr = 1) with total area Ar and uniform average temperature Tr, then the direct refractory augmented exchange area for the black zone pairs is given by ⎯⎯⎯ ⎯⎯⎯ si sr⋅srsj ⎯ ⎯ AiFi,j = AjFj,i = s⎯⎯i ⎯sj +  for 1 ≤ i,j ≤ Mb (5-129) Ar − s⎯⎯r⎯sr For the special case Mb = 2 and Mr = 1, Eq. (5-129) then simplifies to

ε1ε2A1A2 ε21A21 ææ SS = ε ε A A ε22 A22 /[ε1A1 + ε2 A2] 1 2 1 2



5-27



(5-128a)

(5-128b)

where εi ≠ 0. Notice that Eq. (5-128a) appears deceptively similar to Eq. (5-127). Collectively, Eqs. (5-128) along with various formulas to ⎯ compute Fi,j (F-bar) are sometimes called the three-zone source/sink/ refractory SSR model. ⎯ The following formulas permit the calculation of Fi,j from requisite direct exchange areas. For the special case where the enclosure is divided into any number of black source-sink zones, Mb ≥ 2, and the remainder of the enclosure is lumped together into a single refractory

⎯ A1A2 − (s⎯⎯1⎯s2)2 1 A1F1,2 = ⎯s⎯1⎯s2 +  =  ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ 1(A1 − s1s2) + 1(A2 − s2s1) A1 + A2 − 2s⎯⎯1⎯s2

(5-131)

which necessitates the evaluation of only one direct exchange area. Let the Mr refractory zones be numbered last. Then the ⎯ Mb × Mb array of refractory augmented direct exchange areas [AiFi,j] is symmetric and satisfies and the conservation relation ⎯ [Ai⋅Fi,j]◊1M = AI◊1M (5-132a) b

with

b

⎯ F ◊1M = 1M

(5-132b) ⎯⎯⎯ ⎯⎯⎯ Temporarily denote S1S2]R as the value of S1S2 computed from Eq. (5-128a) which⎯⎯assumes εr = 0. It remains to demonstrate the relation⎯ ⎯⎯⎯ ship between S1S2]R and the total exchange area S1S2 computed from the matrix method for M = 3 when zone 3 is an adiabatic refractory for which Q3 = 0 and ε3 ≠ 0. Let Θi = (Ei − E2)/(E1 − E2) denote the dimensionless emissive power where E1 > E2 such that Θ1 = 1 and Θ2 = 0. The dimensionless refractory emissive power may then be calculated ⎯ ⎯⎯ ⎯ ⎯⎯ ⎯ ⎯ ⎯ from Eq. (5-121) as Θ3 = S3S1[S3S1 + S3S2], which when substituted ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ into Eq. (5-122a) leads to S1S2]R = S1S2 + S2S3⋅Θ3 = S1S2 + S1S3⋅S3S2  ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ [S3S1 + S3S2]. Thus S1S2]R is clearly the refractory-aided total exchange ⎯⎯⎯ area between zone 1 and zone 2 ⎯and ⎯⎯ not S1S2 as calculated by the matrix method in general. That is, S1S2]R includes not only the radiant energy originating at zone 1 and arriving at zone 2 directly and by reflection from zones 2 and 3, but also radiation originating at zone 1 that is absorbed by zone 3 and then wholly reemitted to zone 2; that is, H3 = W3 = E3. Evaluation of any total view⎯factor F i,j using the requisite refractory augmented black view factor Fi,j obviously requires that the latter be readily available and/or ⎯ capable of calculation. The refractory augmented view factor Fi,j is documented for a few geometrically simple cases and can be calculated or approximated for others. If A1 and A2 are equal parallel disks, squares, or rectangles, ⎯ connected by nonconducting but reradiating refractory surfaces, then Fi,j is given by Fig. 5-13 in curves 5 to 8. Let A1 represent an infinite plane and A2 represent one or two rows of infinite parallel tubes. If the only other⎯surface is an adiabatic refractory surface located behind the tubes, F2,1 is then given by curve 5 or 6 of Fig. 5-15. Experience has shown that the simple SSR model can yield quite useful results for a host of practical engineering applications without resorting to digital computation. The error due to representation of the source and sink by single zones is often small, even if the views of the enclosure from different parts of the same zone are dissimilar, provided the surface emissivities are near unity. The error is also small if the temperature variation of the refractory is small. Any degree of accuracy can, of course, be obtained via the matrix method for arbitrarily large M and N by using a digital computer. From a computational viewpoint, when M ≥ 4, the matrix method must be used. The matrix method must also be used for finer-scale calculations such as more detailed wall temperature and flux density profiles. The Electrical Network Analog At each surface zone the total radiant flux is proportional to the difference between Ei and Wi, as indicated by the equation Qi = (εiAiρi)(Ei − Wi). The net flux between M ⎯⎯⎯ zones i and j is also given by Qi,j = si sj(Wi − Wj), where Qi = Qi,j, for b

b

j=1

all 1 ≤ i ≤ M, is the total heat flux leaving each zone. These relations

5-28

HEAT AND MASS TRANSFER

Er Ar
r r

E1

W1

A1
1 1

11

1r

Wr 12

rr 2r W2 22

E2 A2
2 2

FIG. 5-19 Generalized electrical network analog for a three-zone enclosure. Here A1 and A2 are gray surfaces and Ar is a radiatively adiabatic surface. (Hottel, H. C., and A. F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, p. 91.)

suggest a visual electrical analog in which Ei and Wi are analogous to voltage potentials. The quantities εiAiρi and ⎯s⎯i ⎯sj are analogous to conductances (reciprocal impedances), and Qi or Qi,j is analogous to electric currents. Such an electrical analog has been developed by Oppenheim [Oppenheim, A. K., Trans. ASME, 78, 725–735 (1956)]. Figure 5-19 illustrates a generalized electrical network analogy for a three-zone enclosure consisting of one refractory zone and two gray zones A1 and A2. The potential points Ei and Wi are separated by conductances εiAiρi. The emissive powers E1, E2 represent potential sources or sinks, while W1, W2, and Wr are internal node points. In this construction the nodal point representing each surface is connected to that of every other surface it can see directly. Figure 5-19 can be used ⎯⎯⎯ to formulate the total exchange area S1S2 for the SSR model virtually by inspection. The refractory zone is first characterized by a floating potential such that Er = Wr. Next, the resistance for the parallel “current paths” between the internal nodes W1 and W2 is defined 1 1 by  ⎯ ≡  which is identical to Eq. (5-130). A1F1,2 ⎯s⎯1⎯s2 + 1(1s⎯⎯1⎯sr + 1s⎯⎯2⎯sr) Finally, the overall impedance between the source E1 and the sink E2 is represented simply by three resistors in series and is thus given by ρ1 ρ2 1 1  ⎯⎯⎯ =  +  ⎯ +  ε1A1 ε2A2 S1S2 A1F1,2 or

1 ⎯⎯⎯ S1S2 =  ρ2 1 ρ1  +  + ⎯ ε1A1 ε2A2 A1F1,2

heating source is two refractory-backed, internally fired tube banks. Clearly the overall geometry for even this common furnace configuration is too complex to be modeled in an expeditious manner by anything other than a simple engineering idealization. Thus the furnace shown in Fig. 5-20 is modeled in Example 10, by partitioning the entire enclosure into two subordinate furnace compartments. The approach first defines an imaginary gray plane A2, located on the inward-facing side of the tube assemblies. Second, the total exchange area between the tubes to this equivalent gray plane is calculated, making full allowance for the reflection from the refractory tube backing. The plane-to-tube view factor is then defined to be the emissivity of the required equivalent gray plane whose temperature is further assumed to be that of the tubes. This procedure guarantees continuity of the radiant flux into the interior radiant portion of the furnace arising from a moderately complicated external source. Example 9 demonstrates classical zoning calculations for radiation pyrometry in furnace applications. Example 10 is a classical furnace design calculation via zoning an enclosure with a diathermanous atmosphere and M = 4. The latter calculation can only be addressed with the matrix method. The results of Example 10 demonstrate the relative insensitivity of zoning to M > 3 and the engineering utility of the SSR model. Example 9: Radiation Pyrometry A long tunnel furnace is heated by electrical resistance coils embedded in the ceiling. The stock travels on a floormounted conveyer belt and has an estimated emissivity of 0.7. The sidewalls are unheated refractories with emissivity 0.55, and the ceiling emissivity is 0.8. The furnace cross section is rectangular with height 1 m and width 2 m. A total radiation pyrometer is sighted on the walls and indicates the following apparent temperatures: ceiling, 1340°C; sidewall readings average about 1145°C; and the load indicates about 900°C. (a) What are the true temperatures of the furnace walls and stock? (b) What is the net heat flux at each surface? (c) How do the matrix method and SSR models compare? Three-zone model, M = 3: Zone 1: Source (top) Zone 2: Sink (bottom) Zone 3: Refractory (lumped sides) Physical constants: W T0 ≡ 273.15 K σ ≡ 5.670400 × 10−8  m2⋅K4 Enclosure input parameters: He := 1 m ε1 := .8

(5-133)

This result is identically that for the SSR model as obtained previously in Eq. (5-128a). This equation is also valid for Mr ≥ 1 as long as Mb = 2. The electrical network analog methodology can be generalized for enclosures having M > 3. Some Examples from Furnace Design The theory of the past several subsections is best understood in the context of two engineering examples involving furnace modeling. The engineering idealization of the equivalent gray plane concept is introduced first. Figure 5-20 depicts a common furnace configuration in which the

We := 2 m ε2 := .7

AI :=



2 AI := 0 0



0 2 0

 

0 0 2

eI :=





eI :=

m2

ss11 := 0

ρ3 := 1 − ε3







rI :=

ρ1 0 0 0 ρ2 0 0 0 ρ3

0.2 0 0 rI := 0 0.3 0 0 0 0.45

ss22 := 0

2 2 ss33 := 2(H  ss33 := 0.4721 m2 e + We − We)Le

From symmetry and conservation, there are three linear simultaneous results for the off-diagonal elements of ss: ⎯⎯ ⎯⎯ −1 1 1 −1 12 A1 − 1 1 1 1 0 ⎯⎯ ⎯⎯ 1 1 −1 1 13 = 1 0 1 A2 − 22 =  2 ⎯⎯ ⎯⎯ −1 1 1 23 A3 − 33 0 1 1

Thus

s⎯⎯s :=

Furnace chamber cross section. To convert feet to meters, multiply



A3 := 2He⋅Le

 

ε1 0 0 0 ε2 0 0 0 ε3

0.8 0 0 0 0.7 0 0 0 0.55

A2 := A1 ρ2 := 1 − ε2

Compute direct exchange areas by using crossed strings (  3):

×

by 0.3048.

ρ1 := 1 − ε1

A1 0 0 0 A2 0 0 0 A3



FIG. 5-20

A1 := We⋅Le

Le := 1 m ε3 := .55



  

⎯⎯ A1 − 11 ⎯⎯ 1 A2 − 22 =  2 ⎯⎯ A3 − 33

  



⎯⎯ ⎯⎯ ⎯⎯ + A1 + A2 − A3 −11 − 22 + 33 ⎯⎯ ⎯⎯ ⎯⎯ + A1 − A2 + A3 −11 + 22 − 33 ⎯⎯ ⎯⎯ ⎯⎯ − A1 + A2 + A3 +11 − 22 − 33



ss12 := 0.5 (A1 + A2 − A3 + ss33) ss13 := 0.5 (A1 − A2 + A3 − ss33) ss23 := 0.5 ( − A1 + A2 + A3 − ss33)

 

ss11

ss12

ss13

ss12

ss22

ss23

ss13

ss23

ss33

s⎯⎯s :=

0

1.2361

1.2361

0.7639

0

0.7639

0.7639 0.7639

0.4721



   1

m2 (AI − s⎯s⎯)

Compute radiosities W from pyrometer temperature readings:

1 1

0

=

0 0

m2

HEAT TRANSFER BY RADIATION

Twc :=

 

1340.0 K 900.0 C W := σ Twc⋅  + T0 C 1145.0





4

W :=

  384.0 107.4 229.4

kW  m2

Matrix wall flux density relations and heat flux calculations based on W: H := AI−1ss◊W Q := AI(W − H) E := (eI◊AI)−1◊Q + H 460.0 154.0 441.5 kW Q := −435.0 kW E := kW 14.2  H := 324.9  m2 m2 −25.0 241.8 219.1

 

 

 

The sidewalls act as near-adiabatic surfaces since the heat loss through each sidewall is only about 2.7 percent of the total heat flux originating at the source. Actual temperatures versus pyrometer readings: 1397.3 E 0.25 C 434.1 C versus Twc = T: =  − T0  T= K σ 1128.9



 



 

1340.0 900.0 C 1145.0

batic, the roof of the furnace is estimated to lose heat to the surroundings with a flux density (W/m2) equal to 5 percent of the source and sink emissive power difference. An estimate of the radiant flux arriving at the sink is required, as well as estimates for the roof and average refractory temperatures in consideration of refractory service life. Part (a): Equivalent Gray Plane Emissivity Algebraically compute the equivalent gray plane emissivity for the refractory-backed tube bank idealized by the imaginary plane A2, depicted in Fig. 5-15. Solution: Let zone 1 represent one tube and zone 2 represent the effective plane 2, that is, the unit cell for the tube bank. Thus A1 = πD and A2 = C are the corresponding zone areas, respectively (per unit vertical dimension). This notation is consistent with Example 3. Also put ε1 = 0.8 with ε2 = 1.0 and define R = C/D = 12/5 = 2.4. The gray plane effective emissivity is then calculated as the total view factor for the effective plane to tubes, that is, F 2,1 ≡ ⎯ε2. For R = 2.4, Fig. 5-15, ⎯ curve 5, yields the refractory augmented view factor F2,1 ≈ 0.81. Then F 2,1 is 1 calculated from Eq. (5-128b) as F 2,1 =  ≈ 0.702. 01 + (2.4π)⋅0.20.8 + 10.81 A more accurate value is obtained via the matrix method as F 2,1 = 0.70295.

Compare SSR model versus matrix method [use Eqs. (5-128a) and (5-130)]: From Eq. (5-130) 1 ssbar12 := ss1,2 +  ssbar12 = 1.6180 m2 1 1  +  ss1,3 ss2,3

Part (b): Radiant Furnace Chamber with Heat Loss Four-zone model, M = 4: Use matrix method. Zone 1: Sink (floor) Zone 2: Source (lumped sides) Zone 3: Refractory (roof) Zone 4: Refractory (ends and floor strips)

And from Eq. (5-128a)

Physical constants:

1 SSR12 := 

SSR12 = 1.0446 m

1 ρ1 ρ2   +   +  ε1A1 ε2A2 ssbar12

With the numerical result

Q12 := SSR12(E1 − E2)

Enclosure input parameters: A1 := 50 ft2 A2 := 120 ft2 A3 := 80 ft2 A4 := 126 ft2 D := 5 in H := 6 ft

Q12 = 446.3 kW

ε1 := .9 ε2 := .70295 ε3 := .5 ε4 := .5 I4:= identity(4)

Compute total exchange areas ( 3 = 0.55):



 

0.2948 0.8284 0.4769 1 ⎯⎯ 0.8284 0.1761 0.3955 m2 (eI◊AI − SS) 1 0.4769 0.3955 0.2277 1

=

0 0 0

m2

Clearly SSR21 and SS1,2 are unequal. But if SS3,1 Θ3 :=  SS3,1 + SS3,2 SSA12 := SS1,2 + SS2,3Θ3 and SSA12 = 1.0446 m2

Er := E2 + (E1 − E2)Θ3





rI := I4 - eI rI =

s⎯s⎯ =



0 1.559 1.575 1.511

1. 559 2.078 2.839 4.673

1.575 2.839 0 3.018

1.511 4.673 3.018 2.503



0.1 0 0 0 0 0.2971 0 0 0 0 0.5 0 0 0 0 0.5

 

kW Er = 247.8  m2

Numerically the matrix method predicts SSA12 = 1.0446 m2 for Q3 = 0 and ε3 = 0.55, which is identical to SSR1,2 for the SSR model. Thus SSR1,2 = SSA12 is the refractory-aided total exchange area between zone 1 and zone 2. The SSR model also predicts Er = 247.8 kW/m2 versus the experimental value E3 = 219.1 kW/m2 (1172.6C vs. 1128.9C), which is also a consequence of the actual 25.0-kW refractory heat loss. (This example was developed as a MATHCAD 14® worksheet. Mathcad is a registered trademark of Parametric Technology Corporation.)

Example 10: Furnace Simulation via Zoning The furnace chamber depicted in Fig. 5-20 is heated by combustion gases passing through 20 vertical radiant tubes which are backed by refractory sidewalls. The tubes have an outside diameter of D = 5 in (12.7 cm) mounted on C = 12 in (4.72 cm) centers and a gray body emissivity of 0.8. The interior (radiant) portion of the furnace is a 6 × 8 × 10 ft rectangular parallelepiped with a total surface area of 376 ft2 (34.932 m2). A 50-ft2 (4.645-m2) sink is positioned centrally on the floor of the furnace. The tube and sink temperatures are measured with embedded thermocouples as 1500 and 1200°F, respectively. The gray refractory emissivity may be taken as 0.5. While all other refractories are assumed to be radiatively adia-

AI =

2

m

Compute total exchange areas: R := (AI - s⎯s⎯⋅rI)-1 ⎯⎯ ⎯⎯ SS := eI◊AI◊R◊ss◊eI SS =

define

Θ3 := 0.5466

eI =

0.9 0 0 0 0 0.7029 0 0 0 0 0.5 0 0 0 0 0.5



Compute direct exchange areas: There are ∑ = 6 unique direct exchange areas. These are obtained from Eqs. (5-114) and view factor algebra. The final array of direct exchange areas is:

R := (AI − s⎯⎯s◊rI)−1



W T0 ≡ 273.15 K σ ≡ 5.6704 × 10−8  m2⋅K4

2

Thus the SSR model produces Q12 = 446.3 kW versus the measured value Q1 = 460.0 kW or a discrepency of about 3.0 percent. Mathematically the SSR model assumes a value of ε3 = 0.0, which precludes the sidewall heat loss of Q3 = −25.0 kW. This assumption accounts for all of the difference between the two values. It remains to compare SSR12 and SS1,2 computed by the matrix method.

⎯⎯ ⎯⎯ SS := eI◊AI◊R◊s⎯⎯s◊eI SS =

5-29



0.405 1.669 0.955 1.152

4.645 0 0 0 0 11.148 0 0 0 0 7.432 0 0 0 0 11.706

1.669 2.272 1.465 2.431

0.955 1.465 0.234 1.063

1.152 2.431 1.063 1.207



m2

Check matrix conservation via row-sums: (AI − s⎯s⎯)

  1 1 1 1

=

0 0 0 0

m

2

⎯⎯ (eI⭈AI - SS)

   1 1 1 1

=

0 0 0 0

m2

Emissive power and wall flux input data:

TF :=

  1200 1500 32 32

F

T=

5 C T :=  (TF − 32 F)  9 F

 

648.9 815.6 C 0.0 0.0

Q3 := − 0.05⋅A3(E2 − E1)

E=

  40.98 79.66 0.32 0.32



kW  m2

Q3 = −14.37 kW

4



K E := σ T⋅  + T0 C

Q4 := 0



m2

5-30

HEAT AND MASS TRANSFER

Compute refractory emissive powers from known flux inputs Q3 and Q4 using partitioned matrix equations [Eq. (5-120b)]: ER :=

ε3⋅A3 − SS3,3 − SS3,4 −SS4,3 ε4⋅A4 − SS4,4



Q3 + SS3,1⋅E1 + SS3,2⋅E2 4 4,1⋅E1 + SS4,2⋅E2

−1

 ⋅  Q + SS





60.68 kW  ER = 65.73 m2

E :=

   E1 E2 ER1 ER2

E=

40.98 79.66 60.68 65.73



⎯⎯⎯ S1S2 =

kW  m2

Q=

T :=

 σ E

1



C − T0  K

0.25

Specular limit

    −111.87 126.24 kW T = −14.37 0.00

648.9 815.6 C 743.9 764.5

ATubes = 14.59 m2

εTubes = 0.2237

Notes: (1) Results for Q and T here are independent of ε3 and ε4 with the exception of T3, which is indeed a function of ε3. (2) The total surface area of the tubes is ATubes = 14.59 m2. Suppose the tubes were totally surrounded by a black enclosure at the temperature of the sink. The hypothetical emissivity of the tubes would then be εTubes = 0.224. (3) A 5 percent roof heat loss is consistent with practical measurement errors. A sensitivity test was performed with M = 3, 4, and 5 with and without roof heat loss. The SSR model corresponds to M = 3 with zero heat loss. For M = 5, zone 4 corresponded to the furnace ends and zone 5 corresponded to the floor strips. The results are summarized in the following table. With the exception of the temperature of the floor strips, the computed results are seen to be remarkably insensitive to M. Effect of Zone Number M on Computed Results Zero roof heat loss

T3 T4 T5

Q1 Q2 Q3 Q4 Q5

M=5

2

1

2

2

S2

1

(5-134)

2

S2

⎯ ⎯⎯ S1S2 =

A1

 ε1 + ε1 − 1 1

Auxiliary calculations for tube area and effective tube emissivity: Q2 ATubes := 20π⋅D⋅H εTubes :=  ATubes(E2 − E1)

A1

 ρ [1−A A ] ε1 + ρεAA +  (1 − ρ )

For ρD1 = ρD2 = 0 (or equivalently ρ1 = ρS1 with ρ2 = ρS2), Eq. (5-134) yields the limiting case for wholly specular reflection, i.e.

Compute flux values and final zone temperatures: ⎯⎯ Q := εI⋅AI⋅E − SS⋅E

ρSi and a specular component ρDi. The method yields analytical results for a number of two surface zone geometries. In particular, the following equation is obtained for exchange between concentric spheres or infinitely long coaxial cylinders for which A1 < A2:

5 percent roof heat loss

M=3

M=4

M=3

M=4

M=5

765.8 NA NA

Temperature, °C 762.0 762.4 756.2 768.4 765.0 NA NA 780.9 NA

743.9 764.5 NA

744.3 761.1 776.7

Heat flux, kW −117.657 −117.275 −116.251 −112.601 −111.870 −110.877 117.657 117.275 116.251 126.975 126.244 125.251 0.000 0.000 0.000 −14.374 −14.374 −14.374 NA 0.000 0.000 NA 0.00 0.00 NA NA 0.000 NA NA 0.00

(This example was developed as a MATHCAD 14® worksheet. Mathcad is a registered trademark of Parametric Technology Corporation.)

Allowance for Specular Reflection If the assumption that all surface zones are diffuse emitters and reflectors is relaxed, the zoning equations become much more complex. Here, all surface parameters become functions of the angles of incidence and reflection of the radiation beams at each surface. In practice, such details of reflectance and emission are seldom known. When they are, the Monte Carlo method of tracing a large number of beams emitted from random positions and in random initial directions is probably the best method of obtaining a solution. Siegel and Howell (op. cit., Chap. 10) and Modest (op. cit., Chap. 20) review the utilization of the Monte Carlo approach to a variety of radiant transfer applications. Among these is the Monte Carlo calculation of direct exchange areas for very complex geometries. Monte Carlo techniques are generally not used in practice for simpler engineering applications. A simple engineering approach to specular reflection is the so-called diffuse plus specular reflection model. Here the total reflectivity ρi = 1 − εi = ρSi + ρDi is represented as the sum of a diffuse component

(5-134a)

2

which is independent of the area ratio, A1/A2. It is important to notice that Eq. (5-124a) is similar to Eq. (5-127b) but the emissivities here are defined as ε1 ≡ 1 − ρS1 and ε2 ≡ 1 − ρS2. When surface reflection is wholly diffuse [ρS1 = ρS2 = 0 or ρ1 = ρD1 with ρ2 = ρD2], Eq. (5-134) results in a formula identical to Eq. (5-127a), viz. Diffuse limit

⎯⎯⎯ S1S2 =

A1

 ε1 + AA ρε 1

1

2

2

2

(5-134b, 5-127a)

For the case of (infinite) parallel flat plates where A1 = A2, Eq. (5-134) leads to a general formula similar to Eq. (5-134a) but with the stipulation here that ε1 ≡ 1 − ρD1 − ρS1 and ε2 ≡ 1 − ρD2 − ρS2. Another particularly interesting limit of Eq. (5-134) occurs when A2 >> A1, which might represent a small sphere irradiated by an infinite surroundings which can reflect radiation originating at A1 back to A1. That is to say, even though A2 → ∞, the “self” total exchange area does not necessarily vanish, to wit ⎯⎯⎯ ε21ρs2A1 S1S1 =  [1 − ρ1ρs2]

⎯ ⎯⎯ ε1(1 − ρs2)A1 and S1S2 =  [1 − ρ1ρs2]

(5-134c,d)

which again exhibit diffuse and specular limits. The diffuse plus specular reflection model becomes significantly more complex for geometries with M ≥ 3 where digital computation is usually required. An Exact Solution to the Integral Equations—The Hohlraum Exact solutions of the fundamental integral equations for radiative transfer are available for only a few simple cases. One of these is the evaluation of the emittance from a small aperture, of area A1, in the surface of an isothermal spherical cavity of radius R. In German, this geometry is termed a hohlraum or hollow space. For this special case the radiosity W is constant over the inner surface of the cavity. It then follows that the ratio W/E is given by ε WE =  1 − ρ[1 − A1(4πR2)]

(5-135)

where ε and ρ = 1 − ε are the diffuse emissivity and reflectivity of the interior cavity surface, respectively. The ratio W/E is the effective emittance of the aperture as sensed by an external narrow-angle receiver (radiometer) viewing the cavity interior. Assume that the cavity is constructed of a rough material whose (diffuse) emissivity is ε = 0.5. As a point of reference, if the cavity is to simulate a blackbody emitter to better than 98 percent of an ideal theoretical blackbody, Eq. (5-135) then predicts that the ratio of the aperture to sphere areas A1(4πR2) must be less than 2 percent. Equation (5-135) has practical utility in the experimental design of calibration standards for laboratory radiometers. RADIATION FROM GASES AND SUSPENDED PARTICULATE MATTER Introduction Flame radiation originates as a result of emission from water vapor and carbon dioxide in the hot gaseous combustion

HEAT TRANSFER BY RADIATION products and from the presence of particulate matter. The latter includes emission from burning of microscopic and submicroscopic soot particles, and from large suspended particles of coal, coke, or ash. Thermal radiation owing to the presence of water vapor and carbon dioxide is not visible. The characteristic blue color of clean natural gas flames is due to chemiluminescence of the excited intermediates in the flame which contribute negligibly to the radiation from combustion products. Gas Emissivities Radiant transfer in a gaseous medium is characterized by three quantities; the gas emissivity, gas absorptivity, and gas transmissivity. Gas emissivity refers to radiation originating within a gas volume which is incident on some reference surface. Gas absorptivity and transmissivity, however, refer to the absorption and transmission of radiation from some external surface radiation source characterized by some radiation temperature T1. The sum of the gas absorptivity and transmissivity must, by definition, be unity. Gas absorptivity may be calculated from an appropriate gas emissivity. The gas emissivity is a function only of the gas temperature Tg while the absorptivity and transmissivity are functions of both Tg and T1. The standard hemispherical monochromatic gas emissivity is defined as the direct volume-to-surface exchange area for a hemispherical gas volume to an infinitesimal area element located at the center of the planar base. Consider monochromatic transfer in a black hemispherical enclosure of radius R that confines an isothermal volume of gas at temperature Tg. The temperature of the bounding surfaces is T1. Let A2 denote the area of the finite hemispherical surface and dA1 denote an infinitesimal element of area located at the center of the planar base. The (dimensionless) monochromatic direct exchange area for exchange between the finite hemispherical surface A2 and dA1 then follows from direct integration of Eq. (5-116a) as π2 e−K R ∂(s⎯⎯1⎯s⎯2)λ cosφ1 2πR2 sinφ1 dφ1 = e−K R (5-136a)   = 2 φ1= 0 πR ∂A1



λ

λ

and from conservation there results ∂(s⎯⎯1⎯g)λ  = 1 − e−K R ∂A1 λ

(5-136b)

Note that Eq. (5-136b) is identical to the expression for the gas emissivity for a column of path length R. In Eqs. (5-136) the gas absorption coefficient is a function of gas temperature, composition, and wavelength, that is, Kλ = Kλ(T,λ). The net monochromatic radiant flux density at dA1 due to irradiation from the gas volume is then given by ∂(s⎯⎯1⎯g)λ q1g,λ =  (E1,λ − Eg,λ) ≡ αg1,λE1,λ − εg,λEg,λ (5-137) ∂A1 In Eq. (5-137), εg,λ(T,λ) = 1 − exp(−KλR) is defined as the monochromatic or spectral gas emissivity and αg,λ(T,λ) = εg,λ(T,λ). If Eq. (5-137) is integrated with respect to wavelength over the entire EM spectrum, an expression for the total flux density is obtained q1,g = αg,1E1 − εgEg where

and

εg(Tg) =



αg,1(T1,Tg) =

∞

λ=0



Eb,λ(Tg,λ) ελ(Tg,λ)⋅  dλ Eb(Tg)

∞

λ=0

Eb,λ(T1,λ) αg,λ(Tg,λ)⋅  dλ Eb(T1)

(5-138) (5-138a)

(5-138b)

define the total gas emissivity and absorptivity, respectively. The notation used here is analogous to that used for surface emissivity and absorptivity as previously defined. For a real gas εg = αg,1 only if T1 = Tg, while for a gray gas mass of arbitrarily shaped volume εg = αg,1 = ∂(s⎯⎯1⎯g)∂A1 is independent of temperature. Because Kλ(T,λ) is also a function of the composition of the radiating species, it is necessary in what follows to define a second absorption coefficient kp,λ, where Kλ = kp,λp. Here p is the partial pressure of the radiating species, and kp,λ, with units of (atm⋅m)−1, is referred to as the monochromatic line absorption coefficient. Mean Beam Lengths It is always possible to represent the emissivity of an arbitrarily shaped volume of gray gas (and thus the corre-

5-31

sponding direct gas-to-surface exchange area) with an equivalent sphere of radius R = LM. In this context the hemispherical radius R = LM is referred to as the mean beam length of the arbitrary gas volume. Consider, e.g., an isothermal gas layer at temperature Tg confined by two infinite parallel plates separated by distance L. Direct integration of Eq. (5-116a) and use of conservation yield a closedform expression for the requisite surface-gas direct exchange area ∂(s⎯⎯1⎯g)  = [1 − 2E3(KL)] ∂A1



(5-139a)

∞

e−z⋅t dt is defined as the nth-order exponential  n t=1 t integral which is readily available. Employing the definition of gas emissivity, the mean beam length between the plates LM is then defined by the expression where En(z) =

εg = [1 − 2E3(KL)] ≡ 1 − e−KL

M

(5-139b)

Solution of Eq. (5-139b) yields KLM = −ln[2E3(KL)], and it is apparent that KLM is a function of KL. Since En(0) = 1(n − 1) for n > 1, the mean beam length approximation also correctly predicts the gas emissivity as zero when K = 0 and K → ∞. In the limit K → 0, power series expansion of both sides of the Eq. (5-139b) leads to KLM → 2KL ≡ KLM0, where LM ≡ LM0 = 2L. Here LM0 is defined as the optically thin mean beam length for radiant transfer from the entire infinite planar gas layer to a differential element of surface area on one of the plates. The optically thin mean beam length for two infinite parallel plates is thus simply twice the plate spacing L. In a similar manner it may be shown that for a sphere of diameter D, LM0 = 2⁄3 D, and for an infinitely long cylinder LM0 = D. A useful default formula for an arbitrary enclosure of volume V and area A is given by LM0 = 4V/A. This expression predicts LM0 = 8⁄9 R for the standard hemisphere of radius R because the optically thin mean beam length is averaged over the entire hemispherical enclosure. Use of the optically thin value of the mean beam length yields values of gas emissivities or exchange areas that are too high. It is thus necessary to introduce a dimensionless constant β ≤ 1 and define some new average mean beam length such that KLM ≡ βKLM0. For the case of parallel plates, we now require that the mean beam length exactly predict the gas emissivity for a third value of KL. In this example we find β = −ln[2E3(KL)]2KL and for KL = 0.193095 there results β = 0.880. The value β = 0.880 is not wholly arbitrary. It also happens to minimize the error defined by the so-called shape correction factor φ = [∂(s⎯1⎯g⎯)∂A1](1 − e−KL ) for all KL > 0. The required average mean beam length for all KL > 0 is then taken simply as LM = 0.88LM0 = 1.76L. The error in this approximation is less than 5 percent. For an arbitrary geometry, the average mean beam length is defined as the radius of a hemisphere of gas which predicts values of the direct exchange area s⎯1⎯g⎯ A1 = [1 − exp(−KLM)], subject to the optimization condition indicated above. It is has been found that the error introduced by using average mean beam lengths to approximate direct exchange areas is sufficiently small to be appropriate for many engineering calculations. When β = LMLM0 is evaluated for a large number of geometries, it is found that 0.8 < β < 0.95. It is recommended here that β = 0.88 be employed in lieu of any further geometric information. For a single-gas zone, all the requisite direct exchange areas can be approximated for engineering purposes in terms of a single appropriately defined average mean beam length. Emissivities of Combustion Products Absorption or emission of radiation by the constituents of gaseous combustion products is determined primarily by vibrational and rotational transitions between the energy levels of the gaseous molecules. Changes in both vibrational and rotational energy states gives rise to discrete spectral lines. Rotational lines accompanying vibrational transitions usually overlap, forming a so-called vibration-rotation band. These bands are thus associated with the major vibrational frequencies of the molecules. M

5-32

HEAT AND MASS TRANSFER

Spectral Emissivity of Gaseous Species ελ

1.0 0.9 0.8 0.7 H2O

0.6 0.5

CO2

0.4 0.3 0.2 0.1 0.0 0

2

4

6

8 10 12 Wavelength λ [µm]

14

16

18

20

FIG. 5-21 Spectral emittances for carbon dioxide and water vapor after RADCAL. pcL = pwL = 0.36 atm⋅m, Tg = 1500 K.

Each spectral line is characterized by an absorption coefficient kp,λ which exhibits a maximum at some central characteristic wavelength or wave number η0 = 1λ0 and is described by a Lorentz* probability distribution. Since the widths of spectral lines are dependent on collisions with other molecules, the absorption coefficient will also depend upon the composition of the combustion gases and the total system pressure. This brief discussion of gas spectroscopy is intended as an introduction to the factors controlling absorption coefficients and thus the factors which govern the empirical correlations to be presented for gas emissivities and absorptivities. Figure 5-21 shows computed values of the spectral emissivity εg,λ ≡ εg,λ(T,pL,λ) as a function of wavelength for an equimolar mixture of carbon dioxide and water vapor for a gas temperature of 1500 K, partial pressure of 0.18 atm, and a path length L = 2 m. Three principal absorption-emission bands for CO2 are seen to be centered on 2.7, 4.3, and 15 µm. Two weaker bands at 2 and 9.7 µm are also evident. Three principal absorption-emission bands for water vapor are also identified near 2.7, 6.6, and 20 µm with lesser bands at 1.17, 1.36, and 1.87 µm. The total emissivity εg and absorptivity αg,1 are calculated by integration with respect to wavelength of the spectral emissivities, using Eqs. (5-138) in a manner similar to the development of total surface properties. Spectral Emissivities Highly resolved spectral emissivities can be generated at ambient temperatures from the HITRAN database (high-resolution transmission molecular absorption) that has been developed for atmospheric models [Rothman, L. S., Chance, K., and Goldman, A., eds., J. Quant. Spectroscopy & Radiative Trans., 82 (1–4), 2003]. This database includes the chemical species: H2O, CO2, O3, N2O, CO, CH4, O2, NO, SO2, NO2, NH3, HNO3, OH, HF, HCl, HBr, ClO, OCS, H2CO, HOCl, N2, HCN, CH3C, HCl, H2O2, C2H2, C2H6, PH3, COF2, SF6, H2S, and HCO2H. These data have been extended to high temperature for CO2 and H2O, allowing for the changes in the population of different energy levels and in the line half width [Denison, M. K., and Webb, B. W., Heat Transfer, 2, 19–24 (1994)]. The resolution in the single-line models of emissivities is far greater than that needed in engineering calculations. A number of models are available that average the emissivities over narrow-wavelength regimes or over the entire band. An extensive set of measurements of narrowband parameters performed at NASA (Ludwig, C., et al., Handbook of Infrared Radiation from Combustion Gases, NASA SP-3080, 1973) has been used to develop the RADCAL computer code to obtain spectral emissivities for CO2, H2O, CH4, CO, and soot (Grosshandler, *Spectral lines are conventionally described in terms of wave number η = 1λ, with each line having a peak absorption at wave number η0. The Lorentz distrbc where S is the integral of kη over all ibution is defined as kηS =  π[b2c + (η − ηo)2] wave numbers. The parameter S is known as the integrated line intensity, and bc is defined as the collision line half-width, i.e., the half-width of the line is onehalf of its peak centerline value. The units of kη are m−1 atm−1.

W. L., “RADCAL,” NIST Technical Note 1402, 1993). The exponential wideband model is available for emissions averaged over a band for H2O, CO2, CO, CH4, NO, SO2, N2O, NH3, and C2H2 [Edwards, D. K., and Menard, W. A., Appl. Optics, 3, 621–625 (1964)]. The line and band models have the advantages of being able to account for complexities in determining emissivities of line broadening due to changes in composition and pressure, exchange with spectrally selective walls, and greater accuracy in formulating fluxes in gases with temperature gradients. These models can be used to generate the total emissivities and absorptivies that will be used in this chapter. RADCAL is a command-line FORTRAN code which is available in the public domain on the Internet. Total Emissivities and Absorptivities Total emissivities and absorptivities for water vapor and carbon dioxide at present are still based on data embodied in the classical Hottel emissivity charts. These data have been adjusted with the more recent measurements in RADCAL and used to develop the correlations of emissivities given in Table 5-5. Two empirical correlations which permit hand calculation of emissivities for water vapor, carbon dioxide, and four mixtures of the two gases are presented in Table 5-5. The first section of Table 5-5 provides data for the two constants b and n in the empirical relation ⎯⎯⎯ εgTg = b[pL − 0.015]n (5-140a) while the second section of Table 5-5 utilizes the four constants in the empirical correlation ⎯⎯⎯ log (εgTg) = a0 + a1 log (pL) + a2 log2 (pL) + a3 log3 (pL) (5-140b) In both cases the empirical constants are given for the three temperatures of 1000, 1500, and 2000 K. Table 5-5 also includes some six values for the partial pressure ratios pWpC of water vapor to carbon dioxide, namely, 0, 0.5, 1.0, 2.0, 3.0, and ∞. These ratios correspond to composition values of pC / (pC + pW) = 1/(1 + pW /pC) of 0, 1/3, 1/2, 2/3, 3/4, and unity. For emissivity calculations at other temperatures and mixture compositions, linear interpolation of the constants is recommended. The absorptivity can be obtained from the emissivity with aid of Table 5-5 by using the following functional equivalence. ⎯⎯⎯⎯⎯ ⎯⎯⎯ T 0.5 αg, 1Tl = [εgT1(pL⋅Tl Tg)] g (5-141) Tl Verbally, the absorptivity computed from Eq. (5-141) by using the correlations in Table 5-5 is based on a value for gas emissivity εg calculated at a temperature T1 and at a partial-pressure path-length product of (pC + pW)LT1/Tg. The absorptivity is then equal to this value of gas emissivity multiplied by (Tg /T1)0.5. It is recommended that spectrally based models such as RADCAL (loc. cit.) be used particularly when extrapolating beyond the temperature, pressure, or partial-pressure-length product ranges presented in Table 5-5. A comparison of the results of the predictions of Table 5-5 with values obtained via the integration of the spectral results calculated from the narrowband model in RADCAL is provided in Fig. 5-22. Here calculations are shown for pCL = pWL = 0.12 atm⋅m and a gas temperature of 1500 K. The RADCAL predictions are 20 percent higher than the measurements at low values of pL and are 5 percent higher at the large values of pL. An extensive comparison of different sources of emissivity data shows that disparities up to 20 percent are to be expected at the current time [Lallemant, N., Sayre, A., and Weber, R., Prog. Energy Combust. Sci., 22, 543–574, (1996)]. However, smaller errors result for the range of the total emissivity measurements presented in the Hottel emissivity tables. This is demonstrated in Example 11.



Example 11: Calculations of Gas Emissivity and Absorptivity Consider a slab of gas confined between two infinite parallel plates with a distance of separation of L = 1 m. The gas pressure is 101.325 kPa (1 atm), and the gas temperature is 1500 K (2240°F). The gas is an equimolar mixture of CO2 and H2O, each with a partial pressure of 12 kPa (pC = pW = 0.12 atm). The radiative flux to one of its bounding surfaces has been calculated by using RADCAL for two cases. For case (a) the flux to the bounding surface is 68.3 kW/m2 when the emitting gas is backed by a black surface at an ambient temperature of 300 K (80°F). This (cold) back surface contributes less than 1 percent to the flux. In case (b), the flux is calculated as 106.2 kW/m2 when the gas is backed by a black surface at a temperature of 1000 K (1340°F). In this example, gas emissivity and

HEAT TRANSFER BY RADIATION TABLE 5-5

5-33

⎯⎯⎯⎯ Emissivity-Temperature Product for CO2-H2O Mixtures, egTg Limited range for furnaces, valid over 25-fold range of pw + cL, 0.046–1.15 m⋅atm (0.15–3.75 ft⋅atm)

pw /pc pw  pw + pc

0

a

1

2

3

∞

0

s(0.3–0.42)

a(0.42–0.5)

w(0.6–0.7)

e(0.7–0.8)

1

Corresponding to (CH6)x, covering future high H2 fuels

H2O only

CO2 only

Corresponding to (CH)x, covering coal, heavy oils, pitch

Corresponding to Corresponding (CH2)x, covering to CH4, covering distillate oils, paraffins, natural gas and olefines refinery gas ⎯⎯⎯⎯ Constants b and n of εgTg = b(pL − 0.015)n, pL = m⋅atm, T = K

Section 1 T, K

b

n

b

n

b

n

b

n

b

n

b

n

1000 1500 2000

188 252 267

0.209 0.256 0.316

384 448 451

0.33 0.38 0.45

416 495 509

0.34 0.40 0.48

444 540 572

0.34 0.42 0.51

455 548 594

0.35 0.42 0.52

416 548 632

0.400 0.523 0.640

⎯⎯⎯⎯ Constants b and n of εgTg = b(pL − 0.05)n, pL = ft⋅atm, T = °R

T, °R

b

n

b

n

b

n

b

n

b

n

b

n

1800 2700 3600

264 335 330

0.209 0.256 0.316

467 514 476

0.33 0.38 0.45

501 555 519

0.34 0.40 0.48

534 591 563

0.34 0.42 0.51

541 600 577

0.35 0.42 0.52

466 530 532

0.400 0.523 0.640

Section 2

Full range, valid over 2000-fold range of pw + cL, 0.005–10.0 m⋅atm (0.016–32.0 ft⋅atm) ⎯⎯⎯ ⎯ Constants of log10 εgTg = a0 + a1 log pL + a2 log2 pL + a3 log3 pL pL = m⋅atm, T = K

pw  pc

pw  pw + pc

0

0

a

s

1

a

2

w

3

e

∞

1

pL = ft⋅atm, T = °R

T, K

a0

a1

a2

a3

T, °R

a0

a1

a2

a3

1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000

2.2661 2.3954 2.4104 2.5754 2.6451 2.6504 2.6090 2.6862 2.7029 2.6367 2.7178 2.7482 2.6432 2.7257 2.7592 2.5995 2.7083 2.7709

0.1742 0.2203 0.2602 0.2792 0.3418 0.4279 0.2799 0.3450 0.4440 0.2723 0.3386 0.4464 0.2715 0.3355 0.4372 0.3015 0.3969 0.5099

−0.0390 −0.0433 −0.0651 −0.0648 −0.0685 −0.0674 −0.0745 −0.0816 −0.0859 −0.0804 −0.0990 −0.1086 −0.0816 −0.0981 −0.1122 −0.0961 −0.1309 −0.1646

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600

2.4206 2.5248 2.5143 2.6691 2.7074 2.6686 2.7001 2.7423 2.7081 2.7296 2.7724 2.7461 2.7359 2.7811 2.7599 2.6720 2.7238 2.7215

0.2176 0.2695 0.3621 0.3474 0.4091 0.4879 0.3563 0.4561 0.5210 0.3577 0.4384 0.5474 0.3599 0.4403 0.5478 0.4102 0.5330 0.6666

−0.0452 −0.0521 −0.0627 −0.0674 −0.0618 −0.0489 −0.0736 −0.0756 −0.0650 −0.0850 −0.0944 −0.0871 −0.0896 −0.1051 −0.1021 −0.1145 −0.1328 −0.1391

0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165

NOTE: pw /(pw + pc) of s, a, w, and e may be used to cover the ranges 0.2–0.4, 0.4–0.6, 0.6–0.7, and 0.7–0.8, respectively, with a maximum error in εg of 5 percent at pL = 6.5 m⋅atm, less at lower pL’s. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data.

absorptivity are to be computed from these flux values and compared with values obtained by using Table 5-5. Case (a): The flux incident on the surface is equal to εg⋅σ⋅T4g = 68.3 kW/m2; therefore, εg = 68,300(5.6704 × 10−8⋅15004) = 0.238. To utilize Table 5-5, the mean beam length for the gas is calculated from the relation LM = 0.88LM0 = 0.88⋅2L = 1.76 m. For Tg = 1500 K and (pC + pW)LM = 0.24(1.76) = 0.422 atm⋅m, the two-constant correlation in Table 5-5 yields εg = 0.230 and the four-constant correlation yields εg = 0.234. These results are clearly in excellent agreement with the predicted value of εg = 0.238 obtained from RADCAL. Case (b): The flux incident on the surface (106.2 kW/m2) is the sum of that contributed by (1) gas emission εg⋅σ⋅T4g = 68.3 kWm2 and (2) emission from the opposing surface corrected for absorption by the intervening gas using the gas transmissivity, that is, τg,1σ⋅T14 where τg,1 = 1 − αg,1. Therefore αg,1 = [1 − (106,200 − 68,300)(5.6704 × 10−8⋅10004)] = 0.332. Using Table 5-5, the two-constant and four-constant gas emissivities evaluated at T1 = 1000 K and pL = 0.4224⋅ (10001500) = 0.282 atm⋅m are εg = 0.2654 and εg = 0.2707, respectively. Multiplication by the factor (Tg / T1)0.5 = (1500 / 1000) 0.5 = 1.225 produces the final values of the two corresponding gas absorptivities αg,1 = 0.325 and αg,1 = 0.332, respectively. Again the agreement with RADCAL is excellent.

Total mixture emissivity ε

1

RADCAL Hottel (Table 5-5) 0.1

0.01 0.001

0.01

0.1

1

10

Partial pressure–path length product (pc + pw)L [atm.m]

FIG. 5-22 Comparison of Hottel and RADCAL total gas emissivities. Equimolal gas mixture of CO2 and H2O with pc = pw = 0.12 atm and Tg = 1500 K.

Other Gases The most extensive available data for gas emissivity are those for carbon dioxide and water vapor because of their importance in the radiation from the products of fossil fuel combustion. Selected data for other species present in combustion gases are provided in Table 5-6.

5-34

HEAT AND MASS TRANSFER

TABLE 5-6 Temperature PxL, atm⋅ft NH3a SO2b CH4c COd NOd HCle

Total Emissivities of Some Gases 1000°R 0.01 0.047 0.020 0.0116 0.0052 0.0046 0.00022

0.1 0.20 0.13 0.0518 0.0167 0.018 0.00079

1600°R 1.0 0.61 0.28 0.1296 0.0403 0.060 0.0020

0.01 0.020 0.013 0.0111 0.0055 0.0046 0.00036

0.1 0.120 0.090 0.0615 0.0196 0.021 0.0013

2200°R 1.0 0.44 0.32 0.1880 0.0517 0.070 0.0033

0.01 0.0057 0.0085 0.0087 0.0036 0.0019 0.00037

0.1 0.051 0.051 0.0608 0.0145 0.010 0.0014

2800°R 1.0 0.25 0.27 0.2004 0.0418 0.040 0.0036

0.01 (0.001) 0.0058 0.00622 0.00224 0.0078 0.00029

0.1 (0.015) 0.043 0.04702 0.00986 0.004 0.0010

1.0 (0.14) 0.20 0.1525 0.02855 0.025 0.0027

NOTE: Figures in this table are taken from plots in Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 6. Values in parentheses are extrapolated. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1). To convert atmosphere-feet to kilopascal-meters, multiply by 30.89. a Total-radiation measurements of Port (Sc.D. thesis in chemical engineering, MIT, 1940) at 1-atm total pressure, L = 1.68 ft, T to 2000°R. b Calculations of Guerrieri (S.M. thesis in chemical engineering, MIT, 1932) from room-temperature absorption measurements of Coblentz (Investigations of Infrared Spectra, Carnegie Institution, Washington, 1905) with poor allowance for temperature. c Estimated using Grosshandler, W.L., “RADCAL: A Narrow-Band Model for Radial Calculations in a Combustion Environment,” NIST Technical Note 1402, 1993. d Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 30 atm⋅ft. e Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 300 atm⋅ft.

Flames and Particle Clouds Luminous Flames Luminosity conventionally refers to soot radiation. At atmospheric pressure, soot is formed in locally fuel-rich portions of flames in amounts that usually correspond to less than 1 percent of the carbon in the fuel. Because soot particles are small relative to the wavelength of the radiation of interest in flames (primary particle diameters of soot are of the order of 20 nm compared to wavelengths of interest of 500 to 8000 nm), the incident radiation permeates the particles, and the absorption is proportional to the volume of the particles. In the limit of rpλ < < 1, the Rayleigh limit, the monochromatic emissivity ελ is given by ελ = 1 − exp(−K⋅ fv⋅Lλ)

(5-142)

where fv is the volumetric soot concentration, L is the path length in the same units as the wavelength λ, and K is dimensionless. The value K will vary with fuel type, experimental conditions, and the temperature history of the soot. The values of K for a wide range of systems are within a factor of about 2 of one another. The single most important variable governing the value of K is the hydrogen/carbon ratio of the soot, and the value of K increases as the H/C ratio decreases. A value of K = 9.9 is recommended on the basis of seven studies involving 29 fuels [Mulholland, G. W., and Croarkin, C., Fire and Materials, 24, 227–230 (2000)]. The total emissivity of soot εS can be obtained by substituting ελ from Eq. (5-142) for ελ in Eq. (5-138a) to yield εS =



∞

λ=0

Eb,λ(Tg,λ) 15 ελ  dλ = 1 −  [Ψ(3)(1 + K⋅ fv⋅L⋅Tc2)] Eb(Tg) 4 ≅ (1 + K⋅fv⋅L⋅Tc2)−4

(5-143)

Here Ψ(3)(x) is defined as the pentagamma function of x and c2 (m⋅K) is again Planck’s second constant. The approximate relation in Eq. (5-143) is accurate to better than 1 percent for arguments yielding values of εS < 0.7. At present, the largest uncertainty in estimating total soot emissivities is in the estimation of the soot volume fraction fv. Soot forms in the fuel-rich zones of flames. Soot formation rates are a function of fuel type, mixing rate, local equivalence ratio Φ, temperature, and pressure. The equivalence ratio is defined as the quotient of the actual to stoichiometric fuel-to-oxidant ratio Φ = [FO]Act[FO]Stoich. Soot formation increases with the aromaticity or C/H ratio of fuels with benzene, α-methyl naphthalene, and acetylene having a high propensity to form soot and methane having a low soot formation propensity. Oxygenated fuels, such as alcohols, emit little soot. In practical turbulent diffusion flames, soot forms on the fuel side of the flame front. In premixed flames, at a given temperature, the rate of soot formation increases rapidly for Φ > 2. For temperatures above

1500 K, soot burns out rapidly (in less than 0.1s) under fuel-lean conditions, Φ < 1. Because of this rapid soot burnout, soot is usually localized in a relatively small fraction of a furnace or combustor volume. Long, poorly mixed diffusion flames promote soot formation while highly backmixed combustors can burn soot-free. In a typical flame at atmospheric pressure, maximum volumetric soot concentrations are found to be in the range 10−7 < fv < 10−6. This corresponds to a soot formation of about 1.5 to 15 percent of the carbon in the fuel. When fv is to be calculated at high pressures, allowance must be made for the significant increase in soot formation with pressure and for the inverse proportionality of fv with respect to pressure. Great progress is being made in the ability to calculate soot in premixed flames. For example, predicted and measured soot concentration have been compared in a well-stirred reactor operated over a wide range of temperatures and equivalence ratios [Brown, N.J. Revzan, K. L., Frenklach, M., Twenty-seventh Symposium (International) on Combustion, pp. 1573–1580, 1998]. Moreover, CFD (computational fluid dynamics) and population dynamics modeling have been used to simulate soot formation in a turbulent non-premixed ethylene-air flame [Zucca, A., Marchisio, D. L., Barresi, A. A., Fox, R. O., Chem. Eng. Sci., 2005]. The importance of soot radiation varies widely between combustors. In large boilers the soot is confined to small volumes and is of only local importance. In gas turbines, cooling the combustor liner is of primary importance so that only small incremental soot radiation is of concern. In high-temperature glass tanks, the presence of soot adds 0.1 to 0.2 to emissivities of oil-fired flames. In natural gas-fired flames, efforts to augment flame emissivities with soot generation have generally been unsuccessful. The contributions of soot to the radiation from pool fires often dominates, and thus the presence of soot in such flames directly impacts the safe separation distances from dikes around oil tanks and the location of flares with respect to oil rigs. Clouds of Large Black Particles The emissivity εM of a cloud of black particles with a large perimeter-to-wavelength ratio is εM = 1 − exp[−(av)L]

(5-144)

where a/v is the projected area of the particles per unit volume of space. If the particles have no negative curvature (the particle does not “see” any of itself) and are randomly oriented, a = a′4, where a′ is the actual surface area. If the particles are uniform, av = cA = cA′4, where A and A′ are the projected and total areas of each particle and c is the number concentration of particles. For spherical particles this leads to εM = 1 − exp[−(π4)cdp2L] = 1 − exp(−1.5fvLdp)

(5-145)

As an example, consider a heavy fuel oil (CH1.5, specific gravity, 0.95) atomized to a mean surface particle diameter of dp burned with

HEAT TRANSFER BY RADIATION

5-35

20 percent excess air to produce coke-residue particles having the original drop diameter and suspended in combustion products at 1204°C (2200°F). The flame emissivity due to the particles along a path of L m, with dp measured in micrometers, is

balances for each volume zone. These N equations are given by the definition of the N-vector for the net radiant volume absorption S′ = [S′j] for each volume zone ⎯⎯ ⎯⎯ S′ = GS◊E + GG◊Eg − 4KVI◊Eg [N × 1] (5-152)

εM = 1 − exp(−24.3Ldp)

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For ⎯⎯ ⎯⎯ K→ 0, GS → 0, and GG → 0 leading to S′ → 0N. When K ≠ 0 and S′ = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would →. Here H = G/4 is the average flux write Si′/Vi = K[G − 4Eg] = −∇q r g density incident on a given volume zone from all other surface and volume zones. The DO and FV methods are currently available options as “RTE-solvers” in complex simulations of combustion systems using computational fluid dynamics (CFD).* Implementation of Eq. (5-152) necessitates the definition of two additional symmetric ⎯ ⎯ ⎯N⎯⎯× N arrays of exchange areas, namely, ⎯ ⎯ gg = [g⎯⎯i ⎯gj] and GG = [Gi Gj]. In Eq. (5-152) VI = [Vj⋅δi,j] is an N × N diagonal matrix of zone volumes. The total exchange areas in Eq. (5-151) are explicit functions of the direct exchange areas as follows: Surface-to-gas exchange ⎯⎯ ⎯ ⎯ T GS = SG [N × M] (5-153a) Gas-to-gas exchange ⎯⎯ ⎯ + s⎯g ⎯T◊ρI◊R◊s⎯g ⎯ [N × M] GG = g⎯g (5-153b) ⎯ ⎯ ⎯⎯⎯ ⎯ ⎯ ⎯ ⎯ ⎯ The matrices gg = [gi gj] and GG = [Gi Gj] must also satisfy the following matrix conservation relations: Direct exchange areas: 4KVI◊1 = g⎯s⎯ ◊1 + g⎯g⎯ ◊1 (5-154a)

(5-146)

For 200-µm particles and L = 3.05 m, the particle contribution to emissivity is calculated as 0.31. Clouds of Nonblack Particles For nonblack particles, emissivity calculations are complicated by multiple scatter of the radiation reflected by each particle. The emissivity εM of a cloud of gray particles of individual emissivity ε1 can be estimated by the use of a simple modification Eq. (5-144), i.e., εM = 1 − exp[−ε1(av)L]

(5-147)

Equation (5-147) predicts that εM → 1 as L → ∞. This is impossible in a scattering system, and use of Eq. (5-147) is restricted to values of the optical thickness (a/v) L < 2. Instead, the asymptotic value of εM is obtained from Fig. 5-12 as εM = εh (lim L → ∞), where the albedo ω is replaced by the particle-surface reflectance ω = 1 − ε1. Particles with perimeter-to-wavelength ratios of 0.5 to 5.0 can be analyzed, with significant mathematical complexity, by use of the the Mie equations (Bohren, C. F., and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, 1998). Combined Gas, Soot, and Particulate Emission In a mixture of emitting species, the emission of each constituent is attenuated on its way to the system boundary by absorption by all other constituents. The transmissivity of a mixture is the product of the transmissivities of its component parts. This statement is a corollary of Beer’s law. For present purposes, the transmissivity of “species k” is defined as τk = 1 − εk. For a mixture of combustion products consisting of carbon dioxide, water vapor, soot, and oil coke or char particles, the total emissivity εT at any wavelength can therefore be obtained from (1 − εT)λ = (1 − εC)λ(1 − εW)λ(1 − εS)λ(1 − εM)λ

(5-148)

where the subscripts denote the four flame species. The total emissivity is then obtained by integrating Eq. (5-148) over the entire EM energy spectrum, taking into account the variability of εC, εW, and εS with respect to wavelength. In Eq. (5-148), εM is independent of wavelength because absorbing char or coke particles are effectively blackbody absorbers. Computer programs for spectral emissivity, such as RADCAL (loc. cit.), perform the integration with respect to wavelength for obtaining total emissivity. Corrections for the overlap of vibration-rotation bands of CO2 and H2O are automatically included in the correlations for εg for mixtures of these gases. The monochromatic soot emissivity is higher at shorter wavelengths, resulting in higher attenuations of the bands at 2.7 µm for CO2 and H2O than at longer wavelengths. The following equation is recommended for calculating the emissivity εg+S of a mixture of CO2, H2O, and soot εg+S = εg + εS − M⋅εgεS

(5-149)

where M can be represented with acceptable error by the dimensionless function M = 1.12 − 0.27⋅(T1000) + 2.7 × 105fv⋅L

Total exchange areas:

(5-151)

with the definition 1 − εg+S ≡ (1 − εC)(1 − εW)(1 − εS).

M

N

⎯⎯ ⎯⎯ 4KVI◊1N = GS ◊1M + GG ◊1N

(5-154b)

The formal integral definition of the direct gas-gas exchange area is ⎯g⎯⎯g = i j

™ ™ Vj

Vi

e−Kr K2 2 dVj dVi πr

(5-155)

Clearly, when K = 0, the two direct exchange areas involving a gas ⎯ vanish. Computationally it is never necessary to make zone g⎯i⎯s⎯j and g⎯i⎯g j resort to Eq. (5-155) for calculation of ⎯g⎯i ⎯gj. This is so because ⎯s⎯i ⎯gj, ⎯g⎯i ⎯sj, and ⎯g⎯⎯gj may all be calculated arithmetically from appropriate values of ⎯s⎯⎯s byi using associated conservation relations and view factor algebra. i j Weighted Sum of Gray Gas (WSGG) Spectral Model Even in simple engineering calculations, the assumption of a gray gas is almost never a good one. The zone method is now further generalized to make allowance for nongray radiative transfer via incorporation of the weighted sum of gray gas (WSGG) spectral model. Hottel has shown that the emissivity εg(T,L) of an absorbing-emitting gas mixture containing CO2 and H2O of known composition can be approximated by a weighted sum of P gray gases P

εg(T,L) ≈ ap(T)(1 − e−K L) p

(5-150)

In Eq. (5-150), T has units of kelvins and L is measured in meters. Since coke or char emissivities are gray, their addition to those of the CO2, H2O, and soot follows simply from Eq. (5-148) as εT = εg+S + εM − εg+SεM

N

(5-156a)

p=1

where

P

a (T) = 1.0 p=1 p

(5-156b)

In Eqs. (5-156), Kp is some gray gas absorption coefficient and L is some appropriate path length. In practice, Eqs. (5-156) usually yield acceptable accuracy for P ≤ 3. For P = 1, Eqs. (5-156) degenerate to the case of a single gray gas.

RADIATIVE EXCHANGE WITH PARTICIPATING MEDIA Energy Balances for Volume Zones—The Radiation Source Term Reconsider a generalized enclosure with N volume zones confining a gray gas. When the N gas temperatures are unknown, an additional set of N equations is required in the form of radiant energy

*To further clarify the mathematical differences between zoning and the DO and FV methods recognize that (neglecting scatter) the matrix expressions H = ⎯E and 4KH = VI−1g⎯⎯sW+VI−1·g⎯g ⎯ · E represent spaAI−1s⎯s⎯W + AI−1s⎯g g g g tial discretizations of the integral form(s) of the RTE applied at any point (zone) on the boundary or interior of an enclosure, respectively, for a gray gas.

5-36

HEAT AND MASS TRANSFER

The Clear plus Gray Gas WSGG Spectral Model In principle, the emissivity of all gases approaches unity for infinite path length L. In practice, however, the gas emissivity may fall considerably short of unity for representative values of pL. This behavior results because of the band nature of real gas spectral absorption and emission whereby there is usually no significant overlap between dominant absorption bands. Mathematically, this physical phenomenon is modeled by defining one of the gray gas components in the WSGG spectral model to be transparent. For P = 2 and path length LM, Eqs. (5-156) yield the following expression for the gas emissivity εg = a1(1 − e−K L ) + a2(1 − e−K L ) 1

M

2

M

(5-157)

In Eq. (5-157) if K1 = 0 and a2 ≠ 0, the limiting value of gas emissivity is εg (T,∞) → a2. Put K1 = 0 in Eq. (5-157), ag = a2, and define τg = e−K L as the gray gas transmissivity. Equation (5-157) then simplifies to 2

εg = ag(1 − τg)

M

(5-158)

It is important to note in Eq. (5-158) that 0 ≤ ag, τg ≤ 1.0 while 0 ≤ εg ≤ ag. Equation (5-158) constitutes a two-parameter model which may be fitted with only two empirical emissivity data points. To obtain the constants ag and τg in Eq. (5-158) at fixed composition and temperature, denote the two emissivity data points as εg,2 = εg(2pL) > εg,1 = εg(pL) and recognize that εg,1 = ag(1 − τg) and εg,2 = ag(1 − τ2g) = ag(1 − τg)(1 + τg) = εg,1(1 + τg). These relations lead directly to the final emissivity fitting equations εg,2 τg =  − 1 εg,1

mathematical software utilities. The clear plus gray WSGG fitting procedure is demonstrated in Example 8. The Zone Method and Directed Exchange Areas Spectral dependence of real gas spectral properties is now introduced into the zone method via the WSGG spectral model. It is still assumed, however, that all surface zones are gray isotropic emitters and absorbers. General Matrix Representation We first define a new set of four q, GS q, and GG q which are denoted directed exchange areas q SS, SG by an overarrow. The directed exchange areas are obtained from the total exchange areas for gray gases by simple matrix multiplication using weighting factors derived from the WSGG spectral model. The directed exchange areas are denoted by an overarrow to indicate the “sending” and “receiving” zone. The a-weighting factors for transfer originating at a gas zone ag,i are derived from WSGG gas emissivity calculations, while those for transfers originating at a surface zone, ai are derived from appropriate WSGG gas absorptivity calculations. Let agIp = [ap,g,iδi,j] and aIp = [ap,iδi,j] represent the P [M × M] and [N × N] diagonal matrices comprised of the appropriate WSGG a constants. The directed exchange areas are then computed from the associated total gray gas exchange areas via simple diagonal matrix multiplication. q P ⎯⎯ SS = SSp◊aIp P q ⎯⎯ SG = SGp◊agIp

(5-161a)

[M × N]

(5-161b)

p=1

q P ⎯⎯ GS = GSp◊aIp

(5-159a)

[M × N]

(5-161c)

p=1

q P ⎯⎯ GG . = GGp◊agIp

and εg,1 ag =  2 − εg,2εg,1

[M × M]

p=1

[N × N]

(5-161d)

p=1

(5-159b) with

The clear plus gray WSGG spectral model also readily leads to values for gas absorptivity and transmissivity, with respect to some appropriate surface radiation source at temperature T1, for example, αg,1 = ag,1(1 − τg)

(5-160a)

τg,1 = ag,1⋅τg

(5-160b)

and In Eqs. (5-160) the gray gas transmissivity τg is taken to be identical to that obtained for the gas emissivity εg. The constant ag,1 in Eq. (5-160a) is then obtained with knowledge of one additional empirical value for αg,1 which may also be obtained from the correlations in Table 5-5. Notice further in the definitions of the three parameters εg, αg,1, and τg,1 that all the temperature dependence is forced into the two WSGG constants ag and ag,1. The three clear plus gray WSGG constants ag, ag,1, and τg are functions of total pressure, temperature, and mixture composition. It is not necessary to ascribe any particular physical significance to them. Rather, they may simply be visualized as three constants that happen to fit the gas emissivity data. It is noteworthy that three constants are far fewer than the number required to calculate gas emissivity data from fundamental spectroscopic data. The two constants ag and ag,1 defined in Eqs. (5-158) and (5-160) can, however, be interpreted physically in a particularly simple manner. Suppose the gas absorption spectrum is idealized by many absorption bands (boxes), all of which are characterized by the identical absorption coefficient K. The a’s might then be calculated from the total blackbody energy fraction Fb (λT) defined in Eqs. (5-105) and (5-106). That is, ag simply represents the total energy fraction of the blackbody energy distribution in which the gas absorbs. This concept may be further generalized to real gas absorption spectra via the wideband stepwise gray spectral box model (Modest, op. cit., Chap. 14). When P ≥ 3, exponential curve-fitting procedures for the WSGG spectral model become significantly more difficult for hand computation but are quite routine with the aid of a variety of readily available

q P KI = KIp◊agIp

[N × N]

(5-161e)

p=1

In contrast to the total exchange areas which are always independent q, GS q, and GG q are of temperature, the four directed arrays q SS , SG dependent on the temperatures of each and every zone, i.e., as in ap,i = ap (Ti). Moreover, in contrast to total exchange areas, the directed arrays T q q are generally not symmetric and GS q ≠ Sq SS and GG G . Finally, since the directed exchange areas are temperature-dependent, iteration may be required to update the aIp and agIp arrays during the course of a calculation. There is a great deal of latitude with regard to fitting the WSGG a constants in these matrix equations, especially if N > 1 and composition variations are to be allowed for in the gas. An extensive discussion of a fitting for N > 1 is beyond the scope of this presentation. Details of the fitting procedure, however, are presented in Example 12 in the context of a single-gas zone. Having formulated the directed exchange areas, the governing matrix equations for the radiative flux equations at each surface zone and the radiant source term are then given as follows: q q Q = εAI◊E − SS ⋅E − SG⋅Eg (5-162a) q q q S′ = GG⋅Eg + GS⋅E − 4KI⋅VI⋅Eg (5-162b) or the alternative forms r q r q Q = [EI⋅SS − SS ⋅EI]◊1M + [EI⋅SG − SG ⋅EgI]◊1N (5-163a) r q r q S′ = −[EgI⋅GS − GS⋅EI]◊1M − [EgI⋅GG − GG ⋅EgI]◊1N (5-163b) It may be proved that the Q and S′ vectors computed from Eqs. (5162) and (5-163) always exactly satisfy the overall (scalar) radiant energy balance 1MT ◊Q = 1NT ◊S′. In words, the total radiant gas emission for all gas zones in the enclosure must always exactly equal the total radiant energy received at all surface zones which comprise the enclosure. In Eqs. (5-162) and (5-163), the following definitions are employed for the four forward-directed exchange areas r qT r qT r qT r qT SS = SS SG = GS GS = SG GG = GG (5-64a,b,c,d)

HEAT TRANSFER BY RADIATION such that formally there are some eight matrices of directed exchange areas. The four backward-directed arrays of directed exchange areas must satisfy the following conservation relations q q SS ◊1M + SG ◊1N = εI⋅AI⋅1M (5-165a) q q q 4KI⋅VI⋅1N = GS⋅1M + GG ⋅1N (5-165b)

general matrix equations as R = [1(A1 − ⎯s⎯1⎯s⎯1⋅ρ1)]. There are two WSCC clear plus gray constants a1 and ag, and only one unique direct exchange area which satisfies the conservation relation ⎯s⎯1⎯s⎯1 + ⎯s⎯1⎯g = A1. The only two physically meaningful directed exchange areas are those between the surface zone A1 and the gas zone ag⋅ε1A1⋅s⎯s1⎯⎯g q (5-169a) S1G =  ε ⋅A + ρ ⋅s⎯⎯⎯g 1

Subject to the restrictions of no scatter and diffuse surface emission and reflection, the above equations are the most general matrix statement possible for the zone method. When P = 1, the directed exchange areas all reduce to the total exchange areas for a single gray gas. If, in addition, K = 0, the much simpler case of radiative transfer in a transparent medium results. If, in addition, all surface zones are black, the direct, total, and directed exchange areas are all identical. Allowance for Flux Zones As in the case of a transparent medium, we now distinguish between source and flux surface zones. Let M = Ms + Mf represent the total number of surface zones where Ms is the number of source-sink zones and Mf is the number of flux zones. The flux zones are the last to be numbered. To accomplish this, partition the E Q surface emissive power and flux vectors as E = 1 and Q = 1 , E2 Q2 where the subscript 1 denotes surface source/sink zones whose emissive power E1 is specified a priori, and subscript 2 denotes surface flux zones of unknown emissive power vector E2 and known radiative flux vector Q2. Suppose the radiative source vector S′ is known. Appropriate partitioning of Eqs. (5-162) then produces q q q Q1 E1 SS εAI1,1 0 SS1,2 ⋅ E1 − SG1 ⋅ E 1,2 ⋅   =  g q q 0 εAI2,2 Q2 E2 q E2 SG1 SS 2 ,1 SS 2,2 (5-166a)

 

  

 

E q q q q S′ = GG ⋅Eg + [GS 1 GS 2] 1 − 4 KI ⋅VI⋅Eg (5-166b) E2 where the definitions of the matrix partitions follow the conventions with respect to Eq. (5-120). Simultaneous solution of the two unknown vectors in Eqs. (5-166) then yields q q q q E2 = RP⋅[SS 2,1 + SG 2⋅PP⋅GS1]⋅E1 + RP⋅[Q2 − SG 2⋅PP⋅S′] (5-167a)

 

and q Eg = PP⋅[GS1

E q GS 2] 1 E2

  − PP⋅S′

(5-167b)

(5-169b)





where ⎯⎯ S1S2 = ε1ε2 A1A2 s⎯⎯1s⎯2 /det R−1 ⎯ SG =



ε1A1[(A2 − ρ2⋅s⎯⎯2⎯s2)⋅s⎯⎯1⎯g + ρ2⋅s⎯⎯1⎯s2⋅s⎯⎯2⎯g] ε2A2[(A1 − ρ1⋅s⎯⎯1⎯s1)⋅s⎯⎯2⎯g + ρ1⋅s⎯⎯1⎯s2⋅s⎯⎯1⎯g]

(5-170b)

/det R

−1

(5-170c)

⎯ ⎯T with GS = SG and the indicated determinate of R−1 is evaluated algebraically as det R−1 = (A1 − s⎯⎯1⎯s1·ρ1)·(A2 − s⎯⎯2 ⎯s2·ρ2) − ρ1·ρ2·s⎯⎯1⎯s22

(5-170d)

⎯⎯ ⎯⎯ For the WSGG clear gas components we denote SS K = 0 ≡ SS0 and ⎯⎯ ⎯⎯ SG K = 0 ≡ SG 0 = 0. Finally the WSGG arrays of directed exchange areas are computed simply from a-weighted sums of the gray gas total exchange areas as









⎯⎯ a1 0 q ⎯⎯ 1− a1 0 SS = SS0 · 0 1−a2 + SS · 0 a2 q ⎯ SG = SG ·ag

(5-168b)

The emissive power vectors E and Eg are then both known quantities for purposes of subsequent calculation. Algebraic Formulas for a Single Gas Zone As shown in Fig. 5-10, the three-zone system with M = 2 and N = 1 can be employed to simulate a surprisingly large number of useful engineering geometries. These include two infinite parallel plates confining an absorbing-emitting medium; any two-surface zone system where a nonconvex surface zone is completely surrounded by a second zone (this includes concentric spheres and cylinders), and the speckled two-surface enclosure. As in the case of a transparent medium, the inverse reflectivity matrix R is capable of explicit matrix inversion for M = 2. This allows derivation of explicit algebraic equations for all the required directed exchange areas for the clear plus gray WSGG spectral model with M = 1 and 2 and N =1. The Limiting Case M = 1 and N = 1 The directed exchange areas for this special case correspond to a single well-mixed gas zone completely surrounded by a single surface zone A1. Here the reflectivity matrix is a 1 × 1 scalar quantity which follows directly from the

1

Directed Exchange Areas for M = 2 and N = 1 For this case there are four WSGG constants, i.e., a1, a2, ag, and τg. There is one required value of K that is readily obtained from the equation K = −ln(τg)/LM, where τg = exp(−KLM). For an enclosure with M = 2, N = 1, and K ≠ 0, only three unique direct exchange areas are required ⎯ ⎯ ⎯ because conservation stipulates A1 = s⎯1s⎯2 + s⎯ 1s2 + s1g and A2 = s1s2 + s2s2 + s⎯ Eqs. (5-118) readily lead to the 2g. For M = 2 and N = 1, the matrix ⎯ ⎯⎯ general gray gas matrix solution for SS and SG with K ≠ 0 as ⎯⎯⎯⎯ ⎯⎯⎯⎯ ⎯ ⎯⎯ ⎯ ⎯⎯ ε1A1 − S1S2 − S1G S1S2 ⎯⎯⎯⎯ ⎯⎯⎯⎯ ⎯⎯⎯ SS = (5-170a) S1S1 ε2A2 − S1S2 − S2G

where two auxiliary inverse matrices RP and PP are defined as q q −1 PP = [4KI ⋅VI − GG ] (5-168a) q q q −1 RP = [εAI2,2 − SS 2,2 − SG2⋅PP⋅GS 2]

1

The total radiative flux Q1 at surface A1 and the radiative source term Q1 = S are given by q q Q1 = GS1⋅E1 − S1G ⋅Eg (5-169)

and

and

1

⎯⎯⎯ a1⋅ε1A1⋅s1g q GS1 =  ε1⋅A1 + ρ1⋅s⎯⎯1⎯g

   

  

5-37



(5-171a,b,c)



q q a1 0 qT GS = GS · 0 a ≠ SG 2 and finally

q q GG = ag⋅4KV − GS ⋅ 1 (5-171d) 1 The results of this development may be further expanded into algebraic form with the aid of Eq. (5-127) to yield the following ε1ε2A1A2⋅s⎯⎯2 ⎯s1]0(1 − a1) ε1ε2A1A2 ⎯s⎯2⎯s1⋅a1  ⎯⎯⎯s ] +  Sq (5-171e) 2S1 = ε ε A A + (ε A ρ + ε A ρ )s 1 1 1 2 1 1 2 2 2 1 2 1 0 det R−1 ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ ⎯⎯⎯ q = ε1A1[(A2 − ρ2⋅s⎯2⎯s⎯2)⋅s⎯1⎯g⎯ + ρ2⋅s⎯1⎯s⎯2⋅s⎯2⎯g⎯]ag det R−1 SG (5-171f) ε2A2[(A1 − ρ1⋅s1s1)⋅s2g + ρ1⋅s1s2⋅s1g]ag





and



q q q GS = GS1 GS2

(5-171g)

5-38

HEAT AND MASS TRANSFER

q whose matrix elements are given by GS 1 ≡ ε1A1[(A2 − ρ2 ·s⎯s⎯1⎯s1)·s⎯⎯1⎯g + ρ2·s⎯⎯1s⎯2· q ⎯s⎯⎯g]a /det R−1 and GS ⎯s⎯⎯s )·s⎯⎯⎯g + ρ ·s⎯⎯⎯s ·s⎯⎯⎯g]a / det R−1. 2 1 2 ≡ ε2A2[(A1 −ρ1·s 1 1 2 1 1 2 1 2 Derivation of the scalar (algebraic) forms for the directed exchange areas here is done primarily for pedagogical purposes. Computationally, the only advantage is to obviate the need for a digital computer to evaluate a [2 × 2] matrix inverse. Allowance for an Adiabatic Refractory with N = 1 and M = 2 Put N = 1 and M = 2, and let zone 2 represent the refractory surface. Let Q2 = 0 and ε2 ≠ 0, and it then follows that we may define a refractory-aided directed exchange area Sq 1GR by q q S1S2 S2G q Sq 1GR = S1G +  q q S1S2 + S2G

(5-172a)

Assuming radiative equilibrium, the emissive power of the refractory may also be calculated from the companion equation q q S2S1⋅E1 + S2G ⋅Eg E2 =  q q S2S1 + S2G

(5-172b)

In this circumstance, all the radiant energy originating in the gas volume is transferred to the sole sink zone A1. Equation (5-172a) is thus tantamount to the statement that Q1 = S′ or that the net emission from the source ultimately must arrive at the sink. Notice that if ε1 = 0, Eq. (5-172a) leads to a physically incongruous statement since all the directed exchange areas would vanish and no sink would exist. Even for the simple case of M = 2, N = 1, the algebraic complexity of Eqs. (5-171) suggests that numerical matrix manipulation of directed exchange areas is to be preferred rather than calculations using algebraic formulas. Engineering Approximations for Directed Exchange Areas Use of the preceding equations for directed exchange areas with M = 2, N = 1 and the WSGG clear plus gray gas spectral approximation requires knowledge of three independent direct exchange areas. It also formally requires evaluation of three WSGG weighting constants a1, a2, and ag with respect to the three temperatures T1, T2, and Tg. Further simplifications may be made by assuming that radiant transfer for the entire enclosure is characterized by the single mean beam length LM = 0.88⋅4⋅VA. The requisite direct exchange areas are then approximated by A1⋅F1,1 A1⋅F1,2 s⎯s⎯ = τg A ⋅F (5-173a) A ⋅F



with

2

2,1

2



2,2

 

⎯ = (1 − τ ) A1 s⎯g g A2

(5-173b)





also with

 

⎯ = (1 − τ ) A1 s⎯g g A2

(5-174a)

σ[εg T4g − αg,1T41] = εmσ(T4g − T41)

(5-176a)

εg − αg,1(T1Tg)4 εm ≡  1 − (T1Tg)4

(5-176b)

The calculation then proceeds by computing two values of εm at the given Tg and T1 temperature pair and the two values of pLM and 2pLM. We thereby obtain the expression εm = am(1 − τm). It is then assumed that a1 = a2 = ag = am for use in Eqs. (5-171). This simplification may be used for M > 2 as long as N = 1. This simplification is illustrated in Example 12. Example 12: WSGG Clear plus Gray Gas Emissivity Calculations Methane is burned to completion with 20 percent excess air (50 percent relative humidity at 298 K or 0.0088 mol water/mol dry air) in a furnace chamber of floor dimensions 3 × 10 m and height 5 m. The entire surface area of the enclosure is a gray sink with emissivity of 0.8 at temperature 1000 K. The confined gas is well stirred at a temperature of 1500 K. Evaluate the clear plus gray WSGG constants and the mean effective gas emissivity, and calculate the average radiative flux density to the enclosure surface. Two-zone model, M = 1, N = 1: A single volume zone completely surrounded by a single sink surface zone. Function definitions: Gas emissivity: Eq. (5-140a)

εgF(Tg, pL, b, n) := b⋅(pL − 0.015)n ÷ Tg

Gas absorptivity: αg1F(Tg, T1, pL, b, n) Εq. (5-141) εgF(T1, pL⋅T1 ÷ Tg, b, n)⋅T1⋅(Tg ÷ T1)0.5 :=  T1 Mean effective gas emissivity: Eq. (5-176a)

εg − αg⋅(T1 ÷ Tg)4 εgm(εg, αg, Tg, T1) :=  1 − (T1 ÷ Tg)4

W σ  5.670400 × 10−8  m2⋅K4 Enclosure input parameters:

Physical constants:

A1 := 190 m2

Tg := 1500 K

T1 := 1000 K

ε1 := 0.8

ρ1 := 1 − ε1

ρ1 = 0.2

Eg := σ⋅T

kW E1 = 56.70  m2

E1 := σ⋅T

4 g

V := 150 m3

kW Eg = 287.06  m2

Stoichiometry yields the following mole table: Mole Table: Basis 1.0 mol Methane

(5-174b)

where again τg is obtained from the WSGG fit of gas emissivity. These approximate formulas clearly obviate the need for exact values of the direct exchange areas and may be used in conjunction with Eqs. (5-171). For engineering calculations, an additional simplification is sometimes warranted. Again characterize the system by a single mean beam length LM = 0.88⋅4⋅VA and employ the identical value of τg = KLM for all surface-gas transfers. The three a constants might then be obtained by a WSGG data-fitting procedure for gas emissivity and gas absorptivity which utilizes the three different temperatures Tg, T1, and T2. For engineering purposes we choose a simpler method, however. First calculate values of εg and αg1 for gas temperature Tg with respect to the dominant (sink) temperature T1. The net radiative flux between an isothermal gas mass at temperature Tg and a black isothermal bounding surface A1 at temperature T1 (the sink) is given by Eq. (5-138) as Q1,g = A1σ(αg,1T41 − εgT4g)

or

4 1

and for the particular case of a speckled enclosure A21 A1⋅A2 τg s⎯s⎯ =  A A22 A1 + A2 1⋅A2

It is clear that transfer from the gas to the surface and transfer from the surface into the gas are characterized by two different constants of proportionality, εg and αg,1. To allow for the difference between gas emissivity and absorptivity, it proves convenient to introduce a single mean gas emissivity defined by

(5-175)

Species

MW

Moles in

Mass in

Moles out

Y out

CH4 O2 N2 CO2 H2O Totals

16.04 32.00 28.01 44.01 18.02 27.742

1.00000 2.40000 9.02857 0.00000 0.10057 12.52914

16.04 76.80 252.93 0.00 1.81 347.58

0.00000 0.40000 9.02857 1.00000 2.10057 1.52914

0.00000 0.03193 0.72061 0.07981 0.16765 1.00000

pW := .16765 atm pW ÷ pC = 2.101

pC := 0.07981 atm

p := pW + pC

p = 0.2475 atm

The mean beam length is approximated by and

LM = 2.7789 m LM := 0.88⋅4⋅V ÷ A1 pLM := p⋅LM pLM = 0.6877 atm⋅m

pLM := 0.6877

The gas emissivities and absorptivities are then calculated from the two constant correlation in Table 5-5 (column 5 with pw /pc = 2.0) as follows: εg1 := εgF(1500, pLM, 540, .42)

εg1 = 0.3048

HEAT TRANSFER BY RADIATION εg2 := εgF(1500, 2pLM, 540, .42)

εg2 = 0.4097

αg11 := αg1F(1500, 1000, pLM, 444, .34)

αg11 = 0.4124

αg12 := αg1F(1500, 1000, 2pLM, 444, .34)

αg12 = 0.5250

Case (a): Compute Flux Density Using Exact Values of the WSGG Constants εg2 τg :=  − 1 εg1

εg1 ag :=  1 − τg

εg := ag ⋅(1 − τg)

αg11 ag1 :=  1 − τg

τg = 0.3442

ag = 0.4647

εg = 0.3048

ag1 = 0.6289

and the WSGG gas absorption coefficient (which is necessary for calculation of −(ln τg) 1 direct exchange areas) is calculated as K1 :=  or K1 = 0.3838  LM m Compute directed exchange areas: Eqs. (5-169) s1g := (1 − τg)⋅ A1

s1g = 124.61 m2

ag⋅ε1⋅A1⋅s1g DS1G :=  ε1⋅A1 + ρ1⋅s1g

ag1⋅ε1⋅A1⋅s1g DGS1 :=  ε1⋅A1 + ρ1⋅s1g

DS1G = 49.75 m2

DGS1 = 67.32 m2

And finally the gas to sink flux density is computed as Q1 kW  = −55.07  A1 m2 Case (b): Compute the Flux Density Using Mean Effective Gas Emissivity Approximation

Q1 := DGS1⋅E1 − DS1G⋅Eg

Q1 = −10464.0 kW

εgm1 := εgm(εg1,αg11,Tg,T1) εgm2 := εgm(εg2,αg12,Tg,T1)

εgm1 = 0.2783 εgm2 = 0.3813

εgm2 τm :=  − 1 εgm1

εgm1 am :=  1 − τm

εgm := am ⋅(1 − τm)

τm = 0.3701

am = 0.4418

εgm = 0.2783

ε1⋅am⋅s1gm⋅A1 S1Gm :=  ε1⋅A1 + ρ1⋅ s1g m) S1Gm⋅(E1 − Eg) q1m :=  A1

S1Gm = 45.68 m2

s1gm := (1 − τm)⋅A1 s1gm = 119.67 m2

kW q1m = −55.38  m2

Q1 kW compared with  = −55.07  A1 m2 The computed flux densities are nearly equal because there is a single sink zone A1. (This example was developed as a MATHCAD 14® worksheet. Mathcad is a registered trademark of Parametric Technology Corporation.)

proportional to the feed rate, we employ the sink area A1 to define ⋅ a dimensionless firing density as NFD = Hf σ T4Ref ⋅A1 where TRef is some characteristic reference temperature. In practice, gross furnace output performance is often described by using one of several furnace efficiencies. The most common is the gas or gas-side furnace efficiency ηg, defined as the total enthalpy transferred to furnace internals divided by the total available feed enthalpy. Here the total available feed enthalpy is defined to include the lower heating value (LHV) of the fuel plus any air preheat above an arbitrary ambient datum temperature. Under certain conditions the definition of furnace efficiency reduces to some variant of the simple equation ηg = (TRef − Tout)(TRef − T0) where again TRef is some reference temperature appropriate to the system in question. The Long Plug Flow Furnace (LPFF) Model If a combustion chamber of cross-sectional area ADuct and perimeter PDuct is sufficiently long in the direction of flow, compared to its mean hydraulic radius, L>>Rh = ADuct /PDuct, the radiative flux from the gas to the bounding surfaces can sometimes be adequately characterized by the local gas temperature. The physical rationale for this is that the magnitudes of the opposed upstream and downstream radiative fluxes through a cross section transverse to the direction of flow are sufficiently large as to substantially balance each other. Such a situation is not unusual in engineering practice and is referred to as the long furnace approximation. As a result, the radiative flux from the gas to the bounding surface may then be approximated using two-dimensional directed exchange q ∂(S1G ) q areas, S1G /A1 ≡ , calculated using methods as described previously. ∂A1 Consider a duct of length L and perimeter P, and assume plug flow in the direction of flow z. Further assume high-intensity mixing at the entrance end of the chamber such that combustion is complete as the combustion products enter the duct. The duct then acts as a long heat exchanger in which heat is transferred to the walls at constant temperature T1 by the combined effects of radiation and convection. Subject to the long furnace approximation, a differential energy balance on the duct then yields q S1G ⎯ dT m˙ Cp g = P  σ (T4g − T 41) + h(Tg − T1) (5-177) A1 dz





⎯ where m˙ is the mass flow rate and Cp is the heat capacity per unit mass. Equation (5-177) is nonlinear with respect to temperature. To solve Eq. (5-177), first linearize the convective heat-transfer term in ⎯ the 4 4 3 right-hand ⎯ side with the approximation ∆T = T2 − T1 ≈ (T2 − T1)4T1,2 where T1,2 = (T1 + T2)2. This linearization underestimates ∆T by no more than 5 percent when T2/T1 < 1.59. Integration of Eq. (5-177) then leads to the solution (Tg,in − Tg,out)⋅T1 (Tg,out − T1)(Tg,in + T1) 4 ln  + 2.0 tan−1  = − (T 21 + Tg,in⋅Tg,out) (Tg,out + T1)(Tg,in − T1) Deff









(5-178)

ENGINEERING MODELS FOR FUEL-FIRED FURNACES Modern digital computation has evolved methodologies for the design and simulation of fuel-fired combustion chambers and furnaces which incorporate virtually all the transport phenomena, chemical kinetics, and thermodynamics studied by chemical engineers. Nonetheless, there still exist many furnace design circumstances where such computational sophistication is not always appropriate. Indeed, a practical need still exists for simple engineering models for purposes of conceptual process design, cost estimation, and the correlation of test performance data. In this section, the zone method is used to develop perhaps the simplest computational template available to address some of these practical engineering needs. Input/Output Performance Parameters for Furnace Operation The term firing density is typically used to define the basic operational input parameter for fuel-fired furnaces. In practice, firing density is often defined as the input fuel feed rate per unit area (or volume) of furnace heat-transfer surface. Thus defined, the firing ⋅ density is a dimensional quantity. Since the feed enthalpy rate Hf is

5-39

The LPFF model is described by only two dimensionless parameters, namely an effective firing density and a dimensionless sink temperature, viz., NFD Deff =  q S1G  A1 + NCR

and Θ1 = T1/Tg,in

(5-178a,b)



Here the dimensionless firing density, NFD, and a dimensionless convection-radiation namber NCR are defined as – m˙ Cp h NFD =  and NCR =  −3 σ·T13 A1 4σ Tg,1

(5-178c,d)

– where A1 = PL is the duct surface area (the sink area), and T g,1 = – (T g + T1)/2 is treated as a constant. This definition of the effective dimensionless firing density, Deff, clearly delineates the relative

5-40

HEAT AND MASS TRANSFER

roles of radiation and convective heat transfer since radiation and convection are identified as parallel (electrical) conductances. In analogy with a conventional heat exchanger, Eq. (5-178) displays two asymptotic limits. First define Tg,in − Tg,out T −T − g,out1  ηf =  (5-179) Tg,in − T1 Tg,in − T1 = 1  as the efficiency of the long furnace. The two asymptotic limits with respect to firing density are then given by Deff << 1,

Tg,out → T1

ηf → 1

(5-179a)

and Deff >> 1,

Tg,out → Tg,in

4 η f →  R−1 R−1 Deff 1 −  − 2   R+1 R2 + 1





(5-179b)

where R  Tg, in T1 = 1Θ1. For low firing rates, the exit temperature of the furnace gases approaches that of the sink; i.e., sufficient residence time is provided for nearly complete heat removal from the gases. When the combustion chamber is overfired, only a small fraction of the available feed enthalpy heat is removed within the furnace. The exit gas temperature then remains essentially that of the inlet temperature, and the furnace efficiency tends asymptotically to zero. It is important to recognize that the two-dimensional exchange area q q S1 G ∂(S1G)  ≡  in the definition of Deff can represent a lumped twoA1 ∂A1 dimensional exchange area of somewhat arbitrary complexity. This quantity also contains all the information concerning furnace geometry and gas and surface emissivities. To compare the relative importance of radiation with respect to ⎯convection, suppose h = 10 Btu(hr⋅ ft2 ⋅°R) = 0.057 kW(K⋅m2) and Tg,1 = 1250 K, which leads to the numerical value NCR = 0.128; or, in general, NCR is of order 0.1 or less. The importance of the radiation contribution is estimated by bounding the magnitude of the dimensionless directed exchange area. For the case of a single gas zone completely surrounded by a black enclosure, Eq. (5-169) reduces to simply Sq 1G/A1 = εg ≤ 1.0, and it is evident that the magnitude of the radiation contribution never exceeds unity. At high temperatures, radiative effects can easily dominate other modes of heat transfer by an order of magnitude or more. When mean beam length calculations are employed, use LM/D = 0.94 for a cylindrical cross section of diameter D, and 2H⋅W LM0 =  H+W for a rectangular duct of height H and width W. The Well-Stirred Combustion Chamber (WSCC) Model Many combustion chambers utilize high-momentum feed conditions with associated high-intensity mixing. The well-stirred combustion chamber (WSCC) model assumes a single gas zone and high-intensity mixing. Moreover, combustion and heat transfer are visualized to occur simultaneously within the combustion chamber. The WSCC model is characterized by some six temperatures which are listed in rank order as T0, Tair, T1, Te, Tg, and Tf. Even though the combustion chamber is well mixed, it is arbitrarily assumed that the gas temperature within the enclosure Tg is not necessarily equal to the gas exit temperature Te. Rather the two temperatures are related by the simple relation ∆Tge ≡ Tg − Te, where ∆Tge ≈ 170 K (as a representative value) is introduced as an adjustable parameter for purposes of data fitting and to make allowance for nonideal mixing. In addition, T0 is the ambient temperature, Tair is the air preheat temperature, and Tf is a pseudoadiabatic flame temperature, as shall be explained in the following development. The condition ∆Tge ≡ 0 is intended to simulate a perfect continuous well-stirred reactor (CSTR).

Dimensional WSCC Approach A macroscopic enthalpy balance on the well-stirred combustion chamber is written as −∆H = HIn − HOut = QRad + QCon + QRef (5-180) q 4 4 Here QRad = S1GRσ(Tg − T1) represents radiative heat transfer to the sink (with due allowance for the presence of any refractory surfaces). And the two terms QCon = h1A1(Tg − T1) and QRef = UAR(Tg − T0) formulate the convective heat transfer to the sink and through the refractory, respectively. Formulation of the left-hand side of Eq. (5-180) requires representative thermodynamic data and information on the combustion stoichiometry. In particular, the former includes the lower heating value of the fuel, the temperature-dependent molal heat capacity of the inlet and outlet streams, and the air preheat temperature Tair. It proves especially convenient now to introduce the definition of a pseudoadiabatic flame temperature Tf, which is not the true adiabatic flame temperature, but rather is an adiabatic flame temperature based on the average heat capacity of the combustion products over the temperature interval T0 < T < Te. The calculation of Tf does not allow for dissociation of chemical species and is a surrogate for the total enthalpy content of the input fuel-air mixture. It also proves to be an especially convenient system reference temperature. Details for the calculation of Tf are illustrated in Example 13. In terms of this particular definition of the pseudoadiabatic flame temperature Tf, the total enthalpy change and gas efficiency are given simply as ⎯ ⎯ · ∆H = H f − m· ⋅ CP, Prod (Te − T0) = m· CP,Prod (Tf − Te) (5-181a,b) · · ⎯ where Hf  m⋅ Cp,Prod (Tf − T0) and Te = Tg − ∆Tge. This particular definition of Tf leads to an especially convenient formulation of furnace efficiency · ⎯ ⋅ m ⋅CP, Prod (Tf − Te) Tf − Te ηg = QHf =  =  (5-182) · ⎯ m ⋅CP,Prod (Tf − T0) Tf − T0 ⎯ In Eq. (5-182), m⋅ is the total mass flow rate and CP,Prod [J(kg⋅K)] is defined as the average heat capacity of the product stream over the temperature interval T0 < T < Te. The final overall enthalpy balance is then written as · ⎯ m⋅C (T − T ) = Sq G σ (T 4 − T 4) + h A (T − T ) + UA (T − T ) P,Prod

f

e

1

R

g

1

1

1

g

1

R

g

0

(5-183) with Te = Tg − ∆Tge. Equation (5-183) is a nonlinear algebraic equation which may be solved by a variety of iterative methods. The sole unknown quantity, however, in Eq. (5-183) is the gas ⎯ temperature Tg. It should be recognized, in particular, that Tf, Te, C P,Prod, and the directed exchange area are all explicit functions of Tg. The method of solution of Eq. (5-183) is demonstrated in some detail in Example 13. Dimensionless WSCC Approach In Eq. (5-183), assume the convective heat loss through the refractory is negligible, and linearize the convective heat transfer to the sink. These approximations lead to the result ⎯ ⎯3 4 4 4 4 m· ⋅CP,Prod (Tf − Tg + ∆Tge) = Sq 1GR σ (Tg − T 1) + h1 A1(Tg − T 1)4T g,1 (5-184) ⎯ where Tg,1 = (Tg + T1)2 is some characteristic average temperature which is taken as constant. Now normalize all temperatures based on the pseudoadiabatic temperature as in Θi = Ti Tf. Equation (5-184) then leads to the dimensionless equation Deff (1 − Θg + ∆* ) = (Θ4g − Θ41) (5-185) where again Deff = NFD (Sq 1 GA1 + NCR) is defined exactly as in the case of the LPFF model, with the proviso that the ⎯ WSCC dimensionless firing density is defined here as NFD = m· CP,Prod /(σ T3f ⋅A1). The dimensionless furnace efficiency follows directly from Eq. (5-182) as 1 − Θg + ∆* 1 − Θe ηg =  =  1 − Θ0 1 − Θ0

(5-186a)

HEAT TRANSFER BY RADIATION We also define a reduced furnace efficiency η′g η′g  (1 − Θ0)ηg = 1 − Θg + ∆*

(5-186b)

Since Eq. (5-186b) may be rewritten as Θg = (1 + ∆* − η′g), combination of Eqs. (5-185) and (5-186b) then yields the final result Deff η′g = (1 + ∆* − η′g)4 − Θ41

(5-187)

Equation (5-187) provides an explicit relation between the modified furnace efficiency and the effective firing density directly in which the gas temperature is eliminated. Equation (5-187) has two asymptotic limits Deff << 1

Θg → Θ1

η′g → 1 − Θ1 + ∆* and

Deff >> 1

Θg →1 + ∆*

occupies 60 percent of the total interior furnace area and is covered with two rows of 5-in (0.127-m) tubes mounted on equilateral centers with a center-tocenter distance of twice the tube diameter. The sink temperature is 1000 K, and the tube emissivity is 0.7. Combustion products discharge from a 10-m2 duct in the roof which is also tube-screen covered and is to be considered part of the sink. The refractory (zone 2) with emissivity 0.6 is radiatively adiabatic but demonstrates a small convective heat loss to be calculated with an overall heat transfer coefficient U. Compute all unknown furnace temperatures, the gas-side furnace efficiency, and the mean heat flux density through the tube surface. Use the dimensional solution approach for the well-stirred combustor model and compare computed results with the dimensionless WSCC and LPFF models. Computed values for mean equivalent gas emissivity obtained from Eq. (5-174b) and Table 5-5 for Tg = 2000 K for LM = 2.7789 m and T1 = 1000 K are found to be

(5-188a)

and

(1 + ∆*)4 − Θ41 η′g →  Deff + 4(1 + ∆*)3

Θe →1 (5-188b)

Figure 5-23 is a plot of η′g versus Deff computed from Eq. (5-187) for the case ∆* = 0. The asymptotic behavior of Eq. (5-189) mirrors that of the LPFF model. Here, however, for low firing densities, the exit temperature of the furnace exit gases approaches Θe = Θ1 − ∆* rather than the sink temperature. Moreover, for Deff << 1 the reduced furnace efficiency adopts the constant value η′g = 1 − Θe = 1 + ∆* − Θ1. Again at very high firing rates, only a very small fraction of the available feed enthalpy heat is recovered within the furnace. Thus the exit gas temperature remains nearly unchanged from the pseudoadiabatic flame temperature [Te ≈ Tf,] and the gas-side efficiency necessarily approaches zero. Example 13: WSCC Furnace Model Calculations Consider the furnace geometry and combustion stoichiometry described in Example 12. The end-fired furnace is 3 m wide, 5 m tall, and 10 m long. Methane at a firing rate of 2500 kg/h is burned to completion with 20 percent excess air which is preheated to 600°C. The speckled furnace model is to be used. The sink (zone 1)

Tg = 1500 K

am = 0.44181

τm = 0.37014

εm = 0.27828

Tg = 2000 K

am = 0.38435

τm = 0.41844

εm = 0.22352

Over this temperature range the gas emissivity may be calculated by linear interpolation. Additional heat-transfer and thermodynamic data are supplied in context. Three-zone speckled furnace model, M = 2, N = 1: Zone 1: Sink (60 percent of total furnace area) Zone 2: Refractory surface (40 percent of total furnace area) Physical constants: W σ ≡ 5.670400 × 10−8  2 4 m ⋅K

Linear interpolation function for mean effective gas emissivity constants: Y1⋅(2000 K − T) + Y2⋅(T − 1500 K) LINTF(T, Y2, Y1):=  (1500 K < T < 2000 K) 500 K Enclosure input parameters: Vtot := 150 m3

Atot := 190 m2 C1 := 0.6

C2 := 1 − C1

A1 := C1⋅Atot A2 := C2⋅Atot Dtube := .127 m Direct exchange areas for WSGG clear gas component (temperature independent): C1 C2 F := C1 C2





AI :=

0 A A1

0

ss1 := AI·F

2

ss1 =

 45.60 68.40

m

45.60 30.40

1

Reduced gas-side furnace efficiency η′ g

Θ1 = 0.0

Θ1 = 0.7 Θ1 = 0.8 0.1

0.01 0.01

Θ1 = 0.9

0.1

5-41

1

10

100

Dimensionless effective firing density Deff FIG. 5-23 Reduced gas-side furnace efficiency versus effective firing density for well-stirred combustion chamber model. ∆* = 0, Θ1 = 0.0, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9.

2

5-42

HEAT AND MASS TRANSFER

Equivalent gray plane emissivity calculations for sink: εtube := 0.7 Fbar := 0.987 (from Fig. 5-13, curve 6 with ratio = 2.0) 1 ε1eq :=  Dtube 1 1 1  − 1 + 2⋅  ⋅  − 1 +  2⋅π⋅Dtube εtube Fbar 1







ε1 := ε1eq ε2 := 0.6

εI :=

ε1 0

17.94 m2 11.96

31.24 67.93

31.24 14.36

m

2

T0 := 298.15 K

∆Tge := 170 K

69.15

DSG := am⋅εI⋅AI⋅R2⋅sg2

kW E 2 = 231.24  m2

E1 := σ⋅T14

σ E

T2 := 2

kW E1 = 56.704  m2

0.25

T2 = 1421.1 K

Mean beam length calculations:

Enthalpy balance: Basis: 1 mole CH4:

Vtot LM0 := 4⋅  LM0 = 3.1579 m L M := 0.88LM0 LM = 2.7789 m Atot Stoichiometric and thermodynamic input data: g kg g MCH4dot := 2500  MWCH4 := 16.04  MWin := 27.742  h mol mol

hin := LHV + (ΣMols − 1)⋅MCpair⋅(Tair − T0)

cal LHV := 191760  mol

Mols := 12.52914





T MCp(T) := 7.01 + 0.875  1000 K

cal MCpair := 7.31  mol⋅K

  mol⋅K cal

(800 K < T < 1200 K)

Mdot is the total mass flowrate and NCH4dot is the molal flowrate of CH4. MCH4dot⋅MWin⋅ΣMols Mdot :=  MWCH4 kg Mdot = 54174.5  h

MCH4dot NCH4dot :=  MWCH4

mol NCH4dot = 155860.3  h

Overall refractory heat-transfer coefficient: Dr := 0.343 m

kW kr := 0.00050  m⋅K

kW h0 := 0.0114·  m2⋅K

Btu h0 = 2.0077  h⋅ft2R

Btu h1 = 2.9939  h⋅ft2R

kW U = 0.001201  2 m ⋅K

kW h1 := 0.0170  m2⋅K

h⋅ft R

Te := Tg − ∆Tge Te = 1589.2 K kW Eg := σ⋅Tg4 Eg = 543.05  m2 Compute temperature-dependent mean effective gas emissivity via linear interpolation: τm := LINTF(Tg, 0.37014, 0.418442) am := LINTF (Tg, 0.44181, 0.38435) εm = 0.2512

Compare interpolated value: εcom := LINTF(Tg, 0.27828, 0.22352)

εcom = 0.2519

Direct and total exchange areas for WSGG gray gas component: ss2 := τm⋅ss1

11

sg2 := (AI − ss2)

hin Tf := T0 +  ΣMols MCp(Te)

hout := ΣMols⋅MCp(Te)⋅(Te − T0)

Tf = 2580.5 K

cal hout = 135880.8  mol

Hin := NCH4dot⋅hin

Hout := NCH4dot⋅hout ∆H := Hout − Hin

Hin = 43543.59 kW

Hout = 24630.51 kW ∆H = −18913.08 kW

Overall enthalpy balance: Q1g := DS1GR⋅(Eg − E1)

Q1g = 17308.45 kW

Q1Con := h1⋅A1⋅(Tg − T1)

Q1Con = 1471.26 kW

Q2Con := U⋅A2⋅(Tg − T0)

Q2Con = 133.37 kW ERROR ∆H

ERROR := Q1g + Q1Con + Q2Con + ∆H

%ERROR1 := 100 

ERROR = −0.0000 kW

%ERROR1 = 0.00000

DS1GR⋅E1 − (∆H + Q1Con + Q2Con) Tgcalc :=  DS1GR⋅σ





0.25

Tgcalc = 1759.16332 K

Tgnew = 1759.1633222935 K

Go to Start

END OF ITERATION LOOP: Final Gas Temperature Tg = 1759.16 K

Btu U = 0.2115  2

εm := am⋅(1 − τm)

Compute pseudoadiabatic flame temperature Tf :

Tg + Tgcalc Tgnew :=  2

Assume Tg := 1759.1633222447 K

am = 0.4141

cal hin = 240219.9  mol

Average assumed and calculated temperatures for next iteration

1 U :=  1 1 Dr  +  +  h0 h1 kr

START OF ITERATION LOOP: Successive Substitution with Tg as the Trial Variable

τm = 0.3934

SS2 := εI⋅AI⋅R2⋅ss2⋅εI

2

DSS2,1⋅E1 + DSG2⋅Eg Equation (5-172b): E2 :=  DSS2,1 + DSG2

Temperature and emissive power input data: Tair := 873.15 K

46.10 m

DSS1,2⋅DSG2 DS1GR := DSG1 +  DS1GR = 35.59 m2 Eq. (5-172a) DSS1,2 + DSG2 Compute refractory temperature (T2); assume radiative equilibrium:

2

T1 := 1000.0 K

sg2 =

DSS := (1 − am)⋅SS1 + am⋅SS2

ρI := identity (2) − εI

SS1 := εI⋅AI⋅R1⋅ss1⋅εI SS1 =



26.91

Refractory augmented directed gas-sink exchange area:

Total exchange areas for WSGG clear gas component: R1 := (AI − ss1⋅ρI)−1

17.94

Compute directed exchange areas:

ε1eq = 0.86988



0 ε

ss2 =

R2 := (AI − ss2⋅ρI)−1

Hin − Hout ηg :=  Hin

or

Tf − Te η1g :=  Tf − T0

ηg = 0.43435

η1g = 0.43435

Heat flux density calculations: Q1g + Q1Con q1 :=  A1

Dtube qtube := q1⋅ 2⋅  2⋅π⋅Dtube

kW q1 = 164.7  m2

kW qtube = 52.44  m2





Note: This example was also solved with ∆Tge = 0. The results were as follows: Tf = 2552.8 K, Tg = Te = 1707.1 K, T2 = 1381.1 K, ηg = 37.51 percent, Deff = 0.53371, and ∆H = 16332.7 kW. The WSCC model with ∆Tge = 0 predicts a lower performance bound. Compare dimensionless WSCC model: T Θ0 := 0 Tf

T1 Θ1 :=  Tf

∆Tge ∆Tstar :=  Tf

T1 + Tg Tg1 :=  2

MASS TRANSFER Θ0 = 0.1155

Θ1 = 0.3875

∆Tstar = 0.06588

MCp(Te) CpProd :=  MWin Cpprod NFD := Mdot⋅  σ⋅Tf3⋅A1 NFD = 0.17176

Tg1 = 1379.58 K h kg⋅K

CpProd = 0.000352 kW⋅ 

h1 NCR :=  4⋅σ⋅Tg13

NCR = 0.02855

ηprime := ηg⋅(1 − Θ0)

NFD Deff :=  DS1GR  + NCR A1 DS1GR Deff = 0.50409  = 0.31218 A1

(1 + ∆Tstar − ηprime)4 − Θ14 D1eff :=  ηprime

D1eff = 0.50350

versus Deff = 0.50409

This small discrepancy is due to linearization and neglect of convective refractory heat losses in the dimensionless WSCC model. Compare dimensionless LPFF model: 1 Rin :=  Θ1

Rin = 2.58050

Trial and error calculation to match effective firing densities: Assume: Tout := 1000.13763 K

Tout Rout :=  T1

R out = 1.00014

(Rout − 1)⋅(Rin + 1) Rin − Rout CLong := −ln  −2⋅atan  (Rin − 1)⋅(Rout + 1) 1 + Rin⋅Rout



Rin − Rout ηLf :=  Rin − 1





ηL f = 0.99991292

4  = 7.93513 Deff

CLong = 7.93514

versus ηg = 0.43435

Note: The long plug flow furnace Model is so efficient that it would be grossly underfired using the computed WSCC effective firing density. Of the two models, the LPFF model always predicts an upper theoretical performance limit. (This example was developed as a MATHCAD 14® worksheet. Mathcad is registered trademark of Parametric Technology Corporation.)

WSCC Model Utility and More Complex Zoning Models Despite its simplicity, the WSCC construct has a wide variety of practical uses and is of significant pedagogical value. Here an engineering situation of inordinate complexity is described by the definition of only eight dimensionless quantities Deff, NFD, Sq 1 GRA1, NCR, ηg, ∆*, Θ0, and Θ1. The first three are related by the simple algebraic definition Deff = NFD(Sq 1 GRA1 + NCR). These dimensionless quantities con-

5-43

tain all the physical input information for the model, namely, furnace parameters and geometry, radiative properties of the combustion products, and the stoichiometry and thermodynamics of the combustion process. The WSCC model leads to a dimensionless two-dimensional plot of reduced effective furnace efficiency versus dimensionless effective firing density (Fig. 5-23), which is characterized by only two additional parameters, namely, ∆* and Θ1. Of the models presented here, the WSCC model with ∆Tge = 0 produces the lowest furnace efficiencies. The long furnace model usually produces the highest furnace efficiency. This is really not a fair statement because two distinctly different pieces of process equipment are compared. In this regard, a more appropriate definition of the dimensionless − /(σ⋅T3 ⋅A ). firing density for the LPFF model might be N′FD = m⋅ C g,in 1 p It may be counterintuitive, but the WSCC and LPFF models generally do not characterize the extreme conditions for the performance of combustors as in the case of chemical reactors. Figure 5-23 has been used to correlate furnace performance data for a multitude of industrial furnaces and combustors. Typical operational domains for a variety of fuel-fired industrial furnaces are summarized in Table 5-7. The WSCC approach (or “speckled” furnace model) is a classic contribution to furnace design methodology which was first due to Hottel [op. cit.]. The WSCC model provides a simple furnace design template which leads to a host of more complex furnace models. These models include an obvious extension to a tanks-in-series model as well as multizone models utilizing empirical cold-flow velocity patterns. For more information on practical furnace design models, reference is made to Hottel and Sarofim (op. cit., Chap. 14). Qualitative aspects of process equipment have been treated in some detail elsewhere (Baukal, C. E., ed., The John Zink Combustion Handbook, CRC Press, Boca Raton, Fla., 2001). TABLE 5-7 Operational Domains for Representative Process Furnaces and Combustors Furnace or combustor type

Domain A

B C D E

Oil processing furnaces; radiant section of oil tube stills and cracking coils Domestic boiler combustion chambers Glass furnaces Soaking pits Gas-turbine combustors

Dimensionless sink temperature

Dimensionless firing density

Θ1 ≈ 0.4

0.1 < Deff < 1.0

Θ1 ≈ 0.2 0.7 < Θ1 < 0.8 Θ1 ≈ 0.6 0.4 < Θ1 < 0.7

0.5 < Deff < 1.1 0.035 < Deff < 0.8 0.7 < Deff < 1.1 4.0 < Deff < 25.0

MASS TRANSFER GENERAL REFERENCES: Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., Wiley, New York, 2002. Cussler, Diffusion: Mass Transfer in Fluid Systems, 2d ed., Cambridge University Press, Cambridge, 1997. Danner and Daubert, Manual for Predicting Chemical Process Design Data, AIChE, New York, 1983. Daubert and Danner, Physical and Thermodynamic Properties of Pure Chemicals, Taylor and Francis, Bristol, Pa., 1989–1995. Gammon, Marsh, and Dewan, Transport Properties and Related Thermodynamic Data of Binary Mixtures, AIChE, New York. Part 1, 1993: Part 2, 1994. Geankoplis, Transport Processes and Separation Process Principles, 5th ed., Prentice-Hall PTR, Upper Saddle River, N.J., 2003. Kirwan, “Mass Transfer Principles,” Chap. 2 in Rousseau, R. W. (ed.), Handbook of Separation Process Technology, Wiley, New York, 1987. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 7th ed., McGraw-Hill, New York, 2006. Middleman, An Introduction to Mass and Heat Transfer, Wiley, New York, 1997. Poling, Prausnitz, and O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001. Schwartzberg and Chao, Food Technol., 36(2), 73 (1982). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, New York, 1975. Skelland, Diffusional Mass Transfer, Wiley, New York, 1974. Taylor and Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. Thompson, Introduction to Transport Phenomena, Prentice-Hall PTR, Upper Saddle River, N.J., 1999. Treybal, MassTransfer Operations, 3d ed., McGraw-Hill, New York, 1980.

REFERENCES FOR DIFFUSIVITIES, TABLES 5-10, 5-13 TO 5-15 1. Asfour and Dullien, Chem. Eng. Sci., 41, 1891 (1986). 2. Blanc, J. Phys., 7, 825 (1908). 3. Bosse and Bart, Ind. Eng. Chem. Res. 45, 1822 (2006). 4. Brokaw, Ind. Eng. Chem. Process Des. and Dev., 8, 2, 240 (1969). 5. Caldwell and Babb. J. Phys. Chem., 60, 51 (1956). 6. Catchpole and King, Ind. Eng. Chem. Res., 33, 1828 (1994). 7. Chen and Chen, Chem. Eng. Sci., 40, 1735 (1985). 8. Cullinan, AIChE J., 31, 1740–1741 (1985). 9. Cussler, AIChE J., 26, 1 (1980). 10. Fuller, Schettler, and Giddings, Ind. Eng. Chem., 58, 18 (1966). 11. Hayduk and Laudie, AIChE J., 20, 3, 611 (1974). 12. Hayduk and Minhas, Can. J. Chem. Eng., 60, 195 (1982). 13. He and Yu, Ind. Eng. Chem. Res. 36, 4430 (1997). 14. Lee and Thodos, Ind. Eng. Chem. Fundam., 22, 17–26 (1983). 15. Lee and Thodos, Ind. Eng. Chem. Res., 27, 992–997 (1988). 16. Leffler and Cullinan, Ind. Eng. Chem. Fundam., 9, 84, 88 (1970). 17. Liu and Ruckenstein, Ind. Eng. Chem. Res. 36, 888 (1997) and corrections 37, 3524 (1998). 18. Mathur and Thodos, AIChE J., 11, 613 (1965). 19. Matthews and Akgerman, AIChE J., 33, 881 (1987).

5-44 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

HEAT AND MASS TRANSFER

Rathbun and Babb, Ind. Eng. Chem. Proc. Des. Dev., 5, 273 (1966). Riazi and Whitson, Ind. Eng. Chem. Res., 32, 3081 (1993). Siddiqi and Lucas, Can. J. Chem. Eng., 64, 839 (1986). Smith and Taylor, Ind. Eng. Chem. Fundam., 22, 97 (1983). Sridhar and Potter, AIChE J., 23, 4, 590 (1977). Sun and Chen, Ind. Eng. Chem. Res., 26, 815 (1987). Tyn and Calus, J. Chem. Eng. Data, 20, 310 (1975). Umesi and Danner, Ind. Eng. Chem. Process Des. Dev., 20, 662 (1981). Vignes, Ind. Eng. Chem. Fundam., 5, 184 (1966). Wilke, Chem. Eng. Prog., 46, 2, 95 (1950). Wilke and Chang, AIChE J., 1, 164 (1955). Wilke and Lee, Ind. Eng. Chem., 47, 1253 (1955).

REFERENCES FOR DIFFUSIVITIES IN POROUS SOLIDS, TABLE 5-16 32. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, 1984. 33. Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970. 34. Suzuki, Adsorption Engineering, Kodansha—Elsevier, 1990. 35. Yang, Gas Separation by Adsorption Processes, Butterworths, 1987. REFERENCES FOR TABLES 5-17 TO 5-24 36. Bahmanyar, et al., Chem. Engr. Rsch. Des., 68, 74 (1990). 37. Baier et al., Chem. Engr. Sci., 54, 343 (1999). 38. Beenackers and van Swaaij, Chem. Engr. Sci., 48, 3109 (1993). 39. Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, 1960. 40. Blatt et al. (eds.), (ed.), Membrane Science and Technology, 47, Plenum, 1970. 41. Bocquet et al., AIChE J. 51, 1067 (2005). 42. Bolles and Fair, Institution Chem. Eng. Symp. Ser., 56, 3/35 (1979). 43. Bolles and Fair, Chem. Eng., 89(14), 109 (July 12, 1982). 44. Bravo and Fair, Ind. Eng. Chem. Process Des. Dev., 21, 162 (1982). 45. Bravo, Rocha and Fair, Hydrocarbon Processing, 91 (Jan. 1985). 46. Brian and Hales, AIChE J., 15, 419 (1969). 47. Calderbank and Moo-Young, Chem. Eng. Sci., 16, 39 (1961). 48. Cavatorta, Bohm, and Chiappori de del Giorgio, AIChE J. 45, 938 (1999). 49. Chaumat et al., Chem. Eng. Sci. 60, 5930 (2005). 50. Chen, Lin, and Liu, Ind. Eng. Chem. Res. 44, 7868 (2005). 51. Chilton and Colburn, Ind. Eng. Chem., 26, 1183 (1934). 52. Chowdiah, Foutch, and Lee, Ind. Eng. Chem. Res. 42, 1485 (2003). 53. Colburn, Trans. AIChE, 29, 174 (1933). 54. Cornell, Knapp, and Fair, Chem. Engr. Prog., 56(7), 68 (1960). 55. Cornet and Kaloo, Proc. 3rd Int’l. Congr. Metallic Corrosion—Moscow, 3, 83 (1966). 56. Crause and Nieuwoudt, Ind. Eng. Chem. Res. 38, 4928 (1999). 57. Deckwer et al., Biotech. Bioeng. (1981). 58. Dudukovic, Milosevic, and Pjanovic, AIChE J. 42, 269 (1996). 59. Dwivedi and Upadhyay, Ind. Eng. Chem. Process Des. Develop, 16, 1657 (1977). 60. Eisenberg, Tobias, and Wilke, Chem. Engr. Prog. Symp. Sec., 51(16), 1 (1955). 61. El-Shazly et al., Ind. Eng. Chem. Res. 41, 5516 (2002). 62. Elzinga and Banchero, Chem. Engr. Progr. Symp. Ser., 55(29), 149 (1959). 63. Fair, “Distillation” in Rousseau (ed.), Handbook of Separation Process Technology, Wiley, 1987. 64. Fan, Yang, and Wen, AIChE J. 6, 482 (1960). 65. Garner and Suckling, AIChE J., 4, 114 (1958). 66. Geankoplis, Transport Processes and Unit Operations, 4th ed., Prentice Hall, 2003. 67. Gibilaro et al., Chem. Engr. Sci., 40, 1811 (1985). 68. Gilliland and Sherwood, Ind. Engr. Chem., 26, 516 (1934). 69. Gostick et al., Ind. Eng. Chem. Res. 42, 3626 (2003). 70. Griffith, Chem. Engr. Sci., 12, 198 (1960). 71. Guedes de Carvalho and Alves. AIChE J. 45, 2495 (1999). 72. Gupta and Thodos, AIChE J., 9, 751 (1963). 73. Gupta and Thodos, Ind. Eng. Chem. Fundam., 3, 218 (1964). 74. Harriott, AIChE J., 8, 93 (1962). 75. Hassan et al., Ind. Eng. Chem. Res. 44, 5761 (2005). 76. Heertjes, Holve, and Talsma, Chem. Engr. Sci., 3, 122 (1954). 77. Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice-Hall, 1985. 78. Hoffer et al., Chem. Eng. Sci. 59, 259 (2004). 79. Houwing, Billiet, and van der Wielin, AIChE J. 49, 1158 (2003). 80. Hsiung and Thodos, Int. J. Heat Mass Transfer, 20, 331 (1977). 81. Hsu, Sato, and Sage, Ind. Engr. Chem., 46, 870 (1954). 82. Hughmark, Ind. Eng. Chem. Fundam., 6, 408 (1967). 83. Johnson, Besic, and Hamielec, Can. J. Chem. Engr., 47, 559 (1969). 84. Johnstone and Pigford, Trans. AIChE, 38, 25 (1942).

85. Kafesjian, Plank, and Gerhard, AIChE J., 7, 463 (1961). 86. Kelly and Swenson, Chem. Eng. Prog., 52, 263 (1956). 87. King, Separation Processes, 2d ed., McGraw-Hill (1980). 88. Kirwan, “Mass Transfer Principles” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. 89. Klein, Ward, and Lacey, “Membrane Processes—Dialysis and ElectroDialysis” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. 90. Kohl, “Absorption and Stripping” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987. 91. Kojima et al., J. Chem. Engng. Japan, 20, 104 (1987). 92. Koloini, Sopcic, and Zumer, Chem. Engr. Sci., 32, 637 (1977). 93. Kreutzer et al., Ind. Eng. Chem. Res. 44, 9646 (2005). 94. Lancia, Musmarra, and Pepe, Ind. Eng. Chem. Res. 36, 3859 (1997). 95. Larachi et al., Ind. Eng. Chem. Res. 42, 222 (2003). 96. Lau et al., Ind. Eng. Chem. Res. 43, 1302 (2004). 97. Lee, Biochemical Engineering, Prentice-Hall, 1992. 98. Lee and Foster, Appl. Catal., 63, 1 (1990). 99. Lee and Holder, Ind. Engr. Chem. Res., 34, 906 (1995). 100. Lee and Lueptow, Separ. Sci. Technol. 39, 539 (2004). 101. Levich, Physicochemical Hydrodynamics, Prentice-Hall, 1962. 102. Levins and Gastonbury, Trans. Inst. Chem. Engr., 50, 32, 132 (1972). 103. Linton and Sherwood, Chem. Engr. Prog., 46, 258 (1950). 104. Ludwig, Applied Process Design for Chemical and Petrochemical Plants, 2d ed., vol. 2, Gulf Pub. Co., 1977. 105. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 7th ed., McGraw-Hill, 2005. 106. Nelson and Galloway, Chem. Engr. Sci., 30, 7 (1975). 107. Notter and Sleicher, Chem. Eng. Sci., 26, 161 (1971). 108. Ohashi et al., J. Chem. Engr. Japan, 14, 433 (1981). 109. Onda, Takeuchi, and Okumoto, J. Chem. Engr. Japan, 1, 56 (1968). 110. Pangarkar et al., Ind. Eng. Chem. Res. 41, 4141 (2002). 111. Pasternak and Gauvin, AIChE J., 7, 254 (1961). 112. Pasternak and Gauvin, Can. J. Chem. Engr., 38, 35 (April 1960). 113. Patil, Deshmukh, and Joshi, Ind. Eng. Chem. Res. 43, 2765 (2004). 114. Pekdemir, Davies, and Sara, Ind. Eng. Chem. Res. 37, 1560 (1998). 115. Perez and Sandall, AIChE J., 20, 770 (1974). 116. Petrovic and Thodos, Ind. Eng. Chem. Fundam., 7, 274 (1968). 117. Pinczewski and Sideman, Chem. Engr. Sci., 29, 1969 (1974). 118. Prasad and Sirkar, AIChE J., 34, 177 (1988). 119. Rahman and Streat, Chem. Engr. Sci., 36, 293 (1981). 120. Ramirez and Davis, AIChE J., 45, 1355 (1999). 121. Ranz and Marshall, Chem. Engr. Prog., 48, 141, 173 (1952). 122. Reiss, Ind. Eng. Chem. Process Des. Develop, 6, 486 (1967). 123. Riet, Ind. Eng. Chem. Process Des. Dev., 18, 357 (1979). 124. Rocha, Bravo, and Fair, Ind. Eng. Chem. Res., 35, 1660 (1996). 125. Rowe, Chem. Engr. Sci., 30, 7 (1975). 126. Rowe, Claxton, and Lewis, Trans. Inst. Chem. Engr. London, 43, 14 (1965). 127. Ruckenstein and Rajagopolan, Chem. Engr. Commun., 4, 15 (1980). 128. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, 1984. 129. Sanger and Deckwer, Chem. Eng. J. 22, 179 (1981). 130. Satterfield, AIChE J., 21, 209 (1975). 131. Schluter and Deckwer, Chem. Engr. Sci., 47, 2357 (1992). 132. Schmitz, Steiff, and Weinspach, Chem. Engng. Technol., 10, 204 (1987). 133. Schugerl et al., Adv. Biochem. Eng. 8, 63 (1978). 134. Scott and Lobato, Ind. Eng. Chem. Res., 42, 5697 (2003). 135. Sedahmed, Zatout, and Zewail, Ind. Eng. Chem. Res., 37, 3481 (1998). 136. Shah et al., AIChE J. 28, 353 (1982). 137. Sherwood et al., Ind. Eng. Chem. Fundam., 4, 113 (1965). 138. Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, 1975. 139. Siegel, Sparrow, and Hallman, Appl. Sci. Res. Sec. A., 7, 386 (1958). 140. Sissom and Pitts, Elements of Transport Phenomena, McGraw-Hill, 1972. 141. Skelland, Diffusional Mass Transfer, Wiley (1974). 142. Skelland and Cornish, AIChE J., 9, 73 (1963). 143. Skelland and Moeti, Ind. Eng. Chem. Res., 29, 2258 (1990). 144. Skelland and Tedder, “Extraction—Organic Chemicals Processing” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987, pp. 405–466. 145. Skelland and Wellek, AIChE J., 10, 491, 789 (1964). 146. Slater, “Rate Coefficients in Liquid-Liquid Extraction Systems” in Godfrey and Slater, Liquid-Liquid Extraction Equipment, Wiley, 1994, pp. 45–94. 147. Steinberger and Treybal, AIChE J., 6, 227 (1960). 148. Taniguchi and Kimura, AIChE J., 47, 1967 (2000). 149. Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993. 150. Tokarz, Millies, and Mewes, AIChE J., 47, 799 (2001). 151. Tournie, Laguerie, and Couderc, Chem. Engr. Sci., 34, 1247 (1979). 152. Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, 1980.

MASS TRANSFER 153. 154. 155. 156. 157. 158. 159. 160. 161. 162.

Vandu, Liu, and Krishna, Chem. Eng. Sci., 60, 6430 (2005). Von Karman, Trans. ASME, 61, 705 (1939). Wakao and Funazkri, Chem. Engr. Sci., 33, 1375 (1978). Wang, Yuan, and Yu, Ind. Eng. Chem. Res., 44, 8715 (2005). Wankat, Separation Process Engineering, 2d ed., Prentice-Hall, 2007. Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966). Wright and Glasser, AIChE J., 47, 474 (2001). Yagi and Yoshida, Ind. Eng. Chem. Process Des. Dev., 14, 488 (1975). Yapici and Ozbahar, Ind. Eng. Chem. Res., 37, 643 (1998). Zaki, Nirdosh, and Sedahmed, Ind. Eng. Chem. Res., 41, 3307 (2002).

INTRODUCTION This part of Sec. 5 provides a concise guide to solving problems in situations commonly encountered by chemical engineers. It deals with diffusivity and mass-transfer coefficient estimation and common flux equations, although material balances are also presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Mass-transfer calculations involve transport properties, such as diffusivities, and other empirical factors that have been found to relate mass-transfer rates to measured “driving forces” in myriad geometries and conditions. The context of the problem dictates whether the fundamental or more applied coefficient should be used. One key distinction is that, whenever there is flow parallel to an interface through which mass transfer occurs, the relevant coefficient is an empirical combination of properties and conditions. Conversely, when diffusion occurs in stagnant media or in creeping flow without transverse velocity gradients, ordinary diffusivities may be suitable for solving the problem. In either case, it is strongly suggested to employ data, whenever available, instead of relying on correlations. Units employed in diffusivity correlations commonly followed the cgs system. Similarly, correlations for mass transfer correlations used the cgs or English system. In both cases, only the most recent correlations employ SI units. Since most correlations involve other properties and physical parameters, often with mixed units, they are repeated here as originally stated. Common conversion factors are listed in Table 1-4. Fick’s First Law This law relates flux of a component to its composition gradient, employing a constant of proportionality called a diffusivity. It can be written in several forms, depending on the units and frame of reference. Three that are related but not identical are dcA dxA dwA J = −DAB  ≈ M JA = −cDAB  ∝ m JA = −ρDAB  dz dz dz

V A

(5-189)

The first equality (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity, while the equality in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity. These must be used with consistent flux expressions for fixed coordinates and for NC components, such as: NC

NC

NC

J + wA ni

m A

i=1

NA = V JA + cA NiV i = M JA + xA Ni =  MA i=1 i=1

(5-190)

In each case, the term containing the summation accounts for conveyance, which is the amount of component A carried by the net flow in the direction of diffusion. Its impact on the total flux can be as much as 10 percent. In most cases it is much less, and it is frequently ignored. Some people refer to this as the “convective” term, but that conflicts with the other sense of convection which is promoted by flow perpendicular to the direction of flux. Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by DAB and is defined by Fick’s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-189). It is analogous to the thermal diffusivity in Fourier’s law and to the kinematic viscosity in Newton’s law. These analogies are flawed because both heat and momentum are conveniently defined with respect to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires infor-

5-45

mation about bulk motion of the medium in which diffusion occurs. For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. When the flux expressions are consistent, as in Eq. (5-190), the diffusivities in Eq. (5-189) are identical. As a result, experimental diffusivities are often measured under constant volume conditions but may be used for applications involving open systems. It turns out that the two versions are very nearly equivalent for gas-phase systems because there is negligible volume change on mixing. That is not usually true for liquids, however. Self-Diffusivity Self-diffusivity is denoted by DA′A and is the measure of mobility of a species in itself; for instance, using a small concentration of molecules tagged with a radioactive isotope so they can be detected. Tagged and untagged molecules presumably do not have significantly different properties. Hence, the solution is ideal, and there are practically no gradients to “force” or “drive” diffusion. This kind of diffusion is presumed to be purely statistical in nature. In the special case that A and B are similar in molecular weight, polarity, and so on, the self-diffusion coefficients of pure A and B will be approximately equal to the mutual diffusivity, DAB. Second, when A and B are the less mobile and more mobile components, respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient. That is, DA′A ≤ DAB ≤ DB′B. Third, it is a common means for evaluating diffusion for gases at high pressure. Self-diffusion in liquids has been studied by many [Easteal, AIChE J. 30, 641 (1984), Ertl and Dullien, AIChE J. 19, 1215 (1973), and Vadovic and Colver, AIChE J. 18, 1264 (1972)]. Tracer Diffusivity Tracer diffusivity, denoted by DA′B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach self-diffusivities at the pure component limit. That is, at the limit of dilute A in B, DA′B → D°AB and DB ′A → DB ′B; likewise at the limit of dilute B in A, DB ′A → D°BA and DA′B → DA′A. Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as order-of-magnitude estimates of mutual diffusivities. Darken’s equation [Eq. (5-230)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory, and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. Problem Solving Methods Most, if not all, problems or applications that involve mass transfer can be approached by a systematic course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., multicomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are

5-46

HEAT AND MASS TRANSFER

Nomenclature and Units—Mass Transfer Symbols a

Definition

Acs A′ ap c ci cP d db dc d drop dimp dpore DA′A DAB D°AB Deff DK DL DS E ES f F g gc G Ga GM h′ hT H H

Effective interfacial mass transfer area per unit volume Cross-sectional area of vessel Constant (see Table 5-24-K) See a Concentration = P/RT for an ideal gas Concentration of component i = xi c at gas-liquid interface Specific heat Characteristic length Bubble diameter Column diameter Sauter mean diameter Impeller diameter Pore diameter Self-diffusivity (= DA at xA = 1) Mutual diffusivity Mutual diffusivity at infinite dilution of A in B Effective diffusivity within a porous solid = εpD/τ Knudson diffusivity for gases in small pores Liquid phase diffusion coefficient Surface diffusivity Energy dissipation rate/mass Activation energy for surface diffusion Friction factor for fluid flow Faraday’s constant Acceleration due to gravity Conversion factor Gas-phase mass flux Dry air flux Molar gas-phase mass flux Heat transfer coefficient Total height of tower packing Compartment height Henry’s law constant

H′

Henry’s law constant

HG HOG HL HOL HTU jD jH jM m JA

J JSi k k k k′

Height of one transfer unit based on gas-phase resistance Height of one overall gas-phase mass-transfer unit Height of one transfer unit based on liquid-phase resistance Height of one overall liquid-phase mass-transfer unit Height of one transfer unit (general) Chilton-Colburn factor for mass transfer, Eq. (5-289) Chilton-Colburn factor for heat transfer See jD Mass flux of A by diffusion with respect to the mean mass velocity Molar flux of A by diffusion with respect to mean molar velocity Molar flux of A with respect to mean volume velocity Molar flux by surface diffusion Boltzmann’s constant Film mass transfer coefficient Thermal conductivity Mass-transfer coefficient for dilute systems

kG

Gas-phase mass-transfer coefficient for dilute systems

k′G

Gas-phase mass-transfer coefficient for dilute systems

kGa kˆ Ga

Volumetric gas-phase mass-transfer Overall volumetric gas-phase mass-transfer coefficient for concentrated systems Liquid phase mass transfer coefficient for pure absorption (no reaction) Liquid-phase mass-transfer coefficient for dilute systems

J

M A

V A

ˆL k° kL k′L kˆ L kLa K K KG

Liquid-phase mass-transfer coefficient for dilute systems Liquid-phase mass-transfer coefficient for concentrated systems Volumetric liquid-phase mass-transfer coefficient for dilute systems Overall mass transfer coefficient α/R = specific conductance Overall gas-phase mass-transfer coefficient for dilute systems

SI units

U.S. Customary units

m2/m3

ft2/ft3

m2 or cm2

ft2

mol/m3 or mol/l or gequiv/l mol/m3 or mol/l or gequiv/l kJ/(kg⋅K) m or cm m m or cm m m m or cm m2/s or cm2/s m2/s or cm2/s m2/s or cm2/s m2/s m2/s or cm2/s m2/s m2/s or cm2/s

lbmol/ft3 lbmol/ft3 Btu/(lb⋅°F) ft ft ft ft ft ft ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h

J/mol or cal/mol Dimensionless 96,487 Coulomb/gequiv m/s2 1.0 kg/(s⋅m2) kg/(s⋅m2) kmol/(s⋅m2) W/(m2⋅K) = J/(s⋅m2⋅K) m m kPa/(mole-fraction solute in liquid phase) kPa/[kmol/(m3 solute in liquid phase)]

Dimensionless

m m m m m Dimensionless Dimensionless

ft/h2 4.17 × 108 lb ft/[lbf⋅h2] lb/(h⋅ft2) lb/(h⋅ft2) (lbmol)/(h⋅ft2) Btu/(h⋅ft2⋅°F) ft ft (lbf/in2)/(mole-fraction solute in liquid phase) (lbf/in2)/[(lbmol)/(ft3 solute in liquid phase)] or atm/[(lbmole)/ (ft3 solute in liquid phase)] ft ft ft ft ft Dimensionless Dimensionless

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m2⋅s) or mol/(cm2⋅s)

lbmol/(ft2⋅h)

kmol/(m ⋅s) kmol/(m2⋅s) or gmol/(cm2⋅s) 8.9308 × 10−10 gequiv ohm/s m/s or cm/s (J⋅m)/(s⋅m2⋅K) kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/[(s⋅m2)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction in gas)] kmol/[(s⋅m3)(mole fraction)] kmol/(s⋅m3)

lbmol/(ft2 ⋅h) lbmol/(ft2⋅h) ft/hr Btu/(h⋅ft⋅°F) lbmol/[(h⋅ft2)(lbmol/ft3)] or ft/hr lbmol/[(h⋅ft2)lbf/in2 solute partial pressure)] lbmol/[(h⋅ft2)(mole fraction in gas)] (lbmol)/[(h⋅ft3)(mole fraction)] lbmol/(h⋅ft3)

kmol/(s⋅m2)

lbmol/(h⋅ft2)

2

2

kmol/[(s⋅m )(mole-fraction solution in liquid)] kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/(s⋅m2)

(lbmol)/[(h⋅ft2)(mole-fraction solute in liquid)] (lbmol)/[(h⋅ft2)(lbmol/ft3)] or ft/h lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole fraction)]

(lbmol)/[(h⋅ft3)(mole fraction)]

m/s or cm/s ohm/cm kmol/[(s⋅m2)(mole fraction)]

ft/h (lbmol)/[(h⋅ft2)(mole fraction)]

MASS TRANSFER

5-47

Nomenclature and Units—Mass Transfer (Continued) Symbols Kˆ G

Definition

KGa

Overall gas-phase mass-transfer coefficient for concentrated systems Overall volumetric gas-phase mass-transfer dilute systems

K′Ga

Overall volumetric gas-phase mass-transfer dilute systems

(Ka)H KL Kˆ L

Overall enthalpy mass-transfer coefficient Overall liquid-phase mass-transfer coefficient Liquid-phase mass-transfer coefficient for concentrated systems Overall volumetric liquid-phase mass-transfer coefficient for dilute systems Overall volumetric liquid-phase mass-transfer coefficient for concentrated systems Liquid-phase mass flux Molar liquid-phase mass flux Slope of equilibrium curve = dy/dx (mole-fraction solute in gas)/(mole-fraction solute in liquid) Molality of solute Molecular weight of species i Mass in a control volume V Valences of cationic and anionic species See Table 5-24-K Mass flux of A with respect to fixed coordinates Impeller speed Number deck levels Interphase mass-transfer rate of solute A per interfacial area with respect to fixed coordinates Number of components Froude Number (dimp N2/g)

KLa Kˆ La L LM m m Mi M |n+ n −| n′ nA N N′ NA Nc NFr

NOG NOL NTU NKn NPr NRe NSc NSh NSt NW p pB,M pi pT P P Pc Per Q rA R

ρ∞ gx3 Grashof number 2  − 1 (µ/ρ) ρo Number of overall gas-phase mass-transfer units Number of overall liquid-phase mass-transfer units Number of transfer units (general) Knudson number = l/dpore Prandtl number (cpµ/k) Reynolds number (Gd/µG) Schmidt number (µG/ρGDAB) or (µL/ρLDL) Sherwood number (ˆkGRTd/DABpT), see also Tables 5-17 to 5-24 Stanton number (ˆkG/GM) or (ˆkL/LM) Weber number (ρcN2d3imp/σ) Solute partial pressure in bulk gas Log mean partial pressure difference of stagnant gas B Solute partial pressure at gas-liquid interface Total system pressure Pressure Power Critical pressure Perimeter/area Volumetric flow rate Radius of dilute spherical solute Gas constant

R Ri s S t tf T Tb Tc Tr u, v uo u∞ uL vs vT vTS V V Vb Vi vi

Solution electrical resistance Radius of gyration of the component i molecule Fractional surface-renewal rate Tower cross-sectional area = πd2/4 Contact time Formation time of drop Temperature Normal boiling point Critical temperature Reduced temperature = T/Tc Fluid velocity Blowing or suction velocity Velocity away from object Superficial liquid velocity in vertical direction Slip velocity Terminal velocity Stokes law terminal velocity Packed volume in tower Control volume Volume at normal boiling point Molar volume of i at its normal boiling point Partial molar volume of i

NGr







SI units

U.S. Customary units

kmol/(s⋅m2)

lbmol/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in gas)] kmol/[(s⋅m3)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction)] kmol/[(s⋅m2)(mole fraction)] kmol/(s⋅m2)

(lbmol)/[(h⋅ft3(mole-fraction solute in gas)] (lbmol)/[(h⋅ft3)(lbf/in2 solute partial pressure)] lb/[(h⋅ft3)(lb water/lb dry air)] (lbmol)/[(h⋅ft2)(mole fraction)] (lbmol)/(h⋅ft2)

kmol/[(s⋅m3)(mole-fraction solute in liquid)] kmol/(s⋅m3)

(lbmol)/[(h⋅ft3)(mole-fraction solute in liquid)] (lbmol)/(h⋅ft3)

kg/(s⋅m2) kmol/(s⋅m2) Dimensionless

lb/(h⋅ft2) (lbmol)/(h⋅ft2) Dimensionless

mol/1000 g solvent kg/kmol or g/mol kg or g Dimensionless Dimensionless kg/(s⋅m2) Revolution/s Dimensionless kmol/(s⋅m2)

lb/lbmol lb Dimensionless Dimensionless lb/(h⋅ft2) Revolution/min Dimensionless (lbmol)/(h⋅ft2)

Dimensionless Dimensionless

Dimensionless Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless kPa Dimensionless kPa kPa Pa Watts Pa m−1 m3/s Å 8.314 J/mol K = 8.314 Pa m3/(mol K) = 82.057 atm cm3/mol K ohm Å s−1 m2 s s K K K Dimensionless m/s or cm/s m/s m/s m/s m/s m/s m/s m3 m3 or cm3 m3/kmol or cm3/mol m3/kmol or cm3/mol m3/kmol or cm3/mol

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 Dimensionless lbf/in2 lbf/in2 lbf/in2 or atm lbf/in2 or atm ft−1 ft3/h 10.73 ft3 psia/lbmol⋅h

h−1 ft2 h h °R °R °R Dimensionless ft/h ft/h ft/h ft/h ft/h ft/h ft/h ft3 ft3 ft3/lbmol ft3/lbmol ft3/lbmol

5-48

HEAT AND MASS TRANSFER

Nomenclature and Units—Mass Transfer (Concluded) Symbols Vmli Vtower w x x xA xo xBM o BM

x

xi y yBM o yBM yi yio z

Definition

SI units

Molar volume of the liquid-phase component i at the melting point Tower volume per area Width of film Length along plate Mole-fraction solute in bulk-liquid phase Mole fraction of component A Mole-fraction solute in bulk liquid in equilibrium with bulk-gas solute concentration y Logarithmic-mean solvent concentration between bulk liquid and interface values Logarithmic-mean inert-solvent concentration between bulk-liquid value and value in equilibrium with bulk gas Mole-fraction solute in liquid at gas-liquid interface Mole-fraction solute in bulk-gas phase Logarithmic-mean inert-gas concentration [Eq. (5-275)] Logarithmic-mean inert-gas concentration Mole fraction solute in gas at interface Mole-fraction solute in gas at interface in equilibrium with the liquid-phase interfacial solute concentration xi Direction of unidimensional diffusion

U.S. Customary units

m3/kmol or cm3/mol

ft3/lbmol

m3/m2 m m (kmol solute)/(kmol liquid) kmole A/kmole fluid (kmol solute)/(kmol liquid)

ft3/ft2 ft ft (lbmol solute)/(lb mol liquid) lbmol A/lb mol fluid (lbmol solute)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solvent)/(kmol liquid)

(lbmol solvent)/(lbmol liquid)

(kmol solute)/(kmol liquid) (kmol solute)/(kmol gas) (kmol inert gas)/(kmol gas) (kmol inert gas)/(kmol gas) (kmole solute)/(kmol gas) (kmol solute)/(kmol gas)

(lbmol solute)/(lbmol liquid) (lbmol solute)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol solute)/(lbmol gas) (lbmol solute)/(lbmol gas)

m

ft

Dimensionless cm−1 Dimensionless m Dimensionless

Dimensionless

m3/m3 Dimensionless Dimensionless Dimensionless Dimensionless cm2/(gequiv⋅ohm) cm2/ohm gequiv cm2/gequiv ohm Debeyes cP or Pa s kg/(s⋅m) kg/(s⋅m) m2/s kg/m3 or g/cm3 kg/m3 or g/cm3 kg/m3 kg/m3 kmol/m3 kg/m3 or g/cm3 Dimensionless

ft3/ft3 Dimensionless Dimensionless Dimensionless Dimensionless

Dimensionless dyn/cm Å dyn/cm Å Dimensionless Dimensionless Radians/s Dimensionless

Dimensionless lbf/ft

Greek Symbols α α β δ ε ε εA εAB γi γ± λ+ λ− Λ Λo µi µi µG µL ν ρ ρc ρc ρG ρ L ρp ρr ψi ψ σ σi σi σAB τ ω ω Ω

1 + NB/NA Conductance cell constant (measured) 5/6 MA1/2 P1/3 c /T c Effective thickness of stagnant-film layer Fraction of discontinuous phase in continuous phase for two-phase flow Void fraction available for gas flow or fractional gas holdup Characteristic Lennard-Jones energy (εAεB)1/2 Activity coefficient of solute i Mean ionic activity coefficient of solute Infinite dilution conductance of cation and anion 1000 K/C = λ+ + λ− = Λo + f(C) Infinite dilution conductance Dipole moment of i Viscosity of pure i Gas-phase viscosity Liquid-phase viscosity Kinematic viscosity = ρ/µ Density of A Critical density of A Density continuous phase Gas-phase density Average molar density of liquid phase Particle density Reduced density = ρ/ρc Parachor of component i = Viσ1/4 Parameter, Table 5-24-G Interfacial tension Characteristic length Surface tension of component i Binary pair characteristic length = (σA + σB)/2 Intraparticle tortuosity Pitzer’s acentric factor = −[1.0 + log10(P*/Pc)] Rotational velocity Diffusion collision integral = f(kT/εAB) Subscripts

A B G m L super

Solute component in liquid or gas phase Inert-gas or inert-solvent component Gas phase Mean value Liquid phase Superficial velocity

*

At equilibrium

Superscript

Dimensionless ft Dimensionless

lb/(h⋅ft) lb/(h⋅ft) lb/(h⋅ft) ft2/h lb/ft3 lb/ft3 lb/ft3 lb/ft3 (lbmol)/ft3 lb/ft3 Dimensionless

Dimensionless Dimensionless

MASS TRANSFER

5-49

FIG. 5-24 Flowchart illustrating problem solving approach using mass-transfer rate expressions in the context of mass conservation.

cases, flow rates through the discrete ports (nozzles) must be related to the mass-transfer rate in the packing. As a result, the mass-transfer rate is determined via flux equations, and the overall material balance incorporates the stream flow rates mi and integrated fluxes. In such instances, it is common to begin with the most general, differential material balance equations. Then, by eliminating terms that are negligible, the simplest applicable set of equations remains to be solved. Table 5-8 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit crosssectional area and constant volume:

applied to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it applies to many mass-transfer problems. CONTINUITY AND FLUX EXPRESSIONS Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometry is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having Nm bulk-flow ports and Nn ports or interfaces through which diffusive flux can occur: Nm Nn dM (5-191)  = mi + ni Acsi dt i = 1 i=1 where M represents the mass in the unit volume V at any time t; mi is the mass flow rate through the ith port; and ni is the mass flux through the ith port, which has a cross-sectional area of Acsi. The corresponding balance equation for individual components includes a reaction term: Nm

∂ρ j  = −∇⋅ nj + rj ∂t

where nj = ρvj. Applying Fick’s law and expressing composition as concentration gives ∂cj  = −v ⋅ ∇cj + Dj∇2cj + rj ∂t

(5-192)

For the jth component, mij = miwij is the component mass flow rate in stream i; wij is the mass fraction of component j in stream i; and rj is the net reaction rate (mass generation minus consumption) per unit volume V that contains mass M. If it is inconvenient to measure mass flow rates, the product of density and volumetric flow rate is used instead. In addition, most situations that involve mass transfer require material balances, but the pertinent area is ambiguous. Examples are packed columns for absorption, distillation, or extraction. In such TABLE 5-8

Equation ∂nyj ∂nzj ∂nxj ∂ρj  = −  +  +  + rj ∂t ∂x ∂y ∂z



Cartesian



(5-194)

∂nzj 1 ∂rnrj 1 ∂nθj ∂ρj  = −   +   +  + rj ∂t ∂z r ∂r r ∂θ



Cylindrical

Spherical

dxA DAB NA = M JA = −DAB c  =  c (xAL − xAR) dz ∆z

Continuity Equation in Various Coordinate Systems

Coordinate System



(5-195)

2 ∂nφj ∂nθj sin θ ∂ρj 1 1 1 ∂r nrj  = − 2  +   +   + rj ∂r ∂t r sin θ ∂θ r sin θ ∂φ r



(5-193b)

Flux Expressions: Simple Integrated Forms of Fick’s First Law Simplified flux equations that arise from Eqs. (5-189) and (5-190) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAL and xAR at the left-hand and right-hand sides of a hypothetical layer having thickness ∆z. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 1. Equimolar counterdiffusion (NA = −NB)

Nn

dMj  = mij + nij Acsi + rj V dt i=1 i=1

(5-193a)



(5-196)

(5-197)

5-50

HEAT AND MASS TRANSFER

FIG. 5-25

and Bird [Ind. Eng. Chem. Res. 38, 2515 (1999)], and Amundson, Pan, and Paulson [AIChE J. 48, 813 (2003)]. Vrentas and Vrentas treated only ternary mixtures, such as restrictions due to the entropy inequality, application of the Onsager reciprocal relations, and stability. Curtis and Bird reconciled the multicomponent Fick’s law approach with the more elegant Stefan-Maxwell theory. They also provided interrelationships of multicomponent diffusivities devised for various situations, i.e., binary, ternary, and quaternary mixtures. Amundson et al. presented numerical methods for coping with mixtures having four or more components, which are nearly intractable via the analytical S-M method, due to the difficult inversion. Related studies were performed by Ghorayeb and Firoozabadi [AIChE J. 46, 883 (2000)] and Firoozabadi, Ghorayeb, and Saukla [AIChE J. 46, 892 (2000)]. The former covered ordinary molecular diffusion as well as pressure and thermal diffusion for multicomponent mixtures. The latter covered thermal diffusion in multicomponent mixtures.

Hypothetical film and boundary conditions.

2. Unimolar diffusion (NA ≠ 0, NB = 0) 1 − xAR DAB NA = M JA + xANA =  c ln  1 − xAL ∆z 3. Steady state diffusion (NA ≠ −NB ≠ 0)

(5-198)

NA  − xAR NA + NB NA DAB NA = M JA + xA(NA + NB) =   c ln  NA NA + NB ∆z  − xAL NA + NB

DIFFUSIVITY ESTIMATION—GASES Whenever measured values of diffusivities are available, they should be used. Typically, measurement errors are less than those associated with predictions by empirical or even semitheoretical equations. A few general sources of data are Sec. 2 of this handbook; e.g., experimental values for gas mixtures are listed in Table 2-371. Estimation methods for some gaseous applications appear in Eqs. (2-150) through (2-154). Other pertinent references are Schwartzberg and Chao; Poling et al.; Gammon et al.; and Daubert and Danner. Many other more restricted sources are listed under specific topics later in this subsection. Before using diffusivities from either data or correlations, it is a good idea to check their reasonableness with respect to values that have been commonly observed in similar situations. Table 5-9 is a compilation of several rules of thumb. These values are not authoritative; they simply represent guidelines based on experience. Diffusivity correlations for gases are outlined in Table 5-10. Specific parameters for individual equations are defined in the specific text regarding each equation. References are given at the beginning of the “Mass Transfer” subsection. The errors reported for Eqs. (5-202) through (5-205) were compiled by Poling et al., who compared the predictions with 68 experimental values of DAB. Errors cited for Eqs. (5-206) to (5-212) were reported by the authors. Binary Mixtures—Low Pressure—Nonpolar Components Many evaluations of correlations are available [Elliott and Watts, Can. J. Chem., 50, 31 (1972); Lugg, Anal. Chem., 40, 1072 (1968); Marrero and Mason, AIChE J., 19, 498 (1973)]. The differences in accuracy of the correlations are minor, and thus the major concern is ease of calculation. The Fuller-Schettler-Giddings equation is usually the simplest correlation to use and is recommended by Poling et al. Chapman-Enskog (Bird et al.) and Wilke and Lee [31] The inherent assumptions of these equations are quite restrictive (i.e., low density, spherical atoms), and the intrinsic potential function is empirical. Despite that, they provide good estimates of DAB for many polyatomic gases and gas mixtures, up to about 1000 K and a maximum of 70 atm. The latter constraint is because observations for many gases indicate that DABP is constant up to 70 atm. The characteristic length is σAB = (σA + σB)/2 in A˚. In order to estimate ΩD for Eq. (5-202) or (5-203), two empirical equations are

(5-199)

The unfortunate aspect of the last relationship is that one must know a priori the ratio of the fluxes to determine the magnitudes. It is not possible to solve simultaneously the pair of equations that apply for components A and B because the equations are not independent. Stefan-Maxwell Equations Following Eq. (5-190), a simple and intuitively appealing flux equation for applications involving Nc components is Nc

Ni = −cDim ∇xi + xi Nj

(5-200)

j=1

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating multicomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simultaneously by Stefan and Maxwell. The problem is to determine the diffusion coefficient Dim. The Stefan-Maxwell equations are simpler in principle since they employ binary diffusivities: Nc

1 ∇xi =  (xiNj − xj Ni) j = 1 cDij

(5-201)

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dij of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AIChE J., 10, 448 and 460 (1964); Stewart and Prober, Ind. Eng. Chem. Fundam., 3, 224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput. Chem. Eng., 5, 61 (1981)]. Useful studies of multicomponent diffusion were presented by Vrentas and Vrentas [Ind. Eng. Chem. Res. 44, 1112 (2005)], Curtis

TABLE 5-9

Rules of Thumb for Diffusivities (See Cussler, Poling et al., Schwartzberg and Chao) Di magnitude

Continuous phase

m2/s −5

cm2/s 0.1

Di range m2/s −4

cm2/s −6

10 –10

1–10−2

Gas at atmospheric pressure

10

Liquid Liquid occluded in solid matrix Polymers and glasses

10−9 10−10 10−12

10−5 10−6 10−8

10−8–10−10 10−8–10−12 10−10–10−14

10−4–10−6 10−4–10−8 10−6–10−10

Solid

10−14

10−10

10−10–10−34

10−6–10−30

Comments Accurate theories exist, generally within 10%; DiP  constant; Di ∝ T1.66 to 2.0 Approximate correlations exist, generally within 25% Hard cell walls: Deff /Di = 0.1 to 0.2. Soft cell walls: Deff /Di = 0.3 to 0.9 Approximate theories exist for dilute and concentrated limits; strong composition dependence Approximate theories exist; strong temperature dependence

MASS TRANSFER TABLE 5-10

5-51

Correlations of Diffusivities for Gases

Authors*

Equation

Error, %

1. Binary Mixtures—Low Pressure—Nonpolar Chapman-Enskog

1/2 0.001858T 3/2 MAB DAB =  Pσ 2AB ΩD

(5-202)

7.3

Wilke-Lee [31]

1/2 1/2 (0.00217 − 0.0005MAB ) T 3/2MAB DAB =  2 PσAB ΩD

(5-203)

7.0

Fuller-Schettler-Giddings [10]

1/2 0.001T1.75MAB DAB =  2 1/3 P [( v)A + ( v)1/3 B ]

(5-204)

5.4

(5-205)

9.0

2. Binary Mixtures—Low Pressure—Polar Brokaw [4]

1/2 0.001858T 3/2MAB DAB =  2 Pσ AB ΩD

Mathur-Thodos [18]

10.7 × 10−5Tr DAA =  {ρr ≤ 1.5} βρr

(5-206)

5

Lee-Thodos [14]

0.77 × 10−5T DAA = r {ρr ≤ 1} ρrδ

(5-207)

0.5

Lee-Thodos [15]

(0.007094G + 0.001916)2.5Tr DAA =  , [ρr > 1, G < 1] δ

(5-208)

17

3. Self-Diffusivity

4. Supercritical Mixtures Sun and Chen [25]

1.23 × 10−10T DAB =  µ0.799VC0.49 A

(5-209)

5

Catchpole and King [6]

(ρ−0.667 − 0.4510) (1 + MA/MB) R DAB = 5.152 DcTr  (1 + (VcB /VcA)0.333)2

(5-210)

10

Liu and Ruckenstein [17]

kT DAB =  fπµAσAB

(5-211)

5.7



1  σAB 1/2 +  2 1 1 +  1 −  θ∞AB 3σΑ 3 2



T , M

DAB = α(VkA − β)







α = 10 −5 0.56392

B

+ 2.1417 exp



−0.95088 M  AVC A PCA



6.9

(5-212)

 M AVCA β = 8.9061 + 0.93858  PCA 2 k =  [1 − 0.28 exp (−0.3 M A ριA)] 3 *References are listed at the beginning of the “Mass Transfer” subsection.

available. The first is: ΩD = (44.54T*−4.909 + 1.911T*−1.575)0.10 (5-213a) where T* = kT/εAB and εAB = (εA εB)1/2. Estimates for σi and εi are given in Table 5-11. This expression shows that ΩD is proportional to temperature roughly to the −0.49 power at low temperatures and to the TABLE 5-11

−0.16 power at high temperature. Thus, gas diffusivities are proportional to temperatures to the 2.0 power and 1.66 power, respectively, at low and high temperatures. The second is: A C E G ΩD =  +  +  +  (5-213b) T*B exp (DT*) exp (FT*) exp (HT*)

Estimates for ei and si (K, Å, atm, cm3, mol)

Critical point

ε/k = 0.75 Tc

1/3 σ = 0.841 V1/3 c or 2.44 (Tc /Pc)

Critical point

ε/k = 65.3 Tc zc3.6

1.866 V1/3 c σ =  z1.2 c

Normal boiling point Melting point

ε/k = 1.15 Tb ε/k = 1.92 Tm

σ = 1.18 V1/3 b σ = 1.222 V1/3 m

Acentric factor

ε/k = (0.7915 + 0.1693 ω) Tc

T σ = (2.3551 − 0.087 ω) c Pc



1/3

NOTE: These values may not agree closely, so usage of a consistent basis is suggested (e.g., data at the normal boiling point).

5-52

HEAT AND MASS TRANSFER

TABLE 5-12 Atomic Diffusion Volumes for Use in Estimating DAB by the Method of Fuller, Schettler, and Giddings [10] Atomic and Structural Diffusion–Volume Increments, vi (cm3/mol) C H O (N)

16.5 1.98 5.48 5.69

(Cl) (S) Aromatic ring Heterocyclic ring

19.5 17.0 −20.2 −20.2

Diffusion Volumes for Simple Molecules, Σvi (cm3/mol) H2 D2 He N2 O2 Air Ar Kr (Xe) Ne

7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 37.9 5.59

CO CO2 N2O NH3 H2O (CCl2F2) (SF5) (Cl2) (Br2) (SO2)

18.9 26.9 35.9 14.9 12.7 114.8 69.7 37.7 67.2 41.1

Parentheses indicate that the value listed is based on only a few data points.

where A = 1.06036, B = 0.15610, C = 0.1930, D = 0.47635, E = 1.03587, F = 1.52996, G = 1.76474, and H = 3.89411. Fuller, Schettler, and Giddings [10] The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of vi used in this equation are in Table 5-12. Binary Mixtures—Low Pressure—Polar Components The Brokaw [4] correlation was based on the Chapman-Enskog equation, but σAB* and ΩD* were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good: an average absolute error of 6.4 percent, considering the complexity of some of the gas pairs [e.g., (CH3)2O and CH3Cl]. Despite that, Poling, (op. cit.) found the average error was 9.0 percent for combinations of mixtures (including several polar-nonpolar gas pairs), temperatures and pressures. In this equation, ΩD is calculated as described previously, and other terms are: ΩD* = ΩD + 0.19 δ 2AB/T* σAB* = (σA* σB*)1/2 δAB = (δA δB)1/2 εAB* = (εA*εB*)1/2

T* = kT/εAB* σi* = [1.585 Vbi /(1 + 1.3 δ2i )]1/3 δi = 1.94 × 103 µ2i /VbiTbi εi* /k = 1.18 (1 + 1.3 δ 2i )Tbi

Binary Mixtures—High Pressure Of the various categories of gas-phase diffusion, this is the least studied. This is so because of the effects of diffusion being easily distorted by even a slight pressure gradient, which is difficult to avoid at high pressure. Harstad and Bellan [Ind. Eng. Chem. Res. 43, 645 (2004)] developed a correspondingstates expression that extends the Chapman-Enskog method, covered earlier. They express the diffusivity at high pressure by accounting for the reduced temperature, and they suggest employing an equation of state and shifting from DoAB = f(T, P) to DAB = g(T, V). Self-Diffusivity Self-diffusivity is a property that has little intrinsic value, e.g., for solving separation problems. Despite that, it reveals quite a lot about the inherent nature of molecular transport, because the effects of discrepancies of other physical properties are eliminated, except for those that constitute isotopic differences, which are necessary to ascertain composition differences. Self-diffusivity has been studied extensively under high pressures, e.g., greater than 70 atm. There are few accurate estimation methods for mutual diffusivities at such high pressures, because composition measurements are difficult. The general observation for gas-phase diffusion DAB P = constant, which holds at low pressure, is not valid at high pressure. Rather, DAB P decreases as pressure increases. In addition, composition effects, which frequently are negligible at low pressure, are very significant at high pressure.

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiempirical equation, based on hard-sphere theory, to estimate selfdiffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. Zielinski and Hanley [AIChE J. 45, 1 (1999)] developed a model to predict multicomponent diffusivities from self-diffusion coefficients and thermodynamic information. Their model was tested by estimated experimental diffusivity values for ternary systems, predicting drying behavior of ternary systems, and reconciling ternary selfdiffusion data measured by pulsed-field gradient NMR. Mathur and Thodos [18] showed that for reduced densities less than unity, the product DAAρ is approximately constant at a given temperature. Thus, by knowing the value of the product at low pressure, it is possible to estimate its value at a higher pressure. They found at higher pressures the density increases, but the product DAAρ decreases rapidly. In their correlation, β = MA1/2PC1/3/T C5/6. Lee and Thodos [14] presented a generalized treatment of selfdiffusivity for gases (and liquids). These correlations have been tested for more than 500 data points each. The average deviation of the first is 0.51 percent, and that of the second is 17.2 percent. δ = 2 5/6 0.1 MA1/2/P1/2 c V c , s/cm , and where G = (X* − X)/(X* − 1), X = ρr /T r , and X* = ρr /T r0.1 evaluated at the solid melting point. Lee and Thodos [15] expanded their earlier treatment of selfdiffusivity to cover 58 substances and 975 data points, with an average absolute deviation of 5.26 percent. Their correlation is too involved to repeat here, but those interested should refer to the original paper. Liu, Silva, and Macedo [Chem. Eng. Sci. 53, 2403 (1998)] present a theoretical approach incorporating hard-sphere, square-well, and Lennard-Jones models. They compared their resulting estimates to estimates generated via the Lee-Thodos equation. For 2047 data points with nonpolar species, the Lee-Thodos equation was slightly superior to the Lennard-Jones fluid-based model, that is, 5.2 percent average deviation versus 5.5 percent, and much better than the square-well fluid-based model (10.6 percent deviation). For over 467 data points with polar species, the Lee-Thodos equation yielded 36 percent average deviation, compared with 25 percent for the LennardJones fluid-based model, and 19 percent for the square-well fluidbased model. Silva, Liu, and Macedo [Chem. Eng. Sci. 53, 2423 (1998)] present an improved theoretical approach incorporating slightly different Lennard-Jones models. For 2047 data points with nonpolar species, their best model yielded 4.5 percent average deviation, while the LeeThodos equation yielded 5.2 percent, and the prior Lennard-Jones fluid-based model produced 5.5 percent. The new model was much better than all the other models for over 424 data points with polar species, yielding 4.3 percent deviation, while the Lee-Thodos equation yielded 34 percent and the Lennard-Jones fluid-based model yielded 23 percent. Supercritical Mixtures Debenedetti and Reid [AIChE J., 32, 2034 (1986) and 33, 496 (1987)] showed that conventional correlations based on the Stokes-Einstein relation (for liquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group DABµ/T was constant. Thus, although no general correlation applies, only one data point is necessary to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SF6 and CO2. Sun and Chen [25] examined tracer diffusion data of aromatic solutes in alcohols up to the supercritical range and found their data correlated with average deviations of 5 percent and a maximum deviation of 17 percent for their rather limited set of data. Catchpole and King [6] examined binary diffusion data of nearcritical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction factor was required (R = 1). That class was comprised of alcohols as solvents with aromatic or aliphatic solutes, or

MASS TRANSFER carbon dioxide as a solvent with aliphatics except ketones as solutes, or ethylene as a solvent with aliphatics except ketones and naphthalene as solutes. For the second class, the correction factor was R = X 0.17. The class was comprised of carbon dioxide with aromatics; ketones and carbon tetrachloride as solutes; and aliphatics (propane, hexane, dimethyl butane), sulfur hexafluoride, and chlorotrifluoromethane as solvents with aromatics as solutes. In addition, sulfur hexafluoride combined with carbon tetrachloride, and chlorotrifluoromethane combined with 2-propanone were included in that class. In all cases, X = (1 + (VCB /VCA)1/3)2/(1 + MA /MB) was in the range of 1 to 10. Liu and Ruckenstein [17] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on the Stokes-Einstein relation and the long-range correlation, respectively. The parameter 2θoAB was estimated from the Peng-Robinson equation of state. In addition, f = 2.72 − 0.3445 TcB/TcA for most solutes, but for C5 through C14 linear alkanes, f = 3.046 − 0.786 TcB/TcA. In both cases Tci is the species critical temperature. They compared their estimates to 33 pairs, with a total of 598 data points, and achieved lower deviations (5.7 percent) than the Sun-Chen correlation (13.3 percent) and the Catchpole-King equation (11.0 percent). He and Yu [13] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on hard-sphere theory. It is limited to ρr . 0.21, where the reduced density is ρr = ρA(T, P)/ρcA. They compared their estimates to 107 pairs, with a total of 1167 data points, and achieved lower deviations (7.8 percent) than the Catchpole-King equation (9.7 percent), which was restricted to ρr . 1. Silva and Macedo [Ind. Eng. Chem. Res. 37, 1490 (1998)] measured diffusivities of ethers in CO2 under supercritical conditions and compared them to the Wilke-Chang [Eq. (5-218)], Tyn-Calus [Eq. (5-219)], Catchpole-King [Eq. (5-210)], and their own equations. They found that the Wilke-Chang equation provided the best fit. Gonzalez, Bueno, and Medina [Ind. Eng. Chem. Res. 40, 3711 (2001)] measured diffusivities of aromatic compounds in CO2 under supercritical conditions and compared them to the Wilke-Chang [Eq. (5-218)], Hayduk-Minhas [Eq. (5-226)], and other equations. They recommended the Wilke-Chang equation (which yielded a relative error of 10.1 percent) but noted that the He-Yu equation provided the best fit (5.5 percent). Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-13. Stefan-Maxwell equations were discussed earlier. Smith and Taylor [23] compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-214) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. Blanc [3] provided a simple limiting case for dilute component i diffusing in a stagnant medium (i.e., N ≈ 0), and the result, Eq. (5-215), is known as Blanc’s law. The restriction basically means that the compositions of all the components, besides component i, are relatively large and uniform. Wilke [29] obtained solutions to the Stefan-Maxwell equations. The first, Eq. (5-216), is simple and reliable under the same conditions as Blanc’s law. This equation applies when component i diffuses through

a stagnant mixture. It has been tested and verified for diffusion of toluene in hydrogen + air + argon mixtures and for diffusion of ethyl propionate in hydrogen + air mixtures [Fairbanks and Wilke Ind. Eng. Chem., 42, 471 (1950)]. When the compositions vary from one boundary to the other, Wilke recommends that the arithmetic average mole fractions be used. Wilke also suggested using the StefanMaxwell equation, which applies when the fluxes of two or more components are significant. In this situation, the mole fractions are arithmetic averages of the boundary conditions, and the solution requires iteration because the ratio of fluxes is not known a priori. DIFFUSIVITY ESTIMATION—LIQUIDS Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restricted to binary diffusion at infinite dilution D°AB or to self-diffusivity DA′A. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits negligible composition effects and deviations from thermodynamic ideality. Conversely, liquid-phase diffusion almost always involves volumetric and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. Experimental values for liquid mixtures are listed in Table 2-325. Stokes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liquid. For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.) kT DAB =  (5-217) 6πrAµB where A refers to the solute and B refers to the solvent. This equation is applicable to very large unhydrated molecules (M > 1000) in lowmolecular-weight solvents or where the molar volume of the solute is greater than 500 cm3/mol (Reddy and Doraiswamy, Ind. Eng. Chem. Fundam., 6, 77 (1967); Wilke and Chang [30]). Despite its intellectual appeal, this equation is seldom used “as is.” Rather, the following principles have been identified: (1) The diffusion coefficient is inversely proportional to the size rA  VA1/3 of the solute molecules. Experimental observations, however, generally indicate that the exponent of the solute molar volume is larger than one-third. (2) The term DABµB /T is approximately constant only over a 10-to-15 K interval. Thus, the dependence of liquid diffusivity on properties and conditions does not generally obey the interactions implied by that grouping. For example, Robinson, Edmister, and Dullien [Ind. Eng. Chem. Fundam., 5, 75 (1966)] found that ln DAB ∝ −1/T. (3) Finally, pressure does not affect liquid-phase diffusivity much, since µB and VA are only weakly pressure-dependent. Pressure does have an impact at very high levels.

TABLE 5-13 Relationships for Diffusivities of Multicomponent Gas Mixtures at Low Pressure Authors* Stefan-Maxwell, Smith and Taylor [23]

Blanc [2]

Wilke [29]

Equation



Dim = 1 − xi Dim = Dim =

5-53

NC

NC

NC

D  N /N / x −  N /  j=1

j

i

xj

NC

ij

(5-214)

i

(5-215)

ij

 j=1 j≠i

xiNi

j

−1

  D j=1

j=1

xj  Dij

−1

*References are listed at the beginning of the “Mass Transfer” subsection.

(5-216)

5-54

HEAT AND MASS TRANSFER The value of φB for water was originally stated as 2.6, although when the original data were reanalyzed, the empirical best fit was 2.26. Random comparisons of predictions with 2.26 versus 2.6 show no consistent advantage for either value, however. Kooijman [Ind. Eng. Chem. Res. 41, 3326 (2002)] suggests replacing VA with θA VA, in which θA = 1 except when A = water, θA = 4.5. This modification leads to an overall error of 8.7 percent for 41 cases he compared. He suggests retaining ΦB = 2.6 when B = water. It has been suggested to replace the exponent of 0.6 with 0.7 and to use an association factor of 0.7 for systems containing aromatic hydrocarbons. These modifications, however, are not recommended by Umesi and Danner [27]. Lees and Sarram [J. Chem. Eng. Data, 16, 41 (1971)] present a comparison of the association parameters. The average absolute error for 87 different solutes in water is 5.9 percent. Tyn-Calus [26] This correlation requires data in the form of molar volumes and parachors ψi = Viσ1/4 i (a property which, over moderate temperature ranges, is nearly constant), measured at the same temperature (not necessarily the temperature of interest). The parachors for the components may also be evaluated at different temperatures from each other. Quale [Chem. Rev. 53, 439 (1953)] has compiled values of ψi for many chemicals. Group contribution methods are available for estimation purposes (Poling et al.). The following suggestions were made by Poling et al.: The correlation is constrained to cases in which µB < 30 cP. If the solute is water or if the solute is an organic acid and the solvent is not water or a short-chain alcohol, dimerization of the solute A should be assumed for purposes of estimating its volume and parachor. For example, the appropriate values for water as solute at 25°C are VW = 37.4 cm3/mol and ψW = 105.2 cm3g1/4/s1/2mol. Finally, if the solute is nonpolar, the solvent volume and parachor should be multiplied by 8µB. According to Kooijman (ibid.), if the Brock-Bird method (described in Poling et al.) is used to

Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand [Science, 174, 490 (1971)] who adapted a theory of viscosity to self-diffusivity. He postulated that DA′A = B(V − Vms)/Vms, where DA′A is the self-diffusion coefficient, V is the molar volume, and Vms is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V − Vms) can be thought of as the free volume, which increases with temperature; and B is a proportionality constant. Ertl and Dullien (ibid.) found that Hildebrand’s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n > 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. Dilute Binary Nonelectrolytes: General Mixtures These correlations are outlined in Table 5-14. Wilke-Chang [30] This correlation for D°AB is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it applies to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. φB is an association factor of solvent B that accounts for hydrogen bonding. Component B

φB

Water Methanol Ethanol Propanol Others

2.26 1.9 1.5 1.2 1.0

TABLE 5-14

Correlations for Diffusivities of Dilute, Binary Mixtures of Nonelectrolytes in Liquids

Authors*

Equation

Error

1. General Mixtures Wilke-Chang [30]

7.4 × 10−8 (φBMB)1/2 T D°AB =  µB VA0.6

(5-218)

20%

Tyn-Calus [26]

8.93 × 10−8 (VA/VB2 )1/6 (ψB/ψA)0.6 T D°AB =  µB

(5-219)

10%

Umesi-Danner [27]

2.75 × 10−8 (RB/RA2/3) T D°AB =  µB

(5-220)

16%

Siddiqi-Lucas [22]

9.89 × 10−8 VB0.265 T D°AB =  V A0.45 µB0.907

(5-221)

13%

  V

(5-222)

18%

(5-223)

6%

2. Gases in Low Viscosity Liquids Sridhar-Potter [24]

Chen-Chen [7]

Vc D°AB = DBB B VcA

2/3

VB

mlB



(βVcB)2/3(RTcB)1/2 T (Vr − 1)  D°AB = 2.018 × 10−9  MA1/6 (MBVcA)1/3 TcB

1/2

3. Aqueous Solutions Hayduk-Laudie [11]

13.16 × 10−5 D°AW =  µ w1.14 VA0.589

(5-224)

18%

Siddiqi-Lucas [22]

D°AW = 2.98 × 10−7 VA−0.5473 µ w−1.026 T

(5-225)

13%

Hayduk-Minhas [12]

A − 0.791) VA−0.71 D°AB = 13.3 × 10−8 T1.47 µ(10.2/V B

(5-226)

5%

Matthews-Akgerman [19]

D°AB = 32.88 M

(5-227)

5%

Riazi-Whitson [21]

(ρDAB)° µ DAB = 1.07   ρ µ°

(5-228)

15%

4. Hydrocarbon Mixtures

−0.61 A

−1.04 D

V

0.5

T

(VB − VD)

−0.27 − 0.38 ω + (−0.05 + 0.1 ω)P r



*References are listed at the beginning of the “Mass Transfer” subsection.

MASS TRANSFER estimate the surface tension, the error is only increased by about 2 percent, relative to employing experimentally measured values. Umesi-Danner [27] They developed an equation for nonaqueous solvents with nonpolar and polar solutes. In all, 258 points were ˚ of the cominvolved in the regression. Ri is the radius of gyration in A ponent molecule, which has been tabulated by Passut and Danner [Chem. Eng. Progress Symp. Ser., 140, 30 (1974)] for 250 compounds. The average absolute deviation was 16 percent, compared with 26 percent for the Wilke-Chang equation. Siddiqi-Lucas [22] In an impressive empirical study, these authors examined 1275 organic liquid mixtures. Their equation yielded an average absolute deviation of 13.1 percent, which was less than that for the Wilke-Chang equation (17.8 percent). Note that this correlation does not encompass aqueous solutions; those were examined and a separate correlation was proposed, which is discussed later. Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids Sridhar and Potter [24] derived an equation for predicting gas diffusion through liquid by combining existing correlations. Hildebrand had postulated the following dependence of the diffusivity for a gas in a liquid: D°AB = DB′B(VcB /VcA)2/3, where DB′B is the solvent selfdiffusion coefficient and Vci is the critical volume of component i, respectively. To correct for minor changes in volumetric expansion, Sridhar and Potter multiplied the resulting equation by VB /VmlB, where VmlB is the molar volume of the liquid B at its melting point and DB′B can be estimated by the equation of Ertl and Dullien (see p. 5-54). Sridhar and Potter compared experimentally measured diffusion coefficients for twenty-seven data points of eleven binary mixtures. Their average absolute error was 13.5 percent, but Chen and Chen [7] analyzed about 50 combinations of conditions and 3 to 4 replicates each and found an average error of 18 percent. This correlation does not apply to hydrogen and helium as solutes. However, it demonstrates the usefulness of self-diffusion as a means to assess mutual diffusivities and the value of observable physical property changes, such as molar expansion, to account for changes in conditions. Chen-Chen [7] Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB /[0.9724 (VmlB + 0.04765)] and VmlB = the liquid molar volume at the melting point, as discussed previously. Their association parameter β [which is different from the definition of that symbol in Eq. (5-229)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are: β = 1.58 and 2.31, respectively. Dilute Binary Mixtures of a Nonelectrolyte in Water The correlations that were suggested previously for general mixtures, unless specified otherwise, may also be applied to diffusion of miscellaneous solutes in water. The following correlations are restricted to the present case, however. Hayduk and Laudie [11] They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on µw compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. Siddiqi and Lucas [227] These authors examined 658 aqueous liquid mixtures in an empirical study. They found an average absolute deviation of 19.7 percent. In contrast, the Wilke-Chang equation gave 35.0 percent and the Hayduk-Laudie correlation gave 30.4 percent. Dilute Binary Hydrocarbon Mixtures Hayduk and Minhas [12] presented an accurate correlation for normal paraffin mixtures that was developed from 58 data points consisting of solutes from C5 to C32 and solvents from C5 to C16. The average error was 3.4 percent for the 58 mixtures. Matthews and Akgerman [19] The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from C8 to C16 and solvents of C6 and C12. The term they referred to as the “diffusion volume” was simply correlated with the critical volume, as VD = 0.308 Vc. We can infer from Table 5-11 that this is approximately related to the volume at the melting point as VD = 0.945 Vm. Their correlation was valid for diffusion of linear alkanes at temperatures

5-55

up to 300°C and pressures up to 3.45 MPa. Matthews, Rodden, and Akgerman [ J. Chem. Eng. Data, 32, 317 (1987)] and Erkey and Akgerman [AIChE J., 35, 443 (1989)] completed similar studies of diffusion of alkanes, restricted to n-hexadecane and n-octane, respectively, as the solvents. Riazi and Whitson [21] They presented a generalized correlation in terms of viscosity and molar density that was applicable to both gases and liquids. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-202)] for the lowpressure diffusivity and the Stiel-Thodos [AIChE J., 7, 234 (1961)] correlation for low-pressure viscosity: 1/2 1/2 xAµ°M A A + xBµ°M B B µ° =  xAMA1/2 + xBMB1/2 i where µ°i ξi = 3.4 × 10−4 Tr0.94 for Tr i < 1.5 or µ°i ξi = 1.778 × 10−4 (4.58 2/3 i /Pc i Tr i − 1.67)5/8 for Tr i > 1.5. In these equations, ξi = Tc1/6 M1/2 i , and units are in cP, atm, K, and mol. For dense gases or liquids, the Chung et al; [Ind. Eng. Chem. Res., 27, 671 (1988)] or Jossi-Stiel-Thodos [AIChE J., 8, 59 (1962)] correlation may be used to estimate viscosity. The latter is:

(µ − µ°) ξ + 10−4 = (0.1023 + 0.023364 ρr + 0.058533 ρ2r − 0.040758 ρ3r + 0.093324 ρ4r)4 where and

(xA TcA + xBTcB)1/6 ξ =  (xAMA + xBMB)1/2 (xA PcA + xB PcB) ρr = (xA VcA + xB VcB)ρ.

Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute Olander [AIChE J., 7, 175 (1961)] modified the WilkeChang equation to adapt it to the infinite dilution diffusivity of water as the solute. The modification he recommended is simply the division of the right-hand side of the Wilke-Chang equation by 2.3. Unfortunately, neither the Wilke-Chang equation nor that equation divided by 2.3 fit the data very well. A reasonably valid generalization is that the Wilke-Chang equation is accurate if water is very insoluble in the solvent, such as pure hydrocarbons, halogenated hydrocarbons, and nitro-hydrocarbons. On the other hand, the Wilke-Chang equation divided by 2.3 is accurate for solvents in which water is very soluble, as well as those that have low viscosities. Such solvents include alcohols, ketones, carboxylic acids, and aldehydes. Neither equation is accurate for higher-viscosity liquids, especially diols. Dilute Dispersions of Macromolecules in Nonelectrolytes The Stokes-Einstein equation has already been presented. It was noted that its validity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford Physical Chemistry of Macromolecules, Wiley, New York, (1961). Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. Hiss and Cussler [AIChE J., 19, 698 (1973)] Their basis is the diffusion of a small solute in a fairly viscous solvent of relatively large molecules, which is the opposite of the Stokes-Einstein assumptions. The large solvent molecules investigated were not polymers or gels but were of moderate molecular weight so that the macroscopic and microscopic viscosities were the same. The major conclusion is that D°AB µ2/3 = constant at a given temperature and for a solvent viscosity from 5 × 10−3 to 5 Pa⋅s or greater (5 to 5 × 103 cP). This observation is useful if D°AB is known in a given high-viscosity liquid (oils, tars, etc.). Use of the usual relation of D°AB ∝ 1/µ for such an estimate could lead to large errors. Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of DAB are summarized in Table 5-15. Most are based on known values of D°AB and D°BA. In fact, a rule of thumb states that, for many binary systems, D°AB and D°BA bound the DAB vs. xA curve. Cullinan’s [8] equation predicts diffusivities even in lieu of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data.

5-56

HEAT AND MASS TRANSFER TABLE 5-15 Correlations of Diffusivities for Concentrated, Binary Mixtures of Nonelectrolyte Liquids Authors*

Equation

Caldwell-Babb [5] Rathbun-Babb [20] Vignes [28] Leffler-Cullinan [16]

DAB = (xA DBA ° + xB DAB ° )βA DAB = (xA DBA ° + xB DAB ° )βAn DAB = DAB ° xB DBA ° xA βA DAB µ mix = (DAB ° µ B)xB (DBA ° µA) xA βA

Cussler [9]

∂ ln xA K DAB = D0 1 +   − 1 xA xB ∂ ln aA

Cullinan [8]

2πxA xB βA kT  DAB =  2πµ mix (V/A)1/3 1 + βA (2πxA xB − 1)

Asfour-Dullien [1]

DAB ° DAB =  µB

Siddiqi-Lucas [22] Bosse and Bart no. 1 [3] Bosse and Bart no. 2 [3]





(5-231) (5-232) (5-233) (5-234) −1/2



(5-235)







1/2

(5-236)

ζµβ  µ xB

° DBA

xA

(5-237)

A

A

DAB = (CB V B D AB ° + CA V A D BA ° )βA gE ∞ XB ∞ XA DAB = (DAB) (DAB) exp −  RT



(5-238)





gE µDAB = (µβ D∞AB)XB (µAD∞BA)XA exp −  RT

(5-239)



(5-240)

Relative errors for the correlations in this table are very dependent on the components of interest and are cited in the text. *See the beginning of the “Mass Transfer” subsection for references.

Since the infinite dilution values D°AB and D°BA are generally unequal, even a thermodynamically ideal solution like γA = γB = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true “driving force” for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is: ∂ ln γA βA = 1 +  (5-229) ∂ ln xA Caldwell and Babb [5] Darken [Trans. Am. Inst. Mining Met. Eng., 175, 184 (1948)] observed that solid-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractions to the mutual diffusivity: DAB = (xA DB + xB DA) βA

(5-230)

Caldwell and Babb used virtually the same equation to evaluate the mutual diffusivity for concentrated mixtures of common liquids. Van Geet and Adamson [J. Phys. Chem. 68, 238 (1964)] tested that equation for the n-dodecane (A) and n-octane (B) system and found the average deviation of DAB from experimental values to be −0.68 percent. In addition, that equation was tested for benzene + bromobenzene, n-hexane + n-dodecane, benzene + CCl 4, octane + decane, heptane + cetane, benzene + diphenyl, and benzene + nitromethane with success. For systems that depart significantly from thermodynamic ideality, it breaks down, sometimes by a factor of eight. For example, in the binary systems acetone + CCl 4, acetone + chloroform, and ethanol + CCl 4, it is not accurate. Thus, it can be expected to be fairly accurate for nonpolar hydrocarbons of similar molecular weight but not for polar-polar mixtures. Siddiqi, Krahn, and Lucas [J. Chem. Eng. Data, 32, 48 (1987)] found that this relation was superior to those of Vignes and Leffler and Cullinan for a variety of mixtures. Umesi and Danner [27] found an average absolute deviation of 13.9 percent for 198 data points. Rathbun and Babb [20] suggested that Darken’s equation could be improved by raising the thermodynamic correction factor βA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult’s law and found n = 0.3. Furthermore, for polarnonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas [22] followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polar-nonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi, Krahn, and Lucas (ibid.) examined a few other mixtures and

found that n = 1 was probably best. Thus, this approach is, at best, highly dependent on the type of components being considered. Vignes [28] empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl, Ghai, and Dollon [AIChE J., 20, 1 (1974)] evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was very nonideal. Leffler and Cullinan [16] modified Vignes’ equation using some theoretical arguments to arrive at Eq. (5-234), which the authors compared to Eq. (5-233) for the 12 systems mentioned above. The average absolute maximum deviation was only 6 percent. Umesi and Danner [27], however, found an average absolute deviation of 11.4 percent for 198 data points. For normal paraffins, it is not very accurate. In general, the accuracies of Eqs. (5-233) and (5-234) are not much different, and since Vignes’ is simpler to use, it is suggested. The application of either should be limited to nonassociating systems that do not deviate much from ideality (0.95 < βA < 1.05). Cussler [9] studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. Do is a reference diffusion coefficient discussed hereafter; aA is the activity of component A; and K is a constant. By assuming that Do could be predicted by Eq. (5-233) with β = 1, K was found to be equal to 0.5 based on five binary systems and validated with a sixth binary mixture. The limitations of Eq. (5-235) using Do and K defined previously have not been explored, so caution is warranted. Gurkan [AIChE J., 33, 175 (1987)] showed that K should actually be closer to 0.3 (rather than 0.5) and discussed the overall results. Cullinan [8] presented an extension of Cussler’s cluster diffusion theory. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel [AIChE J., 37, 470 (1991)], and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally valid at higher concentrations). Asfour and Dullien [1] developed a relation for predicting alkane diffusivities at moderate concentrations that employs:



Vfm ζ=  Vf xAVf xB



2/3

MxAMxB  Mm

(5-241)

MASS TRANSFER x

where Vfxi = Vfi i; the fluid free volume is Vf i = Vi − Vml i for i = A, B, and m, in which Vml i is the molar volume of the liquid at the melting point and



xA2 2 xA xB xB2 Vmlm =  + + VmlA VmlAB VmlB

−1



1/3 3 V1/3 mlA + V mlB VmlAB =  2 and µ is the mixture viscosity; Mm is the mixture mean molecular weight; and βA is defined by Eq. (5-229). The average absolute error of this equation is 1.4 percent, while the Vignes equation and the Leffler-Cullinan equation give 3.3 percent and 6.2 percent, respectively. Siddiqi and Lucas [22] suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolarnonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. Bosse and Bart added a term to account for excess Gibbs free energy, involved in the activation energy for diffusion, which was previously omitted. Doing so yielded minor modifications of the Vignes and Leffler-Cullinan equations [Eqs. (5-233) and (5-234), respectively]. The UNIFAC method was used to assess the excess Gibbs free energy. Comparing predictions of the new equations with data for 36 pairs and 326 data points yielded relative deviations of 7.8 percent and 8.9 percent, respectively, but which were better than the closely related Vignes (12.8 percent) and Leffler-Cullinan (10.4 percent) equations. Binary Electrolyte Mixtures When electrolytes are added to a solvent, they dissociate to a certain degree. It would appear that the solution contains at least three components: solvent, anions, and cations. If the solution is to remain neutral in charge at each point (assuming the absence of any applied electric potential field), the anions and cations diffuse effectively as a single component, as for molecular diffusion. The diffusion of the anionic and cationic species in the solvent can thus be treated as a binary mixture. Nernst-Haskell The theory of dilute diffusion of salts is well developed and has been experimentally verified. For dilute solutions of a single salt, the well-known Nernst-Haskell equation (Poling et al.) is applicable:

and



   



   

1 1 1 1  +   +  n− n n RT n+ + −  = 8.9304 × 10−10 T  D°AB =  1 1 F 2 1 + 1 +     λ0+ λ0− λ0+ λ0−

(5-242)

5-57

decreases rapidly from D°AB. As concentration is increased further, however, DAB rises steadily, often becoming greater than D°AB. Gordon proposed the following empirical equation, which is applicable up to concentrations of 2N: ln γ 1 µB DAB = D°AB   1+ (5-244) ln m CBVB µ where D°AB is given by the Nernst-Haskell equation. References that tabulate γ as a function of m, as well as other equations for DAB, are given by Poling et al. Morgan, Ferguson, and Scovazzo [Ind. Eng. Chem. Res. 44, 4815 (2005)] They studied diffusion of gases in ionic liquids having moderate to high viscosity (up to about 1000 cP) at 30°C. Their range was limited, and the empirical equation they found was







1 DAB = 3.7 × 10−3  2 µ0.59 B VAρB



(5-245)

which yielded a correlation coefficient of 0.975. Of the estimated diffusivities 90 percent were within ±20 percent of the experimental values. The exponent for viscosity approximately confirmed the observation of Hiss and Cussler (ibid). Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent liquiddiffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binary diffusion coefficients have significantly limited the utility of the theories (see Poling et al.). The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to liquids since the coefficients are so dependent on conditions. That is, in liquids, each Dij can be strongly composition dependent in binary mixtures and, moreover, the binary Dij is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each Dij is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B + C). Umesi and Danner [27] compared the three equations given below for 49 ternary systems. All three equations were equivalent, giving average absolute deviations of 25 percent. Perkins and Geankoplis [Chem. Eng. Sci., 24, 1035 (1969)] n

where D°AB = diffusivity based on molarity rather than normality of dilute salt A in solvent B, cm2/s. The previous definitions can be interpreted in terms of ionicspecies diffusivities and conductivities. The latter are easily measured and depend on temperature and composition. For example, the equivalent conductance Λ is commonly tabulated in chemistry handbooks as the limiting (infinite dilution) conductance Λo and at standard concentrations, typically at 25°C. Λ = 1000K/C = λ+ + λ− = Λo + f(C), (cm2/ohm gequiv); K = α/R = specific conductance, (ohm cm)−1; C = solution concentration, (gequiv/ᐉ); α = conductance cell constant (measured), (cm−1); R = solution electrical resistance, which is measured (ohm); and f(C) = a complicated function of concentration. The resulting equation of the electrolyte diffusivity is |z+| + |z−| DAB =  (|z−| / D+) + (|z+| / D−)

(5-243)

where |z| represents the magnitude of the ionic charge and where the cationic or anionic diffusivities are D = 8.9304 × 10−10 Tλ /|z| cm2/s. The coefficient is kN0 /F 2 = R/F 2. In practice, the equivalent conductance of the ion pair of interest would be obtained and supplemented with conductances of permutations of those ions and one independent cation and anion. This would allow determination of all the ionic conductances and hence the diffusivity of the electrolyte solution. Gordon [J. Phys. Chem. 5, 522 (1937)] Typically, as the concentration of a salt increases from infinite dilution, the diffusion coefficient

Dam µ m0.8 = xj D°A j µ0.8 j

(5-246)

j=1 j≠A

Cullinan [Can. J. Chem. Eng. 45, 377 (1967)] This is an extension of Vignes’ equation to multicomponent systems: n

Dam =

 (D° ) Aj

xj

(5-247)

j=1 j≠A

Leffler and Cullinan [16] They extended their binary relation to an arbitrary multicomponent mixture, as follows: n

Dam µm =

 (D° µ ) Aj

j

xj

(5-248)

j=1 j≠A

where DAj is the dilute binary diffusion coefficient of A in j; DAm is the dilute diffusion of A through m; xj is the mole fraction; µj is the viscosity of component j; and µm is the mixture viscosity. Akita [Ind. Eng. Chem. Fundam., 10, 89 (1981)] Another case of multicomponent dilute diffusion of significant practical interest is that of gases in aqueous electrolyte solutions. Many gas-absorption processes use electrolyte solutions. Akita presents experimentally tested equations for this case. Graham and Dranoff [Ind. Eng. Chem. Fundam., 21, 360 and 365 (1982)] They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion.

5-58

HEAT AND MASS TRANSFER

Pinto and Graham [AIChE J. 32, 291 (1986) and 33, 436 (1987)] They studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M. Anderko and Lencka [Ind. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. DIFFUSION OF FLUIDS IN POROUS SOLIDS Diffusion in porous solids is usually the most important factor controlling mass transfer in adsorption, ion exchange, drying, heterogeneous catalysis, leaching, and many other applications. Some of the applications of interest are outlined in Table 5-16. Applications of these equations are found in Secs. 16, 22, and 23. Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-249). The tortuosity τ that expresses this hindrance has been estimated from geometric arguments. Unfortunately,

TABLE 5-16

measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity Deff (and hence τ) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for silica gel, alumina, and other porous solids is 2 ≤ τ ≤ 6, but for activated carbon, 5 ≤ τ ≤ 65. In small pores and at low pressures, the mean free path ᐉ of the gas molecule (or atom) is significantly greater than the pore diameter dpore. Its magnitude may be estimated from 3.2 µ RT 1/2 ᐉ=  m P 2πM As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinary diffusion and Knudsen diffusion may be important [see Eqs. (5-252) and (5-253)]. For gases and vapors that adsorb on the porous solid, surface diffusion may be important, particularly at high surface coverage [see Eqs. (5-254) and (5-257)]. The mechanism of surface diffusion may be viewed as molecules hopping from one surface site to another. Thus, if adsorption is too strong, surface diffusion is impeded, while if adsorption is too weak, surface diffusion contributes insignificantly to the overall rate. Surface diffusion and bulk diffusion usually occur in parallel [see Eqs. (5-258) and (5-259)]. Although Ds is expected to be less than Deff, the solute flux due to surface diffusion may be larger





Relations for Diffusion in Porous Solids

Mechanism

Equation

Bulk diffusion in pores

ε pD Deff =  τ

Knudsen diffusion

T DK = 48.5 dpore  M

Applies to



1/2

in m2/s

References*

(5-249)

Gases or liquids in large pores. NK n = ᐉ/d pore < 0.01

[33]

(5-250)

Dilute (low pressure) gases in small pores. NK n = ᐉ/d pore > 10

Geankoplis, [34, 35]

(5-251)

"

"

"

"

(5-252)

"

"

"

"

ε pDK DKeff =  τ dC Ni = −DK i dz Combined bulk and Knudsen diffusion

1 − α xA 1 Deff =  +  Deff DKeff



−1



NB α=1+ NA



1 1 Deff =  +  Deff DKeff Surface diffusion

−1



(5-253)

NA = NB

dq JSi = −DSeff ρp i dz

(5-254)

Adsorbed gases or vapors

ε p DS DSeff =  τ

(5-255)

"

DSθ = 0 DSθ =  (1 − θ)

(5-256)

θ = fractional surface coverage ≤ 0.6

(5-257)

"

"

"

"

(5-258)

"

"

"

"

(5-259)

"

"

"

"

(5-260)

"

"

"

"



−ES DS = DS′ (q) exp  RT



Parallel bulk and surface diffusion

 

Geankoplis, [32, 35]

NA ≠ NB

 

dp dq J = − Deff i + DSeff ρp i dz dz



dp J = −Dapp i dz



dqi Dapp = Deff + DSeff ρp  dpi

*See the beginning of the “Mass Transfer” subsection for references.

"

"

[32, 34, 35]

"

[34]

MASS TRANSFER than that due to bulk diffusion if ∂qi /∂z >> ∂Ci /∂z. This can occur when a component is strongly adsorbed and the surface coverage is high. For all that, surface diffusion is not well understood. The references in Table 5-16 should be consulted for further details.

solute in bulk-gas phase, yi = mole-fraction solute in gas at interface, x = mole-fraction solute in bulk-liquid phase, and xi = mole-fraction solute in liquid at interface. The mass-transfer coefficients defined by Eqs. (5-261) and (5-262) are related to each other as follows:

INTERPHASE MASS TRANSFER Transfer of material between phases is important in most separation processes in which two phases are involved. When one phase is pure, mass transfer in the pure phase is not involved. For example, when a pure liquid is being evaporated into a gas, only the gas-phase mass transfer need be calculated. Occasionally, mass transfer in one of the two phases may be neglected even though pure components are not involved. This will be the case when the resistance to mass transfer is much larger in one phase than in the other. Understanding the nature and magnitudes of these resistances is one of the keys to performing reliable mass transfer. In this section, mass transfer between gas and liquid phases will be discussed. The principles are easily applied to the other phases. Mass-Transfer Principles: Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-liquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equilibrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a liquid are brought into contact. For systems in which the solute concentrations in the gas and liquid phases are dilute, the rate of transfer may be expressed by equations which predict that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus NA = k′G(p − pi) = k′L(ci − c)

(5-261)

where NA = mass-transfer rate, k′G = gas-phase mass-transfer coefficient, k′L = liquid-phase mass-transfer coefficient, p = solute partial pressure in bulk gas, pi = solute partial pressure at interface, c = solute concentration in bulk liquid, and ci = solute concentration in liquid at interface. The mass-transfer coefficients k′G and k′L by definition are equal to the ratios of the molal mass flux NA to the concentration driving forces (p − pi) and (ci − c) respectively. An alternative expression for the rate of transfer in dilute systems is given by NA = kG(y − yi) = kL(xi − x) (5-262) where NA = mass-transfer rate, kG = gas-phase mass-transfer coefficient, kL = liquid-phase mass-transfer coefficient, y = mole-fraction

5-59

kG = k′G pT kL = k′L ρ L

(5-263) (5-264)

where pT = total system pressure employed during the experimental determinations of k′G values and ρ L = average molar density of the liquid phase. The coefficient kG is relatively independent of the total system pressure and therefore is more convenient to use than k′G, which is inversely proportional to the total system pressure. The above equations may be used for finding the interfacial concentrations corresponding to any set of values of x and y provided the ratio of the individual coefficients is known. Thus (y − yi)/(xi − x) = kL /kG = k′Lρ L/k′G pT = LMHG /GMHL (5-265) where LM = molar liquid mass velocity, GM = molar gas mass velocity, HL = height of one transfer unit based on liquid-phase resistance, and HG = height of one transfer unit based on gas-phase resistance. The last term in Eq. (5-265) is derived from Eqs. (5-284) and (5-286). Equation (5-265) may be solved graphically if a plot is made of the equilibrium vapor and liquid compositions and a point representing the bulk concentrations x and y is located on this diagram. A construction of this type is shown in Fig. 5-27, which represents a gasabsorption situation. The interfacial mole fractions yi and xi can be determined by solving Eq. (5-265) simultaneously with the equilibrium relation y°i = F(xi) to obtain yi and xi. The rate of transfer may then be calculated from Eq. (5-262). If the equilibrium relation y°i = F(xi) is sufficiently simple, e.g., if a plot of y°i versus xi is a straight line, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference is y − y°, and another is x° − x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. The rate of mass transfer may be defined by the equation NA = KG(y − y°) = kG(y − yi) = kL(xi − x) = KL(x° − x) (5-266) where KG = overall gas-phase mass-transfer coefficient, KL = overall liquid-phase mass-transfer coefficient, y° = vapor composition in equilibrium with x, and x° = liquid composition in equilibrium with vapor of composition y. This equation can be rearranged to the formula 1 1 y − y° 1 1 yi − y° 1 1 yi − y° =  =+  =+  KG kG y − yi kG kG y − yi kG kL xi − x













(5-267)

Identification of concentrations at a point in a countercurrent absorption tower.

FIG. 5-27 FIG. 5-26

Concentration gradients near a gas-liquid interface.

5-60

HEAT AND MASS TRANSFER

in view of Eq. (5-265). Comparison of the last term in parentheses with the diagram of Fig. 5-27 shows that it is equal to the slope of the chord connecting the points (x,y°) and (xi,yi). If the equilibrium curve is a straight line, then this term is the slope m. Thus 1/KG = (1/kG + m/kL) (5-268) When Henry’s law is valid (pA = HxA or pA = H′CA), the slope m can be computed according to the relationship m = H/pT = H′ρ (5-269) L/pT where m is defined in terms of mole-fraction driving forces compatible with Eqs. (5-262) through (5-268), i.e., with the definitions of kL, kG, and KG. If it is desired to calculate the rate of transfer from the overall concentration difference based on bulk-liquid compositions (x° − x), the appropriate overall coefficient KL is related to the individual coefficients by the equation 1/KL = [1/kL + 1/(mkG)] (5-270) Conversion of these equations to a k′G, k′L basis can be accomplished readily by direct substitution of Eqs. (5-263) and (5-264). Occasionally one will find k′L or K′L values reported in units (SI) of meters per second. The correct units for these values are kmol/ [(s⋅m2)(kmol/m3)], and Eq. (5-264) is the correct equation for converting them to a mole-fraction basis. When k′G and K′G values are reported in units (SI) of kmol/[(s⋅m2) (kPa)], one must be careful in converting them to a mole-fraction basis to multiply by the total pressure actually employed in the original experiments and not by the total pressure of the system to be designed. This conversion is valid for systems in which Dalton’s law of partial pressures (p = ypT) is valid. Comparison of Eqs. (5-268) and (5-270) shows that for systems in which the equilibrium line is straight, the overall mass transfer coefficients are related to each other by the equation KL = mKG

(5-271)

When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the equilibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences (y − y°) and (x° − x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., (xi − x) and (y − yi). In most types of separation equipment such as packed or spray towers, the interfacial area that is effective for mass transfer cannot be accurately determined. For this reason it is customary to report experimentally observed rates of transfer in terms of transfer coefficients based on a unit volume of the apparatus rather than on a unit of interfacial area. Such volumetric coefficients are designated as KGa, kLa, etc., where a represents the interfacial area per unit volume of the apparatus. Experimentally observed variations in the values of these volumetric coefficients with variations in flow rates, type of packing, etc., may be due as much to changes in the effective value of a as to changes in k. Calculation of the overall coefficients from the individual volumetric coefficients is made by means of the equations 1/KGa = (1/kGa + m/kLa)

(5-272)

1/KLa = (1/kLa + 1/mkGa)

(5-273)

Because of the wide variation in equilibrium, the variation in the values of m from one system to another can have an important effect on the overall coefficient and on the selection of the type of equipment to use. For example, if m is large, the liquid-phase part of the overall resistance might be extremely large where kL might be relatively small. This kind of reasoning must be applied with caution, however, since species with different equilibrium characteristics are separated under different

operating conditions. Thus, the effect of changes in m on the overall resistance to mass transfer may partly be counterbalanced by changes in the individual specific resistances as the flow rates are changed. Mass-Transfer Principles: Concentrated Systems When solute concentrations in the gas and/or liquid phases are large, the equations derived above for dilute systems no longer are applicable. The correct equations to use for concentrated systems are as follows: NA = kˆ G(y − yi)/yBM = kˆ L(xi − x)/xBM = Kˆ G(y − y°)/y°BM = Kˆ L(x° − x)/x°BM

(5-274)

(1 − y) − (1 − yi) yBM =  ln [(1 − y)/(1 − yi)]

(5-275)

where (NB = 0)

(1 − y) − (1 − y°) y°BM =  ln [(1 − y)/(1 − y°)] (1 − x) − (1 − xi) xBM =  ln [(1 − x)/(1 − xi)]

(5-276) (5-277)

(1 − x) − (1 − x°) x°BM =  (5-278) ln [(1 − x)/(1 − x°)] and where kˆ G and kˆ L are the gas-phase and liquid-phase mass-transfer coefficients for concentrated systems and Kˆ G and Kˆ L are the overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems. These coefficients are defined later in Eqs. (5-281) to (5-283). The factors yBM and xBM arise from the fact that, in the diffusion of a solute through a second stationary layer of insoluble fluid, the resistance to diffusion varies in proportion to the concentration of the insoluble stationary fluid, approaching zero as the concentration of the insoluble fluid approaches zero. See Eq. (5-198). The factors y°BM and x°BM cannot be justified on the basis of masstransfer theory since they are based on overall resistances. These factors therefore are included in the equations by analogy with the corresponding film equations. In dilute systems the logarithmic-mean insoluble-gas and nonvolatileliquid concentrations approach unity, and Eq. (5-274) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-197). Substitution of Eqs. (5-275) through (5-278) into Eq. (5-274) results in the following simplified formula: NA = kˆ G ln [(1 − yi)/(1 − y)] = Kˆ G ln [(1 − y°)/(1 − y)] = kˆ L ln [(1 − x)/(1 − xi)] = Kˆ L ln [(1 − x)/(1 − x°)]

(5-279) Note that the units of kˆ G, Kˆ G, kˆ L, and Kˆ L are all identical to each other, i.e., kmol/(s⋅m2) in SI units. The equation for computing the interfacial gas and liquid compositions in concentrated systems is (y − yi)/(xi − x) = kˆ LyBM / kˆ GxBM = LMHGyBM /GMHL x BM = kL /kG (5-280) This equation is identical to the one for dilute systems since kˆ G = kGyBM and kˆ L = kLxBM. Note, however, that when kˆ G and kˆ L are given, the equation must be solved by trial and error, since xBM contains xi and yBM contains yi. The overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems are computed according to the following equations: 1 yBM 1 xBM 1 yi − y° =+   Kˆ G y°BM kˆ G y°BM kˆ L xi − x





(5-281)

1 xBM 1 yBM 1 x° − xi =+  (5-282)  Kˆ L x°BM kˆ L x°BM kˆ G y − yi When the equilibrium curve is a straight line, the terms in parentheses can be replaced by the slope m or 1/m as before. In this case the





MASS TRANSFER overall mass-transfer coefficients for concentrated systems are related to each other by the equation (5-283) Kˆ L = m Kˆ G(x°BM /y°BM) All these equations reduce to their dilute-system equivalents as the inert concentrations approach unity in terms of mole fractions of inert concentrations in the fluids. HTU (Height Equivalent to One Transfer Unit) Frequently the values of the individual coefficients of mass transfer are so strongly dependent on flow rates that the quantity obtained by dividing each coefficient by the flow rate of the phase to which it applies is more nearly constant than the coefficient itself. The quantity obtained by this procedure is called the height equivalent to one transfer unit, since it expresses in terms of a single length dimension the height of apparatus required to accomplish a separation of standard difficulty. The following relations between the transfer coefficients and the values of HTU apply: (5-284) H = G /k ay = G /kˆ a G

M

G

BM

M

G

HOG = GM /KGay°BM = GM/Kˆ Ga H = L /k ax = L /kˆ a

(5-286)

HOL = LM /KLax°BM = LM/Kˆ La

(5-287)

L

M

L

BM

M

(5-285)

L

The equations that express the addition of individual resistances in terms of HTUs, applicable to either dilute or concentrated systems, are yBM mGM xBM (5-288) HOG =  HG +   HL y°BM LM y°BM xBM LM yBM HOL =  HL +  (5-289)  HG x°BM mGM x°BM These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to carry out the same separation in a stagewise or platetype apparatus. For equimolal counterdiffusion, such as in a binary distillation, the number of overall gas-phase transfer units NOG required for changing the composition of the vapor stream from y1 to y2 is y1 dy NOG = (5-290)  y2 y − y° When diffusion is in one direction only, as in the absorption of a soluble component from an insoluble gas,



NOG =



y1

y2

y°BM dy  (1 − y)(y − y°)

(5-291)

The total height of packing required is then hT = HOGNOG

(5-292)

When it is known that HOG varies appreciably within the tower, this term must be placed inside the integral in Eqs. (5-290) and (5-291) for accurate calculations of hT. For example, the packed-tower design equation in terms of the overall gas-phase mass-transfer coefficient for absorption would be expressed as follows: hT =

y° dy G    K ay°  (1 − y)(y − y°) y1

M

y2

G

BM

(5-293)

BM

where the first term under the integral can be recognized as the HTU term. Convenient solutions of these equations for special cases are discussed later. Definitions of Mass-Transfer Coefficients kˆ G and kˆ L The mass-transfer coefficient is defined as the ratio of the molal mass flux NA to the concentration driving force. This leads to many different ways of defining these coefficients. For example, gas-phase masstransfer rates may be defined as (5-294) N = k (y − y ) = k′ (p − p ) = kˆ (y − y )/y A

G

i

G

i

G

i

BM

where the units (SI) of kG are kmol/[(s⋅m2)(mole fraction)], the units of

5-61

k′G are kmol/[(s⋅m2)(kPa)], and the units of kˆ G are kmol/(s⋅m2). These coefficients are related to each other as follows: (5-295) kG = kGyBM = k′G pT yBM where pT is the total system pressure (it is assumed here that Dalton’s law of partial pressures is valid). In a similar way, liquid-phase mass-transfer rates may be defined by the relations (5-296) N = k (x − x) = k′ (c − c) = kˆ (x − x)/x A

L

i

L

i

L

i

BM

where the units (SI) of kL are kmol/[(s⋅m2)(mole fraction)], the units of k′L are kmol/[(s⋅m2)(kmol/m3)] or meters per second, and the units of kˆ L are kmol/(s⋅m2). These coefficients are related as follows: ρLxBM (5-297) kˆ L = kLxBM = k′L where  ρL is the molar density of the liquid phase in units (SI) of kilomoles per cubic meter. Note that, for dilute solutions where xBM ⬟ 1, kL and kˆ L will have identical numerical values. Similarly, for dilute gases kˆ G ⬟ kG. Simplified Mass-Transfer Theories In certain simple situations, the mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend these theoretical calculations to more complex situations. Although these theories are often not accurate, they are useful to provide a physical picture for variations in the mass-transfer coefficient. For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as 1 − yi DABpT (y − yi) DABpT NA =   =  ln  RT δG yBM RT δG 1−y

(5-298)

where DAB = the diffusion coefficient or “diffusivity,” δG = the “effective” thickness of a stagnant-gas layer which would offer a resistance to molecular diffusion equal to the experimentally observed resistance, and R = the gas constant. [Nernst, Z. Phys. Chem., 47, 52 (1904); Whitman, Chem. Mat. Eng., 29, 149 (1923), and Lewis and Whitman, Ind. Eng. Chem., 16, 1215 (1924)]. The film thickness δG depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables: (5-299) NSh = kˆ GRTd/DABpT = f(NRe,NSc) where NSh is the Sherwood number, NRe (= Gd/µG) is the Reynolds number based on the characteristic length d appropriate to the geometry of the particular system; and NSc (= µG /ρGDAB) is the Schmidt number. According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form (y − yi)/yBM, and the most appropriate mass-transfer coefficient is therefore kˆ G. This concept is to be found in all the key equations for the design of mass-transfer equipment. The Sherwood-number relation for gas-phase mass-transfer coefficients as represented by the film diffusion model in Eq. (5-299) can be rearranged as follows: NSh = (kˆ G /GM)NReNSc = NStNReNSc = f (NRe,NSc) (5-300) where NSt = kˆ G /GM = k′G pBM /GM is known as the Stanton number. This equation can now be stated in the alternative functional forms (5-301) NSt = kˆ G /GM = g(NRe,NSc) 2 /3 (5-302) jD = NSt ⋅ NSc where j is the Chilton-Colburn “j factor” for mass transfer (discussed later). The important point to note here is that the gas-phase masstransfer coefficient kˆ G depends principally upon the transport properties of the fluid (NSc) and the hydrodynamics of the particular system involved (NRe). It also is important to recognize that specific masstransfer correlations can be derived only in conjunction with the

5-62

HEAT AND MASS TRANSFER

investigator’s particular assumptions concerning the numerical values of the effective interfacial area a of the packing. The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant; i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Inst. Chem. Eng., 31, 365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theory. If the velocity of the flowing stream is uniform over a very deep region of liquid (total thickness, δT >>  Dt), the time-averaged masstransfer coefficient according to penetration theory is given by

 t k′L = 2D L/π

(5-303)

where k′L = liquid-phase mass-transfer coefficient, DL = liquid-phase diffusion coefficient, and t = contact time. In practice, the contact time t is not known except in special cases in which the hydrodynamics are clearly defined. This is somewhat similar to the case of the stagnant-film theory in which the unknown quantity is the thickness of the stagnant layer δ (in film theory, the liquid-phase mass-transfer coefficient is given by k′L = DL /δ). The penetration theory predicts that k′L should vary by the square root of the molecular diffusivity, as compared with film theory, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a w power. Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and the latter method normally is preferred. Danckwerts [Ind. Eng. Chem., 42, 1460 (1951)] proposed an extension of the penetration theory, called the surface renewal theory, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danckwerts model gives

s k′L = D

(5-304)

where s = fractional rate of surface renewal. Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do δ and t in the film and penetration models. The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surfacerenewal models. Thus, in view of its relative simplicity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for making a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-24 must be employed. Mass-Transfer Correlations Because of the tremendous importance of mass transfer in chemical engineering, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Some of these studies are summarized in Tables 5-17 to 5-24. Each table is for a specific geometry or type of contactor, starting with flat plates, which have the simplest geometry (Table 5-17); then wetted wall columns (Table 5-18); flow in pipes and ducts (Table 5-19); submerged objects (Table 5-20); drops and

bubbles (Table 5-21); agitated systems (Table 5-22); packed beds of particles for adsorption, ion exchange, and chemical reaction (Table 523); and finishing with packed bed two-phase contactors for distillation, absorption and other unit operations (Table 5-24). Although extensive, these tables are not meant to be encyclopedic, but a variety of different configurations are shown to provide a flavor of the range of correlations available. These correlations include transfer to or from one fluid and either a second fluid or a solid. Many of the correlations are for kL and kG values obtained from dilute systems where xBM ≈ 1.0 and yBM ≈ 1.0. The most extensive source for older mass-transfer correlations in a variety of geometries is Skelland (Diffusional Mass Transfer, 1974). The extensive review of bubble column systems (see Table 5-21) by Shah et al. [AIChE J. 28, 353 (1982)] includes estimation of bubble size, gas holdup, interfacial area kLa, and liquid dispersion coefficent. For correlations for particle-liquid mass transfer in stirred tanks (part of Table 5-22) see the review by Pangarkar et al. [Ind. Eng. Chem. Res. 41, 4141 (2002)]. For mass transfer in distillation, absorption, and extraction in packed beds (Table 5-24), see also the appropriate sections in this handbook and the review by Wang, Yuan, and Yu [Ind. Eng. Chem. Res. 44, 8715 (2005)]. For simple geometries, one may be able to determine a theoretical (T) form of the mass-transfer correlation. For very complex geometries, only an empirical (E) form can be found. In systems of intermediate complexity, semiempirical (S) correlations where the form is determined from theory and the coefficients from experiment are often useful. Although the major limitations and constraints in use are usually included in the tables, obviously many details cannot be included in this summary form. Readers are strongly encouraged to check the references before using the correlations in important situations. Note that even authoritative sources occasionally have typographical errors in the fairly complex correlation equations. Thus, it is a good idea to check several sources, including the original paper. The references will often include figures comparing the correlations with data. These figures are very useful since they provide a visual picture of the scatter in the data. Since there are often several correlations that are applicable, how does one choose the correlation to use? First, the engineer must determine which correlations are closest to the current situation. This involves recognizing the similarity of geometries, which is often challenging, and checking that the range of parameters in the correlation is appropriate. For example, the Bravo, Rocha, and Fair correlation for distillation with structured packings with triangular cross-sectional channels (Table 5-24-H) uses the Johnstone and Pigford correlation for rectification in vertical wetted wall columns (Table 5-18-F). Recognizing that this latter correlation pertains to a rather different application and geometry was a nontrivial step in the process of developing a correlation. If several correlations appear to be applicable, check to see if the correlations have been compared to each other and to the data. When a detailed comparison of correlations is not available, the following heuristics may be useful: 1. Mass-transfer coefficients are derived from models. They must be employed in a similar model. For example, if an arithmetic concentration difference was used to determine k, that k should only be used in a mass-transfer expression with an arithmetic concentration difference. 2. Semiempirical correlations are often preferred to purely empirical or purely theoretical correlations. Purely empirical correlations are dangerous to use for extrapolation. Purely theoretical correlations may predict trends accurately, but they can be several orders of magnitude off in the value of k. 3. Correlations with broader data bases are often preferred. 4. The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 5. More recent data is often preferred to older data, since end effects are better understood, the new correlation often builds on earlier data and analysis, and better measurement techniques are often available. 6. With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kap or of HTU are more accurate than individual correlations of k and

MASS TRANSFER TABLE 5-17

5-63

Mass-Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid Situation

A. Laminar, local, flat plate, forced flow

Laminar, average, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′x NSh,x =  = 0.323(NRe,x)1/2(NSc)1/3 D Coefficient 0.332 is a better fit.

k′m L NSh,avg =  = 0.646(NRe,L)1/2(NSc)1/3 D

References*

[T] Low M.T. rates. Low mass-flux, constant property systems. NSh,x is local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius’ approximate solution.

[77] p. 183 [87] p. 526 [138] p. 79 [140] p. 518

xu ∞ ρ NRe,x =  , x = length along plate µ

[141] p. 110

Lu ∞ ρ NRe,L =  , 0.664 (Polhausen) µ

[91] p. 480

Coefficient 0.664 is a better fit. k′m is mean mass-transfer coefficient for dilute systems. j-factors

B. Laminar, local, flat plate, blowing or suction and forced flow

C. Laminar, local, flat plate, natural convection vertical plate

D. Laminar, stationary disk

Laminar, spinning disk

E. Laminar, inclined, plate

f jD = jH =  = 0.664(NRe,L)−1/2 2 k′x NSh,x =  = (Slope)y = 0 (NRe,x)1/2(NSc)1/3 D

k′x −1/4 1/4 NSh,x =  = 0.508N1/2 N Gr Sc (0.952 + NSc) D gx 3 ρ∞ NGr = 2  − 1 (µ/ρ) ρ0





[T] Low MT rates. Dilute systems, ∆ρ/ρ << 1. NGr NSc < 108. Use with arithmetic concentration difference. x = length from plate bottom.

k′d disk 1/3 NSh =  = 0.879N 1/2 Re N Sc D NRe < ∼ 104

[T] Asymptotic solution for large NSc. u = ωddisk/2, ω = rotational speed, rad/s. Rotating disks are often used in electrochemical research.

3 2 1/3 x ρ g sinα NSh,avg = 0.783N1/9  Re,film N Sc µ2





2/9

[141] p. 271

[77] p. 185

[140] p. 271 [141] p. 120

[138] p. 240

[101] p. 60 [138] p. 240

[T] Constant-property liquid film with low masstransfer rates. Use arithmetic concentration difference. Newtonian fluid. Solute does not penetrate past region of linear velocity profile. Differences between theory and experiment. w = width of plate, δ f = film thickness, α = angle of inclination, x = distance from start soluble surface.

[141] p. 130 [138] p. 209

k′x 0.8 NSh,x =  = 0.0292 N Re,x , D

[S] Low mass-flux with constant property system. Use with arithmetic concentration difference. NSc = 1.0, NRe,x > 105

[77] p. 191 [138] p. 201 [141] p. 221

k′L 0.8 NSh,avg =  = 0.0365N Re,L D

Based on Prandtl’s 1/7-power velocity law,

f −0.2 jD = jH =  = 0.037 N Re,L 2

[E] Chilton-Colburn analogies, NSc = 1.0, (gases), f = drag coefficient. Corresponds to item 5-17-F and refers to same conditions. 8000 < NRe < 300,000. Can apply analogy, jD = f/2, to entire plate (including laminar portion) if average values are used.



3µQ δ film =  wρg sinα

G. Laminar and turbulent, flat plate, forced flow

[T] Blowing is positive. Other conditions as above. uo    N Re,x u∞ 0.6 0.5 0.25 0.0 −2.5   (Slope)y = 0 0.01 0.06 0.17 0.332 1.64

[T] Stagnant fluid. Use arithmetic concentration difference.

k′m x NSh,avg =  D

Turbulent, average, flat plate, forced flow

[S] Analogy. NSc = 1.0, f = drag coefficient. jD is defined in terms of k′m.

k′d disk 8 NSh =  =  D π

4Qρ NRe,film =  < 2000 µ2

F. Turbulent, local flat plate, forced flow

is a better fit for NSc > 0.6, NRe,x < 3 × 105.

2/3 jD = (kG/GM)NSc 2/3 jH = (h′CpG) NPr



1/3

= film thickness



y u =  u∞ δ

1/7

[77] p. 193 [88] p. 112 [138] p. 201 [141] p. 271 [80] [53]

5-64

HEAT AND MASS TRANSFER

TABLE 5-17

Mass-Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid (Concluded) Situation

H. Laminar and turbulent, flat plate, forced flow

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.8 NSh,avg = 0.037N 1/3 Sc (N Re,L − 15,500) to NRe,L = 320,000

References*

[E] Use arithmetic concentration difference.

NSh,avg = 0.037N 1/3 Sc





0.664 1/2 0.8 0.8 × N Re,L − N Re,Cr +  N Re,Cr 0.037

k′m L NSh,avg =  , NSc > 0.5 D Entrance effects are ignored. NRe,Cr is transition laminar to turbulent.

[88] p. 112 [138] p. 201

in range 3 × 105 to 3 × 106. I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate

J. Perforated flat disk Perforated vertical plate. Natural convection.

K. Turbulent, vertical plate

k′x 2/5 7/15 NSh,x =  = 0.0299N Gr N Sc D 2/3 −2/5 Sc

× (1 + 0.494N )

2/5 7/15 2/3 −2/5 NSh,avg = 0.0249N Gr N Sc × (1 + 0.494 N Sc )

gx3 ρ∞ NGr = 2  − 1 , (µ/ρ) ρ0







0.04

Characteristic length = L, electrode height

Average deviation ± 10%

k′m x xρg 2/9 NSh,avg =  = 0.327N Re,film N 1/3 Sc  D µ2

[E] See 5-17-E for terms.



3 2



2/9

Solute remains in laminar sublayer.

NSh = cN N a Re

[141] p. 229

4Qρ > 2360 NRe,film =  wµ2

1/3

13 Sc

k ′ddisk 1.1 1/3 NSh =  = 5.6N Re N Sc D 6 × 10 < NRe < 2 × 10 120 < NSc < 1200

6

k′dtank b c NSh =  = aN Re N Sc D a depends on system. a = 0.0443 [40]; b is often 0.65–0.70 [89]. If 2 ωd tank ρ NRe =  µ

O. Spiral type RO (seawater desalination)

[162]

[E]1 × 1010 < NScNGr < 5 × 1013 and 1939 < NSc < 2186

13 13 NGr NSh = 0.1NSc

5

N. Mass transfer to a flat plate membrane in a stirred vessel

[E]6 × 109 < NScNGr < 1012 and 1943 < NSc < 2168 dh = hole diameter

Characteristic length = disk diameter d



M. Turbulent, spinning disk

[141] p. 225

k′m L NSh,avg =  D

0.35 dh NSh = 0.059N0.35  Sc NGr d

Q2 δ film = 0.172  w 2g L. Cross-corrugated plate (turbulence promoter for membrane systems)

[S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. NGr > 1010 Assumes laminar boundary layer is small fraction of total.

[E] Entrance turbulent channel For parallel flow and corrugations: NSc = 1483, a = 0.56, c = 0.268 NSc = 4997, a = 0.50, c = 0.395 Corrugations perpendicular to flow: NSc = 1483, a = 0.57, c = 0.368 NSc = 4997, a = 0.52, c = 0.487

[134]

[E] Use arithmetic concentration difference. u = ωddisk /2 where ω = rotational speed, radians/s. NRe = ρωd 2/2µ.

[55] [138] p. 241

[E] Use arithmetic concentration difference. ω = stirrer speed, radians/s. Useful for laboratory dialysis, R.O., U.F., and microfiltration systems.

[40] [89] p. 965

b = 0.785 [40]. c is often 0.33 but other values have been reported [89].

14 NSh = 0.210 N23 Re NSc

[E] Polyamide membrane.

Or with slightly larger error,

p = 6.5 MPa and TDS rejection = 99.8%. Recovery ratio 40%.

N14 NSh = 0.080 N0.875 Re Sc *See the beginning of the “Mass Transfer” subsection for references.

[148]

MASS TRANSFER

5-65

TABLE 5-18 Mass-Transfer Correlations for Falling Films with a Free Surface in Wetted Wall Columns— Transfer between Gas and Liquid Situation A. Laminar, vertical wetted wall column

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′m x x NSh,avg =  ≈ 3.41  D δfilm (first term of infinite series)



3µQ δfilm =  wρg



1/3

[T] Low rates M.T. Use with log mean concentration difference. Parabolic velocity distribution in films. w = film width (circumference in column)

= film thickness

Derived for flat plates, used for tubes if

4Qρ NRe,film =  < 20 wµ

B. Turbulent, vertical wetted wall column

rtube

C. Turbulent, very short column

k′m dt 0.83 0.44 N Sc NSh,avg =  = 0.023N Re D

[152] p. 50

> 3.0. σ = surface tension

[E] Use with log mean concentration difference for correlations in B and D. NRe is for gas. NSc for vapor in gas. 2000 < NRe ≤ 35,000, 0.6 ≤ NSc ≤ 2.5. Use for gases, dt = tube diameter.

0.5 NSh,avg = 0.0318N0.790 Rc NSc

[S] Reevaluated data

0.08 NSh = 0.00283NRe,gN0.5 Sc,gNRe,liq

[E] Evaporation data

[56]

NSh = kg(dtube − 2)D

NSh,g = 11 to 65, NRe,g = 2400 to 9100

NRe,g = gug(dtube − 2)g

NRe,liq = 110 to 480, NSc,g = 0.62 to 1.93  = film thickness



k′m d t 0.83 0.44 4Qρ N Sc  NSh,avg =  = 0.00814N Re D wµ





0.15



4Qρ 30 ≤  < 1200 wµ

k′m d t 0.8 1/3 N Sc NSh,avg =  = 0.023N Re D E. Turbulent, with ripples

[141] p. 137

[68] [77] p.181 [138] p. 211 [141] p. 265 [149] p. 212 [152] p. 71 [58]

NRe,liq = liqQliq[#(dtube − 2)] D. Turbulent, vertical wetted wall column with ripples



1/2

[138] p. 78

If NRe,film > 20, surface waves and rates increase. An approximate solution Dapparent can be used. Ripples are suppressed with a wetting agent good to NRe = 1200.

A coefficient 0.0163 has also been reported using NRe′, where v = v of gas relative to liquid film. Better fit

ρg  2σ

References*



2 NSh =  #

0.5 . 0.5 N0.5 Re,ε NSc

[E] For gas systems with rippling.





4Qρ Fits 5-18-B for  = 1000 wµ

[85] [138] p. 213

[E] “Rounded” approximation to include ripples. Includes solid-liquid mass-transfer data to find s 0.83 coefficient on NSc. May use N Re . Use for liquids. See also Table 5-19. . ∂vsy ∂vcx [E] ε = dilation rate of surface =  +  ∂x ∂y

[150]

. NRe,ε. = εL2 F. Rectification in vertical wetted wall column with turbulent vapor flow, Johnstone and Pigford correlation

k′G dcol pBM 0.33 NSh,avg =  = 0.0328(N′Re) 0.77 N Sc Dv p 3000 < N R′ e < 40,000, 0.5 < NSc < 3 d col vrel ρv N′Re =  , v rel = gas velocity relative to µv 3 liquid film =  uavg in film 2

[E] Use logarithmic mean driving force at two ends of column. Based on four systems with gas-side resistance only. pBM = logarithmic mean partial pressure of nondiffusing species B in binary mixture. p = total pressure Modified form is used for structured packings (See Table 5-24-H).

[84] [138] p. 214 [156]

*See the beginning of the “Mass Transfer” subsection for references.

ap since the measurements are simpler to determine the product kap or HTU. 7. Finally, if a mass-transfer coefficient looks too good to be true, it probably is incorrect. To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-23-A. For packed beds in distillation, absorption, and so on in Table 5-24, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated

from the dispersed phase holdup and mean drop size correlations. Godfrey, Obi, and Reeve [Chem. Engr. Prog. 85, 61 (Dec. 1989)] summarize these correlations. For many systems,  ddrop/dimp = −0.6 (const)NWe where NWe = ρc N 2d 3imp /σ. Piché, Grandjean, and Larachi [Ind. Eng. Chem. Res. 41, 4911 (2002)] developed two correlations for reconciling the gas-liquid mass-transfer coefficient and interfacial area in randomly packed towers. The correlation for the interfacial area was a function of five dimensionless groups, and yielded a relative error of 22.5 percent for 325 data points. That equation, when combined with a correlation for NSh as a function of four dimensionless groups, achieved a relative error of 24.4 percent, for 3455 data points for the product k′Ga.

5-66

HEAT AND MASS TRANSFER

TABLE 5-19

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer Is from Wall to Fluid

Situation A. Tubes, laminar, fully developed parabolic velocity profile, developing concentration profile, constant wall concentration

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation 0.0668(d t /x)NRe NSc k′d t NSh =  = 3.66 +  1 + 0.04[(d t /x)NRe NSc]2/3 D

References*

[T] Use log mean concentration difference. For x/d t  < 0.10, NRe < 2100. NRe NSc x = distance from tube entrance. Good agreement with experiment at values

[77] p. 176 [87] p. 525 [141] p. 159

π d 104 >  t NReNSc > 10 4 x Fully developed concentration profile

B. Tubes, approximate solution

k′d t NSh =  = 3.66 D



k′d t d NSh,x =  = 1.077 t D x

(NRe NSc)1/3



D. Laminar, fully developed parabolic velocity profile, constant mass flux at wall

[T] For arithmetic concentration difference.

[141] p. 166

W  > 400 ρDx Leveque’s approximation: Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well.

1/3

(NRe NSc)1/3

 

∞



−2 a 2j (x/rt) 1 − 4 a −2 j exp  1 dt NRe NSc j=1 NSh,avg =   NRe NSc  2 L ∞ −2 a 2j (x/rt) −2 1 + 4 a j exp  NRe NSc j=1 Graetz solution for heat transfer written for M.T.





11 1 ∞ exp [−λ 2j (x/rt)/(NRe NSc)] NSh, x =  −   48 2 j = 1 Cj λ 4j λ 2j 25.68 83.86 174.2 296.5 450.9

j 1 2 3 4 5

[141] p. 165

1/3

k′d t d NSh,avg =  = 1.615 t D L

C. Tubes, laminar, uniform plug velocity, developing concentration profile, constant wall concentration

x/d t [T]  > 0.1 NRe NSc

−1



cj 7.630 × 10−3 2.058 × 10−3 0.901 × 10−3 0.487 × 10−3 0.297 × 10−3



[T] Use arithmetic concentration difference. Fits W gas data well, for  < 50 (fit is fortuitous). Dρx NSh,avg = (k′m d t)/D. a1 = 2.405, a 2 = 5.520, a 3 = 8.654, a4 = 11.792, a 5 = 14.931. Graphical solutions are in references. [T] Use log mean concentration difference. NRe < 2100

[103] [141] p. 150

[139] [141] p. 167

k′d t NSh,x =  D vd t ρ NRe =  µ

E. Laminar, alternate

0.023(dt /L)NRe NSc NSh = 4.36 +  1 + 0.0012(dt/L)NRe NSc

k′d t [T] Nsh =  , Use log mean concentration D difference. NRe < 2100

F. Laminar, fully developed concentration and velocity profile

k′dt 48 NSh =  =  = 4.3636 D 11

[T] Use log mean concentration difference. NRe < 2100

[141] p. 167

G. Vertical tubes, laminar flow, forced and natural convection

(NGr NSc d/L)3/4 1/3 1  0.0742  NSh,avg = 1.62N Gz NGz

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other. Good agreement with experiment.

[127]





1/3

[77] p. 176

g∆ρd 3 NRe NSc d NGz =  , NGr =  ρν 2 L H. Hollow-fiber extraction inside fibers

NSh = 0.5NGz,NGz < 6

I. Tubes, laminar, RO systems

k′m d t ud 2t NSh,avg =  = 1.632  D DL

[E] Use arithmetic concentration difference.

[41]

Use arithmetic concentration difference. Thin concentration polarization layer, not fully developed. NRe < 2000, L = length tube.

[40]

NSh = 1.62N ,NGz ≥ 6 0.5 Gz



1/3

J. Tubes and parallel plates, laminar RO

Graphical solutions for concentration polarization. Uniform velocity through walls.

[T]

[137]

K. Rotating annulus for reverse osmosis

For nonvortical flow: d 0.5 NSh = 2.15 NTa  ri

[E,S] NTa = Taylor number = riωdν

[100]

  

For vortical flow: d NSh = 1.05 NTa  ri

0.18 13 Sc

N

ri = inner cylinder radius

  N

ω = rotational speed, rads

0.5

13 Sc

d = gap width between cylinders

MASS TRANSFER TABLE 5-19

5-67

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer Is from Wall to Fluid (Continued)

Situation

Correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

L. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant wall concentration

Graphical solution

[T] Low transfer rates.

[141] p. 176

L′. 5-19-L, fully developed

k′(2h) NSh =  = 7.6 D

[T] h = distance between plates. Use log mean concentration difference.

[141] p. 177

NRe NSc  < 20 x/(2h) M. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant mass flux at wall N. 5-19-M, fully developed

O. Laminar flow, vertical parallel plates, forced and natural convection

Graphical solution

[T] Low transfer rates.

k′(2h) NSh =  = 8.23 D

[T] Use log mean concentration difference. NRe NSc  < 20 x/(2h)



(NGr NSc h/L)3/4 NSh,avg = 1.47N 1/3 Gz 1  0.0989  NGz



[141] p. 176

[141] p. 177

1/3

[T] Approximate solution. Use minus sign if forced and natural convection oppose each other. Good agreement with experiment.

[127]

g∆ρh3 NRe NSc h NGz =  , NGr =  ρν 2 L



P. Parallel plates, laminar, RO systems

k′(2Hp) uH 2p NSh,avg =  = 2.354  D DL

Q. Tubes, turbulent

k′m d t 1/3 NSh,avg =  = 0.023N 0.83 Re N Sc D

1/3

2100 < NRe < 35,000 0.6 < NSc < 3000

k′m d t 0.44 NSh,avg =  = 0.023N 0.83 Re N Sc D

Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference.

[40]

[E] Use with log mean concentration difference at two ends of tube. Good fit for liquids.

[77] p. 181 [103] [152] p. 72

From wetted wall column and dissolution data— see Table 5-18-B.

[E] Evaporation of liquids. Use with log mean concentration difference. Better fit for gases.

2000 < NRe < 35,000 0.6 < NSc < 2.5 R. Tubes, turbulent

k′d t 0.346 NSh =  = 0.0096 N 0.913 Re N Sc D

S. Tubes, turbulent, smooth tubes, Reynolds analogy

k′d t f NSh =  =  NRe NSc D 2 f = Fanning friction faction

T. Tubes, turbulent, smooth tubes, Chilton-Colburn analogy



f jD = jH /  2 NSh f −0.2 If  = 0.023N −0.2 = 0.023N Re Re , jD =  2 NRe N 1/3 Sc

[E] 430 < NSc < 100,000. Dissolution data. Use for high NSc.

[105] p. 668

[T] Use arithmetic concentration difference. NSc near 1.0 Turbulent core extends to wall. Of limited utility.

[66] p. 474 [77] p. 171 [141] p. 239 [149] p. 250

[E] Use log-mean concentration difference. Relating jD to f/2 approximate. NPr and NSc near 1.0. Low concentration. Results about 20% lower than experiment. 3 × 104 < NRe < 106

[39] pp. 400, 647 [51][53]

k′d t NSh =  , Sec. 5-17-G D jD = jH = f(NRe, geometry and B.C.) U. Tubes, turbulent, smooth tubes, constant surface concentration, Prandtl analogy

k′d t ( f /2)NRe NSc NSh =  =  D 2(NSc − 1) 1 + 5f/ f −0.25  = 0.04NRe 2

[68][77] p. 181 [88] p. 112 [138] p. 211

[E] Good over wide ranges. [T] Use arithmetic concentration difference. Improvement over Reynolds analogy. Best for NSc near 1.0.

[141] p. 264 [149] p. 251 [66] p. 475 [39] p. 647 [51]

[77] p. 173 [141] p. 241

5-68

HEAT AND MASS TRANSFER

TABLE 5-19

Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer Is from Wall to Fluid (Concluded)

Situation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation

References*

V. Tubes, turbulent, smooth tubes, Constant surface concentration, Von Karman analogy

( f/2)NReNSc  NSh = 5 2 (NSc − 1) + ln 1 +  (NSc − 1) 1 + 5f/ 6 f −0.25  = 0.04N Re 2

[T] Use arithmetic concentration difference. NSh = k′dt /D. Improvement over Prandtl, NSc < 25.

[77] p. 173 [141] p. 243 [149] p. 250 [154]

W. Tubes, turbulent, smooth tubes, constant surface concentration

For 0.5 < NSc < 10:

[S] Use arithmetic concentration difference. Based on partial fluid renewal and an infrequently replenished thin fluid layer for high Nsc. Good fit to available data.

[77] p. 179 [117]








NSh,avg = 0.0097N N −1/3 −1/6 × (1.10 + 0.44N Sc − 0.70N Sc ) 9/10 Re

1/2 Sc

u bulk d t NRe =  ν

For 10 < NSc < 1000: NSh,avg 9/10 1/2 −1/3 −1/6 0.0097N Re N Sc (1.10 + 0.44 N Sc − 0.70N Sc ) =  1/2 −1/3 −1/6 1 + 0.064 N Sc (1.10 + 0.44N Sc − 0.70N Sc )

k′avg d t NSh,avg =  D

9/10 1/3 N Sc For NSc > 1000: N Sh,avg = 0.0102 N Re

X. Turbulent flow, tubes

NSh NSh −0.12 −2/3 NSt =  = = 0.0149N Re N Sc NPe NRe NSc

[E] Smooth pipe data. Data fits within 4% except at NSc > 20,000, where experimental data is underpredicted. NSc > 100, 105 > NRe > 2100

[107]

Y. Turbulent flow, noncircular ducts

Use correlations with

Can be suspect for systems with sharp corners.

[141] p. 289

4 cross-sectional area d eq =  wetted perimeter

Z. Decaying swirling flow in pipe

Parallel plates: 2hw d eq = 4  2w + 2h

−0.400 0.759 NSh,avg = 0.3508N13 × (1 + tanθ)0.271 Sc NRe (xd)

NRe = 1730 to 8650, NSc = 1692

[E,S] x = axial distance, d = diameter, θ = vane angle (15° to 60°) Regression coefficient = 0.9793. Swirling increases mass transfer.

[161]

*See the beginning of the “Mass Transfer” subsection for references.

Effects of Total Pressure on kˆ G and kˆ L The influence of total system pressure on the rate of mass transfer from a gas to a liquid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-298), where (5-305) kˆ G = DABpT /RT δG Since the quantity DAB pT is known to be relatively independent of the pressure, it follows that the rate coefficients kˆ G, kGyBM, and k′G pTyBM (= k′G pBM) do not depend on the total pressure of the system, subject to the limitations discussed later. Investigators of tower packings normally report k′Ga values measured at very low inlet-gas concentrations, so that yBM = 1, and at total pressures close to 100 kPa (1 atm). Thus, the correct rate coefficient for use in packed-tower designs involving the use of the driving force (y − yi)/yBM is obtained by multiplying the reported k′Ga values by the value of pT employed in the actual test unit (e.g., 100 kPa) and not the total pressure of the system to be designed. From another point of view one can correct the reported values of k′Ga in kmol/[(s⋅m3)(kPa)], valid for a pressure of 101.3 kPa (1 atm), to some other pressure by dividing the quoted values of k′Ga by the design pressure and multiplying by 101.3 kPa, i.e., (k′Ga at design pressure pT) = (k′Ga at 1 atm) × 101.3/pT. One way to avoid a lot of confusion on this point is to convert the experimentally measured k′Ga values to values of kˆ Ga straightaway, before beginning the design calculations. A design based on the rate coefficient kˆ Ga and the driving force (y − yi)/yBM will be independent of the total system pressure with the following limitations: caution should be employed in assuming that kˆ Ga is independent of total pressure for systems having significant vapor-phase nonidealities, for systems that operate in the vicinity of the critical

point, or for total pressures higher than about 3040 to 4050 kPa (30 to 40 atm). Experimental confirmations of the relative independence of kˆ G with respect to total pressure have been widely reported. Deviations do occur at extreme conditions. For example, Bretsznajder (Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford, 1971, p. 343) discusses the effects of pressure on the DABpT product and presents experimental data on the self-diffusion of CO2 which show that the D-p product begins to decrease at a pressure of approximately 8100 kPa (80 atm). For reduced temperatures higher than about 1.5, the deviations are relatively modest for pressures up to the critical pressure. However, deviations are large near the critical point (see also p. 5-52). The effect of pressure on the gas-phase viscosity also is negligible for pressures below about 5060 kPa (50 atm). For the liquid-phase mass-transfer coefficient kˆ L, the effects of total system pressure can be ignored for all practical purposes. Thus, when using kˆ G and kˆ L for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then kˆ G, kˆ L, and a can all change significantly. Effects of Temperature on kˆ G and kˆ L The Stanton-number relationship for gas-phase mass transfer in packed beds, Eq. (5-301), indicates that for a given system geometry the rate coefficient kˆ G depends only on the Reynolds number and the Schmidt number. Since the Schmidt number for a gas is approximately independent of temperature, the principal effect of temperature upon kˆ G arises from changes in the gas viscosity with changes in temperature. For normally encountered temperature ranges, these effects will be small owing to the fractional powers involved in Reynolds-number terms (see Tables 5-17 to 5-24). It thus can be concluded that for all

TABLE 5-20

Mass-Transfer Correlations for Flow Past Submerged Objects

Situation A. Single sphere

B. Single sphere, creeping flow with forced convection

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′G pBLM RTd s 2r NSh =  =  PD r − rs 5 10 50 ∞ (asymptotic limit) r/rs 2 NSh 4.0 2.5 2.22 2.04 2.0 k′d NSh =  = [4.0 + 1.21(NRe NSc)2/3]1/2 D

k′d NSh =  = a(NRe NSc)1/3 D a = 1.00  0.01 C. Single spheres, molecular diffusion, and forced convection, low flow rates

1/2 1/3 N Sh = 2.0 + AN Re N Sc A = 0.5 to 0.62

A = 0.60.

[T] Use with log mean concentration difference. r = distance from sphere, rs, ds = radius and diameter of sphere. No convection.

[141] p. 18

[T] Use with log mean concentration difference. Average over sphere. Numerical calculations. (NRe NSc) < 10,000 NRe < 1.0. Constant sphere diameter. Low mass-transfer rates.

[46][88] p. 114 [105] [138] p. 214

[T] Fit to above ignoring molecular diffusion.

[101] p. 80

1000 < (NReNSc) < 10,000.

[138] p. 215

[E] Use with log mean concentration difference. Average over sphere. Frössling Eq. (A = 0.552), 2 ≤ NRe ≤ 800, 0.6 ≤ Nsc ≤ 2.7. NSh lower than experimental at high NRe. [E] Ranz and Marshall 2 ≤ NRe ≤ 200, 0.6 ≤ Nsc ≤ 2.5. Modifications recommended [110] See also Table 5-23-O.

[39]

[E] Liquids 2 ≤ NRe ≤ 2,000. Graph in Ref. 138, p. 217–218. [E] 100 ≤ NRe ≤ 700; 1,200 ≤ NSc ≤ 1525.

A = 0.95. A = 0.95. A = 0.544.

References*

[E] Use with arithmetic concentration difference. NSc = 1; 50 ≤ NRe ≤ 350.

[77], p. 194 [88] p. 114 [141] p. 276 [39] p. 409, 647 [121] [110] [138] p. 217 [141] p. 276 [65][66] p. 482 [138] p. 217 [126][141] p. 276 [81][141] p. 276

D. Same as 5-20-C

k′ds 0.35 NSh =  = 2.0 + 0.575N 1/2 Re N Sc D

[E] Use with log mean concentration difference. NSc ≤ 1, NRe < 1.

[70][141] p. 276

E. Same as 5-20-C

k′ds 0.53 1/3 NSh =  = 2.0 + 0.552 N Re N Sc D

[E] Use with log mean concentration difference. 1.0 < NRe ≤ 48,000 Gases: 0.6 ≤ NSc ≤ 2.7.

[66] p. 482

F. Single spheres, forced concentration, any flow rate

k′L d s E1/3d p4/3ρ NSh =  = 2.0 + 0.59  D µ

[S] Correlates large amount of data and compares to published data. vr = relative velocity between fluid and sphere, m/s. CDr = drag coefficient for single particle fixed in fluid at velocity vr. See 5-23-F for calculation details and applications. E1/3d p4/3 ρ 2 <  < 63,000 µ

[108]

[E] Use with arithmetic concentration difference. Liquids, 2000 < NRe < 17,000. High NSc, graph in Ref. 138, p. 217–218. [E] 1500 ≤ NRe ≤ 12,000.

[66] p. 482 [147] [138] p. 217 [141] p. 276

k′d s = 0.43 N 0.56 N 1/3 NSh =  Re Sc D

[E] 200 ≤ NRe ≤ 4 × 104, “air” ≤ NSc ≤ “water.”

[141] p. 276

k′d s 0.514 1/3 NSh =  = 0.692 N Re N Sc D

[E] 500 ≤ NRe ≤ 5000.

[112] [141] p. 276





0.57 1/3 N Sc

Energy dissipation rate per unit mass of fluid (ranges 570 < NSc < 1420):    d s

CDr E=  2 G. Single spheres, forced convection, high flow rates, ignoring molecular diffusion

H. Single sphere immersed in bed of smaller particles. For gases.

v 3r



m2 3

p

k′d s 0.62 1/3 NSh =  = 0.347N Re N Sc D k′d 0.6 1/3 NSh = s = 0.33 N Re N Sc D





4 4 kd1 NSh,avg =  = ε 4 +  N23 Pe′ +  NPe′ D′ π 5

1 + 91 N

12

Pe′

Limit NPe′→0,NSh,avg = 2ε I. Single cylinders, perpendicular flow

k′d s 1/3 NSh =  = AN 1/2 Re N Sc , A = 0.82 D A = 0.74 A = 0.582 jD = 0.600(N Re)−0.487 k′d cyl NSh =  D



uod1 [T] Compared to experiment. NPe′ =  , D′ D′ = Dτ, D = molecular diffusivity, d1 = diameter large particle, τ = tortuosity. Arithmetic conc. difference fluid flow in inert bed follows Darcy’s law.

[71]

[E] 100 < NRe ≤ 3500, NSc = 1560.

[141] p. 276

[E] 120 ≤ NRe ≤ 6000, NSc = 2.44.

[141] p. 276

[E] 300 ≤ NRe ≤ 7600, NSc = 1200.

[142]

[E] Use with arithmetic concentration difference.

[141] p. 276

50 ≤ NRe ≤ 50,000; gases, 0.6 ≤ NSc ≤ 2.6; liquids; 1000 ≤ NSc ≤ 3000. Data scatter  30%.

[66] p. 486 5-69

5-70

HEAT AND MASS TRANSFER

TABLE 5-20

Mass-Transfer Correlations for Flow Past Submerged Objects (Concluded)

Situation J. Rotating cylinder in an infinite liquid, no forced flow

Correlation k′ 0.644 −0.30 j′D =  N Sc = 0.0791N Re v Results presented graphically to NRe = 241,000. ωdcyl vdcyl µ NRe =  where v =  = peripheral velocity 2 ρ

K. Stationary or rotating cylinder for air

Stationary: c S13 NSh,avg = ANRe c 2.0 × 104 ≤ NRe ≤ 2.5 × 105; dH = 0.3, Tu = 0.6% A = 0.0539, c = 0.771 [114] A and c depend on geometry [37] Rotating in still air: NSh,avg = 0.169N23 Re,ω

Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

[E] Used with arithmetic concentration difference. Useful geometry in electrochemical studies.

[60]

112 < NRe ≤ 100,000. 835 < NSc < 11490

[138] p. 238

k′ = mass-transfer coefficient, cm/s; ω = rotational speed, radian/s. [E] Reasonable agreement with data of other investigators. d = diameter of cylinder, H = height of wind tunnel, Tu of = turbulence level, NRew = rotational Reynold’s number = uωdρµ, uω = cylinder surface velocity. Also correlations for two-dimensional slot jet flow [114]. For references to other correlations see [37].

[37]

[E] Used with arithmetic concentration difference. 120 ≤ NRe ≤ 6000; standard deviation 2.1%. Eccentricities between 1:1 (spheres) and 3:1. Oblate spheroid is often approximated by drops.

[141] p. 284

[E] Used with arithmetic concentration difference. Agrees with cylinder and oblate spheroid results, 15%. Assumes molecular diffusion and natural convection are negligible.

[88] p. 115 [141] p. 285

500 ≤ N Re, p ≤ 5000. Turbulent.

[111] [112]

[T] Use with arithmetic concentration difference. Hard to reach limits in experiments.  Spheres and cubes A = 2, tetrahedrons A = 26 . octahedrons 22

[88] p. 114

[E] Use with logarithmic mean concentration difference.

[118]

[114]

1.0E4 ≤ NRe,ω ≤1.0E5; NSc ≈2.0; NGr ≈2.0 × 106 L. Oblate spheroid, forced convection

NSh −0.5 jD =  = 0.74 N Re NRe N 1/3 Sc dch vρ total surface area NRe =  , dch =  µ perimeter normal to flow

[142]

e.g., for cube with side length a, dch = 1.27a. k′d ch NSh =  D M. Other objects, including prisms, cubes, hemispheres, spheres, and cylinders; forced convection

N. Other objects, molecular diffusion limits

O. Shell side of microporous hollow fiber module for solvent extraction

v d ch ρ −0.486 jD = 0.692N Re,p , N Re,p =  µ Terms same as in 5-20-J.

k′d ch NSh =  = A D

0.33 NSh = β[d h(1 − ϕ)/L]N 0.6 Re N Sc

K dh NSh =  D d h vρ K = overall mass-transfer coefficient N Re =  ,  µ β = 5.8 for hydrophobic membrane. β = 6.1 for hydrophilic membrane.

See Table 5-23 for flow in packed beds. *See the beginning of the “Mass Transfer” subsection for references.

dh = hydraulic diameter 4 × cross-sectional area of flow =  wetted perimeter ϕ = packing fraction of shell side. L = module length. Based on area of contact according to inside or outside diameter of tubes depending on location of interface between aqueous and organic phases. Can also be applied to gas-liquid systems with liquid on shell side.

TABLE 5-21

Mass-Transfer Correlations for Drops, Bubbles, and Bubble Columns

Conditions A. Single liquid drop in immiscible liquid, drop formation, discontinuous (drop) phase coefficient

Correlations ρd kˆ d,f = A  Md

 

24 A =  (penetration theory) 7



24 A =  (0.8624) (extension by fresh surface 7 elements) B. Same as 5-21-A

kˆ df = 0.0432 d p ρd ×  tf Md

−0.334

µd   c ρ d d σ p g

   

C. Single liquid drop in immiscible liquid, drop formation, continuous phase coefficient D. Same as 5-21-C

av

0.089

uo  dp g

G. Same as 5-21-E, continuous phase coefficient, stagnant drops, spherical H. Single bubble or drop with surfactant. Stokes flow.



c

av

f

  

F. Same as 5-21-E

−0.601

kL,c = 0.386 ρc ×  Mc

E. Single liquid drop in immiscible liquid, free rise or fall, discontinuous phase coefficient, stagnant drops

d p2  t f Dd

D   πt

ρc kˆ cf = 4.6  Mc

Dc  tf

av

−dp ρd kL,d,m =   6t M d

0.5

ρcσgc  ∆ρgt fµ c

 0.407

0.148

gt f2  dp

−Dd j 2π 2 t

 ln π6 j1 exp  (d /2) 


−d p ρd kˆ L,d,m =   6t Md

∞

2

av

2

j=1

p

2

πD t  ln 1 −  d /2  1/2 1/2 d

av

p

ρc kL,c,m dc NSh =  = 0.74  Mc Dc

 N

(NSc)1/3

1/2 Re

av

[E] Use arithmetic mole fraction difference. Based on 23 data points for 3 systems. Average absolute deviation 26%. Use with surface area of drop after detachment occurs. uo = velocity through nozzle; σ = interfacial tension.

[141] p. 401

[T] Use arithmetic mole fraction difference. Based on rate of bubble growth away from fixed orifice. Approximately three times too high compared to experiments.

[141] p. 402

[E] Average absolute deviation 11% for 20 data points for 3 systems.

[141] p. 402 [144] p. 434

[T] Use with log mean mole fraction differences based on ends of column. t = rise time. No continuous phase resistance. Stagnant drops are likely if drop is very viscous, quite small, or is coated with surface active agent. kL,d,m = mean dispersed liquid M.T. coefficient.

[141] p. 404 [144] p. 435

[S] See 5-21-E. Approximation for fractional extractions less than 50%.

[141] p. 404 [144] p. 435

vs d p ρc [E] NRe =  , vs = slip velocity between µc drop and continuous phase.

[141] p. 407 [142][144] p. 436

[T] A = surface retardation parameter

A 5.49 α =  +  A + 6.10 A + 28.64

A = BΓorµDs = NMaNPe,s

0.35A + 17.21 β =  A + 34.14

Γ = surfactant surface conc.

[120]

NPe,s = surface Peclet number = ur/Ds Ds = surface diffusivity

0.0026 < NPe,s < 340, 2.1 < NMa < 1.3E6

NPe = bulk Peclet number

NPe = 1.0 to 2.5 × 104,

For A >> 1 acts like rigid sphere:

NRe = 2.2 × 10−6 to 0.034

β → 0.35, α → 12864 = 0.035

ρc kL,c,m d3 NSh =  = 0.74  Mc Dc

[144] p. 434

NMa = BΓoµu = Marangoni no.

 (N

Re,3

)1/2(NSc,c)1/3

[E] Used with log mean mole fraction. Differences based on ends of extraction column; 100 measured values 2% deviation. Based on area oblate spheroid.

av

vsd3ρc NRe,3 =  µc

J. Single liquid drop in immiscible liquid, Free rise or fall, discontinuous phase coefficient, circulating drops

[141] p. 399

NSh = 2.0 + αNβPe, NSh = 2rkD

2r = 2 to 50 µm, A = 2.8E4 to 7.0E5

I. 5-21-E, oblate spheroid

[T,S] Use arithmetic mole fraction difference. Fits some, but not all, data. Low mass transfer rate. Md = mean molecular weight of dispersed phase; tf = formation time of drop. kL,d = mean dispersed liquid phase M.T. coefficient kmole/[s⋅m2 (mole fraction)].

A = 1.31 (semiempirical value)



References*

1/2

Dd  πt f

av

Comments E = Empirical, S = Semiempirical, T = Theoretical

[141] p. 285, 406, 407

total drop surface area vs = slip velocity, d3 =  perimeter normal to flow



dp 3 kdr,circ = −  ln  6θ 8

∞

B

j=1

2 j

λ j64Ddθ exp −  d p2





Eigenvalues for Circulating Drop k d d p /Dd

λ1

λ2

3.20 10.7 26.7 107 320 ∞

0.262 0.680 1.082 1.484 1.60 1.656

0.424 4.92 5.90 7.88 8.62 9.08

λ3

B1

B2

B3

15.7 19.5 21.3 22.2

1.49 1.49 1.49 1.39 1.31 1.29

0.107 0.300 0.495 0.603 0.583 0.596

0.205 0.384 0.391 0.386

[T] Use with arithmetic concentration difference.

[62][76][141] p. 405

θ = drop residence time. A more complete listing of eigenvalues is given by Refs. 62 and 76.

[152] p. 523

k′L,d,circ is m/s.

5-71

5-72

HEAT AND MASS TRANSFER

TABLE 5-21

Mass-Transfer Correlations for Drops, Bubbles, and Bubble Columns (Continued)

Conditions K. Same as 5-21-J

L. Same as 5-21-J

d p ρd  kˆ L,d,circ = −  6θ Md

1/2 R 1/2πD1/2 d θ ln 1 −  d p /2 av

 



ˆk L,d,circ d p NSh =  Dd ρd = 31.4  Mf

−0.34

  d

M. Liquid drop in immiscible liquid, free rise or fall, continuous phase coefficient, circulating single drops

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

4Dd t 2 p

av

2 −0.125 d p v s ρc N Sc,d  σg c



−0.37



k′L,c d p NSh,c =  Dd





F

1/3 0.072

dp g 0.484 = 2 + 0.463N Re,drop N 0.339  Sc,c D c2/3

K=N

N. Same as 5-21-M, circulating, single drop O. Same as 5-21-M, circulating swarm of drops P. Liquid drops in immiscible liquid, free rise or fall, discontinuous phase coefficient, oscillating drops

µc  µd

µcvs  σg c

[E] Used with mole fractions for extraction less than 50%, R ≈ 2.25.

[141] p. 405

[E] Used with log mean mole fraction difference. dp = diameter of sphere with same volume as drop. 856 ≤ NSc ≤ 79,800, 2.34 ≤ σ ≤ 4.8 dynes/cm.

[144] p. 435 [145]

[E] Used as an arithmetic concentration difference.

[82]

d pv sρc NRe,drop =  µc

F = 0.281 + 1.615K + 3.73K 2 − 1.874 K 1/8 Re,drop

References*

Solid sphere form with correction factor F.

  1/4

ρc k L,c d p NSh =  = 0.6  Mc Dc

1/6

 N

ρc k L,c = 0.725  Mc

 N

1/2 Re,drop

1/2 N Sc,c

av

−0.43 Re,drop

−0.58 N Sc,c v s (1 − φ d)

av

k L,d,osc d p NSh =  Dd ρd = 0.32  Md

−0.14

t   d 4Dd 2 p

av

σ 3g c3 ρ2c 0.68 N Re,drop  gµ 4c∆ρ





0.10

[E] Used as an arithmetic concentration difference. Low σ.

[141] p. 407

[E] Used as an arithmetic concentration difference. Low σ, disperse-phase holdup of drop swarm. φ d = volume fraction dispersed phase.

[141] p. 407 [144] p. 436

[E] Used with a log mean mole fraction difference. Based on ends of extraction column.

[141] p. 406

d pvsρc NRe,drop =  , 411 ≤ NRe ≤ 3114 µc d p = diameter of sphere with volume of drop. Average absolute deviation from data, 10.5%.

[144] p. 435 [145]

Low interfacial tension (3.5–5.8 dyn), µc < 1.35 centipoise. Q. Same as 5-21-P

0.00375v k L,d,osc = s 1 + µ d /µ c

[T] Use with log mean concentration difference. Based on end of extraction column. No continuous phase resistance. kL,d,osc in cm/s, vs = drop velocity relative to continuous phase.

[138] p. 228 [141] p. 405

R. Single liquid drop in immiscible liquid, range rigid to fully circulating

kcdp 0.5 0.33 N Sc NSh,c,rigid =  = 2.43 + 0.774N Re Dc

[E] Allows for slight effect of wake. Rigid drops: 104 < NPe,c < 106 Circulating drops: 10 < NRe < 1200, 190 < NSc < 241,000, 103 < NPe,c < 106

[146] p. 58

[E] Used with log mean mole fraction difference. 23 data points. Average absolute deviation 25%. t f = formation time.

[141] p. 408

[E] Used with log mean mole fraction difference. 20 data points. Average absolute deviation 22%.

[141] p. 409

0.33 + 0.0103NReN Sc

 

2 NSh,c,fully circular =  N 0.5 Pe,c π 0.5 Drops in intermediate range: NSh,c − NSh,c,rigid  = 1 − exp [−(4.18 × 10−3)N 0.42 Pe,c] NSh,c,fully circular − NSh,c,rigid S. Coalescing drops in immiscible liquid, discontinuous phase coefficient

T. Same as 5-21-S, continuous phase coefficient

ρd d kˆ d,coal = 0.173 p  t f Md

µd

−1.115

  ρ D

∆ρgd p2 ×  σg c



av

d

d

 D v s2 t f

1.302

0.146

d

ρ kˆ c,coal = 5.959 × 10−4  M



ρdu3s

av

d p2 ρc ρd v3s

   g µ  µ σg

D × c tf

0.5

c

0.332

d

c

0.525

MASS TRANSFER TABLE 5-21

5-73

Mass-Transfer Correlations for Drops, Bubbles, and Bubble Columns (Continued)

Conditions U. Single liquid drops in gas, gas side coefficient

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations kˆg Mg d p P 1/2 N 1/3  = 2 + AN Re,g Sc,g Dgas ρg

[E] Used for spray drying (arithmetic partial pressure difference).

A = 0.552 or 0.60.

vs = slip velocity between drop and gas stream.

d pρgvs NRe,g =  µg V. Single water drop in air, liquid side coefficient



DL kL = 2  πt

X. Same as 5-21-W, medium to large bubbles Y. Same as 5-21-X

Z. Taylor bubbles in single capillaries (square or circular)

1/2

, short contact times

k′c d b NSh =  = 1.0(NReNSc)1/3 Dc k′c d b NSh =  = 1.13(NReNSc)1/2 Dc



k′c d b db NSh =  = 1.13(NReNSc)1/2  Dc 0.45 + 0.2d b 500 ≤ NRe ≤ 8000



DuG kLa = 4.5  Luc



12





1  dc

uG + uL Applicable  Lslug AA. Gas-liquid mass transfer in monoliths



0.5

> 3s−0.5

AC. Same as 5-21-AB, large bubbles

14

k′c d b 1/3 NSh =  = 2 + 0.31(NGr)1/3N Sc , d b < 0.25 cm Dc d b3|ρG − ρL|g NRa =  = Raleigh number µ L DL k′c d b NSh =  = 0.42 (NGr)1/3N 1/2 Sc , d b > 0.25 cm Dc 6 Hg Interfacial area  = a =  volume db

AD. Bubbles in bubble columns. Hughmark correlation

dg13 kLd 0.779  NSh =  = 2 + bN0.546 Sc N Re D23 D



b = 0.061 single gas bubbles;

[90] p. 389

[T] Solid-sphere Eq. (see Table 5-20-B). d b < 0.1 cm, k′c is average over entire surface of bubble.

[105] [138] p. 214

[T] Use arithmetic concentration difference. Droplet equation: d b > 0.5 cm.

[138] p. 231

[S] Use arithmetic concentration difference. Modification of above (X), db > 0.5 cm. No effect SAA for dp > 0.6 cm.

[83][138] p. 231

[E] Air-water

[153]

For most data kLa ± 20%.



P kLa ≈ 0.1  V

[T] Use arithmetic concentration difference. Penetration theory. t = contact time of drop. Gives plot for k G a also. Air-water system.

Luc = unit cell length, Lslug = slug length, dc = capillary i.d.

P/V = power/volume (kW/m3), range = 100 to 10,000 AB. Rising small bubbles of gas in liquid, continuous phase. Calderbank and Moo-Young correlation

[90] p. 388 [121]

Sometimes written with MgP/ρg = RT.

DL k L = 10  , long contact times dp W. Single bubbles of gas in liquid, continuous phase coefficient, very small bubbles

References*



[E] Each channel in monolith is a capillary. Results are in expected order of magnitude for capillaries based on 5-21-Z.

[93]

kL is larger than in stirred tanks. [E] Use with arithmetic concentration difference. Valid for single bubbles or swarms. Independent of agitation as long as bubble size is constant. Recommended by [136]. Note that NRa = NGr NSc.

[47][66] p. 451

[E] Use with arithmetic concentration difference. For large bubbles, k′c is independent of bubble size and independent of agitation or liquid velocity. Resistance is entirely in liquid phase for most gas-liquid mass transfer. Hg = fractional gas holdup, volume gas/total volume.

[47][66] p. 452 [88] p. 119 [97] p. 249 [136]

[E] d = bubble diameter

[55] [82] [152] p. 144

[88] p. 119 [152] p. 156 [136]

0.116

Air–liquid. Recommended by [136, 152]. For swarms, calculate

b = 0.0187 swarms of bubbles,

NRe with slip velocity Vs.

Vg VL Vs =  −  0G 1− 0G

0G = gas holdup VG = superficial gas velocity Col. diameter = 0.025 to 1.1 m ρ′L = 776 to 1696 kg/m3 µL = 0.0009 to 0.152 Pa⋅s

AE. Bubbles in bubble column

−0.84 kLa = 0.00315uG0.59 µeff

[E] Recommended by [136].

[57]

5-74

HEAT AND MASS TRANSFER

TABLE 5-21

Mass-Transfer Correlations for Drops, Bubbles, and Bubble Columns (Concluded)

Conditions

Correlations 0.15D ν kL =   D dVs



AF. Bubbles in bubble column

AG. High-pressure bubble column

AH. Three phase (gas-liquid-solid) bubble column to solid spheres

Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

12

N34 Re

kLa = 1.77σ−0.22 exp(1.65ul − 65.3µl)ε1.2 g 790 < ρL < 1580 kg/m3 0.00036 < µl < 0.0383 Pa⋅s 0.0232 < σl < 0.0726 Nm 0.028 < ug < 0.678 ms 0 < ul < 0.00089 ms



ksdp ed4p 0.264 Nsh =  = 2.0 + 0.545N13 Sc  D ν3 NSc = 137 to 50,000 (very wide range) dp = particle diameter (solids)

[E] dVs = Sauter mean bubble diameter, NRe = dVsuGρLµL. Recommended by [49] based on experiments in industrial system.

[49] [133]

[E] Pressure up to 4.24 MPa.

[96]

T up to 92°C. εg = gas holdup. Correlation to estimate εg is given. 0.045 < dcol < 0.45 m, dcolHcol > 5 0.97 < ρg < 33.4 kgm3 [E] e = local energy dissipation rate/unit mass, e = ugg

[129] [136]

NSc = µL(ρLD) Recommended by [136].

See Table 5-22 for agitated systems. *See the beginning of the “Mass Transfer” subsection for references.

practical purposes kˆ G is independent of temperature and pressure in the normal ranges of these variables. For modest changes in temperature the influence of temperature upon the interfacial area a may be neglected. For example, in experiments on the absorption of SO2 in water, Whitney and Vivian [Chem. Eng. Prog., 45, 323 (1949)] found no appreciable effect of temperature upon k′Ga over the range from 10 to 50°C. With regard to the liquid-phase mass-transfer coefficient, Whitney and Vivian found that the effect of temperature upon kLa could be explained entirely by variations in the liquid-phase viscosity and diffusion coefficient with temperature. Similarly, the oxygen-desorption data of Sherwood and Holloway [Trans. Am. Inst. Chem. Eng., 36, 39 (1940)] show that the influence of temperature upon HL can be explained by the effects of temperature upon the liquid-phase viscosity and diffusion coefficients (see Table 5-24-A). It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be very large and therefore must be carefully accounted for when using kˆ L or HL data. For liquids the mass-transfer coefficient kˆ L is correlated as either the Sherwood number or the Stanton number as a function of the Reynolds and Schmidt numbers (see Table 5-24). Typically, the general form of the correlation for HL is (Table 5-24) a HL = bNRe N1/2 Sc

(5-306)

where b is a proportionality constant and the exponent a may range from about 0.2 to 0.5 for different packings and systems. The liquidphase diffusion coefficients may be corrected from a base temperature T1 to another temperature T2 by using the Einstein relation as recommended by Wilke [Chem. Eng. Prog., 45, 218 (1949)]: D2 = D1(T2 /T1)(µ1/µ2)

(5-307)

The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures: NSc2 = NSc1(T1 /T2)(ρ1 /ρ2)(µ2 /µ1)2

(5-308)

Substitution of this relation into Eq. (5-306) shows that for a given geometry the effect of temperature on HL can be estimated as HL2 = HL1(T1 /T2)1/2(ρ1 /ρ2)1/2(µ2 /µ1)1 − a

(5-309)

In using these relations it should be noted that for equal liquid flow rates (5-310) HL2 /HL1 = (kˆ La)1/(kˆ La)2

ˆ G and k ˆ L When Effects of System Physical Properties on k designing packed towers for nonreacting gas-absorption systems for which no experimental data are available, it is necessary to make corrections for differences in composition between the existing test data and the system in question. The ammonia-water test data (see Table 5-24-B) can be used to estimate HG, and the oxygen desorption data (see Table 5-24-A) can be used to estimate HL. The method for doing this is illustrated in Table 5-24-E. There is some conflict on whether the value of the exponent for the Schmidt number is 0.5 or 2/3 [Yadav and Sharma, Chem. Eng. Sci. 34, 1423 (1979)]. Despite this disagreement, this method is extremely useful, especially for absorption and stripping systems. It should be noted that the influence of substituting solvents of widely differing viscosities upon the interfacial area a can be very large. One therefore should be cautious about extrapolating kˆ La data to account for viscosity effects between different solvent systems. ˆ G and k ˆ L As disEffects of High Solute Concentrations on k cussed previously, the stagnant-film model indicates that kˆ G should be independent of yBM and kG should be inversely proportional to yBM. The data of Vivian and Behrman [Am. Inst. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model’s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use in concentrated systems is kˆ G and the proper driving-force term is of the form (y − yi)/yBM. The use of the rate coefficient kˆ L and the driving force (xi − x)/xBM is believed to be appropriate. For many practical situations the liquidphase solute concentrations are low, thus making this assumption unimportant. ˆ G and k ˆ L When a chemInfluence of Chemical Reactions on k ical reaction occurs, the transfer rate may be influenced by the chemical reaction as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reaction kinetics on the absorption rate in addition to accounting for the effects of gas solubility, diffusivity, and system hydrodynamics. There is no sharp dividing line between pure physical absorption and absorption controlled by the rate of a chemical reaction. Most cases fall in an intermediate range in which the rate of absorption is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equilibria between the various diffusing species involved in the reaction may affect the rate of absorption.

MASS TRANSFER TABLE 5-22

5-75

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems

Situation A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation k′LT d p 1/3  = 2 + 0.6N 1/2 Re,T N Sc D Replace vslip with v T = terminal velocity. Calculate Stokes’ law terminal velocity d p2|ρ p − ρc|g v Ts =  18µ c

10 0.65

100 0.37

1,000 0.17

10,000 0.07

100,000 0.023

Approximate: k′L = 2k′LT B. Solid, neutrally buoyant particles, continuous phase coefficient

[74][138] p. 220–222 [110]

(Reynolds number based on Stokes’ law.) v T d p ρc NRe,T =  µc

and correct: NRe,Ts 1 v T /v Ts 0.9

[S] Use log mean concentration difference. v Ts d p ρc Modified Frossling equation: NRe,Ts =  µc

References*



k′Ld p d imp 0.36 NSh =  = 2 + 0.47N 0.62  Re,p N Sc D d tank

(terminal velocity Reynolds number.) k′L almost independent of d p. Harriott suggests different correction procedures. Range k′L /k′LT is 1.5 to 8.0.

[74]

0.17

Graphical comparisons are in Ref. 88, p. 116.

[E] Use log mean concentration difference. Density unimportant if particles are close to neutrally buoyant. Also used for drops. Geometric effect (d imp/d tank) is usually unimportant. Ref. 102 gives a variety of references on correlations.

[88] p. 115 [102] p. 132 [152] p. 523

[E] E = energy dissipation rate per unit mass fluid E1/3d p4/3 Pgc =  , P = power, NRe,p =  Vtank ρc ν C. Same as 22-B, small particles D. Solid particles with significant density difference

E. Small solid particles, gas bubbles or liquid drops, dp < 2.5 mm. Aerated mixing vessels F. Highly agitated systems; solid particles, drops, and bubbles; continuous phase coefficient

G. Liquid drops in baffled tank with flat six-blade turbine

0.52 NSh = 2 + 0.52N Re,p N 1/3 Sc , NRe,p < 1.0



k′L d p d pv slip NSh =  = 2 + 0.44  D ν

0.38 N Sc

k′L d p d 3p |ρp − ρ c| NSh =  = 2 + 0.31  D µ cD





(P/Vtank)µ cg c 2/3 k′LN Sc = 0.13  ρc2







[E] Use log mean concentration difference. NSh standard deviation 11.1%. vslip calculated by methods given in reference.

[102] [110]

[E] Use log mean concentration difference. g = 9.80665 m/s 2. Second term RHS is free-fall or rise term. For large bubbles, see Table 5-21-AC.

[46][67] p. 487 [97] p. 249

[E] Use arithmetic concentration difference. Use when gravitational forces overcome by agitation. Up to 60% deviation. Correlation prediction is low (Ref. 102). (P/Vtank) = power dissipated by agitator per unit volume liquid.

[47] [66] p. 489 [110]

[E] Use arithmetic concentration difference. Studied for five systems.

[144] p. 437

1/4

1.582

2 NRe = d imp Nρc /µ c , NOh = µ c /(ρc d impσ)1/2

1.025 N 1.929 Re N Oh

φ = volume fraction dispersed phase. N = impeller speed (revolutions/time). For dtank = htank, average absolute deviation 23.8%.

k′c d p 1/3 NSh =  = 1.237 × 10 −5 N Sc N 2/3 D ρd d p2

   D  σ

5/12 d imp × N Fr  dp

[88] p. 116

1/3

(ND)1/2 k′c a = 2.621 × 10−3  d imp d imp × φ 0.304  d tank

H. Liquid drops in baffled tank, low volume fraction dispersed phase



[E] Terms same as above.

1/2

dp

1/2

5/4

φ−1/2

tank

Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for φ ≤ 0.06 [Ref. 146] a = 6φ/dˆ 32, where dˆ 32 is Sauter mean diameter when 33% mass transfer has occurred.

[E] 180 runs, 9 systems, φ = 0.01. kc is timeaveraged. Use arithmetic concentration difference.







2 d impN 2 d imp NSc NRe =  , NFr =  µc g



d p = particle or drop diameter; σ = interfacial tension, N/m; φ = volume fraction dispersed phase; a = interfacial volume, 1/m; and kcαD c2/3 implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 143.

[143] [146] p. 78

5-76

HEAT AND MASS TRANSFER

TABLE 5-22

Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems (Concluded)

Situation I. Gas bubble swarms in sparged tank reactors

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation ν 1/3 P/VL a qG ν 1/3 b k′L a 2 =C   2 g ρ(νg 4)1/3 VL g Rushton turbines: C = 7.94 × 10−4, a = 0.62, b = 0.23. Intermig impellers: C = 5.89 × 10−4, a = 0.62, b = 0.19.





  

References*

[E] Use arithmetic concentration difference. Done for biological system, O2 transfer. htank /Dtank = 2.1; P = power, kW. VL = liquid volume, m3. qG = gassing rate, m3/s. k′L a = s −1. Since a = m2/m3, ν = kinematic viscosity, m2/s. Low viscosity system. Better fit claimed with qG /VL than with uG (see 5-22-J to N).

[131]



u 0.5 G

[E] Use arithmetic concentration difference. Ion free water VL < 2.6, uG = superficial gas velocity in m/s. 500 < P/VL < 10,000. P/VL = watts/m3, VL = liquid volume, m3.

[98] [123]



u 0.2 G

[E] Use arithmetic concentration difference. Water with ions. 0.002 < VL < 4.4, 500 < P/VL < 10,000. Same definitions as 5-22-I.

[98] [101]

[E] Air-water. Same definitions as 5-22-I. 0.005 < uG < 0.025, 3.83 < N < 8.33, 400 < P/VL < 7000 h = Dtank = 0.305 or 0.610 m. VG = gas volume, m3, N = stirrer speed, rpm. Method assumes perfect liquid mixing.

[67] [98]

[E] Use arithmetic concentration difference. CO2 into aqueous carboxyl polymethylene. Same definitions as 5-22-L. µeff = effective viscosity from power law model, Pa⋅s. σ = surface tension liquid, N/m.

[98] [115]

[E] Use arithmetic concentration difference. O2 into aqueous glycerol solutions. O2 into aqueous millet jelly solutions. Same definitions as 5-22-L.

[98] [160]

[E] Use arithmetic concentration difference. Solids are glass beads, d p = 320 µm. ε s = solids holdup m3/m3 liquid, (k′L a)o = mass transfer in absence of solids. Ionic salt solution— noncoalescing.

[38] [132]

[E] Three impellers: Pitched blade downflow turbine, pitched blade upflow turbine, standard disk turbine. Baffled cylindrical tanks 1.0- and 1.5-m ID and 8.2 × 8.2-m square tank. Submergence optimized all cases. Good agreement with data. N = impeller speed, s−1; d = impeller diameter, m; H = liquid height, m; V = liquid volume, m3; kLa = s−1, g = acceleration gravity = 9.81 m/s2

[113]

[E] Same tanks and same definitions as in 5-22-P. VA = active volume = p/(πρgNd).

[113]

[E] Hydrogenation with Raney-type nickel catalyst in stirred autoclave. Used varying T, p, solvents. dst = stirrer diameter.

[78]

J. Same as 5-22-I

P k′L a = 2.6 × 10−2  VL

K. Same as 5-22-J

P k′L a = 2.0 × 10−3  VL

L. Same as 5-22-I, baffled tank with standard blade Rushton impeller

P k′L a = 93.37  VL

M. Same as 5-22-L

d 2imp µ eff k′L a  = 7.57  D ρD



0.4

0.7

0.76

u G0.45

µG

   µ 



0.5

0.694

eff

d 2impNρL ×  µ eff

d   σ 1.11

uG

0.447

d imp = impeller diameter, m; D = diffusivity, m2/s N. Same as 5-22-L, bubbles

O. Gas bubble swarm in sparged stirred tank reactor with solids present

P. Surface aerators for air-water contact



k′Lad 2imp d 2impNρ  = 0.060  µ eff D

d N µ u   g  σ 2 imp

2

0.19

eff

k′La  = 1 − 3.54(εs − 0.03) (k′La)o 300 ≤ P/Vrx < 10,000 W/m3, 0.03 ≤ εs ≤ 0.12 0.34 ≤ uG ≤ 4.2 cm/s, 5 < µ L < 75 Pa⋅s −0.54

−1.08

  d

k La H 0.82 N0.48  = bN0.71  p Fr NRe N d

V

3

b = 7 × 10−6, Np = P/(ρN3d5) NRe = Nd2ρliq/µliq NFr = N2d/g, P/V = 90 to 400 W/m3

Q. Gas-inducing impeller for air-water contact



VA kLaV(v/g2)1/3 d3 = ANBFr  V

C

G

0.6

Single impeller: A = 0.00497, B = 0.56, C = 0.32 Multiple impeller: A = 0.00746, B = 0.54, C = 0.38 R. Gas-inducing impeller with dense solids

kLad2st 0.9 −0.1 = (1.26 × 10−5) N1.8 ShGL =  Re NSc NWe D NRe = ρNd2St/µ, NSc = µ/(ρD), NWe = ρN2d2St/σ

See also Table 5-21. *See the beginning of the “Mass Transfer” subsection for references.

MASS TRANSFER TABLE 5-23

5-77

Mass-Transfer Correlations for Fixed and Fluidized Beds

Transfer is to or from particles Situation A. For gases, fixed and fluidized beds, Gupta and Thodos correlation

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation

References*

2.06 jH = jD =  , 90 ≤ NRe ≤ A εN 0.575 Re

v superd p ρ [E] For spheres. NRe =  µ

[72] [73]

Equivalent:

A = 2453 [Ref. 141], A = 4000 [Ref. 77]. For NRe > 1900, j H = 1.05j D. Heat transfer result is in absence of radiation.

[77] p. 195 [141]

2.06 0.425 1/3 N Sc NSh =  N Re ε For other shapes: ε jD  = 0.79 (cylinder) or 0.71 (cube) (ε j D)sphere

k′d s NSh =  D Graphical results are available for NRe from 1900 to 10,300. surface area a =  = 6(1 − ε)/d p volume For spheres, dp = diameter. ar t. u .A re a For nonspherical: d p = 0.567 P Srf

B. For gases, for fixed beds, Petrovic and Thodos correlation

0.357 0.641 1/3 NSh =  N Re N Sc ε 3 < NRe < 900 can be extrapolated to NRe < 2000.

C. For gases and liquids, fixed and fluidized beds

0.4548 jD =  , 10 ≤ NRe ≤ 2000 εN 0.4069 Re k′d s NSh jD =  , NSh =  D NReN 1/3 Sc

[E] Packed spheres, deep beds. Corrected for axial dispersion with axial Peclet number = 2.0. Prediction is low at low NRe. NRe defined as in 5-23-A.

[116][128] p. 214 [155]

[E] Packed spheres, deep bed. Average deviation 20%, NRe = dpvsuperρ/µ. Can use for fluidized beds. 10 ≤ NRe ≤ 4000.

[60][66] p. 484

D. For gases, fixed beds

0.499 jD =  0.382 εN Re

[E] Data on sublimination of naphthalene spheres dispersed in inert beads. 0.1 < NRe < 100, NSc = 2.57. Correlation coefficient = 0.978.

[80]

E. For liquids, fixed bed, Wilson and Geankoplis correlation

1.09 , 0.0016 < NRe < 55 jD =  2/3 εN Re 165 ≤ NSc ≤ 70,600, 0.35 < ε < 0.75 Equivalent:

[E] Beds of spheres,

[66] p. 484

1.09 1/3 NSh =  N 1/3 Re N Sc ε

Deep beds.

d pVsuperρ NRe =  µ

0.25 jD =  , 55 < NRe < 1500, 165 ≤ NSc ≤ 10,690 εN 0.31 Re 0.25 0.69 1/3 N Sc Equivalent: NSh =  N Re ε F. For liquids, fixed beds, Ohashi et al. correlation

k′d s E 1/3d 4/3 p ρ NSh =  = 2 + 0.51  D µ





0.60

N 1/3 Sc

E = Energy dissipation rate per unit mass of fluid



v 3r = 50(1 − ε)ε 2 CDo  , m2/s 3 dp 50(1 − ε)CD =  ε





v 3super  dp



General form:

[141] p. 287 [158]

[S] Correlates large amount of published data. Compares number of correlations, v r = relative velocity, m/s. In packed bed, v r = v super /ε. CDo = single particle drag coefficient at v super cal−m culated from CDo = AN Re . i NRe 0 to 5.8 5.8 to 500 >500

A 24 10 0.44

m 1.0 0.5 0

Ranges for packed bed:

α E 1/3 D 4/3 p ρ β NSh = 2 + K  N Sc µ



[77] p. 195

k′d s NSh =  D



applies to single particles, packed beds, two-phase tube flow, suspended bubble columns, and stirred tanks with different definitions of E.

0.001 < NRe < 1000, 505 < NSc < 70,600, E 1/3d p4/3ρ 0.2 <  < 4600 µ Compares different situations versus general correlation. See also 5-20-F.

[108]

5-78

HEAT AND MASS TRANSFER

TABLE 5-23

Mass Transfer Correlations for Fixed and Fluidized Beds (Continued)

Situation G. Electrolytic system. Pall rings. Transfer from fluid to rings.

Correlation Full liquid upflow: 1/3 Nsh = kLde/D = 4.1N0.39 Re NSc NRedeu/ν = 80 to 550 Irrigated liquid downflow (no gas flow): 1/3 NSh = 5.1N0.44 Re NSc

H. For liquids, fixed and fluidized beds

1.1068 ε jD =  , 1.0 < N Re ≤ 10 N 0.72 Re k′d s NSh ε jD =  , NSh =  D NRe N 1/3 Sc

I. For gases and liquids, fixed and fluidized beds, Dwivedi and Upadhyay correlation

J. For gases and liquids, fixed bed

0.765 0.365 ε jD =  + N 0.82 N 0.386 Re Re Gases: 10 ≤ N Re ≤ 15,000. Liquids: 0.01 ≤ N Re ≤ 15,000. k′d s d pv superρ NRe =  , NSh =  D µ −0.415 jD = 1.17N Re , 10 ≤ NRe ≤ 2500

k′ pBM 2/3 jD =   NSc vav P

Comments E = Empirical, S = Semiempirical, T = Theoretical

References*

[E] de = diameter of sphere with same surface area as Pall ring. Full liquid upflow agreed with literature values. Schmidt number dependence was assumed from literature values. In downflow, NRe used superficial fluid velocity.

[69]

[E] Spheres:

[59][66] p. 484

d pv superρ NRe =  µ [E] Deep beds of spheres, NSh jD =  N Re N 1/3 Sc Best fit correlation at low conc. [52] Based on 20 gas studies and 17 liquid studies. Recommended instead of 5-23-C or E. [E] Spheres: Variation in packing that changes ε not allowed for. Extensive data referenced. 0.5 < NSc < 15,000. Comparison with other results are shown.

[59] [77] p. 196 [52]

[138] p. 241

d pv superρ NRe =  µ K. For liquids, fixed and fluidized beds, Rahman and Streat correlation L. Size exclusion chromatography of proteins

0.86 NSh =  NReN1/3 Sc , 2 ≤ NRe ≤ 25 ε 1.903 kLd 1/3 NSh =  =  N1/3 Re N Sc D ε

M. Liquid-free convection with fixed bed Raschig rings. Electrochemical.

NSh = kd/D = 0.15 (NSc NGr)0.32 NGr = Grashof no. = gd3∆ρ/(ν2ρ) If forced convection superimposed, NSh, overall = (N3Sh,forced + N3Sh,free)1/3

N. Oscillating bed packed with Raschig rings. Dissolution of copper rings.

Batch (no net solution flow): 0.7 0.35 NSh = 0.76N0.33 Sc NRe,v(dc/h) 503 < NRe,v < 2892 960 < NSc < 1364, 2.3 < dc/h < 7.6

O. For liquids and gases, Ranz and Marshall correlation

k′d 1/2 NSh =  = 2.0 + 0.6N 1/3 Sc N Re D d pv superρ NRe =  µ

P. For liquids and gases, Wakao and Funazkri correlation

0.6 NSh = 2.0 + 1.1N 1/3 Sc N Re , 3 < NRe < 10,000

k′film d p ρf vsuperρ NSh =  , NRe =  D µ εDaxial  = 10 + 0.5NScNRe D

Q. Acid dissolution of limestone in fixed bed R. Semifluidized or expanded bed. Liquid-solid transfer.

0.44 1/3 NSh = 1.77 N0.56 Re NSc (1 − ε) 20 < NRe < 6000

k film d p 1/3 NSh =  = 2 + 1.5 (1 − εL)N1/3 Re NSc D NRe = ρpdpu/µεL; NSc = µ/ρD

[E] Can be extrapolated to NRe = 2000. NRe = dpvsuperρ/µ. Done for neutralization of ion exchange resin.

[119]

[E] Slow mass transfer with large molecules. Aqueous solutions. Modest increase in NSh with increasing velocity.

[79]

[E] d = Raschig ring diameter, h = bed height 1810 < NSc < 2532, 0.17 < d/h < 1.0

[135]

10.6 × 106 < NScNGr < 21 × 107 [E] NSh = kdc/D, NRe,v = vibrational Re = ρvvdc/µ vv = vibrational velocity (intensity) dc = col. diameter, h = column height Average deviation is ± 12%.

[61]

[E] Based on freely falling, evaporating spheres (see 5-20-C). Has been applied to packed beds, prediction is low compared to experimental data. Limit of 2.0 at low NRe is too high. Not corrected for axial dispersion.

[121][128] p. 214 [155] [110]

[E] Correlate 20 gas studies and 16 liquid studies. Corrected for axial dispersion with: Graphical comparison with data shown [128], p. 215, and [155]. Daxial is axial dispersion coefficient.

[128] p. 214 [155]

[E] Best fit was to correlation of Chu et al., Chem. Eng. Prog., 49(3), 141(1953), even though no reaction in original.

[94]

[E] εL = liquid-phase void fraction, ρp = particle density, ρ = fluid density, dp = particle diameter. Fits expanded bed chromatography in viscous liquids.

[64] [159]

MASS TRANSFER TABLE 5-23

5-79

Mass Transfer Correlations for Fixed and Fluidized Beds (Concluded)

Situation S. Mass-transfer structured packing and static mixers. Liquid with or without fluidized particles. Electrochemical

T. Liquid fluidized beds

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlation Fixed bed: 0.572 j′ = 0.927NRe′ , N′Re < 219 −0.435 , 219 < N′Re < 1360 j′ = 0.443NRe′ Fluidized bed with particles: j = 6.02N−0.885 , or Re j′ = 16.40N−0.950 Re′ Natural convection: NSh = 0.252(NScNGr)0.299 Bubble columns: Structured packing: NSt = 0.105(NReNFrN2Sc)−0.268 Static mixer: NSt = 0.157(NReNFrN2Sc)−0.298





(2ξ/εm)(1 − ε)1/2 2ξ/ε m +  − 2 tan h (ξ/ε m) [1 − (1 − ε)1/3]2 NSh =  ξ/ε m  − tan h (ξ/ε m) 1 − (1 − ε1/2) where α 1 1/2 − 1  N 1/3 ξ=  Sc N Re 2 (1 − ε)1/3





This simplifies to: α2

ε 1 − 2m 1 NSh =  −1 (1 − ε)1/3 (1 − ε)1/3



U. Liquid fluidized beds

 2 N

0.306 ρ s − ρ NSh = 0.250N 0.023  Re N Ga ρ



NSh = 0.304N

−0.057 Re

N



0.332 Ga



Re

(NRe < 0.1)

2/3 N Sc

0.282

ρs − ρ  ρ

(ε < 0.85)

0.410 N Sc



0.297

N

(ε > 0.85)

0.404 Sc

This can be simplified (with slight loss in accuracy at high ε) to 0.323 ρ s − ρ NSh = 0.245N Ga  ρ





0.300 0.400 N Sc

References*

k cos β [E] Sulzer packings, j′ =  N2/3 Sc , v β = corrugation incline angle. NRe′ = v′ d′hρ/µ, v′ = vsuper /(ε cos β), d′h = channel side width. Particles enhance mass transfer in laminar flow for natural convection. Good fit with correlation of Ray et al., Intl. J. Heat Mass Transfer, 41, 1693 (1998). NGr = g ∆ ρZ3ρ/µ2, Z = corrugated plate length. Bubble column results fit correlation of Neme et al., Chem. Eng. Technol., 20, 297 (1997) for structured packing. NSt = Stanton number = kZ/D NFr = Froude number = v2super/gz

[48]

[S] Modification of theory to fit experimental data. For spheres, m = 1, NRe > 2.

[92] [106] [125]

k′L d p NSh =  , D

Vsuper d pξ NRe =  µ

m = 1 for NRe > 2; m = 0.5 for NRe < 1.0; ε = voidage; α = const. Best fit data is α = 0.7. Comparison of theory and experimental ion exchange results in Ref. 92.

[E] Correlate amount of data from literature. Predicts very little dependence of NSh on velocity. Compare large number of published correlations.

[151]

k′L d p d p ρvsuper d p3 ρ 2g NSh =  , NRe =  , NGa =  , D µ µ2 µ NSc =  ρD 1.6 < NRe < 1320, 2470 < NGa < 4.42 × 106 ρs − ρ 0.27 <  < 1.114, 305 < NSc < 1595 ρ

V. Liquid film flowing over solid particles with air present, trickle bed reactors, fixed bed

W. Supercritical fluids in packed bed

kL 1/3 NSh =  = 1.8N 1/2 Re N Sc , 0.013 < NRe < 12.6 aD two-phases, liquid trickle, no forced flow of gas. 1/2 1/3 NSh = 0.8N Re N Sc , one-phase, liquid only.



1/3 NSh (N 1/2 Re N Sc ) = 0.5265   (NSc NGr)1/4 (NSc NGr)1/4



2 N 1/3 N Re Sc + 2.48  NGr





0.6439

1.6808

− 0.8768

L [E] NRe =  , irregular granules of benzoic acid, aµ 0.29 ≤ dp ≤ 1.45 cm. L = superficial liquid flow rate, kg/m2s. a = surface area/col. volume, m2/m3. [E] Natural and forced convection. 0.3 < NRe < 135.



Downflow in trickle bed and upflow in bubble columns.

Literature review and meta-analysis. Analyzed both downflow and upflow. Recommendations for best mass- and heat-transfer correlations (see reference).

Y. Liquid-solid transfer. Electrochemical reaction. Lessing rings. Transfer from liquid to solid

Liquid only:

[E] Electrochemical reactors only. d = Lessing ring diameter, 1 < d < 1.4 cm, NRe = ρvsuper d/µ, Deviation ±7% for both cases. NRe,gas = ρgasVsuper,gasd/µgas Presence of gas enhances mass transfer.

NSh = kd/D = 1.57N N 1390 < NSc < 4760, 166 < NRe < 722 Cocurrent two-phase (liquid and gas) in packed bubble column: 0.34 0.11 NSh = 1.93N1/3 Sc NRe NRe,gas 60 < NRe,gas < 818, 144 < NRe < 748 0.46 Re

[99]

1.553

X. Cocurrent gas-liquid flow in fixed beds

1/3 Sc

[130]

[95]

[75]

NOTE: For NRe < 3 convective contributions which are not included may become important. Use with logarithmic concentration difference (integrated form) or with arithmetic concentration difference (differential form). *See the beginning of the “Mass Transfer” subsection for references.

5-80

HEAT AND MASS TRANSFER

TABLE 5-24 Mass-Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) Situation A. Absorption, counter-current, liquid-phase coefficient HL, Sherwood and Holloway correlation for random packings

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations



L 0.5 HL = a L  N Sc,L , L = lb/hr ft 2 µL Ranges for 5-24-B (G and L) Packing

aG

b

c

G

L

aL

n

Raschig rings 3/8 inch 1 1 2

2.32 7.00 6.41 3.82

0.45 0.39 0.32 0.41

0.47 0.58 0.51 0.45

32.4 0.30 0.811 0.30 1.97 0.36 5.05 0.32

0.74 0.24 0.40 0.45

200–500 200–800 200–600 200–800

500–1500 0.00182 0.46 400–500 0.010 0.22 500–4500 — — 500–4500 0.0125 0.22

Berl saddles 1/2 inch 1/2 1 1.5

B. Absorption counter-current, gasphase coefficient HG, for random packing

C. Absorption and and distillation, counter-current, gas and liquid individual coefficients and wetted surface area, Onda et al. correlation for random packings

References*

n

200–700 200–800 200–800 200–1000

500–1500 0.0067 0.28 400–4500 — — 400–4500 0.0059 0.28 400–4500 0.0062 0.28

0.5 a G(G) bN Sc,v GM HG =  =  kˆ G a (L) c



k′G RT G =A  a pDG a pµ G ρL k L′  µ Lg



1/3



−2.0 N 1/3 Sc,G (a pd p′ )





2/3

−1/2 N Sc,L (a p d′p)0.4

k′L = lbmol/hr ft 2 (lbmol/ft 3) [kgmol/s m2 (kgmol/m3)]



[104] p. 187 [105] [138] p. 606 [157] [156]

[E] Based on ammonia-water-air data in Fellinger’s 1941 MIT thesis. Curves: Refs. 104, p. 186 and 138, p. 607. Constants given in 5-24A. The equation is dimensional. G = lb/hr ft 2 , G M = lbmol/hr ft 2, kˆ G = lbmol/hr ft 2.

[104] p. 189 [138] p. 607 [157]

[E] Gas absorption and desorption from water and organics plus vaporization of pure liquids for Raschig rings, saddles, spheres, and rods. d′p = nominal packing size, a p = dry packing surface area/volume, a w = wetted packing surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. σ = surface tension, dynes/cm. A = 5.23 for packing ≥ 1/2 inch (0.012 m) A = 2.0 for packing < 1/2 inch (0.012 m) k′G = lbmol/hr ft 2 atm [kg mol/s m2 (N/m2)]

[44]

0.7

L = 0.0051  aw µL

aw  = 1 − exp ap

[E] From experiments on desorption of sparingly soluble gases from water. Graphs [Ref. 138], p. 606. Equation is dimensional. A typical value of n is 0.3 [Ref. 66] has constants in kg, m, and s units for use in 5-24-A and B with kˆ G in kgmole/s m2 and kˆ L in kgmole/s m2 (kgmol/m3). Constants for other packings are given by Refs. 104, p. 187 and 152, p. 239. LM HL =  kˆ L a L M = lbmol/hr ft 2, kˆ L = lbmol/hr ft 2, a = ft 2/ft 3, µ L in lb/(hr ft). Range for 5-24-A is 400 < L < 15,000 lb/hr ft2

σ −1.45 c σ




  aµ

L2a p ×  ρ L2 g

0.75

 −0.05

0.1

L

p

L

L  ρLσa p

0.2

[90] p. 380 [109][149] p. 355 [156]

Critical surface tensions, σ C = 61 (ceramic), 75 (steel), 33 (polyethylene), 40 (PVC), 56 (carbon) dynes/cm. L 4 <  < 400 aw µ L G 5 <  < 1000 ap µG Most data ± 20% of correlation, some ± 50%. Graphical comparison with data in Ref. 109.

D. Distillation and absorption, counter-current, random packings, modification of Onda correlation, Bravo and Fair correlation to determine interfacial area

Use Onda’s correlations (5-24-C) for k′G and k′L. Calculate: G L HG =  , HL =  , HOG = HG + λHL k G′ aePMG k′LaeρL m λ= LM/GM σ0.5 ae = 0.498ap  (NCa,LNRe,G)0.392 Z0.4



LµL 6G NRe,G =  , NCa,L =  (dimensionless) ρLσgc apµG

[E] Use Bolles & Fair (Ref. 43) database to determine new effective area ae to use with Onda et al. (Ref. 109) correlation. Same definitions as 5-24-C. P = total pressure, atm; MG = gas, molecular weight; m = local slope of equilibrium curve; LM/GM = slope operating line; Z = height of packing in feet. Equation for ae is dimensional. Fit to data for effective area quite good for distillation. Good for absorption at low values of (Nca,L × NRe,G), but correlation is too high at higher values of (NCa,L × NRe,G).

[44]

MASS TRANSFER

5-81

TABLE 5-24 Mass-Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Continued) Situation E. Absorption and distillation, countercurrent gas-liquid flow, random and structured packing. Determine HL and HG

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations





  372  6.782/0.0008937

0.357 HL =  fP

NSc  0.660 NSc



−0.5



0.226 HG =  fp

b

Gx  6.782

0.5



Gy  0.678



0.35

0.3

Gx/µ

Relative transfer coefficients [91], fp values are in table: Size, in.

Ceramic Raschig rings

Ceramic Berl saddles

Metal Pall rings

Metal Intalox

Metal Hypac

0.5 1.0 1.5 2.0

1.52 1.20 1.00 0.85

1.58 1.36 — —

— 1.61 1.34 1.14

— 1.78 — 1.27

— 1.51 — 1.07

Norton Intalox structured: 2T, fp = 1.98; 3T, fp = 1.94.

F. Absorption, cocurrent downward flow, random packings, Reiss correlation

Air-oxygen-water results correlated by k′La = 0.12EL0.5. Extended to other systems.



DL k′La = 0.12EL0.5 5 2.4 × 10 ∆p EL =  ∆L





0.5

vL

2-phase

∆p  = pressure loss in two-phase flow = lbf/ft2 ft ∆L k′G a = 2.0 + 0.91EG2/3 for NH3 ∆p Eg =  vg ∆L 2-phase vg = superficial gas velocity, ft/s



G. Absorption, stripping, distillation, counter-current, HL, and HG, random packings, Bolles and Fair correlation

For Raschig rings, Berl saddles, and spiral tile: φCflood Z HL =  N 0.5 Sc,L  3.28 3.05



0.15

Cflood = 1.0 if below 40% flood—otherwise, use figure in [54] and [157]. Aψ(d′col)mZ0.33N0.5 Sc,G  0.16 µ ρwater 1.25 σwater 0.8 n HG = L L    µwater ρL σL









Figures for φ and ψ in [42 and 43] Ranges: 0.02 <φ> 0.300; 25 < ψ < 190 m.

H. Distillation and absorption. Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation

Equivalent channel:



1 1 deq = Bh  +  B + 2S 2S



Use modified correlation for wetted wall column (See 5-18-F) k′vdeq 0.333 NSh,v =  = 0.0338N0.8 Re,vNSc,v Dv deqρv(Uv,eff + UL,eff) NRe,v =  µv Calculate k′L from penetration model (use time for liquid to flow distance s). k′L = 2(DLUL,eff /πS)1/2.

[S] HG based on NH3 absorption data (5–28B) for which HG, base = 0.226 m with NSc, base = 0.660 at Gx, base = 6.782 kg/(sm2) and Gy, base = 0.678 kg/(sm2) with 11⁄2 in. ceramic Raschig rings. The exponent b on NSc is reported as either 0.5 or as 2⁄3.

References* [66] p. 686, 659 [138] [156]

HG for NH3 with 11⁄2 Raschig rings fp =  HG for NH3 with desired packing HL based on O2 desorption data (5-24-A). Base viscosity, µbase = 0.0008937 kg/(ms). HL in m. Gy < 0.949 kg/(sm2), 0.678 < Gx < 6.782 kg/(sm2). Best use is for absorption and stripping. Limited use for organic distillation [156]. [E] Based on oxygen transfer from water to air 77°F. Liquid film resistance controls. (Dwater @ 77°F = 2.4 × 10−5). Equation is dimensional. Data was for thin-walled polyethylene Raschig rings. Correlation also fit data for spheres. Fit 25%. See [122] for graph. k′La = s−1 DL = cm/s EL = ft, lbf/s ft3 vL = superficial liquid velocity, ft/s

[122] [130] p. 217

[E] Ammonia absorption into water from air at 70°F. Gas-film resistance controls. Thin-walled polyethylene Raschig rings and 1-inch Intalox saddles. Fit 25%. See [122] for fit. Terms defined as above.

[122]

[E] Z = packed height, m of each section with its own liquid distribution. The original work is reported in English units. Cornell et al. (Ref. 54) review early literature. Improved fit of Cornell’s φ values given by Bolles and Fair (Refs. [42], [43]) and [157].

[42, 43, 54] [77] p. 428 [90] p. 381 [141] p. 353 [157] [156]

A = 0.017 (rings) or 0.029 (saddles) d′col = column diameter in m (if diameter > 0.6 m, use d′col = 0.6) m = 1.24 (rings) or 1.11 (saddles) n = 0.6 (rings) or 0.5 (saddles) L = liquid rate, kg/(sm2), µwater = 1.0 Pa⋅s, ρwater = 1000 kg/m3, σwater = 72.8 mN/m (72.8 dyn/cm). HG and HL will vary from location to location. Design each section of packing separately. [T] Check of 132 data points showed average deviation 14.6% from theory. Johnstone and Pigford [Ref. 84] correlation (5-18-F) has exponent on NRe rounded to 0.8. Assume gauze packing is completely wet. Thus, aeff = ap to calculate HG and HL. Same approach may be used generally applicable to sheet-metal packings, but they will not be completely wet and need to estimate transfer area. L = liquid flux, kg/s m2, G = vapor flux, kg/s m2. Fit to data shown in Ref. [45]. G L HG =  , HL =  k′vapρv k′LapρL effective velocities 3Γ ρL2 g Uv,super Uv,eff =  , UL,eff =   2ρL 3µLΓ ε sin θ



Perimeter 4S + 2B Per =  =  Bh Area



0.333

L ,Γ= Per

[45] [63] p. 310, 326 [149] p. 356, 362 [156]

5-82

HEAT AND MASS TRANSFER

TABLE 5-24 Mass-Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Concluded) Situation I. Distillation and absorption, countercurrent flow. Structured packing with corrugations. Rocha, Bravo, and Fair correlation.

Comments E = Empirical, S = Semiempirical, T = Theoretical

Correlations

[E, T] Modification of Bravo, Rocha, and Fair (5-24-H). Same definitions as in (5-24-H) unless defined differently here. Recommended [156]. hL = fractional hold-up of liquid CE = factor for slow surface renewal CE ~ 0.9 ae = effective area/volume (1/m) ap = packing surface area/volume (1/m)

kgS 0.33 NSh,G =  = 0.054 N0.8 Re N Sc Dg uliq,super ug,super uv,eff =  , uL,eff =  , ε(1 − hL)sin θ εhL sin θ DL CE uL,eff kL = 2  πS ug,super λuL,super HOG = HG + λ HL =  +  kg ae kL ae





γ = contact angle; for sheet metal, cos γ = 0.9 for σ < 0.055 N/m cos γ = 5.211 × 10−16.8356, σ > 0.055 N/m m dy λ =  , m =  from equilibrium L/V dx

Packing factors:

J. Rotating packed bed (Higee)

ap

ε

FSE

θ

233 233 213 350

0.95 0.95 0.95 0.93

0.350 0.344 0.415 0.350

45º 45º 45º 45º



[124], [156]

FSE = surface enhancement factor

Interfacial area: 29.12 (NWeNFr)0.15 S0.359 ae  = FSE  0.6 N0.2 (1 − 0.93 cos γ)(sin θ)0.3 ap Re,L ε

Flexi-pac 2 Gempak 2A Intalox 2T Mellapak 350Y

References*



Vi kLa dp Vo  1 − 0.93  − 1.13  = 0.65 N0.5 Sc Dap Vt Vt

[E] Studied oxygen desorption from water into N2. Packing 0.22-mm-diameter stainless-steel mesh. ε = 0.954, ap = 829 (1/m), hbed = 2 cm. a = gas-liquid area/vol (1/m) L = liquid mass flux, kg/(m2S) ac = centrifugal accel, m2/S Vi, Vo, Vt = volumes inside inner radius, between outer radius and housing, and total, respectively, m3. Coefficient (0.3) on centrifugal acceleration agrees with literature values (0.3–0.38).

[50]

(Ka)HVtower L −n′  = 0.07 + A′N′  L Ga A′ and n′ depend on deck type (Ref. 86), 0.060 ≤ A′ ≤ 0.135, 0.46 ≤ n′ ≤ 0.62. General form fits the graphical comparisons (Ref. 138).

[E] General form. Ga = lb dry air/hr ft2. L = lb/h ft2, N′ = number of deck levels. (Ka)H = overall enthalpy transfer coefficient =

[86][104] p. 220 [138] p. 286

L. Liquid-liquid extraction, packed towers

Use k values for drops (Table 5-21). Enhancement due to packing is at most 20%.

[E] Packing decreases drop size and increases interfacial area.

[146] p. 79

M.Liquid-liquid extraction in rotating-disc contactor (RDC)

kc,RDC N  = 1.0 + 2.44  kc NCr

kc, kd are for drops (Table 5-21) Breakage occurs when N > NCr. Maximum enhancement before breakage was factor of 2.0. N = impeller speed H = compartment height, Dtank = tank diameter, σ = interfacial tension, N/m. Done in 0.152 and 0.600 m RDC.

[36][146] p. 79



dp3ρ2ac

L2

  µ  ρa σ 0.17

L ×  apµ

0.3

2

0.3

p

500 ≤ NSc ≤ 1.2 E5; 0.0023 ≤ L/(apµ) ≤ 8.7 120 ≤ (d3pρ2ac)/µ2 ≤ 7.0 E7; 3.7 E − 6 ≤ L2/(ρapσ) ≤ 9.4 E − 4 kLa dp 9.12 ≤  ≤ 2540 Dap K. High-voidage packings, cooling towers, splash-grid packings





σ NCr = 7.6 × 10−4  ddrop µc



2.5

H  D tank



kd,RDC N H  = 1.0 + 1.825   kd NCr Dtank N. Liquid-liquid extraction, stirred tanks

See Table 5-22-E, F, G, and H.





lb water lb/(h)(ft3)  lb dry air Vtower = tower volume, ft3/ft2. If normal packings are used, use absorption masstransfer correlations.

[E]

See also Sec. 14. *See the beginning of the “Mass Transfer” subsection for references.

The gas-phase rate coefficient kˆ G is not affected by the fact that a chemical reaction is taking place in the liquid phase. If the liquidphase chemical reaction is extremely fast and irreversible, the rate of absorption may be governed completely by the resistance to diffusion in the gas phase. In this case the absorption rate may be estimated by

knowing only the gas-phase rate coefficient kˆ G or else the height of one gas-phase transfer unit HG = GM /(kˆ Ga). It should be noted that the highest possible absorption rates will occur under conditions in which the liquid-phase resistance is negligible and the equilibrium back pressure of the gas over the solvent is zero.

MASS TRANSFER Such situations would exist, for instance, for NH3 absorption into an acid solution, for SO2 absorption into an alkali solution, for vaporization of water into air, and for H2S absorption from a dilute-gas stream into a strong alkali solution, provided there is a large excess of reagent in solution to consume all the dissolved gas. This is known as the gas-phase mass-transfer limited condition, when both the liquid-phase resistance and the back pressure of the gas equal zero. Even when the reaction is sufficiently reversible to allow a small back pressure, the absorption may be gas-phase-controlled, and the values of kˆ G and HG that would apply to a physical-absorption process will govern the rate. The liquid-phase rate coefficient kˆ L is strongly affected by fast chemical reactions and generally increases with increasing reaction rate. Indeed, the condition for zero liquid-phase resistance (m/kˆ L) implies that either the equilibrium back pressure is negligible, or that kˆ L is very large, or both. Frequently, even though reaction consumes the solute as it is dissolving, thereby enhancing both the mass-transfer coefficient and the driving force for absorption, the reaction rate is slow enough that the liquid-phase resistance must be taken into account. This may be due either to an insufficient supply of a second reagent or to an inherently slow chemical reaction. In any event the value of kˆ L in the presence of a chemical reaction normally is larger than the value found when only physical absorption occurs, kˆ L0 . This has led to the presentation of data on the effects of chemical reaction in terms of the “reaction factor” or “enhancement factor” defined as (5-311) φ = kˆ L / kˆ L0 ≥ 1 where kˆ L = mass-transfer coefficient with reaction and kˆ L0 = masstransfer coefficient for pure physical absorption. It is important to understand that when chemical reactions are involved, this definition of kˆ L is based on the driving force defined as the difference between the concentration of unreacted solute gas at the interface and in the bulk of the liquid. A coefficient based on the total of both unreacted and reacted gas could have values smaller than the physical-absorption mass-transfer coefficient kˆ L0 . When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of kˆ L or HL. Extrapolation to widely different concentration ranges or operating conditions should be made with caution, since the mass-transfer coefficient kˆ L may vary in an unexpected fashion, owing to changes in the apparent chemical-reaction mechanism. Generalized prediction methods for kˆ L and HL do not apply when chemical reaction occurs in the liquid phase, and therefore one must use actual operating data for the particular system in question. A discussion of the various factors to consider in designing gas absorbers and strippers when chemical reactions are involved is presented by Astarita, Savage, and Bisio, Gas Treating with Chemical Solvents, Wiley (1983) and by Kohl and Nielsen, Gas Purification, 5th ed., Gulf (1997). Effective Interfacial Mass-Transfer Area a In a packed tower of constant cross-sectional area S the differential change in solute flow per unit time is given by −d(GMSy) = NAa dV = NAaS dh

(5-312)

where a = interfacial area effective for mass transfer per unit of packed volume and V = packed volume. Owing to incomplete wetting of the packing surfaces and to the formation of areas of stagnation in the liquid film, the effective area normally is significantly less than the total external area of the packing pieces. The effective interfacial area depends on a number of factors, as discussed in a review by Charpentier [Chem. Eng. J., 11, 161 (1976)]. Among these factors are (1) the shape and size of packing, (2) the packing material (for example, plastic generally gives smaller interfacial areas than either metal or ceramic), (3) the liquid mass velocity, and (4), for small-diameter towers, the column diameter. Whereas the interfacial area generally increases with increasing liquid rate, it apparently is relatively independent of the superficial gas mass velocity below the flooding point. According to Charpentier’s review, it appears valid to assume that the interfacial area is independent of the column height when specified in terms of unit packed volume (i.e., as a). Also, the existing data for chemically reacting gas-liquid systems (mostly aqueous electrolyte solutions) indicate that

5-83

the interfacial area is independent of the chemical system. However, this situation may not hold true for systems involving large heats of reaction. Rizzuti et al. [Chem. Eng. Sci., 36, 973 (1981)] examined the influence of solvent viscosity upon the effective interfacial area in packed columns and concluded that for the systems studied the effective interfacial area a was proportional to the kinematic viscosity raised to the 0.7 power. Thus, the hydrodynamic behavior of a packed absorber is strongly affected by viscosity effects. Surface-tension effects also are important, as expressed in the work of Onda et al. (see Table 5-24-C). In developing correlations for the mass-transfer coefficients kˆ G and kˆ L, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La Experimental determinations of the individual mass-transfer coefficients kˆ G and kˆ L and of the effective interfacial area a involve the use of extremely difficult techniques, and therefore such data are not plentiful. More often, column experimental data are reported in terms of overall volumetric coefficients, which normally are defined as follows: and

K′Ga = nA /(hTSpT ∆y°1m)

(5-313)

KLa = nA /(hTS ∆x°1m)

(5-314)

where K′Ga = overall volumetric gas-phase mass-transfer coefficient, KLa = overall volumetric liquid-phase mass-transfer coefficient, nA = overall rate of transfer of solute A, hT = total packed depth in tower, S = tower cross-sectional area, pT = total system pressure employed during the experiment, and ∆x°1m and ∆y°1m are defined as

and

(y − y°)1 − (y − y°)2 ∆y°1m =  ln [(y − y°)1/(y − y°)2]

(5-315)

(x° − x)2 − (x° − x)1 ∆x°1m =  ln [(x° − x)2/(x° − x)1]

(5-316)

where subscripts 1 and 2 refer to the bottom and top of the tower respectively. Experimental K′Ga and KLa data are available for most absorption and stripping operations of commercial interest (see Sec. 14). The solute concentrations employed in these experiments normally are very low, so that KLa ⬟ Kˆ La and K′GapT ⬟ Kˆ Ga, where pT is the total pressure employed in the actual experimental-test system. Unlike the individual gas-film coefficient kˆ Ga, the overall coefficient Kˆ Ga will vary with the total system pressure except when the liquid-phase resistance is negligible (i.e., when either m = 0, or kˆ La is very large, or both). Extrapolation of KGa data for absorption and stripping to conditions other than those for which the original measurements were made can be extremely risky, especially in systems involving chemical reactions in the liquid phase. One therefore would be wise to restrict the use of overall volumetric mass-transfer-coefficient data to conditions not too far removed from those employed in the actual tests. The most reliable data for this purpose would be those obtained from an operating commercial unit of similar design. Experimental values of HOG and HOL for a number of distillation systems of commercial interest are also readily available. Extrapolation of the data or the correlations to conditions that differ significantly from those used for the original experiments is risky. For example, pressure has a major effect on vapor density and thus can affect the hydrodynamics significantly. Changes in flow patterns affect both masstransfer coefficients and interfacial area. Chilton-Colburn Analogy On occasion one will find that heattransfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy [90, 53] (see Tables 5-17-G and 5-19-T) provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental jM or jH data obtained with gases to predict liquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer of f/2. The Chilton-Colburn analogy can be used for simultaneous heat and mass transfer as long as the concentration and temperature fields are independent [Venkatesan and Fogler, AIChE J. 50, 1623 (2004)].

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Section 6

Fluid and Particle Dynamics

James N. Tilton, Ph.D., P.E. Principal Consultant, Process Engineering, E. I. du Pont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware)

FLUID DYNAMICS Nature of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible and Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . Streamlines, Pathlines, and Streaklines . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow, Reynolds Number . . . . . . . . . . . . . . . . Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroscopic and Microscopic Balances . . . . . . . . . . . . . . . . . . . . . . . Macroscopic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance, Bernoulli Equation . . . . . . . . . . . . . . . . Microscopic Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance, Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Momentum and Navier-Stokes Equations . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Force Exerted on a Reducing Bend. . . . . . . . . . . . . . . . . Example 2: Simplified Ejector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Venturi Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Plane Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incompressible Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Factor and Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entrance and Exit Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncircular Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonisothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Newtonian Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Pipe Diameter, Turbulent Flow . . . . . . . . . . . . . . . . . . . . . Economic Pipe Diameter, Laminar Flow . . . . . . . . . . . . . . . . . . . . . . Vacuum Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-4 6-4 6-4 6-4 6-5 6-5 6-5 6-5 6-5 6-5 6-5 6-6 6-6 6-6 6-6 6-6 6-6 6-7 6-7 6-7 6-7 6-7 6-8 6-8 6-8 6-9 6-9 6-9 6-9 6-9 6-10 6-10 6-11 6-11 6-11 6-12 6-12 6-13 6-13 6-14 6-15 6-15 6-15

Slip Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Losses in Pipeline Elements . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Length and Velocity Head Methods . . . . . . . . . . . . . . . . . Contraction and Entrance Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Entrance Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion and Exit Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fittings and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Losses with Fittings and Valves . . . . . . . . . . . . . . . . . . . . Curved Pipes and Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow through Orifices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mach Number and Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Gas Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . Adiabatic Frictionless Nozzle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Flow through Frictionless Nozzle . . . . . . . . . . . . . . . . . . Adiabatic Flow with Friction in a Duct of Constant Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Compressible Flow with Friction Losses . . . . . . . . . . . . . Convergent/Divergent Nozzles (De Laval Nozzles) . . . . . . . . . . . . . . Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids and Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solids and Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated-Pipe Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Pipe Distributor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slot Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning Vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated Plates and Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flow Straightening Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stirred Tank Agitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipeline Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Beds of Granular Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-15 6-16 6-16 6-16 6-16 6-17 6-17 6-18 6-19 6-20 6-20 6-22 6-22 6-22 6-22 6-23 6-23 6-24 6-24 6-24 6-26 6-26 6-30 6-30 6-32 6-32 6-33 6-33 6-33 6-34 6-34 6-34 6-34 6-35 6-36 6-36 6-36 6-37 6-37 6-39 6-39 6-39 6-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

6-2

FLUID AND PARTICLE DYNAMICS

Tower Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Layer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Plate, Zero Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylindrical Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Flat Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Wetting Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Surface Traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic Transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: Response to Instantaneous Valve Closing . . . . . . . . . . . Pulsating Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-40 6-40 6-40 6-40 6-41 6-41 6-41 6-41 6-42 6-43 6-43 6-43 6-43 6-44 6-44 6-44 6-44 6-44 6-45

Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eddy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-45 6-46 6-46 6-46 6-47 6-47 6-49

PARTICLE DYNAMICS Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminal Settling Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonspherical Rigid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hindered Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-dependent Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-51 6-51 6-51 6-52 6-53 6-53 6-54 6-55 6-55 6-56

Nomenclature and Units* In this listing, symbols used in this section are defined in a general way and appropriate SI units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Some symbols have more than one definition; the appropriate one is identified at the place of application. Symbol

Definition

SI units

U.S. customary units

a A b b c cf C Ca C0 CD d D De Dij

Pressure wave velocity Area Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force Cumulative residence time distribution Froude number Acceleration of gravity Mass flux Enthalpy per unit mass Liquid depth Ratio of specific heats Kinetic energy of turbulence Power law coefficient Viscous losses per unit mass Length Mass flow rate Mass Mach number Morton number Molecular weight Power law exponent Blend time number Best number Power number Pumping number Pressure Entrained flow rate Volumetric flow rate Throughput (vacuum flow) Heat input per unit mass Radial coordinate Radius Ideal gas universal constant Volume fraction of phase i Reynolds number Density ratio

m/s m2 m m m/s Dimensionless m3/s Dimensionless Dimensionless Dimensionless m m Dimensionless 1/s

ft/s ft2 in ft ft/s Dimensionless ft3/s Dimensionless Dimensionless Dimensionless ft ft Dimensionless 1/s

Pa J/s Dimensionless Dimensionless 1/s N Dimensionless

lbf/in2 ft ⋅ lbf/s Dimensionless Dimensionless 1/s lbf Dimensionless

Dimensionless m/s2 kg/(m2 ⋅ s) J/kg m Dimensionless J/kg kg/(m ⋅ s2 − n) J/kg m kg/s kg Dimensionless Dimensionless kg/kgmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Pa m3/s m3/s Pa ⋅ m3/s J/kg m m J/(kgmole ⋅ K) Dimensionless Dimensionless Dimensionless

Dimensionless ft/s2 lbm/(ft2 ⋅ s) Btu/lbm ft Dimensionless ft ⋅ lbf/lbm lbm/(ft ⋅ s2 − n) ft ⋅ lbf/lbm ft lbm/s lbm Dimensionless Dimensionless lbm/lbmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 ft3/s ft3/s lbf ⋅ ft3/s Btu/lbm ft ft Btu/(lbmole ⋅ R) Dimensionless Dimensionless Dimensionless

E E˙ v Eo f f F F Fr g G h h k k K lv L m ˙ M M M Mw n Nb ND NP NQ p q Q Q δQ r R R Ri Re s

Symbol

Definition

SI units

U.S. customary units

s S S S St t t T u u U v V V We ˙s W δWs x y z z

Entropy per unit mass Slope Pumping speed Surface area per unit volume Strouhal number Time Force per unit area Absolute temperature Internal energy per unit mass Velocity Velocity Velocity Velocity Volume Weber number Rate of shaft work Shaft work per unit mass Cartesian coordinate Cartesian coordinate Cartesian coordinate Elevation

J/(kg ⋅ K) Dimensionless m3/s l/m Dimensionless s Pa K J/kg m/s m/s m/s m/s m3 Dimensionless J/s J/kg m m m m

Btu/(lbm ⋅ R) Dimensionless ft3/s l/ft Dimensionless s lbf/in2 R Btu/lbm ft/s ft/s ft/s ft/s ft3 Dimensionless Btu/s Btu/lbm ft ft ft ft

α α β β γ˙ Γ

Velocity profile factor Included angle Velocity profile factor Bulk modulus of elasticity Shear rate Mass flow rate per unit width Boundary layer or film thickness Kronecker delta Pipe roughness Void fraction Turbulent dissipation rate Residence time Angle Mean free path Viscosity Kinematic viscosity Density Surface tension Cavitation number Components of total stress tensor Shear stress Time period Components of deviatoric stress tensor Energy dissipation rate per unit volume Angle of inclination Vorticity

Greek Symbols

δ δij % % % θ θ λ µ ν ρ σ σ σij τ τ τij Φ φ ω

Dimensionless Radians Dimensionless Pa l/s kg/(m ⋅ s)

Dimensionless Radians Dimensionless lbf/in2 l/s lbm/(ft ⋅ s)

m

ft

Dimensionless m Dimensionless J/(kg ⋅ s) s Radians m Pa ⋅ s m2/s kg/m3 N/m Dimensionless Pa

Dimensionless ft Dimensionless ft ⋅ lbf/(lbm ⋅ s) s Radians ft lbm/(ft ⋅ s) ft2/s lbm/ft3 lbf/ft Dimensionless lbf/in2

Pa s Pa

lbf/in2 s lbf/in2

J/(m3 ⋅ s)

ft ⋅ lbf/(ft3 ⋅ s)

Radians 1/s

Radians 1/s

* Note that with U.S. Customary units, the conversion factor gc may be required to make equations in this section dimensionally consistent; gc = 32.17 (lbm⋅ft)/(lbf⋅s2). 6-3

6-4

FLUID AND PARTICLE DYNAMICS

FLUID DYNAMICS

Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation. The force is directly proportional to the area of the plate; the shear stress is τ = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient γ˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, µ. τ µ= γ˙

(6-1)

The SI units of viscosity are kg/(m ⋅ s) or Pa ⋅ s (pascal second). The cgs unit for viscosity is the poise; 1 Pa ⋅ s equals 10 poise or 1000 centipoise (cP) or 0.672 lbm/(ft ⋅ s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (6-1). Kinematic viscosity ν  µ/ρ is the ratio of viscosity to density. The SI units of kinematic viscosity are m2/s. The cgs stoke is 1 cm2/s. Rheology In general, fluid flow patterns are more complex than the one shown in Fig. 6-1, as is the relationship between fluid deformation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitutive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 6-1. Complete definitions require extension to multidimensional flow. For more information, several good references are available, including Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977); Metzner (“Flow of Non-Newtonian Fluids” in Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971); and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967).

A

du τ = µγ˙ = µ  dy

Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behavior. Shear-thinning fluids are those for which the slope of the rheogram decreases with increasing shear rate. These fluids have also been called pseudoplastic, but this terminology is outdated and discouraged. Many polymer melts and solutions, as well as some solids suspensions, are shear-thinning. Shear-thinning fluids without yield stresses typically obey a power law model over a range of shear rates. (6-4) τ = Kγ˙ n The apparent viscosity is µ = Kγ˙ n − 1

F V

y

τy

lat Di

t an

an toni Ne w Shear rate |du/dy|

H

Deformation of a fluid subjected to a shear stress.

ti c as m tic ha las ng op i B pl

x FIG. 6-1

(6-2)

All fluids for which the viscosity varies with shear rate are nonNewtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. Non-Newtonian fluids include those for which a finite stress τy is required before continuous deformation occurs; these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material; its rheogram has a constant slope µ∞, called the infinite shear viscosity. (6-3) τ = τy + µ∞γ˙

Ps eu d

NATURE OF FLUIDS

Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 6-2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is independent of shear rate, and may depend only on temperature and perhaps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant:

Shear stress τ

GENERAL REFERENCES: Batchelor, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967; Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., Wiley, New York, 2002; Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Mass., 1967; Denn, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980; Landau and Lifshitz, Fluid Mechanics, 2d ed., Pergamon, 1987; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Panton, Incompressible Flow, Wiley, New York, 1984; Schlichting, Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987; Shames, Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992; Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971; Streeter and Wylie, Fluid Mechanics, 8th ed., McGraw-Hill, New York, 1985; Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975; Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981.

FIG. 6-2

Shear diagrams.

(6-5)

FLUID DYNAMICS The factor K is the consistency index or power law coefficient, and n is the power law exponent. The exponent n is dimensionless, while K is in units of kg/(m ⋅ s2 − n). For shear-thinning fluids, n < 1. The power law model typically provides a good fit to data over a range of one to two orders of magnitude in shear rate; behavior at very low and very high shear rates is often Newtonian. Shear-thinning power law fluids with yield stresses are sometimes called Herschel-Bulkley fluids. Numerous other rheological model equations for shear-thinning fluids are in common use. Dilatant, or shear-thickening, fluids show increasing viscosity with increasing shear rate. Over a limited range of shear rate, they may be described by the power law model with n > 1. Dilatancy is rare, observed only in certain concentration ranges in some particle suspensions (Govier and Aziz, pp. 33–34). Extensive discussions of dilatant suspensions, together with a listing of dilatant systems, are given by Green and Griskey (Trans. Soc. Rheol, 12[1], 13–25 [1968]); Griskey and Green (AIChE J., 17, 725–728 [1971]); and Bauer and Collins (“Thixotropy and Dilatancy,” in Eirich, Rheology, vol. 4, Academic, New York, 1967). Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Collins (ibid.). Rheopectic behavior is the opposite of thixotropy. Shear stress increases with time at constant shear rate. Rheopectic behavior has been observed in bentonite sols, vanadium pentoxide sols, and gypsum suspensions in water (Bauer and Collins, ibid.) as well as in some polyester solutions (Steg and Katz, J. Appl. Polym. Sci., 9, 3, 177 [1965]). Viscoelastic fluids exhibit elastic recovery from deformation when stress is removed. Polymeric liquids comprise the largest group of fluids in this class. A property of viscoelastic fluids is the relaxation time, which is a measure of the time required for elastic effects to decay. Viscoelastic effects may be important with sudden changes in rates of deformation, as in flow startup and stop, rapidly oscillating flows, or as a fluid passes through sudden expansions or contractions where accelerations occur. In many fully developed flows where such effects are absent, viscoelastic fluids behave as if they were purely viscous. In viscoelastic flows, normal stresses perpendicular to the direction of shear are different from those in the parallel direction. These give rise to such behaviors as the Weissenberg effect, in which fluid climbs up a shaft rotating in the fluid, and die swell, where a stream of fluid issuing from a tube may expand to two or more times the tube diameter. A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For very large Deborah numbers, the behavior closely resembles that of an elastic solid. Analysis of viscoelastic flows is very difficult. Simple constitutive equations are unable to describe all the material behavior exhibited by viscoelastic fluids even in geometrically simple flows. More complex constitutive equations may be more accurate, but become exceedingly difficult to apply, especially for complex geometries, even with advanced numerical methods. For good discussions of viscoelastic fluid behavior, including various types of constitutive equations, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, vol. 2: Kinetic Theory, Wiley, New York, 1977); Middleman (The Flow of High Polymers, Interscience (Wiley) New York, 1968); or Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). Polymer processing is the field which depends most on the flow of non-Newtonian fluids. Several excellent texts are available, including Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977) and Tadmor and Gogos (Principles of Polymer Processing, Wiley, New York, 1979).

6-5

There is a wide variety of instruments for measurement of Newtonian viscosity, as well as rheological properties of non-Newtonian fluids. They are described in Van Wazer, Lyons, Kim, and Colwell (Viscosity and Flow Measurement, Interscience, New York, 1963); Coleman, Markowitz, and Noll (Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966); Dealy and Wissbrun (Melt Rheology and Its Role in Plastics Processing, Van Nostrand Reinhold, 1990). Measurement of rheological behavior requires well-characterized flows. Such rheometric flows are thoroughly discussed by Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGrawHill, New York, 1974). KINEMATICS OF FLUID FLOW Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components vx, vy, and vz. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly constant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see following). Compressible fluids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. Streamlines, Pathlines, and Streaklines These are curves in a flow field which provide insight into the flow pattern. Streamlines are tangent at every point to the local instantaneous velocity vector. A pathline is the path followed by a material element of fluid; it coincides with a streamline if the flow is steady. In unsteady flow the pathlines generally do not coincide with streamlines. Streaklines are curves on which are found all the material particles which passed through a particular point in space at some earlier time. For example, a streakline is revealed by releasing smoke or dye at a point in a flow field. For steady flows, streamlines, pathlines, and streaklines are indistinguishable. In two-dimensional incompressible flows, streamlines are contours of the stream function. One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction; velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor Dij has nine components. In Cartesian coordinates, ∂vi ∂vj Dij =  +  (6-6) ∂xj ∂xi





where the subscripts i and j refer to the three coordinate directions. Some authors define the deformation rate tensor as one-half of that given by Eq. (6-6). Vorticity The relative motion between two points in a fluid can be decomposed into three components: rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a tensor ωij = ∂vi /∂xj − ∂vj /∂xi. The vector of vorticity given by one-half the

6-6

FLUID AND PARTICLE DYNAMICS

curl of the velocity vector is another measure of rotation. In twodimensional flow in the x-y plane, the vorticity ω is given by 1 ∂v ∂v ω =  y − x (6-7) 2 ∂x ∂y Here ω is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irrotational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fluid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU ρ/µ where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally.





CONSERVATION EQUATIONS Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and conservation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equations. Macroscopic Equations An arbitrary control volume of finite size Va is bounded by a surface of area Aa with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitrary control volume. Mass Balance Applied to the control volume, the principle of conservation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) d ρ dV + ρ(v − w) ⋅ n dA = 0 (6-8)  Aa dt Va This equation is also known as the continuity equation.





Area Aa

Volume Va

n outwardly directed unit normal vector

w boundary velocity v fluid velocity FIG. 6-3

V2 2

Arbitrary control volume for application of conservation equations.

V1 1 FIG. 6-4

Fixed control volume with one inlet and one outlet.

Simplified forms of Eq. (6-8) apply to special cases frequently found in practice. For a control volume fixed in space with one inlet of area A1 through which an incompressible fluid enters the control volume at an average velocity V1, and one outlet of area A2 through which fluid leaves at an average velocity V2, as shown in Fig. 6-4, the continuity equation becomes V1 A1 = V2 A2

(6-9)

The average velocity across a surface is given by V = (1/A)

 v dA A

where v is the local velocity component perpendicular to the inlet surface. The volumetric flow rate Q is the product of average velocity and the cross-sectional area, Q = VA. The average mass velocity is G = ρV. For steady flows through fixed control volumes with multiple inlets and/or outlets, conservation of mass requires that the sum of inlet mass flow rates equals the sum of outlet mass flow rates. For incompressible flows through fixed control volumes, the sum of inlet flow rates (mass or volumetric) equals the sum of exit flow rates, whether the flow is steady or unsteady. Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). d ρv dV + ρv(v − w) ⋅ n dA = ρg dV + tn dA (6-10)  Aa Va Aa dt Va Here g is the gravity vector and tn is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integral on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m ˙ through a control volume fixed in space with one inlet and one outlet (Fig. 6-4), with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors V1 and V2, the momentum equation becomes









m(β ˙ 2V2 − β1V1) = −p1A1 − p2A2 + F + Mg

(6-11)

where M is the total mass of fluid in the control volume. The factor β arises from the averaging of the velocity across the area of the inlet or outlet surface. It is the ratio of the area average of the square of velocity magnitude to the square of the area average velocity magnitude. For a uniform velocity, β = 1. For turbulent flow, β is nearly unity, while for laminar pipe flow with a parabolic velocity profile, β = 4/3. The vectors A1 and A 2 have magnitude equal to the areas of the inlet and outlet surfaces, respectively, and are outwardly directed normal to the surfaces. The vector F is the force exerted on the fluid by the nonflow boundaries of the control volume. It is also assumed that the stress vector tn is normal to the inlet and outlet surfaces, and that its magnitude may be approximated by the pressure p. Equation (6-11) may be generalized to multiple inlets and/or outlets. In such cases, the mass flow rates for all the inlets and outlets are not equal. A distinct flow rate m ˙ i applies to each inlet or outlet i. To generalize the equation, pA terms for each inlet and outlet, − mβV ˙ terms for each inlet, and mβV ˙ terms for each outlet are included.

FLUID DYNAMICS Balance equations for angular momentum, or moment of momentum, may also be written. They are used less frequently than the linear momentum equations. See Whitaker (Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). Total Energy Balance The total energy balance derives from the first law of thermodynamics. Applied to the arbitrary control volume of Fig. 6-3, it leads to an equation for the rate of change of the sum of internal, kinetic, and gravitational potential energy. In this equation, u is the internal energy per unit mass, v is the magnitude of the velocity vector v, z is elevation, g is the gravitational acceleration, and q is the heat flux vector: v2 d v2 ρ u +  + gz dV + ρ u +  + gz (v − w) ⋅ n dA  Aa dt Va 2 2

 

  =  (v ⋅ t ) dA − 



n

Aa

Aa

(q ⋅ n) dA (6-12)

The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundary. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers; this work is called ˙ S. shaft work; its rate is W A useful simplification of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m ˙ through a single planar entrance and a single planar exit (Fig. 6-4), to which the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tn is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + ρgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. V 12 V 22 h1 + α1  + gz1 = h2 + α2  + gz2 − δQ − δWS 2 2

(6-13)

Here, h is the enthalpy per unit mass, h = u + p/ρ. The shaft work per ˙ s /m. unit of mass flowing through the control volume is δWS = W ˙ Similarly, δQ is the heat input per unit of mass. The factor α is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, α = 1. In turbulent flow, α is usually assumed to equal unity; in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a parabolic velocity profile, α = 2. Mechanical Energy Balance, Bernoulli Equation A balance equation for the sum of kinetic and potential energy may be obtained from the momentum balance by forming the scalar product with the velocity vector. The resulting equation, called the mechanical energy balance, contains a term accounting for the dissipation of mechanical energy into thermal energy by viscous forces. The mechanical energy equation is also derivable from the total energy equation in a way that reveals the relationship between the dissipation and entropy generation. The macroscopic mechanical energy balance for the arbitrary control volume of Fig. 6-3 may be written, with p = thermodynamic pressure, as d v2 v2 ρ  + gz dV + ρ  + gz (v − w) ⋅ n dA  Aa dt Va 2 2

  =

Va

  p  ⋅ v dV +  (v ⋅ t ) dA − 



n

Aa

Va

Φ dV

(6-14)

The last term is the rate of viscous energy dissipation to internal energy, E˙ v = Va Φ dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function Φ for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such

6-7

cases additional information, which may come from empirical correlations, is needed. For the same special conditions as for Eq. (6-13), the mechanical energy equation is reduced to p 2 dp V12 V 22 α1  + gz1 + δWS = α2  + gz2 + (6-15)  + lv p 2 2 ρ 1 Here lv = E˙ v /m ˙ is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes p2 p1 V12 V22 (6-16)  + α1  + gz1 + δWS =  + α2  + gz2 + lv ρ ρ 2 2 The Bernoulli equation can be written for incompressible, inviscid flow along a streamline, where no shaft work is done.



p1 V12 p2 V22  +  + gz1 =  +  + gz2 ρ 2 ρ 2

(6-17)

Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generalized to multiple inlets or outlets. Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as ∂ρ ∂ρvx ∂ρvy ∂ρvz +++=0 ∂t ∂x ∂y ∂z

(6-18)

In terms of the substantial derivative, D/Dt, Dρ ∂ρ ∂ρ ∂ρ ∂ρ ∂vx ∂vy ∂vz    + vx  + vy  + vz  = −ρ  +  +  Dt ∂t ∂x ∂y ∂z ∂x ∂y ∂z





(6-19)

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be expressed as Dρ (6-20)  = −ρ∇ ⋅ v Dt For incompressible flow, ∂v ∂v ∂v ∇ ⋅ v = x + y + z = 0 (6-21) ∂x ∂y ∂z Stress Tensor The stress tensor is needed to completely describe the stress state for microscopic momentum balances in multidimensional flows. The components of the stress tensor σij give the force in the j direction on a plane perpendicular to the i direction, using a sign convention defining a positive stress as one where the fluid with the greater i coordinate value exerts a force in the positive i direction on the fluid with the lesser i coordinate. Several references in fluid mechanics and continuum mechanics provide discussions, to various levels of detail, of stress in a fluid (Denn; Bird, Stewart, and Lightfoot; Schlichting; Fung [A First Course in Continuum Mechanics, 2d. ed., Prentice-Hall, Englewood Cliffs, N.J., 1977]; Truesdell and Toupin [in Flügge, Handbuch der Physik, vol. 3/1, Springer-Verlag, Berlin, 1960]; Slattery [Momentum, Energy and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981]). The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the threedimensional generalization of Eq. (6-2): τij = µDij

(6-22)

6-8

FLUID AND PARTICLE DYNAMICS

The total stress is σij = (−p + λ∇ ⋅ v)δij + τij

(6-23)

The identity tensor δij is zero for i ≠ j and unity for i = j. The coefficient λ is a material property related to the bulk viscosity, κ = λ + 2µ/3. There is considerable uncertainty about the value of κ. Traditionally, Stokes’ hypothesis, κ = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to σij = −pδij + τij

(6-24)

Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fluid mechanics texts (e.g., Slattery [ibid.]; Denn; Whitaker; and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) leads to the Navier-Stokes equations, whose three Cartesian components are ∂v ∂v ∂v ∂v ρ x + vx x + vy x + vz x ∂t ∂x ∂y ∂z





∂p ∂2vx ∂2vx ∂2vx =−+µ  + + + ρgx (6-25) ∂x ∂x2 ∂y2 ∂z2



∂v ∂v ∂v ∂v ρ y + vx y + vy y + vz y ∂t ∂x ∂y ∂z







∂p ∂2vy ∂2vy ∂2vy =−+µ  + + + ρgy (6-26) ∂y ∂x2 ∂y2 ∂z2



∂v ∂v ∂v ∂v ρ z + vx z + vy z + vz z ∂t ∂x ∂y ∂z







∂p ∂2vz ∂2vz ∂2vz =−+µ  + + + ρgz (6-27) ∂z ∂x2 ∂y2 ∂z2





In vector notation, Dv ∂v ρ  =  + (v ⋅ ∇)v = −∇p + µ∇2v + ρg (6-28) Dt ∂t The pressure and gravity terms may be combined by replacing the pressure p by the equivalent pressure P = p + ρgz. The left-hand side terms of the Navier-Stokes equations are the inertial terms, while the terms including viscosity µ are the viscous terms. Limiting cases under which the Navier-Stokes equations may be simplified include creeping flows in which the inertial terms are neglected, potential flows (inviscid or irrotational flows) in which the viscous terms are neglected, and boundary layer and lubrication flows in which certain terms are neglected based on scaling arguments. Creeping flows are described by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1965); potential flows by Lamb (Hydrodynamics, 6th ed., Dover, New York, 1945) and Milne-Thompson (Theoretical Hydrodynamics, 5th ed., Macmillan, New York, 1968); boundary layer theory by Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987); and lubrication theory by Batchelor (An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967) and Denn (Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980). Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity component) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather

than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-Stokes equations, Dirichlet and Neumann, or essential and natural, boundary conditions may be satisfied by different means. Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, ∇p = ρg. Letting z be directed vertically upward, so that gz = −g where g is the gravitational acceleration (9.806 m2/s), the pressure field is given by dp/dz = −ρg

(6-29)

This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of ρ with z. The force exerted on a submerged planar surface of area A is given by F = pc A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows. Example 1: Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity V1 is given and the pressures p1 and p2 are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exit velocity V2 = V1A1/A2. The mass flow rate is obtained by m ˙ = ρV1A1. Assume that the velocity profile is nearly uniform so that β is approximately unity. The force exerted on the fluid by the bend has x and y components; these can be found from Eq. (6-11). The x component gives Fx = m(V ˙ 2x − V1x) + p1A1x + p2 A2x while the y component gives Fy = m(V ˙ 2y − V1y) + p1 A1y + p2 A2y The velocity components are V1x = V1, V1y = 0, V2x = V2 cos θ, and V2y = V2 sin θ. The area vector components are A1x = −A1, A1y = 0, A 2x = A 2 cos θ, and A 2y = A 2 sin θ. Therefore, the force components may be calculated from Fx = m(V ˙ 2 cos θ − V1) − p1A1 + p2A2 cos θ Fy = mV ˙ 2 sin θ + p2A2 sin θ The force acting on the fluid is F; the equal and opposite force exerted by the fluid on the bend is F.

V2 θ

V1 F y x Force at a reducing bend. F is the force exerted by the bend on the fluid. The force exerted by the fluid on the bend is F.

FIG. 6-5

FLUID DYNAMICS

6-9

y H

x

FIG. 6-8

FIG. 6-6

Draft-tube ejector.

Example 2: Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may be applied on the control volume with inlet and outlet surfaces 1 and 2 as indicated in the figure. The cross-sectional area is uniform, A1 = A2 = A. Let the mass flow rates and velocities of the primary and secondary fluids be m ˙ p, m ˙ s, Vp and Vs. Assume for simplicity that the density is uniform. Conservation of mass gives m˙2 = m ˙p + m ˙ s. The exit velocity is V2 = m ˙ 2 /(ρA). The principle momentum exchange in the ejector occurs between the two fluids. Relative to this exchange, the force exerted by the walls of the device are found to be small. Therefore, the force term F is neglected from the momentum equation. Written in the flow direction, assuming uniform velocity profiles, and using the extension of Eq. (6-11) for multiple inlets, it gives the pressure rise developed by the device: (p2 − p1)A = (m ˙p+m ˙ s)V2 − m ˙ pVp − m ˙ sVs Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook.

Example 3: Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives p1 V 12 p2 V 22 +=+ ρ 2 ρ 2 The continuity equation gives V2 = V1A/a, and V1 = Q/A. The pressure drop measured by the manometer is p1 − p2 = (ρm − ρ)g∆z. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 2(ρm − ρ)g∆z 1 Q =   A ρ[(A/a)2 − 1]



Example 4: Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks; the solution presented here closely follows Denn.

Plane Poiseuille flow.

This problem requires use of the microscopic balance equations because the velocity is to be determined as a function of position. The boundary conditions for this flow result from the no-slip condition. All three velocity components must be zero at the plate surfaces, y = H/2 and y = −H/2. Assume that the flow is fully developed, that is, all velocity derivatives vanish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and ∂vx /∂x = 0, reduces to dvy =0 dy Since vy = 0 at y = H/2, the continuity equation integrates to vy = 0. This is a direct result of the assumption of fully developed flow. The Navier-Stokes equations are greatly simplified when it is noted that vy = vz = 0 and ∂vx /∂x = ∂vx /∂z = ∂vx /∂t = 0. The three components are written in terms of the equivalent pressure P: ∂P ∂2vx 0=−+µ ∂x ∂y2 ∂P 0=− ∂y ∂P 0=− ∂z The latter two equations require that P is a function only of x, and therefore ∂P/∂x = dP/dx. Inspection of the first equation shows one term which is a function only of x and one which is only a function of y. This requires that both terms are constant. The pressure gradient −dP/dx is constant. The x-component equation becomes d 2vx 1 dP =  dy2 µ dx Two integrations of the x-component equation give 1 dP vx =   y2 + C1y + C2 2µ dx where the constants of integration C1 and C2 are evaluated from the boundary conditions vx = 0 at y = H/2. The result is



  

2y 2 H2 dP vx =  −  1 −  8µ dx H This is a parabolic velocity distribution. The average velocity V = (1/H)  −H/2 vx dy is H/2 dP H2 V= − 12µ dx





This flow is one-dimensional, as there is only one nonzero velocity component, vx, which, along with the pressure, varies in only one coordinate direction.

1

2

∆z

INCOMPRESSIBLE FLOW IN PIPES AND CHANNELS Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to p2 p1 (6-30)  + gz1 =  + gz 2 + lv ρ ρ In terms of the equivalent pressure, P  p + ρgz, P1 − P2 = ρlv

FIG. 6-7

Venturi flowmeter.

(6-31)

The pressure drop due to frictional losses lv is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity ∆P  P1 − P2.

6-10

FLUID AND PARTICLE DYNAMICS

FIG. 6-9 Fanning Friction Factors. Reynolds number Re = DVρ/µ, where D = pipe diameter, V = velocity, ρ = fluid density, and µ = fluid viscosity. (Based on Moody, Trans. ASME, 66, 671 [1944].)

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length ∆P/L to the pipe diameter D, density ρ, viscosity , and average velocity V through two dimensionless groups, the Fanning friction factor f and the Reynolds number Re. D∆P f (6-32) 2ρV 2L DVρ Re   (6-33) µ For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness %/D also affects the friction factor. Figure 6-9 plots f as a function of Re and %/D. Values of % for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction factor used by Moody (Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor: ρV 2 τw f  2

(6-34)

Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar; over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. Reliable correlations for the friction factor in transitional flow are not available. For laminar flow, the Hagen-Poiseuille equation 16 f  Re

Re ≤ 2,100

(6-35)

TABLE 6-1 Values of Surface Roughness for Various Materials* Material Drawn tubing (brass, lead, glass, and the like) Commercial steel or wrought iron Asphalted cast iron Galvanized iron Cast iron Wood stove Concrete Riveted steel

Surface roughness %, mm 0.00152 0.0457 0.122 0.152 0.259 0.183–0.914 0.305–3.05 0.914–9.14

* From Moody, Trans. Am. Soc. Mech. Eng., 66, 671–684 (1944); Mech. Eng., 69, 1005–1006 (1947). Additional values of ε for various types or conditions of concrete wrought-iron, welded steel, riveted steel, and corrugated-metal pipes are given in Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, pp. 6-12–6-13. To convert millimeters to feet, multiply by 3.281 × 10−3.

may be derived from the Navier-Stokes equation and is in excellent agreement with experimental data. It may be rewritten in terms of volumetric flow rate, Q = VπD2/4, as π∆PD4 Q= Re ≤ 2,100 (6-36) 128µL For turbulent flow in smooth tubes, the Blasius equation gives the friction factor accurately for a wide range of Reynolds numbers. 0.079 f= 4,000 < Re < 105 (6-37) Re0.25

FLUID DYNAMICS The Colebrook formula (Colebrook, J. Inst. Civ. Eng. [London], 11, 133–156 [1938–39]) gives a good approximation for the f-Re-(%/D) data for rough pipes over the entire turbulent flow range: 1 1.256 %  = −4 log  +  3.7D Ref f





Re > 4,000



 













In laminar flow, f is independent of %/D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydraulically smooth flow). For greater Reynolds numbers, f deviates from the smooth pipe curve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. Two common pipe flow problems are calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given the pressure drop. When flow rate is given, the Reynolds number may be calculated directly to determine the flow regime, so that the appropriate relations between f and Re (or pressure drop and flow rate or velocity) can be selected. When flow rate is specified and the flow is turbulent, Eq. (6-39) or (6-40), being explicit in f, may be preferable to Eq. (6-38), which is implicit in f and pressure drop. When the pressure drop is given and the velocity and flow rate are to be determined, the Reynolds number cannot be computed directly, since the velocity is unknown. Instead of guessing and checking the flow regime, it may be useful to observe that the quantity Ref (D3/2/) ρP/(2 L) , appearing in the Colebrook equation (6-38), does not include velocity and so can be computed directly. The upper limit Re 2,100 for laminar flow and use of Eq. (6-35) corresponds to Ref 183. For smooth pipe, the lower limit Re 4,000 for the Colebrook equation corresponds to Ref 400. Thus, at least for smooth pipes, the flow regime can be determined without trial and error from P/L, µ, ρ, and D. When pressure drop is given, Eq. (6-38), being explicit in velocity, is preferable to Eqs. (6-39) and (6-40), which are implicit in velocity. As Fig. 6-9 suggests, the friction factor is uncertain in the transition range, and a conservative choice should be made for design purposes. Velocity Profiles In laminar flow, the solution of the NavierStokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. r2 v = 2V 1 −  (6-41) R2 In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between.





r z

(6-38)

Equation (6-38) was used to construct the curves in the turbulent flow regime in Fig. 6-9. An equation by Churchill (Chem. Eng., 84[24], 91–92 [Nov. 7, 1977]) approximating the Colebrook formula offers the advantage of being explicit in f: 1 0.27% 7 0.9  = −4 log  +  Re > 4,000 (6-39) Re D  f Churchill also provided a single equation that may be used for Reynolds numbers in laminar, transitional, and turbulent flow, closely fitting f 16/Re in the laminar regime, and the Colebrook formula, Eq. (6-38), in the turbulent regime. It also gives unique, reasonable values in the transition regime, where the friction factor is uncertain. 1/12 8 12 1 (6-40) f 2    Re (A  B)3/2 where 16 1 A 2.457 ln  0.9 (7/Re)  0.27ε /D and 37,530 16 B  Re

6-11

(

2 v = 2V 1 – r 2 R

R

(

v max = 2V FIG. 6-10

Parabolic velocity profile for laminar flow in a pipe, with average

velocity V.

Viscous sublayer u+ = y+

for

y+ < 5

(6-42)

u+ = 5.00 ln y+ − 3.05

for

5 < y+ < 30

(6-43)

for

y+ > 30

(6-44)

Buffer zone Turbulent core u+ = 2.5 ln y+ + 5.5

Here, u+ = v/u* is the dimensionless, time-averaged axial velocity, u* =  τw  /ρ is the friction velocity and τw = fρV 2/2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu*ρ/µ. The universal velocity profile is valid in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, τw = D∆P/4L where ∆P is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are valid only near the pipe wall. For rough pipes, the velocity profile in the turbulent core is given by u+ = 2.5 ln y/% + 8.5

for

y+ > 30

(6-45)

when the dimensionless roughness %+ = %u*ρ/µ is greater than 5 to 10; for smaller %+, the velocity profile in the turbulent core is unaffected by roughness. For velocity profiles in the transition region, see Patel and Head (J. Fluid Mech., 38, part 1, 181–201 [1969]) where profiles over the range 1,500 < Re < 10,000 are reported. Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fully developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara, and Riza, Can. J. Chem. Engr., 71, 472–476 [1993]) Lent /D = 0.370 exp(−0.148Re) + 0.0550Re + 0.260

(6-46)

In turbulent flow, the entrance length is about Lent /D = 40

(6-47)

The frictional losses in the entrance region are larger than those for the same length of fully developed flow. (See the subsection, “Frictional Losses in Pipeline Elements,” following.) At the pipe exit, the velocity profile also undergoes rearrangement, but the exit length is much shorter than the entrance length. At low Re, it is about one pipe radius. At Re > 100, the exit length is essentially 0. Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(θ) is given by θ F(θ) = 0 for θ < avg  2 2 1 θ θ F(θ) = 1 −  avg for θ ≥ avg (6-48)   4 θ 2



where F(θ) is the fraction of material which resides in the pipe for less than time θ and θavg is the average residence time, θ = V/L.

6-12

FLUID AND PARTICLE DYNAMICS

The residence time distribution in long transfer lines may be made narrower (more uniform) with the use of flow inverters or static mixing elements. These devices exchange fluid between the wall and central regions. Variations on the concept may be used to provide effective mixing of the fluid. See Godfrey (“Static Mixers,” in Harnby, Edwards, and Nienow, Mixing in the Process Industries, 2d ed., Butterworth Heinemann, Oxford, 1992); Etchells and Meyer (“Mixing in Pipelines, in Paul, Atiemo-Obeng, and Kresta, Handbook of Industrial Mixing, Wiley Interscience, Hoboken, N.J., 2004). A theoretically derived equation for laminar flow in helical pipe coils by Ruthven (Chem. Eng. Sci., 26, 1113–1121 [1971]; 33, 628–629 [1978]) is given by θavg 1 θavg 2.81 F(θ) = 1 −   for 0.5 <  < 1.63 (6-49) 4 θ θ and was substantially confirmed by Trivedi and Vasudeva (Chem. Eng. Sci., 29, 2291–2295 [1974]) for 0.6 < De < 6 and 0.0036 < D/Dc < 0.097 where De = Re D /Dc is the Dean number and Dc is the diameter of curvature of the coil. Measurements by Saxena and Nigam (Chem. Eng. Sci., 34, 425–426 [1979]) indicate that such a distribution will hold for De > 1. The residence time distribution for helical coils is narrower than for straight circular pipes, due to the secondary flow which exchanges fluid between the wall and center regions. In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93) gives DL = 1.01νRe0.875 for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253–278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D, DH = D − d, for a rectangular duct of sides a, b, DH = ab/[2(a + b)]. The hydraulic radius RH is defined as one-fourth of the hydraulic diameter. With the hydraulic diameter subsititued for D in f and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in f and Re is the actual average velocity V = Q/A; for noncircular pipes; it is not Q/(πDH2 /4). The pressure drop should be calculated from the friction factor for noncircular pipes. Equations relating Q to ∆P and D for circular pipes may not be used for noncircular pipes with D replaced by DH because V ≠ Q/(πDH2 /4). Turbulent flow in noncircular channels is generally accompanied by secondary flows perpendicular to the axial flow direction (Schlichting). These flows may cause the pressure drop to be slightly greater than that computed using the hydraulic diameter method. For data on pressure drop in annuli, see Brighton and Jones (J. Basic Eng., 86, 835–842 [1964]); Okiishi and Serovy (J. Basic Eng., 89, 823–836 [1967]); and Lawn and Elliot (J. Mech. Eng. Sci., 14, 195–204 [1972]). For rectangular ducts of large aspect ratio, Dean (J. Fluids Eng., 100, 215–233 [1978]) found that the numerator of the exponent in the Blasius equation (6-37) should be increased to 0.0868. Jones (J. Fluids Eng., 98, 173–181 [1976]) presents a method to improve the estimation of friction factors for rectangular ducts using a modification of the hydraulic diameter–based Reynolds number. The hydraulic diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters DE defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36); that is, DE 

  

(128QµL/π∆P)1/4. Equivalent diameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V ≠ Q/(πDE/4). Equivalent diameter DE is not to be used in the friction factor and Reynolds number; f ≠ 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter DH used for turbulent flow. Ellipse, semiaxes a and b (Lamb, Hydrodynamics, 6th ed., Dover, New York, 1945, p. 587):









32a3b3 1/4 (6-50) DE =  a2 + b2 Rectangle, width a, height b (Owen, Trans. Am. Soc. Civ. Eng., 119, 1157–1175 [1954]): 128ab3 DE =  πK a/b = K=

1 28.45

1.5 20.43

2 17.49

3 15.19

1/4

4 14.24

(6-51) 5 13.73

10 12.81

∞ 12

Annulus, inner diameter D1, outer diameter D2 (Lamb, op. cit., p. 587): 1/4 D22 − D12 DE = (D22 − D12) D22 + D12 −  (6-52) ln (D2 /D1)








For isosceles triangles and regular polygons, see Sparrow (AIChE J., 8, 599–605 [1962]), Carlson and Irvine (J. Heat Transfer, 83, 441–444 [1961]), Cheng (Proc. Third Int. Heat Transfer Conf., New York, 1, 64–76 [1966]), and Shih (Can. J. Chem. Eng., 45, 285–294 [1967]). The critical Reynolds number for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1,900 < Rec < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558–561 [1966]). In triangular ducts, 1,600 < Rec < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., 11, 106–117 [1972]; Bandopadhayay and Hinwood, J. Fluid Mech., 59, 775–783 [1973]). Nonisothermal Flow For nonisothermal flow of liquids, the friction factor may be increased if the liquid is being cooled or decreased if the liquid is being heated, because of the effect of temperature on viscosity near the wall. In shell and tube heat-exchanger design, the recommended practice is to first estimate f using the bulk mean liquid temperature over the tube length. Then, in laminar flow, the result is divided by (µa/µw)0.23 in the case of cooling or (µa/µw)0.38 in the case of heating. For turbulent flow, f is divided by (µa/µw)0.11 in the case of cooling or (µa /µw)0.17 in case of heating. Here, µa is the viscosity at the average bulk temperature and µw is the viscosity at the average wall temperature (Seider and Tate, Ind. Eng. Chem., 28, 1429–1435 [1936]). In the case of rough commercial pipes, rather than heat-exchanger tubing, it is common for flow to be in the “complete” turbulence regime where f is independent of Re. In such cases, the friction factor should not be corrected for wall temperature. If the liquid density varies with temperature, the average bulk density should be used to calculate the pressure drop from the friction factor. In addition, a (usually small) correction may be applied for acceleration effects by adding the term G2[(1/ρ2) − (1/ρ1)] from the mechanical energy balance to the pressure drop ∆P = P1 − P2, where G is the mass velocity. This acceleration results from small compressibility effects associated with temperature-dependent density. Christiansen and Gordon (AIChE J., 15, 504–507 [1969]) present equations and charts for frictional loss in laminar nonisothermal flow of Newtonian and non-Newtonian liquids heated or cooled with constant wall temperature. Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by ∆T = ∆P/Cv ρ for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about

FLUID DYNAMICS 6°C per 10 MPa. The temperature rise in laminar flow is highly nonuniform, being concentrated near the pipe wall where most of the dissipation occurs. This may result in significant viscosity reduction near the wall, and greatly increased flow or reduced pressure drop, and a flattened velocity profile. Compensation should generally be made for the heat effect when ∆P exceeds 1.4 MPa (203 psi) for adiabatic walls or 3.5 MPa (508 psi) for isothermal walls (Gerard, Steidler, and Appeldoorn, Ind. Eng. Chem. Fundam., 4, 332–339 [1969]). Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of sufficient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to lv = g(z1 − z2). In terms of the friction factor and hydraulic diameter or hydraulic radius, 2 f V 2L f V 2L lv =  =  = g(z1 − z2) (6-53) DH 2RH The hydraulic radius is the cross-sectional area divided by the wetted perimeter, where the wetted perimeter does not include the free surface. Letting S = sin θ = channel slope (elevation loss per unit length of channel, θ = angle between channel and horizontal), Eq. (6-53) reduces to 2gSRH V=  (6-54) f The most often used friction correlation for open channel flows is due to Manning (Trans. Inst. Civ. Engrs. Ireland, 20, 161 [1891]) and is equivalent to 29n2 f= (6-55) RH1/3 where n is the channel roughness, with dimensions of (length)1/6. Table 6-2 gives roughness values for several channel types. For gradual changes in channel cross section and liquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x.



dh h db fV 2(b + 2h)  (1 − Fr) − Fr   = S −  dx b dx 2gbh



(6-56)

For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile TABLE 6-2 Eq. (6-55)

Average Values of n for Manning Formula, Surface

n, m1/6

n, ft1/6

Cast-iron pipe, fair condition Riveted steel pipe Vitrified sewer pipe Concrete pipe Wood-stave pipe Planed-plank flume Semicircular metal flumes, smooth Semicircular metal flumes, corrugated Canals and ditches Earth, straight and uniform Winding sluggish canals Dredged earth channels Natural-stream channels Clean, straight bank, full stage Winding, some pools and shoals Same, but with stony sections Sluggish reaches, very deep pools, rather weedy

0.014 0.017 0.013 0.015 0.012 0.012 0.013 0.028

0.011 0.014 0.011 0.012 0.010 0.010 0.011 0.023

0.023 0.025 0.028

0.019 0.021 0.023

0.030 0.040 0.055 0.070

0.025 0.033 0.045 0.057

SOURCE: Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, p. 7-22. For detailed information, see Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959, pp. 110–123.

6-13

h(x). The dimensionless Froude number is Fr = V 2/gh. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c =  gh; they cannot propagate upstream in supercritical flows. The specific energy Esp is nearly constant. V2 Esp = h +  2g

(6-57)

This equation is cubic in liquid depth. Below a minimum value of Esp there are no real positive roots; above the minimum value there are two positive real roots. At this minimum value of Esp the flow is critical; that is, Fr = 1, V =  gh, and Esp = (3/2)h. Near critical flow conditions, wave motion and sudden depth changes called hydraulic jumps are likely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959) discusses the numerous surface profile shapes which may exist in nonuniform open channel flows. For flow over a sharp-crested weir of width b and height L, from a liquid depth H, the flow rate is given approximately by 2 Q =  Cd b2g(H − L)3/2 3

(6-58)

where Cd ≈ 0.6 is a discharge coefficient. Flow through notched weirs is described under flow meters in Sec. 10 of the Handbook. Non-Newtonian Flow For isothermal laminar flow of timeindependent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate–pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length ∆P/L, the flow rate is given by πD3τw 4τ τ y4 Q= 1 − y +  32µ∞ 3τw 3τ w4





(6-59)

where the wall stress is τw = D∆P/(4L). The velocity profile consists of a central nondeforming plug of radius rP = 2τy /(∆P/L) and an annular deforming region. The velocity profile in the annular region is given by 1 ∆P vz =   (R2 − r2) − τy(R − r) rP ≤ r ≤ R (6-60) µ∞ 4L where r is the radial coordinate and R is the pipe radius. The velocity of the central, nondeforming plug is obtained by setting r = rP in Eq. (6-60). When Q is given and Eq. (6-59) is to be solved for τw and the pressure drop, multiple positive roots for the pressure drop may be found. The root corresponding to τw < τy is physically unrealizable, as it corresponds to rp > R and the pressure drop is insufficient to overcome the yield stress. For a power law fluid, Eq. (6-4), with constant properties K and n, the flow rate is given by





∆P Q=π  2KL



n R  1 + 3n 1/n

(1 + 3n)/n

(6-61)

− r (1 + n)/n]

(6-62)

and the velocity profile by ∆P vz =  2KL



n [R  1 + n 1/n

(1 + n)/n

Similar relations for other non-Newtonian fluids may be found in Govier and Aziz and in Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977). For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the RabinowitschMooney relations give a general relationship for the shear rate at the pipe wall. 8V 1 + 3n′ (6-63) γ˙ w =   D 4n′





where n′ is the slope of a plot of D∆P/(4L) versus 8V/D on logarithmic coordinates, d ln [D∆P/(4L)] n′ =  (6-64) d ln (8V/D)

6-14

FLUID AND PARTICLE DYNAMICS

By plotting capillary viscometry data this way, they can be used directly for pressure drop design calculations, or to construct the rheogram for the fluid. For pressure drop calculation, the flow rate and diameter determine the velocity, from which 8V/D is calculated and D∆P/(4L) read from the plot. For a Newtonian fluid, n′ = 1 and the shear rate at the wall is γ˙ = 8V/D. For a power law fluid, n′ = n. To construct a rheogram, n′ is obtained from the slope of the experimental plot at a given value of 8V/D. The shear rate at the wall is given by Eq. (6-63) and the corresponding shear stress at the wall is τw = D∆P/(4L) read from the plot. By varying the value of 8V/D, the shear rate versus shear stress plot can be constructed. The generalized approach of Metzner and Reed (AIChE J., 1, 434 [1955]) for time-independent non-Newtonian fluids defines a modified Reynolds number as Dn′V 2 − n′ρ ReMR   (6-65) K′8n′ − 1 where K′ satisfies D∆P 8V n′ (6-66)  = K′  4L D With this definition, f = 16/ReMR is automatically satisfied at the value of 8V/D where K′ and n′ are evaluated. Equation (6-66) may be obtained by integration of Eq. (6-64) only when n′ is a constant, as, for example, the cases of Newtonian and power law fluids. For Newtonian fluids, K′ = µ and n′ = 1; for power law fluids, K′ = K[(1 + 3n)/ (4n)]n and n′ = n. For Bingham plastics, K′ and n′ are variable, given as a function of τw (Metzner, Ind. Eng. Chem., 49, 1429–1432 [1957]).



µ∞ K = τ 1w− n′  1 − 4τy /3τw + (τy /τw)4/3





n′

(6-67)

1 − 4τy /(3τw) + (τy /τw) /3 n′ =  1 − (τy /τw)4 4

(6-68)

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter (AIChE J., 7, 445–448 [1961]), Wheeler and Wissler (AIChE J., 11, 207–212 [1965]), Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977), and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). Steady-state, fully developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate–pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. The transition to turbulent flow begins at ReMR in the range of 2,000 to 2,500 (Metzner and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K′ and n′ must be evaluated for the τw condition in question in order to determine ReMR and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963]; 14, 691 [1968]; Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967]; and Govier and Aziz, pp. 213–215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzner and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of ReMR increased to beyond 10,000 for some materials. For turbulent flow of non-Newtonian fluids, the design chart of Dodge and Metzner (AIChE J., 5, 189 [1959]), Fig. 6-11, is most widely used. For Bingham plastic materials in turbulent flow, it is generally assumed that stresses greatly exceed the yield stress, so that the friction factor–Reynolds number relationship for Newtonian fluids applies, with µ∞ substituted for µ. This is equivalent to setting n′ = 1 and τy /τw = 0 in the Dodge-Metzner method, so that ReMR = DVρ/µ∞. Wilson and Thomas (Can. J. Chem. Eng., 63, 539–546 [1985]) give friction factor equations for turbulent flow of power law fluids and Bingham plastic fluids. Power law fluids: 1−n 1+n 1 1  =  + 8.2  + 1.77 ln  1+n 2 fN f 





(6-69)

FIG. 6-11 Fanning friction factor for non-Newtonian flow. The abscissa is defined in Eq. (6-65). (From Dodge and Metzner, Am. Inst. Chem. Eng. J., 5, 189 [1959].)

where fN is the friction factor for Newtonian fluid evaluated at Re = DVρ/µeff where the effective viscosity is 3n + 1 µeff = K  4n



n−1

n−1

 D 8V

(6-70)

Bingham fluids: (1 − ξ)2 1 1  =  + 1.77 ln  + ξ(10 + 0.884ξ) 1+ξ fN f 





(6-71)

where fN is evaluated at Re = DVρ/µ∞ and ξ = τy /τw. Iteration is required to use this equation since τw = fρV 2/2. Drag reduction in turbulent flow can be achieved by adding soluble high molecular weight polymers in extremely low concentration to Newtonian liquids. The reduction in friction is generally believed to be associated with the viscoelastic nature of the solutions effective in the wall region. For a given polymer, there is a minimum molecular weight necessary to initiate drag reduction at a given flow rate, and a critical concentration above which drag reduction will not occur (Kim, Little, and Ting, J. Colloid Interface Sci., 47, 530–535 [1974]). Drag reduction is reviewed by Hoyt (J. Basic Eng., 94, 258–285 [1972]); Little, et al. (Ind. Eng. Chem. Fundam., 14, 283–296 [1975]) and Virk (AIChE J., 21, 625–656 [1975]). At maximum possible drag reduction in smooth pipes, 1 50.73  = −19 log  (6-72) Ref f



or, approximately,

0.58 f= Re0.58



(6-73)

for 4,000 < Re < 40,000. The actual drag reduction depends on the polymer system. For further details, see Virk (ibid.). Economic Pipe Diameter, Turbulent Flow The economic optimum pipe diameter may be computed so that the last increment of investment reduces the operating cost enough to produce the required minimum return on investment. For long cross-country pipelines, alloy pipes of appreciable length and complexity, or pipelines with control valves, detailed analyses of investment and operating costs should be made. Peters and Timmerhaus (Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991) provide a detailed method for determining the economic optimum size. For pipelines of the lengths usually encountered in chemical plants and petroleum refineries, simplified selection charts are often adequate. In many cases there is an economic optimum velocity that is nearly independent of diameter, which may be used to estimate the economic diameter from the flow rate. For low-viscosity liquids in schedule 40 steel pipe, economic optimum velocity is typically in the range of 1.8 to 2.4 m/s (5.9 to 7.9 ft/s). For gases with density ranging

FLUID DYNAMICS from 0.2 to 20 kg/m3 (0.013 to 1.25 lbm/ft3), the economic optimum velocity is about 40 m/s to 9 m/s (131 to 30 ft/s). Charts and rough guidelines for economic optimum size do not apply to multiphase flows. Economic Pipe Diameter, Laminar Flow Pipelines for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated by operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid., Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtonian fluids, see Skelland (Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by collisions between molecules. An approximate expression based on Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]) for the mean free path is 2µ 8RT λ=  (6-74)  p πMw The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = λ/D. Molecular flow is characterized by Kn > 1.0; continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. Vacuum flow is usually described with flow variables different from those used for normal pressures, which often leads to confusion. Pumping speed S is the actual volumetric flow rate of gas through a flow cross section. Throughput Q is the product of pumping speed and absolute pressure. In the SI system, Q has units of Pa⋅m3/s.



Q = Sp

(6-75)

The mass flow rate w is related to the throughput using the ideal gas law. Mw w=Q (6-76) RT Throughput is therefore proportional to mass flow rate. For a given mass flow rate, throughput is independent of pressure. The relation between throughput and pressure drop ∆p = p1 − p2 across a flow element is written in terms of the conductance C. Resistance is the reciprocal of conductance. Conductance has dimensions of volume per time. Q = C∆p (6-77)

TABLE 6-3

6-15

Constants for Circular Annuli

D2 /D1

K

D2 /D1

K

0 0.259 0.500

1.00 1.072 1.154

0.707 0.866 0.966

1.254 1.430 1.675

Conductance equations for several other geometries are given by Ryans and Roper (Process Vacuum System Design and Operation, Chap. 2, McGraw-Hill, New York, 1986). For a circular annulus of outer and inner diameters D1 and D2 and length L, the method of Guthrie and Wakerling (Vacuum Equipment and Techniques, McGrawHill, New York, 1949) may be written (D1 − D2)2(D1 + D2) C = 0.42K  L

RT  M

(6-83)

w

where K is a dimensionless constant with values given in Table 6-3. For a short pipe of circular cross section, the conductance as calculated for an orifice from Eq. (6-82) is multiplied by a correction factor K which may be approximated as (Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938, pp. 306–308) 1 K =  for 0 ≤ L/D ≤ 0.75 (6-84) 1 + (L/D) 1 + 0.8(L/D) K = 2 for L/D > 0.75 (6-85) 1 + 1.90(L/D) + 0.6(L/D) For L/D > 100, the error in neglecting the end correction by using the fully developed pipe flow equation (6-81) is less than 2 percent. For rectangular channels, see Normand (Ind. Eng. Chem., 40, 783–787 [1948]). Yu and Sparrow ( J. Basic Eng., 70, 405–410 [1970]) give a theoretically derived chart for slot seals with or without a sheet located in or passing through the seal, giving mass flow rate as a function of the ratio of seal plate thickness to gap opening. Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]), which uses the parameter X=

2µ RT  π8 Dλ =  p D M

(6-86)

m

where pm is the arithmetic mean absolute pressure. A correction factor F, read from Fig. 6-12 as a function of X, is applied to the conductance

The conductance of a series of flow elements is given by 1 1 1 1 =+++⋅⋅⋅ C C1 C2 C3

(6-78)

while for elements in parallel, C = C1 + C2 + C3 + ⋅ ⋅ ⋅

(6-79)

For a vacuum pump of speed Sp withdrawing from a vacuum vessel through a connecting line of conductance C, the pumping speed at the vessel is SpC S= (6-80) Sp + C Molecular Flow Under molecular flow conditions, conductance is independent of pressure. It is proportional to  T /Mw, with the proportionality constant a function of geometry. For fully developed pipe flow, πD3 RT (6-81) C=  8L Mw For an orifice of area A, RT C = 0.40A  (6-82) Mw





FIG. 6-12 Correction factor for Poiseuille’s equation at low pressures. Curve A: experimental curve for glass capillaries and smooth metal tubes. (From Brown, et al., J. Appl. Phys., 17, 802 [1946].) Curve B: experimental curve for iron pipe (From Riggle, courtesy of E. I. du Pont de Nemours & Co.)

6-16

FLUID AND PARTICLE DYNAMICS

for viscous flow. πD4pm C=F (6-87) 128µL For slip flow through square channels, see Milligan and Wilkerson (J. Eng. Ind., 95, 370–372 [1973]). For slip flow through annuli, see Maegley and Berman (Phys. Fluids, 15, 780–785 [1972]). The pump-down time θ for evacuating a vessel in the absence of air in-leakage is given approximately by V p1 − p0 θ = t ln  (6-88) S0 p2 − p0 where Vt = volume of vessel plus volume of piping between vessel and pump; S0 = system speed as given by Eq. (6-80), assumed independent of pressure; p1 = initial vessel pressure; p2 = final vessel pressure; and p0 = lowest pump intake pressure attainable with the pump in question. See Dushman and Lafferty (Scientific Foundations of Vacuum Technique, 2d ed., Wiley, New York, 1962). The amount of inerts which has to be removed by a pumping system after the pump-down stage depends on the in-leakage of air at the various fittings, connections, and so on. Air leakage is often correlated with system volume and pressure, but this approach introduces uncertainty because the number and size of leaks does not necessily correlate with system volume, and leakage is sensitive to maintenance quality. Ryans and Roper (Process Vacuum System Design and Operation, McGraw-Hill, New York, 1986) present a thorough discussion of air leakage.

 



a fixed quantity, independent of D. This approach tends to be most accurate for a single fitting size and loses accuracy with deviation from this size. For laminar flows, Le/D correlations normally have a size dependence through a Reynolds number term. The other method is the velocity head method. The term V 2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V 2/2g. Thus H is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to ρV 2 where V is a characteristic velocity of the flow. In the velocity head method, the losses are reported as a number of velocity heads K. Then, the engineering Bernoulli equation for an incompressible fluid can be written ρV 2 ρV22 ρV12 p1 − p2 = α2  − α1  + ρg(z2 − z1) + K  2 2 2

where V is the reference velocity upon which the velocity head loss coefficient K is based. For a section of straight pipe, K = 4 fL/D. Contraction and Entrance Losses For a sudden contraction at a sharp-edged entrance to a pipe or sudden reduction in crosssectional area of a channel, as shown in Fig. 6-13a, the loss coefficient based on the downstream velocity V2 is given for turbulent flow in Crane Co. Tech Paper 410 (1980) approximately by



A K = 0.5 1 − 2 A1

FRICTIONAL LOSSES IN PIPELINE ELEMENTS The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contractions, elbows, and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to frictional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. Losses lv for incompressible flow in sections of straight pipe of constant diameter may be calculated as previously described using the Fanning friction factor: ∆P 2 fV 2L lv =  =  (6-89) ρ D where ∆P = drop in equivalent pressure, P = p + ρgz, with p = pressure, ρ = fluid density, g = acceleration of gravity, and z = elevation. Losses in the fittings of a piping network are frequently termed minor losses or miscellaneous losses. These descriptions are misleading because in process piping fitting losses are often much greater than the losses in straight piping sections. Equivalent Length and Velocity Head Methods Two methods are in common use for estimating fitting loss. One, the equivalent length method, reports the losses in a piping element as the length of straight pipe which would have the same loss. For turbulent flows, the equivalent length is usually reported as a number of diameters of pipe of the same size as the fitting connection; Le/D is given as

(a) FIG. 6-13

(b)

(6-90)



(6-91)

Example 5: Entrance Loss Water, ρ = 1,000 kg/m3, flows from a large vessel through a sharp-edged entrance into a pipe at a velocity in the pipe of 2 m/s. The flow is turbulent. Estimate the pressure drop from the vessel into the pipe. With A2 /A1 ∼ 0, the viscous loss coefficient is K = 0.5 from Eq. (6-91). The mechanical energy balance, Eq. (6-16) with V1 = 0 and z2 − z1 = 0 and assuming uniform flow (α2 = 1) becomes ρV22 ρV22 p1 − p2 =  + 0.5  = 4,000 + 2,000 = 6,000 Pa 2 2 Note that the total pressure drop consists of 0.5 velocity heads of frictional loss contribution, and 1 velocity head of velocity change contribution. The frictional contribution is a permanent loss of mechanical energy by viscous dissipation. The acceleration contribution is reversible; if the fluid were subsequently decelerated in a frictionless diffuser, a 4,000 Pa pressure rise would occur.

For a trumpet-shaped rounded entrance, with a radius of rounding greater than about 15 percent of the pipe diameter (Fig. 6-13b), the turbulent flow loss coefficient K is only about 0.1 (Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975, pp. 420–421). Rounding of the inlet prevents formation of the vena contracta, thereby reducing the resistance to flow. For laminar flow the losses in sudden contraction may be estimated for area ratios A2 /A1 < 0.2 by an equivalent additional pipe length Le given by Le /D = 0.3 + 0.04Re

(c)

(6-92)

(d)

Contractions and enlargements: (a) sudden contraction, (b) rounded contraction, (c) sudden enlargement, and (d) uniformly diverging duct.

FLUID DYNAMICS where D is the diameter of the smaller pipe and Re is the Reynolds number in the smaller pipe. For laminar flow in the entrance to rectangular ducts, see Shah (J. Fluids Eng., 100, 177–179 [1978]) and Roscoe (Philos. Mag., 40, 338–351 [1949]). For creeping flow, Re < 1, of power law fluids, the entrance loss is approximately Le/D = 0.3/n (Boger, Gupta, and Tanner, J. Non-Newtonian Fluid Mech., 4, 239–248 [1978]). For viscoelastic fluid flow in circular channels with sudden contraction, a toroidal vortex forms upstream of the contraction plane. Such flows are reviewed by Boger (Ann. Review Fluid Mech., 19, 157–182 [1987]). For creeping flow through conical converging channels, inertial acceleration terms are negligible and the viscous pressure drop ∆p = ρlv may be computed by integration of the differential form of the Hagen-Poiseuille equation Eq. (6-36), provided the angle of convergence is small. The result for a power law fluid is 3n + 1 ∆p = 4K  4n



8V  1 −   
 D  6n tan (α/2)  D n

2

n

1

D2

2

3n

(6-93)

1

where D1 = inlet diameter D2 = exit diameter V2 = velocity at the exit α = total included angle Equation (6-93) agrees with experimental data (Kemblowski and Kiljanski, Chem. Eng. J. (Lausanne), 9, 141–151 [1975]) for α < 11°. For Newtonian liquids, Eq. (6-93) simplifies to





  


32V 1 D 3 ∆p = µ 2  1 − 2 (6-94) D2 6 tan (α/2) D1 For creeping flow through noncircular converging channels, the differential form of the Hagen-Poiseulle equation with equivalent diameter given by Eqs. (6-50) to (6-52) may be used, provided the convergence is gradual. Expansion and Exit Losses For ducts of any cross section, the frictional loss for a sudden enlargement (Fig. 6-13c) with turbulent flow is given by the Borda-Carnot equation: A V12 V12 − V22 lv =  = 1 − 1 2 2 A2





2

(6-95)

where V1 = velocity in the smaller duct V2 = velocity in the larger duct A1 = cross-sectional area of the smaller duct A2 = cross-sectional area of the larger duct Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed ( J. Eng. Power, 98, 327–334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanical energy balance Eq. (6-90), noting that p1 = p2. This exit loss is due to the dissipation of the discharged jet; there is no pressure drop at the exit. For creeping Newtonian flow (Re < 1), the frictional loss due to a sudden enlargement should be obtained from the same equation for a sudden contraction (Eq. [6-92]). Note, however, that Boger, Gupta, and Tanner (ibid.) give an exit friction equivalent length of 0.12 diameter, increasing for power law fluids as the exponent decreases. For laminar flows at higher Reynolds numbers, the pressure drop is twice that given by Eq. (6-95). This results from the velocity profile factor α in the mechanical energy balance being 2.0 for the parabolic laminar velocity profile. If the transition from a small to a large duct of any cross-sectional shape is accomplished by a uniformly diverging duct (see Fig. 6-13d) with a straight axis, the total frictional pressure drop can be computed by integrating the differential form of Eq. (6-89), dlv /dx = 2 f V 2/D over the length of the expansion, provided the total angle α between the diverging walls is less than 7°. For angles between 7 and 45°, the loss coefficient may be estimated as 2.6 sin(α/2) times the loss coefficient for a sudden expansion; see Hooper (Chem. Eng., Nov. 7, 1988). Gibson (Hydraulics and Its Applications, 5th ed., Constable, London 1952, p. 93) recommends multiplying the sudden enlargement loss by 0.13 for 5° < α < 7.5° and by 0.0110α1.22 for 7.5° < α <

6-17

35°. For angles greater than 35 to 45°, the losses are normally considered equal to those for a sudden expansion, although in some cases the losses may be greater. Expanding flow through standard pipe reducers should be treated as sudden expansions. Trumpet-shaped enlargements for turbulent flow designed for constant decrease in velocity head per unit length were found by Gibson (ibid., p. 95) to give 20 to 60 percent less frictional loss than straight taper pipes of the same length. A special feature of expansion flows occurs when viscoelastic liquids are extruded through a die at a low Reynolds number. The extrudate may expand to a diameter several times greater than the die diameter, whereas for a Newtonian fluid the diameter expands only 10 percent. This phenomenon, called die swell, is most pronounced with short dies (Graessley, Glasscock, and Crawley, Trans. Soc. Rheol., 14, 519–544 [1970]). For velocity distribution measurements near the die exit, see Goulden and MacSporran (J. Non-Newtonian Fluid Mech., 1, 183–198 [1976]) and Whipple and Hill (AIChE J., 24, 664–671 [1978]). At high flow rates, the extrudate becomes distorted, suffering melt fracture at wall shear stresses greater than 105 N/m2. This phenomenon is reviewed by Denn (Ann. Review Fluid Mech., 22, 13–34 [1990]). Ramamurthy (J. Rheol., 30, 337–357 [1986]) has found a dependence of apparent stick-slip behavior in melt fracture to be dependent on the material of construction of the die. Fittings and Valves For turbulent flow, the frictional loss for fittings and valves can be expressed by the equivalent length or velocity head methods. As fitting size is varied, K values are relatively more constant than Le/D values, but since fittings generally do not achieve geometric similarity between sizes, K values tend to decrease with increasing fitting size. Table 6-4 gives K values for many types of fittings and valves. Manufacturers of valves, especially control valves, express valve capacity in terms of a flow coefficient Cv, which gives the flow rate through the valve in gal/min of water at 60°F under a pressure drop of 1 lbf/in2. It is related to K by C1d 2 Cv =  (6-96) K  where C1 is a dimensional constant equal to 29.9 and d is the diameter of the valve connections in inches. For laminar flow, data for the frictional loss of valves and fittings are meager (Beck and Miller, J. Am. Soc. Nav. Eng., 56, 62–83 [1944]; Beck, ibid., 56, 235–271, 366–388, 389–395 [1944]; De Craene, Heat. Piping Air Cond., 27[10], 90–95 [1955]; Karr and Schutz, J. Am. Soc. Nav. Eng., 52, 239–256 [1940]; and Kittredge and Rowley, Trans. ASME, 79, 1759–1766 [1957]). The data of Kittredge and Rowley indicate that K is constant for Reynolds numbers above 500 to 2,000, but increases rapidly as Re decreases below 500. Typical values for K for laminar flow Reynolds numbers are shown in Table 6-5. Methods to calculate losses for tee and wye junctions for dividing and combining flow are given by Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990), including effects of Reynolds number, angle between legs, area ratio, and radius. Junctions with more than three legs are also discussed. The sources of data for the loss coefficient charts are Blaisdell and Manson (U.S. Dept. Agric. Res. Serv. Tech. Bull. 1283 [August 1963]) for combining flow and Gardel (Bull. Tech. Suisses Romande, 85[9], 123–130 [1957]; 85[10], 143–148 [1957]) together with additional unpublished data for dividing flow. Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-section bends of constant area, as shown in Fig. 6-14a, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is K = K*CReCoCf

(6-97)

where K*, given in Fig. 6-14b, is the loss coefficient for a smoothwalled bend at a Reynolds number of 106. The Reynolds number correction factor CRe is given in Fig. 6-14c. For 0.7 < r/D < 1 or for K* < 0.4, use the CRe value for r/D = 1. Otherwise, if r/D < 1, obtain CRe from K* CRe =  K* + 0.2(1 − CRe, r/D = 1)

(6-98)

6-18

FLUID AND PARTICLE DYNAMICS

TABLE 6-4 Additional Frictional Loss for Turbulent Flow through Fittings and Valvesa

Type of fitting or valve 45° ell, standardb,c,d,e,f 45° ell, long radiusc 90° ell, standardb,c,e,f,g,h Long radius b,c,d,e Square or miter h 180° bend, close returnb,c,e Tee, standard, along run, branch blanked off e Used as ell, entering rung,i Used as ell, entering branchc,g,i Branching flowi,j,k Coupling c,e Unione Gate valve,b,e,m open e open a open d open Diaphragm valve, open e open a open d open Globe valve,e,m Bevel seat, open a open Composition seat, open a open Plug disk, open e open a open d open Angle valve,b,e open Y or blowoff valve,b,m open Plug cock θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Butterfly valve θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Check valve,b,e,m swing Disk Ball Foot valvee Water meter,h disk Piston Rotary (star-shaped disk) Turbine-wheel a

Additional friction loss, equivalent no. of velocity heads, K 0.35 0.2 0.75 0.45 1.3 1.5 0.4 1.0 1.0 1l 0.04 0.04 0.17 0.9 4.5 24.0 2.3 2.6 4.3 21.0 6.0 9.5 6.0 8.5 9.0 13.0 36.0 112.0 2.0 3.0 0.05 0.29 1.56 17.3 206.0 0.24 0.52 1.54 10.8 118.0 2.0 10.0 70.0 15.0 7.0 15.0 10.0 6.0

Lapple, Chem. Eng., 56(5), 96–104 (1949), general survey reference. “Flow of Fluids through Valves, Fittings, and Pipe,” Tech. Pap. 410, Crane Co., 1969. c Freeman, Experiments upon the Flow of Water in Pipes and Pipe Fittings, American Society of Mechanical Engineers, New York, 1941. d Giesecke, J. Am. Soc. Heat. Vent. Eng., 32, 461 (1926). e Pipe Friction Manual, 3d ed., Hydraulic Institute, New York, 1961. f Ito, J. Basic Eng., 82, 131–143 (1960). g Giesecke and Badgett, Heat. Piping Air Cond., 4(6), 443–447 (1932). h Schoder and Dawson, Hydraulics, 2d ed., McGraw-Hill, New York, 1934, p. 213. i Hoopes, Isakoff, Clarke, and Drew, Chem. Eng. Prog., 44, 691–696 (1948). j Gilman, Heat. Piping Air Cond., 27(4), 141–147 (1955). k McNown, Proc. Am. Soc. Civ. Eng., 79, Separate 258, 1–22 (1953); discussion, ibid., 80, Separate 396, 19–45 (1954). For the effect of branch spacing on junction losses in dividing flow, see Hecker, Nystrom, and Qureshi, Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 103(HY3), 265–279 (1977). l This is pressure drop (including friction loss) between run and branch, based on velocity in the mainstream before branching. Actual value depends on the flow split, ranging from 0.5 to 1.3 if mainstream enters run and from 0.7 to 1.5 if mainstream enters branch. m Lansford, Loss of Head in Flow of Fluids through Various Types of 1a-in. Valves, Univ. Eng. Exp. Sta. Bull. Ser. 340, 1943. b

TABLE 6-5 Additional Frictional Loss for Laminar Flow through Fittings and Valves Additional frictional loss expressed as K Type of fitting or valve

Re = 1,000

500

100

50

90° ell, short radius Gate valve Globe valve, composition disk Plug Angle valve Check valve, swing

0.9 1.2 11 12 8 4

1.0 1.7 12 14 8.5 4.5

7.5 9.9 20 19 11 17

16 24 30 27 19 55

SOURCE: From curves by Kittredge and Rowley, Trans. Am. Soc. Mech. Eng., 79, 1759–1766 (1957).

The correction Co (Fig. 6-14d) accounts for the extra losses due to developing flow in the outlet tangent of the pipe, of length Lo. The total loss for the bend plus outlet pipe includes the bend loss K plus the straight pipe frictional loss in the outlet pipe 4fLo /D. Note that Co = 1 for Lo /D greater than the termination of the curves on Fig. 6-14d, which indicate the distance at which fully developed flow in the outlet pipe is reached. Finally, the roughness correction is frough Cf =  fsmooth

(6-99)

where frough is the friction factor for a pipe of diameter D with the roughness of the bend, at the bend inlet Reynolds number. Similarly, fsmooth is the friction factor for smooth pipe. For Re > 106 and r/D ≥ 1, use the value of Cf for Re = 106. Example 6: Losses with Fittings and Valves It is desired to calculate the liquid level in the vessel shown in Fig. 6-15 required to produce a discharge velocity of 2 m/s. The fluid is water at 20°C with ρ = 1,000 kg/m3 and µ = 0.001 Pa ⋅ s, and the butterfly valve is at θ = 10°. The pipe is 2-in Schedule 40, with an inner diameter of 0.0525 m. The pipe roughness is 0.046 mm. Assuming the flow is turbulent and taking the velocity profile factor α = 1, the engineering Bernoulli equation Eq. (6-16), written between surfaces 1 and 2, where the pressures are both atmospheric and the fluid velocities are 0 and V = 2 m/s, respectively, and there is no shaft work, simplifies to V2 gZ =  + lv 2 Contributing to lv are losses for the entrance to the pipe, the three sections of straight pipe, the butterfly valve, and the 90° bend. Note that no exit loss is used because the discharged jet is outside the control volume. Instead, the V 2/2 term accounts for the kinetic energy of the discharging stream. The Reynolds number in the pipe is DVρ 0.0525 × 2 × 1000 Re =  =  = 1.05 × 105 µ 0.001 From Fig. 6-9 or Eq. (6-38), at %/D = 0.046 × 10−3/0.0525 = 0.00088, the friction factor is about 0.0054. The straight pipe losses are then 4fL V 2 lv(sp) =   D 2



4 × 0.0054 × (1 + 1 + 1) V 2 =   0.0525 2





V2 = 1.23  2 The losses from Table 6-4 in terms of velocity heads K are K = 0.5 for the sudden contraction and K = 0.52 for the butterfly valve. For the 90° standard radius (r/D = 1), the table gives K = 0.75. The method of Eq. (6-94), using Fig. 6-14, gives K = K*CReCoCf



0.0054 = 0.24 × 1.24 × 1.0 ×  0.0044 = 0.37



This value is more accurate than the value in Table 6-4. The value fsmooth = 0.0044 is obtainable either from Eq. (6-37) or Fig. 6-9. The total losses are then V2 V2 lv = (1.23 + 0.5 + 0.52 + 0.37)  = 2.62  2 2

FLUID DYNAMICS

(a)

(b)

(c)

(d)

6-19

Loss coefficients for flow in bends and curved pipes: (a) flow geometry, (b) loss coefficient for a smooth-walled bend at Re = 106, (c) Re correction factor, (d) outlet pipe correction factor. (From D. S. Miller, Internal Flow Systems, 2d ed., BHRA, Cranfield, U.K., 1990.)

FIG. 6-14

and the liquid level Z is





V2 V2 1 V2 Z =   + 2.62  = 3.62  g 2 2 2g 3.62 × 2 =  = 0.73 m 2 × 9.81 2

V2 = 2 m/s m 1

2

90° horizontal bend 1m FIG. 6-15

Tank discharge example.

1m





1 Z

Curved Pipes and Coils For flow through curved pipe or coil, a secondary circulation perpendicular to the main flow called the Dean effect occurs. This circulation increases the friction relative to straight pipe flow and stabilizes laminar flow, delaying the transition Reynolds number to about D Recrit = 2,100 1 + 12  (6-100) Dc where Dc is the coil diameter. Equation (6-100) is valid for 10 < Dc / D < 250. The Dean number is defined as Re De =  (6-101) (Dc /D)1/2 In laminar flow, the friction factor for curved pipe fc may be expressed in terms of the straight pipe friction factor f = 16/Re as (Hart, Chem. Eng. Sci., 43, 775–783 [1988]) De1.5 fc /f = 1 + 0.090  (6-102) 70 + De





6-20

FLUID AND PARTICLE DYNAMICS

For turbulent flow, equations by Ito (J. Basic Eng, 81, 123 [1959]) and Srinivasan, Nandapurkar, and Holland (Chem. Eng. [London] no. 218, CE113-CE119 [May 1968]) may be used, with probable accuracy of 15 percent. Their equations are similar to 0.0073 0.079 fc =  + (6-103) Re0.25 (D c/D ) The pressure drop for flow in spirals is discussed by Srinivasan, et al. (loc. cit.) and Ali and Seshadri (Ind. Eng. Chem. Process Des. Dev., 10, 328–332 [1971]). For friction loss in laminar flow through semicircular ducts, see Masliyah and Nandakumar (AIChE J., 25, 478– 487 [1979]); for curved channels of square cross section, see Cheng, Lin, and Ou (J. Fluids Eng., 98, 41–48 [1976]). For non-Newtonian (power law) fluids in coiled tubes, Mashelkar and Devarajan (Trans. Inst. Chem. Eng. (London), 54, 108–114 [1976]) propose the correlation fc = (9.07 − 9.44n + 4.37n2)(D/Dc)0.5(De′)−0.768 + 0.122n (6-104) where De′ is a modified Dean number given by 1 6n + 2 De′ =   8 n



D  Re D n

MR

(6-105)

c

where ReMR is the Metzner-Reed Reynolds number, Eq. (6-65). This correlation was tested for the range De′ = 70 to 400, D/Dc = 0.01 to 0.135, and n = 0.35 to 1. See also Oliver and Asghar (Trans. Inst. Chem. Eng. [London], 53, 181–186 [1975]). Screens The pressure drop for incompressible flow across a screen of fractional free area α may be computed from ρV 2 ∆p = K  2

(6-106)

where ρ = fluid density V = superficial velocity based upon the gross area of the screen K = velocity head loss 1−α   α

1 K = 2 C

2

2

(6-107)

The discharge coefficient for the screen C with aperture Ds is given as a function of screen Reynolds number Re = Ds(V/α)ρ/µ in Fig. 6-16 for plain square-mesh screens, α = 0.14 to 0.79. This curve fits most of the data within 20 percent. In the laminar flow region, Re < 20, the discharge coefficient can be computed from

e C = 0.1R

FIG. 6-16

& Co.)

(6-108)

Coefficients greater than 1.0 in Fig. 6-16 probably indicate partial pressure recovery downstream of the minimum aperture, due to rounding of the wires. Grootenhuis (Proc. Inst. Mech. Eng. [London], A168, 837–846 [1954]) presents data which indicate that for a series of screens, the total pressure drop equals the number of screens times the pressure drop for one screen, and is not affected by the spacing between screens or their orientation with respect to one another, and presents a correlation for frictional losses across plain square-mesh screens and sintered gauzes. Armour and Cannon (AIChE J., 14, 415–420 [1968]) give a correlation based on a packed bed model for plain, twill, and “dutch” weaves. For losses through monofilament fabrics see Pedersen (Filtr. Sep., 11, 586–589 [1975]). For screens inclined at an angle θ, use the normal velocity component V ′ V′ = V cos θ

(6-109)

(Carothers and Baines, J. Fluids Eng., 97, 116–117 [1975]) in place of V in Eq. (6-106). This applies for Re > 500, C = 1.26, α ≤ 0.97, and 0 < θ < 45°, for square-mesh screens and diamond-mesh netting. Screens inclined at an angle to the flow direction also experience a tangential stress. For non-Newtonian fluids in slow flow, friction loss across a square-woven or full-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between adjacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Sci., 18, 408–415 [1978]). For screen stacks, the losses of individual screens should be summed. JET BEHAVIOR A free jet, upon leaving an outlet, will entrain the surrounding fluid, expand, and decelerate. To a first approximation, total momentum is conserved as jet momentum is transferred to the entrained fluid. For practical purposes, a jet is considered free when its cross-sectional area is less than one-fifth of the total cross-sectional flow area of the region through which the jet is flowing (Elrod, Heat. Piping Air Cond., 26[3], 149–155 [1954]), and the surrounding fluid is the same as the jet fluid. A turbulent jet in this discussion is considered to be a free jet with Reynolds number greater than 2,000. Additional discussion on the relation between Reynolds number and turbulence in jets is given by Elrod (ibid.). Abramowicz (The Theory of Turbulent Jets, MIT Press, Cambridge, 1963) and Rajaratnam (Turbulent Jets, Elsevier, Amsterdam, 1976) provide thorough discourses on turbulent jets. Hussein, et al. (J. Fluid Mech., 258, 31–75 [1994]) give extensive

Screen discharge coefficients, plain square-mesh screens. (Courtesy of E. I. du Pont de Nemours

FLUID DYNAMICS TABLE 6-6

6-21

Turbulent Free-Jet Characteristics

Where both jet fluid and entrained fluid are air Rounded-inlet circular jet Longitudinal distribution of velocity along jet center line*† x Vc D0 for 7 <  < 100  = K V0 D0 x K=5 for V0 = 2.5 to 5.0 m/s K = 6.2 for V0 = 10 to 50 m/s Radial distribution of longitudinal velocity† FIG. 6-17

Configuration of a turbulent free jet.





Vc r log  = 40  Vr x

2

x for 7 <  < 100 D0

Jet angle°†

velocity data for a free jet, as well as an extensive discussion of free jet experimentation and comparison of data with momentum conservation equations. A turbulent free jet is normally considered to consist of four flow regions (Tuve, Heat. Piping Air Cond., 25[1], 181–191 [1953]; Davies, Turbulence Phenomena, Academic, New York, 1972) as shown in Fig. 6-17: 1. Region of flow establishment—a short region whose length is about 6.4 nozzle diameters. The fluid in the conical core of the same length has a velocity about the same as the initial discharge velocity. The termination of this potential core occurs when the growing mixing or boundary layer between the jet and the surroundings reaches the centerline of the jet. 2. A transition region that extends to about 8 nozzle diameters. 3. Region of established flow—the principal region of the jet. In this region, the velocity profile transverse to the jet is self-preserving when normalized by the centerline velocity. 4. A terminal region where the residual centerline velocity reduces rapidly within a short distance. For air jets, the residual velocity will reduce to less than 0.3 m/s, (1.0 ft/s) usually considered still air. Several references quote a length of 100 nozzle diameters for the length of the established flow region. However, this length is dependent on initial velocity and Reynolds number. Table 6-6 gives characteristics of rounded-inlet circular jets and rounded-inlet infinitely wide slot jets (aspect ratio > 15). The information in the table is for a homogeneous, incompressible air system under isothermal conditions. The table uses the following nomenclature: B0 = slot height D0 = circular nozzle opening q = total jet flow at distance x q0 = initial jet flow rate r = radius from circular jet centerline y = transverse distance from slot jet centerline Vc = centerline velocity Vr = circular jet velocity at r Vy = velocity at y Witze (Am. Inst. Aeronaut. Astronaut. J., 12, 417–418 [1974]) gives equations for the centerline velocity decay of different types of subsonic and supersonic circular free jets. Entrainment of surrounding fluid in the region of flow establishment is lower than in the region of established flow (see Hill, J. Fluid Mech., 51, 773–779 [1972]). Data of Donald and Singer (Trans. Inst. Chem. Eng. [London], 37, 255–267 [1959]) indicate that jet angle and the coefficients given in Table 6-6 depend upon the fluids; for a water system, the jet angle for a circular jet is 14° and the entrainment ratio is about 70 percent of that for an air system. Most likely these variations are due to Reynolds number effects which are not taken into account in Table 6-6. Rushton (AIChE J., 26, 1038–1041 [1980]) examined available published results for circular jets and found that the centerline velocity decay is given by D0 Vc (6-110)  = 1.41Re0.135  V0 x where Re = D0V0ρ/µ is the initial jet Reynolds number. This result corresponds to a jet angle tan α/2 proportional to Re−0.135.



x for  < 100 D0

α  20° Entrainment of surrounding fluid‡

x for 7 <  < 100 D0

q x  = 0.32  q0 D0

Rounded-inlet, infinitely wide slot jet Longitudinal distribution of velocity along jet centerline‡



x B0 0.5 Vc for 5 <  < 2,000 and V0 = 12 to 55 m/s  = 2.28  V0 B0 x Transverse distribution of longitudinal velocity‡





y Vc log  = 18.4  Vx x

2

x for 5 <  < 2,000 B0

Jet angle‡ α is slightly larger than that for a circular jet Entrainment of surrounding fluid‡



q x  = 0.62  q0 B0

0.5

x for 5 <  < 2,000 B0

*Nottage, Slaby, and Gojsza, Heat, Piping Air Cond., 24(1), 165–176 (1952). †Tuve, Heat, Piping Air Cond., 25(1), 181–191 (1953). ‡Albertson, Dai, Jensen, and Rouse, Trans. Am. Soc. Civ. Eng., 115, 639–664 (1950), and Discussion, ibid., 115, 665–697 (1950).

Characteristics of rectangular jets of various aspect ratios are given by Elrod (Heat., Piping, Air Cond., 26[3], 149–155 [1954]). For slot jets discharging into a moving fluid, see Weinstein, Osterle, and Forstall (J. Appl. Mech., 23, 437–443 [1967]). Coaxial jets are discussed by Forstall and Shapiro (J. Appl. Mech., 17, 399–408 [1950]), and double concentric jets by Chigier and Beer (J. Basic Eng., 86, 797–804 [1964]). Axisymmetric confined jets are described by Barchilon and Curtet (J. Basic Eng., 777–787 [1964]). Restrained turbulent jets of liquid discharging into air are described by Davies (Turbulence Phenomena, Academic, New York, 1972). These jets are inherently unstable and break up into drops after some distance. Lienhard and Day (J. Basic Eng. Trans. AIME, p. 515 [September 1970]) discuss the breakup of superheated liquid jets which flash upon discharge. Density gradients affect the spread of a single-phase jet. A jet of lower density than the surroundings spreads more rapidly than a jet of the same density as the surroundings, and, conversely, a denser jet spreads less rapidly. Additional details are given by Keagy and Weller (Proc. Heat Transfer Fluid Mech. Inst., ASME, pp. 89–98, June 22–24 [1949]) and Cleeves and Boelter (Chem. Eng. Prog., 43, 123–134 [1947]). Few experimental data exist on laminar jets (see Gutfinger and Shinnar, AIChE J., 10, 631–639 [1964]). Theoretical analysis for velocity distributions and entrainment ratios are available in Schlichting and in Morton (Phys. Fluids, 10, 2120–2127 [1967]). Theoretical analyses of jet flows for power law non-Newtonian fluids are given by Vlachopoulos and Stournaras (AIChE J., 21, 385–388 [1975]), Mitwally (J. Fluids Eng., 100, 363 [1978]), and Sridhar and Rankin (J. Fluids Eng., 100, 500 [1978]).

6-22

FLUID AND PARTICLE DYNAMICS Vena contracta

.90 .85 Co, orifice number

Pipe area A Orifice area A o FIG. 6-18

Flow through an orifice.

.80

Data scatter ±2%

.75 .70

FLOW THROUGH ORIFICES

.65

Section 10 of this Handbook describes the use of orifice meters for flow measurement. In addition, orifices are commonly found within pipelines as flow-restricting devices, in perforated pipe distributing and return manifolds, and in perforated plates. Incompressible flow through an orifice in a pipeline, as shown in Fig. 6-18, is commonly described by the following equation for flow rate Q in terms of the pressures P1, P2, and P3; the orifice area A o; the pipe cross-sectional area A; and the density ρ. 2(P1P2) Q vo Ao Co Ao  [1  (A o /A)2]



Co A o



(1  A /A) [1(A /A) ] 2(P1P3) o

o

2

(6-111)

The velocity based on the hole area is vo. The pressure P1 is the pressure upstream of the orifice, typically about 1 pipe diameter upstream, the pressure P2 is the pressure at the vena contracta, where the flow passes through a minimum area which is less than the orifice area, and the pressure P3 is the pressure downstream of the vena contracta after pressure recovery associated with deceleration of the fluid. The velocity of approach factor 1  (A o /A)2 accounts for the kinetic energy approaching the orifice, and the orifice coefficient or discharge coefficient Co accounts for the vena contracta. The location of the vena contracta varies with A0 /A, but is about 0.7 pipe diameter for Ao /A , 0.25. The factor 1  Ao /A accounts for pressure recovery. Pressure recovery is complete by about 4 to 8 pipe diameters downstream of the orifice. The permanent pressure drop is P1  P3. When the orifice is at the end of pipe, discharging directly into a large chamber, there is negligible pressure recovery, the permanent pressure drop is P1  P2, and the last equality in Eq. (6-111) does not apply. Instead, P2 P3. Equation (6-111) may also be used for flow across a perforated plate with open area A o and total area A. The location of the vena contracta and complete recovery would scale not with the vessel or pipe diameter in which the plate is installed, but with the hole diameter and pitch between holes. The orifice coefficient has a value of about 0.62 at large Reynolds numbers (Re = DoVoρ/µ > 20,000), although values ranging from 0.60 to 0.70 are frequently used. At lower Reynolds numbers, the orifice coefficient varies with both Re and with the area or diameter ratio. See Sec. 10 for more details. When liquids discharge vertically downward from a pipe of diameter Dp, through orifices into gas, gravity increases the discharge coefficient. Figure 6-19 shows this effect, giving the discharge coefficient in terms of a modified Froude number, Fr = ∆p/(gDp). The orifice coefficient deviates from its value for sharp-edged orifices when the orifice wall thickness exceeds about 75 percent of the orifice diameter. Some pressure recovery occurs within the orifice and the orifice coefficient increases. Pressure drop across segmental orifices is roughly 10 percent greater than that for concentric circular orifices of the same open area. COMPRESSIBLE FLOW Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows

0

50

100 150 ∆p , Froude number ρgDp

200

Orifice coefficient vs. Froude number. (Courtesy E. I. duPont de Nemours & Co.)

FIG. 6-19

containing gases. Liquid flows are normally considered incompressible, except for certain calculations involved in hydraulic transient analysis (see following) where compressibility effects are important even for nearly incompressible liquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vols. I and II, Ronald Press, New York [1953]) and Zucrow and Hofmann (Gas Dynamics, vols. I and II, Wiley, New York [1976]). In chemical process applications, one-dimensional gas flows through nozzles or orifices and in pipelines are the most important applications of compressible flow. Multidimensional external flows are of interest mainly in aerodynamic applications. Mach Number and Speed of Sound The Mach number M = V/c is the ratio of fluid velocity, V, to the speed of sound or acoustic velocity, c. The speed of sound is the propagation velocity of infinitesimal pressure disturbances and is derived from a momentum balance. The compression caused by the pressure wave is adiabatic and frictionless, and therefore isentropic. c=

∂p  

∂ρ

(6-112)

s

The derivative of pressure p with respect to density ρ is taken at constant entropy s. For an ideal gas, ∂p

=  ∂ρ M kRT

s

where

w

k = ratio of specific heats, Cp /Cv R = universal gas constant (8,314 J/kgmol K) T = absolute temperature Mw = molecular weight

Hence for an ideal gas, c=

kRT  M

(6-113)

w

Most often, the Mach number is calculated using the speed of sound evaluated at the local pressure and temperature. When M = 1, the flow is critical or sonic and the velocity equals the local speed of sound. For subsonic flow M < 1 while supersonic flows have M > 1. Compressibility effects are important when the Mach number exceeds 0.1 to 0.2. A common error is to assume that compressibility effects are always negligible when the Mach number is small. The proper assessment of whether compressibility is important should be based on relative density changes, not on Mach number. Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibility

FLUID DYNAMICS effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with ρ = pMw /RT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant friction factor f over a length L of a channel of constant cross section and hydraulic diameter DH, yields,



 

RT 4 fL p1 p21 − p22 = G2   + 2 ln  Mw DH p2

(6-114)

where the mass velocity G = w/A = ρV is the mass flow rate per unit cross-sectional area of the channel. The logarithmic term on the righthand side accounts for the pressure change caused by acceleration of gas as its density decreases, while the first term is equivalent to the calculation of frictional losses using the density evaluated at the average pressure (p1 + p2)/2. Solution of Eq. (6-114) for G and differentiation with respect to p2 reveals a maximum mass flux Gmax = p2 Mw/ (R T) and a corresponding exit velocity V2,max =  R T /Mw and exit Mach number M2 = 1/k. This apparent choking condition, though often cited, is not physically meaningful for isothermal flow because at such high velocities, and high rates of expansion, isothermal conditions are not maintained. Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. Figure 6-20 illustrates adiabatic discharge of a perfect gas through a frictionless nozzle from a large chamber where velocity is effectively zero. A perfect gas obeys the ideal gas law ρ = pMw /RT and also has constant specific heat. The subscript 0 refers to the stagnation conditions in the chamber. More generally, stagnation conditions refer to the conditions which would be obtained by isentropically decelerating a gas flow to zero velocity. The minimum area section, or throat, of the nozzle is at the nozzle exit. The flow through the nozzle is isentropic because it is frictionless (reversible) and adiabatic. In terms of the exit Mach number M1 and the upstream stagnation conditions, the flow conditions at the nozzle exit are given by k − 1 2 k / (k − 1) p0 (6-115)  = 1 +  M1 p1 2





T0 k−1 2  = 1 +  M1 T1 2

(6-116)

ρ0 k − 1 2 1 / (k − 1) (6-117)  = 1 +  M1 ρ1 2 The mass velocity G = w/A, where w is the mass flow rate and A is the nozzle exit area, at the nozzle exit is given by







kMw M1 G = p0  (6-118)  RT0 k − 1 2 (k + 1) / 2(k − 1) 1 +  M1 2 These equations are consistent with the isentropic relations for a per-





6-23

fect gas p/p0 = (ρ/ρ0)k, T/T0 = (p/p0)(k − 1)/k. Equation (6-116) is valid for adiabatic flows with or without friction; it does not require isentropic flow. However, Eqs. (6-115) and (6-117) do require isentropic flow. The exit Mach number M1 may not exceed unity. At M1 = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by *; sonic exit conditions are found by substituting M1 = M*1 = 1 into Eqs. (6-115) to (6-118).



p* 2 =  p0 k+1

k/(k − 1)



(6-119)

T* 2 = T0 k + 1

(6-120)

ρ* 2 =  ρ0 k+1



G* = p0

1/(k − 1)



(6-121)

2 kM   

k + 1





RT (k + 1)/(k − 1)

w

(6-122)

0

Note that under choked conditions, the exit velocity is V = V* = c* =

kR T */ M w, not kR T 0/M w. Sonic velocity must be evaluated at the

exit temperature. For air, with k = 1.4, the critical pressure ratio p*/p0 is 0.5285 and the critical temperature ratio T*/T0 = 0.8333. Thus, for air discharging from 300 K, the temperature drops by 50 K (90 R). This large temperature decrease results from the conversion of internal energy into kinetic energy and is reversible. As the discharged jet decelerates in the external stagant gas, it recovers its initial enthalpy. When it is desired to determine the discharge rate through a nozzle from upstream pressure p0 to external pressure p2, Equations (6-115) through (6-122) are best used as follows. The critical pressure is first determined from Eq. (6-119). If p2 > p*, then the flow is subsonic (subcritical, unchoked). Then p1 = p2 and M1 may be obtained from Eq. (6-115). Substitution of M1 into Eq. (6-118) then gives the desired mass velocity G. Equations (6-116) and (6-117) may be used to find the exit temperature and density. On the other hand, if p2 ≤ p*, then the flow is choked and M1 = 1. Then p1 = p*, and the mass velocity is G* obtained from Eq. (6-122). The exit temperature and density may be obtained from Eqs. (6-120) and (6-121). When the flow is choked, G = G* is independent of external downstream pressure. Reducing the downstream pressure will not increase the flow. The mass flow rate under choking conditions is directly proportional to the upstream pressure. Example 7: Flow through Frictionless Nozzle Air at p0 and temperature T0 = 293 K discharges through a frictionless nozzle to atmospheric pressure. Compute the discharge mass flux G, the pressure, temperature, Mach number, and velocity at the exit. Consider two cases: (1) p0 = 7 × 105 Pa absolute, and (2) p0 = 1.5 × 105 Pa absolute. 1. p0 = 7.0 × 105 Pa. For air with k = 1.4, the critical pressure ratio from Eq. (6-119) is p*/p0 = 0.5285 and p* = 0.5285 × 7.0 × 105 = 3.70 × 105 Pa. Since this is greater than the external atmospheric pressure p2 = 1.01 × 105 Pa, the flow is choked and the exit pressure is p1 = 3.70 × 105 Pa. The exit Mach number is 1.0, and the mass flux is equal to G* given by Eq. (6-118). G* = 7.0 × 105 ×

2 1.4 × 29 = 1,650 kg/m ⋅ s  

 1.4 + 1





8314 × 293 (1.4 + 1)/(1.4 − 1)

2

The exit temperature, since the flow is choked, is

p0

p1

p2

 



T* 2 T* =  T0 =  × 293 = 244 K T0 1.4 + 1

T */ M w = 313 m/s. The exit velocity is V = Mc = c* = kR 2. p0 = 1.5 × 105 Pa. In this case p* = 0.79 × 105 Pa, which is less than p2. 5 Hence, p1 = p2 = 1.01 × 10 Pa. The flow is unchoked (subsonic). Equation (6-115) is solved for the Mach number. 1.5 × 105 1.4 − 1 5 = 1 +  M 21 1.01 × 10 2



FIG. 6-20

Isentropic flow through a nozzle.

M1 = 0.773

1.4/(1.4 − 1)



6-24

FLUID AND PARTICLE DYNAMICS

Substitution into Eq. (6-118) gives G. G = 1.5 × 105 ×

1.4 × 29

 8,314 × 293

0.773 ×  = 337 kg/m2 ⋅ s (1.4 + 1)/2(1.4 − 1) 1.4 − 1 1 +  × 0.7732 2 The exit temperature is found from Eq. (6-116) to be 261.6 K or −11.5°C. The exit velocity is

 

V = Mc = 0.773 ×





1.4 × 8314 × 261.6  = 250 m/s

29

Adiabatic Flow with Friction in a Duct of Constant Cross Section Integration of the differential forms of the continuity, momentum, and total energy equations for a perfect gas, assuming a constant friction factor, leads to a tedious set of simultaneous algebraic equations. These may be found in Shapiro (Dynamics and Thermodynamics of Compressible Fluid Flow, vol. I, Ronald Press, New York, 1953) or Zucrow and Hofmann (Gas Dynamics, vol. I, Wiley, New York, 1976). Lapple’s (Trans. AIChE., 39, 395–432 [1943]) widely cited graphical presentation of the solution of these equations contained a subtle error, which was corrected by Levenspiel (AIChE J., 23, 402–403 [1977]). Levenspiel’s graphical solutions are presented in Fig. 6-21. These charts refer to the physical situation illustrated in Fig. 6-22, where a perfect gas discharges from stagnation conditions in a large chamber through an isentropic nozzle followed by a duct of length L. The resistance parameter is N = 4fL/DH, where f = Fanning friction factor and DH = hydraulic diameter. The exit Mach number M2 may not exceed unity. M2 = 1 corresponds to choked flow; sonic conditions may exist only at the pipe exit. The mass velocity G* in the charts is the choked mass flux for an isentropic nozzle given by Eq. (6-118). For a pipe of finite length, the mass flux is less than G* under choking conditions. The curves in Fig. 6-21 become vertical at the choking point, where flow becomes independent of downstream pressure. The equations for nozzle flow, Eqs. (6-114) through (6-118), remain valid for the nozzle section even in the presence of the discharge pipe. Equations (6-116) and (6-120), for the temperature variation, may also be used for the pipe, with M2, p2 replacing M1, p1 since they are valid for adiabatic flow, with or without friction. The graphs in Fig. 6-21 are based on accurate calculations, but are difficult to interpolate precisely. While they are quite useful for rough estimates, precise calculations are best done using the equations for one-dimensional adiabatic flow with friction, which are suitable for computer programming. Let subscripts 1 and 2 denote two points along a pipe of diameter D, point 2 being downstream of point 1. From a given point in the pipe, where the Mach number is M, the additional length of pipe required to accelerate the flow to sonic velocity (M = 1) is denoted Lmax and may be computed from

4fLmax 1 − M 2 k + 1 +  ln = D kM 2 2k



k+1  M 2 2  k−1 1 +  M 2 2



(6-123)

With L = length of pipe between points 1 and 2, the change in Mach number may be computed from



4fL 4 fLmax =  D D

−  D 4fLmax

1

(6-124)

2

Equations (6-116) and (6-113), which are valid for adiabatic flow with friction, may be used to determine the temperature and speed of sound at points 1 and 2. Since the mass flux G = ρv = ρcM is constant, and ρ = PMw /RT, the pressure at point 2 (or 1) can be found from G and the pressure at point 1 (or 2). The additional frictional losses due to pipeline fittings such as elbows may be added to the velocity head loss N = 4fL/DH using the same velocity head loss values as for incompressible flow. This works well for fittings which do not significantly reduce the channel crosssectional area, but may cause large errors when the flow area is greatly

reduced, as, for example, by restricting orifices. Compressible flow across restricting orifices is discussed in Sec. 10 of this Handbook. Similarly, elbows near the exit of a pipeline may choke the flow even though the Mach number is less than unity due to the nonuniform velocity profile in the elbow. For an abrupt contraction rather than rounded nozzle inlet, an additional 0.5 velocity head should be added to N. This is a reasonable approximation for G, but note that it allocates the additional losses to the pipeline, even though they are actually incurred in the entrance. It is an error to include one velocity head exit loss in N. The kinetic energy at the exit is already accounted for in the integration of the balance equations. Example 8: Compressible Flow with Friction Losses Calculate the discharge rate of air to the atmosphere from a reservoir at 106 Pa gauge and 20°C through 10 m of straight 2-in Schedule 40 steel pipe (inside diameter = 0.0525 m), and 3 standard radius, flanged 90° elbows. Assume 0.5 velocity heads lost for the elbows. For commercial steel pipe, with a roughness of 0.046 mm, the friction factor for fully rough flow is about 0.0047, from Eq. (6-38) or Fig. 6-9. It remains to be verified that the Reynolds number is sufficiently large to assume fully rough flow. Assuming an abrupt entrance with 0.5 velocity heads lost, 10 N = 4 × 0.0047 ×  + 0.5 + 3 × 0.5 = 5.6 0.0525 The pressure ratio p3 /p0 is 1.01 × 105 = 0.092  (1 × 106 + 1.01 × 105) From Fig. 6-21b at N = 5.6, p3 /p0 = 0.092 and k = 1.4 for air, the flow is seen to be choked. At the choke point with N = 5.6 the critical pressure ratio p2 /p0 is about 0.25 and G/G* is about 0.48. Equation (6-122) gives G* = 1.101 × 106 ×

2 1.4 × 29 = 2,600 kg/m ⋅ s  

 1.4 + 1





8,314 × 293.15 (1.4 + 1)/(1.4 − 1)

2

Multiplying by G/G* = 0.48 yields G = 1,250 kg/m2 ⋅ s. The discharge rate is w = GA = 1,250 × π × 0.05252/4 = 2.7 kg/s. Before accepting this solution, the Reynolds number should be checked. At the pipe exit, the temperature is given by Eq. (6-120) since the flow is choked. Thus, T2 = T* = 244.6 K. The viscosity of air at this temperature is about 1.6 × 10−5 Pa ⋅ s. Then DVρ DG 0.0525 × 1,250 Re =  =  =  = 4.1 × 106 µ µ 1.6 × 10−5 At the beginning of the pipe, the temperature is greater, giving greater viscosity and a Reynolds number of 3.6 × 106. Over the entire pipe length the Reynolds number is very large and the fully rough flow friction factor choice was indeed valid.

Once the mass flux G has been determined, Fig. 6-21a or 6-21b can be used to determine the pressure at any point along the pipe, simply by reducing 4fL/DH and computing p2 from the figures, given G, instead of the reverse. Charts for calculation between two points in a pipe with known flow and known pressure at either upstream or downstream locations have been presented by Loeb (Chem. Eng., 76[5], 179–184 [1969]) and for known downstream conditions by Powley (Can. J. Chem. Eng., 36, 241–245 [1958]). Convergent/Divergent Nozzles (De Laval Nozzles) During frictionless adiabatic one-dimensional flow with changing crosssectional area A the following relations are obeyed: dA dp 1 − M 2 dρ dV (6-125)  = 2 (1 − M 2) =   = −(1 − M 2)  A ρV M2 ρ V Equation (6-125) implies that in converging channels, subsonic flows are accelerated and the pressure and density decrease. In diverging channels, subsonic flows are decelerated as the pressure and density increase. In subsonic flow, the converging channels act as nozzles and diverging channels as diffusers. In supersonic flows, the opposite is true. Diverging channels act as nozzles accelerating the flow, while converging channels act as diffusers decelerating the flow. Figure 6-23 shows a converging/diverging nozzle. When p2 /p0 is less than the critical pressure ratio (p*/p0), the flow will be subsonic in the converging portion of the nozzle, sonic at the throat, and supersonic in the diverging portion. At the throat, where the flow is critical and the velocity is sonic, the area is denoted A*. The cross-sectional

FLUID DYNAMICS

6-25

(a)

(b) Design charts for adiabatic flow of gases; (a) useful for finding the allowable pipe length for given flow rate; (b) useful for finding the discharge rate in a given piping system. (From Levenspiel, Am. Inst. Chem. Eng. J., 23, 402 [1977].) FIG. 6-21

area and pressure vary with Mach number along the converging/ diverging flow path according to the following equations for isentropic flow of a perfect gas:





p0 k−1  = 1 +  M2 p 2



(k + 1) / 2(k − 1)



A 1 2 k−1  =   1 +  M2 A* M k + 1 2

L p0

p2

p3

D

k / (k − 1)



p1

(6-126) (6-127)

FIG. 6-22

entrance.

Adiabatic compressible flow in a pipe with a well-rounded

6-26

FIG. 6-23

FLUID AND PARTICLE DYNAMICS

Converging/diverging nozzle.

The temperature obeys the adiabatic flow equation for a perfect gas, k−1 T0 (6-128)  = 1 +  M2 T 2 Equation (6-128) does not require frictionless (isentropic) flow. The sonic mass flux through the throat is given by Eq. (6-122). With A set equal to the nozzle exit area, the exit Mach number, pressure, and temperature may be calculated. Only if the exit pressure equals the ambient discharge pressure is the ultimate expansion velocity reached in the nozzle. Expansion will be incomplete if the exit pressure exceeds the ambient discharge pressure; shocks will occur outside the nozzle. If the calculated exit pressure is less than the ambient discharge pressure, the nozzle is overexpanded and compression shocks within the expanding portion will result. The shape of the converging section is a smooth trumpet shape similar to the simple converging nozzle. However, special shapes of the diverging section are required to produce the maximum supersonic exit velocity. Shocks result if the divergence is too rapid and excessive boundary layer friction occurs if the divergence is too shallow. See Liepmann and Roshko (Elements of Gas Dynamics, Wiley, New York, 1957, p. 284). If the nozzle is to be used as a thrust device, the diverging section can be conical with a total included angle of 30° (Sutton, Rocket Propulsion Elements, 2d ed., Wiley, New York, 1956). To obtain large exit Mach numbers, slot-shaped rather than axisymmetric nozzles are used. MULTIPHASE FLOW Multiphase flows, even when restricted to simple pipeline geometry, are in general quite complex, and several features may be identified which make them more complicated than single-phase flow. Flow pattern description is not merely an identification of laminar or turbulent flow. The relative quantities of the phases and the topology of the interfaces must be described. Because of phase density differences, vertical flow patterns are different from horizontal flow patterns, and horizontal flows are not generally axisymmetric. Even when phase equilibrium is achieved by good mixing in two-phase flow, the changing equilibrium state as pressure drops with distance, or as heat is added or lost, may require that interphase mass transfer, and changes in the relative amounts of the phases, be considered. Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969) and Govier and Aziz present mass, momentum, mechanical energy, and total energy balance equations for two-phase flows. These equations are based on one-dimensional behavior for each phase. Such equations, for the most part, are used as a framework in which to interpret experimental data. Reliable prediction of multiphase flow behavior generally requires use of data or correlations. Two-fluid modeling, in which the full three-dimensional microscopic (partial differential) equations of motion are written for each phase, treating each as a continuum, occupying a volume fraction which is a continuous function of position, is a rapidly developing technique made possible by improved computational methods. For some relatively simple examples not requiring numerical computation, see Pearson (Chem. Engr. Sci., 49, 727–732 [1994]). Constitutive equations for two-fluid models are not yet sufficiently robust for accurate general-purpose two-phase flow computation, but may be quite good for particular classes of flows.

Liquids and Gases For cocurrent flow of liquids and gases in vertical (upflow), horizontal, and inclined pipes, a very large literature of experimental and theoretical work has been published, with less work on countercurrent and cocurrent vertical downflow. Much of the effort has been devoted to predicting flow patterns, pressure drop, and volume fractions of the phases, with emphasis on fully developed flow. In practice, many two-phase flows in process plants are not fully developed. The most reliable methods for fully developed gas/liquid flows use mechanistic models to predict flow pattern, and use different pressure drop and void fraction estimation procedures for each flow pattern. Such methods are too lengthy to include here, and are well suited to incorporation into computer programs; commercial codes for gas/liquid pipeline flows are available. Some key references for mechanistic methods for flow pattern transitions and flow regime– specific pressure drop and void fraction methods include Taitel and Dukler (AIChE J., 22, 47–55 [1976]), Barnea, et al. (Int. J. Multiphase Flow, 6, 217–225 [1980]), Barnea (Int. J. Multiphase Flow, 12, 733–744 [1986]), Taitel, Barnea, and Dukler (AIChE J., 26, 345–354 [1980]), Wallis (One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969), and Dukler and Hubbard (Ind. Eng. Chem. Fundam., 14, 337–347 [1975]). For preliminary or approximate calculations, flow pattern maps and flow regime–independent empirical correlations, are simpler and faster to use. Such methods for horizontal and vertical flows are provided in the following. In horizontal pipe, flow patterns for fully developed flow have been reported in numerous studies. Transitions between flow patterns are gradual, and subjective owing to the visual interpretation of individual investigators. In some cases, statistical analysis of pressure fluctuations has been used to distinguish flow patterns. Figure 6-24 (Alves, Chem. Eng. Progr., 50, 449–456 [1954]) shows seven flow patterns for horizontal gas/liquid flow. Bubble flow is prevalent at high ratios of liquid to gas flow rates. The gas is dispersed as bubbles which move at velocity similar to the liquid and tend to concentrate near the top of the pipe at lower liquid velocities. Plug flow describes a pattern in which alternate plugs of gas and liquid move along the upper part of the pipe. In stratified flow, the liquid flows along the bottom of the pipe and the gas flows over a smooth liquid/gas interface. Similar to stratified flow, wavy flow occurs at greater gas velocities and has waves moving in the flow direction. When wave crests are sufficiently high to bridge the pipe, they form frothy slugs which move at much greater than the average liquid velocity. Slug flow can cause severe and sometimes dangerous vibrations in equipment because of impact of the high-velocity slugs against bends or other fittings. Slugs may also flood gas/liquid separation equipment. In annular flow, liquid flows as a thin film along the pipe wall and gas flows in the core. Some liquid is entrained as droplets in the gas

FIG. 6-24 Gas/liquid flow patterns in horizontal pipes. (From Alves, Chem. Eng. Progr., 50, 449–456 [1954].)

FLUID DYNAMICS

6-27

FIG. 6-25 Flow-pattern regions in cocurrent liquid/gas flow through horizontal pipes. To convert lbm/(ft2 ⋅ s) to kg/(m2 ⋅ s), multiply by 4.8824. (From Baker, Oil Gas J., 53[12], 185–190, 192, 195 [1954].)

core. At very high gas velocities, nearly all the liquid is entrained as small droplets. This pattern is called spray, dispersed, or mist flow. Approximate prediction of flow pattern may be quickly done using flow pattern maps, an example of which is shown in Fig. 6-25 (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). The Baker chart remains widely used; however, for critical calculations the mechanistic model methods referenced previously are generally preferred for their greater accuracy, especially for large pipe diameters and fluids with physical properties different from air/water at atmospheric pressure. In the chart, (6-129) λ = (ρ′G ρ′L)1/2 1 µ′L 1/3 ψ =  2 (6-130) σ′ (ρ′L ) GL and GG are the liquid and gas mass velocities, µ′L is the ratio of liquid viscosity to water viscosity, ρ′G is the ratio of gas density to air density, ρ′L is the ratio of liquid density to water density, and σ′ is the ratio of liquid surface tension to water surface tension. The reference properties are at 20°C (68°F) and atmospheric pressure, water density 1,000 kg/m3 (62.4 lbm/ft3), air density 1.20 kg/m3 (0.075 lbm/ft3), water viscosity 0.001 Pa ⋅ s (1.0 cP), and surface tension 0.073 N/m (0.0050 lbf/ft). The empirical parameters λ and ψ provide a crude accounting for physical properties. The Baker chart is dimensionally inconsistent since the dimensional quantity GG /λ is plotted against a dimensionless one, GLλψ/GG, and so must be used with GG in lbm/(ft2 ⋅ s) units on the ordinate. To convert to kg/(m2 ⋅ s), multiply by 4.8824. Rapid approximate predictions of pressure drop for fully developed, incompressible horizontal gas/liquid flow may be made using the method of Lockhart and Martinelli (Chem. Eng. Prog., 45, 39–48 [1949]). First, the pressure drops that would be expected for each of the two phases as if flowing alone in single-phase flow are calculated. The Lockhart-Martinelli parameter X is defined in terms of the ratio of these pressure drops: (∆p/L)L 1/2 X=  (6-131) (∆p/L)G The two-phase pressure drop may then be estimated from either of the single-phase pressure drops, using ∆p ∆p (6-132)  = YL  L TP L L













∆Lp



∆p = YG  (6-133) L G where YL and YG are read from Fig. 6-26 as functions of X. The curve labels refer to the flow regime (laminar or turbulent) found for each of or

TP

Parameters for pressure drop in liquid/gas flow through horizontal pipes. (Based on Lockhart and Martinelli, Chem. Engr. Prog., 45, 39 [1949].)

FIG. 6-26

the phases flowing alone. The common turbulent-turbulent case is approximated well by 20 1 YL = 1 +  + 2 (6-134) X X Lockhart and Martinelli (ibid.) correlated pressure drop data from pipes 25 mm (1 in) in diameter or less within about 50 percent. In general, the predictions are high for stratified, wavy, and slug flows and low for annular flow. The correlation can be applied to pipe diameters up to about 0.1 m (4 in) with about the same accuracy. The volume fraction, sometimes called holdup, of each phase in two-phase flow is generally not equal to its volumetric flow rate fraction, because of velocity differences, or slip, between the phases. For each phase, denoted by subscript i, the relations among superficial velocity Vi, in situ velocity vi, volume fraction Ri, total volumetric flow rate Qi, and pipe area A are Qi = Vi A = vi Ri A (6-135) Vi vi =  (6-136) Ri The slip velocity between gas and liquid is vs = vG − vL . For two-phase gas/liquid flow, RL + RG = 1. A very common mistake in practice is to assume that in situ phase volume fractions are equal to input volume fractions. For fully developed incompressible horizontal gas/liquid flow, a quick estimate for RL may be obtained from Fig. 6-27, as a function of the Lockhart-Martinelli parameter X defined by Eq. (6-131). Indications are that liquid volume fractions may be overpredicted for liquids more viscous than water (Alves, Chem. Eng. Prog., 50, 449–456 [1954]), and underpredicted for pipes larger than 25 mm diameter (Baker, Oil Gas J., 53[12], 185–190, 192–195 [1954]). A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisenberg and Weinberger (AIChE J., 25, 240–245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70, 431–437 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans. Inst. Chem. Engrs., 60, 292–305, 323–333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian

6-28

FLUID AND PARTICLE DYNAMICS

Liquid volume fraction in liquid/gas flow through horizontal pipes. (From Lockhart and Martinelli, Eng. Prog., 45, 39 [1949].) FIG. 6-27

liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. Pressure drop data for a 1-in feed tee with the liquid entering the run and gas entering the branch are given by Alves (Chem. Eng. Progr., 50, 449–456 [1954]). Pressure drop and division of two-phase annular flow in a tee are discussed by Fouda and Rhodes (Trans. Inst. Chem. Eng. [London], 52, 354–360 [1974]). Flow through tees can result in unexpected flow splitting. Further reading on gas/liquid flow through tees may be found in Mudde, Groen, and van den Akker (Int. J. Multiphase Flow, 19, 563–573 [1993]); Issa and Oliveira (Computers and Fluids, 23, 347–372 [1993]) and Azzopardi and Smith (Int. J. Multiphase Flow, 18, 861–875 [1992]). Results by Chenoweth and Martin (Pet. Refiner, 34[10], 151–155 [1955]) indicate that single-phase data for fittings and valves can be used in their correlation for two-phase pressure drop. Smith, Murdock, and Applebaum (J. Eng. Power, 99, 343–347 [1977]) evaluated existing correlations for two-phase flow of steam/water and other gas/liquid mixtures through sharp-edged orifices meeting ASTM standards for flow measurement. The correlation of Murdock (J. Basic Eng., 84, 419–433 [1962]) may be used for these orifices. See also Collins and Gacesa (J. Basic Eng., 93, 11–21 [1971]), for measurements with steam and water beyond the limits of this correlation. For pressure drop and holdup in inclined pipe with upward or downward flow, see Beggs and Brill (J. Pet. Technol., 25, 607–617 [1973]); the mechanistic model methods referenced above may also be applied to inclined pipes. Up to 10° from horizontal, upward pipe inclination has little effect on holdup (Gregory, Can. J. Chem. Eng., 53, 384–388 [1975]). For fully developed incompressible cocurrent upflow of gases and liquids in vertical pipes, a variety of flow pattern terminologies and descriptions have appeared in the literature; some of these have been summarized and compared by Govier, Radford, and Dunn (Can. J. Chem. Eng., 35, 58–70 [1957]). One reasonable classification of patterns is illustrated in Fig. 6-28. In bubble flow, gas is dispersed as bubbles throughout the liquid, but with some tendency to concentrate toward the center of the pipe. In slug flow, the gas forms large Taylor bubbles of diameter nearly equal to the pipe diameter. A thin film of liquid surrounds the Taylor bubble. Between the Taylor bubbles are liquid slugs containing some bubbles. Froth or churn flow is characterized by strong intermittency and intense mixing, with neither phase easily described as continuous or dispersed. There remains disagreement in the literature as to whether churn flow is a real fully developed flow pattern or is an indication of large entry length for developing slug flow (Zao and Dukler, Int. J. Multiphase Flow, 19, 377–383 [1993]; Hewitt and Jayanti, Int. J. Multiphase Flow, 19, 527–529 [1993]). Ripple flow has an upward-moving wavy layer of liquid on the pipe wall; it may be thought of as a transition region to annular, annular mist, or film flow, in which gas flows in the core of the pipe while an annulus of liquid flows up the pipe wall. Some of the liquid is

FIG. 6-28 Flow patterns in cocurrent upward vertical gas/liquid flow. (From Taitel, Barnea, and Dukler, AIChE J., 26, 345–354 [1980]. Reproduced by permission of the American Institute of Chemical Engineers © 1980 AIChE. All rights reserved.)

entrained as droplets in the gas core. Mist flow occurs when all the liquid is carried as fine drops in the gas phase; this pattern occurs at high gas velocities, typically 20 to 30 m/s (66 to 98 ft/s). The correlation by Govier, et al. (Can. J. Chem. Eng., 35, 58–70 [1957]), Fig. 6-29, may be used for quick estimate of flow pattern. Slip, or relative velocity between phases, occurs for vertical flow as well as for horizontal. No completely satisfactory, flow regime– independent correlation for volume fraction or holdup exists for vertical flow. Two frequently used flow regime–independent methods are those by Hughmark and Pressburg (AIChE J., 7, 677 [1961]) and Hughmark (Chem. Eng. Prog., 58[4], 62 [April 1962]). Pressure drop in upflow may be calculated by the procedure described in Hughmark (Ind. Eng. Chem. Fundam., 2, 315–321 [1963]). The mechanistic, flow regime–based methods are advisable for critical applications. For upflow in helically coiled tubes, the flow pattern, pressure drop, and holdup can be predicted by the correlations of Banerjee, Rhodes, and Scott (Can. J. Chem. Eng., 47, 445–453 [1969]) and

Flow-pattern regions in cocurrent liquid/gas flow in upflow through vertical pipes. To convert ft/s to m/s, multiply by 0.3048. (From Govier, Radford, and Dunn, Can. J. Chem. Eng., 35, 58–70 [1957].)

FIG. 6-29

FLUID DYNAMICS

6-29

used one-dimensional model for flashing flow through nozzles and pipes is the homogeneous equilibrium model which assumes that both phases move at the same in situ velocity, and maintain vapor/ liquid equilibrium. It may be shown that a critical flow condition, analogous to sonic or critical flow during compressible gas flow, is given by the following expression for the mass flux G in terms of the derivative of pressure p with respect to mixture density ρm at constant entropy: ∂p Gcrit = ρm (6-139)  ∂ρm s





Critical head for drain and overflow pipes. (From Kalinske, Univ. Iowa Stud. Eng., Bull. 26 [1939–1940].) FIG. 6-30

Akagawa, Sakaguchi, and Ueda (Bull JSME, 14, 564–571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles (Can. J. Chem. Eng., 52, 25–35 [1974]) and Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 741–744 [1982]). Use of drift flux theory for void fraction modeling in downflow is presented by Clark and Flemmer (Chem. Eng. Sci., 39, 170–173 [1984]). Downward inclined two-phase flow data and modeling are given by Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 735–740 [1982]). Data for downflow in helically coiled tubes are presented by Casper (Chem. Ing. Tech., 42, 349–354 [1970]). The entrance to a drain is flush with a horizontal surface, while the entrance to an overflow pipe is above the horizontal surface. When such pipes do not run full, considerable amounts of gas can be drawn down by the liquid. The amount of gas entrained is a function of pipe diameter, pipe length, and liquid flow rate, as well as the drainpipe outlet boundary condition. Extensive data on air entrainment and liquid head above the entrance as a function of water flow rate for pipe diameters from 43.9 to 148.3 mm (1.7 to 5.8 in) and lengths from about 1.22 to 5.18 m (4.0 to 17.0 ft) are reported by Kalinske (Univ. Iowa Stud. Eng., Bull. 26, pp. 26–40 [1939–1940]). For heads greater than the critical, the pipes will run full with no entrainment. The critical head h for flow of water in drains and overflow pipes is given in Fig. 6-30. Kalinske’s results show little effect of the height of protrusion of overflow pipes when the protrusion height is greater than about one pipe diameter. For conservative design, McDuffie (AIChE J., 23, 37–40 [1977]) recommends the following relation for minimum liquid height to prevent entrainment. h 2 Fr ≤ 1.6  (6-137) D where the Froude number is defined by VL Fr   (6-138) g( ρL − ρ D /ρL  G)



where g = acceleration due to gravity VL = liquid velocity in the drain pipe ρL = liquid density ρG = gas density D = pipe inside diameter h = liquid height For additional information, see Simpson (Chem. Eng., 75[6], 192–214 [1968]). A critical Froude number of 0.31 to ensure vented flow is widely cited. Recent results (Thorpe, 3d Int. Conf. Multi-phase Flow, The Hague, Netherlands, 18–20 May 1987, paper K2, and 4th Int. Conf. Multi-phase Flow, Nice, France, 19–21 June 1989, paper K4) show hysteresis, with different critical Froude numbers for flooding and unflooding of drain pipes, and the influence of end effects. Wallis, Crowley, and Hagi (Trans. ASME J. Fluids Eng., 405–413 [June 1977]) examine the conditions for horizontal discharge pipes to run full. Flashing flow and condensing flow are two examples of multiphase flow with phase change. Flashing flow occurs when pressure drops below the bubble point pressure of a flowing liquid. A frequently

p /∂ ρ  The corresponding acoustic velocity (∂ m)s is normally much less than the acoustic velocity for gas flow. The mixture density is given in terms of the individual phase densities and the quality (mass flow fraction vapor) x by 1 x 1−x (6-140) =+ ρm ρG ρL Choked and unchoked flow situations arise in pipes and nozzles in the same fashion for homogeneous equilibrium flashing flow as for gas flow. For nozzle flow from stagnation pressure p0 to exit pressure p1, the mass flux is given by p1 dp G2 = −2ρ2m1 (6-141)  p0 ρm The integration is carried out over an isentropic flash path: flashes at constant entropy must be carried out to evaluate ρm as a function of p. Experience shows that isenthalpic flashes provide good approximations unless the liquid mass fraction is very small. Choking occurs when G obtained by Eq. (6-141) goes through a maximum at a value of p1 greater than the external discharge pressure. Equation (6-139) will also be satisfied at that point. In such a case the pressure at the nozzle exit equals the choking pressure and flashing shocks occur outside the nozzle exit. For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to dp G2 1 dx′ G2 =0 (6-142)  + g dz +  d  + 2f   ρm ρm ρm D ρm2 where x′ is distance along the pipe. Integration over a length L of pipe assuming constant friction factor f yields





p

−

2

p1

ρm dp − g



z2

z1

ρm2 dz

G2 =  ln (ρm1 /ρm2) + 2 fL/D

(6-143)

Frictional pipe flow is not isentropic. Strictly speaking, the flashes must be carried out at constant h + V 2/2 + gz, where h is the enthalpy per unit mass of the two-phase flashing mixture. The flash calculations are fully coupled with the integration of the Bernoulli equation; the velocity V must be known at every pressure p to evaluate ρm. Computational routines, employing the thermodynamic and material balance features of flowsheet simulators, are the most practical way to carry out such flashing flow calculations, particularly when multicompent systems are involved. Significant simplification arises when the mass fraction liquid is large, for then the effect of the V 2/2 term on the flash splits may be neglected. If elevation effects are also negligible, the flash computations are decoupled from the Bernoulli equation integration. For many horizontal flashing flow calculations, this is satisfactory and the flash computatations may be carried out first, to find ρm as a function of p from p1 to p2, which may then be substituted into Eq. (6-143). With flashes carried out along the appropriate thermodynamic paths, the formalism of Eqs. (6-139) through (6-143) applies to all homogeneous equilibrium compressible flows, including, for example, flashing flow, ideal gas flow, and nonideal gas flow. Equation (6-118), for example, is a special case of Eq. (6-141) where the quality x = 1 and the vapor phase is a perfect gas. Various nonequilibrium and slip flow models have been proposed as improvements on the homogeneous equilibrium flow model. See, for example, Henry and Fauske (Trans. ASME J. Heat Transfer, 179–187 [May 1971]). Nonequilibrium and slip effects both increase

6-30

FLUID AND PARTICLE DYNAMICS

computed mass flux for fixed pressure drop, compared to homogeneous equilibrium flow. For flow paths greater than about 100 mm, homogeneous equilibrium behavior appears to be the best assumption (Fischer, et al., Emergency Relief System Design Using DIERS Technology, AIChE, New York [1992]). For shorter flow paths, the best estimate may sometimes be given by linearly interpolating (as a function of length) between frozen flow (constant quality, no flashing) at 0 length and equilibrium flow at 100 mm. In a series of papers by Leung and coworkers (AIChE J., 32, 1743–1746 [1986]; 33, 524–527 [1987]; 34, 688–691 [1988]; J. Loss Prevention Proc. Ind., 2[2], 78–86 [April 1989]; 3[1], 27–32 [January 1990]; Trans. ASME J. Heat Transfer, 112, 524–528, 528–530 [1990]; 113, 269–272 [1991]) approximate techniques have been developed for homogeneous equilibrium calculations based on pseudo–equation of state methods for flashing mixtures. Relatively less work has been done on condensing flows. Slip effects are more important for condensing than for flashing flows. Soliman, Schuster, and Berenson (J. Heat Transfer, 90, 267–276 [1968]) give a model for condensing vapor in horizontal pipe. They assume the condensate flows as an annular ring. The LockhartMartinelli correlation is used for the frictional pressure drop. To this pressure drop is added an acceleration term based on homogeneous flow, equivalent to the G2d(1/ρm) term in Eq. (6-142). Pressure drop is computed by integration of the incremental pressure changes along the length of pipe. For condensing vapor in vertical downflow, in which the liquid flows as a thin annular film, the frictional contribution to the pressure drop may be estimated based on the gas flow alone, using the friction factor plotted in Fig. 6-31, where ReG is the Reynolds number for the gas flowing alone (Bergelin et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, June 22–24, 1949, pp. 19–28). dp 2f′GρGVG2 −  =  (6-144) dz D To this should be added the GG2 d(1/ρG)/dx term to account for velocity change effects. Gases and Solids The flow of gases and solids in horizontal pipe is usually classified as either dilute phase or dense phase flow. Unfortunately, there is no clear dilineation between the two types of flow, and the dense phase description may take on more than one meaning, creating some confusion (Knowlton et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]). For dilute phase flow, achieved at low solids-to-gas weight ratios (loadings), and high gas velocities, the solids may be fully suspended and fairly uniformly dispersed over the pipe cross section (homogeneous flow), particularly for low-density or small particle size solids. At lower gas velocities, the solids may

FIG. 6-31 Friction factors for condensing liquid/gas flow downward in vertical pipe. In this correlation Γ/ρL is in ft2/h. To convert ft2/h to m2/s, multiply by 0.00155. (From Bergelin et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, 1949, p. 19.)

bounce along the bottom of the pipe. With higher loadings and lower gas velocities, the particles may settle to the bottom of the pipe, forming dunes, with the particles moving from dune to dune. In dense phase conveying, solids tend to concentrate in the lower portion of the pipe at high gas velocity. As gas velocity decreases, the solids may first form dense moving strands, followed by slugs. Discrete plugs of solids may be created intentionally by timed injection of solids, or the plugs may form spontaneously. Eventually the pipe may become blocked. For more information on flow patterns, see Coulson and Richardson (Chemical Engineering, vol. 2, 2d ed., Pergamon, New York, 1968, p. 583); Korn (Chem. Eng., 57[3], 108–111 [1950]); Patterson (J. Eng. Power, 81, 43–54 [1959]); Wen and Simons (AIChE J., 5, 263–267 [1959]); and Knowlton et al. (Chem. Eng. Progr., 90[4], 44–54 [April 1994]). For the minimum velocity required to prevent formation of dunes or settled beds in horizontal flow, some data are given by Zenz (Ind. Eng. Chem. Fundam., 3, 65–75 [1964]), who presented a correlation for the minimum velocity required to keep particles from depositing on the bottom of the pipe. This rather tedious estimation procedure may also be found in Govier and Aziz, who provide additional references and discussion on transition velocities. In practice, the actual conveying velocities used in systems with loadings less than 10 are generally over 15 m/s, (49 ft/s) while for high loadings (>20) they are generally less than 7.5 m/s (24.6 ft/s) and are roughly twice the actual solids velocity (Wen and Simons, AIChE J., 5, 263–267 [1959]). Total pressure drop for horizontal gas/solid flow includes acceleration effects at the entrance to the pipe and frictional effects beyond the entrance region. A great number of correlations for pressure gradient are available, none of which is applicable to all flow regimes. Govier and Aziz review many of these and provide recommendations on when to use them. For upflow of gases and solids in vertical pipes, the minimum conveying velocity for low loadings may be estimated as twice the terminal settling velocity of the largest particles. Equations for terminal settling velocity are found in the “Particle Dynamics” subsection, following. Choking occurs as the velocity is dropped below the minimum conveying velocity and the solids are no longer transported, collapsing into solid plugs (Knowlton, et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]). See Smith (Chem. Eng. Sci., 33, 745–749 [1978]) for an equation to predict the onset of choking. Total pressure drop for vertical upflow of gases and solids includes acceleration and frictional affects also found in horizontal flow, plus potential energy or hydrostatic effects. Govier and Aziz review many of the pressure drop calculation methods and provide recommendations for their use. See also Yang (AIChE J., 24, 548–552 [1978]). Drag reduction has been reported for low loadings of small diameter particles (<60 µm diameter), ascribed to damping of turbulence near the wall (Rossettia and Pfeffer, AIChE J., 18, 31–39 [1972]). For dense phase transport in vertical pipes of small diameter, see Sandy, Daubert, and Jones (Chem. Eng. Prog., 66, Symp. Ser., 105, 133–142 [1970]). The flow of bulk solids through restrictions and bins is discussed in symposium articles (J. Eng. Ind., 91[2] [1969]) and by Stepanoff (Gravity Flow of Bulk Solids and Transportation of Solids in Suspension, Wiley, New York, 1969). Some problems encountered in discharge from bins include (Knowlton et al., Chem. Eng. Progr., 90[4], 44–54 [April 1994]) flow stoppage due to ratholing or arching, segregation of fine and coarse particles, flooding upon collapse of ratholes, and poor residence time distribution when funnel flow occurs. Solid and Liquids Slurry flow may be divided roughly into two categories based on settling behavior (see Etchells in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 12, Butterworth-Heinemann, Oxford, 1993). Nonsettling slurries are made up of very fine, highly concentrated, or neutrally buoyant particles. These slurries are normally treated as pseudohomogeneous fluids. They may be quite viscous and are frequently non-Newtonian. Slurries of particles that tend to settle out rapidly are called settling slurries or fast-settling slurries. While in some cases positively buoyant solids are encountered, the present discussion will focus on solids which are more dense than the liquid. For horizontal flow of fast-settling slurries, the following rough description may be made (Govier and Aziz). Ultrafine particles, 10 µm

FLUID DYNAMICS

Flow pattern regimes and pressure gradients in horizontal slurry flow. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

6-31

FIG. 6-32

or smaller, are generally fully syspended and the particle distributions are not influenced by gravity. Fine particles 10 to 100 µm (3.3 × 10−5 to 33 × 10−5 ft) are usually fully suspended, but gravity causes concentration gradients. Medium-size particles, 100 to 1000 µm, may be fully suspended at high velocity, but often form a moving deposit on the bottom of the pipe. Coarse particles, 1,000 to 10,000 µm (0.0033 to 0.033 ft), are seldom fully suspended and are usually conveyed as a moving deposit. Ultracoarse particles larger than 10,000 µm (0.033 ft) are not suspended at normal velocities unless they are unusually light. Figure 6-32, taken from Govier and Aziz, schematically indicates four flow pattern regions superimposed on a plot of pressure gradient vs. mixture velocity VM = VL + VS = (QL + QS)/A where VL and VS are the superficial liquid and solid velocities, QL and QS are liquid and solid volumetric flow rates, and A is the pipe cross-sectional area. VM4 is the transition velocity above which a bed exists in the bottom of the pipe, part of which is stationary and part of which moves by saltation, with the upper particles tumbling and bouncing over one another, often with formation of dunes. With a broad particle-size distribution, the finer particles may be fully suspended. Near VM4, the pressure gradient rapidly increases as VM decreases. Above VM3, the entire bed moves. Above VM2, the solids are fully suspended; that is, there is no deposit, moving or stationary, on the bottom of the pipe. However, the concentration distribution of solids is asymmetric. This flow pattern is the most frequently used for fast-settling slurry transport. Typical mixture velocities are in the range of 1 to 3 m/s (3.3 to 9.8 ft/s). The minimum in the pressure gradient is found to be near VM2. Above VM1, the particles are symmetrically distributed, and the pressure gradient curve is nearly parallel to that for the liquid by itself. The most important transition velocity, often regarded as the minimum transport or conveying velocity for settling slurries, is VM2. The Durand equation (Durand, Minnesota Int. Hydraulics Conf., Proc., 89, Int. Assoc. for Hydraulic Research [1953]; Durand and Condolios, Proc. Colloq. On the Hyd. Transport of Solids in Pipes, Nat. Coal Board [UK], Paper IV, 39–35 [1952]) gives the minimum transport velocity as VM2 = FL[2gD(s − 1)]0.5 where

(6-145)

g = acceleration of gravity D = pipe diameter s = ρS /ρL = ratio of solid to liquid density FL = a factor influenced by particle size and concentration

Probably FL is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand’s empirical correlation for FL as a function of particle diameter and the input, feed volume fraction

FIG. 6-33 Durand factor for minimum suspension velocity. (From Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972.)

solids, CS = QS /(QS + QL). The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667–1670 [1987]). No single correlation for pressure drop in horizontal solid/liquid flow has been found satisfactory for all particle sizes, densities, concentrations, and pipe sizes. However, with reference to Fig. 6-32, the following simplifications may be considered. The minimum pressure gradient occurs near VM2 and for conservative purposes it is generally desirable to exceed VM2. When VM2 is exceeded, a rough guide for pressure drop is 25 percent greater than that calculated assuming that the slurry behaves as a psuedohomogeneous fluid with the density of the mixture and the viscosity of the liquid. Above the transition velocity to symmetric suspension, VM1, the pressure drop closely approaches the pseuodohomogeneous pressure drop. The following correlation by Spells (Trans. Inst. Chem. Eng. [London], 33, 79–84 [1955]) may be used for VM1. DVM1ρM 0.775 2 VM1 = 0.075  gDS(s − 1) (6-146) µ where D = pipe diameter DS = particle diameter (such that 85 percent by weight of particles are smaller than DS) ρM = the slurry mixture density µ = liquid viscosity s = ρS /ρL = ratio of solid to liquid density





Between VM2 and VM1 the concentration of solids gradually becomes more uniform in the vertical direction. This transition has been modeled by several authors as a concentration gradient where turbulent diffusion balances gravitational settling. See, for example, Karabelas (AIChE J., 23, 426–434 [1977]). Published correlations for pressure drop are frequently very complicated and tedious to use, may not offer significant accuracy advantages over the simple guide given here, and many of them are applicable only for velocities above VM2. One which does include the effect of sliding beds is due to Gaessler (Doctoral Dissertation, Technische Hochshule, Karlsruhe, Germany [1967]; reproduced by Govier and Aziz, pp. 668–669). Turian and Yuan (AIChE J., 23, 232–243 [1977]; see also Turian and Oroskar, AIChE J., 24, 1144 [1978]) segregated a large body of data into four flow regime groups

6-32

FLUID AND PARTICLE DYNAMICS

and developed empirical correlations for predicting pressure drop in each flow regime. Pressure drop data for the flow of paper stock in pipes are given in the data section of Standards of the Hydraulic Institute (Hydraulic Institute, 1965). The flow behavior of fiber suspensions is discussed by Bobkowicz and Gauvin (Chem. Eng. Sci., 22, 229–241 [1967]), Bugliarello and Daily (TAPPI, 44, 881–893 [1961]), and Daily and Bugliarello (TAPPI, 44, 497–512 [1961]). In vertical flow of fast-settling slurries, the in situ concentration of solids with density greater than the liquid will exceed the feed concentration C = QS /(QS + QL) for upflow and will be smaller than C for downflow. This results from slip between the phases. The slip velocity, the difference between the in situ average velocities of the two phases, is roughly equal to the terminal settling velocity of the solids in the liquid. Specification of the slip velocity for a pipe of a given diameter, along with the phase flow rates, allows calculation of in situ volume fractions, average velocities, and holdup ratios by simple material balances. Slip velocity may be affected by particle concentration and by turbulence conditions in the liquid. Drift-flux theory, a framework incorporating certain functional forms for empirical expressions for slip velocity, is described by Wallis (One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969). Minimum transport velocity for upflow for design purposes is usually taken as twice the particle settling velocity. Pressure drop in vertical pipe flow includes the effects of kinetic and potential energy (elevation) changes and friction. Rose and Duckworth (The Engineer, 227[5,903], 392 [1969]; 227[5,904], 430 [1969]; 227[5,905], 478 [1969]; see also Govier and Aziz, pp. 487–493) have developed a calculation procedure including all these effects, which may be applied not only to vertical solid/liquid flow, but also to gas/solid flow and to horizontal flow. For fast-settling slurries, ensuring conveyance is usually the key design issue while pressure drop is somewhat less important. For nonsettling slurries conveyance is not an issue, because the particles do not separate from the liquid. Here, viscous and rheological behavior, which control pressure drop, take on critical importance. Fine particles, often at high concentration, form nonsettling slurries for which useful design equations can be developed by treating them as homogeneous fluids. These fluids are usually very viscous and often non-Newtonian. Shear-thinning and Bingham plastic behavior are common; dilatancy is sometimes observed. Rheology of such fluids must in general be empirically determined, although theoretical results are available for some very limited circumstances. Further discussion of both fast-settling and nonsettling slurries may be found in Shook (in Shamlou, Processing of Solid-Liquid Suspensions, Chap. 11, Butterworth-Heinemann, Oxford, 1993). FLUID DISTRIBUTION Uniform fluid distribution is essential for efficient operation of chemicalprocessing equipment such as contactors, reactors, mixers, burners, heat exchangers, extrusion dies, and textile-spinning chimneys. To obtain optimum distribution, proper consideration must be given to flow behavior in the distributor, flow conditions upstream and downstream of the distributor, and the distribution requirements of the equipment. Even though the principles of fluid distribution have been well developed for more than three decades, they are frequently overlooked by equipment designers, and a significant fraction of process equipment needlessly suffers from maldistribution. In this subsection, guides for the design of various types of fluid distributors, taking into account only the flow behavior within the distributor, are given. Perforated-Pipe Distributors The simple perforated pipe or sparger (Fig. 6-34) is a common type of distributor. As shown, the flow

distribution is uniform; this is the case in which pressure recovery due to kinetic energy or momentum changes, frictional pressure drop along the length of the pipe, and pressure drop across the outlet holes have been properly considered. In typical turbulent flow applications, inertial effects associated with velocity changes may dominate frictional losses in determining the pressure distribution along the pipe, unless the length between orifices is large. Application of the momentum or mechanical energy equations in such a case shows that the pressure inside the pipe increases with distance from the entrance of the pipe. If the outlet holes are uniform in size and spacing, the discharge flow will be biased toward the closed end. Disturbances upstream of the distributor, such as pipe bends, may increase or decrease the flow to the holes at the beginning of the distributor. When frictional pressure drop dominates the inertial pressure recovery, the distribution is biased toward the feed end of the distributor. For turbulent flow, with roughly uniform distribution, assuming a constant friction factor, the combined effect of friction and inertial (momentum) pressure recovery is given by 4fL ρVi2 ∆p =  − 2K  (discharge manifolds) (6-147) 3D 2 where ∆p = net pressure drop over the length of the distributor L = pipe length D = pipe diameter f = Fanning friction factor Vi = distributor inlet velocity





The factor K would be 1 in the case of full momentum recovery, or 0.5 in the case of negligible viscous losses in the portion of flow which remains in the pipe after the flow divides at a takeoff point (Denn, pp. 126–127). Experimental data (Van der Hegge Zijnen, Appl. Sci. Res., A3, 144–162 [1951–1953]; and Bailey, J. Mech. Eng. Sci., 17, 338–347 [1975]), while scattered, show that K is probably close to 0.5 for discharge manifolds. For inertially dominated flows, ∆p will be negative. For return manifolds the recovery factor K is close to 1.0, and the pressure drop between the first hole and the exit is given by ρVe2 4 fL ∆p =  + 2K  (return manifolds) (6-148) 3D 2 where Ve is the pipe exit velocity. One means to obtain a desired uniform distribution is to make the average pressure drop across the holes ∆po large compared to the pressure variation over the length of pipe ∆p. Then, the relative variation in pressure drop across the various holes will be small, and so will be the variation in flow. When the area of an individual hole is small compared to the cross-sectional area of the pipe, hole pressure drop may be expressed in terms of the discharge coefficient Co and the velocity across the hole Vo as





1 ρV o2 ∆po =   C o2 2

(6-149)

Provided Co is the same for all the holes, the percent maldistribution, defined as the percentage variation in flow between the first and last holes, may be estimated reasonably well for small maldistribution by (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]) ∆p − |∆p|   ∆p

Percent maldistribution = 100 1 −

o

(6-150)

o

This equation shows that for 5 percent maldistribution, the pressure drop across the holes should be about 10 times the pressure drop over the length of the pipe. For discharge manifolds with K = 0.5 in Eq. (6-147), and with 4fL/3D << 1, the pressure drop across the holes should be 10 times the inlet velocity head, ρVi2/2 for 5 percent maldistribution. This leads to a simple design equation. Discharge manifolds, 4fL/3D << 1, 5% maldistribution:

FIG. 6-34

Perforated-pipe distributor.

Vo Ap  =  = 10Co Vi Ao

(6-151)

FLUID DYNAMICS Here Ap = pipe cross-sectional area and Ao is the total hole area of the distributor. Use of large hole velocity to pipe velocity ratios promotes perpendicular discharge streams. In practice, there are many cases where the 4fL/3D term will be less than unity but not close to zero. In such cases, Eq. (6-151) will be conservative, while Eqs. (6-147), (6-149), and (6-150) will give more accurate design calculations. In cases where 4fL/(3D) > 2, friction effects are large enough to render Eq. (6-151) nonconservative. When significant variations in f along the length of the distributor occur, calculations should be made by dividing the distributor into small enough sections that constant f may be assumed over each section. For return manifolds with K = 1.0 and 4fL/(3D) << 1, 5 percent maldistribution is achieved when hole pressure drop is 20 times the pipe exit velocity head. Return manifolds, 4fL/3D << 1, 5% maldistribution: Vo Ap (6-152)  =  = 20Co Ve Ao When 4fL/3D is not negligible, Eq. (6-152) is not conservative and Eqs. (6-148), (6-149), and (6-150) should be used. One common misconception is that good distribution is always provided by high pressure drop, so that increasing flow rate improves distribution by increasing pressure drop. Conversely, it is mistakenly believed that turndown of flow through a perforated pipe designed using Eqs. (6-151) and (6-152) will cause maldistribution. However, when the distribution is nearly uniform, decreasing the flow rate decreases ∆p and ∆po in the same proportion, and Eqs. (6-151) and (6-152) are still satisfied, preserving good distribution independent of flow rate, as long as friction losses remain small compared to inertial (velocity head change) effects. Conversely, increasing the flow rate through a distributor with severe maldistribution will not generally produce good distribution. Often, the pressure drop required for design flow rate is unacceptably large for a distributor pipe designed for uniform velocity through uniformly sized and spaced orifices. Several measures may be taken in such situations. These include the following: 1. Taper the diameter of the distributor pipe so that the pipe velocity and velocity head remain constant along the pipe, thus substantially reducing pressure variation in the pipe. 2. Vary the hole size and/or the spacing between holes to compensate for the pressure variation along the pipe. This method may be sensitive to flow rate and a distributor optimized for one flow rate may suffer increased maldistribution as flow rate deviates from design rate. 3. Feed or withdraw from both ends, reducing the pipe flow velocity head and required hole pressure drop by a factor of 4. The orifice discharge coefficient Co is usually taken to be about 0.62. However, Co is dependent on the ratio of hole diameter to pipe diameter, pipe wall thickness to hole diameter ratio, and pipe velocity to hole velocity ratio. As long as all these are small, the coefficient 0.62 is generally adequate. Example 9: Pipe Distributor A 3-in schedule 40 (inside diameter 7.793 cm) pipe is to be used as a distributor for a flow of 0.010 m3/s of water (ρ = 1,000 kg/m3, µ = 0.001 Pa ⋅ s). The pipe is 0.7 m long and is to have 10 holes of uniform diameter and spacing along the length of the pipe. The distributor pipe is submerged. Calculate the required hole size to limit maldistribution to 5 percent, and estimate the pressure drop across the distributor. The inlet velocity computed from Vi = Q/A = 4Q/(πD2) is 2.10 m/s, and the inlet Reynolds number is DVi ρ 0.07793 × 2.10 × 1,000 Re =  =  = 1.64 × 105 µ 0.001 For commercial pipe with roughness % = 0.046 mm, the friction factor is about 0.0043. Approaching the last hole, the flow rate, velocity, and Reynolds number are about one-tenth their inlet values. At Re = 16,400 the friction factor f is about 0.0070. Using an average value of f = 0.0057 over the length of the pipe, 4fL/3D is 0.068 and may reasonably be neglected so that Eq. (6-151) may be used. With Co = 0.62, Vo Ap  =  = 10Co = 10 × 0.62 = 1.96 Vi Ao With pipe cross-sectional area Ap = 0.00477 m2, the total hole area is 0.00477/1.96 = 0.00243 m2. The area and diameter of each hole are then

6-33

0.00243/10 = 0.000243 m2 and 1.76 cm. With Vo /Vi = 1.96, the hole velocity is 1.96 × 2.10 = 4.12 m/s and the pressure drop across the holes is obtained from Eq. (6-149). 1,000(4.12)2 1 ρVo2 1 ∆po =   = 2 ×  = 22,100 Pa Co2 2 0.62 2 Since the hole pressure drop is 10 times the pressure variation in the pipe, the total pressure drop from the inlet of the distributor may be taken as approximately 22,100 Pa.

Further detailed information on pipe distributors may be found in Senecal (Ind. Eng. Chem., 49, 993–997 [1957]). Much of the information on tapered manifold design has appeared in the pulp and paper literature (Spengos and Kaiser, TAPPI, 46[3], 195–200 [1963]; Madeley, Paper Technology, 9[1], 35–39 [1968]; Mardon, et al., TAPPI, 46[3], 172–187 [1963]; Mardon, et al., Pulp and Paper Magazine of Canada, 72[11], 76–81 [November 1971]; Trufitt, TAPPI, 58[11], 144–145 [1975]). Slot Distributors These are generally used in sheeting dies for extrusion of films and coatings and in air knives for control of thickness of a material applied to a moving sheet. A simple slotted pipe for turbulent flow conditions may give severe maldistribution because of nonuniform discharge velocity, but also because this type of design does not readily give perpendicular discharge (Koestel and Tuve, Heat. Piping Air Cond., 20[1], 153–157 [1948]; Senecal, Ind. Eng. Chem., 49, 993–997 [1957]; Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). For slots in tapered ducts where the duct cross-sectional area decreases linearly to zero at the far end, the discharge angle will be constant along the length of the duct (Koestel and Young, ibid.). One way to ensure an almost perpendicular discharge is to have the ratio of the area of the slot to the cross-sectional area of the pipe equal to or less than 0.1. As in the case of perforatedpipe distributors, pressure variation within the slot manifold and pressure drop across the slot must be carefully considered. In practice, the following methods may be used to keep the diameter of the pipe to a minimum consistent with good performance (Senecal, Ind. Eng. Chem., 49, 993–997 [1957]): 1. Feed from both ends. 2. Modify the cross-sectional design (Fig. 6-35); the slot is thus farther away from the influence of feed-stream velocity. 3. Increase pressure drop across the slot; this can be accomplished by lengthening the lips (Fig. 6-35). 4. Use screens (Fig. 6-35) to increase overall pressure drop across the slot. Design considerations for air knives are discussed by Senecal (ibid.). Design procedures for extrusion dies when the flow is laminar, as with highly viscous fluids, are presented by Bernhardt (Processing of Thermoplastic Materials, Rheinhold, New York, 1959, pp. 248–281). Turning Vanes In applications such as ventilation, the discharge profile from slots can be improved by turning vanes. The tapered duct is the most amenable for turning vanes because the discharge angle remains constant. One way of installing the vanes is shown in Fig. 6-36. The vanes should have a depth twice the spacing (Heating, Ventilating, Air Conditioning Guide, vol. 38, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1960, pp. 282–283) and a curvature at the upstream end of the vanes of a circular arc which is tangent to the discharge angle θ of a slot without vanes and perpendicular at the downstream or discharge end of the vanes (Koestel and Young, Heat. Piping Air Cond., 23[7], 111–115 [1951]). Angle θ can be estimated from Cd As cot θ =  (6-153) Ad

FIG. 6-35

Modified slot distributor.

6-34

FLUID AND PARTICLE DYNAMICS

FIG. 6-38 FIG. 6-36

Smoothing out a nonuniform profile in a channel.

Turning vanes in a slot distributor.

where A s = slot area A d = duct cross-sectional area at upstream end Cd = discharge coefficient of slot Vanes may be used to improve velocity distribution and reduce frictional loss in bends, when the ratio of bend turning radius to pipe diameter is less than 1.0. For a miter bend with low-velocity flows, simple circular arcs (Fig. 6-37) can be used, and with high-velocity flows, vanes of special airfoil shapes are required. For additional details and references, see Ower and Pankhurst (The Measurement of Air Flow, Pergamon, New York, 1977, p. 102); Pankhurst and Holder (Wind-Tunnel Technique, Pitman, London, 1952, pp. 92–93); Rouse (Engineering Hydraulics, Wiley, New York, 1950, pp. 399–401); and Jorgensen (Fan Engineering, 7th ed., Buffalo Forge Co., Buffalo, 1970, pp. 111, 117, 118). Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker (Ind. Eng. Chem., 38, 622–624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile:



V2,max (V1,max /V)2 + α2 − α1 + α2K (6-154)  =  V 1+K Here, V is the area average velocity, K is the number of velocity heads of pressure drop provided by the uniform resistance, ∆p = KρV 2/2, and α is the velocity profile factor used in the mechanical energy balance, Eq. (6-13). It is the ratio of the area average of the cube of the velocity, to the cube of the area average velocity V. The shape of the exit velocity profile appears twice in Eq. (6-154), in V2,max /V and α2. Typically, K is on the order of 10, and the desired exit velocity profile

is fairly uniform so that α2 ∼ 1.0 may be appropriate. Downstream of the resistance, the velocity profile will gradually reestablish the fully developed profile characteristic of the Reynolds number and channel shape. The screen or perforated plate open area required to produce the resistance K may be computed from Eqs. (6-107) or (6-111). Screens and other flow restrictions may also be used to suppress stream swirl and turbulence (Loehrke and Nagib, J. Fluids Eng., 98, 342–353 [1976]). Contraction of the channel, as in a venturi, provides further reduction in turbulence level and flow nonuniformity. Beds of Solids A suitable depth of solids can be used as a fluid distributor. As for other types of distribution devices, a pressure drop of 10 velocity heads is typically used, here based on the superficial velocity through the bed. There are several substantial disadvantages to use of particle beds for flow distribution. Heterogeneity of the bed may actually worsen rather than improve distribution. In general, uniform flow may be found only downstream of the point in the bed where sufficient pressure drop has occurred to produce uniform flow. Therefore, inefficiency results when the bed also serves reaction or mass transfer functions, as in catalysts, adsorbents, or tower packings for gas/liquid contacting, since portions of the bed are bypassed. In the case of trickle flow of liquid downward through column packings, inlet distribution is critical since the bed itself is relatively ineffective in distributing the liquid. Maldistribution of flow through packed beds also arises when the ratio of bed diameter to particle size is less than 10 to 30. Other Flow Straightening Devices Other devices designed to produce uniform velocity or reduce swirl, sometimes with reduced pressure drop, are available. These include both commercial devices of proprietary design and devices discussed in the literature. For pipeline flows, see the references under flow inverters and static mixing elements previously discussed in the “Incompressible Flow in Pipes and Channels” subsection. For large area changes, as at the entrance to a vessel, it is sometimes necessary to diffuse the momentum of the inlet jet discharging from the feed pipe in order to produce a more uniform velocity profile within the vessel. Methods for this application exist, but remain largely in the domain of proprietary, commercial design. FLUID MIXING

FIG. 6-37

Miter bend with vanes.

Mixing of fluids is a discipline of fluid mechanics. Fluid motion is used to accelerate the otherwise slow processes of diffusion and conduction to bring about uniformity of concentration and temperature, blend materials, facilitate chemical reactions, bring about intimate contact of multiple phases, and so on. As the subject is too broad to cover fully, only a brief introduction and some references for further information are given here. Several texts are available. These include Paul, Atiemo-Obeng, and Kresta (Handbook of Industrial Mixing, Wiley-Interscience, Hoboken N.J., 2004); Harnby, Edwards, and Nienow (Mixing in the Process Industries, 2d ed., Butterworths, London, 1992); Oldshue (Fluid Mixing Technology, McGraw-Hill, New York, 1983); Tatterson (Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill, New York, 1991); Uhl and Gray (Mixing, vols. I–III, Academic, New York, 1966, 1967, 1986); and Nagata (Mixing: Principles and Applications, Wiley, New York, 1975). A good overview of stirred tank agitation is given in the series of articles from Chemical Engineering (110–114, Dec. 8, 1975; 139–145, Jan. 5, 1976; 93–100, Feb. 2, 1976; 102–110,

FLUID DYNAMICS

Typical stirred tank configurations, showing time-averaged flow patterns for axial flow and radial flow impellers. (From Oldshue, Fluid Mixing Technology, McGraw-Hill, New York, 1983.)

FIG. 6-39

Apr. 26, 1976; 144–150, May 24, 1976; 141–148, July 19, 1976; 89–94, Aug. 2, 1976; 101–108, Aug. 30, 1976; 109–112, Sept. 27, 1976; 119–126, Oct. 25, 1976; 127–133, Nov. 8, 1976). Process mixing is commonly carried out in pipeline and vessel geometries. The terms radial mixing and axial mixing are commonly used. Axial mixing refers to mixing of materials which pass a given point at different times, and thus leads to backmixing. For example, backmixing or axial mixing occurs in stirred tanks where fluid elements entering the tank at different times are intermingled.

6-35

Mixing of elements initially at different axial positions in a pipeline is axial mixing. Radial mixing occurs between fluid elements passing a given point at the same time, as, for example, between fluids mixing in a pipeline tee. Turbulent flow, by means of the chaotic eddy motion associated with velocity fluctuation, is conducive to rapid mixing and, therefore, is the preferred flow regime for mixing. Laminar mixing is carried out when high viscosity makes turbulent flow impractical. Stirred Tank Agitation Turbine impeller agitators, of a variety of shapes, are used for stirred tanks, predominantly in turbulent flow. Figure 6-39 shows typical stirred tank configurations and timeaveraged flow patterns for axial flow and radial flow impellers. In order to prevent formation of a vortex, four vertical baffles are normally installed. These cause top-to-bottom mixing and prevent mixing-ineffective swirling motion. For a given impeller and tank geometry, the impeller Reynolds number determines the flow pattern in the tank: D2Nρ ReI =  (6-155) µ where D = impeller diameter, N = rotational speed, and ρ and µ are the liquid density and viscosity. Rotational speed N is typically reported in revolutions per minute, or revolutions per second in SI units. Radians per second are almost never used. Typically, ReI > 104 is required for fully turbulent conditions throughout the tank. A wide transition region between laminar and turbulent flow occurs over the range 10 < ReI < 104. The power P drawn by the impeller is made dimensionless in a group called the power number: P NP =  (6-156) ρN 3D5 Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In

FIG. 6-40 Dimensionless power number in stirred tanks. (Reprinted with permission from Bates, Fondy, and Corpstein, Ind. Eng. Chem. Process Design Develop., 2, 310 [1963].)

6-36

FLUID AND PARTICLE DYNAMICS

turbulent flow, where inertial forces dominate, the power number is nearly constant. Impellers are sometimes viewed as pumping devices; the total volumetric flow rate Q discharged by an impeller is made dimensionless in a pumping number: Q NQ = 3 (6-157) ND Blend time tb, the time required to achieve a specified maximum standard deviation of concentration after injection of a tracer into a stirred tank, is made dimensionless by multiplying by the impeller rotational speed: Nb = tb N

(6-158)

Dimensionless pumping number and blend time are independent of Reynolds number under fully turbulent conditions. The magnitude of concentration fluctuations from the final well-mixed value in batch mixing decays exponentially with time. The design of mixing equipment depends on the desired process result. There is often a tradeoff between operating cost, which depends mainly on power, and capital cost, which depends on agitator size and torque. For some applications bulk flow throughout the vessel is desired, while for others high local turbulence intensity is required. Multiphase systems introduce such design criteria as solids suspension and gas dispersion. In very viscous systems, helical ribbons, extruders, and other specialized equipment types are favored over turbine agitators. Pipeline Mixing Mixing may be carried out with mixing tees, inline or motionless mixing elements, or in empty pipe. In the latter case, large pipe lengths may be required to obtain adequate mixing. Coaxially injected streams require lengths on the order of 100 pipe diameters. Coaxial mixing in turbulent single-phase flow is characterized by the turbulent diffusivity (eddy diffusivity) DE which determines the rate of radial mixing. Davies (Turbulence Phenomena, Academic, New York, 1972) provides an equation for DE which may be rewritten as DE ∼ 0.015DVRe−0.125 where

(6-159)

D = pipe diameter V = average velocity Re = pipe Reynolds number, DVρ/µ ρ = density µ = viscosity

Properly designed tee mixers, with due consideration given to main stream and injected stream momentum, are capable of producing high degrees of uniformity in just a few diameters. Forney (Jet Injection for Optimum Pipeline Mixing, in “Encyclopedia of Fluid Mechanics,” vol. 2., Chap. 25, Gulf Publishing, 1986) provides a thorough discussion of tee mixing. Inline or motionless mixers are generally of proprietary commercial design, and may be selected for viscous or turbulent, single or multiphase mixing applications. They substantially reduce required pipe length for mixing. TUBE BANKS Pressure drop across tube banks may not be correlated by means of a single, simple friction factor—Reynolds number curve, owing to the variety of tube configurations and spacings encountered, two of which are shown in Fig. 6-41. Several investigators have allowed for configuration and spacing by incorporating spacing factors in their friction factor expressions or by using multiple friction factor plots. Commercial computer codes for heat-exchanger design are available which include features for estimating pressure drop across tube banks. Turbulent Flow The correlation by Grimison (Trans. ASME, 59, 583–594 [1937]) is recommended for predicting pressure drop for turbulent flow (Re ≥ 2,000) across staggered or in-line tube banks for tube spacings [(a/Dt), (b/Dt)] ranging from 1.25 to 3.0. The pressure drop is given by 2 4fNrρVmax ∆p =  2

(6-160)

FIG. 6-41

where

Tube-bank configurations.

f = friction factor Nr = number of rows of tubes in the direction of flow ρ = fluid density Vmax = fluid velocity through the minimum area available for flow

For banks of staggered tubes, the friction factor for isothermal flow is obtained from Fig. (6-42). Each “fence” (group of parametric curves) represents a particular Reynolds number defined as DtVmaxρ Re =  (6-161) µ where Dt = tube outside diameter and µ = fluid viscosity. The numbers along each fence represent the transverse and inflow-direction spacings. The upper chart is for the case in which the minimum area for flow is in the transverse openings, while the lower chart is for the case in which the minimum area is in the diagonal openings. In the latter case, Vmax is based on the area of the diagonal openings and Nr is the number of rows in the direction of flow minus 1. A critical comparison of this method with all the data available at the time showed an average deviation of the order of 15 percent (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For tube spacings greater than 3 tube diameters, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) can be used as an approximation. As an approximation, the pressure drop can be taken as 0.72 velocity head (based on Vmax per row of tubes for tube spacings commonly encountered in practice (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For banks of in-line tubes, f for isothermal flow is obtained from Fig. 6-43. Average deviation from available data is on the order of 15 percent. For tube spacings greater than 3Dt, the charts of Gram, Mackey, and Monroe (Trans. ASME, 80, 25–35 [1958]) can be used. As an approximation, the pressure drop can be taken as 0.32 velocity head (based on Vmax) per row of tubes (Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1954). For turbulent flow through shallow tube banks, the average friction factor per row will be somewhat greater than indicated by Figs. 6-42 and 6-43, which are based on 10 or more rows depth. A 30 percent increase per row for 2 rows, 15 percent per row for 3 rows, and 7 percent per row for 4 rows can be taken as the maximum likely to be encountered (Boucher and Lapple, Chem. Eng. Prog., 44, 117–134 [1948]). For a single row of tubes, the friction factor is given by Curve B in Fig. 6-44 as a function of tube spacing. This curve is based on the data of several experimenters, all adjusted to a Reynolds number of 10,000. The values should be substantially independent of Re for 1,000 < Re < 100,000. For extended surfaces, which include fins mounted perpendicularly to the tubes or spiral-wound fins, pin fins, plate fins, and so on, friction data for the specific surface involved should be used. For

FLUID DYNAMICS

6-37

FIG. 6-42 Upper chart: Friction factors for staggered tube banks with minimum fluid flow area in transverse openings. Lower chart: Friction factors for staggered tube banks with minimum fluid flow area in diagonal openings. (From Grimison, Trans. ASME, 59, 583 [1937].)

details, see Kays and London (Compact Heat Exchangers, 2d ed., McGraw-Hill, New York, 1964). If specific data are unavailable, the correlation by Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation. When a large temperature change occurs in a gas flowing across a tube bundle, gas properties should be evaluated at the mean temperature Tm = Tt + K ∆Tlm

(6-162)

where Tt = average tube-wall temperature K = constant ∆Tlm = log-mean temperature difference between the gas and the tubes Values of K averaged from the recommendations of Chilton and Genereaux (Trans. AIChE, 29, 151–173 [1933]) and Grimison (Trans. ASME, 59, 583–594 [1937]) are as follows: for in-line tubes, 0.9 for cooling and −0.9 for heating; for staggered tubes, 0.75 for cooling and −0.8 for heating. For nonisothermal flow of liquids across tube bundles, the friction factor is increased if the liquid is being cooled and decreased if the liquid is being heated. The factors previously given for nonisothermal

flow of liquids in pipes (“Incompressible Flow in Pipes and Channels”) should be used. For two-phase gas/liquid horizontal cross flow through tube banks, the method of Diehl and Unruh (Pet. Refiner, 37[10], 124–128 [1958]) is available. Transition Region This region extends roughly over the range 200 < Re < 2,000. Figure 6-45 taken from Bergelin, Brown, and Doberstein (Trans. ASME, 74, 953–960 [1952]) gives curves for friction factor fT for five different configurations. Pressure drop for liquid flow is given by 2 µs 0.14 4 fT Nr ρVmax ∆p =   (6-163) 2 µb where Nr = number of major restrictions encountered in flow through the bank (equal to number of rows when minimum flow area occurs in transverse openings, and to number of rows minus 1 when it occurs in the diagonal openings); ρ = fluid density; Vmax = velocity through minimum flow area; µs = fluid viscosity at tube-surface temperature and µb = fluid viscosity at average bulk temperature. This method gives the friction factor within about 25 percent. Laminar Region Bergelin, Colburn, and Hull (Univ. Delaware Eng. Exp. Sta. Bull., 2 [1950]) recommend the following equations for



6-38

FLUID AND PARTICLE DYNAMICS

FIG. 6-43

Friction factors for in-line tube banks. (From Grimison, Trans. ASME, 59, 583 [1937].)

pressure drop with laminar flow (Rev < 100) of liquids across banks of plain tubes with pitch ratios P/Dt of 1.25 and 1.50: 2 280N D 1.6 µs m ρV max ∆p = r t (6-164)   Rev P µb 2

  



arrangements), and other quantities are as defined following Eq. (6-163). Bergelin, et al. (ibid.) show that pressure drop per row is independent of the number of rows in the bank with laminar flow. The pressure drop is predicted within about 25 percent.

0.57 m= (6-165) (Rev)0.25 where Rev = DvVmaxρ/µb; Dv = volumetric hydraulic diameter [(4 × free-bundle volume)/(exposed surface area of tubes)]; P = pitch (= a for in-line arrangements, = a or c [whichever is smaller] for staggered

Friction factors for transition region flow across tube banks. (Pitch is the minimum center-to-center tube spacing.) (From Bergelin, Brown, and Doberstein, Trans. ASME, 74, 953 [1952].)

FIG. 6-45

Friction factors vs. transverse spacing for single row of tubes. (From Boucher and Lapple, Chem. Eng. Prog., 44, 117 [1948].) FIG. 6-44

FLUID DYNAMICS

FIG. 6-46

6-39

Friction factor for beds of solids. (From Leva, Fluidization, McGraw-Hill, New York, 1959, p. 49.)

The validity of extrapolating Eq. (6-164) to pitch ratios larger than 1.50 is unknown. The correlation of Gunter and Shaw (Trans. ASME, 67, 643–660 [1945]) may be used as an approximation in such cases. For laminar flow of non-Newtonian fluids across tube banks, see Adams and Bell (Chem. Eng. Prog., 64, Symp. Ser., 82, 133–145 [1968]). Flow-induced tube vibration occurs at critical fluid velocities through tube banks, and is to be avoided because of the severe damage that can result. Methods to predict and correct vibration problems may be found in Eisinger (Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]) and Chen (J. Sound Vibration, 93, 439–455 [1984]). BEDS OF SOLIDS Fixed Beds of Granular Solids Pressure-drop prediction is complicated by the variety of granular materials and of their packing arrangement. For flow of a single incompressible fluid through an incompressible bed of granular solids, the pressure drop may be estimated by the correlation given in Fig. 6-46 (Leva, Chem. Eng., 56[5], 115–117 [1949]), or Fluidization, McGraw-Hill, New York, 1959). The modified friction factor and Reynolds number are defined by Dpρφ3s − n%3|∆p| fm   2G2L(1 − %)3 − n

(6-166)

DpG Re′   (6-167) µ where −∆p = pressure drop L = depth of bed Dp = average particle diameter, defined as the diameter of a sphere of the same volume as the particle % = void fraction n = exponent given in Fig. 6-46 as a function of Re′ φs = shape factor defined as the area of sphere of diameter Dp divided by the actual surface area of the particle G = fluid superficial mass velocity based on the empty chamber cross section ρ = fluid density µ = fluid viscosity

As for any incompressible single-phase flow, the equivalent pressure P = p + ρgz where g = acceleration of gravity z = elevation, may be used in place of p to account for gravitational effects in flows with vertical components. In creeping flow (Re′ < 10), 100 fm =  (6-168) Re′ At high Reynolds numbers the friction factor becomes nearly constant, approaching a value of the order of unity for most packed beds. In terms of S, particle surface area per unit volume of bed, 6(1 − %) Dp =  φsS

(6-169)

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same qualitative behavior as that given by Leva’s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. Creeping flow (Re′ <∼ 1) through porous media is often described in terms of the permeability k and Darcy’s law: −∆P µ (6-170) =V L k where V = superficial velocity. The SI units for permeability are m2. Creeping flow conditions generally prevail in geological porous media. For multidimensional flows through isotropic porous media, the superficial velocity V and pressure gradient ∇P vectors replace the corresponding one-dimensional variables in Eq. (6-170). µ ∇P = −  V (6-171) k

6-40

FLUID AND PARTICLE DYNAMICS

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation: ∇ 2P = 0

(6-172)

For anisotropic or oriented porous media, as are frequently found in geological media, permeability varies with direction and a permeability tensor K, with nine components Kij giving the velocity compenent in the i direction due to a pressure gradient in the j direction, may be introduced. For further information, see Slattery (Momentum, Energy and Mass Transfer in Continua, Krieger, Huntington, New York, 1981, pp. 193–218). See also Dullien (Chem. Eng. J. [Laussanne], 10, 1,034 [1975]) for a review of pressure-drop methods in single-phase flow. Solutions for Darcy’s law for several geometries of interest in petroleum reservoirs and aquifers, for both incompressible and compressible flows, are given in Craft and Hawkins (Applied Petroleum Reservoir Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1959). See also Todd (Groundwater Hydrology, 2nd ed., Wiley, New York, 1980). For granular solids of mixed sizes the average particle diameter may be calculated as 1 xi (6-173) =  Dp Dp,i i where xi = weight fraction of particles of size Dp,i. For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived: 2ZRG T v 2fmL(1 − %) p21 − p22 =  ln 2 +  Mw v1 φ 3s − n%3Dp 2

3−n





(6-174)

where p1 = upstream absolute pressure p2 = downstream absolute pressure R = gas constant T = absolute temperature Mw = molecular weight v1 = upstream specific volume of gas v2 = downstream specific volume of gas For creeping flow of power law non-Newtonian fluids, the method of Christopher and Middleton (Ind. Eng. Chem. Fundam., 4, 422–426 [1965]) may be used: 150HLV n(1 − %)2 −∆p =  (6-175) Dp2 φ 2s %3 K 3 n Dp2 φ2s%4 (1 − n)/2 H= 9+ (6-176)  12 n (1 − %)2 where V = G/ρ = superficial velocity, K, n = power law material constants, and all other variables are as defined in Eq. (6-166). This correlation is supported by data from Christopher and Middleton (ibid.), Gregory and Griskey (AIChE J., 13, 122–125 [1967]), Yu, Wen, and Bailie (Can. J. Chem. Eng., 46, 149–154 [1968]), Siskovic, Gregory, and Griskey (AIChE J., 17, 176–187 [1978]), Kemblowski and Mertl (Chem. Eng. Sci., 29, 213–223 [1974]), and Kemblowski and Dziuminski (Rheol. Acta, 17, 176–187 [1978]). The measurements cover the range n = 0.50 to 1.60, and modified Reynolds number Re′ = 10−8 to 10, where







DpV 2 − nρ Re′ =  (6-177) H For the case n = 1 (Newtonian fluid), Eqs. (6-175) and (6-176) give a pressure drop 25 percent less than that given by Eqs. (6-166) through (6-168). For viscoelastic fluids see Marshall and Metzner (Ind. Eng. Chem. Fundam., 6, 393–400 [1967]), Siskovic, Gregory, and Griskey (AIChE J., 13, 122–125 [1967]) and Kemblowski and Dziubinski (Rheol. Acta, 17, 176–187 [1978]). For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the “Vacuum Flow” subsection) may be applied when the pore is of the same or

smaller order as the mean free path, as described by Monet and Vermeulen (Chem. Eng. Prog., 55, Symp. Ser., 25 [1959]). Tower Packings For the flow of a single fluid through a bed of tower packing, pressure drop may be estimated using the preceding methods. See also Sec. 14 of this Handbook. For countercurrent gas/liquid flow in commercial tower packings, both structured and unstructured, several sources of data and correlations for pressure drop and flooding are available. See, for example, Strigle (Random Packings and Packed Towers, Design and Applications, Gulf Publishing, Houston, 1989; Chem. Eng. Prog., 89[8], 79–83 [August 1993]), Hughmark (Ind. Eng. Chem. Fundam., 25, 405–409 [1986]), Chen (Chem. Eng. Sci., 40, 2139–2140 [1985]), Billet and Mackowiak (Chem. Eng. Technol., 11, 213–217 [1988]), Krehenwinkel and Knapp (Chem. Eng. Technol., 10, 231–242 [1987]), Mersmann and Deixler (Ger. Chem. Eng., 9, 265–276 [1986]), and Robbins (Chem. Eng. Progr., 87[5], 87–91 [May 1991]). Data and correlations for flooding and pressure drop for structured packings are given by Fair and Bravo (Chem. Eng. Progr., 86[1], 19–29 [January 1990]). Fluidized Beds When gas or liquid flows upward through a vertically unconstrained bed of particles, there is a minimum fluid velocity at which the particles will begin to move. Above this minimum velocity, the bed is said to be fluidized. Fluidized beds are widely used, in part because of their excellent mixing and heat and mass transfer characteristics. See Sec. 17 of this Handbook for detailed information. BOUNDARY LAYER FLOWS Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the solid surface is called the boundary layer. The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundary layer thickness is conventionally taken to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simplified by scaling arguments. Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987) is the most comprehensive source for information on boundary layer flows. Flat Plate, Zero Angle of Incidence For flow over a wide, thin flat plate at zero angle of incidence with a uniform free-stream velocity, as shown in Fig. 6-47, the critical Reynolds number at which the boundary layer becomes turbulent is normally taken to be xVρ Rex =  = 500,000 (6-178) µ where V = free-stream velocity ρ = fluid density µ = fluid viscosity x = distance from leading edge of the plate Uniform free-stream velocity V

y δ(x ) x L FIG. 6-47

Boundary layer on a flat plate at zero angle of incidence.

FLUID DYNAMICS However, the transition Reynolds number depends on free-stream turbulence and may range from 3 × 105 to 3 × 106. The laminar boundary layer thickness δ is a function of distance from the leading edge: δ ≈ 5.0xRe −0.5 x

(6-179)

The total drag on the plate of length L and width b for a laminar boundary layer, including the drag on both surfaces, is: FD = 1.328bLρV 2ReL−0.5

(6-180)

For non-Newtonian power law fluids (Acrivos, Shah, and Peterson, AIChE J., 6, 312–317 [1960]; Hsu, AIChE J., 15, 367–370 [1969]), FD = CbLρV 2ReL′− 1/(1 + n)

(6-181)

n = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C = 2.075 1.958 1.838 1.727 1.627 1.538 1.460 1.390 1.328 where Re′L = ρV 2 − nLn/K and K and n are the power law material constants (see Eq. [6-4]). For a turbulent boundary layer, the thickness may be estimated as δ ≈ 0.37xRe −0.2 x

(6-182)

and the total drag force on both sides of the plate of length L is 0.455 1,700 −  ρbLV 2 5 × 105 < ReL < 109 (6-183) FD =  (log ReL)2.58 ReL Here the second term accounts for the laminar leading edge of the boundary layer and assumes that the critical Reynolds number is 500,000. Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and Lighthill (Proc. R. Soc. [London], 230A, 188–203 [1955]), Jaffe and Okamura (Z. Angew. Math. Phys., 19, 564–574 [1968]), and Stewartson (Q. Appl. Math., 13, 113–122 [1955]). For a turbulent boundary layer, the total drag may be estimated as





FD = cjπrLρV 2

(6-184)

where r = cylinder radius, L = cylinder length, and the average friction coefficient is given by (White, J. Basic Eng., 94, 200–206 [1972]) L 0.4 ReL−1/3 (6-185) cj = 0.0015 + 0.30 + 0.015  r for ReL = 106 to 109 and L/r < 106. Continuous Flat Surface Boundary layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48a shows the continuous flat surface (Sakiadis, AIChE J., 7, 26–28,



 

6-41

221–225, 467–472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. Fluid Mech., 26, 145–161 [1966]). For a laminar boundary layer, the thickness is given by δ = 6.37xRe x−0.5

(6-186)

and the total drag exerted on the two surfaces is FD = 1.776bLρV 2ReL−0.5

(6-187)

The total flow rate of fluid entrained by the surface is q = 3.232bLVReL−0.5

(6-188)

The theoretical velocity field was experimentally verified by Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) and Szeri, Yates, and Hai ( J. Lubr. Technol., 98, 145–156 [1976]). For non-Newtonian power law fluids see Fox, Erickson, and Fan (AIChE J., 15, 327–333 [1969]). For a turbulent boundary layer, the thickness is given by δ = 1.01xRe −0.2 x

(6-189)

and the total drag on both sides by FD = 0.056bLρV 2ReL−0.2

(6-190)

and the total entrainment by q = 0.252bLVReL−0.2

(6-191)

When the laminar boundary layer is a significant part of the total length of the object, the total drag should be corrected by subtracting a calculated turbulent drag for the length of the laminar section and then adding the laminar drag for the laminar section. Tsou, Sparrow, and Goldstein (Int. J. Heat Mass Transfer, 10, 219–235 [1967]) give an improved analysis of the turbulent boundary layer; their data indicate that Eq. (6-190) underestimates the drag by about 15 percent. Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48b is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26–28, 221–225, 467– 472 [1961]); Vasudevan and Middleman, AIChE J., 16, 614 [1970]). The critical-length Reynolds number for transition is Rex 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49. The normalized boundary layer thickness and integral friction coefficient are from Vasudevan and Middleman, who used a similarity solution of the boundary layer equations. The drag force over a length x is given by — V 2 FD C f x  2#ro 2

(6-192)

The entrainment flow rate is from Sakiadis, who used an integral momentum approximation rather than the exact similarity solution. q V

(a)

(6-193)

Further laminar boundary layer analysis is given by Crane (Z. Angew. Math. Phys., 23, 201–212 [1972]). For a turbulent boundary layer, the total drag may be roughly estimated using Eqs. (6-184) and (6-185) for finite cylinders. Measured forces by Kwon and Prevorsek (J. Eng. Ind., 101, 73–79 [1979]) are greater than predicted this way. The laminar boundary layer on deforming continuous surfaces with velocity varying with axial position is discussed by Vleggaar (Chem. Eng. Sci., 32, 1517–1525 [1977] and Crane (Z. Angew. Math. Phys., 26, 619–622 [1975]). VORTEX SHEDDING

(b) Continuous surface: (a) continuous flat surface, (b) continuous cylindrical surface. (From Sakiadis, Am. Inst. Chem. Eng. J., 7, 221, 467 [1961].) FIG. 6-48

When fluid flows past objects or through orifices or similar restrictions, vortices may periodically be shed downstream. Objects such as smokestacks, chemical-processing columns, suspended pipelines, and electrical transmission lines can be subjected to damaging vibrations and forces due to the vortices, especially if the shedding frequency is close to a natural vibration frequency of the object. The shedding can

6-42

FLUID AND PARTICLE DYNAMICS For 40 < Re < 200 the vortices are laminar and the Strouhal number has a nearly constant value of 0.2 for flow past a cylinder. Between Re = 200 and 400 the Strouhal number is no longer constant and the wake becomes irregular. Above about Re = 400 the vortices become turbulent, the wake is once again stable, and the Strouhal number remains constant at about 0.2 up to a Reynolds number of about 10 5. Above Re = 105 the vortex shedding is difficult to see in flow visualization experiments, but velocity measurements still show a strong spectral component at St = 0.2 (Panton, p. 392). Experimental data suggest that the vortex street disappears over the range 5 × 105 < Re < 3.5 × 106, but is reestablished at above 3.5 × 106 (Schlichting). Vortex shedding exerts alternating lateral forces on a cylinder, perpendicular to the flow direction. Such forces may lead to severe vibration or mechanical failure of cylindrical elements such as heatexchanger tubes, transmission lines, stacks, and columns when the vortex shedding frequency is close to resonant bending frequency. According to Den Hartog (Proc. Nat. Acad. Sci., 40, 155–157 [1954]), the vortex shedding and cylinder vibration frequency will shift to the resonant frequency when the calculated shedding frequency is within 20 percent of the resonant frequency. The well-known Tacoma Narrows bridge collapse resulted from resonance between a torsional oscillation and vortex shedding (Panton, p. 392). Spiral strakes are sometimes installed on tall stacks so that vortices at different axial positions are not shed simultaneously. The alternating lateral force FK, sometimes called the von Kármán force, is given by (Den Hartog, Mechanical Vibrations, 4th ed., McGraw-Hill, New York, 1956, pp. 305–309):

Boundary layer parameters for continuous cylindrical surfaces. — (∆/#r 2o is from Sakiadis, Am. Inst. Chem. Engr. J., 7, 467 [1961]; C f x/2ro and /ro are from Vasudevan and Middleman, Am. Inst. Chem. Eng. J., 16, 614 [1970].)

ρV 2 FK = CK A  2

FIG 6-49

also produce sound. See Krzywoblocki (Appl. Mech. Rev., 6, 393–397 [1953]) and Marris (J. Basic Eng., 86, 185–196 [1964]). Development of a vortex street, or von Kármán vortex street is shown in Fig. 6-50. Discussions of the vortex street may be found in Panton (pp. 387–393). The Reynolds number is DVρ Re =  µ

(6-194)

where D = diameter of cylinder or effective width of object V = free-stream velocity ρ = fluid density µ = fluid viscosity For flow past a cylinder, the vortex street forms at Reynolds numbers above about 40. The vortices initially form in the wake, the point of formation moving closer to the cylinder as Re is increased. At a Reynolds number of 60 to 100, the vortices are formed from eddies attached to the cylinder surface. The vortices move at a velocity slightly less than V. The frequency of vortex shedding f is given in terms of the Strouhal number, which is approximately constant over a wide range of Reynolds numbers. fD St   V

FIG. 6-50

Vortex street behind a cylinder.

(6-195)

(6-196)

where CK = von Kármán coefficient A = projected area perpendicular to the flow ρ = fluid density V = free-stream fluid velocity For a cylinder, CK = 1.7. For a vibrating cylinder, the effective projected area exceeds, but is always less than twice, the actual cylinder projected area (Rouse, Engineering Hydraulics, Wiley, New York, 1950). The following references pertain to discussions of vortex shedding in specific structures: steel stacks (Osker and Smith, Trans. ASME, 78, 1381–1391 [1956]; Smith and McCarthy, Mech. Eng., 87, 38–41 [1965]); chemical-processing columns (Freese, J. Eng. Ind., 81, 77–91 [1959]); heat exchangers (Eisinger, Trans. ASME J. Pressure Vessel Tech., 102, 138–145 [May 1980]; Chen, J. Sound Vibration, 93, 439–455 [1984]; Gainsboro, Chem. Eng. Prog., 64[3], 85–88 [1968]; “Flow-Induced Vibration in Heat Exchangers,” Symp. Proc., ASME, New York, 1970); suspended pipe lines (Baird, Trans. ASME, 77, 797–804 [1955]); and suspended cable (Steidel, J. Appl. Mech., 23, 649–650 [1956]). COATING FLOWS In coating flows, liquid films are entrained on moving solid surfaces. For general discussions, see Ruschak (Ann. Rev. Fluid Mech., 17, 65–89 [1985]), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992), and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). It is generally important to control the thickness and uniformity of the coatings. In dip coating, or free withdrawal coating, a solid surface is withdrawn from a liquid pool, as shown in Fig. 6-51. It illustrates many of the features found in other coating flows, as well. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) provide an early review of the theory of dip coating. The coating flow rate and film thickness are controlled by the withdrawal rate and the flow behavior in the meniscus region. For a withdrawal velocity V and an angle of inclination from the horizontal φ, the film thickness h may be

FLUID DYNAMICS

6-43

V Γ = mass flow rate per unit width of surface

δ

FIG. 6-52

FIG. 6-51

Eng. [London], 45, 345–352 [1967]), Stainthorp and Allen (Trans. Inst. Chem. Eng. [London], 43, 85–91 [1967]) and Watanabe, et al. (J. Chem. Eng. [Japan], 8[1], 75 [1975]). Laminar Flow For films falling down vertical flat surfaces, as shown in Fig. 6-52, or vertical tubes with small film thickness compared to tube radius, laminar flow conditions prevail for Reynolds numbers less than about 2,000, where the Reynolds number is given by

Dip coating.

estimated for low withdrawal velocities by ρg h  σ



where

1/2

0.944 =  Ca2/3 (1 − cos φ)1/2

Falling film.

4Γ Re =  µ

(6-197)

g = acceleration of gravity Ca = µV/σ = capillary number µ = viscosity σ = surface tension

Equation (6-197) is asymptotically valid as Ca → 0 and agrees with experimental data up to capillary numbers in the range of 0.01 to 0.03. In practice, where high production rates require high withdrawal speeds, capillary numbers are usually too large for Eq. (6-197) to apply. Approximate analytical methods for larger capillary numbers have been obtained by numerous investigators, but none appears wholly satisfactory, and some are based on questionable assumptions (Ruschak, Ann. Rev. Fluid Mech., 17, 65–89 [1985]). With the availability of high-speed computers and the development of the field of computational fluid dynamics, numerical solutions accounting for two-dimensional flow aspects, along with gravitational, viscous, inertial, and surface tension forces are now the most effective means to analyze coating flow problems. Other common coating flows include premetered flows, such as slide and curtain coating, where the film thickness is an independent parameter that may be controlled within limits, and the curvature of the mensiscus adjusts accordingly; the closely related blade coating; and roll coating and extrusion coating. See Ruschak (ibid.), Cohen and Gutoff (Modern Coating and Drying Technology, VCH Publishers, New York, 1992), and Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977). For dip coating of wires, see Taughy (Int. J. Numerical Meth. Fluids, 4, 441–475 [1984]). Many coating flows are subject to instabilities that lead to unacceptable coating defects. Three-dimensional flow instabilities lead to such problems as ribbing. Air entrainment is another common defect. FALLING FILMS Minimum Wetting Rate The minimum liquid rate required for complete wetting of a vertical surface is about 0.03 to 0.3 kg/m ⋅ s for water at room temperature. The minimum rate depends on the geometry and nature of the vertical surface, liquid surface tension, and mass transfer between surrounding gas and the liquid. See Ponter, et al. (Int. J. Heat Mass Transfer, 10, 349–359 [1967]; Trans. Inst. Chem.

(6-198)

where Γ = liquid mass flow rate per unit width of surface and µ = liquid viscosity. For a flat film surface, the following equations may be derived. The film thickness δ is



3Γµ δ=  ρ2g

1/3

(6-199)

The average film velocity is Γ gρδ2 V== ρδ 3µ

(6-200)

The downward velocity profile u(x) where x = 0 at the solid surface and x = δ at the liquid/gas interface is given by



 

x 2x u = 1.5V  −  δ δ

2

(6-201)

These equations assume that there is no drag force at the gas/liquid interface, such as would be produced by gas flow. For a flat surface inclined at an angle θ with the horizontal, the preceding equations may be modified by replacing g by g sin θ. For films falling inside vertical tubes with film thickness up to and including the full pipe radius, see Jackson (AIChE J., 1, 231–240 [1955]). These equations have generally given good agreement with experimental results for low-viscosity liquids (<0.005 Pa ⋅ s) (< 5 cP) whereas Jackson (ibid.) found film thicknesses for higher-viscosity liquids (0.01 to 0.02 Pa⋅s (10 to 20 cP) were significantly less than predicted by Eq. (6-197). At Reynolds numbers of 25 or greater, surface waves will be present on the liquid film. West and Cole (Chem. Eng. Sci., 22, 1388– 1389 [1967]) found that the surface velocity u(x = δ) is still within 7 percent of that given by Eq. (6-201) even in wavy flow. For laminar non-Newtonian film flow, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977, p. 215, 217), Astarita, Marrucci, and Palumbo (Ind. Eng. Chem. Fundam., 3, 333–339 [1964]) and Cheng (Ind. Eng. Chem. Fundam., 13, 394–395 [1974]). Turbulent Flow In turbulent flow, Re > 2,000, for vertical surfaces, the film thickness may be estimated to within 25 percent using Γ1.75µ0.25 δ = 0.304  ρ2g





1/3

(6-202)

6-44

FLUID AND PARTICLE DYNAMICS

Replace g by g sin θ for a surface inclined at angle θ to the horizontal. The average film velocity is V = Γ/ρδ. Tallmadge and Gutfinger (Ind. Eng. Chem., 59[11], 19–34 [1967]) discuss prediction of drainage rates from liquid films on flat and cylindrical surfaces. Effect of Surface Traction If a drag is exerted on the surface of the film because of motion of the surrounding fluid, the film thickness will be reduced or increased, depending upon whether the drag acts with or against gravity. Thomas and Portalski (Ind. Eng. Chem., 50, 1081–1088 [1958]), Dukler (Chem. Eng. Prog., 55[10], 62–67 [1959]), and Kosky (Int. J. Heat Mass Transfer, 14, 1220–1224 [1971]) have presented calculations of film thickness and film velocity. Film thickness data for falling water films with cocurrent and countercurrent air flow in pipes are given by Zhivaikin (Int. Chem. Eng., 2, 337–341 [1962]). Zabaras, Dukler, and Moalem-Maron (AIChE J., 32, 829–843 [1986]) and Zabaras and Dukler (AIChE J., 34, 389–396 [1988]) present studies of film flow in vertical tubes with both cocurrent and countercurrent gas flow, including measurements of film thickness, wall shear stress, wave velocity, wave amplitude, pressure drop, and flooding point for countercurrent flow. Flooding With countercurrent gas flow, a condition is reached with increasing gas rate for which flow reversal occurs and liquid is carried upward. The mechanism for this flooding condition has been most often attributed to waves either bridging the pipe or reversing direction to flow upward at flooding. However, the results of Zabaras and Dukler (ibid.) suggest that flooding may be controlled by flow conditions at the liquid inlet and that wave bridging or upward wave motion does not occur, at least for the 50.8-mm diameter pipe used for their study. Flooding mechanisms are still incompletely understood. Under some circumstances, as when the gas is allowed to develop its normal velocity profile in a “calming length” of pipe beneath the liquid draw-off, the gas superficial velocity at flooding will be increased, and increases with decreasing length of wetted pipe (Hewitt, Lacy, and Nicholls, Proc. Two-Phase Flow Symp., University of Exeter, paper 4H, AERE-4 4614 [1965]). A bevel cut at the bottom of the pipe with an angle 30° from the vertical will increase the flooding velocity in small-diameter tubes at moderate liquid flow rates. If the gas approaches the tube from the side, the taper should be oriented with the point facing the gas entrance. Figures 6-53 and 6-54 give correlations for flooding in tubes with square and slant bottoms (courtesy Holmes, DuPont Co.) The superficial mass velocities of gas and liquid GG and GL, and the physical property parameters λ and ψ are the same as those defined for the Baker chart (“Multiphase Flow” subsection, Fig. 6-25). For tubes larger than 50 mm (2 in), flooding velocity appears to be relatively insensitive to diameter and the flooding curves for 1.98-in diameter may be used. HYDRAULIC TRANSIENTS Many transient flows of liquids may be analyzed by using the full timedependent equations of motion for incompressible flow. However,

FIG. 6-54 Flooding in vertical tubes with square top and slant bottom. To convert lbm/(ft2 ⋅s) to kg/(m2 ⋅s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.)

there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. Water Hammer When liquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected; a pounding of the line commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the fluid and elasticity of the pipe limit the pressure rise to a finite value. The Joukowski formula gives the maximum pressure rise as ∆p = ρa∆V where

(6-203)

ρ = liquid density ∆V = change in liquid velocity a = pressure wave velocity

The wave velocity is given by β /ρ  a =  +( β /E )( D /b) 1 where

(6-204)

β = liquid bulk modulus of elasticity E = elastic modulus of pipe wall D = pipe inside diameter b = pipe wall thickness

The numerator gives the wave velocity for perfectly rigid pipe, and the denominator corrects for wall elasticity. This formula is for thin-walled pipes; for thick-walled pipes, the factor D/b is replaced by Do2 + Di2 2  Do2 − Di2 where Do = pipe outside diameter Di = pipe inside diameter Example 10: Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2.0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 × 109 Pa. Repeat for a plastic pipe of the same dimensions, with E = 1.4 × 109 Pa. The liquid is water with β = 2.2 × 109 Pa and ρ = 1,000 kg/m3. For the steel pipe, D = 102.3 mm, b = 6.02 mm, and the wave speed is /ρ  β  a = 1  )( D /b ) +β (/E

Flooding in vertical tubes with square top and square bottom. To convert lbm/(ft2 ⋅ s) to kg/(m2 ⋅ s), multiply by 4.8824; to convert in to mm, multiply by 25.4. (Courtesy of E. I. du Pont de Nemours & Co.) FIG. 6-53

 ×109/1000 2.2 =  9  02.3 /6 .0 2) +(2.2 ×10/207 ×109)(1 1 = 1365 m/s

FLUID DYNAMICS The maximum pressure rise is ∆p = ρa∆V = 1,000 × 1,365 × 2.0 = 2.73 × 106 Pa For the plastic pipe,

 ×109/1000 2.2 a =  9 9           .4 ×10 ) (102 .3 /6 .02) 1 + (2.2 × 10 /1 = 282 m/s ∆p = ρa∆V = 1,000 × 282 × 2.0 = 5.64 × 105 Pa

The maximum pressure surge is obtained when the valve closes in less time than the period τ required for the pressure wave to travel from the valve to the pipe inlet and back, a total distance of 2L. 2L τ= (6-205) a The pressure surge will be reduced when the time of flow stoppage exceeds the pipe period τ, due to cancellation between direct and reflected waves. Wood and Jones (Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 99, (HY1), 167–178 [1973]) present charts for reliable estimates of water-hammer pressure for different valve closure modes. Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978) describe several solution methods for hydraulic transients, including the method of characteristics, which is well suited to computer methods for accurate solutions. A rough approximation for the peak pressure for cases where the valve closure time tc exceeds the pipe period τ is (Daugherty and Franzini, Fluid Mechanics with Engineering Applications, McGraw-Hill, New York, 1985): τ ∆p ≈  ρa∆V (6-206) tc Successive reflections of the pressure wave between the pipe inlet and the closed valve result in alternating pressure increases and decreases, which are gradually attenuated by fluid friction and imperfect elasticity of the pipe. Periods of reduced pressure occur while the reflected pressure wave is traveling from inlet to valve. Degassing of the liquid may occur, as may vaporization if the pressure drops below the vapor pressure of the liquid. Gas and vapor bubbles decrease the wave velocity. Vaporization may lead to what is often called liquid column separation; subsequent collapse of the vapor pocket can result in pipe rupture. In addition to water hammer induced by changes in valve setting, including closure, numerous other hydraulic transient flows are of interest, as, for example (Wylie and Streeter, Hydraulic Transients, McGraw-Hill, New York, 1978), those arising from starting or stopping of pumps; changes in power demand from turbines; reciprocating pumps; changing elevation of a reservoir; waves on a reservoir; turbine governor hunting; vibration of impellers or guide vanes in pumps, fans, or turbines; vibration of deformable parts such as valves; draft-tube instabilities due to vortexing; and unstable pump or fan characteristics. Tube failure in heat exhangers may be added to this list. Pulsating Flow Reciprocating machinery (pumps and compressors) produces flow pulsations, which adversely affect flow meters and process control elements and can cause vibration and equipment failure, in addition to undesirable process results. Vibration and damage can result not only from the fundamental frequency of the pulse producer but also from higher harmonics. Multipiston double-acting units reduce vibrations. Pulsation dampeners are often added. Damping methods are described by M. W. Kellogg Co. (Design of Piping Systems, rev. 2d ed., Wiley, New York, 1965). For liquid phase pulsation damping, gas-filled surge chambers, also known as accumulators, are commonly used; see Wylie and Streeter (Hydraulic Transients, McGraw-Hill, New York, 1978). Software packages are commercially available for simulation of hydraulic transients. These may be used to analyze piping systems to reveal unsatisfactory behavior, and they allow the assessment of design changes such as increases in pipe-wall thickness, changes in valve actuation, and addition of check valves, surge tanks, and pulsation dampeners.



6-45

Cavitation Loosely regarded as related to water hammer and hydraulic transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the liquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernoulli equation; see the preceding “Conservation Equations” subsection) as in eddies or vortices, or near boundary contours; by rapid vibration of a boundary; by separation of liquid during water hammer; or by an overall reduction in static pressure, as due to pressure drop in the suction line of a pump. Collapse of vapor bubbles once they reach zones where the pressure exceeds the vapor pressure can cause objectionable noise and vibration and extensive erosion or pitting of the boundary materials. The critical cavitation number at inception of cavitation, denoted σi, is useful in correlating equipment performance data: p − pv σi =  ρV 2/2 where

(6-207)

p = static pressure in undisturbed flow pv = vapor pressure ρ = liquid density V = free-stream velocity of the liquid

The value of the cavitation number for incipient cavitation for a specific piece of equipment is a characteristic of that equipment. Cavitation numbers for various head forms of cylinders, for disks, and for various hydrofoils are given by Holl and Wislicenus (J. Basic Eng., 83, 385–398 [1961]) and for various surface irregularities by Arndt and Ippen (J. Basic Eng., 90, 249–261 [1968]), Ball (Proc. ASCE J. Constr. Div., 89(C02), 91–110 [1963]), and Holl (J. Basic Eng., 82, 169–183 [1960]). As a guide only, for blunt forms the cavitation number is generally in the range of 1 to 2.5, and for somewhat streamlined forms the cavitation number is in the range of 0.2 to 0.5. Critical cavitation numbers generally depend on a characteristic length dimension of the equipment in a way that has not been explained. This renders scale-up of cavitation data questionable. For cavitation in flow through orifices, Fig. 6-55 (Thorpe, Int. J. Multiphase Flow, 16, 1023–1045 [1990]) gives the critical cavitation number for inception of cavitation. To use this cavitation number in Eq. (6-207), the pressure p is the orifice backpressure downstream of the vena contracta after full pressure recovery, and V is the average velocity through the orifice. Figure 6-55 includes data from Tullis and Govindarajan (ASCE J. Hydraul. Div., HY13, 417–430 [1973]) modified to use the same cavitation number definition; their data also include critical cavitation numbers for 30.50- and 59.70-cm pipes

Critical cavitation number vs. diameter ratio β. (Reprinted from Thorpe, “Flow regime transitions due to cavitation in the flow through an orifice,” Int. J. Multiphase Flow, 16, 1023–1045. Copyright © 1990, with kind permission from Elsevier Science, Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, United Kingdom.) FIG. 6-55

6-46

FLUID AND PARTICLE DYNAMICS

(12.00- to 23.50-in). Very roughly, compared with the 15.40-cm pipe, the cavitation number is about 20 percent greater for the 30.50-cm (12.01-in) pipe and about 40 percent greater for the 59.70-cm (23.50-in) diameter pipe. Inception of cavitation appears to be related to release of dissolved gas and not merely vaporization of the liquid. For further discussion of cavitation, see Eisenberg and Tulin (Streeter, Handbook of Fluid Dynamics, Sec. 12, McGraw-Hill, New York, 1961).

velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = x, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side: 3

TURBULENCE Turbulent flow occurs when the Reynolds number exceeds a critical value above which laminar flow is unstable; the critical Reynolds number depends on the flow geometry. There is generally a transition regime between the critical Reynolds number and the Reynolds number at which the flow may be considered fully turbulent. The transition regime is very wide for some geometries. In turbulent flow, variables such as velocity and pressure fluctuate chaotically; statistical methods are used to quantify turbulence. Time Averaging In turbulent flows it is useful to define timeaveraged and fluctuation values of flow variables such as velocity components. For example, the x-component velocity fluctuation v′x is the difference between the actual instantaneous velocity vx and the timeaveraged velocity vx: v′x(x, y, z, t) = vx(x, y, z, t) − vx(x, y, z)

(6-208)

The actual and fluctuating velocity components are, in general, functions of the three spatial coordinates x, y, and z and of time t. The timeaveraged velocity vx is independent of time for a stationary flow. Nonstationary processes may be considered where averages are defined over time scales long compared to the time scale of the turbulent fluctuations, but short compared to longer time scales over which the time-averaged flow variables change due, for example, to timevarying boundary conditions. The time average over a time interval 2T centered at time t of a turbulently fluctuating variable ζ(t) is defined as 1 t+T ζ(t) =  ζ(τ) dτ (6-209) 2T t − T where τ = dummy integration variable. For stationary turbulence, ζ does not vary with time. 1 t+T lim  ζ(τ) dτ (6-210) ζ = T→∞ 2T t − T





The time average of a fluctuation ζ′ = ζ  ζ = 0. Fluctuation magnitudes are quantified by root mean squares. x′ v˜ ′x = (v ) 2

(6-211)

In isotropic turbulence, statistical measures of fluctuations are equal in all directions. v˜ ′x = v˜ ′y = v˜ ′z

(6-212)

∂τ(t) ji

 ∂x

j=1

j

with j components also being x, y, z. The Reynolds stresses are given by τ(t) vi′ v′j ij = −ρ

(6-215)

The Reynolds stresses are nonzero because the velocity fluctuations i  in different coordinate directions are correlated so that v′ v′j in general is nonzero. Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semiempirical models to express the Reynolds stresses in terms of timeaveraged velocities. This is the closure problem of turbulence. In all but the simplest geometries, numerical methods are required. Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simplified here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity µt . ∂vi ∂vj −ρ vi′ v′j = µt  +  (6-216) ∂xj ∂xi





An additional turbulence pressure term equal to −wkδij, where k = turbulent kinetic energy and δi j = 1 if i = j and δij = 0 if i ≠ j, is sometimes included in the right-hand side. To solve the equations of motion using the Boussinesq approximation, it is necessary to provide equations for the single scalar unknown µt (and k, if used) rather than the nine unknown tensor components τ (t) ij . With this approximation, and using the effective viscosity µeff = µ + µt, the time-averaged momentum equation is similar to the original Navier-Stokes equation, with time-averaged variables and µeff replacing the instantaneous variables and molecular viscosity. However, solutions to the time-averaged equations for turbulent flow are not identical to those for laminar flow because µeff is not a constant. The universal turbulent velocity profile near the pipe wall presented in the preceding subsection “Incompressible Flow in Pipes and Channels” may be developed using the Prandtl mixing length approximation for the eddy viscosity,

 

In homogeneous turbulence, turbulence properties are independent of spatial position. The kinetic energy of turbulence k is given by

dvx µt = ρ lP2  dy

1 k =  (˜v′x2 + v˜ ′y2 + v˜ ′z2) 2

where lP is the Prandtl mixing length. The turbulent core of the universal velocity profile is obtained by assuming that the mixing length is proportional to the distance from the wall. The proportionality constant is one of two constants adjusted to fit experimental data. The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD; see the following subsection), is the k–% model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio k2/%.

(6-213)

Turbulent velocity fluctuations ultimately dissipate their kinetic energy through viscous effects. Macroscopically, this energy dissipation requires pressure drop, or velocity decrease. The energy dissipation rate per unit mass is usually denoted %. For steady flow in a pipe, the average energy dissipation rate per unit mass is given by 2f V3 %= D where

(6-214)

ρ = fluid density f = Fanning friction factor D = pipe inside diameter

When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged

(6-217)

k2 µt = ρCµ  (6-218) % where the value Cµ = 0.09 is normally used. Semiempirical partial differential conservation equations for k and % derived from the

FLUID DYNAMICS Navier-Stokes equations with simplifying closure assumptions are coupled with the equations of continuity and momentum: ∂ ∂  (ρk) +  (ρvik) ∂t ∂xi









∂ ∂  (ρ%) +  (ρvi%) ∂t ∂xi ρ%2 ∂ µ ∂% %µ ∂vi ∂vj ∂vi =  t  + C1% t  +   − C2%  ∂x ∂x ∂x j i j ∂xi σ% ∂xi k k





where ν = kinematic viscosity and % = energy dissipation per unit mass. The size of the Kolmogorov eddy scale is lK = (ν3/%)1/4

∂vi ∂vj ∂vi ∂ µ ∂k =  t  + µ t  +   − ρ% (6-219) ∂xi ∂xj ∂xj ∂xi σk ∂xi





(6-220)

In these equations summations over repeated indices are implied. The values for the empirical constants C1% = 1.44, C2% = 1.92, σk = 1.0, and σ% = 1.3 are widely accepted (Launder and Spaulding, The Numerical Computation of Turbulent Flows, Imperial Coll. Sci. Tech. London, NTIS N74-12066 [1973]). The k–% model has proved reasonably accurate for many flows without highly curved streamlines or significant swirl. It usually underestimates flow separation and overestimates turbulence production by normal straining. The k–% model is suitable for high Reynolds number flows. See Virendra, Patel, Rodi, and Scheuerer (AIAA J., 23, 1308–1319 [1984]) for a review of low Reynolds number k–% models. More advanced models, more complex and computationally intensive, are being developed. For example, the renormalization group theory (Yakhot and Orszag, J. Scientific Computing, 1, 1–51 [1986]; Yakhot, Orszag, Thangam, Gatski, and Speziale, Phys. Fluids A, 4, 1510–1520 [1992]) modification of the k–% model provides theoretical values of the model constants and provides substantial improvement in predictions of flows with stagnation, separation, normal straining, transient behavior such as vortex shedding, and relaminarization. Stress transport models provide equations for all nine Reynolds stress components, rather than introducing eddy viscosity. Algebraic closure equations for the Reynolds stresses are available, but are no longer in common use. Differential Reynolds stress models (e.g., Launder, Reece, and Rodi, J. Fluid Mech., 68, 537–566 [1975]) use differential conservation equations for all nine Reynolds stress components. In direct numerical simulation of turbulent flows, the solution of the unaveraged equations of motion is sought. Due to the extreme computational intensity, solutions to date have been limited to relatively low Reynolds numbers in simple geometries. Since computational grids must be sufficiently fine to resolve even the smallest eddies, the computational difficulty rapidly becomes prohibitive as Reynolds number increases. Large eddy simulations use models for subgrid turbulence while solving for larger-scale fluctuations. Eddy Spectrum The energy that produces and sustains turbulence is extracted from velocity gradients in the mean flow, principally through vortex stretching. At Reynolds numbers well above the critical value there is a wide spectrum of eddy sizes, often described as a cascade of energy from the largest down to the smallest eddies. The largest eddies are of the order of the equipment size. The smallest are those for which viscous forces associated with the eddy velocity fluctuations are of the same order as inertial forces, so that turbulent fluctuations are rapidly damped out by viscous effects at smaller length scales. Most of the turbulent kinetic energy is contained in the larger eddies, while most of the dissipation occurs in the smaller eddies. Large eddies, which extract energy from the mean flow velocity gradients, are generally anisotropic. At smaller length scales, the directionality of the mean flow exerts less influence, and local isotropy is approached. The range of eddy scales for which local isotropy holds is called the equilibrium range. Davies (Turbulence Phenomena, Academic, New York, 1972) presents a good discussion of the spectrum of eddy lengths for well-developed isotropic turbulence. The smallest eddies, usually called Kolmogorov eddies (Kolmogorov, Compt. Rend. Acad. Sci. URSS, 30, 301; 32, 16 [1941]), have a characteristic velocity fluctuation v′ ˜ K given by v˜ K′ = (ν%)1/4

(6-221)

6-47

(6-222)

The Reynolds number for the Kolmogorov eddy, ReK = lK v˜ ′k /ν, is equal to unity by definition. In the equilibrium range, which exists for well-developed turbulence and extends from the medium eddy sizes down to the smallest, the energy dissipation at the smaller length scales is supplied by turbulent energy drawn from the bulk flow and passed down the spectrum of eddy lengths according to the scaling rule (˜v′)3 %= l

(6-223)

which is consistent with Eqs. (6-221) and (6-222). For the medium, or energy-containing, eddy size, (˜ve′)3 %= le

(6-224)

τw /ρ used earlier For turbulent pipe flow, the friction velocity u* =  in describing the universal turbulent velocity profile may be used as an estimate for v˜ e′. Together with the Blasius equation for the friction factor from which % may be obtained (Eq. 6-214), this provides an estimate for the energy-containing eddy size in turbulent pipe flow: le = 0.05DRe−1/8

(6-225)

where D = pipe diameter and Re = pipe Reynolds number. Similarly, the Kolmogorov eddy size is lK = 4DRe−0.78

(6-226)

Most of the energy dissipation occurs on a length scale about 5 times the Kolmogorov eddy size. The characteristic fluctuating velocity for these energy-dissipating eddies is about 1.7 times the Kolmogorov velocity. The eddy spectrum is normally described using Fourier transform methods; see, for example, Hinze (Turbulence, McGraw-Hill, New York, 1975), and Tennekes and Lumley (A First Course in Turbulence, MIT Press, Cambridge, 1972). The spectrum E(κ) gives the fraction of turbulent kinetic energy contained in eddies of wavenumber ∞ between κ and κ + dκ, so that k = 0 E(κ) dκ. The portion of the equilibrium range excluding the smallest eddies, those which are affected by dissipation, is the inertial subrange. The Kolmogorov law gives E(κ) ∝ κ−5/3 in the inertial subrange. Several texts are available for further reading on turbulent flow, including Pope (Turbulent Flows, Cambridge University Press, Cambridge, U.K., 2000), Tennekus and Lumley (ibid.), Hinze (Turbulence, McGraw-Hill, New York, 1975), Landau and Lifshitz (Fluid Mechanics, 2d ed., Chap. 3, Pergamon, Oxford, 1987) and Panton (Incompressible Flow, Wiley, New York, 1984). COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations of continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conservation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. Textbooks include Fletcher (Computational Techniques for Fluid Dynamics, vol. 1: Fundamental and General Techniques, and vol. 2: Specific Techniques for Different Flow Categories, Springer-Verlag, Berlin, 1988), Hirsch (Numerical Computation of Internal and External Flows, vol. 1: Fundamentals of Numerical Discretization, and vol. 2: Computational Methods for Inviscid and Viscous Flows, Wiley, New York, 1988), Peyret and Taylor (Computational Methods for Fluid

6-48

FLUID AND PARTICLE DYNAMICS

Computational fluid dynamic simulation of flow over a square cylinder, showing one vortex shedding period. (From Choudhury et al., Trans. ASME Fluids Div., TN-076 [1994].)

FIG. 6-56

FLUID DYNAMICS Flow, Springer-Verlag, Berlin, 1990), Canuto, Hussaini, Quarteroni, and Zang (Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988), Anderson, Tannehill, and Pletcher (Computational Fluid Mechanics and Heat Transfer, Hemisphere, New York, 1984), and Patankar (Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C., 1980). A wide variety of numerical methods has been employed, but three basic steps are common. 1. Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or curves which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 2. Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. The finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of the nodal unknowns at each interior node. Finite volume methods, related to finite difference methods, may be derived by a volume integration of the equations of motion, with application of the divergence theorem, reducing by one the order of the differential equations. Equivalently, macroscopic balance equations are written on each cell. Finite element methods are weighted residual techniques in which the unknown dependent variables are expressed in terms of basis functions interpolating among the nodal values. The basis functions are substituted into the equations of motion, resulting in error residuals which are multiplied by the weighting functions, integrated over the control volume, and set to zero to produce algebraic equations in terms of the nodal unknowns. Selection of the weighting functions defines the various finite element methods. For example, Galerkin’s method uses the nodal interpolation basis functions as weighting functions. Each method also has its own method for implementing boundary conditions. The end result after discretization of the equations and application of the boundary conditions is a set of algebraic equations for the nodal unknown variables. Discretization in time is also required for the ∂/∂t time derivative terms in unsteady flow; finite differencing in time is often used. The discretized equations represent an approximation of the exact equations, and their solution gives an approximation for the flow variables. The accuracy of the solution improves as the grid is refined; that is, as the number of nodal points is increased. 3. Solution of the algebraic equations. For creeping flows with constant viscosity, the algebraic equations are linear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. A CFD method called the lattice Boltzmann method is based on modeling the fluid as a set of particles moving with discrete velocities on a discrete grid or lattice, rather than on discretization of the governing continuum partial differential equations. Lattice Boltzmann approximations can be constructed that give the same macroscopic behavior as the Navier-Stokes equations. The method is currently used mainly in academic and research codes, rather than in general-purpose commercial CFD codes. There appear to be significant computational advantages to the lattice Boltzmann method. Lattice Boltzmann simulations incorporating turbulence models, and of multiphase flows and flows with heat transfer, species diffusion, and reaction, have been carried out. For a review of the method, see Chen and Doolen [Ann. Rev. Fluid Mech., 30, 329 (1998)].

6-49

CFD solutions, especially for complex three-dimensional flows, generate very large quantities of solution data. Computer graphics have greatly improved the ability to examine CFD solutions and visualize flow. CFD methods are used for incompressible and compressible, creeping, laminar and turbulent, Newtonian and non-Newtonian, and isothermal and nonisothermal flows. Chemically reacting flows, particularly in the field of combustion, have been simulated. Solution accuracy must be considered from several perspectives. These include convergence of the algorithms for solving the nonlinear discretized equations and convergence with respect to refinement of the mesh so that the discretized equations better approximate the exact equations and, in some cases, so that the mesh more accurately fits the true geometry. The possibility that steady-state solutions are unstable must always be considered. In addition to numerical sources of error, modeling errors are introduced in turbulent flow, where semiempirical closure models are used to solve time-averaged equations of motion, as discussed previously. Most commercial CFD codes include the k–% turbulence model, which has been by far the most widely used. More accurate models, such as differential Reynolds stress and renormalization group theory models, are also becoming available. Significant solution error is known to result in some problems from inadequacy of the turbulence model. Closure models for nonlinear chemical reaction source terms may also contribute to inaccuracy. Large eddy simulation (LES) methods for turbulent flow are available in some commercial CFD codes. LES methods are based on filtering fluctuating variables, so that lower-frequency eddies, with scales larger than the grid spacing, are resolved, while higher-frequency eddies, the subgrid fluctuations, are filtered out. The subgrid-scale Reynolds stress is estimated by a turbulence model. The Smagorinsky model, a one-equation mixing length model, is used in most commercial codes that offer LES options and is also used in many academic and research CFD codes. See Wilcox (Turbulence Modeling for CFD, ~ada, Calif., 1998). 2d ed., DCW Industries, La Can In its general sense, multiphase flow is not currently solvable by computational fluid dynamics. However, in certain cases reasonable solutions are possible. These include well-separated flows where the phases are confined to relatively well-defined regions separated by one or a few interfaces and flows in which a second phase appears as discrete particles of known size and shape whose motion may be approximately computed with drag coefficient formulations, or rigorously computed with refined meshes applying boundary conditions at the particle surface. Two-fluid modeling, in which the phases are treated as overlapping continua, with each phase occupying a volume fraction that is a continuous function of position (and time) is a useful approximation which is becoming available in commercial software. See Elghobashi and Abou-Arab ( J. Physics Fluids, 26, 931–938 [1983]) for a k–% model for two-fluid systems. Figure 6-56 gives an example CFD calculation for time-dependent flow past a square cylinder at a Reynolds number of 22,000 (Choudhury, et al., Trans. ASME Fluids Div., Lake Tahoe, Nev. [1994]). The computation was done with an implementation of the renormalization group theory k–% model. The series of contour plots of stream function shows a sequence in time over about 1 vortex-shedding period. The calculated Strouhal number (Eq. [6-195]) is 0.146, in excellent agreement with experiment, as is the time-averaged drag coefficient, CD = 2.24. Similar computations for a circular cylinder at Re = 14,500 have given excellent agreement with experimental measurements for St and CD (Introduction to the Renormalization Group Method and Turbulence Modeling, Fluent, Inc., 1993). DIMENSIONLESS GROUPS For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly found in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may be found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46–60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71–78 [1968]).

6-50

FLUID AND PARTICLE DYNAMICS

TABLE 6-7

Dimensionless Groups and Their Significance

Name

Symbol

Formula

Archimedes number

Ar

gL (ρp − ρ)ρ  µ2

inertial forces × buoyancy forces  (viscous forces)2

Particle settling

Bingham number

Bm

τ yL  µ ∞V

yield stress  viscous stress

Flow of Bingham plastics = yield number, Y

Bingham Reynolds number

ReB

LVρ  µ∞

inertial force  viscous force

Flow of Bingham plastics

Blake number

B

Vρ  µ(1 − %)s

inertial force  viscous force

Beds of solids

Bond number

Bo

(ρL − ρG)L2g  σ

gravitational force  surface-tension force

Atomization = Eotvos number, Eo

Capillary number

Ca

µV  σ

viscous force  surface-tension force

Two-phase flows, free surface flows

Cauchy number

C

ρV 2  β

inertial force  compressibility force

Compressible flow, hydraulic transients

Cavitation number

σ

p − pv  ρV 2/2

excess pressure above vapor pressure  velocity head

Cavitation

Dean number

De

Re  (Dc/D)1/2

inertial force Reynolds number ×  centrifugal force

Flow in curved channels

Deborah number

De

λ

fluid relaxation time  flow characteristic time

Viscoelastic flow

Drag coefficient

CD

FD  AρV 2/2

drag force  projected area × velocity head

Flow around objects, particle settling

Elasticity number

El

λµ 2 ρL

elastic force  inertial force

Viscoelastic flow

Euler number

Eu

∆p 2 ρV

frictional pressure loss  2 × velocity head

Fluid friction in conduits

Fanning friction factor

f

D∆p 2τ w =  2ρV 2L ρV 2

wall shear stress  velocity head

Fluid friction in conduits Darcy friction factor = 4f

Froude number

Fr

V2  gL

inertial force  gravity force

 Often defined as Fr = V/gL

Densometric Froude number

Fr′

ρV 2  (ρd − ρ)gL

inertial force  gravity force

or

Hedstrom number

He

L2 τYρ  µ ∞2

Bingham Reynolds number × Bingham number

Flow of Bingham plastics

Hodgson number

H

V′ω∆p  q p 

time constant of system  period of pulsation

Pulsating gas flow

Mach number

M

V  c

fluid velocity  sonic velocity

Compressible flow

Newton number

Ne

PD  V2L

2  Fanning friction factor

Ohnesorge number

Z

µ  (ρLσ)1/2

viscous force  (inertial force × surface tension force)1/2

Weber number Atomization =  Reynolds number

Peclet number

Pe

LV  D

convective transport  diffusive transport

Heat, mass transfer, mixing

Pipeline parameter

Pn

aVo  2gH

maximum water-hammer pressure rise  2 × static pressure

Water hammer

3

Physical interpretation

Comments

V Fr′ =  d−  L /ρ  ρ)g (ρ

PARTICLE DYNAMICS TABLE 6-7

Dimensionless Groups and Their Significance (Concluded)

Name

Symbol

Formula

Physical interpretation impeller drag force  inertial force

Comments

Power number

Po

P  ρN 3L5

Prandtl velocity ratio

v+

v  (τ w /ρ)1/2

velocity normalized by friction velocity

Reynolds number

Re

LVρ  µ

inertial force  viscous force

Strouhal number

St

f′L  V

vortex shedding frequency × characteristic flow time scale

Weber number

We

ρV 2L  σ

inertial force  surface tension force SI Units Nomenclature

Nomenclature a A c D Dc D′ f′ FD g H L N p pv p  ∆p

6-51

Wave speed Projected area Sonic velocity Diameter of pipe Diameter of curvature Diffusivity Vortex shedding frequency Drag force Acceleration of gravity Static head Characteristic length Rotational speed Pressure Vapor pressure Average static pressure Frictional pressure drop

m/s m m/s m m m2/s 1/s N m/s m m 1/s Pa Pa Pa Pa

Agitation

P q  s v V V′  % λ µ µ∞ ρ ρG, ρL ρd σ ω

Turbulent flow near a wall, friction velocity =  τw /ρ

Vortex shedding, von Karman vortex streets Bubble, drop formation SI Units

Power Average volumetric flow rate Particle area/particle volume Local fluid velocity Characteristic or average fluid velocity System volume Bulk modulus Void fraction Fluid relaxation time Fluid viscosity Infinite shear viscosity (Bingham plastics) Fluid density Gas, liquid densities Dispersed phase density Surface tension Characteristic frequency or reciprocal time scale of flow

Watts m3/s 1/m m/s m/s m3 Pa m3 s Pa ⋅ s Pa ⋅ s kg/m3 kg/m3 kg/m3 N/m 1/s

PARTICLE DYNAMICS GENERAL REFERENCES: Brodkey, The Phenomena of Fluid Motions, AddisonWesley, Reading, Mass., 1967; Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951; Levich, Physicochemical Hydrodynamics, PrenticeHall, Englewood Cliffs, N.J., 1962; Orr, Particulate Technology, Macmillan, New York, 1966; Shook and Roco, Slurry Flow, Butterworth-Heinemann, Boston, 1991; Wallis, One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969.

forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces. TERMINAL SETTLING VELOCITY A particle falling under the action of gravity will accelerate until the drag force balances gravitational force, after which it falls at a constant terminal or free-settling velocity ut, given by ut =

DRAG COEFFICIENT Whenever relative motion exists between a particle and a surrounding fluid, the fluid will exert a drag upon the particle. In steady flow, the drag force on the particle is CD APρu2 FD =  (6-227) 2 where FD = drag force CD = drag coefficient AP = projected particle area in direction of motion ρ = density of surrounding fluid u = relative velocity between particle and fluid The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some solid bodies, such as aerofoils, a lift force component perpendicular to the liquid velocity is also exerted. For free-falling particles, lift

where

2gm (ρ − ρ)  ρρ A C p

p

p

P

(6-228)

D

g = acceleration of gravity m p = particle mass ρp = particle density

and the remaining symbols are as previously defined. Settling particles may undergo fluctuating motions owing to vortex shedding, among other factors. Oscillation is enhanced with increasing separation between the mass and geometric centers of the particle. Variations in mean velocity are usually less than 10 percent. The drag force on a particle fixed in space with fluid moving is somewhat lower than the drag force on a particle freely settling in a stationary fluid at the same relative velocity. Spherical Particles For spherical particles of diameter dp, Eq. (6-228) becomes 4gdp(ρp − ρ) ut =  (6-229) 3ρCD



6-52

FLUID AND PARTICLE DYNAMICS

Drag coefficients for spheres, disks, and cylinders: Ap = area of particle projected on a plane normal to direction of motion; C = overall drag coefficient, dimensionless; Dp = diameter of particle; Fd = drag or resistance to motion of body in fluid; Re = Reynolds number, dimensionless; u = relative velocity between particle and main body of fluid; µ = fluid viscosity; and ρ = fluid density. (From Lapple and Shepherd, Ind. Eng. Chem., 32, 605 [1940].)

FIG. 6-57

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Rep = dpρu/µ where µ = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes’ law gives 24 CD =  Rep < 0.1 (6-230) Rep which may also be written FD = 3πµ udp

Rep < 0.1

(6-231)

and gives for the terminal settling velocity gdp2 (ρp − ρ) ut =  18µ

Rep < 0.1

(6-232)

In the intermediate regime (0.1 < Rep < 1,000), the drag coefficient may be estimated within 6 percent by

 

24 CD =  1 + 0.14Re 0.70 p Rep



0.1 < Rep < 1,000

(6-233)

In the Newton’s law regime, which covers the range 1,000 < Rep < 350,000, CD = 0.445, within 13 percent. In this region, Eq. (6-227) becomes gd (ρ − ρ)  ρ

ut = 1.73

p

p

1,000 < Rep < 350,000

(6-234)

Between about Rep = 350,000 and 1 × 106, the drag coefficient drops dramatically in a drag crisis owing to the transition to turbulent flow in the boundary layer around the particle, which delays aft separation, resulting in a smaller wake and less drag. Beyond Re = 1 × 106, the drag coefficient may be estimated from (Clift, Grace, and Weber): 8 × 104 CD = 0.19 −  Rep

Rep > 1 × 106

(6-235)

Drag coefficients may be affected by turbulence in the free-stream flow; the drag crisis occurs at lower Reynolds numbers when the free

stream is turbulent. Torobin and Guvin (AIChE J., 7, 615–619 [1961]) found that the drag crisis Reynolds number decreases with increasing free-stream turbulence, reaching a value of 400 when the relative turbulence intensity, defined as u′/U R is 0.4. Here u′  is the rms fluctuating velocity and  UR is the relative velocity between the particle and the fluid. For computing the terminal settling velocity, correlations for drag coefficient as a function of Archimedes number gd3p () Ar  (6-236) 2 may be more convenient than CD-Re correlations, because the latter are implicit in terminal velocity, and the settling regime is unknown. Karamanev [Chem. Eng Comm. 147, 75 (1996)] provided a correlation for drag coefficient for settling solid spheres in terms of Ar. 432 0.517 CD  (1  0.0470Ar 2/3)   (6-237)  Ar 1  154Ar−1/3 This equation reduces to Stokes’ law CD = 24/Re in the limit Ar —>0 and is a fit to data up to about Ar = 2 × 1010, where it gives CD 0.50, slightly greater than the Newton’s law value above. For rising light spheres, which exhibit more energy dissipating lateral motion than do falling dense spheres, Karamanev found that Eq. (6-237) is followed up to Ar = 13,000 and that for Ar 13,000, the drag coefficient is CD = 0.95. For particles settling in non-Newtonian fluids, correlations are given by Dallon and Christiansen (Preprint 24C, Symposium on Selected Papers, part III, 61st Ann. Mtg. AIChE, Los Angeles, Dec. 1–5, 1968) for spheres settling in shear-thinning liquids, and by Ito and Kajiuchi (J. Chem. Eng. Japan, 2[1], 19–24 [1969]) and Pazwash and Robertson (J. Hydraul. Res., 13, 35–55 [1975]) for spheres settling in Bingham plastics. Beris, Tsamopoulos, Armstrong, and Brown (J. Fluid Mech., 158 [1985]) present a finite element calculation for creeping motion of a sphere through a Bingham plastic. Nonspherical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the

PARTICLE DYNAMICS TABLE 6-8

6-53

Free-Fall Orientation of Particles

Reynolds number*

Orientation

0.1–5.5

All orientations are stable when there are three or more perpendicular axes of symmetry. Stable in position of maximum drag. Unpredictable. Disks and plates tend to wobble, while fuller bluff bodies tend to rotate. Rotation about axis of least inertia, frequently coupled with spiral translation.

5.5–200 200–500 500–200,000 SOURCE:

From Becker, Can. J. Chem. Eng., 37, 85–91 (1959). *Based on diameter of a sphere having the same surface area as the particle.

direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direction of motion) are given in Fig. 6-57 as a function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton’s law region is reported by Knudsen and Katz (Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). Pettyjohn and Christiansen (Chem. Eng. Prog., 44, 157–172 [1948]) present correlations for the effect of particle shape on freesettling velocities of isometric particles. For Re < 0.05, the terminal or free-settling velocity is given by gd s2(ρp − ρ) ut = K1  (6-238) 18µ ψ K1 = 0.843 log  (6-239) 0.065 where ψ = sphericity, the surface area of a sphere having the same volume as the particle, divided by the actual surface area of the particle; ds = equivalent diameter, equal to the diameter of the equivalent sphere having the same volume as the particle; and other variables are as previously defined. In the Newton’s law region, the terminal velocity is given by



ut =



4d (ρ − ρ)g  3K ρ s

p

(6-240)

3

K3 = 5.31 − 4.88ψ

(6-241)

Equations (6-238) to (6-241) are based on experiments on cubeoctahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity ψ ranges from 0.906 to 0.670, respectively. See also Clift, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with ψ as a parameter may be found in Brown, et al. (Unit Operations, Wiley, New York, 1950) and in Govier and Aziz. For particles with ψ < 0.67, the correlations of Becker (Can. J. Chem. Eng., 37, 85–91 [1959]) should be used. Reference to this paper is also recommended for intermediate region flow. Settling characteristics of nonspherical particles are discussed by Clift, Grace, and Weber, Chaps. 4 and 6. The terminal velocity of axisymmetric particles in axial motion can be computed from Bowen and Masliyah (Can. J. Chem. Eng., 51, 8–15 [1973]) for low–Reynolds number motion: V′ gDs2(ρp − ρ) (6-242) ut =   K2 18µ K2 = 0.244 + 1.035) − 0.712)2 + 0.441)3

(6-243)

where Ds = diameter of sphere with perimeter equal to maximum particle projected perimeter V′ = ratio of particle volume to volume of sphere with diameter Ds ) = ratio of surface area of particle to surface area of a sphere with diameter Ds and other variables are as defined previously.

Values of exponent n for use in Eq. (6-242). (From Maude and Whitmore, Br. J. Appl. Phys., 9, 481 [1958]. Courtesy of the Institute of Physics and the Physical Society.)

FIG. 6-58

Hindered Settling When particle concentration increases, particle settling velocities decrease because of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered settling is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore (Br. J. Appl. Phys., 9, 477–482 [1958]) give, for uniformly sized spheres, ut = ut0 (1 − c)n

(6-244)

where ut = terminal settling velocity ut0 = terminal velocity of a single sphere (infinite dilution) c = volume fraction solid in the suspension n = function of Reynolds number Rep = dput0 ρ/µ as given Fig. 6-58 In the Stokes’ law region (Rep < 0.3), n = 4.65 and in the Newton’s law region (Rep > 1,000), n = 2.33. Equation (6-244) may be applied to particles of any size in a polydisperse system, provided the volume fraction corresponding to all the particles is used in computing terminal velocity (Richardson and Shabi, Trans. Inst. Chem. Eng. [London], 38, 33–42 [1960]). The concentration effect is greater for nonspherical and angular particles than for spherical particles (Steinour, Ind. Eng. Chem., 36, 840–847 [1944]). Theoretical developments for low–Reynolds number flow assemblages of spheres are given by Happel and Brenner (Low Reynolds Number Hydrodynamics, PrenticeHall, Englewood Cliffs, N.J., 1965) and Famularo and Happel (AIChE J., 11, 981 [1965]) leading to an equation of the form ut0 ut =  1 + γc1/3

(6-245)

where γ is about 1.3. As particle concentration increases, resulting in interparticle contact, hindered settling velocities are difficult to predict. Thomas (AIChE J., 9, 310 [1963]) provides an empirical expression reported to be valid over the range 0.08 < ut /ut0 < 1: ut ln  = −5.9c (6-246) ut0



Time-dependent Motion The time-dependent motion of particles is computed by application of Newton’s second law, equating the rate of change of particle motion to the net force acting on the particle. Rotation of particles may also be computed from the net torque. For large particles moving through low-density gases, it is usually sufficient to compute the force due to fluid drag from the

6-54

FLUID AND PARTICLE DYNAMICS

relative velocity and the drag coefficient computed for steady flow conditions. For two- and three-dimensional problems, the velocity appearing in the particle Reynolds number and the drag coefficient is the amplitude of the relative velocity. The drag force, not the relative velocity, is to be resolved into vector components to compute the particle acceleration components. Clift, Grace, and Weber (Bubbles, Drops and Particles, Academic, London, 1978) discuss the complexities that arise in the computation of transient drag forces on particles when the transient nature of the flow is important. Analytical solutions for the case of a single particle in creeping flow (Rep = 0) are available. For example, the creeping motion of a sphericial particle released from rest in a stagnant fluid is described by dU ρ dU ρpV  = g(ρp − ρ)V − 3πµdpU −  V  dt 2 dt



3  −  d 2p  π ρµ 2

(dU/dt) ds   s t− t

t=s

(6-247)

0

Here, U = particle velocity, positive in the direction of gravity, and V = particle volume. The first term on the right-hand side is the net gravitational force on the particle, accounting for buoyancy. The second is the steady-state Stokes drag (Eq. 6-231). The third is the added mass or virtual mass term, which may be interpreted as the inertial effect of the fluid which is accelerated along with the particle. The volume of the added mass of fluid is half the particle volume. The last term, the Basset force, depends on the entire history of the transient motion, with past motions weighted inversely with the square root of elapsed time. Clift, et al. provide integrated solutions. In turbulent flows, particle velocity will closely follow fluid eddy velocities when (Clift et al.) d p2 [(2ρp /ρ) + 1] τ0 >>  (6-248) 36ν where τ0 = oscillation period or eddy time scale, the right-hand side expression is the particle relaxation time, and ν = kinematic viscosity. Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(ρL − ρG)de/σ, where ρL = liquid density, ρG = gas density, de = bubble diameter, σ = surface tension, and the equivalent diameter de is the diameter of a sphere with volume equal to that of

the bubble. Small bubbles (<1-mm [0.04-in] diameter) remain spherical and rise in straight lines. The presence of surface active materials generally renders small bubbles rigid, and they rise roughly according to the drag coefficient and terminal velocity equations for spherical solid particles. Bubbles roughly in the range 2- to 8-mm (0.079- to 0.32-in) diameter assume flattened, ellipsoidal shape, and rise in a zigzag or spiral pattern. This motion increases dissipation and drag, and the rise velocity may actually decrease with increasing bubble diameter in this region, characterized by rise velocities in the range of 20 to 30 cm/s (0.7 to 1.0 ft/s). Large bubbles, >8-mm (0.32-in) diameter, are greatly deformed, assuming a mushroomlike, spherical cap shape. These bubbles are unstable and may break into smaller bubbles. Carefully purified water, free of surface active materials, allows bubbles to freely circulate even when they are quite small. Under creeping flow conditions Reb = dburρL /µL < 1, where ur = bubble rise velocity and µL = liquid viscosity, the bubble rise velocity may be computed analytically from the Hadamard-Rybczynski formula (Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1962, p. 402). When µG /µL << 1, which is normally the case, the rise velocity is 1.5 times the rigid sphere Stokes law velocity. However, in practice, most liquids, including ordinary distilled water, contain sufficient surface active materials to render small bubbles rigid. Larger bubbles undergo deformation in both purified and ordinary liquids; however, the variation in rise velocity for large bubbles with degree of purity is quite evident in Fig. 6-59. For additional discussion, see Clift, et al., Chap. 7. Karamanev [op. cit.] provided equations for bubble rise velocity based on the Archimedes number and on use of the bubble projected diameter dh in the drag coefficient and the bubble equivalent diameter in Ar. The Archimedes number is as defined in Eq. (6-236) except that the density difference is liquid density minus gas density, and dp is replaced by de. d ut 40.3 e dh

d (#d /6) V   40.3  d C C 1/3 D

h

0.517 432 CD  (10.0470Ar 2/3)   1154Ar1/3 Ar CD 0.95

3 e

e

(6-249)

D

Ar , 13,000

Ar 13,000

FIG. 6-59 Terminal velocity of air bubbles in water at 20°C. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978).

1/3

(6-250)

(6-251)

PARTICLE DYNAMICS

6-55

Note that the terminal velocity may be evaluated explicitly from µ u =  M −0.149(J − 0.857) ρd

(6-259)

In Eq. (6-257), µ = viscosity of continuous liquid and µw = viscosity of water, taken as 0.9 cP (0.0009 Pa ⋅ s). For drop velocities in non-Newtonian liquids, see Mhatre and Kinter (Ind. Eng. Chem., 51, 865–867 [1959]); Marrucci, Apuzzo, and Astarita (AIChE J., 16, 538–541 [1970]); and Mohan, et al. (Can. J. Chem. Eng., 50, 37–40 [1972]). Liquid Drops in Gases Liquid drops falling in stagnant gases appear to remain spherical and follow the rigid sphere drag relationships up to a Reynolds number of about 100. Large drops will deform,

Drag coefficient for water drops in air and air bubbles in water. Standard drag curve is for rigid spheres. (From Clift, Grace, and Weber, Bubbles, Drops and Particles, Academic, New York, 1978.)

FIG. 6-60

de  (1  0.163Eo0.757)1/3 Eo , 40 dh de  0.62 dh

Eo 40

(6-252)

(6-253)

Applied to air bubbles in water, these expressions give reasonable agreement with the contaminated water curve in Fig. 6-59. Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815–820 [1965]); Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235–256 [1970]); and Acharya, Mashelkar, and Ulbrecht (Chem. Eng. Sci., 32, 863–872 [1977]). Liquid Drops in Liquids Very small liquid drops in immisicibile liquids behave like rigid spheres, and the terminal velocity can be approximated by use of the drag coefficient for solid spheres up to a Reynolds number of about 10 (Warshay, Bogusz, Johnson, and Kintner, Can. J. Chem. Eng., 37, 29–36 [1959]). Between Reynolds numbers of 10 and 500, the terminal velocity exceeds that for rigid spheres owing to internal circulation. In normal practice, the effect of drop phase viscosity is neglected. Grace, Wairegi, and Nguyen (Trans. Inst. Chem. Eng., 54, 167–173 [1976]; Clift, et al., op. cit., pp. 175–177) present a correlation for terminal velocity valid in the range Eo < 40 Re > 0.1 (6-254) M < 10−3 where M = Morton number = gµ4∆ρ/ρ2σ3 Eo = Eotvos number = g∆ρd 2/σ Re = Reynolds number = duρ/µ ∆ρ = density difference between the phases ρ = density of continuous liquid phase d = drop diameter µ = continuous liquid viscosity σ = surface tension u = relative velocity The correlation is represented by

where

J = 0.94H0.757

(2 < H ≤ 59.3)

(6-255)

J = 3.42H0.441

(H > 59.3)

(6-256)

4 µ H =  EoM−0.149  3 µw

−0.14



(6-257)

J = ReM 0.149 + 0.857

(6-258)

Terminal velocities of spherical particles of different densities settling in air and water at 70°F under the action of gravity. To convert ft/s to m/s, multiply by 0.3048. (From Lapple, et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 292.)

FIG. 6-61

6-56

FLUID AND PARTICLE DYNAMICS

with a resulting increase in drag, and in some cases will shatter. The largest water drop which will fall in air at its terminal velocity is about 8 mm (0.32 in) in diameter, with a corresponding velocity of about 9 m/s (30 ft/s). Drops shatter when the Weber number defined as ρGu d We =  σ 2

(6-260)

exceeds a critical value. Here, ρG = gas density, u = drop velocity, d = drop diameter, and σ = surface tension. A value of Wec = 13 is often cited for the critical Weber number. Terminal velocities for water drops in air have been correlated by Berry and Prnager (J. Appl. Meteorol., 13, 108–113 [1974]) as Re = exp [−3.126 + 1.013 ln ND − 0.01912(ln ND)2]

β*

kw

β

kw

0.0 0.05 0.1 0.2 0.3

1.000 0.885 0.792 0.596 0.422

0.4 0.5 0.6 0.7 0.8

0.279 0.170 0.0945 0.0468 0.0205

SOURCE: From Haberman and Sayre, David W. Taylor Model Basin Report 1143, 1958. *β = particle diameter divided by vessel diameter.

(6-261)

for 2.4 < ND < 107 and 0.1 < Re < 3,550. The dimensionless group ND (often called the Best number [Clift et al.]) is given by 4ρ∆ρgd 3 ND =  3µ2

TABLE 6-9 Wall Correction Factor for Rigid Spheres in Stokes’ Law Region

(6-262)

and is proportional to the similar Archimedes and Galileo numbers. Figure 6-61 gives calculated settling velocities for solid spherical particles settling in air or water using the standard drag coefficient curve for spherical particles. For fine particles settling in air, the Stokes-Cunningham correction has been applied to account for particle size comparable to the mean free path of the gas. The correction is less than 1 percent for particles larger than 16 µm settling in air. Smaller particles are also subject to Brownian motion. Motion of particles smaller than 0.1 µm is dominated by Brownian forces and gravitational effects are small. Wall Effects When the diameter of a settling particle is significant compared to the diameter of the container, the settling velocity is

reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor kw is multiplied by the settling velocity obtained from Stokes’ law to obtain the corrected settling rate. For values of diameter ratio β = particle diameter/vessel diameter less than 0.05, kw = 1/(1 + 2.1β) (Zenz and Othmer, Fluidization and Fluid-Particle Systems, Reinhold, New York, 1960, pp. 208–209). In the range 100 < Re < 10,000, the computed terminal velocity for rigid spheres may be multiplied by k′w to account for wall effects, where k′w is given by (Harmathy, AIChE J., 6, 281 [1960]) 1 − β2 k′w = 4 (6-263) + β 1 For gas bubbles in liquids, there is little wall effect for β < 0.1. For β > 0.1, see Uto and Kintner (AIChE J., 2, 420–424 [1956]), Maneri and Mendelson (Chem. Eng. Prog., 64, Symp. Ser., 82, 72–80 [1968]), and Collins (J. Fluid Mech., 28, part 1, 97–112 [1967]).

Section 7

Reaction Kinetics*

Tiberiu M. Leib, Ph.D. Principal Consultant, DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company; Fellow, American Institute of Chemical Engineers Carmo J. Pereira, Ph.D., MBA DuPont Fellow, DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company; Fellow, American Institute of Chemical Engineers

REFERENCES BASIC CONCEPTS Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Concentration on Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat of Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion, Extent of Reaction, Selectivity, and Yield . . . . . . . . . . . . . . Concentration Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stoichiometric Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Catalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-5 7-5 7-5 7-6 7-6 7-6 7-6 7-7 7-7 7-8 7-8 7-8 7-9 7-9

IDEAL REACTORS Ideal Batch Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactor (BR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semibatch Reactor (SBR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Continuous Stirred Tank Reactor (CSTR) . . . . . . . . . . . . . . . . . . . Plug Flow Reactor (PFR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Recycle Reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples for Some Simple Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-11 7-11 7-12 7-12 7-12 7-12 7-13

KINETICS OF COMPLEX HOMOGENEOUS REACTIONS Chain Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phosgene Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ozone Conversion to Oxygen in Presence of Chlorine. . . . . . . . . . . . Hydrogen Bromide Synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonchain Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Homogeneous Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-14 7-14 7-14 7-15 7-15 7-15 7-15

*The contributions of Stanley M. Walas, Ph.D., Professor Emeritus, Department of Chemical and Petroleum Engineering, University of Kansas (Fellow, American Institute of Chemical Engineers), author of this section in the seventh edition, are acknowledged. The authors of the present section would like to thank Dennie T. Mah, M.S.Ch.E., Senior Consultant, DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company (Senior Member, American Institute of Chemical Engineers; Member, Industrial Electrolysis and Electrochemical Engineering; Member, The Electrochemical Society), for his contributions to the “Electrochemical Reactions” subsection; and John Villadsen, Ph.D., Senior Professor, Department of Chemical Engineering, Technical University of Denmark, for his contributions to the “Biochemical Reactions” subsection. We acknowledge comments from Peter Harriott, Ph.D., Fred H. Rhodes Professor of Chemical Engineering (retired), School of Chemical and Biomolecular Engineering, Cornell University, on our original outline and on the subject of heat transfer in packed-bed reactors. The authors also are grateful to the following colleagues for reading the manuscript and for thoughtful comments: Thomas R. Keane, DuPont Fellow (retired), DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company (Senior Member, American Institute of Chemical Engineers); Güray Tosun, Ph.D., Senior Consultant, DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company (Senior Member, American Institute of Chemical Engineers); and Nitin H. Kolhapure, Ph.D., Senior Consulting Engineer, DuPont Engineering Research and Technology, E. I. du Pont de Nemours and Company (Senior Member, American Institute of Chemical Engineers). 7-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

7-2

REACTION KINETICS

Acid-Catalyzed Isomerization of Butene-1 . . . . . . . . . . . . . . . . . . . . . Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-15 7-15 7-16

INTRINSIC KINETICS FOR FLUID-SOLID CATALYTIC REACTIONS Adsorption Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-16 Dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 Different Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 Change in Number of Moles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 Reactant in the Gas Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 Chemical Equilibrium in Gas Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17 No Rate-Controlling Step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18 Liquid-Solid Catalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18 Biocatalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-18 FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER Gas-Solid Catalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19 External Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-19 Intraparticle Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-20 Intraparticle Diffusion and External Mass-Transfer Resistance. . . . . 7-22 Heat-Transfer Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-22 Catalyst Deactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-22 Gas-Solid Noncatalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23 Sharp Interface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-23 Volume Reaction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-25 GAS-LIQUID REACTIONS Reaction-Diffusion Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-27

GAS-LIQUID-SOLID REACTIONS Gas-Liquid-Solid Catalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymerization Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bead Polymerization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emulsion Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution Polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymer Characterization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-28 7-29 7-29 7-29 7-29 7-29 7-29

Chain Homopolymerization Mechanism and Kinetics . . . . . . . . . . . . Step Growth Homopolymerization Mechanism and Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Copolymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biochemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monod-Type Empirical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemostat with Empirical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrochemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ohmic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .DETERMINATION OF MECHANISM AND KINETICS Laboratory Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiphase Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Catalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bioreactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Half-Life Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Models in Parameters, Single Reaction . . . . . . . . . . . . . . . . . . Nonlinear Models in Parameters, Single Reaction . . . . . . . . . . . . . . . Network of Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction of Mechanism and Kinetics . . . . . . . . . . . . . . . . . . . . . . . . Lumping and Mechanism Reduction . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Steady States, Oscillations, and Chaotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Software Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-30 7-30 7-30 7-30 7-31 7-31 7-32 7-32 7-32 7-33 7-33 7-33

7-33 7-34 7-35 7-35 7-35 7-35 7-35 7-35 7-36 7-36 7-36 7-36 7-37 7-37 7-37 7-38 7-38 7-38 7-38 7-38 7-39 7-39

Nomenclature and Units The component A is identified by the subscript a. Thus, the number of moles is na; the fractional conversion is Xa; the extent of reaction is ζa; the partial pressure is p; the rate of consumption is ra; the molar flow rate is Na; the volumetric flow rate is q; reactor volume is Vr or simply V for batch reactors; the volumetric concentration is Ca = na /V or Ca = Na /q; the total pressure is P; and the temperature is T. Throughout this section, equations are presented without specification of units. Use of any consistent unit set is appropriate. Following is a listing of typical nomenclature expressed in SI and U.S. Customary System units. Symbol

Definition

A, B, C, . . . A a BR b

Names of substances Free radical, as CH3 Activity Batch reactor Estimate of kinetic parameters, vector Concentration of substance A Continuous stirred tank reactor Initial concentration Heat capacity at constant pressure Heat capacity change in a reaction Diffusivity, dispersion coefficient Effective diffusivity Knudsen diffusivity Degree of polymerization Activation energy, enhancement factor for gas-liquid mass transfer with reaction, electrochemical cell potential Faraday constant, F statistic Efficiency of initiation in polymerization Ca /Ca0 or na /na0, fraction of A remaining unconverted Hatta number Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution Number of stages in a CSTR battery, reaction order, number of electrons in electrochemical reaction, number of experiments Number of moles of A present Total number of moles Total pressure, live polymer species Pure error sum of squares Plug flow reactor Number of kinetic parameters

Ca CSTR C0 cp ∆cp D De DK DP E

F f fa Ha He ∆G ∆Hr I j Ka Ke k L LFSS M m N NCLD NMWD n

na nt P PESS PFR p

SI units

U.S. Customary System units

Symbol

Definition

pa q Q R kg⋅molm3

lb⋅molft3

kg⋅molm3 kJ(kg⋅ K)

lb⋅molft3 Btu(lbm⋅°F)

kJ(kg⋅ K)

Btu(lbm⋅°F)

2

m /s m2/s m2/s

kJkg⋅mol kJkg⋅ mol

ft2/s ft2/s ft2/s

Btulb⋅mol Btulb⋅mol

2

A/m

m

Partial pressure of substance A Volumetric flow rate Electric charge Radial position, radius, universal gas constant Re Reynolds number RgSS Regression sum of squares RSS Residual sum of squares ra Rate of reaction of A per unit volume S Selectivity, stoichiometric matrix, objective function for parameter estimation SBR Semibatch reactor Sc Schmidt number Sh Sherwood number ∆S Entropy change s Estimate of variance t Time, t statistic u Linear velocity V Volume of reactor, variancecovariance matrix v Molar volume WCLD Weight chain length distribution WMMD Weight molecular weight distribution X Linear model matrix for parameter estimation, fractional conversion Xa 1 − fa = 1 − CaCa0 or 1 − naa0, fraction of A converted x Axial position in a reactor, mole fraction in liquid Y Yield; yield coefficient for biochemical reactions y Mole fraction in gas, predicted dependent variable z x/L, normalized axial position

ft

lb⋅mol β δ

δ(t) kg⋅mol kg⋅mol

lb⋅mol lb⋅mol

Pa m3/s Coulomb

U.S. Customary System units psi ft3/s

kJ(kg⋅ mol⋅K) Btu(lb⋅mol⋅ Rr) m/s

ft/s

m3kg⋅mol

ft3lb⋅mol

Variable

Greek letters α

kg⋅mol

SI units

ε

Φ φ

Fraction of initial catalyst activity, probability of propagation for chain polymerization, confidence level r/R, normalized radial position, fraction of poisoned catalyst, kinetic parameter vector Film thickness or boundary layer thickness, relative change in number of moles by reaction Unit impulse input, Dirac function Fraction void space in a packed bed, relative change in number of moles by reaction, residual error, porosity, current efficiency Weisz Prater parameter Thiele modulus 7-3

7-4

REACTION KINETICS

Nomenclature and Units (Concluded) Symbol

Definition

SI units

U.S. Customary System units

η

ζ

Effectiveness factor of porous catalyst, overpotential in electrochemical reactions Parameter for instantaneous gas-liquid reaction, moments in polymer chain length Viscosity, biomass growth rate, average chain length in polymers µρ, kinematic viscosity, stoichiometric coefficient, fraction of surface covered by adsorbed species Dimensionless time Density Variance Residence time, tortuosity factor Extent of reaction

act anode B cathode cell current, j D

Activation At anode Bed At cathode Electrochemical cell Current, species j Diffusion, dispersion

λ µ ν

θ ρ σ τ

Subscripts d e f G i j L m max

Greek letters

kg/m3

lbm/ft3

Subscripts

n o obs p projected r S s surf v x 0 1

⁄2

Deactivation Equilibrium Forward reaction, final, formation Gas Component i Reaction j Liquid Based on mass, mass transfer Maximum biomass growth, maximum extent of reaction Chain length in polymers Oxidized observed Particle Electrode projected area Reverse reaction, reduced Substrate Solid or catalyst, saturation, surface Surface Based on volume Biomass At initial or inlet conditions, as in Ca0, na0, V′0, at reference temperature Half-life Superscripts

eq o T

Equilibrium At reference temperature Transposed matrix

REFERENCES GENERAL REFERENCES: Amundson, Mathematical Methods in Chemical Engineering—Matrices and Their Application, Prentice-Hall International, New York, 1966; Aris, Elementary Chemical Reactor Analysis, Prentice-Hall, 1969; Astarita, Mass Transfer with Chemical Reaction, Elsevier, New York, 1967; Bamford and Tipper (eds.), Comprehensive Chemical Kinetics, Elsevier, 1969; Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., Wiley, New York, 2002; Boudart, Kinetics of Chemical Processes, Prentice-Hall, 1968; Boudart and Djega-Mariadassou, Kinetics of Heterogeneous Catalytic Reactions, Princeton University Press, Princeton, N.J., 1984; Brotz, Fundamentals of Chemical Reaction Engineering, AddisonWesley, 1965; Butt, Reaction Kinetics and Reactor Design, Prentice-Hall, 1980; Butt and Petersen, Activation, Deactivation and Poisoning of Catalysts, Academic Press, 1988; Capello and Bielski, Kinetic Systems: Mathematical Description of Kinetics in Solution, Wiley, 1972; Carberry, Chemical and Catalytic Reaction Engineering, McGraw-Hill, 1976; Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987; Chen, Process Reactor Design, Allyn & Bacon, 1983; Churchill, The Interpretation and Use of Rate Data: The Rate Concept, McGraw-Hill, New York, 1974; Cooper and Jeffreys, Chemical Kinetics and Reactor Design, Prentice-Hall, 1971; Cremer and Watkins (eds.), Chemical Engineering Practice, vol. 8: Chemical Kinetics, Butterworths, 1965; Davis and Davis, Fundamentals of Chemical Reaction Engineering, McGraw-Hill, 2003; Delmon and Froment, Catalyst Deactivation, Elsevier, 1980; Denbigh and Turner, Chemical Reactor Theory, Cambridge, 1971; Denn, Process Modeling, Langman, New York, 1986; Fogler, Elements of Chemical Reaction Engineering, 4th ed., Prentice-Hall, 2006; Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990; Froment and Hosten, “Catalytic Kinetics—Modeling,” in Catalysis—Science and Technology, Springer Verlag, New York, 1981; Harriott, Chemical Reactor Design, Dekker, 2003; Hill, An Introduction to Chemical Engineering Kinetics and Reactor Design, 2d ed., Wiley, 1990; Holland and Anthony, Fundamentals of Chemical Reaction Engineering, Prentice-Hall, 1989; Kafarov, Cybernetic Methods in Chemistry and Chemical Engineering, Mir Publishers, 1976; Laidler, Chemical Kinetics, Harper & Row, 1987; Lapidus and Amundson (eds.), Chemical Reactor Theory— A Review, Prentice-Hall, 1977; Levenspiel, Chemical Reaction Engineering, 3d ed., Wiley, 1999; Lewis (ed.), Techniques of Chemistry, vol. 4: Investigation of Rates and Mechanisms of Reactions, Wiley, 1974; Masel, Chemical Kinetics and

Catalysis, Wiley, 2001; Naumann, Chemical Reactor Design, Wiley, 1987; Panchenkov and Lebedev, Chemical Kinetics and Catalysis, Mir Publishers, 1976; Petersen, Chemical Reaction Analysis, Prentice-Hall, 1965; Rase, Chemical Reactor Design for Process Plants: Principles and Case Studies, Wiley, 1977; Rose, Chemical Reactor Design in Practice, Elsevier, 1981; Satterfield, Heterogeneous Catalysis in Practice, McGraw-Hill, 1991; Schmidt, The Engineering of Chemical Reactions, Oxford University Press, 1998; Smith, Chemical Engineering Kinetics, McGraw-Hill, 1981; Steinfeld, Francisco, and Hasse, Chemical Kinetics and Dynamics, Prentice-Hall, 1989; Ulrich, Guide to Chemical Engineering Reactor Design and Kinetics, Ulrich, 1993; Van Santen and Neurock, Molecular Heterogeneous Catalysis: A Conceptual and Computational Approach, Wiley, 2006; Van Santen and Niemantsverdriet, Chemical Kinetics and Catalysis, Fundamental and Applied Catalysis, Plenum Press, New York, 1995; van’t Riet and Tramper, Basic Bioreactor Design, Dekker, 1991; Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; reprint, Butterworths, 1989; Walas, Chemical Reaction Engineering Handbook of Solved Problems, Gordon & Breach Publishers, 1995; Westerterp, van Swaaij, and Beenackers, Chemical Reactor Design and Operation, Wiley, 1984. REFERENCES FOR LABORATORY REACTORS: Berty, Laboratory reactors for catalytic studies, in Leach, ed., Applied Industrial Catalysis, vol. 1, Academic, 1983, pp. 41–57; Berty, Experiments in Catalytic Reaction Engineering, Elsevier, 1999; Danckwerts, Gas-Liquid Reactions, McGraw-Hill, 1970; Hoffmann, Industrial Process Kinetics and parameter estimation, in ACS Advances in Chemistry 109:519–534 (1972); Hoffman, Kinetic data analysis and parameter estimation, in de Lasa (ed.), Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, pp. 69–105; Horak and Pasek, Design of Industrial Chemical Reactors from Laboratory Data, Heiden, Philadelphia, 1978; Rase, Chemical Reactor Design for Process Plants, Wiley, 1977, pp. 195–259; Shah, Gas-LiquidSolid Reactor Design, McGraw-Hill, 1979, pp. 149–179; Charpentier, Mass Transfer Rates in Gas-Liquid Absorbers and Reactors, in Drew et al., eds., Advances in Chemical Engineering, vol. 11, Academic Press, 1981.

BASIC CONCEPTS The mechanism and corresponding kinetics provide the rate at which the chemical or biochemical species in the reactor system react at the prevailing conditions of temperature, pressure, composition, mixing, flow, heat, and mass transfer. Observable kinetics represent the true intrinsic chemical kinetics only when competing phenomena such as transport of mass and heat are not limiting the rates. The intrinsic chemical mechanism and kinetics are unique to the reaction system. Knowledge of the intrinsic kinetics therefore facilitates reactor selection, choice of optimal operating conditions, and reactor scale-up and design, when combined with understanding of the associated physical and transport phenomena for different reactor scales and types.

7-5

This section covers the following key aspects of reaction kinetics: • Chemical mechanism of a reaction system and its relation to kinetics • Intrinsic rate data using equations that can be correlative, lumped, or based on detailed elementary kinetics • Catalytic kinetics • Effect of mass transfer on kinetics in heterogeneous systems • Intrinsic kinetic rates from experimental data and/or from theoretical calculations • Kinetic parameter estimation The use of reaction kinetics for analyzing and designing suitable reactors is discussed in Sec. 19.

BASIC CONCEPTS MECHANISM The mechanism describes the reaction steps and the relationship between the reaction rates of the chemical components. A single chemical reaction includes reactants A, B, . . . and products R, S, . . .

νaA + νb B + … ⇔ νr R + νs S + …

(7-1)

where νi are the stoichiometric coefficients of components A, B, . . . , i.e., the relative number of molecules of A, B, . . . that participate in the reaction. For instance, the HBr synthesis has the global stoichiometry H2 + Br2 ⇔ 2HBr. The stoichiometry of the reaction defines the reaction elemental balance (atoms of H and Br, for instance) and therefore relates the number of molecules of reactants and products participating in the reaction. The stoichiometric coefficients are not unique for a given reaction, but their ratios are unique. For instance, for the HBr synthesis above we could have written the stoichiometric equation 1 ⁄2H2 + 1⁄2Br2 ⇔ HBr as well. Often several reactions occur simultaneously, resulting in a network of reactions. When the network is broken down into elementary or single-event steps (such as a single electron transfer), the network represents the true mechanism of the chemical transformations leading from initial reactants to final products through intermediates. The intermediates can be molecules, ions, free radicals, transition state complexes, and other moieties. A network of global reactions, with each reaction representing the combination of a number of elementary steps, does not represent the true mechanism of the chemical transformation but is still useful for global reaction rate calculations, albeit empirically. The stoichiometry can only be written in a unique manner for elementary reactions, since as shown later, the reaction rate for elementary reactions is determined directly by the stoichiometry through the concept of the law of mass action. REACTION RATE The specific rate of consumption or production of any reaction species i, ri, is the rate of change of the number of molecules of species i with time per unit volume of reaction medium: 1 dn ri =  i V dt

(7-2)

The rate is negative when i represents a reactant (dni /dt is negative since ni is decreasing with time) and positive when i represents a product

(dni /dt positive since ni is increasing with time). The specific rate of a reaction, e.g., that in Eq. (7-1) is defined as r = −ri  νI

for reactants

r = ri νI

for products

(7-3)

By this definition, the specific rate of reaction is uniquely defined, and its sign is always positive. Inversely, the rate of reaction of each component or species participating in the reaction is the specific reaction rate multiplied by the species’ stoichiometric coefficient with the corrected sign (negative for reactants, positive for products). CLASSIFICATION OF REACTIONS Reactions can be classified in several ways. On the basis of mechanism they may be 1. Irreversible, i.e., the reverse reaction rate is negligible: A + B ⇒ C + D, e.g., CO oxidation CO + 12 O2 ⇒ CO2 2. Reversible: A + B ⇔ C + D, e.g., the water-gas shift CO + H2O ⇔ CO2 + H2 3. Equilibrium, a special case with zero net rate, i.e., with the forward and reverse reaction rates of a reversible reaction being equal. All reversible reactions, if left to go to completion, end in equilibrium. 4. Networks of simultaneous reactions, i.e., consecutive, parallel, complex (combination of consecutive and parallel reactions): A+B⇒C+D

C+E⇒F+G

e.g., two-step hydrogenation of acetylene to ethane CHCH + H2 ⇒ CH2=CH2

CH2=CH2 + H2 ⇒ CH3CH3

A further classification is from the point of view of the number of reactant molecules participating in the reaction, or the molecularity: 1. Unimolecular: A ⇒ B, e.g., isomerization of ortho-xylene to para-xylene, O-xylene ⇒ P-xylene, or A ⇒ B + C, e.g., decomposition CaCO3 ⇒ CaO + CO2 2. Bimolecular: A + B ⇒ C or 2A ⇒ B or A + B ⇒ C + D, e.g., C2H4 + H2 ⇒ C2H6 3. Trimolecular: A + B + C ⇒ D or 3A ⇒ B This last classification has fundamental meaning only when considering elementary reactions, i.e., reactions that constitute a single chemical transformation or a single event, such as a single electron transfer. For elementary reactions, molecularity is rarely higher than 2. Often elementary reactions are not truly unimolecular, since in order for the reaction to occur, energy is required and it is obtained through collision with other molecules such as an inert solvent or gas.

7-6

REACTION KINETICS

Thus the unimolecular reaction A ⇒ B could in reality be represented as a bimolecular reaction A + X ⇒ B + X, i.e., A collides with X to produce B and X, and thus no net consumption of X occurs. Reactions can be further classified according to the phases present. Examples for the more common cases are 1. Homogeneous gas, e.g., methane combustion 2. Homogeneous liquid, e.g., acid/base reactions to produce soluble salts 3. Heterogeneous gas-solid, e.g., HCN synthesis from NH3, CH4, and air on a solid catalyst 4. Heterogeneous gas-liquid, e.g., absorption of CO2 in amine solutions 5. Heterogeneous liquid-liquid, e.g., reaction in immiscible organic and aqueous phases such as synthesis of adipic acid from cyclohexanone and nitric acid 6. Heterogeneous liquid-solid, e.g., reaction of limestone with sulfuric acid to make gypsum 7. Heterogeneous solid-solid, e.g., self-propagating, high-temperature synthesis of inorganic pure oxides (SHS) 8. Heterogeneous gas-liquid-solid, e.g., catalytic Fischer-Tropsch synthesis of hydrocarbons from CO and H2 9. Heterogeneous gas-liquid-liquid, e.g., oxidations or hydrogenations with phase transfer catalysts Reactions can also be classified with respect to the mode of operation in the reaction system as 1. Isothermal constant volume (batch) 2. Isothermal constant pressure (continuous) 3. Adiabatic 4. Nonisothermal temperature-controlled (by cooling or heating), batch or continuous EFFECT OF CONCENTRATION ON RATE The concentration of the reaction components determines the rate of reaction. For instance, for the irreversible reaction pA + qB ⇒ r C + sD

(7-4)

the rate can be represented empirically as a power law function of the reactant concentrations such as n r = kCaaCbb Ci = i (7-5) V The exponents a and b represent the order of the reaction with respect to components A and B, and the sum a + b represents the overall order of the reaction. The order can be a positive, zero, or negative number indicating that the rate increases, is independent of, or decreases with an increase in a species concentration, respectively. The exponents can be whole (integral order) or fraction (fractional order). In Eq. (7-5) k is the specific rate constant of the reaction, and it is independent of concentrations for elementary reactions only. For global reactions consisting of several elementary steps, k may still be constant over a narrow range of compositions and operating conditions and therefore can be considered constant for limited practical purposes. A further complexity arises for nonideal chemical solutions where activities have to be used instead of concentrations. In this case the rate constant can be a function of composition even for elementary steps (see, for instance, Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). When Eq. (7-4) represents a global reaction combining a number of elementary steps, then rate equation (7-5) represents an empirical correlation of the global or overall reaction rate. In this case exponents a and b have no clear physical meaning other than indicating the overall effect of the various concentrations on rate, and they do not have any obvious relationship to the stoichiometric coefficients p and q. This is not so for elementary reactions, as shown in the next subsection. Also, as shown later, power law and rate expressions other than power law (e.g., hyperbolic) can be developed for specific reactions by starting with the mechanism of the elementary steps and making simplifying assumptions that are valid under certain conditions.

LAW OF MASS ACTION As indicated above, the dependence of rate on concentration can be shown to be of the general form r = kf(Ca, Cb, . . .)

(7-6)

For elementary reactions, the law of mass action states that the rate is proportional to the concentrations of the reactants raised to the power of their respective molecularity. Thus for an elementary irreversible reaction such as (7-4) the rate equation is r = kCpa Cbq

(7-7)

Hence, the exponents p and q of Eq. (7-7) are the stoichiometric coefficients when the stoichiometric equation truly represents the mechanism of reaction, i.e., when the reactions are elementary. As discussed above, the exponents a and b in Eq. (7-5) identify the order of the reaction, while the stoichiometric coefficients p and q in Eq. (7-7) also identify the molecularity—for elementary reactions these are the same. EFFECT OF TEMPERATURE The Arrhenius equation relates the specific rate constant to the absolute temperature



E k = k0 exp −  RT



(7-8)

where E is called the activation energy and k0 is the preexponential factor. As seen from Eq. (7-8), the rate can be a very strongly increasing (exponential) function of temperature, depending on the magnitude of the activation energy E. This equation works well for elementary reactions, and it also works reasonably well for global reactions over a relatively narrow range of temperatures in the absence of mass-transfer limitations. The Arrhenius form represents an energy barrier on the reaction pathway between reactants and products that has to be overcome by the reactant molecules. The Arrhenius equation can be derived from theoretical considerations using either of two competing theories, the collision theory and the transition state theory. A more accurate form of Eq. (7-8) includes an additional temperature factor



E k = k0Tm exp −  RT



0<m<1

(7-9)

but the Tm factor is often neglected because of the usually much stronger dependence on temperature of the exponential factor in Eq. (7-9), as m is usually small. When m is larger, as it can be for complex molecules, then the Tm term has to be taken into consideration. For more details, see Masel, Chemical Kinetics and Catalysis, Wiley, 2001; Levenspiel, Chemical Reaction Engineering, 3d ed., Wiley, 1999). HEAT OF REACTION Chemical reactions are accompanied by evolution or absorption of energy. The enthalpy change (difference between the total enthalpy of formation of the products and that of the reactants) is called the heat of reaction ∆Hr: ∆Hr = (νr Hfr + νs Hfs + . . . ) − (νa Hfa + νb Hfb + . . . )

(7-10)

where Hfi are the enthalpies of formation of components i. The reaction is exothermic if heat is produced by the reaction (negative heat of reaction) and endothermic if heat is consumed (positive heat of reaction). The magnitude of the effective heat of reaction depends upon temperature and the phases of the reactants and product. To estimate the dependence of the heat of reaction on temperature relative to a

BASIC CONCEPTS reference temperature T0, the following expression can be used, provided there is no phase change:

 ∆c dT T

∆Hr(T) = ∆Hr(T0) +

p

T0

∆cp = (νrcpr + νscps + . . .) − (νacpa + νbcpb + . . .)

(7-11)

where cpi are the heat capacities of component i. The heat of reaction can be measured by using calorimetry, or it can be calculated by using a variety of thermodynamic methods out of the scope of this chapter (see relevant sections of this handbook, thermodynamic texts, and Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley & Sons, New York, 2002). It is important to accurately capture the energy balance and its relation to the heat of reaction and heat capacities (see also Denn, Process Modeling, Langman, New York, 1986, for correct formulations). The coupling of the heat of reaction with the reaction rate through the Arrhenius equation often has a dominating effect on reactor selection and control, and on the laboratory reactor setup required to obtain accurate intrinsic kinetics and mechanism. More on this can be found in Sec. 19. CHEMICAL EQUILIBRIUM Often reactions or reaction steps in a network of reactions are at chemical equilibrium; i.e., the rate of the forward reaction equals the rate of the reverse reaction. For instance, for the reversible reaction kf νa A + νb B ⇔ ν r R + νsS kr

(7-12)

with mass action kinetics, the rate may be written as r = rf − rr = kf Cνa Cνb − krCνr Cνs a

b

r

s

(7-13)

At chemical equilibrium the forward and reverse reaction rates are equal according to the principle of microscopic reversibility: r = rf − rr = 0

(7-14)

The equilibrium constant Ke (based on volumetric concentrations) is defined as the ratio of the forward and reverse rate constants and is related to the composition at equilibrium as follows: ν Crν,eCs,e k K e = f =  ν ν kr Ca,eCb,e r

a

s

(7-15)

b

Ke can be calculated from the free energy change of the reaction. Using the van’t Hoff relation, we obtain the dependence of Ke on temperature: ∆Hr d(ln K e)  = 2 RT dT

(7-16)

Integrating with respect to temperature, we obtain a form similar to the Arrhenius expression of the rate constant for a narrow range of temperature: ∆Hr K e = K e0 exp  RT





(7-17)

A more general integral form of Eq. (7-16) is



1 ln K e(T) = ln K e (T0) +  T R

T

∆Hr(T0) +

 ∆c dT T

p

T

0  dt T2 T0

(7-18)

7-7

When a reversible reaction is not at equilibrium, knowledge of Ke can be used to eliminate the rate constant of the reverse reaction by using Eq. (7-15) as follows: r = kf (Cνaa Cνbb − Cνrr Cνs sK e)

(7-19)

When several reversible reactions occur simultaneously, each reaction rj is characterized by its equilibrium constant Kej. When the Kej are known, the composition at equilibrium can be calculated from a set of equations such as Eq. (7-15) for each reaction. CONVERSION, EXTENT OF REACTION, SELECTIVITY, AND YIELD Conversion of a reactant is the number of moles converted per initial or feed moles of a reactant. Thus for component A na Xa = 1−  na0

(7-20)

A limiting reactant is a reactant whose concentration at the start of the reaction is the least of all reactants relative to the required stoichiometric amount needed for complete conversion. For instance, for the single reaction (7-12), A is the limiting reactant if the initial molar ratio of concentrations of A and B is less than the ratio of their stoichiometric coefficients: na0 νa  νb nb0 < 

(7-21)

Once the limiting reactant is depleted, the respective reaction stops even though other (nonlimiting) reactants may still be abundant. For each reaction or each step in a network of reactions, a unique extent of reaction ξ that relates the composition of components that participate in the reaction to each other can be defined. For instance, for the single reaction (7-1): na0 − na nb0 − nb nr0 − nr ns0 − ns ξ =  =  = . . . = − = − = . . . νa νb νr νs

(7-22)

The extent of reaction is related to conversion as follows: Xanao Xbnbo . . . ξ=  =  = νa νb

(7-23)

When A is the limiting reactant as in Eq. (7-21), the maximum extent of reaction (with A fully converted) is na0 ξmax =  νa

(7-24)

For multiple reactions with reactants participating in more than one reaction, it is more difficult to determine the limiting reactant and often it is necessary to calculate the concentration as the reactions proceed to determine which reactant is consumed first. When the limiting reactant is depleted, all reactions that use this component as reactant stop, and the corresponding rates become zero. Selectivity S of a product is the ratio of the rate of production of that product to the rate of production of all products combined. For a single reaction selectivity is trivial—if more than one product occurs, then the selectivity of each product is the ratio of the stoichiometric coefficient of that product to the sum of stoichiometric coefficients of all the products. Thus for reaction (7-1)

νr Sr =  νr + νs + . . .

(7-25)

7-8

REACTION KINETICS

The selectivity of product R for a network of reactions, with all the reactions making the various products included, is rr Sr = 

ri

and for gas yi ni Ci C pi p yi =  =  = i =  = i ∑ni ∑Ci C ∑pi P

(7-26)

(7-35)

The last two terms are only valid for an ideal gas.

all products i

1

For instance, for the network of reactions A + B ⇒ C + D 2 C + E ⇒ F + G, the selectivity to product C is

STOICHIOMETRIC BALANCES Single Reactions Equation (7-22) shows that for a single reaction, the number of moles and concentration of all other components can be calculated from the extent of reaction ζ or the conversion based on the limiting reactant, say A, Xa. In terms of number of moles ni,

r1 − r2 r1 − r2 rc Sc =  =  =  (r1 − r2) + r1 + r2 + r2 2r1 + r2 rc + rd + rf + rg The yield Y of a product R with respect to a reactant A is the ratio of the rate of production of R to that of consumption of A: rr Yr =  −ra

na = na0 − νaξ = na0(1 − Xa)

νb nb = nb0 − νbξ = nb0(1 − Xb) = nb0 −  na0 Xa νa ...

(7-27)

νr nr = nr0 + νrξ = nr0 −  na0 Xa νa νs ns = ns0 + νsξ = ns0 −  na0 Xa νa ...

For a single reaction the yield is trivial and Eq. (7-27) simplifies to the ratio of the respective stoichiometric coefficients:

νr Yr =  νa

(7-28)

The yield quantifies the efficiency of the respective reactant utilization to make the desired products.

Similarly the number of moles of each component in terms of moles of A, na, is

νb nb = nb0 −  (na0 − na) νa ... νr nr = nr0 +  (na0 − na) νa

CONCENTRATION TYPES Different concentration types are used for different reaction systems. For gas-phase reactions, volumetric concentration or partial pressures are equally useful and can be related by the thermodynamic equation of state. For instance, for ideal gases (approximation valid for gases at very low pressure) PV = nRT

When applied to individual components in a constant-volume system, (7-30)

Using Eq. (7-5), we obtain the relationship between the volumetric concentrations and partial pressures: pi Ci =  RT

(7-31)

For an ideal gas the total concentration is P C=  RT

Change in number of moles by the reaction and change in temperature, pressure, and density affect the translation of stoichiometric balances from number of moles to volumetric concentrations. These relationship are different for gases and liquids. For instance, for constant density systems (such as many liquid-phase isothermal reactions) or for constant temperature, constant pressure gas reaction with no change in number of moles, Eqs. (7-36) and (7-37) can be changed to volumetric concentration Ci by dividing each equation by the constant reaction volume V (e.g., in a batch reactor) and using Eq. (7-5). For example, for the single reaction (7-4) with rate (7-5):

(7-32)

q Cb = Cb0 −  (Ca0 − Ca) p

For higher pressure and nonideal gases, a compressibility factor zi can be used:

r Cr = Cr0 +  (Ca0 − Ca) p

piV = zi ni RT

and

pi Ci =  zi RT

ni Ci C xi =  =  = i ∑ni ∑Ci C

(7-34)

(7-38)

s Cs = Cs0 +  (Ca0 − Ca) p

(7-33)

Other relevant equations of state can also be used for both gases and liquids. This aspect is not in the scope of this section, and the reader is referred to the relevant section of this handbook. Other concentration units include mole fractions for liquid xi:

(7-37)

νs ns = ns0 +  (na0 − na) νa ...

(7-29)

piV = ni RT

(7-36)







b q q r = kCaa Cb0 −  (Ca0 − Ca) = k[Ca0(1 − Xa)]a Cb0 −  Ca0 Xa p p



b

(7-39)

It is best to represent all concentrations in terms of that of the limiting reactant. Often there is a change in total number of moles due to reaction. Taking the general reaction (7-1), in the gas phase the change in

BASIC CONCEPTS number of moles relative to moles of component A converted, δa, and the total number of moles can be calculated as follows:

The stoichiometric matrix S for this network is S=

νq + νs + . . . − νa − νb − . . . δa =  νa n0 = ∑ni0

n = ∑ni

(7-40)

n  = 1 + ya0 δ a Xa = 1 + εaXa n0 εa = ya0 δa Using the ideal gas law, Eq. (7-29), the volume change depends on conversion as follows: V T P0 n T P0  =    =   (1 + εa Xa) V0 T0 P n0 T0 P

Ca0(1 − Xa) n Ca = a =  1 + εaXa V

(7-42)

Applying this to reaction (7-4) and rate (7-5) gives Cb0 − (q/p)Ca0 Xa

    1+εX a

a

b

(7-43)

a

Compare this to Eq. (7-39) where there is no change in number of moles. Reaction Networks The analysis for single reactions can be extended to a network of reactions by defining an extent of reaction for each reaction, or by choosing a representative reactant concentration for each reaction step. For a complex network, the number of independent extents of reaction required to calculate the concentration of all components is equal to the number of independent reactions, which is less than or equal to the total number of reactions in the network. To calculate the number of independent reactions, and to form a set of independent reactions and corresponding independent set of concentrations or extents of reaction, we need to construct the stoichiometric matrix and determine its rank. The stoichiometric matrix is used to derive a relationship between the concentrations and the independent extents of reaction similar to that of a single reaction. The stoichiometric matrix is the matrix of the stoichiometric coefficients of the reaction network with negative signs for reactants and positive signs for products. For instance, the hydrodechlorination of Freon 12 (CF2Cl2) can proceed with the following consecutive mechanism [Bonarowska et al., “Hydrodechlorination of CCl2F2 (CFC-12) over Silica-Supported Palladium-Gold Catalysts,” Appl. Catal. B: Environmental, 30:187–193, 2001]: CF2Cl2 + H2 ⇒ CF2ClH + HCl

1 0 −1 1

0 −1 1 −1 1

S1 =

 −10 −11

CF2ClH + H2 ⇒ CF2H2 + HCl



det(S1) = 1 ≠ 0

Hence the two reactions are independent. Now if we add another step, which converts Freon 12 directly into the final hydrofluorocarbon CF2H2; CF2Cl2 + 2H2 ⇒ CF2H2 + 2HCl, then the stoichiometric matrix becomes



s Cs0 − ννC a0 Xa nb a Cs =  =  V 1 + εa Xa ...



−1

S=

b Cb0 − ννC a0 Xa nb a Cb =  =  V 1 + εa Xa ... r Cr0 − ννC a0 Xa nb a Cr =  =  V 1 + εa Xa

Ca0(1 − Xa) r = k  1 + εaXa



The first row refers to the first reaction and the second row to the second reaction. The columns (species) are in the following order: 1-CF2Cl2, 2-CF2ClH, 3-CF2H2, 4-H2, and 5-HCl. The rank of a matrix is the largest square submatrix obtained by deleting rows and columns, whose determinant is not zero. The rank equals the number of independent reactions. This is also equivalent to stating that there are reactions in the network that are linear combinations of the independent reactions. The rank of S above is 2, since the determinant of the first 2 × 2 submatrix is not zero (there are other 2 × 2 submatrices that are not zero as well but it is sufficient to have at least one that is not zero):

(7-41)

Hence, for an isothermal constant-pressure ideal gas reaction system,

7-9

−1

1 0 −1

0 −1 1 −1 −1

0 1 −2

1 1 2



Since the last reaction is a linear combination of the first two (sum), it can be easily proved that the rank remains unchanged at 2. So to conclude, the concentrations of all components in this network can be expressed in terms of two, say H2 and Freon 12, and the first two reactions form an independent reaction set. In case of more complicated networks it may be difficult to determine the independent reactions by observation alone. In this case the Gauss-Jordan decomposition leads to a set of independent reactions (see, e.g., Amundson, Mathematical Methods in Chemical Engineering—Matrices and Their Application, Prentice-Hall International, New York, 1966). For a network of reactions the general procedure is as follows: 1. Generate the reaction network by including all known reaction steps. 2. Generate the corresponding stoichiometric matrix. 3. Calculate the rank of the stoichiometric matrix which equals the number of independent reactions and independent component concentrations required to calculate all the remaining component concentrations. 4. For relatively simple networks, observation allows selection of reactions that are independent—for more complex systems use the Gauss-Jordan elimination to reduce the network to a set of independent (nonzero rows) reactions. 5. Select the independent concentration variables and independent reactions, and use these to calculate all other concentrations and reaction rates. CATALYSIS A catalyst is a material that increases the rate of both the forward and reverse reactions of a reaction step, with no net consumption or generation of catalyst by the reaction. A catalyst does not affect the reaction thermodynamics, i.e., the equilibrium composition or the heat of reaction. It does, however, affect the temperature sensitivity of the reaction rate by lowering the activation energy or the energy barrier on the reaction pathway from reactants to products. This allows the reaction to occur faster than the corresponding uncatalyzed reaction at a given temperature. Alternatively, catalytic reactions can proceed at lower temperatures than the corresponding noncatalytic reactions. For a network of reactions, the catalyst is often used to speed up desired reactions and/or to slow down undesired reactions for improved selectivity. On the basis of catalysis, reactions can be further classified into 1. Noncatalytic reactions, e.g., free-radical gas-phase reactions such as combustion of hydrocarbons.

7-10

REACTION KINETICS

2. Homogeneous catalytic reactions with the catalyst being dissolved in the same phase as the reactants and products in a homogeneous reaction medium. Here the catalyst is uniformly distributed throughout the system, e.g., the hydroformylation of olefins in the presence of dissolved Co or Rh carbonyls. 3. Heterogeneous catalytic reactions, with the catalyst, for instance, being a solid in contact with reactants and products in a gassolid, gas-liquid-solid, or a liquid-solid reaction system. Here the catalyst is not uniformly distributed, and the reaction occurring on the catalyst surface requires, for instance, adsorption of reactants and desorption of products from the solid surface, e.g., the catalytic cracking of gasoil to gasoline and lighter hydrocarbons. Table 7-1 illustrates the enhancement of the reaction rates by the catalyst—this enhancement can be of many orders of magnitude.

TABLE 7-1 The Rate of Enhancement of Some Reactions in the Presence of a Catalyst Reaction Ortho H2 ⇒ para H2 2NH3 ⇒ N2  3H2 C2H4  H2 ⇒ C2H6 H2  Br2 ⇒ 2HBr 2NO  2H2 ⇒ N2  2H2O CH3COH ⇒ CH4  CO CH3CH3 ⇒ C2H4  H2 (CH3)3COH ⇒ (CH3)2 CH2CH2  H2O SOURCE:

Catalyst

Rate enhancement

Temperature, K

Pt (solid) Mo (solid) Pt (solid) Pt (solid) Ru (solid) I2 (gas) NO2 (gas) HBr (gas)

1040 1020 1042 1  108 3  1016 4  106 1  109 3  108

300 600 300 300 500 500 750 750

Masel, Chemical Kinetics and Catalysis, Wiley, 2001, Table 12.1.

IDEAL REACTORS Reactions occur in reactors, and in addition to the intrinsic kinetics, observed reaction rates depend on the reactor type, scale, geometry, mode of operation, and operating conditions. Similarly, understanding of the reactor system used in the kinetic experiments is required to determine the reaction mechanism and intrinsic kinetics. In this section we address the effect of reactor type on observed rates. In Sec. 19 the effect of reactor type on performance (rates, selectivity, yield) is discussed in greater detail. Material, energy, and momentum balances are essential to fully describe the performance of reactors, and often simplifying assumptions and phenomenological assumptions are needed especially for energy and momentum terms, as indicated in greater detail in Sec. 19

(see also Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley & Sons, New York, 2002). Ideal reactors allow us to simplify the energy, momentum, and material balances, thus focusing the analysis on intrinsic kinetics. A useful classification of ideal reactor types is in terms of their concentration distributions versus reaction time and space. Three types of ideal reactors are considered in this section: 1. Ideal batch reactors (BRs) including semibatch reactors (SBRs) 2. Ideal continuously stirred tank reactor (CSTR), including single and multiple stages 3. Plug flow reactor (PFR) with and without recycle Figure 7-1 shows these types of ideal reactors. Other types of ideal and nonideal reactors are treated in detail in Sec. 19.

Reactants

(a)

(b)

(c) FIG. 7-1 Types of ideal reactors: (a) Batch or semibatch. (b) CSTR or series of CSTRs.

(c) Plug flow.

IDEAL REACTORS The general form of a balance equation is Input + sources − outputs = accumulation

(7-44)

balance is valid for each component since we can multiply each side of the equation by the component molecular weight to obtain the true mass balance equation. In terms of conversion, Eq. (7-45) can be rewritten as

IDEAL BATCH REACTOR Batch Reactor (BR) Ideal batch reactors (Fig. 7-1a) are tanks provided with agitation for uniform composition and temperature at all times. An ideal batch reactor can be operated under isothermal conditions (constant temperature), temperature-programmed mode (by controlling cooling rate according to a protocol), or adiabatic mode with no heat crossing the reactor boundaries). In adiabatic mode the temperature is increasing, decreasing, or constant as the reaction proceeds for exothermic, endothermic, and thermally neutral reactions, respectively. In the ideal batch reactor, all the reactants are loaded into the reactor and well mixed by agitation before the conditions for reaction initiation (temperature and pressure) are reached; as the reaction proceeds, the concentration varies with time, but at any one time it is uniform throughout due to agitation. Laboratory batch reactors can be single-phase (e.g., gas or liquid), multiphase (e.g., gas-liquid or gas-liquid-solid), and catalytic or noncatalytic. In this section we limit the discussion to operation at isothermal conditions. This eliminates the need to consider energy, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable. An ideal isothermal single-phase batch reactor in which a general reaction network takes place has the following general material balance equation: dni  = V νijrj dt j

ni = ni0

at t = 0

(7-45)

ni0 dXi  = −V νij rj dt

τBR = −ni0

(7-46)

dXi  V νij rj

Xif

Xi0

(7-47)

Ci = Ci0

at t = 0

(7-48)

A typical concentration profile versus time for a reactant is shown in Fig. 7-2a. Integration of Eq. (7-48) gives the batch residence time τBR =



Cif

Ci0

dCi 

νij rj

(7-49)

For instance, for a single reaction, Eq. (7-43) can be used to describe the reaction rate ri in terms of one reactant concentration. For reaction networks integration of a system of ordinary differential equations is required.

(b)

Concentration

(a)

(c)



dCi  = νij rj dt

Time

2 3 4 5 Stage number

at t = 0

To integrate we need to represent all reaction rates rj in terms of the conversion Xi. For a single reaction this is straightforward [see, e.g., Eq. (7-43)]. However, for a network of reactions, integration of a system of often nonlinear differential equations is required using implicit or semi-implicit integration. For references please see the relevant section of this handbook or any textbook on ordinary differential equations. A special case of batch reactors is constant-volume or constantdensity operation typical of liquid-phase reactions, with volume invariant with time:

Time

Concentration 1

ni = ni0

and we can integrate this equation to get the batch reaction time or batch residence time τBR required to obtain a conversion Xi, starting with initial conversion Xi0 and ending with final conversion Xif:

Concentration

Concentration

The left-hand side is the accumulation term in moles per second of component i, and the right-hand side is the source term due to chemical reaction also in moles per second, which includes all reactions j that consume or produce component i, and the corresponding stoichiometric coefficients are represented in matrix form as νij with a positive sign for products and a negative sign for reactants. This molar

7-11

Distance along reactor (d)

FIG. 7-2 Concentration profiles in batch and continuous flow: (a) Batch

time profile. (b) Semibatch time profile. (c) Five-stage CSTRs distance profile. (d) Tubular flow distance profile.

7-12

REACTION KINETICS

Semibatch Reactor (SBR) In semibatch operation, a gas of limited solubility or a liquid reactant may be fed in gradually as it is used up. An ideal isothermal single-phase semibatch reactor in which a general reaction network takes place has the following general material balance equation: dni  = Ni0 + V νijrj dt

ni = ni0

Ni0 = Ni0(t)

for ts0i ≤ t ≤ ts1i

at t = 0 (7-50)

The first term on the right-hand side of Eq. (7-50) is the molar feed rate of the components, which can be different for each component, hence the subscript i, and can vary with time. A typical concentration profile versus time for a reactant whose concentration is kept constant initially by controlling the feed rate is shown in Fig. 7-2b. Knowledge of the reaction kinetics allows these ordinary differential equations to be integrated to obtain the reactor composition versus time. IDEAL CONTINUOUS STIRRED TANK REACTOR (CSTR) In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-1b). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable: dni  = Ni0 − Ni + V νijrj dt

ni = ni0

at t = 0

Ni0 = q0Ci0

Ni = qCi

Ci0 − (q /q0)Ci τCSTR = 

νijrj

In a plug flow reactor all portions of the feed stream move with the same radially uniform velocity along parallel streamlines and therefore have the same residence time; that is, there is no mixing in the axial direction but complete mixing radially (Fig. 7-1c). As the reaction proceeds, the concentration falls off with distance. A PFR can be operated under either transient conditions or steady-state conditions. In this section we limit the discussion to steady-state conditions. This eliminates the need to consider partial differential equations in time and space. We further limit the discussion to isothermal operation, which together with the defined plug flow also eliminates the need for energy and momentum balance equations. Due to the radially uniform composition, the component material balances are simple ordinary differential equations with axial distance from inlet as the independent variable. An isothermal single-phase steady-state PFR in which a general reaction network takes place has the following general material balance equation: dNi  = νijrj dV

Ni = Ni0

(7-54)

Equation (7-54) allows calculation of the residence time required to achieve a given conversion or effluent composition. In the case of a network of reactions, knowing the reaction rates as a function of volumetric concentrations allows solution of the set of often nonlinear algebraic material balance equations using an implicit solver such as the multi variable Newton-Raphson method to determine the CSTR effluent concentration as a function of the residence time. As for batch reactors, for a single reaction all compositions can be expressed in terms of a component conversion or volumetric concentration, and Eq. (7-54) then becomes a single nonlinear algebraic equation solved by the Newton-Raphson method (for more details on this method see the relevant section this handbook).

at V = 0

(7-56)

Note the similarity between the ideal batch and the plug flow reactors, Eqs. (7-45) and (7-56), respectively. In terms of conversion, Eq. (7-56) can be written as dX Ni0 i = − νijrj dV

(7-57)

Equation (7-57) can be integrated to calculate the reactor volume required to achieve a given conversion Xi: V = −Ni0



Xif

Xi0

dXi



νijrj

(7-58)

An apparent residence time based on feed conditions can be defined for a single-phase PFR as follows: V τPFR =  q0

(7-53)

In Eq. (7-53) the feed and effluent molar rates Ni0 and Ni are expressed in terms of volumetric flow rates q0 and q (inlet and outlet, respectively) and concentrations. Thus Eq. (7-52) can be rewritten as

j

PLUG FLOW REACTOR (PFR)

(7-52)

Equation (7-52) can be expressed in terms of volumetric concentration or in terms of conversions just as we did with the batch reactor. An apparent residence time based on feed conditions can be defined for a single-phase CSTR as follows: V τCSTR =  q0

ij

(7-51)

At steady state the differential equations simplify to algebraic equations as indicated below: Ni0 − Ni V=− 

νijrj

A special case of Eq. (7-54) is a constant-density system (e.g., a liquid-phase reaction), with the true average residence time τCSTR Ci0 − Ci τCSTR =  q = q0 (7-55)

ν r When a number of such CSTRs are employed in series, the concentration profile is step-shaped if the abscissa is the total residence time or the stage number as indicated by a typical reactant concentration profile in Fig. 7-2c.

(7-59)

Equation (7-58) becomes τPFR = − Ci0



Xif

Xi0

dXi 

νijrj

(7-60)

Equation (7-60) is identical to that of the ideal batch reactor, Eq. (7-47), and the two reactor systems can be modeled in identical fashion. For a constant-density system with no change in number of moles, with the true residence time τPFR: τPFR =



Cif

Ci0

dCi 

νijrj

(7-61)

This is identical to the corresponding ideal batch reactor, Eq. (7-49). Ideal Recycle Reactor All reactor modes can sometimes be advantageously operated with recycling part of the product or intermediate streams. Heated or cooled recycle streams serve to moderate undesirable temperature gradients, and they can be processed for changes in composition such as separating products to remove equilibrium limitations, before being returned. Say the recycle flow rate in

IDEAL REACTORS

7-13

TABLE 7-2 Consecutive and Parallel First-Order Reactions in an Isothermal Constant-Volume Ideal Batch or Plug Flow Reactor. The independent variable t is either the batch time or the plug flow residence time. Reaction network 1

Material balances

2

A ⇒B ⇒C

1

A ⇒B

Concentration profiles

dCa  k1Ca dt

Ca Ca0ek t

dCb  k1Ca  k2Cb dt

k1Ca0 Cb Cb0ek2t   (ek1t  ek2t) k2  k1

dCc  k2Cb dt

Cc Ca0  Cb0  Cc0  Ca  Cb

dCa  (k1  k2)Ca dt

Ca Ca0e (k

dcb  k1Ca dt

k1Ca0 r Cb Cb0   (1  e(k1k2) ) k2  k1

dCc  k2Ca dt

Cc Ca0  Cb0  Cc0  Ca  Cb

1

1

 k2)r

2

A ⇒C

a PFR is qR and the fresh feed rate is q0. With a fresh feed concentration of C0 and a product concentration of C2, the composite reactor feed concentration C1 and the recycle ratio R are C0 + RC2 C1 =  1+R

qR R=  q0

(7-62)



τPFR = (1 + R)

Cif

Ci0

dCi 

νijrj

(7-63)

Accordingly, the change in concentration (or in temperature) across the reactor can be made as small as desired by increasing the recycle ratio. Eventually, the reactor can become a well-mixed unit with essentially constant concentration and temperature, while substantial differences in composition will concurrently arise between the fresh feed inlet and the product withdrawal outlet, similar to a CSTR. Such an operation is useful for obtaining experimental data for analysis of rate

EXAMPLES FOR SOME SIMPLE REACTIONS Table 7-2 and Figs. 7-3 and 7-4 show the analytical solution of the integrals for two simple first-order reaction systems in an isothermal constant-volume batch reactor or plug flow reactor. Table 7-3 shows the analytical solution for the same reaction systems in an isothermal constant-density CSTR. Section 19 provides discussion about advantages and disadvantages of CSTRs versus PFR and BR for various reaction systems.

TABLE 7-3 Consecutive and Parallel First-Order Reactions in an Isothermal Constant-Volume Ideal CSTR Reaction network 1

2

A ⇒B ⇒C

1

A ⇒B

(7-64)

Hence, recycling increases the residence time or reactor size required to achieve a given conversion, since 1 + R >1.

The change in concentration across the reactor becomes C2 − C0 ∆C = C1 − C2 =  1+R

equations. In the simplest case, where the product is recycled without change in composition, the flow reactor equation at constant density is

Material balances

Concentration profiles

Ca0  Ca  τk1Ca 0

Ca0 Ca  1  τk1

Cb0  Cb  τ(k1Ca  k2Cb)

Cb0 τk1Ca0 Cb    1  τk2 (1  τk1)(1  τk2)

Cc0  Cc  τk2Cb 0

Cc Ca0  Cb0  Cc0  Ca  Cb

Ca0  Ca  τ(k1  k2)Ca 0

Ca0 Ca  1  τ (k1  k2)

2

A ⇒C Cb0  Cb  τk1Ca 0

τ k1Ca0 Cb Cb0  1  τ(k1  k2)

Cc0  Cc  τk2Ca 0

Cc Ca0  Cb0  Cc0  Ca  Cc

7-14

REACTION KINETICS

1

1

k1 = 2 h-1

0.8

k2 = 1 h-1

A

Concentrations, kmol/m3

Concentrations, kmol/m3

k1 = 2 h-1

B

0.6

0.4

C

0.2

k2 = 1 h-1

0.8

A

B

0.6

C

0.4

0.2

0

0 0

1

2

3

4

5

0

1

2

k1t FIG. 7-3 Concentration profiles for the reaction A→B→C.

3

4

5

k1t FIG. 7-4 Concentration profiles for the reaction A→B and A→C.

KINETICS OF COMPLEX HOMOGENEOUS REACTIONS Global or complex reactions are not usually well represented by mass action kinetics because the rate results from the combined effect of several simultaneous elementary reactions (each subject to mass action kinetics) that underline the global reaction. The elementary steps include short-lived and unstable intermediate components such as free radicals, ions, molecules, transition complexes, etc. The reason many global reactions between stable reactants and products have complex mechanisms is that these unstable intermediates have to be produced in order for the reaction to proceed at reasonable rates. Often simplifying assumptions lead to closed-form kinetic rate expressions even for very complex global reactions, but care must be taken when using these since the simplifying assumptions are valid over limited ranges of compositions, temperature, and pressure. These assumptions can fail completely—in that case the full elementary reaction network has to be considered, and no closed-form kinetics can be derived to represent the complex system as a global reaction. Typical simplifying assumptions include these: • Pseudo-steady-state approximation for the unstable intermediate; i.e., the concentration of these does not change during reaction • Equilibrium for certain fast reversible reactions and completion of very fast irreversible steps • Rate-determining step(s); i.e., the global reaction rate is determined by the rate(s) of the slowest step(s) in the reaction network composing the overall or global reaction These simplifying assumptions allow elimination of some reaction steps, and representation of free radical and short-lived intermediates concentrations in terms of the concentration of the stable measurable components, resulting in complex non–mass action rate expressions. Complex reactions can proceed through chain or nonchain mechanisms. In a chain reaction, the active unstable components are produced in an initiation step and are repeatedly regenerated through propagation steps, and only a small fraction of these are converted to stable components through a termination step. Free radicals are examples of such unstable components frequently encountered in chain reactions: free radicals are molecular fragments having one or more unpaired electrons, are usually short-lived (milliseconds), and are highly reactive. They are detectable spectroscopically, and some have been isolated. They occur as initiators and intermediates in such

basic phenomena as oxidation, combustion, photolysis, and polymerization. Several examples of free radical mechanisms possessing nonintegral power law or hyperbolic rate equations are cited below. In a nonchain reaction, the unstable intermediate, such as an activated complex or transition state complex, reacts further to produce the products, and it is not regenerated through propagation but is continually made from reactants in stoichiometric quantities. CHAIN REACTIONS Phosgene Synthesis The global reaction CO + Cl2 ⇒ COCl2 proceeds through the following free radical mechanism: Cl2 ⇔ 2Cl• Cl• + CO ⇔ COCl• COCl• + Cl2 ⇒ COCl2 + Cl• Assuming the first two reactions are in equilibrium, expressions are found for the concentrations of the free radicals Cl• and COCl• in terms of the species CO, Cl2, and COCL2, and when these are substituted into the mass action rate expression of the third reaction, the rate becomes rCOCl = k(CO)(Cl2)32 2

(7-65)

Ozone Conversion to Oxygen in Presence of Chlorine The Cl2 global reaction 2O3 ⇒ 3O2 in the presence of Cl2 proceeds through the following sequence: Cl2 + O3 ⇒ ClO• + ClO2• ClO2• + O3 ⇒ ClO3• + O2 ClO3• + O3 ⇒ ClO2• + 2O ClO3• + ClO3• ⇒ Cl2 + 3O2

KINETICS OF COMPLEX HOMOGENEOUS REACTIONS The chain carriers ClO•, ClO2•, and ClO3• are assumed to attain pseudo-steady state. Then, rO = k(Cl2)12(O3)32

The rates of formation of the free radicals R• and M• reach pseudosteady states, resulting in the following polymerization rate:



k rp = kp(M)(M∗) = kp I kt

(7-66)

3

7-15

Hydrogen Bromide Synthesis The global reaction H2 + Br2 ⇒ 2HBr proceeds through the following chain of reactions:

12

(M)(I)12

(7-68)

NONCHAIN REACTIONS

Br2 ⇔ 2Br•

Nonchain reactions proceed through an active intermediate to the products. Many homogeneous nonchain reactions are also homogeneously catalyzed reactions, discussed below.

Br• + H2 ⇔ HBr + H• H• + Br2 ⇒ HBr + Br•

HOMOGENEOUS CATALYSIS Assuming pseudo-steady state for the concentrations of the free radicals H• and Br•, the global rate equation becomes k1(H2)(Br2)12 rHBr =  k2 + HBr/Br2

Homogeneous catalysts proceed through an activated or transition state complex between reactant(s) and catalysts, which decomposes into products. Homogeneous catalysts are dissolved in the homogeneous reaction mixture and include among others acids/bases, metal salts, radical initiators, solvents, and enzymes. Acid-Catalyzed Isomerization of Butene-1 Butene-1 isomerizes to butene-2 in the presence of an acid according to the global reaction

(7-67)

Chain Polymerization For free radical polymerization, the following generic mechanism can be postulated: Initiation:

H+

CH3CH2HC=CH2 ⇒ CH3HC=CHCH2

1

1 ⇒ 2R•

Even though this appears to be a monomolecular reaction, it is not, as it proceeds through the following mechanism:

2

R• + M ⇒ RM•

H

Propagation: 1

kp

CH3CH2HC=CH2 + H+ ⇔ {CH3CH2HC–CH2}+

kp

⇒ CH3HC=CHCH2 + H+

RM• + M ⇒ RM2•

2

RMn• + M ⇒ RMn+1•

Assuming reaction 1 is in equilibrium, the reaction rate is

Termination:

r = k2K1[H+][CH3CH2HC=CH2]

kt

RMn• + RMm• ⇒ RMn+m

Enzyme Kinetics Enzymes are homogeneous catalysts for cellular and enzymatic reactions. The enzyme E and the reactant S are

kt

RMn• + RMm• ⇒ RMn + RMm 1

Ci/(Ca0+Cp0)

0.8

P

0.6

Ca0/Cp0 = 2

0.4 A 0.2

0 0

1

(7-69)

2

3

4

5

6

7

k(Ca0+Cp0)t FIG. 7-5 Product concentration profile for the autocatalytic reaction A  P ⇒ 2P with rate r = kCaCp.

7-16

REACTION KINETICS

assumed to form a complex ES that then dissociates into product P and releases the enzyme: S + E ⇔ ES

r = kCaCp

ES ⇒ E + P Assuming equilibrium for the first step results in the following rate, developed by Michaelis and Menten [Biochem. Zeit., 49:333 (1913)] and named Michaelis-Menten kinetics, k(S) rp = −rs =  Km + (S)

initially in a batch or in the feed of a continuous reactor. Examples are cell growth in fermentation and combustion of fuels. For instance, the irreversible elementary reaction A + P ⇒ 2P has the mass action kinetics

(7-70)

Here Km is the inverse of the equilibrium constant for the first reaction. AUTOCATALYSIS In an autocatalytic reaction, a reactant reacts with a product to make more product. For the reaction to proceed, therefore, product must be present

(7-71)

For an ideal batch reactor (see, e.g., Steinfeld, Francisco, and Hase, Chemical Kinetics and Dynamics, Prentice-Hall, 1989): dCa  = −kCaCp dt

dCp  = kCaCp dt

Ca + Cp = Ca0 + Cp0 (7-72)

Integration results in the following concentration profile: Ca0 + Cp0 CP =  1 + (Ca0/Cp0)e −k(Ca0 + Cp0)t

Ca = Ca0 + Cp0 − Cp (7-73)

Figure 7-5 illustrates the dimensionless concentration profile for the reactant A and product P, Ci /(Ca0 + Cp0), for Ca0 /Cp0 = 2, indicative of a maximum rate at the inflexion point (maximum in slope of the concentration-time curve), typical to autocatalytic reactions.

INTRINSIC KINETICS FOR FLUID-SOLID CATALYTIC REACTIONS There are a large number of fluid-solid catalytic reactions, mostly gassolid, including catalytic cracking, oxidation of polluting gases in automotive and power generation catalytic converters, partial oxidation synthesis reactions such as HCN synthesis, chemical vapor deposition, etc. (see, e.g., Sec. 19 for more examples). Examples of solid catalysts include, among others, supported metals, transition metal oxides and sulfides, solid acids and bases, and immobilized homogeneous catalysts and enzymes. Solid catalysts can be a fine powder (suspended in a liquid or fluidized by a flowing gas), cylindrical, spherical, and morecomplex-shaped particles (in a packed bed), a thin layer of active components (on the walls of a monolith or a foam) and gauzes. The solid catalyst can be porous with active component distributed throughout the particle volume, or nonporous with active component present on the exposed catalyst external surface alone. The analysis of Langmuir [J. Am. Chem. Soc. 40:1361 (1918)] and Hinshelwood (Kinetics of Chemical Change, Oxford, 1940) form the basis for the simplified treatment of kinetics on heterogeneous catalysts. For a solid catalyzed reaction between gas phase reactants A and B, the postulated mechanism consists of the following steps in series: 1. The reactants from the gas adsorb to bond to active sites on the catalyst surface as molecules or dissociated atoms. The rate of adsorption is proportional to the partial pressure of reactants and to the fraction of uncovered surface sites ϑv. More than one type of active site can be present. The adsorption isotherms such as the Langmuir isotherm relate the partial pressure of an adsorbed species to its surface coverage, and the form of this relationship is indicative of the type of adsorption process taking place (see, for more details, Masel, Chemical Kinetics and Catalysis, Wiley, 2001). 2. The adsorbed species react on the surface to form adsorbed products. The rate of reaction between adsorbed species is proportional to their adsorbed concentrations on the surface. 3. The adsorbed products desorb into the gas. The rate of desorption of species A is proportional to the fraction of the surface covered by A, ϑa. For instance, for the simple irreversible reaction A + B ⇒ C + D, the postulated mechanism is A + σ ⇔ Aσ B + σ ⇔Bσ Aσ + Bσ ⇒ Cσ + Dσ Cσ ⇔ C + σ Dσ ⇔ D + σ

(7-74)

Aσ, Bσ, Cσ, and Dσ above are adsorbed species on the catalyst surface, and σ is an available active site. We will consider a variety of possible scenarios for this simple solid-catalyzed system. Note that the intrinsic specific reaction rate for such systems is often expressed on a unit mass catalyst basis (rm) instead of unit reaction volume basis (rV), and the latter is related to the former through the catalyst loading (mass catalyst/reaction volume) or bed density: rm Mcat rv =  = rmρB V

(7-75)

ADSORPTION EQUILIBRIUM Assuming equilibrium for all adsorption steps (e.g., the surface reaction is rate-limiting), the net rates of adsorption of reactants and product are all zero. ri = ki pi1v − k−i1i ⇒ 0

i = a, b, c, d

(7-76)

A material balance on all sites yields 1v = 1 − 1a − 1b − 1c − 1d

(7-77)

and solving for the surface coverages gives



kb 1b =  pb1v = K b pb1v k−b



kd 1d =  pd 1v = K d pd 1v k−d

ka 1a =  pa1v = K a pa1v k−a kc 1c =  pc1v = K c pc1v k−c





(7-78)

The fraction of surface not covered is 1 1v =  1 + K a pa + K b pb + K c pc + K d pd

(7-79)

In the denominator, terms may be added for adsorbed inerts (e.g., K1p1) that may be present, and analogous expressions for the other participants. The rate of reaction or the rate-determining step is that between adsorbed reactant species: kpa pb r = kpa pb12v = 2 (1 + K a pa + K b pb + K c pc + K d pd)

(7-80)

INTRINSIC KINETICS FOR FLUID-SOLID CATALYTIC REACTIONS

7-17

DISSOCIATION

and the rate is

A diatomic molecule A2 may adsorb dissociatively as atoms

k(pa − pm pn /K) pmpn 1m1n r = k 1a1v −  = pa −  12v = 2 (1 + K a pa + K m pm + K n pn) K K

A2 + 2σ ⇒ 2Aσ



(7-81)





(7-88) with the result

REACTANT IN THE GAS PHASE  Kapa

1a =  =  Kapa1v 1 +  Kapa + Kbpb + . . .

When A in the gas phase reacts directly with adsorbed B:

(7-82)

A + Bσ ⇒ products

and the rate-determining step and its rate are 2Aσ + Bσ ⇒ products + 3σ

r = k′1 1b = kpapb1 2 a

2 v

kpa pb r = kpa1b = kpa pb1v =  1 + K i pi

(7-83)

(7-89)

This mechanism is called the Ely-Rideal kinetics. DIFFERENT SITES CHEMICAL EQUILIBRIUM IN GAS PHASE

When A and B adsorb on chemically different sites σ1 and σ2, the rate of the reaction, with surface reaction controlling, A + B ⇒ unadsorbed products

When A is not in adsorbtive equilibrium but it is in chemical equilibrium in the gas phase according to

(7-84)

A+B⇔M+N

is

kpa pb r =  (1 + Ka pa)(1 + Kbpb)

(7-85)

kpm pn /K e r = kpa pb12v =  (7-91) 1 + Ka(pm pn /K e pb) + K b pb + K m pm + K npn

When the numbers of moles of product is larger than that of the reactants, extra sites are required: (7-86)

Aσ + σ ⇒ Mσ + Nσ

(7-87)

(7-90)

this expression is substituted for pa wherever it appears in the rate equation. If the rate-determining step is the surface reaction between adsorbed species, then

CHANGE IN NUMBER OF MOLES

A⇔M+N

pm pn pa =  K e pb

Table 7-4 summarizes some examples of reactions where all substances are in adsorptive equilibrium and the surface reaction controls the rate. In Table 7-5, substance A is not in adsorptive equilibrium, and its adsorption rate is controlling. Details of the derivations of these and some other equations are presented by Yang and Hougen [Chem. Eng.

TABLE 7-4 Surface-Reaction Controlling. Adsorptive equilibrium maintained for all participants. Reaction 1. A → M + N A→M+N A→M+N

Special condition General case Sparsely covered surface Fully covered surface

2. A A M

Basic rate equation

Driving force

Adsorption term

r = kθa r = kθa r = kθa

pa pa 1

1 + Kapa + Kmpm + Knpn 1 1

r = k1θa − k−1θm

pm pa −  K

1 + Kapa + Kmpm (1 + Kapa + Kmpm + Knpn)2

3. A A M + N

Adsorbed A reacts with vacant site

r = k1θaθv − k−1θmθn

pmpn pa −  K

4. A2 A M

Dissociation of A2 upon adsorption

r = k1θa2 − k−1θmθv

5. A + B → M + N A+B→M+N

Adsorbed B reacts with A in gas but not with adsorbed A

r = kθaθb r = kpaθb

pm pa −  K papb papb

2  (1 + K apa + Kmpm)

(1 + Kapa + Kbpb + Kmpm + Knpn)2 1 + Kapa + Kbpb + Kmpm + Knpn

6. A + B A M

r = k1θaθb − k−1θmθv

pm papb −  K

(1 + Kapa + Kbpb + Kmpm)2

7. A + B A M + N

r = k1θaθb − k−1θmθn

pmpn papb −  K

(1 + Kapa + Kbpb + Kmpm + Knpn)2

r = k1θa2θb − k−1θmθnθv

pmpn papb −  K

3  (1 + K apa + Kbpb + Kmpm + Knpn)

8. A2 + B A M + N NOTE:

Dissociation of A2 upon adsorption

The rate equation is: k (driving force) r =  adsorption term When an inert substance I is adsorbed, the term KI pI is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

7-18

REACTION KINETICS

TABLE 7-5

Adsorption-Rate Controlling (Rapid Surface Reaction)

Reaction

Special condition

Basic rate equation

Driving force

Adsorption term

pa

Kapmpn 1+ + Kmpm + Knpn K



pm pa −  K

Kapm 1+ + Kmpm K





pmpn pa −  K

Kapmpn 1+ + Kmpm + Knpn K





pm pa −  K

Kp +K p 1 + K

pa

Kapmpn 1 +  + Kbpb + Kmpm + Knpn Kpb

1. A → M + N

r = kpaθv

2. A A M

θa r = k paθv −  Ka

3. A A M + N

θa r = k paθv −  Ka



4. A2 A M

Dissociation of A2 upon adsorption

θa2 r = k paθv2 −  Ka

5. A + B → M + N

Unadsorbed A reacts with adsorbed B

r = kpaθv

2

a m

m m

6. A + B A M

θa r = k paθv −  Ka



pm pa −  Kpb

Kn pm 1 +  + Kbpb + Kmpm Kpb

7. A + B A M + N

θa r = k paθv −  Ka



pmpn pa −  Kpb

Kapmpn 1 +  + Kbpb + Kmpm + Knpn Kpb

θa2 r = k paθv2 −  Ka

pmpn pa −  Kpb

+Kp +K p 1 + Kp

8. A2 + B A M + N NOTES:

 

Dissociation of A2 upon adsorption





Kapmpn

b b

m m

b



+ Knpn

2

The rate equation is:

k (driving force) r =  adsorption term Adsorption rate of substance A is controlling in each case. When an inert substance I is adsorbed, the term Kipi is to be added to the adsorption term. SOURCE: From Walas, Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989.

Prog. 46:146 (1950)], Walas (Reaction Kinetics for Chemical Engineers, McGraw-Hill, 1959; Butterworths, 1989, pp. 153–164), and Rase (Chemical Reactor Design for Process Plants, vol. 1, Wiley, 1977, pp. 178–191). NO RATE-CONTROLLING STEP All the relations developed above assume that only one step is controlling. In a reaction system, changing the operating conditions may shift the control from one step to another. It is therefore also obvious that at certain conditions there is no single step controlling. In that case all the reactions and their respective rates have to be considered, and the adsorbed species cannot be eliminated from the rate expressions to obtain a single closed-form kinetic rate. LIQUID-SOLID CATALYTIC REACTIONS An analogous treatment for liquid-solid catalysis can be derived, with partial pressures replaced by liquid volumetric concentrations. Otherwise all the analyses hold. BIOCATALYSIS Biochemical reactions such as aerobic and anaerobic fermentations occur in the presence of living organisms or cells, such as bacteria, algae, and yeast. These reactions can be considered as biocatalyzed by the organism. Thus in a typical bioreactor a substrate (such as glucose) is fed into the fermenter or bioreactor in the presence of an initial amount of cells. The desired product can be the cells themselves or a secreted chemical called a metabolite. In either case the cells multiply in the presence of the substrate, and the rate of production of cells is proportional to the concentration of the cells— hence this process is autocatalytic. In a batch reactor with ample

supply of substrate, this results in exponential growth of the culture. A typical cell or biomass growth rate function, called the Monod kinetics, is identical in form to the Michaelis-Menten enzyme kinetics in Eq. (7-70): µ maxCs µ=  Ks + Cs

(7-92)

In Eq. (7-92) µ is the specific growth rate of the culture. It is measured in units of reciprocal time (h−1). Growth rate µ is related to the volumetric growth rate rx of the culture: rx = Cxµ. This means that the true unit of µ is, e.g., (g biomass formed/h)/(g biomass present), where g biomass is the dry-weight (DW) of the biomass, obtained after evaporation of the water content of the cell (which constitutes about 80 percent of the wet biomass weight). Similarly Cx has the unit, e.g., (g DW)/(L medium volume). The variable Cs in Eq. (7-92) is the concentration of the limiting substrate in the medium (g/L). There are many substrates (including micronutrients) in the medium, but there is usually just one that determines the specific growth rate. This substrate is often a sugar (most likely glucose) but it could also be a metal ion (Mg2+ etc.), or PO43−, NH4+, . . . , or perhaps a hormone. The limiting substrate may easily change during a fermentation, and then the rate expression will change. The two parameters in Eq. (7-92) are the maximum specific growth rate µmax (h−1) and the saturation constant Ks (g substrate/L). The value of Ks is obtained as the substrate concentration at which µ = 1/2 µmax (see Fig. 7-6). The form of Eq. (7-92) is entirely empirical, but it incorporates two important features: (1) At high substrate concentration the whole cell machinery is involved in cell synthesis, and the specific growth rate reaches a maximum µmax; (2) at low substrate concentration formation of biomass is a first-order rate process (as in any other chemical reaction) and µ → (µmaxKs)Cs. Note that for many commonly used microorganisms Ks is much

FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER

7-19

0.3 0.25 0.2 µ, h

-1

µmax = 0.3 h-1 0.15

Ks = 0.15 g/L

0.1 0.05 0 0

0.5

1

1.5 Cs, g/L

2

2.5

3

FIG. 7-6 The effect of substrate concentration on specific growth rate.

smaller than the substrate concentration Cs.Thus in batch cultivations Ks is several orders of magnitude smaller than Cs until the very end of the batch, and this is what gives the well-known exponential growth [see Eq. (7-93)]. Equation (7-93) applies for batch cultures after an initial lag phase when cell machinery is synthesized. Typical values for Ks are 150 mg/L (Saccharomyces cerevisiae), 5 to 10 mg/L (lactic bacteria and E. coli), and less than 1 mg/L (filamentous fungi).

dCx  = µCx dt

Cx = Cx0 e µt

(7-93)

Equation (7-93) may have to be modified by subtraction of a deathrate term µdCx. µd may well increase during the batch fermentation in which case the net growth rate of (viable) cells eventually becomes negative, and the concentration of (viable) cells will start to decrease.

FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER GAS-SOLID CATALYTIC REACTIONS The Langmuir-Hinshelwood mechanism of adsorption/reaction described above allowed us to relate the gas concentrations and partial pressures in the vicinity of the catalyst surface to the adsorbed species concentration at the active sites, which in turn determined the surface reaction rates. In practice, two additional mass-transfer processes may need to be considered: 1. Diffusion to and from the bulk gas to the external catalyst surface, represented as an external mass-transfer process across a film or boundary layer concentration gradient. For nonporous catalyst this is the only mass-transfer step. 2. Diffusion to and from the catalyst external surface through pores in a porous catalyst particle to active sites inside the catalyst particle where the adsorption and reaction occur, represented as intraparticle diffusion and modeled as a diffusion-reaction process. External Mass Transfer In a reactor, the solid catalyst is deposited on the surface of narrow tubes (such as monolith or foams), is packed as particles in a tube, or is suspended in slurry or in a fluidized bed as fine particles. For these systems, the bulk concentration of the gas phase approaches that on the catalyst surface if the masstransfer rate from bulk to surface is substantially larger than the reaction rates on the surface. This, however, is often not the case. The mechanism of mass transfer and reaction on the external catalyst surface includes the following consecutive steps:

1. Mass transfer of gas reactants from the bulk gas to the solid catalyst surface, also called external mass transfer 2. Adsorption, reaction on the surface, and desorption of products, e.g., Langmuir-Hinshelwood kinetics 3. Mass transfer of products from the catalyst surface to the bulk gas At steady state all these rates are equal. For example, for a first-order irreversible reaction A ⇒ B, the rate of mass transfer equals the rate of intrinsic reaction: ks as(Ca − Cas) = kCas

(7-94)

Here as is, for instance, the external particle surface area/volume of reactor. Eliminating the surface concentration Cas in terms of the observable bulk gas concentration Ca yields the overall specific rate of consumption of A: Ca robs = kCa =  1/(ks as) + 1/k 1 k obs =  1/(ksas) + 1/k

(7-95)

Hence the observable overall rate constant kobs is actually a combination of the mass-transfer coefficient and the intrinsic rate

7-20

REACTION KINETICS

coefficient; or in terms of resistances (in units of time) the overall resistance is the sum of the mass transfer and intrinsic kinetic resistance. For this first-order rate case, the overall behavior remains first order in bulk gas concentration. The two limiting cases are mass transfer and reaction control, respectively: kobs = ksas

and robs = ksasCa

kobs = k

and robs = kCa

for ksas << k Mass transfer control (7-96) for ksas >> k Kinetic control

The mass-transfer coefficient depends on the geometry of the solid surface, on the hydrodynamic conditions in the vicinity of the catalyst (which are a function, e.g., of the reactor type, geometry, operating conditions, flow regime), and it also depends on the diffusivity of the gas species. Correlations for the mass-transfer coefficient are a large topic and outside the scope of this section. For more details see Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley & Sons, New York, 2002, and relevant sections in this handbook. For non-first-order kinetics a closed-form relationship such as the series of resistances cannot always be derived, but the steady-state assumption of the consecutive mass and reaction steps still applies. Intraparticle Diffusion As indicated above, the larger the catalyst surface area per unit reaction volume as, the larger the overall reaction rate. For a fixed mass of catalyst, decreasing the particle size increases the total external surface area available for the reaction. Another way to increase the surface area is by providing a porous catalyst with lots of internal surface area. The internal structure of the catalyst determines how accessible these internal sites are to the gas-phase reactant and how easily can the products escape back to the gas. The analysis is based on the pseudo-homogeneous reaction diffusion equation, with the gas reactant diffusing through the pores and reacting at active sites inside the catalyst particle. For a first-order irreversible reaction of species A in an infinite slab geometry, the diffusion-reaction equations describe the decreasing reactant concentration from the external surface to the center of the slab: d 2Cay Dea  − kCay = 0 dy2

Cay(L) = Cas

dCay  (0) dy

r = ηkCas



(7-99)

The effectiveness factor can be written as a function of a dimensionless independent variable called the Thiele modulus, which for a firstorder reaction is defined below together with the corresponding effectiveness factor derived by integration of the corresponding diffusion-reaction equation (7-97):



tanhφslab η=  φslab

 D (C ) r (C ) dC 1  2

Cas

Cae

ea

ay

a

ay

(7-102)

ay

In Eq. (7-102) component A is the limiting reactant. For example, for an nth-order irreversible reaction Vp φ=  Spx

(7-98)

L

k  Dea

(7-101)

The parameters that describe the pore structure are the porosity εs, accounting for the fact that diffusion only occurs through the gasfilled part of the particle, and the tortuosity τ accounting for the effect of diffusion path length and contraction/expansion of pores along the diffusion path. The diffusion regime depends on the diffusing molecule, pore size, and operating conditions (concentration, temperature, pressure), and this can be visualized in Fig. 7-7. As indicated, the effective diffusion coefficient ranges over many orders of magnitude from very low values in the configurational regime (e.g., in zeolites) to high values in the regular regime. There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g., Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley & Sons, New York, 2002; Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993; and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. The larger the characteristic length L, the larger the Thiele modulus, the smaller the effectiveness factor, and the steeper the reactant concentration profile in the catalyst particle. A generalized characteristic length definition Vp/Spx (particle volume/external particle surface area) brings together the η-φ curves for a variety of particle shapes, as illustrated in Table 7-6 and Fig. 7-8 for for slabs, cylinders, and spheres. Here I0 and I1 are the corresponding modified Bessel functions of the first kind. Further generalization of the Thiele modulus and effectiveness factor for a general global reaction and various shapes is

(7-97)

Hence the effectiveness factor is the ratio of the actual rate to that if the reactions were to occur at the external surface concentration, i.e., in absence of intraparticle diffusion resistance: (1/L) r(Cay) dy 0 rate with pore diffusion resistance η =  =  r(Cas) rate at external surface conditions

ε Dea = s Da τ

(Vp/Spx)ra(Cas) φ = 

The concept of effectiveness factor has been developed to calculate the overall reaction rate in terms of the concentration at the external surface Cas:

φslab = L

between the gas-filled pores and the solid parts of the particle. For most catalysts (except for straight channel monoliths), the diffusion path is not straight and has varying cross section. Hence, the effective diffusivity of A is defined based on the catalyst internal structure and the gas diffusivity of A in the gas mixture as follows:

(n + 1) kC  

2 D n−1 as

(7-103)

ea

This generalized Thiele modulus works well with the effectiveness factors for low and high values of the Thiele modulus, but it is not as accurate for intermediate values. However, these differences are not significant, given the uncertainties associated with measuring some of the other key parameters that go into the calculation of the Thiele modulus, e.g., the effective diffusivity and the intrinsic rate constant. Effect of Intraparticle Diffusion on Observed Order and Activation Energy Taking the nth-order reaction case in the limit of intraparticle diffusion control, i.e., large Thiele modulus, the effectiveness factor is 1 η=  φ

(7-104)

the observed rate is (7-100)

Since the model is pseudo-homogeneous, there is no distinction

Spx robs = ηr =  Vp

C (n + 1) 2Deak

(n + 1)/2 as

(7-105)

FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER D cm2/s 1

7-21

Regular 1 bar Gases 10 bar

10−2 Knudsen 10−4

Liquids

10−6

10−8 Configurational 10−10

10−12

10−14

0.1

1

10

100 r, nm

1000

10000

FIG. 7-7 Diffusion regimes in heterogeneous catalysts. [From Weisz, Trans. Fara. Soc. 69:

1696–1705 (1973); Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990, Figure 3.5.1-1.]

and the observed rate constant is Spx kobs = ηk =  Vp

e ke D n + 1

2

ea0

(ED /RT)

0

(E/RT)

(7-106)

Hence, the observed order and activation energy differ from those of the intrinsic nth-order kinetics: n+1 n obs =  2

E E + ED E obs =  M  2 2

(7-107)

Here ED is the activation energy for diffusion. TABLE 7-6 Effectiveness Factors for Different Shapes for a First-Order Reaction Vp /Spx

Effectiveness factor η

R

tanh φ  φ

Infinite cylinder

R/2

I1(2φ)  φ I0(2φ)

Sphere

R/3

1 3 1     φ tanh 3φ φ

Shape Infinite slab





The observed and intrinsic reaction order is the same under intraparticle diffusion control only for a first-order reaction. Weisz and Prater [“Interpretation of Measurements in Experimental Catalysis,” Adv. Catal. 6: 144 (1954)] developed general estimates for the observed order and activation energy over the entire range of φ: n − 1 d ln η nobs = n +   2 d ln φ

E − ED d ln η Eobs = E +   2 d ln φ

(7-108)

Weisz and Prater [“Interpretation of Measurements in Experimental Catalysis,” Adv. Catal. 6: 144 (1954)] also developed a general criterion for diffusion limitations, which can guide the lab analysis of rate data:



3V If Φ = p Spx



2

robs  >> 1 DeaCas << 1

then diffusion-limited (7-109) then no diffusional resistance

Effect of Intraparticle Diffusion for Reaction Networks For multiple reactions, intraparticle diffusion resistance can also affect the observed selectivity and yield. For example, for consecutive reactions intraparticle diffusion resistance reduces the yield of the intermediate (often desired) product if both reactions have the same order. For parallel reactions diffusion resistance reduces the selectivity to the higher-order reaction. For more details see, e.g., Carberry, Chemical and Catalytic Reaction Engineering, McGrawHill, 1976; and Levenspiel, Chemical Reaction Engineering, 3d ed., Wiley, 1999.

7-22

REACTION KINETICS 1.0 0.8 P 0.6

C

0.4

S

η 0.2

0.1 0.1

0.2

0.4

0.6

0.8

1

2

4

6

8

10

Λ FIG. 7-8 Effectiveness factors for a slab (P), a cylinder (C), and a sphere (S). [Adapted from Fig. 1 in Aris and Rester, “The Effect of Shape on the Effectiveness Factor,” Chem. Eng. Sci. 24: 793 (1969).]

1 1 φ2  =  +  ηt η 3Sh′

(7-110)

εsksR Sh′ =  Dea

(7-111)

As indicated above, intraparticle diffusion lowers the apparent activation energy. The apparent activation energy is even further lowered under external mass-transfer control. Figure 7-9 illustrates how the rate-controlling step changes with temperature, and as a result the dependence of the apparent first-order rate constant on temperature also changes, from a very strong dependence under kinetic control to virtual independence under external mass-transfer control. Note that in the limit of external diffusion control, the activation energy Eobs→ 0, as can be shown when substituting Eq. (7-110) in Eq. (7108). For more details on how to represent the combined effect of external and intraparticle diffusion on effectiveness factor for more complex systems, see Luss, “Diffusion-Rection Interactions in Catalyst Pellets.” Heat-Transfer Resistances A similar analysis regarding external and intraparticle heat-transfer limitations leads to temperature

gradients which add further complexity to the behavior of heterogeneous catalytic systems, including steady-state multiplicity. More details are given in Sec. 19. Catalyst Deactivation The catalyst life ranges from seconds to minutes to a few days to several years, as the active surface of a catalyst is degraded by chemical, thermal, or mechanical factors. Chemical deactivation occurs due to feed or product poisoning or masking. Poisoning may be due to compounds such as P, S, As, Na, and Bi that have free electron pairs and is generally considered irreversible. In some cases a reduced life is simply accepted, as in the case of slow accumulation of trace metals from feed to catalytic cracking; but in other cases the deactivation is too rapid. Sulfur and water are removed from feed to ammonia synthesis, sulfur from feed to platinum reforming, and arsenic from feed to SO2 oxidation with platinum. Masking may be due to covering of the active sites by contaminants in either the feed or products. Examples of feed masking agents can include Si (from organic silicons) and rust. An example of product masking is coking. Reactivation sometimes is done in place;

106 k

For more complex reactions, the effect of intraparticle diffusion resistance on rate, selectivity, and yield depends on the particulars of the network. Also, the use of the Thiele modulus–effectiveness factor relationships is not as easily applicable, and numerical solution of the diffusion-reaction equations may be required. Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, “Diffusion-Reaction Interactions in Catalyst Pellets,” in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor ηt can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction:

220 Oxidation of acetylene on manganese dioxide 200 catalyst 180 External diffusion 160 140 Pore diffusion 120 100 80 60 Chemical rate 40 20 10 12 14 16 18 20 22 24 26 28 30 10,000/T

FIG. 7-9 Dependence of the rate-controlling step on temperature.

FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER for instance, coke is burned off cracking catalyst or off nickel and nickel-molybdenum catalysts in a fluidized reactor/regenerator system. Thermal deactivation is primarily due to rearrangement of the active sites at high temperature due to sintering. Sintering results in agglomeration of active ingredients (lower dispersion). In most cases sintering is irreversible; however Pt/Al2O3 catalysts have been regenerated in place by Cl2 treatment. The catalyst also can be modified by additives, for instance, chromia to nickel to prevent sintering, rhenium to platinum to reduce coking, and so on. Mechanical deactivation may be caused by attrition or erosion and subsequent loss of catalyst as fines. The attrition resistance of catalysts is related to the nature of the support and its porosity. For additional references, see, e.g., Thomas, Catalytic Processes and Proven Catalysts, Academic Press, 1970; Butt and Petersen, Activation, Deactivation and Poisoning of Catalysts, Academic Press, 1988; and Delmon and Froment, Catalyst Deactivation, Elsevier, 1980. The activity α at any time on stream may be simply defined as the ratio of the rate at time t to the rate with fresh catalyst r(t) α=  r(t = 0)

(7-112)

The rate of destruction of active sites and pore structure can be expressed as a kinetic relation that can be generally first- or secondorder. For instance, for a second-order dependence, dα  = −kdα2 dt

(7-113)

the corresponding integral is 1 α=  1 + kd t

(7-114)

This type of deactivation mechanism often applies catalyst sintering and coke deactivation. The deactivation rate constant is expected to have an Arrhenius dependence on temperature. When the feedstock contains constant proportions of reactive impurities, the rate of decline may also depend on the concentration of the main reactant, e.g., for a power law rate dα  = −kdαpCq dt

(7-115)

Such a differential equation must be solved simultaneously with a rate equation for the main reactant. The deactivation rate constants are estimated by methods like those for finding constants of any rate equation, given suitable (α, t) data. There are different chemical deactivation mechanisms—two of the most common are described below. For more details see Butt and Petersen, Activation, Deactivation and Poisoning of Catalysts, Academic Press, 1988; and Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990. In uniform deactivation, the poison is distributed uniformly throughout the pellet and degrades it gradually. In pore mouth (shell progressive) poisoning, the poison is so effective that it kills the active site as it enters the pore; hence complete deactivation begins at the mouth and moves gradually inward. Uniform Deactivation When uniform deactivation occurs, the specific rate declines by a factor 1 − β, where β is the fractional poisoning. β is calculated from the poisoning rate, and it is often assumed to be proportional to the concentration of the poison in the bulk fluid. Then a power law rate equation becomes r = k(1 − β)ηCns

(7-116)

The effectiveness depends on β through the Thiele modulus



φ=L

k(1 − β)Cns − 1  De

(7-117)

7-23

To find the effectiveness under poisoned conditions, this form of the Thiele modulus is substituted into the appropriate relation for effectiveness. For example, for a first-order reaction in slab geometry, the effectiveness factor is tanh[L k(1  − β) /De] tanh φ η =  =  L k(1  − β) /De] φ

(7-118)

Figure 7-10a shows the ratio of the effectiveness factor with uniform poisoning to that without poisoning versus the fraction poisoned for the above case of first-order reaction in a slab. Pore Mouth (or Shell Progressive) Poisoning This mechanism occurs when the poisoning of a pore surface begins at the mouth of the pore and moves gradually inward. This is a moving boundary problem, and the pseudo-steady-state assumption is made that the boundary moves slowly compared with diffusion of poison and reactants and reaction on the active surface. β is the fraction of the pore that is deactivated. The poison diffuses through the dead zone and deposits at the interface between the dead and active zones. The reactants diffuse across the dead zone without reaction, followed by diffusion-reaction in the active zone. Figure 7-10b shows simulation results for the ratio of the effectiveness factor with pore mouth poisoning to that without poisoning for a first-order reaction in a slab. GAS-SOLID NONCATALYTIC REACTIONS Examples of gas-solid noncatalytic reactions include production of iron from iron ores, roasting of sulfide oxides, combustion of solid fuels, chlorination of Ti ores to make TiCl4 in the production of TiO2 pigments, incineration of waste, decomposition of solids to produce gases, e.g., solid propellants and explosives. The kinetic treatment of these reactions has to take into consideration external mass transfer and intraparticle diffusion just as in the case of gas-solid catalytic reactions. However there are major differences, the primary one being consumption of the solid reactant, making the conditions inside the solid particle transient in nature, including change in unreacted particle size, particle density, porosity, etc. For more details see, e.g., Wen [“Noncatalytic Heterogeneous Solid-Fluid Reaction Models,” Ind. Eng. Chem. 60(9):34–54 (1968)], Szekely (in Lapidus and Amundson (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977), Doraiswamy and Kulkarni [in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987], and Levenspiel (Chemical Reaction Engineering, 3d ed., Wiley, 1999). The basic steps are identical to those of catalytic gas-solid reactions. However, as indicated above, the process is transient (non-steadystate) due to change in particle size and properties as the reaction progresses. Several models that describe gas-solid noncatalytic reactions are summarized in Table 7-7. The first two, the sharp interface and volume reaction models, are pseudo-homogeneous, form part of the class of shrinking core models, and can be treated by using the Thiele modulus and effectiveness factor concept. The last three are heterogeneous models. Sharp Interface Model For a first-order reaction in gas reactant, A(g) + bB(s) ⇒ products

(7-119)

a rate of conversion of the solid B per unit particle volume of rb = kCasCs

(7-120)

and assuming pseudo-steady state for the gas-phase component, it can be shown that Ca0 robs =  R2p Rp b 1   +   − 1 +  kpap Dea Rs kCs0





(7-121)

7-24

REACTION KINETICS 1.0 φ

0.9

1

Ratio of effectivenesses

0.8 0.7 0.6

2

0.5 3

0.4

5

0.3 0.2 0.1 0.0 0.0

10 Uniform 0.1

0.2

0.3

0.4 0.5 0.6 Fraction poisoned

0.7

0.8

0.9

1.0

0.7

0.8

0.9

1.0

(a)

1.0 0.9

Ratio of effectivenesses

0.8 0.7 φ

0.6

1

0.5 0.4

3

0.3

5

0.2

10

0.1

Pore mouth

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Fraction poisoned (b) FIG. 7-10 Poisoning of a first-order reaction. (a) Uniform poisoning. (b) Pore mouth poisoning.

where Rp and Rs are the particle and solid core radii, ap is the external particle diameter-based interfacial area per particle volume, and k is based on particle volume. Equation (7-121) represents three resistances in series—external mass transfer, diffusion in the reacted (ash) zone, and reaction at the unreacted solid-ash interface. The conversion of the solid reactant B is obtained from integration of the pseudo-steady-state diffusion model with reaction at the boundary:

1 ∂ ∂Cas 2  Dear 2  = 0 r ∂r ∂r





∂Cas 1 Dea  =  kCasCs ∂r b ∂Cas Dea  = ks(Ca0 − Cas) ∂r

at r = Rs at r = Rp

(7-122)

FLUID-SOLID REACTIONS WITH MASS AND HEAT TRANSFER TABLE 7-7

7-25

Noncatalytic Gas-Solid Reaction-Diffusion Models

Model

Main features

Sharp interface model (SIM)

Volume reaction model Grain model

Crackling core model Nucleation model

Reacting solid is nonporous. Reacted solid ash is porous. Reaction occurs at the ash-unreacted solid interface. Reacting solid is also porous. Reaction occurs everywhere in the particle. Particle is divided into identical solid spherical grains. Each grain reacts according to the sharp interface model. Combination of SIM and grain model. Nucleation of metals in metal reduction reactions.

CA

CS0

C sAs

CSs

The moving boundary radius Rc is determined from a material balance that relates the unreacted solid volume to the reaction rate. Integration gives the time τ required to achieve a given conversion of the solid B, Xb:

 










RpCs0 1 1 Rp Rp τ =    −  Xb +  1 − (1 − Xb)23 bCa0 3 ks Dea 2Dea 3b +  1 − (1 − Xb)13 Cs0Rpk



R Xb = 1 − s Rp

3

 Fluid film

Solid reactant (7-123)

Similar solutions can be obtained for other shapes (Doraiswamy and Kulkarni, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987). Figure 7-11 shows typical concentration profiles for this case.

CA

CS 0

C sA

FIG. 7-12 Typical concentration profiles for the volume reaction model. [From Wen, “Noncatalytic Heterogeneous Solid-Fluid Reaction Models,” Ind. Eng. Chem. 60(9): 34–54 (1968), Fig. 3.]

Volume Reaction Model A typical concentration profile for the volume reaction model is shown in Fig. 7-12. A general transient model of diffusion-reaction that uses the effective diffusivity concept described for gas-solid catalytic reactions can be derived here as well, e.g., for a spherical particle:

s

∂(εsCas) 1 ∂ ∂Cas  = 2  Dear2  − ra ∂t r ∂r ∂r



Cas = Cas0 s

CS

Cs = Cs0

∂Cs  = −rs ∂t



∂Cas  =0 ∂r

at t = 0

∂Cas Dea  r = R = ks(Ca − Casr = R) ∂r



C εs = εs0 + Cs0(vs − vp) 1 − s Cs0



εs β Dea  =  , β=2−3 εs0 Dea0



at r = 0 (7-124)

(7-125)

Here the porosity and the diffusivity vary with conversion of solid; vs and vp are the reactant and product molar volumes. A Thiele modulus φ and dimensionless time θ can be defined, e.g., for a rate secondorder in A and first-order in S: Fluid film

r = kC2asCs



φ=R

FIG. 7-11 Sharp interface model—concentration profiles. [From Wen, “Non-

catalytic Heterogeneous Solid-Fluid Reaction Models,” Ind. Eng. Chem. 60(9): 34–54 (1968), Fig. 1.]

kCas(Rs)Cs0  Dea0

(7-126) θ = bkC2as0 t

(7-127)

For the given rate expression, equations (7-124) to (7-127) can be numerically integrated, e.g., in Fig. 7-13 for reaction control and Fig. 7-14 for intraparticle diffusion control, both with negligible external mass-transfer resistance; x is the fractional conversion.

REACTION KINETICS

FIG. 7-13 Concentration profiles with reaction control φ 1, in absence of gas particle mass-transfer resistance. [From Wen, “Noncat-

alytic Heterogeneous Solid-Fluid Reaction Models,” Ind. Eng. Chem. 60(9): 34–54 (1968), Fig. 11.]

1.0

1.0

1 2

0.8

3 4

0.6

6

5

0.6

5

6

4

0.4

0.4 3

2

1

0.2

0.0 0

CAs / CAsO

0.8

Cs / Cso

7-26

0.2

0.2

0.4

0.6

0.8

1.0

0

0.2

r/R

0.4

0.6

0.8

0.0 1.0

r/R Curve 1 2 3 4 5 6

θ 0 1.84 4.88 10.16 16.40 20.56

x 0 0.151 0.417 0.708 0.892 0.957

FIG. 7-14 Concentration profiles with intraparticle diffusion control, φ 70, in absence of gas particle mass-transfer resistance.

[From Wen, “Noncatalytic Heterogeneous Solid-Fluid Reaction Models,” Ind. Eng. Chem. 60(9): 34–54 (1968), Fig. 12.]

GAS-LIQUID REACTIONS

7-27

GAS-LIQUID REACTIONS Many industrial processes employ gas-liquid reactions that can be either noncatalytic or homogeneously catalyzed. These include, for instance, absorption of acid gases (SO3, NO2, CO2), chlorinations (benzene, dodecane, toluene), oxidations (P-Xylene to terephthalic acid, cyclohexane to cyclohenanone and cyclohexanol, acetaldehyde to acetic acid), hydrogenations (olefins, esters to fatty acids), and hydroformylation of olefins to alcohols, to name a few. See also Sec. 19 and Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979). These reactions include gas reactants dissolving in a liquid and reacting there with a liquid reactant. When determining the kinetics of such reactions from lab data, one needs to understand the mechanism and the controlling steps, just as was the case for heterogeneous gassolid reactions. The simplest model is the two-film model, and it involves the following consecutive steps for the gaseous reactant: 1. Mass transfer of gas reactant from bulk gas to the gas-liquid interface across the gas film. 2. At the gas-liquid interface, the liquid and gas concentrations of the gaseous reactant are assumed to be at thermodynamic equilibrium. 3. Mass transfer of the dissolved gas reactant to the bulk liquid across the liquid film—if the reaction is fast, the reaction will occur both in the liquid film (in parallel with diffusion) and in the bulk liquid. For a volatile liquid reactant or a volatile product, these steps are essentially reversed. For a nonvolatile liquid reactant or product, only the reaction and diffusion in the liquid take place. Figure 7-15 describes the absorbing gas concentration profiles in a gas-liquid system. For a general gas-liquid reaction: A(g) + bB(l) → products

(7-128)

the two-film pseudo-steady-state model is described by the following fluxes across the interface for the gaseous reactant A: DaG DaL NaG =  (Pa − Pai) = kG (Pa − Pai) = NaL =  (CLai − CLa) δG δL = kL (CLai − CLa) (7-129) Here the subscript L denotes liquid, G denotes gas, i denotes the gasliquid interface (where the gas and liquid concentrations are in equilibrium). The thickness of the liquid and gas films is not a directly measurable quantity, and instead mass-transfer coefficients are

defined as indicated above. These depend on the diffusivity of the molecule, geometry, flow, and operating conditions; and typical values can be viewed in Sec. 19. In addition to the two-film steady-state model, other more accurate, non-steady-state models have also been developed such as the surface renewal and penetration models (see, e.g., Astarita, Mass Transfer with Chemical Reaction, Elsevier, 1967). In many industrial cases of interest, mass-transfer resistance in the gasfilm is negligible, especially considering that gas-phase diffusivities are 2 to 3 orders of magnitude larger for the same species than those in the liquid. Hence we drop the subscripts L and G from the concentrations since the concentrations considered are in the liquid phase only. REACTION-DIFFUSION REGIMES Depending on the relative rates of diffusion and reaction, the following diffusion-reaction regimes occur: tD << tr

slow reaction regime with reaction control

tD >> tr

fast reaction regime with diffusion control

tD  tr

both reaction and diffusion are important Cai − Cae tr =  r(Cai)

Da tD =  k2L

Here tD and tr are the diffusion and reaction times, respectively, and kL is the mass-transfer coefficient in the absence of reaction. For the fast reaction regime, diffusion and reaction occur in parallel in the liquid film, while for the slow reaction regime, there is no reaction in the liquid film and the mass transfer can be considered to occur independently of reaction in a consecutive manner. For the slow reaction regime, the following subregimes can be defined: tm << tr

slow reaction kinetic control

tm >> tr

slow reaction mass-transfer control

tm  tr 1 tm =  kLa

both reaction and mass transfer are important

ci catalyst

ci

Liquid film

(7-130)

Gas film

Bulk liquid

Bulk gas

T

Liquid film

T

FIG. 7-15 Absorbing gas concentration and temperature profiles (exothermic reaction) in gas-liquid and gas-liquid-solid reactions.

(7-131)

7-28

REACTION KINETICS Bl A*

Bl A*

0

δ

0

very slow or kinetically controlled

Bl A*

0

δ

Fast pseudo-mth order

δ

slow or mass transfer controlled

Reaction plane

Bl A*

Bl

A*

0

δ

Fast (m, n)th order

0

δ Instantaneous

FIG. 7-16 Concentration profiles for the general reaction A(g)  bB(l) → products with the rate

r = kCamCbn. [Adapted from Mills, Ramachandran, and Chaudhari, “Multiphase Reaction Engineering for Fine Chemicals and Pharmaceuticals,” Rev. Chem. Eng. 8(1–2):1 (1992), Figs. 19 and 20.]

Here tm is the mass-transfer time. Only under slow reaction kinetic control regime can intrinsic kinetics be derived directly from lab data. Otherwise the intrinsic kinetics have to be extracted from the observed rate by using the mass-transfer and diffusion-reaction equations, in a manner similar to those defined for catalytic gas-solid reactions. For instance, in the slow reaction regime, Cai ra,obs =  Hea /kG a + 1/kLa + 1/k 1 kobs =  Hea /kG a + 1/kLa + 1/k

(7-132) (7-133)

(7-134)

Solving the diffusion-reaction equation in the liquid, the enhancement factor can be related to the Hatta number Ha, which is similar to the Thiele modulus defined for heterogeneous gas-solid catalysts. Thus, the Hatta number and its relation to the controlling regime are tD Ha =  = tR

slow reaction regime

Ha >> 1

fast reaction regime



maximum mass transfer rate through film maximum reaction rate in the film

(7-135)

For instance, for a first-order reaction in the gaseous reactant A (e.g., with large excess of liquid reactant B), the following relates the enhancement factor to the Hatta number: Ha = δL

 =  k D

for Cb >> Cai

(7-136)





(7-137)

 kDa

k

L0

a

Here Hea is the Henry constant for the solute a. For the fast reaction regime, instead of the effectiveness factor adjustment for the intrinsic reaction rate, it is customary to define an enhancement factor for mass-transfer enhancement by the reaction, defined as the ratio of mass transfer in presence of reaction in the liquid, to mass transfer in absence of reaction: E = kLkL0

Ha << 1

Ha 1 C E =  1 − a  > 1 tanh Ha Cai cosh Ha

When both A and B have comparable concentrations, then the enhancement factor is an increasing function of an additional parameter: DbCb λ=  bDaCai

(7-138)

In the limit of an instantaneous reaction, the reaction occurs at a plane where the concentration of both reactants A and B is zero and the flux of A equals the flux of B. The criterion for an instantaneous reaction is Cb Ha12 >>  bCai

E∞ = 1 + λ >> 1

(7-139)

Figure 7-16 illustrates typical concentration profiles of A and B for the various diffusion-reaction regimes.

GAS-LIQUID-SOLID REACTIONS GAS-LIQUID-SOLID CATALYTIC REACTIONS Many solid catalyzed reactions take place with one of the reactants absorbing from the gas phase into the liquid and reacting with a liquid reactant on the surface or inside the pores of a solid catalyst (see Fig. 7-15). Examples include the Fischer-Tropsch synthesis of hydrocarbons from synthesis gas (CO and H2) in the presence of Fe or Co-

based heterogeneous catalysts, methanol synthesis from synthesis gas (H2 + CO) in the presence of heterogeneous CuO/ZnO catalyst, and a large number of noble metal catalyzed hydrogenations among others. For a slow first-order reaction of a gaseous reactant, the concept of resistances in series can be expanded as follows, e.g., for a slurry reactor with fine catalyst powder:

GAS-LIQUID-SOLID REACTIONS Cai ra,obs =  He 1 1 1 +   +  +  kGa kLa ksas k

1 kobs =  He 1 1 1  +  +  +  kGa kLa ksas k (7-140)

Intraparticle diffusion resistance may become important when the particles are larger than the powders used in slurry reactors, such as for catalytic packed beds operating in trickle flow mode (gas and liquid downflow), in upflow gas-liquid mode, or countercurrent gas-liquid mode. For these the effectiveness factor concept for intraparticle diffusion resistance has to be considered in addition to the other resistances present. See more details in Sec. 19. POLYMERIZATION REACTIONS Polymers are high-molecular-weight compounds assembled by the linking of small molecules called monomers. Most polymerization reactions involve two or three phases, as indicated below. There are several excellent references dealing with polymerization kinetics and reactors, including Ray in Lapidus and Amundson, (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977; Tirrel et al. in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987; and Meyer and Keurentjes (eds.), Handbook of Polymer Reaction Engineering, Wiley, 2005. Polymerization can be classified according to the main phase in which the reaction occurs as liquid (most polymerizations), vapor (e.g., Ziegler Natta polymerization of olefins), and solid phase (e.g., finishing of melt step polymerization). Polymerization reactions occur in liquid phase and can be further subclassified into 1. Bulk mass polymerization: a. Polymer soluble in monomer b. Polymer insoluble in monomer c. Polymer swollen by monomer 2. Solution polymerization a. Polymer soluble in solvent b. Polymer insoluble in solvent 3. Suspension polymerization with initiator dissolved in monomer 4. Emulsion polymerization with initiator dissolved in dispersing medium Polymerization can be catalytic or noncatalytic, and can be homogeneously or heterogeneously catalyzed. Polymers that form from the liquid phase may remain dissolved in the remaining monomer or solvent, or they may precipitate. Sometimes beads are formed and remain in suspension; sometimes emulsions form. In some processes solid polymers precipitate from a fluidized gas phase. Polymerization processes are also characterized by extremes in temperature, viscosity, and reaction times. For instance, many industrial polymers are mixtures with molecular weights of 104 to 107. In polymerization of styrene the viscosity increased by a factor of 106 as conversion increased from 0 to 60 percent. The adiabatic reaction temperature for complete polymerization of ethylene is 1800 K (3240°R). Initiators of the chain reactions have concentration as low as 10−8 g⋅molL, so they are highly sensitive to small concentrations of poisons and impurities. Polymerization mechanism and kinetics require special treatment and special mathematical tools due to the very large number of similar reaction steps. Some polymerization types are briefly described next. Bulk Polymerization The monomer and initiators are reacted with or without mixing, e.g., without mixing to make useful shapes directly. Because of viscosity limitations, stirred bulk polymerization is not carried to completion. For instance, for addition polymerization conversions as low as 30 to 60 percent are achieved, with the remaining monomer stripped out and recycled (e.g., in the case of polystyrene). Bead Polymerization Bulk reaction proceeds in droplets of 10to 1000-µm diameter suspended in water or other medium and insulated from each other by some colloid. A typical suspending agent is polyvinyl alcohol dissolved in water. The polymerization can be done to high conversion. Temperature control is easy because of the moderating thermal effect of the water and its low viscosity. The suspensions sometimes are unstable and agitation may be critical. Examples

7-29

are polyvinyl acetate in methanol, copolymers of acrylates and methacrylates, and polyacrylonitrile in aqueous ZnCl2 solution. Emulsion Polymerization Emulsions have particles of 0.05- to 5.0µm diameter. The product is a stable latex, rather than a filterable suspension. Some latexes are usable directly, as in paints, or they may be coagulated by various means to produce very high-molecular-weight polymers. Examples are polyvinyl chloride and butadiene-styrene rubber. Solution Polymerization These processes may retain the polymer in solution or precipitate it. Examples include polyethylene, the copolymerization of styrene and acrylonitrile in methanol, the aqueous solution of acrylonitrile to precipitate polyacrylonitrile. Polymer Characterization The physical properties of polymers depend largely on the molecular weight distribution (MWD), which can cover a wide range. Since it is impractical to fractionate the products and reformulate them into desirable ranges of molecular weights, immediate attainment of desired properties must be achieved through the correct choice of reactor type and operating conditions, notably of distributions of residence time and temperature. High viscosities influence those factors. For instance, high viscosities prevalent in bulk and melt polymerizations can be avoided with solution, bead, or emulsion operations. The interaction between the flow pattern in the reactor and the type of reaction affects the MWD. If the period during which the molecule is growing is short compared with the residence time in the reactor, the MWD in a batch reactor is broader than in a CSTR. This situation holds for many free radical and ionic polymerization processes where the reaction intermediates are very short lived. In cases where the growth period is the same as the residence time in the reactor, the MWD is narrower in batch than in CSTR. Polymerizations that have no termination step—for instance, polycondensations—are of this type. This topic is treated by Denbigh [J. Applied Chem., 1:227(1951)]. Four types of MWD can be defined: (1) The number chain length distribution (NCLD), relating the chain length distribution to the number of molecules per unit volume; (2) the weight chain length distribution (WCLD) relating the chain length distribution to the weight of molecules per unit volume; (3) the number molecular weight distribution (NMWD) relating the chain length distribution to molecular weight; and (4) the weight molecular weight distribution (WMWD) relating the weight distribution to molecular weight. Two average molecular weights and corresponding average chain lengths are typically defined: the number average molecular weight Mn and the corresponding number average chain length µn; and the weight average molecular weight Mw and the corresponding weight average chain length µw. Their ratio is called polydispersity and describes the width of the molecular weight distribution. ∞

w jPj j=1 ∞

Mn = 

Pj j=1

∞

∞

jP

j=1

j

µn =  ∞

Pj j=1

w j2Pj j=1 ∞

Mw = 

jPj j =1

µw Mw polydispersity =  =  µn Mn

∞

jP

j=1 2

j

µw =  ∞

jPj j =1

(7-141)

The average chain lengths can be related to the moments λ k of the distribution as follows: λ1 µn =  λ0

λ2 µw =  λ1

λ 0λ2 polydispersity =  λ21

∞

λ k = j kPj j =1

(7-142) Here Pj is the concentration of the polymer with chain length j—the same symbol is also used for representing the polymer species Pj; w is the molecular weight of the repeating unit in the chain. A factor in addition to the residence time distribution and temperature distribution that affects the molecular weight distribution is the type of the chemical reaction (e.g., step or addition polymerization). Two major polymerization mechanisms are considered: chain growth and step growth. In addition, polymerization can be homopolymerization—a single monomer is used—and copolymerization usually with two monomers with complementary functional groups.

7-30

REACTION KINETICS

Chain Homopolymerization Mechanism and Kinetics Free radical and ionic polymerizations proceed through this type of mechanism, such as styrene polymerization. Here one monomer molecule is added to the chain in each step. The general reaction steps and corresponding rates can be written as follows: kd

I → 2f R

initiation

ki

R + M → P1 kp

Pj + M → Pj+1

n = 1, 2, . . .

propagation

kf

Pj + M → P1 + Mn

(7-143)

Pj + Pk → Mj + Mk

termination

ktc

Pj + Pk → Mj+k Here Pj is the growing or live polymer, and Mj is the dead or product polymer. Assuming reaction steps independent of chain length and assuming pseudo-steady-state approximation for the radicals lead to the following rates for monomer and initiator conversion and live polymer distribution. The growing chains distribution is the most probable distribution [see, e.g., Ray in Lapidus and Amundson (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977; Tirrel et al. in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987]: kpM α =  (kp + kf) M + (ktc + ktd)P

Pn = (1 − α)Pα n−1



2fkdI P=  ktc + ktd



dM 2fkd r =  = − kp  dt ktc + ktd





12

dI  = −kdI dt

12

(7-144)

kpM DPinst n =  P (0.5ktc + ktd)

I12M

Here r is the rate of polymerization, α is the probability of propagation, DPninst is the instantaneous degree of polymerization, i.e., the number of monomer units on the dead polymer, and f is the initiation efficiency. Compare r in Eq. (7-144) with the simpler Eq. (7-68). When chain transfer is the primary termination mechanism, such as in anionic polymerization, then the polydispersity is 2. Mathematically, the infinite set of equations describing the rate of each chain length can be solved by using the z transform method (a discrete method), continuous variable approximation method, or the method of moments [see, e.g., Ray in Lapidus and Amundson (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977]. Typical ranges of the kinetic parameters for low conversion homopolymerization are given in Table 7-8. For more details see Hutchenson in Meyer and Keurentjes (eds.), Handbook of Polymer Reaction Engineering, Wiley, 2005. Step Growth Homopolymerization Mechanism and Kinetics Here any two growing chains can react with each other. The propagation mechanism is an infinite set of reactions: kp

Pj + Pk→ Pj+ k nm

(7-145)

TABLE 7-8 Typical Ranges of Kinetic Parameters Coefficient/concentration kd, 1/s f kp, L /(mol/s) kt, L /(mol/s) ktr /kp I, mol/L M, mol/L

P10 (P10τ)n−1 Pn =  (P10τ + 1)n+1 2−α µn =  1−α

n≥1

kpnm = kp for all n, m

2 µw =  (1 − α)(2 − α)

Typical range 106–104 0.4–0.9 102–104 106–108 106–104 104–102 1–10

SOURCE: Hutchenson, “Typical Ranges of Kinetic Parameters,” in Handbook of Reaction Engineering, Wiley, 2005, Table 4.1.



t

τ = kpM dt 0

µw 2 polydispersity =  = 2 µn (2 − α)

P10τ α=  P10τ + 1

transfer

kid

For instance, some nylons are produced through this mechanism. This is usually modeled under the simplifying assumption that the rate constants are independent of chain length. This assumption was proved pretty accurate, and by using the z transform it results in the Flory distribution:

(7-146)

Copolymerization Copolymerization involves more than one monomer, usually two comonomers, as opposed to the single monomer involved in the chain growth and step homopolymerization schemes above. Examples are some nylons, polyesters, and aramids. Here as well there are step growth and chain growth mechanisms, and these are much more complex [see, e.g., Ray in Lapidus and Amundson (eds.), Chemical Reactor Theory—A Review, Prentice-Hall, 1977]. BIOCHEMICAL REACTIONS Mechanism and kinetics in biochemical systems describe the cellular reactions that occur in living cells. Biochemical reactions involve two or three phases. For example, aerobic fermentation involves gas (air), liquid (water and dissolved nutrients), and solid (cells), as described in the “Biocatalysis” subsection above. Bioreactions convert feeds called substrates into more cells or biomass (cell growth), proteins, and metabolic products. Any of these can be the desired product in a commercial fermentation. For instance, methane is converted to biomass in a commercial process to supply fish meal to the fish farming industry. Ethanol, a metabolic product used in transportation fuels, is obtained by fermentation of corn-based or sugar-cane-based sugars. There is a substantial effort to develop genetically modified biocatalysts that produce a desired metabolite at high yield. Bioreactions follow the same general laws that govern conventional chemical reactions, but the complexity of the mechanism is higher due to the close coupling of bioreactions and enzymes that are turned on (expressed) or off (repressed) by the cell depending on the conditions in the fermenter and in the cell. Thus the rate expression (7-92) can mainly be used to design bioreaction processes when the culture is in balanced growth, i.e., for steady-state cultivations or batch growth for as long as the substrate concentration is much higher than Cs. After a sudden process upset (e.g., a sudden change in substrate concentration or pH), the control network of the cell that lies under the mass flow network is activated, and dramatic changes in the kinetics of product formation can occur. Table 7-9 summarizes key differences between biochemical and conventional chemical systems [see, e.g., Leib, Pereira, and Villadsen, “Bioreactors, A Chemical Engineering Perspective,” Chem. Eng. Sci. 56: 5485–5497 (2001)]. TABLE 7-9 Biological versus Chemical Systems • There is tighter control on conditions (e.g., pH, temperature, substrate and product concentrations, dissolved O2 concentration, avoidance of contamination by foreign organisms). • Pathways can be turned on/off by the microorganism through expression of certain enzymes depending on the substrate type and concentration and operating conditions, leading to a richness of behavior unparalleled in chemical systems. • The global stoichiometry changes with operating conditions and feed composition; kinetics and stoichiometry obtained from steady-state (chemostat) data cannot be used reliably over a wide range of conditions, unless fundamental models are employed. • Long-term adaptations (mutations) may occur in response to environment changes that can alter completely the product distribution. • Only the substrates that maximize biomass growth are utilized even in the presence of multiple substrates. • Cell energy balance requirements pose additional constraints on the stoichiometry that can make it very difficult to predict flux limitations.

GAS-LIQUID-SOLID REACTIONS TABLE 7-10

Heirarchy of Kinetic Models in Biological Systems

• Stoichiometric black box models (similar to a single global chemical reaction) represent the biochemistry by a single global reaction with fixed stoichiometric or yield coefficients (limited to a narrow range of conditions). Black box models can be used over a wider range of conditions by establishing different sets of yield coefficient for different conditions. These are also needed to establish the quantitative amounts of various nutrients needed for the completion of the bioreaction. • Unstructured models view the cell as a single component interacting with the fermentation medium, and each bioreaction is considered to be a global reaction, with a corresponding empirical rate expression. • Structured models include information on individual reactions or groups of reactions occurring in the cell, and cell components such as DNA, RNA, and proteins are included in addition to the primary metabolites and substrates (see, e.g., the active cell model of Nielsen and Villadsen, Bioreaction Engineering Principles, 2d ed., Kluwer Academic/Plenum Press, 2003). • Fundamental models include cell dimensions, transport of substrates and metabolites across the cell membrane, and the elementary cell bioreaction steps and their corresponding enzyme induction mechanism. In recent years further kinetic steps have been added to the above models which are based on the conversion of substrates to metabolites. Thus the kinetics of protein synthesis by transcription and translation from the genome add much further complexity to cell kinetics.

The network of bioreactions is called the metabolic network, the series of consecutive steps between key intermediates in the network are called metabolic pathways, and the determination of the mechanism and kinetics is called metabolic flux analysis. As for chemical systems, there are several levels of mechanistic and kinetic representation and analysis, listed in order of increasing complexity in Table 7-10. Additional complexity can be included through cell population balances that account for the distribution of cell generation present in the fermenter through use of stochastic models. In this section we limit the discussion to simple black box and unstructured models. For more details on bioreaction systems, see, e.g., Nielsen, Villadsen, and Liden, Bioreaction Engineering Principles, 2d ed., Kluwer, Academic/Plenum Press, 2003; Bailey and Ollis, Biochemical Engineering Fundamentals, 2d ed., McGraw-Hill, 1986; Blanch and Clark, Biochemical Engineering, Marcel Dekker, 1997; and Sec. 19. Mechanism Stoichiometric balances are done on a C atom basis called C-moles, e.g., relative to the substrate (denoted by subscript s), and the corresponding stoichiometric coefficients Ysi (based on Cmole of the primary substrate) are called yield coefficients. For instance, CH2O + YsoO2 + YsnNH3 + Yss1S1 + . . . ⇒ Ysx X + YscCO2 + Ysp1P1 + . . . + YswH2O

(7-147)

Here the reactants (substrates) are glucose (CH2O), O2, NH3, and a sulfur-providing nutrient S1, and the products are biomass X, CO2, metabolic product P1, and H2O. The products of bioreactions can be reduced or oxidized, and all feasible pathways have to be redox neutral. There are several cofactors that transfer redox power in a pathway or between pathways, each equivalent to the reducing power of a molecule of H2, e.g., nicotinamide adenine dinucleotide (NADH), and these have to be included in the stoichiometric balances as H equivalents through redox balancing. For instance, for the reaction of glucose to glycerol (CH8/3O), 13 NADH equivalent is consumed: 1 CH2O +  NADH ⇒ CH8/3O 3

(7-148)

The stoichiometry in the biochemical literature often does not show H2O produced by the reaction; however, for complete elemental balance, water has to be included, and this is easily done once an O2 requirement has been determined based on a redox balance. Likewise for simplicity, the other form of the cofactor [e.g., the oxidized form of the cofactor NADH in Eq. (7-148)] is usually left out. In

7-31

addition to C balances, for aerobic systems cell respiration has to be accounted for as well through a stoichiometric equation: NADH + 0.5O2 ⇒ H2O + γ ATP

(7-149)

The associated free energy produced or consumed in each reaction is captured in units of adenosine triphosphate (ATP). The ATP stoichiometry is usually obtained from biochemical tables since the energy has to be also balanced for the cell. Thus for Eq. (7-148) the stoichiometric ATP requirement to convert one C-mole of glucose to one C-mole of glycerol is 13. In calculations of the carbon flux distribution in different pathways this ATP requirement has to be added on the left-hand side of the equation. Again the other form of the cofactor ATP is usually left out to simplify the reaction equation. There are several metabolic pathways that are repeated for many living cells, and these are split into two: catabolic or energy-producing and anabolic or energy-consuming, the later producing building blocks such as amino acids and the resulting macromolecules such as proteins. Of course the energy produced in catabolic steps has to be balanced by the energy consumed in anabolic steps. Catabolic pathways include the well-studied glycolysis, TCA cycle, oxidative phosphorylation, and fermentative pathways. For more details see Stephanopoulos, Aristidou, and Nielsen, Metabolic Engineering: Principles and Methodologies, Academic Press, 1998; and Nielsen, Villadsen, and Liden, Bioreaction Engineering Principles, 2d ed., Kluwer, Academic/Plenum Press, 2003; Bailey and Ollis, Biochemical Engineering Fundamentals, 2d ed., McGraw-Hill, 1986. Monod-Type Empirical Kinetics Many bioreactions show increased biomass growth rate with increasing substrate concentration at low substrate concentration for the limiting substrate, but no effect of substrate concentration at high concentrations. This behavior can be represented by the Monod equation (7-92). Additional variations on the Monod equation are briefly illustrated below. For two essential substrates the Monod equation can be modified as µmaxCs1Cs2 µ =  (Ks1 + Cs2)(Ks2 + Cs2)

(7-150)

This type of rate expression is often used in models for water treatment, and many environmental factors can be included (the effect of, e.g., phosphate, ammonia, volatile fatty acids, etc.). The correlation between parameters in such complicated models is, however, severe, and very often a simple Monod model (7-92) with only one limiting substrate is sufficient. When substrate inhibition occurs, µmaxCs µ = 2 Ks + Cs + K1/Cs

(7-151)

O2 is typically a substrate that in high concentrations leads to substrate inhibition, but a high concentration of the carbon source can also be inhibiting (e.g., in bioremediation of toxic waste a high concentration of the organic substrate can well lead to severe inhibition or death of the microorganism). When product inhibition is present, µ maxCs Cp µ=  1−  Ks + Cs Cpmax





(7-152)

Here the typical example is the inhibitor effect of ethanol on yeast growth. Considerable efforts are made by the biocompanies to develop yeast strains that are tolerant to high ethanol concentrations since this will give considerable savings in, e.g., production of biofuel by fermentation. The various component reaction rates for a single reaction can be related to the growth rate by using the stoichiometric (yield) coefficients, e.g., from Eq. (7-147): Ysi ri = YxiµCx =  µCx Ysx

(7-153)

7-32

REACTION KINETICS

Chemostat with Empirical Kinetics Using the CSTR equation (7-54) for a constant-volume single reaction [Eq. (7-147)], the substrate, biomass, and product material balances are 1  µCx + D(Cs0 − Cs) = 0 Ysx µCx − DCx = 0 → D = µ

(7-154)

Ysp  µCx − DCp = 0 Ysx Here Cs0 is the feed substrate concentration, and D is the dilution rate, which at steady-state constant volume is equal to both the feed and effluent volumetric flow rates and to the specific growth rate. The effluent concentrations of substrate, biomass, and products can be calculated by using a suitable expression for the specific growth rate µ such as one of the relevant variants of the Monod kinetics described above. ELECTROCHEMICAL REACTIONS Electrochemical reactions involve coupling between chemical reactions and electric charge transfer and may have two or three phases, for instance, a gas (e.g., H2 or O2 evolved at the electrodes or fed as reactants), a liquid (the electrolyte solution), and solids (electrodes). Electrocatalysts may be employed to enhance the reaction for a particular desired product. Hence, electrochemical reactions are heterogeneous reactions that occur at the surface of electrodes and involve the transfer of charge in the form of electrons as part of a chemical reaction. The electrochemical reaction can produce a chemical change by passing an electric current through the system (e.g., electrolysis), or reversely a chemical change can produce electric energy (e.g., using a battery or fuel cell to power an appliance). There are a variety of practical electrochemical reactions, some occurring naturally, such as corrosion, and others used in production of chemicals (e.g., the decomposition of HCl to produce Cl2 and H2, the production of caustic soda and chlorine, the smelting of aluminum), electroplating, and energy generation (e.g., fuel cells and photovoltaics). Electrochemical reactions are reversible and can be generally written as a reduction-oxidation (redox) couple: → ←

O + ne−

R

where O is an oxidized and R is a reduced species. For instance, the corrosion process includes oxidation at the anode: Fe

→ ←

O2 + H2O + 4e

← →

I × εcurrent,i × MWi mass  =  m =  nF time (7-156)

I current j=  =  Aprojected area

Since electrochemical reactions are heterogeneous at electrode surfaces, the current I is generally normalized by dividing it by the geometric or projected area of the electrode, resulting in the quantity known as the current density j, in units of kA/m2. The overall electrochemical cell equilibrium potential Eocell, as measured between the cathode and the anode, is related to the Gibbs free energy change for the overall electrochemical reaction: o ∆Go = ∆Ho − T ∆So = −nFEcell o ∆G o o Ecell = −  = Eocathode − Eanode nF

i

4OH

The overall electrochemical reaction is the stoichiometric sum of the anode and cathode reactions: 2Fe2+ + 4OH− (four electron transfer process, n = 4)

The anode and cathode reactions are close coupled in that the electric charge is conserved; therefore, the overall production rate is a direct function of the electric charge passed per unit time, the electric current I. For references on electrochemical reaction kinetics and mechanism, see, e.g., Newman and Thomas-Alvea, Electrochemical Systems, 3d ed., Wiley Interscience, 2004; Bard and Faulkner, Electrochemical Methods: Fundamentals and Applications, 2d ed., Wiley, 2001; Bethune and Swendeman, “Table of Electrode Potentials and Temperature Coefficients,” Encyclopedia of Electrochemistry, Van Nostrand Reinhold, New York 1964, pp. 414–424; and Bethune and Swendeman, Standard Aqueous Electrode Potentials and Temperature Coefficients, C. A. Hampel Publisher, 1964.

(7-157)

Each electrode reaction, anode and cathode, or half-cell reaction has an associated energy level or electrical potential (volts) associated with it. Values of the standard equilibrium electrode reduction potentials Eo at unit activity and 25°C may be obtained from the literature (de Bethune and Swendeman Loud, Encyclopedia of Electrochemistry, Van Nostrand Reinhold, 1964). The overall electrochemical cell equilibrium potential either can be obtained from ∆G values or is equal to the cathode half-cell potential minus the anode half-cell potential, as shown above. The Nernst equation allows one to calculate the equilibrium potential Eeq when the activity of the reactants or products is not at unity:

i ν M −

(7-155)

where n is the number of equivalents per mole, m is the number of moles, F is the Faraday constant, Q is the charge, and t is time. The total current passed may represent several parallel electrochemical reactions; therefore, we designate a current efficiency for each chemical species. The chemical species production rate (mass/time) is related to the total current passed I, the species current efficiency εcurrent,i, and the molecular weight of the chemical species MWi:

ni i

→ ne−

RT Eeq = Eo −  ln(Πaνi ) nF i

−

→ ←

Q = nmF F = 96,485 Cequiv Q charge I=  =  A t time

Fe2+ + 2e−

and reduction at the cathode:

2Fe + O2 + H2O

Faraday’s law relates the charge transferred by ions in the electrolyte and electrons in the external circuit, to the moles of chemical species reacted (Newman and Thomas-Alvea, Electrochemical Systems, 3d ed., Wiley Interscience, 2004):

∂E

 ∂T

P

(7-158)

∆S

 nF

where νi is the stoichiometric coefficient of chemical species i (positive for products; negative for reactants), Mi is the symbol for species i, ni is the charge number of the species, ai is the activity of the chemical species, E is the formal potential, and ∏ represents the product of all respective activities raised to their stiochiometric powers as required by the reaction. Please note that if the value of the equilibrium potential is desired at another temperature, Eo must also be evaluated at the new temperature as indicated. Kinetic Control In 1905, Julius Tafel experimentally observed that when mass transport was not limiting, the current density j of electrochemical reactions exhibited the following behavior: j a′eηact/ b′

or

ηact a  b log j

where the quantity ηact is known as the activation overpotential E  E eq, and is the difference between the actual electrode potential

DETERMINATION OF MECHANISM AND KINETICS E and the reversible equilibrium potential of the electrochemical reaction Eeq. Thus the driving force for the electrochemical reaction is not the absolute potential; it is the activation overpotential η act. This relationship between the current density and activation overpotential has been further developed and resulted in the ButlerVolmer equation: j r  kf Co  krCr j j0(e−(αnF/RT)ηact  e[(1α)nF/RT]ηact) nF ηact E  Eeq (7-159) Here the reaction rate r is defined per unit electrode area, moles per area per time, j0 is the equilibrium exchange current when E = Eeq, ηact is the activation overpotential, and α is the transfer coefficient. For large activation overpotentials, the Tafel empirical equation applies: ηact a  b log j

for ηact 100 mV, b Tafel slope

(7-160)

For small activation overpotentials, linearization gives nF j j0  ηact RT

(7-161)

Mass-Transfer Control The surface concentration at the electrodes differs significantly from the bulk electrolyte concentration. The Nernst equation applies to the surface concentrations (or activities in case of nonideal solutions): RT Eeq Eo   ln(∏ aνi,surf) nF i

(7-162)

7-33

If mass transfer is limiting, then a limiting current is obtained for each chemical species i: nFDiCi ji,lim  nFkL,iCi δ

(7-163)

where Di is the diffusion coefficient, δ is the boundary layer thickness, and kL,i is the mass-transfer coefficient of species i. The effect of mass transfer is included as follows:









j j j j0 1   e −(αnF/RT)ηact  1   e [(1α)nF/RT]ηact ja,lim jc,lim



Ci,surf j  1  Ci ji,lim



i o,r



(7-164)

Ohmic Control The overall electrochemical reactor cell voltage may be dependent on the kinetic and mass-transfer aspects of the electrochemical reactions; however, a third factor is the potential lost within the electrolyte as current is passing through this phase. The potential drops may become dominant and limit the electrochemical reactions requiring an external potential to be applied to drive the reactions or significantly lower the delivered electrical potential in power generation applications such as batteries and fuel cells. Multiple Reactions With multiple reactions, the total current is the sum of the currents from the individual reactions with anodic currents positive and cathodic currents negative. This is called the mixed potential principle. For more details see Bard and Faulkner, Electrochemical Methods: Fundamentals and Applications, 2d ed., Wiley, 2001.

DETERMINATION OF MECHANISM AND KINETICS Laboratory data are the predominant source for reaction mechanism and kinetics in industrial practice. However, often laboratory data intended for scoping and demonstration studies rather than for kinetic evaluation have to be used, thus reducing the effectiveness and accuracy of the resulting kinetic model. The following are the steps required to obtain kinetics from laboratory data: 1. Develop initial guesses on mechanism, reaction time scale, and potential kinetic models from the literature, scoping experiments, similar chemistries, and computational chemistry calculations, when possible. 2. Select a suitable laboratory reactor type and scale, and analytical tools for kinetic measurements. 3. Develop à priori factorial experimental design or sequential experimental design. 4. When possible, provide ideal reactor conditions, e.g., good mechanical agitation in batch and CSTR, high velocity flow in PFR. 5. Estimate the limiting diffusion-reaction regimes under the prevailing lab reactor conditions for heterogeneous reactions, and use the appropriate lab reactor model. When possible, operate the reactor under kinetic control. 6. Discriminate between competing mechanisms and kinetic rates by forcing maximum differentiation between competing hypotheses through the experimental design, and by obtaining the best fit of the kinetic data to the proposed kinetic forms. LABORATORY REACTORS Selection of the laboratory reactor type and size, and associated feed and product handling, control, and analytical schemes depends on the type of reaction, reaction time scales, and type of analytical methods required. The criteria for selection include equipment cost, ease of operation, ease of data analysis, accuracy, versatility, temperature uniformity, and controllability, suitability for mixed phases, and scale-up

feasibility. Many configurations of laboratory reactors have been employed. Rase (Chemical Reactor Design for Process Plants, Wiley, 1977) and Shah (Gas-Liquid-Solid Reactor Design, McGraw-Hill, 1979) each have about 25 sketches, and Shah’s bibliography has 145 items classified into 22 categories of reactor types. Jankowski et al. [Chemische Technik 30: 441–446 (1978)] illustrate 25 different kinds of gradientless laboratory reactors for use with solid catalysts. Laboratory reactors are of two main types: 1. Reactors used to obtain fundamental data on intrinsic chemical rates free of mass-transfer resistances or other complications. Some of the gas-liquid lab reactors, for instance, employ known interfacial areas, thus avoiding the uncertainty regarding the area for gas to liquid mass transfer. When ideal behavior cannot be achieved, intrinsic kinetic estimates need to account for mass- and heat-transfer effects. 2. Reactors used to obtain scale-up data due to their similarity to the reactor intended for the pilot or commercial plant scale. How to scale down from the conceptual commercial or pilot scale to lab scale is a difficult problem in itself, and it is not possible to maintain all key features while scaling down. The first type is often the preferred one—once the intrinsic kinetics are obtained at “ideal” lab conditions, scale-up is done by using models or correlations that describe large-scale reactor hydrodynamics coupled with the intrinsic kinetics. However, in some cases ideal conditions cannot be achieved, and the laboratory reactor has to be adequately modeled to account for mass and heat transfer and nonideal mixing effects to enable extraction of intrinsic kinetics. In addition, with homogeneous reactions, attention must be given to prevent wall-catalyzed reactions, which can result in observed kinetics that are fundamentally different from intrinsic homogeneous kinetics. This is a problem for scale-up, due to the high surface/volume ratio in small reactors versus the low surface/volume ratio in large-scale systems, resulting in widely different contributions of wall effects at different scales. Similar issues arise in bioreactors with the potential of

7-34

REACTION KINETICS

undesirable wall growth of the biocatalyst cells masking the homogeneous growth kinetics. In catalytic reactions certain reactor configurations may enhance undesirable homogeneous reactions, and the importance of these reactions may be different at larger scale, causing potential scale-up pitfalls. The reaction rate is expressed in terms of chemical compositions of the reacting species, so ultimately the variation of composition with time or space must be found. The composition is determined in terms of a property that is measured by some instrument and calibrated. Among the measures that have been used are titration, pressure, refractive index, density, chromatography, spectrometry, polarimetry, conductimetry, absorbance, and magnetic resonance. Therefore, batch or semibatch data are converted to composition as a function of time (C, t), or to composition and temperature as functions of time (C, T, t), to prepare for kinetic analysis. In a steady CSTR and PFR, the rate and compositions in the effluent are observed as a function of residence time. When a reaction has many reactive species (which may be the case even for apparently simple processes such as pyrolysis of ethane or synthesis of methanol), a factorial or sequential experimental design should be developed and the data can be subjected to a response surface analysis (Box, Hunter, and Hunter, Statistics for Experimenters, 2d ed., Wiley Interscience, 2005; Davies, Design and Analysis of Industrial Experiments, Oliver & Boyd, 1954). This can result in a black box correlation or statistical model, such as a quadratic (limited to first- and second-order effects) for the variables x1, x2, and x3: r k1x1  k2x2  k3x3  k12x1x2  k13x1x3  k23x2x3 Analysis of such statistical correlations may reveal the significant variables and interactions and may suggest potential mechanisms and kinetic models, say, of the Langmuir-Hinshelwood type, that could be analyzed in greater detail by a regression process. The variables xi could be various parameters of heterogeneous processes as well as concentrations. An application of this method to isomerization of npentane is given by Kittrel and Erjavec [Ind. Eng. Chem. Proc. Des. Dev. 7: 321 (1968)]. Table 7-11 summarizes laboratory reactor types that approach the three ideal concepts BR, CSTR and PFR, classified according to reaction types.

TABLE 7-11 Laboratory Reactors Reaction Homogeneous gas Homogeneous liquid Catalytic gas-solid

Noncatalytic gas-solid Liquid-solid Gas-liquid

Gas-liquid-solid Solid-solid

Reactor Isothermal U-tube in temperature-controlled batch Mechanically agitated batch or CSTR with jacketed cooling/heating Packed tube in furnace Isothermal U-tube in temperature-controlled bath Rotating basket with jacketed cooling/heating Internal recirculation (Berty) reactor with jacketed cooling/heating Packed tube in furnace Packed tube in furnace CSTR with jacketed cooling/heating Fixed interface CSTR Wetted wall Laminar jet Slurry CSTR with jacketed cooling/heating Packed bed with downflow, upflow, or countercurrent Packed tube in furnace

For instance, Fig. 7-17 summarizes laboratory reactor types and hydrodynamics for gas-liquid reactions. Batch Reactors In the simplest kind of investigation, reactants can be loaded into a number of sealed tubes, kept in a thermostatic bath for various periods, shaken mechanically to maintain uniform composition, and analyzed. In terms of cost and versatility, the stirred batch reactor is the unit of choice for homogeneous or heterogeneous slurry reactions including gas-liquid and gas-liquid-solid systems. For multiphase systems the reactants can be semibatch or continuous. The BR is especially suited to reactions with half-lives in excess of 10 min. Samples are taken at time intervals, and the reaction is stopped by cooling, by dilution, or by destroying a residual reactant such as an acid or base; analysis can then be made at a later time. Analytic methods that do not necessitate termination of reaction include nonintrusive measurements of (1) the amount of gas produced, (2) the gas pressure in a constantvolume vessel, (3) absorption of light, (4) electrical or thermal conductivity, (5) polarography, (6) viscosity of polymerization, (7) pH and DO probes, and so on. Operation may be isothermal, with the important effect of temperature determined from several isothermal runs, or the composition and temperature may be recorded simultaneously and the

FIG. 7-17 Principal types of laboratory reactors for gas-liquid reactions. [From Fig. 8 in J. C. Charpentier, “Mass Transfer Rates in Gas-

Liquid Absorbers and Reactors,” in Drew et al. (eds.), Advances in Chemical Engineering, vol. 11, Academic Press, 1981.]

DETERMINATION OF MECHANISM AND KINETICS data regressed. On the laboratory scale, it is essential to ensure that a BR is stirred to uniform composition, and for critical cases such as high viscosities this should be checked with tracer tests. Flow Reactors CSTRs and other devices that require flow control are more expensive and difficult to operate. However, CSTRs and PFRs are the preferred laboratory reactors for steady operation. One of the benefits of CSTRs is their isothermicity and the fact that their mathematical representation is algebraic, involving no differential equations, thus making data analysis simpler. For laboratory research purposes, CSTRs are considered feasible for holding times of 1 to 4000 s, reactor volumes of 2 to 1000 cm3 (0.122 to 61 in3), and flow rates of 0.1 to 2.0 cm3/s. Fast reactions and those in the gas phase are generally done in tubular flow reactors, just as they are often done on the commercial scale. Usually it is not possible to measure compositions along a PFR, although temperatures can be measured using a thermowell with fixed or mobile thermocouple bundle. PFRs can be kept at nearly constant temperatures; small-diameter tubes immersed in a fluidized sand bed or molten salt can hold quite constant temperatures of a few hundred degrees. Other PFRs are operated at near adiabatic conditions by providing dual radial temperature control to minimize the radial heat flux, with multiple axial zones. A recycle unit can be operated as a differential reactor with arbitrarily small conversion and temperature change. Test work in a tubular flow unit may be desirable if the intended commercial unit is of that type. Multiphase Reactors Reactions between gas-liquid, liquid-liquid, and gas-liquid-solid phases are often tested in CSTRs. Other laboratory types are suggested by the commercial units depicted in appropriate sketches in Sec. 19 and in Fig. 7-17 [Charpentier, Mass Transfer Rates in Gas-Liquid Absorbers and Reactors, in Drew et al. (eds.), Advances in Chemical Engineering, vol. 11, Academic Press, 1981]. Liquids can be reacted with gases of low solubilities in stirred vessels, with the liquid charged first and the gas fed continuously at the rate of reaction or dissolution. Some of these reactors are designed to have known interfacial areas. Most equipment for gas absorption without reaction is adaptable to absorption with reaction. The many types of equipment for liquid-liquid extraction also are adaptable to reactions of immiscible liquid phases. Solid Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable, and some of these were discussed above. For more details see Mears [Ind. Eng. Chem. Proc. Des. Devel. 10: 541–547 (1971); Ind. Eng. Chem. Fund. 15: 20–23 (1976)] and Satterfield (Heterogeneous Catalysis in Practice, McGraw-Hill, 1991, p. 491). For catalytic investigations, the rotating basket or fixed basket with internal recirculation is the standard device, usually more convenient and less expensive than equipment with external recirculation. In the fixed-basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients (see, e.g., Berty, Experiments in Catalytic Reaction Engineering, Elsevier, 1999). A unit holding 50 cm3 (3.05 in3) of catalyst can operate up to 800 K (1440°R) and 50 bar (725 psi). When deactivation occurs rapidly (in a few seconds during catalytic cracking, for instance), the fresh activity can be maintained with a transport reactor through which both reactants and fresh catalyst flow without slip and with short contact time. Since catalysts often are sensitive to traces of impurities, the time deactivation of the catalyst usually can be evaluated only with commercial feedstock. Physical properties of catalysts also may need to be checked periodically, including pellet size, specific surface, porosity, pore size and size distribution, effective diffusivity, and active metals content and dispersion. The effectiveness of a porous catalyst is found by measuring conversions with successively smaller pellets until no further change occurs. These topics are touched on by Satterfield (Heterogeneous Catalysis in Industrial Practice, McGraw-Hill, 1991). To determine the deactivation kinetics, long-term deactivation studies at constant conditions and at different temperatures are required. In some cases, accelerated aging can be induced to reduce the time required for the experimental work, by either increasing the feed flow

7-35

rate (if the deactivation is a result of feed or product poisoning) or increasing the temperature above the standard reaction temperature. These require a good understanding of how the higher-temperature or rate-accelerated deactivation correlates with deactivation at the operating reaction temperature and rate. Bioreactors There are several types of laboratory bioreactors used with live organisms as biocatalysts: 1. Mechanically agitated batch/semibatch with pH control and nutrients or other species either fed at the start or added continuously based on a recipe or protocol. 2. CSTR to maintain a constant dilution rate (the feed rate). These require some means to separate the biocatalyst from the product and recycle to the reactor, such as centrifuge or microfiltration: a. Chemostat controls the flow to maintain a constant fermentation volume. b. Turbidostat controls the biomass or cells concentration. c. pH-auxostat controls pH in the effluent (same as pH in reactor). d. Productostat controls the effluent concentration of one of the metabolic products. The preferred reactor for kinetics is the chemostat, but semibatch reactors are more often used owing to their simpler operation. Calorimetry Another category of laboratory systems that can be used for kinetics includes calorimeters. These are primarily used to establish temperature effects and thermal runaway conditions, but can also be employed to determine reaction kinetics. Types of calorimeters are summarized in Table 7-12; for more details see Reid, “Differential Microcalorimeters,” J. Physics E: Scientific Instruments, 9 (1976). Additional methods of laboratory data acquisition are described in Masel, Chemical Kinetics and Catalysis, Wiley, 2001. KINETIC PARAMETERS The kinetic parameters are constants that appear in the intrinsic kinetic rate expressions and are required to describe the rate of a reaction or reaction network. For instance, for the simple global nth-order reaction with Arrhenius temperature dependence: A⇒B

r kCna

k k0eE/RT

(7-165)

The kinetic parameters are k0, E, and n, and knowledge of these parameters and the prevailing concentration and temperature fully determines the reaction rate. For a more complex expression such as the Langmuir-Hinshelwood rate for gas reaction on heterogeneous catalyst surface with equilibrium adsorption of reactants A and B on two different sites and nonadsorbing products, Eq. (7-85) can be rewritten as k0 eE/RTPaPb r  (1  K a0 eE /RT)(1  K b0 eE /RT) aa

ab

(7-166)

and the kinetic parameters are k0, E, Ka0, Eaa, Kb0, and Eab. A number of factors limit the accuracy with which parameters needed for the design of commercial equipment can be determined. The kinetic parameters may be affected by inaccurate accounting for laboratory reactor heat and mass transport, and hydrodynamics; correlations for these are typically determined under nonreacting conditions at ambient temperature and pressure and with nonreactive model fluids and may not be applicable or accurate at reaction conditions. Experimental uncertainty including errors in analysis, measurement, TABLE 7-12

Calorimetric Methods

Adiabatic Accelerating rate calorimeter (ARC) Vent sizing package (VSP) calorimeter PHI-TEC Dewar Automatic pressure tracking adiabatic calorimeter (APTAC)

Nonadiabatic Reaction calorimeter (RC1) + IR Differential scaning calorimeter (DSC) Thermal gravitometry (TG) Isothermal calorimetry Differential thermal analysis (DTA) Differential microcalorimeters Advanced reaction system screening tool (ARSST)

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REACTION KINETICS

and control is also a contributing factor (see, e.g., Hoffman, “Kinetic Data Analysis and Parameter Estimation,” in de Lasa (ed.), Chemical Reactor Design and Technology, Martinus Nijhoff, 1986.

E ln r ln k0    n ln Ca RT

ln C FIG. 7-18 Determination of the rate constant and reaction order.

ln k

In this section we focus on the three main types of ideal reactors: BR, CSTR, and PFR. Laboratory data are usually in the form of concentrations or partial pressures versus batch time (batch reactors), concentrations or partial pressures versus distance from reactor inlet or residence time (PFR), or rates versus residence time (CSTR). Rates can also be calculated from batch and PFR data by differentiating the concentration versus time or distance data, usually by numerical curve fitting first. It follows that a general classification of experimental methods is based on whether the data measure rates directly (differential or direct method) or indirectly (integral of indirect method). Table 7-13 shows the pros and cons of these methods. Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. Differential Data Analysis As indicated above, the rates can be obtained either directly from differential CSTR data or by differentiation of integral data. A common way of evaluating the kinetic parameters is by rearrangement of the rate equation, to make it linear in parameters (or some transformation of parameters) where possible. For instance, using the simple nth-order reaction in Eq. (7-165) as an example, taking the natural logarithm of both sides of the equation results in a linear relationship between the variables ln r, 1/T, and ln Ca:

ln r

DATA ANALYSIS METHODS

l/T FIG. 7-19 Determination of the activation energy.

(7-167) Ca0 ln 

kτ Ca

Multilinear regression can be used to find the constants k0, E, and n. For constant-temperature (isothermal) data, Eq. (7-167) can be simplified by using the Arrhenius form as ln r ln k  n ln Ca

(7-168)

and the kinetic parameters n and k can be determined as the intercept and slope of the best straight-line fit to the data, respectively, as shown in Fig. 7-18. The preexponential k0 and activation energy E can be obtained from multiple isothermal data sets at different temperatures by using the linearized form of the Arrhenius equation E ln k ln k0   RT

 C Ca0

for n ≠ 1

a

ln 2 τ1/2  k

(7-169)

for n 1

2n1  1 τ1/2  (n  1) kCn−1 a0

(7-171) for n ≠ 1

Indirect method

Advantages Get rate equation directly Easy to fit data to a rate law High confidence on final rate equation Disadvantages Difficult experiment Need many runs

Disadvantages Must infer rate equation Hard to analyze rate data Low confidence on final rate equation

Not suitable for very fast or very slow reactions

Advantages Easier experiment Can do a few runs and get important information Suitable for all reactions including very fast or very slow ones

Masel, Chemical Kinetics and Catalysis, Wiley, 2001, Table 3.2.

ln C

Comparison of Direct and Indirect Methods

Direct method

SOURCE:

(7-170)

1  kτ(n  1)Cn−1 a0

For the first-order case, the rate constant k can be obtained directly from the slope of the graph of the left-hand side of Eq. (7-170) versus batch time, as shown in Fig. 7-20. For orders other than first, plotting the natural log of Eq. (7-170) can at least indicate if the order is larger or smaller than 1, as shown in Fig. 7-21. The Half-Life Method The half-life is the batch time required to get 50 percent conversion. For an nth-order reaction,

as shown in Fig. 7-19. Integral Data Analysis Integral data such as from batch and PFR relate concentration to time or distance. Integration of the BR equation for an nth-order homogeneous constant-volume reaction yields TABLE 7-13

n1

for n 1

t FIG. 7-20 Determination of first-order rate constant from integral data.

DETERMINATION OF MECHANISM AND KINETICS

der

r or de d

Se co n

In(CA0/CA)

0.8

Firs t or

Half o

rder

1

0.6

r

rde

do

ir Th

0.4 0.2 0

0

1

0.5

1.5

Time FIG. 7-21 Reaction behavior for nth-order reaction. (Masel, Chemical Kinet-

ics and Catalysis, Wiley, 2001, Fig. 3.15.)

Thus for first-order reactions, the half-life is constant and independent of the initial reactant concentration and can be used directly to calculate the rate constant k. For non-first-order reactions, Eq. (7-171) can be linearized as follows: 2n1  1 lnτ1/2 ln   (n  1) ln Ca0 (n  1)k

for n ≠ 1 (7-172)

The reaction order n can be obtained from the slope and the rate constant k from the intercept of the plot of Eq. (7-172), shown in Fig. 7-22. Complex Rate Equations The examples above are for special cases amenable to simple treatment. Complex rate equations and reaction networks with complex kinetics require individual treatment, which often includes both numerical solvers for the differential and algebraic equations describing the laboratory reactor used to obtain the data and linear or nonlinear parameter estimation.

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Chemical Reactor Design and Technology, Martinus Nijhoff, 1985, pp. 69–105] are also very useful. As indicated above, the acquisition of kinetic data and parameter estimation can be a complex endeavor. It includes statistical design of experiments, laboratory equipment, computer-based data acquisition, complex analytical methods, and statistical evaluation of the data. Regression is the procedure used to estimate the kinetic parameters by fitting kinetic model predictions to experimental data. When the parameters can be made to appear linear in the kinetic model (through transformations, grouping of parameters, and rearrangement), the regression is linear, and an accurate fit to data can be obtained, provided the form of the kinetic model represents well the reaction kinetics and the data provide enough width in temperature, pressure, and composition for statistically significant estimates. Often such linearization is not possible. Linear Models in Parameters, Single Reaction We adopt the terminology from Froment and Hosten, “Catalytic Kinetics—Modeling,” in Catalysis—Science and Technology, Springer-Verlag, New York, 1981. For n observations (experiments) of the concentration vector y for a model linear in the parameter vector β of length p < n, the residual error ε is the difference between the kinetic modelpredicted values and the measured data values: ε y  Xβ y  y^

(7-173)

The linear model is represented as a linear transformation of the parameter vector β through the model matrix X. Estimates b of the true parameters β are obtained by minimizing the objective function S(β), the sum of squares of the residual errors, while varying the values of the parameters: n

β

S(β) εTε (y  y^) 2 → Min

(7-174)

i 1

This linear optimization problem, subject to constraints on the possible values of the parameters (e.g., requiring positive preexponentials, activation energies, etc.) can be solved to give the estimated parameters: b (X TX)1XTy

(7-175)

When the error is normally distributed and has zero mean and variance σ 2, then the variance-covariance matrix V(b) is defined as

PARAMETER ESTIMATION The straightforward method to obtain kinetic parameters from data is the numerical fitting of the concentration data (e.g., from BR or PFR) to integral equations, or the rate data (e.g., from a CSTR or from differentiation of BR or PFR) to rate equations. This is done by parameter estimation methods described here. An excellent reference for experimental design and parameter estimation (illustrated for heterogeneous gas-solid reactions) is the review paper of Froment and Hosten, “Catalytic Kinetics—Modeling,” in Catalysis—Science and Technology, Springer-Verlag, New York, 1981. Two previous papers devoted to this topic by Hofmann [in Chemical Reaction Engineering, ACS Advances in Chemistry, 109: 519–534 (1972); in de Lasa (ed.),

V(b) (XTX)1σ 2

(7-176)

An estimate for σ2, denoted s2, is n

(y  y) i 1

^ 2 i

s2  np

(7-177)

When V(b) is known from experimental observations, a weighted objective function should be used for optimization of the objective function: β

S(β) εTV1ε → Min

(7-178)

and the estimates b are obtained as ln t1/2

b (XTV1X)1XTV1y

(7-179)

The parameter fit is adequate if the F test is satisfied, that is, Fc, the calculated F, is larger than the tabulated statistical one at the confidence level of 1  α:

ln C0 FIG. 7-22 Determination of reaction order and rate constant from half-life data.

LFSS  n  p  ne  1 Fc  ≥ F(n  p  ne  1, ne  1;1  α) PESS  (7-180) ne  1

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REACTION KINETICS n

ne

i 1

i 1

LFSS (yi  y^i)2  (yi  yi)2

ne

PESS (yi  yi)2 i 1

Here yi are the averaged values of the data for replicates. Equation (7180) is valid if there are n replicate experiments and the pure error sum of squares (PESS) is known. Without replicates,

THEORETICAL METHODS

RgSS  p Fc  ≥ F(p,n  p;1  α) RSS  np n

RSS (yi  y^ i)2

n

RgSS y^2i i 1

(7-181)

i 1

The bounds on the parameter estimates are given by the t statistics: α α bi  t n  p;1   ≤ βi ≤ bi  t n  p;1   2 2









(7-182)

An example of a linear model in parameters is Eq. (7-167), where the parameters are ln k0, E, and n, and the linear regression can be used directly to estimate these. Nonlinear Models in Parameters, Single Reaction In practice, the parameters appear often in nonlinear form in the rate expressions, requiring nonlinear regression. Nonlinear regression does not guarantee optimal parameter estimates even if the kinetic model adequately represents the true kinetics and the data width is adequate. Further, the statistical tests of model adequacy apply rigorously only to models linear in parameters, and can only be considered approximate for nonlinear models. For a general nonlinear model f(xi, β), where x is the vector of the independent model variables and β is the vector of parameters, ε y  f(x, β)

centration [e.g., Eq. 7-39]. With a network of reactions, there are a number of dependent variables equal to the number of stoichiometrically independent reactions, also called responses. In this case the objective function has to be modified. For details see Froment and Hosten, “Catalytic Kinetics—Modeling,” in Catalysis—Science and Technology, Springer-Verlag, New York, 1981.

(7-183)

An example of a model nonlinear in parameters is Eq. (7-166). Here it is not possible through any number of transformations to obtain a linear form in all the parameters k0, E, Ka0, Eaa, Kb0, Eab. Note that for some Langmuir-Hinshelwood rate expressions it is possible to linearize the model in parameters at isothermal conditions and obtain the kinetic constants for each temperature, followed by Arrheniustype plots to obtain activation energies (see, e.g., Churchill, The Interpretation and Use of Rate Data: The Rate Concept, McGrawHill, 1974). Minimization of the sum of squares of residuals does not result in a closed form for nonlinear parameter estimates as for the linear case; rather it requires an iterative numerical solution, and having a reasonable initial estimate for the parameter values and their feasible ranges is critical for success. Also, the minima in the residual sum of squares are local and not global. To obtain global minima that better represent the kinetics over a wide range of conditions, parameter estimation has to be repeated with a wide range of initial parameter guesses to increase the chance of reaching the global minimum. The nonlinear regression procedure typically involves a steepest descent optimization search combined with Newton’s linearization method when a minimum is approached, enhancing the convergence speed [e.g., the Marquardt-Levenberg or Newton-Gauss method; Marquardt, J. Soc. Ind. Appl. Math. 2: 431 (1963)]. An integral part of the parameter estimation methodology is mechanism discrimination, i.e., selection of the best mechanism that would result in the best kinetic model. Nonlinear parameter estimation is an extensive topic and will not be further discussed here. For more details see Froment and Hosten, “Catalytic Kinetics —Modeling,” in Catalysis—Science and Technology, Springer-Verlag, New York, 1981. Network of Reactions The statistical parameter estimation for multiple reactions is more complex than for a single reaction. As indicated before, a single reaction can be represented by a single con-

Prediction of Mechanism and Kinetics Reaction mechanisms for a variety of reaction systems can be predicted to some extent by following a set of heuristic rules derived from experience with a wide range of chemistries. For instance, Masel, Chemical Kinetics and Catalysis, Wiley, 2001, chapter 5, enumerates the rules for gas-phase chain and nonchain reactions including limits on activation energies for various elementary steps. Other reaction systems such as ionic reactions, and reactions on metal and acid surfaces, are also discussed by Masel, although these mechanisms are not as well understood. Nevertheless, the rules can lead to computer-generated mechanisms for complex systems such as homogeneous gas-phase combustion and partial oxidation of methane and higher hydrocarbons. Developments in computational chemistry methods allow, in addition to the derivation of most probable elementary mechanisms, prediction of thermodynamic and kinetic reaction parameters for relatively small molecules in homogeneous gas-phase and liquid-phase reactions, and even for some heterogeneous catalytic systems. This is especially useful for complex kinetics where there is no easily discernible rate-determining step, and therefore no simple closed-form global reaction rate can be determined. In particular, estimating a large number of kinetic parameters from laboratory data requires a large number of experiments and use of intermediate reaction components that are not stable or not readily available. The nonlinear parameter estimation with many parameters is difficult, with no assurance that global minima are actually obtained. For such complex systems, computational chemistry estimates are an attractive starting point, requiring experimental validation. Computational chemistry includes a wide range of methods of varying accuracy and complexity, summarized in Table 7-14. Many of these methods have been implemented as software packages that require high-speed supercomputers or parallel computers to solve realistic reactions. For more details on computational chemistry, see, e.g., Cramer, Essentials of Computational Chemistry: Theories and Models, 2d ed., Wiley, 2004. Lumping and Mechanism Reduction It is often useful to reduce complex reaction networks to a smaller reaction set which still maintains the key features of the detailed reaction network but with a much smaller number of representative species, reactions, and kinetic parameters. Simple examples were already given above for reducing simple networks into global reactions through assumptions such as pseudo-steady state, rate-limiting step, and equilibrium reactions. In general, mechanism reduction can only be used over a limited range of conditions for which the simplified system simulates the original complete reaction network. This reduces the number of kinetic parameters that have to be either estimated from data or calculated by TABLE 7-14 Computational Chemistry Methods Abinitio methods (no empirical parameters) Electronic structure determination (time-independent Schrodinger equation) Hartree-Fock (HF) with corrections Quantum Monte Carlo (QMT) Density functional theory (DFT) Chemical dynamics determination (time-dependent Schrodinger equation) Split operator technique Multiconfigurational time-dependent Hartree-Fock method Semiclassical method Semiempirical methods (approximate parts of HF calculations such as twoelectron integrals) Huckel Extended Huckel Molecular mechanics (avoids quantum mechanical calculations) Empirical methods (group contributions) Polanyi linear approximation of activation energy

DETERMINATION OF MECHANISM AND KINETICS using computational chemistry. The simplified system also reduces the computation load for reactor scale-up, design, and optimization. A type of mechanism reduction called lumping is typically performed on a reaction network that consists of a large number of similar reactions occurring between similar species, such as homologous series or molecules having similar functional groups. Such situations occur, for instance, in the oil refining industry, examples including catalytic reforming, catalytic cracking, hydrocracking, and hydrotreating. Lumping is done by grouping similar species, or molecules with similar functional groups, into pseudo components called lumped species. The behavior of the lumped system depends on the initial composition, the distribution of the rate constants in the detailed system, and the form of the rate equation. The two main issues in lumping are 1. Determination of the lump structure that simulates the detailed system over the required range of conditions 2. Determination of the kinetics of the lumped system from general knowledge about the type of kinetics and the overall range of parameters of the detailed system Lumping has been applied extensively to first-order reaction networks [e.g., Wei and Kuo, “A Lumping Analysis in Monomolecular Reaction Systems,” I&EC Fundamentals 8(1): 114–123 (1969); Golikeri and Luss, “Aggregation of Many Coupled Consecutive First Order Reactions,” Chem. Eng. Sci. 29: 845–855 (1974)]. For instance, it has been shown that a lumped reaction network of first-order reactions can behave under certain conditions as a global second-order reaction. Where analytical solutions were not available, others, such as Golikeri and Luss, “Aggregation of Many Coupled Consecutive First Order Reactions,” Chem. Eng. Sci. 29: 845–855 (1974), developed bounds that bracketed the behavior of the lump for first-order reactions as a function of the initial composition and the rate constant distribution. Lumping has not been applied as successfully to nonlinear or higher-order kinetics. More recent applications of lumping were published, including structure-oriented lumping that lumps similar structural groups, by Quann and Jaffe, “Building Useful Models of Complex Reaction Systems in Petroleum Refining,” Chem. Eng. Sci. 51(10): 1615–1635 (1996). For other types of systems such as highly branched reaction networks for homogeneous gas-phase combustion and combined homogeneous and catalytic partial oxidation, mechanism reduction involves pruning branches and pathways of the reaction network that do not contribute significantly to the overall reaction. This pruning is done by using sensitivity analysis. See, e.g., Bui et al., “Hierarchical Reduced Models for Catalytic Combustion: H2/Air Mixtures near Platinum Surfaces,” Combustion Sci. Technol. 129(1–6):243–275 (1997). Multiple Steady States, Oscillations, and Chaotic Behavior There are reaction systems whose steady-state behavior depends on

7-39

the initial or starting conditions; i.e., for different starting conditions, different steady states can be reached at the same final operating conditions. This behavior is called steady-state multiplicity and is often the result of the interaction of kinetic and transport phenomena having distinct time scales. For some cases, the cause of the multiplicity is entirely reaction-related, as shown below. Associated with steady-state multiplicity is hysteresis, and higher-order instabilities such as selfsustained oscillations and chaotic behavior. The existence of multiple steady states may be relevant to analysis of laboratory data, since faster of slower rates may be observed at the same conditions depending on how the lab reactor is started up. For example, CO oxidation on heterogeneous Rh catalyst exhibits hysteresis and multiple steady states, and one of the explained causes is the existence of two crystal structures for Rh, each with a different reactivity (Masel, Chemical Kinetics and Catalysis, Wiley, 2001, p. 38). Another well-known example of chemistry-related instability includes the oscillatory behavior of the Bhelousov-Zhabotinsky reaction of malonic acid and bromate in the presence of homogeneous Ce catalyst having the overall reaction Ce4

HOOCCH2COOH  HBrO ⇒ products Ce can be in two oxidation states, Ce3+ and Ce4+, and there are competing reaction pathways. Complex kinetic models are required to predict the oscillatory behavior, the most well known being that of Noyes [e.g., Showalter, Noyes, and Bar-Eli, J. Chem. Phys. 69(6): 2514–2524 (1978)]. A large body of work has been done to develop criteria that determine the onset of chemistry and transport chemistry-based instabilities. More details and transport-reaction coupling-related examples are discussed in Sec. 19. SOFTWARE TOOLS There are a number of useful software packages that enable efficient analysis of laboratory data for developing the mechanism and kinetics of reactions and for testing the kinetics by using simple reactor models. The reader is referred to search the Internet as some of these software packages change ownership or name. Worth mentioning are the Aspen Engineering Suite (Aspen), the Thermal Safety Software suite (Cheminform St. Petersburg), the Matlab suite (Mathworks), the Chemkin software suite (Reaction Design), the NIST Chemical Kinetics database (NIST), and Gepasi for biochemical kinetics (freeware). The user is advised to experiment and validate any software package with known data and kinetics to ensure robustness and reliability.

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Section 8

Process Control

Thomas F. Edgar, Ph.D. Professor of Chemical Engineering, University of Texas—Austin (Section Editor, Advanced Control Systems, Process Measurements) Cecil L. Smith, Ph.D. Principal, Cecil L. Smith Inc. (Batch Process Control, Telemetering and Transmission, Digital Technology for Process Control, Process Control and Plant Safety) F. Greg Shinskey, B.S.Ch.E. Consultant (retired from Foxboro Co.) (Fundamentals of Process Dynamics and Control, Unit Operations Control) George W. Gassman, B.S.M.E. Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc. (Controllers, Final Control Elements, and Regulators) Andrew W. R. Waite, P.Eng. Principal Process Control Consultant, EnTech Control, a Division of Emerson Electric Canada (Controllers, Final Control Elements, and Regulators) Thomas J. McAvoy, Ph.D. Professor of Chemical Engineering, University of Maryland— College Park (Fundamentals of Process Dynamics and Control) Dale E. Seborg, Ph.D. Professor of Chemical Engineering, University of California— Santa Barbara (Advanced Control Systems)

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL The General Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5 Computer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5 Process Dynamics and Mathematical Models . . . . . . . . . . . . . . . . . . . . . 8-5 Open-Loop versus Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . . . 8-5 Physical Models versus Empirical Models . . . . . . . . . . . . . . . . . . . . . 8-6 Nonlinear versus Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Simulation of Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 Transfer Functions and Block Diagrams . . . . . . . . . . . . . . . . . . . . . . . 8-8 Continuous versus Discrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8-8 Process Characteristics in Transfer Functions. . . . . . . . . . . . . . . . . . . 8-9 Fitting Dynamic Models to Experimental Data . . . . . . . . . . . . . . . . . 8-12 Feedback Control System Characteristics. . . . . . . . . . . . . . . . . . . . . . . . 8-12 Closing the Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 On/Off Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Proportional Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-14 Proportional-plus-Integral (PI) Control . . . . . . . . . . . . . . . . . . . . . . . 8-14 Proportional-plus-Integral-plus-Derivative (PID) Control . . . . . . . . 8-15 Controller Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16

Controller Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tuning Methods Based on Known Process Models . . . . . . . . . . . . . . Tuning Methods When Process Model Is Unknown . . . . . . . . . . . . . Set-Point Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ADVANCED CONTROL SYSTEMS Benefits of Advanced Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Control Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cascade Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Delay Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selective and Override Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy Logic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expert Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control Strategies for Multivariable Control. . . . . . . . . . . . . . . . . . . . Decoupling Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pairing of Controlled and Manipulated Variables . . . . . . . . . . . . . . . . RGA Method for 2 × 2 Control Problems . . . . . . . . . . . . . . . . . . . . . . RGA Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages and Disadvantages of MPC . . . . . . . . . . . . . . . . . . . . . . .

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Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

8-2

PROCESS CONTROL

Economic Incentives for Automation Projects . . . . . . . . . . . . . . . . . . Basic Features of MPC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of MPC and Online Optimization . . . . . . . . . . . . . . . . . . Real-Time Process Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential Features of Optimization Problems. . . . . . . . . . . . . . . . . . . Development of Process (Mathematical) Models . . . . . . . . . . . . . . . . Formulation of the Objective Function. . . . . . . . . . . . . . . . . . . . . . . . Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Variable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Western Electric Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUSUM Control Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Capability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Six-Sigma Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariate Statistical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .

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UNIT OPERATIONS CONTROL Piping and Instrumentation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . Control of Heat Exchangers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steam-Heated Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange of Sensible Heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distillation Column Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of a Single Product. . . . . . . . . . . . . . . . . . . . . . . . Controlling Quality of Two Products . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controlling Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drying Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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BATCH PROCESS CONTROL Batch versus Continuous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batches and Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Routing and Production Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . Production Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Automation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Device States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulatory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Reactor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Batch Production Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equipment Suite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Unit or Batch Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Item of Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structured Batch Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Technology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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PROCESS MEASUREMENTS General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy and Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics of Process Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filled-System Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bimetal Thermometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pyrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Element Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orifice Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Venturi Meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Meter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex-Shedding Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultrasonic Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Mass Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Mass Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Level Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Float-Actuated Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sonic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laser Level Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radar Level Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Property Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density and Specific Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dielectric Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Composition Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chromatographic Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infrared Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ultraviolet and Visible-Radiation Analyzers . . . . . . . . . . . . . . . . . . . . Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electroanalytical Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conductometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific-Ion Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dew Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitance Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oxide Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photometric Moisture Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hall Effect Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling Systems for Process Analyzers . . . . . . . . . . . . . . . . . . . . . . . . . Selecting the Sampling Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Withdrawal from Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TELEMETERING AND TRANSMISSION Analog Signal Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pulse Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serial Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microprocessor-Based Transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . Transmitter/Actuator Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering and Smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-65 8-65 8-65 8-65 8-66 8-66 8-66 8-66 8-67

DIGITAL TECHNOLOGY FOR PROCESS CONTROL Hierarchy of Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Devices and Final Control Elements. . . . . . . . . . . . . . Safety and Environmental/Equipment Protection . . . . . . . . . . . . . . . Regulatory Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real-Time Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corporate Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Hardware in Process Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Loop Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Programmable Logic Controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Personal Computer Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distributed Database and the Database Manager . . . . . . . . . . . . . . . . . Data Historian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Field Communications and Field Bus. . . . . . . . . . . . . . . . . . . . . Internodal Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Control Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-68 8-68 8-68 8-68 8-68 8-68 8-69 8-69 8-69 8-69 8-69 8-69 8-70 8-70 8-70 8-70 8-71

PROCESS CONTROL CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS Pneumatic, Electronic, and Digital Controllers . . . . . . . . . . . . . . . . . . . 8-71 Pneumatic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-71 Electronic (Digital) Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-72 Control Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-74 Valve Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-74 Special Application Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-76 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-76 Other Process Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-78 Valves for On/Off Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-78 Pressure Relief Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-78 Check Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-79 Valve Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-79 Materials and Pressure Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-79 Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-79 Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-81 Cavitation and Flashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-82 Seals, Bearings, and Packing Systems . . . . . . . . . . . . . . . . . . . . . . . . . 8-82 Flow Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-83 Valve Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-84

8-3

Valve Positioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Booster Relays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solenoid Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trip Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit Switches and Stem Position Transmitters . . . . . . . . . . . . . . . . . Fire and Explosion Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjustable-Speed Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pilot-Operated Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overpressure Protection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-84 8-89 8-90 8-91 8-91 8-91 8-91 8-91 8-91 8-92 8-92 8-93 8-94

PROCESS CONTROL AND PLANT SAFETY Role of Automation in Plant Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrity of Process Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . Considerations in Implementation of Safety Interlock Systems. . . . . . . Interlocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-94 8-95 8-95 8-96 8-96

8-4

PROCESS CONTROL

Nomenclature Symbol A Aa Ac Av Al b B B*i cA C Cd Ci C*i CL C0 Cp Cr CV D D*i e E f F, f FL gc gi G Gc Gd Gf Gm Gp Gt Gv hi h1 H i Ii j J k kf kr K Kc Kd Km Kp Ku L Lp M mc Mv Mr Mw n N p pc pd pi pu q qb Q rc R RT R1

Definition Area Actuator area Output amplitude limits Amplitude of controlled variable Cross-sectional area of tank Controller output bias Bottoms flow rate Limit on control Concentration of A Cumulative sum Discharge coefficient Inlet concentration Limit on control move Specific heat of liquid Integration constant Process capability Heat capacity of reactants Valve flow coefficient Distillate flow rate, disturbance Limit on output Error Economy of evaporator Function of time Feed flow rate Pressure recovery factor Unit conversion constant Algebraic inequality constraint Transfer function Controller transfer function Disturbance transfer function Feedforward controller transfer function Sensor transfer function Process transfer function Transmitter transfer function Valve transfer function Algebraic equality constraints Liquid head in tank Latent heat of vaporization, control limit or threshold Summation index Impulse response coefficient Time index Objective function or performance index Time index Flow coefficient Kinetic rate constant Gain, slack parameter Controller gain Disturbance transfer function gain Measurement gain Process gain Ultimate controller gain (stability) Load variable Sound pressure level Manipulated variable Number of constraints Mass flow Mass of reactants Molecular weight Number of data points, number of stages or effects Number of inputs/outputs, model horizon Proportional band (%) Vapor pressure Actuator pressure Pressure Proportional band (ultimate) Radiated energy flux Energy flux to a black body Flow rate Number of constraints Equal-percentage valve characteristic Resistance in temperature sensor Valve resistance

Symbol

Definition

s s Si t T T(s) Tb Tf TR U u, U V Vs w wi W x ⎯x xi xT X y, Y Ysp z zi Z

Laplace transform variable Search direction Step response coefficient Time Temperature, target Decoupler transfer function Base temperature Exhaust temperature Reset time Heat-transfer coefficient Manipulated variable, controller output Volume Product value Mass flow rate Weighting factor Steam flow rate Mass fraction Sample mean Optimization variable Pressure drop ratio factor Transform of deviation variable Process output, controlled variable, valve travel Set point Controller tuning law, expansion factor Feed mole fraction (distillation) Compressibility factor

α aT β γ δ ∆q ∆t ∆T ∆u ε ζ θ λ Λ

ρ σ Σt τ τd D τF τI τP τo φPI

Digital filter coefficient Temperature coefficient of resistance Resistance thermometer parameter Ratio of specific heats Move suppression factor, shift in target value Load step change Time step Temperature change Control move Spectral emissivity, step size Damping factor (second-order system) Time delay Relative gain array parameter, wavelength Relative gain array Deviation variable Density Stefan-Boltzmann constant, standard deviation Total response time Time constant Natural period of closed loop, disturbance time constant Derivative time (PID controller) Filter time constant Integral time (PID controller) Process time constant Period of oscillation Phase lag

A

Species A Best Controller Disturbance Effective Feedforward Initial, inlet Load, disturbance Measurement or sensor Process Steady state Set-point value Transmitter Ultimate Valve

Greek Symbols

Subscripts b c d eff f i L m p s sp t u v

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL THE GENERAL CONTROL SYSTEM

by the proper selection of control modes to satisfy the requirements of the process and, second, by the appropriate tuning of those modes. Feedforward Control A feedforward system uses measurements of disturbance variables to position the manipulated variable in such a way as to minimize any resulting deviation. The disturbance variables could be either measured loads or the set point, the former being more common. The feedforward gain must be set precisely to reduce the deviation of the controlled variable from the set point. Feedforward control is usually combined with feedback control to eliminate any offset resulting from inaccurate measurements and calculations and unmeasured load components. The feedback controller can be used as a bias on the feedforward controller or in a multiplicative form. Computer Control Computers have been used to replace analog PID controllers, either by setting set points of lower-level controllers in supervisory control or by driving valves directly in direct digital control. Single-station digital controllers perform PID control in one or two loops, including computing functions such as mathematical operations, characterization, lags, and dead time, with digital logic and alarms. Distributed control systems provide all these functions, with the digital processor shared among many control loops; separate processors may be used for displays, communications, file servers, and the like. A host computer may be added to perform high-level operations such as scheduling, optimization, and multivariable control. More details on computer control are provided later in this section.

A process is shown in Fig. 8-1 with a manipulated input U, a load input D, and a controlled output Y, which could be flow, pressure, liquid level, temperature, composition, or any other inventory, environmental, or quality variable that is to be held at a desired value identified as the set point Ysp. The load may be a single variable or an aggregate of variables either acting independently or manipulated for other purposes, affecting the controlled variable much as the manipulated variable does. Changes in load may occur randomly as caused by changes in weather, diurnally with ambient temperature, manually when operators change production rate, stepwise when equipment is switched into or out of service, or cyclically as the result of oscillations in other control loops. Variations in load will drive the controlled variable away from the set point, requiring a corresponding change in the manipulated variable to bring it back. The manipulated variable must also change to move the controlled variable from one set point to another. An open-loop system positions the manipulated variable either manually or on a programmed basis, without using any process measurements. This operation is acceptable for well-defined processes without disturbances. An automated transfer switch is provided to allow manual adjustment of the manipulated variable in case the process or the control system is not performing satisfactorily. A closed-loop system uses the measurement of one or more process variables to move the manipulated variable to achieve control. Closedloop systems may include feedforward, feedback, or both. Feedback Control In a feedback control loop, the controlled variable is compared to the set point Ysp, with the error E acted upon by the controller to move U in such a way as to minimize the error. This action is specifically negative feedback, in that an increase in error moves U so as to decrease the error. (Positive feedback would cause the error to expand rather than diminish and therefore does not regulate.) The action of the controller is selectable to allow use on process gains of both signs. The controller has tuning parameters related to proportional, integral, derivative, lag, dead time, and sampling functions. A negative feedback loop will oscillate if the controller gain is too high; but if it is too low, control will be ineffective. The controller parameters must be properly related to the process parameters to ensure closed-loop stability while still providing effective control. This relationship is accomplished, first,

PROCESS DYNAMICS AND MATHEMATICAL MODELS GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004; Marlin, Process Control, McGraw-Hill, New York, 2000; Ogunnaike and Ray, Process Dynamics Modeling and Control, Oxford University Press, New York, 1994; Smith and Corripio, Principles and Practices of Automatic Process Control, Wiley, New York, 1997.

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reactor in order to control conditions in the reactor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reactor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is Kc. If a small change in the temperature of the inlet stream occurs, then

Disturbance, D Feedforward Controller

Set Point, Ysp

Error

Feedback Controller

Manipulated Variable, U

Process

Controlled Variable, Y

Feedback Loop FIG. 8-1 Block diagram for feedforward and feedback control.

8-5

8-6

PROCESS CONTROL

FIG. 8-2 Reactor control system.

depending on the value of Kc, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (Kc = 0), which is called the open loop, or the normal dynamic response of the process by itself. As Kc increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have as small an effect as possible on the process under study. As the gain is increased further, eventually a point is reached where the reactor temperature oscillates indefinitely, which is undesirable. This point is called the stability limit, where Kc = Ku, the ultimate controller gain. Increasing Kc further causes the magnitude of the oscillations to increase, with the result that the control valve will cycle between full open and closed. The responses shown in Fig. 8-3 are typical of the vast majority of regulatory loops encountered in the process industries. Figure 8-3

FIG. 8-3 Typical control system responses.

shows that there is an optimal choice for Kc, somewhere between 0 (no control) and Ku (stability limit). If one has a dynamic model of a process, then this model can be used to calculate controller settings. In Fig. 8-3, no time scale is given, but rather the figure shows relative responses. A well-designed controller might be able to speed up the response of a process by a factor of roughly 2 to 4. Exactly how fast the control system responds is determined by the dynamics of the process itself. Physical Models versus Empirical Models In developing a dynamic process model, there are two distinct approaches that can be taken. The first involves models based on first principles, called physical or first principles models, and the second involves empirical models. The conservation laws of mass, energy, and momentum form the basis for developing physical models. The resulting models typically involve sets of differential and algebraic equations that must be solved simultaneously. Empirical models, by contrast, involve postulating the form of a dynamic model, usually as a transfer function, which is discussed below. This transfer function contains a number of parameters that need to be estimated from data. For the development of both physical and empirical models, the most expensive step normally involves verification of their accuracy in predicting plant behavior. To illustrate the development of a physical model, a simplified treatment of the reactor, shown in Fig. 8-2, is used. It is assumed that the reactor is operating isothermally and that the inlet and exit volumetric flows and densities are the same. There are two components, A and B, in the reactor, and a single first-order reaction of A→B takes place. The inlet concentration of A, which we call ci, varies with time. A dynamic mass balance for the concentration of A, denoted cA, can be written as follows: dcA V = Fci − FcA − krVcA dt

(8-1)

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL In Eq. (8-1), the flow in of component A is Fci, the flow out is FcA, and the loss via reaction is krVcA, where V = reactor volume and kr = kinetic rate constant. In this example, ci is the input, or forcing, variable and cA is the output variable. If V, F, and kr are constant, Eq. (8-1) can be rearranged by dividing by F + krV so that it contains only two groups of parameters. The result is dcA τ = Kci − cA dt

dcA V = Fci − FcA − krVc2A dt

(8-3)

Since cA appears in this equation to the second power, the equation is nonlinear. The difference between linear systems and nonlinear systems can be seen by considering the steady-state behavior of Eq. (8-1) compared to Eq. (8-3) (the left-hand side is zero; that is, dcA/dt = 0). For a given change in ci, ∆ci, the change in cA calculated from Eq. (8-1), ∆cA, is always proportional to ∆ci, and the proportionality constant is K [see Eq. (8-2)]. The change in the output of a system divided by a change in the input to the system is called the process gain. Linear systems have constant process gains for all changes in the input. By contrast, Eq. (8-3) gives a ∆cA that varies with ∆ci, which is a function of the concentration levels in the reactor. Thus, depending on the reactor operating conditions, a change in ci produces different changes in cA. In this case, the process has a nonlinear gain. Systems with nonlinear gains are more difficult to control than linear systems that have constant gains. Simulation of Dynamic Models Linear dynamic models are particularly useful for analyzing control system behavior. The insight gained through linear analysis is invaluable. However, accurate dynamic process models can involve large sets of nonlinear equations. Analytical solution of these models is not possible. Thus, in these cases, one must turn to simulation approaches to study process dynamics and the effect of process control. Equation (8-3) will be used to illustrate the simulation of nonlinear processes. If dcA/dt on the left-hand side of Eq. (8-3) is replaced with its finite difference approximation, one gets cA(t) + ∆t ⋅ [Fci(t) − FcA(t) − krVc2A(t)] cA(t + ∆t) =  V

Define

(8-4)

Starting with an initial value of cA and given ci(t), Eq. (8-4) can be solved for cA(t + ∆t). Once cA(t + ∆t) is known, the solution process can be repeated to calculate cA(t + 2∆t), and so on. This approach is called the Euler integration method; while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. As discussed in Sec. 3, more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations:

Fci(t) − FcA − krVc2A f(cAt) =  V

(8-5)

cA(t + ∆t) = cA(t) + ∆t(m1 + 2m2 + 2m3 + m4)

(8-6)

m1 = f [cA(t), t]

(8-7)

then

(8-2)

where τ = V(F + krV) and K = F/(F + krV). For this example, the resulting model is a first-order differential equation in which τ is called the time constant and K the process gain. As an alternative to deriving Eq. (8-2) from a dynamic mass balance, one could simply postulate a first-order differential equation to be valid (empirical modeling). Then it would be necessary to estimate values for τ and K so that the postulated model described the reactor’s dynamic response. The advantage of the physical model over the empirical model is that the physical model gives insight into how reactor parameters affect the values of τ and K, which in turn affects the dynamic response of the reactor. Nonlinear versus Linear Models If V, F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives appear to only the first power. If the rate of reaction were secondorder, then the resulting dynamic mass balance would be

8-7

with

m1 ∆t ∆t m2 = f cA(t) + 2  , t +  2 2





(8-8)

m2 ∆t ∆t m3 = f cA(t) + 2  , t +  2 2



(8-9)

m4 = f [cA(t) + m3 ∆t, t + ∆t]

(8-10)



In this method, the mi’s are calculated sequentially in order to take a step in time. Even though this method requires calculation of the four additional mi values, for equivalent accuracy the fourth-order Runge Kutta method can result in a faster numerical solution, because it permits a larger step ∆t to be taken. Increasingly sophisticated simulation packages are being used to calculate the dynamic behavior of processes and to test control system behavior. These packages have good user interfaces, and they can handle stiff systems where some variables respond on a time scale that is much much faster or slower than that of other variables. A simple Euler approach cannot effectively handle stiff systems, which frequently occur in chemical process models. See Sec. 3 of this handbook for more details. Laplace Transforms When mathematical models are used to describe process dynamics in conjunction with control system analysis, the models generally involve linear differential equations. Laplace transforms are very effective for solving linear differential equations. The key advantage of using Laplace transforms is that they convert differential equations to algebraic equations. The resulting algebraic equations are easier to solve than the original differential equations. When the Laplace transform is applied to a linear differential equation in time, the result is an algebraic equation in a new variable s, called the Laplace variable. To get the solution to the original differential equation, one needs to invert the Laplace transform. Table 8-1 gives a number of useful Laplace transform pairs, and more extensive tables are available (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004). To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subject to the initial condition that cA = ci = 0 at t = 0. If cA were not initially zero, one would define a deviation variable between cA and its initial value cA0. Then the transfer function would be developed by using this deviation variable. If ci changes from zero to c⎯i, taking the Laplace transform of both sides of Eq. (8-2) gives τ dcA £  = £(K ⎯ci) − £(cA) dt





TABLE 8-1 Frequently Used Laplace Transforms Time function f(t)

Transform F(s)

A At Ae−at A(1 − e−tτ) A sin ωt f(t − θ) df/dt f(t)dt

A/s A/s2 A/(s + a) A[s(τs + 1)] Aω(s2 + ω2) e−θF(s) sF(s) − f(0) F(s)/s

(8-11)

8-8

PROCESS CONTROL

Denoting £(CA) as CA(s) and using the relationships in Table 8-1 give Kc⎯i τ sCA(s) =  s − CA(s)

(8-12)

Equation (8-12) can be solved for CA to give Kc⎯i s CA(s) =  τs + 1

(8-13)

By using the entries in Table 8-1, Eq. (8-13) can be inverted to give the transient response of cA as cA(t) = (Kc⎯i)(1 − e−1τ)

(8-14)

Equation (8-14) shows that cA starts from 0 and builds up exponentially to a final concentration of Kci. Note that to get Eq. (8-14), it was only necessary to solve the algebraic Eq. (8-12) and then find the inverse of CA(s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher-order differential equations with Laplace transforms. Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use of transfer functions and block diagrams. A transfer function can be obtained by starting with a physical model, as discussed previously. If the physical model is nonlinear, first it needs to be linearized around an operating point. The resulting linearized model is then approximately valid in a region around this operating point. To illustrate how transfer functions are developed, Eq. (8-2) will again be used. First, one defines deviation variables, which are the process variables minus their steady-state values at the operating point. For Eq. (8-2), there would be deviation variables for both cA and ci, and these are defined as ξ = cA − c⎯A

(8-15)

ξi = ci − c⎯i

(8-16)

where the overbar stands for steady state. Substitution of Eqs. (8-15) and (8-16) into Eq. (8-2) gives dξ τ  = Kξi − ξ + (K ⎯ci − ⎯cA) dt

(8-17)

The term in parentheses in Eq. (8-17) is zero at steady state, and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation is solved. By denoting X(s) as the Laplace transform of ξ and Xi(s) as the transform of ξi, the final transfer function can be written as X K  =  Xi τs + 1

FIG. 8-4 First-order transfer function.

time. Most real processes have variables that are continuous, such as temperature, pressure, and flow. However, some processes involve discrete events, such as the starting or stopping of a pump. In addition, modern plants are controlled by digital computers, which are discrete. In controlling a process, a digital system samples variables at a fixed rate, and the resulting system is a sampled data system. From one sampling instant until the next, variables are assumed to remain fixed at their sampled values. Similarly, in controlling a process, a digital computer sends out signals to control elements, usually valves, at discrete instants of time. These signals remain fixed until the next sampling instant. Figure 8-5 illustrates the concept of sampling a continuous function. At integer values of the sampling rate ∆t, the value of the variable to be sampled is measured and held until the next sampling instant. To deal with sampled data systems, the z transform has been developed. The z transform of the function given in Fig. 8-5 is defined as ∞

Z(f) = f(n ∆t)z−n

In an analogous manner to Laplace transforms, one can develop transfer functions in the z domain as well as block diagrams. Tables of z transform pairs have been published (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004) so that the discrete transfer functions can be inverted back to the time domain. The inverse gives the value of the function at the discrete sampling instants. Sampling a continuous variable results in a loss of information. However, in practical applications, sampling is fast enough that the loss is typically insignificant and the difference between continuous and discrete modeling is small in terms of its effect on control. Increasingly, model predictive controllers that make use of discrete dynamic models are being used in the process industries. The purpose of these controllers is to guide a process to optimum operating points. These model predictive control algorithms are typically run at much slower sampling rates than are used for basic control loops such as flow control or pressure control. The discrete dynamic models used are normally developed from data generated

(8-18)

Equation (8-18) is an example of a first-order transfer function. As mentioned above, an alternative to formally deriving Eq. (8-18) involves simply postulating its form and then identifying its two parameters, the process gain K and time constant τ, to fit the process under study. In fitting the parameters, data can be generated by forcing the process. If step forcing is used, then the resulting response is called the process reaction curve. Often transfer functions are placed in block diagrams, as shown in Fig. 8-4. Block diagrams show how changes in an input variable affect an output variable. Block diagrams are a means of concisely representing the dynamics of a process under study. Since linearity is assumed in developing a block diagram, if more than one variable affects an output, the contributions from each can be added. Continuous versus Discrete Models The preceding discussion has focused on systems where variables change continuously with

(8-19)

n=0

FIG. 8-5 Sampled data example.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

8-9

FIG. 8-6 Single tank with exit valve.

from plant testing, as discussed hereafter. For a detailed discussion of modeling sampled data systems, the interested reader is referred to textbooks on digital control (Astrom and Wittenmark, Computer Controlled Systems, Prentice-Hall, Englewood Cliffs, N.J., 1997). Process Characteristics in Transfer Functions In many cases, process characteristics are expressed in the form of transfer functions. In the previous discussion, a reactor example was used to illustrate how a transfer function could be derived. Here, another system involving flow out of a tank, shown in Fig. 8-6, is considered. Proportional Element First, consider the outflow through the exit valve on the tank. If the flow through the line is turbulent, then Bernoulli’s equation can be used to relate the flow rate through the valve to the pressure drop across the valve as f1 = kf Av 2gc(h1  − h0)

(8-20)

where f1 = flow rate, kf = flow coefficient, Av = cross-sectional area of the restriction, gc = constant, h1 = liquid head in tank (pressure at the base of the tank), and h0 = atmospheric pressure. This relationship between flow and pressure drop across the valve is nonlinear, and it can be linearized around a particular operating point to give ⎯ ⎯ 1 f1 − f1 =  (h1 − h1) R1



(8-21)

—

where R1 = f1(gck2f A2v) is called the resistance of the valve in analogy with an electrical resistance. The transfer function relating changes in flow to changes in head is shown in Fig. 8-7, and it is an example of a pure gain system with no dynamics. In this case, the process gain is K = 1/R1. Such a system has an instantaneous dynamic response, and for a step change in head, there is an immediate step change in flow, as shown in Fig. 8-8. The exact magnitude of the step in flow depends on — the operating flow f 1 as the definition of R1 shows. First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives dh1 A1  = fi − f1 dt

(8-22)

where Al is the cross-sectional area of the tank and fi is the inlet flow. By substituting Eq. (8-21) into Eq. (8-22) and following the approach

FIG. 8-7 Proportional element transfer function.

FIG. 8-8

Response of proportional element.

discussed above for deriving transfer functions, one can develop the transfer function relating changes in h1 to changes in fi. The resulting transfer function is another example of a first-order system, shown in Fig. 8-4, and it has a gain K = R1 and a time constant τ1 = R1Al. For a step change⎯ in fi, h1 follows a decaying exponential response from its initial ⎯ value h1 to a final value of h1 + R1 ∆fi (Fig. 8-9). At a time equal to τ1, the transient in h1 is 63 percent finished; and at 3τ1, the response is 95 percent finished. These percentages are the same for all first-order processes. Thus, knowledge of the time constant of a first-order process gives insight into how fast the process responds to sudden input changes. Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except that fl no longer depends on h1. For changes in fi, the transfer function relating changes in h1 to changes in fi is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross-sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held constant, then the level builds up linearly, as shown in Fig. 8-11. Eventually the liquid will overflow the tank. Second-Order Element Because of their linear nature, transfer functions can be combined in a straightforward manner. Consider the two-tank system shown in Fig. 8-12. For tank 1, the transfer function relating changes in f1 to changes in fi is F1(s) 1  =  Fi(s) A1R1 + 1

(8-23)

Since fl is the inlet flow to tank 2, the transfer function relating changes in h2 to changes in fl has the same form as that given in Fig. 8-4: H2(s) R2  =  F1(s) A2R2s + 1

(8-24)

Equations (8-23) and (8-24) can be multiplied to give the final transfer function relating changes in h2 to changes in fi, as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R2 and two time constants A1R1 and A2R2. For two tanks with equal areas, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when ζ < 1, the response oscillates; and when ζ < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response; most exhibit an S-shaped step response.

8-10

PROCESS CONTROL

FIG. 8-9

Response of first-order system.

Distance-Velocity Lag (Dead-Time Element) The dead-time or time-delay element, commonly called a distance-velocity lag, is often encountered in process systems. For example, if a temperaturemeasuring element is located downstream from a heat exchanger, a time delay occurs before the heated fluid leaving the exchanger

FIG. 8-10

Pure capacity or integrating transfer function.

FIG. 8-11

Response of pure capacity system.

arrives at the temperature measurement point. If some element of a system produces a dead time of θ time units, then an input to that unit f(t) will be reproduced at the output as f(t − θ). The transfer function for a pure dead-time element is shown in Fig. 8-17, and the transient response of the element is shown in Fig. 8-18.

FIG. 8-12

Two tanks in series.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-13

FIG. 8-14

Second-order transfer function.

FIG. 8-17

Dead-time transfer function.

FIG. 8-18

Response of dead-time system.

8-11

Response of second-order system.

Higher-Order Lags If a process is described by a series of n firstorder lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by K G(s) = n (τs + 1)

(8-25)

The step response of this transfer function is shown in Fig. 8-19. Note that all curves reach about 60 percent of their final value at t = nτ. FIG. 8-15

General second-order transfer function.

FIG. 8-16

Response of general second-order system.

FIG. 8-19

Response of nth-order lags.

8-12

PROCESS CONTROL

U1

G11

Y1

G21

G12

U2

FIG. 8-20

G22

Y2

Example of 2 × 2 transfer function.

Higher-order systems can be approximated by a first- or secondorder plus dead-time system for control system design. Multi-input, Multioutput Systems The dynamic systems considered up to this point have been examples of single-input, singleoutput (SISO) systems. In chemical processes, one often encounters systems where one input can affect more than one output. For example, assume that one is studying a distillation tower in which both reflux and boil-up are manipulated for control purposes. If the output variables are the top and bottom product compositions, then each input affects both outputs. For this distillation example, the process is referred to as a 2 × 2 system to indicate the number of inputs and outputs. In general, multi-input, multioutput (MIMO) systems can have n inputs and m outputs with n ≠ m, and they can be nonlinear. Such a system would be called an n × m system. An example of a transfer function for a 2 × 2 linear system is given in Fig. 8-20. Note that since linear systems are involved, the effects of the two inputs on each output are additive. In many process control systems, one input is selected to control one output in a MIMO system. For m outputs there would be m such selections. For this type of control strategy, one needs to consider which inputs and outputs to couple, and this problem is referred to as loop pairing. Another important issue that arises involves interaction between control loops. When one loop makes a change in its manipulated variable, the change affects the other loops in the system. These changes are the direct result of the multivariable nature of the process. In some cases, the interaction can be so severe that overall control system performance is drastically reduced. Finally, some of the modern approaches to process control tackle the MIMO problem directly, and they simultaneously use all manipulated variables to control all output variables rather than pair one input to one output (see later section on multivariable control). Fitting Dynamic Models to Experimental Data In developing empirical transfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The x’s in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first-order with dead time and have the transfer function Y(s) G(s) =  = K exp (−θs)τs + 1 U(s)

(8-26)

The response y(t), produced by Eq. (8-26) can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of

model parameters K, τ, and θ fitted to the data. These parameters are calculated by using optimization to minimize the squared difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by yj (measured response), tj (time of measured response), j = 1 to n. Then the least squares problem can be formulated as n

minτ,θ,κ [yj − ^y(tj)]2

(8-27)

j=1

^ where y(t j) is the predicted value of y at time tj and n is the number of data points. This optimization problem can be solved to calculate the optimal values of K, τ, and θ. A number of software packages such as Excel Solver are available for minimizing Eq. (8-27). One operational problem caused by step forcing is the fact that the process under study is moved away from its steady-state operating point. Plant managers may be reluctant to allow large steadystate changes, since normal production will be disturbed by the changes. As a result, alternative methods of forcing actual processes have been developed, and these included pulse testing and pseudorandom binary signal (PRBS) forcing, both of which are illustrated in Fig. 8-22. With pulse forcing, one introduces a step, and then after a period of time the input is returned to its original value. The result is that the process dynamics are excited, but after the forcing the process returns to its original steady state. PRBS forcing involves a series of pulses of fixed height and random duration, as shown in Fig. 8-22. The advantage of PRBS is that forcing can be concentrated on particular frequency ranges that are important for control system design. Transfer function models are linear, but chemical processes are known to exhibit nonlinear behavior. One could use the same type of optimization objective as given in Eq. (8-27) to determine parameters in nonlinear first-principles models, such as Eq. (8-3) presented earlier. Also, nonlinear empirical models, such as neural network models, have recently been proposed for process applications. The key to the use of these nonlinear empirical models is to have highquality process data, which allows the important nonlinearities to be identified.

FEEDBACK CONTROL SYSTEM CHARACTERISTICS GENERAL REFERENCES: Shinskey, Process Control Systems, 4th ed., McGrawHill, New York, 1996; Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 1989.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-21

8-13

Plot of experimental data and first-order model fit.

There are two objectives in applying feedback control: (1) regulate the controlled variable at the set point following changes in load and (2) respond to set-point changes, the latter called servo operation. In fluid processes, almost all control loops must contend with variations in load, and therefore regulation is of primary importance. While most loops will operate continuously at fixed set points, frequent changes in set points can occur in flow loops and in batch production. The most common mechanism for achieving both objectives is feedback control, because it is the simplest and most universally applicable approach to the problem. Closing the Loop The simplest representation of the closed feedback loop is shown in Fig. 8-23. The load is shown entering the process at the same point as the manipulated variable because that is the most common point of entry, and because, lacking better information, the elements in the path of the manipulated variable are the best estimates of those in the load path. The load rarely impacts directly on the controlled variable without passing through the dominant lag in the process. Where the load is unmeasured, its current value can be observed as the controller output required to keep the controlled variable Y at set point Ysp. If the loop is opened—either by placing the controller in manual operation or by setting its gains to zero—the load will have complete influence over the controlled variable, and the set point will have none. Only by closing the loop with controller gain as high as possible will the influence of the load be minimized and that of the set point be maximized. There is a practical limit to the controller gain, however, at the point where the controlled variable develops a uniform oscillation. This is defined as the limit of stability, and it is reached when the product of gains in the loop ΠG = GcGvGp is equal to 1.0 at the period of the oscillation. If the gain of any element in the loop increases from this condition, oscillations will expand, creating a dangerous situation where safe limits of operation could be exceeded in a few cycles. Consequently, control loops should be left in a condition where the loop gain is less than 1.0 by a safe margin that allows for possible variations in process parameters. Figure 8-24 describes a load response under PID (proportional-integral-derivative) control where the loop is well damped at a loop gain of 0.56; loop gain is then increased to 0.93 and to 1.05, creating a lightly damped and then an expanding cycle, respectively. In controller tuning, a choice must be made between performance and robustness. Performance is a measure of how well a given con-

troller with certain parameter settings regulates a variable, relative to the best response that can be achieved for that particular process. Robustness is a measure of how small a change in a process parameter is required to bring the loop from its current state to the limit of stability (ΠG = 1.0). The well-damped loop in Fig. 8-24 has a robustness of 79 percent, in that increasing the gain of any element in the loop by a factor of 1/0.56, or 1.79, would bring the loop to the limit of stability. Increasing controller performance by raising its gain can therefore be expected to decrease robustness. Both performance and robustness are functions of the dynamics of the process being controlled, the selection of the controller, and the tuning of the controller parameters. On/Off Control An on/off controller is used for manipulated variables having only two states. They commonly control temperatures in homes, electric water heaters and refrigerators, and pressure and liquid level in pumped storage systems. On/off control is satisfactory where slow cycling is acceptable, because it always leads to cycling when the load lies between the two states of the manipulated variable. The cycle will be positioned symmetrically about the set point only if the load happens to be equidistant between the two states of the manipulated variable. The period of the symmetric cycle will be approximately 4θ, where θ is the dead time in the loop. If the load is not centered between the states of the manipulated variable, the period will tend to increase and the cycle will follow a sawtooth pattern. Every on/off controller has some degree of dead band, also known as lockup, or differential gap. Its function is to prevent erratic switching between states, thereby extending the life of contacts and motors. Instead of changing states precisely when the controlled variable crosses the set point, the controller will change states at two different points for increasing and decreasing signals. The difference between these two switching points is the dead band (see Fig. 8-25); it increases the amplitude and period of the cycle, similar to the effects of dead time. A three-state controller is used to drive either a pair of independent two-state actuators, such as heating and cooling valves, or a bidirectional motorized actuator. The controller is comprised of two on/off controllers, each with dead band, separated by a dead zone. While the controlled variable lies within the dead zone, neither output is energized. This controller can drive a motorized valve to the point where the manipulated variable matches the load, thereby avoiding cycling.

8-14

PROCESS CONTROL

FIG. 8-22

Rectangular pulse response and PRBS testing.

Proportional Control A proportional controller moves its output proportional to the deviation e between the controlled variable y and its set point ysp: 100 u = Kc e + b =  e + b P

(8-28)

where e = ±(y − ysp), the sign selected to produce negative feedback. In some controllers, proportional gain Kc is introduced as a pure number; in others, it is set as 100/P, where P is the proportional band in percent. The output bias b of the controller is also known as manual reset. The proportional controller is not a good regulator, because any change in output required to respond to a change in load results in a corresponding change in the controlled variable. To minimize the resulting offset, the bias

should be set at the best estimate of the load, and the proportional band set as low as possible. Processes requiring a proportional band of more than a few percent may control with unacceptably large values of offset. Proportional control is most often used to regulate liquid level, where variations in the controlled variable carry no economic penalty and where other control modes can easily destabilize the loop. It is actually recommended for controlling the level in a surge tank when manipulating the flow of feed to a critical downstream process. By setting the proportional band just under 100 percent, the level is allowed to vary over the full range of the tank capacity as inflow fluctuates, thereby minimizing the resulting rate of change of manipulated outflow. This technique is called averaging level control. Proportional-plus-Integral (PI) Control Integral action eliminates the offset described above by moving the controller output at a

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL

FIG. 8-23

8-15

Both load regulation and set-point response require high gains for the feedback controller.

rate proportional to the deviation from set point—the output will then not stop moving until the deviation is zero. Although available alone in an integral controller, it is most often combined with proportional action in a PI controller:





100 1 u =  e +  edt + C0 P τI

(8-29)

where τI is the integral time constant in minutes; in some controllers it is introduced as integral gain or reset rate 1τI in repeats per minute. The last term in the equation is the constant of integration, the value of the controller output when integration begins. The PI controller is by far the most commonly used controller in the process industries. Because the integral term lags the proportional term by 90° in phase, the PI controller then always produces a phase lag between 0° and 90°: τo φPI = −tan  2πτI −1



1 τD,eff =  1τD + 1τI τD 100 Kc =  1 +  τI P



Set

0.93

Time FIG. 8-24

(8-32)



The performance of the interacting controller is almost as high as that of the noninteracting controller on most processes, but the tuning rules differ because of the above relationships. Both controllers are in common use in digital systems. There is always a gain limit placed upon the derivative vector—a value of 10 is typical. However, interaction decreases the derivative

1.05

increases.

(8-31)

τI,eff = τI + τD

∏G = 0.56

Controlled variable



where τD is the derivative time constant. Note that derivative action is applied to the controlled variable rather than to the deviation, as it should not be applied to the set point; the selection of the sign for the derivative term must be consistent with the action of the controller. In some PID controllers, the integral and derivative terms are combined serially rather than in parallel, as done in the last equation. This results in interaction between these modes, such that the effective values of the controller parameters differ from their set values as follows:

(8-30)

where τo is the period of oscillation of the loop. The phase angle should be kept between 15° for lag-dominant processes and 45° for dead-time-dominant processes for optimum results. Proportional-plus-Integral-plus-Derivative (PID) Control The derivative mode moves the controller output as a function of the rate of change of the controlled variable, which adds phase lead to the controller, increasing its speed of response. It is normally combined with proportional and integral modes. The noninteracting or ideal form of the PID controller appears functionally as



100 dy 1 u =  e +  e dt ± τD  + C0 P τI dt

Transition from well-damped load response to instability develops as loop gain

8-16

PROCESS CONTROL

FIG. 8-25

On/off controller characteristics.

gain below this value by the factor 1 + τDτI, which is the reason for the decreased performance of the interacting PID controller. Sampling in a digital controller has a similar effect, limiting derivative gain to the ratio of derivative time to the sample interval of the controller. Noise on the controlled variable is amplified by the derivative gain, preventing its use in controlling flow and liquid level. Derivative action is recommended for control of temperature and composition in multiple-capacity processes with little measurement noise. Controller Comparison Figure 8-26 compares the step load response of a distributed lag without control, and with P, PI, and interacting PID control. A distributed lag is a process whose resistance and capacity are distributed throughout its length—a heat exchanger is characteristic of this class, its heat-transfer surface and heat capacity being uniformly distributed. Other examples include imperfectly stirred tanks and distillation columns—both trayed and packed. The signature of a distributed lag is its open-loop (uncontrolled) step response, featuring a relatively short dead time followed by a dominant lag called Στ, which is the time required to reach 63.2 percent complete response. The proportional controller is unable to return the controlled variable to the set point following the step load change, as a deviation is required to sustain its output at a value different from its fixed bias b. The amount of proportional offset produced as a fraction of the uncontrolled offset is 1/(1 + KKc), where K is the steady-state process

FIG. 8-26

gain—in Fig. 8-26, that fraction is 0.13. Increasing Kc can reduce the offset, but with an accompanying loss in damping. The PI and PID controller were tuned to produce a minimum integrated absolute error (IAE). Their response curves are similar in appearance to a gaussian distribution curve, but with a damped cycle in the trailing edge. The peak deviation of the PID response curve is only 0.12 times the uncontrolled offset, occurring at 0.36Στ; the peak deviation of the PI response curve is 0.21 times the uncontrolled offset, occurring at 0.48Στ. These values can be used to predict the load response of any distributed lag whose parameters K and Στ are known or can be estimated as described below. CONTROLLER TUNING The performance of a controller depends as much on its tuning as on its design. Tuning must be applied by the end user to fit the controller to the controlled process. There are many different approaches to controller tuning, based on the particular performance criteria selected, whether load or set-point changes are more important, whether the process is lag- or dead-time-dominant, and the availability of information about the process dynamics. The earliest definitive work in this field was done at the Taylor Instrument Company by Ziegler and Nichols (Trans. ASME, p. 759, 1942), tuning PI and interacting PID

Minimum-IAE tuning gives very satisfactory load response for a distributed lag.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL controllers for optimum response to step load changes applied to lagdominant processes. While these tuning rules are still in use, they do not apply to set-point changes, dead-time-dominant processes, or noninteracting PID controllers (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004). Controller Performance Criteria The most useful measures of controller performance in an industrial setting are the maximum deviation in the controlled variable resulting from a disturbance, and its integral. The disturbance could be to the set point or to the load, depending on the variable being controlled and its context in the process. The size of the deviation and its integral are proportional to the size of the disturbance (if the loop is linear at the operating point). While actual disturbances arising in a plant may appear to be random, the controller needs a reproducible test to determine how well it is tuned. The disturbance of choice for test purposes is the step, because it can be applied manually, and by containing all frequencies including zero it exercises all modes of the controller. (The step actually has the same frequency distribution as integrated white noise, a “random walk.”) When tuned optimally for step disturbances, the controller will be optimally tuned for most other disturbances as well. A step change in set point, however, may be a poor indicator of a loop’s load response. For example, a liquid-level controller does not have to integrate to follow a set-point change, as its steady-state output is independent of the set point. Stepping a flow controller’s set point is

an effective test of its tuning, however, as its steady-state output is proportional to its set point. Other loops should be load-tested: simulate a load change from a steady state at zero deviation by transferring the controller to manual and stepping its output, and then immediately transferring back to automatic before a deviation develops. Figure 8-27a and b shows variations in the response of a distributed lag to a step change in load for different combinations of proportional and integral settings of a PI controller. The maximum deviation is the most important criterion for variables that could exceed safe operating levels, such as steam pressure, drum level, and steam temperature in a boiler. The same rule can apply to product quality if violating specifications causes it to be rejected. However, if the product can be accumulated in a downstream storage tank, its average quality is more important, and this is a function of the deviation integrated over the residence time of the tank. Deviation in the other direction, where the product is better than specification, is safe but increases production costs in proportion to the integrated deviation because quality is given away. For a PI or PID controller, the integrated deviation—better known as integrated error IE—is related to the controller settings PτI IE = ∆u  100

Poptx3

Poptx2 Controlled variable

Popt

Set

Popt/1.5

0

1

2

3

4

Time, t/ ∑τ

(a)

τI optx2

Controlled variable

τI optx3

τI opt

Set I opt/1.5

0

1

2

3

4

Time, t/ ∑τ

(b) The optimum settings produce minimum-IAE load response. (a) The proportional band primarily affects damping and peak deviation. (b) Integral time determines overshoot.

FIG. 8-27

8-17

(8-33)

8-18

PROCESS CONTROL

where ∆u is the difference in controller outputs between two steady states, as required by a change in load or set point. The proportional band P and integral time τI are the indicated settings of the controller for PI and both interacting and noninteracting PID controllers. Although the derivative term does not appear in the relationship, its use typically allows a 50 percent reduction in integral time and therefore in IE. The integral time in the IE expression should be augmented by the sample interval if the controller is digital, the time constant of any filter used, and the value of any dead-time compensator. It would appear, from the above, that minimizing IE is simply a matter of minimizing the P and τI settings of the controller. However, settings will be reached that produce excessive oscillations, such as shown in the lowest two response curves in Fig. 8-27a and b. It is preferable instead to find a combination of controller settings that minimizes integrated absolute error IAE, which for both load and setpoint changes is a well-damped response with minimal overshoot. The curves designated Popt and τI,opt in Fig. 8-27 are the same minimumIAE response to a step change in load for a distributed-lag process under PI control. Because of the very small overshoot, the IAE will be only slightly larger than the IE. Loops that are tuned to minimize IAE tend to give responses that are close to minimum IE and with minimum peak deviation. The other curves in Fig. 8-27a and b describe the effects of individual adjustments to P and τI, respectively, around those optimum values and can serve as a guide to fine-tuning a PI controller. The performance of a controller (and its tuning) must be based on what is achievable for a given process. The concept of best practical IE (IEb) for a step change in load ∆L to a process consisting of dead time and one or two lags can be estimated (Shinskey, Process Control Systems, 4th ed., McGraw-Hill, New York, 1996) IEb = ∆LKLτL(1 − e−θτ ) L

(8-34)

where KL is the gain and τL the primary time constant in the load path, and θ the dead time in the manipulated path to the controlled variable. If the load or its gain is unknown, ∆u and K (= KvKp) may be substituted. If the process is non-self-regulating (i.e., an integrator), the relationship is ∆Lθ2 IEb =  τ1

(8-35)

where τ1 is the time constant of the process integrator. The peak deviation of the best practical response curve is IEb eb =  θ + τ2

(8-36)

where τ2 is the time constant of a common secondary lag (e.g., in the measuring device).

TABLE 8-2 Tuning Rules Using Known Process Parameters Process Dead-time-dominant Lag-dominant

Non-self-regulating

Distributed lags

NOTE:

P 250K

τI 0.5θ

PI PIDn PIDi

106Kθτm 77Kθτm 106Kθτm

4.0θ 1.8θ 1.5θ

0.45θ 0.55θ

PI PIDn PIDi

106θτ1 78θτ1 108θτ1

4.0θ 1.9θ 1.6θ

0.48θ 0.58θ

PI PIDn PIDi

20K 10K 15K

0.50 Στ 0.30 Στ 0.25 Στ

0.09 Στ 0.10Στ

Controller PI

τD

n = noninteracting, i = interacting controller modes.

The performance of any controller can be measured against this standard by comparing the IE it achieves in responding to a load change with the best practical IE. Maximum performance levels for PI controllers on lag-dominant processes lie in the 20 to 30 percent range, while for PID controllers they fall between 40 and 60 percent, varying with secondary lags. Tuning Methods Based on Known Process Models The most accurate tuning rules for controllers have been based on simulation, where the process parameters can be specified and IAE and IE can be integrated during the simulation as an indication of performance. Controller settings are then iterated until a minimum IAE is reached for a given disturbance. Next these optimum settings are related to the parameters of the simulated process in tables, graphs, or equations, as a guide to tuning controllers for processes whose parameters are known (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004). This is a multidimensional problem, however, in that the relationships change as a function of process type, controller type, and source of disturbance. Table 8-2 summarizes these rules for minimum-IAE load response for the most common controllers. The process gain and time constant τm are obtained from the product of Gv and Gp in Fig. 8-23. Derivative action is not effective for dead-time-dominant processes. Any secondary lag, sampling interval, or filter time constant should be added to dead time θ. The principal limitation to using these rules is that the true process parameters are often unknown. Steady-state gain K can be calculated from a process model, or determined from the steady-state results of a step test as ∆c∆u, as shown in Fig. 8-28. The test will not be viable, however if the time constant of the process τm is longer than a few minutes, since five time constants must elapse to approach a steady state within 1 percent, and unexpected disturbances may intervene. Estimated dead time θ is the time from the step to the intercept of a

FIG. 8-28 If a steady state can be reached, gain K and time constant τ can be estimated from a step response; if not, use τ1 instead.

FUNDAMENTALS OF PROCESS DYNAMICS AND CONTROL TABLE 8-3 Tuning Rules Using Slope and Intercept Controller

P

τI

PI PIDn PIDi

150θτ1 75θτ1 113θτ1

3.5θ 2.1θ 1.8θ

NOTE:

TABLE 8-4 Tuning Rules Using Proportional Cycle τD

Controller

P

τI

τD

0.63θ 0.70θ

PI PIDn PIDi

1.70Pu 1.30Pu 1.80Pu

0.81τn 0.48τn 0.38τn

0.11τn 0.14τn

n = noninteracting, i = interacting controller modes.

NOTE:

straight line tangent to the steepest part of the response curve. The estimated time constant τ is the time from that point to 63 percent of the complete response. In the presence of a significant secondary lag, these results will not be completely accurate, however. The time for 63 percent response may be more accurately calculated as the residence time of the process: its volume divided by current volumetric flow rate. Tuning Methods When Process Model Is Unknown Ziegler and Nichols developed two tuning methods for processes with unknown parameters. The open-loop method uses a step test without waiting for a steady state to be reached and is therefore applicable to very slow processes. Dead time is estimated from the intercept of the steepest tangent to the response curve in Fig. 8-28, whose slope is also used. If the process is non-self-regulating, the controlled variable will continue to follow this slope, changing by an amount equal to ∆u in a time equal to its time constant τ1. This time estimate τ1 is used along with θ to tune controllers according to Table 8-3, applicable to lag-dominant processes. If the process is known to be a distributed lag, such as a heat exchanger, distillation column, or stirred tank, then better results will be obtained by converting the estimated values of θ and τ1 to K and Στ and using Table 8-2. The conversion factors are K = 7.5θτ1 and Στ = 7.0θ. The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control, by setting the integral time to maximum and derivative time to zero and reducing the proportional band until a constant-amplitude cycle results. The natural period τn of the cycle (the proportional controller contributes no phase shift to alter it) is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band Pu which was found to produce the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced by using on-off control to cycle the manipulated variable between two limits. The period of the cycle will be close to τn if the cycle is symmetric; the peak-to-peak ampli-

Set-point tuning

n = noninteracting, i = interacting controller modes.

tude Ac of the controlled variable divided by the difference between the output limits Au is a measure of process gain at that period and is therefore related to Pu for the proportional cycle: π A Pu = 100  c 4 Au

(8-37)

The factor π4 compensates for the square wave in the output. Tuning rules are given in Table 8-4. Set-Point Response All the above tuning methods are intended to minimize IAE for step load changes. When applied to lag-dominant processes, the resulting controller settings produce excessive overshoot of set-point changes. This behavior has led to the practice of tuning to optimize set-point response, which unfortunately degrades the load response of lag-dominant loops. An option has been available with some controllers to remove proportional action from set-point changes, which eliminates set-point overshoot but lengthens settling time. A preferred solution to this dilemma is available in many modern controllers which feature an independent gain adjustment for the set point, through which set-point response can be optimized after the controller has been tuned to optimize load response. Figure 8-29 shows set-point and load responses of a distributed lag for both set-point and load tuning, including the effects of fractional set-point gain Kr. The set point was stepped at time zero, and the load stepped at time 2.4. With full set-point gain, the PI controller was tuned for minimum-IAE set-point response with P = 29K and τI = Στ, compared to P = 20K and τI = 0.50Στ for minimum-IAE load response. These settings increase its IE for load response by a factor of 2.9, and its peak deviation by 20 percent, over optimum load tuning. However, with optimum load tuning, that same set-point overshoot can be obtained with set-point gain Kr = 0.54. The effects of full set-point gain (1.0) and no set-point gain (0) are shown for comparison.

Load tuning

Kr = 1.0

Controlled Set variable

0.54

0

0.00

0.80

1.60

2.40

3.20

4.00

4.80

Time, t/ Tuning proportional and integral settings to optimize set-point response degrades load response; using a separate set-point gain adjustment allows both responses to be optimized.

FIG. 8-29

8-19

8-20

PROCESS CONTROL

ADVANCED CONTROL SYSTEMS BENEFITS OF ADVANCED CONTROL The economics of most processes are determined by the steady-state operating conditions. Excursions from these steady-state conditions usually have a less important effect on the economics of the process, except when the excursions lead to off-specification products. To enhance the economic performance of a process, the steady-state operating conditions must be altered in a manner that leads to more efficient process operation. The hierarchy shown in Fig. 8-30 indicates that process control activities consist of the following five levels: Level 1: Measurement devices and actuators Level 2: Safety, environmental/equipment protection Level 3: Regulatory control Level 4: Real-time optimization Level 5: Planning and scheduling

Levels 4 and 5 clearly affect the process economics, as both levels are directed to optimizing the process in some manner. In contrast, levels 1, 2, and 3 would appear to have no effect on process economics. Their direct effect is indeed minimal, although indirectly they can have a major effect. Basically, these levels provide the foundation for all higher levels. A process cannot be optimized until it can be operated consistently at the prescribed targets. Thus, satisfactory regulatory control must be the first goal of any automation effort. In turn, the measurements and actuators provide the process interface for regulatory control. For most processes, the optimum operating point is determined by a constraint. The constraint might be a product specification (a product stream can contain no more than 2 percent ethane); violation of this constraint causes off-specification product. The constraint might be an equipment limit (vessel pressure rating is 300 psig); violation of this constraint causes the equipment protection mechanism (pressure

The five levels of process control and optimization in manufacturing. Time scales are shown for each level. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

FIG. 8-30

ADVANCED CONTROL SYSTEMS relief device) to activate. As the penalties are serious, violation of such constraints must be very infrequent. If the regulatory control system were perfect, the target could be set exactly equal to the constraint (i.e., the target for the pressure controller could be set at the vessel relief pressure). However, no regulatory control system is perfect. Therefore, the value specified for the target must be on the safe side of the constraint, thus allowing the control system some operating margin. How much depends on the following: 1. The performance of the control system (i.e., how effectively it responds to disturbances). The faster the control system reacts to a disturbance, the closer the process can be operated to the constraint. 2. The magnitude of the disturbances to which the control system must respond. If the magnitude of the major disturbances can be reduced, the process can be operated closer to the constraint. One measure of the performance of a control system is the variance of the controlled variable from the target. Both improving the control system and reducing the disturbances will lead to a lower variance in the controlled variable. In a few applications, improving the control system leads to a reduction in off-specification product and thus improved process economics. However, in most situations, the process is operated sufficiently far from the constraint that very little, if any, off-specification product results from control system deficiencies. Management often places considerable emphasis on avoiding off-specification production, so consequently the target is actually set far more conservatively than it should be. In most applications, simply improving the control system does not directly lead to improved process economics. Instead, the control system improvement must be accompanied by shifting the target closer to the constraint. There is always a cost of operating a process in a conservative manner. The cost may be a lower production rate, a lower process efficiency, a product giveaway, or other. When management places undue emphasis on avoiding off-specification production, the natural reaction is to operate very conservatively, thus incurring other costs. The immediate objective of an advanced control effort is to reduce the variance in an important controlled variable. However, this effort must be coupled with a commitment to adjust the target for this controlled variable so that the process is operated closer to the constraint. In large-throughput (commodity) processes, very small shifts in operating targets can lead to large economic returns.

ADVANCED CONTROL TECHNIQUES GENERAL REFERENCES: Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York. 2004. Stephanopoulos, Chemical Process Control: An Introduction to Theory and Practice, Prentice-Hall, Englewood Cliffs, N.J., 1984. Shinskey, Process Control Systems, 4th ed., McGraw-Hill, New York, 1996. Ogunnaike and Ray, Process Dynamics, Modeling, and Control, Oxford University Press, New York, 1994.

While the single-loop PID controller is satisfactory in many process applications, it does not perform well for processes with slow dynamics, time delays, frequent disturbances, or multivariable interactions. We discuss several advanced control methods below that can be implemented via computer control, namely, feedforward control, cascade control, time-delay compensation, selective and override control, adaptive control, fuzzy logic control, and statistical process control. Feedforward Control If the process exhibits slow dynamic response and disturbances are frequent, then the application of feedforward control may be advantageous. Feedforward (FF) control differs from feedback (FB) control in that the primary disturbance or load (D) is measured via a sensor and the manipulated variable (U) is adjusted so that deviations in the controlled variable from the set point are minimized or eliminated (see Fig. 8-31). By taking control action based on measured disturbances rather than controlled variable error, the controller can reject disturbances before they affect the controlled variable Y. To determine the appropriate settings for the manipulated variable, one must develop mathematical models that relate

FIG. 8-31

8-21

Simplified block diagrams for feedforward and feedback control.

1. The effect of the manipulated variable U on the controlled variable Y 2. The effect of the disturbance D on the controlled variable Y These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exact, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the ability to predict the appropriate control action). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standard practice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. FF control therefore attempts to eliminate the effects of measurable disturbances, while FB control would correct for unmeasurable disturbances and modeling errors. This latter case is often referred to as feedback trim. These controllers have become widely accepted in the chemical process industries since the 1960s. Design Based on Material and Energy Balances Consider a heat exchanger example (see Fig. 8-32) to illustrate the use of FF and FB control. The control objective is to maintain T2, the exit liquid temperature, at the desired value (or set point) T2sp despite variations in the inlet liquid flow rate F and inlet liquid temperature Tl. This is done by manipulating W, the steam flow rate. A feedback control scheme would entail measuring T2, comparing T2 to T2sp, and then adjusting W. A feedforward control scheme requires measuring F and Tl, and adjusting W (knowing T2sp), in order to control exit temperature T2. Figure 8-33a and b shows the control system diagrams for FB and FF control. A feedforward control algorithm can be designed for the heat exchanger in the following manner. Using a steady-state energy balance and assuming no heat loss from the heat exchanger, WH = FC(T2 − T1)

(8-38)

where H = latent heat of vaporization and C = specific heat of liquid C W =  F(T2 − T1) H

(8-39)

W = K1 F(T2 − T1)

(8-40)

or

8-22

PROCESS CONTROL TABLE 8-5 Relative Advantages and Disadvantages of Feedforward and Feedback Advantages

Disadvantages Feedforward

• Acts before the effect of a disturbance has been felt by the system • Is good for systems with large time constant or dead time • Does not introduce instability in the closed-loop response

• Requires direct measurement of all possible disturbances • Cannot cope with unmeasured disturbances • Is sensitive to process/model error Feedback

• Does not require identification and measurement of any disturbance for corrective action • Does not require an explicit process model • Is possible to design controller to be robust to process/model errors

• Control action not taken until the effect of the disturbance has been felt by the system • Is unsatisfactory for processes with large time constants and frequent disturbances • May cause instability in the closed-loop response

with CL K1 =  H

(8-41)

W = K1F(Tset − T1)

(8-42)

For disturbance rejection [D(s) ≠ 0] we require that Y(s) = 0, or zero error. Solving Eq. (8-43) for Gf gives −GL Gf =  GtGvGp

Replace T2 by T2sp

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties C and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect C or H). Therefore, Kl can be designated as an adjustable parameter that can be tuned. The use of a physical model for FF control is desirable because it provides a physical basis for the control law and gives an a priori estimate of what the tuning parameters should be. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and T2sp are multiplied]. Block Diagram Analysis One shortcoming of this feedforward design procedure is that it is based on the steady-state characteristics of the process and, as such, neglects process dynamics (i.e., how fast the controlled variable responds to changes in the load and manipulated variables). Thus, it is often necessary to include “dynamic compensation” in the feedforward controller. The most direct method of designing the FF dynamic compensator is to use a block diagram of a general process, as shown in Fig. 8-34, where Gt represents the disturbance transmitter, Gf is the feedforward controller, Gd relates the disturbance to the controlled variable, Gv is the valve, Gp is the process, Gm is the output transmitter, and Gc is the feedback controller. All blocks correspond to transfer functions (via Laplace transforms). Using block diagram algebra and Laplace transform variables, the controlled variable Y(s) is given by GtGfL(s) + GdD(s) Y(s) =  1 + GmGcGvGp

(8-44)

(a)

(8-43)

(b) (a) Feedback control of a heat exchanger. (b) Feedforward control of a heat exchanger.

FIG. 8-33 FIG. 8-32

A heat exchanger diagram.

ADVANCED CONTROL SYSTEMS

8-23

FIG. 8-34 A block diagram of a feedforward-feedback control system. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

Suppose the dynamics of Gd and Gp are first-order; in addition, assume that Gc = Kc and Gt = Kt (constant gains for simplicity). Kd Y(s) Gd(s) =  =  τds + 1 D(s)

(8-45)

Kp Y(s) Gp(s) =  =  τps + 1 U(s)

(8-46)

τ p s + 1 −Kd −K(τ p s + 1) Gf (s) =  .  =  τ d s + 1 Kp Kc Kt τds + 1

(8-47)

Using Eq. (8-44),

where K is the overall ratio of the gains in Eq. (8-47).

(a) FIG. 8-35

control.

The above FF controller can be implemented by using a digital computer. Figure 8-35a and b compares typical responses for PID FB control, steady-state FF control (s = 0), dynamic FF control, and combined FF/FB control. In practice, the engineer can tune K, τp, and τd in the field to improve the performance of the FF controller. The feedforward controller can also be simplified to provide steady-state feedforward control. This is done by setting s = 0 in Gf (s). This might be appropriate if there is uncertainty in the dynamic models for Gd and Gp. Other Considerations in Feedforward Control The tuning of feedforward and feedback control systems can be performed independently. In analyzing the block diagram in Fig. 8-34, note that Gf is chosen to cancel out the effects of the disturbance D(s), as long as there are no model errors. For the feedback loop, therefore, the effects of D(s) can also be ignored, which for the servo case is Y(s) GcGvGpKm  =  Ysp(s) 1 + GcGvGpGm

(8-48)

(b)

(a) Comparison of FF (steady-state model) and PID FB control for disturbance change. (b) Comparison of FF (dynamic model) and combined FF/FB

8-24

PROCESS CONTROL

FIG. 8-36

Cascade control of an exothermic chemical reactor. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

Note that the characteristic equation will be unchanged for the FF + FB system; hence system stability will be unaffected by the presence of the FF controller. In general, the tuning of the FF controller can be less conservative than for the case of FB alone, because smaller excursions from the set point will result. This in turn would make the dynamic model Y(s) more accurate. The tuning of the controller in the feedback loop can be theoretically performed independently of the feedforward loop (i.e., the feedforward loop does not introduce instability in the closed-loop response). For more information on feedforward/feedback control applications and design of such controllers, refer to the general references. Cascade Control One of the disadvantages of using conventional feedback control for processes with large time lags or delays is that disturbances are not recognized until after the controlled variable deviates from its set point. In these processes, correction by feedback control is generally slow and results in long-term deviation from the set point. One way to improve the dynamic response to load changes is by using a secondary measurement point and a secondary controller; the secondary measurement point is located so that it recognizes the upset condition before the primary controlled variable is affected. One such approach is called cascade control, which is routinely used in most modern computer control systems. Consider a chemical reactor, where reactor temperature is to be controlled by coolant flow to the jacket of the reactor. The reactor temperature can be influenced by changes in disturbance variables such as feed rate or feed temperature; a feedback controller could be employed to compensate for such disturbances by adjusting a valve on the coolant flow to the reactor jacket. However, suppose an increase occurs in the coolant temperature as a result of changes in the plant coolant system. This will cause a change in the reactor temperature measurement, although such a change will not occur quickly, and the corrective action taken by the controller will be delayed. Cascade control is one solution to this problem (see Fig. 8-36). Here the jacket temperature is measured, and an error signal is sent from this point to the coolant control valve; this reduces coolant flow,

maintaining the heat-transfer rate to the reactor at a constant level and rejecting the disturbance. The cascade control configuration will also adjust the setting of the coolant control valve when an error occurs in the reactor temperature. The cascade control scheme shown in Fig. 8-36 contains two controllers. The primary controller is the reactor temperature controller. It measures the reactor temperature, compares it to the set point, and computes an output, which is the set point for the coolant flow rate controller. The secondary controller compares this set point to the coolant temperature measurement and adjusts the valve. The principal advantage of cascade control is that the secondary measurement (jacket temperature) is located closer to a potential disturbance in order to improve the closed-loop response. Figure 8-37 shows the block diagram for a general cascade control system. In tuning of a cascade control system, the secondary controller (in the inner loop) is tuned first with the primary controller in manual. Often only a proportional controller is needed for the secondary loop, because offset in the secondary loop can be treated by using proportional-plus-integral action in the primary loop. When the primary controller is transferred to automatic, it can be tuned by using the techniques described earlier in this section. For more information on theoretical analysis of cascade control systems, see the general references for a discussion of applications of cascade control. Time-Delay Compensation Time delays are a common occurrence in the process industries because of the presence of recycle loops, fluid-flow distance lags, and dead time in composition measurements resulting from use of chromatographic analysis. The presence of a time delay in a process severely limits the performance of a conventional PID control system, reducing the stability margin of the closed-loop control system. Consequently, the controller gain must be reduced below that which could be used for a process without delay. Thus, the response of the closed-loop system will be sluggish compared to that of the system with no time delay. To improve the performance of time-delay systems, special control algorithms have been developed to provide time-delay compensation. The Smith predictor technique is the best-known algorithm; a related method is called the analytical predictor. Various investigators have

ADVANCED CONTROL SYSTEMS

8-25

FIG. 8-37 Block diagram of the cascade control system. For a chemical reactor Gd1 would correspond to a feed temperature or composition disturbance, while Gd2 would be a change in the cooling water temperature. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

found that, based on integral squared error, the performance of the Smith predictor can be better than that for a conventional controller, as long as the time delay is known accurately. The Smith predictor is a model-based control strategy that involves a more complicated block diagram than that for a conventional feedback controller, although a PID controller is still central to the control strategy (see Fig. 8-38). The key concept is based on better coordination of the timing of manipulated variable action. The loop configuration takes into account the fact that the current controlled variable measurement is not a result of the current manipulated variable action, but the value taken θ time units earlier. Time-delay compensation can yield excellent performance; however, if the process model parameters change (especially the time delay), the Smith predictor

FIG. 8-38

performance will deteriorate and is not recommended unless other precautions are taken. Selective and Override Control When there are more controlled variables than manipulated variables, a common solution to this problem is to use a selector to choose the appropriate process variable from among a number of available measurements. Selectors can be based on multiple measurement points, multiple final control elements, or multiple controllers, as discussed below. Selectors are used to improve the control system performance as well as to protect equipment from unsafe operating conditions. One type of selector device chooses as its output signal the highest (or lowest) of two or more input signals. This approach is often referred to as auctioneering. On instrumentation diagrams, the symbol

Block diagram of the Smith predictor. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

8-26

PROCESS CONTROL

> denotes a high selector and < a low selector. For example, a high selector can be used to determine the hot-spot temperature in a fixedbed chemical reactor. In this case, the output from the high selector is the input to the temperature controller. In an exothermic catalytic reaction, the process may run away due to disturbances or changes in the reactor. Immediate action should be taken to prevent a dangerous rise in temperature. Because a hot spot may potentially develop at one of several possible locations in the reactor, multiple (redundant) measurement points should be employed. This approach minimizes the time required to identify when a temperature has risen too high at some point in the bed. The use of high or low limits for process variables is another type of selective control, called an override. The feature of antireset windup in feedback controllers is a type of override. Another example is a distillation column with lower and upper limits on the heat input to the column reboiler. The minimum level ensures that liquid will remain on the trays, while the upper limit is determined by the onset of flooding. Overrides are also used in forced-draft combustion control systems to prevent an imbalance between airflow and fuel flow, which could result in unsafe operating conditions. Other types of selective systems employ multiple final control elements or multiple controllers. In some applications, several manipulated variables are used to control a single process variable (also called split-range control). Typical examples include the adjustment of both inflow and outflow from a chemical reactor to control reactor pressure or the use of both acid and base to control pH in wastewater treatment. In this approach, the selector chooses among several controller outputs which final control element should be adjusted. Adaptive Control Process control problems inevitably require online tuning of the controller constants to achieve a satisfactory degree of control. If the process operating conditions or the environment changes significantly, the controller may have to be retuned. If these changes occur quite frequently, then adaptive control techniques should be considered. An adaptive control system is one in which the controller parameters are adjusted automatically to compensate for changing process conditions. The subject of adaptive control is one of current interest. New algorithms continue to be developed, but these need to be field-tested before industrial acceptance can be expected. An adaptive controller is inherently nonlinear and therefore more complicated than the conventional PID controller. Fuzzy Logic Control The application of fuzzy logic to process control requires the concepts of fuzzy rules and fuzzy inference. A fuzzy rule, also known as a fuzzy IF-THEN statement, has the form If x then y if input1 = high

changing dynamic characteristics or exhibits nonlinearities, fuzzy logic control should offer a better alternative to using constant PID settings. Most fuzzy logic software begins building its information base during the autotune function. In fact, the majority of the information used in the early stages of system start-up comes from the autotune solutions. In addition to single-loop process controllers, products that have benefited from the implementation of fuzzy logic are camcorders, elevators, antilock braking systems, and televisions with automatic color, brightness, and sound control. Sometimes fuzzy logic controllers are combined with pattern recognition software such as artificial neural networks (Passino and Yurkovich, Fuzzy Control, Addison-Wesley, Reading, Mass., 1998; Kosko, Neural Networks and Fuzzy Systems, Prentice-Hall, Englewood Cliffs, N.J., 1992). EXPERT SYSTEMS An expert system is a computer program that uses an expert’s knowledge in a particular domain to solve a narrowly focused, complex problem. An offline system uses information entered manually and produces results in visual form to guide the user in solving the problem at hand. An online system uses information taken directly from process measurements to perform tasks automatically or instruct or alert operating personnel to the status of the plant. Each expert system has a rule base created by the expert to respond as the expert would to sets of input information. Expert systems used for plant diagnostics and management usually have an open rule base, which can be changed and augmented as more experience accumulates and more tasks are automated. The system begins as an empty shell with an assortment of functions such as equation solving, logic, and simulation, as well as input and display tools to allow an expert to construct a proprietary rule base. The “expert” in this case would be the person or persons having the deepest knowledge about the process, its problems, its symptoms, and remedies. Converting these inputs to meaningful outputs is the principal task in constructing a rule base. First-principles models (deep knowledge) produce the most accurate results, although heuristics are always required to establish limits. Often modeling tools such as artificial neural nets are used to develop relationships among the process variables. A number of process control vendors offer comprehensive, objectoriented software environments for building and deploying expert systems. Advantages of such software include transforming complex real-time data to useful information through knowledge-based reasoning and analysis, monitoring for potential problems before they adversely impact operations, diagnosing root causes of time-critical problems to speed up resolution, and recommending or taking corrective actions to help ensure successful recovery.

and input2 = low

MULTIVARIABLE CONTROL

then output = medium

GENERAL REFERENCES: Shinskey, Process Control Systems, 4th ed., McGrawHill, New York, 1996. Seborg, Edgar, and Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York, 2004. McAvoy, Interaction Analysis, lSA, Research Triangle Park, N.C., 1983.

Three functions are required to perform logical inferencing with fuzzy rules. The fuzzy AND is the product of a rule’s input membership values, generating a weight for the rule’s output. The fuzzy OR is a normalized sum of the weights assigned to each rule that contributes to a particular decision. The third function used is defuzzification, which generates a crisp final output. In one approach, the crisp output is the weighted average of the peak element values. With a single feedback control architecture, information that is readily available to the algorithm includes the error signal, difference between the process variable and the set-point variable, change in error from previous cycles to the current cycle, changes to the setpoint variable, change of the manipulated variable from cycle to cycle, and change in the process variable from past to present. In addition, multiple combinations of the system response data are available. As long as the irregularity lies in that dimension wherein fuzzy decisions are being based or associated, the result should be enhanced performance. This enhanced performance should be demonstrated in both the transient and steady-state response. If the system tends to have

Process control books and journal articles tend to emphasize problems with a single controlled variable. In contrast, many processes require multivariable control with many process variables to be controlled. In fact, for virtually any important industrial process, at least two variables must be controlled: product quality and throughput. In this section, strategies for multivariable control are considered. Three examples of simple multivariable control systems are shown in Fig. 8-39. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate wA or wB affects both of the controlled variables w and x. For the pH neutralization process in Fig. 8-39b, liquid level h and exit stream pH are to be controlled by adjusting the acid and base flow rates wA and wB. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit

ADVANCED CONTROL SYSTEMS strong process interactions. In contrast, the process interactions for the gas-liquid separator in Fig. 8-39c are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirect effect on one of the controlled variables, pressure P. Strong process interactions can cause serious problems if a conventional multiloop feedback control scheme (e.g., either PI or PID controllers) is employed. The process interactions can produce undesirable control loop interactions where the controllers fight each other. Also, it may be difficult to determine the best pairing of controlled and manipulated variables. For example, in the in-line blending process in Fig. 8-39a, should w be controlled with wA and x with wB, or vice versa? Control Strategies for Multivariable Control If a conventional multiloop control strategy performs poorly due to control loop interactions, a number of solutions are available: 1. Detune one or more of the control loops. 2. Choose different controlled or manipulated variables (or their pairings).

8-27

3. Use a decoupling control system. 4. Use a multivariable control scheme (e.g., model predictive control). Detuning a controller (e.g., using a smaller controller gain or a larger reset time) tends to reduce control loop interactions by sacrificing the performance for the detuned loops. This approach may be acceptable if some of the controlled variables are faster or less important than others. The selection of controlled and manipulated variables is of crucial importance in designing a control system. In particular, a judicious choice may significantly reduce control loop interactions. For the blending process in Fig. 8-39a, a straightforward control strategy would be to control x by adjusting wA, and w by adjusting wB. But physical intuition suggests that it would be better to control x by adjusting the ratio wA/(wA + wB) and to control product flow rate w by the sum wA + wB. Thus, the new manipulated variables would be U1 = wA/(wA + wB) and U2 = wA + wB. In this control scheme, U1 affects only x, and U2 affects only w. Thus, the control loop interactions have been eliminated. Similarly, for the pH neutralization process in Fig. 8-39b, the control loop interactions would be greatly reduced if pH were controlled by Ul = wA (wA + wB) and liquid level h were controlled by U2 = wA + wB. Decoupling Control Systems Decoupling control systems provide an alternative approach for reducing control loop interactions. The basic idea is to use additional controllers called decouplers to compensate for undesirable process interactions. As an illustrative example, consider the simplified block diagram for a representative decoupling control system shown in Fig. 8-40. The two controlled variables Yl and Y2 and two manipulated variables Ul and U2 are related by the four process transfer functions Gp11, Gp12, and so on. For example, Gp11 denotes the transfer function between U1 and Yl: Y1(s)  = Gp11(s) U1(s)

(8-49)

Figure 8-40 includes two conventional feedback controllers: Gc1 controls Y1 by manipulating U1, and Gc2 controls Y2 by manipulating U2. The output signals from the feedback controllers serve as input signals to the two decouplers T12 and T21. The block diagram is in a simplified form because the disturbance variables and transfer functions for the final control elements and sensors have been omitted. The function of the decouplers is to compensate for the undesirable process interactions represented by Gp12 and Gp21. Suppose that the process transfer functions are all known. Then the ideal design equations are

FIG. 8-39

Physical examples of multivariable control problems.

Gp12(s) T12(s) = −  Gp11(s)

(8-50)

Gp21(s) T21(s) = −  Gp22(s)

(8-51)

These decoupler design equations are very similar to the ones for feedforward control in an earlier section. In fact, decoupling can be interpreted as a type of feedforward control where the input signal is the output of a feedback controller rather than a measured disturbance variable. In principle, ideal decoupling eliminates control loop interactions and allows the closed-loop system to behave as a set of independent control loops. But in practice, this ideal behavior is not attained for a variety of reasons, including imperfect process models and the presence of saturation constraints on controller outputs and manipulated variables. Furthermore, the ideal decoupler design equations in (850) and (8-51) may not be physically realizable and thus would have to be approximated. In practice, other types of decouplers and decoupling control configurations have been employed. For example, in partial decoupling, only a single decoupler is employed (i.e., either T12 or T21 in Fig. 8-40

8-28

PROCESS CONTROL

FIG. 8-40

A decoupling control system.

is set equal to zero). This approach tends to be more robust than complete decoupling and is preferred when one of the controlled variables is more important than the other. Static decouplers can be used to reduce the steady-state interactions between control loops. They can be designed by replacing the transfer functions in Eqs. (8-50) and (851) with the corresponding steady-state gains K p12 T12(s) = −  K p11

(8-52)

K p21 T21(s) = −  K p22

(8-53)

The advantage of static decoupling is that less process information is required, namely, only steady-state gains. Nonlinear decouplers can be used when the process behavior is nonlinear. Pairing of Controlled and Manipulated Variables A key decision in multiloop control system design is the pairing of manipulated and controlled variables. This is referred to as the controller pairing problem. Suppose there are N controlled variables and N manipulated variables. Then N! distinct control configurations exist. For example, if N = 5, then there are 120 different multiloop control schemes. In practice, many would be rejected based on physical insight or previous experience. But a smaller number (say, 5 to 15) may appear to be feasible, and further analysis would be warranted. Thus, it is very useful to have a simple method for choosing the most promising control configuration. The most popular and widely used technique for determining the best controller pairing is the relative gain array (RGA) method (Bristol, “On a New Measure of Process Interaction,” IEEE Trans. Auto. Control, AC-11: 133, 1966). The RGA method provides two important items of information: 1. A measure of the degree of process interactions between the manipulated and controlled variables 2. A recommended controller pairing An important advantage of the RGA method is that it requires minimal process information, namely, steady-state gains. Another advantage is that the results are independent of both the physical units used and the scaling of the process variables. The chief disadvantage of the RGA method is that it neglects process dynamics, which can be an important factor in the pairing decision. Thus, the RGA analysis

should be supplemented with an evaluation of process dynamics. Although extensions of the RGA method that incorporate process dynamics have been reported, these extensions have not been widely applied. RGA Method for 2 × 2 Control Problems To illustrate the use of the RGA method, consider a control problem with two inputs and two outputs. The more general case of N × N control problems is considered elsewhere (McAvoy, Interaction Analysis, ISA, Research Triangle Park, N.C., 1983). As a starting point, it is assumed that a linear, steady-state process model in Eqs. (8-54) and (8-55) is available, where U1 and U2 are steady-state values of the manipulated inputs; Y1 and Y2 are steady-state values of the controlled outputs; and the K values are steady-state gains. The Y and U variables are deviation variables from nominal steady-state values. This process model could be obtained in a variety of ways, such as by linearizing a theoretical model or by calculating steady-state gains from experimental data or a steady-state simulation. Y1 = K11U1 + K12U2

(8-54)

Y2 = K21U1 + K22U2

(8-55)

By definition, the relative gain λij between the ith manipulated variable and the jth controlled variable is defined as open-loop gain between Yi and Uj λij =  closed-loop gain between Yi and Uj

(8-56)

where the open-loop gain is simply Kij from Eqs. (8-54) and (8-55). The closed-loop gain is defined to be the steady-state gain between Uj and Yi when the other control loop is closed and no offset occurs in the other controlled variable due to the presence of integral control action. The RGA for the 2 × 2 process is denoted by Λ=



λ11

λ12

λ21

λ22



(8-57)

The RGA has the important normalization property that the sum of the elements in each row and each column is exactly 1. Consequently, the RGA in Eq. (8-57) can be written as

ADVANCED CONTROL SYSTEMS



λ

1−λ

Λ= 1−λ

λ



(8-58)

where λ can be calculated from the following formula: 1 λ =  1 − K12K21/(K11K22)

(8-59)

Ideally, the relative gains that correspond to the proposed controller pairing should have a value of 1 because Eq. (8-56) implies that the open- and closed-loop gains are then identical. If a relative gain equals 1, the steady-state operation of this loop will not be affected when the other control loop is changed from manual to automatic, or vice versa. Consequently, the recommendation for the best controller pairing is to pair the controlled and manipulated variables so that the corresponding relative gains are positive and close to 1. RGA Example To illustrate use of the RGA method, consider the following steady-state version of a transfer function model for a pilot-scale, methanol-water distillation column (Wood and Berry, “Terminal Composition Control of a Binary Distillation Column,” Chem. Eng. Sci., 28: 1707, 1973): K11 = 12.8, K12 = −18.9, K21 = 6.6, and K22 = −19.4. It follows that λ = 2 and Λ=

 −1 2

−1 2



(8-60)

Thus it is concluded that the column is fairly interacting and the recommended controller pairing is to pair Y1 with U1 and Y2 with U2. MODEL PREDICTIVE CONTROL GENERAL REFERENCES: Qin and Badgwell, Control Eng. Practice, 11: 773, 2003. Rawlings, IEEE Control Systems Magazine, 20(3): 38, 2000). Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004. Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002. Seborg, Edgar, and Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York, 2004, Chap. 20.

The model-based control strategy that has been most widely applied in the process industries is model predictive control (MPC). It is a general method that is especially well suited for difficult multi-input, multioutput (MIMO) control problems where there are significant interactions between the manipulated inputs and the controlled outputs. Unlike other model-based control strategies, MPC can easily accommodate inequality constraints on input and output variables such as upper and lower limits and rate-of-change limits. A key feature of MPC is that future process behavior is predicted by using a dynamic model and available measurements. The controller outputs are calculated so as to minimize the difference between the predicted process response and the desired response. At each sampling instant, the control calculations are repeated and the predictions updated based on current measurements. In typical industrial applications, the set point and target values for the MPC calculations are updated by using online optimization based on a steady-state model of the process. The current widespread interest in MPC techniques was initiated by pioneering research performed by two industrial groups in the 1970s. Shell Oil (Houston, Tex.) reported its Dynamic Matrix Control (DMC) approach in 1979, while a similar technique, marketed as IDCOM, was published by a small French company ADERSA in 1978. Since then, there have been thousands of applications of these and related MPC techniques in oil refineries and petrochemical plants around the world. Thus, MPC has had a substantial impact and is currently the method of choice for difficult multivariable control problems in these industries. However, relatively few applications have been reported in other process industries, even though MPC is a very general approach that is not limited to a particular industry.

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Advantages and Disadvantages of MPC Model predictive control offers a number of important advantages in comparison with conventional multiloop PID control: 1. It is a general control strategy for MIMO processes with inequality constraints on input and output variables. 2. It can easily accommodate difficult or unusual dynamic behavior such as large time delays and inverse responses. 3. Because the control calculations are based on optimizing control system performance, MPC can be readily integrated with online optimization strategies to optimize plant performance. 4. The control strategy can be easily updated online to compensate for changes in process conditions, constraints, or performance criteria. But current versions of MPC have significant disadvantages in comparison with conventional multiloop control: 1. The MPC strategy is very different from conventional multiloop control strategies and thus initially unfamiliar to plant personnel. 2. The MPC calculations can be relatively complicated [e.g., solving a linear programming (LP) or quadratic programming (QP) problem at each sampling instant] and thus require a significant amount of computer resources and effort. These optimization strategies are described in the next section. 3. The development of a dynamic model from plant data is timeconsuming, typically requiring days, or even weeks, of around-theclock plant tests. 4. Because empirical models are generally used, they are valid only over the range of conditions considered during the plant tests. 5. Theoretical studies have demonstrated that MPC can perform poorly for some types of process disturbances, especially when output constraints are employed. Because MPC has been widely used and has had considerable impact, there is a broad consensus that its advantages far outweigh its disadvantages. Economic Incentives for Automation Projects Industrial applications of advanced process control strategies such as MPC are motivated by the need for improvements regarding safety, product quality, environmental standards, and economic operation of the process. One view of the economics incentives for advanced automation techniques is illustrated in Fig. 8-41. Distributed control systems (DCS) are widely used for data acquisition and conventional singleloop (PID) control. The addition of advanced regulatory control systems such as selective controls, gain scheduling, and time-delay compensation can provide benefits for a modest incremental cost. But

FIG. 8-41

Economic incentives for automation projects in the process industries.

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PROCESS CONTROL

experience has indicated that the major benefits can be obtained for relatively small incremental costs through a combination of MPC and online optimization. The results in Fig. 8-41 are shown qualitatively, rather than quantitatively, because the actual costs and benefits are application-dependent. A key reason why MPC has become a major commercial and technical success is that there are numerous vendors who are licensed to market MPC products and install them on a turnkey basis. Consequently, even medium-sized companies are able to take advantage of this new technology. Payout times of 3 to 12 months have been widely reported. Basic Features of MPC Model predictive control strategies have a number of distinguishing features: 1. A dynamic model of the process is used to predict the future outputs over a prediction horizon consisting of the next P sampling periods. 2. A reference trajectory is used to represent the desired output response over the prediction horizon. 3. Inequality constraints on the input and output variables can be included as an option. 4. At each sampling instant, a control policy consisting of the next M control moves is calculated. The control calculations are based on minimizing a quadratic or linear performance index over the prediction horizon while satisfying the constraints. 5. The performance index is expressed in terms of future control moves and the predicted deviations from the reference trajectory. 6. A receding horizon approach is employed. At each sampling instant, only the first control move (of the M moves that were calculated) is actually implemented. 7. Then the predictions and control calculations are repeated at the next sampling instant. These distinguishing features of MPC will now be described in greater detail. Dynamic Model A key feature of MPC is that a dynamic model of the process is used to predict future values of the controlled outputs. There is considerable flexibility concerning the choice of the dynamic model. For example, a physical model based on first principles (e.g., mass and energy balances) or an empirical model developed from data could be employed. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonlinear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical linear models expressed in either step response or impulse response form. For simplicity, we consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipulated variable u can be expressed as N−1

^ y(k + 1) = y0 + Si ∆u(k − i + 1) + SN u(k − N + 1)

(8-61)

FIG. 8-42

Step response for a unit step change in the input.

ber of “bump tests” are required to compensate for unanticipated disturbances, process nonlinearities, and noisy measurements. Horizons The step response model in Eq. (8-61) is equivalent to the following impulse response model: N

y^(k) = hi u(k − i) + y0

(8-63)

i=1

where the impulse response coefficients hi are related to the step response coefficients by hi = Si − Si−1. Step and impulse response models typically contain a large number of parameters because the model horizon N is usually quite large (30 < N < 120). In fact, these models are often referred to as nonparametric models. The receding horizon feature of MPC is shown in Fig. 8-43 with the current sampling instant denoted by k. Past and present input signals [u(i) for i ≤ k] are used to predict the output at the next P sampling instants [y(k + i) for i = 1, 2, . . . , P]. The control calculations are performed to generate an M-step control policy [u(k), u(k + 1), . . . , u(k + M − 1)], which optimizes a performance index. The first control action u(k) is implemented. Then at the next sampling instant k + 1, the prediction and control calculations are repeated in order to determine u(k + 1). In Fig. 8-43 the reference trajectory (or target) is considered to be constant. Other possibilities include a gradual or step set-point change that can be generated by online optimization. Performance Index The performance index for MPC applications is usually a linear or quadratic function of the predicted errors and calculated future control moves. For example, the following quadratic performance index has been widely used:

i=1

^ where y(k + 1) is the predicted value of y at the k + 1 sampling instant, u(k) is the value of the manipulated input at time k, and the model parameters Si are referred to as the step response coefficients. The initial value y0 is assumed to be known. The change in the manipulated input from one sampling instant to the next is denoted by

∆u(k) = u (k) − u(k − 1)

(8-62)

The step response model is also referred to as a discrete convolution model. In principle, the step response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-42. The step response coefficients Si are simply the values of the output variable at the sampling instants, after the initial value y0 has been subtracted. Theoretically, they can be determined from a single step response, but, in practice, a num-

P

M

i=1

i=1

min J = Qie2(k + i) + Ri ∆u2(k + i − 1)

∆u(k)

(8-64)

The value e(k + i) denotes the predicted error at time k + i, e(k + i) = ysp(k + i) − yˆ (k + i)

(8-65)

where ysp(k + i) is the set point at time k + i and ∆u(k) denotes the column vector of current and future control moves over the next M sampling instants: ∆u(k) = [∆u(k), ∆u(k + 1), . . . , ∆u(k + M − 1)]T

(8-66)

Equation (8-64) contains two types of design parameters that can also be used for tuning purposes. Weighting factor Ri penalizes large control moves, while weighting factor Qi allows the predicted errors to be weighed differently at each time step, if desired.

ADVANCED CONTROL SYSTEMS

8-31

The “moving horizon” approach of model predictive control. (Seborg, Edgar, and Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

FIG. 8-43

Inequality Constraints Inequality constraints on the future inputs or their rates of change are widely used in the MPC calculations. For example, if both upper and lower limits are required, the constraints could be expressed as u− (k) ≤ u(k + j) ≤ u+ (k) ∆u− (k) ≤ ∆u(k + j) ≤ ∆u+ (k)

for j = 0, 1, . . . , M − 1

(8-67)

for j = 0, 1, . . . , M − 1 (8-68)

where Bi and Ci are constants. Constraints on the predicted outputs are sometimes included as well: y− (k) ≤ y^(k + j) ≤ y+ (k)

for j = 0, 1, . . . , P

(8-69)

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004; Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. Implementation of MPC For a new MPC application, a cost/benefit analysis is usually performed prior to project approval. Then the steps involved in the implementation of MPC can be summarized as follows (Hokanson and Gerstle, Dynamic Matrix Control Multivariable Controllers, in Practical Distillation Control, Luyben (ed.), Van Nostrand Reinhold, New York, 1992, p. 248; Qin and Badgwell, Control Eng. Practice, 11: 773, 2003). Step 1: Initial Controller Design The first step in MPC design is to select the controlled, manipulated, and measured disturbance variables. These choices determine the structure of the MPC system and should be based on process knowledge and control objectives. In typical

applications the number of controlled variables ranges from 5 to 40, and the number of manipulated variables is typically between 5 and 20. Step 2: Pretest Activity During the pretest activity, the plant instrumentation is checked to ensure that it is working properly and to decide whether additional sensors should be installed. The pretest data can be used to estimate the steady-state gain and approximate settling times for each input/output pair. This information is used to plan the full plant tests of step 3. As part of the pretest, it is desirable to benchmark the performance of the existing control system for later comparison with MPC performance (step 8). Step 3: Plant Tests The dynamic model for the MPC calculations is developed from data collected during special plant tests. The excitation for the plant tests usually consists of changing an input variable or a disturbance variable (if possible) from one value to another, using either a series of step changes with different durations or a pseudorandom binary sequence (PRBS). To ensure that sufficient data are obtained for model identification, each input variable is typically moved to a new value, 8 to 15 times during a plant test (Qin and Badgwell, Control Eng. Practice, 11: 773, 2003). Identification during closed-loop operation is becoming more common. Step 4: Model Development The dynamic model is developed from the plant test data by selecting a model form (e.g., a step response model) and then estimating the model parameters. However, first it is important to eliminate periods of test data where plant upsets or other abnormal situations have occurred. Decisions to omit portions of the test data are based on visual inspection of the data, knowledge of the process, and experience. Parameter estimation is usually based on least squares estimation. Step 5: Control System Design and Simulation The preliminary control system design from step 1 is critically evaluated and modified, if necessary. Then the MPC design parameters are selected including the sampling periods, weighting factors, and control and prediction horizons. Next, the closed-loop system is simulated, and the MPC design parameters are adjusted, if necessary, to obtain satisfactory control system performance and robustness over the specified range of operating conditions.

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PROCESS CONTROL

Step 6: Operator Interface Design and Operator Training Operator training is important because MPC concepts such as predictive control, multivariable interactions, and constraint handling are very different from conventional regulatory control concepts. Thus, understanding why the MPC system responds the way that it does, especially for unusual operating conditions, can be a challenge for both operators and engineers. Step 7: Installation and Commissioning After an MPC control system is installed, it is first evaluated in a “prediction mode.” Model predictions are compared with measurements, but the process continues to be controlled by the existing control system. After the output predictions are judged to be satisfactory, the calculated MPC control moves are evaluated to determine if they are reasonable. Finally, the MPC software is evaluated during closed-loop operation with the calculated control moves implemented as set points to the DCS control loops. The MPC design parameters are tuned, if necessary. The commissioning period typically requires some troubleshooting and can take as long as, or even longer than, the plant tests of step 3. Step 8: Measuring Results and Monitoring Performance The evaluation of MPC system performance is not easy, and widely accepted metrics and monitoring strategies are not available. However, useful diagnostic information is provided by basic statistics such as the means and standard deviations for both measured variables and calculated quantities, such as control errors and model residuals. Another useful statistic is the relative amount of time that an input is saturated or a constraint is violated, expressed as a percentage of the total time the MPC system is in service. Integration of MPC and Online Optimization As indicated in Fig. 8-41, significant potential benefits can be realized by using a combination of MPC and online optimization. At present, most commercial MPC packages integrate the two methodologies in a hierarchical configuration such as the one shown in Fig. 8-44. The MPC calcula-

FIG. 8-44

Hierarchical control configuration for MPC and online optimization.

tions are performed quite often (e.g., every 1 to 10 min) and implemented as set points for PID control loops at the DCS level. The targets and constraints for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model. These calculations may be performed as often as the MPC calculations. As an option, the targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. These calculations tend to be performed less frequently (e.g., every 1 to 24 h) due to the complexity of the calculations and the process models. The combination of MPC and frequent online optimization has been successfully applied in oil refineries and petrochemical plants around the world. REAL-TIME PROCESS OPTIMIZATION GENERAL REFERENCES: Biegler, Grossmann, and Westerberg, Systematic Methods of Chemical Process Design, Prentice-Hall, Upper Saddle River, N.J., 1997. Darby and White, On-line Optimization of Complex Process Units, Chem. Engr. Prog., 84(10): 51, 1998. Edgar, Himmelblau, and Lasdon, Optimization of Chemical Processes, 2d ed., McGraw-Hill, New York, 2001. Forbes, Marlin, and MacGregor, Model Selection Criteria for Economics-Based Optimizing Control, Comp. Chem. Engng., 18: 497, 1994; Marlin and Hrymak, RealTime Optimization of Continuous Processes, Chem. Proc. Cont V, AIChE Symp. Ser., 93(316): 156, 1997. Narashimhan and Jordache, Data Reconciliation and Gross Error Detection, Gulf Publishing, Houston, Tex., 2000. Nash and Sofer, Linear and Nonlinear Programming, McGraw-Hill, New York, 1996. Shobrys and White, Planning, Scheduling, and Control Systems: Why They Cannot Work Together, Comp. Chem. Engng., 26: 149, 2002. Timmons, Jackson, and White, Distinguishing On-line Optimization Benefits from Those of Advanced Controls, Hydrocarb Proc., 79(6): 69, 2000.

The chemical industry has undergone significant changes during the past 20 years due to the increased cost of energy and raw materials, more stringent environmental regulations, and intense worldwide competition. Modifications of both plant design procedures and plant operating conditions have been implemented to reduce costs and meet constraints. One of the most important engineering tools that can be employed in such activities is optimization. As plant computers have become more powerful, the size and complexity of problems that can be solved by optimization techniques have correspondingly expanded. A wide variety of problems in the operation and analysis of chemical plants (as well as many other industrial processes) can be solved by optimization. Real-time optimization means that the process operating conditions (set points) are evaluated on a regular basis and optimized, as shown earlier in level 4 in Fig. 8-28. Sometimes this is called steady-state optimization or supervisory control. This section examines the basic characteristics of optimization problems and their solution techniques and describes some representative benefits and applications in the chemical and petroleum industries. Typical problems in chemical engineering process design or plant operation have many possible solutions. Optimization is concerned with selecting the best among the entire set of solutions by efficient quantitative methods. Computers and associated software make the computations involved in the selection manageable and cost-effective. Engineers work to improve the initial design of equipment and strive for enhancements in the operation of the equipment once it is installed in order to realize the greatest production, the greatest profit, the maximum cost, the least energy usage, and so on. In plant operations, benefits arise from improved plant performance, such as improved yields of valuable products (or reduced yields of contaminants), reduced energy consumption, higher processing rates, and longer times between shutdowns. Optimization can also lead to reduced maintenance costs, less equipment wear, and better staff utilization. It is helpful to systematically identify the objective, constraints, and degrees of freedom in a process or a plant if such benefits as improved quality of designs, faster and more reliable troubleshooting, and faster decision making are to be achieved. Optimization can take place at many levels in a company, ranging from a complex combination of plants and distribution facilities down through individual plants, combinations of units, individual pieces of equipment, subsystems in a piece of equipment, or even smaller entities.

ADVANCED CONTROL SYSTEMS Problems that can be solved by optimization can be found at all these levels. While process design and equipment specification are usually performed prior to the implementation of the process, optimization of operating conditions is carried out monthly, weekly, daily, hourly, or even every minute. Optimization of plant operations determines the set points for each unit at the temperatures, pressures, and flow rates that are the best in some sense. For example, the selection of the percentage of excess air in a process heater is quite critical and involves a balance on the fuel/air ratio to ensure complete combustion and at the same time make the maximum use of the heating potential of the fuel. Typical day-to-day optimization in a plant minimizes steam consumption or cooling water consumption, optimizes the reflux ratio in a distillation column, or allocates raw materials on an economic basis [Latour, Hydro. Proc., 58(6): 73, 1979, and 58(7): 219, 1979]. A real-time optimization (RTO) system determines set-point changes and implements them via the computer control system without intervention from unit operators. The RTO system completes all data transfer, optimization calculations, and set-point implementation before unit conditions change that may invalidate the computed optimum. In addition, the RTO system should perform all tasks without upsetting plant operations. Several steps are necessary for implementation of RTO, including determination of the plant steady state, data gathering and validation, updating of model parameters (if necessary) to match current operations, calculation of the new (optimized) set points, and the implementation of these set points. To determine if a process unit is at steady state, a program monitors key plant measurements (e.g., compositions, product rates, feed rates, and so on) and determines if the plant is close enough to steady state to start the sequence. Only when all the key measurements are within the allowable tolerances is the plant considered steady and the optimization sequence started. Tolerances for each measurement can be tuned separately. Measured data are then collected by the optimization computer. The optimization system runs a program to screen the measurements for unreasonable data (gross error detection). This validity checking automatically modifies the model updating calculation to reflect any bad data or when equipment is taken out of service. Data validation and reconciliation (online or offline) is an extremely critical part of any optimization system. The optimization system then may run a parameter-fitting case that updates model parameters to match current plant operation. The integrated process model calculates such items as exchanger heattransfer coefficients, reactor performance parameters, furnace efficiencies, and heat and material balances for the entire plant. Parameter fitting allows for continual updating of the model to account for plant deviations and degradation of process equipment. After completion of the parameter fitting, the information regarding the current plant constraints, control status data, and economic values for feed products, utilities, and other operating costs is collected. The economic values are updated by the planning and scheduling department on a regular basis. The optimization system then calculates the optimized set points. The steady-state condition of the plant is rechecked after the optimization case is successfully completed. If the plant is still steady, then the values of the optimization targets are transferred to the process control system for implementation. After a line-out period, the process control computer resumes the steadystate detection calculations, restarting the cycle. Essential Features of Optimization Problems The solution of optimization problems involves the use of various tools of mathematics, which is discussed in detail in Sec. 3. The formulation of an optimization problem requires the use of mathematical expressions. From a practical viewpoint, it is important to mesh properly the problem statement with the anticipated solution technique. Every optimization problem contains three essential categories: 1. An objective function to be optimized (revenue function, cost function, etc.) 2. Equality constraints (equations) 3. Inequality constraints (inequalities) Categories 2 and 3 comprise the model of the process or equipment; category 1 is sometimes called the economic model.

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TABLE 8-6 Six Steps Used to Solve Optimization Problems 1 Analyze the process itself so that the process variables and specific characteristics of interest are defined (i.e., make a list of all the variables). 2 Determine the criterion for optimization, and specify the objective function in terms of the above variables together with coefficients. This step provides the performance model (sometimes called the economic model, when appropriate). 3 Develop via mathematical expressions a valid process or equipment model that relates the input/output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 4 If the problem formulation is too large in scope, (a) break it up into manageable parts and/or (b) simplify the objective function and model. 5 Apply a suitable optimization technique to the mathematical statement of the problem. 6 Check the answers and examine the sensitivity of the result to changes in the coefficients in the problem and the assumptions.

No single method or algorithm of optimization exists that can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on (1) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar, Himmelblau, and Lasdon, Optimization of Chemical Processes, 2d ed., McGraw-Hill, New York, 2001). You do not have to follow the cited order exactly, but you should cover all the steps at some level of detail. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps 1, 2, and 3 deal with the mathematical definition of the problem: identification of variables, specification of the objective function, and statement of the constraints. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objective function (the economic model) in typical RTO applications. Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust computational methods for optimization. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique can depend on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. Development of Process (Mathematical) Models Constraints in optimization problems arise from physical bounds on the variables, empirical relations, physical laws, and so on. The mathematical relations describing the process also comprise constraints. Two general categories of models exist: 1. Those based on physical theory 2. Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input/output data without any physiochemical analysis of the process. For these models, optimization is often used to fit a model to process data, using a procedure called parameter estimation. The well-known least squares curve-fitting procedure is based on optimization theory, assuming that the model parameters are contained linearly in the model. One example is the yield matrix, where the percentage yield of each product in a unit operation is estimated for each feed component

8-34

PROCESS CONTROL

by using process data rather than employing a mechanistic set of chemical reactions. Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective to mathematical terms. In the chemical process industries, the objective function often is expressed in units of currency per unit time (e.g., U.S. dollars per week, month, or year) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. A typical economic model involves the costs of raw materials, values of products, and costs of production as functions of operating conditions, projected sales figures, and the like. An objective function can be expressed in terms of these quantities; e.g., annual operating profit ($/yr) might be expressed as J = FsVs − FrCr − OC N

(8-70)

r

J = profit/time

where

s FsVs = sum of product flow rates times respective product values (income)

r F C r

r

= sum of feed flows times respective unit costs

OC = operating costs/time Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, the availability of efficient unconstrained optimization techniques is important because these techniques must be applied in real time, and iterative calculations may require excessive computer time. Two classes of unconstrained techniques are single-variable optimization and multivariable optimization. Single-Variable Optimization Many real-time optimization problems can be reduced to the variation of a single-variable so as to maximize profit or some other overall process objective function. Some examples of single-variable optimization include optimizing the reflux ratio in a distillation column or the air/fuel ratio in a furnace. While most processes actually are multivariable processes with several operating degrees of freedom, often we choose to optimize only the most important variable in order to keep the strategy uncomplicated. One characteristic implicitly required in a single-variable optimization problem is that the objective function J be unimodal in the variable x. The selection of a method for one-dimensional search is based on the tradeoff between the number of function evaluations and computer time. We can find the optimum by evaluating the objective function at many values of x, using a small grid spacing (∆x) over the allowable range of x values, but this method is generally inefficient. There are three classes of techniques that can be used efficiently for one-dimensional search: indirect, region elimination, and interpolation. Indirect methods seek to solve the necessary condition dJ/dx = 0 by iteration, but these methods are not as popular as the second two classes. Region elimination methods include equal interval search, dichotomous search (or bisecting), Fibonacci search, and golden section. These methods do not use information on the shape of the function (other than its being unimodal) and thus tend to be rather conservative. The third class of techniques uses repeated polynomial fitting to predict the optimum. These interpolation methods tend to converge rapidly to the optimum without being very complicated. Two interpolation methods, quadratic and cubic interpolation, are used in many optimization packages. Multivariable Optimization The numerical optimization of general nonlinear multivariable objective functions requires that efficient and robust techniques be employed. Efficiency is important since iteration is employed. For example, in multivariable “grid” search for a

problem with four independent variables, an equally spaced grid for each variable is prescribed. For 10 values of each of the four variables, 104 total function evaluations would be required to find the best answer for the grid intersections, but this result may not be close enough to the true optimum and would require further search. A larger number of variables (say, 20) would require exponentially more computation, so grid search is a very inefficient method for most problems. In multivariable optimization, the difficulty of dealing with multivariable functions is usually resolved by treating the problem as a series of one-dimensional searches. For a given starting point, a search direction s is specified, and the optimum is found by searching along that direction. The step size ε is the distance moved along s. Then a new search direction is determined, followed by another one-dimensional search. The algorithm used to specify the search direction depends on the optimization method selected. There are two basic types of unconstrained optimization algorithms: (1) those requiring function derivatives and (2) those that do not. Here we give only an overview and refer the reader to Sec. 3 or the references for more details. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an actual process measurement (such as yield) can be the objective function, and no mathematical model for the process is required. Methods that do not require derivatives are called direct methods and include sequential simplex (Nelder-Meade) and Powell’s method. The sequential simplex method is quite satisfactory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell’s method is more efficient than the simplex method and is based on the concept of conjugate search directions. This class of methods can be used in special cases but is not recommended for optimization involving more than 6 to 10 variables. The second class of multivariable optimization techniques in principle requires the use of partial derivatives of the objective function, although finite difference formulas can be substituted for derivatives. Such techniques are called indirect methods and include the following classes: 1. Steepest descent (gradient) method 2. Conjugate gradient (Fletcher-Reeves) method 3. Newton’s method 4. Quasi-newton methods The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction in which the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton’s method requires second-derivative information but is very efficient, while quasi-newton retains most of the benefits of Newton’s method but utilizes only first-derivative information. All these techniques are also used with constrained optimization. Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed than those described above, because the unconstrained optimum may correspond to unrealistic values of the operating variables. The general form of a nonlinear programming problem allows for a nonlinear objective function and nonlinear constraints, or Minimize

J(x1, x2, . . . , xn)

Subject to

hi(x1, x2, . . . , xn) = 0

i = 1, rc

gi(x1, x2, . . . , xn) > 0

i = 1, mc

(8-71)

In this case, there are n process variables with rc equality constraints and mc inequality constraints. Such problems pose a serious challenge to performing optimization calculations in a reasonable amount of time. Typical constraints in chemical process optimization include operating conditions (temperatures, pressures, and flows have limits), storage capacities, and product purity specifications. An important class of constrained optimization problems is one in which both the objective function and the constraints are linear. The solution of these problems is highly structured and can be obtained

ADVANCED CONTROL SYSTEMS rapidly. The accepted procedure, linear programming (LP), has become quite popular in the past 20 years, solving a wide range of industrial problems. It is increasingly being used for online optimization. For processing plants, there are several different kinds of linear constraints that may arise, making the LP method of great utility. 1. Production limitation due to equipment throughput restrictions, storage limits, or market constraints 2. Raw material (feedstock) limitation 3. Safety restrictions on allowable operating temperatures and pressures 4. Physical property specifications placed on the composition of the final product. For blends of various products, we usually assume that a composite property can be calculated through the mass-averaging of pure-component physical properties 5. Material and energy balances of the steady-state model The optimum in linear programming lies at the constraint intersections, which was generalized to any number of variables and constraints by George Dantzig. The simplex algorithm is a matrix-based numerical procedure for which many digital computer codes exist, for both mainframe and microcomputers (Edgar, Himmelblau, and Lasdon, Optimization of Chemical Processes, 2d ed., McGraw-Hill, New York, 2001; Nash and Sofer, Linear and Nonlinear Programming, McGraw-Hill, New York, 1996). The algorithm can handle virtually any number of inequality constraints and any number of variables in the objective function, and utilizes the observation that only the constraint boundaries need to be examined to find the optimum. In some instances, nonlinear optimization problems even with nonlinear constraints can be linearized so that the LP algorithm can be employed to solve them (called successive linear programming, or SLP). In the process industries, LP and SLP have been applied to a wide range of RTO problems, including refinery scheduling, olefins production, the optimal allocation of boiler fuel, and the optimization of a total plant. Nonlinear Programming The most general case for optimization occurs when both the objective function and the constraints are nonlinear, a case referred to as nonlinear programming. While the ideas behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. All the methods discussed below have been utilized to solve nonlinear programming problems in the field of chemical engineering design and operations. Nonlinear programming is now used extensively in the area of real-time optimization. One of the older and most accessible NLP algorithms uses iterative linearization and is called the generalized reduced gradient (GRG) algorithm. The GRG algorithm employs linear or linearized constraints and uses slack variables to convert all constraints to equality constraints. The GRG algorithm is used in the Excel Solver. CONOPT is a reduced gradient algorithm that works well for large-scale problems and nonlinear constraints. CONOPT and GRG work best for problems in which the number of degrees of freedom is small (the number of constraints is nearly equal to the number of variables). Successive quadratic programming (SQP) solves a sequence of quadratic programs that approach the solution of the original NLP by linearizing the constraints and using a quadratic approximation to the objective function. Lagrange multipliers are introduced to handle constraints, and the search procedure generally employs some variation of Newton’s method, a second-order method that approximates the hessian matrix using first derivatives (Biegler et al., 1997; Edgar et al., 2001). MINOS and NPSOL are software packages that are suitable for problems with large numbers of variables (more variables than equations) and constraints that are linear or nearly linear. Successive linear programming (SLP) is used less often for solving RTO problems. It requires linear approximations of both the objective function and the constraints but sometimes exhibits poor convergence to optima that are not located at constraint intersections. One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are linear. While the solution is iterative, it can be obtained quickly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predictive control, which is heavily used in the refining industry. See the earlier subsection on model predictive control for more details.

8-35

Figure 8-45 gives an overview of which optimization algorithms are appropriate for certain types of RTO problems. Software libraries such as GAMS (General Algebraic Modeling System) or NAG (Numerical Algorithms Group) offer one or more NLP algorithms, but rarely are all algorithms available from a single source. Also there are quite a few good optimization software programs that are free and can be found by a web search. No single NLP algorithm is best for every problem, so several solvers should be tested on a given application. See Edgar, Himmelblau, and Lasdon, Optimization of Chemical Processes, McGraw-Hill, New York, 2001. Linear and nonlinear programming solvers have been interfaced to spreadsheet software for desktop computers. The spreadsheet has become a popular user interface for entering and manipulating numeric data. Spreadsheet software increasingly incorporates analytic tools that are accessible from the spreadsheet interface and permit access to external databases. For example, Microsoft Excel incorporates an optimization-based routine called Solver that operates on the values and formulas of a spreadsheet model. Current versions (4.0 and later) include LP and NLP solvers and mixed integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. Referring to Fig. 8-30, the highest level of process control, planning and scheduling, also employs optimization extensively, often with variables that are integer. Level 5 sets production goals to meet supply and logistics constraints and addresses time-varying capacity and workforce utilization decisions. Enterprise resource planning (ERP) and supply chain management (SCM) in level 5 refer to the links in a web of relationships involving retailing (sales), distribution, transportation, and manufacturing. Planning and scheduling usually operate over relatively long time scales and tend to be decoupled from the rest of the activities in lower levels. For example, all of the refineries owned by an oil company are usually included in a comprehensive planning and scheduling model. This model can be optimized to obtain target levels and prices for interrefinery transfers, crude oil and product allocations to each refinery, production targets, inventory targets, optimal operating conditions, stream allocations, and blends for each refinery. Some planning and scheduling problems are mixed-integer optimization problems that involve both continuous and integer problems; whether or not to operate or use a piece of equipment is a binary (on/off) decision that arises in batch processing. Solution techniques for this type of problem include branch and bound methods and global search. This latter approach handles very complex problems with multiple optima by using algorithms such as tabu search, scatter search, simulated annealing, and genetic evolutionary algorithms (see Edgar, Himmelblau, and Lasdon). STATISTICAL PROCESS CONTROL In industrial plants, large numbers of process variables must be maintained within specified limits in order for the plant to operate properly. Excursions of key variables beyond these limits can have significant consequences for plant safety, the environment, product quality, and plant profitability. Statistical process control (SPC), also called statistical quality control (SQC), involves the application of statistical techniques to determine whether a process is operating normally or abnormally. Thus, SPC is a process monitoring technique that relies on quality control charts to monitor measured variables, especially product quality. The basic SPC concepts and control chart methodology were introduced by Walter Shewhart in the 1930s. The current widespread interest in SPC techniques began in the 1950s when they were successfully applied first in Japan and then elsewhere. Control chart methodologies are now widely used to monitor product quality and other variables that are measured infrequently or irregularly. The basic SPC methodology is described in introductory statistics textbooks (e.g., Montgomery and Runger, Applied Statistics and Probability for Engineers, 3d ed., Wiley, New York, 2002) and some process control textbooks (e.g., Seborg, Edgar, and Mellichamp, Process Dynamics and Control, 2d ed., Wiley, New York, 2004).

8-36

PROCESS CONTROL

FIG. 8-45

Diagram for selection of optimization techniques with algebraic constraints and objective function.

An example of the most common control chart, the Shewhart chart, is shown in Fig. 8-46. It merely consists of measurements plotted versus sample number with control limits that indicate the range for normal process operation. The plotted data are either an individual measurement x or the sample mean ⎯x if more than one sample is measured at each sampling instant. The sample mean for k samples is cal-

culated as, 1 ⎯x =  n

n

x j =1

j

(8-72)

The Shewhart chart in Fig. 8-46 has a target (T), an upper control limit (UCL), and a lower control limit (LCL). The target (or centerline) is the

ADVANCED CONTROL SYSTEMS

8-37

Shewhart chart for sample mean x⎯. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

FIG. 8-46

desired (or expected) value for ⎯x while the region between UCL and LCL defines the range of normal variability. If all the ⎯x data are within the control limits, the process operation is considered to be normal or “in a state of control.” Data points outside the control limits are considered to be abnormal, indicating that the process operation is out of control. This situation occurs for the 21st sample. A single measurement slightly beyond a control limit is not necessarily a cause for concern. But frequent or large chart violations should be investigated to determine the root cause. The major objective in SPC is to use process data and statistical techniques to determine whether the process operation is normal or abnormal. The SPC methodology is based on the fundamental assumption that normal process operation can be characterized by random variations around a mean value. The random variability is caused by the cumulative effects of a number of largely unavoidable phenomena such as electrical measurement noise, turbulence, and random fluctuations in feedstock or catalyst preparation. If this situation exists, the process is said to be in a state of statistical control (or in control), and the control chart measurements tend to be normally distributed about the mean value. By contrast, frequent control chart violations would indicate abnormal process behavior or an out-of-control situation. Then a search would be initiated to attempt to identify the assignable cause or the special cause of the abnormal behavior

The control limits in Fig. 8-46 (UCL and LCL) are based on the assumption that the measurements follow a normal distribution. Figure 8-47 shows the probability distribution for a normally distributed random variable x with mean µ and standard deviation σ. There is a very high probability (99.7 percent) that any measurement is within 3 standard deviations of the mean. Consequently, the control ˆ where σˆ is an estimate limits for x are typically chosen to be T ± 3σ, of σ. This estimate is usually determined from a set of representative data for a period of time when the process operation is believed to be typical. For the common situation in which the plotted variable is the sample mean, its standard deviation is estimated. Shewhart control charts enable average process performance to be monitored, as reflected by the sample mean. It is also advantageous to monitor process variability. Process variability within a sample of k measurements can be characterized by its range, standard deviation, or sample variance. Consequently, control charts are often used for one of these three statistics. Western Electric Rules Shewhart control charts can detect abnormal process behavior by comparing individual measurements with control chart limits. But the pattern of measurements can also provide useful information. For example, if 10 consecutive measurements are all increasing, then it is very unlikely that the process is in a

Probabilities associated with the normal distribution. (Source: Montgomery and Runger, Applied Statistics and Probability for Engineers, 3d ed., Wiley, New York, 2002.)

FIG. 8-47

8-38

PROCESS CONTROL

state of control. A wide variety of pattern tests (also called zone rules) can be developed based on the properties of the normal distribution. For example, the following excerpts from the Western Electric Rules (Western Electric Company, Statistical Quality Control Handbook, Delmar Printing Company, Charlotte, N.C., 1956; Montgomery and Runger, Applied Statistics and Probability for Engineers, 3d ed., Wiley, New York, 2002) indicate that the process is out of control if one or more of the following conditions occur: 1. One data point outside the 3σ control limits 2. Two out of three consecutive data points beyond a 2σ limit 3. Four out of five consecutive data points beyond a 1σ limit and on one side of the centerline 4. Eight consecutive points on one side of the centerline Note that the first condition is the familiar Shewhart chart limits. Pattern tests can be used to augment Shewhart charts. This combination enables out-of-control behavior to be detected earlier, but the false-alarm rate is higher than that for Shewhart charts alone. CUSUM Control Charts Although Shewhart charts with 3σ limits can quickly detect large process changes, they are ineffective for small, sustained process changes (e.g., changes smaller than 1.5σ). Alternative control charts have been developed to detect small changes such as the CUSUM control chart. They are often used in conjunction with Shewhart charts. The cumulative sum (CUSUM) is defined to be a running summation of the deviations of the plotted variable from its target. If the sample mean is plotted, the cumulative sum at sampling instant k, C(k), is C(k) =

k

[x⎯ ( j) − T] j=1

(8-73)

where T is the target for ⎯x. During normal process operation, C(k) fluctuates about zero. But if a process change causes a small shift in x⎯, C(k) will drift either upward or downward. The CUSUM control chart was originally developed using a graphical approach based on V masks. However, for computer calculations, it is more convenient to use an equivalent algebraic version that consists of two recursive equations

+

C+(k) = max[0, ⎯x(k) − (T + K) + C+(k − 1)]

(8-74)

C−(k) = max[0, (T − K) − ⎯x(k) + C−(k − 1)]

(8-75)

−

where C and C denote the sums for the high and low directions, and K is a constant, the slack parameter. The CUSUM calculations are initialized by setting C+(0) = C − 0 = 0. A deviation from the target that is larger than K increases either C+ or C−. A control limit violation occurs when either C+ or C− exceeds a specified control limit (or threshold) H. After a limit violation occurs, that sum is reset to zero or to a specified value. The selection of the threshold H can be based on considerations of average run length (ARL), the average number of samples required to detect a disturbance of specified magnitude. For example, suppose that the objective is to be able to detect if the sample mean ⎯x has shifted from the target by a small amount δ. The slack parameter K is usually specified as K = 0.5δ. For the ideal situation (e.g., normally distributed, uncorrelated disturbances), ARL values have been tabulated for different values of δ, K, and H. Table 8-7 summarizes ARL TABLE 8-7 Average Run Lengths for CUSUM Control Charts Shift from target (in multiples of ) 0 0.25 0.50 0.75 1.00 2.00 3.00

ARL for H 4 168. 74.2 26.6 13.3 8.38 3.34 2.19

ARL for H 5 465. 139. 38.0 17.0 10.4 4.01 2.57

Adapted from Ryan, Statistical Methods for Quality Improvement, 2d ed., Wiley, New York, 2000.

values for two values of H and different values of δ. (The values of δ are usually expressed as multiples of the standard deviation σ.) The ARL values indicate the average number of samples before a change of δ is detected. Thus the ARL values for δ = 0 indicate the average time between false alarms, i.e., the average time between successive CUSUM alarms when no shift in ⎯x has occurred. Ideally, we would like the ARL value to be very large for δ = 0 and small for δ ≠ 0. Table 8-7 shows that as the magnitude of the shift δ increases, ARL decreases and thus the CUSUM control chart detects the change faster. Increasing the value of H from 4σ to 5σ increases all the ARL values and thus provides a more conservative approach. The relative performance of the Shewhart and CUSUM control charts is compared in Fig. 8-48 for a set of simulated data for the tensile strength of a resin. It is assumed that the tensile strength x is normally distributed with a mean of µ = 70 MPa and a standard deviation of σ = 3 MPa. A single measurement is available at each sampling instant. A constant (σ = 0.5σ = 1.5) was added to x(k) for k ≥ 10 in order to evaluate each chart’s ability to detect a small process shift. The CUSUM chart was designed using K = 0.5σ and H = 5σ. The Shewhart chart fails to detect the 0.5σ shift in x at k = 10. But the CUSUM chart quickly detects this change because a limit violation occurs at k = 20. The mean shift can also be detected by applying the Western Electric Rules in the previous section. Process Capability Indices Also known as process capability ratios, these provide a measure of whether an in-control process is meeting its product specifications. Suppose that a quality variable x must have a volume between an upper specification limit (USL) and a lower specification limit (LSL) in order for product to satisfy customer requirements. The capability index Cp is defined as USL − LSL Cp =  6σ

(8-76)

where σ is the standard deviation of x. Suppose that Cp = 1 and x is normally distributed. Based on the normal distribution, we would expect that 99.7 percent of the measurements satisfy the specification limits, or equivalently, we would expect that only 2700 out of 1 million measurements would lie outside the specification limits. If Cp < 1, the product specifications are satisfied; for Cp > 1, they are not. However, capability indices are applicable even when the data are not normally distributed. A second capability index Cpk is based on average process performance ⎯x as well as process variability σ. It is defined as Min[x⎯ − LSL, USL − ⎯x] Cpk =  3σ

(8-77)

Although both Cp and Cpk are used, we consider Cpk to be superior to Cp for the following reason. If ⎯x = T, the process is said to be “centered” and Cpk = Cp. But for ⎯x ≠ T, Cp does not change, even though the process performance is worse, while Cpk does decrease. For this reason, Cpk is preferred. If the standard deviation σ is not known, it is replaced by an estimate σ^ in Eqs. (8-76) and (8-77). For situations where there is only a single specification limit, either USL or LSL, the definitions of Cp and Cpk can be modified accordingly. In practical applications, a common objective is to have a capability index of 2.0 while a value greater than 1.5 is considered to be acceptable. If the Cpk value is too low, it can be improved by making a change that either reduces process variability or causes ⎯x to move closer to the target. These improvements can be achieved in a number of ways that include better process control, better process maintenance, reduced variability in raw materials, improved operator training, and process changes. Six-Sigma Approach Product quality specifications continue to become more stringent as a result of market demands and intense worldwide competition. Meeting quality requirements is especially difficult for products that consist of a very large number of components and for manufacturing processes that consist of hundreds of individual steps. For example, the production of a microelectronic device typically requires 100 to 300 batch processing steps. Suppose

UNIT OPERATIONS CONTROL

8-39

Comparison of Shewhart and CUSUM control charts for the resin example. (Source: Seborg et al., Process Dynamics and Control, 2d ed., Wiley, New York, 2004.)

FIG. 8-48

that there are 200 steps and that each one must meet a quality specification for the final product to function properly. If each step is independent of the others and has a 99 percent success rate, the overall yield of satisfactory product is (0.99)200 = 0.134, or only 13.4 percent. This low yield is clearly unsatisfactory. Similarly, even when a processing step meets 3σ specifications (99.73 percent success rate), it will still result in an average of 2700 “defects” for every 1 million produced. Furthermore, the overall yield for this 200-step process is still only 58.2 percent. The six-sigma approach was pioneered by the Motorola Corporation in the early 1980s as a strategy for achieving both six-sigma quality and continuous improvement. Since then, other large corporations have adopted companywide programs that apply the six-sigma approach to all their business operations, both manufacturing and nonmanufacturing. Thus, although the six-sigma approach is “data-driven” and based on statistical techniques, it has evolved into a broader management philosophy that has been implemented successfully by many large corpora-

tions. The six-sigma programs have also had a significant financial impact. Multivariate Statistical Techniques For common SPC monitoring problems, two or more quality variables are important, and they can be highly correlated. For these situations, multivariable (or multivariate) SPC techniques can offer significant advantages over the single-variable methods discussed earlier. Multivariate monitoring based on the classical Hotelling’s T 2 statistic (Montgomery, Introduction to Statistical Quality Control, 4th ed., Wiley, New York, 2001) can be effective if the data are not highly correlated and the number of variables p is not large (for example, p < 10). Fortunately, alternative multivariate monitoring techniques such as principalcomponent analysis (PCA) and partial least squares (PLS) methods have been developed that are very effective for monitoring problems with large numbers of variables and highly correlated data [Piovoso and Hoo (eds.), Special Issue of IEEE Control Systems Magazine, 22(5), 2002].

UNIT OPERATIONS CONTROL PIPING AND INSTRUMENTATION DIAGRAMS GENERAL REFERENCES: Shinskey, Process Control Systems, 4th ed., McGraw-Hill, New York, 1996. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992. Luyben, Tyreus, and Luyben, Plantwide Process Control, McGraw-Hill, New York, 1998.

The piping and instrumentation (P&I) diagram provides a graphical representation of the control configuration of the process. P&I diagrams illustrate the measuring devices that provide inputs to the con-

trol strategy, the actuators that will implement the results of the control calculations, and the function blocks that provide the control logic. They may also include piping details such as line sizes and the location of hand valves and condensate traps. The symbology for drawing P&I diagrams generally follows standards developed by the Instrumentation, Systems, and Automation Society (ISA). The chemicals, refining, and food industries generally follow this standard. The standards are updated from time to time, primarily because the continuing evolution in control system hardware and software provides additional capabilities for implementing

8-40

PROCESS CONTROL

TC 102

CONTROL OF HEAT EXCHANGERS

TY 102

TT 102

Steam

Steam-Heated Exchangers Steam, the most common heating medium, transfers its latent heat in condensing, causing heat flow to be proportional to steam flow. Thus a measurement of steam flow is essentially a measure of heat transfer. Consider raising a liquid from temperature T1 to T2 by condensing steam: Q = WH = MC(T2 − T1)

Liquid Condensate FIG. 8-49

Example of a simplified piping and instrumentation diagram.

control schemes. The ISA symbols are simple and represent a device or function as a circle containing its tag number and identifying the type of variable being controlled, e.g., pressure, and the function performed, e.g., control: PC-105. Examples of extensions of the ISA standard appear on the pages following. Figure 8-49 presents a simplified P&I diagram for a temperature control loop that applies the ISA symbology. The measurement devices and most elements of the control logic are shown as circles: 1. TT102 is the temperature transmitter. 2. TC102 is the temperature controller. 3. TY102 is the current-to-pneumatic (I/P) transducer. The symbol for the control valve in Fig. 8-49 is for a pneumatic modulating valve without a valve positioner. Electronic (4- to 20-mA) signals are represented by dashed lines. In Fig. 8-49, these include the signal from the transmitter to the controller and the signal from the controller to the I/P transducer. Pneumatic signals are represented by solid lines with double crosshatching at regular intervals. The signal from the I/P transducer to the valve actuator is pneumatic. The ISA symbology provides different symbols for different types of actuators. Furthermore, variations for the controller symbol distinguish control algorithms implemented in distributed control systems from those in panel-mounted single-loop controllers.

where W and H are the mass flow of steam and its latent heat, M and C are the mass flow and specific heat of the liquid, and Q is the rate of heat transfer. The response of controlled temperature to steam flow is linear: dT2 H (8-79)  =  dW MC However, the steady-state process gain described by this derivative varies inversely with liquid flow: adding a given increment of heat flow to a smaller flow of liquid produces a greater temperature rise. Dynamically, the response of liquid temperature to a step in steam flow is that of a distributed lag, shown in Fig. 8-26 (uncontrolled). The time required to reach 63 percent complete response, Στ, is essentially the residence time of the fluid in the exchanger, which is its volume divided by its flow. The residence time then varies inversely with flow. Table 8-2 gives optimum settings for PI and PID controllers for distributed lags, the proportional band varying directly with steadystate gain, and integral and derivative settings directly with Στ. Since both these parameters vary inversely with liquid flow, fixed settings for the temperature controller are optimal at only one flow rate. The variability of the process parameters with flow causes variability in load response, as shown in Fig. 8-50. The PID controller was tuned for optimum (minimum-IAE) load response at 50 percent flow. Each curve represents the response of exit temperature to a 10 percent step in liquid flow, culminating at the stated flow. The 60 percent curve is overdamped and the 40 percent curve is underdamped. The differences in gain are reflected in the amplitude of the deviation, and the differences in dynamics are reflected in the period of oscillation. If steam flow is linear with controller output, as it is in Fig. 8-50, undamped oscillations will be produced when the flow decreases by

The response of a heat exchanger varies with flow in both gain and dynamics; here the PID temperature controller was tuned for optimum response at 50 percent flow. FIG. 8-50

(8-78)

UNIT OPERATIONS CONTROL

8-41

FIG. 8-51 Heat-transfer rate in sensible-heat exchange varies nonlinearly with flow of the manipulated fluid, requiring equal-percentage valve characterization.

one-third from the value at which the controller was optimally tuned— in this example at 33 percent flow. The stable operating range can be extended to one-half the original flow by using an equal-percentage (logarithmic) steam valve, whose gain varies directly with steam flow, thereby compensating for the variable process gain. Further extension requires increasing the integral setting and reducing the derivative setting from their optimum values. The best solution is to adapt all three PID settings to change inversely with measured flow, thereby keeping the controller optimally tuned for all flow rates. Feedforward control can also be applied, as described previously under “Advanced Control Techniques.” The feedforward system solves Eq. (8-78) for the manipulated set point to the steam flow controller, first by subtracting inlet temperature T1 from the output of the outlet temperature controller (in place of T2), and then by multiplying the result by the dynamically compensated liquid flow measurement. If the inlet temperature is not subject to rapid or wide variation, it can be left out of the calculation. Feedforward is capable of a reduction in integrated error as much as a 100-fold, but requires the use of a steam flow loop and lead-lag compensator to approach this effectiveness. Multiplication of the controller output by liquid flow varies the feedback loop gain directly proportional to flow, extending the stable operating range of the feedback loop much as the equal-percentage steam valve did without feedforward. (This system is eminently applicable to control of fired heaters in oil refineries, which commonly provide a circulating flow of hot oil to several distillation columns and are therefore subject to frequent disturbances). Steam flow is sometimes controlled by manipulating a valve in the condensate line rather than the steam line, because it is smaller and hence less costly. Heat transfer then is changed by raising or lowering the level of condensate flooding the heat-transfer surface, an operation that is slower than manipulating a steam valve. Protection also needs to be provided against an open condensate valve blowing steam into the condensate system. Exchange of Sensible Heat When there is no change in phase, the rate of heat transfer is no longer linear with the flow of the manipulated stream, but is a function of the mean temperature difference ∆Tm: Q = UA ∆Tm = MHCH(TH1 − TH2) = MCCC(TC2 − TC1) (8-79a) where U and A are the overall heat-transfer coefficient and area and subscripts H and C refer to the hot and cold fluids, respectively. An example would be a countercurrent cooler, where the hot-stream outlet temperature is controlled. Using the logarithmic mean tempera-

ture difference and solving for TH2 give 1 − MHCH MCCC TH2 = TC1 + (TH1 − TC1)  ε − MHCH MCCC

(8-79b)

where

 



1 1 ε = exp −UA  −  MCCC MHCH

(8-79c)

At a given flow of hot fluid, the heat-transfer rate is plotted as a function of coolant flow in Fig. 8-51, as a percentage of its maximum value (corresponding to TC2 = TC1). The extreme nonlinearity of this relationship requires the use of an equal-percentage coolant valve for gain compensation. The variable dynamics of the distributed lag also apply, limiting the stable operating range in the same way as for the steam-heated exchanger. Sensible-heat exchangers are also subject to variations in the temperature of the manipulated stream, an increasingly common problem where heat is being recovered at variable temperatures for reuse. Figure 8-52 shows a temperature controller (TC) setting a heat flow controller (QC) in cascade. A measurement of the manipulated flow is multiplied by its temperature difference across the heat exchanger to calculate the current heat-transfer rate, by using the right side of Eq. (8-78). Variations in supply temperature then appear as variations in calculated heat-transfer rate, which the QC can quickly correct by adjusting the manipulated flow. An equal-percentage valve is still required to linearize the secondary loop, but the primary loop of temperature-setting heat flow is linear. Feedforward can be added by multiplying the dynamically compensated flow measurement of the other fluid by the output of the temperature controller. When a stream is manipulated whose flow is independently determined, such as the flow of a product or of a heat-transfer fluid from a fired heater, a three-way valve is used to divert the required flow to the heat exchanger. This does not alter the linearity of the process or its sensitivity to supply variations, and it even adds the possibility of independent flow variations. The three-way valve should have equal-percentage characteristics, and heat flow control may be even more beneficial. DISTILLATION COLUMN CONTROL Distillation columns have four or more closed loops—increasing with the number of product streams and their specifications—all of which interact with one another to some extent. Because of this interaction, there are many possible ways to pair manipulated and controlled variables through controllers and other mathematical functions, with widely differing

8-42

PROCESS CONTROL

FIG. 8-52

Manipulating heat flow linearizes the loop and protects against variations in supply temperature.

degrees of effectiveness. Columns also differ from one another, so that no single rule of configuring control loops can be applied successfully to all. The following rules apply to the most common separations. Controlling Quality of a Single Product If one of the products of a column is far more valuable than the other(s), its quality should be controlled to satisfy given specifications, and its recovery should be maximized by minimizing losses of its principal component in other streams. This is achieved by maximizing the reflux ratio consistent with flooding limits on trays, which means maximizing the flow of internal reflux or vapor, whichever is limiting. The same rule should be followed when heating and cooling have little value. A typical example is the separation of high-purity propylene from much lower-valued propane, usually achieved with the waste heat of quench water from the cracking reactors. The most important factor affecting product quality is the material balance. In separating a feed stream F into distillate D and bottom B products, an overall mole flow balance must be maintained F=D+B

FIG. 8-53

(8-80)

as well as a balance on each component Fzi = Dyi + Bxi

(8-81)

where z, y, and x are mole fractions of component i in the respective streams. Combining these equations gives a relationship between the composition of the products and their relative portion of the feed: zi − xi D B  =1−  =  yi − xi F F

(8-82)

From the above, it can be seen that control of either xi or yi requires both product flow rates to change with feed rate and feed composition. Figure 8-53 shows a propylene-propane fractionator controlled at maximum boil-up by the differential pressure controller (DPC) across the trays. This loop is fast enough to reject upsets in the temperature

The quality of high-purity propylene should be controlled by manipulating the material balance.

UNIT OPERATIONS CONTROL

FIG. 8-54

8-43

Depropanizers require control of the quality of both products, here using reflux ratio and boil-up ratio manipulation.

of the quench water quite easily. Pressure is controlled by manipulating the heat-transfer surface in the condenser through flooding. If the condenser should become overloaded, pressure will rise above the set point, but this has no significant effect on the other control loops. Temperature measurements on this column are not helpful, as the difference between the component boiling points is too small. Propane content in the propylene distillate is measured by a chromatographic analyzer sampling the overhead vapor for fast response, and it is controlled by the analyzer controller (AC) manipulating the ratio of distillate to feed rates. The feedforward signal from feed rate is dynamically compensated by f(t) and nonlinearly characterized by f(x) to account for variations in propylene recovery as the feed rate changes. Distillate flow can be measured and controlled more accurately than reflux flow by a factor equal to the reflux ratio, which in this column is typically between 10 and 20. Therefore reflux flow is placed under accumulator level control (LC). Yet composition responds to the difference between internal vapor and reflux flow rates. To eliminate the lag inherent in the response of the accumulator level controller, reflux flow is driven by the subtractor in the direction opposite to distillate flow—this is essential to fast response of the composition loop. The gain of converting distillate flow changes to reflux flow changes can even be increased beyond −1, thereby changing the accumulator level loop from a lag into a dominant lead. Controlling Quality of Two Products Where the two products have similar values, or where heating and cooling costs are comparable to product losses, the compositions of both products should be controlled. This introduces the possibility of strong interaction between the two composition loops, as they tend to have similar speeds of response. Interaction in most columns can be minimized by controlling distillate composition with reflux ratio and bottom composition with boil-up, or preferably boil-up/bottom flow ratio. These loops are insensitive to variations in feed rate, eliminating the need for feedforward control, and they also reject heat balance upsets quite effectively.

Figure 8-54 shows a depropanizer controlled by reflux and boil-up ratios. The actual mechanism through which these ratios are manipulated is as D/(L + D) and B/(V + B), where L is reflux flow and V is vapor boil-up, which decouples the temperature loops from the liquid-level loops. Column pressure here is controlled by flooding both the condenser and accumulator; however, there is no level controller on the accumulator, so this arrangement will not function with an overloaded condenser. Temperatures are used as indications of composition in this column because of the substantial difference in boiling points between propane and butanes. However, off-key components such as ethane do affect the accuracy of the relationship, so that an analyzer controller is used to set the top temperature controller (TC) in cascade. If the products from a column are especially pure, even this configuration may produce excessive interaction between the composition loops. Then the composition of the less pure product should be controlled by manipulating its own flow; the composition of the remaining product should be controlled by manipulating the reflux ratio if it is the distillate, or the boil-up ratio if it is the bottom product. Most sidestream columns have a small flow dedicated to removing an off-key impurity entering the feed, and that stream must be manipulated to control its content in the major product. For example, an ethylene fractionator separates its feed into a high-purity ethylene sidestream, an ethane-rich bottom product, and a small flow of methane overhead. This small flow must be withdrawn to control the methane content in the ethylene product. The key impurities may then be controlled in the same way as in a two-product column. Most volatile mixtures have a relative volatility that varies inversely with column pressure. Therefore, their separation requires less energy at lower pressure, and savings in the range of 20 to 40 percent have been achieved. Column pressure can be minimized by floating on the condenser, i.e., by operating the condenser with minimal or no restrictions. In some columns, such as the propylene-propane splitter, pressure can be left uncontrolled. Where it cannot, the set point of the

8-44

PROCESS CONTROL

pressure controller can be indexed by an integral-only controller acting to slowly drive the pressure control valve toward a position just short of maximum cooling. In the case of a flooded condenser, the degree of reflux subcooling can be controlled in place of condenser valve position. Where column temperatures are used to indicate product composition, their measurements must be pressure-compensated. CHEMICAL REACTORS Composition Control The first requirement for successful control of a chemical reactor is to establish the proper stoichiometry, i.e., to control the flow rates of the reactants in the proportions needed to satisfy the reaction chemistry. In a continuous reactor, this begins by setting ingredient flow rates in ratio to one another. However, because of variations in the purity of the feed streams and inaccuracy in flow metering, some indication of excess reactant such as pH or a composition measurement should be used to trim the ratios. Many reactions are incomplete, leaving one or more reactants unconverted. They are separated from the products of the reaction and recycled to the reactor, usually contaminated with inert components. While reactants can be recycled to complete conversion (extinction), inerts can accumulate to the point of impeding the reaction and must be purged from the system. Inerts include noncondensible gases that must be vented and nonvolatiles from which volatile products must be stripped. If one of the reactants differs in phase from the others and the product(s), it may be manipulated to close the material balance on that phase. For example, a gas reacting with liquids to produce a liquid product may be added as it is consumed to control reactor pressure; a gaseous purge would be necessary. Similarly, a liquid reacting with a gas to produce a gaseous product could be added as it is consumed to control the liquid level in the reactor; a liquid purge would be required. Where a large excess of one reactant A is used to minimize side reactions, the unreacted excess is sent to a storage tank for recycling. Its flow from the recycle storage tank is set in the desired ratio to the flow of reactant B, with the flow of fresh A manipulated to control the recycle tank level if the feed is a liquid, or tank pressure if it is a gas. Some catalysts travel with the reactants and must be recycled in the same way. With batch reactors, it may be possible to add all reactants in their proper quantities initially, if the reaction rate can be controlled by injection of initiator or adjustment of temperature. In semibatch operation, one key ingredient is flow-controlled into the batch at a rate that sets the production. This ingredient should not be manipulated for temperature control of an exothermic reactor, as the loop includes two dominant lags—concentration of the reactant and heat capacity of the reaction mass—and can easily go unstable. It also presents the unfavorable dynamic of inverse response—increasing feed rate may lower temperature by its sensible heat before the increased reaction rate raises temperature. Temperature Control Reactor temperature should always be controlled by heat transfer. Endothermic reactions require heat and therefore are eminently self-regulating. Exothermic reactions produce heat, which tends to raise reaction temperature, thereby increasing the reaction rate and producing more heat. This positive feedback is countered by negative feedback in the cooling system, which removes more heat as the reactor temperature rises. Most continuous reactors have enough heat-transfer surface relative to reaction mass that negative feedback dominates and they are self-regulating. But most batch reactors do not, and they are therefore steady-state unstable. Unstable reactors can be controlled if their temperature controller gain can be set high enough, and if their cooling system has enough margin to accommodate the largest expected disturbance in heat load. Stirred-tank reactors are lag-dominant, and their dynamics allow a high controller gain, but plug flow reactors are dead-timedominant, preventing their temperature controller from providing enough gain to overcome steady-state instability. Therefore unstable plug flow reactors are also uncontrollable, their temperature tending to limit-cycle in a sawtooth wave. A stable reactor can become unstable as its heat-transfer surface fouls, or as the production rate is increased beyond a critical point (Shinskey, “Exothermic Reactors: The Stable, the Unstable, and the Uncontrollable,” Chem. Eng., pp. 54–59, March 2002).

The stirred-tank reactor temperature controller sets the coolant outlet temperature in cascade, with primary integral feedback taken from the secondary temperature measurement.

FIG. 8-55

Figure 8-55 shows the recommended system for controlling the temperature of an exothermic stirred-tank reactor, either continuous or batch. The circulating pump on the coolant loop is absolutely essential to effective temperature control in keeping dead time minimum and constant—without it, dead time varies inversely with cooling load, causing limit-cycling at low loads. Heating is usually required to raise the temperature to reaction conditions, although it is often locked out of a batch reactor once the initiator is introduced. The valves are operated in split range, the heating valve opening from 50 to 100 percent of controller output and the cooling valve opening from 50 to 0 percent. The cascade system linearizes the reactor temperature loop, speeds its response, and protects it from disturbances in the cooling system. The flow of heat removed per unit of coolant flow is directly proportional to the temperature rise of the coolant, which varies with both the temperature of the reactor and the rate of heat transfer from it. Using an equal-percentage cooling valve helps compensate for this nonlinearity, but incompletely. The flow of heat across the heat-transfer surface is linear with both temperatures, leaving the primary loop with a constant gain. Using the coolant exit temperature as the secondary controlled variable as shown in Fig. 8-55 places the jacket dynamics in the secondary loop, thereby reducing the period of the primary loop. This is dynamically advantageous for a stirred-tank reactor because of the slow response of its large heat capacity. However, a plug flow reactor cooled by an external heat exchanger lacks this heat capacity, and requires the faster response of the coolant inlet temperature loop. Performance and robustness are both improved by using the secondary temperature measurement as the feedback signal to the integral mode of the primary controller. (This feature may be available only with controllers that integrate by positive feedback.) This places the entire secondary loop in the integral path of the primary controller, effectively pacing its integral time to the rate at which the secondary temperature is able to respond. It also permits the primary controller to be left in the automatic mode at all times without integral windup. The primary time constant of the reactor is MrCr τ1 =  UA

(8-83)

where Mr and Cr are the mass and heat capacity of the reactants and U and A are the overall heat-transfer coefficient and area, respectively.

UNIT OPERATIONS CONTROL The control system of Fig. 8-55 was tested on a pilot reactor where the heat-transfer area and mass could both be changed by a factor of 2, changing τ1 by a factor of 4 as confirmed by observations of the rates of temperature rise. Yet neither controller required retuning as τ1 varied. The primary controller should be PID and the secondary controller at least PI in this system (if the secondary controller has no integral mode, the primary will control with offset). Set-point overshoot in batch reactor control can be avoided by setting the derivative time of the primary controller higher than its integral time, but this is effective only with interacting PID controllers.

M0 x0 = Mnxn

(8-84)

where M0 and x0 are the mass flow and solid fraction of the feed and Mn and xn are their values in the product after n effects of evaporation. The total solvent evaporated from all the effects must then be 0



x − Mn = M0 1 − 0 xn



(8-85)

For a steam-heated evaporator, each unit of steam W0 applied produces a known amount of evaporation, based on the number of effects and their fractional economy E:

W = nEW

0

(8-86)

(A comparable statement can be made with regard to the power applied to a mechanical recompression evaporator.) In summary, the steam flow required to increase the solid content of the feed from x0

FIG. 8-56

M0(1 − x0 xn) W0 =  nE

(8-87)

The usual measuring device for feed flow is a magnetic flowmeter, which is a volumetric device whose output F must be multiplied by density ρ to produce mass flow M0. For most aqueous solutions fed to evaporators, the product of density and the function of solid content appearing above is linear with density:



The most important consideration in controlling the quality of concentrate from an evaporator is the forcing of the vapor withdrawal rate to match the flow of excess solvent entering the feed. The mass flow rates of solid material entering and leaving are equal in the steady state

W = M

to xn is



x Fρ 1 − 0 ≈ F[1 − m(ρ − 1)] xn

CONTROLLING EVAPORATORS

8-45

(8-88)

where slope m is determined by the desired product concentration and density is in grams per milliliter. The required steam flow in pounds per hour for feed measured in gallons per minute is then 500F[1 − m(ρ − 1)] W0 =  nE

(8-89)

where the factor of 500 converts gallons per minute of water to pounds per hour. The factor nE is about 1.74 for a double-effect evaporator and 2.74 for a triple-effect. Using a thermocompressor (ejector) driven with 150 lb/in2 steam on a single-effect evaporator gives an nE of 2.05; it essentially adds the equivalent of one effect to the evaporator train. A cocurrent evaporator train with its controls is illustrated in Fig. 8-56. The control system applies equally well to countercurrent or mixed-feed evaporators, the principal difference being the tuning of the dynamic compensator f(t), which must be done in the field to minimize the short-term effects of changes in feed flow on product quality. Solid concentration in the product is usually measured as density; feedback trim is applied by the analyzer controller AC adjusting the slope m of the density function, which is the only term related to xn. This recalibrates the system whenever xn must move to a new set point. The accuracy of the system depends on controlling heat flow; therefore if steam pressure varies, compensation must be applied to correct

Controlling the evaporators requires matching steam flow and evaporative load, here using feedforward control.

8-46

PROCESS CONTROL

for both steam density and enthalpy as a function of pressure. Some evaporators must use unreliable sources of low-pressure steam. In this case, the measurement of pressure-compensated steam flow can be used to set feed flow by solving the last equation for F, using W0 as a variable. The steam flow controller would be set for a given production rate, but the dynamically compensated steam flow measurement would be the input signal to calculate the feed flow set point. Both of these configurations are widely used in controlling corn syrup concentrators. DRYING OPERATIONS Controlling dryers is much different from controlling evaporators, because online measurements of feed rate and composition and product composition are rarely available. Most dryers transfer moisture from wet feed into hot dry air in a single pass. The process is generally very self-regulating, in that moisture becomes progressively harder to remove from the product as it dries: this is known as falling-rate drying. Controlling the temperature of the air leaving a cocurrent dryer tends to regulate the moisture in the product, as long as the rate and moisture content of the feed and air are reasonably constant. However, at constant outlet air temperature, product moisture tends to rise with all three of these disturbance variables. In the absence of moisture analyzers, regulation of product quality can be improved by raising the temperature of the exhaust air in proportion to the evaporative load. The evaporative load can be estimated by the loss in temperature of the air passing through the dryer in the steady state. Changes in load are first observed in upsets in exhaust temperature at a given inlet temperature; the controller then responds by returning the exhaust air to its original temperature by changing that of the inlet air. Figure 8-57 illustrates the simplest application of this principle as the linear relationship T0 = Tb + K ∆T

(8-90)

where T0 is the set point for exhaust temperature elevated above a base temperature Tb corresponding to zero-load operation and ∆T is the drop in air temperature from inlet to outlet. Coefficient K must be set to regulate product moisture over the expected range of evaporative load. If K is set too low, product moisture will increase with increasing load; if K is set too high, product moisture will decrease with increasing load. While K can be estimated from the model of a dryer, it does depend on the rate-of-drying curve for the product, its mean particle size, and whether the load variations are due primarily to changes in feed rate or feed moisture.

FIG. 8-57

It is important to have the most accurate measurement of exhaust temperature attainable. Note that Fig. 8-57 shows the sensor inserted into the dryer upstream of the rotating seal, because air infiltration there could cause the temperature in the exhaust duct to read low— even lower than the wet-bulb temperature, an impossibility without either substantial heat loss or outside-air infiltration. The calculation of the exhaust temperature set point forms a positive feedback loop capable of destabilizing the dryer. For example, an increase in evaporative load causes the controller to raise the inlet temperature, which will in turn raise the calculated set point, calling for a further increase in inlet temperature. The gain in the set-point loop K is typically well below the gain of the exhaust temperature measurement responding to the same change in inlet temperature. Negative feedback then dominates in the steady state, but the response of the exhaust temperature measurement is delayed by the dryer. A compensating lag f(t) is shown inserted in the set-point loop to prevent positive feedback from dominating in the short term, which could cause cycling. Lag time can be safely set equal to the integral time of the outlet-air temperature controller. If product moisture is measured offline, analytical results can be used to adjust K and Tb manually. If an online analyzer is used, the analyzer controller would be most effective in adjusting the bias Tb, as is done in the figure. While a rotary dryer is shown, commonly used for grains and minerals, this control system has been successfully applied to fluid-bed drying of plastic pellets, air-lift drying of wood fibers, and spray drying of milk solids. The air may be steam-heated as shown or heated by direct combustion of fuel, provided that a representative measurement of inlet air temperature can be made. If it cannot, then evaporative load can be inferred from a measurement of fuel flow, which then would replace ∆T in the set-point calculation. If the feed flows countercurrent to the air, as is the case when drying granulated sugar, the exhaust temperature does not respond to variations in product moisture. For these dryers, the moisture in the product can better be regulated by controlling its temperature at the point of discharge. Conveyor-type dryers are usually divided into a number of zones, each separately heated with recirculation of air, which raises its wet-bulb temperature. Only the last two zones may require indexing of exhaust air temperature as a function of ∆T. Batch drying, used on small lots such as pharmaceuticals, begins operation by blowing air at constant inlet temperature through saturated product in constant-rate drying, where ∆T is constant at its maximum value ∆Tc. When product moisture reaches the point where falling-rate drying begins, the exhaust temperature begins to rise. The desired product moisture will be reached at a corresponding exhaust

Product moisture from a cocurrent dryer can be regulated through temperature control indexed to heat load.

BATCH PROCESS CONTROL temperature Tf, which is related to the temperature Tc observed during constant-rate drying, as well as to ∆Tc: Tf = Tc + K ∆Tc

(8-91)

The control system requires that the values of Tc and ∆Tc observed during the first minutes of operation be stored as the basis for the above calculation of endpoint. When the exhaust temperature then reaches

8-47

the calculated value of Tf, drying is terminated. Coefficient K can be estimated from models, but requires adjustment online to reach product specifications repeatedly. Products having different moisture specifications or particle size will require different settings of K, but the system does compensate for variations in feed moisture, batch size, air moisture, and inlet temperature. Some exhaust air may be recirculated to control the dew point of the inlet air, thereby conserving energy toward the end of the batch and when the ambient air is especially dry.

BATCH PROCESS CONTROL GENERAL REFERENCES: Fisher, Batch Control Systems: Design, Application, and Implementation, ISA, Research Triangle Park, N.C., 1990. Rosenof and Ghosh, Batch Process Automation, Van Nostrand Reinhold, New York, 1987.

BATCH VERSUS CONTINUOUS PROCESSES When one is categorizing process plants, the following two extremes can be identified: 1. Commodity plants. These plants are custom-designed to produce large amounts of a single product (or a primary product plus one or more secondary products). An example is a chlorine plant, where the primary product is chlorine and the secondary products are hydrogen and sodium hydroxide. Usually the margins (product value less manufacturing costs) for the products from commodity plants are small, so the plants must be designed and operated for best possible efficiencies. Although a few are batch, most commodity plants are continuous. Factors such as energy costs are life-and-death issues for such plants. 2. Specialty plants. These plants are capable of producing small amounts of a variety of products. Such plants are common in fine chemicals, pharmaceuticals, foods, and so on. In specialty plants, the margins are usually high, so factors such as energy costs are important but not life-and-death issues. As the production amounts are relatively small, it is not economically feasible to dedicate processing equipment to the manufacture of only one product. Instead, batch processing is utilized so that several products (perhaps hundreds) can be manufactured with the same process equipment. The key issue in such plants is to manufacture consistently each product in accordance with its specifications. The above two categories represent the extremes in process configurations. The term semibatch designates plants in which some processing is continuous but other processing is batch. Even processes that are considered to be continuous can have a modest amount of batch processing. For example, the reformer unit within a refinery is thought of as a continuous process, but the catalyst regeneration is normally a batch process. In a continuous process, the conditions within the process are largely the same from one day to the next. Variations in feed composition, plant utilities (e.g., cooling water temperature), catalyst activities, and other variables occur, but normally these changes either are about an average (e.g., feed compositions) or exhibit a gradual change over an extended period (e.g., catalyst activities). Summary data such as hourly averages, daily averages, and the like are meaningful in a continuous process. In a batch process, the conditions within the process are continually changing. The technology for making a given product is contained in the product recipe that is specific to that product. Such recipes normally state the following: 1. Raw material amounts. This is the stuff needed to make the product. 2. Processing instructions. This is what must be done with the stuff to make the desired product. This concept of a recipe is quite consistent with the recipes found in cookbooks. Sometimes the term recipe is used to designate only the raw material amounts and other parameters to be used in manufacturing a batch. Although appropriate for some batch processes, this concept is far too restrictive for others. For some products, the differ-

ences from one product to the next are largely physical as opposed to chemical. For such products, the processing instructions are especially important. The term formula is more appropriate for the raw material amounts and other parameters, with recipe designating the formula and the processing instructions. The above concept of a recipe permits the following three different categories of batch processes to be identified: 1. Cyclical batch. Both the formula and the processing instructions are the same from batch to batch. Batch operations within processes that are primarily continuous often fall into this category. The catalyst regenerator within a reformer unit is a cyclical batch process. 2. Multigrade. The processing instructions are the same from batch to batch, but the formula can be changed to produce modest variations in the product. In a batch PVC plant, the different grades of PVC are manufactured by changing the formula. In a batch pulp digester, the processing of each batch or cook is the same, but at the start of each cook, the process operator is permitted to change the formula values for chemical-to-wood ratios, cook time, cook temperature, and so on. 3. Flexible batch. Both the formula and the processing instructions can change from batch to batch. Emulsion polymerization reactors are a good example of a flexible batch facility. The recipe for each product must detail both the raw materials required and how conditions within the reactor must be sequenced to make the desired product. Of these, the flexible batch is by far the most difficult to automate and requires a far more sophisticated control system than either the cyclical batch or the multigrade batch facility. Batches and Recipes Each batch of product is manufactured in accordance with a product recipe, which contains all information (formula and processing instructions) required to make a batch of the product (see Fig. 8-58). For each batch of product, there will be one and only one product recipe. However, a given product recipe is normally used to make several batches of product. To uniquely identify a batch of product, each batch is assigned a unique identifier called the batch ID. Most companies adopt a convention for generating the batch ID, but this convention varies from one company to the next. In most batch facilities, more than one batch of product will be in some stage of production at any given time. The batches in progress may or may not be using the same recipe. The maximum number of batches that can be in progress at any given time is a function of the equipment configuration for the plant. The existence of multiple batches in progress at a given time presents numerous opportunities for the process operator to make errors, such as charging a material to the wrong batch. Charging a material to the wrong batch is almost always detrimental to the batch to which the material is incorrectly charged. Unless this error is recognized quickly so that the proper charge can be made, the error is also detrimental to the batch to which the charge was supposed to have been made. Such errors usually lead to an off-specification batch, but the consequences could be more serious and could result in a hazardous condition. Recipe management refers to the assumption of such duties by the control system. Each batch of product is tracked throughout its production, which may involve multiple processing operations on various pieces of processing equipment. Recipe management ensures that all actions specified in the product recipe are performed on each batch of product made in accordance with that recipe. As the batch proceeds from one piece of processing equipment to the next, recipe management

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PROCESS CONTROL

FIG. 8-58

Batch control overview.

is also responsible for ensuring that the proper type of process equipment is used and that this processing equipment is not currently in use by another batch. By assuming such responsibilities, the control system greatly reduces the incidences where operator error results in off-specification batches. Such a reduction in error is essential to implement justin-time production practices, where each batch of product is manufactured at the last possible moment. When a batch (or batches) is made today for shipment by overnight truck, there is insufficient time for producing another batch to make up for an off-specification batch. Routing and Production Monitoring In some facilities, batches are individually scheduled. However, in most facilities, production is scheduled by product runs (also called process orders), where a run is the production of a stated quantity of a given product. From the stated quantity and the standard yield of each batch, the number of batches can be determined. As this is normally more than one batch of product, a production run is normally a sequence of some number of batches of the same product. In executing a production run, the following issues must be addressed (see Fig. 8-58): 1. Processing equipment must be dedicated to making the run. More than one run is normally in progress at a given time. The maximum number of runs simultaneously in progress depends on the equipment configuration of the plant. Routing involves determining which processing equipment will be used for each production run. 2. Raw material must be utilized. When a production run is scheduled, the necessary raw materials must be allocated to the production run. As the individual batches proceed, the consumption of raw materials must be monitored for consistency with the allocation of raw materials to the production run. 3. The production quantity for the run must be achieved by executing the appropriate number of batches. The number of batches is determined from a standard yield for each batch. However, some batches may achieve yields higher than the standard yield, but other

batches may achieve yields lower than the standard yield. The actual yields from each batch must be monitored, and significant deviations from the expected yields must be communicated to those responsible for scheduling production. The last two activities are key components of production monitoring, although production monitoring may also involve other activities such as tracking equipment utilization. Production Scheduling In this regard, it is important to distinguish between scheduling runs (sometimes called long-term scheduling) and assigning equipment to runs (sometimes called routing or short-term scheduling). As used here production scheduling refers to scheduling runs and is usually a corporate-level as opposed to a plantlevel function. Short-term scheduling or routing was previously discussed and is implemented at the plant level. The long-term scheduling is basically a material resources planning (MRP) activity involving the following: 1. Forecasting. Orders for long-delivery raw materials are issued at the corporate level based on the forecast for the demand for products. The current inventory of such raw materials is also maintained at the corporate level. This constitutes the resources from which products can be manufactured. Functions of this type are now incorporated into supply chain management. 2. Orders for products. Orders are normally received at the corporate level and then assigned to individual plants for production and shipment. Although the scheduling of some products is based on required product inventory levels, scheduling based on orders and shipping directly to the customer (usually referred to as just-in-time) avoids the costs associated with maintaining product inventories. 3. Plant locations and capacities. While producing a product at the nearest plant usually lowers transportation costs, plant capacity limitations sometimes dictate otherwise. Any company competing in the world economy needs the flexibility to accept orders on a worldwide basis and then assign them to individual plants to be filled. Such a function is logically implemented within the corporate-level information technology framework.

BATCH PROCESS CONTROL BATCH AUTOMATION FUNCTIONS Automating a batch facility requires a spectrum of functions. Interlocks Some of these are provided for safety and are properly called safety interlocks. However, others are provided to avoid mistakes in processing the batch. When safety is not involved, terms such as permissives and process actions are sometimes used in lieu of interlocks. Some understand the term interlock to have a connection to safety (interlock will be subsequently defined as a protective response initiated on the detection of a process hazard). Discrete Device States Discrete devices such as two-position valves can be driven to either of two possible states. Such devices can be optionally outfitted with limit switches that indicate the state of the device. For two-position valves, the following combinations are possible: 1. No limit switches 2. One limit switch on the closed position 3. One limit switch on the open position 4. Two limit switches In process control terminology, the discrete device driver is the software routine that generates the output to a discrete device such as a valve and also monitors the state feedback information to ascertain that the discrete device actually attains the desired state. Given the variety of discrete devices used in batch facilities, this logic must include a variety of capabilities. For example, valves do not instantly change states; instead each valve exhibits a travel time for the change from one state to another. To accommodate this characteristic of the field device, the processing logic within the discrete device driver must provide for a user-specified transition time for each field device. When equipped with limit switches, the potential states for a valve are as follows: 1. Open. The valve has been commanded to open, and the limit switch inputs are consistent with the open state. 2. Closed. The valve has been commanded to close, and the limit switch inputs are consistent with the closed state. 3. Transition. This is a temporary state that is only possible after the valve has been commanded to change state. The limit switch inputs are not consistent with the commanded state, but the transition time has not expired. 4. Invalid. The transition time has expired, and the limit switch inputs are not consistent with the commanded state for the valve. The invalid state is an abnormal condition that is generally handled in a manner similar to process alarms. The transition state is not considered to be an abnormal state but may be implemented in either of the following ways: 1. Drive and wait. Further actions are delayed until the device attains its commanded state. 2. Drive and proceed. Further actions are initiated while the device is in the transition state. The latter is generally necessary for devices with long travel times, such as flush-fitting reactor discharge valves that are motor-driven. Closing of such valves is normally done via drive and wait; however, drive and proceed is usually appropriate when opening the valve. Although two-state devices are most common, the need occasionally arises for devices with three or more states. For example, an agitator may be on high speed, on slow speed, or off. Process States Batch processing usually involves imposing the proper sequence of states on the process. For example, a simple blending sequence might be as follows: 1. Transfer specified amount of material from tank A to tank R. The process state is “transfer from A.” 2. Transfer specified amount of material from tank B to tank R. The process state is “transfer from B.” 3. Agitate for specified time. The process state is “agitate without cooling.” 4. Cool (with agitation) to specified target temperature. The process state is “agitate with cooling.” For each process state, the various discrete devices are expected to be in a specified device state. For process state “transfer from A,” the device states might be as follows: 1. Tank A discharge valve: open 2. Tank R inlet valve: open

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3. Tank A transfer pump: running 4. Tank R agitator: off 5. Tank R cooling valve: closed For many batch processes, process state representations are a very convenient mechanism for representing the batch logic. A grid or table can be constructed, with the process states as rows and the discrete device states as columns (or vice versa). For each process state, the state of every discrete device is specified to be one of the following: 1. Device state 0, which may be valve closed, agitator off, and so on 2. Device state 1, which may be valve open, agitator on, and so on 3. No change or don’t care This representation is easily understandable by those knowledgeable about the process technology and is a convenient mechanism for conveying the process requirements to the control engineers responsible for implementing the batch logic. Many batch software packages also recognize process states. A configuration tool is provided to define a process state. With such a mechanism, the batch logic does not need to drive individual devices but can simply command that the desired process state be achieved. The system software then drives the discrete devices to the device states required for the target process state. This normally includes the following: 1. Generating the necessary commands to drive each device to its proper state. 2. Monitoring the transition status of each device to determine when all devices have attained their proper states. 3. Continuing to monitor the state of each device to ensure that the devices remain in their proper states. Should any discrete device not remain in its target state, failure logic must be initiated. Regulatory Control For most batch processes, the discrete logic requirements overshadow the continuous control requirements. For many batch processes, the continuous control can be provided by simple loops for flow, pressure, level, and temperature. However, very sophisticated advanced control techniques are occasionally applied. As temperature control is especially critical in reactors, the simple feedback approach is replaced by model-based strategies that rival, if not exceed, the sophistication of advanced control loops in continuous plants. In some installations, alternative approaches for regulatory control may be required. Where a variety of products are manufactured, the reactor may be equipped with alternative heat removal capabilities, including the following: 1. Jacket filled with cooling water. Most such jackets are oncethrough, but some are recirculating. 2. Heat exchanger in a pump-around loop. 3. Reflux condenser. The heat removal capability to be used usually depends on the product being manufactured. Therefore, regulatory loops must be configured for each possible option, and sometimes for certain combinations of the possible options. These loops are enabled and disabled depending on the product being manufactured. The interface between continuous controls and sequence logic (discussed shortly) is also important. For example, a feed might be metered into a reactor at a variable rate, depending on another feed or possibly on reactor temperature. However, the product recipe calls for a specified quantity of this feed. The flow must be totalized (i.e., integrated), and when the flow total attains a specified value, the feed must be terminated. The sequence logic must have access to operational parameters such as controller modes. That is, the sequence logic must be able to switch a controller to manual, automatic, or cascade. Furthermore, the sequence logic must be able to force the controller output to a specified value. Sequence Logic Sequence logic must not be confused with discrete logic. Discrete logic is especially suitable for interlocks or permissives; e.g., the reactor discharge valve must be closed for the feed valve to be opened. Sequence logic is used to force the process to attain the proper sequence of states. For example, a feed preparation might be to first charge A, then charge B, next mix, and finally cool. Although discrete logic can be used to implement sequence logic, other alternatives are often more attractive.

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PROCESS CONTROL

Sequence logic is often, but not necessarily, coupled with the concept of a process state. Basically, the sequence logic determines when the process should proceed from the current state to the next and sometimes what the next state should be. Sequence logic must encompass both normal and abnormal process operations. Thus, sequence logic is often viewed as consisting of two distinct but related parts: 1. Normal logic. This sequence logic provides for the normal or expected progression from one process state to another. 2. Failure logic. This logic provides for responding to abnormal conditions, such as equipment failures. Of these, the failure logic can easily be the most demanding. The simplest approach is to stop or hold on any abnormal condition and let the process operator sort things out. However, this is not always acceptable. Some failures lead to hazardous conditions that require immediate action; waiting for the operator to decide what to do is not acceptable. The appropriate response to such situations is best determined in conjunction with the process hazards analysis. No single approach has evolved as the preferred way to implement sequence logic. The approaches utilized include the following: 1. Discrete logic. Although sequence logic is different from discrete logic, sequence logic can be implemented using discrete logic capabilities. Simple sequences are commonly implemented as ladder diagrams in programmable logic controllers (PLCs). Sequence logic can also be implemented using the boolean logic functions provided by a distributed control system (DCS), although this approach is now infrequently pursued. 2. Programming languages. Traditional procedural languages do not provide the necessary constructs for implementing sequence logic. This necessitates one of the following: a. Special languages. The necessary extensions for sequence logic are provided by extending the syntax of the programming language. This is the most common approach within distributed control systems. The early implementations used BASIC as the starting point for the extensions; the later implementations used C as the starting point. A major problem with this approach is portability, especially from one manufacturer to the next but sometimes from one product version to the next within the same manufacturer’s product line. b. Subroutine or function libraries. The facilities for sequence logic are provided via subroutines or functions that can be referenced from programs written in FORTRAN or C. This requires a generalpurpose program development environment and excellent facilities to trap the inevitable errors in such programs. Operating systems with such capabilities have long been available on the larger computers, but not for the microprocessors utilized within DCSs. However, such operating systems are becoming more common within DCSs. 3. State machines. This technology is commonly applied within the discrete manufacturing industries. However, its migration to process batch applications has been limited. 4. Graphical implementations. For sequence logic, the flowchart traditionally used to represent the logic of computer programs must be extended to provide parallel execution paths. Such extensions have been implemented in a graphical representation generally referred to as a sequential function chart, which is a derivative of an earlier technology known as Grafcet. As process engineers have demonstrated a strong dislike for ladder logic, most PLC manufacturers now provide sequential function charts either in addition to or as an alternative to ladder logic. Many DCS manufacturers also provide sequential function charts either in addition to or as an alternative to special sequence languages. INDUSTRIAL APPLICATIONS An industrial example requiring simple sequence logic is the effluent tank with two sump pumps illustrated in Fig. 8-59. There are two sump pumps, A and B. The tank is equipped with three level switches, one for low level (LL), one for high level (LH), and one for high-high level (LHH). All level switches actuate on rising level. The logic is to be as follows: 1. When level switch LH actuates, start one sump pump. This must alternate between the sump pumps. If pump A is started on this occasion, then pump B must be started on the next occasion.

Discharge MS

MS

Waste Streams

LHH

Effluent Tank

LH

LL

FIG. 8-59

Sump Pumps A

B

Effluent tank process.

2. When level switch LHH actuates, start the other sump pump. 3. When level switch LL deactuates, stop all sump pumps. Once a sump pump is started, it is not stopped until level switch LL deactuates. With this logic, one, both, or no sump pump may be running when the level is between LL and LH. Either one or both sump pumps may be running when the level is between LH and LHH. Figure 8-60a presents the ladder logic implementation of the sequence logic. Ladder diagrams were originally developed for representing hardwired logic, but are now widely used in PLCs. The vertical bar on the left provides the source of power; the vertical bar on the right is ground. If a coil is connected between the power source and ground, the coil will be energized. If a circuit consisting of a set of contacts is inserted between the power source and the coil, the coil will be energized only if power can flow through the circuit. This will depend on the configuration of the circuit and the states of the contacts within the circuit. Ladder diagrams are constructed as rungs, with each rung consisting of a circuit of contacts and an output coil. Contacts are represented as vertical bars. A vertical bar represents a normally open contact; power flows through this contact only if the device with which the contact is associated is actuated (energized). Vertical bars separated by a slash represent a normally closed contact; power flows through this contact only if the device with which the contact is associated is not actuated. The level switches actuate on rising level. If the vessel level is below the location of the switch, the normally open contact is open and the normally closed contact is closed. If the level is above the location of the switch, the normally closed contact is closed and the normally open contact is open. The first rung in Fig. 8-60a is for pump A. It will run if one (or more) of the following conditions is true: 1. Level is above LH and pump A is the lead pump. A coil (designated as LeadIsB) will be subsequently provided to designate the pump to be started next (called the lead pump). If this coil is energized, pump B is the lead pump. Hence, pump A is to be started at LH if this coil is not energized, hence the use of the normally closed contact on coil LeadIsB in the rung of ladder logic for pump A. 2. Level is above LHH. 3. Pump A is running and the level is above LL. The second rung is an almost identical circuit for pump B. The difference is the use of the normally open contact on the coil LeadIsB. When implemented as hardwired logic, ladder diagrams are truly parallel logic; i.e., all circuits are active at all instants of time. But when ladder diagrams are implemented in PLCs, the behavior is slightly different. The ladder logic is scanned very rapidly (on the order of 100 times per second), which gives the appearance of parallel logic. But within a scan of ladder logic, the rungs are executed sequentially. This

BATCH PROCESS CONTROL LH

PumpA

LeadIsB

Rung 1 LHH PumpA

LL

LH

LeadIsB

PumpB Rung 2

LHH PumpB

LL

LL

LL1

OneShot Rung 3 LL1

LL

Rung 4 OneShot LeadIsB

LeadIsB Rung 5

LeadIsB OneShot

(a)

PumpA = Stopped PumpB = Stopped LeadIsB = false LH = true if (LeadIsB) PumpB = Running else PumpA = Running

LL = false PumpA = Stopped PumpB = Stopped LeadIsB = !LeadIsB

LHH = true PumpA = Running PumpB = Running LL = false PumpA = Stopped PumpB = Stopped LeadIsB = !LeadIsB

(b) FIG. 8-60

(a) Ladder logic. (b) Sequence logic for effluent tank sump pumps.

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permits constructs within ladder logic for PLCs that make no sense in hardwired circuits. One such construct is for a “one-shot.” Some PLCs provide this as a built-in function, but here it will be presented in terms of separate components. The one-shot is generated by the third rung of ladder logic in Fig. 8-60a. But first examine the fourth rung. The input LL drives the output coil LL1. This coil provides the state of level switch LL on the previous scan of ladder logic. This is used in the third rung to produce the one-shot. Output coil OneShot is energized if 1. LL is not actuated on this scan of ladder logic (note the use of the normally closed contact for LL) 2. LL was actuated on the previous scan of ladder logic (note the use of the normally open contact for LL1) When LL deactuates, coil OneShot is energized for one scan of ladder logic. OneShot does not energize when LL actuates (a slight modification of the circuit would give a one-shot when LL actuates). The one-shot is used in the fifth rung of ladder logic to toggle the lead pump. The output coil LeadIsB is energized provided that 1. LeadIsB is energized and OneShot is not energized. Once LeadIsB is energized, it remains energized until the next “firing” of the one-shot. 2. LeadIsB is not energized and OneShot is energized. This causes coil LeadIsB to change states each time the one-shot fires. Ladder diagrams are ideally suited for representing discrete logic, such as required for interlocks. Sequence logic can be implemented via ladder logic, but usually with some tricks or gimmicks (the oneshot in Fig. 8-60a is such a gimmick). These are well known to those “skilled in the art” of PLC programming. But to others, they can be quite confusing. Figure 8-60b provides a sequential function chart for the pumps. Sequential function charts consist of steps and transitions. A step consists of actions to be performed, as represented by statements. A transition consists of a logical expression. As long as the logical expression is false, the sequence logic remains at the transition. When the logical expression is true, the sequence logic proceeds to the step following the transition. The basic constructs of sequential function charts are presented in Fig. 8-61. The basic construct of a sequential function chart is the step-transition-step. But also note the constructs for OR and AND. At the divergent OR, the logic proceeds on only one of the possible paths, specifically, the one whose transition is the first to attain the true condition. At the divergent AND, the logic proceeds on all paths simultaneously, and all must complete to proceed beyond the convergent AND. This enables sequential function charts to provide parallel logic. In the sequential function chart in Fig. 8-60b for the pumps, the logic is initiated with both pumps stopped and pump A as the lead pump. When LH actuates, the lead pump is started. A divergent OR is used to create two paths: 1. If LL deactuates, both pumps are stopped and the lead pump is swapped. 2. If LHH actuates, both pumps are started (one is already running). Both remain running until LL deactuates, at which time both are stopped. The logic then loops to the transition for LH actuating. Although not illustrated here, programming languages (either custom sequence languages or traditional languages extended by libraries of real-time functions) are a viable alternative for implementing the logic for the pumps. Graphical constructs such as ladder logic and sequential function charts are appealing to those uncomfortable with traditional programming languages. But in reality, these are programming methodologies. BATCH REACTOR CONTROL The reactors in flexible batch chemical plants usually present challenges. Many reactors have multiple mechanisms for heating and/ or cooling. The reactor in Fig. 8-62 has three mechanisms: 1. Heat with steam. 2. Cool with cooling tower water. 3. Cool with chilled water.

8-52

PROCESS CONTROL Vent Inerts

Initial Step

MS Vent

Exhaust Vacuum Pump

Step

Feed(s)

Chilled Water Supply

Terminal Step

Transition

Cooling Water Supply

Condition

Condensate

Divergent OR

Convergent OR

Divergent AND

Convergent AND

Elements of sequential function charts.

Sometimes glycol is an option; occasionally liquid nitrogen is used to achieve even lower temperatures. Some jacket configurations require sequences for switching between the various modes for heating and cooling (the jacket has to be drained or depressurized before another medium can be admitted). The reactor in Fig. 8-62 has three mechanisms for pressure control: 1. Vacuum 2. Atmospheric (using the vent and inert valves) 3. Pressure Some reactors are equipped with multiple vacuum pumps, with different operating modes for moderate vacuum versus high vacuum. Sequence logic is usually required to start vacuum pumps and establish vacuum. With three options for heating/cooling and three options for pressure, the reactor in Fig. 8-62 has nine combinations of operating modes.

FIG. 8-62

Chilled Water Return

Cooling Water Return

Discharge

Trap

FIG. 8-61

Steam

MS

Drain

Chemical reactor schematic.

In practice, this is actually a low number. This number increases with features such as 1. Recirculations or pump-arounds containing a heater and/or cooler 2. Reflux condensers that can be operated at total reflux (providing only cooling) or such that some component is removed from the reacting system These further increase the number of possible combinations. Some combinations may not make sense, may not be used in current production operations, or otherwise can be eliminated. However, the net number of combinations that must be supported tends to be large. The order in which the systems are activated usually depends on the product being manufactured. Sometimes heating/cooling and pressure control are established simultaneously; sometimes heating/cooling is established first and then pressure control, and sometimes pressure control is established first and then heating/cooling. One has to be very careful when imposing restrictions. Suppose no current products establish pressure control first and then establish heating/cooling. But what about the next product to be introduced? After all, this is a flexible batch facility. Such challenging applications for recipe management and sequence logic require a detailed analysis of the production equipment and the operations conducted within that production equipment. While applications such as the sump pumps in the effluent tank can be pursued without it, a structured approach is essential in flexible batch facilities. BATCH PRODUCTION FACILITIES Especially for flexible batch applications, the batch logic must be properly structured in order to be implemented and maintained in a reasonable manner. An underlying requirement is that the batch process equipment be properly structured. The following structure is appropriate for most batch production facilities. Plant A plant is the collection of production facilities at a geographic site. The production facilities at a site normally share warehousing, utilities, and the like. Equipment Suite An equipment suite is the collection of equipment available for producing a group of products. Normally, this

BATCH PROCESS CONTROL group of products is similar in certain respects. For example, they might all be manufactured from the same major raw materials. Within the equipment suite, material transfer and metering capabilities are available for these raw materials. The equipment suite contains all the necessary types of processing equipment (reactors, separators, and so on) required to convert the raw materials to salable products. A plant may consist of only one suite of equipment, but large plants usually contain multiple equipment suites. Process Unit or Batch Unit A process unit is a collection of processing equipment that can, at least at certain times, be operated in a manner completely independent of the remainder of the plant. A process unit normally provides a specific function in the production of a batch of product. For example, a process unit might be a reactor complete with all associated equipment (jacket, recirculation pump, reflux condenser, and so on). However, each feed preparation tank is usually a separate process unit. With this separation, preparation of the feed for the next batch can be started as soon as the feed tank is emptied for the current batch. All but the very simplest equipment suites contain multiple process units. The minimum number of process units is one for each type of processing equipment required to make a batch of product. However, many equipment suites contain multiple process units of each type. In such equipment suites, multiple batches and multiple production runs can be in progress at a given time. Item of Equipment An item of equipment is a hardware item that performs a specific purpose. Examples are pumps, heat exchangers, agitators, and the like. A process unit could consist of a single item of equipment, but most process units consist of several items of equipment that must be operated in harmony to achieve the function expected of the process unit. Device A device is the smallest element of interest to batch logic. Examples of devices include measurement devices and actuators. STRUCTURED BATCH LOGIC Flexible batch applications must be pursued by using a structured approach to batch logic. In such applications, the same processing equipment is used to make a variety of products. In most facilities, little or no proprietary technology is associated with the equipment itself; the proprietary technology is how this equipment is used to produce each of the products. The primary objective of the structured approach is to separate cleanly the following two aspects of the batch logic: Product Technology Basically, this encompasses the product technology, such as how to mix certain molecules to make other molecules. This technology ultimately determines the chemical and physical properties of the final product. The product recipe is the principal source for the product technology. Process Technology The process equipment permits certain processing operations (e.g., heat to a specified temperature) to be undertaken. Each processing operation will involve certain actions (e.g., opening appropriate valves). The need to keep these two aspects separated is best illustrated by a situation where the same product is to be made at different plants. While it is possible that the processing equipment at the two plants is identical, this is rarely the case. Suppose one plant uses steam for heating its vessels, but the other uses a hot oil system as the source of heat. When a product recipe requires that material be heated to a specified temperature, each plant can accomplish this objective, but each will go about it in quite different ways. The ideal case for a product recipe is as follows: 1. It contains all the product technology required to make a product. 2. It contains no equipment-dependent information, i.e., no process technology. In the previous example, such a recipe would simply state that the product must be heated to a specified temperature. Whether heating is undertaken with steam or hot oil is irrelevant to the product technology. By restricting the product recipe to a given product technology, the same product recipe can be used to make products at

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different sites. At a given site, the specific approach to be used to heat a vessel is important. The traditional approach is for an engineer at each site to expand the product recipe into a document that explains in detail how the product is to be made at the specific site. This document goes by various names, although standard operating procedure or SOP is a common one. Depending on the level of detail to which it is written, the SOP could specify exactly which valves must be opened to heat the contents of a vessel. Thus, the SOP is site-dependent and contains both product technology and process technology. In structuring the logic for a flexible batch application, the following organization permits product technology to be cleanly separated from process technology: • A recipe consists of a formula and one or more processing operations. Ideally, only product technology is contained in a recipe. • A processing operation consists of one or more phases. Ideally, only product technology is contained in a processing operation. • A phase consists of one or more actions. Ideally, only process technology is contained in a phase. In this structure, the recipe and processing operations would be the same at each site that manufactures the product. However, the logic that comprises each phase would be specific to a given site. In the heating example from above, each site would require a phase to heat the contents of the vessel. However, the logic within the phase at one site would accomplish the heating by opening the appropriate steam valves, while the logic at the other site would accomplish the heating by opening the appropriate hot oil valves. Usually the critical part of structuring batch logic is the definition of the phases. There are two ways to approach this: 1. Examine the recipes for the current products for commonality, and structure the phases to reflect this commonality. 2 Examine the processing equipment to determine what processing capabilities are possible, and write phases to accomplish each possible processing capability. There is the additional philosophical issue of whether to have a large number of simple phases with few options each, or a small number of complex phases with numerous options. The issues are analogous to structuring a complex computer program into subprograms. Each possible alternative has advantages and disadvantages. As the phase contains no product technology, the implementation of a phase must be undertaken by those familiar with the process equipment. Furthermore, they should undertake this on the basis that the result will be used to make a variety of products, not just those that are initially contemplated. The development of the phase logic must also encompass all equipment-related safety issues. The phase should accomplish a clearly defined objective, so the implementers should be able to thoroughly consider all relevant issues in accomplishing this objective. The phase logic is defined in detail, implemented in the control system, and then thoroughly tested. Except when the processing equipment is modified, future modifications to the phase should be infrequent. The result should be a very dependable module that can serve as a building block for batch logic. Even for flexible batch applications, a comprehensive menu of phases should permit most new products to be implemented by using currently existing phases. By reusing existing phases, numerous advantages accrue: 1. The engineering effort to introduce a new recipe at a site is reduced. 2. The product is more likely to be on-spec the first time, thus avoiding the need to dispose of off-spec product. 3. The new product can be supplied to customers sooner, hopefully before competitors can supply the product. There is also a distinct advantage in maintenance. When a problem with a phase is discovered and the phase logic is corrected, the correction is effectively implemented in all recipes that use the phase. If a change is implemented in the processing equipment, the affected phases must be modified accordingly and then thoroughly tested. These modifications are also effectively implemented in all recipes that use these phases.

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PROCESS MEASUREMENTS GENERAL REFERENCES: Baker, Flow Measurement Handbook, Cambridge University Press, New York, 2000. Connell, Process Instrumentation Applications Manual, McGraw-Hill, New York, 1966. Dakin, and Culshaw (eds), Optical Fiber Sensors: Applications, Analysis, and Future Trends, vol. IV, Artech House, Norwood, Mass., 1997. Dolenc, “Choose the Right Flow Meter,” Chem. Engr. Prog., 92(1): 22, 1996. Johnson, Process Control Instrumentation Technology, 6th ed., Prentice-Hall, Upper Saddle River, N.J., 2000. Liptak (ed.), Instrument Engineers Handbook, 3d ed., vol. 1: Process Measurement, Chilton Books, Philadelphia, 2000. Nichols, On-Line Process Analyzers, Wiley, New York, 1988. Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004. Soloman, Sensors Handbook, McGraw-Hill, New York, 1999. Spitzer, Flow Measurement, 2d ed., ISA, Research Triangle Park, N.C., 2001.

GENERAL CONSIDERATIONS Process measurements encompass the application of the principles of metrology to the process in question. The objective is to obtain values for the current conditions within the process and to make this information available in a form usable by the control system, process operators, or management information systems. The term measured variable or process variable designates the process condition that is being determined. Process measurements fall into two categories: 1. Continuous measurements. An example of a continuous measurement is a level measurement device that determines the liquid level in a tank (e.g., in meters). 2. Discrete measurements. An example of a discrete measurement is a level switch that indicates the presence or absence of liquid at the location at which the level switch is installed. In continuous processes, most process control applications rely on continuous measurements. In batch processes, many of the process control applications utilize discrete as well as continuous measurements. In both types of processes, the safety interlocks and process interlocks rely largely on discrete measurements. Continuous Measurements In most applications, continuous measurements provide more information than discrete measurements. Basically, discrete measurements involve a yes/no decision, whereas continuous measurements may entail considerable signal processing. The components of a typical continuous measurement device are as follows: 1. Sensor. This component produces a signal that is related in a known manner to the process variable of interest. The sensors in use today are primarily of the electrical analog variety, and the signal is in the form of a voltage, a resistance, a capacitance, or some other directly measurable electrical quantity. Prior to the mid-1970s, instruments tended to use sensors whose signal was mechanical and thus compatible with pneumatic technology. Since that time, the fraction of sensors that are digital has grown considerably, often eliminating the need for analog-to-digital conversion. 2. Signal processing. The signal from a sensor is usually related in a nonlinear fashion to the process variable of interest. For the output of the measurement device to be linear with respect to the process variable of interest, linearization is required. Furthermore, the signal from the sensor might be affected by variables other than the process variable. In this case, additional variables must be sensed, and the signal from the sensor compensated to account for the other variables. For example, reference junction compensation is required for thermocouples (except when used for differential temperature measurements). 3. Transmitter. The measurement device output must be a signal that can be transmitted over some distance. Where electronic analog transmission is used, the low range on the transmitter output is 4 mA, and the upper range is 20 mA. Microprocessor-based transmitters (often referred to as smart transmitters) are usually capable of transmitting the measured variable digitally in engineering units. Accuracy and Repeatability Definitions of terminology pertaining to process measurements can be obtained from standards available from the Instrumentation, Systems, and Automation Society

(ISA) and from the Scientific Apparatus Makers Association [now Measurement, Control, and Automation Association (MCAA)], both of which are updated periodically. An appreciation of accuracy and repeatability is especially important. Some applications depend on the accuracy of the instrument, but other applications depend on repeatability. Excellent accuracy implies excellent repeatability; however, an instrument can have poor accuracy but excellent repeatability. In some applications, this is acceptable, as discussed below. Range and Span A continuous measurement device is expected to provide credible values of the measured value between a lower range and an upper range. The difference between the upper range and the lower range is the span of the measurement device. The maximum value for the upper range and the minimum value for the lower range depend on the principles on which the measurement device is based and on the design chosen by the manufacturer of the measurement device. If the measured variable is greater than the upper range or less than the lower range, the measured variable is said to be out of range or the measurement device is said to be overranged. Accuracy Accuracy refers to the difference between the measured value and the true value of the measured variable. Unfortunately, the true value is never known, so in practice accuracy refers to the difference between the measured value and an accepted standard value for the measured variable. Accuracy can be expressed in four ways: 1. As an absolute difference in the units of the measured variable 2. As a percent of the current reading 3. As a percent of the span of the measured variable 4. As a percent of the upper range of the span For process measurements, accuracy as a percent of span is the most common. Manufacturers of measurement devices always state the accuracy of the instrument. However, these statements always specify specific or reference conditions at which the measurement device will perform with the stated accuracy, with temperature and pressure most often appearing in the reference conditions. When the measurement device is applied at other conditions, the accuracy is affected. Manufacturers usually also provide some statements on how accuracy is affected when the conditions of use deviate from the referenced conditions in the statement of accuracy. Although appropriate calibration procedures can minimize some of these effects, rarely can they be totally eliminated. It is easily possible for such effects to cause a measurement device with a stated accuracy of 0.25 percent of span at reference conditions to ultimately provide measured values with accuracies of 1 percent or less. Microprocessor-based measurement devices usually provide better accuracy than do the traditional electronic measurement devices. In practice, most attention is given to accuracy when the measured variable is the basis for billing, such as in custody transfer applications. However, whenever a measurement device provides data to any type of optimization strategy, accuracy is very important. Repeatability Repeatability refers to the difference between the measurements when the process conditions are the same. This can also be viewed from the opposite perspective. If the measured values are the same, repeatability refers to the difference between the process conditions. For regulatory control, repeatability is of major interest. The basic objective of regulatory control is to maintain uniform process operation. Suppose that on two different occasions, it is desired that the temperature in a vessel be 800°C. The regulatory control system takes appropriate actions to bring the measured variable to 800°C. The difference between the process conditions at these two times is determined by the repeatability of the measurement device. In the use of temperature measurement for control of the separation in a distillation column, repeatability is crucial but accuracy is not. Composition control for the overhead product would be based on a measurement of the temperature on one of the trays in the rectifying section. A target would be provided for this temperature. However, at

PROCESS MEASUREMENTS periodic intervals, a sample of the overhead product is analyzed in the laboratory and the information is provided to the process operator. Should this analysis be outside acceptable limits, the operator would adjust the set point for the temperature. This procedure effectively compensates for an inaccurate temperature measurement; however, the success of this approach requires good repeatability from the temperature measurement. Dynamics of Process Measurements Especially where the measurement device is incorporated into a closed-loop control configuration, dynamics are important. The dynamic characteristics depend on the nature of the measurement device, and on the nature of components associated with the measurement device (e.g., thermowells and sample conditioning equipment). The term measurement system designates the measurement device and its associated components. The following dynamics are commonly exhibited by measurement systems: • Time constants. Where there is a capacity and a throughput, the measurement device will exhibit a time constant. For example, any temperature measurement device has a thermal capacity (mass times heat capacity) and a heat flow term (heat-transfer coefficient and area). Both the temperature measurement device and its associated thermowell will exhibit behavior typical of time constants. • Dead time. Probably the best example of a measurement device that exhibits pure dead time (time delay) is the chromatograph, because the analysis is not available for some time after a sample is injected. Additional dead time results from the transportation lag within the sample system. Even continuous analyzer installations can exhibit dead time from the sample system. • Underdamped. Measurement devices with mechanical components often have a natural harmonic and can exhibit underdamped behavior. The displacer type of level measurement device is capable of such behavior. While the manufacturers of measurement devices can supply some information on the dynamic characteristics of their devices, interpretation is often difficult. Measurement device dynamics are quoted on varying bases, such as rise time, time to 63 percent response, settling time, etc. Even where the time to 63 percent response is quoted, it might not be safe to assume that the measurement device exhibits first-order behavior. Where the manufacturer of the measurement device does not supply the associated equipment (thermowells, sample conditioning equipment, etc.), the user must incorporate the characteristics of these components to obtain the dynamics of the measurement system. An additional complication is that most dynamic data are stated for configurations involving reference materials such as water and air. The nature of the process material will affect the dynamic characteristics. For example, a thermowell will exhibit different characteristics when immersed in a viscous organic emulsion than when immersed in water. It is often difficult to extrapolate the available data to process conditions of interest. Similarly, it is often impossible, or at least very difficult, to experimentally determine the characteristics of a measurement system under the conditions where it is used. It is certainly possible to fill an emulsion polymerization reactor with water and determine the dynamic characteristics of the temperature measurement system. However, it is not possible to determine these characteristics when the reactor is filled with the emulsion under polymerization conditions. The primary impact of unfavorable measurement dynamics is on the performance of closed-loop control systems. This explains why most control engineers are very concerned with minimizing measurement dynamics, even though the factors considered in dynamics are often subjective. Selection Criteria The selection of a measurement device entails a number of considerations given below, some of which are almost entirely subjective. 1. Measurement span. The measurement span required for the measured variable must lie entirely within the instrument’s envelope of performance. 2. Performance. Depending on the application, accuracy, repeatability, or perhaps some other measure of performance is appropriate.

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Where closed-loop control is contemplated, speed of response must be included. 3. Reliability. Data available from the manufacturers can be expressed in various ways and at various reference conditions. Often, previous experience with the measurement device within the purchaser’s organization is weighted most heavily. 4. Materials of construction. The instrument must withstand the process conditions to which it is exposed. This encompasses considerations such as operating temperatures, operating pressures, corrosion, and abrasion. For some applications, seals or purges may be necessary. 5. Prior use. For the first installation of a specific measurement device at a site, training of maintenance personnel and purchases of spare parts might be necessary. 6. Potential for releasing process materials to the environment. Fugitive emissions are receiving ever-increasing attention. Exposure considerations, both immediate and long-term, for maintenance personnel are especially important when the process fluid is either corrosive or toxic. 7. Electrical classification. Article 500 of the National Electric Code provides for the classification of the hazardous nature of the process area in which the measurement device will be installed. If the measurement device is not inherently compatible with this classification, suitable enclosures must be purchased and included in the installation costs. 8. Physical access. Subsequent to installation, maintenance personnel must have physical access to the measurement device for maintenance and calibration. If additional structural facilities are required, they must be included in the installation costs. 9. Invasive or noninvasive. The insertion of a probe can result in fouling problems and a need for maintenance. Probe location must be selected carefully for good accuracy and minimal fouling. 10. Cost. There are two aspects of the cost: a. Initial purchase and installation (capital cost). b. Recurring costs (operational expense). This encompasses instrument maintenance, instrument calibration, consumables (e.g., titrating solutions must be purchased for automatic titrators), and any other costs entailed in keeping the measurement device in service. Calibration Calibration entails the adjustment of a measurement device so that the value from the measurement device agrees with the value from a standard. The International Standards Organization (ISO) has developed a number of standards specifically directed to calibration of measurement devices. Furthermore, compliance with the ISO 9000 standards requires that the working standard used to calibrate a measurement device be traceable to an internationally recognized standard such as those maintained by the National Institute of Standards and Technology (NIST). Within most companies, the responsibility for calibrating measurement devices is delegated to a specific department. Often, this department may also be responsible for maintaining the measurement device. The specific calibration procedures depend on the type of measurement device. The frequency of calibration is normally predetermined, but earlier action may be dictated if the values from the measurement device become suspect. Calibration of some measurement devices involves comparing the measured value with the value from the working standard. Pressure and differential pressure transmitters are calibrated in this manner. Calibration of analyzers normally involves using the measurement device to analyze a specially prepared sample whose composition is known. These and similar approaches can be applied to most measurement devices. Flow is an important measurement whose calibration presents some challenges. When a flow measurement device is used in applications such as custody transfer, provision is made to pass a known flow through the meter. However, such a provision is costly and is not available for most in-process flowmeters. Without such a provision, a true calibration of the flow element itself is not possible. For orifice meters, calibration of the flowmeter normally involves calibration of the differential pressure transmitter, and the orifice plate is usually only inspected for deformation, abrasion, etc. Similarly, calibration of a magnetic flowmeter normally involves calibration of the voltage

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TABLE 8-8 Online Measurement Options for Process Control Temperature Flow Pressure Thermocouple Resistance temperature detector (RTD) Filled-system thermometer Bimetal thermometer Pyrometer Total radiation Photoelectric Ratio Laser Surface acoustic wave Semiconductor

Orifice Venturi Rotameter Turbine Vortex-shedding Ultrasonic Magnetic Thermal mass Coriolis Target

Level

Liquid column Elastic element Bourdon tube Bellow Diaphragm Strain gauges Piezoresistive transducers Piezoelectric transducers Optical fiber

measurement circuitry, which is analogous to calibration of the differential pressure transmitter for an orifice meter. In the next section we cover the major types of measurement devices used in the process industries, principally the “big five” measurements: temperature, flow rate, pressure, level, and composition, along with online physical property measurement techniques. Table 8-8 summarizes the different options under each of the principal measurements. TEMPERATURE MEASUREMENTS Measurement of the hotness or coldness of a body or fluid is commonplace in the process industries. Temperature-measuring devices utilize systems with properties that vary with temperature in a simple, reproducible manner and thus can be calibrated against known references (sometimes called secondary thermometers). The three dominant measurement devices used in automatic control are thermocouples, resistance thermometers, and pyrometers, and they are applicable over different temperature regimes. Thermocouples Temperature measurements using thermocouples are based on the discovery by Seebeck in 1821 that an electric current flows in a continuous circuit of two different metallic wires if the two junctions are at different temperatures. The thermocouple may be represented diagrammatically as shown in Fig. 8-63. There A and B are the two metals, and Tl and T2 are the temperatures of the junctions. Let Tl and T2 be the reference junction (cold junction) and the measuring junction, respectively. If the thermoelectric current i flows in the direction indicated in Fig. 8-63, metal A is customarily referred to as thermoelectrically positive to metal B. Metal pairs used for thermocouples include platinum-rhodium (the most popular and accurate), chromel-alumel, copper-constantan, and iron-constantan. The thermal emf is a measure of the difference in temperature between T2 and Tl. In control systems the reference junction is usually located at the emf-measuring device. The reference junction may be held at constant temperature such as in an ice bath or a thermostated oven, or it may be at ambient temperature but electrically compen-

Composition

Float-activated Chain gauge Lever Magnetically coupled Head devices Bubble tube Electrical (conductivity) Sonic Laser Radiation Radar

Gas-liquid chromatography (GLC) Mass spectrometry (MS) Magnetic resonance analysis (MRA) Infrared (IR) spectroscopy Raman spectroscopy Ultraviolet (uv) spectroscopy Thermal conductivity Refractive index (RI) Capacitance probe Surface acoustic wave Electrophoresis Electrochemical Paramagnetic Chemi/bioluminescence Tunable diode laser absorption

sated (cold junction compensated circuit) so that it appears to be held at a constant temperature. Resistance Thermometers The resistance thermometer depends upon the inherent characteristics of materials to change in electrical resistance when they undergo a change in temperature. Industrial resistance thermometers are usually constructed of platinum, copper, or nickel, and more recently semiconducting materials such as thermistors are being used. Basically, a resistance thermometer is an instrument for measuring electrical resistance that is calibrated in units of temperature instead of in units of resistance (typically ohms). Several common forms of bridge circuits are employed in industrial resistance thermometry, the most common being the Wheatstone bridge. A resistance thermometer detector (RTD) consists of a resistance conductor (metal) which generally shows an increase in resistance with temperature. The following equation represents the variation of resistance with temperature (°C): RT = R0(1 + a1T + a2T 2 + . . . + anT n)

(8-92)

R0 = resistance at 0°C The temperature coefficient of resistance, αT is expressed as 1 dRT αT =   RT dT

(8-93)

For most metals αT is positive. For many pure metals, the coefficient is essentially constant and stable over large portions of their useful range. Typical resistance versus temperature curves for platinum, copper, and nickel are given in Fig. 8-64, with platinum usually the metal of choice. Platinum has a useful range of −200 to 800°C, while nickel (−80 to 320°C) and copper (−100 to 100°C) are more limited. Detailed resistance versus temperature tables are available from the National Institute of Standards and Technology (NIST) and suppliers of resistance thermometers. Table 8-9 gives recommended temperature measurement ranges for thermocouples and RTDs. Resistance thermometers are receiving increased usage because they are about 10 times more accurate than thermocouples. Thermistors Thermistors are nonlinear temperature-dependent resistors, and normally only the materials with negative temperature coefficient of resistance (NTC type) are used. The resistance is related to temperature as



1 1 RT = RT exp β  −  T Tr r



(8-94)

where αT is a reference temperature, which is generally 298 K. Thus

FIG. 8-63

Basic circuit of Seebeck effect.

1 dRT αT =   RT dT

(8-95)

PROCESS MEASUREMENTS

8-57

FIG. 8-64 Typical resistance thermometer curves for platinum, copper, and nickel wire, where RT = resistance at temperature T and R0 = resistance at 0°C.

The value of β is on the order of 4000, so at room temperature (298 K), αT = −0.045 for thermistor and 0.0035 for 100-Ω platinum RTD. Compared with RTDs, NTC-type thermistors are advantageous in that the detector dimension can be made small, resistance value is higher (less affected by the resistances of the connecting leads), and it has higher temperature sensitivity and low thermal inertia of the sensor. Disadvantages of thermistors to RTDs include nonlinear characteristics and low measuring temperature range. Filled-System Thermometers The filled-system thermometer is designed to provide an indication of temperature some distance removed from the point of measurement. The measuring element (bulb) contains a gas or liquid that changes in volume, pressure, or vapor pressure with temperature. This change is communicated through a capillary tube to a Bourdon tube or other pressure- or volume-sensitive device. The Bourdon tube responds so as to provide a motion related to the bulb temperature. Those systems that respond to volume changes are completely filled with a liquid. Systems that respond to TABLE 8-9 Recommended Temperature Measurement Ranges for RTDs and Thermocouples Resistance Thermometer Detectors (RTDs) −200–+850°C −80–+320°C

100V Pt 120V Ni Thermocouples Type B Type E Type J Type K Type N Type R Type S Type T

700–+1820°C −175–+1000°C −185–+1200°C −175–+1372°C 0–+1300°C 125–+1768°C 150–+1768°C −170–+400°C

pressure changes either are filled with a gas or are partially filled with a volatile liquid. Changes in gas or vapor pressure with changes in bulb temperatures are carried through the capillary to the Bourdon. The latter bulbs are sometimes constructed so that the capillary is filled with a nonvolatile liquid. Fluid-filled bulbs deliver enough power to drive controller mechanisms and even directly actuate control valves. These devices are characterized by large thermal capacity, which sometimes leads to slow response, particularly when they are enclosed in a thermal well for process measurements. Filled-system thermometers are used extensively in industrial processes for a number of reasons. The simplicity of these devices allows rugged construction, minimizing the possibility of failure with a low level of maintenance, and inexpensive overall design of control equipment. In case of system failure, the entire unit must be replaced or repaired. As they are normally used in the process industries, the sensitivity and percentage of span accuracy of these thermometers are generally the equal of those of other temperature-measuring instruments. Sensitivity and absolute accuracy are not the equal of those of short-span electrical instruments used in connection with resistance-thermometer bulbs. Also the maximum temperature is somewhat limited. Bimetal Thermometers Thermostatic bimetal can be defined as a composite material made up of strips of two or more metals fastened together. This composite, because of the different expansion rates of its components, tends to change curvature when subjected to a change in temperature. With one end of a straight strip fixed, the other end deflects in proportion to the temperature change, the square of the length, and inversely as the thickness, throughout the linear portion of the deflection characteristic curve. If a bimetallic strip is wound into a helix or a spiral and one end is fixed, the other end will rotate when heat is applied. For a thermometer with uniform scale divisions, a bimetal must be designed to have linear deflection over the desired temperature range. Bimetal thermometers are used at temperatures ranging from 580 down to −180°C and lower. However, at the low temperatures

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PROCESS CONTROL

the rate of deflection drops off quite rapidly. Bimetal thermometers do not have long-time stability at temperatures above 430°C. Pyrometers Planck’s distribution law gives the radiated energy flux qb(λ, T)d λ in the wavelength range λ to λ + dλ from a black surface: 1 C1 qb(λ, T) =   λ5 eC λT − 1 2



(8-97)

∞

0

(8-98)

−C2 Sλ = KC1ελ1λ−5 1 exp  λ1T



(8-100)

−C2  Sλ = KC1ε λ2λ−5 2 exp λ2T



(8-101)





2

The ratio of the signals Sλ and Sλ is 1

2

Sλ ελ λ2 5 C2 1 1  =   exp   −  Sλ ελ λ1 T λ2 λ1 1

1

2

2

  



(8-99)

where q(T) is the radiated energy flux from a real body with emissivity εT. Total Radiation Pyrometers In total radiation pyrometers, the thermal radiation is detected over a large range of wavelengths from the object at high temperature. The detector is normally a thermopile, which is built by connecting several thermocouples in series to increase the temperature measurement range. The pyrometer is calibrated for black bodies, so the indicated temperature Tp should be converted for non–black body temperature. Photoelectric Pyrometers Photoelectric pyrometers belong to the class of band radiation pyrometers. The thermal inertia of thermal radiation detectors does not permit the measurement of rapidly changing temperatures. For example, the smallest time constant of a thermal detector is about 1 ms while the smallest time constant of a photoelectric detector can be about 1 or 2 s. Photoelectric pyrometers may use photoconductors, photodiodes, photovoltaic cells, or vacuum photocells. Photoconductors are built from glass plates with thin film coatings of 1-µm thickness, using PbS, CdS, PbSe, or PbTe. When the incident radiation has the same wavelength as the materials are able to absorb, the captured incident photons free photoelectrons, which form an electric current. Photodiodes in germanium or silicon are operated with a reverse bias voltage applied. Under the influence of the incident radiation, their conductivity as well as their reverse saturation current is proportional to the intensity of the radiation within the spectral response band from 0.4 to 1.7 µm for Ge and from 0.6 to 1.1 µm for Si. Because of the above characteristics, the operating range of a photoelectric pyrometer can be either spectral or in specific band. Photoelectric pyrometers can be applied for a specific choice of the wavelength. Disappearing Filament Pyrometers Disappearing filament pyrometers can be classified as spectral pyrometers. The brightness of a lamp filament is changed by adjusting the lamp current until the filament disappears against the background of the target, at which point the temperature is measured. Because the detector is the human eye, it is difficult to calibrate for online measurements. Ratio Pyrometers The ratio pyrometer is also called the twocolor pyrometer. Two different wavelengths are utilized for detecting

(8-102)

Nonblack or nongray bodies are characterized by the wavelength dependence of their spectral emissivity. Let Tc be defined as the temperature of the body corresponding to the temperature of a black body. If the ratio of its radiant intensities at wavelengths λ1 and λ2 equals the ratio of the radiant intensities of the non–black body, whose temperature is to be measured at the same wavelength, then Wien’s law gives ελ exp(−C2λ1T) exp(−C2λ1Tc)  =  ελ exp(−C2λ2T) exp(−C2λ2Tc) 1

where σ is the Stefan-Boltzmann constant. Similar to Eq. (8-93), the emissivity εT for the total radiation is q(T) εT =  qb(T)

2

1

where q(λ,T) is the radiated energy flux from a real body in the wavelength range λ to λ + dλ and 0 < ελ,T < 1. Integrating Eq. (8-96) over all wavelengths gives the Stefan-Boltzmann equation qb(T) = qb(λ, T)dλ = σT4

1

(8-96)

where C1 = 3.7418 × 1010 µW⋅µm4⋅cm−2 and C2 = 14,388 µm⋅K. If the target object is a black body and if the pyrometer has a detector which measures the specific wavelength signal from the object, the temperature of the object can be exactly estimated from Eq. (8-96). While it is possible to construct a physical body that closely approximates black body behavior, most real-world objects are not black bodies. The deviation from a black body can be described by the spectral emissivity q(T) εT =  qb(T)

the radiated signal. If one uses Wien’s law for small values of AT, the detected signals from spectral radiant energy flux emitted at wavelengths λ1 and λ2 with emissivities ελ and ελ are

(8-103)

2

where T is the true temperature of the body. Rearranging Eq. (8-103) gives T=

ln ελ ελ

1

−1

+   C T (1λ −1λ ) 1

1

2

2

2

(8-104)

c

For black or gray bodies, Eq. (8-104) reduces to λ2 Sλ  =  λ1 Sλ

 exp T  λ − λ 

1

2

5

C2

1

1

2

1

(8-105)

Thus by measuring Sλ and Sλ , the temperature T can be estimated. Accuracy of Pyrometers Most of the temperature estimation methods for pyrometers assume that the object is either a gray body or has known emissivity values. The emissivity of the non–black body depends on the internal state or the surface geometry of the objects. Also the medium through which the thermal radiation passes is not always transparent. These inherent uncertainties of the emissivity values make the accurate estimation of the temperature of the target objects difficult. Proper selection of the pyrometer and accurate emissivity values can provide a high level of accuracy. 1

2

PRESSURE MEASUREMENTS Pressure, defined as force per unit area, is usually expressed in terms of familiar units of weight-force and area or the height of a column of liquid which produces a like pressure at its base. Process pressuremeasuring devices may be divided into three groups: (1) those based on the measurement of the height of a liquid column, (2) those based on the measurement of the distortion of an elastic pressure chamber, and (3) electrical sensing devices. Liquid-Column Methods Liquid-column pressure-measuring devices are those in which the pressure being measured is balanced against the pressure exerted by a column of liquid. If the density of the liquid is known, the height of the liquid column is a measure of the pressure. Most forms of liquid-column pressure-measuring devices are commonly called manometers. When the height of the liquid is

PROCESS MEASUREMENTS observed visually, the liquid columns are contained in glass or other transparent tubes. The height of the liquid column may be measured in length units or calibrated in pressure units. Depending on the pressure range, water and mercury are the liquids most frequently used. Because the density of the liquid used varies with temperature, the temperature must be taken into account for accurate pressure measurements. Elastic Element Methods Elastic element pressure-measuring devices are those in which the measured pressure deforms some elastic material (usually metallic) within its elastic limit, the magnitude of the deformation being approximately proportional to the applied pressure. These devices may be loosely classified into three types: Bourdon tube, bellows, and diaphragm. Bourdon Tube Elements Probably the most frequently used process pressure-indicating device is the C-spring Bourdon tube pressure gauge. Gauges of this general type are available in a wide variety of pressure ranges and materials of construction. Materials are selected on the basis of pressure range, resistance to corrosion by the process materials, and effect of temperature on calibration. Gauges calibrated with pressure, vacuum, compound (combination pressure and vacuum), and suppressed-zero ranges are available. Bellows Element The bellows element is an axially elastic cylinder with deep folds or convolutions. The bellows may be used unopposed, or it may be restrained by an opposing spring. The pressure to be measured may be applied either to the inside or to the space outside the bellows, with the other side exposed to atmospheric pressure. For measurement of absolute pressure either the inside or the space outside of the bellows can be evacuated and sealed. Differential pressures may be measured by applying the pressures to opposite sides of a single bellows or to two opposing bellows. Diaphragm Elements Diaphragm elements may be classified into two principal types: those that utilize the elastic characteristics of the diaphragm and those that are opposed by a spring or other separate elastic element. The first type usually consists of one or more capsules, each composed of two diaphragms bonded together by soldering, brazing, or welding. The diaphragms are flat or corrugated circular metallic disks. Metals commonly used in diaphragm elements include brass, phosphor bronze, beryllium copper, and stainless steel. Ranges are available from fractions of an inch of water to over 800 in (200 kPa) gauge. The second type of diaphragm is used for containing the pressure and exerting a force on the opposing elastic element. The diaphragm is a flexible or slack diaphragm of rubber, leather, impregnated fabric, or plastic. Movement of the diaphragm is opposed by a spring which determines the deflection for a given pressure. This type of diaphragm is used for the measurement of extremely low pressure, vacuum, or differential pressure. Electrical Methods Electrical methods for pressure measurement include strain gauges, piezoresistive transducers, and piezoelectric transducers. Strain Gauges When a wire or other electrical conductor is stretched elastically, its length is increased and its diameter is decreased. Both of these dimensional changes result in an increase in the electrical resistance of the conductor. Devices utilizing resistancewire grids for measuring small distortions in elastically stressed materials are commonly called strain gauges. Pressure-measuring elements utilizing strain gauges are available in a wide variety of forms. They usually consist of one of the elastic elements described earlier to which one or more strain gauges have been attached to measure the deformation. There are two basic strain gauge forms: bonded and unbonded. Bonded strain gauges are bonded directly to the surface of the elastic element whose strain is to be measured. The unbonded strain gauge transducer consists of a fixed frame and an armature that moves with respect to the frame in response to the measured pressure. The strain gauge wire filaments are stretched between the armature and frame. The strain gauges are usually connected electrically in a Wheatstone bridge configuration. Strain gauge pressure transducers are manufactured in many forms for measuring gauge, absolute, and differential pressures and vacuum. Full-scale ranges from 25.4 mm of water to 10,134 MPa are available. Strain gauges bonded directly to a diaphragm pressure-sensitive element usually have an extremely fast response time and are suitable for high-frequency dynamic pressure measurements.

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Piezoresistive Transducers A variation of the conventional strain gauge pressure transducer uses bonded single-crystal semiconductor wafers, usually silicon, whose resistance varies with strain or distortion. Transducer construction and electrical configurations are similar to those using conventional strain gauges. A permanent magnetic field is applied perpendicular to the resonating sensor. An alternating current causes the resonator to vibrate, and the resonant frequency is a function of the pressure (tension) of the resonator. The principal advantages of piezoresistive transducers are a much higher bridge voltage output and smaller size. Full-scale output voltages of 50 to 100 mV/V of excitation are typical. Some newer devices provide digital rather than analog output. Piezoelectric Transducers Certain crystals produce a potential difference between their surfaces when stressed in appropriate directions. Piezoelectric pressure transducers generate a potential difference proportional to a pressure-generated stress. Because of the extremely high electrical impedance of piezoelectric crystals at low frequency, these transducers are usually not suitable for measurement of static process pressures. FLOW MEASUREMENTS Flow, defined as volume per unit of time at specified temperature and pressure conditions, is generally measured by positive displacement or rate meters. The term positive displacement meter applies to a device in which the flow is divided into isolated measured volumes when the number of fillings of these volumes is counted in some manner. The term rate meter applies to all types of flowmeters through which the material passes without being divided into isolated quantities. Movement of the material is usually sensed by a primary measuring element that activates a secondary device. The flow rate is then inferred from the response of the secondary device by means of known physical laws or from empirical relationships. The principal classes of flow-measuring instruments used in the process industries are variable-head, variable-area, positive-displacement, and turbine instruments; mass flowmeters; vortex-shedding and ultrasonic flowmeters; magnetic flowmeters; and more recently, Coriolis mass flowmeters. Head meters are covered in detail in Sec. 5. Orifice Meter The most widely used flowmeter involves placing a fixed-area flow restriction (an orifice) in the pipe carrying the fluid. This flow restriction causes a pressure drop which can be related to flow rate. The sharp-edge orifice is popular because of its simplicity, low cost, and the large amount of research data on its behavior. For the orifice meter, the flow rate Qa for a liquid is given by CdA2 Qa =  ⋅ 2  1 − (A 2A1)

2(p − p )  ρ 1

2

(8-106)

where p1 − p2 is the pressure drop, ρ is the density, A1 is the pipe crosssectional area, A2 is the orifice cross-sectional area, and Cd is the discharge coefficient. The discharge coefficient Cd varies with the Reynolds number at the orifice and can be calibrated with a single fluid, such as water (typically Cd ≈ 0.6). If the orifice and pressure taps are constructed according to certain standard dimensions, quite accurate (about 0.4 to 0.8 percent error) values of Cd may be obtained. Also note that the standard calibration data assume no significant flow disturbances such as elbows and valves for a certain minimum distance upstream of the orifice. The presence of such disturbances close to the orifice can cause errors of as much as 15 percent. Accuracy in measurements limits the meter to a flow rate range of 3:1. The orifice has a relatively large permanent pressure loss that must be made up by the pumping machinery. Venturi Meter The venturi tube operates on exactly the same principle as the orifice [see Eq. (8-102)]. Discharge coefficients of venturis are larger than those for orifices and vary from about 0.94 to 0.99. A venturi gives a definite improvement in power losses over an orifice and is often indicated for measuring very large flow rates, where power losses can become economically significant. The initial higher cost of a venturi over an orifice may thus be offset by reduced operating costs.

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Rotameter A rotameter consists of a vertical tube with a tapered bore in which a float changes position with the flow rate through the tube. For a given flow rate, the float remains stationary because the vertical forces of differential pressure, gravity, viscosity, and buoyancy are balanced. The float position is the output of the meter and can be made essentially linear with flow rate by making the tube area vary linearly with the vertical distance. Turbine Meter If a turbine wheel is placed in a pipe containing a flowing fluid, its rotary speed depends on the flow rate of the fluid. A turbine can be designed whose speed varies linearly with flow rate. The speed can be measured accurately by counting the rate at which turbine blades pass a given point, using magnetic pickup to produce voltage pulses. By feeding these pulses to an electronic pulse rate meter one can measure flow rate by summing the pulses during a timed interval. Turbine meters are available with full-scale flow rates ranging from about 0.1 to 30,000 gal/min for liquids and 0.1 to 15,000 ft3/min for air. Nonlinearity can be less than 0.05 percent in the larger sizes. Pressure drop across the meter varies with the square of flow rate and is about 3 to 10 psi at full flow. Turbine meters can follow flow transients quite accurately since their fluid/mechanical time constant is on the order of 2 to 10 ms. Vortex-Shedding Flowmeters These flowmeters take advantage of vortex shedding, which occurs when a fluid flows past a nonstreamlined object (a blunt body). The flow cannot follow the shape of the object and separates from it, forming turbulent vortices or eddies at the object’s side surfaces. As the vortices move downstream, they grow in size and are eventually shed or detached from the object. Shedding takes place alternately at either side of the object, and the rate of vortex formation and shedding is directly proportional to the volumetric flow rate. The vortices are counted and used to develop a signal linearly proportional to the flow rate. The digital signals can easily be totaled over an interval of time to yield the flow rate. Accuracy can be maintained regardless of density, viscosity, temperature, or pressure when the Reynolds number is greater than 10,000. There is usually a low flow cutoff point below which the meter output is clamped at zero. This flowmeter is recommended for use with relatively clean, low-viscosity liquids, gases, and vapors, and rangeability of 10:1 to 20:1 is typical. A sufficient length of straight-run pipe is necessary to prevent distortion in the fluid velocity profile. Ultrasonic Flowmeters An ultrasonic flowmeter is based upon the variable time delays of received sound waves which arise when a flowing liquid’s rate of flow is varied. Two fundamental measurement techniques, depending upon liquid cleanliness, are generally used. In the first technique two opposing transducers are inserted in a pipe so that one transducer is downstream from the other. These transducers are then used to measure the difference between the velocity at which the sound travels with the direction of flow and the velocity at which it travels against the direction of flow. The differential velocity is measured either by (1) direct time delays using sound wave burst or (2) frequency shifts derived from beat-together, continuous signals. The frequency measurement technique is usually preferred because of its simplicity and independence of the liquid static velocity. A relatively clean liquid is required to preserve the uniqueness of the measurement path. In the second technique, the flowing liquid must contain scatters in the form of particles or bubbles which will reflect the sound waves. These scatters should be traveling at the velocity of the liquid. A Doppler method is applied by transmitting sound waves along the flow path and measuring the frequency shift in the returned signal from the scatters in the process fluid. This frequency shift is proportional to liquid velocity. Magnetic Flowmeters The principle behind these flowmeters is Faraday’s law of electromagnetic inductance. The magnitude of the voltage induced in a conductive medium moving at right angles through a magnetic field is directly proportional to the product of the magnetic flux density, the velocity of the medium, and the path length between the probes. A minimum value of fluid conductivity is required to make this approach viable. The pressure of multiple phases or undissolved solids can affect the accuracy of the measurement if the velocities of the phases are different from that for straight-run pipe. Magmeters are very accurate over wide flow ranges and are especially

accurate at low flow rates. Typical applications include metering viscous fluids, slurries, or highly corrosive chemicals. Because magmeters should be filled with fluid, the preferred installation is in vertical lines with flow going upward. However, magmeters can be used in tight piping schemes where it is impractical to have long pipe runs, typically requiring lengths equivalent to five or more pipe diameters. Coriolis Mass Flowmeters Coriolis mass flowmeters utilize a vibrating tube in which Coriolis acceleration of a fluid in a flow loop can be created and measured. They can be used with virtually any liquid and are extremely insensitive to operating conditions, with pressure ranges over 100:1. These meters are more expensive than volumetric meters and range in size from 116 to 6 in. Due to the circuitous path of flow through the meter, Coriolis flowmeters exhibit higher than average pressure changes. The meter should be installed so that it will remain full of fluid, with the best installation in a vertical pipe with flow going upward. There is no Reynolds number limitation with this meter, and it is quite insensitive to velocity profile distortions and swirl, hence there is no requirement for straight piping upstream. Thermal Mass Flowmeters The trend in the chemical process industries is toward increased usage of mass flowmeters that are independent of changes in pressure, temperature, viscosity, and density. Thermal mass meters are widely used in semiconductor manufacturing and in bioprocessing for control of low flow rates (called mass flow controllers, or MFCs). MFCs measure the heat loss from a heated element, which varies with flow rate, with an accuracy of ± 1 percent. Capacitance probes measure the dielectric constant of the fluid and are useful for flow measurements of slurries and other two-phase flows. LEVEL MEASUREMENTS The measurement of level can be defined as the determination of the location of the interface between two fluids, separable by gravity, with respect to a fixed reference plane. The most common level measurement is that of the interface between a liquid and a gas. Other level measurements frequently encountered are the interface between two liquids, between a granular or fluidized solid and a gas, and between a liquid and its vapor. A commonly used basis for classification of level devices is as follows: float-actuated, displacer, and head devices, and a miscellaneous group which depends mainly on fluid characteristics. Float-Actuated Devices Float-actuated devices are characterized by a buoyant member which floats at the interface between two fluids. Because a significant force is usually required to move the indicating mechanism, float-actuated devices are generally limited to liquid-gas interfaces. By properly weighting the float, they can be used to measure liquid-liquid interfaces. Float-actuated devices may be classified on the basis of the method used to couple the float motion to the indicating system, as discussed below. Chain or Tape Float Gauge In these types of gauges, the float is connected to the indicating mechanism by means of a flexible chain or tape. These gauges are commonly used in large atmospheric storage tanks. The gauge-board type is provided with a counterweight to keep the tape or chain taut. The tape is stored in the gauge head on a spring-loaded reel. The float is usually a pancake-shaped hollow metal float, with guide wires from top to bottom of the tank to constrain it. Lever and Shaft Mechanisms In pressurized vessels, floatactuated lever and shaft mechanisms are frequently used for level measurement. This type of mechanism consists of a hollow metal float and lever attached to a rotary shaft which transmits the float motion to the outside of the vessel through a rotary seal. Magnetically Coupled Devices A variety of float-actuated level devices which transmit the float motion by means of magnetic coupling have been developed. Typical of this class of devices are magnetically operated level switches and magnetic-bond float gauges. A typical magnetic-bond float gauge consists of a hollow magnet-carrying float which rides along a vertical nonmagnetic guide tube. The follower magnet is connected and drives an indicating dial similar to that on a conventional tape float gauge. The float and guide tube are in contact with the measured fluid and come in a variety of materials for resistance to corrosion and to withstand high pressures or vacuum. Weighted floats for liquid-liquid interfaces are available.

PROCESS MEASUREMENTS Head Devices A variety of devices utilize hydrostatic head as a measure of level. As in the case of displacer devices, accurate level measurement by hydrostatic head requires an accurate knowledge of the densities of both heavier-phase and lighter-phase fluids. The majority of this class of systems utilize standard pressure and differential pressure measuring devices. Bubble Tube Systems The commonly used bubble tube system sharply reduces restrictions on the location of the measuring element. To eliminate or reduce variations in pressure drop due to the gas flow rate, a constant differential regulator is commonly employed to maintain a constant gas flow rate. Because the flow of gas through the bubble tube prevents entry of the process liquid into the measuring system, this technique is particularly useful with corrosive or viscous liquids, liquids subject to freezing, and liquids containing entrained solids. Electrical Methods Two electrical characteristics of fluids, conductivity and dielectric constant, are frequently used to distinguish between two phases for level measurement purposes. An application of electrical conductivity is the fixed-point level detection of a conductive liquid such as high and low water levels. A voltage is applied between two electrodes inserted into the vessel at different levels. When both electrodes are immersed in the liquid, a current flows. Capacitance-type level measurements are based on the fact that the electrical capacitance between two electrodes varies with the dielectric constant of the material between them. A typical continuous level measurement system consists of a rod electrode positioned vertically in a vessel, the other electrode usually being the metallic vessel wall. The electrical capacitance between the electrodes is a measure of the height of the interface along the rod electrode. The rod is usually conductively insulated from process fluids by a coating of plastic. The dielectric constants of most liquids and solids are markedly higher than those of gases and vapors (by a factor of 2 to 5). The dielectric constant of water and other polar liquids is 10 to 20 times that of hydrocarbons and other nonpolar liquids. Thermal Methods Level-measuring systems may be based on the difference in thermal characteristics between the fluids, such as temperature or thermal conductivity. A fixed-point level sensor based on the difference in thermal conductivity between two fluids consists of an electrically heated thermistor inserted into the vessel. The temperature of the thermistor and consequently its electrical resistance increase as the thermal conductivity of the fluid in which it is immersed decreases. Because the thermal conductivity of liquids is markedly higher than that of vapors, such a device can be used as a point level detector for liquid-vapor interface. Sonic Methods A fixed-point level detector based on sonic propagation characteristics is available for detection of a liquid-vapor interface. This device uses a piezoelectric transmitter and receiver, separated by a short gap. When the gap is filled with liquid, ultrasonic energy is transmitted across the gap, and the receiver actuates a relay. With a vapor filling the gap, the transmission of ultrasonic energy is insufficient to actuate the receiver. Laser Level Transmitters These are designed for bulk solids, slurries, and opaque liquids. A laser near the vessel top fires a short pulse of light down to the surface of the process liquid, where it reflects back to a detector at the vessel top. A timing circuit measures the elapsed time and calculates the fluid depth. Lasers are attractive because lasers have no false echoes and can be directed through tight spaces. Radar Level Transmitters Radar systems operate by beaming microwaves downward, from either a horn or parabolic dish located on top of the vessel. The signal reflects off the fluid surface back to the source after it detects a change in dielectric constant from the vapor to the fluid. The round-trip time is proportional to the distance to the fluid level. Guided-wave radar systems provide a rigid probe or flexible cable to guide the microwave down the height of the tank and back. Guided-wave radar is much more efficient than open-air radar because the guide provides a more focused energy path. PHYSICAL PROPERTY MEASUREMENTS Physical property measurements are sometimes equivalent to composition analyzers, because the composition can frequently be inferred from the measurement of a selected physical property.

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Density and Specific Gravity For binary or pseudobinary mixtures of liquids or gases or a solution of a solid or gas in a solvent, the density is a function of the composition at a given temperature and pressure. Specific gravity is the ratio of the density of a noncompressible substance to the density of water at the same physical conditions. For nonideal solutions, empirical calibration will give the relationship between density and composition. Several types of measuring devices are described below. Liquid Column Density may be determined by measuring the gauge pressure at the base of a fixed-height liquid column open to the atmosphere. If the process system is closed, then a differential pressure measurement is made between the bottom of the fixed-height liquid column and the vapor over the column. If vapor space is not always present, the differential pressure measurement is made between the bottom and top of a fixed-height column with the top measurement being made at a point below the liquid surface. Displacement There are a variety of density measurement devices based on displacement techniques. A hydrometer is a constant-weight, variable-immersion device. The degree of immersion, when the weight of the hydrometer equals the weight of the displaced liquid, is a measure of the density. The hydrometer is adaptable to manual or automatic usage. Another modification includes a magnetic float suspended below a solenoid, the varying magnetic field maintaining the float at a constant distance from the solenoid. Change in position of the float, resulting from a density change, excites an electrical system which increases or decreases the current through the solenoid. Direct Mass Measurement One type of densitometer measures the natural vibration frequency and relates the amplitude to changes in density. The density sensor is a V-shaped tube held stationary at its node points and allowed to vibrate at its natural frequency. At the curved end of the V is an electrochemical device that periodically strikes the tube. At the other end of the V the fluid is continuously passed through the tube. Between strikes, the tube vibrates at its natural frequency. The frequency changes directly in proportion to changes in density. A pickup device at the curved end of the V measures the frequency and electronically determines the fluid density. This technique is useful because it is not affected by the optical properties of the fluid. However, particulate matter in the process fluid can affect the accuracy. Radiation-Density Gauges Gamma radiation may be used to measure the density of material inside a pipe or process vessel. The equipment is basically the same as for level measurement, except that here the pipe or vessel must be filled over the effective, irradiated sample volume. The source is mounted on one side of the pipe or vessel and the detector on the other side with appropriate safety radiation shielding surrounding the installation. Cesium 137 is used as the radiation source for path lengths under 610 mm (24 in) and cobalt 60 above 610 mm. The detector is usually an ionization gauge. The absorption of the gamma radiation is a function of density. Since the absorption path includes the pipe or vessel walls, an empirical calibration is used. Appropriate corrections must be made for the source intensity decay with time. Viscosity Continuous viscometers generally measure either the resistance to flow or the drag or torque produced by movement of an element (moving surface) through the fluid. Each installation is normally applied over a narrow range of viscosities. Empirical calibration over this range allows use on both newtonian and nonnewtonian fluids. One such device uses a piston inside a cylinder. The hydrodynamic pressure of the process fluid raises the piston to a preset height. Then the inlet valve closes, the piston is allowed to free-fall, and the time of travel (typically a few seconds) is a measure of viscosity. Other geometries include the rotation of a spindle inside a sample chamber and a vibrating probe immersed in the fluid. Because viscosity depends on temperature, the viscosity measurement must be thermostated with a heater or cooler. Refractive Index When light travels from one medium (e.g., air or glass) into another (e.g., a liquid), it undergoes a change of velocity and, if the angle of incidence is not 90°, a change of direction. For a given interface, angle, temperature, and wavelength of light, the amount of deviation or refraction will depend on the composition of

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the liquid. If the sample is transparent, the normal method is to measure the refraction of light transmitted through the glass-sample interface. If the sample is opaque, the reflectance near the critical angle at a glass-sample interface is measured. In an online refractometer the process fluid is separated from the optics by a prism material. A beam of light is focused on a point in the fluid which creates a conic section of light at the prism, striking the fluid at different angles (greater than or less than the critical angle). The critical angle depends on the species concentrations; as the critical angle changes, the proportions of reflected and refracted light change. A photodetector produces a voltage signal proportional to the light refracted, when compared to a reference signal. Refractometers can be used with opaque fluids and in streams that contain particulates. Dielectric Constant The dielectric constant of material represents its ability to reduce the electric force between two charges separated in space. This property is useful in process control for polymers, ceramic materials, and semiconductors. Dielectric constants are measured with respect to vacuum (1.0); typical values range from 2 (benzene) to 33 (methanol) to 80 (water). The value for water is higher than that for most plastics. A measuring cell is made of glass or some other insulating material and is usually doughnut-shaped, with the cylinders coated with metal, which constitute the plates of the capacitor. Thermal Conductivity All gases and vapor have the ability to conduct heat from a heat source. At a given temperature and physical environment, radiation and convection heat losses will be stabilized, and the temperature of the heat source will mainly depend on the thermal conductivity and thus the composition of the surrounding gases. Thermal conductivity analyzers normally consist of a sample cell and a reference cell, each containing a combined heat source and detector. These cells are normally contained in a metal block with two small cavities in which the detectors are mounted. The sample flows through the sample cell cavity past the detector. The reference cell is an identical cavity with a detector through which a known gas flows. The combined heat source and detectors are normally either wire filaments or thermistors heated by a constant current. Because their resistance is a function of temperature, the sample detector resistance will vary with sample composition while the reference detector resistance will remain constant. The output from the detector bridge will be a function of sample composition. CHEMICAL COMPOSITION ANALYZERS Chemical composition is generally the most challenging online measurement. Before the era of online analyzers, messengers were required to deliver samples to the laboratory for analysis and to return the results to the control room. The long time delay involved prevented process adjustment from being made, affecting product quality. The development of online analyzers has automated this approach and reduced the analysis time. However, manual sampling is still frequently employed, especially in the specialty chemical industry where few instruments are commercially available. It is not unusual for a chemical composition analysis system to cost over $100,000, so it is important to assess the payback of such an investment versus the cost of manual sampling. Potential quality improvements can be an important consideration. A number of composition analyzers used for process monitoring and control require chemical conversion of one or more sample components preceding quantitative measurement. These reactions include formation of suspended solids for turbidimetric measurement, formation of colored materials for colorimetric detection, selective oxidation or reduction for electrochemical measurement, and formation of electrolytes for measurement by electrical conductance. Some nonvolatile materials may be separated and measured by gas chromatography after conversion to volatile derivatives. Chromatographic Analyzers These analyzers are widely used for the separation and measurement of volatile compounds and of compounds that can be quantitatively converted to volatile derivatives. The compounds to be measured are separated by placing a portion of the sample in a chromatographic column and carrying the compounds through the column with a gas stream, called gas chroma-

tography, or GC. As a result of the different affinities of the sample components for the column packing, the compounds emerge successively as binary mixtures with the carrier gas. A detector at the column outlet measures a specific physical property that can be related to the concentrations of the compounds in the carrier gas. Both the concentration peak height and the peak height-time integral, i.e., peak area, can be related to the concentration of the compound in the original sample. The two detectors most commonly used for process chromatographs are the thermal conductivity detector and the hydrogen flame ionization detector. Thermal conductivity detectors, discussed earlier, require calibration for the thermal response of each compound. Hydrogen flame ionization detectors are more complicated than thermal conductivity detectors but are capable of 100 to 10,000 times greater sensitivity for hydrocarbons and organic compounds. For ultrasensitive detection of trace impurities, carrier gases must be specially purified. Typically, all components can be analyzed in a 5- to 10-min time period (although miniaturized GCs are faster). High-performance liquid chromatography (HPLC) can be used to measure dissolved solute levels, including proteins. Infrared Analyzers Many gaseous and liquid compounds absorb infrared radiation to some degree. The degree of absorption at specific wavelengths depends on molecular structure and concentration. There are two common detector types for nondispersive infrared analyzers. These analyzers normally have two beams of radiation, an analyzing and a reference beam. One type of detector consists of two gas-filled cells separated by a diaphragm. As the amount of infrared energy absorbed by the detector gas in one cell changes, the cell pressure changes. This causes movement in the diaphragm, which in turn causes a change in capacitance between the diaphragm and a reference electrode. This change in electrical capacitance is measured as the output. The second type of detector consists of two thermopiles or two bolometers, one in each of the two radiation beams. The infrared radiation absorbed by the detector is measured by a differential thermocouple output or a resistance thermometer (bolometer) bridge circuit. There are two common detector types for nondispersive analyzers. These analyzers normally have two beams of radiation, an analyzing and a reference beam. One type of detector consists of two gas-filled cells separated by a diaphragm. As the amount of infrared energy absorbed by the detector gas in one cell changes, the cell pressure changes. This causes movement in the diaphragm, which in turn causes a change in capacitance between the diaphragm and a reference electrode. This change in electrical capacitance is measured as the output. The second type of detector consists of two thermopiles or two bolometers, one in each of the two radiation beams. The infrared radiation absorbed by the detector is measured by a differential thermocouple output or a resistance thermometer (bolometer) bridge circuit. With gas-filled detectors, a chopped light system is normally used in which one side of the detector sees the source through the analyzing beam and the other side sees through the reference beam, alternating at a frequency of a few hertz. Ultraviolet and Visible-Radiation Analyzers Many gas and liquid compounds absorb radiation in the near-ultraviolet or visible region. For example, organic compounds containing aromatic and carbonyl structural groups are good absorbers in the ultraviolet region. Also many inorganic salts and gases absorb in the ultraviolet or visible region. In contrast, straight chain and saturated hydrocarbons, inert gases, air, and water vapor are essentially transparent. Process analyzers are designed to measure the absorbance in a particular wavelength band. The desired band is normally isolated by means of optical filters. When the absorbance is in the visible region, the term colorimetry is used. A phototube is the normal detector. Appropriate optical filters are used to limit the energy reaching the detector to the desired level and the desired wavelength region. Because absorption by the sample is logarithmic if a sufficiently narrow wavelength region is used, an exponential amplifier is sometimes used to compensate and produce a linear output. Paramagnetism A few gases including O2, NO, and NO2 exhibit paramagnetic properties as a result of unpaired electrons. In a nonuniform magnetic field, paramagnetic gases, because of their magnetic susceptibility, tend to move toward the strongest part of the

PROCESS MEASUREMENTS field, thus displacing diamagnetic gases. Paramagnetic susceptibility of these gases decreases with temperature. These effects permit measurement of the concentration of the strongest paramagnetic gas, oxygen. An oxygen analyzer uses a dumbbell suspended in the magnetic field which is repelled or attracted toward the magnetic field depending on the magnetic susceptibility of the gas. Other Analyzers Mass spectroscopy (MS) determines the partial pressures of gases in a mixture of directing ionized gases into a detector under a vacuum (10−6 torr), and the gas phase composition is then monitored more or less continuously based on the molecular weight of the species (Nichols, 1988). Sometimes GC is combined with MS to obtain a higher level of discrimination of the components present. Fiber-optic sensors are attractive options (although highercost) for acquiring measurements in harsh environments such as high temperature or pressure. The transducing technique used by these sensors is optical and does not involve electric signals, so they are immune to electromagnetic interference. Raman spectroscopy uses fiber optics and involves pulsed light scattering by molecules. It has a wide variety of applications in process control. Workman, Koch, and Veltkamp, Anal. Chem., 75: 2859, 2003. Significant advances have occurred during the past decade to miniaturize the size of the measurement system in order to make online analysis economically feasible and to reduce the time delays that often are present in analyzers. Recently, chemical sensors have been placed on microchips, even those requiring multiple physical, chemical, and biochemical steps (such as electrophoresis) in the analysis. This device has been called lab-on-a-chip. The measurements of chemical composition can be direct or indirect, the latter case referring to applications where some property of the process stream is measured (such as refractive index) and then related to composition of a particular component. ELECTROANALYTICAL INSTRUMENTS Conductometric Analysis Solutions of electrolytes in ionizing solvents (e.g., water) conduct current when an electrical potential is applied across electrodes immersed in the solution. Conductance is a function of ion concentration, ionic charge, and ion mobility. Conductance measurements are ideally suited for measurement of the concentration of a single strong electrolyte in dilute solutions. At higher concentrations conductance becomes a complex, nonlinear function of concentration requiring suitable calibration for quantitative measurements. Measurement of pH The primary detecting element in pH measurement is the glass electrode. A potential is developed at the pH-sensitive glass membrane as a result of differences in hydrogen ion activity in the sample and a standard solution contained within the electrode. This potential measured relative to the potential of the reference electrode gives a voltage which is expressed as pH. Instrumentation for pH measurement is among the most widely used process measurement devices. Rugged electrode systems and highly reliable electronic circuits have been developed for this use. After installation, the majority of pH measurement problems are sensor-related, mostly on the reference side, including junction plugging, poisoning, and depletion of electrolyte. For the glass (measuring electrode), common difficulties are broken or cracked glass, coating, and etching or abrasion. Symptoms such as drift, sluggish response, unstable readings, and inability to calibrate are indications of measurement problems. Online diagnostics such as impedance measurements, wiring checks, and electrode temperature are now available in most instruments. Other characteristics that can be measured offline include efficiency or slope and asymmetry potential (offset), which indicate whether the unit should be cleaned or changed [Nichols, Chem. Engr. Prog., 90(12):64, 1994; McMillan, Chem. Engr. Prog., 87(12):30, 1991]. Specific-Ion Electrodes In addition to the pH glass electrode specific for hydrogen ions, a number of electrodes which are selective for the measurement of other ions have been developed. This selectivity is obtained through the composition of the electrode membrane (glass, polymer, or liquid-liquid) and the composition of the electrode. These electrodes are subject to interference from other ions, and the

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response is a function of the total ionic strength of the solution. However, electrodes have been designed to be highly selective for specific ions, and when properly used, these provide valuable process measurements. MOISTURE MEASUREMENT Moisture measurements are important in the process industries because moisture can foul products, poison reactions, damage equipment, or cause explosions. Moisture measurements include both absolute moisture methods and relative-humidity methods. The absolute methods provide a primary output that can be directly calibrated in terms of dew point temperature, molar concentration, or weight concentration. Loss of weight on heating is the most familiar of these methods. The relative-humidity methods provide a primary output that can be more directly calibrated in terms of percentage of saturation of moisture. Dew Point Method For many applications the dew point is the desired moisture measurement. When concentration is desired, the relation between water content and dew point is well known and available. The dew point method requires an inert surface whose temperature can be adjusted and measured, a sample gas stream flowing past the surface, a manipulated variable for adjusting the surface temperature to the dew point, and a means of detecting the onset of condensation. Although the presence of condensate can be detected electrically, the original and most often used method is the optical detection of change in light reflection from an inert metallic-surface mirror. Some instruments measure the attenuation of reflected light at the onset of condensation. Others measure the increase of light dispersed and scattered by the condensate instead of, or in addition to, the reflectedlight measurement. Surface cooling is obtained with an expendable refrigerant liquid, conventional mechanical refrigeration, or thermoelectric cooling. Surface-temperature measurement is usually made with a thermocouple or a thermistor. Piezoelectric Method A piezoelectric crystal in a suitable oscillator circuit will oscillate at a frequency dependent on its mass. If the crystal has a stable hygroscopic film on its surface, the equivalent mass of the crystal varies with the mass of water sorbed in the film. Thus the frequency of oscillation depends on the water in the film. The analyzer contains two such crystals in matched oscillator circuits. Typically, valves alternately direct the sample to one crystal and a dry gas to the other on a 30-s cycle. The oscillator frequencies of the two circuits are compared electronically, and the output is the difference between the two frequencies. This output is then representative of the moisture content of the sample. The output frequency is usually converted to a variable dc voltage for meter readout and recording. Multiple ranges are provided for measurement from about 1 ppm to near saturation. The dry reference gas is preferably the same as the sample except for the moisture content of the sample. Other reference gases which are adsorbed in a manner similar to the dried sample gas may be used. The dry gas is usually supplied by an automatic dryer. The method requires a vapor sample to the detector. Mist striking the detector destroys the accuracy of measurement until it vaporizes or is washed off the crystals. Water droplets or mist may destroy the hygroscopic film, thus requiring crystal replacement. Vaporization or gasliquid strippers may sometimes be used for the analysis of moisture in liquids. Capacitance Method Several analyzers utilize the high dielectric constant of water for its detection in solutions. The alternating electric current through a capacitor containing all or part of the sample between the capacitor plates is measured. Selectivity and sensitivity are enhanced by increasing the concentration of moisture in the cell by filling the capacitor sample cell with a moisture-specific sorbent as part of the dielectric. This both increases the moisture content and reduces the amount of other interfering sample components. Granulated alumina is the most frequently used sorbent. These detectors may be cleaned and recharged easily and with satisfactory reproducibility if the sorbent itself is uniform. Oxide Sensors Aluminum oxide can be used as a sensor for moisture analysis. A conductivity call has one electrode node of aluminum,

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which is anodized to form a thin film of aluminum oxide, followed by coating with a thin layer of gold (the opposite electrode). Moisture is selectively adsorbed through the gold layer and into the hygroscopic aluminum oxide layer, which in turn determines the electrical conductivity between gold and aluminum oxide. This value can be related to ppm water in the sample. This sensor can operate between near vacuum to several hundred atmospheres, and it is independent of flow rate (including static conditions). Temperature, however, must be carefully monitored. A similar device is based on phosphorous pentoxide. Moisture content influences the electric current between two inert metal electrodes, which are fabricated as a helix on the inner wall of a tubular nonconductive sample cell. For a constant dc voltage applied to the electrodes, a current flows that is proportional to moisture. The moisture is absorbed into the hygroscopic phosphorous pentoxide, where the current electrolyzes the water molecules into hydrogen and oxygen. This sensor will handle moisture up to 1000 ppm and 6-atm pressure. As with the aluminum oxide ion, temperature control is very important. Photometric Moisture Analysis This analyzer requires a light source, a filter wheel rotated by a synchronous motor, a sample cell, a detector to measure the light transmitted, and associated electronics. Water has two absorption bands in the near-infrared region at 1400 and 1900 nm. This analyzer can measure moisture in liquid or gaseous samples at levels from 5 ppm up to 100 percent, depending on other chemical species in the sample. Response time is less than 1 s, and samples can be run up to 300°C and 400 psig. OTHER TRANSDUCERS Other types of transducers used in process measurements include mechanical drivers such as gear trains and electrical drivers such as a differential transformer or a Hall effect (semiconductor-based) sensor. Gear Train Rotary motion and angular position are easily transduced by various types of gear arrangements. A gear train in conjunction with a mechanical counter is a direct and effective way to obtain a digital readout of shaft rotations. The numbers on the counter can mean anything desired, depending on the gear ratio and the actuating device used to turn the shaft. A pointer attached to a gear train can be used to indicate a number of revolutions or a small fraction of a revolution for any specified pointer rotation. Differential Transformer These devices produce an ac electrical output from linear movement of an armature. They are very versatile in that they can be designed for a full range of output with any range of armature travel up to several inches. The transformers have one or two primaries and two secondaries connected to oppose each other. With an ac voltage applied to the primary, the output voltage depends on the position of the armature and the coupling. Such devices produce accuracies of 0.5 to 1.0 percent of full scale and are used to transmit forces, pressures, differential pressures, or weights up to 1500 m. They can also be designed to transmit rotary motion. Hall Effect Sensors Some semiconductor materials exhibit a phenomenon in the presence of a magnetic field which is adaptable to sensing devices. When a current is passed through one pair of wires attached to a semiconductor, such as germanium, another pair of wires properly attached and oriented with respect to the semiconductor will develop a voltage proportional to the magnetic field present and the current in the other pair of wires. Holding the exciting current constant and moving a permanent magnet near the semiconductor produce a voltage output proportional to the movement of the magnet. The magnet may be attached to a process variable measurement device which moves the magnet as the variable changes. Hall effect devices provide high speed of response, excellent temperature stability, and no physical contact. SAMPLING SYSTEMS FOR PROCESS ANALYZERS The sampling system consists of all the equipment required to present a process analyzer with a clean representative sample of a process stream and to dispose of that sample. When the analyzer is part of an automatic control loop, the reliability of the sampling system is as important as the reliability of the analyzer or the control equipment.

Sampling systems have several functions. The sample must be withdrawn from the process, transported, conditioned, introduced to the analyzer, and disposed. Probably the most common problem in sample system design is the lack of realistic information concerning the properties of the process material at the sampling point. Another common problem is the lack of information regarding the conditioning required so that the analyzer may utilize the sample without malfunction for long periods. Some samples require enough conditioning and treating that the sampling systems become equivalent to miniature online processing plants. These systems possess many of the same fabrication, reliability, and operating problems as small-scale pilot plants except that the sampling system must generally operate reliably for much longer periods of time. Selecting the Sampling Point The selection of the sampling point is based primarily on supplying the analyzer with a sample whose composition or physical properties are pertinent to the control function to be performed. Other considerations include selecting locations that provide representative homogeneous samples with minimum transport delay, locations which collect a minimum of contaminating material, and locations which are accessible for test and maintenance procedures. Sample Withdrawal from Process A number of considerations are involved in the design of sample withdrawal devices which will provide representative samples. For example, in a horizontal pipe carrying process fluid, a sample point on the bottom of the pipe will collect a maximum count of rust, scale, or other solid materials being carried along by the process fluid. In a gas stream, such a location will also collect a maximum amount of liquid contaminants. A sample point on the top side of a pipe will, for liquid streams, collect a maximum amount of vapor contaminants being carried along. Bends in the piping which produce swirls or cause centrifugal concentration of the denser phase may cause maximum contamination to be at unexpected locations. Two-phase process materials are difficult to sample for a total-composition representative sample. A typical method for obtaining a sample of process fluid well away from vessel or pipe walls is an eduction tube inserted through a packing gland. This sampling method withdraws liquid sample and vaporizes it for transporting to the analyzer location. The transport lag time from the end of the probe to the vaporizer is minimized by using tubing having a small internal volume compared with pipe and valve volumes. This sample probe may be removed for maintenance and reinstalled without shutting down the process. The eduction tube is made of material which will not corrode so that it will slide through the packing gland even after long periods of service. There may be a small amount of process fluid leakage until the tubing is withdrawn sufficiently to close the gate valve. A swaged ferrule on the end of the tube prevents accidental ejection of the eduction tube prior to removal of the packing gland. The section of pipe surrounding the eduction tube and extending into the process vessel provides mechanical protection for the eduction tube. Sample Transport Transport time—the time elapsed between sample withdrawal from the process and its introduction into the analyzer—should be minimized, particularly if the analyzer is an automatic analyzer-controller. Any sample transport time in the analyzer-controller loop must be treated as equivalent to process dead time in determining conventional feedback controller settings or in evaluating controller performance. Reduction in transport time usually means transporting the sample in the vapor state. Design considerations for sample lines are as follows: 1. The structural strength or protection must be compatible with the area through which the sample line runs. 2. Line size and length must be small enough to meet transport time requirements without excessive pressure drop or excessive bypass of sample at the analyzer input. 3. Line size and internal surface quality must be adequate to prevent clogging by the contaminants in the sample. 4. The prevention of a change of state of the sample may require installation, refrigeration, or heating of the sample line. 5. Sample line material must be such as to minimize corrosion due to sample or the environment.

TELEMETERING AND TRANSMISSION Sample Conditioning Sample conditioning usually involves the removal of contaminants or some deleterious component from the sample mixture and/or the adjustment of temperature, pressure, and flow rate of the sample to values acceptable to the analyzer. Some of the more common contaminants which must be removed are rust, scale, corrosion products, deposits due to chemical reactions, and tar. In sampling some process streams, the material to be removed may include the primary process product such as polymer or the main constituent of the stream such as oil. In other

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cases the material to be removed is present in trace quantities, e.g., water in an online chromatograph sample which can damage the chromatographic column packing. When contaminants or other materials which will hinder analysis represent a large percentage of the stream composition, their removal may significantly alter the integrity of the sample. In some cases, removal must be done as part of the analysis function so that removed material can be accounted for. In other cases, proper calibration of the analyzer output will suffice.

TELEMETERING AND TRANSMISSION ANALOG SIGNAL TRANSMISSION Modern control systems permit the measurement device, the control unit, and the final control element to be physically separated by several hundred meters, if necessary. This requires the transmission of the measured variable from the measurement device to the control unit, and the transmission of the controller output from the control unit to the final control element. In each case, transmission of a single value in only one direction is required. Such requirements can be met by analog signal transmission. A measurement range is defined for the value to be transmitted, and the value is basically transmitted as a percent of the span. For the measured variable, the logical span is the measurement span. For the controller output, the logical span is the range of the final control element (e.g., valve fully closed to valve fully open). For pneumatic transmission systems, the signal range used for the transmission is 3 to 15 psig. In each pneumatic transmission system, there can be only one transmitter, but there can be any number of receivers. When most measurement devices were pneumatic, pneumatic transmission was the logical choice. However, with the displacement of pneumatic measurement devices by electronic devices, pneumatic transmission is now limited to applications where the unique characteristics of pneumatics make it the logical choice. In order for electronic transmission systems to be less susceptible to interference from magnetic fields, current is used for the transmission signal instead of voltage. The signal range is 4 to 20 mA. In each circuit or “current loop,” there can be only one transmitter. There can be more than one receiver, but not an unlimited number. For each receiver, a 250-Ω range resistor is inserted into the current loop, which provides a 1- to 5-V input to the receiving device. The number of receivers is limited by the power available from the transmitter. Both pneumatic and electronic transmission use a live zero. This enables the receiver to distinguish a transmitted value of 0 percent of span from a transmitter or transmission system failure. Transmission of 0 percent of span provides a signal of 4 mA in electronic transmission. Should the transmitter or the transmission system fail (i.e., an open circuit in a current loop), the signal level would be 0 mA. For most measurement variable transmissions, the lower range corresponds to 4 mA and the upper range corresponds to 20 mA. On an open circuit, the measured variable would fail to its lower range. In some applications, this is undesirable. For example, in a fired heater that is heating material to a target temperature, failure of the temperature measurement to its lower range value would drive the output of the combustion control logic to the maximum possible firing rate. In such applications, the analog transmission signal is normally inverted, with the upper range corresponding to 4 mA and the lower range corresponding to 20 mA. On an open circuit, the measured variable would fail to its upper range. For the fired heater, failure of the measured variable to its upper range would drive the output of the combustion control logic to the minimum firing rate. DIGITAL SYSTEMS With the advent of the microprocessor, digital technology began to be used for data collection, feedback control, and all other information processing requirements in production facilities. Such systems must

acquire data from a variety of measurement devices, and control systems must drive final control elements. Analog Inputs and Outputs Analog inputs are generally divided into two categories: 1. High level. Where the source is a process transmitter, the range resistor in the current loop converts the 4- to 20-mA signal to a 1- to 5-V signal. The conversion equipment can be unipolar (i.e., capable of processing only positive voltages). 2. Low level. The most common low-level signals are inputs from thermocouples. These inputs rarely exceed 30 mV and could be zero or even negative. The conversion equipment must be bipolar (i.e., capable of processing positive and negative voltages). Ultimately, such signals are converted to digital values via an analog-to-digital (A/D) converter. However, the A/D converter is normally preceded by two other components: 1. Multiplexer. This permits one A/D converter to service multiple analog inputs. The number of inputs to a multiplexer is usually between 8 and 256. 2. Amplifier. As A/D converters require high-level signals, a highgain amplifier is required to convert low-level signals to high-level signals. One of the important parameters for the A/D converter is its resolution. The resolution is stated in terms of the number of significant binary digits (bits) in the digital value. As the repeatability of most process transmitters is around 0.1 percent, the minimum acceptable resolution for a bipolar A/D converter is 12 bits, which translates to 11 data bits plus 1 bit for the sign. With this resolution, the analog input values can be represented to 1 part in 211, or 1 part in 2048. Normally, a 5-V input is converted to a digital value of 2000, which effectively gives a resolution of 1 part in 2000, or 0.05 percent. Very few process control systems utilize resolutions higher than 14 bits, which translates to a resolution of 1 part in 8000, or 0.0125 percent. For 4- to 20-mA inputs, the resolution is not quite as good as stated above. For a 12-bit bipolar A/D converter, 1 V converts to a digital value of 400. Thus, the range for the digital value is 400 to 2000, making the effective input resolution 1 part in 1600, or 0.0625 percent. Sometimes this is expressed as a resolution of 7.4 bits, where 27.4 = 1600. On the output side, dedicated digital-to-analog (D/A) converters are provided for each analog output. Outputs are normally unipolar and require a lower resolution than inputs. A 10-bit resolution is normally sufficient, giving a resolution of 1 part in 1000, or 0.1 percent. Pulse Inputs Where the sensor within the measurement device is digital, analog-to-digital conversion can be avoided. For rotational devices, the rotational element can be outfitted with a shaft encoder that generates a known number of pulses per revolution. The digital system can process such inputs in any of the following ways: 1. Count the number of pulses over a fixed interval of time. 2. Determine the time for a specified number of pulses. 3. Determine the duration of time between the leading (or trailing) edges of successive pulses. Of these, the first option is the most commonly used in process applications. Turbine flowmeters are probably the most common example where pulse inputs are used. Another example is a watthour meter. Basically any measurement device that involves a rotational element can be interfaced via pulses. Occasionally, a nonrotational measurement device can generate pulse outputs. One example is the

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vortex-shedding meter, where a pulse can be generated when each vortex passes over the detector. Serial Interfaces Some very important measurement devices cannot be reasonably interfaced via either analog or pulse inputs. Two examples are the following: 1. Chromatographs can perform a total composition analysis for a sample. It is possible but inconvenient to provide an analog input for each component. Furthermore, it is often desirable to capture other information, such as the time that the analysis was made (normally the time the sample was injected). 2. Load cells are capable of resolutions of 1 part in 100,000. A/D converters for analog inputs cannot even approach such resolutions. One approach to interfacing with such devices involves serial interfaces. This has two aspects: 1. Hardware interface. The RS-232 interface standard is the basis for most serial interfaces. 2. Protocol. This is interpreting the sequence of characters transmitted by the measurement device. There are no standards for protocols, which means that custom software is required. One advantage of serial interfaces is that two-way communication is possible. For example, a “tare” command can be issued to a load cell. Microprocessor-Based Transmitters The cost of microprocessor technology has declined to the point where it is economically feasible to incorporate a microprocessor into each transmitter. Such microprocessor-based transmitters are often referred to as smart transmitters. As opposed to conventional or dumb transmitters, the smart transmitters offer the following capabilities: 1. They check on the internal electronics, such as verifying that the voltage levels of internal power supplies are within specifications. 2. They check on environmental conditions within the instruments, such as verifying that the case temperature is within specifications. 3. They perform compensation of the measured value for conditions within the instrument, such as compensating the output of a pressure transmitter for the temperature within the transmitter. Smart transmitters are much less affected by temperature and pressure variations than are conventional transmitters. 4. They perform compensation of the measured value for other process conditions, such as compensating the output of a capacitance level transmitter for variations in process temperature. 5. They linearize the output of the transmitter. Functions such as square root extraction of the differential pressure for a head-type flowmeter can be done within the instrument instead of within the control system. 6. They configure the transmitter from a remote location, such as changing the span of the transmitter output. 7. They do automatic recalibration of the transmitter. Although this is highly desired by users, the capabilities, if any, in this respect depend on the type of measurement. Due to these capabilities, smart transmitters offer improved performance over conventional transmitters. Transmitter/Actuator Networks With the advent of smart transmitters and smart actuators, the limitations of the 4- to 20-mA analog signal transmission retard the full utilization of the capabilities of smart devices. For smart transmitters, the following capabilities are required: 1. Transmission of more than one value from a transmitter. Information beyond the measured variable is available from the smart transmitter. For example, a smart pressure transmitter can also report the temperature within its housing. Knowing that this temperature is above normal values permits corrective action to be taken before the device fails. Such information is especially important during the initial commissioning of a plant. 2. Bidirectional transmission. Configuration parameters such as span, engineering units, and resolution must be communicated to the smart transmitter. Similar capabilities are required for smart actuators. Eventually this technology, generally referred to as field bus, will replace all other forms of transmission (analog, pulse, and serial). Its acceptance by the user community has been slow, mainly due to the absense of a standard. Smart transmitters and smart valves have gained widespread acceptance, but initially the signal transmission continued to be via current loops. During this interim, several manu-

facturers introduced products based on proprietary communications technologies. Most manufacturers released sufficient information for others to develop products using their communications technologies, but those considering doing so found the costs to be an impediment. A standard is now available, specifically, IEC 61158, “Digital Data Communications for Measurement and Control.” In most industries, a standard defines only one way of doing something, but not in the computer industry. IEC 61158 defines five different communications technologies. Several manufacturers had selected a direction, and standards are not immune to commercial considerations. The net result is that one person’s “field bus” might not be the same as another’s. Most plants select one technology for their field bus, although it is possible to have multiple field bus interfaces on a control system and not have all the same type. Although there will be winners and losers, most companies find the appealing features of field bus technologies too great to continue with the older technologies. U.S. manufacturers have generally chosen a field bus technology known as Foundation Fieldbus. In Europe, a field bus technology known as Profibus has gained wide acceptance. Probably the major advantages of these technologies lie in installation and commissioning. There are fewer wires and connections in a field bus installation. In most plants, the cost of the wiring is proving to be a minor issue. With fewer wires and connections, errors in field bus installations are fewer and can be located more quickly. The major cost savings prove to be in configuring and commissioning the system. With time, field bus will largely replace analog, pulse, and serial signal transmissions within industrial facilities. FILTERING AND SMOOTHING A signal received from a process transmitter generally contains the following components distinguished by their frequency [frequencies are measured in hertz (Hz), with 60-cycle ac being a 60-Hz frequency): 1. Low-frequency process disturbances. The control system is expected to react to these disturbances. 2. High-frequency process disturbances. The frequency of these disturbances is beyond the capability of the control system to effectively react. 3. Measurement noise. 4. Stray electrical pickup, primarily 50- or 60-cycle ac. The objective of filtering and smoothing is to remove the last three components, leaving only the low-frequency process disturbances. Normally this has to be accomplished by using the proper combination of analog and digital filters. Sampling a continuous signal results in a phenomenon often referred to as aliasing or foldover. To represent a sinusoidal signal, a minimum of four samples are required during each cycle. Consequently, when a signal is sampled at a frequency ωs, all frequencies higher than (π2)ωs cannot be represented at their original frequency. Instead, they are present in the sampled signal with their original amplitude but at a lower-frequency harmonic. Because of the aliasing or foldover issues, a combination of analog and digital filtering is usually required. The sampler (i.e., the A/D converter) must be preceded by an analog filter that rejects those highfrequency components such as stray electrical pickup that would result in foldover when sampled. In commercial products, analog filters are normally incorporated into the input processing hardware by the manufacturer. The software then permits the user to specify digital filtering to remove any undesirable low-frequency components. On the analog side, the filter is often the conventional resistorcapacitor or RC filter. However, other possibilities exist. For example, one type of A/D converter is called an integrating A/D because the converter basically integrates the input signal over a fixed interval of time. By making the interval 610 s, this approach provides excellent rejection of any 60-Hz electrical noise. On the digital side, the input processing software generally provides for smoothing via the exponentially weighted moving average, which is the digital counterpart to the RC network analog filter. The smoothing equation is as follows: yi = α xi + (1 − α)yi−1 where xi = current value of input yi = current output from filter

(8-107)

TELEMETERING AND TRANSMISSION yi−1 = previous output from filter α = filter coefficient The degree of smoothing is determined by the filter coefficient α, with α = 1 being no smoothing and α = 0 being infinite smoothing (no effect of new measurements). The filter coefficient α is related to the filter time constant τF and the sampling interval ∆t by −∆t α = 1 − exp  τF





(8-108)

or by the approximation ∆t α=  ∆t + τF

(8-109)

Another approach to smoothing is to use the arithmetic moving average, which is represented by n

x j=1

i+1− j

yi =  n

(8-110)

The term moving is used because the filter software maintains a storage array with the previous n values of the input. When a new value is received, the oldest value in the storage array is replaced with the new value, and the arithmetic average is recomputed. This permits the filtered value to be updated each time a new input value is received. In process applications, determining τF (or α) for the exponential filter and n for the moving average filter is often done merely by observing the behavior of the filtered value. If the filtered value is “bouncing,” the degree of smoothing (that is, τF or n) is increased. This can easily lead to an excessive degree of filtering, which will limit the performance of any control system that uses the filtered value. The degree of filtering is best determined from the frequency spectrum of the measured input, but such information is rarely available for process measurements. ALARMS The purpose of an alarm is to alert the process operator to a process condition that requires immediate attention. An alarm is said to occur whenever the abnormal condition is detected and the alert is issued. An alarm is said to return to normal when the abnormal condition no longer exists. Analog alarms can be defined on measured variables, calculated variables, controller outputs, and the like. For analog alarms, the following possibilities exist: 1. High/low alarms. A high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. A low alarm is generated when the value is less than or equal to the value specified for the low-alarm limit. 2. Deviation alarms. An alarm limit and a target are specified. A high-deviation alarm is generated when the value is greater than or equal to the target plus the deviation alarm limit. A low-deviation alarm is generated when the value is less than or equal to the target minus the deviation alarm limit. 3. Trend or rate-of-change alarms. A limit is specified for the maximum rate of change, usually specified as a change in the measured value per minute. A high-trend alarm is generated when the rate of change of the variable is greater than or equal to the value specified for the trend alarm limit. A low-trend alarm is generated when the rate of change of the variable is less than or equal to the negative of the value specified for the trend alarm limit. Most systems permit multiple alarms of a given type to be configured for a given value. For example, configuring three high alarms provides a high alarm, a high-high alarm, and a high-high-high alarm. One operational problem with analog alarms is that noise in the variable can cause multiple alarms whenever its value approaches a limit. This can be avoided by defining a dead band on the alarm. For example, a high alarm would be processed as follows:

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1. Occurrence. The high alarm is generated when the value is greater than or equal to the value specified for the high-alarm limit. 2. Return to normal. The high-alarm return to normal is generated when the value is less than or equal to the high-alarm limit less the dead band. As the degree of noise varies from one input to the next, the dead band must be individually configurable for each alarm. Discrete alarms can be defined on discrete inputs, limit switch inputs from on/off actuators, and so on. For discrete alarms, the following possibilities exist: 1. Status alarms. An expected or normal state is specified for the discrete value. A status alarm is generated when the discrete value is other than its expected or normal state. 2. Change-of-state alarm. A change-of-state alarm is generated on any change of the discrete value. The expected sequence of events on an alarm is basically as follows: 1. The alarm occurs. This usually activates an audible annunciator. 2. The alarm occurrence is acknowledged by the process operator. When all alarms have been acknowledged, the audible annunciator is silenced. 3. Corrective action is initiated by the process operator. 4. The alarm condition returns to normal. However, additional requirements are imposed at some plants. Sometimes the process operator must acknowledge the alarm’s return to normal. Some plants require that the alarm occurrence be reissued if the alarm remains in the occurred state longer than a specified time. Consequently, some “personalization” of the alarming facilities is done. When alarms were largely hardware-based (i.e., the panel alarm systems), the purchase and installation of the alarm hardware imposed a certain discipline on the configuration of alarms. With digital systems, the suppliers have made it extremely easy to configure alarms. In fact, it is sometimes easier to configure alarms on a measured value than not to configure the alarms. Furthermore, the engineer assigned the responsibility for defining alarms is often paranoid that an abnormal process condition will go undetected because an alarm has not been configured. When alarms are defined on every measured and calculated variable, the result is an excessive number of alarms, most of which are duplicative and unnecessary. The accident at the Three Mile Island nuclear plant clearly demonstrated that an alarm system can be counterproductive. An excessive number of alarms can distract the operator’s attention from the real problem that needs to be addressed. Alarms that merely tell the operator something that is already known do the same. In fact, a very good definition of a nuisance alarm is one that informs the operator of a situation of which the operator is already aware. The only problem with applying this definition lies determining what the operator already knows. Unless some discipline is imposed, engineering personnel, especially where contractors are involved, will define far more alarms than plant operations require. This situation may be addressed by simply setting the alarm limits to values such that the alarms never occur. However, changes in alarms and alarm limits are changes from the perspective of the process safety management regulations. It is prudent to impose the necessary discipline to avoid an excessive number of alarms. Potential guidelines are as follows: 1. For each alarm, a specific action is expected from the process operator. Operator actions such as call maintenance are inappropriate with modern systems. If maintenance needs to know, modern systems can inform maintenance directly. 2. Alarms should be restricted to abnormal situations for which the process operator is responsible. A high alarm on the temperature in one of the control system cabinets should not be issued to the process operator. Correcting this situation is the responsibility of maintenance, not the process operator. 3. Process operators are expected to be exercising normal surveillance of the process. Therefore, alarms are not appropriate for situations known to the operator either through previous alarms or through normal process surveillance. The “sleeping operator” problem can be addressed by far more effective means than the alarm system. 4. When the process is operating normally, no alarms should be triggered. Within the electric utility industry, this design objective is

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known as darkboard. Application of darkboard is especially important in batch plants, where much of the process equipment is operated intermittently. Ultimately, guidelines such as those above will be taken seriously only if production management pays attention to the process alarms. The consequences of excessive and redundant alarms will be felt primarily by those responsible for production operations. Therefore, production management must make adequate resources available for reviewing and analyzing the proposed alarm configurations. Another serious distraction to a process operator is the multiplealarm event, where a single event within the process results in multiple alarms. When the operator must individually acknowledge each alarm, considerable time can be lost in silencing the obnoxious annunciator before the real problem is addressed. Air-handling systems are especially vulnerable to this, where any fluctuation in pressure (e.g., resulting from a blower trip) can cause a number of pressure alarms to occur. Point alarms (high alarms, low alarms, status alarms, etc.) are especially vulnerable to the multiple-alarm event. This can be addressed in one of two ways: 1. Ganging alarms. Instead of individually issuing the point alarms, all alarms associated with a certain aspect of the process

are configured to give a single trouble alarm. The responsibility rests entirely with the operator to determine the nature of the problem. 2. Intelligent alarms. Logic is incorporated into the alarm system to determine the nature of the problem and then issue a single alarm to the process operator. While the intelligent alarm approach is clearly preferable, substantial process analysis is required to support intelligent alarming. Meeting the following two objectives is quite challenging: 1. The alarm logic must consistently detect abnormal conditions within the process. 2. The alarm logic must not issue an alert to an abnormal condition when in fact none exists. Often the latter case is more challenging than the former. Logically, the intelligent alarm effort must be linked to the process hazards analysis. Developing an effective intelligent alarming system requires substantial commitments of effort, involving process engineers, control systems engineers, and production personnel. Methodologies such as expert systems can facilitate the implementation of an intelligent alarming system, but they must still be based on a sound analysis of the potential process hazards.

DIGITAL TECHNOLOGY FOR PROCESS CONTROL GENERAL REFERENCES: Auslander and Ridgely, Design and Implementation of Real-Time Software for the Control of Mechanical Systems, Prentice-Hall, Upper Saddle River, N.J., 2002. Herb, Understanding Distributed Processor Systems for Control, ISA, Research Triangle Park, N.C., 1999. Hughes, Programmable Controllers, ISA, Research Triangle Park, N.C., 1997. Johnson, Process Control Instrumentation Technology, 6th ed., Prentice-Hall, Upper Saddle River, N.J., 2000. Liptak, Instrument Engineers Handbook, Chilton Book Company, Philadelphia, 1995. Webb and Reis, Programmable Logic Controllers, 4th ed., Prentice-Hall, Upper Saddle River, N.J., 2002.

Since the 1970s, process controls have evolved from pneumatic analog technology to electronic analog technology to microprocessor-based controls. Electronic and pneumatic controllers have now virtually disappeared from process control systems, which are dominated by programmable electronic systems based on microprocessor technology. HIERARCHY OF INFORMATION SYSTEMS Coupling digital controls with networking technology permits information to be passed from level to level within a corporation at high rates of speed. This technology is capable of presenting the measured variable from a flow transmitter installed in a plant in a remote location anywhere in the world to the company headquarters in less than 1 s. A hierarchical representation of the information flow within a company leads to a better understanding of how information is passed from one layer to the next. Such representations can be developed in varying degrees of detail, and most companies have developed one that describes their specific practices. The following hierarchy consists of five levels, as shown in Fig. 8-30. Measurement Devices and Final Control Elements This lowest layer couples the control and information systems to the process. The measurement devices provide information on the current conditions within the process. The final control elements permit control decisions to be imposed on the process. Although traditionally analog, smart transmitters and smart valves based on microprocessor technology are now beginning to dominate this layer. Safety and Environmental/Equipment Protection The level 2 functions play a critical role by ensuring that the process is operating safely and satisfies environmental regulations. Process safety relies on the principle of multiple protection layers that involve groupings of equipment and human actions. One layer includes process control functions, such as alarm management during abnormal situations, and safety instrumented systems for emergency shutdowns. The safety equipment (including sensors and block valves) operates independently of the regular instrumentation used for regulatory control in level 3. Sensor validation techniques can be employed to confirm that the sensors are functioning properly.

Regulatory Controls The objective of this layer is to operate the process at or near the targets supplied by a higher layer in the hierarchy. To achieve consistent process operations, a high degree of automatic control is required from the regulatory layer. The direct result is a reduction in variance in the key process variables. More uniform product quality is an obvious benefit. However, consistent process operation is a prerequisite for optimizing the process operations. To ensure success for the upper-level functions, the first objective of any automation effort must be to achieve a high degree of regulatory control. Real-Time Optimization Determining the most appropriate targets for the regulatory layer is the responsibility of the RTO layer. Given the current production and quality targets for a unit, RTO determines how the process can be best operated to meet them. Usually this optimization has a limited scope, being confined to a single production unit or possibly even a single unit operation within a production unit. RTO translates changes in factors such as current process efficiencies, current energy costs, cooling medium temperatures, and so on to changes in process operating targets so as to optimize process operations. Production Controls The nature of the production control logic differs greatly between continuous and batch plants. A good example of production control in a continuous process is refinery optimization. From the assay of the incoming crude oil, the values of the various possible refined products, the contractual commitments to deliver certain products, the performance measures of the various units within a refinery, and the like, it is possible to determine the mix of products that optimizes the economic return from processing this crude. The solution of this problem involves many relationships and constraints and is solved with techniques such as linear programming. In a batch plant, production control often takes the form of routing or short-term scheduling. For a multiproduct batch plant, determining the long-term schedule is basically a manufacturing resource planning (MRP) problem, where the specific products to be manufactured and the amounts to be manufactured are determined from the outstanding orders, the raw materials available for production, the production capacities of the process equipment, and other factors. The goal of the MRP effort is the long-term schedule, which is a list of the products to be manufactured over a specified period of time (often one week). For each product on the list, a target amount is also specified. To manufacture this amount usually involves several batches. The term production run often refers to the sequence of batches required to make the target amount of product, so in effect the longterm schedule is a list of production runs. Most multiproduct batch plants have more than one piece of equipment of each type. Routing refers to determining the specific pieces of equipment that will be used to manufacture each run on the long-term

DIGITAL TECHNOLOGY FOR PROCESS CONTROL production schedule. For example, the plant might have five reactors, eight neutralization tanks, three grinders, and four packing machines. For a given run, a rather large number of routes are possible. Furthermore, rarely is only one run in progress at a given time. The objective of routing is to determine the specific pieces of production equipment to be used for each run on the long-term production schedule. Given the dynamic nature of the production process (equipment failures, insertion/deletion of runs into the long-term schedule, etc.), the solution of the routing problem continues to be quite challenging. Corporate Information Systems Terms such as management information systems (MIS), enterprise resource planning (ERP), supply chain management (SCM), and information technology (IT) are frequently used to designate the upper levels of computer systems within a corporation. From a control perspective, the functions performed at this level are normally long-term and/or strategic. For example, in a processing plant, long-term contracts are required with the providers of the feedstocks. A forecast must be developed for the demand for possible products from the plant. This demand must be translated to needed raw materials, and then contracts executed with the suppliers to deliver these materials on a relatively uniform schedule. DIGITAL HARDWARE IN PROCESS CONTROL Digital control technology was first applied to process control in 1959, using a single central computer (and analog backup for reliability). In the mid-1970s, a microcomputer-based process control architecture referred to as a distributed control system (DCS) was introduced and rapidly became a commercial success. A DCS consists of some number of microprocessor-based nodes that are interconnected by a digital communications network, often called a data highway. Today the DCS is still dominant, but there are other options for carrying out computer control, such as single-loop controllers, programmable logic controllers, and personal computer controllers. A brief review of each type of controller device is given below; see the following section “Controllers, Final Control Elements, and Regulators” for more details on controller hardware options. Single-Loop Controllers The single-loop controller (SLC) is the digital equivalent of analog single-loop controllers. It is a self-contained microprocessor-based unit that can be rack-mounted. Many manufacturers produce single processor units that handle cascade control or multiple loops, typically 4, 8, or 16 loops per unit, and incorporate self-tuning or auto-tuning PID control algorithms.

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Programmable Logic Controllers Programmable logic controllers (PLCs) are simple digital devices that are widely used to control sequential and batch processes, although they now have additional functions that implement PID control and other mathematical operations. PLCs can be utilized as stand-alone devices or in conjunction with digital computer control systems. Because the logical functions are stored in main memory, one measure of a PLC’s capability is its memory scan rate. Most PLCs are equipped with an internal timing capability to delay an action by a prescribed amount of time, to execute an action at a prescribed time, and so on. Newer PLC models often are networked to serve as one component of a DCS, with operator I/O provided by a separate component in the network. A distinction is made between configurable and programmable PLCs. The term configurable implies that logical operations (performed on inputs to yield a desired output) are located in PLC memory, perhaps in the form of ladder diagrams by selecting from a PLC menu or by direct interrogation of the PLC. Most control engineers prefer the simplicity of configuring the PLC to the alternative of programming it. However, some batch applications, particularly those involving complex sequencing, are best handled by a programmable approach. Personal Computer Controllers In comparison with PLCs, PCs have the advantages of lower purchase cost, graphics output, large memory, large selection of software products (including databases and development tools), more programming options (use of C or Java versus ladder logic), richer operating systems, and open networking. PLCs have the following advantages: lower maintenance cost, operating system and hardware optimized for control, fast boot times, ruggedness, low failure rate, longer support for product models, and self-contained units. A number of vendors have introduced so-called scalable process control systems. Scalable means that the size of the control and instrumentation systems is easily expanded by simply adding more devices. This feature is possible because of the trend toward more openness (i.e., “plug and play” between devices), smaller size, lower cost, greater flexibility, and more off-the-shelf hardware and software in digital control systems. A typical system includes personal computers, an operating system, object-oriented database technology, modular field-mounted controllers, and plug-and-play integration of both system and intelligent field devices. New devices are automatically recognized and configured with the system. Advanced control algorithms can be executed at the PC level. Distributed Control System Figure 8-65 depicts a representative distributed control system. The DCS consists of many commonly

FIG. 8-65 A DCS using a broadband (high-bandwidth) data highway and field bus connected to a single remote control unit that operates smart devices and single-loop controllers.

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used components, including multiplexers (MUXs), single-loop and multiple-loop controllers, PLCs, and smart devices. A system includes some of or all the following components: 1. Control network. The control network is the communication link between the individual components of a network. Coaxial cable and, more recently, fiber-optic cable have often been used, sometimes with Ethernet protocols. A redundant pair of cables (dual redundant highway) is normally supplied to reduce the possibility of link failure. 2. Workstations. Workstations are the most powerful computers in the system, capable of performing functions not normally available in other units. A workstation acts both as an arbitrator unit to route internodal communications and as the database server. An operator interface is supported, and various peripheral devices are coordinated through the workstations. Computationally intensive tasks, such as real-time optimization or model predictive control, are implemented in a workstation. Operators supervise and control processes from these workstations. Operator stations may be connected directly to printers for alarm logging, printing reports, or process graphics. 3. Remote control units (RCUs). These components are used to implement basic control functions such as PID control. Some RCUs may be configured to acquire or supply set points to single-loop controllers. Radio telemetry (wireless) may be installed to communicate with MUX units located at great distances. 4. Application stations. These separate computers run application software such as databases, spreadsheets, financial software, and simulation software via an OPC interface. OPC is an acronym for object linking and embedding for process control, a software architecture based on standard interfaces. These stations can be used for email and as web servers, for remote diagnosis configuration, and even for operation of devices that have an IP (Internet Protocol) address. Applications stations can communicate with the main database contained in online mass storage systems. Typically hard disk drives are used to store active data, including online and historical databases and nonmemory resident programs. Memory resident programs are also stored to allow loading at system start-up. 5. Field buses and smart devices. An increasing number of fieldmounted devices are available that support digital communication of the process I/O in addition to, or in place of, the traditional 4- to 20mA current signal. These devices have greater functionality, resulting in reduced setup time, improved control, combined functionality of separate devices, and control valve diagnostic capabilities. Digital communication also allows the control system to become completely distributed where, e.g., a PID control algorithm could reside in a valve positioner or in a sensor/transmitter. DISTRIBUTED DATABASE AND THE DATABASE MANAGER A database is a centralized location for data storage. The use of databases enhances system performance by maintaining complex relations between data elements while reducing data redundancy. A database may be built based on the relational model, the entity relationship model, or some other model. The database manager is a system utility program or programs acting as the gatekeeper to the databases. All functions retrieving or modifying data must submit a request to the manager. Information required to access the database includes the tag name of the database entity, often referred to as a point, the attributes to be accessed, and the values, if modifying. The database manager maintains the integrity of the databases by executing a request only when not processing other conflicting requests. Although a number of functions may read the same data item at the same time, writing by a number of functions or simultaneous read and write of the same data item is not permitted. To allow flexibility, the database manager must also perform point addition or deletion. However, the ability to create a point type or to add or delete attributes of a point type is not normally required because, unlike other data processing systems, a process control system normally involves a fixed number of point types and related attributes. For example, analog and binary input and output types are required for process I/O points. Related attributes for these point types include tag names, values, and hardware addresses. Different

system manufacturers may define different point types using different data structures. We will discuss other commonly used point types and attributes as they appear. Data Historian A historical database is built similar to an online database. Unlike their online counterparts, the information stored in a historical database is not normally accessed directly by other subsystems for process control and monitoring. Periodic reports and longterm trends are generated based on the archived data. The reports are often used for planning and system performance evaluations such as statistical process (quality) control. The trends may be used to detect process drifts or to compare process variations at different times. The historical data are sampled at user-specified intervals. A typical process plant contains a large number of data points, but it is not feasible to store data for all points at all times. The user determines if a data point should be included in the list of archive points. Most systems provide archive-point menu displays. The operators are able to add or delete data points to the archive point lists. The sampling periods are normally some multiples of their base scan frequencies. However, some systems allow historical data sampling of arbitrary intervals. This is necessary when intermediate virtual data points that do not have the scan frequency attribute are involved. The archive point lists are continuously scanned by the historical database software. Online databases are polled for data, and the times of data retrieval are recorded with the data obtained. To conserve storage space, different data compression techniques are employed by various manufacturers. DIGITAL FIELD COMMUNICATIONS AND FIELD BUS Microprocessor-based equipment, such as smart instruments and single-loop controllers with digital communications capability, are now used extensively in process plants. A field bus, which is a low-cost protocol, is necessary to perform efficient communication between the DCS and devices that may be obtained from different vendors. Figure 8-65 illustrates a LAN-based DCS with field buses and smart devices connected to a data highway. Presently, there are several regional and industry-based field bus standards, including the French standard (FIP), the German standard (Profibus), and proprietary standards by DCS vendors, generally in the United States, led by the Fieldbus Foundation, a not-for-profit corporation. As of 2002, international standards organizations had adopted all these field bus standards rather than a single unifying standard. However, there will likely be further developments in field bus standards in the future. A benefit of standardizing the field bus is that it has encouraged third-party traditional equipment manufacturers to enter the smart equipment market, resulting in increased competition and improved equipment quality. Several manufacturers have made available field bus controllers that reside in the final control element or measurement transmitter. A suitable communications modem is present in the device to interface with a proprietary PC-based, or hybrid analog/digital bus network. At present, field bus controllers are single-loop controllers containing 8and 16-bit microprocessors that support the basic PID control algorithm as well as other functionalities. Case studies in implementing such digital systems have shown significant reductions in cost of installation (mostly cabling and connections) versus traditional analog field communication. A general movement has also begun in the direction of using the high-speed Ethernet standard (100 Mbit/s or higher), allowing data transfer by TCP/IP that is used pervasively in computer networking. This would allow any smart device to communicate directly with others in the network or to be queried by the operator regarding its status and settings. INTERNODAL COMMUNICATIONS For a group of computers to become a network, intercomputer communication is required. Prior to the 1980s, each system vendor used a proprietary protocol to network its computers. Ad hoc approaches were sometimes used to connect third-party equipment but were not costeffective with regard to system maintenance, upgrade, and expansion.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS The recent introduction of standardized communication protocols has led to a decrease in capital cost. Most current DCS network protocol designs are based on the ISO-OSI seven-layer model with physical, data link, network, transport, session, presentation, and application layers. An effort in standardizing communication protocols for plant automation was initiated by General Motors in the early 1980s. This work culminated in the Manufacturing Automation Protocol (MAP), which adopted the ISO-OSI standards as its basis. MAP specifies a broadband backbone local-area network (LAN) that incorporates a selection of existing standard protocols suitable for discrete component manufacturing. MAP was intended to address the integration of DCSs used in process control. Subsequently, TCP/IP (Transmission Control Protocol/Internet Protocol) was adopted for communication between nodes that have different operating systems. Communication programs also act as links to the database manager. When data are requested from a remote node, the database manager transfers the request to the remote node database manager via the communication programs. The remote node communication programs then relay the request to the resident database manager and return the requested data. The remote database access and the existence of communications equipment and software are transparent to the user. PROCESS CONTROL LANGUAGES Originally, software for process control utilized high-level programming languages such as FORTRAN and BASIC. Some companies have incorporated libraries of software routines for these languages, but others have developed specialty languages characterized by natural language statements. The most widely adopted user-friendly approach is the fill-in-the-forms or table-driven process control languages (PCLs). Typical PCLs include function block diagrams, ladder logic, and programmable logic. The core of these languages is a num-

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ber of basic function blocks or software modules, such as analog in, digital in, analog out, digital out, PID, summer, and splitter. Using a module is analogous to calling a subroutine in conventional FORTRAN or C programs. In general, each module contains one or more inputs and an output. The programming involves connecting outputs of function blocks to inputs of other blocks via the graphical-user interface. Some modules may require additional parameters to direct module execution. Users are required to fill in templates to indicate the sources of input values, the destinations of output values, and the parameters for forms/tables prepared for the modules. The source and destination blanks may specify process I/O channels and tag names when appropriate. To connect modules, some systems require filling in the tag names of modules originating or receiving data. Many DCSs allow users to write custom code (similar to BASIC) and attach it to data points, so that the code is executed each time the point is scanned. The use of custom code allows many tasks to be performed that cannot be carried out by standard blocks. All process control languages contain PID control blocks of different forms. Other categories of function blocks include 1. Logical operators. AND, OR, and exclusive OR (XOR) functions. 2. Calculations. Algebraic operations such as addition, multiplication, square root extraction, or special function evaluation. 3. Selectors. Min and max functions, transferring data in a selected input to the output or the input to a selected output. 4. Comparators. Comparison of two analog values and transmission of a binary signal to indicate whether one analog value exceeds the other. 5. Timers. Delayed activation of the output for a programmed duration after activation by the input signal. 6. Process dynamics. Emulation of a first-order process lag (or lead) and time delay.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS GENERAL REFERENCES: Driskell, Control-Valve Selection and Sizing, ISA, 1983. McMillan, Process/Industrial Instruments and Controls Handbook, 5th ed., McGraw-Hill, 1999. Hammitt, Cavitation and Multiphase Flow Phenomena, McGraw-Hill, 1980. ANSI/ISA-75.25.01, Test Procedure for Control Valve Response Measurement from Step Inputs, The Instrumentation Systems and Automation Society, 2000. Norton and Karczub, Fundamentals of Noise and Vibration Analysis for Engineers, 2d ed., Cambridge University Press, 2003. Ulanski, Valve and Actuator Technology, McGraw-Hill, 1991. Kinsler et al., Fundamentals of Acoustics, 4th ed., Wiley. National Electrical Code Handbook, 9th ed., National Fire Protection Association, Inc.

External control of the process is achieved by devices that are specially designed, selected, and configured for the intended process control application. The text that follows covers three very common function classifications of process control devices: controllers, final control elements, and regulators. The process controller is the “master” of the process control system. It accepts a set point and other inputs and generates an output or outputs that it computes from a rule or set of rules that is part of its internal configuration. The controller output serves as an input to another controller or, more often, as an input to a final control element. The final control element typically is a device that affects the flow in the piping system of the process. The final control element serves as an interface between the process controller and the process. Control valves and adjustable speed pumps are the principal types discussed. Regulators, though not controllers or final control elements, perform the combined function of these two devices (controller and final control element) along with the measurement function commonly associated with the process variable transmitter. The uniqueness, control performance, and widespread usage of the regulator make it deserving of a functional grouping of its own.

PNEUMATIC, ELECTRONIC, AND DIGITAL CONTROLLERS Pneumatic Controllers The pneumatic controller is an automatic controller that uses variable air pressure as input and output signals. An air supply is also required to “power” the mechanical components of the controller and provide an air source for the controller output signal. Pneumatic controllers were first available in the early 1940s but are now rarely used for large-scale industrial control projects. Pneumatic controllers are still used where cost, ruggedness, or the installation requires an all-pneumatic solution. Pneumatic process transmitters are used to produce a pressure signal that is proportional to the calibrated range of the measuring element. Of the transmitter range 0 to 100 percent is typically represented by a 0.2- to 1.0-bar (3- to 15-psig) pneumatic signal. This signal is sent through tubing to the pneumatic controller process variable feedback connection. The process variable feedback can also be sensed directly in cases where the sensing element has been incorporated into the controller design. Controllers with integral sensing elements are available that sense pressure, differential pressure, temperature, and level. The pneumatic controller is designed so that 0 to 100 percent output is also represented by 0.2 to 1.0 bar (3 to 15 psig). The output signal is sent through tubing to the control valve or other final control element. Most pneumatic controllers provide a manual control mode where the output pressure is manually set by operating personnel. The controller design also provides a mechanism to adjust the set point. Early controller designs required “balancing” of the controller output prior to switching to or from automatic and manual modes. This procedure minimized inadvertent disturbance to the process caused by potentially large differences between the automatic and manual output levels. Later designs featured “bumpless” or “procedureless” automatic-to-manual transfer.

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Although the pneumatic controller is often used in single-loop control applications, cascade strategies can be implemented where the controller design supports input of external or remote set-point signals. A balancing procedure is typically required to align the remote set point with the local set point before the controller is switched into cascade mode. Almost all pneumatic controllers include indicators for process variable, set point, and output. Many controller designs also feature integral chart recorders. There are versions of the pneumatic controller that support combinations of proportional, integral, and derivative actions. The pneumatic controller can be installed into panel boards that are adjacent to the process being controlled or in a centrally located control room. Field-mountable controllers can be installed directly onto the control valve, a nearby pipestand, or wall in close proximity to the control valve and/or measurement transmitter. If operated on clean, dry plant air, pneumatic controllers offer good performance and are extremely reliable. In many cases, however, plant air is neither clean nor dry. A poor-quality air supply will cause unreliable performance of pneumatic controllers, pneumatic field measurement devices, and final control elements. The main shortcoming of the pneumatic controller is its lack of flexibility when compared to modern electronic controller designs. Increased range of adjustability, choice of alternative control algorithms, the communication link to the control system, and other features and services provided by the electronic controller make it a superior choice in most of today’s applications. Controller performance is also affected by the time delay induced by pneumatic tubing runs. For example, a 100-m run of 6.35-mm ( 14 -in) tubing will typically cause 5 s of apparent process dead time, which will limit the control performance of fast processes such as flows and pressures. Pneumatic controllers continue to be used in areas where it would be hazardous to use electronic equipment, such as locations with flammable or explosive atmospheres or other locations where compressed air is available but where access to electrical services is limited or restricted. Electronic (Digital) Controllers Almost all the electronic process controllers used today are microprocessor-based, digital devices. In the transition from pneumatic to electronic controllers, a number of analog controller designs were available. Due to the inflexible nature of the analog designs, these controllers have been almost completely replaced by digital designs. The microprocessor-based controllers contain, or have access to, input/output (I/O) interface electronics that allow various types of signals to enter and leave the controller’s processor. The resolution of the analog I/O channels of the controller varies by manufacturer and age of the design. The 12- to 14-bit conversion resolution of the analog input channels is quite common. Conversion resolution of the analog output channels is typically 10- to 12-bit. Newer designs support up to 16 bit input and output resolution. Although 10bit output resolution had been considered satisfactory for many years, it has recently been identified as a limitation of control performance. This limitation has emerged as the performance of control valve actuators has improved and the use of other high-resolution field devices, such as variable-speed pump drives, has become more prevalent. These improvements have been driven by the need to deliver higher operating efficiencies and improved product specifications through enhanced process control performance. Sample rates for the majority of digital controllers are adjustable and range from 1 sample every 5 s to 10 samples per second. Some controller designs have fixed sample rates that fall within the same range. Hardwired low-pass filters are usually installed on the analog inputs to the controller to help protect the sampler from aliasing errors. The real advantage of digital controllers is the substantial flexibility offered by a number of different configuration schemes. The simplest form of configuration is seen in a controller design that features a number of user-selectable control strategies. These strategies are customized by setting “tunable” parameters within the strategy. Another common configuration scheme uses a library of function blocks that can be selected and combined to form the desired control strategy. Each function block has adjustable parameters. Additional configuration schemes include text-based scripting languages, higher-level languages such as Basic or C, and ladder logic. Some digital controller designs allow the execution rates of control strategy elements to be set independently of each other and indepen-

dently of the I/O subsystem sample rate. Data passed from control element to subsystems that operate at slower sample or execution rates present additional opportunities for timing and aliasing errors. Distributed Control Systems Some knowledge of the distributed control system (DCS) is useful in understanding electronic controllers. A DCS is a process control system with sufficient performance to support large-scale, real-time process applications. The DCS has (1) an operations workstation with input devices, such as a keyboard, mouse, track ball, or other similar device, and a display device, such as a CRT or LCD panel; (2) a controller subsystem that supports various types of controllers and controller functions; (3) an I/O subsystem for converting analog process input signals to digital data and digital data back to analog output signals; (4) a higher-level computing platform for performing process supervision, historical data trending and archiving functions, information processing, and analysis; and (5) communication networks to tie the DCS subsystems, plant areas, and other plant systems together. The component controllers used in the controller subsystem portion of the DCS can be of various types and include multiloop controllers, programmable logic controllers, personal computer controllers, single-loop controllers, and field bus controllers. The type of electronic controller utilized depends on the size and functional characteristic of the process application being controlled. Personal computers are increasingly being used as DCS operations workstations or interface stations in place of custom-built machines. This is due to the low cost and high performance of the PC. See the earlier section “Digital Technology for Process Control.” Multiloop Controllers The multiloop controller is a DCS network device that uses a single 32-bit microprocessor to provide control functions to many process loops. The controller operates independently of the other devices on the DCS network and can support from 20 to 500 loops. Data acquisition capability for 1000 analog and discrete I/O channels or more can also be provided by this controller. The I/O is typically processed through a subsystem that is connected to the controller through a dedicated bus or network interface. The multiloop controller contains a variety of function blocks (for example, PID, totalizer, lead/lag compensator, ratio control, alarm, sequencer, and boolean) that can be “soft-wired” together to form complex control strategies. The multiloop controller also supports additional configuration schemes including text-based scripting languages, higher-level languages such as Basic or C, and, to a limited extent, ladder logic. The multiloop controller, as part of a DCS, communicates with other controllers and human/machine interface (HMI) devices also on the DCS network. Programmable Logic Controllers The programmable logic controller (PLC) originated as a solid-state, and far more flexible, replacement for the hardwired relay control panel and was first used in the automotive industry for discrete manufacturing control. Today, PLCs are primarily used to implement boolean logic functions, timers, and counters. Some PLCs offer a limited number of math functions and PID control. PLCs are often used with on/off input and output devices such as limit or proximity switches, solenoid-actuated process control valves, and motor switch gear. PLCs vary greatly in size with the smallest supporting less than 128 I/O channels and the largest supporting more than 1023 I/O channels. Very small PLCs combine processor, I/O, and communications functions into a single, self-contained unit. For larger PLC systems, hardware modules such as the power supply, processor module, I/O modules, communication module, and backplane are specified based on the application. These systems support multiple I/O backplanes that can be chained together to increase the I/O count available to the processor. Discrete I/O modules are available that support high-current motor loads and generalpurpose voltage and current loads. Other modules support analog I/O and special-purpose I/O for servomotors, stepping motors, high-speed pulse counting, resolvers, decoders, displays, and keyboards. PLC I/O modules often come with indicators to determine the status of key I/O channels. When used as an alternative to a DCS, the PLC is programmed with a handheld or computer-based loader. The PLC is typically programmed with ladder logic or a high-level computer language such as BASIC, FORTRAN, or C. Programmable logic controllers use 16- or 32-bit microprocessors and offer some form of point-to-point serial communications such as RS-232C, RS-485, or

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS networked communication such as Ethernet with proprietary or open protocols. PLCs typically execute the boolean or ladder logic configuration at high rates; 10-ms execution intervals are common. This does not necessarily imply that the analog I/O or PID control functions are executed at the same rate. Many PLCs execute the analog program at a much slower rate. Manufacturers’ specifications must be consulted. Personal Computer Controller Because of its high performance at low cost and its unexcelled ease of use, application of the personal computer (PC) as a platform for process controllers is growing. When configured to perform scan, control, alarm, and data acquisition (SCADA) functions and combined with a spreadsheet or database management application, the PC controller can be a low-cost, basic alternative to the DCS or PLC. Using the PC for control requires installation of a board into the expansion slot in the computer, or the PC can be connected to an external I/O module by using a standard communication port on the PC. The communication is typically achieved through a serial interface (RS232, RS-422, or IEEE-488), universal serial bus (USB), or Ethernet. The controller card/module supports 16- or 32-bit microprocessors. Standardization and high volume in the PC market have produced a large selection of hardware and software tools for PC controllers. The PC can also be interfaced to a DCS to perform advanced control or optimization functions that are not available within the standard DCS function library. Single-Loop Controller The single-loop controller (SLC) is a process controller that produces a single output. SLCs can be pneumatic, analog electronic, or microprocessor-based. Pneumatic SLCs are discussed in the pneumatic controller section, and analog electronic SLC is not discussed because it has been virtually replaced by the microprocessor-based design. The microprocessor-based SLC uses an 8- or 16-bit microprocessor with a small number of digital and analog process input channels with control logic for the I/O incorporated within the controller. Analog inputs and outputs are available in the standard ranges (1 to 5 V dc and 4 to 20 mA dc). Direct process inputs for temperature sensors (thermistor RTD and thermocouple types) are available. Binary outputs are also available. The face of the SLC has some form of visible display and pushbuttons that are used to view or adjust control values and configuration. SLCs are available for mounting in panel openings as small as 48 × 48 mm (1.9 × 1.9 in). The processor-based SLC allows the user to select from a set of predefined control strategies or to create a custom control strategy by using a set of control function blocks. Control function blocks include PID, on/off, lead/lag, adder/subtractor, multiply/divide, filter functions, signal selector, peak detector, and analog track. SLCs feature auto/manual transfer switching, multi-set-point self-diagnostics, gain scheduling, and perhaps also time sequencing. Most processor-based SLCs have self-tuning or auto-tuning PID control algorithms. Sample times for the microprocessor-based SLCs vary from 0.1 to 0.5 s. Lowpass analog electronic filters are usually installed on the process inputs to minimize aliasing errors caused by high-frequency content in the process signal. Input filter time constants are typically in the range of 0.1 to 1 s. Microprocessor-based SLCs may be made part of a DCS by using the communication port (RS-488 is common) on the controller or may be operated in a stand-alone mode independently of the DCS. Field Bus Controller Field bus technology is a network-based communications system that interconnects measurement and control equipment such as sensors, actuators, and controllers. Advanced field bus systems, intended for process control applications, such as Foundation Fieldbus, enable digital interoperability among these devices and have a built-in capability to distribute the control application across the network. Several manufacturers have made available Foundation Fieldbus devices that support process controller functionality. These controllers, known as field bus controllers, typically reside in the final control element or measurement transmitter, but can be designed into any field bus device. A suitable communications interface connects the Fieldbus segment to the distributed control system. When configuring the control strategy, all or part of the strategy may be loaded into the field bus devices. The remaining part of the control strategy would reside in the DCS itself. The distribution of the control function depends on the processing capacity of the field bus devices, the control strategy, and where it makes sense to perform these functions. Linearization of a control valve could be performed in the digital valve positioner (controller), for

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example. Temperature and pressure compensation of a flow measurement could be performed in the flow transmitter processor. The capability of field bus devices varies greatly. Some devices will allow instances of control system function blocks to be loaded and executed, while other devices allow the use of only preconfigured function blocks. Field bus controllers are typically configured as single-loop PID controllers, but cascade or other complex control strategies can be configured depending on the capability of the field bus device. Field bus devices that have native support for process control functions do not necessarily implement the PID algorithm in the same way. It is important to understand these differences so that the controller tuning will deliver the desired closed-loop characteristics. The functionality of field bus devices is projected to increase as the controller market develops. Controller Reliability and Application Trends Critical process control applications demand a high level of reliability from the electronic controller. Some methods that improve the reliability of electronic controllers include (1) focusing on robust circuit design using quality components; (2) using redundant circuits, modules, or subsystems where necessary; (3) using small backup systems when needed; (4) reducing repair time and using more powerful diagnostics; and (5) distributing functionality to more independent modules to limit the impact of a failed module. Currently, the trend in process control is away from centralized process control and toward an increased number of small distributed control or PLC systems. This trend will put emphasis on the evolution of the field bus controller and continued growth of the PC-based controller. Also, as hardware and software improve, the functionality of the controller will increase, and the supporting hardware will be physically smaller. Hence, the traditional lines between the DCS and the PLC will become less distinct as systems become capable of supporting either function set. Controller Performance and Process Dynamics The design of a control loop must take the control objectives into account. What do you want this loop to do? And under what operating conditions? There may be control applications that require a single control objective and others that have several control objectives. Control objectives may include such requirements as minimum variance control at steady state, maximum speed of recovery from a major disturbance, maximum speed of set-point response where overshoot and ringing are acceptable, critically damped set-point response with no overshoot, robustness issues, and special start-up or shutdown requirements. The control objectives will define not only the tuning requirements of the controller, but also, to a large extent, the allowable dynamic parameters of the field instruments and process design. Process dynamics alone can prevent control objectives from being realized. Tuning of the controller cannot compensate for an incompatible process or unrealistic control objectives. For most controllers, the difference between the set-point and process feedback signal, the error, is the input to the PID algorithm. The calculated PID output is sent back to the final control element. Every component between the controller output and the process feedback is considered by the controller as the “process” and will directly affect the dynamics and ultimately the performance of the system. This includes not only the dynamics of the physical process, but also the dynamics of the field instruments, signal conditioning equipment, and controller signal processing elements such as filters, scaling, and linearization routines. The choice of final control element can significantly affect the dynamics of the system. If the process dynamics are relatively slow, with time constants of a few minutes or longer, most control valves are fast enough that their contribution to the overall process time response will be negligible. In cases where the process time constants are only a few seconds, the control valve dynamics may become the dominant lag in the overall response. Excessive filtering in the field-sensing devices may also mask the true process dynamics and potentially limit control performance. Often, the design of a control loop and the tuning of the controller are a compromise between a number of different control objectives. When a compromise is unacceptable, gain scheduling or other adaptive tuning routine may be necessary to match the controller response to the most appropriate control objective. When one is tuning a controller, the form of the PID algorithm must be known. The three common forms of the PID algorithm are parallel

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PROCESS CONTROL

or noninteracting, classical or interacting, and the ISA Standard form. In most cases, a controller with any of these PID forms can be tuned to produce the desired closed-loop response. The actual tuning parameters will be different. The units of the tuning parameters also affect their value. The controller gain parameter is typically represented as a pure gain (Kc), acting on the error, or as proportional band (PB). In cases where the proportional band parameter is used, the equivalent controller gain is equal to 100 divided by the proportional band and represents the percent span that the error must traverse to produce a 100 percent change in the controller output. The proportional band is always applied to the controller error in terms of percent of span and percent of output. Controllers that use a gain tuning parameter commonly scale the error into percent span and use a percent output basis. In some controllers, the error is scaled by using a separate parameter into percent span prior to the PID algorithm. The gain parameter can also be applied to the error in engineering units. Even though most controller outputs are scaled as a percent, in cascade strategies the controller output may need to be scaled to the same span at the slave loop set point. In this case, the controller gain may in fact be required to calculate the controller output in terms of the slave loop engineering units. The execution rate of a digital controller should be sufficiently fast, compared to the process dynamics, to produce a response that closely approximates that of an analog controller with the same tuning. A general rule of thumb is that the execution interval should be at least 3 times faster than the dominant lag of the process or about 10 times faster than the closed-loop time constant. The controller can be used when the sample rates are slower than this recommendation, but the controller output will consist of a series of coarse steps as compared to a smooth response. This may create stability problems. Some integral and derivative algorithms may not be accurate when the time-based tuning parameters approach the controller execution interval. The analog inputs of the controller are typically protected from aliasing errors through the use of one- or two-pole analog filters. Faster sample rates allow a smaller antialiasing filter and improved input response characteristics. Some controllers or I/O subsystems oversample the analog inputs with respect to the controller execution interval and then process the input data through a digital filter. This technique can produce a more representative measurement with less quantization noise. Differences in the PID algorithm, controller parameters, units, and other fundamental control functions highlight the importance of understanding the structure of the controller and the requirement of sufficiently detailed documentation. This is especially important for the controller but is also important for the field instruments, final control elements, and device that have the potential to affect the signal characteristics. CONTROL VALVES A control valve consists of a valve, an actuator, and usually one or more valve control devices. The valves discussed in this section are applicable to throttling control (i.e., where flow through the valve is regulated to any desired amount between maximum and minimum limits). Other valves such as check, isolation, and relief valves are addressed in the next subsection. As defined, control valves are automatic control devices that modify the fluid flow rate as specified by the controller. Valve Types Types of valves are categorized according to their design style. These styles can be grouped into type of stem motion— linear or rotary. The valve stem is the rod, shaft, or spindle that connects the actuator with the closure member (i.e., a movable part of the valve that is positioned in the flow path to modify the rate of flow). Movement of either type of stem is known as travel. The major categories are described briefly below. Globe and Angle The most common linear stem-motion control valve is the globe valve. The name comes from the globular cavities around the port. In general, a port is any fluid passageway, but often the reference is to the passage that is blocked off by the closure member when the valve is closed. In globe valves, the closure member is called a plug. The plug in the valve shown in Fig. 8-66 is guided by a large-diameter port and moves within the port to provide the flow control orifice of the valve. A very popular alternate construction is a cage-guided plug, as illustrated in Fig. 8-67. In many such designs,

Actuator yoke mounting boss

Spring-loaded PTFE V-ring packing

Bonnet Guide bushing Plug

Body

Seat ring

Gasket

FIG. 8-66 Post-guided contour plug globe valve with metal seat and raisedface flange end connections. (Courtesy Fisher Controls International LLC.)

Stem

Graphite packing Bonnet gasket

Backup ring Seal ring

Cage

Valve plug

Seat

PTFE disk

Metal disk retainer

FIG. 8-67 Cage-guided balanced plug globe valve with polymer seat and plug seal. (Courtesy Fisher Controls International LLC.)

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS openings in the cage provide the flow control orifices. The valve seat is the zone of contact between the moving closure member and the stationary valve body, which shuts off the flow when the valve is closed. Often the seat in the body is on a replaceable part known as a seat ring. This stationary seat can also be designed as an integral part of the cage. Plugs may also be port-guided by wings or a skirt that fits snugly into the seat-ring bore. One distinct advantage of cage guiding is the use of balanced plugs in single-port designs. The unbalanced plug depicted in Fig. 8-66 is subjected to a static pressure force equal to the port area times the valve pressure differential (plus the stem area times the downstream pressure) when the valve is closed. In the balanced design (Fig. 8-67), note that both the top and bottom of the plug are subjected to the same downstream pressure when the valve is closed. Leakage via the plug-to-cage clearance is prevented by a plug seal. Both plug types are subjected to hydrostatic force due to internal pressure acting on the stem area and to dynamic flow forces when the valve is flowing. The plug, cage, seat ring, and associated seals are known as the trim. A key feature of globe valves is that they allow maintenance of the trim via a removable bonnet without removing the valve body from the line. Bonnets are typically bolted on but may be threaded in smaller sizes. Angle valves are an alternate form of the globe valve. They often share the same trim options and have the top-entry bonnet style. Angle valves can eliminate the need for an elbow but are especially useful when direct impingement of the process fluid on the body wall is to be avoided. Sometimes it is not practical to package a long trim within a globe body, so an angle body is used. Some angle bodies are self-draining, which is an important feature for dangerous fluids. Butterfly The classic design of butterfly valves is shown in Fig. 8-68. Its chief advantage is high capacity in a small package and a very low initial cost. Much of the size and cost advantage is due to the wafer body design, which is clamped between two pipeline flanges. In the simplest design, there is no seal as such, merely a small clearance gap between the disc OD and the body ID. Often a true seal is provided by a resilient material in the body that is engaged via an interference fit with the disc. In a lined butterfly valve, this material covers the entire body ID and extends around the body ends to eliminate the need for pipeline joint gaskets. In a fully lined valve, the disc is also coated to minimize corrosion or erosion. A high-performance butterfly valve has a disc that is offset from the shaft centerline. This eccentricity causes the seating surface to move away from the seal once the disc is out of the closed position, reducing friction and seal wear. It is also known as an eccentric disc valve; the advantage of the butterfly valve is improved shutoff while maintaining

Stem PTFE molded to inside diameter and sides of backup ring Bearing Shaft

Body

Special disc design to reduce dynamic torque

Partial cutaway of wafer-style lined butterfly valve. (Courtesy Fisher Controls International LLC.)

FIG. 8-68

high ultimate capacity at a reasonable cost. This cost advantage relative to other design styles is particularly true in sizes above 6-in nominal pipe size (NPS). Improved shutoff is due to advances in seal technologies, including polymer, flexing metal, combination metal with polymer inserts, and so on, many utilizing pressure assist. Ball Ball valves get their name from the shape of the closure member. One version uses a full spherical member with a cylindrical bore through it. The ball is rotated one-quarter turn from the fullclosed to the full-open position. If the bore is the same diameter as the mating-pipe fitting ID, the valve is referred to as full-bore. If the hole is undersized, the ball valve is considered to be a venturi style. A segmented ball is a portion of a hollow sphere that is large enough to block the port when closed. Segmented balls often have a V-shaped contour along one edge, which provides a desirable flow characteristic (see Fig. 8-69). Both full ball and segmented ball valves are known for

V-notch ball Ball seal

Seal protector ring Gasket

Actuator mounting Follower shaft

Groove pin Body

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Drive shaft Bearing Taper key

Packing follower

FIG. 8-69 Segmented ball valve. Partial view of actuator mounting shown 90° out of position. (Courtesy Fisher Controls International LLC.)

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PROCESS CONTROL

their low resistance to flow when full open. Shutoff leakage is minimized through the use of flexing or spring-loaded elastomeric or metal seals. Bodies are usually in two or three pieces or have a removable retainer to facilitate installing seals. End connections are usually flanged or threaded in small sizes, although segmented ball valves are offered in wafer style also. Plug There are two substantially different rotary valve design categories referred to as plug valves. The first consists of a cylindrical or slightly conical plug with a port through it. The plug rotates to vary the flow much as a ball valve does. The body is top-entry but is geometrically simpler than a globe valve and thus can be lined with fluorocarbon polymer to protect against corrosion. These plug valves have excellent shutoff but are generally not for modulating service due to high friction. A variation of the basic design (similar to the eccentric butterfly disc) only makes sealing contact in the closed position and is used for control. The other rotary plug design is portrayed in Fig. 8-70. The seating surface is substantially offset from the shaft, producing a ball-valvelike motion with the additional cam action of the plug into the seat when closing. In reverse flow, high-velocity fluid motion is directed inward, impinging on itself and only contacting the plug and seat ring. Multiport This term refers to any valve or manifold of valves with more than one inlet or outlet. For throttling control, the three-way body is used for blending (two inlets, one outlet) or as a diverter (one inlet, two outlets). A three-way valve is most commonly a special globelike body with special trim that allows flow both over and under the plug. Two rotary valves and a pipe tee can also be used. Special three-, four-, and five-way ball valve designs are used for switching applications. Special Application Valves Digital Valves True digital valves consist of discrete solenoidoperated flow ports that are sized according to binary weighing. The valve can be designed with sharp-edged orifices or with streamlined nozzles that can be used for flow metering. Precise control of the throttling control orifice is the strength of the digital valve. Digital valves are mechanically complicated and expensive, and they have considerably reduced maximum flow capacities compared to the globe and rotary valve styles. Cryogenic Service Valves designed to minimize heat absorption for throttling liquids and gases below 80 K are called cryogenic service

Eccentric plug valve shown in erosion-resistant reverse flow direction. Shaded components can be made of hard metal or ceramic materials. (Courtesy Fisher Controls International LLC.)

FIG. 8-70

valves. These valves are designed with small valve bodies to minimize heat absorption and long bonnets between the valve and actuator to allow for extra layers of insulation around the valve. For extreme cases, vacuum jacketing can be constructed around the entire valve to minimize heat influx. High Pressure Valves used for pressures nominally above 760 bar (11,000 psi, pressures above ANSI Class 4500) are often customdesigned for specific applications. Normally, these valves are of the plug type and use specially hardened plug and seat assemblies. Internal surfaces are polished, and internal corners and intersecting bores are smoothed to reduce high localized stresses in the valve body. Steam loops in the valve body are available to raise the body temperature to increase ductility and impact strength of the body material. High-Viscous Process Used most extensively by the polymer industry, the valve for high-viscous fluids is designed with smooth finished internal passages to prevent stagnation and polymer degradation. These valves are available with integral body passages through which a heat-transfer fluid is pumped to keep the valve and process fluid heated. Pinch The industrial equivalent of controlling flow by pinching a soda straw is the pinch valve. Valves of this type use fabric-reinforced elastomer sleeves that completely isolate the process fluid from the metal parts in the valve. The valve is actuated by applying air pressure directly to the outside of the sleeve, causing it to contract or pinch. Another method is to pinch the sleeve with a linear actuator with a specially attached foot. Pinch valves are used extensively for corrosive material service and erosive slurry service. This type of valve is used in applications with pressure drops up to 10 bar (145 psi). Fire-Rated Valves that handle flammable fluids may have additional safety-related requirements for minimal external leakage, minimal internal (downstream) leakage, and operability during and after a fire. Being fire-rated does not mean being totally impervious to fire, but a sample valve must meet particular specifications such as those of American Petroleum Institute (API) 607, Factory Mutual Research Corp. (FM) 7440, or the British Standard 5146 under a simulated fire test. Due to very high flame temperature, metal seating (either primary or as a backup to a burned-out elastomer) is mandatory. Solids Metering The control valves described earlier are primarily used for the control of fluid (liquid or gas) flow. Sometimes these valves, particularly the ball, butterfly, or sliding gate valves, are used to throttle dry or slurry solids. More often, special throttling mechanisms such as venturi ejectors, conveyers, knife-type gate valves, or rotating vane valves are used. The particular solids-metering valve hardware depends on the volume, density, particle shape, and coarseness of the solids to be handled. Actuators An actuator is a device that applies the force (torque) necessary to cause a valve’s closure member to move. Actuators must overcome pressure and flow forces as well as friction from packing, bearings or guide surfaces, and seals; and must provide the seating force. In rotary valves, maximum friction occurs in the closed position, and the moment necessary to overcome it is referred to as breakout torque. The rotary valve shaft torque generated by steady-state flow and pressure forces is called dynamic torque. It may tend to open or close the valve depending on valve design and travel. Dynamic torque per unit pressure differential is largest in butterfly valves at roughly 70° open. In linear stem-motion valves, the flow forces should not exceed the available actuator force, but this is usually accounted for by default when the seating force is provided. Actuators often provide a fail-safe function. In the event of an interruption in the power source, the actuator will place the valve in a predetermined safe position, usually either full-open or full-closed. Safety systems are often designed to trigger local fail-safe action at specific valves to cause a needed action to occur, which may not be a complete process or plant shutdown. Actuators are classified according to their power source. The nature of these sources leads naturally to design features that make their performance characteristics distinct. Pneumatic Despite the availability of more sophisticated alternatives, the pneumatically driven actuator is still by far the most popular type. Historically the most common has been the spring and diaphragm design (Fig. 8-71). The compressed air input signal fills a

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

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Air connection Diaphragm casing

O-ring seal

Diaphragm and stem shown in up position

Piston Diaphragm plate Lower diaphragm casing

Piston rod

Cylinder Sliding seal

Actuator spring Actuator stem Spring seat

Lever Spring adjustor Stem connector

Yoke Travel indicator disk Indicator scale FIG. 8-71 Spring and diaphragm actuator with an “up” fail-safe mode. Spring adjuster allows slight alteration of bench set. (Courtesy Fisher Controls International LLC.)

chamber sealed by an elastomeric diaphragm. The pressure force on the diaphragm plate causes a spring to be compressed and the actuator stem to move. This spring provides the fail-safe function and contributes to the dynamic stiffness of the actuator. If the accompanying valve is “push down to close,” the actuator depicted in Fig. 8-71 will be described as “air to close” or synonymously as fail-open. A slightly different design yields “air to open” or fail-closed action. The spring is typically precompressed to provide a significant available force in the failed position (e.g., to provide seating load). The spring also provides a proportional relationship between the force generated by air pressure and stem position. The pressure range over which a spring and diaphragm actuator strokes in the absence of valve forces is known as the bench set. The chief advantages of spring and diaphragm actuators are their high reliability, low cost, adequate dynamic response, and fail-safe action—all of which are inherent in their simple design. Alternately, the pressurized chamber can be formed by a circular piston with a seal on its outer edge sliding within a cylindrical bore. Higher operating pressure [6 bar (~ 90 psig) is typical] and longer strokes are possible. Piston actuators can be spring-opposed but many times are in a dual-acting configuration (i.e., compressed air is applied to both sides of the piston with the net force determined from the pressure difference—see Fig. 8-72). Dynamic stiffness is usually higher with piston designs than with spring and diaphragm actuators; see “Positioner/Actuator Stiffness.” Fail-safe action, if necessary, is achieved without a spring through the use of additional solenoid valves, trip valves, or relays. See “Valve Control Devices.” Motion Conversion Actuator power units with translational output can be adapted to rotary valves that generally need 90° or less rotation. A lever is attached to the rotating shaft, and a link with pivoting means on the end connects to the linear output of the power unit, an arrangement similar to an internal combustion engine crankshaft, connecting rod, and piston. When the actuator piston, or more commonly the diaphragm plate, is designed to tilt, one pivot can be eliminated (see Fig. 8-72). Scotch yoke and rack-and-pinion arrangements are also commonly used, especially with piston power units.

FIG. 8-72 Double-acting piston rotary actuator with lever and tilting piston for motion conversion. (Courtesy Fisher Controls International LLC.)

Friction and the changing mechanical advantage of these motion conversion mechanisms mean the available torque may vary greatly with travel. One notable exception is vane-style rotary actuators whose offset “piston” pivots, giving direct rotary output. Hydraulic The design of typical hydraulic actuators is similar to that of double-acting piston pneumatic types. One key advantage is the high pressure [typically 35 to 70 bar (500 to 1000 psi)], which leads to high thrust in a smaller package. The incompressible nature of the hydraulic oil means these actuators have very high dynamic stiffness. The incompressibility and small chamber size connote fast stroking speed and good frequency response. The disadvantages include high initial cost, especially when considering the hydraulic supply. Maintenance is much more difficult than with pneumatics, especially on the hydraulic positioner. Electrohydraulic actuators have similar performance characteristics and cost/maintenance ramifications. The main difference is that they contain their own electric-powered hydraulic pump. The pump may run continuously or be switched on when a change in position is required. Their main application is remote sites without an air supply when a fail-safe spring return is needed. Electric The most common electric actuators use a typical motor—three-phase ac induction, capacitor-start split-phase induction, or dc. Normally the motor output passes through a large gear reduction and, if linear motion output is required, a ball screw or thread. These devices can provide large thrust, especially given their size. Lost motion in the gearing system does create backlash, but if not operating across a thrust reversal, this type of actuator has very high stiffness. Usually the gearing system is self-locking, which means that forces on the closure member cannot move it by spinning a nonenergized motor. This behavior is called a lock-in-lastposition fail-safe mode. Some gear systems (e.g., low-reduction spur gears) can be backdriven. A solenoid-activated mechanical brake or locking current to motor field coils is added to provide lock-in-lastposition fail-safe mode. A battery backup system for a dc motor can guard against power failures. Otherwise, an electric actuator is not acceptable if fail-open/closed action is mandatory. Using electric power requires environmental enclosures and explosion protection, especially in hydrocarbon processing facilities; see the full discussion in “Valve Control Devices.”

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PROCESS CONTROL

Unless sophisticated speed control power electronics is used, position modulation is achieved via bang-zero-bang control. Mechanical inertia causes overshoot, which is (1) minimized by braking and/or (2) hidden by adding dead band to the position control. Without these provisions, high starting currents would cause motors to overheat from constant “hunting” within the position loop. Travel is limited with power interruption switches or with force (torque) electromechanical cutouts when the closed position is against a mechanical stop (e.g., a globe valve). Electric actuators are often used for on/off service. Stepper motors can be used instead, and they, as their name implies, move in fixed incremental steps. Through gear reduction, the typical number of increments for 90° rotation ranges from 5000 to 10,000; hence positioning resolution at the actuator is excellent. An electromagnetic solenoid can be used to directly actuate the plug on very small linear stem-motion valves. A solenoid is usually designed as a two-position device, so this valve control is on/off. Special solenoids with position feedback can provide proportional action for modulating control. Force requirements of medium-sized valves can be met with piloted plug designs, which use process pressure to assist the solenoid force. Piloted plugs are also used to minimize the size of common pneumatic actuators, especially when there is need for high seating load. Manual A manually positioned valve is by definition not an automatic control valve, but it may be involved with process control. For rotary valves, the manual operator can be as simple as a lever, but a wheel driving a gear reduction is necessary in larger valves. Linear motion is normally created with a wheel turning a screw-type device. A manual override is usually available as an option for the powered actuators listed above. For spring-opposed designs, an adjustable travel stop will work as a one-way manual input. In more complex designs, the handwheel can provide loop control override via an engagement means. Some gear reduction systems of electric actuators allow the manual positioning to be independent of the automatic positioning without declutching. OTHER PROCESS VALVES In addition to the throttling control valve, other types of process valves can be used to manipulate a process. Valves for On/Off Applications Valves are often required for service that is primarily nonthrottling. Valves in this category, depending on the service requirements, may be of the same design as the types used for throttling control or, as in the case of gate valves, different in design. Valves in this category usually have tight shutoff when they are closed and low pressure drops when they are wide open. The on/off valve can be operated manually, such as by handwheel or lever; or automatically, with pneumatic or electric actuators. Batch Batch process operation is an application requiring on/off valve service. Here the valve is opened and closed to provide reactant, catalyst, or product to and from the batch reactor. Like the throttling control valve, the valve used in this service must be designed to open and close thousands of times. For this reason, valves used in this application are often the same valves used in continuous throttling applications. Ball valves are especially useful in batch operations. The ball valve has a straight-through flow passage that reduces pressure drop in the wide-open state and provides tight shutoff capability when closed. In addition, the segmented ball valve provides for shearing action between the ball and the ball seat that promotes closure in slurry service. Isolation A means for pressure-isolating control valves, pumps, and other piping hardware for installation and maintenance is another common application for an on/off valve. In this application, the valve is required to have tight shutoff so that leakage is stopped when the piping system is under repair. As the need to cycle the valve in this application is far less than that of a throttling control valve, the wear characteristics of the valve are less important. Also, because many are required in a plant, the isolation valve needs to be reliable, simple in design, and simple in operation. The gate valve, shown in Fig. 8-73, is the most widely used valve in this application. The gate valve is composed of a gatelike disc that moves perpendicular to the flow stream. The disc is moved up and

FIG. 8-73

Gate valve. (Courtesy Crane Valves.)

down by a threaded screw that is rotated to effect disc movement. Because the disc is large and at right angles to the process pressure, a large seat loading for tight shutoff is possible. Wear produced by high seat loading during the movement of the disc prohibits the use of the gate valve for throttling applications. Pressure Relief Valves The pressure relief valve is an automatic pressure-relieving device designed to open when normal conditions are exceeded and to close again when normal conditions are restored. Within this class there are relief valves, pilot-operated pressure relief valves, and safety valves. Relief valves (see Fig. 8-74) have springloaded discs that close a main orifice against a pressure source. As pressure rises, the disc begins to rise off the orifice and a small amount of fluid passes through the valve. Continued rise in pressure above the opening pressure causes the disc to open the orifice in a proportional fashion. The main orifice reduces and closes when the pressure returns to the set pressure. Additional sensitivity to overpressure conditions can be improved by adding an auxiliary pressure relief valve (pilot) to the basic pressure relief valve. This combination is known as a pilot-operated pressure relief valve. The safety valve is a pressure relief valve that is designed to open fully, or pop, with only a small amount of pressure over the rated limit. Where conventional safety valves are sensitive to downstream pressure and may have unsatisfactory operating characteristics in variable backpressure applications, pressure-balanced safety relief valve designs are available to minimize the effect of downstream pressure on performance. Application and sizing of pressure relief valves, pilot-operated pressure relief valves, and safety valves for use on pressure vessels are found in the ASME Boiler and Pressure Vessel Code, Section VIII, Division 1, “Rules for Construction of Pressure Vessels,” Paragraphs UG-125 through UG-137.

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

FIG. 8-74

Relief valve. (Courtesy of Teledyne Fluid Systems, Farris Engineering.)

Check Valves The purpose of a check valve is to allow relatively unimpeded flow in the desired direction but to prevent flow in the reverse direction. Two common designs are swing-type and lift-type check valves, the names of which denote the motion of the closure member. In the forward direction, flow forces overcome the weight of the member or a spring to open the flow passage. With reverse pressure conditions, flow forces drive the closure member into the valve seat, thus providing shutoff. VALVE DESIGN CONSIDERATIONS Functional requirements and the properties of the controlled fluid determine which valve and actuator types are best for a specific application. If demands are modest and no unique valve features are required, the valve design style selection may be determined solely by

8-79

cost. If so, general-purpose globe or angle valves provide exceptional value, especially in sizes less than 3-in NPS and hence are very popular. Beyond type selection, there are many other valve specifications that must be determined properly to ultimately yield improved process control. Materials and Pressure Ratings Valves must be constructed from materials that are sufficiently immune to corrosive or erosive action by the process fluid. Common body materials are cast iron, steel, stainless steel, high-nickel alloys, and copper alloys such as bronze. Trim materials need better corrosion and erosion resistance due to the higher fluid velocity in the throttling region. High hardness is desirable in erosive and cavitating applications. Heat-treated and precipitationhardened stainless steels are common. High hardness is also good for guiding, bearing, and seating surfaces; cobalt-chromium alloys are utilized in cast or wrought form and frequently as welded overlays called hard facing. In less stringent situations, chrome plating, heat-treated nickel coatings, and nitriding are used. Tungsten carbide and ceramic trim are warranted in extremely erosive services. See Sec. 25, “Materials of Construction,” for specific material properties. Since the valve body is a pressurized vessel, it is usually designed to comply with a standardized system of pressure ratings. Two common systems are described in the standards ASME B16.34 and EN 12516. Internal pressure limits under these standards are divided into broad classes, with specific limits being a function of material and temperature. Manufacturers also assign their own pressure ratings based on internal design rules. A common insignia is 250 WOG, which means a pressure rating of 250 psig (~17 bar) in water, oil, or gas at ambient temperature. “Storage and Process Vessels” in Sec. 10 provides introductory information on compliance of pressure vessel design to industry codes (e.g., ASME Boiler and Pressure Vessel Code, Section VIII; ASME B31.3 Chemical Plant and Petroleum Refinery Piping). Valve bodies are also standardized to mate with common piping connections: flanged, butt-welded end, socket-welded end, and screwed end. Dimensional information for some of these joints and class pressure-temperature ratings are included in Sec. 10, “Process Plant Piping.” Control valves have their own standardized face-to-face dimensions that are governed by ANSI/ISA Standards S75.08 and S75.22. Butterfly valves are also governed by API 609 and Manufacturers Standardization Society (MSS) SP-67 and SP-68. Sizing Throttling control valves must be selected to pass the required flow rate, given expected pressure conditions. Sizing is not merely matching the end connection size with surrounding piping; it is a key step in ensuring that the process can be properly controlled. Sizing methods range from simple models based on elementary fluid mechanics to very complex models when unusual thermodynamics or nonideal behaviors occur. Basic sizing practices have been standardized (for example, ANSI-75.01.01) and are implemented as PC-based programs by manufacturers. The following is a discussion of very basic sizing equations and the associated physics. Regardless of the particular process variable being controlled (e.g., temperature, level, pH), the output of a control valve is the flow rate. The throttling valve performs its function of manipulating the flow rate by virtue of being an adjustable resistance to flow. Flow rate and pressure conditions are normally known when a process is designed, and the valve resistance range must be matched accordingly. In the tradition of orifice and nozzle discharge coefficients, this resistance is embodied in the valve flow coefficient CV. By applying the principles of conversation of mass and energy, the mass flow rate w kg/h is given for a liquid by w = 27.3CVρ(p  p2) 1 −

(8-111)

where p1 and p2 are upstream and downstream static pressure, bar, respectively. The density of the fluid ρ is expressed in kilograms per cubic meter. This equation is valid for nonvaporizing, turbulent flow conditions for a valve with no attached fittings. While the above equation gives the relationship between pressure and flow from a macroscopic point of view, it does not explain what is going on inside the valve. Valves create a resistance to flow by restricting the cross-sectional area of the flow passage and by forcing the fluid to change direction as it passes through the body and trim.

8-80

PROCESS CONTROL

The conservation of mass principle dictates that, for steady flow, the product of density, average velocity, and cross-sectional area remain a constant. The average velocity of the fluid stream at the minimum restriction in the valve is therefore much higher than that at the inlet. Note that due to the abrupt nature of the flow contraction that forms the minimum passage, the main fluid stream may separate from the passage walls and form a jet that has an even smaller cross section, the so-called vena contracta. The ratio of minimum stream area to the corresponding passage area is called the contraction coefficient. As the fluid expands from the minimum cross-sectional area to the full passage area in the downstream piping, large amounts of turbulence are generated. Direction changes can also induce significant amounts of turbulence. Figure 8-75 is an illustration of how the mean pressure changes as fluid moves through a valve. Some of the potential energy that was stored in the fluid by pressurizing it (e.g., the work done by a pump) is first converted to the kinetic energy of the fast-moving fluid at the vena contracta. Some of that kinetic energy turns into the kinetic energy of turbulence. As the turbulent eddies break down into smaller and smaller structures, viscous effects ultimately convert all the turbulent energy to heat. Therefore, a valve converts fluid energy from one form to another. For many valve constructions, it is reasonable to approximate the fluid transition from the valve inlet to the minimum cross section of the flow stream as an isentropic or lossless process. Using this approximation, the minimum pressure pVC can be estimated from the

Bernoulli relationship. See Sec. 6 (“Fluid and Particle Mechanics”) for more background information. Downstream of the vena contracta, the flow is definitely not lossless due to all the turbulence that is generated. As the flow passage area increases and the fluid slows down, some of the kinetic energy of the fluid is converted back to pressure energy as pressure recovers. The energy that is permanently lost via turbulence accounts for the permanent pressure or head loss of the valve. The relative amount of pressure that is recouped determines whether the valve is considered to be high- or low-recovery. The flow passage geometry at and downstream of the vena contracta primarily determines the amount of recovery. The amount of recovery is quantified by the liquid pressure recovery factor FL FL =

p −p  p −p 1

2

1

(8-112)

VC

Under some operating conditions, sufficient pressure differential may exist across the valve to cause the vena contracta pressure to drop to the saturation pressure (also known as the vapor pressure) of the liquid. If this occurs, a portion of the liquid will vaporize, forming a two-phase, compressible mixture within the valve. If sufficient vapor forms, the flow may choke. When a flow is choked, any increase in pressure differential across the valve no longer produces an increase in flow through the valve. The vena contracta condition at choked flow for pure liquids has been shown to be pVC = FFpv

(8-113)

where FF = 0.96 − 0.28

 p pv

(8-114)

c

and pVC is the absolute vena contracta pressure under choked conditions, FF is the liquid critical pressure ratio factor, pv is the absolute vapor pressure of the liquid at inlet temperature, and pc is the absolute thermodynamic critical pressure of the fluid. Equations (8-112) and (8-113) can be used together to determine the pressure differential across the valve at the point where the flow chokes ∆pchoked = FL2(p1 − FFpv)

(8-115)

The pressure recovery factor is a constant for any given valve at a given opening. The value of this factor can be established by flow test and is published by the valve manufacturer. If the actual pressure differential across the valve is greater than the choked pressure differential of Eq. (8-115), then ∆pchoked should be used in Eq. (8-111) to determine the correct valve size. A more complete presentation of sizing relationships is given in ANSI 75.01.01, including provisions for pipe reducers and Reynolds number effects. Equations (8-111) to (8-115) are restricted to incompressible fluids. For gases and vapors, the fluid density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal gas law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z (see Sec. 2, “Physical and Chemical Data,” for definitions and data for common fluids). For compressible fluids  TZ

w = 94.8CV p1 Y

xMω

(8-116)

1

Generic depictions of average pressure at subsequent cross sections throughout a control valve. The FL values selected for illustration are 0.9 and 0.63 for low and high recovery, respectively. Internal pressure in the high-recovery valve is shown as a dashed line for flashing conditions (p2 < pv) with pv = B. FIG. 8-75

where p1 is in bar absolute, T1 is inlet temperature in kelvins, Mw is the molecular weight, and x is the dimensionless pressure drop ratio (p1 − p2)/p1. The expansion factor Y accounts for changes in the fluid density as the fluid passes through the valve. It is dependent on pressure drop and valve geometry. Experimental data have shown that for

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS small departures in the ratio of specific heat from that of air (1.4), a simple linear relationship can be used to represent the expansion factor: 1.4x Y=1−  3xTγ

xTγ x≤  1.4

for

xTγ x>  1.4

and

Y = 0.67

(8-118)

Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflected, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through solids in a complex array of types of elastic waves. Sound is characterized by its amplitude, frequency, phase, and direction of propagation. Sound strength is therefore location-dependent and is often quantified as a sound pressure level Lp in decibels based on the root mean square (rms) sound pressure value pS, where pS Lp = 10 log  preference





2

usually random, broad-spectrum noise. The total sound pressure level from two such statistically uncorrelated sources is (in decibels) (pS1)2 + (pS2)2 Lp = 10 log  (preference)2

(8-117)

where γ is the ratio of specific heats and xT is an experimentally determined factor for a specific valve and is the largest value of x that contributes to flow (i.e., values of x greater than xT do not contribute to flow). The terminal value of x, xT, results from a phenomenon known as choking. Given a nozzle geometry with fixed inlet conditions, the mass flow rate will increase as p2 is decreased up to a maximum amount at the critical pressure drop. The velocity at the vena contracta has reached sonic, and a standing shock has formed. This shock causes a step change in pressure as flow passes through it, and further reduction in p2 does not increase mass flow. Thus xT relates to the critical pressure drop ratio and also accounts for valve geometry effects. The value of xT varies with flow path geometry and is supplied by the valve manufacturer. In the choked case,

(8-120)

For example, two sources of equal strength combine to create an Lp that is 3 dB higher. While noise is annoying to listen to, the real reasons for being concerned about noise relate to its impact on people and equipment. Hearing loss can occur due to long-term exposure to moderately high, or even short exposure to very high, noise levels. The U.S. Occupational Safety and Health Act (OSHA) has specific guidelines for permissible levels and exposure times. The human ear has a frequency-dependent sensitivity to sound. When the effect on humans is the criterion, Lp measurements are weighted to account for the ear’s response. This socalled A-weighted scale is defined in ANSI S1.4 and is commonly reported as LpA. Figure 8-76 illustrates the difference between actual and perceived airborne sound pressure levels. At sufficiently high levels, noise and the associated vibration can damage equipment. There are two approaches to fluid-generated noise control—source or path treatment. Path treatment means absorbing or blocking the transmission of noise after it has been created. The pipe itself is a barrier. The sound pressure level inside a standard schedule pipe is roughly 40 to 60 dB higher than on the outside. Thicker-walled pipe reduces levels somewhat more, and adding acoustical insulation on the outside of the pipe reduces ambient levels up to 10 dB per inch of thickness. Since noise propagates relatively unimpeded inside the

(8-119)

For airborne sound, the reference pressure is 2 × 10−5 Pa (29 × 10−1 psi), which is nominally the human threshold of hearing at 1000 Hz. The corresponding sound pressure level is 0 dB. A voice in conversation is about 50 dB, and a jackhammer operator is subject to 100 dB. Extreme levels such as a jet engine at takeoff might produce 140 dB at a distance of 3 m, which is a pressure amplitude of 200 Pa (29 × 10−3 psi). These examples demonstrate both the sensitivity and the wide dynamic range of the human ear. Traveling sound waves carry energy. Sound intensity I is a measure of the power passing through a unit area in a specified direction and is related to pS. Measuring sound intensity in a process plant gives clues to the location of the source. As one moves away from the source, the fact that the energy is spread over a larger area requires that the sound pressure level decrease. For example, doubling one’s distance from a point source reduces Lp by 6 dB. Viscous action from the induced fluid motion absorbs additional acoustic energy. However, in free air, this viscous damping is negligible over short distances (on the order of 1 m). Noise is a group of sounds with many nonharmonic frequency components of varying amplitudes and random phase. The turbulence generated by a throttling valve creates noise. As a valve converts potential energy to heat, some of the energy becomes acoustic energy as an intermediate step. Valves handling large amounts of compressible fluid through a large pressure change create the most noise because more total power is being transformed. Liquid flows are noisy only under special circumstances, as will be seen in the next subsection. Due to the random nature of turbulence and the broad distribution of length and velocity scales of turbulent eddies, valve-generated sound is

8-81

FIG. 8-76

Valve-generated sound pressure level spectrums.

8-82

PROCESS CONTROL

pipe, barrier approaches require the entire downstream piping system to be treated in order to be totally effective. In-line silencers place absorbent material inside the flow stream, thus reducing the level of the internally propagating noise. Noise reductions up to 25 dB can be achieved economically with silencers. The other approach to valve noise problems is the use of quiet trim. Two basic strategies are used to reduce the initial production of noise—dividing the flow stream into multiple paths and using several flow resistances in series. Sound pressure level Lp is proportional to mass flow and is dependent on vena contracta velocity. If each path is an independent source, it is easy to show from Eq. (8-120) that p S2 is inversely proportional to the number of passages; additionally, smaller passage size shifts the predominate spectral content to higher frequencies, where structural resonance may be less of a problem. Series resistances or multiple stages can reduce maximum velocity and/or produce backpressure to keep jets issuing from multiple passages from acting independently. While some of the basic principles are understood, predicting noise for a particle flow passage requires some empirical data as a basis. Valve manufacturers have developed noise prediction methods for the valves they build. ANSI/ISA-75.17 is a public-domain methodology for standard (non-low-noise) valve types, although treatment of some multistage, multipath types is underway. Low-noise hardware consists of special cages in linear stem valves, perforated domes or plates and multichannel inserts in rotary valves, and separate devices that use multiple fixed restrictions. Cavitation and Flashing From the discussion of pressure recovery it was seen that the pressure at the vena contracta can be much lower than the downstream pressure. If the pressure on a liquid falls below its vapor pressure pv, the liquid will vaporize. Due to the effect of surface tension, this vapor phase will first appear as bubbles. These bubbles are carried downstream with the flow, where they collapse if the pressure recovers to a value above pv. This pressure-driven process of vapor bubble formation and collapse is known as cavitation. Cavitation has three negative side effects in valves—noise and vibration, material removal, and reduced flow. The bubble collapse process is a violent asymmetric implosion that forms a high-speed microjet and induces pressure waves in the fluid. This hydrodynamic noise and the mechanical vibration that it can produce are far stronger than other noise generation sources in liquid flows. If implosions occur adjacent to a solid component, minute pieces of material can be removed, which, over time, will leave a rough, cinderlike surface. The presence of vapor in the vena contracta region puts an upper limit on the amount of liquid that will pass through a valve. A mixture of vapor and liquid has a lower density than that of the liquid alone. While Eq. (8-111) is not applicable to two-phase flows because pressure changes are redistributed due to varying density and the two phases do not necessarily have the same average velocity, it does suggest that lower density reduces the total mass flow rate. Figure 8-77 illustrates a typical flow rate/pressure drop relationship. As with compressible gas flow at a given p1, flow increases as p2 is decreased until the flow chokes (i.e., no additional fluid will pass). The transition between incompressible and choked flow is gradual because, within the convoluted flow passages of valves, the pressure is actually an uneven distribution at each cross section and consequently vapor formation zones increase gradually. In fact, isolated zones of bubble formation or incipient cavitation often occur at pressure drops well below that at which a reduction in flow is noticeable. The similarity between liquid and gas choking is not serendipitous; it is surmised that the two-phase fluid is traveling at the mixture’s sonic velocity in the throat when choked. Complex fluids with components having varying vapor pressures and/or entrained noncondensable gases (e.g., crude oil) will exhibit soft vaporization/implosion transitions. There are several methods to reduce cavitation or at least its negative side effects. Material damage is slowed by using harder materials and by directing the cavitating stream away from passage walls (e.g., with an angle body flowing down). Sometimes the system can be designed to place the valve in a higher p2 location or add downstream resistance, which creates backpressure. A low recovery valve has a higher minimum pressure for a given p2 and so is a means to eliminate the cavitation itself, not just its side effects. In Fig. 8-75, if pv < B, neither valve will cavitate substantially. For pv > B but pv < A, the high

FIG. 8-77

Liquid flow rate versus pressure drop (assuming constant p1 and pv).

recovery valve will cavitate substantially, but the low recovery valve will not. Special anticavitation trims are available for globe and angle valves and more recently for some rotary valves. These trims use multiple contraction/expansion stages or other distributed resistances to boost FL to values sometimes near unity. If p2 is below pv, the two-phase mixture will continue to vaporize in the body outlet and/or downstream pipe until all liquid phase is gone, a condition known as flashing. The resulting huge increase in specific volume leads to high velocities, and any remaining liquid droplets acquire much of the higher vapor-phase velocity. Impingement of these droplets can produce material damage, but it differs from cavitation damage because it exhibits a smooth surface. Hard materials and directing the two-phase jets away from solid surfaces are means to avoid this damage. Seals, Bearings, and Packing Systems In addition to their control function, valves often need to provide shutoff. FCI 70-2-1998 and IEC 60534-4 recognize six standard classifications and define their asshipped qualification tests. Class I is an amount agreed to by user and supplier with no test needed. Classes II, III, and IV are based on an air test with maximum leakage of 0.5 percent, 0.1 percent, and 0.01 percent of rated capacity, respectively. Class V restricts leakage to 5 × 10−6 mL of water per second per millimeter of port diameter per bar differential. Class VI allows 0.15 to 6.75 mL/min of air to escape depending on port size; this class implies the need for interference-fit elastomeric seals. With the exception of class V, all classes are based on standardized pressure conditions that may not represent actual conditions. Therefore, it is difficult to estimate leakage in service. Leakage normally increases over time as seals and seating surfaces become nicked or worn. Leak passages across the seat-contact line, known as

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

8-83

wire drawing, may form and become worse over time—even in hard metal seats under sufficiently high-pressure differentials. Polymers used for seat and plug seals and internal static seals include PTFE (polytetrafluoroethylene) and other fluorocarbons, polyethylene, nylon, polyether-ether-ketone, and acetal. Fluorocarbons are often carbon- or glass-filled to improve mechanical properties and heat resistance. Temperature and chemical compatibility with the process fluid are the key selection criteria. Polymer-lined bearings and guides are used to decrease friction, which lessens dead band and reduces actuator force requirements. See Sec. 25, “Materials of Construction,” for properties. Packing forms the pressure-tight seal, where the stem protrudes through the pressure boundary. Packing is typically made from PTFE or, for high temperature, a bonded graphite. If the process fluid is toxic, more sophisticated systems such as dual packing, live-loaded, or a flexible metal bellows may be warranted. Packing friction can significantly degrade control performance. Pipe, bonnet, and internal-trim joint gaskets are typically a flat sheet composite. Gaskets intended to absorb dimensional mismatch are typically made from filled spiralwound flat stainless-steel wire with PTFE or graphite filler. The use of asbestos in packing and gaskets has been largely eliminated. Flow Characteristics The relationship between valve flow and valve travel is called the valve flow characteristic. The purpose of flow characterization is to make loop dynamics independent of load, so that a single controller tuning remains optimal for all loads. Valve gain is one factor affecting loop dynamics. In general, gain is the ratio of change in output to change in input. The input of a valve is travel y, and the output is flow w. Since pressure conditions at the valve can depend on flow (hence travel), valve gain is dw ∂w dCV ∂w dp1 ∂w dp2  =  +  +  dy ∂CV dy ∂p1 dy ∂p2 dy

(8-121)

An inherent valve flow characteristic is defined as the relationship between flow rate and travel, under constant-pressure conditions. Since the rightmost two terms in Eq. (8-121) are zero in this case, the inherent characteristic is necessarily also the relationship between flow coefficient and travel. Figure 8-78 shows three common inherent characteristics. A linear characteristic has a constant slope, meaning the inherent valve gain is a constant. The most popular characteristic is equal-percentage, which gets its name from the fact that equal changes in travel produce equal-percentage changes in the existing flow coefficient. In other words, the slope of the curve is proportional to CV, or equivalently that inherent valve gain is proportional to flow. The equal-percentage characteristic can be expressed mathematically by CV(y) = (rated CV) exp

− 1 ln R   rated y y

(8-122)

This expression represents a set of curves parameterized by R. Note that CV (y = 0) equals (rated CV)/R rather than zero; real equal-percentage characteristics deviate from theory at some small travel to meet shutoff requirements. An equal-percentage characteristic provides perfect compensation for a process where the gain is inversely proportional to flow (e.g., liquid pressure). Quick opening does not have a standardized mathematical definition. Its shape arises naturally from high-capacity plug designs used in on/off service globe valves. Frequently, pressure conditions at the valve will change with flow rate. This so-called process influence [the rightmost two terms on the right-hand side of Eq. (8-121)] combines with inherent gain to express the installed valve gain. The flow versus travel relationship for a specific set of conditions is called the installed flow characteristic. Typically, valve ∆p decreases with load, since pressure losses in the piping system increase with flow. Figure 8-79 illustrates how allocation of total system head to the valve influences the installed flow characteristics. For a linear or quick-opening characteristic, this transition toward a concave down shape would be more extreme. This effect of typical process pressure variation, which causes equal-percentage

FIG. 8-78

Typical inherent flow characteristics.

characteristics to have fairly constant installed gain, is one reason the equal-percentage characteristic is the most popular. Due to clearance flow, flow force gradients, seal friction, and the like, flow cannot be throttled to an arbitrarily small value. Installed rangeability is the ratio of maximum to minimum controllable flow. The actuator and positioner, as well as the valve, influence the installed rangeability. Inherent rangeability is defined as the ratio of the largest to the smallest CV within which the characteristic meets specified criteria (see ISA 75.11). The R value in the equal-percentage definition is a theoretical rangeability only. While high installed rangeability is desirable, it is also important not to oversize a valve; otherwise, turndown (ratio of maximum normal to minimum controllable flow) will be limited. Sliding stem valves are characterized by altering the contour of the plug when the port and plug determine the minimum (controlling) flow area. Passage area versus travel is also easily manipulated in characterized cage designs. Inherent rangeability varies widely, but typical values are 30 for contoured plugs and 20 to 50 for characterized cages. While these types of valves can be characterized, the degree to which manufacturers conform to the mathematical ideal is revealed by plotting measured CV versus travel. Note that ideal equal-percentage will plot as a straight line on a semilog graph. Custom characteristics that compensate for a specific process are possible. Rotary stem-valve designs are normally offered only in their naturally occurring characteristic, since it is difficult to appreciably alter this. If additional characterization is required, the positioner or controller may be characterized. However, these approaches are less direct, since it is possible for device nonlinearity and dynamics to distort the compensation.

8-84

PROCESS CONTROL

Installed flow characteristic as a function of percent of total system head allocated to the control valve (assuming constant-head pump, no elevation head loss, and an R equal to 30 equal-percentage inherent characteristic).

FIG. 8-79

FIG. 8-80 Valve and actuator with valve positioner attached. (Courtesy Fisher Controls International LLC.)

VALVE CONTROL DEVICES Devices mounted on the control valve that interface various forms of input signals, monitor and transmit valve position, or modify valve response are valve control devices. In some applications, several auxiliary devices are used together on the same control valve. For example, mounted on the control valve, one may find a current-to-pressure transducer, a valve positioner, a volume booster relay, a solenoid valve, a trip valve, a limit switch, a process controller, and/or a stem position transmitter. Figure 8-80 shows a valve positioner mounted on the yoke leg of a spring and diaphragm actuator. As most throttling control valves are still operated by pneumatic actuators, the control valve device descriptions that follow relate primarily to devices that are used with pneumatic actuators. The functions of hydraulic and electrical counterparts are very similar. Specific details on a particular valve control device are available from the vendor of the device. Valve Positioners The valve positioner, when combined with an appropriate actuator, forms a complete closed-loop valve position control system. This system makes the valve stem conform to the input signal coming from the process controller in spite of force loads that the actuator may encounter while moving the control valve. Usually, the valve positioner is contained in its own enclosure and is mounted on the control valve. The key parts of the positioner/actuator system, shown in Fig. 8-81a, are (1) an input conversion network, (2) a stem position feedback network, (3) a summing junction, (4) an amplifier network, and (5) an actuator.

The input conversion network shown is the interface between the input signal and the summer. This block converts the input current or pressure (from an I/P transducer or a pneumatic process controller) to a voltage, electric current, force, torque, displacement, or other particular variable that can be directly used by the summer. The input conversion usually contains a means to adjust the slope and offset of the block to provide for a means of spanning and zeroing the positioner during calibration. In addition, means for changing the sense (known as “action”) of the input/output characteristic are often addressed in this block. Also exponential, logarithmic, or other predetermined characterization can be put in this block to provide a characteristic that is useful in offsetting or reinforcing a nonlinear valve or process characteristic. The stem position feedback network converts stem travel to a useful form for the summer. This block includes the feedback linkage which varies with actuator type. Depending on positioner design, the stem position feedback network can provide span and zero and characterization functions similar to that described for the input conversion block. The amplifier network provides signal conversion and suitable static and dynamic compensation for good positioner performance. Control from this block usually reduces to a form of proportional or proportional plus derivative control. The output from this block in the case of a pneumatic positioner is a single connection to the spring and diaphragm actuator or two connections for push/pull operation of a

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

(a)

(b) Positioner/actuators. (a) Generic block diagram. (b) Example of a pneumatic positioner/actuator.

FIG. 8-81

springless piston actuator. The action of the amplifier network and the action of the stem position feedback can be reversed together to provide for reversed positioner action. By design, the gain of the amplifier network shown in Fig. 8-81a is made very large. Large gain in the amplifier network means that only a small proportional deviation will be required to position the actuator through its active range of travel. This means that the signals into the summer track very closely and that the gain of the input conversion block and the stem position feedback block determine the closed-loop relationship between the input signal and the stem travel. Large amplifier gain also means that only a small amount of additional stem travel deviation will result when large external force loads are applied to the actuator stem. For example, if the positioner’s amplifier network has a gain of 50 (and assuming that high packing box friction loads require 25 percent of the actuator’s range of thrust to move the actuator), then only 25 percent/50 (or 0.5 percent deviation) between input signal and output travel will result due to valve friction. Figure 8-81b is an example of a pneumatic positioner/actuator. The input signal is a pneumatic pressure that (1) moves the summing beam, which (2) operates the spool valve amplifier, which (3) provides flow to and from the piston actuator, which (4) causes the actuator to move and continue moving until (5) the feedback force returns the beam to its original position and stops valve travel at a new position. Typical positioner operation is thereby achieved. Static performance measurements related to positioner/actuator operation include the conformity, measured accuracy, hysteresis, dead band, repeatability, and locked stem pressure gain. Definitions and standardized test procedures for determining these measurements can be found in ISA-S75.13, “Method of Evaluating the Performance of Positioners with Analog Input Signals and Pneumatic Output.”

8-85

Dynamics of Positioner-Based Control Valve Assemblies Control valve assemblies are complete, functional units that include the valve body, actuator, positioner, if so equipped, associated linkages, and any auxiliary equipment such as current to pneumatic signal transducers and air supply pressure regulators. Although performance information such as frequency response, sensitivity, and repeatability data may be available for a number of these components individually, it is the performance of the entire assembly that will ultimately determine how well the demand signal from the controller output is transferred through the control valve to the process. The valve body, actuator, and positioner combination is typically responsible for the majority of the control valve assembly’s dynamic behavior. On larger actuators, the air supply pressure regulator capacity or other airflow restrictions may limit the control valve assembly’s speed of response. The control valve assembly response can usually be characterized quite well by using a first-order plus dead-time response model. The control valve assembly will also exhibit backlash, stiction, and other nonlinear behavior. During normal operation of a control loop, the controller usually makes small output changes from one second to the next. Typically this change is less than 1 percent. With very small controller output changes, e.g., less than 0.1 percent, the control valve assembly may not move at all. As the magnitude of the controller output change increases, eventually the control valve will move. At the threshold of movement, the positional accuracy and repeatability of the control valve are usually quite poor. The speed of response may be quite slow and may occur after a number of seconds of dead time. This poor performance is due to the large backlash and stiction effects relative to the requested movement and the small output change of the positioner. With a further increase in the magnitude of the controller output steps, the behavior of the control valve typically becomes more repeatable and “linear.” Dead time usually drops to only a fraction of a second, and the first-order time constant becomes faster. For much larger steps in the controller output, e.g., over 10 percent, the positioner and air supply equipment may be unable to deliver the necessary air volume to maintain the first-order response. In this case, the control valve will exhibit very little dead time, but will be rate-limited and will ramp toward the requested position. It is within the linear region of motion that the potential for the best control performance exists. When one is specifying a control valve for process control applications, in addition to material, style, and size information, the dynamic response characteristics and maximum allowable dead band (sum of backlash, stiction, and hysteresis effects) must be stated. The requirement for the control valve assembly’s speed of response is ultimately determined by the dynamic characteristics of the process and the control objectives. Typically, the equivalent first-order time constant specified for the control valve assembly should be at least 5 times faster than the desired controller closed-loop time constant. If this requirement is not met, the tuning of the control loop must be slowed down to accommodate the slow control valve response, otherwise, control robustness and stability may be compromised. The dead band of the control valve assembly is typically the determining factor for control resolution and frequently causes control instability in the form of a “limit” cycle. The controller output will typically oscillate across a range that is 1 to 2 times the magnitude of the control valve dead band. This is very dependent on the nature of the control valve nonlinearities, the process dynamics, and the controller tuning. The magnitude of the process limit cycle is determined by the size of the control valve dead band multiplied by the installed gain of the control valve. For this reason, a high-performance control valve assembly, e.g., with only 0.5 percent dead band, may cause an unacceptably large process limit cycle if the valve is oversized and has a high installed gain. For typical process control applications, the installed gain of the control valve should be in the range of 0.5 to 2 percent of the process variable span per percent of the controller output. The total dead band of the control valve assembly should be less than 1 percent. For applications that require more precise control, the dead band and possibly the installed gain of the control valve must be reduced. Specialized actuators are available that are accurate down to 0.1 percent or less. At this level of performance, however, the design of the valve body, bearings, linkages, and seals starts to become a significant source of dead band.

8-86

PROCESS CONTROL

10000

Stiffness, kN/m

Actuator with positioner 1000

100

10 0.01

Actuator without positioner

0.1

1 Frequency, Hz

10

100

Actuator stiffness as a function of frequency for a 69-in2 spring and diaphragm pneumatic actuator. Actuator with positioner exhibits higher stiffness over the lower frequency range compared to that of the pneumatic actuator without a positioner.

FIG. 8-82

Positioner/Actuator Stiffness Minimizing the effect of dynamic loads on valve stem travel is an important characteristic of the positioner/actuator. Stem position must be maintained in spite of changing reaction forces caused by valve throttling. These forces can be random (buffeting force) or can result from a negative-slope force/stem travel characteristic (negative gradient); either could result in valve stem instability and loss of control. To reduce and eliminate the effect of these forces, the effective stiffness of the positioner/actuator must be made sufficiently high to maintain stationary control of the valve stem. The stiffness characteristic of the positioner/actuator varies with the forcing frequency. Figure 8-82 indicates the stiffness of the positioner/actuator is increased at low frequencies and is directly related to the locked-stem pressure gain provided by the positioner. As frequency increases, a dip in the stiffness curve results from dynamic gain attenuation in the pneumatic amplifiers in the positioner. The value at the bottom of the dip is the sum of the mechanical stiffness of the spring in the actuator and the air spring effect produced by air enclosed in the actuator casing. At yet higher frequencies, actuator inertia dominates and causes a corresponding rise in system stiffness. The air spring effect results from adiabatic expansion and compression of air in the actuator casing. Numerically, the small perturbation value for air spring stiffness in newtons per meter is given by γpa A2a Air spring rate =  V

(8-123)

where γ is the ratio of specific heats (1.4 for air), pa is the actuator pressure in pascals absolute, Aa is the actuator pressure area in square meters, and V is the internal actuator volume in cubic meters. Positioner Application Positioners are widely used on pneumatic valve actuators. Often they provide improved process loop control because they reduce valve-related nonlinearity. Dynamically, positioners maintain their ability to improve control valve performance for sinusoidal input frequencies up to about one-half of the positioner bandwidth. At input frequencies greater than this, the attenuation in the positioner amplifier network gets large, and valve nonlinearity begins to affect final control element performance more significantly. Because of this, the most successful use of the positioner occurs when the positioner response bandwidth is greater than twice that of the most dominant time lag in the process loop. Some typical examples in which the dynamics of the positioner are sufficiently fast to improve process control are the following: 1. In a distributed control system (DCS) process loop with an electronic transmitter. The DCS controller and the electronic transmitter have time constants that are dominant over the positioner response. Positioner operation is therefore beneficial in reducing valve-related nonlinearity.

2. In a process loop with a pneumatic controller and a large process time constant. Here the process time constant is dominant, and the positioner will improve the linearity of the final control element. Some common processes with large time constants that benefit from positioner application are liquid level, temperature, large-volume gas pressure, and mixing. 3. Additional situations in which valve positioners are used: a. On springless actuators where the actuator is not usable for throttling control without position feedback. b. When split ranging is required to control two or more valves sequentially. In the case of two valves, the smaller control valve is calibrated to open in the lower half of the input signal range, and a larger valve is calibrated to open in the upper half of the input signal range. Calibrating the input command signal range in this way is known as split-range operation and increases the practical range of throttling process flows over that of a single valve. c. In open-loop control applications where best static accuracy is needed. On occasion, positioner use can degrade process control. Such is the case when the process controller, process, and process transmitter have time constants that are similar to or smaller than that of the positioner/actuator. This situation is characterized by low process controller proportional gain (gain < 0.5), and hunting or limit cycling of the process variable is observed. Improvements here can be made by doing one of the following: 1. Install a dominant first-order, low-pass filter in the loop ahead of the positioner and retune the process loop. This should allow increased proportional gain in the process loop and reduce hunting. Possible means for adding the filter include adding it to the firmware of the DCS controller, by adding an external RC network on the output of the process controller or by enabling the filter function in the input of the positioner, if it is available. Also, some transducers, when connected directly to the actuator, form a dominant first-order lag that can be used to stabilize the process loop. 2. Select a positioner with a faster response characteristic. Processor-Based Positioners When designed around an electronic microcontroller, the valve positioner [now commonly referred to as a digital valve controller (DVC)] takes on additional functionality that provides convenience and performance enhancements over the traditional design. The most common form of processor-based positioner, shown in Fig. 8-81, is a digitally communicating stem position controller that operates by using the fundamental blocks shown in Fig. 8-81a. A local display is part of the positioner and provides tag information, command input and travel, servo tuning parameters, and diagnostic information. Often auxiliary sensors are integrated into the device to provide increased levels of functionality and performance. Sensed variables can include actuator pressure, relay input pressure, relay valve position, board temperature, or a discrete input. A 4- to 20-mA valve travel readback circuit is also common. The travel sensor is based on a potentiometer or can be a noncontacting type such as a variable capacitance sensor, Hall effect sensor, or GMR device. Some positioners require a separate connection to an ac or dc supply voltage, but the majority of the designs are “loop-powered,” which means that they receive power either through the current input (for positioners that require a 4- to 20-mA analog input signal) or through the digital communications link when the control signal is a digital signal. Processor-based positioners support automatic travel calibration and automatic response tuning for quick commissioning of the final control element. Features of this type of valve positioner include compensators for improved static and dynamic travel response; diagnostics for evaluating positioner, actuator, and valve health; and the capability to be polled from remote locations through a PC-based application or through a handheld communicator attached to the field wiring. Capability to support custom firmware for special valve applications, such as emergency safety shutdown, is also a characteristic of the processor-based design. Digital Field Communications To provide increased data transmission capability between valve-mounted devices and the host control system, manufacturers are providing digital network means in their devices. The field networks, commonly known as field buses, compete fiercely in the marketplace and have varying degrees of flexibility and specific application strengths. A prospective field bus

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS customer is advised to study the available bus technologies and to make a selection based on needs, and not be seduced by the technology itself. Generally, a field bus protocol must be nonproprietary (“open”) so that different vendors of valve devices can design their bus interface to operate properly on the selected field bus network. Users demand that the devices be “interoperable” so that the device will work with other devices on the same segment or can be substituted with a device from an alternate manufacturer. International standardization of some of the protocols is currently underway (for example, IEC 61158) whereas others are sponsored by user groups or foundations that provide democratic upgrades to the standard and provide network compliance testing. The physical wiring typically used is the plant standard twisted-pair wiring for 4- to 20-mA instrumentation. Because of the networking capability of the bus, more than one device can be supported on a single pair of wires, and thus wiring requirements are reduced. Compared to a host level bus such as Ethernet, field buses exhibit slower communication rates, have longer transmission distance capability (1 to 2 km), use standard two-wire installation, are capable of multidrop busing, can support bus-powered devices, do not have redundant modes of bus operation, and are available for intrinsically safe installations. Devices on the field bus network may be either powered by the bus itself or powered separately. The simplest digital networks available today support discrete sensors and on/off actuators, including limit switches and motor starters. Networks of this type have fast cycle times and are often used as an alternative to PLC discrete I/O. More sophisticated field networks are designed to support process automation, more complex process transmitters, and throttling valve actuators. These process-level networks are fundamentally continuous and analoglike in operation, and data computation is floating-point. They support communication of materials of construction, calibration and commissioning, device and loop level diagnostics (including information displays outlining corrective action), and unique manufacturer-specific functionality. Some process networks are able to automatically detect, identify, and assign an address to a new device added to the network, thus reducing labor, eliminating addressing errors, and indicating proper network function immediately after the connection is made. Final control elements operated by the process-level network include I/P transducers, motorized valves, digital valve controllers, and transmitters. A particular field network protocol known as HART®* (Highway Addressable Remote Transducer) is the most widely used field network protocol. It is estimated that as of 2004 there are more than 14 million HART-enabled devices installed globally and that 70 percent *HART is a registered trademark of the HART Communication Foundation.

Valve Travel Readback

Loop +

Current Sense 4–20-mA Input

Anti-alias Filter, Sample & Hold

Comm. Modem Local Keypad

Loop − FIG. 8-83

Microcontroller

of all processor-based process measurement and control instruments installed each year use HART communications. HART’s popularity is based on its similarity to the traditional 4- to 20-mA field signaling and thus represents a safe, controlled transition to digital field communications without the risk often associated with an abrupt change to a totally digital field bus. With this protocol, the digital communications occur over the same two wires that provide the 4- to 20-mA process control signal without disrupting the process signal. The protocol uses the frequency-shift keying (FSK) technique (see Fig. 8-83) where two individual frequencies, one representing the mark and the other representing the space, are superimposed on the 4- to 20-mA current signal. As the average value of the signals used is zero, there is no dc offset value added to the 4- to 20-mA signal. The HART protocol is principally a master/slave protocol which means that a field device (slave) speaks only when requested by a master device. In this mode of operation, the slave can update the master at a rate of twice per second. An optional communication mode, burst mode, allows a HART slave device to continuously broadcast updates without stimulus requests from the master device. Update rates of 3 to 4 updates per second are typical in the burst mode of operation. HART-enabled devices are provided by the valve device manufacturer at little or no additional cost. The HART network is compatible with existing 4- to 20-mA applications using current plant personnel and practices, provides for a gradual transition from analog to fully digital protocols, and is provided by the valve device manufacturer at little or no additional cost. Contact the HART Communication Foundation for additional information. Wireless digital communication to and from the final control element is not yet commercially available but is presently being investigated by more than one device manufacturer. The positive attribute of a wireless field network is the reduced cost of a wireless installation compared to a wired installation. Hurdles for wireless transmissions include security from nonnetwork sources, transmission reliability in the plant environment, limited bus speed, and the conservative nature of the process industry relative to change. Initial installations of wireless networks will support secondary variables and diagnostics, then primary control of processes with large time constants, and finally general application to process control. Both point-to-point and mesh architectures are being evaluated for commercialization at the device level. Mesh architectures rely on the other transmitting devices in the area to receive and then pass on any data transmission, thus rerouting communications around sources of interference. Two unlicensed spread spectrum radio bands are the main focus for current wireless development: 900 MHz and 2.4 GHz. The 900-MHz band is unique to North America and has better propagation and penetrating properties than the 2.4-GHz band. The 2.4-GHz band is a worldwide band and has wider channels, allowing much higher data rates. The spread

Local Display

D/A Convert

Anti-alias Filter, Sample & Hold

V+ Power Supply

D/A Convert

Generic loop powered digital valve controller.

8-87

Anti-alias Filter, Sample & Hold

Supply Pressure

Current to Pressure

To Pneumatic Pneumatic Actuator Relay

Auxiliary Sensor Inputs Travel Sensor

Valve Travel

8-88

PROCESS CONTROL

spectrum technique uses multiple frequencies within the radio band to transmit data. Spread spectrum is further divided into the direct sequence technique, where the device changes frequency many times per data bit, and the frequency-hopping technique, where the device transmits a data packet on one frequency and then changes to a different frequency. Because of the rapid growth expected in this decade, the prospective wireless customer is encouraged to review up-to-date literature to determine the state of field wireless commercialization as it applies to her or his specific application. Diagnostic Capability The rapid proliferation of communicating, processor-based digital valve controllers over the last decade has led to a corresponding rise in diagnostic capability at the control valve. Diagnosing control valve health is critical to plant operation as maintenance costs can be reduced by identifying the valves that are candidates for repair. Less time is spent during plant shutdown repairing valves that do not need repair, which ultimately results in increased online operating time. Valve diagnostics can detect and flag a failed valve more quickly than by any other means, and can be configured to cause the valve to move to its fail-safe position on detection of specified fault conditions. The diagnostic-enabled positioner, when used with its host-based software application, can pinpoint exact components in a given final control element that have failed, and can recommend precise maintenance procedures to follow to remedy the fault condition. The state variables that provide valve position control are used to diagnose the health of the final control element. In addition, some digital valve controller designs integrate additional sensors into their construction to provide increased diagnostic capability. For example, pressure sensors are provided to detect supply pressure, actuator pressure (upper and lower cylinder pressures in the case of a springless piston actuator), and internal pilot pressure. Also, the position of the pneumatic relay valve is available in some designs to provide quiescent flow data used for leak detection in the actuator. Valve diagnostics are divided into two types: online and offline. Offline diagnostics are those diagnostics that occur when the control valve is bypassed or otherwise isolated from the process. The offline diagnostic routine manipulates the travel command to the valve and records the corresponding valve travel, actuator pressure, and servodrive value. These parameters are plotted in various combinations to provide hysteresis plus dead-band information, actuator operating pressure as a function of travel, valve friction, servodrive performance, valve seating load, positioner calibration endpoints, and dynamic response traces. Small- and large-amplitude step inputs as well as large slow ramps (exceeding 100 percent of the input range) are common offline test waveforms generated by the diagnostic as command inputs for offline diagnostic tests. Figure 8-84 is an example of one offline diagnostic test performed on a small globe valve actuated by a spring and diaphragm actuator. During this test the command input, travel, actuator pressure, and servodrive level are recorded and plotted as they result from a command input that is slowly ramped by the diagnostic routine (Fig. 8-85a). This diagnostic is extremely useful in detecting problems with the valve/actuator system and can flag potential problems with the final control element before catastrophic failure occurs. For example, Fig. 8-85b indicates the overall tracking capability of the control valve, and Fig. 8-85c indicates the pressure operating range of the actuator and the amount of frictional force resulting from the combined effects of valve packing and valve plug contact. Figure 8-85d displays the level of servodrive required to stroke the valve from one end of travel to the other. The composite operative health of the control valve is determined through comparison of the empirical levels presented in Fig. 8-85 with the manufacturers’ recommendations. Recommended maintenance actions result from this comparison. Online diagnostics are diagnostics that monitor and evaluate conditions at the control valve during normal throttling periods (i.e., during valve-in-service periods). Online diagnostics monitor mean levels and disturbances generated in the normal operation of the valve and typically do not force or generate disturbances on the valve’s operation. For example, an online diagnostic can calculate travel deviation relative to the input command and flag a condition where the valve travel has deviated beyond a preset band. Such an event, if it exists for more than a short time, indicates that the valve has lost its ability to track the input

Hybrid point-to-point communications between the control room and the control valve device.

FIG. 8-84

command within specified limits. Additional diagnostics could suggest that the feedback linkage has ceased functioning, or that the valve has stuck, or that some other specific malfunction is the cause of excess travel deviation. The manufacturer of the positioner diagnostic incorporates default limits into the host software application that are used to determine the relative importance of a specific deviation. To quickly indicate the severity of a problem detected by a diagnostic routine, a red, yellow, or green, or “advise, maintenance now, or failed,” indication is presented on the user-interface screen for the valve problem diagnosed. Help notes and recommended remedial action are available by pointing and clicking on the diagnostic icon presented on the user’s display. Event-triggered recording is an online diagnostic technique supported in digital valve controllers (DVCs). Functionally a triggering event, such as a valve coming off a travel stop or a travel deviation alert, starts a time-series recording of selected variables. A collection of variables such as the input command, stem travel, actuator pressure, and drive command are stored for several minutes before and after the triggered event. These variables are then plotted as time series for immediate inspection or are stored in memory for later review. Event-triggering diagnostics are particularly useful in diagnosing valves that are closed or full-open for extended periods. In this case the event-triggered diagnostic focuses on diagnostic rich data at the time the valve is actually in operation and minimizes the recording of flat-line data with little diagnostic content. Other online diagnostics detected by DVC manufacturers include excess valve friction, supply pressure failure, relay operation failure, broken actuator spring, current to pressure module failure, actuator diaphragm leaking, and shifted travel calibration. Safety shutdown valves, which are normally wide open and operate infrequently, are expected to respond to a safety trip command reliably and without fault. To achieve the level of reliability required in this application, the safety valve must be periodically tested to ensure positive operation under safety trip conditions. To test the operation of the shutdown system without disturbing the process, the traditional method is to physically lock the valve stem in the wide-open position and then to electrically operate the pneumatic shutdown solenoid valve. Observing that the pneumatic solenoid valve has properly vented the actuator pressure to zero, the actuator is seen as capable of applying sufficient spring force to close the valve, and a positive safety valve test is indicated. The

8-89

120 100 80

Travel, %

Input and Travel, %

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

60 40 20 Command Input Travel

0 -20 -50

0

120 100 80 60 40 20 0 -20

50 100 150 200 250 300 350 Time, s

-20

20

(b)

(a)

40 60 Input, %

80

100 120

90

30

80 Drive, %

Actuator Pressure, psi

0

20

70 60

10

50

-20 (c)

0

20

40

60

80

100 120

Travel, %

-20

(d)

0

20

40 60 Travel, %

80

100

120

FIG. 8-85 Offline valve diagnostic scan showing results of a diagnostic ramp. (a) The command input and resulting travel. (b) The dynamic scan. (c) The valve signature. (d) The servodrive versus travel plot. The hysteresis shown in the valve signature results from sliding friction due to valve packing and valve plug contact.

pneumatic solenoid valve is then returned to its normal electrical state, the actuator pressure returns to full supply pressure, and the valve stem lock mechanism is removed. This procedure, though necessary to enhance process safety, is time-consuming and takes the valve out of service during the locked stem test. Digital valve controllers are able to validate the operation of a safety shutdown valve by using an online diagnostic referred to as a partial stroke test. The partial stroke test is substituted for the traditional test method described above and does not require the valve to be locked in the wide-open position to perform the test. In a fashion similar to that shown in Fig. 8-85a (the partial stroke diagnostic), the system physically ramps the command input to the positioner from the wide-open position to a new position, pauses at the new position for a few seconds, and then ramps the command input back to the wide-open position (see Fig. 8-86a). During this time, the valve travel measurement is monitored and compared to the input command. If the travel measurement deviates for the input by more than a fixed amount for the configured period of time, the valve is considered to have failed the test and a failed-test message is communicated to the host system. Also during this test, the actuator pressure required to move the valve is detected via a dedicated pressure sensor (see Fig. 8-86b). If the thrust (pressure) required to move the valve during the partial stroke test exceeds the predefined thrust limit for this test, the control valve is determined to have a serious sticking problem, the test is immediately aborted, and the valve is flagged as needing maintenance. The partial stroke test can be automated to perform on a periodic basis, for instance, once a week; or it can be initialized by operator request at any time. The amount of valve travel that occurs during the partial stroke test is typically limited to a minimum valve position of 70 percent open or greater. This limit is imposed to prevent the partial stroking of the safety valve from significantly affecting the process flow through the valve.

Comparison of partial stroke curves from past tests can indicate the gradual degradation of valve components. Use of “overlay” graphics, identification of unhealthy shifts in servodrive, increases in valve friction, and changes in dynamic response provide information leading to a diagnosis of needed maintenance. In addition to device-level diagnostics, networked final control elements, process controllers, and transmitters can provide “loop” level diagnostics that can detect loops that are operating below expectations. Process variability, time in a limit (saturated) condition, and time in the wrong control mode are metrics used to detect problems in process loop operation. Transducers The current-to-pressure transducer (I/P transducer) is a conversion interface that accepts a standard 4- to 20-mA input current from the process controller and converts it to a pneumatic output in a standard pneumatic pressure range [normally 0.2 to 1.0 bar (3 to 15 psig) or, less frequently, 0.4 to 2.0 bar (6 to 30 psig)]. The output pressure generated by the transducer is connected directly to the pressure connection on a spring-opposed diaphragm actuator or to the input of a pneumatic valve positioner. Figure 8-87a is the schematic of a basic I/P transducer. The transducer shown is characterized by (1) an input conversion that generates an angular displacement of the beam proportional to the input current, (2) a pneumatic amplifier stage that converts the resulting angular displacement to pneumatic pressure, and (3) a pressure area that serves as a means to return the beam to very near its original position when the new output pressure is achieved. The result is a device that generates a pressure output that tracks the input current signal. The transducer shown in Fig. 8-88a is used to provide pressure to small load volumes (normally 4.0 in3 or less), such as a positioner or booster input. With only one stage of pneumatic amplification, the flow

8-90

PROCESS CONTROL

Actuator Pressure, psi

Input and Travel, %

105 Input

100

Travel 95 90 85 0

10 Time, s

20

30 25 20 15 10 85

30

(a)

90

95 Travel, %

100

105

(b)

FIG. 8-86 Online partial stroke diagnostic used to validate the operability of a pneumatically operated safety shutdown valve. (a) Input command generated by the diagnostic and resulting travel. (b) Actuator pressure measured over the tested range of travel.

capacity of this transducer is limited and not sufficient to provide responsive load pressure directly to a pneumatic actuator. The flow capacity of the transducer can be increased by adding a booster relay such as the one shown in Fig. 8-87b. The flow capacity of the booster relay is nominally 50 to 100 times that of the nozzle amplifier shown in Fig. 8-87a and makes the combined transducer/booster suitably responsive to operate pneumatic actuators. This type of transducer is stable for all sizes of load volume and produces measured accuracy (see ANSI/ISA-51.1, “Process Instrumentation Terminology,” for the definition of measured accuracy) of 0.5 to 1.0 percent of span. Better measured accuracy results from the transducer design shown in Fig. 8-87c. In this design, pressure feedback is taken at the output of the booster relay stage and fed back to the main summer. This allows the transducer to correct for errors generated in the pneumatic booster as well as errors in the I/P conversion stage. Also, particularly with the new analog electric and digital versions of this design, PID control is used in the transducer control network to give extremely good static accuracy, fast dynamic response, and reasonable stability into a wide range of load volumes (small instrument bellows to large actuators). Also environmental factors such as temperature change, vibration, and supply pressure fluctuation affect this type of transducer the least. Even a perfectly accurate I/P transducer cannot compensate for stem position errors generated by friction, backlash, and varying force loads coming from the actuator and valve. To do this compensation, a different control valve device—the valve positioner—is required. Booster Relays The booster relay is a single-stage power amplifier having a fixed gain relationship between the input and output pressures. The device is packaged as a complete stand-alone unit with pipe thread connections for input, output, and supply pressure. The booster amplifier shown in Fig. 8-87b shows the basic construction of the booster relay. Enhanced versions are available that provide specific features such as (1) variable gain to split the output range of a pneumatic controller to operate more than one valve or to provide additional actuator force; (2) low hysteresis for relaying measurement and control signals; (3) high flow capacity for increased actuator stroking speed; and (4) arithmetic, logic, or other compensation functions for control system design. A particular type of booster relay, called a dead-band booster, is shown in Fig. 8-88. This booster is designed to be used exclusively between the output of a valve positioner and the input to a pneumatic actuator. It is designed to provide extra flow capacity to stroke the actuator faster than with the positioner alone. The dead-band booster is designed intentionally with a large dead band (approximately 5 percent of the input span), elastomer seats for tight shutoff, and an adjustable bypass valve connected between the input and output of the booster. The bypass valve is tuned to provide the best compromise between increased actuator stroking speed and positioner/actuator stability.

With the exception of the dead-band booster, the application of booster relays has diminished somewhat by the increased use of current-to-pressure transducers, electropneumatic positioners, and electronic control systems. Transducers and valve positioners serve much the same functionality as the booster relay in addition to interfacing with the electronic process controller.

(a)

(b)

(c) Current-to-pressure transducer component parts. (a) Direct-current–pressure conversion. (b) Pneumatic booster amplifier (relay). (c) Block diagram of a modern I/P transducer.

FIG. 8-87

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS Input signal Diaphragms

Exhaust port

Bypass valve adjusting screw Adjustable restriction

Exhaust

Supply port

Supply

FIG. 8-88

Output to actuator

Dead-band booster relay. (Courtesy Fisher Controls International

LLC.)

Solenoid Valves The electric solenoid valve has two output states. When sufficient electric current is supplied to the coil, an internal armature moves against a spring to an extreme position. This motion causes an attached pneumatic or hydraulic valve to operate. When current is removed, the spring returns the armature and the attached solenoid valve to the deenergized position. An intermediate pilot stage is sometimes used when additional force is required to operate the main solenoid valve. Generally, solenoid valves are used to pressurize or vent the actuator casing for on/off control valve application and safety shutdown applications. Trip Valves The trip valve is part of a system used where a specific valve action (i.e., fail up, fail down, or lock in last position) is required when pneumatic supply pressure to the control valve falls below a preset level. Trip systems are used primarily on springless piston actuators requiring fail-open or fail-closed action. An air storage or “volume” tank and a check valve are used with the trip valve to provide power to stroke the valve when supply pressure is lost. Trip valves are designed with hysteresis around the trip point to avoid instability when the trip pressure and the reset pressure settings are too close to the same value. Limit Switches and Stem Position Transmitters Travel limit switches, position switches, and valve position transmitters are devices that detect the component’s relative position, when mounted on the valve, actuator, damper, louver, or other throttling element. The switches are used to operate alarms, signal lights, relays, solenoid valves, or discrete inputs into the control system. The valve position transmitter generates a 4- to 20-mA output that is proportional to the position of the valve. FIRE AND EXPLOSION PROTECTION Electrical equipment and wiring methods can be sources of ignition in environments with combustible concentrations of gas, liquid, dust, fibers, or flyings. Most of the time it is possible to locate the electronic equipment away from these hazardous areas. However, where electric or electronic valve-mounted instruments must be used in areas where there is a hazard of fire or explosion, the equipment and installation must meet requirements for safety. Articles 500 through 504 of the National Electrical Code cover the definitions and requirements for electrical and electronic equipment used in the class I (flammable gases or vapors), divisions 1 and 2; class II (combustible dust), divisions 1 and 2; and class III (ignitable fibers or flyings), divisions 1 and 2. Division 1 locations are locations with hazardous concentrations of gases, vapors, or combustible dust under normal operating conditions;

8-91

hazardous concentration of gases, vapors, or combustible dust that occur frequently due to repair, maintenance, or leakage; or hazardous due to the presence of easily ignitable fibers or materials producing combustible flyings during handling, manufacturing, or use. Division 2 locations are locations that normally do not have ignitable concentrations of gases, vapors, or combustible dust. Division 2 locations might become hazardous through failure of ventilating equipment; adjacent proximity to a class I, division 1 location where ignitable concentrations of gases or vapors might occasionally exist; through dust accumulations on or in the vicinity of the electrical equipment sufficient to interfere with the safe dissipation of heat or by abnormal operation or failure of electrical equipment; or when easily ignitable fibers are stored or handled other than in the process of manufacture. An alternate method used for class I hazardous locations is the European “zone” method described in IEC 60079-10, “Electrical Apparatus for Explosive Gas Atmospheres.” The zone designation for class I locations has been adapted by the NEC as an alternate method and is defined in Article 505 of the NEC. Acceptable protection techniques for electrical and electronic valve accessories used in specific class and division locations include explosionproof enclosures; intrinsically safe circuits; nonincendive circuits, equipment, and components; dust-ignition-proof enclosures; dusttight enclosures; purged and pressurized enclosures; oil immersion for current-interrupting contacts; and hermetically sealed equipment. Details of these techniques can be found in the National Electrical Code Handbook, available from the National Fire Protection Association. Certified testing and approval for control valve devices used in hazardous locations is normally procured by the manufacturer of the device. The manufacturer typically goes to a third-party laboratory for testing and certification. Applicable approval standards are available from CSA, CENELEC, FM, SAA, and UL. Environmental Enclosures Enclosures for valve accessories are sometimes required to provide protection from specific environmental conditions. The National Electrical Manufacturers Association (NEMA) provides descriptions and test methods for equipment used in specific environmental conditions in NEMA 250. IEC 60529, “Degrees of Protection Provided by Enclosures (IP Code),” describes the European system for classifying the degrees of protection provided by the enclosures of electrical equipment. Rain, windblown dust, hose-directed water, and external ice formation are examples of environmental conditions that are covered by these enclosure standards. Of growing importance is the electronic control valve device’s level of immunity to, and emission of, electromagnetic interference in the chemical valve environment. Electromagnetic compatibility (EMC) for control valve devices is presently mandatory in the European Community and is specified in International Electrotechnical Commission (IEC) 61326, “Electrical Equipment for Measurement Control and Laboratory Use—EMC Requirements.” Test methods for EMC testing are found in the series IEC 61000-4, “EMC Compatibility (EMC), Testing and Measurement Techniques.” Somewhat more stringent EMC guidelines are found in the German document NAMUR NE21, “Electromagnetic Compatibility of Industrial Process and Laboratory Control Equipment.” ADJUSTABLE-SPEED PUMPS An alternative to throttling a process with a process control valve and a fixed-speed pump is by adjusting the speed of the process pump and not using a throttling control valve at all. Pump speed can be varied by using variable-speed prime movers such as turbines, motors with magnetic or hydraulic couplings, and electric motors. Each of these methods of modulating pump speed has its own strengths and weaknesses, but all offer energy savings and dynamic performance advantages over throttling with a control valve. The centrifugal pump directly driven by a variable-speed electric motor is the most commonly used hardware combination for adjustablespeed pumping. The motor is operated by an electronic motor speed controller whose function is to generate the voltage or current waveform required by the motor to make the speed of the motor track the input command signal from the process controller.

8-92

PROCESS CONTROL

Pressure, flow, and power for throttling a process using a control valve and a constant-speed pump compared to throttling with an adjustable-speed pump.

FIG. 8-89

The most popular form of motor speed control for adjustable-speed pumping is the voltage-controlled pulse-width-modulated (PWM) frequency synthesizer and ac squirrel-cage induction motor combination. The flexibility of application of the PWM motor drive and its 90+ percent electrical efficiency along with the proven ruggedness of the traditional ac induction motor makes this combination popular. From an energy consumption standpoint, the power required to maintain steady process flow with an adjustable-speed-pump system (three-phase PWM drive and a squirrel-cage induction motor driving a centrifugal pump on water) is less than that required with a conventional control valve and a fixed-speed pump. Figure 8-89 shows this to be the case for a system where 100 percent of the pressure loss is due to flow velocity losses. At 75 percent flow, the figure shows that using the constant-speed pump/control valve results in a 10.1-kW rate, while throttling with the adjustable-speed pump and not using a control valve results in a 4.1-kW rate. This trend of reduced energy consumption is true for the entire range of flows, although amounts vary. From a dynamic response standpoint, the electronic adjustablespeed pump has a dynamic characteristic that is more suitable in process control applications than those characteristics of control valves. The small amplitude response of an adjustable-speed pump does not contain the dead band or the dead time commonly found in the small amplitude response of the control valve. Nonlinearities associated with friction in the valve and discontinuities in the pneumatic portion of the control valve instrumentation are not present with electronic variable-speed drive technology. As a result, process control with the adjustable-speed pump does not exhibit limit cycles, problems related to low controller gain, and generally degraded process loop performance caused by control valve nonlinearities. Unlike the control valve, the centrifugal pump has poor or nonexistent shutoff capability. A flow check valve or an automated on/off valve may be required to achieve shutoff requirements. This requirement may be met by automating an existing isolation valve in retrofit applications. REGULATORS A regulator is a compact device that maintains the process variable at a specific value in spite of disturbances in load flow. It combines the functions of the measurement sensor, controller, and final control element into one self-contained device. Regulators are available to control pressure, differential pressure, temperature, flow, liquid level, and other basic process variables. They are used to control the differential across a filter press, heat exchanger, or orifice plate. Regulators are used for monitoring pressure variables for redundancy, flow check, and liquid surge relief.

Regulators may be used in gas blanketing systems to maintain a protective environment above any liquid stored in a tank or vessel as the liquid is pumped out. When the temperature of the vessel is suddenly cooled, the regulator maintains the tank pressure and protects the walls of the tank from possible collapse. Regulators are known for their fast dynamic response. The absence of time delay that often comes with more sophisticated control systems makes the regulator useful in applications requiring fast corrective action. Regulators are designed to operate on the process pressures in the pipeline without any other sources of energy. Upstream and downstream pressures are used to supply and exhaust the regulator. Exhausting is connected back to the downstream piping so that no contamination or leakage to the external environment occurs. This makes regulators useful in remote locations where power is not available or where external venting is not allowed. The regulator is limited to operating on processes with clean, nonslurry process fluids. The small orifice and valve assemblies contained in the regulator can plug and malfunction if the process fluid that operates the regulator is not sufficiently clean. Regulators are normally not suited to systems that require constant set-point adjustment. Although regulators are available with capability to respond to remote set-point adjustment, this feature adds complexity to the regulator and may be better addressed by a control-valve-based system. In the simplest of regulators, tuning of the regulator for best control is accomplished by changing a spring, an orifice, or a nozzle. Self-Operated Regulators Self-operated regulators are the simplest form of regulator. This regulator (see Fig. 8-90a) is composed of a main throttling valve, a diaphragm or piston to sense pressure, and a spring. The self-contained regulator is completely operated by the process fluid, and no outside control lines or pilot stage is used. In general, self-operated regulators are simple in construction, are easy to operate and maintain, and are usually stable devices. Except for some of the pitot-tube types, self-operated regulators have very good dynamic response characteristics. This is so because any change in the controlled variable registers directly and immediately upon the main diaphragm to produce a quick response to the disturbance. The disadvantage of the self-operated regulator is that it is not generally capable of maintaining a set point as load flow is increased. Because of the proportional nature of the spring and diaphragmthrottling effect, offset from set point occurs in the controlled variable as flow increases. Figure 8-91 shows a typical regulation curve for the self-contained regulator. Reduced set-point offset with increasing load flow can be achieved by adding a pitot tube to the self-operated regulator. The

CONTROLLERS, FINAL CONTROL ELEMENTS, AND REGULATORS

8-93

Spring Main throttling valve Diaphragm

(a) FIG. 8-90

(b)

Regulators. (a) Self-operated. (b) Pilot-operated. (Courtesy Fisher Controls International LLC.)

tube is positioned somewhere near the vena contracta of the main regulator valve. As flow though the valve increases, the measured feedback pressure from the pitot tube drops below the control pressure. This causes the main valve to open or boost more than it would if the static value of control pressure were acting on the diaphragm. The resultant effect keeps the control pressure closer to the set point and thus prevents a large drop in process pressure during high-loadflow conditions. Figure 8-91 shows the improvement that the pitot-

FIG. 8-91

tube regulator provides over the regulator without the tube. A side effect of adding a pitot-tube method is that the response of the regulator can be slowed due to the restriction provided by the pitot tube. Pilot-Operated Regulators Another category of regulators uses a pilot stage to provide the load pressure on the main diaphragm. This pilot is a regulator itself that has the ability to multiply a small change in downstream pressure into a large change in pressure applied to the regulator diaphragm. Due to this high-gain feature, pilot-operated

Pressure regulation curves for three regulator types.

8-94

PROCESS CONTROL

regulators can achieve a dramatic improvement in steady-state accuracy over that achieved with a self-operated regulator. Figure 8-91 shows for regulation at high flows the pilot-operated regulator is the best of the three regulators shown. The main limitation of the pilot-operated regulator is stability. When the gain in the pilot amplifier is raised too much, the loop can become unstable and oscillate or hunt. The two-path pilot regulator (see Fig. 890b) is also available. This regulator combines the effects of self-operated and the pilot-operated styles and mathematically produces the equivalent of proportional plus reset control of the process pressure. Overpressure Protection Figure 8-91 shows a characteristic rise in control pressure that occurs at low or zero flow. This lockup

tail is due to the effects of imperfect plug and seat alignment and the elastomeric effects of the main throttle valve. If, for some reason, the main throttle valve fails to completely shut off, or if the valve shuts off but the control pressure continues to rise for other reasons, the lockup tail could get very large, and the control pressure could rise to extremely high values. Damage to the regulator or the downstream pressure volume could occur. To avoid this situation, some regulators are designed with a built-in overpressure relief mechanism. Overpressure relief circuits usually are composed of a spring-opposed diaphragm and valve assembly that vents the downstream piping when the control pressure rises above the set-point pressure.

PROCESS CONTROL AND PLANT SAFETY GENERAL REFERENCE: Guidelines for Safe Automation of Chemical Processes, AIChE Center for Chemical Process Safety, New York, 1993.

Accidents in chemical plants make headline news, especially when there is loss of life or the general public is affected in even the slightest way. This increases the public’s concern and may lead to government action. The terms hazard and risk are defined as follows: • Hazard. A potential source of harm to people, property, or the environment. • Risk. Possibility of injury, loss, or an environmental accident created by a hazard. Safety is the freedom from hazards and thus the absence of any associated risks. Unfortunately, absolute safety cannot be realized. The design and implementation of safety systems must be undertaken with a view to two issues: • Regulatory. The safety system must be consistent with all applicable codes and standards as well as “generally accepted good engineering practices.” • Technical. Just meeting all applicable regulations and “following the crowd” do not relieve a company of its responsibilities. The safety system must work. The regulatory environment will continue to change. As of this writing, the key regulatory instrument is OSHA 29 CFR 1910.119, “Process Safety Management of Highly Hazardous Chemicals,” which pertains to process safety management within plants in which certain chemicals are present. In addition to government regulation, industry groups and professional societies are producing documents ranging from standards to guidelines. Two applicable standards are IEC 61508, “Functional Safety of Electrical/Electronic/Programmable Electronic Safetyrelated Systems,” and ANSI/ISA S84.01, “Application of Safety Instrumented Systems for the Process Industries.” Guidelines for Safe Automation of Chemical Processes from the American Institute of Chemical Engineers’ Center for Chemical Process Safety (1993) provides comprehensive coverage of the various aspects of safety; and although short on specifics, it is very useful to operating companies developing their own specific safety practices (i.e., it does not tell you what to do, but it helps you decide what is proper for your plant). The ultimate responsibility for safety rests with the operating company; OSHA 1910.119 is clear on this. Each company is expected to develop (and enforce) its own practices in the design, installation, testing, and maintenance of safety systems. Fortunately, some companies make these documents public. Monsanto’s Safety System Design Practices was published in its entirety in the proceedings of the International Symposium and Workshop on Safe Chemical Process Automation, Houston, Texas, September 27–29, 1994 (available from the American Institute of Chemical Engineers’ Center for Chemical Process Safety).

ROLE OF AUTOMATION IN PLANT SAFETY As microprocessor-based controls displaced hardwired electronic and pneumatic controls, the impact on plant safety has definitely been positive. When automated procedures replace manual procedures for routine operations, the probability of human errors leading to hazardous situations is lowered. The enhanced capability for presenting information to the process operators in a timely manner and in the most meaningful form increases the operator’s awareness of current conditions in the process. Process operators are expected to exercise due diligence in the supervision of the process, and timely recognition of an abnormal situation reduces the likelihood that the situation will progress to the hazardous state. Figure 8-92 depicts the layers of safety protection in a typical chemical plant. Although microprocessor-based process controls enhance plant safety, their primary objective is efficient process operation. Manual operations are automated to reduce variability, to minimize the time required, to increase productivity, and so on. Remaining competitive in the world market demands that the plant be operated in the best manner possible, and microprocessor-based process controls provide numerous functions that make this possible. Safety is never compromised in the effort to increase competitiveness, but enhanced safety is a by-product of the process control function and is not a primary objective. By attempting to maintain process conditions at or near their design values, the process controls also attempt to prevent abnormal conditions from developing within the process.

FIG. 8-92

Layers of safety protection in chemical plants.

PROCESS CONTROL AND PLANT SAFETY Although process controls can be viewed as a protective layer, this is really a by-product and not the primary function. Where the objective of a function is specifically to reduce risk, the implementation is normally not within the process controls. Instead, the implementation is within a separate system specifically provided to reduce risk. This system is generally referred to as the safety interlock system. As safety begins with the process design, an inherently safe process is the objective of modern plant designs. When this cannot be achieved, process hazards of varying severity will exist. Where these hazards put plant workers and/or the general public at risk, some form of protective system is required. Process safety management addresses the various issues, ranging from assessment of the process hazard to ensuring the integrity of the protective equipment installed to cope with the hazard. When the protective system is an automatic action, it is incorporated into the safety interlock system, not within the process controls. INTEGRITY OF PROCESS CONTROL SYSTEMS Ensuring the integrity of process controls involves hardware issues, software issues, and human issues. Of these, the hardware issues are usually the easiest to assess and the software issues the most difficult. The hardware issues are addressed by providing various degrees of redundancy, by providing multiple sources of power and/or an uninterruptible power supply, and the like. The manufacturers of process controls provide a variety of configuration options. Where the process is inherently safe and infrequent shutdowns can be tolerated, nonredundant configurations are acceptable. For more-demanding situations, an appropriate requirement might be that no single component failure be able to render the process control system inoperable. For the very critical situations, triple-redundant controls with voting logic might be appropriate. The difficulty lies in assessing what is required for a given process. Another difficulty lies in assessing the potential for human errors. If redundancy is accompanied with increased complexity, the resulting increased potential for human errors must be taken into consideration. Redundant systems require maintenance procedures that can correct problems in one part of the system while the remainder of the system is in full operation. When maintenance is conducted in such situations, the consequences of human errors can be rather unpleasant. The use of programmable systems for process control presents some possibilities for failures that do not exist in hardwired electromechanical implementations. Probably of greatest concern are latent defects or “bugs” in the software, either the software provided by the supplier or the software developed by the user. The source of this problem is very simple. There is no methodology available that can be applied to obtain absolute assurance that a given set of software is completely free of defects. Increased confidence in a set of software is achieved via extensive testing, but no amount of testing results in absolute assurance that there are no defects. This is especially true of real-time systems, where the software can easily be exposed to a sequence of events that was not anticipated. Just because the software performs correctly for each event individually does not mean that it will perform correctly when two (or more) events occur at nearly the same time. This is further complicated by the fact that the defect may not be in the programming; it may be in how the software was designed to respond to the events. The testing of any collection of software is made more difficult as the complexity of the software increases. Software for process control has become progressively complex, mainly because the requirements have become progressively demanding. To remain competitive in the world market, processes must be operated at higher production rates, within narrower operating ranges, closer to equipment limits, and so on. Demanding applications require sophisticated control strategies, which translate to more-complex software. Even with the best efforts of both supplier and user, complex software systems are unlikely to be completely free of defects.

8-95

CONSIDERATIONS IN IMPLEMENTATION OF SAFETY INTERLOCK SYSTEMS Where hazardous conditions can develop within a process, a protective system of some type must be provided. Sometimes this is in the form of process hardware such as pressure relief devices. However, sometimes logic must be provided for the specific purpose of taking the process to a state where the hazardous condition cannot exist. The term safety interlock system is normally used to designate such logic. The purpose of the logic within the safety interlock system is very different from that of the logic within the process controls. Fortunately, the logic within the safety interlock system is normally much simpler than the logic within the process controls. This simplicity means that a hardwired implementation of the safety interlock system is usually an option. Should a programmable implementation be chosen, this simplicity means that latent defects in the software are less likely to be present. Most safety systems only have to do simple things, but they must do them very, very well. The difference in the nature of process controls and safety interlock systems leads to the conclusion that these two should be physically separated (see Fig. 8-92). That is, safety interlocks should not be piggybacked onto a process control system. Instead, the safety interlocks should be provided by equipment, either hardwired or programmable, that is dedicated to the safety functions. As the process controls become more complex, faults are more likely. Separation means that faults within the process controls have no consequences in the safety interlock system. Modifications to the process controls are more frequent than modifications to the safety interlock system. Therefore, physically separating the safety interlock system from the process controls provides the following benefits: 1. The possibility of a change to the process controls leading to an unintentional change to the safety interlock system is eliminated. 2. The possibility of a human error in the maintenance of the process controls having consequences for the safety interlock system is eliminated. 3. Management of change is simplified. 4. Administrative procedures for software version control are more manageable. Separation also applies to the measurement devices and actuators. Although the traditional point of reference for safety interlock systems is a hardwired implementation, a programmed implementation is an alternative. The potential for latent defects in software implementation is a definite concern. Another concern is that solid-state components are not guaranteed to fail to the safe state. The former is addressed by extensive testing; the latter is addressed by manufacturer-supplied and/or user-supplied diagnostics that are routinely executed by the processor within the safety interlock system. Although issues must be addressed in programmable implementations, the hardwired implementations are not perfect either. Where a programmed implementation is deemed to be acceptable, the choice is usually a programmable logic controller that is dedicated to the safety function. PLCs are programmed with the traditional relay ladder diagrams used for hardwired implementations. The facilities for developing, testing, and troubleshooting PLCs are excellent. However, for PLCs used in safety interlock systems, administrative procedures must be developed and implemented to address the following issues: 1. Version controls for the PLC program must be implemented and rigidly enforced. Revisions to the program must be reviewed in detail and thoroughly tested before implemention in the PLC. The various versions must be clearly identified so that there can be no doubt as to what logic is provided by each version of the program. 2. The version of the program that is currently being executed by the PLC must be known with absolute certainty. It must be impossible for a revised version of the program undergoing testing to be downloaded to the PLC. Constant vigilance is required to prevent lapses in such administrative procedures.

8-96

PROCESS CONTROL

INTERLOCKS An interlock is a protective response initiated on the detection of a process hazard. The interlock system consists of the measurement devices, logic solvers, and final control elements that recognize the hazard and initiate an appropriate response. Most interlocks consist of one or more logic conditions that detect out-of-limit process conditions and respond by driving the final control elements to the safe states. For example, one must specify that a valve fails open or fails closed. The potential that the logic within the interlock could contain a defect or bug is a strong incentive to keep it simple. Within process plants, most interlocks are implemented with discrete logic, which means either hardwired electromechanical devices or programmable logic controllers. The discrete logic within process plants can be broadly classified as follows: 1. Safety interlocks. These are designed to protect the public, the plant personnel, and possibly the plant equipment from process hazards. These are implemented within the safety interlock system. 2. Process actions. These are designed to prevent process conditions that would unduly stress equipment (perhaps leading to minor damage), lead to off-specification product, and so on. Basically, the process actions address hazards whose consequences essentially lead to a monetary loss, possibly even a short plant shutdown. Although sometimes referred to as interlocks, process actions address situations that are not deemed to be process hazards. Implementation of process actions within process control systems is perfectly acceptable. Furthermore, it is also permissible (and probably advisable) for responsible operations personnel to be authorized to bypass or ignore a process action. Safety interlocks must be implemented within the separate safety interlock system. Bypassing or ignoring safety interlocks by operations personnel is simply not permitted. When this is necessary for actions such as verifying that the interlock continues to be functional, such situations must be infrequent and incorporated into the design of the interlock. Safety interlocks are assigned to categories that reflect the severity of the consequences, should the interlock fail to perform as intended. The specific categories used within a company are completely at the discretion of the company. However, most companies use categories that distinguish among the following: 1. Hazards that pose a risk to the public. Complete redundancy is normally required. 2. Hazards that could lead to injury of company personnel. Partial redundancy is often required (e.g., redundant measurements but not redundant logic). 3. Hazards that could result in major equipment damage and consequently lengthy plant downtime. No redundancy is normally required for these, although redundancy is always an option. Situations resulting in minor equipment damage that can be quickly repaired do not generally require a safety interlock; however, a process action might be appropriate. A process hazards analysis is intended to identify the safety interlocks required for a process and to provide the following for each: 1. The hazard that is to be addressed by the safety interlock 2. The classification of the safety interlock 3. The logic for the safety interlock, including inputs from measurement devices and outputs to actuators

The process hazards analysis is conducted by an experienced, multidisciplinary team that examines the process design, plant equipment, operating procedures, and so on, using techniques such as hazard and operability studies (HAZOP), failure mode and effect analysis (FMEA), and others. The process hazards analysis recommends appropriate measures to reduce the risk, including (but not limited to) the safety interlocks to be implemented in the safety interlock system. Diversity is recognized as a useful approach to reduce the number of defects. The team that conducts the process hazards analysis does not implement the safety interlocks but provides the specifications for the safety interlocks to another organization for implementation. This organization reviews the specifications for each safety interlock, seeking clarifications as necessary from the process hazards analysis team and bringing any perceived deficiencies to the attention of the process hazards analysis team. Diversity can be used to further advantage in redundant configurations. Where redundant measurement devices are required, different technology can be used for each. Where redundant logic is required, one can be programmed and one hardwired. Reliability of the interlock systems has two aspects: 1. It must react, should the hazard arise. 2. It must not react when there is no hazard. Emergency shutdowns often pose risks in themselves, and therefore they should be undertaken only when truly appropriate. The need to avoid extraneous shutdowns is not motivated by a desire simply to avoid disruption in production operations. Although safety interlocks can inappropriately initiate shutdowns, the process actions are usually the major source of problems. It is possible to configure so many process actions that it is not possible to operate the plant. TESTING As part of the detailed design of each safety interlock, written test procedures must be developed for the following purposes: 1. Ensure that the initial implementation complies with the requirements defined by the process hazards analysis team. 2. Ensure that the interlock (hardware, software, and I/O) continues to function as designed. The design must also determine the time interval over which this must be done. Often these tests must be done with the plant in full operation. The former is the responsibility of the implementation team and is required for the initial implementation and following any modification to the interlock. The latter is the responsibility of plant maintenance, with plant management responsible for seeing that it is done at the specified interval of time. Execution of each test must be documented, showing when it was done, by whom, and the results. Failures must be analyzed for possible changes in the design or implementation of the interlock. These tests must encompass the complete interlock system, from the measurement devices through the final control elements. Merely simulating inputs and checking the outputs is not sufficient. The tests must duplicate the process conditions and operating environments as closely as possible. The measurement devices and final control elements are exposed to process and ambient conditions and thus are usually the most likely to fail. Valves that remain in the same position for extended periods may stick in that position and not operate when needed. The easiest component to test is the logic; however, this is the least likely to fail.

Section 9

Process Economics

James R. Couper, D.Sc. Professor Emeritus, The Ralph E. Martin Department of Chemical Engineering, University of Arkansas—Fayetteville (Section Editor) Darryl W. Hertz, B.S. Manager, Front-End Loading and Value-Improving Practices Group, KBR (Front-End Loading, Value-Improving Practices) (Francis) Lee Smith, Ph.D., M.Eng. Principal, Wilcrest Consulting Associates, Houston, Texas (Front-End Loading, Value-Improving Practices)

GENERAL COMMENTS ACCOUNTING AND FINANCIAL CONSIDERATIONS Principles of Accounting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balance Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Income Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulated Retained Earnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Financial Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship Between Balance Sheets and Income Statements. . . . . . . Financing Assets by Debt and/or Equity . . . . . . . . . . . . . . . . . . . . . . . . Cost of Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inventory Evaluation and Cost Control. . . . . . . . . . . . . . . . . . . . . . . . . . Budgets and Cost Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAPITAL COST ESTIMATION Total Capital Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Land . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Capital Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Use of Cost Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Equipment Sizing and Costing. . . . . . . . . . . . . . . . . . . . . Estimation of Fixed Capital Investment . . . . . . . . . . . . . . . . . . . . . . . Example 4: Seven-Tenths Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Fixed Capital Investment Using the Lang, Hand, and Wroth Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments on Significant Cost Items . . . . . . . . . . . . . . . . . . . . . . . . . Computerized Cost Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contingency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Offsite Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allocated Capital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Start-up Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Capital Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9-10 9-10 9-10 9-13 9-13 9-13 9-13 9-14 9-14 9-16 9-17 9-17 9-17 9-17 9-17 9-17 9-17

MANUFACTURING-OPERATING EXPENSES Raw Material Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operating Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Payroll Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Direct Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Control Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plant Indirect Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Manufacturing Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packaging and Shipping Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Product Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Overhead Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Operating Expense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapid Manufacturing Expense Estimation . . . . . . . . . . . . . . . . . . . . . . Scale-up of Manufacturing Expenses . . . . . . . . . . . . . . . . . . . . . . . . . . FACTORS THAT AFFECT PROFITABILITY Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Value of Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compounding-Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Effective Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: After-Tax Cash Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cash Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative Cash Position Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Cumulative Cash Position Table (Time Zero at Start-up) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-18 9-18 9-18 9-18 9-18 9-18 9-18 9-18 9-18 9-20 9-20 9-20 9-20 9-20 9-20 9-20 9-20 9-20 9-21

9-21 9-22 9-22 9-22 9-23 9-23 9-23 9-23 9-23 9-25 9-27 9-27 9-27 9-27 9-28 9-1

Copyright © 2008, 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGraw-Hill Companies, Inc. Click here for terms of use.

9-2

PROCESS ECONOMICS

Cumulative Cash Position Plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Zero at Start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-28 9-28

PROFITABILITY Quantitative Measures of Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . Payout Period Plus Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net Present Worth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discounted Cash Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Profitability Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . Qualitative Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Break-Even Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strauss Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tornado Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relative Sensitivity Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feasibility Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-30 9-30 9-30 9-30 9-32 9-32 9-32 9-32 9-32 9-32 9-32 9-32 9-34

OTHER ECONOMIC TOPICS Comparison of Alternative Investments . . . . . . . . . . . . . . . . . . . . . . . . . Net Present Worth (NPW) Method. . . . . . . . . . . . . . . . . . . . . . . . . . . Cash Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-35 9-36 9-36

Uniform Annual Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: Choice among Alternatives . . . . . . . . . . . . . . . . . . . . . . . Replacement Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Replacement Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Opportunity Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12: Optimum Number of Evaporator Effects . . . . . . . . . . . Interactive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-36 9-36 9-36 9-38 9-39 9-39 9-39 9-41

CAPITAL PROJECT EXECUTION AND ANALYSIS Front-End Loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of FEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical FEL Deliverables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Value-Improving Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIP Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIP Planning and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . VIPs That Apply the Value Methodology . . . . . . . . . . . . . . . . . . . . . . Sources of Expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-41 9-41 9-42 9-47 9-48 9-48 9-50 9-52 9-53 9-53

GLOSSARY

PROCESS ECONOMICS

9-3

Nomenclature and Units Symbol A1 ATC B C CB CE (CFC)BL (CEQ)DEL CHE CL COE CP cP D DCFROR EBIT e F F FB FCI FE FEL FIFO FOB FM FP f1, f2, f3 f´ I IRS i ieff K

Definition Annual conversion expense at production rate 1 Annual capital outlay Constant Cost of equipment Base cost of carbon steel exchanger Chemical Engineering cost index Battery-limits fixed capital investment Delivered equipment cost Purchased equipment cost, heat exchanger Cost of labor Cash operating expenses Equipment cost in base year Viscosity Depreciation Discounted cash flow rate of return Earnings before interest and taxes Naperian logarithm base Future value, future worth, future amount Heat exchanger efficiency factor Heat exchanger design type Fixed capital investment Fixed expenses Front-end loading First in, first out (inventory) Free on board Material of construction cost factor Design pressure cost factor Inflation factors for years 1, 2, and 3 Declining-balance factor Investment Internal Revenue Service Nominal interest Effective interest Factor for cost index

Units $ $ Dimensionless $ $ Dimensionless $ $ $ $ $ $ cP $ %

Symbol LIFO M MACRS m m, n, p, q N n P PC POP POP + I Q R1, R2 S

$ 2.718 $ Dimensionless Dimensionless $ $ lb Dimensionless Dimensionless Dimensionless Dimensionless $ % % Dimensionless

SL Sp gr TE Tc Tf Ts U UAC UD Ve Vi VE VIP X Y

Definition Last in, last out (inventory) Annual raw material expense Modified Accelerated Cost Recovery System Number of interest periods per year constants or exponent Annual labor requirements Number of years, depreciation Principal, present value, present worth Personal computer Payout period (no interest) Payout period plus interest Energy transferred Annual production rates Salvage value or equipment capacity Straight-line depreciation Specific gravity Total expenses Combined incremental tax rate Incremental federal income tax rate Incremental state income tax rate Annual utility expenses Uniform annual cost Overall heat-transfer coefficient Asset value at end of year Asset value at beginning of year Variable expenses Value-improving practice Plant capacity Operating labor

Units lb $ Varies Dimensionless Operators per shift per year Years $ Years Years Btu/h lb/yr Various Dimensionless $ % % % $ $ Btu/(hft2!F) $ $ $ tons/day operator-hour/ton per processing step

GENERAL REFERENCES: Allen, D. H., Economic Evaluation of Projects, 3d ed., Institution of Chemical Engineers, Rugby, England, 1991. Baasel, W. D., Chemical Engineering Plant Design, 2d ed., Van Nostrand Reinhold, New York, 1989. Brown, T. R., Hydrocarbon Processing, October 2000, pp. 93–100. Canada, J. R., and J. A. White, Capital Investment Decision: Analysis for Management and Engineering, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1980. Chemical Engineering (ed.), Modern Cost Engineering, McGraw-Hill, New York, 1979. Couper, J. R., and W. H. Rader, Applied Finance and Economic Analysis for Scientists and Engineers, Van Nostrand Reinhold, New York, 1986. Couper, J. R., and O. T. Beasley, The Chemical Process Industries: Function and Economics, Dekker, New York, 2001. Couper, J. R., Process Engineering Economics, Dekker, New York, 2003. Garrett, D. E., Chemical Engineering Economics, Van Nostrand Reinhold, New York, 1989. Grant, E. L., and W. G. Ireson, Engineering Economy, 2d ed., Wiley, New York, 1950. Grant, E. L, W. G. Ireson, and R. S. Leavenworth, Engineering Economy, 8th ed., Wiley, New York, 1990. Hackney, J. W., and K. K. Humphreys (eds.), Control and Management of Capital Projects, 2d ed., McGraw-Hill, New York, 1992. Hill, D. A., and L. E. Rockley, Secrets of Successful Financial Management, Heinemann, London, 1990. Holland, F. A., F. A. Watson, and J. E. Wilkerson, Introduction to Process Economics, 2d ed., Wiley, London, 1983. K. K. Humphreys, F. C. Jelen, and J. H. Black (eds.), Cost and

Optimization Engineering, 3d ed., McGraw-Hill, New York, 1991. A Guide to Capital Cost Estimation, 3d ed., Institution of Chemical Engineers, Rugby, England, 1990. Kharbanda, O. P., and E. A. Stallworthy, Capital Cost Estimating in the Process Industries, 2d ed., Butterworth-Heinemann, London, 1988. How to Read an Annual Report, Merrill Lynch, New York, 1997. Nickerson, C. B., Accounting Handbook for Non Accountants, 2d ed., CBI Publishing, Boston, 1979. Ostwald, P. F., Engineering Cost Estimating, 3d ed., Prentice-Hall, Englewood Cliffs, N.J., 1991. Park, W. R., and D. E. Jackson, Cost Engineering Analysis, 2d ed., Wiley, New York, 1984. Peters, M. S., and K. D. Timmerhaus, Plant Design and Economics for Chemical Engineers, 6th ed., McGraw-Hill, New York, 2003. Popper, H. (ed.), Modern Cost Estimating Techniques, McGraw-Hill, New York, 1970. Rose, L. M., Engineering Investment Decisions: Planning under Uncertainty, Elsevier, Amsterdam, 1976. Thorne, H. C., and J. B. Weaver (eds.), Investment Appraisal for Chemical Engineers, American Institute of Chemical Engineers, New York, 1991. Ulrich, G., and P. T. Vasudevan, Chemical Engineering Process Design and Economics, CRC Press, Boca Raton, Fla., 2004. ValleRiestra, J. F., Project Evaluation in the Chemical Process Industries, McGraw-Hill, New York, 1983. Wells, G. L., Process Engineering with Economic Objectives, Wiley, New York, 1973. Woods, D. R., Process Design and Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1993.

GENERAL COMMENTS One of the most confusing aspects of process engineering economics is the nomenclature used by various authors and companies. In this part of Sec. 9, generic, descriptive terms have been used. Further, an attempt has been made to bring together most of the methods currently in use for project evaluation and to present them in such a way as to make them amenable to modern computational techniques. Most of the calculations can be performed on handheld calculators equipped with scientific function keys. For calculations requiring greater sophistication than that of handheld calculators, algorithms may be solved by using such programs as MATHCAD, TKSOLVER, etc. Spreadsheets are also used whenever the solution to a problem lends itself to this technique. The nomenclature in process economics has been developed by accountants, engineers, and others such that there is no one correct set of nomenclature. Often it seems confusing, but one must question

what is meant by a certain term since companies have adopted their own language. A glossary of terms is included at the end of this section to assist the reader in understanding the nomenclature. Further, abbreviations of terms such as DCFRR (discounted cash flow rate of return) are used to reduce the wordiness. The number of letters and numbers used to define a variable has been limited to five. The parentheses are removed whenever the letter group is used to define a variable for a computer. Also, a general symbol is defined for a type variable and is modified by mnemonic subscript, e.g., an annual cash quantity, annual capital outlay ATC, $/year. Wherever a term like this is introduced, it is defined in the text. It is impossible to allow for all possible variations of equation requirements, but it is hoped that the nomenclature presented will prove adequate for most purposes and will be capable of logical extension to other more specialized requirements.

ACCOUNTING AND FINANCIAL CONSIDERATIONS PRINCIPLES OF ACCOUNTING Accounting has been defined as the art of recording business transactions in a systematic manner. It is the language of business and is used to communicate financial information. Conventions that govern accounting are fairly simple, but their application is complex. In this section, the basic principles are illustrated by a simple example and applied to analyzing a company report. The fair allocation of costs requires considerable technical knowledge of operations, so a close liaison between process engineers and accountants in a company is desirable. In simplest terms, assets that are the economic resources of a company are balanced against equities that are claims against the firm. In equation form, Assets = Equities or

Assets = Liabilities + Owners’ Equity

This dual aspect has led to the double-entry bookkeeping system in use today. Any transaction that takes place causes changes in the accounting equation. An increase in assets must be accompanied by one of the following: • An increase in liabilities • An increase in stockholders’ equity • An increase in assets 9-4

A change in one part of the equation due to an economic transaction must be accompanied by an equal change in another place— therefore, the term double-entry bookkeeping. On a page of an account, the left-hand side is designated the debit side and the right-hand side is the credit side. This convention holds regardless of the type of account. Therefore, for every economic transaction, there is an entry on the debit side balanced by the same entry on the credit side. All transactions in their original form (receipts and invoices) are recorded chronologically in a journal. The date of the transaction together with an account title and a brief description of the transaction is entered. Table 9-1 is an example of a typical journal page for a company. Journal entries are transferred to a ledger in a process called posting. Separate ledger accounts, such as a revenue account, expense account, liability account, or asset account, may be set up for each major transaction. Table 9-2 shows an example of a typical ledger page. The number of ledger accounts depends on the information that management needs to make decisions. Periodically, perhaps on a monthly basis but certainly on a yearly basis, the ledger sheets are closed and balanced. The ledger sheets are then intermediate documents between journal records and balance sheets, income statements, and retained earnings statements, and they provide information for management and various government reports. For example, a consolidated income statement can be prepared for the ledger, revenue, and expense accounts. In like manner, the asset and liability accounts provide information for balance sheets.

ACCOUNTING AND FINANCIAL CONSIDERATIONS TABLE 9-1 Date 200X Mar 1 Mar 4 Mar 11 Mar 13 Apr 4

Typical Journal Page Explanation

LP

Debit

Cash J. Jones, Capital Property Cash Mortgage Remodeling Bldg. Cash Equipment Cash Note Payable To J. Jones Cash

1 2 4 1 3 5 1 6 1 3 2 1

$95,000

Credit

$95,000 5,000 3,000 2,000 7,800 7,800 62,300 10,000 52,300 2,500 2,500

SOURCE:

J. R. Couper, Process Engineering Economics, Dekker, New York, 2003. By permission of Taylor & Francis Books, Inc., Boca Raton, Fla.

FINANCIAL STATEMENTS A basic knowledge of accounting and financial statements is necessary for a chemical professional to be able to analyze a firm’s operation and to communicate with accountants, financial personnel, and managers. Financial reports of a company are important sources of information used by management, owners, creditors, investment bankers, and financial analysts. All publicly held companies are required to submit annual reports to the Securities and Exchange Commission. As with any field a certain basic nomenclature is used to be able to understand the financial operation of a company. It should be emphasized that companies may also have their own internal nomenclature, but some terms are universally accepted. In this section, the common terminology is used. A financial report contains two important documents—the balance sheet and the income statement. Two other documents that appear in the financial report are the accumulated retained earnings and the changes in working capital. All these documents are discussed in the following sections using a fictitious company. Balance Sheet The balance sheet represents an accounting view of the financial status of a company on a particular date. Table 9-3 is an example of a balance sheet for a company. The date frequently used by corporations is December 31 of any given year, although some companies are now using June 30 or September 30 as the closing date. It is as if the company’s operation were frozen in time on that date. The term consolidated means that all the balance sheet and income statement data include information from the parent as well as subsidiary operations. The balance sheet consists of two parts: assets are the items that the company owns, and liabilities and stockholders’ equity are what the TABLE 9-2

9-5

company owes to creditors and stockholders. Although the balance sheet has two sides, it is not part of the double-entry accounting system. The balance sheet is not an account but a statement of claims against company assets on the date of the reporting period. The claims are the creditors and the stockholders. Therefore, the total assets must equal the total liabilities plus the stockholders’ equity. Assets are classified as current, fixed, or intangibles. Current assets include cash, cash equivalents, marketable securities, accounts receivable, inventories, and prepaid expenses. Cash and cash equivalents are those items that can be easily converted to cash. Marketable securities are securities that a company holds that also may be converted to cash. Accounts receivable are the amounts due a company from customers from material that has been delivered but has not been collected as yet. Customers are given 30, 60, or 90 days in which to pay; however, some customers fail to pay bills on time or may not be able to pay at all. An allowance is made for doubtful accounts. The amount is deducted from the accounts receivables. Inventories include the cost of raw materials, goods in process, and product on hand. Prepaid expenses include insurance premiums paid, charges for leased equipment, and charges for advertising that are paid prior to the receipt of the benefit from these items. The sum of all the above items is the total current assets. The term current refers to the fact that these assets are easily converted within a year, or more likely in a shorter time, say, 90 days. Fixed assets are items that have a relatively long life such as land, buildings, and manufacturing equipment. The sum of these items is the total property, plant, and equipment. From this total, accumulated depreciation is subtracted and the result is net property and equipment. Last, an item referred to as intangibles includes a variety of items such as patents, licenses, intellectual capital, and goodwill. Intangibles are difficult to evaluate since they have no physical existence; e.g., goodwill is the value of the company’s name and reputation. The sum of the total current assets, net property, and intangibles is the total assets. Liabilities are the obligations that the company owes to creditors and stockholders. Current liabilities are obligations that come due within a year and include accounts payable (money owed to creditors for goods and services), notes payable (money owed to banks, corporations, or other lenders), accrued expenses (salaries and wages to employees, interest on borrowed funds, fees due to professionals, etc.), income taxes payable, current part of long-term debt, and other current liabilities due within the year. Long-term liabilities are the amounts due after 1 year from date of the financial report. They include deferred income taxes that a company is permitted to postpone due to accelerated depreciation to encourage investment, (but they must be paid sometime in the future) and bonds and notes that do not have to be paid within the year but at some later date. The sum of the current and long-term liabilities is the total liabilities.

Typical Ledger Page Cash: Account 01

200X Mar 1 Capital

J-1

$95,000

J-1

$2,500

Mar 1 Mar 11 Mar 13 Apr 4

Property Remodeling Equipment J. Jones

J-1 J-1 J-1 J-1

$3,000 7,800 10,000 2,500

Capital

J-1

$95,000

Mortgage Note Payable

J-1 J-1

$2,000 52,300

Capital: Account 02 Apr 4 Cash to J. Jones

Mar 1 Accounts Payable: Account 03 Mar 4 Mar 13 Property and Building: Account 04

Mar 4 Mar 11

J-1 J-1

$5,000 7,800

Mar 13

J-1

$62,300

Equipment: Account 05 SOURCE:

J. R. Couper, Process Engineering Economics, Dekker, New York, 2003. By permission of Taylor & Francis Books, Inc., Boca Raton, Fla.

9-6

PROCESS ECONOMICS

TABLE 9-3

TABLE 9-4

Consolidated Balance Sheeta (December 31)

Assets Current assets Cash Marketable securities Accounts receivableb Inventories Prepaid expenses Total current assets Fixed assets Land Buildings Machinery Office equipment Total fixed assets Less accumulated depreciation Net fixed assets Intangibles Total assets Liabilities Current liabilities Accounts payable Notes payable Accrued expenses payable Federal income taxes payable Total current liabilities Long-term liabilities Debenture bonds, 10.3% due in 2015 Debenture bonds, 11.5% due in 2007 Deferred income taxes Total liabilities Stockholder’s equity Preferred stock, 5% cumulative $5 par value—200,000 shares Common stock, $1 par value 2000 28,000,000 shares 2000X 32,000,000 shares Capital surplus Accumulated retained earnings Total stockholder’s equity Total liabilities and stockholder’s equity

2005

Consolidated Income Statement (December 31) 2005

2004

$932,000

$850,000

692,000 40,000 113,500 $86,500

610,000 36,000 110,000 $94,000

10,000 (22,000) $74,500 24,500 $50,000

7,000 (22,000) $79,000 26,000 $53,000

2004

$63,000 41,000 135,000 149,000 3,200 $391,200

$51,000 39,000 126,000 153,000 2,500 $371,500

35,000 101,000 278,000 24,000 $438,000 128,000 $310,000 4,500 $705,700

35,000 97,500 221,000 19,000 $372,500 102,000 $270,500 4,500 $646,500

2005

2004

$92,300 67,500 23,200 18,500 $201,500

$81,300 59,500 26,300 17,500 $184,600

110,000 125,000 11,600 $448,100

110,000 125,000 10,000 $429,600

$10,000

$10,000

32,000

28,000

8,000 207,600 $257,600 $705,700

6,000 172,900 $216,900 $646,500

a

All amounts in thousands of dollars. Includes an allowance for doubtful accounts. SOURCE: J. R. Couper, Process Engineering Economics, Dekker, New York, 2003. By permission of Taylor & Francis Books, Inc., Boca Raton, Fla. b

Stockholders’ equity is the interest that all stockholders have in a company and is a liability with respect to the company. This category includes preferred and common stock as well as additional paid-in capital (the amount that stockholders paid above the par value of the stock) and retained earnings. These are earnings from accumulated profit that a company earns and are used for reinvestment in the company. The sum of these items is the stockholders’ equity. On a balance sheet, the sum of the total liabilities and the stockholders’ equity must equal the total assets, hence the term balance sheet. Comparing balance sheets for successive years, one can follow changes in various items that will indicate how well the company manages its assets and meets its obligations. Income Statement An income statement shows the revenue and the corresponding expenses for the year and serves as a guide for how the company may do in the future. Often income statements may show how the company performed for the last two or three years. Table 9-4 is an example of a consolidated income statement. Net sales are the primary source of revenue from goods and services. This figure includes the amount reported after returned goods, discounts, and allowances for price reductions are taken into account. Cost of sales represents all the expenses to convert raw materials to finished products. The major components of these expenses are direct material, direct labor, and overhead. If the cost of sales is subtracted from net sales, the result is the gross margin. One of the most important items on the income statement is depreciation and amortization. Depreciation is an allowance the federal government permits for the

Net sales (revenue) Cost of sales and operating expenses Cost of goods sold Depreciation and amortization Sales, general, and administrative expenses Operating profit Other income (expenses) Dividends and interest income Interest expense Income before provision for income taxes Provision for federal income taxes Net profit for year

SOURCE: J. R. Couper, Process Engineering Economics, Dekker, New York, 2003. By permission of Taylor & Francis Books, Inc., Boca Raton, Fla.

wear and tear as well as the obsolescence of plant and equipment and is treated as an expense. Amortization is the decline in value of intangible assets such as patents, franchises, and goodwill. Selling, general, and administrative expenses include the marketing salaries, advertising expenses, travel, executive salaries, as well as office and payroll expenses. When depreciation, amortization, and the sales and administrative expenses are subtracted from the gross margin, the result is the operating income. Dividends and interest income received by the company are then added. Next interest expense earned by the stockholders and income taxes are subtracted, yielding the term income before extraordinary loss. It is the expenses a company may incur for unusual and infrequent occasions. When all the above items are added or subtracted from the operating income, net income (or loss) is obtained. This latter term is the “bottom line” often referred to in various reports. Accumulated Retained Earnings This is an important part of the financial report because it shows how much money has been retained for growth and how much has been paid as dividends to stockholders. When the accumulated retained earnings increase, the company has greater value. The calculation of this value of the retained earnings begins with the previous year’s balance. To that figure add the net profit after taxes for the year. Dividends paid to stockholders are then deducted, and the result is the accumulated retained earnings for the year. See Table 9-5. Concluding Remarks One of the most important sections of an annual report is the “notes.” These contain any liabilities that a company may have due to impending litigation that could result in charges or expenses not included in the annual report. OTHER FINANCIAL TERMS Profit margin is the ratio of net income to total sales, expressed as a percentage or sometimes quoted as the ratio of profit before interest and taxes to sales, expressed as a percentage. Operating margin is obtained by subtracting operating expenses from gross profit expressed as a percentage of sales. Net worth is the difference between total assets and total liabilities plus stockholders’ equity. Working capital is the difference between total current assets and current liabilities. TABLE 9-5 Accumulated Retained Earnings Statementa (December 31) Balance as of January 1 Net profit for year Total for year Less dividends paid on: Preferred stock Common stock Balance December 31 a

2005

2004

$172,900 50,000 $222,900

$141,850 53,000 $194,850

700 14,600 $207,600

700 21,250 $172,900

All amounts in thousands of dollars. J. R. Couper, Process Engineering Economics, Dekker, New York, 2003. By permission of Taylor & Francis Books, Inc., Boca Raton, Fla. SOURCE:

ACCOUNTING AND FINANCIAL CONSIDERATIONS FINANCIAL RATIOS

TABLE 9-6

There are many financial ratios of interest to financial analysts. A brief discussion of some of these ratios follows; however, a more complete discussion may be found in Couper (2003). Liquidity ratios are a measure of a company’s ability to pay its shortterm debts. Current ratio is obtained by dividing the current assets by the current liabilities. Depending on the economic climate, this ratio is 1.5 to 2.0 for the chemical process industries, but some companies operate closer to 1.0. The quick ratio is another measure of liquidity and is cash plus marketable securities divided by the current liabilities and is slightly greater than 1.0. Leverage ratios are an indication of the company’s overall debt burden. The debt/total assets ratio is determined by dividing the total debt by total assets expressed as a percentage. The industry average is 35 percent. Debt/equity ratio is another such ratio. The higher these ratios, the greater the financial risk since if an economic downturn did occur, it might be difficult for a company to meet the creditors’ demands. The times interest earned is a measure of the extent to which profit could decline before a company is unable to pay interest charges. The ratio is calculated by dividing the earnings before interest and taxes (EBIT) by interest charges. The fixed-charge coverage is obtained by dividing the income available for meeting fixed charges by the fixed charges. Activity ratios are a measure of how effectively a firm manages its assets. There are two inventory/turnover ratios in common use today. The inventory/sales ratio is found by dividing the inventory by the sales. Another method is to divide the cost of sales by inventory. The average collection period measures the number of days that customers’ invoices remain unpaid. Fixed assets and total assets turnover indicate how well the fixed and total assets of the firm are being used. Profitability ratios are used to determine how well income is being managed. The gross profit margin is found by dividing the gross profits by the net sales, expressed as a percentage. The net operating margin is equal to the earnings before interest and taxes divided by net sales. Another measure, the profit margin on sales, is calculated by dividing the net profit after taxes by net sales. The return on total assets ratio is the net profit after taxes divided by the total assets expressed as a percentage. The return on equity ratio is the net income after taxes and interest divided by stockholders’ equity. Table 9-6 shows the financial ratios for Tables 9-3 and 9-4. Table 9-7 is a summary of selected financial ratios and industry averages.

Liquidity

RELATIONSHIP BETWEEN BALANCE SHEETS AND INCOME STATEMENTS There is a relationship between these two documents because information obtained from each is used to calculate the returns on assets and equity. Figure 9-1 is an operating profitability tree for a fictitious TABLE 9-7

Financial Ratios for Tables 9-3 and 9-4

Current ratio = $391,200/$201,500 = 1.94 Cash ratio = $391,200 − 149,000/$201,500 = 1.20 Leverage Debt/assets ratio = [($448,100 − 201,500)/$705,700] × 100 = 35% Times interest earned = $74,500 − 22,000/$22,000 = 4.39 Fixed-charge coverage = $86,500/$22,000 = 3.93 Activity Inventory turnover = $932,000/$149,000 = 6.25 Average collection period = $135,000/($932,000/365) = 52.8 days Fixed-assets turnover = $932,000/$438,000 = 2.13 Total-assets turnover = $932,000/$705,700 = 1.32 Profitability Gross profit margin = [($932,000 − 692,000)/$932,000] × 100 = 25.8% Net operating margin = $74,500/$932,000 × 100 = 7.99% Profit margin on sales = $50,000/$932,000 × 100 = 5.36% Return on net worth (return on equity) = [$50,000/($705,700 − 448,100)] × 100 = 19.4% Return on total assets = ($50,000/$705,700) × 100 = 7.09%

company and contains the fixed and variable expenses as reported on internal company reports, such as the manufacturing expense sheet. Figure 9-2 is a financial family tree for the same company depicting the relationship between values in the income statement and the balance sheet. FINANCING ASSETS BY DEBT AND/OR EQUITY The various options for obtaining funds to finance new projects are not a simple matter. Significant factors such as the state of the economy, inflation, a company’s present indebtedness, and the cost of capital will affect the decision. Should a company incur more long-term debt, or should it seek new venture capital from equity sources? A simple yes or no answer will not suffice because the financial decision is complex. One consideration is the company’s position with respect to leverage. If a company has a large proportion of its debt in bonds and preferred stock, the common stock is highly leveraged. Should the earnings decline, say, by 10 percent, the dividends available to common stockholders might be wiped out. The company also might not be able to cover the interest on its bonds without dipping into the accumulated earnings. A high debt/equity ratio illustrates the fundamental weakness of companies with a large amount of debt. When low-interest financing is available, such as for large government projects, the return-on-equity evaluations are used. Such leveraging is tantamount to transferring money from one pocket to another; or, to

Selected Financial Ratios Item

Liquidity Current ratio Cash ratio Leverage Debt to total assets Times interest earned Fixed-charge coverage Activity Inventory turnover Average collection period Fixed assets turnover Total assets turnover Profitability Gross profit margin Net operating margin Profit margin on sales Return on net worth (return on equity) Return on total assets

9-7

Equation for calculation

Industry average

Current assets/current liabilities Current assets − inventory/current liabilities

1.5–2.0 1.0–1.5

Total debt/total assets Profit before taxes plus interest charges/interest charges Income available for meeting fixed charges/fixed charges

30–40% 7.0–8.0 6.0

Sales or revenue/inventory Receivables/sales per day Sales/fixed assets Sales/total assets Net sales − cost of goods sold/sales Net operating profit before taxes/sales Net profit after taxes/sales Net profit after taxes/net worth Net profit after taxes/total assets

7.0 40–60 days 2–4 1–2 25–40% 10–15% 5–8% 15% 7–10%

9-8

+ −

Net income

+

+ ÷

+

Sales expenses

+ Operating profit

−

+

Income taxes

Interest

−

Return on assets

Sales

General and administrative

+

+

Cost of goods sold

+

Labor

Operating expense

Nonoperating income

+

Income statement

Fixed expenses

Manufacturing overhead

Depreciation

+

Total assets

Operating profitability tree. (Source: Adapted from Couper, 2003.)

Direct expenses

+

+

FIG. 9-1

Raw material

Variable expenses

Internal reports

ACCOUNTING AND FINANCIAL CONSIDERATIONS +

9-9

Sales

+ Income Statement

Net income

Income taxes

−

Interest

+

Return on equity

÷

−

−

Nonoperating income

Total assets

Balance Sheet

Financial family tree. (Source: Adapted from Couper, 2003.)

put it another way, a company may find itself borrowing from itself. In the chemical process industries, debt/equity ratios of 0.3 to 0.5 are common for industries that are capital-intensive (Couper et al., 2001). Much has been written on the strategies of financing a corporate venture. The correct strategy has to be evaluated from the standpoint of what is best for the company. It must maintain a debt/equity ratio similar to those of successful companies in the same line of business. COST OF CAPITAL The cost of capital is what it costs a company to borrow money from all sources, such as loans, bonds, and preferred and common stock. It is an important consideration in determining a company’s minimum acceptable rate of return on an investment. A company must make more than the cost of capital to pay its debts and make a profit. From profits, a company pays dividends to the stockholders. If a company ignores the cost of capital to increase dividends to the stockholders, then management is not meeting its obligations to pay off outstanding debts. A sample calculation of the after-tax weighted cost of capital is found in Table 9-8. Each debt item is divided by the total debt, and TABLE 9-8

Operating expenses

Return on assets

÷

Total liabilities

Stockholders' equity

FIG. 9-2

Operating profit

Cost of Capital Illustration

Balance sheet 12/31/XX

Debt, $M

Long-term debt Revolving account 438% debentures 612% debentures 634% debentures 712% debentures 938% loan Other Total long-term debt Deferred taxes Reserves Preferred stock Shareholders’ equity Total debt

5.0 12.0 3.4 9.4 74.5 125.0 23.2 252.5 67.7 16.1 50.0 653.9 1,040.2

After-tax yield to maturity, %

After-tax weighted average cost, %

4.5 4.0 4.7 4.2 4.2 4.4 4.4

0.02 0.05 0.02 0.04 0.30 0.53 0.10 1.06 0 0 0.42 9.80 11.28

0.0 0.0 8.6 15.6

Each debt item in $M divided by the total debt times the after-tax yield to maturity equals the after-tax weighted average cost contributing to the cost of capital. SOURCE: Private communication.

that result is multiplied by the after-tax yield to maturity that equals the after-tax weighted average cost of that debt item contributing to the cost of capital. The information to estimate the cost of capital may be obtained from the annual report, the 10K, or the 10Q reports. WORKING CAPITAL The accounting definition of working capital is total current assets minus total current liabilities. This information can be found from the balance sheet. Current assets consist chiefly of cash, marketable securities, accounts receivable, and inventories; current liabilities include accounts payable, short-term debts, and the part of the long-term debt currently due. The accounting definition is in terms of the entire company. For economic evaluation purposes, another definition of working capital is used. It is the funds, in addition to the fixed capital, that a company must contribute to a project. It must be adequate to get the plant in operation and to meet subsequent obligations when they come due. Working capital is not a one-time investment that is known at the project inception, but varies with the sales level and other factors. The relationship of working capital to other project elements may be viewed in the cash flow model (see Fig. 9-9). Estimation of an adequate amount of working capital is found in the section “Capital Investment.” INVENTORY EVALUATION AND COST CONTROL Under ordinary circumstances, inventories are priced (valued) at some form of cost. The problem in valuating inventory lies in “determining what costs are to be identified with inventories in a given situation” (Nickerson, 1979). Valuation of materials can be made by using the • Cost of a specific lot • Average cost • Standard cost Under “cost of a specific lot,” those lots to be valuated must be identified by referring to related invoices. Many companies use the average cost for valuating inventories. The average used should be weighted by the quantities purchased rather than by an average purchase price. Average cost method tends to spread the effects of short-run price changes and has a tendency to level out profits in those industries that use raw materials whose prices are volatile. For many manufacturing companies, inventory valuation is an important

9-10

PROCESS ECONOMICS

consideration varying in degree of importance. Inventories that are large are subject to significant fluctuations from time to time in size and mix and in prices, costs, and values. Materials are valuated in accordance with their acquisition. Some companies use the first-in, first-out (FIFO) basis. Materials are used in order of their acquisition to minimize losses from deterioration. Another method is last-in, first-out (LIFO) in which materials coming in are the first to leave storage for use. The method used depends on a number of factors. Accounting texts discuss the pros and cons of each method, often giving numerical examples. Some items to consider are income tax considerations and cash flow that motivate management to adopt conservative valuation policies. Tax savings may accrue using one method compared to the other, but they may not be permanent. Whatever method is selected, consistency is important so that comparability of reported figures may be maintained from one time period to another. It is management’s responsibility to make the decision regarding the method used. In some countries, government regulations control the method to be used. There are several computer software programs that permit the user to organize, store, search, and manage inventory from a desktop computer. BUDGETS AND COST CONTROL A budget is an objective expressed in monetary terms for planning and controlling the resources of a company. Budgeted numbers are

objectives, not achievements. A comparison of actual expenses with budgeted (cost standards) figures is used for control at the company, plant, departmental, or project level. A continuing record of performance should be maintained to provide the data for preparing future budgets (Nickerson, 1979). Often when a company compares actual results with cost standards or budgeted figures, a need for improving operations will surface. For example, if repairs to equipment continuously exceed the budgeted amount, perhaps it is time to consider replacement of that older equipment with a newer, more efficient model. Budgets are usually developed for a 1-year period; however, budgets for various time frames are frequently prepared. For example, in planning future operations, an intermediate time period of, say, 5 years may be appropriate, or for long-range planning the time period selected may be 10 years. A cost control system is used • To provide early warning of uneconomical or excessive costs in operations • To provide relevant feedback to the personnel responsible for developing budgets • To develop cost standards • To promote a sense of cost consciousness • To summarize progress Budgetary models based upon mathematical equations are available to determine the effect of changes in variables. There are numerous sources extant in the literature for these models.

CAPITAL COST ESTIMATION TOTAL CAPITAL INVESTMENT The total capital investment includes funds required to purchase land, design and purchase equipment, structures, and buildings as well as to bring the facility into operation (Couper, 2003). The following is a list of items constituting the total capital investment: Land Fixed capital investment Offsite capital Allocated capital Working capital Start-up expenses Other capital items (interest on borrowed funds prior to start-up; catalysts and chemicals; patents, licenses, and royalties; etc.) Land Land is often acquired by a company some time prior to the building of a manufacturing facility. When a project is committed to be built on this land, the value of the land becomes part of that facility’s capital investment. Fixed Capital Investment When a firm considers the manufacture of a product, a capital cost estimate is prepared. These estimates are required for a variety of reasons such as feasibility studies, the selection of alternative processes or equipment, etc., to provide information for planning capital appropriations or to enable a contractor to bid on a project. Included in the fixed capital investment is the cost of purchasing, delivery, and installation of manufacturing equipment, piping, automatic controls, buildings, structures, insulation, painting, site preparation, environmental control equipment, and engineering and construction costs. The fixed capital investment is significant in developing the economics of a process since this figure is used in estimating operating expenses and calculating depreciation, cash flow, and project profitability. The estimating method used should be the best, most accurate means consistent with the time and money available to prepare the estimate. Classification of Estimates There are two broad classes of estimates: grass roots and battery limits. Grass-roots estimates include the entire facility, starting with site preparation, buildings and structures, processing equipment, utilities, services, storage facilities, railroad yards, docks, and plant roads. A battery-limits estimate is one in which an imaginary boundary is drawn around the proposed facility to be estimated. It is assumed that all materials, utilities, and services are available in the quality and quantity required to manufacture a product. Only costs within the boundary are estimated.

Quality of Estimates Capital cost estimation is more art than science. An estimator must use considerable judgment in preparing the estimate, and as the estimator gains experience, the accuracy of the estimate improves. There are several types of fixed capital cost estimates: • Order-of-magnitude (ratio estimate). Rule-of-thumb methods based on cost data from similar-type plants are used. The probable accuracy is −30 percent to +50 percent. • Study estimate (factored estimate). This type requires knowledge of preliminary material and energy balances as well as major equipment items. It has a probable accuracy of −25 to +30 percent. • Preliminary estimate (budget authorization estimate). More details about the process and equipment, e.g., design of major plant items, are required. The accuracy is probably −20 to +25 percent. • Definitive estimate (project control estimate). The data needed for this type of estimate are more detailed than those for a preliminary estimate and include the preparation of specifications and drawings. The probable accuracy is −10 to +15 percent. • Detailed estimate (firm estimate). Complete specifications, drawings, and site surveys for the plant construction are required, and the estimate has an accuracy of −5 to +10 percent. Detailed information requirements for each type of estimate may be found in Fig. 9-3. In periods of high inflation, the results of various estimates and accuracy may overlap. At such times, four categories may be more suitable, namely, study, preliminary, definitive, and detailed categories. At present, some companies employing the front-end loading (FEL) process for project definition and execution use three categories: Project stage

Accuracy

Conceptual Feasibility Definition

±/40% ±/25% ±/10%

For more information on the FEL process, see “Capital Project Execution and Analysis” near the end of Sec 9. Scope The scope is a document that defines a project. It contains words, drawings, and costs. A scope should answer the following questions clearly: What product is being manufactured? How much is being produced?

CAPITAL COST ESTIMATION

9-11

ESTIMATING INFORMATION GUIDE Information Either Required or Available Detailed (firm) Definitive (project control) Estimate types

Preliminary (budget authorization) Study (factored) Order of magnitude (ratio) Location General description Site survey Geotechnical report Site plot plan and contours Well-developed site facilities

• •

Rough sketches Preliminary Engineered

•

Rough sizes and construction Engineered specifications Vessel data sheets General arrangement Final arrangement

•

Buildings and structures

Rough sizes and construction Foundation sketches Architectural and construction Preliminary structural design General arrangements and elevations Detailed drawings

•

Utilities and services

Rough quantities Preliminary heat balance Preliminary flow sheets Engineered heat balance Engineered flow sheets Detailed drawings

•

Piping and insulation

Preliminary flow sheets Engineered flow sheets Piping layouts and schedules Insulation rough specifications Insulation applications Insulation details

•

Instrumentation

Preliminary list Engineered list Detail drawings •

Electrical

Rough motor list and sizes Engineered list and sizes Substation number and size Preliminary specifications Distribution specifications Preliminary interlocks and controls Engineered single-line diagrams Detailed drawings Engineering and drafting Construction supervision Craft labor Product, capacity, location, utilities, and services Building requirements, process, storage, and handling

•

Site

Process flow Equipment

Work-hours Project scope

• • • •

• • • • •

• • • • • •

•

•

• • •

• •

• • •

• • • • •

• • • •

•

• •

• • •

•

• •

•

• •

•

• •

• • •

• •

• •

• •

• •

•

• • •

• • •

•

•

•

•

•

• •

•

• • •

•

•

FIG. 9-3 Information guide for preparing estimates. (Source: Perry’s Chemical Engineers’ Handbook, 5th ed., McGraw-Hill, New York, 1973.)

What is the quality of the product? Where is the product to be produced? What is the quality of the estimate? What is the basis for the estimate? What are the knowns and unknowns with respect to the project? Before an estimate can be prepared, it is essential to prepare a scope. It may be as simple as a single page, such as for an order-of-

magnitude estimate, or several large manuals, for a detailed estimate. As the project moves forward from inception to a detailed estimate, the scope must be revised and updated to provide the latest information. Changes during the progress of a project are inevitable, but a well-defined scope prepared in advance can help minimize costly changes. If a scope is properly defined, the following results:

PROCESS ECONOMICS

An understanding between those who prepared the scope (engineering) and those who accept it (management) A document that indicates clearly what is provided in terms of technology, quality, schedule, and cost A basis in enough detail to be used in controlling the project and its costs to permit proper evaluation of any proposed changes A device to permit subsequent evaluation of the performance compared to the intended performance A document to control the detailed estimate for the final design, construction, and design Equipment Cost Data The foundation of a fixed capital investment estimate is the equipment cost data. From this information, through the application of factors or percentages based upon the estimator’s experience, the fixed capital investment is developed. These cost data are reported as purchased, delivered, or installed cost. Purchased cost is the price of the equipment FOB at the manufacturer’s plant, whereas delivered cost is the purchased price plus the delivery charge to the purchaser’s plant FOB. Installed cost means the equipment has been purchased, delivered, uncrated, and placed on a foundation in the purchaser’s operating department but does not include piping, electrical, instrumentation, insulation, etc., costs. Perhaps a better name might be set-in-place cost. It is essential to have reliable cost data since the engineer producing the estimate starts with this information and develops the fixed capital cost estimate. The estimator must know the source of the data, the basis for the data, its date, potential errors, and the range over which the data apply. There are many sources of graphical equipment cost data in the literature, but some are old and the latest published data were in the early 1990s. There have been no significant cost data published recently. To obtain current cost data, one should solicit bids from vendors; however, it is essential to impress on the vendor that the information is to be used for preliminary estimates. A disadvantage of using vendor sources is that there is a chance of compromising proprietary information. Cost-capacity plots of equipment indicate a straight-line relationship on a log-log plot. Figure 9-4 is an example of such a plot. A convenient method of presenting these data is in equation format:



S n C2 = C1 2 S1 where C1 = cost of equipment of capacity S1 C2 = cost of equipment of capacity S2

(9-1)

Purchased cost, dollars

106

105

304 SS mixing tank Carbon steel mixing tank 304 SS storage tank Carbon steel storage tank Mixing tanks include agitator and drive

FIG. 9-4

Cost-capacity plot.

0.7 0.6 0.5 0.4

Capacity FIG. 9-5

Variation of n on cost-capacity plot.

n = exponent that may vary between 0.4 and 1.2 depending on type of equipment Equation (9-1) is known as the six-tenths rule since the average value for all equipment is about 0.6. D. S. Remer and L. H. Chai (Chemical Engineering Progress, August 1990, pp. 77–82) published an extensive list of six-tenths data. Figure 9-5 shows how the exponent may vary from 0.4 to 0.9 for a given equipment item. Data accuracy is the highest in the narrow, middle-range of capacity, but at either end of the plot, the error is great. These errors occur when one correlates cost data with one independent variable when more than one variable is necessary to represent the data, or when pressure, temperature, materials of construction, or design features vary considerably. A convenient way to display cost-capacity data is by algorithms. They are readily adaptable for computerized cost estimation programs. Algorithm modifiers in equation format may be used to account for temperature, pressure, material of construction, equipment type, etc. Equation (9-2) is an example of obtaining the cost of a shell-and-tube heat exchanger by using such modifiers. (9-2)

where CHE = purchased equipment cost K = factor for cost index based upon a base year CB = base cost of a carbon-steel floating-head exchanger, 150-psig design pressure FD = design-type cost factor if different from that in CB FM = material-of-construction cost factor FP = design pressure cost factor

Jan, 1985 103 104 Capacity, gallons

0.6

0.8

CHE = KCBFDFMFP

104

103 2 10

0.9

Cost

9-12

105

Each cost factor is obtained from equations or tables from Couper, 2003, App. C, and have been updated to third-quarter 2002. Cost Indices Cost data are given as of a specific date and can be converted to more recent costs through the use of cost indices. In general, the indices are based upon constant dollars in a base year and actual dollars in a specific year. In this way, with the proper application of the index, the effect of inflation (or deflation) and price increases by multiplying the historical cost by the ratio of the present cost index divided by the index applicable in the historical year. Labor, material, construction costs, energy prices, and product prices all change at different rates. Most cost indices represent national averages, and local averages may vary considerably. Table 9-9 is a list of selected values of three cost indices of significance in the chemical process industries.

CAPITAL COST ESTIMATION Ci = (1 + f1) (1 + f2) (1 + f3) CP

TABLE 9-9

Selected Cost Indices

Year Base

M&S Index(1)a 1926 = 100

CE index(1) 1957 − 1959 = 100

Nelson-Farrar index(2) 1946 = 100

915.1 943.1 964.2 1039.1 1061.9 1089.0 1093.9 1104.2 1123.6 1124.7

357.6 358.2 368.1 381.7 389.5 394.1 394.3 395.6 402.0 457.4

1225.7 1277.3 1349.7 1418.9 1477.6 1542.7 1579.7 1642.2 1710.4 1856.1

1990 1992 1994 1996 1998 2000 2001 2002 2003 2004 (3Q)

9-13

(1) From 1990 onward, the M&S and CE indices are from Chemical Engineering magazine. (2) The Nelson-Farrar indices from 1990 onward are found in Oil and Gas Journal. a Process industry average instead of all industry average.

= (1.030) (1.042) (1.047) ($475,000) = $533,800

Equipment Sizing Before equipment costs can be obtained, it is necessary to determine equipment size from material and energy balances. For preliminary estimates, rules of thumb may be used; but for definitive and detailed estimates, detailed equipment calculations must be made. Example 3: Equipment Sizing and Costing Oil at 490,000 lb/h is to be heated from 100 to 170!F with 145,000 lb/h of kerosene initially at 390!F from another section of a plant. The oil enters at 20 psig and the kerosene at 25 psig. The physical properties are Oil—0.85 sp gr, 3.5 cP at 135!F, 0.49 sp ht Kerosene—0.82 sp gr, 0.45 cP, 0.61 sp ht Estimate the cost of an all-carbon-steel exchanger in late 2004. Assume a counterflow shell-and-tube exchanger. Solution: Energy required to heat oil stream (490,000)(0.49)(170 − 100) = 16,807,000 Btu/h

The chemical engineering (CE) index and the Marshall and Swift index are found in each issue of the magazine Chemical Engineering. The Oil and Gas Journal reports the Nelson-Farrar Refinery indices in the first issue of each quarter. The base years selected for each index are generally periods of low inflation so that the index is stable. The derivation of base values is referred to in the respective publications. A cost index is used to project a cost from a base year to another selected year. The following equation is used: index at Θ2 Cost at Θ2 = cost at Θ2  (9-3) index at Θ2





!

= 200 F 220 − 100 LMTD =  ln 2.2 = 152! F Calculate the exchanger efficiency factor, F. 170 − 100 P =  = 0.241 390 − 100



Example 1: Use of Cost Index A centrifuge cost $95,000 in 1999. What is the cost of the same centrifuge in third quarter of 2004? Use the CE index. Solution:

390 − 200 R =  = 2.71 170 − 100 From Perry F = 0.88. Since the factor must be greater than 0.75, the exchanger is satisfactory. Therefore, T = (F)(LMTD) = (0.88)(152) = 134!F. Assume UD = 50 Btu/(hft2!F).

CE index in 1999 = 390.6

Q = UD A T = 16,800,000 = (50)(A)(134)

CE index in 3d quarter 2004 = 457.4

A = 2510 ft2

Cost in 2004 = cost in 1999 (CE index in 3d quarter 2004/ CE index in 1999)





490,000 0.49 Exit kerosene temperature T = 390 −   (170 − 100) 145,000 0.61

= exp [8.821 − 0.30863(7.83) + 0.0681(61.3)] = $39,300 base cost

Inflation When costs are to be projected into the future due to inflation, it is a highly speculative exercise, but it is necessary for estimating investment costs, operating expenses, etc. Inflation is the increase in price of goods without a corresponding increase in productivity. A method for estimating an inflated cost is Ci = (1 + f1) (1 + f2) (1 + f3) CP

Use the cost algorithm cited above. CB = exp [8.821 − 0.30863 ln A + 0.0681(ln A)2]



457.4 = $95,000  = $111,200 390.6

(9-4)

where Ci = inflated cost f1 = inflation rate the first year f2 = inflation rate the second year f3 = inflation rate the third year CP = cost in a base year The assumed inflation factors f are obtained from federal economic reports, financial sources such as banks and investment houses, and news media. These factors must be reviewed periodically to update estimates. Example 2: Inflation A dryer today costs $475,000. The projected inflation rates for the next 3 years are 3, 4.2, and 4.7 percent. Calculate the projected cost in 3 years. Solution:

FD = 1.0 FP = 1.00

FM = for cs/cs material = 1.0 since this exchange is operating below 4 bar



463 K = 1.218 (CE index 4th qtr 2004/CE index 1st qtr 2003) = 1.218   406 = 1.389 Therefore, CHE = KCBFDFMFP = (1.389)(39,300)(1.0)(1.0)(1.0) = $54,600.

Estimation of Fixed Capital Investment Order-of-Magnitude Methods The ratio method will give the fixed capital investment per gross annual sales; however, most of these data are from the 1960s, and no recent data have been published. The ratio above is called the capital ratio, often used by financial analysts. The reciprocal of the capital ratio is the turnover ratio that for various businesses ranges from 4 to 0.3. The chemical industry has an average of about 0.4 to 0.5. The ratio method of obtaining fixed capital investment is rapid but suitable only for order-of-magnitude estimates. The exponential method may be used to obtain a rapid capital cost for a plant based upon existing company data or from published sources such as those of D. S. Remer and L. H. Chai, Chemical Engineering,

9-14

PROCESS ECONOMICS

April 1990, pp. 138–175. In the method known as the seven-tenths rule, the cost-capacity data for process plants may be correlated by a logarithmic plot similar to the six-tenths plot for equipment. Remer and Chai compiled exponents for a variety of processes and found that the exponents ranged from 0.6 to 0.8. When the data are used to obtain a capital cost for a different-size plant, the estimated capital must be for the same process. The equation is capacity of plant B 0.7 Cost of plant B = cost of plant A  (9-5)  capacity of plant A





Cost indices may be used to correct costs for time changes. Example 4: Seven-Tenths Rule A company is considering the manufacture of 150,000 tons annually of ethylene oxide by the direct oxidation of ethylene. According to Remer and Chai (1990), the cost capacity exponent for such a plant is 0.67. A subsidiary of the company built a 100,000-ton annual capacity plant for $70 million fixed capital investment in 1996. Using the seven-tenths rule, estimate the cost of the proposed new facility in the third quarter 2004. Solution:



Cap150 Cost150 = Cost100  Cap100

  CE index 1996 0.67

CE index 3Q 2004



 = $110,000,000  381.7

150,000 = ($70,000,000)  100,000

0.67

457.4

Study Method The single-factor method begins with collecting the delivered cost of various items of equipment and applying one factor to obtain the battery-limits (BL) fixed capital (FC) investment or total capital investment as follows: (CFC)BL = f )(CEQ)DEL where

(9-6)

(CFC)BL = battery-limits fixed capital investment or total capital investment (CEQ)DEL = delivered equipment costs

The single factors include piping, automatic controls, insulation, painting, electrical work, engineering costs, etc. (Couper, 2003). Table 9-10 shows the Lang factors for various types of processing plants. The boundaries between the classifications are not clear-cut, and considerable judgment is required in the selection of the appropriate factors. Preliminary Estimate Methods A refinement of the Lang factor method is the Hand method. The Hand factors are found in Table 9-11. Equipment is grouped in categories, such as heat exchangers and pumps, and then a factor is applied to each group to obtain the installed cost; finally the groups are summed to give the battery-limits installed cost. Wroth compiled a more detailed list of installation factors; a selection of these can be found in Table 9-12. The Lang and Hand methods start with purchased equipment costs whereas the Wroth method begins with delivered equipment costs, so delivery charges must be included in the Lang and Hand methods. At best the Lang and Hand methods will yield study quality estimates, and the Wroth method might yield a preliminary quality estimate. Example 5: Fixed Capital Investment Using the Lang, Hand, and Wroth Methods The following is a list of the purchased equipment costs for a proposed processing unit: Heat exchangers Distillation towers and internals Receivers TABLE 9-10

$620,000 975,000 320,000

Lang Factors Lang factors

Type of plant

Fixed capital investment

Total capital investment

Solid processing Solid-fluid processing Fluid processing

4.0 4.3 5.0

4.7 5.0 6.0

Adapted from M. S. Peters, K. D. Timmerhaus, and R. West, Plant Design and Economics for Chemical Engineers, 5th ed., McGraw-Hill, New York, 2004.

TABLE 9-11

Hand Factors

Equipment type

Factor

Fractionating columns Pressure vessels Heat exchangers Fired heaters Pumps Compressors Instruments Miscellaneous equipment

4.0 4.0 3.5 2.0 4.0 2.5 4.0 2.5

Adapted from W. E. Hand, Petroleum Refiner, September 1958, pp. 331–334.

Accumulator drum Pumps and motors Automatic controls Miscellaneous equipment

125,000 220,000 275,000 150,000

Assume delivery charges are 5 percent of the purchased price. Estimate the fixed capital investment 2 years into the future, using the Lang, Hand, and Wroth methods. The inflation rates are 3.5 percent for the first year and 4.0 percent for the second. Solution:

Equipment

Purchased equipment cost

Delivered equipment cost

Heat exchangers Distillation towers, internals Receivers Accumulator drum Pumps and motors Automatic controls Miscellaneous equipment Total

$620,000 975,000 320,000 125,000 220,000 275,000 150,000 $2,685,000

$651,000 1,024,000 336,000 131,000 231,000 289,000 158,000 $2,820,000

Lang method: The Lang factor for a fluid processing unit starting with purchased equipment costs is 5.0. Therefore, fixed capital investment is $2,820,000 × 5.0 × 1.035 × 1.040 = $15,177,000. Hand method: The Hand method begins with purchased equipment costs, and factors are applied from Table 9-11.

TABLE 9-12

Selected Wroth Factors Equipment

Factor

Blender Blowers and fans Centrifuge Compressors Centrifugal (motor-driven) Centrifugal (steam-driven, including turbine) Reciprocating (steam and gas) Reciprocating (motor-driven less motor) Ejectors, vacuum Furnaces (packaged units) Heat exchangers Instruments Motors, electric Pumps Centrifugal (motor-driven less motor) Centrifugal (steam-driven including turbine) Positive-displacement (less motor) Reactors (factor as appropriate, equivalent-type equipment) Refrigeration (packaged units) Tanks Process Storage Fabricated and field-erected 50,000+ gal Towers (columns)

2.0 2.5 2.0 2.0 2.0 2.3 2.3 2.5 2.0 4.8 4.1 3.5 7.0 6.5 5.0 — 2.5 4.1 3.5 2.0 4.0

Abstracted from W. F. Wroth, Chemical Engineering, October 17, 1960, p. 204.

CAPITAL COST ESTIMATION Hand method: Purchased equipment cost

Equipment Heat exchangers Distillation towers, internals Receivers Accumulator drum Pumps and motors Automatic controls Miscellaneous TOTAL

$620,000 975,000 320,000 125,000 220,000 275,000 150,000 $2,685,000

Hand factor 3.5 4.0 2.5* 2.5* 4.0 4.0 2.5

Purchased equipment installed cost $2,170,000 3,900,000 800,000 313,000 880,000 1,100,000 375,000 $9,538,000

The asterisk on the receivers and accumulators indicates that if these vessels are pressure vessels, a factor of 4.0 should be used instead of 2.5. The total purchased equipment installed is $9,538,000 for non–pressure vessels and the delivered cost is $10,015,000. Therefore, the fixed capital investment installed would be $10,015,000 × 1.035 × 1.040 = $10,780,000. Using pressure vessels increases the total purchased equipment cost $667,000; therefore, the fixed capital investment for this case including inflation would be $10,780,000 × 1.05 × 1.035 × 1.04 = $11,534,000. Wroth method:

Equipment

Delivered equipment cost

Heat exchangers Distillation towers, internals Receivers Accumulator drum Pumps and motors Automatic controls Miscellaneous TOTAL

$651,000 1,024,000 336,000 131,000 231,000 289,000 158,000 $2,820,000

Wroth factor 4.8 4.0 3.5 3.5 7.0 4.1 4.0 (assumed)

Delivered equipment installed cost $3,125,000 4,096,000 1,176,000 459,000 1,617,000 1,185,000 632,000 $12,290,000

The total delivered installed equipment cost is the fixed capital investment and, corrected for 2 years of inflation, will be $12,290,000 × 1.035 × 1.040 = $13,229,000. Therefore, the summary of the fixed capital investment by the various methods is Lang $15,177,000 Hand 11,534,000 Wroth 13,229,000

Experience has shown that the fixed capital investment by the Lang method is generally higher than that of the other methods. Whatever figure is reported to management, it is advisable to state the potential accuracy of these methods. Brown developed guidelines for the preparation of order-of-magnitude and study capital cost estimates based upon the Lang and Hand methods. Brown modified Lang and Hand methods for materials of construction, instrumentation, and location factors. He found that the modified Hand and Garrett module factor methods gave results within 3.5 percent. TABLE 9-13

9-15

Other multiple-factor methods that have been published in the past are those by C. E. Chilton, Cost Estimation in the Process Industries, McGraw-Hill, New York, 1960; M. S. Peters, K. D. Timmerhaus, and R. E. West, Plant Design and Economics for Chemical Engineers, 5th ed., McGraw-Hill, New York, 2003; C. A. Miller, Chemical Engineering, Sept. 13, 1965, pp. 226–236; and F. A. Holland, F. A. Watson, and V. K. Wilkinson, Chem. Eng., Apr. 1, 1974, pp. 71–76. These methods produced preliminary quality estimates. Most companies have developed their own in-house multiple-factor methods for preliminary cost estimation. Step-counting methods are based upon a number of processing steps or “functional units.” The concept was first introduced by H. E. Wessel, Chem. Eng., 1952, p. 209. Subsequently, R. D. Hill, Petrol. Refin., 35(8):106–110, August 1956; F. C. Zevnik and R. L. Buchanan, Chem. Eng. Progress, 59(2):70–77, February 1963; and J. H. Taylor, Eng. Process Econ., 2:259–267, 1977, further developed the stepcounting method. A step or functional unit is a significant process step including all process equipment and ancillary equipment necessary for operating the unit. A functional unit may be a unit operation, unit process, or separation in which mass and energy are transferred. The sum of all functional units is the total fixed capital investment. Pumping and heat exchangers are considered as part of a functional unit. In-process storage is generally ignored except for raw materials, intermediates, or products. Difficulties are encountered in applying the method due to defining a step. This takes practice and experience. If equipment has been omitted from a step, the resulting estimate is seriously affected. These methods are reported to yield estimates of study quality or at best preliminary quality. Definitive Estimate Methods Modular methods are an extension of the multiple-factor methods and have been proposed by several authors. One of the most comprehensive methods and one of the earliest was that of K. M. Guthrie, Chem. Eng., 76:114–142, Mar. 24, 1969. It began with equipment FOB equipment costs, and through the use of factors listed in Table 9-13, the module material cost was obtained. Labor for erection and setting equipment was added to the material cost as well as indirect costs for freight, insurance, engineering, and field expenses to give a total module cost. Such items as contingencies, contractors’ fees, auxiliaries, site development land, and industrial buildings were added if applicable. Since any plant consists of equipment modules, these are summed to give the total fixed capital investment. Unfortunately, the factors and data are old but the concept is useful. Garrett (1989) developed a similar method based upon a variety of equipment modules, starting with purchased equipment costs obtained from plots and applying factors for materials of construction, instrumentation, and plant location. The method provides for all supporting and connecting equipment to make the equipment installation operational. See Table 914. T. R. Brown, Hydrocarbon Processing, October 2000, pp. 93–100, made modifications to the Garrett method. Another method, called the discipline method, mentioned by L. R. Dysert, Cost Eng. 45(6), June 6, 2003, is similar to the models of

Guthrie Method Factors* Exchangers Details

Furnaces

FOB equipment Piping Concrete Steel Instruments Electrical Insulation Paint Total materials = M Erection and setting (L) +, excluding site preparation and auxiliaries (M + L) Freight, insurance, taxes, engineering, home office, construction Overhead or field expense Total module factor

1.00 0.18 0.10 0.04 0.02 1.34 0.30 1.64 0.60 2.24

Shell and tube 1.00 0.46 0.05 0.03 0.10 0.02 0.05 1.71 0.63 2.34 0.08 0.95 3.37

Vessels Air-cooled

Vertical

Horizontal

Pump and driver

Compressor and driver

1.00 0.18 0.02

1.00 0.61 0.10 0.08 0.12 0.05 0.08 0.01 2.05 0.95 3.00 0.08 1.12 4.20

1.00 0.42 0.06

1.00 0.30 0.04

1.00 0.21 0.12

0.06 0.05 0.05 0.01 1.65 0.59 2.24 0.08 0.92 3.24

0.03 0.31 0.03 0.01 1.72 0.70 2.42 0.08 0.97 3.47

0.08 0.16 0.03 0.01 1.61 0.58 2.19 0.08 0.97 3.24

0.05 0.12 0.01 1.38 0.38 1.76 0.70 2.46

*From K. M. Guthrie, Chem. Eng., 76, 114–142 (Mar. 24, 1969). Based on FOB equipment cost = 100 (carbon steel).

Tanks 1.00

1.20 0.13 1.33 0.08 1.41

9-16

PROCESS ECONOMICS

TABLE 9-14

Selected Garrett Module Factors

Equipment type (carbon steel unless otherwise noted) Agitators: dual-bladed turbines/single-blade propellers Agitated tanks Air conditioning Blender, ribbon Blowers, centrifugal Centrifuges: solid-bowl, screen-bowl, pusher, stainless steel Columns: distillation, absorption, etc. Horizontal Vertical Compressors: low-, medium-, high-pressure Coolers, quenchers Crystallizers Drives/motors Electric, for fans, compressors, pumps Electric for other units Gasoline Turbine: gas and steam Dryers Fluid bed, spray Rotary Dust collectors Bag filters Cyclones, multiclones Evaporators, single-effect stainless steel Falling film Forced circulation Fans Filters Belt, rotary drum and leaf, tilting pan Others Furnaces Heat exchangers Air-cooled Double-pipe Shell-and-tube Mills Hammer Ball, rod Pumps Centrifugal Reciprocating Turbine Reactors, jacketed, no agitator 304 SS Glass-lined Mild steel Vacuum equipment SOURCE:

TABLE 9-15 Module factor 2.0 2.5 1.46 2.0 2.5 2.0 3.05 4.16 2.6 avg. 2.7 2.6 avg. 1.5 2.0 2.0 3.5 2.7 2.3 2.2 3.0 2.3 2.9 2.2 2.4 2.8 2.1 2.2 1.8 3.2 2.8 2.3 avg. 5.0 3.3 1.8 1.8 2.1 2.3 2.2

Adapted from Garrett (1989).

Direct capital cost account titles

010 020 030 040 050 060 070 080 090 100 110 120 130 140 150 160 170 180 190 200 210 220 230

Equipment items Instrument items Setting and testing equipment Setting and testing instruments Piling Excavation Foundations Supports, platforms, and structures Other building items Fire protection and sprinklers Piping Ductwork Electrical and wiring Site preparation Sewers, drains, and plumbing Underground piping Yards, roads, and fencing Railroads Insulation Painting Walls, masonry, roofs, and roofing Spares Lump-sum contracts

500 510 530 550 580 670 740 750 760 790 800 810 870 880 890

Site burden Direct labor burden Construction equipment, tools, and supplies Rental and servicing construction equipment and tools Premium wages and overtime—contractor Temporary facilities Cancellation charges Abandoned design Self-insured losses Unvouchered liabilities In-house engineering Outside engineering Undeveloped design allowances Distributives transferred to expense Contingencies—capital items

900 910 920 930 940 990

Dismantling Sales and use taxes Repairs expense Relocation and modification expense Start-up relocation and modification expense Contingencies

Distributives

Expense

SOURCE:

Guthrie and Garrett. It uses equipment factors to generate separate costs for each of the “disciplines” associated with the installation of equipment, such as installation labor, concrete, structural steel, and piping, to obtain direct field costs for each type of equipment, e.g., heat exchangers, towers, and reactors. Modular methods, depending on the amount of detail provided, will yield preliminary quality estimates. Detailed Estimate Method For estimates in the detailed category, a code of account needs to be used to prevent oversight of certain significant items in the capital cost. See Table 9-15. Each item in the code is estimated and provides the capital cost estimate; then this estimate serves for cost control during the construction phase of a project. Comments on Significant Cost Items Piping This cost includes the cost of the pipe, installation labor, valves, fittings, supports, and miscellaneous items necessary for complete installation of all pipes in the process. The accuracy of the estimates can be seriously in error by the improper application of estimating techniques to this component. Many pipe estimating methods are extant in the literature. Two general methods have been used to estimate piping costs when detailed flow sheets are not available. One method is to use a percentage of the FOB equipment costs or a percentage of the fixed capital investment. Typical figures are 80 to 100 percent of the FOB equipment costs or 20 to 30 percent of the fixed capital investment. This method is used

Code of Accounts

Category number

Private communication.

for preliminary estimates. Another group of methods such as the Dickson “N” method (R. A. Dickson, Chem. Eng., 57:123–135, Nov. 1947), estimating by weight, estimating by cost per joint, etc., requires a detailed piping takeoff from either PID or piping drawings with piping specifications, material costs, labor expenses, etc. These methods are used for definitive or detailed estimates where accuracy of 10 to 15 percent is required. The takeoff methods must be employed with great care and accuracy by an experienced engineer. A detailed breakdown by plant type for process piping costs is presented in Peters et al. (2003) and in Perry’s Chemical Engineers’ Handbook, 6th ed., 1984. Electrical This item consists of transformers, wiring, switching gear, as well as instrumentation and control wiring. The installed costs of the electrical items may be estimated as 20 to 40 percent of the delivered equipment costs or 5 to 10 percent of the fixed capital investment for preliminary estimates. As with piping estimation, the process design must be well along toward completion before detailed electrical takeoffs can be made. Buildings and Structures The cost of the erection of buildings and structures in a chemical process plant as well as the plumbing, heating and ventilation, and miscellaneous building service items may be estimated as 20 to 50 percent of delivered equipment costs or as 10 to 20 percent of the fixed capital investment for a preliminary estimate.

CAPITAL COST ESTIMATION Yards, Railroad Sidings, Roads, etc. This investment includes roads, railroad spurs, docks, and fences. A reasonable figure for preliminary estimates is 15 to 20 percent of the FOB equipment cost or 3 to 7 percent of the fixed capital investment for a preliminary estimate. Service Facilities For a process plant, utility services such as steam, water, electric power, fuel, compressed air, shop facilities, and a cafeteria require capital expenditures. The cost of these facilities lumped together may be 10 to 20 percent of the fixed capital investment for a preliminary estimate. (Note: Buildings, yards, and service facilities must be well defined to obtain a definitive or detailed estimate.) Environmental Control and Waste Disposal These items are treated as a separate expenditure and are difficult to estimate due to the variety and complexity of the process requirements. Pollution control equipment is generally included as part of the process design. Couper (2003) and Peters et al. (2003) mention that at present there are no general guidelines for estimating these expenditures. Computerized Cost Estimation With the advent of powerful personal computers (PCs) and software packages, capital cost estimates advanced from large mainframe computers to the PCs. The reasons for using computer cost estimation and economic evaluation packages are time saved on repetitive calculations and reduction in mathematical errors. Numerous computer simulation software packages have been developed over the past two decades. Examples of such software are those produced by ASPEN, ICARUS, CHEMCAD, SUPERPRO, PRO II, HYSYS, etc.; but most do not contain cost estimation software packages. ICARUS developed a PC cost estimation and economic evaluation package called Questimate. This system built a cost estimate from design and equipment cost modules, bulk items, site construction, piping and ductwork, buildings, electrical equipment, instruments, etc., developing worker-hours for engineering and fieldwork costs. This process is similar to quantity takeoff methods to which unit costs are applied. A code of accounts is also provided. ASPEN acquired ICARUS in 2000 and developed Process Evaluator based on Questimate that is used for conceptual design, known as front-end loading (FEL). More information on FEL and valueimproving process (VIP) is found later in Sec. 9. Basic and detailed estimates are coupled with a business decision framework in ASPENTECH ICARUS 2000. EstPro is a process plant cost estimation package for conceptual cost estimation for conceptual design only. It may be obtained from Gulf Publishing, Houston, Tex. Many companies have developed their own factored estimates using computer spreadsheets based upon their in-house experience and cost database information that they have built from company project history. For detailed estimates, the job is outsourced to designconstruction companies that have the staff to perform those estimates. Whatever package is used, it is recommended that computergenerated costs be spot-checked for reasonable results using a handheld calculator since errors do occur. Some commercial software companies will develop cost estimation databases in cooperation with a company for site-specific costs. Contingency This is a provision for unforeseen events that experience has demonstrated are likely to occur. Contingencies are of two types: process and project contingency. In the former, there are uncertainties in Equipment and performance Integration of old and new process steps Scaling up to a large-scale plant size Accurate definition of certain process parameters, such as severity of process conditions, number of recycles, process blocks and equipment, multiphase streams, and unusual separations No matter how much time and effort are spent preparing estimates, there is a chance of errors occurring due to Engineering errors and omissions Cost and labor rate changes Construction problems Estimating inaccuracies Miscellaneous “unforeseens” Weather-related problems Strikes by fabricators, transportation, and construction personnel

9-17

For preliminary estimates, a 15 to 20 percent project contingency should be applied if the process information is firm. As the quality of the estimate moves to definitive and detailed, the contingency value may be lowered to 10 to 15 percent and 5 to 10 percent, respectively. Experience has shown that the smaller the dollar value of the project, the higher the contingency should be. Offsite Capital These facilities include all structures, equipment, and services that do not enter into the manufacture of a product but are important to the functioning of the plant. Such capital items might be steam-generating and electrical-generating and distribution facilities, well-water cooling tower, and pumping stations for water distribution, etc. Service capital might be auxiliary buildings, such as warehouses, service roads, railroad spurs, material storage, fire protection equipment, and security systems. For estimating purposes, the following percentages of the fixed capital investment might be used: Small modification of offsites, 1 to 5 percent Restructuring of offsites, 5 to 15 percent Major expansion of offsites, 15 to 45 percent Grass-roots plants, 45 to 150 percent Allocated Capital This is capital that is shared due to its proportionate share use in a new facility. Such items include intermediate chemicals, utilities, services and sales, administration, research, and engineering overhead. Working Capital Working capital is the funds necessary to conduct day-to-day company business. These are funds required to purchase raw materials, supplies, etc. It is continuously liquidated and rejuvenated from the sale of products or services. If an adequate amount of working capital is available, management has the necessary flexibility to cover expenses in case of strikes, delays, fires, etc. Several methods are available for estimating an adequate amount of working capital. They may be broadly classified into percentage and inventory methods. The percentage methods are satisfactory for study and preliminary capital estimates. The percentage methods are of two types: percentage based on capital investment and percentage based upon sales. In the former method, 15 to 25 percent of the total capital investment may be sufficient for preliminary estimates. In the case of certain specialty chemicals where the raw materials are expensive, it is perhaps better to use the percentage of sales method. Such chemicals as flavors, fragrances, perfumes, etc., are in this category. Experience has shown that 15 to 45 percent of sales has been used with 30 to 35 percent being a reasonable average value. Start-up Expenses Start-up expenses are defined as the total costs directly related to bringing a new manufacturing facility onstream. Start-up time is the time span between the end of construction and the beginning of normal operation. Normal operation is operation at a certain percentage of design capacity or a specified number of days of continuous operation or the ability to make product of a specified purity. Start-up costs are part of the total capital investment and include labor, materials, and overhead for design modifications or changes due to errors on the part of engineering, contractors, costs of tests, final alterations and adjustments. These items cannot be included as contingency because it is known that such work will be necessary before the project is completed. Experience has shown that start-up costs are a percentage of the battery-limits fixed capital investment of the order on average of 3 percent. Depending on the tax laws in effect, not all start-up costs can be expensed and a portion must be capitalized. Start-up costs can reduce the after-tax earnings during the early years of a project because of a delay in the start-up of production causing a loss of earnings. Construction changes are items of capital cost, and production start-up costs are expensed as an operating expense. Other Capital Items Paid-up royalties and licenses are considered part of the capital investment since these are replacements for capital to perform process research and development. The initial catalyst and chemical charge, especially for noble metal catalysts and/or in electrolytic processes, is a large amount. These materials are considered to have a life of 1 year. If funds must be borrowed for a new facility, then the interest on borrowed funds during the construction period is capitalized; otherwise, the interest is part of the operating expense.

9-18

PROCESS ECONOMICS

MANUFACTURING-OPERATING EXPENSES The estimation of manufacturing expenses has received less attention in the open literature than the estimation of capital requirements. Operating expenses are estimated from proprietary company files. In this section, methods for estimating the elements that constitute operating expenses are presented. Operating expenses consist of the expense of manufacturing a product, packaging and shipping, as well as general overhead expense. These are described later in this section. Figure 9-6 shows an example of a typical manufacturing expense sheet.

labor rates and any potential labor rate increases in the near future. One should not forget shift differential and overtime charges. Once the number of operators per shift has been established, the annual labor expense and unit expense may be estimated. Wessel (Chem. Eng., 59:209–210, July 1952) developed a method for estimating labor requirements for various types of chemical processes in the United States. The equation is applicable for a production rate of 2 to 2000 tons/day (2000 lb/ton). log Y = −0.783 log X + 1.252 + B

RAW MATERIAL EXPENSE Estimates of the amount of raw material consumed can be obtained from the process material balance. Normally, the raw material expense is the largest expense item in the manufacture of a product. Since yields in a chemical reaction determine the quantity of raw materials consumed, assumed yields may be used to obtain approximate exploratory estimates if possible ranges are given. The prices of the raw materials are published in various trade journals that list material according to form, grade, method of delivery, unit of measure, and cost per unit. The Chemical Marketing Reporter is a typical source of these prices. The prices are generally higher than quotations from suppliers, and these latter should be used whenever possible. It may be possible for a company to negotiate the price of a raw material based upon large-quantity use on a long-term basis. With the amount of material used from the material balance and the price of the raw material, the following information can be obtained: annual material consumption, annual material expense, as well as the consumption and expense per unit of product. Occasionally, by-products may be produced, and if there is a market for these materials, a credit can be given. By-products are treated in the same manner as raw materials and are entered into the manufacturing expense sheet as a credit. If by-products are intermediates for which no market exists, they may be credited to downstream or subsequent operations at a value equivalent to their value as a replacement, or no credit may be obtained. DIRECT EXPENSES These are the expenses that are directly associated with the manufacture of a product, e.g., utilities, labor, and maintenance. Utilities The utility requirements are obtained from the material and energy balances. Utilities include steam, electricity, cooling water, fuel, compressed air, and refrigeration. The current utility prices can be obtained from company plant accounting or from the plant utility supervisor. This person might be able to provide information concerning rate prices for the near future. As requirements increase, the unit cost declines. If large incremental amounts are required, e.g., electricity, it may be necessary to tie the company’s utility line to a local utility as a floating source. With the current energy demands increasing, the unit costs of all utilities are increasing. Any prices quoted need to be reviewed periodically to determine their effect on plant operations. A company utility supervisor is a good source of future price trends. Unfortunately, there are no shortcuts for estimating and projecting utility prices. Utilities are the third largest expense item in the manufacture of a product, behind raw materials and labor. Operating Labor The most reliable method for estimating labor requirements is to prepare a table of shift, weekend, and vacation coverage. For round-the-clock operation of a continuous process, one operator per shift requires 4.2 operators, if it is assumed that 21 shifts cover the operation and each operator works five, 8-h shifts per week. For batch or semicontinuous operation, it is advisable to prepare a labor table, listing the number of tasks and the number of operators required per task, paying particular attention to primary processing steps such as filtration and distillation that may have several items of equipment per step. Labor rates may be obtained from the union contract or from a company labor relation supervisor. This person will know the current

where

(9-7)

Y = operating labor, operator h/ton per processing step X = plant capacity, tons/day B = constant depending upon type of process + 0.132 (for batch operations that have minimum labor requirements) + 0 (for operations with average labor requirements) − 0.167 (for a well-instrumented continuous process)

A processing step is one in which a unit operation occurs; e.g., a filtration step might consist of a feed (precoat) tank, pump, filter, and receiver so a processing step may have several items of equipment. By using a flow sheet, the number of processing steps may be counted. The Wessel equation does not take into account changes in labor productivity, but this information can be obtained from each issue of Chemical Engineering. Labor productivity varies widely in various sections of this country but even more widely in foreign countries. Ulrich (1984) developed a table for estimating labor requirements from flow sheets and drawings of the process. Consideration is given to the type and arrangement of equipment, multiplicity of units, and amount of process control equipment. This method is easier to use than the Wessel method and has been updated in a new edition of the original text. Supervision The approximate expense for supervision of operations depends on process complexity, but 15 to 30 percent of the operating labor expense is reasonable. Payroll Charges This item includes workers’ compensation, social security premiums, unemployment taxes, paid vacations, holidays, and some part of health and dental insurance premiums. The figure has steadily declined from 1980 and now is 30 to 40 percent of operating labor plus supervision expenses. Maintenance The maintenance expense consists of two components, namely, materials and labor, approximately 60 and 40 percent, respectively. Company records are the best information sources, however, a value of 6 to 10 percent of the fixed capital investment is a reasonable figure. Processes with a large amount of rotating equipment or that operate at extremes of temperature and/or pressure have higher maintenance requirements. Miscellaneous Direct Expenses These items include operating supplies, clothing and laundry, laboratory expenses, royalties, environmental control expenses, etc. Item Operating supplies Clothing and laundry Laboratory expenses Royalties and patents

Basis

Percentage

Operating labor Operating labor Operating labor Sales

5–7 10–15 10–20 1–5

Environmental Control Expense Wastes from manufacturing operations must be disposed of in an environmentally acceptable manner. This direct expense is borne by each manufacturing department. Some companies have their own disposal facilities, or they may contract with a firm that handles the disposal operation. However the wastes are handled, there is an expense. Published data are found in the open literature, some of which have been published by Couper (2003).

MANUFACTURING-OPERATING EXPENSES TOTAL OPERATING EXPENSES ======================================================================================================= PRODUCT: PLASTICIZER X TOTAL SALES ($/YR): 7200000 RATED CAPACITY (MM LBS/YR): 12 LOCATION: FIXED CAPITAL INVESTMENT: 800000 LAND 25000 WORKING CAPITAL 120000 OPERATING HOURS (HRS/YR): DATE: BY: RAW MATERIALS: MATERIAL A AND B

UNIT LB

ANNUAL QUANTITY 12000000

$/UNIT .23

$/YEAR 2760000 0 0 0 2760000

UNIT

ANNUAL QUANTITY

$/UNIT

BY-PRODUCT CREDIT

$/YEAR 0 0 0

NET MATERIAL EXPENSE

2760000

GROSS MATERIAL EXPENSE BY-PRODUCTS:

DIRECT EXPENSES: UNIT UTILITIES: steam, low pressure steam, medium pressure steam, high pressure GROSS STEAM EXPENSE STEAM CREDIT NET STEAM EXPENSE electricity cooling water fuel gas other: city water TOTAL UTILITIES COST LABOR: men per shift annual labor rate per shift TOTAL LABOR COSTS SUPERVISION: % total of labor expense SUPERVISION EXPENSE= PAYROLL CHARGES, FRINGE BENEFITS: % total of labor expense PAYROLL EXPENSE MAINTENANCE % of fixed capital investment MAINTENANCE EXPENSE SUPPLIES: % of operating labor SUPPLIES EXPENSE LABORATORY: laboratory hours per year cost per hour TOTAL LABORATORY EXPENSE FIG. 9-6

Total operating expense sheet.

ANNUAL QUANTITY

$/UNIT

LB

60000000

.003

KWH GALLONS

3000000 72000000

.035 .000045

GALLONS

360000000

.0002

4 25000

$/YEAR 0 0 180000 180000 0 180000 105000 3240 0 0 72000 540240

25000 100000 0 18000

40 47200 8 64000 0 1800 900 30 27000

9-19

9-20

PROCESS ECONOMICS

ROYALTIES WASTE DISPOSAL: tons per year waste charge per ton WASTE DISPOSAL EXPENSE OTHER: laundry

0 0 0 6000

6000

TOTAL DIRECT EXPENSE TOTAL DIRECT + NET MATERIAL COSTS INDIRECT EXPENSES: DEPRECIATION % of fixed capital investment life of project (yrs) DEPRECIATION PLANT INDIRECT EXPENSES % of fixed capital investment PLANT INDIRECT EXPENSES

804240 3564240

100 7 114000 5 40000

TOTAL INDIRECT EXPENSES TOTAL MANUFACTURING EXPENSE: PACKAGING, SHIPPING EXPENSE rated capacity per dollars per unit PACKAGING AND SHIPPING EXPENSE TOTAL PRODUCTION EXPENSE GENERAL OVERHEAD EXPENSES percent of annual sales GENERAL OVERHEAD EXPENSES TOTAL OPERATING EXPENSE FIG. 9-6

154000 3718240 12000000 .005 60000 3778240 5 360000 4138240

(Continued)

INDIRECT EXPENSES These indirect expenses consist of two major items; depreciation and plant indirect expenses. Depreciation The Internal Revenue Service allows a deduction for the “exhaustion, wear and tear and normal obsolescence of equipment used in the trade or business.” (This topic is treated more fully later in this section.) Briefly, for manufacturing expense estimates, straight-line depreciation is used, and accelerated methods are employed for cash flow analysis and profitability calculations. Plant Indirect Expenses These expenses cover a wide range of items such as property taxes, personal and property liability insurance premiums, fire protection, plant safety and security, maintenance of plant roads, yards and docks, plant personnel staff, and cafeteria expenses (if one is available). A quick estimate of these expenses based upon company records is on the order of 2 to 4 percent of the fixed capital investment. Hackney presented a method for estimating these expenses based upon a capital investment factor, and a labor factor, but the result is high.

various amounts to numerous destinations. Often these expenses come under the heading of freight allowed in the sale of a product. TOTAL PRODUCT EXPENSE The sum of the total manufacturing expense and the packaging and inplant shipping expense is the total product expense. GENERAL OVERHEAD EXPENSE This expense is often separated from the manufacturing expenses. It includes the expense of maintaining sales offices throughout the country, staff engineering departments, and research and development facilities and administrative offices. All manufacturing departments are expected to share in these expenses so an appropriate charge is made for each product varying between 6 and 15 percent of the product’s annual revenue. The wide range in percentage will vary depending on the amount of customer service required due to the nature of the product.

TOTAL MANUFACTURING EXPENSE

TOTAL OPERATING EXPENSE

The total manufacturing expense for a product is the sum of the raw materials and direct and indirect expenses.

The sum of the total product expense and the general overhead expense is the total operating expense. This item ultimately becomes part of the operating expense on the income statement.

PACKAGING AND SHIPPING EXPENSES The packaging expense depends on how the product is sold. The package may vary from small containers to fiberpacks to leverpacks, or the product may be shipped via tank truck, tank car, or pipeline. Each product must be considered and the expense of the container included on a case-by-case basis. The shipping expense includes the in-plant movement to warehousing facilities. Product delivery expenses are difficult to estimate because products are shipped in

RAPID MANUFACTURING EXPENSE ESTIMATION Holland et al. (1953) developed an expression for estimating annual manufacturing expenses for production rates other than the base case based upon fixed capital investment, labor requirements, and utility expense. A1 = mCfci + ncL N1 + pU1

(9-8)

FACTORS THAT AFFECT PROFITABILITY

TABLE 9-16 Typical Labor Requirements for Various Equipment

1.0 Log conversion expense, $/lb

Equipment

0.1

1.0

10.0

Log annual production, 106 lb FIG. 9-7

9-21

Laborers per unit per shift

Blowers and compressor Centrifuge Crystallizer, mechanical Dryers Rotary Spray Tray Evaporator Filters Vacuum Plate and frame Rotary and belt Heat exchangers Process vessels, towers (including auxiliary pumps and exchangers) Reactors Batch Continuous

0.1–0.2 0.25–0.50 0.16 0.5 1.0 0.5 0.25 0.125–0.25 1.0 0.1 0.1 0.2–0.5 1.0 0.5

Adapted from G. D. Ulrich, A Guide to Chemical Engineering Process Design and Economics, Wiley, New York, 1984.

Annual conversion expense as a function of production rate.

where Cfci = fixed capital investment, $ CL = cost of labor, $ per operator per shift N1 = annual labor requirements, operators/shift/year at rate 1 U1 = annual utility expenses at production rate 1 A1 = annual conversion expense at rate 1 m, n, p = constants obtained from company records in consistent units Equation (9-8) can be modified to include raw materials by adding a term qM1, where q = a constant and M1 = annual raw material expense at rate 1. SCALE-UP OF MANUFACTURING EXPENSES If it is desired to estimate the annual manufacturing expense at some rate other than a base case, the following modification may be made:









R 0.7 R 0.25 R2 R A2 = mCfci 2 + nCLN1 2 + pU1  + qM1 2 (9-9) R1 R1 R1 R1 where A2 = annual manufacturing expense at production rate 2 R1 = production rate 1 R2 = production rate 2 Equation (9-9) may also be used to calculate data for a plot of manufacturing expense as a function of annual production rate, as shown in Fig. 9-7. Plots of these data show that the manufacturing expense per unit of production decreases with increasing plant size. The first term in Eq. (9-9) reflects the increase in the capital investment by using the 0.7 power for variations in production rates. Labor varies as the 0.25 power for continuous operations based upon experience. Utilities and raw materials are essentially in direct proportion to the amount of product manufactured, so the exponent of these terms is unity.

FACTORS THAT AFFECT PROFITABILITY DEPRECIATION According to the Internal Revenue Service (IRS), depreciation is defined as an allowance for the decrease in value of a property over a period of time due to wear and tear, deterioration, and normal obsolescence. The intent is to recover the cost of an asset over a period of time. It begins when a property is placed in a business or trade for the production of income and ends when the asset is retired from service or when the cost of the asset is fully recovered, whichever comes first. Depreciation and taxes are irrevocably tied together. It is essential to be aware of the latest tax law changes because the rules governing depreciation will probably change. Over the past 70 years, there have been many changes in the tax laws of which depreciation is a major component. Couper (2003) discussed the history and development of depreciation accounting. Accelerated depreciation was introduced in the early 1950s to stimulate investment and the economy. It allowed greater depreciation rates in the early years of a project when markets were not well established, manufacturing facilities were coming onstream, and expenses were high due to bringing the facility up to design capacity. The current methods for determining annual depreciation charges are the straight-line depreciation and the Modified Accelerated Cost Recovery System (MACRS). In the straight-line method, the cost of an asset is distributed over its expected useful life such that the annual charge is I+S D =  $ per year n

(9-10)

where D = annual depreciation charge I = investment n = number of years S = salvage value The MACRS went into effect in January 1987 (Couper, 2003) with six asset recovery periods: 3, 5, 7, 10, 15, and 20 years. It is based upon the declining-balance method. The equation for the declining-balance method is Ve = Vi (1 − f )

(9-11)

where Vi = value of asset at beginning of year Ve = value of asset at end of year f = declining-balance factor For 150 percent declining balance f = 1.5, and for 200 percent f = 2.0. These factors are applied to the previous year’s remaining balance. It is evident that the declining-balance method will not recover the asset that the IRS permits. Therefore, a combination of the declining-balance and straight-line methods forms the basis for the MACRS. Class lives for selected industries are found in Couper (2003), but most chemical processing equipment falls in the 5-year category and petroleum processing equipment in the 7-year category. For those assets with class lives less than 10 years, a 200 percent declining-balance

9-22

PROCESS ECONOMICS

TABLE 9-17 Depreciation Class Lives and MACRS Recovery Periods Asset class 00.12 00.4 13.3 20.3 20.5 22.4 22.5 26.1 28.0 30.1 30.2 32.1 32.2 32.3 33.2 32.4 49.223 49.25

Class life, yr

Description of asset Information systems Industrial steam and electric generation and/or distribution systems Petroleum refining Manufacture of vegetable oils and vegetable oil products Manufacture of food and beverages Manufacture of textile yams Manufacture of nonwoven fabrics Manufacture of pulp and paper Manufacture of chemicals and allied products Manufacture of rubber products Manufacture of finished plastic products Manufacture of glass products Manufacture of cement Manufacture of other stone and clay products Manufacture of primary nonferrous metals Manufacture of primary steel mill products Substitute natural gas-coal gasification Liquefied natural gas plant

MACRS recovery period, yr

6 22

5 15

16 18

10 10

4 8 10 13 9.5

3 5 7 7 5

14 11 14 20 15

7 7 7 15 7

14

7

15

7

18 22

10 15

DEPLETION Depletion is concerned with the diminution of natural resources. Generally depletion does not enter into process economic studies. Rules for determining the amount of depletion are found in the IRS Publication 535. AMORTIZATION

SOURCE: “How to Depreciate Property,” Publication 946, Internal Revenue Service, U.S. Department of Treasury, Washington, 1999.

method with a switch to straight-line in the later years is used. The IRS adopted a half-year convention for both depreciation methods. Under this convention, a property placed in service is considered to be only onehalf year irrespective of when during the year the property was placed in service. Table 9-17 is a listing of the class lives, and Table 9-18 contains factors with the half-year convention for both the MACRS and straightline methods. Depreciation is entered as an indirect expense on the manufacturing expense sheet based upon the straight-line method. However, when one is determining the after-tax cash flow, straight-line depreciation is removed from the manufacturing expense and the MACRS depreciation is entered. This is illustrated under the section on cash flow. TABLE 9-18

There are certain terms that apply to depreciation: • Depreciation reserve is the accumulated depreciation at a specific time. • Book value is the original investment minus the accumulated depreciation. • Service life is the time period during which an asset is in service and is economically feasible. • Salvage value is the net amount of money obtained from the sale of a used property over and above any charges involved in the removal and sale of the property. • Scrap value implies that the asset has no further useful life and is sold for the amount of scrap material in it. • Economic life is the most likely period of successful operation before a need arises for subsequent investment in additional equipment as the result of product or process obsolescence or equipment due to wear and tear.

Amortization is the ratable deduction for the cost of an intangible property over its useful life, perhaps a 15-year life, via straight-line calculations. An example of an intangible property is a franchise, patent, trademark, etc. Two IRS publications, Form 4562 and Publication 535 (1999), established the regulations regarding amortization. TAXES Corporations pay an income tax based upon gross earnings, as shown in Table 9-19. Most major corporations pay the federal tax rate of 34 percent on their annual gross earnings. In addition, some states have a stepwise corporate income tax rate. State income tax is deductible as an expense item before the calculation of the federal tax. If Ts is the incremental tax rate and Tf is the incremental federal tax, both expressed as decimals, then the combined incremental rate Tc is

Depreciation Rates for Straight-Line and MACRS Methods Straight-line half-year convention

Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

3 16.67% 33.33 33.33 16.67

5

7

10

15

20

Year

3

5

10.00% 20.00 20.00 20.00 20.00 10.00

7.14% 14.29 14.29 14.28 14.29 14.28 14.29 7.14

5.0% 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 5.0

3.33% 6.67 6.67 6.67 6.67 6.67 6.67 6.66 6.67 6.66 6.67 6.66 6.67 6.66 6.67 3.33

2.5% 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 2.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

33.33% 44.45 14.81 7.41

20.00% 32.00 19.20 11.52 11.52 5.76

MACRS* half-year convention 7 10 14.29% 24.49 17.49 12.49 8.93 8.92 8.93 4.46

10.00% 18.00 14.40 11.52 9.22 7.37 6.55 6.55 6.56 6.55 3.28

*General depreciation system. Declining-balance switching to straight-line. Recovery periods 3, 5, 7, 10, 15, and 20 years. “How to Depreciate Property,” Publication 946, Internal Revenue Service, U.S. Department of Treasury, Washington, 1999.

SOURCE:

15

20

5.00% 9.50 8.55 7.70 6.93 6.23 5.90 5.90 5.91 5.90 5.91 5.90 5.91 5.90 5.91 2.95

3.750% 7.219 6.677 6.177 5.713 5.285 4.888 4.522 4.462 4.461 4.462 4.461 4.462 4.461 4.462 4.461 4.462 4.461 4.462 4.461 2.231

FACTORS THAT AFFECT PROFITABILITY TABLE 9-19

Year 2: P + Pi(1 + i) = P(1 + i)2 = F2

Corporate Federal Income Tax Rates Taxable income

Tax rate, %

Annual gross earnings less than $50,000 Annual gross earnings greater than $50,000 but not over $75,000 Annual gross earnings greater than $75,000 plus 5% of gross earnings over $100,000 or $11,750, whichever is greater Corporations with gross earnings of at least $335,000 pay a flat rate of 34%

15 25 34

9-23 (9-13)

P(1 + i) n = F

Year n:

An interest rate quoted on an annual basis is called nominal interest. However, interest may be payable on a semiannual, quarterly, monthly, or daily basis. To determine the amount compounded, the following equation applies:



i F = P 1 +  m

SOURCE:

U.S. Corporate Income Tax Return, Form 1120, Internal Revenue Service, U. S. Department of Treasury, Washington, 1999.



mn

(9-14)

where m = number of interest periods per year n = number of years i = nominal interest Tc = Ts + (1 − Ts)Tf

(9-12)

If the federal rate is 34 percent and the state rate is 7 percent, then the combined rate is Tc = 0.07 + (1 − 0.07)(0.34) = 0.39 Therefore, the combined tax rate is 39 percent. TIME VALUE OF MONEY In business, money is either borrowed or loaned. If money is loaned, there is the risk that it may not be repaid. From the lender’s standpoint, the funds could have been invested somewhere else and made a profit; therefore, the interest charged for the loan is compensation for the forgone profit. The borrower may look upon this interest as the cost of renting money. The amount of interest charged depends on the scarcity of money, the size of the loan, the length of the loan period, the risk that the lender feels that the loan may not be repaid, and the prevailing economic conditions. Engineers involved in the presentation and/or the evaluation of an investment of funds in a venture, therefore, need to understand the time value of money and how it is applied in the evaluation of projects. The amount of the loan is called the principal P. The longer the time for which the money is loaned, the greater the total amount of interest paid. The future amount of the money F is greater than the principal or present worth P. The relationship between F and P depends upon the type of interest used. Table 9-20 is a summary of the nomenclature used in time value of money calculations. Simple Interest The relationship between F and P is F = P(1 + in). The interest is charged on the original loan and not on the unpaid balance (Couper and Rader, 1986). The interest is paid at the end of each time interval. Although the simple interest concept still exists, it is seldom used in business. Discrete Compound Interest In financial transactions, loans or deposits are made using compound interest. The interest is not withdrawn but is added to the principal for that time period. In the next time period, the interest is calculated upon the principal plus the interest from the preceding time period. This process illustrates compound interest. In equation format,

Interest calculated for a given time period is known as discrete compound interest, with discrete referring to a discrete time period. Table 9-21 contains 5 and 6 percent discrete interest factors. Examples of the use of discrete factors for various applications are found in Table 9-22, assuming that the present time is when the first funds are expended. Continuous Compound Interest In some companies, namely, petroleum, petrochemical, and chemical companies, money transactions occur hourly or daily, or essentially continuously. The receipts from sales and services are invested immediately upon receipt. The interest on this cash flow is continuously compounded. To use continuous compounding when evaluating projects or investments, one assumes that cash flows continuously. In continuous compounding, the year is divided into an infinite number of periods. Mathematically, the limit of the interest term is



r lim n → ∞ 1 +  m where



mn

= ern

(9-15)

n = number of years m = number of interest periods per year r = nominal interest rate e = base for naperian logarithms

The numerical difference between discrete compound interest and continuous compound interest is small, but when large sums of money are involved, the difference may be significant. Table 9-23 is an abbreviated continuous interest table, assuming that time zero is when start-up occurs. A summary of the equations for discrete compound and continuous compound interest is found in Table 9-24. Compounding-Discounting When money is moved forward in time from the present to a future time, the process is called compounding. The effect of compounding is that the total amount of money increases with time due to interest. Discounting is the reverse process, i.e., a sum of money moved backward in time. Figure 9-8 is a

Year 1: P + Pi = P(1 + i) = F1 0 TABLE 9-20 Symbol F

P

A

1

2

3 4 Periods of time

Interest Nomenclature Definition Future sum Future value Future worth Future amount Principal Present worth Present value Present amount End of period payment in a uniform series

P Compounding

Discounting FIG. 9-8

Compounding-discounting diagram.

5

6

TABLE 9-21

Discrete Compound Interest Factors*

9-24

Single payment

Uniform annual series

Single payment

Uniform annual series

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

Compoundamount factor

Presentworth factor

Sinkingfund factor

Capitalrecovery factor

Compoundamount factor

Presentworth factor

n

Given P, to find F (1 + i)n

Given F, to find P 1 n (1 + i)

Given F, to find A i  (1 + i)n − 1

Given P, to find A i(1 + i)n  (1 + i)n − 1

Given A, to find F (1 + i)n − 1  i

Given A, to find P (1 + i)n − 1  i(1 + i)n

Given P, to find F (1 + i)n

Given F, to find P 1 n (1 + i)

Given F, to find A i  (1 + i)n − 1

Given P, to find A i(1 + i)n  (1 + i)n − 1

Given A, to find F (1 + i)n − 1  i

Given A, to find P (1 + i)n − 1  i(1 + i)n

1 2 3 4 5

1.050 1.103 1.158 1.216 1.276

0.9524 .9070 .8638 .8227 .7835

1.00000 0.48780 .31721 .23201 .18097

1.05000 0.53780 .36721 .28201 .23097

1.000 2.050 3.153 4.310 5.526

0.952 1.859 2.723 3.546 4.329

1.060 1.124 1.191 1.262 1.338

0.9434 .8900 .8396 .7921 .7473

1.00000 0.48544 .31411 .22859 .17740

1.06000 0.54544 .37411 .28859 .23740

1.000 2.060 3.184 4.375 5.637

0.943 1.833 2.673 3.465 4.212

1 2 3 4 5

6 7 8 9 10

1.340 1.407 1.477 1.551 1.629

.7462 .7107 .6768 .6446 .6139

.14702 .12282 .10472 .09069 .07940

.19702 .17282 .15472 .14069 .12950

6.802 8.142 9.549 11.027 12.578

5.076 5.786 6.463 7.108 7.722

1.419 1.504 1.594 1.689 1.791

.7050 .6651 .6274 .5919 .5584

.14336 .11914 .10104 .08702 .07587

.20336 .17914 .16104 .14702 .13587

6.975 8.394 9.897 11.491 13.181

4.917 5.582 6.210 6.802 7.360

6 7 8 9 10

11 12 13 14 15

1.710 1.796 1.886 1.980 2.079

.5847 .5568 .5303 .5051 .4810

.07039 .06283 .05646 .05102 .04634

.12039 .11283 .10646 .10102 .09634

14.207 15.917 17.713 19.599 21.579

8.306 8.863 9.394 9.899 10.380

1.898 2.012 2.133 2.261 2.397

.5268 .4970 .4688 .4423 .4173

.06679 .05928 .05296 .04758 .04296

.12679 .11928 .11296 .10758 .10296

14.972 16.870 18.882 21.015 23.276

7.887 8.384 8.853 9.295 9.712

11 12 13 14 15

16 17 18 19 20

2.183 2.292 2.407 2.527 2.653

.4581 .4363 .4155 .3957 .3769

.04227 .03870 .03555 .03275 .03024

.09227 .08870 .08555 .08275 .08024

23.657 25.840 28.132 30.539 33.066

10.838 11.274 11.690 12.085 12.462

2.540 2.693 2.854 3.026 3.207

.3936 .3714 .3503 .3305 .3118

.03895 .03544 .03236 .02962 .02718

.09895 .09544 .09236 .08962 .08718

25.673 28.213 30.906 33.760 36.786

10.106 10.477 10.828 11.158 11.470

16 17 18 19 20

21 22 23 24 25

2.786 2.925 3.072 3.225 3.386

.3589 .3418 .3256 .3101 .2953

.02800 .02597 .02414 .02247 .02095

.07800 .07597 .07414 .07247 .07095

35.719 38.505 41.430 44.502 47.727

12.821 13.163 13.489 13.799 14.094

3.400 3.604 3.820 4.049 4.292

.2942 .2775 .2618 .2470 .2330

.02500 .02305 .02128 .01968 .01823

.08500 .08305 .08128 .07968 .07823

39.993 43.392 46.996 50.816 54.865

11.764 12.042 12.303 12.550 12.783

21 22 23 24 25

26 27 28 29 30

3.556 3.733 3.920 4.116 4.322

.2812 .2678 .2551 .2429 .2314

.01956 .01829 .01712 .01605 .01505

.06956 .06829 .06712 .06605 .06505

51.113 54.669 58.403 62.323 66.439

14.375 14.643 14.898 15.141 15.372

4.549 4.822 5.112 5.418 5.743

.2198 .2074 .1956 .1846 .1741

.01690 .01570 .01459 .01358 .01265

.07690 .07570 .07459 .07358 .07265

59.156 63.706 68.528 73.640 79.058

13.003 13.211 13.406 13.591 13.765

26 27 28 29 30

31 32 33 34 35

4.538 4.765 5.003 5.253 5.516

.2204 .2099 .1999 .1904 .1813

.01413 .01328 .01249 .01176 .01107

.06413 .06328 .06249 .06176 .06107

70.761 75.299 80.064 85.067 90.320

15.593 15.803 16.003 16.193 16.374

6.088 6.453 6.841 7.251 7.686

.1643 .1550 .1462 .1379 .1301

.01179 .01100 .01027 .00960 .00897

.07179 .07100 .07027 .06960 .06897

84.802 90.890 97.343 104.184 111.435

13.929 14.084 14.230 14.368 14.498

31 32 33 34 35

40 45 50

7.040 8.985 11.467

.1420 .1113 .0872

.00828 .00626 .00478

.05828 .05626 .05478

120.800 159.700 209.348

17.159 17.774 18.256

10.286 13.765 18.420

.0972 .0727 .0543

.00646 .00470 .00344

.06646 .06470 .06344

154.762 212.744 290.336

15.046 15.456 15.762

40 45 50

55 60 65 70 75

14.636 18.679 23.840 30.426 38.833

.0683 .0535 .0419 .0329 .0258

.00367 .00283 .00219 .00170 .00132

.05367 .05283 .05219 .05170 .05132

272.713 353.584 456.798 588.529 756.654

18.633 18.929 19.161 19.343 19.485

24.650 32.988 44.145 59.076 79.057

.0406 .0303 .0227 .0169 .0126

.00254 .00188 .00139 .00103 .00077

.06254 .06188 .06139 .06103 .06077

394.172 533.128 719.083 967.932 1,300.949

15.991 16.161 16.289 16.385 16.456

55 60 65 70 75

80 85 90 95 100

49.561 63.254 80.730 103.035 131.501

.0202 .0158 .0124 .0097 .0076

.00103 .00080 .00063 .00049 .00038

.05103 .05080 .05063

971.229 1,245.087 1,594.607 2,040.694 2,610.025

19.596 19.684 19.752 19.806 19.848

105.796 141.579 189.465 253.546 339.302

.0095 .0071 .0053 .0039 .0029

.00057 .00043 .00032 .00024 .00018

.06057 .06043 .06032 .06024 .06018

1,746.600 2,342.982 3,141.075 4,209.104 5,638.368

16.509 16.549 16.579 16.601 16.618

80 85 90 95 100

5% Compound Interest Factors

.05038

n

6% Compound Interest Factors

*Factors presented for two interest rates only. By using the appropriate formulas, values for other interest rates may be calculated.

FACTORS THAT AFFECT PROFITABILITY TABLE 9-22

9-25

Examples of the Use of Compound Interest Table

Given: $2500 is invested now at 5 percent. Required: Accumulated value in 10 years (i.e., the amount of a given principal). Solution:

F = P(1 + i)n = $2500 × 1.0510 Compound-amount factor = (1 + i)n = 1.0510 = 1.629 F = $2500 × 1.629 = $4062.50

Given: $19,500 will be required in 5 years to replace equipment now in use. Required: With interest available at 3 percent, what sum must be deposited in the bank at present to provide the required capital (i.e., the principal which will amount to a given sum)? 1 1 Solution: P = F n = $19,500 5 (1 + i) 1.03 Present-worth factor = 1/(1 + i)n = 1/1.035 = 0.8626 P = $19,500 × 0.8626 = $16,821 Given: $50,000 will be required in 10 years to purchase equipment. Required: With interest available at 4 percent, what sum must be deposited each year to provide the required capital (i.e., the annuity which will amount to a given fund)? i 0.04 Solution: A = F  = $50,000  (1 + i)n − 1 1.0410 − 1 i 0.04 Sinking-fund factor =  =  = 0.08329 (1 + i)n − 1 1.0410 − 1 A = $50,000 × 0.08329 = $4,164 Given: $20,000 is invested at 10 percent interest. Required: Annual sum that can be withdrawn over a 20-year period (i.e., the annuity provided by a given capital). i(1 + i)n 0.10 × 1.1020 Solution: A = P  = $20,000  n (1 + i) − 1 1.1020 − 1 i(1 + i)n 0.10 × 1.1020 Capital-recovery factor =  =  = 0.11746 n (1 + i) − 1 1.1020 − 1 A = $20,000 × 0.11746 = $2349.20 Given: $500 is invested each year at 8 percent interest. Required: Accumulated value in 15 years (i.e., amount of an annuity). (1 + i)n − 1 1.0815 − 1 Solution: F = A  = $500  i 0.08 (1 − i)n − 1 1.0815 − 1 Compound-amount factor =  =  = 27.152 i 0.08 F = $500 × 27.152 = $13,576 Given: $8000 is required annually for 25 years. Required: Sum that must be deposited now at 6 percent interest. (1 + i)n − 1 1.0625 − 1 Solution: P = A  = $8000  n i(1 + i) 0.06 × 1.0625 (1 + i)n − 1 1.0625 − 1 Present-worth factor =  =  = 12.783 n i(1 + i) 0.06 × 1.0625 P = $8000 × 12.783 = $102,264

sketch of this process. The time periods are years, and the interest is normally on an annual basis using end-of-year money flows. The longer the time before money is received, the less it is worth at present. Effective Interest Rates When an interest rate is quoted, it is nominal interest that is stated. These quotes are on an annual basis, however, when compounding occurs that is not the actual or effective interest. According to government regulations, an effec-

tive rate APY must be stated also. The effective interest is calculated by



i ieff = 1 +  m



(m)(1)

−1

(9-16)

The time period for calculating the effective interest rate is 1 year.

9-26

TABLE 9-23

Condensed Continuous Interest Table*

Factors for determining zero-time values for cash flows which occur at other than zero time. Compounding of Cash Flows Which Occur:

1%

5%

10%

15%

20%

25%

30%

35%

40%

50%

60%

70%

80%

90%

100%

1.005 1.010 1.015 1.020 1.030

1.025 1.051 1.078 1.105 1.162

1.051 1.105 1.162 1.221 1.350

1.078 1.162 1.252 1.350 1.568

1.105 1.221 1.350 1.492 1.822

1.133 1.284 1.455 1.649 2.117

1.162 1.350 1.568 1.822 2.460

1.191 1.419 1.690 2.014 2.858

1.221 1.492 1.822 2.226 3.320

1.284 1.649 2.117 2.718 4.482

1.350 1.822 2.460 3.320 6.050

1.419 2.014 2.858 4.055 8.166

1.492 2.226 3.320 4.953 11.023

1.568 2.460 3.857 6.050 14.880

1.649 2.718 4.482 7.389 20.086

1.002 1.005 1.008 1.010 1.015

1.013 1.025 1.038 1.052 1.079

1.025 1.052 1.079 1.107 1.166

1.038 1.079 1.121 1.166 1.263

1.052 1.107 1.166 1.230 1.370

1.065 1.136 1.213 1.297 1.489

1.079 1.166 1.263 1.370 1.622

1.093 1.197 1.315 1.448 1.769

1.107 1.230 1.370 1.532 1.933

1.136 1.297 1.489 1.718 2.321

1.166 1.370 1.622 1.933 2.805

1.197 1.448 1.769 2.182 3.412

1.230 1.532 1.933 2.471 4.176

1.263 1.622 2.117 2.805 5.141

1.297 1.718 2.321 3.194 6.362

.990 .980 .970 .961 .951

.951 .905 .861 .819 .779

.905 .819 .741 .670 .606

.861 .741 .638 .549 .472

.819 .670 .549 .449 .368

.779 .606 .472 .368 .286

.741 .549 .407 .301 .223

.705 .497 .350 .247 .174

.670 .449 .301 .202 .135

.606 .368 .223 .135 .082

.549 .301 .165 .091 .050

.497 .247 .122 .061 .030

.449 .202 .091 .041 .018

.407 .165 .067 .027 .011

.360 .135 .050 .018 .007

.905 .861 .819 .779

.606 .472 .368 .286

.368 .223 .135 .082

.223 .105 .050 .024

.135 .050 .018 .007

.082 .024 .007 .002

.050 .011 .002 .001

.030 .005 .001 —

.018 .002 — —

.007 .001 — —

.002 — — —

.001 — — —

— — — —

— — — —

— — — —

.995 .985 .975 .966 .956

.975 .928 .883 .840 .799

.952 .861 .779 .705 .638

.929 .799 .688 .592 .510

.906 .742 .608 .497 .407

.885 .689 .537 .418 .326

.864 .640 .474 .351 .260

.844 .595 .419 .295 .208

.824 .552 .370 .248 .166

.787 .477 .290 .176 .106

.752 .413 .226 .124 .068

.719 .357 .177 .088 .044

.688 .309 .139 .062 .028

.659 .268 .109 .044 .018

.632 .232 .086 .032 .012

.946 .937 .928 .918 .909

.760 .723 .687 .654 .622

.577 .522 .473 .428 .387

.439 .378 .325 .280 .241

.333 .273 .224 .183 .150

.254 .197 .154 .120 .093

.193 .143 .108 .078 .058

.147 .103 .073 .051 .036

.112 .075 .050 .034 .022

.065 .039 .024 .014 .009

.037 .020 .011 .006 .003

.022 .011 .005 .003 .001

.013 .006 .002 .001 —

.007 .003 .001 — —

.004 .002 .001 — —

E. Uniformly over 5-Year Periods 1st 5 years 6th through 10th year 11th through 15th year 16th through 20th year 21st through 25th year

.975 .928 .883 .840 .799

.885 .689 .537 .418 .326

.787 .477 .290 .176 .106

.704 .332 .157 .074 .035

.632 .232 .086 .032 .012

.571 .164 .047 .013 .004

.518 .116 .026 .006 .001

.472 .082 .014 .002 —

.432 .058 .008 .001 —

.367 .030 .002 — —

.317 .016 .001 — —

.277 .008 — — —

.245 .004 — — —

.220 .002 — — —

.199 .001 — — —

F. Declining to Nothing at Constant Rate 1st 5 years ” 10 ” ” 15 ” ” 20 ” ” 25 ”

.983 .968 .952 .936 .922

.922 .852 .791 .736 .687

.852 .736 .643 .568 .506

.791 .643 .536 .456 .394

.736 .568 .456 .377 .320

.687 .506 .394 .320 .269

.643 .456 .347 .278 .231

.603 .413 .309 .245 .203

.568 .377 .278 .219 .180

.506 .320 .231 .180 .147

.456 .278 .198 .153 .124

.413 .245 .172 .133 .108

.377 .219 .153 .117 .095

.347 .198 .137 .105 .085

.320 .180 .124 .095 .077

A. In an Instant 1  2

1 112 2 3

year before ” ” ” ” ” ” ” ”

B. Uniformly until Zero Time From 12 year before to 0 time ” 1 ” ” ” ” ” ” 112 ” ” ” ” ” ” 2 ” ” ” ” ” ” 3 ” ” ” ” ” Discounting of Cash Flows Which Occur: C. In an Instant 1 year later 2 ” ” 3 ” ” 4 ” ” 5 ” ” 10 years later 15 ” ” 20 ” ” 25 ” ” D. Uniformly over Individual Years 1st year 2nd ” 3rd ” 4th ” 5th ” 6th year 7th ” 8th ” 9th ” 10th ”

*From tables compiled by J. C. Gregory, The Atlantic Refining Co.

FACTORS THAT AFFECT PROFITABILITY TABLE 9-24

Summary of Discrete and Compound Interest Equations

Factor

Find

Given

Single payment Compound amount

F

P

F = P(1 + i)n

P

F

1 P = F n (1 + i)

F

A

Sinking fund

A

F

Present worth

P

A

(1 + i)n−1 P = A  i(1 + i)n



A(F/A i,n)

ern − 1 P = A  ern(er − 1)

Capital recovery

A

P

i(1 + i)n A = P  (1 + i)n − 1

P(A/P i,n)

ern (er − 1) A = P  ern − 1

Present worth Uniform series Compound amount

Discrete compounding



1+i ieff =  m





(m)(1)

1 + 0.833 =  12





(12)(1)

 



− 1 = (1.00694)12 − 1

= 1.0865 − 1 = 0.0865

CASH FLOW Cash flow is the amount of funds available to a company to meet current operating expenses. Cash flow may be expressed on a before- or after-tax basis. After-tax cash flow is defined as the net profit (income) after taxes plus depreciation. It is an integral part of the net present worth (NPW) and discounted cash flow profitability calculations. The cash flow diagram, also referred to as a cash flow model (Fig. 9-9), shows the relationship between revenue, cash operating expenses, depreciation, and profit. This diagram is similar in many respects to a process flow diagram, but it is in dollars. Revenue is generated from the

Cash operating expenses

Operating income Depreciation

After taxes FIG. 9-9

Cash flow model.

P(F/P i,n)

F = P(ern)

P(F/P r,n)∞

F(P/F i,n)

P = F(e−rn)

F(P/F r,n)∞

A(F/A i,n) F(A/F i,n)

ern − 1 F=A   er − 1 er − 1 A=F   ern − 1

 



 

F(F/A r,n)∞ F(A/F r,n)∞

 

A(P/A r,n)∞ A(P/A r,n)∞

sale of a product manufactured in “operations.” Working capital is replenished from sales and may be considered to be in dynamic equilibrium with operations. Leaving the operations box is a stream, “cash operating expenses.” It includes all the cash expenses incurred in the operation but does not include the noncash item depreciation. Since depreciation is an allowance, it is reported on the operating expense sheet, in accordance with the tax laws, as an operating expense item. (See the section “Operating Expense Estimation.”) Depreciation is an internal expense, and this allowance is retained within the company. If the cash operating expenses are subtracted from the revenue, the result is the operating income. If depreciation is subtracted from the operating income, the net profit before taxes results. Federal income taxes are then deducted from the net profit before taxes, giving the net profit after taxes. When depreciation and net profit after taxes are summed, the result is the after-tax cash flow. The terminology in Fig. 9-9 is consistent with that found in most company income statements in company annual reports. An equation can be developed for cash flow as follows: CF = (R − C − D)(1 − t) + D where

(9-17)

R = revenue C = cash operating expenses D = depreciation t = tax rate CF = after-tax cash flow

CF = t(D) + (1 − t)(R) − (1 − t)(C)

Operations

Net income



Continuous compounding

Equation (9-17) can be rearranged algebraically to yield Eq. (9-18)

Revenue (sales)

Depletion

 



The effective interest rate is 8.65 percent.

Working capital



(1 + i)n−1 F = A  i i A = F  (1 + i)n − 1

Example 6: Effective Interest Rate A person is quoted an 8.33 percent nominal interest rate on a 4-year loan compounded monthly. Determine the effective interest rate. Solution:

Cash flow

9-27

Gross profit

(9-18)

The term t × D is only the result of an algebraic manipulation, and no interpretation should be assumed. This term t × D is the contribution to cash flow from depreciation, and (1 − t) × R and (1 − t) × C are the contributions to cash flow from revenues and cash operating expenses, respectively. Example 7 is a sample calculation of the after-tax cash flow and the tabulated results. Example 7: After-Tax Cash Flow The revenue from the manufacture of a product in the first year of operation is $9.0 million, and the cash operating expenses are $4.5 million. Depreciation on the invested capital is $1.7 million. If the federal income tax rate is 35 percent, calculate the after-tax cash flow. Solution: The resulting after-tax cash flow is $3.52 million. See Fig. 9-10.

Federal income taxes Cumulative Cash Position Table To organize cash flow calculations, it is suggested that a cumulative cash position table be prepared by using an electronic spreadsheet. For this discussion,

9-28

PROCESS ECONOMICS Revenue $9.00M

Working capital

Year

Operating income Depreciation $4.50M Cash flow $3.52

Gross income $2.80M Federal income tax $0.98M

After tax $1.82M FIG. 9-10

40 42 45 48 50 50 47 45 40 35

20,000 21,000 23,400 24,960 27,500 28,000 23,500 21,600 18,800 15,750

10,320 10,800 11,520 12,240 13,470 13,970 13,175 12,645 11,320 9,995

Cumulative Cash Position Plot A pictorial representation of the cumulative cash flows as a function of time is the cumulative cash position plot. All expenditures for capital as well as revenue from sales are plotted as a function of time. Figure 9-11 is such an idealized plot showing time zero at start-up in part a and time zero when the first funds are expended in part b. It should be understood that the plots have been idealized for illustration purposes. Expenditures are usually stepwise, and accumulated cash flow from sales is seldom a straightline but more likely a curve with respect to time. Time Zero at Start-up Prior to time zero, expenditures are made for land, fixed capital investment, and working capital. It is assumed that land had been purchased by the company at some time in the past, and a parcel is allocated for the project under consideration. Land is allocated instantaneously to the project sometime prior to the purchase of equipment and construction of the plant. The fixed capital investment is purchased and installed over a period of time prior to start-up. For the purpose of this presentation, it is assumed that it occurs uniformly over a period of time. Both land and fixed capital investment are compounded to time zero by using the appropriate compound interest factors. At time zero, working capital is charged to the project. Start-up expenses are entered in the first year of operation after start-up. After time zero, start-up occurs and then manufacturing begins and income is generated, so cash flow begins to accumulate if the process is sound. At the end of the project life, land and working capital are recovered instantaneously.

Cash flow model for Example 7. M = million.

time zero is assumed to be at project start-up. Expenditures for land and equipment occurred prior to time zero and represent negative cash flows. At time zero, working capital is charged to the project as a negative cash flow. Start-up expenses are charged in the first year, and positive cash flow from the sale of product as net income after taxes plus depreciation begins, reducing the negative cash position. This process continues until the project is terminated. At that time, adjustments are made to recover land and working capital. An example of a cumulative cash position table is Table 9-25. When equipment is added for plant expansions to an existing facility, it may be more convenient to use time zero when the first expenditures occur. The selection of either time base is satisfactory for economic analysis as long as consistency is maintained. Example 8: Cumulative Cash Position Table (Time Zero at

Start-up) A specialty chemical company is considering the manufacture of an additive for use in the plastics industry. The following is a list of production, sales, and cash operating expenses.

TABLE 9-25

Cash operating expenses, $1,000

Land for the project is available at $300,000. The fixed capital investment was estimated to be $12,000,000. A working capital of $1,800,000 is needed initially for the venture. Start-up expenses based upon past experience are estimated to be $750,000. The project qualifies under IRS guidelines as a 5-year class life investment. The company uses MACRS depreciation with the half-year convention. At the conclusion of the project, the land and working capital are returned to management. Develop a cash flow analysis for this project, using a cumulative cash position table (Table 9-25).

$1.70M

Net profit

Sales, $1,000

1 2 3 4 5 6 7 8 9 10

Cash operating expenses $4.50M

Operations

Production, Mlb

Cash Flow Analysis for Example 8

Year Production Mlb/yr All money is $1,000 Land Fixed capital investment $1000 Working capital Start-up expenses Total capital investment Sales Cash operating expenses Operating income Depreciation Net income before taxes Income tax Net income after taxes Depreciation After-tax cash flow Cumulative cash flow Capital recovery End of project value

−2

−2 to 0

0

1

2

3

4

5

6

7

8

9

10

End 10

40

42

45

48

50

50

47

45

40

35

20,000 10,320 9,680 2,400 7,280 3,640 3,640 2,400 6,040 −8,810

21,000 10,800 10,200 3,840 6,360 2,226 4,134 3,840 7,974 −836

23,400 11,520 11,880 2,300 9,580 3,353 6,227 2,300 8,527 7,691

24,960 12,240 12,720 1,390 11,330 3,966 7,364 1,390 8,754 16,445

27,500 13,470 14,030 1,380 12,650 4,428 8,222 1,380 9,602 26,047

28,000 13,970 14,030 690 13,340 4,669 8,671 690 9,361 35,408

23,500 13,175 10,325 0 10,325 3,614 6,711 0 6,711 42,119

21,600 12,645 8,955 0 8,955 3,134 5,821 0 5,821 47,940

18,800 11,320 7,480 0 7,480 2,618 4,862 0 4,882 52,802

15,750 9,995 5,755 0 5,755 2,014 3,741 0 3,741 56,543

−300 −12,000

−300 −300

−12,000 −12,300

−1,800 −750 14,250

−2,550 −14,850

2,100 58,643

Cumulative cash position, $

FACTORS THAT AFFECT PROFITABILITY

Land

0

Recovery of working capital and land

Net income after taxes

Time

Start-up at time zero

Fixed capital investment Annual cash flow Working capital

(a)

Recovery of working capital and land

Cumulative cash position, $

Net income after taxes

0

Land

Fixed capital investment

Time

Annual cash flow

Working capital (b) FIG. 9-11 Typical cumulative cash position plot. (a) Time zero is start-up. (b) Time zero occurs when first funds are spent.

9-29

9-30

PROCESS ECONOMICS

PROFITABILITY In the free enterprise system, companies are in business to make a profit. Management has the responsibility of investing in ventures that are financially attractive by increasing earnings, providing attractive rates of return, and increasing value added. Every viable business has limitations on the capital available for investment; therefore, it will invest in the most economically attractive ventures. The objectives and goals of a company are developed by management. Corporate objectives may include one or several of the following: maximize return on investment, maximize return on stockholders’ equity, maximize aggregate earnings, maximize common stock prices, increase market share, increase economic value, increase earnings per share of stock, and increase market value added. These objectives are the ones most frequently listed by executives. To determine the worthiness of a venture, quantitative and qualitative measures of profitability are considered. QUANTITATIVE MEASURES OF PROFITABILITY When a company invests in a venture, the investment must earn more than the cost of capital for it to be worthwhile. A profitability estimate is an attempt to quantify the desirability of taking a risk in a venture. The minimum acceptable rate of return (MARR) for a venture depends on a number of factors such as interest rate, cost of capital, availability of capital, degree of risk, economic project life, and other competing projects. Management will adjust the MARR depending on any of the above factors to screen out the more attractive ventures. When a company invests in a venture, the investment must earn more than the cost of capital and should be able to pay dividends. Although there have been many quantitative measures suggested through the years, some did not take into account the time value of money. In today’s economy, the following measures are the ones most companies use: Payout period (POP) plus interest Net present worth (NPW) Discounted cash flow (DCFROR) Payout Period Plus Interest Payout period (POP) is the time that will be required to recover the depreciable fixed capital investment from the accrued after-tax cash flow of a project with no interest considerations. In equation format depreciable fixed capital investment Payout period =  (9-19) after-tax cash flow

FIG. 9-12

This model does not take into account the time value of money, and no consideration is given to cash flows that occur in a project’s later years after the depreciable investment has been recovered. A variation on this method includes interest, called payout period plus interest (POP + I); and the net effect is to increase the payout period. This variation accounts for the time value of money. Payout period plus interest (POP + I) =

  after-tax cash flow depreciable fixed capital investment

(9-20)

i

Neither of these methods makes provision for including land and working capital, and no consideration is given to cash flows that occur in a project’s later years after the depreciable fixed investment has been recovered for projects that earn most of their profit in the early years. Net Present Worth In the net present worth method, an arbitrary time frame is selected as the basis of calculation. This method is the measure many companies use, as it reflects properly the time value of money and its effect on profitability. In equation form Net present worth (NPW) = present worth of all cash inflows−present worth of all investment items (9-21) When the NPW is calculated according to Eq. (9-21), if the result is positive, the venture will earn more than the interest (discount) rate used; conversely, if the NPW is negative, the venture earns less than that rate. Discounted Cash Flow In the discounted cash flow method, all the yearly after-tax cash flows are discounted or compounded to time zero depending upon the choice of time zero. The following equation is used to solve for the interest rate i, which is the discounted cash flow rate of return (DCFROR). n

DCFROR = (after-tax cash flows) = 0 0

(9-22)

Equation (9-22) may be solved graphically or analytically by an iterative trial-and-error procedure for the value of i, which is the discounted cash flow rate of return. It has also been known as the profitability index. For a project to be profitable, the interest rate must exceed the cost of capital. The effect of interest on the cash position of a project is shown in Fig. 9-12. As interest increases, the time to recover the capital expenditures is increased.

Effect of interest rate on cash flow (time zero occurs when first funds are expanded).

PROFITABILITY TABLE 9-26 Time

9-31

Profitability Analysis for Example 9 Cash flow

−2 −300 −2 to 0 −12,000 0 −2,250 1 6,040 2 7,974 3 8,527 4 8,754 5 9,602 6 9,361 7 6,711 8 5,821 9 4,862 10 3,471 End 10 2,100 Net present worth Discounted cash flow rate of return

20% interest factors

Present worth 20% interest

30% interest factors

Present worth 30% interest

40% interest factors

Present worth 40% interest

1.492 1.230 1.000 0.906 0.742 0.608 0.497 0.407 0.333 0.273 0.224 0.183 0.150 0.135

−448 −14,760 −2,250 5,472 5,917 5,184 4,351 3,908 3,117 1,832 1,304 890 561 284 15,362

1.822 1.370 1.000 0.864 0.640 0.474 0.351 0.260 0.193 0.143 0.106 0.078 0.058 0.050

−547 −16,440 −2,250 5,219 5,103 4,042 3,073 2,497 1,807 960 617 379 217 105 4,782

2.226 1.532 1.000 0.824 0.552 0.370 0.248 0.166 0.112 0.075 0.050 0.034 0.022 0.018

−668 −18,384 −2,250 4,977 4,402 3,155 2,171 1,594 1,048 503 291 165 82 284 −2,360

33.90%

13 12 11 10 F. I. T. 9

Re ve nu e

$ millions/year

8 7

Net income after taxes Net income before taxes

6 5

Break-even point op tal

s

nse

xpe ge

tin

era

To

4

Variable expenses 3 2 1

Fixed expenses

Shutdown point 0

0

10

20

30

40

50

60

70

Production rate (% of capacity) FIG. 9-13

Break-even plot.

80

90

100

9-32

PROCESS ECONOMICS

In the chemical business, operating net profit and cash flow are received on a nearly continuous basis, therefore, there is justification for using the condensed continuous interest tables, such as Table 9-23, in discounted cash flow calculations.

ordinal or a ranking system, but most have had little or no success. Couper (2003) discussed in greater detail the effect of qualitative measures on the decision-making process.

Example 9: Profitability Calculations Example 8 data are used to demonstrate these calculations. Calculate the following: a. Payout period (POP) b. Payout period with interest (POP + I) c. NPW at a 30 percent interest rate d. DCF rate of return Solution: a. From Table 9-26, the second column is the cash flow by years with no interest. The payout period occurs where the cumulative cash flow is equal to the fixed capital investment, $12,000,000 or 1.7 years. b. In Table 9-26, the payout period at 30 percent interest occurs at 2.4 years. c. The results of the present worth calculations for 20, 30, and 40 percent interest rates are tabulated. At 30 percent interest, the net present worth is $4,782,000, and since it is a positive figure, this means the project will earn more than 30 percent interest. d. Discounted cash flow rate of return is determined by interpolating in Table 9-26. At 30 percent interest the net present worth is positive, and at 40 percent interest it is negative. By definition, the DCFROR occurs when the summation of the net present worth equals zero. This occurs at an interest of 33.9 percent.

SENSITIVITY ANALYSIS Whenever an economic study is prepared, the marketing, capital investment, and operating expense data used are estimates, and therefore a degree of uncertainty exists. Questions arise such as, What if the capital investment is 15 percent greater than the value reported? A sensitivity analysis is used to determine the effect of percentage changes in pertinent variables on the profitability of the project. Such an analysis indicates which variables are most susceptible to change and need further study. Break-Even Analysis Break-even analysis is a simple form of sensitivity analysis and is a useful concept that can be of value to managers. Break-even refers to the point in an operation where income just equals expenses. Figure 9-13 is a pictorial example of the results of a break-even analysis, showing that the break-even point is at 26 percent of production capacity. Management wants to do better than just break even; therefore, such plots can be used as a profit planning tool, for product pricing, production operating level, incremental equipment costs, etc. Another significant point is the shutdown point where revenue just equals the fixed expenses. Therefore, if a proposed operation can’t make fixed expenses, it should be shut down. Strauss Plot R. Strauss (Chem. Eng., pp. 112–116, Mar. 25, 1968) developed a sensitivity plot, in Fig. 9-14, in which the ordinate is a measure of profitability and the abscissa is the change in a variable greater than (or less than) the value used in the base case. Where the abscissa crosses the ordinate is the result of the base case of NPW, return, annual worth, etc. The slope of a line on this “spider” plot is the degree of change in profitability resulting from a change in a

QUALITATIVE MEASURES In addition to quantitative measures, there are certain qualitative measures or intangible factors that may affect the ultimate investment decision. Those most frequently mentioned by management are employee morale, employee safety, environmental constraints, legal constraints, product liability, corporate image, and management goals. Attempts have been made to quantify these intangibles by using an

NPW ($ millions) 26

24

22

20

18 −20

−10

+10

% variation +20

16

14

12

Sales price Sales volume

10

8 FIG. 9-14

Strauss plot.

Raw material price Fixed capital investment

PROFITABILITY variable, selling price, sales volume, investment, etc. The length of the line represents the sensitivity of the variable and its degree of uncertainty. Positive-slope lines are income-related, and negative-slope lines are expense-related. A spreadsheet is useful in developing data for this “what if” plot since numerous scenarios must be prepared to develop the plot. Tornado Plot Another graphical sensitivity analysis is the “tornado” plot. Its name is derived from the shape of the resulting envelope. As in other methods, a base case is solved first, usually expressing the profitability as the net present worth. In Fig. 9-15, the NPW is a vertical line, and variations in each selected variable above and below the base case are solved and plotted. In this figure, the variables of selling price, sales volume, operating expenses, raw material expenses, share of the market, and investment are plotted. It is apparent that the selling price and sales volume are the critical factors affecting the profitability. A commercial computer program known as @RISK® developed by the Palisade Corporation, Newfield, N.Y., may be used to prepare a tornado plot. Relative Sensitivity Plot Another type of analysis developed by J. C. Agarwal and I. V. Klumpar (Chem. Eng., pp. 66–72, Sept. 29, 1975) is the relative sensitivity plot. The variables studied are related to those in the base case, and the resulting plot is the relative profitability. Although sensitivity analyses are easy to prepare and they yield useful information for management, there is a serious disadvantage. Only one variable at a time can be studied. Frequently, there are synergistic effects among variables; e.g., in marketing, the variables such as sales volume, selling price, and market share may have a synergistic effect, and that effect cannot be taken into account. Other interrelated variables such as fixed capital investment, maintenance, and other investment-based items also cannot be represented properly. These disadvantages lead to another management tool—uncertainty analysis.

NPW at 30% interest

−25%

Selling price

−10%

Market share

Operating expenses

Determine profitability Repeat process many times

Print results FIG. 9-16

Schematic diagram of Monte Carlo simulation.

Investment

+8%

+5%

Typical tornado plot. (Source: Adapted from Couper, 2003.)

Select random values

Probability

+10%

Raw materials

−8%

Investment

+15%

Operating expenses

−5%

Market share

+20%

Sales volume

−15%

FIG. 9-15

+25%

Selling price

−20%

UNCERTAINTY ANALYSIS This analysis allows the user to account for variable interaction that is another level of sophistication. Two terms need clarification—

9-33

9-34

PROCESS ECONOMICS 100

Probability of achieving indicated return or less

90

80

70

60

50

40

30

20

10

0 0 FIG. 9-17

5 10 15 20 25 Percent DCFROR on investment

30

35

Probability curve for Monte Carlo simulation.

uncertainty and risk. Uncertainty is exactly what the word means— not certain. Risk, however, implies that the probability of achieving a specific outcome is known within certain confidence limits. Since sensitivity analysis has the shortcoming of being able to inspect only one variable at a time, the next step is to use probability risk analysis, generally referred to as the Monte Carlo technique. R. C. Ross (Chem. Eng., pp. 149–155, Sept. 20, 1971), P. Macalusa (BYTE, pp. 179–192, March 1984), and D. B. Hertz (Harvard Bus. Rev., pp. 96–108, Jan-Feb 1968) have written classic articles on the use of the Monte Carlo technique in uncertainty analysis. These articles incorporate subjective probabilities and assumptions of the distribution of errors into the analysis. Each variable is represented by a probability distribution model. Figure 9-16 is a pictorial representation of the steps in the Monte Carlo simulation. The first step is to gather enough data to develop a reasonable probability model. Not all variables follow the normal distribution curve, but perhaps sales volume and salesrelated variables do. Studies have shown that capital investment estimates are best represented by a beta distribution. Next the task is to select random values from the various models by using a random number generator and from these data calculate a profitability measure such as NPW or rate of return. The procedure is repeated a number of times to generate a plot of the probability of achieving a given profitability versus profitability. Figure 9-17 is a typical plot. Once the analysis has been performed, the next task is to interpret the results. Management must understand what the results mean and the reliability of the results. Experience can be gained only by performing uncertainty analyses, not just one or two attempts, to develop confidence in

the process. The stakes may be high enough to spend time and learn the method. Software companies such as @RISK or SAS permit the user to develop probability models and perform the Monte Carlo analysis. The results may be plotted as the probability of achieving at least a given return or of achieving less than the desired profitability. FEASIBILITY ANALYSIS A feasibility analysis is prepared for the purpose of determining that a proposed investment meets the minimum requirements established

TABLE 9-27 Checklist of Required Information for a Feasibility Analysis Fixed capital investment Working capital requirements Total capital investment Total manufacturing expense Packaging and in-plant expense Total product expense General overhead expense Total operating expense Marketing data Cash flow analysis Project profitability Sensitivity analysis Uncertainty analysis

OTHER ECONOMIC TOPICS TABLE 9-28

9-35

Marketing Data Template

Project title: Basis: Sales and marketing data are not inflated (20__ dollars) Amount

20__ %Total

Amount

20__ %Total

Amount

20__ %Total

Total market Units Average realistic price, $/unit Value, $ Estimated product sales (with AR) Units Average realistic price, $/unit Value, $ Current product sales (without AR) Units Average realistic price, $/unit Value, $ Incremental product sales Units Average realistic price, $/unit Value, $ Current product sales displaced by improved product sales Units Value, $ Total improved product sales Units Value, $ NOTE:

Table extends to the right to accommodate the number of project years. AR = appropriation request.

TABLE 9-29

Cash Flow Analysis Template Cash flow summary 200X

200Y

200Z, etc.*

Investment Land Fixed capital investment Offsite capital Allocated capital Working capital Start-up expenses Interest Catalysts and chemicals Licenses, patents, etc. Total capital investment Income statement Income Expenses Cash operating expenses Depreciation Total operating expenses Operating income Net income before taxes Federal income taxes Net income after taxes Cash flow Capital recovery Cumulative cash flow *Table may be extended to the right to accommodate the number of years of the project.

by management. It should be in sufficient detail to provide management with the facts required to make an investment decision. All the basic information has been discussed in considerable detail in the earlier parts of Sec. 9. The minimum information required should include, but not be limited to, that in Table 9-27. Forms and spreadsheets are the most succinct method to present the information. The forms should state clearly the fund amounts and the date that each estimate was performed. The forms may be developed so that data for other scenarios may be reported by extending the tables to the right of the page. It is suggested that blank lines be included for any additional information. Finally the engineer preparing the feasibility analysis should make recommendations based upon management’s guidelines. The development of the information required for Table 9-27 was discussed previously in Sec. 9 with the exception of marketing information. An important document for a feasibility analysis is the marketing data so that the latest income projections can be included for management’s consideration. As a minimum, the tabulation of sales volume, sales prices, and market share both domestically and globally should be included. Table 9-28 shows a sample of such marketing information. Other templates may be prepared for total capital investment, working capital, total product expense, general overhead expense, and cash flow. Table 9-29 may be used to organize cash flow data by showing investment, operating expenses, cash flow, and cumulative cash flow.

OTHER ECONOMIC TOPICS COMPARISON OF ALTERNATIVE INVESTMENTS Engineers are often confronted with making choices between alternative equipment, designs, procedures, plans or methods. The courses of action require different amounts of capital and different operating expenses. Some basic concepts must be considered before attempting

to use mathematical methods for a solution. It is necessary to clearly define the alternatives and their merits. Flow of money takes the form of expenditures or income. Savings from operations are considered as income or as a reduction in operating expenses. Income taxes and inflation as well as a reasonable return on the investment must be included. Money spent is negative and money earned or saved is positive.

9-36

PROCESS ECONOMICS

Expenditures are of two kinds; instantaneous like land, working capital and capital recovery or uniformly continuous for plant investment, operating expenses, etc. A methodology involving after-tax cash flow is developed to reduce all the above to a manageable format. In an earlier part of this section, after-tax cash flow was defined as CF = (R − C − D)(1 − t) + D where

(9-17)

CF = after-tax cash flow D = depreciation t = tax rate S = sales or revenue C = cash operating expenses (COE)

For the situation in which each case will produce the same revenue or the same benefit, R will equal 0. Rearranging Eq. (9-17) algebraically yields or

CF = (t)(D) + (1 − t) (S − C)

(9-17a)

CF = (t)(D) + (1 − t) (−C)

(9-17b)

CF = (t)(D) − (1 − t) (C)

(9-17c)

This expression is applied to each alternative. [Note: As mentioned under cash flow, the first term in the above equations, (t)(D), is the result of an algebraic rearrangement, and no other significance should be assumed.] Several methods are available for determining the choice among alternatives: Net present worth Rate of return Capitalized cost Cash flow Uniform annual cost Humphreys in Jelen and Black, Cost and Optimization Engineering (1991), has shown that each of these methods would result in the same decision, but the numerical results will differ. Net Present Worth Method The NPW method allows the conversion of all money flows to be discounted to the present time. Appropriate interest factors are applied depending on how and when the cash flow enters a venture. They may be instantaneous, as in the purchase of capital equipment, or uniform, as in operating expenses. The alternative with the more positive NPW is the one to be preferred. In some instances, the alternatives may have different lives so the cost analysis must be for the least common multiple number of years. For example, if alternative A has a 2-year life and alternative B has a 3-year life, then 6 years is the least common multiple. The rate of return, capitalized cost, cash flow, and uniform annual cost methods avoid this complication. Rate of return and capitalized cost methods are discussed at length in Humphreys (1991). Cash Flow Method Cash flows for each case are determined, and the case that generates the greater cash flow is the preferred one. Uniform Annual Cost (UAC) Method In the uniform annual cost method, the cost is determined over the entire estimated project life. The least common multiple does not have to be calculated, as in the NPW method. This is the advantage of the UAC method; however, the result obtained by this method is more meaningful than the results obtained by other methods. The UAC method begins with a calculation for each alternative. If discrete interest is used, the annual cost C is found by multiplying the present worth P by the appropriate discrete interest factor, found in Table 9-21, for the number of years n and the interest rate i. If continuous interest is preferred, the UAC equation is NPW UAC =  (years of life) (continuous interest factor)

(9-23)

The continuous interest factor may be found from continuous interest equations or from the continuous interest table, Table 9-30. In this table time zero is the present, and all cash flows are discounted back to the present. Note that there are three sections to this table,

depending on the cash flow: uniform, instantaneous, or declining uniformly to zero. One enters the table with the argument R × T, where R is the interest rate expressed as a whole number and T is the time in years to obtain a factor. This factor is then used to calculate the present worth of the cash flow item. All cash flows are summed algebraically, giving the net present worth which is substituted in Eq. (9-23). This procedure is followed for both alternatives, and the alternative that yields the more positive UAC (or the least negative) value is the preferred alternative. In Eq. (9-23) the “factor” is always the uniform factor that annualizes all the various cash flows. This method of comparing alternatives is demonstrated in Example 10. Example 10: Choice among Alternatives Two filters are considered for installation in a process to remove solids from a liquid discharge stream to meet environmental requirements. The equipment is to be depreciated over a 7-year period by the straight-line method. The income tax rate is 35 percent, and 15 percent continuous interest is to be used. Assume that the service life is 7 years and there is no capital recovery. Data for the two systems are as follows: System

B

C

Fixed investment Annual operating expenses

$18,000 14,200

$30,000 4,800

Which alternative is preferred? Solution: System B: Year 0 0–7 0–7

Item

Cash flow, $

Factor

PW, $

Investment Contribution to cash flow from depreciation Contribution to cash flow from operating expense

−18,000 (0.35)(18,000)

1.0 0.6191

−18,000 +3,900

(1 − 0.35)(7)(14,200)

0.6191

−40,000

NPW B

−54,100

−$54,100 NPW UACB =  =  = −$12,484 (years of life)(uniform factor) (7)(0.6191) System C: Year

Item

Cash flow, $

Factor

PW, $

0 0–7

Investment Contribution to cash flow from depreciation Contribution to cash flow from operating expense

−30,000 (0.35)(30,000)

1.0 0.6191

−30,000 +6,500

(1 − 0.35)(10)(4,800)

0.6191

−19,316

NPW C

−42,816

0–10

−$42,816 NPW UACC =  =  = −$6,916 (years of life)(uniform factor) (10)(0.6191)

Alternative C is preferred because it has the more positive UAC. REPLACEMENT ANALYSIS During the lifetime of a physical asset, continuation of its use may make it a candidate for replacement. In this type of analysis, a replacement is intended to supplant a similar item performing the same service without plant or equipment expansion. In a chemical plant, replacement usually refers to a small part of the processing equipment such as a heat exchanger, filter, or compressor. If the replacement is required due to “physical” deterioration, there is no question of whether to replace the item, but the entire plant may be shut down if it is not replaced. The problem then becomes whether the equipment

OTHER ECONOMIC TOPICS TABLE 9-30 R×T

9-37

Factors for Continuous Discounting 0

1

2

3

4

5

6

7

8

9

.9754 .9286 .8848 .8437 .8053 .7692 .7353 .7035 .6736 .6455 .6191 .5942 .5708 .5487 .5279 .5082 .4897 .4721 .4555 .4399 .4250 .4109 .3976 .3849 .3729 .3615 .3507 .3404 .3306 .3212 .3123 .3039 .2958 .2880 .2807 .2736 .2669 .2604 .2542 .2483 .2426

.9706 .9241 .8806 .8398 .8016 .7657 .7320 .7004 .6707 .6428 .6166 .5918 .5685 .5466 .5259 .5063 .4879 .4704 .4539 .4383 .4236 .4096 .3963 .3837 .3718 .3604 .3496 .3394 .3296 .3203 .3115 .3030 .2950 .2873 .2799 .2729 .2662 .2598 .2536 .2477 .2421

.9658 .9196 .8764 .8359 .7979 .7622 .7288 .6974 .6679 .6401 .6140 .5894 .5663 .5444 .5239 .5044 .4861 .4687 .4523 .4368 .4221 .4082 .3950 .3825 .3706 .3593 .3486 .3384 .3287 .3194 .3106 .3022 .2942 .2865 .2792 .2722 .2655 .2591 .2530 .2471 .2415

.9610 .9152 .8722 .8319 .7942 .7588 .7256 .6944 .6650 .6374 .6115 .5871 .5640 .5423 .5219 .5025 .4843 .4671 .4507 .4353 .4207 .4069 .3937 .3813 .3695 .3582 .3476 .3374 .3277 .3185 .3098 .3014 .2934 .2858 .2785 .2715 .2649 .2585 .2524 .2466 .2410

.9563 .9107 .8681 .8281 .7906 .7554 .7224 .6913 .6622 .6348 .6090 .5847 .5618 .5402 .5199 .5007 .4825 .4654 .4492 .4338 .4193 .4055 .3925 .3801 .3683 .3571 .3465 .3364 .3268 .3176 .3089 .3006 .2926 .2850 .2778 .2709 .2642 .2579 .2518 .2460 .2404

.9512 .8607 .7788 .7047 .6376 .5769 .5220 .4724 .4274 .3867 .3499 .3166 .2865 .2592 .2346 .2122 .1920 .1738 .1572 .1423 .1287 .1165 .1054 .0954 .0863 .0781 .0707 .0639 .0578 .0523 .0474 .0429

.9418 .8521 .7711 .6977 .6313 .5712 .5169 .4677 .4232 .3829 .3465 .3135 .2837 .2567 .2322 .2101 .1901 .1720 .1557 .1409 .1275 .1153 .1044 .0944 .0854 .0773 .0699 .0633 .0573 .0518 .0469 .0424

.9324 .8437 .7634 .6907 .6250 .5655 .5117 .4630 .4190 .3791 .3430 .3104 .2808 .2541 .2299 .2080 .1882 .1703 .1541 .1395 .1262 .1142 .1033 .0935 .0846 .0765 .0693 .0627 .0567 .0513 .0464 .0420

.9231 .8353 .7558 .6839 .6188 .5599 .5066 .4584 .4148 .3753 .3396 .3073 .2780 .2516 .2276 .2060 .1864 .1686 .1526 .1381 .1249 .1130 .1023 .0926 .0837 .0758 .0686 .0620 .0561 .0508 .0460 .0416

.9139 .8270 .7483 .6771 .6126 .5543 .5016 .4538 .4107 .3716 3362 .3042 .2753 .2491 .2254 .2039 .1845 .1670 .1511 .1367 .1237 .1119 .1013 .0916 .0829 .0750 .0679 .0614 .0556 .0503 .0455 .0412

Uniform 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400

1.0000 .9516 .9063 .8639 .8242 .7869 .7520 .7192 .6883 .6594 .6321 .6065 .5823 .5596 .5381 .5179 .4988 .4808 .4637 .4476 .4323 .4179 .4042 .3912 .3789 .3672 .3560 .3455 .3354 .3259 .3167 .3080 .2998 .2919 .2843 .2771 .2702 .2636 .2573 .2512 .2454

.9950 .9470 .9020 .8598 .8204 .7833 .7486 .7160 .6854 .6566 .6295 .6040 .5800 .5574 .5361 .5160 .4970 .4790 .4621 .4460 .4309 .4165 .4029 .3899 .3777 .3660 .3550 .3445 .3344 .3249 .3158 .3072 .2990 .2911 .2836 .2764 .2695 .2629 .2567 .2506 .2449

.9901 .9423 .8976 .8558 .8166 .7798 .7453 .7128 .6824 .6538 .6269 .6015 .5777 .5552 .5340 .5140 .4951 .4773 .4604 .4445 .4294 .4151 .4015 .3887 .3765 .3649 .3539 .3434 .3335 .3240 .3150 .3064 .2982 .2903 .2828 .2757 .2688 .2623 .2560 .2500 .2443

.9851 .9377 .8933 .8517 .8128 .7762 .7419 .7097 .6795 .6510 .6243 .5991 .5754 .5530 .5320 .5121 .4933 .4756 .4588 .4429 .4279 .4137 .4002 .3874 .3753 .3638 .3528 .3424 .3325 .3231 .3141 .3055 .2974 .2896 .2821 .2750 .2682 .2617 .2554 .2495 .2437

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310

1.0000 .9048 .8187 .7408 .6703 .6065 .5488 .4966 .4493 .4066 .3679 .3329 .3012 .2725 .2466 .2231 .2019 .1827 .1653 .1496 .1353 .1225 .1108 .1003 .0907 .0821 .0743 .0672 .0608 .0550 .0498 .0450

.9900 .8958 .8106 .7334 .6637 .6005 .5434 .4916 .4449 .4025 .3642 .3296 .2982 .2698 .2441 .2209 .1999 .1809 .1637 .1481 .1340 .1212 .1097 .0993 .0898 .0813 .0735 .0665 .0602 .0545 .0493 .0446

.9802 .8869 .8025 .7261 .6570 .5945 .5379 .4868 .4404 .3985 .3606 .3263 .2952 .2671 .2417 .2187 .1979 .1791 .1620 .1466 .1327 .1200 .1086 .0983 .0889 .0805 .0728 .0659 .0596 .0539 .0488 .0442

.9704 .8781 .7945 .7189 .6505 .5886 .5326 .4819 .4360 .3946 .3570 .3230 .2923 .2645 .2393 .2165 .1959 .1773 .1604 .1451 .1313 .1188 .1075 .0973 .0880 .0797 .0721 .0652 .0590 .0534 .0483 .0437

.9803 .9332 .8891 .8477 .8090 .7727 .7386 .7066 .6765 .6483 .6217 .5967 .5731 .5509 .5299 .5101 .4915 .4738 .4572 .4414 .4265 .4123 .3989 .3862 .3741 .3627 .3518 .3414 .3315 .3222 .3132 .3047 .2966 .2888 .2814 .2743 .2675 .2610 .2548 .2489 .2432 Instantaneous .9608 .8694 .7866 .7118 .6440 .5827 .5273 .4771 .4317 .3906 .3535 .3198 .2894 .2618 .2369 .2144 .1940 .1755 .1588 .1437 .1300 .1177 .1065 .0963 .0872 .0789 .0714 .0646 .0584 .0529 .0478 .0433

9-38

PROCESS ECONOMICS

TABLE 9-30 R×T

Factors for Continuous Discounting (Concluded) 0

1

2

3

4

5

6

7

8

9

.0388 .0351 .0317 .0287 .0260 .0235 .0213 .0193 .0174

.0384 .0347 .0314 .0284 .0257 .0233 .0211 .0191 .0172

.0380 .0344 .0311 .0282 .0255 .0231 .0209 .0189 .0171

.0376 .0340 .0308 .0279 .0252 .0228 .0207 .0187 .0169

.0373 .0337 .0305 .0276 .0250 .0226 .0204 .0185 .0167

.9836 .9518 .9216 .8929 .8655 .8393 .8144 .7906 .7679 .7462 .7255 .7057 .6867 .6686 .6512 .6345 .6186 .6033 .5886 .5745 .5610 .5480 .5355 .5234 .5119 .5008 .4900 .4797 .4698 .4602 .4509 .4420 .4334 .4251 .4170 .4092 .4017 .3945 .3874 .3806 .3740

.9803 .9487 .9187 .8901 .8628 .8368 .8120 .7883 .7657 .7441 .7235 .7038 .6849 .6668 .6495 .6329 .6170 .6018 .5872 .5731 .5596 .5467 .5342 .5223 .5108 .4997 .4890 .4787 .4688 .4592 .4500 .4411 .4325 .4242 .4162 .4085 .4010 .3937 .3867 .3799 .3734

.9771 .9457 .9158 .8873 .8601 .8343 .8096 .7860 .7635 .7420 .7215 .7018 .6830 .6650 .6478 .6313 .6155 .6003 .5857 .5718 .5583 .5454 .5330 .5211 .5096 .4986 .4879 .4777 .4678 .4583 .4491 .4402 .4317 .4234 .4154 .4077 .4003 .3930 .3860 .3793 .3727

.9739 .9426 .9129 .8845 .8575 .8317 .8072 .7837 .7613 .7399 .7195 .6999 .6812 .6633 .6461 .6297 .6139 .5988 .5843 .5704 .5570 .5442 .5318 .5199 .5085 .4975 .4869 .4767 .4669 .4574 .4482 .4394 .4308 .4226 .4147 .4070 .3995 .3923 .3854 .3786 .3721

.9707 .9396 .9100 .8817 .8549 .8292 .8048 .7814 .7591 .7378 .7175 .6980 .6794 .6615 .6445 .6281 .6124 .5973 .5829 .5690 .5557 .5429 .5306 .5188 .5074 .4964 .4859 .4757 .4659 .4564 .4473 .4385 .4300 .4218 .4139 .4062 .3988 .3916 .3847 .3779 .3714

Instantaneous 320 330 340 350 360 370 380 390 400

.0408 .0369 .0334 .0302 .0273 .0247 .0224 .0202 .0183

.0404 .0365 .0330 .0299 .0271 .0245 .0221 .0200 .0181

.0400 .0362 .0327 .0296 .0268 .0242 .0219 .0198 .0180

.0396 .0358 .0324 .0293 .0265 .0240 .0217 .0196 .0178

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400

1.0000 .9675 .9365 .9071 .8790 .8522 .8267 .8024 .7792 .7570 .7358 .7155 .6961 .6776 .6598 .6428 .6265 .6109 .5959 .5815 .5677 .5544 .5417 .5294 .5176 .5063 .4953 .4848 .4747 .4649 .4555 .4464 .4376 .4292 .4210 .4131 .4055 .3981 .3909 .3840 .3773

.9968 .9643 .9335 .9042 .8763 .8496 .8242 .8000 .7769 .7548 .7337 .7135 .6942 .6757 .6581 .6411 .6249 .6093 .5944 .5801 .5663 .5531 .5404 .5282 .5165 .5052 .4943 .4838 .4737 .4640 .4546 .4455 .4368 .4283 .4202 .4123 .4047 .3973 .3902 .3833 .3766

.9934 .9612 .9305 .9013 .8735 .8470 .8218 .7977 .7746 .7526 .7316 .7115 .6923 .6739 .6563 .6395 .6233 .6078 .5929 .5787 .5650 .5518 .5392 .5270 .5153 .5041 .4932 .4828 .4727 .4630 .4537 .4446 .4359 .4275 .4194 .4115 .4040 .3966 .3895 .3826 .3760

.9902 .9580 .9275 .8985 .8708 .8445 .8193 .7953 .7724 .7505 .7296 .7096 .6905 .6721 .6546 .6378 .6217 .6063 .5915 .5773 .5636 .5505 .5379 .5258 .5142 .5029 .4922 .4818 .4717 .4621 .4527 .4438 .4351 .4267 .4186 .4108 .4032 .3959 .3888 .3820 .3753

.0392 .0354 .0321 .0290 .0263 .0238 .0215 .0194 .0176

Declining Uniformly to 0

SOURCE:

.9867 .9549 .9246 .8957 .8681 .8419 .8169 .7930 .7702 .7484 .7275 .7076 .6886 .6704 .6529 .6362 .6202 .6048 .5900 .5759 .5623 .5492 .5367 .5246 .5130 .5018 .4911 .4807 .4707 .4611 .4518 .4429 .4342 .4259 .4178 .4100 .4025 .3952 .3881 .3813 .3747

Adapted and abridged from Couper, 2003.

should be replaced like for like or whether an alternative should be chosen that may be different in cost and/or efficiency. If the replacement is due to technical obsolescence, the timing of the replacement may be important, especially if a plant expansion may be imminent in the near future. Whatever the situation, the replaced item should not present a bottleneck to the processing. The engineer should understand replacement theory to determine if alternative equipment is adequate for the job but with different costs and timing. Certain terminology has been developed to identify the equipment under consideration. The item in place is called the defender, and the candidate for replacement is called the challenger. This terminology and methodology was reported by E. L. Grant and W. G. Ireson in Engineering Economy, Wiley, New York, 1950. To apply this method, there are certain rules. The value of the defender asset is a sunk cost and is irrelevant except insofar as it affects cash flow from depreciation for the rest of its life and a tax credit for the book loss if it is

replaced sooner than its depreciation life. A capital cost for the defender is the net capital recovery forgone and the tax credit from the book loss of the defender asset that was not realized. The UAC method will be used and will be computed for each case, using the time period most favorable to each. For the defender it is 1 year and for the challenger it is the full economic life. The UAC for the challenger is handled in the same manner as in the comparison of alternatives. The method is demonstrated in Example 11. Example 11: Replacement Analysis A 3-year-old reciprocating compressor is being considered for replacement. Its original cost was $150,000, and it was being depreciated over a 7-year period by the straight-line method. If it is replaced now, the net proceeds from its sale are $50,000, and it is believed that 1 year from now they will be $35,000. A new centrifugal compressor can be installed for $160,000, which would save the company $2000 per year in operating expenses for the 10-year life. At the end of the 10th year, its net proceeds are estimated to be zero. The 7-year depreciation applies also to the centrifugal

OTHER ECONOMIC TOPICS compressor. A 35 percent tax rate may be assumed. The company requires a 15 percent after-tax return on an investment of this type. Should the present compressor be replaced now? Solution: The UAC method will be used as a basis for comparison. It is assumed that all money flows are continuous, and continuous interest will be used. Defender case: The basis for this unit will be 1 year. If it is not replaced now, the rules listed above indicate that there is an equivalent of a capital cost for two benefits forgone (given up). They are 1. Net proceeds now at 3 years of $50,000 2. Tax credit for the loss not realized Thus net loss forgone = book value at the end of 3 years minus net capital recovery, or NLF3 = BV3 − NCR3 where

NLF3 = net loss forgone at end of 3 years BV3 = book value at end of 3 years NCR3 = net capital recovery at end of 3 years





1 NLF3 = $150,000 1 −  − $50,000 = $35,714 7



1 Depreciation for 4th year = $150,000  = $21,429 7





4 NLF4 = BV4 − NCR4 = $150,000 1 −  − $35,000 = $29,286 7 Year

Item

At 0a At 0a 0–1 0–1 End 1 End 1

Cash flow, $

Tax credit for net loss (0.35)(−35,714) forgone Net cash recovery forgone −50,000 Contribution to CF from (0.35)(−21,429) depreciation Contribution to CF from (1 − 0.35)(−15,000) operting expense Tax credit for net loss (0.35)(29,286) Net cash 35,000

Factor

PW, $

1.0

−12,500

1.0 0.9286b

−50,000 6,965

0.9286b

−9,054

c

0.8607 8,822 0.8607c 30,125 NPW −25,642 a For the defender case, 0 year is the end of the third year. b From Table 9-30, uniform section, for the argument R × T = 15 × 1 = 15. c From Table 9-30, instantaneous section, for the argument R × T = 15 × 1 = 15. NPW −$25,642 UAC =  =  = −$27,614 (years of life)(uniform factor) (1)(0.9286) Challenger case:

Year 0 0–7 0–10 d e

Item

Cash flow, $

Factor

PW, $

First cost Contribution to CF from depreciation Contribution

−160,000 (0.35)(160,000)

1.0 0.6191d

−160,000 56,000

(1 − 0.35)(10)(−13,000)

0.5179e NPW

−43,763 −147,763

From Table 9-30, uniform section, R × T = 15 × 7 = 105. From Table 9-30, uniform section, R × T = 15 × 10 = 150.

−$147,763 NPW UAC =  =  = −$28,531 (years of life)(uniform factor) (10)(0.5179) The UAC for the defender case is less negative (more positive) than that for the challenger case; therefore, the defender should not be replaced now. But there will be a time in the near future when the defender should be replaced, as maintenance and deterioration will increase.

OPPORTUNITY COST Opportunity cost refers to the cost or value that is forgone or given up because a proposed investment is undertaken, often used as a base case. Perhaps the term should be lost opportunity. For example, the profit from production in obsolete facilities is an opportunity cost of

9-39

replacing them with more efficient ones. In cost analysis on investments, an incremental approach is often used, and if it is applied properly, the correct cost analysis will result. ECONOMIC BALANCE An engineering cost analysis can be used to find either a minimum cost or a maximum profit for a venture. This analysis is called an economic balance since it involves the balancing of economic factors to determine optimum design or optimum operating conditions. Such an analysis involves engineering tradeoffs. It may be more beneficial to invest more capital to reduce operating expenses or, conversely, incur more operating expenses without the addition of costly capital. An economic balance, then, is a study of all costs, expenses, revenues, and savings that pertain to an operation or perhaps an equipment item size. In this presentation, certain terminology is used; e.g., the term cost refers to a one-time purchase of capital equipment. A recurring expense is called an operating expense, such as utilities, labor, and maintenance. All costs and operating expenses are related to an arbitrarily designated controllable variable such as heat exchanger area, thickness of insulation, or number of units. There are certain practical considerations that must be recognized in attempting to find the best or optimum condition. Occasionally, a solution may lead to a result for which industrial equipment is not available in the optimum size; therefore, engineering judgment must be exercised. A smaller-diameter pipe might lead to a higher pumping expense but lower pipe costs while a larger-diameter pipe would result in lower pumping expense but higher capital cost. The engineer then encounters an engineering tradeoff that must be resolved. The essential elements of an economic balance are • Fixed and variable expenses • An allowance for depreciation • An acceptable return on the investment A total expense equation is TE = FE + VE where TE = total expenses FE = fixed expenses VE = variable expenses

(9-24)

The guidelines for solving a single-variable economic balance consist of the following steps: 1. Determine all expenses that vary as the controllable variable changes and that need to be considered in the balance. 2. Determine whether any operating limitations exist, such as pressure drop in columns where flooding occurs or limiting heads for pipelines for gravity flow. 3. Mathematically express the expenses as a function of the variables related to the equipment; otherwise use variables that define the operation such as temperature, pressure, and concentration. The final expression should include all pertinent expenses, eliminating those that are not significant. Frequently only one variable is used. 4. Ascertain if the optimum size must be one of a number of discrete sizes commercially available or whether it can be any size. 5. Solve the total expense equation either analytically or graphically. The solution to Eq. (9-24) may be found analytically or graphically. In the analytical method, the fixed and variable expenses are related in equation format to the controllable variable. Example 12 is an engineering balance example of this method. Equation (9-24) can be differentiated with respect to the controllable variable and that result set equal to zero to find the optimum condition. A graphical method involves determining the fixed and variable expenses for a range of equipment sizes. A plot of TE, FE, and VE as a function of the controllable sizes yields a plot identifying the optimum. Figure 9-18 is a graphical solution to the TE equation. Should the optimum result fall between two commercial sizes, then the engineer must exercise judgment in the tradeoff. Example 12: Optimum Number of Evaporator Effects Determine the number of evaporator effects for the minimum total annual operating

9-40

PROCESS ECONOMICS Basis: 1 year of operation, duty 108 Btu/day. Let N equal the number of effects. Solution: Analytical method:

140

Annual fixed expenses = FE

120



FE = $40,000N 0.143 + 0.100 +

0.15  1 − 0.35

= $18,960N

dep. maint. profit (return)

TE

Expenses, $

100

Annual variable expenses VE in this example are essentially the steam expenses, and all other variable expenses do not enter into the equation. Therefore $112,500 VE = 108 Btu/day × 300 day/yr × $3.00/106 Btu × 1/(0.8N) =  N

80

$112,500 TE = FE + VE = $18,960N +  N

VE

Take the derivative of TE with respect to N and set the result equal to zero.

FE

60

d (TE) 112,500 =0  = 18,960 −  dN N2

N = 2.44 effects

40

Therefore 3 effects are required. Graphical method: Assume the number of effects and calculate the fixed, variable, and total expenses for effect.

20

N

18,960N

112,500/N

TE

1 2 3 4

$18,960 37,920 56,880 75,840

$112,500 56,250 37,500 28,125

$131,460 94,210 94,380 103,965

1

2

3

4

5

Number of effects FIG. 9-18

The minimum TE occurs between 2 and 3 effects, but 3 effects are recommended to evaporate the colloidal solution. Figure 9-18 is a plot of the graphical solution.

Optimum number of evaporator effects in Example 12.

expense of a small evaporator system to concentrate a colloidal suspension. Steam costs $3.00 per million Btu (MBtu), and each pound of steam will evaporate 0.8N lb water, with N being the number of effects. The total capital cost of each effect is $40,000 and has an estimated life of 10 years. The annual maintenance expense is 10 percent of the capital cost. Labor and other expenses not mentioned may be considered to be independent of the number of effects. The system will operate 300 days/yr with 100,000,000 Btu/day evaporator duty. Depreciation on the equipment is by 7-year straight-line method, and the tax rate is 35 percent. Annual net profit after taxes on the investment must be 15 percent for the installed equipment.

In another example involving the reclaiming of a product using an evaporator and a dryer in series, the product is pumped from the discharge of the evaporator to the dryer. See Fig. 9-19. An economic analysis indicated that a 55 percent slurry is optimum, but perhaps such a slurry is too thick to be pumped. Therefore, engineering judgment must be exercised. The slurry was pumped not at 55 percent but at a lesser concentration of 50 percent, although it was not optimum. When more than one controllable variable affects the economic balance, the solution approach is essentially the same as that for the single-variable case, but determining the optimum is tedious.

6000 lb/day 15 wt% aqueous solution; s = 0.15

Evaporator

Water evaporated 6000 − (900/s) lb/day

Water vaporized in dryer (900/s) − 900 lb/day

900 lb/day dry product 900/s lb/day FIG. 9-19

Dryer

Evaporator–dryer system. (Source: Adapted from Couper, 2003.)

CAPITAL PROJECT EXECUTION AND ANALYSIS INTERACTIVE SYSTEMS If the TE does not pass through a minimum or maximum, but continues to decline or to increase with the number of equipment items or equipment size, the next step is to look at the flow sheet for equipment upstream or downstream from the selected item. It may be necessary to group two or more items and treat them as one in the analysis. Such a system is said to be interactive, since more than one item affects the optimum results. An example of such an interactive system is the

9-41

removal of nitrogen from helium in a natural gas stream. Carbon adsorption is a method for removing nitrogen, but compressors are also required since this is a high-pressure process. If one attempts to find the optimum operating pressure, optimizing on compressor pressure will not result in an optimum condition; and conversely, optimizing on the size of the carbon bed will not yield an optimum. This is an example of an interactive system. Therefore, to find the optimum pressure, both the size of the carbon bed and the compressor pressure must be considered together.

CAPITAL PROJECT EXECUTION AND ANALYSIS The demands made on business organizations with the arrival of global free trade have made sophisticated management of capital projects, for the purposes of minimizing capital costs and maximizing project profitability, a necessity. Two elements of such sophisticated project management practice are front-end loading (FEL) and valueimproving practices (VIPs). These two management practices are, and must be, closely integrated activities, as seen in Fig. 9-20. As shown, they are performed during the early stages of a project life cycle, when they can be, and are, effective at influencing a project’s profitability. However, they have very different characteristics, as are detailed in the following sections. Properly performed together, FEL and VIPs maximize project profitability by ensuring that all matters that influence project profitability are considered in the most productive manner and at the most optimal time. FRONT-END LOADING GENERAL REFERENCES: Porter, James B., E. I. DuPont de Nemours and Company, DuPont’s Role in Capital Projects, Proceedings of Government/Industry

Forum: The Owner’s Role in Project Management and Pre-project Planning, 2002. Smith, C. C., Improved Project Definition Insures Value-added Performance—Part 1, Hydrocarbon Processing, August 2000, pp. 95–99. KBR Frontend Loading Program, data compiled from selected large projects from 1993 through 2003, http://www.halliburton.com/kbr/index.jsp. Merrow, E. W., Independent Project Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000. Merrow, E. W., Independent Project Analysis, Inc., 30th Annual Engineering & Construction Contracting Conference, September 1998. HPI Impact, Hydrocarbon Processing, August 2002, p. 23, data obtained from Independent Project Analysis, Inc. PDRI: Project Definition Rating Index— Industrial Projects, p. 5, Construction Industry Institute (CII), University of Texas at Austin, http://construction-institute.org, July 1996. Merrow, E. W., Independent Project Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000.

Introduction Front-end loading (FEL) is the process by which a company develops a detailed definition of the scope of a capital project that meets corporate business objectives. The term frontend loading was first coined by the DuPont company in 1987 and has been used throughout the chemical, refining, and oil and gas industries ever since (Porter, James B., E. I. DuPont de Nemours and

FRONT-END-LOADING PHASE

Potential to Influence Project Costs


Classes of Facility Quality
Technology Selection
Constructability: FEL-1
Value Engineering: FEL-1


Value Engineering
Design to Capacity: Equip. Level
Constructability: FEL-3
Process Reliability Simulation (RAM): Equip. Level
3D-CAD


Constructability: EPC
Customization of Standards & Specifications
Design to Capacity FEL-2: Unit Operation Level
Process Simplification
Constructability: FEL-2
Energy Optimization
Waste Minimization
Predictive Maintenance
Process Reliability Simulation (RAM): Block Level

Conceptual FEL-1 FIG. 9-20

EPC PHASE

Feasibility FEL-2

Definition FEL-3

Detailed Design

Front-end loading and the implementation of value-improving practices.

Construction Turnover & Start-up

9-42

PROCESS ECONOMICS

Front-End Loading (FEL)

Conceptual Phase (FEL-1)

Feasibility Phase (FEL-2)

Definition Phase (FEL-3)

= Decision Gates FIG. 9-21

Engineering, Procurement, Construction (EPC) Phase

Operate Phase

= Project Authorization

FEL in the capital project life cycle.

Company). The product of the FEL process is a design-basis package of customized information used to support the production of detailed engineering design documents. Completion of the FEL design-basis package typically coincides with project AFE (Authorization for Expenditure) or project authorization. Project authorization is that point in the project life cycle where the owner organization commits the majority of the project’s capital investment and contracts. FEL starts when an idea for a project is first conceived by a research and development group, project engineering group, plant group, or business unit. FEL activity continues until the project is authorized. After initial conception of an idea, organized interaction is required among the various project stakeholders to assemble the project design-basis package for subsequent authorization. Within the FEL phases, decision points are formally established by the operating company authorizing the initiation of a capital project development effort. These formal decision gates allow for continuity across the enterprise for authorization of additional funding for the next phase of engineering and project definition. Figure 9-21 illustrates the typical decision gates or stage gates for capital projects. When the level of project definition is sufficient to support a definitive cost estimate for both the entire project and its projected rate of return, major project funding is authorized for expenditure. This is the conclusion of the FEL process and any significant involvement of the process design engineer. Not until the conclusion of the engineering, procurement, and construction (EPC) phase does the process engineer again become involved. At this time, commissioning and start-up become the focus where the validation occurs for all that was done in the FEL phases many months earlier. Differing terminology used by companies, engineers, and project management teams is often a point of confusion. Most people seem to think they know what all the terms mean. This is never the case. Confirmation of which terminology will be used by all involved in the project is a must. Nearly every operating company and engineering contractor use differing terminology. FEL terminology is often misunderstood and further confused by differing references to which FEL phase the project is actually in. Figure 9-22 provides some idea of the differing terminology for each project phase used by only a few major oil and gas and chemical companies today. These terms change periodically, so diligence in confirming such terminology is a key task for the process engineers to finish, before beginning their work. The influence of changes on capital projects is considerably affected by when those changes occur. The earlier a change is considered and incorporated into the project scope, the greater its potential influence on the project’s profitability and the greater the ease of

incorporating the change. This means that late changes (e.g., in the EPC phase) are far more expensive to implement and are considered very undesirable. Late changes which are potentially advantageous are often not implemented because the cost to implement the change exceeds the benefits of doing so. Conversely, the cost to implement a change at the earlier phases of the project is far lower than making the same change after detailed engineering is underway. Figure 9-23 shows how quickly this influence curve changes as the typical project progresses (Smith, C. C., Improved Project Definition Insures Value-added Performance—Part 1). This is why proactively seeking changes during FEL is far more advantageous to profitability than is allowing those needed changes to be “discovered” during later project phases. This also means that potentially beneficial changes (value improvements) must be sought during FEL, or else they stand a good chance of not being cost-effective to implement during the EPC phase. This is also why seeking operations, maintenance, and construction experience during FEL offers significant profitability advantages over practices which bring such experience onto the project team following FEL. Characteristics of FEL Front-end loading is a specialized and adaptable work process. This work process translates financial and marketing opportunities to a technical reality in the form of a capital project. It is particularly important that the project be defined in sufficient detail by the engineering deliverables, which are generated by the FEL work process, prior to the point where major funds are authorized. In this manner, overall project risks are identified and sufficiently mitigated to have project funding approved. To achieve this important level of definition, critical decisions must be made and adhered to. In addition, the FEL project team should proactively seek value improvement alternatives and challenge the project premises, scope, and design until such time as implementation of those alternatives loses their profitability and/or technical advantage. By doing so, such value improvements will not develop into costly corrections, which surface later, during the EPC phase. Goals and Objectives of FEL The FEL work process must enable nearly constant consideration of changes as the work progresses. FEL phases must consider the long-term implications of every aspect of the design. Predictability of equipment and process system life cycle costs must always be balanced with operations and maintenance preferences, as well as the need for the project to maintain its profitability or ROI (return on investment). Additional important goals and objectives of FEL projects are as follows: • Develop a well-defined and acceptably profitable project. • Define the primary technical and financial drivers for capital project investment.

CAPITAL PROJECT EXECUTION AND ANALYSIS

9-43

FRONT-END LOADING COMPANY A APPRAISE

SELECT

DEFINE

EXECUTE

OPERATE

EVALUATE ALTERNATIVES

DEVELOPMENT

EXECUTE

OPERATE / EVALUATE

DEVELOPMENT

VALIDATION

IMPLEMENTATION

OPERATION

FEASIBILITY

AFD (AUTHORITY FOR DEVELOPMENT)

EPC / START-UP

OPERATION

FEASIBILITY

SCOPE FINALIZATION

EPC / START-UP

OPERATION

COMPANY B ASSESS OPPORTUNITIES

COMPANY C ASSESSMENT

COMPANY D APPRAISAL / CONCEPTUAL

COMPANY E CONCEPT DEVELOPMEN T FIG. 9-22

Project life cycle terminology.

• Challenge baseline premises, and purposely seek out and evaluate alternatives and opportunities. • Minimize changes during the EPC, turnover, and start-up phases. • Reduce project schedule and capital cost. • Reduce the business and project execution risk. • Balance project technical, financial, and operational profitability drivers.

RAPIDLY DECREASING INFLUENCE

LOW INFLUENCE

COST

INFLUENCE

MAJOR INFLUENCE

Comparison of FEL Projects with EPC Projects FEL projects are very different from EPC projects. Engineers and project managers having significant experience only with projects in the EPC phase often are unfamiliar with the significant differences between the philosophies and challenges of the FEL phase and the EPC phase of projects. One of the most important (but most subtle) aspects of FEL is the demand during FEL for more highly experienced staff and

Conceptual Feasibility Phase Phase FEL-1 FEL-2 FIG. 9-23

Definition Phase FEL-3

Project life cycle cost-influence curve.

Engineering, Procurement, Construction (EPC) Phase

9-44

PROCESS ECONOMICS

FEL

EPC

Undefined

Defined

Actively Seeks Changes

Actively Resists Changes

Impact of Change

Low

High

Opportunity for Change

High

Low

Typically Reimbursable

Typically Lump Sum

Value Improvement Potential

High

Low

Client Participation

Encouraged

Discouraged

Information-Driven

Deliverable-Driven

Project State Changes

Contract Type

Philosophy FIG. 9-24

FEL projects versus EPC projects.

more sophisticated analysis tools, as compared to EPC projects which have achieved a well-defined project prior to authorization. This is so because of the need in FEL to create, analyze, and implement improvements to what many might consider a “good” design. In spite of its relatively short duration, FEL proactively seeks to implement the best possible design. The nearly constantly changing environment requires people of many different disciplines and functions to work together to communicate effectively. A well-integrated team always seems to perform best during FEL, if FEL has wellestablished, informal, and personal interfaces between project groups and organizations. The following describes how the FEL phase is distinguished from the EPC phase: • FEL proactively seeks data, resources, support, and decision making. • Projects in the FEL phase place a higher level of importance on close and effective owner-contractor management interfaces. • FEL demands continuous realignment of client desires and requirements with contractor needs. • FEL requires greater development of personal relationships that result in respect and trust. • FEL demands significantly higher frequency of feedback of owner satisfaction. • FEL emphasizes elimination of low- or zero-value scope. • FEL improves the capital productivity of projects by using bestavailable technology. • FEL focuses on overall project profitability rather than on only cost, schedule, and workhours. • FEL focuses almost entirely on the owner’s business needs. Figure 9-24 lists further differences between FEL and EPC projects. Understanding these many differences is very important to the process engineer, in that awareness of them, and the driving forces behind them, will prepare the chemical engineer for the challenging and rewarding environment of FEL projects. Parameters of FEL Phases Important aspects of each phase of FEL are cost estimate accuracy, cumulative engineering hours spent, and the contingency assigned to the cost estimate. Figure 9-25 lists the typical parameters encountered industrywide (KBR Front-end Loading Program, data compiled from selected large projects from 1993 through 2003). For the capital cost estimate, each operating company may request a slightly different accuracy, which is often project-specific. What is important is the level of engineering required to support such estimating accuracy. This determination is the responsibility of both the owner and the engineering contractor. Agreement on this is critical prior to initiating project work. The engineering hours spent during each phase of FEL vary widely between small and large projects. This is also true for those projects where new or emerging technology is being applied or where higher

throughput capacities are being applied than previously commercially demonstrated. Projects such as these may require additional engineering to achieve the desired estimate accuracy and project contingency. FEL Project Performance Characteristics Overall project performance can be enhanced by ensuring that the following characteristics are emphasized during the FEL phases. • Methodical business and project execution planning is necessary. • Effective integration of workforce between owner and contractor staff is important. • Projects with an integrated management team (owner and engineering contractor) have the lowest number of design changes at any project stage. • Engineering contractor should be brought into project in early FEL phases. • Clear roles must exist for project team members that relate to the expertise of both owner and contractor staff. • Effective personal communication is required between owner and contractor organizations and their project team representatives, ensuring extensive site and manufacturing input. • Schedule and cost goals are set by integrated business and technical project team composed of owner and contractor representatives. Figure 9-26 illustrates the benefit of good FEL performance on project costs (Merrow, E. W., Independent Project Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000). Figure 9-27 illustrates the benefit of good FEL performance on critical path schedule (Merrow, E. W., Independent

Cost Estimate Accuracy

FEL-1

FEL-2

FEL-3

(CONCEPTUAL)

(FEASIBILITY)

(DEFINITION)

± 40%

± 25%

±10%

Cumulative Engineering Hours Spent

1–5%

5–15%

15–30%

Contingency

15–20%

10–15%

8–12%

FIG. 9-25

Parameters of FEL phases.

CAPITAL PROJECT EXECUTION AND ANALYSIS

9-45

Relative Capital Cost (Industry Average = 1.0)

1.2

1.1

t = 6.3 p < 0.0001

1.0

Industry Average Cost

0.9

0.8 Best

Fair

Good

Poor

Screening

Level of FEL FIG. 9-26

FEL drives better cost performance.

Project Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000). IPA statistics indicate that significant project financial and schedule benefits can be realized by implementing a thorough FEL effort prior to the EPC phase. Figure 9-28 presents the benefits of having an integrated project team during FEL on the overall project performance. This performance impacts overall project costs as well as schedule and operability. An integrated project team produces fewer late changes. This means lower capital costs, better and more predictable schedules, and a slightly better operability, as compared statistically to similar projects lacking an integrated management team (Merrow, E. W., Independent Project Analysis, Inc., 30th Annual Engineering & Construction Contracting Conference, September 1998). In addition, project data indicate that a well-integrated FEL team can produce significantly better project performance in terms of lower capital investment, as compared to projects where FEL teams were not properly integrated. This illustrates the benefits for each engineering team member working closely together with each other team member, to produce the most profitable project results. Although project teams, once integrated and functioning with clear roles and responsibilities, perform better, this edge can be quickly lost

if key members of that team are changed. The impacts of changes of project managers to a well-integrated FEL team are shown in Fig. 9-29 (HPI Impact, Hydrocarbon Processing, p. 23, August 2002, data obtained from Independent Project Analysis, Inc.). Investment in FEL for Best Project Performance The cost and schedule required to optimally complete the FEL phase of a project are always under pressure and must be justified. This is especially true for “fast-track” projects where the time pressures can be significant. The Construction Industry Institute (CII) has shown that higher levels of preproject planning (i.e., front-end loading) effort can result in significant cost and schedule savings, as seen in Fig. 9-30 (PDRI: Project Definition Rating Index—Industrial Projects, Construction Industry Institute, University of Texas at Austin, July 1996). The process engineer produces the best project performance, when he or she strives, with the entire integrated FEL project team, to define the overall project (not just the process design) as well as possible, prior to AFE. The level of definition of a project during the FEL phases has a direct influence on the project’s ultimate outcome in terms of the number and impacts of changes in the EPC phase. This level of FEL performance translates to fewer major changes in engineering, construction, and during start-up (Merrow, E. W., Independent Project

Execution Schedule Index

1.4 1.2

t = 5.4 p < 0.0001

1.0 0.8 0.6 0.4 Best

Good

Fair

Level of FEL FIG. 9-27

Good FEL speeds execution time.

Poor

Screening

PROCESS ECONOMICS

Cost Index

Operability Index

0.5

0.5

1.5

0.6

0.6

1.4

0.7

0.7

1.3

0.8

0.8

1.2

0.9

1.1

0.9

1.0

Integrated Teams

Nonintegrated Teams

1.1

Nonintegrated Teams

SCREENING FIG. 9-28

Schedule Index

Integrated Teams

POOR

FAIR

GOOD

BEST

FEL Index

Integrated Teams

1.0

1.0

Nonintegrated Teams

1.1

0.9

1.2

1.2

0.8

1.3

1.3

0.7

1.4

1.4

0.6

1.5

1.5

0.5

Integrated Teams Nonintegrated Teams

BETTER

9-46

Integrated teams result In better FEL and better overall performance.

Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000). These conclusions are depicted by Fig. 9-31. A major late change is defined by IPA’s data to mean changes made after the start of detailed engineering and involving impacts greater than either 0.5 percent of the total project capital investment or 1 month in critical path project schedule. These graphs illustrate why better project performance is produced through proactively seeking profit-improving changes as early

as possible. One of the reasons for this observation is that operation, maintenance, and construction expertise is incorporated into the project at the very beginning—during FEL. This means that the process design engineer should be working closely with these real-world experts as they design processes and their support systems. This also means that to improve overall project performance, achieving the best practical or highest level of definition during FEL is critical. Finally, this high level of definition results in a

NO TURNOVERS

1 TURNOVER

2 TURNOVERS

3 TURNOVERS 30

20

10

0 Std. Dev.

FIG. 9-29

Effects of project manager turnover on cost growth.

10

20

30  Std. Dev.

40

50

CAPITAL PROJECT EXECUTION AND ANALYSIS LEVEL OF FEL EFFORT

COST

SCHEDULE

High

4%

13%

Medium

2%

8%

16%

26%

20%

39%

Low Difference FIG. 9-30

• • • • • • • • • •

Project performance versus level of FEL effort.

reduced number of changes during the EPC phase. These observations should be the critical goals of all project teams. The size of capital project also has an influence on FEL outcome based on IPA statistics. IPA’s data indicate that small projects benefit more from better project definition prior to the EPC phase than do major projects. The data also indicate that small projects typically have more late changes than do larger projects. Figure 9-32 illustrates the effect of large projects versus small projects in terms of the impact of late changes (Merrow, E. W., Independent Project Analysis, Inc., 32d Annual Engineering & Construction Contracting Conference, Sept. 28–29, 2000). Figure 9-32 also illustrates that the level of FEL performance directly impacts the number of, and the consequences of, late changes made in projects of any size. Typical FEL Deliverables Every process engineer assigned to a project should be acutely aware of which deliverables or end products are required by those who must commission their work. This should be very well understood by all parties prior to starting the work. Further, the splits of work (who will do which aspect of the work) must be well understood. Today, it is very common to have multiple operating companies form a joint venture to authorize major projects. It is also common for multiple engineering contractors to form joint ventures to execute the engineering for the FEL phases of the project. Typical Conceptual Phase (FEL-1) Deliverables These are listed below. Each project will customize these deliverables to suit the particular needs of the project. There is no such thing as a “standard” FEL. Therefore, the process engineer must understand what the details are for each deliverable needed, what the minimum level is for the engineering required to meet those requirements, and in which formats that information and data will be needed.

PERCENTAGE OF PROJECTS WITH MAJOR CHANGES

60 50

Strategic business assessment Key technology selected and risk identified Market assessment for feed, products, and capacity Potential sites identified and under evaluation Cost estimate (±40 percent) Preliminary project milestone schedule Block flow diagrams completed Process cases identified Critical long-lead equipment identified Value-improving practices reports Typical Feasibility Phase (FEL-2) Deliverables These are listed below. In this phase, emphasis is on determining the best technical and economic flow scheme, as well as the support systems required to provide the necessary annual production rate at the sales quality required. The focus for the process engineer should be on confirming the number and type of process and technology studies needed, as well as the number of alternate cases required to be evaluated and/or simulated. • Strategic business assessment • Project schedule level 1 • Cost estimate (±25 percent) • Overall project execution strategy • Contracting and purchasing strategies • Permitting and regulatory compliance plan • Soil survey and report • Project alternatives analysis • Process flow diagrams for selected option(s) • Preliminary utility flow diagrams and balances • Preliminary equipment list and equipment load sheets • Materials of construction • Process hazards analysis report • Value-improving practices reports Typical Definition Phase (FEL-3) Deliverables These are listed below. In this phase, emphasis is typically on optimizing the best flow scheme and support systems combination. This optimum includes consideration of the plot plan and equipment arrangements for the entire facility. Process optimization cannot be done in isolation. Significant and continuous interaction with operations, maintenance, and construction experts always produces the best results. The emphasis in this phase is on achieving the best practical level of project definition and a good-quality project estimate of +/−10 percent. This level of project definition and cost estimate quality is normally required in order to present to management a candidate project which has the right combination of overall risk and projected economic performance, and thereby secure an AFE. • Strategic business assessment • Detailed EPC phase project execution plan • Detailed EPC phase project master schedule

Change in Engineering Change in Construction Change in Start-up

40 30 20 10 0 Best

Fair

Good

LEVEL OF FEL FIG. 9-31

Good FEL drives late changes down.

9-47

Poor

Screening

PERCENTAGE OF PROJECTS EXPERIENCING LATE CHANGES

9-48

PROCESS ECONOMICS Introduction Value-improving practices (VIPs) are formal structured practices applied to capital projects to improve profitability (or “value”) above that which is attained through the application of proven good engineering and project management practices. VIPs are formal analyses of project characteristics and features performed by small multidisciplinary teams at identified optimum times during the engineering design and development of capital projects. Application of VIPs to capital projects has been statistically proved to significantly improve project profitability according to Independent Project Analysis, Inc. (IPA) and the Construction Industry Institute (CII). IPA data presented in Fig. 9-33 have been gathered from many capital projects since 1987. These data indicate that about 2.5 percent reduction in the relative capital cost can be expected for high-performing projects due to implementation of good front-end loading work processes. The bestperforming projects are often referred to as “Best Practical” or “Best in Class” projects and represent the upper 20 percent of projects. However, when FEL improvement is combined with rigorous application of VIPs, the project performance improves to about 10 percent in reduction of relative capital cost (Lavingia, Jr., N. J., Improve Profitability Through Effective Project Management and TCM). Experience of at least one major engineering contractor indicates that about a 20 percent capital cost improvement can be expected through judicious use of their modified VIPs (KBR Value Improving Practices Program, 1995 through 2005). These improved results come about from continual adaptation and improvement of the VIPs themselves to maintain their relevance and ability to improve projects above what the project teams can accomplish by themselves. The VIPs that have been statistically verified by IPA benchmarking of capital projects are listed below. Each has a different purpose and focus, but all produce project profitability improvements that the project team cannot achieve on its own. • Classes of facility quality • Technology selection • Process simplification • Constructability • Customization of standards and specifications • Energy optimization • Predictive maintenance • Waste minimization • Process reliability simulation • Value engineering • Design to capacity • 3D-CAD Selection of the most applicable VIPs to be performed during a specific project is the focus of the VIP planning session, which should be held just following project kickoff. Figure 9-34 presents the optimal times during a large project for consideration of VIPs. The duration for FEL has been assumed to be 12 months. Every project will have a

100% Major Projects

Small Projects

80% 60% 40% 20% 0%

Best

Good

Fair

Poor

Screening

LEVEL OF FEL FIG. 9-32

Rate of late major changes is higher for small projects.

• Completed environment permit submittal • Project plan/project execution plan a. Cash flow plan for EPC phase b. Training, commissioning, and start-up plans c. Contracting plans d. Materials management plan e. Safety process and quality management plan • Cost estimate (±10 percent) • Finalized utility flow diagrams and balances • P&ID’s—issue IPL (issue for plant layout) • Plot plans and critical equipment layouts • Equipment list and equipment data sheets • Single-line electrical diagrams • Control system summary and control room layout • Materials of construction • VIPs reports • Hazard and operability studies (HAZOP) report VALUE-IMPROVING PRACTICES GENERAL REFERENCES: Independent Project Analysis, Inc. (IPA), http://www.ipaglobal.com. The Construction Industry Institute (CII), University of Texas at Austin, http://construction-institute.org. Lavingia, Jr., N. J., Improve Profitability Through Effective Project Management and TCM, 36th Annual Engineering & Construction Contracting Conference, Sept. 4, 2003. KBR Value Improving Practices Program, 1995 through 2005, http://www.halliburton.com/ kbr/index.jsp. PDRI: Project Definition Rating Index—Industrial Projects, Construction Industry Institute (CII), University of Texas at Austin, http://construction-institute.org, July 1996. Society of American Value Engineers International (SAVE), http://www.value-eng.org. KBR experience.

Relative Capital Cost

1.20 1.15 1.10 1.05

FEL Improvement Only Industry Average Cost

1.00 0.95 0.90 0.85

FEL Improvement + Value Improving Practices

Best Practical

Good

Fair

FEL Rating FIG. 9-33

FEL and VIPs drive lower capital investment.

Poor

FIG. 9-34

Typical VIP implementation relationship.

9-49

9-50

PROCESS ECONOMICS

unique duration for FEL with each phase dictated by the owner organization. The best times to conduct the VIP workshops should be considered at the project outset. Key anchor points for VIPs are the first appearance of the process flow diagrams (PFDs). The second anchor point for VIPs is the issue of preliminary piping and instrumentation diagrams (P&IDs). When capital projects are benchmarked by third-party organizations such as IPA or through CII’s Project Definition Rating Index (PDRI: Project Definition Rating Index—Industrial Projects, Construction Industry Institute), the implementation of applicable VIPs to the project is a part of their analysis. All the VIPs, when properly implemented, focus on producing better project definition and resultant economic improvements. The level of project definition achieved during FEL phases of the project is the focus of such benchmarking efforts. VIPs as a group of practices are often described by their characteristics: • Out-of-the-ordinary practices are used to improve cost, schedule, and/or reliability of capital projects. • They are used primarily during FEL project phases. • Formal and documented practices involve repeatable work processes. • All involve a formal facilitated workshop to confirm the value gained to the project and to formally approve VIP team recommendations. • All involve stated explicit support from the owner’s corporate executive team. • VIPs must be performed by a trained experienced VIP facilitator— someone who is not a member of the project team • VIPs are characterized by statistical links between the use of the practice and better project performance which are demonstrated, systematic, repeatable, and proven correlations VIPs are also further clarified by what they are not, as below. • Just “good engineering” • Simple brainstorming or strategy sessions • Business as usual • A special look at some aspect of the project • Cost reduction or scope reduction exercises • PFD or P&ID reviews • Safety reviews • Audits • Project readiness reviews VIP Descriptions Classes of Facility Quality VIP The class of facility quality VIP determines the appropriate classes of facility quality that would produce the highest value or profitability in terms of • Capital investment (CAPEX) • Planned facility life • Expandability • Level of automation • Equipment selection, • Operating expense (OPEX) • Environmental controls • Capacity • Technology This VIP individually confirms the best overall design philosophy for the project team, for each of the parameters listed above. Here, the designer first learns how aggressive the owner organization wants the facility design and operation to be in terms of overall risk. For example, if the plant is to have the lowest possible OPEX, then the designer will incorporate greater levels of automation, instrumentation, and robustness of mechanical design in the overall facility. The results of this VIP are used by the project management team to update its project execution plan for each FEL phase. The class of facility quality VIP provides the best results when conducted prior to executing any other VIP effort in the conceptual phase (FEL-1) of the project. Technology Selection VIP The technology selection VIP is the application of evaluation criteria aligned with the project’s business objectives to identify manufacturing and processing technology that may be superior to that currently used. The goal is to ensure that the technology suite finally selected is the most competitive available. This requires a systematic search, both inside and outside the operating company’s organization, to identify emerging technology alternatives. This formal facilitated process is also meant to ensure due diligence for all parties involved and that all emerging and near-commercial alternative

technologies for accomplishing a particular processing function are objectively considered. This VIP is most commonly applied at the unit operation level, although it has also been successfully applied down to the major equipment level (KBR Value Improving Practices Program, 2000 through 2005). This VIP is particularly effective for combating the NIH syndrome (“not-invented-here”). The goals of this VIP are to document which technology evaluation criteria are applicable and then to conduct a formal technology screening and evaluation assessment. The result is a prioritized listing of technology options for each selected application for the project. The preferred time to execute this VIP is the midpoint in the conceptual phase (FEL-1). Process Simplification VIP The process simplification VIP uses the value methodology and is a formal, rigorous process to search for opportunities to eliminate or combine process and utility system steps or equipment, ultimately resulting in the reduction of investment and operating costs. The focus is the reduction of installed costs and critical path schedule while balancing these value improvements with expected facility operability, flexibility, and overall life cycle costs. The process simplification VIP does far more than just evaluate and simplify processing steps. This very productive VIP ensures that low- or zero-value functions or equipment included in the project scope are challenged by experienced world-class experts and eliminated, if possible. This VIP tries to systematically differentiate “wants” from “needs” and remove the “wants.” It can be especially effective for providing a neutral professional environment for identifying and challenging “sacred cows” and then removing them. Removal of these low- or zero-value functions yields significant profitability improvements to the overall project. Process simplification results in • Reduced capital costs (CAPEX) • Improved critical path schedule • Reduced process inventory • Increased yields • Reduced operating and maintenance costs (OPEX) • Increased productivity • Incremental capacity gains • Reduced utility and support systems requirements • Reduced waste generation Process simplification is executed in a formal workshop with a trained experienced facilitator. This VIP should always include key participants from each of the project owner’s organizations, the engineering contractor organization, key third-party technology licensors, and equipment or systems vendors, where possible. One or more “cold eyes” reviewers or subject matter experts, who have extensive experience, should be included to provide an objective and unbiased perspective. This VIP also provides a means for integrating overall plantwide systems. The process simplification VIP is typically performed during the feasibility phase (FEL-2) after the preliminary PFDs and heat and material balances become available. However, for very large and complex projects, considerable value has been gained by also performing this VIP at the midpoint or later in the conceptual phase (FEL-1). Constructability VIP The constructability VIP is the systematic implementation of the latest engineering, procurement and construction concepts and lessons learned, which are consistent with the facility’s operations and maintenance requirements. The goal is to enhance construction safety, scope, cost, schedule, and quality. Since the constructability VIP has seen widespread implementation in industry for capital projects over the last 20 years, in order for this VIP to remain consistent with the definition of a VIP (i.e., above what project teams can do on their own), at least one large engineering and construction company has enhanced this VIP to include a formal facilitated workshop that seeks profitability improvements above those already identified by the project team in the course of its normal work (KBR Value Improving Practices Program, 2001 through 2005). Both work processes described below are mutually additive, flexible, and compatible. The traditional constructability work process includes the following characteristics: • Starts at the FEL-1 phase and continues through facility start-up • Is an ongoing structured program

CAPITAL PROJECT EXECUTION AND ANALYSIS • Optimizes the combined use of operations, maintenance, engineering, procurement, key vendors, and construction knowledge and experience • Enhances the achievement of project objectives • Has construction experts working with the engineering and procurement process that results in construction safety, cost, schedule, and quality savings • Uses on-project and off-project expertise The enhanced constructability VIP adds the following to the traditional approach: • Includes a formal facilitated workshop • Is held in every engineering phase of the project with a focus on the pertinent aspects of that phase • Identifies value improvements and their benefits above those already being considered by the traditional constructability work process • Focuses on the systematic implementation of the latest engineering, procurement, construction concepts, and lessons learned • Involves a detail review of planning, design, procurement, fabrication, and installation functions to achieve the best overall project safety performance, lowest CAPEX, and the shortest reasonable schedule • Applies operations and maintenance requirements and expertise • Includes considerations for operability and maintainability • Enhances construction safety, scope, cost, schedule, and quality A formal constructability VIP workshop conducted in the conceptual phase (FEL-1) should focus on the overall project construction strategies regarding site layout, construction and turnaround laydown areas, access to the site for large equipment and modules, modularization, sequencing of heavy lifts, limitations regarding procurement, limitations regarding fabrication and transport, area labor limitations, and coordination with any existing or nearby structures or facilities. A formal constructability VIP workshop conducted in the feasibility phase (FEL-2) should focus on more specific topics of layout optimization, using a preliminary plot plan and equipment layout for the project. Considerations should include optimum site layout in terms of construction laydown areas; optimum equipment arrangement to reduce piping and steel for structures and piperack; specific sizes and weights for modules; which components will be included in each module; crane locations for heavy lifts; equipment requiring early purchase to allow project schedule to be achieved; further analysis of limitations regarding procurement, fabrication, and area labor availability; and precommissioning, commissioning, and start-up considerations. A formal constructability VIP workshop conducted in the definition phase (FEL-3) focuses on even greater detail for what was discussed above. In the detailed engineering stage (EPC phase), considerable detail will be reviewed to evaluate how the project can best construct what will be needed. Here, significant application of detailed lessons learned is reviewed and considered. Constructability VIP workshops should be formal facilitated workshops drawing on personnel from operations, maintenance, and construction in addition to project and owner organization representation. Customization of Standards and Specifications VIP The customization of standards and specifications VIP is a direct and systematic method to improve project value by selecting the most appropriate codes, standards, and specifications for the project. The goal is to make helpful changes to meet the actual project requirements, ensuring that the codes, standards, and specifications selected do not exceed those required for the project, and maximizing the use of specifications from equipment vendors to obtain the best overall value. This VIP is beyond typical good engineering practices and should not be confused with ongoing systematic improvements in corporate standards and specifications, or with required identification of applicable procurement specifications to be used for the project. This formal VIP takes a combination of project owner and engineering contractor corporate specifications and aggressively seeks profitability improvements consistent with the project’s goals and limitations. This VIP maximizes the procurement of off-the-shelf equipment over equipment customized for the project. Industry experience indicates that project-specific “Fit for Purpose” standards and specifications on the average cost less than the general application of traditional standards. This VIP is best performed early in the feasibility phase (FEL-2), and should include project team members involved from both the project owner and

9-51

engineering contractor, as well as appropriate suppliers of major packaged subsystems, modularized equipment, etc. Energy Optimization VIP The energy optimization VIP is the systematic process for the evaluation of the thermal efficiency of a process (or multiple subunits within a larger process or facility). The goal is to improve the economic utilization of energy. This optimization starts by using the “pinch” technology branch of process energy integration (energy pinch) to identify better process energy exchange options. Energy pinch (usually just called pinch) is a methodology for the conceptual design of process heating, utility, and power systems. Pinch allows the maximization of energy utilization within a process, while minimizing the use of plant utilities. Such minimization is achieved by reusing energy, via process stream–to–process stream heat exchange. A pinch analysis is performed by analyzing the tradeoff between the energy which can be recovered and the additional capital costs which must be added to do so. It includes the project’s design and thermodynamic constraints (performance targets) available during this preliminary design phase. The benefits of pinch technology include lower operating costs, occasionally reduced capital cost, improved operability/flexibility, increased throughput, and site-specific process optimization and reduced emissions. Pinch technology can be applied for both grass-roots and retrofit applications. Typical applications include process heat integration as well as sitewide heat and power integration. However, this methodology is profitably applied to the optimization of high-value complex mass flow problems, such as refinery hydrogen network optimization (hydrogen pinch) and wastewater minimization (water pinch). Once the minimum theoretical energy requirements and applicable process options have been determined, a formal facilitated workshop follows to modify the process or facility to bring the design closer to the thermodynamic optimum within project economic constraints. The energy optimization VIP is most beneficial for processes where energy and related capital expense are a relatively large fraction of the total operating cost. The benefits result in reduced energy requirements and environmental emissions in balance with project economics. This VIP should be implemented in the feasibility phase (FEL-2) when preliminary PFDs and heat and material balances are available. Predictive Maintenance VIP The predictive maintenance VIP is the proactive use of sensors and associated controls to monitor the machinery mechanical “health,” using both current state and historical trends, to optimize effective planning of all shutdowns and maintenance, thereby detecting equipment abnormalities and diagnosing potential problems before they cause permanent equipment damage. Examples include real-time corrosion monitoring and equipment vibration monitoring. This additional instrumentation is generally economically justified in the case of critical equipment items and key operations. Predictive maintenance reduces maintenance costs, improves the confidence of extending time between turnarounds, improves reliability, and provides a more predictable maintenance schedule for key process equipment. It also minimizes the amount of remaining equipment life that is lost through using only preventive maintenance practices. Preventive maintenance is an older practice which is limited to periodic inspections and repairs to avoid unplanned equipment breakdowns. For the predictive maintenance VIP to be effective, maintenance personnel from the project owner’s organization must be involved in determining key predictive maintenance requirements. Suppliers of critical equipment items (i.e., compressors) are also important participants in this process. The predictive maintenance VIP is considered by some operating companies and engineering contractors to have become standard practice. For those where it is not already standard practice, this VIP should be initiated in the feasibility phase (FEL-2) and concluded with a formal facilitated workshop and report of recommendations to the project management team. Waste Minimization VIP The waste minimization VIP involves a formal process stream-by-stream analysis to identify ways to eliminate or reduce the generation of wastes or nonuseful streams within the chemical process itself. For those streams not eliminated or converted to salable by-products, it provides the method for managing the resulting wastes. This VIP incorporates environmental requirements into the facility design and combines life cycle environmental

9-52

PROCESS ECONOMICS

benefits and positive economic returns through energy reductions, reduced end-of-pipe treatment requirements, and improved raw material yields. The waste reduction hierarchy is to • Eliminate or minimize the generation of waste through source reduction • Recycle by use, reuse, or reclamation those potential waste materials that cannot be eliminated or minimized • Treat all waste that is nevertheless generated to reduce volume, toxicity, or mobility prior to storage or disposal This VIP is considered by some engineering contractors to have become standard practice. For those where it is not standard practice, the waste minimization VIP should be executed in a formal workshop with an experienced facilitator with project owner and engineering contractor representatives always involved. A “cold eyes” reviewer with extensive experience should also be included to add a nonbiased perspective. The waste minimization VIP should be implemented at the feasibility phase (FEL-2) when preliminary PFDs and heat and material balances are available. Process Reliability Simulation VIP The process reliability simulation VIP is the use of reliability, availability, and maintainability (RAM) computer simulation modeling of the process and the mechanical reliability of the facility. A principal goal is to optimize the engineering design in terms of life cycle cost, thereby maximizing the project’s potential profitability. The objective is to determine the optimum relationships between maximum production rates and design and operational factors. Process reliability simulation is also applied for safety purposes, since it considers the consequences of specific equipment failures and failure modes. This VIP is typically led by an engineer experienced in plant operations and the use of the RAM simulation modeling software. The VIP should also directly involve the project owner since that organization would most often supply the historical operating and maintenance information required for the development of the simulation model. This process provides the project team with a more effective means of assessing, early in the design, the cost/benefit impact of changes in design, identification of bottlenecks in the system, simulation of key operating scenarios, determination of equipment-sparing needs, training and maintenance requirements of a facility. The process reliability simulation VIP should be initiated in the feasibility phase (FEL-2) to produce a block-level RAM model. Based on the results of that model, a more detailed equipment-level RAM model should be developed starting in the definition phase (FEL-3). Value Engineering VIP The value engineering VIP is a flexible, organized, multidisciplinary team effort directed at analyzing the functions, issues, and essential characteristics of a project, process, technology, or system. The goal is to satisfy those functions, issues, and essential characteristics at the lowest life cycle cost. The value engineering VIP rigorously examines what is needed to meet the business objectives of a project and the elimination of non-value-adding investment. An open-minded attitude by participants is required to effectively remove unneeded scope and in doing so reduce the installed costs of the project. This VIP tries to systematically differentiate “wants” from “needs” and remove the “wants.” Tests for non-income-producing investments include redundancy, overdesign, manufacturing add-ons, upgraded materials of construction, and customized design versus vendor standards. The value engineering VIP also ensures that low- or zero-value functions or equipment included in the project scope are challenged to be the highest value possible for the project. Removal of these lowor zero-value functions from the project scope, if possible, will most likely yield significant profitability improvements to the overall project. These can encompass the following: • Misalignment of unit or system capacity or operations capability with respect to the overall facility • Overly conservative assumptions of the basic design data • Overly conservative interpretation of how the facilities will be used during peak, seasonal, or upset conditions • Preinvestment included in the project scope that may not be value added • Overdesign of equipment or systems to provide uneconomic added flexibility The value engineering VIP is executed in a formal workshop with a trained experienced technical workshop facilitator. Both the project

owner and the engineering contractor are always involved. Third-party licensors and equipment/system vendors should be included where applicable. One (or more) cold-eyes reviewer with extensive experience should also be included to provide an unbiased perspective. This VIP leverages the growing accumulation of more detailed project knowledge to test the value of earlier, more generalized scope assumptions. It also tests the presumed added value of different stakeholder requirements, which have influenced the evolution of the project scope. This highly adaptable VIP results in reduced capital costs (CAPEX), improved critical path schedule, reduced process inventory, increased yields, reduced operating and maintenance costs (OPEX), increased productivity, incremental capacity gains, reduced utility and support systems requirements, and reduced waste generation. The value engineering VIP should be conducted in the definition phase (FEL-3) when the first issue of P&IDs is available. Design to Capacity VIP The design to capacity VIP systematically evaluates the maximum capacity of major equipment, ancillary piping, valving, instrumentation, and associated engineering calculations and guidelines. The goal is to improve life cycle costs (profitability) by eliminating preinvestment and overdesign. This VIP requires the systematic and formal evaluation of the maximum capacity of each piece of equipment instead of the traditional practice of designing with an extra safety factor or margin to allow for additional catch-up capacity or some future production increase. The goal is also to eliminate overdesign in both calculations and engineering guidelines. This VIP is conducted as a facilitated workshop with both project owner and engineering contractor representation. This VIP reduces capital investment by confirming minimum required capacities and flexibility necessary only to meet current project business objectives. The workshop drills down to each specific system and subsystem and finally scrutinizes the design of each equipment item. This workshop is often combined with the value engineering VIP, which overlaps significantly. The design to capacity VIP should be conducted in the definition phase (FEL-3) when the first issue of P&IDs is available. 3D-CAD VIP The 3D-CAD VIP is the creation of a detailed three-dimensional (3D) computer model depicting the proposed process and associated equipment along with the optimized plant layout and specific equipment arrangements and orientations. The 3D model can then be used to generate computerized interference checks of bulk material configurations and equipment and extraction of error-free fabrication drawings and material quantities. The goals of this VIP are to reduce engineering and construction rework, improve operability and maintainability, and confirm the incorporation into the design of advantageous human factors (a.k.a. ergonomics) focused on ease of operation and maintenance. A number of industry-accepted state-of-the-art 3D computer-aided design (CAD) systems have been used for this purpose. The specification-driven 3D-CAD system allows a computer model to be built to allow extraction of drawings from the model for fabrication and erection. The extracted drawings are enhanced in their accuracy by the computer interference detection system which greatly reduces field rework. The principal benefit of utilizing 3D-CAD is the ability to produce an electronic model that accurately resembles the completed facility. This enables project teams, clients, and constructors to review and agree on the plant design before construction starts. The model can then be used to generate interference checks of bulk material configurations and equipment, as well as the extraction of error-free fabrication drawings and material quantities. The system also utilizes its design review capabilities to confirm proposed designs and obtain approvals from key project stakeholders and owners. This VIP is considered by most major engineering contractors to have become standard practice. The 3D-CAD VIP model development should be initiated in the feasibility phase (FEL-2) after the plot plan has been finalized and the first issue of P&IDs is available. VIP Planning and Implementation Each VIP has a unique character, and it should be performed at a certain time and in a certain way to produce the best results for the project. Part of the power of VIPs is that they can be used to improve the overall economics of the project without the need for inordinate additional time or expense. Ironically, the return on investment (ROI) for the cost of implementing each VIP is usually much greater than that ROI for the overall proposed project. For one engineering contractor, the typical ROI for

CAPITAL PROJECT EXECUTION AND ANALYSIS implementing VIPs is at least one order of magnitude higher (KBR Value Improving Practices Program, 2000 through 2005). It is important to reiterate that the benefits of VIPs cannot be realized by just executing “good engineering.” The application and implementation of VIPs to any project must have the explicit commitment of the owner’s corporation executives. VIP execution must be deliberately and carefully planned in the initial phase of the project. For all projects, this VIP planning meeting should take place immediately following project kickoff. The project management team and the selected VIP facilitator should • Confirm which VIPs should be applied to the project and when • Incorporate the planned VIPs into the project scope of work and schedule • Determine the required workshop resources and best combination of engineering, operations, maintenance, construction, and other expertise for each selected VIP workshop team VIPs That Apply the Value Methodology Nearly all VIPs are conducted only once in a project at a “sweet spot” where maximum benefit is found. For example, the process simplification VIP is anchored at the first appearance of the preliminary PFDs, while the value engineering VIP and the design to capacity VIP are anchored at the first issue of the P&IDs. Both of these apply the value methodology [Society of American Value Engineers International (SAVE)] that has produced excellent results in industry for more than 55 years. The typical approach and steps for these three unique VIPs are presented below. Preparation and Planning Before the VIP is begun, the goals, objectives, and scheduled time for the formal workshop must be agreed upon by the integrated project management team. The workshop facilitator must ensure that all the information required for the workshop is available and that the workshop team members have been fully briefed on the VIP’s objectives, methodology, and expectations. The Formal Workshop The formal workshop is always structured to make maximum use of the multi-disciplinary team’s time and effort. Such workshops typically require no less than 2 days and as many as 5 days depending on the size and complexity of the project. The required workshop length should be determined by the VIP facilitator. A typical process simplification VIP, value engineering VIP, and design to capacity VIP workshop includes the following phases of a typical “job plan” that are supported by the Society of American Value Engineers International. The information phase In this phase, team members review important background materials and confirm their understanding of the basis for the design of the project, the constraints, and the sensitivity of the relevant capital and operating costs. Here, incorporating important unresolved project issues into the workshop produces more meaningful financial and technical results. Discussion of the issues’ validity and basis are determined during the first day of the workshop. These issues very often become some of the best brainstorming targets for cost and schedule reduction ideas. A very specific and structured methodology is used which is known as the Function Analysis System Technique (FAST). This function analysis diagramming illustrates the logical or functional relationships and dependencies between different process systems and project activities. These diagrams are then reviewed and critiqued together with the associated costs of selected groups of process functions or project functions. The function analysis can be performed at this stage, but often time can be saved by preparing a draft of these FAST diagrams prior to the workshop with a small group of the workshop team members. The speculation phase Once the pertinent information and issues have been reviewed and the important functions of each process and project step identified, the team is encouraged to speculate on alternative methods to perform each function and to solve each major project issue. Brainstorming sessions within a creative environment encourage the team to strive for new and innovative ideas. The conceptual phase The team then reviews the ideas against relevant project criteria such as potential impact on long-term project economics, impact on operations and maintenance costs, effect on the capital cost for the project, validity to the project scope of work, technical risks associated with implementation of the new concept, impact on project schedule, and cost required to implement the improvement. Each study has specific criteria against which proposed alterna-

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tives are judged. The ideas are weighted, sorted, grouped, linked, and ranked so that the best of the technically viable ideas are efficiently identified for further detailed study. The feasibility phase The ideas with the most merit are developed into preliminary two-page written proposals with potential benefits approximated as part of the workshop. Performing this important activity following the formal workshop has been shown to often result in significant loss of potential for implementation. The VIP team expands the ideas ranked highest to obtain additional technical and economic insights and information to support the idea. The proposals are then presented internally, to the assembled VIP workshop team, and discussed to determine whether the ideas retain sufficient technical and economic merit to be recommended by the VIP team to a separate steering committee or project management team. Experience indicates that having the VIP team perform this stage within the formal workshop produces the best results. The presentation phase The VIP team formally presents the profitability recommendations consistent with the objectives and constraints of the workshop and their implementation plans to the steering committee or the project management team. The steering committee then approves those recommendations that pass muster and authorizes the project team to begin the implementation effort. Often, this approval is conditional on early validation by subject matter experts within the project owner’s organization, but not present within the workshop. This external feasibility check is meant to provide support to the project team for any additional resources and schedule time needed to fully incorporate the improvements into the project scope of work. Report and follow-up After completing the intensive VIP workshop, the workshop facilitator completes the written VIP final report for the project record. During this time, the project management team assigns each approved recommendation to a member of the project team, estimates the engineering time and resources required to incorporate the improvement into the project scope of work, and communicates the results of the VIP within the integrated project team. This follow-up action plan creates a very positive and cost-conscious attitude within the project team that leads to further improvements in project value. Sources of Expertise VIP workshops should be planned and led by a trained experienced facilitator who has significant experience in effectively conducting such VIP workshops. Technical expertise for VIP workshops should be a combination of senior project team members and subject matter experts from the operating company’s organization, the engineering contractor’s organization, licensed technology providers, and any key fabrication or installation subcontractors to be used. Figure 9-35 illustrates the best balance of expertise for VIP workshops (KBR Value Improving Practices Program, 1995 through 2005).

Project Team Experts (<50%) • Operating Company Experts • Engineering Contractor Experts • Technology Licensor Experts

Non–Project Team Experts (>50%) • Operating Company Experts • Engineering Contractor Experts • Outside Subject Matter Experts

FIG. 9-35

The ideal VIP team makeup.

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GLOSSARY Accounts payable The value of purchased goods and services that are being used but have not been paid. Accounts receivable Credit extended to customers, usually on a 30-day basis. Cash is set aside to take care of the probability that some customers may not pay their bills. Accrual basis The accounting method that recognizes revenues and disbursement of funds by receipt of bills or orders and not by cash flow, distinguished from cash basis. Administrative expense An overhead expense due to the general direction of a company beyond the plant level. It includes administrative and office salaries, rent, auditing, accounting, legal, central purchasing and engineering, etc., expenses. Allocation of expenses A procedure whereby overhead expenses and other indirect charges are assigned back to processing units or to products on what is expected to be an equitable basis. All allocations are somewhat arbitrary. Amortization Often used interchangeably with depreciation, but there is a slight difference depending on whether the life of an asset is known. If the period of time is known to be usually more than a year, this annual expense is amortization; however, if the life is estimated, then it is depreciation. Annual net sales Pounds of product sold times the net selling price. Net means that any allowances have been subtracted from the gross selling price. Annual report Management’s report to the stockholders and other interested parties at the end of a year of operation showing the status of the company, its activities, funds, income, profits, expenses, and other information. Appurtenances The auxiliaries to either process or nonprocess equipment: piping, electrical, insulation, instrumentation, etc. Assets The list of money on hand, marketable securities, monies due, investments, plants, properties, intellectual property, inventory, etc., at cost or market value, whichever is smaller. The assets are what a company (or person) owns. Balance sheet This is an accounting, historical tabulation of assets, liabilities, and stockholders’ equity for a company. The assets must equal the liabilities plus the stockholders’ equity. Battery limit A geographic boundary defining the coverage of a specific project. Usually it takes in the manufacturing area of a proposed plant, including all process equipment but excluding provision for storage, site preparation, utilities, administrative buildings, or auxiliary facilities. Bonds When one purchases a bond, the company (or person) acquires an interest in debt and becomes a creditor of the company. The purchaser receives the right to receive regular interest payments and the subsequent repayment of the principal. Book value Current investment value on the company books as the original installed cost less depreciation accruals. Book value of common stock Net worth of a firm divided by the number of shares of common stock issued at the time of a report. Break-even chart An economic production chart depicting total revenue and total expenses as functions of operation of a processing facility. Break-even point The percentage of capacity at which income equals all fixed and variable expenses at that level of operation. By-product A product made as a consequence of the production of a main product. The by-product may have a market value or a value as a raw material. Capacity The estimated maximum level of production on a sustained basis. Capital ratio Ratio of capital investment to sales dollars; the reciprocal of capital turnover. Capital recovery The process by which original investment in a project is recovered over its life. Capital turnover The ratio of sales dollars to capital investment; the reciprocal of capital ratio. Cash Money that is on hand to pay for operating expenses, e.g., wages, salaries, raw materials, supplies, etc., to maintain a liquid financial position.

Cash basis The accounting basis whereby revenue and expense are recorded when cash is received and paid, distinguished from accrual basis. Cash flow Net income after taxes plus depreciation (and depletion) flowing into the company treasury. Code of accounts A system in which items of expense or fixed capital such as equipment and material are identified with numerical figures to facilitate accounting and cost control. Common stock Money paid into a corporation for the purchase of shares of common stock that becomes the permanent capital of the firm. Common stockholders have the right to transfer ownership and may sell the stock to individuals or firms. Common stockholders have the right to vote at annual meetings on company business or may do so by proxy. Compound interest The interest charges under the condition that interest is charged on previous interest plus principal. Contingencies An allowance for unforeseeable elements of cost in fixed investment estimates that previous experience has shown to exist. Continuous compounding A mathematical procedure for evaluating compound interest based upon continuous interest function rather than discrete interest periods. Conversion expense The expense of converting raw materials to finished product. Corporation In 1819, defined by Chief Justice Marshall of the Supreme Court as “an artificial being, invisible, intangible and existing only in contemplation of law.” It exists by the grace of a state, and the laws of a state govern the procedure for its formation. Cost of capital The cost of borrowing money from all sources, namely, loans, bonds, and preferred and common stock. It is expressed as an interest rate. Cost center For accounting purposes, a grouping of equipment and facilities comprising a product manufacturing system. Cost of sales The sum of the fixed and variable (direct and indirect) expenses for manufacturing a product and delivering it to a customer. Decision or decision making A program of action undertaken as a result of (1) an established policy or (2) an analysis of variables that can be altered to influence a final result. Depletion A provision in the tax regulations that allows a business to charge as current expense a noncash expense representing the portion of limited natural resources consumed in the conduct of business. Depreciation A reasonable allowance by the Internal Revenue Service for the exhaustion, wear and tear, and normal obsolescence of equipment used in a trade or business. The property must have a useful life of more than 1 year. Depreciation is a noncash expense deductible from income for tax purposes. Design to cost A management technique to achieve system designs that meet cost parameters. Cost as a design parameter is considered on a continuous basis as part of a system’s development and production processes. Direct expense An expense directly associated with the production of a product such as utilities, labor, and maintenance. Direct labor expense The expense of labor involved in the manufacture of a product or in the production of a service. Direct material expense The expense associated with materials consumed in the manufacture of a product or the production of a service. Distribution expense Expense including advertising, preparation of samples, travel, entertainment, freight, warehousing, etc., to distribute a sample or product. Dollar volume Dollar worth of a product manufactured per unit of time. Earnings The difference between income and operating expenses. Economic life The period of commercial use of a product or facility. It may be limited by obsolescence, physical life of equipment, or changing economic conditions.

GLOSSARY Economic value added The period dollar profit above the cost of capital. It is a means to measure an organization’s value and a way to determine how management’s decisions contribute to the value of a company. Effective interest The true value of interest computed by equations for the compound interest rate for a period of 1 year. Equity The owner’s actual capital held by a company for its operations. Escalation A provision in actual or estimated cost for an increase in equipment cost, material, labor, expenses, etc., over those specified in an original estimate or contract due to inflation. External funds Capital obtained by selling stocks or bonds or by borrowing. FEL (front-end loading) The process by which a company develops a detailed definition of the scope of a capital project that meets corporate business objectives. FIFO (first in, first out) The valuation of raw material and supplies inventory, meaning first into the company or process is the first used or out. Financial expense The charges for use of borrowed funds. Fixed assets The real or material facilities that represent part of the capital in an economic venture. Fixed capital Item including the equipment and buildings. Fixed expense An expense that is independent of the rate of output, e.g., depreciation and plant indirect expenses. Fringe benefits Employee welfare benefits; expenses of employment over and above compensation for actual time worked, such as holidays, vacations, sick leave, insurance. Full cost accounting Method of pricing goods and services to reflect their true costs, including production, use recycling, and disposal. Future worth The expected value of capital in the future according to some predetermined method of computation. Goods manufactured, cost of Total expense (direct and indirect expenses) including overhead charges. Goods-in-process inventory The holdup of product in a partially finished state. Goods sold, cost of The total of all expenses before income taxes that is deducted from income (revenue). Grass-roots plant A complete plant erected on new site including land, site preparation, battery-limits facilities, and auxiliary facilities. Gross domestic product An indicator of a country’s economic activity. It is the sum of all goods and services produced by a nation within its borders. Gross margin (profit) Total revenue minus cost of goods manufactured. Gross national product An economic indicator of a country’s economic activity. It is the sum of all the goods and services produced by a nation both within and outside its borders. Income Profit before income taxes or gross income from sales before deduction of expenses. Income statement The statement of earnings of a firm as approximated by accounting practices, usually covering a 1-year period. Income tax The tax imposed on corporate profits by the federal and/or state governments. Indirect expenses Part of the manufacturing expense of a product not directly related to the amount of product manufactured, e.g., depreciation, local taxes, and insurance. Internal funds Capital available from depreciation and accumulated retained earnings. Inventory The quantity of raw materials and/or supplies held in a process or in storage. Last in, first out (LIFO) The valuation of raw materials and supplies, meaning the last material into a process or storage is the first used or out. Leverage The influence of debt on the earning rate of a company. Liabilities An accounting term for capital owed by a company. Life cycle cost Cost of development, acquisition, support, and disposal of a system over its full life. Manufacturing expense The sum of the raw material, labor, utilities, maintenance, depreciation, local taxes, etc., expenses. It is the sum of the direct and indirect (fixed and variable) manufacturing expenses.

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Marginal cost The incremental cost of making one additional unit without additional investment in facilities Market capitalization The product of the number of shares of common stock outstanding and the share price. Market value added A certain future economic value added for a company. It is the present value of the future economic value (EVA) generated by a company. It is a measure of how much value a firm has created. Minimum acceptable rate of return (MARR) The level of return on investment, at or above the cost of capital, chosen as acceptable for discounting or cutoff purposes. Net sales price Gross sales price minus freight adjustments. Net worth The sum of the stockholders’ investment plus surplus, or total assets minus total liabilities. Nominal interest The number applied loosely to describe the annual interest rate. Obsolescence The occurrence of decreasing value of physical assets due to technological changes rather than physical deterioration. Operating expense The sum of the manufacturing expense for a product and the general, administrative, and selling expenses. Operating margin The gross margin minus the general, administrative, and selling expenses. Opportunity cost The estimate of values that are forgone by undertaking one alternative instead of another one. Payout time (payback period) The time to recover the fixed capital investment from profit plus depreciation. It is usually after taxes but not always. Preferred stock Stock having claims that it commands over common stock, with the preference related to dividends. The holders of such stock receive dividends before any distribution is made to common stockholders. Preferred stockholders usually do not have voting rights as common stockholders do. Present worth The value at some datum time (present time) of expenditures, costs, profits, etc., according to a predetermined method of computation. It is the current value of cash flow obtained by discounting. Production rate The amount of product manufactured in a given time period. Profitability A term generally applied in a broad sense to the economic feasibility of a proposed venture or an ongoing operation. It is generally considered to be related to return on investment. Rate of return on investment The efficiency ratio relating profit or cash flow to investment. Replacement A new facility that takes the place of an older facility with no increase in capacity. Revenue The net sales received from the sale of a product or a service to a customer. Sales, administration, research, and engineering expenses (SARE) Overhead expenses incurred as a result of maintaining sales offices and administrative offices and the expense of maintaining research and engineering departments. This item is usually expressed as a percentage of annual net sales. Sales volume The amount of sales expressed in pounds, gallons, tons, cubic feet, etc., per unit of time. Salvage value The value that can be realized from equipment or other facilities when taken out of service and sold. Selling expense Salaries and commissions paid to sales personnel. Simple interest The interest charges in any time period that is only charged on the principal. Sinking fund An accounting procedure computed according to a specified procedure to provide capital to replace an asset. Surplus The excess of earnings over expenses that is not distributed to stockholders. Tax credit The amount available to a firm as part of its annual return because of deductible expenses for tax purposes. Examples have been research and development expenses, energy tax credit, etc. Taxes In a manufacturing cost statement, usually property taxes. In an income statement, usually federal and state income taxes. Time value of money The expected interest rate that capital should or would earn. Money has value with respect to time.

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Total operating investment The fixed capital investment, backup capital, auxiliary capital, utilities and services capital, and working capital. Utilities and services capital Electrical substations, plant sewers, water distribution facilities, and occasionally the steam plant. Value added The difference between the raw material expense and the selling price of that product. Value-improving practices (VIPs) Formal structured practices applied to capital projects to improve profitability (“or value”) above

that which is attained through the application of proven good engineering and project management practices. Variable expense Any expense that varies directly with production output. Working capital In the accounting sense, the current assets minus the current liabilities. It consists of the total amount of money invested in raw materials, supplies, goods in process, product inventories, accounts receivable, and cash minus those liabilities due within 1 year.

Section 10

Transport and Storage of Fluids

Meherwan P. Boyce, Ph.D., P.E. Chairman and Principal Consultant, The Boyce Consultancy Group, LLC; Fellow, American Society of Mechanical Engineers; Registered Professional Engineer (Texas) (Section Editor, Measurement of Flow, Pumps and Compressors) Victor H. Edwards, Ph.D., P.E. Process Director, Aker Kvaerner, Inc.; Fellow, American Institute of Chemical Engineers; Member, American Association for the Advancement of Science, American Chemical Society, National Society of Professional Engineers; Life Member, New York Academy of Sciences; Registered Professional Engineer (Texas) (Section Editor, Measurement of Flow) Terry W. Cowley, B.S., M.A. Consultant, DuPont Engineering; Member, American Society of Mechanical Engineers, American Welding Society, National Association of Corrosion Engineers (Polymeric Materials) Timothy Fan, P.E., M.Sc. Chief Project Engineer, Foster Wheeler USA; Member, American Society of Mechanical Engineers, Registered Professional Engineer (Massachusetts and Texas) (Piping) Hugh D. Kaiser, P.E., B.S., MBA Principal Engineer, PB Energy Storage Services, Inc.; Senior Member, American Institute of Chemical Engineers; Registered Professional Engineer (Texas) (Underground Storage of Liquids and Gases, Cost of Storage Facilities, Bulk Transport of Fluids) Wayne B. Geyer, P.E. Executive Vice President, Steel Tank Institute and Steel Plate Fabricators Association; Registered Professional Engineer (Atmospheric Tanks) David Nadel, P.E., M.S. Senior Principal Mechanical Engineer, Aker Kvaerner, Inc.; Registered Professional Engineer (Pressure Vessels) Larry Skoda, P.E. Principal Piping Engineer, Aker Kvaerner, Inc.; Registered Professional Engineer (Texas) (Piping) Shawn Testone Product Manager, De Dietrich Process Systems (Glass Piping and GlassLined Piping) Kenneth L. Walter, Ph.D. Process Manager—Technology, Aker Kvaerner, Inc.; Senior Member, American Institute of Chemical Engineers, Sigma Xi, Tau Beta Pi (Storage and Process Vessels)

MEASUREMENT OF FLOW Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties and Behavior of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistive Thermal Detectors (RTDs) . . . . . . . . . . . . . . . . . . . . . . . . . Static Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dry- and Wet-Bulb Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Column Manometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Tube Size for Manometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplying Gauges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Pressure Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration of Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average Static Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifications for Piezometer Taps . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10-2

TRANSPORT AND STORAGE OF FLUIDS

Variables Affecting Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Profile Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flow Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitot Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traversing for Mean Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Industry Guidelines and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Pressure Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volumetric Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable-Area Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Channel Flow Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Pressure Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orifice Meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venturi Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Nozzles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Flow Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elbow Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anemometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow Transverse-Momentum Mass Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inferential Mass Flowmeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coriolis Mass Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable-Area Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Phase Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Solid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Liquid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid-Solid Mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowmeter Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-11 10-11 10-11 10-11 10-13 10-13 10-14 10-14 10-14 10-14 10-14 10-14 10-14 10-14 10-14 10-15 10-15 10-16 10-18 10-19 10-19 10-20 10-20 10-21 10-21 10-21 10-21 10-21 10-21 10-21 10-22 10-22 10-22 10-22 10-22 10-23 10-23 10-23 10-23 10-23

PUMPS AND COMPRESSORS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Dynamic Head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Suction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Suction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Discharge Head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Static Discharge Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Work Performed in Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net Positive Suction Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suction Limitations of a Pump. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NPSH Requirements for Other Liquids . . . . . . . . . . . . . . . . . . . . . . . Example 1: NPSH Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive-Displacement Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocating Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piston Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diaphragm Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gear Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screw Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid-Displacement Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Casings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Action of a Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-24 10-25 10-25 10-25 10-25 10-25 10-25 10-25 10-25 10-25 10-25 10-26 10-26 10-26 10-27 10-27 10-27 10-27 10-27 10-27 10-27 10-27 10-27 10-28 10-28 10-28 10-28 10-28 10-30 10-30 10-31 10-31 10-32 10-32 10-32 10-33 10-33

Centrifugal Pump Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Process Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sealing the Centrifugal Chemical Pump . . . . . . . . . . . . . . . . . . . . . . . Double-Suction Single-Stage Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . Close-Coupled Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Canned-Motor Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sump Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multistage Centrifugal Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propeller and Turbine Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow (Propeller) Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbine Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regenerative Pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetic Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compression of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocating Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fans and Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forward-Curved Blade Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Backward-Curved Blade Blowers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fan Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous-Flow Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impeller Fabrication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial-Flow Compressors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive-Displacement Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . Rotary Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ejector Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uses of Ejectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacuum Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sealing of Rotating Shafts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncontact Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Labyrinth Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ring Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Seal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Floating Seal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packing Seal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Face Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Seal Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal and External Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Throttle Bushings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bearings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust Bearings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust-Bearing Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Centrifugal Compressor Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Fouling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Failures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impeller Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Thrust Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Journal Bearing Failures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thrust Bearing Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Seal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotor Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-33 10-34 10-34 10-34 10-35 10-35 10-35 10-36 10-36 10-37 10-37 10-37 10-37 10-38 10-38 10-39 10-39 10-40 10-40 10-42 10-42 10-42 10-44 10-45 10-49 10-49 10-49 10-50 10-50 10-52 10-52 10-52 10-53 10-54 10-54 10-56 10-56 10-57 10-57 10-58 10-58 10-58 10-59 10-59 10-59 10-62 10-62 10-62 10-62 10-63 10-63 10-64 10-64 10-65 10-65 10-65 10-66 10-67 10-67 10-68 10-69 10-69 10-69 10-70 10-70 10-70 10-70 10-71 10-72

PROCESS PLANT PIPING Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Codes and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units: Pipe and Tubing Sizes and Ratings . . . . . . . . . . . . . . . . . . . . . . Pressure-Piping Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . National Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Government Regulations: OSHA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . International Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Code Contents and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of Pipe System Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-73 10-73 10-73 10-73 10-73 10-73 10-74 10-74 10-74

TRANSPORT AND STORAGE OF FLUIDS General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Material Considerations—Metals . . . . . . . . . . . . . . . . . . . . . Specific Material Considerations—Nonmetals . . . . . . . . . . . . . . . . . . Metallic Piping System Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . Seamless Pipe and Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welded Pipe and Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Joining Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flanged Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ring Joint Flanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bolting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Mechanical Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Fittings and Bends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cast Iron, Ductile Iron, and High-Silicon Iron Piping Systems. . . . . . . Cast Iron and Ductile Iron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-Silicon Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonferrous Metal Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aluminum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Copper and Copper Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nickel and Nickel Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Titanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flexible Metal Hose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Pipe and Metallic Piping Systems with Nonmetallic Linings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cement-Lined Carbon-Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concrete Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glass Pipe and Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Glass-Lined Steel Pipe and Fittings. . . . . . . . . . . . . . . . . . . . . . . . . . . Fused Silica or Fused Quartz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic-Lined Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rubber-Lined Steel Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reinforced-Thermosetting-Resin (RTR) Pipe . . . . . . . . . . . . . . . . . . Design of Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safeguarding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Fluid Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Category D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Category M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Support, Anchor, and Terminal Movements . . . . . . . . . . . Reduced Ductility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Condensation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Criteria: Metallic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limits of Calculated Stresses due to Sustained Loads and Displacement Strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Design of Metallic Components . . . . . . . . . . . . . . . . . . . . . . Test Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Expansion and Flexibility: Metallic Piping . . . . . . . . . . . . . . Reactions: Metallic Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Criteria: Nonmetallic Pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabrication, Assembly, and Erection . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welding, Brazing, or Soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-74 10-75 10-76 10-76 10-76 10-76 10-77 10-77 10-81 10-85 10-85 10-87 10-89 10-93 10-98 10-98 10-99 10-99 10-99 10-100 10-100 10-101 10-101 10-103 10-103 10-104 10-104 10-105 10-105 10-105 10-106 10-106 10-107 10-107 10-107 10-107 10-107 10-107 10-107 10-108 10-108 10-108 10-108 10-108 10-111 10-111 10-113 10-114 10-120 10-122 10-123 10-123 10-123

10-3

Bending and Forming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preheating and Heat Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joining Nonmetallic Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assembly and Erection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination, Inspection, and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination and Inspection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examination Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type and Extent of Required Examination . . . . . . . . . . . . . . . . . . . . . Impact Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost Comparison of Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces of Piping on Process Machinery and Piping Vibration . . . . . . . . Heat Tracing of Piping Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Heat-Tracing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing the Best Tracing System. . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-126 10-126 10-126 10-126 10-126 10-126 10-128 10-131 10-133 10-133 10-135 10-135 10-135 10-137 10-140

STORAGE AND PROCESS VESSELS Storage of Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmospheric Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shop-Fabricated Storage Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . USTs versus ASTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aboveground Storage Tanks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Tank Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Container Materials and Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pond Storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Underground Cavern Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Holders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of Gases in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Storage in Pressure Vessels, Bottles, and Pipe Lines . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavern Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cost of Storage Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bulk Transport of Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipe Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tank Cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tank Trucks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marine Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials of Construction for Bulk Transport . . . . . . . . . . . . . . . . . . . Pressure Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Code Administration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ASME Code Section VIII, Division 1 . . . . . . . . . . . . . . . . . . . . . . . . . ASME Code Section VIII, Division 2 . . . . . . . . . . . . . . . . . . . . . . . . . Additional ASME Code Considerations . . . . . . . . . . . . . . . . . . . . . . . Other Regulations and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessels with Unusual Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . ASME Code Developments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessel Codes Other than ASME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vessel Design and Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Care of Pressure Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure-Vessel Cost and Weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-140 10-140 10-140 10-140 10-140 10-144 10-144 10-145 10-146 10-146 10-148 10-148 10-148 10-148 10-149 10-149 10-149 10-149 10-149 10-149 10-150 10-151 10-151 10-151 10-151 10-151 10-152 10-155 10-155 10-157 10-157 10-158 10-158 10-158 10-158 10-159

10-4

TRANSPORT AND STORAGE OF FLUIDS

Nomenclature and Units In this listing, symbols used in the section are defined in a general way and appropriate SI and U.S. customary units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Symbol A A A∞ a a a B b b C C C

C C Ca C1 cp cv D D, D0 d E E Ea Ec Ej Em F F F f f f G g gc H H, h Had h h i i ii io I J K

U.S. customary units

Symbol

m2

ft2

K

2

2

Definition Area Factor for determining minimum value of R1 Free-stream speed of sound Area Duct or channel width Coefficient, general Height Duct or channel height Coefficient, general Coefficient, general Conductance Sum of mechanical allowances (thread or groove depth) plus corrosion or erosion allowances Cold-spring factor Constant Capillary number Estimated self-spring or relaxation factor Constant-pressure specific heat Constant-volume specific heat Diameter Outside diameter of pipe Diameter Modulus of elasticity Quality factor As-installed Young’s modulus Casting quality factor Joint quality factor Minimum value of Young’s modulus Force Friction loss Correction factor Frequency Friction factor Stress-range reduction factor Mass velocity Local acceleration due to gravity Dimensional constant Depth of liquid Head of fluid, height Adiabatic head Flexibility characteristic Height of truncated cone; depth of head Specific enthalpy Stress-intensification factor In-plane stressintensification factor Out-plane stress intensification factor Electric current Mechanical equivalent of heat Index, constant or flow parameter

SI units

K1 m m

ft ft

m m

ft ft

m3/s mm

ft3/s in

k k k L L L M Mi, mi Mo

Dimensionless

Dimensionless

Mt M∞ m m N

J/(kg⋅K)

Btu/(lb⋅°R)

N N

J/(kg⋅K)

Btu/(lb⋅°R)

m mm m N/m2

ft in ft lbf/ft2

MPa

kip/in2 (ksi)

MPa

2

kip/in (ksi)

N (N⋅m)/kg Dimensionless Hz Dimensionless

lbf (ft⋅lbf)/lb Dimensionless l/s Dimensionless

NS NDe NFr NRe NWe NPSH n n n n P Pad p p Q Q Q Q

kg/(s⋅m2) m/s2

lb/(s⋅ft2) ft/s2

q R

1.0 (kg⋅m)/(N⋅s2) m m N⋅m/kg

32.2 (lb⋅ft)/ (lbf⋅s2) ft ft lbf⋅ft/lbm

R R R R

m

in

J/kg

Btu/lb

R R Ra

Rm A 1.0 (N⋅m)/J

A 778 (ft⋅lbf)/ Btu R1

Definition Fluid bulk modulus of elasticity Constant in empirical flexibility equation Ratio of specific heats Flexibility factor Adiabatic exponent cp /cv Length Developed length of piping between anchors Dish radius Molecular weight In-plane bending moment Out-plane bending moment Torsional moment Free stream Mach number Mass Thickness Number of data points or items Frictional resistance Equivalent full temperature cycles Strouhal number Dean number Froude number Reynolds number Weber number Net positive suction head Polytropic exponent Pulsation frequency Constant, general Number of items Design gauge pressure Adiabatic power Pressure Power Heat Volume Volume rate of flow (liquids) Volume rate of flow (gases) Volume flow rate Gas constant Radius Electrical resistance Head reading Range of reaction forces or moments in flexibility analysis Cylinder radius Universal gas constant Estimated instantaneous reaction force or moment at installation temperature Estimated instantaneous maximum reaction force or moment at maximum or minimum metal temperature Effective radius of miter bend

SI units

U.S. customary units

N/m2

lbf/ft2

Dimensionless

Dimensionless

m m

ft ft

m kg/mol N⋅mm N⋅mm

in lb/mol in⋅lbf in⋅lbf

N⋅mm

in⋅lbf

kg m Dimensionless

lb ft Dimensionless

Dimensionless

Dimensionless

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless m

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless ft

Hz

1/s

Dimensionless kPa kW Pa kW J m3 m3/h

Dimensionless lbf/in2 hp lbf/ft2 hp Btu ft3 gal/min

m3/h

ft3/min (cfm)

m3/s 8314 J/ (K⋅mol) m Ω m N or N⋅mm

ft3/s 1545 (ft⋅lbf)/ (mol⋅°R) ft Ω ft lbf or in⋅lbf

m J/(kg⋅K) N or N⋅mm

ft (ft⋅lbf)/(lbm⋅°R) lbf or in⋅lbf

N or N⋅mm

lbf or in⋅lbf

mm

in

TRANSPORT AND STORAGE OF FLUIDS

10-5

Nomenclature and Units (Concluded) Symbol r r rc rk r2 S S S S S SA SE SL ST Sb Sc Sh St s s T Ts T 苶 T 苶b T 苶h t t t tm

tr U u u V V

Definition Radius Pressure ratio Critical pressure ratio Knuckle radius Mean radius of pipe using nominal wall thickness T 苶 Specific surface area Fluid head loss Specific energy loss Speed Basic allowable stress for metals, excluding factor E, or bolt design stress Allowable stress range for displacement stress Computed displacementstress range Sum of longitudinal stresses Allowable stress at test temperature Resultant bending stress Basic allowable stress at minimum metal temperature expected Basic allowable stress at maximum metal temperature expected Torsional stress Specific gravity Specific entropy Temperature Effective branch-wall thickness Nominal wall thickness of pipe Nominal branch-pipe wall thickness Nominal header-pipe wall thickness Head or shell radius Pressure design thickness Time Minimum required thickness, including mechanical, corrosion, and erosion allowances Pad or saddle thickness Straight-line distance between anchors Specific internal energy Velocity Velocity Volume

SI units

U.S. customary units

Symbol

m Dimensionless

ft Dimensionless

v W W w x x x

m mm

in in

m2/m3 Dimensionless m/s2 m3/s MPa

ft2/ft3 Dimensionless lbf/lb ft3/s kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa

kip/in2 (ksi)

MPa MPa

kip/in2 (ksi) kip/in2 (ksi)

α

MPa

kip/in2 (ksi)

α σ α, β, θ β

MPa

kip/in2 (ksi)

J/(kg⋅K) K (°C) mm

Btu/(lb⋅°R) °R (°F) in

mm

in

mm

in

mm

in

mm mm s mm

in in s in

mm m

in ft

J/kg m/s m/s m3

Btu/lb ft/s ft/s ft3

Y y y Z Z Ze Z z

Definition

SI units

Specific volume Work Weight Weight flow rate Weight fraction Distance or length Value of expression [(p2 /p1)(k − 1/k) − 1] Expansion factor Distance or length Resultant of total displacement strains Section modulus of pipe Vertical distance Effective section modulus for branch Gas-compressibility factor Vertical distance

U.S. customary units

m3/kg N⋅m kg kg/s Dimensionless m

ft3/lb lbf⋅ft lb lb/s Dimensionless ft

Dimensionless m mm

Dimensionless ft in

mm3 m mm3

in3 ft in3

Dimensionless

Dimensionless

m

ft

Greek symbols

β Γ Γ δ ε ε η ηad ηp θ λ µ ν ρ σ σc τ φ φ φ ψ ψ

Viscous-resistance coefficient Angle Half-included angle Angles Inertial-resistance coefficient Ratio of diameters Liquid loading Pulsation intensity Thickness Wall roughness Voidage—fractional free volume Viscosity, nonnewtonian fluids Adiabatic efficiency Polytropic efficiency Angle Molecular mean free-path length Viscosity Kinematic viscosity Density Surface tension Cavitation number Shear stress Shape factor Angle Flow coefficient Pressure coefficient Sphericity

1/m2

1/ft2

° ° ° 1/m

° ° ° 1/ft

Dimensionless kg/(s⋅m) Dimensionless m m Dimensionless

Dimensionless lb/(s⋅ft) Dimensionless ft ft Dimensionless

Pa⋅s

lb/(ft⋅s)

° m

° ft

Pa⋅s m2/s kg/m3 N/m Dimensionless N/m2 Dimensionless °

lb/(ft⋅s) ft2/s lb/ft3 lbf/ft Dimensionless lbf/ft2 Dimensionless °

Dimensionless

Dimensionless

MEASUREMENT OF FLOW GENERAL REFERENCES: ASME, Performance Test Code on Compressors and Exhausters, PTC 10-1997, American Society of Mechanical Engineers (ASME), New York, 1997. Norman A. Anderson, Instrumentation for Process Measurement and Control, 3d ed., CRC Press, Boca Raton, Fla., 1997. Roger C. Baker, Flow Measurement Handbook: Industrial Designs, Operating Principles, Performance, and Applications, Cambridge University Press, Cambridge, United Kingdom, 2000. Roger C. Baker, An Introductory Guide to Flow Measurement, ASME, New York, 2003. Howard S. Bean, ed., Fluid Meters—Their Theory and Application—Report of the ASME Research Committee on Fluid Meters, 6th ed., ASME, New York, 1971. Douglas M. Considine, Editor-in-Chief, Process/Industrial Instruments and Controls Handbook, 4th ed., McGraw-Hill, New York, 1993. Bela G. Liptak, Editor-in-Chief, Process Measurement and Analysis, 4th ed., CRC Press, Boca Raton, Fla., 2003. Richard W. Miller, Flow Measurement Engineering Handbook, 3d ed., McGraw-Hill, New York, 1996. Ower and Pankhurst, The Measurement of Air Flow, Pergamon, Oxford, United Kingdom, 1966. Brian Price et al., Engineering Data Book, 12th ed., Gas Processors Suppliers Association, Tulsa, Okla., 2004. David W. Spitzer, Flow Measurement, 2d ed., Instrument Society of America, Research Triangle Park, N.C., 2001. David W. Spitzer, Industrial Flow Measurement, 3d ed., Instrument Society of America, Research Triangle Park, N.C., 2005.

INTRODUCTION The flow rate of fluids is a critical variable in most chemical engineering applications, ranging from flows in the process industries to environmental flows and to flows within the human body. Flow is defined as mass flow or volume flow per unit of time at specified temperature and pressure conditions for a given fluid. This subsection deals with the techniques of measuring pressure, temperature, velocities, and flow rates of flowing fluids. For more detailed discussion of these variables, consult Sec. 8. Section 8 introduces methods of measuring flow rate, temperature, and pressure. This subsection builds on the coverage in Sec. 8 with emphasis on measurement of the flow of fluids. PROPERTIES AND BEHAVIOR OF FLUIDS Transportation and the storage of fluids (gases and liquids) involves the understanding of the properties and behavior of fluids. The study of fluid dynamics is the study of fluids and their motion in a force field. Flows can be classified into two major categories: (a) incompressible and (b) compressible flow. Most liquids fall into the incompressibleflow category, while most gases are compressible in nature. A perfect fluid can be defined as a fluid that is nonviscous and nonconducting. Fluid flow, compressible or incompressible, can be classified by the ratio of the inertial forces to the viscous forces. This ratio is represented by the Reynolds number (NRe). At a low Reynolds number, the flow is considered to be laminar, and at high Reynolds numbers, the TABLE 10-1

flow is considered to be turbulent. The limiting types of flow are the inertialess flow, sometimes called Stokes flow, and the inviscid flow that occurs at an infinitely large Reynolds number. Reynolds numbers (dimensionless) for flow in a pipe is given as: ρVD µ

NRe = ᎏ

where ρ is the density of the fluid, V the velocity, D the diameter, and µ the viscosity of the fluid. In fluid motion where the frictional forces interact with the inertia forces, it is important to consider the ratio of the viscosity µ to the density ρ. This ratio is known as the kinematic viscosity (ν). Tables 10-1 and 10-2 give the kinematic viscosity for several fluids. A flow is considered to be adiabatic when there is no transfer of heat between the fluid and its surroundings. An isentropic flow is one in which the entropy of each fluid element remains constant. To fully understand the mechanics of flow, the following definitions explain the behavior of various types of fluids in both their static and flowing states. A perfect fluid is a nonviscous, nonconducting fluid. An example of this type of fluid would be a fluid that has a very small viscosity and conductivity and is at a high Reynolds number. An ideal gas is one that obeys the equation of state: P ᎏ = RT ρ

(10-2)

where P = pressure, ρ = density, R is the gas constant per unit mass, and T = temperature. A flowing fluid is acted upon by many forces that result in changes in pressure, temperature, stress, and strain. A fluid is said to be isotropic when the relations between the components of stress and those of the rate of strain are the same in all directions. The fluid is said to be Newtonian when this relationship is linear. These pressures and temperatures must be fully understood so that the entire flow picture can be described. The static pressure in a fluid has the same value in all directions and can be considered as a scalar point function. It is the pressure of a flowing fluid. It is normal to the surface on which it acts and at any given point has the same magnitude irrespective of the orientation of the surface. The static pressure arises because of the random motion in the fluid of the molecules that make up the fluid. In a diffuser or nozzle, there is an increase or decrease in the static pressure due to the change in velocity of the moving fluid. Total pressure is the pressure that would occur if the fluid were brought to rest in a reversible adiabatic process. Many texts and engineers use the words total and stagnation to describe the flow characteristics interchangeably. To be accurate, the stagnation pressure

Density, Viscosity, and Kinematic Viscosity of Water and Air in Terms of Temperature Air at a pressure of 760 mm Hg (14.696 lbf/in2)

Water Temperature (°C)

(°F)

Density ␳ (lbf sec2/ft4)

−20 −10 0 10 20 40 60 80 100

−4 14 32 50 68 104 140 176 212

— — 1.939 1.939 1.935 1.924 1.907 1.886 1.861

Viscosity µ × 106 (lbf sec/ft2)

Kinematic viscosity ν × 106 (ft2/sec)

— — 37.5 27.2 21.1 13.68 9.89 7.45 5.92

— — 19.4 14.0 10.9 7.11 5.19 3.96 3.19

Conversion factors: 1 kp sec2/m4 = 0.01903 lbf sec2/ft4 (= slug/ft3) 1 lbf sec2/ft4 = 32.1719 lb/ft3 (lb = lb mass; lbf = lb force) 1 kp sec2/m4 = 9.80665 kg/m3 (kg = kg mass; kp = kg force) 3 3 1 kg/m = 16.02 lb/ft 10-6

(10-1)

Density ␳ (lbf sec2/ft4)

Viscosity µ × 106 (lbf sec/ft2)

Kinematic viscosity ν × 106 (ft2/sec)

0.00270 0.00261 0.00251 0.00242 0.00234 0.00217 0.00205 0.00192 0.00183

0.326 0.338 0.350 0.362 0.375 0.399 0.424 0.449 0.477

122 130 140 150 160 183 207 234 264

MEASUREMENT OF FLOW TABLE 10-2

Kinematic Viscosity Temperature

Liquid

°C

°F

ν × 106 (ft2/s)

Glycerine Mercury Mercury Lubricating oil Lubricating oil Lubricating oil

20 0 100 20 40 60

68 32 212 68 104 140

7319 1.35 0.980 4306 1076 323

is the pressure that would occur if the fluid were brought to rest adiabatically or diabatically. Total pressure will only change in a fluid if shaft work or work of extraneous forces are introduced. Therefore, total pressure would increase in the impeller of a compressor or pump; it would remain constant in the diffuser. Similarly, total pressure would decrease in the turbine impeller but would remain constant in the nozzles. Static temperature is the temperature of the flowing fluid. Like static pressure, it arises because of the random motion of the fluid molecules. Static temperature is in most practical installations impossible to measure since it can be measured only by a thermometer or thermocouple at rest relative to the flowing fluid that is moving with the fluid. Static temperature will increase in a diffuser and decrease in a nozzle. Total temperature is the temperature that would occur when the fluid is brought to rest in a reversible adiabatic manner. Just like its counterpart total pressure, total and stagnation temperatures are used interchangeably by many test engineers. Dynamic temperature and pressure are the difference between the total and static conditions. Pd = PT − Ps

(10-3)

Td = TT − Ts

(10-4)

where subscript d refers to dynamic, T to total, and s to static. Another helpful formula is: 1 PK = ᎏ ρV2 2

(10-5)

For incompressible fluids, PK = Pd. TOTAL TEMPERATURE For most points requiring temperature monitoring, either thermocouples or resistive thermal detectors (RTDs) can be used. Each type of temperature transducer has its own advantages and disadvantages, and both should be considered when temperature is to be measured. Since there is considerable confusion in this area, a short discussion of the two types of transducers is necessary. Thermocouples The various types of thermocouples provide transducers suitable for measuring temperatures from −330 to 5000°F (−201 to 2760°C). Thermocouples function by producing a voltage proportional to the temperature differences between two junctions of dissimilar metals. By measuring this voltage, the temperature difference can be determined. It is assumed that the temperature is known at one of the junctions; therefore, the temperature at the other junction can be determined. Since the thermocouples produce a voltage, no external power supply is required to the test junction; however, for accurate measurement, a reference junction is required. For a temperature monitoring system, reference junctions must be placed at each thermocouple or similar thermocouple wire installed from the thermocouple to the monitor where there is a reference junction. Properly designed thermocouple systems can be accurate to approximately ±2°F (±1°C). Resistive Thermal Detectors (RTDs) RTDs determine temperature by measuring the change in resistance of an element due to temperature. Platinum is generally utilized in RTDs because it remains mechanically and electrically stable, resists contaminations, and can be highly refined. The useful range of platinum RTDs is

10-7

−454–1832°F (−270−1000°C). Since the temperature is determined by the resistance in the element, any type of electrical conductor can be utilized to connect the RTD to the indicator; however, an electrical current must be provided to the RTD. A properly designed temperature monitoring system utilizing RTDs can be accurate ±0.02°F (±0.01°C). STATIC TEMPERATURE Since this temperature requires the thermometer or thermocouple to be at rest relative to the flowing fluid, it is impractical to measure. It can be, however, calculated from the measurement of total temperature and total and static pressure. TO TS = ᎏᎏ PO (k − 1)/k ᎏᎏ PS

冢 冣

(10-6)

DRY- AND WET-BULB TEMPERATURES The moisture content or humidity of air has an important effect on the properties of the gaseous mixture. Steam in air at any relative humidity less than 100 percent must exist in a superheated condition. The saturation temperature corresponding to the actual partial pressure of the steam in air is called the dew point. This term arose from the fact that when air at less than 100 percent relative humidity is cooled to the temperature at which it becomes saturated, the air has reached the minimum temperature to which it can be cooled without precipitation of the moisture (dew). Dew point can also be defined as that temperature at which the weight of steam associated with a certain weight of dry air is adequate to saturate that weight of air. The dry-bulb temperature of air is the temperature that is indicated by an ordinary thermometer. When an air temperature is stated without any modifying term, it is always taken to be the dry-bulb temperature. In contrast to dry-bulb, or air, temperature, the term wet-bulb temperature of the air, or simply wet-bulb temperature, is employed. When a thermometer, with its bulb covered by a wick wetted with water, is moved through air unsaturated with water vapor, the water evaporates in proportion to the capacity of the air to absorb the evaporated moisture, and the temperature indicated by the thermometer drops below the dry-bulb, or air, temperature. The equilibrium temperature finally reached by the thermometer is known as the wet-bulb temperature. The purpose in measuring both the dry-bulb and wet-bulb temperature of the air is to find the exact humidity characteristics of the air from the readings obtained, either by calculation or by use of a psychrometric chart. Instruments for measuring wet-bulb and dry-bulb temperatures are known as psychrometers. A sling psychrometer consists of two thermometers mounted side by side on a holder, with provision for whirling the whole device through the air. The dry-bulb thermometer is bare, and the wet bulb is covered by a wick which is kept wetted with clean water. After being whirled a sufficient amount of time, the wet-bulb thermometer reaches its equilibrium point, and both the wet-bulb and dry-bulb thermometers are then quickly read. Rapid relative movement of the air past the wet-bulb thermometer is necessary to get dependable readings. For other methods of measuring the moisture content of gases, see Sec. 8. PRESSURE MEASUREMENTS Pressure is defined as the force per unit area. Pressure devices measure with respect to the ambient atmospheric pressure: The absolute pressure Pa is the pressure of the fluid (gauge pressure) plus the atmospheric pressure. Process pressure-measuring devices may be divided into three groups: 1. Those that are based on the height of a liquid column (manometers) 2. Those that are based on the measurement of the distortion of an elastic pressure chamber (mechanical pressure gauges such as Bourdontube gauges and diaphragm gauges)

10-8

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-3

(a) FIG. 10-1

Open manometers.

3. Electric sensing devices (strain gauges, piezoresistive transducers, and piezoelectric transducers) This subsection contains an expanded discussion of manometric methods. See Sec. 8 for other methods. Liquid-Column Manometers The height, or head, pn = ρhgⲐgc to which a fluid rises in an open vertical tube attached to an apparatus containing a liquid is a direct measure of the pressure at the point of attachment and is frequently used to show the level of liquids in tanks and vessels. This same principle can be applied with U-tube gauges (Fig. 10-1a) and equivalent devices (such as that shown in Fig. 10-1b) to measure pressure in terms of the head of a fluid other than the one under test. Most of these gauges may be used either as open or as differential manometers. The manometric fluid that constitutes the measured liquid column of these gauges may be any liquid immiscible with the fluid under pressure. For high vacuums or for high pressures and large pressure differences, the gauge liquid is a high-density liquid, generally mercury; for low pressures and small pressure differences, a low-density liquid (e.g., alcohol, water, or carbon tetrachloride) is used. The open U tube (Fig. 10-1a) and the open gauge (Fig. 10-1b) each show a reading hM m (ft) of manometric fluid. If the interface of the manometric fluid and the fluid of which the pressure is wanted is K m (ft) below the point of attachment, A, ρA is the density of the latter fluid at A, and ρM is that of the manometric fluid, then gauge pressure pA (lbf/ft2) at A is pA = (hMρM − KρA)(gⲐgc)

(10-7)*

where g = local acceleration due to gravity and gc = dimensional constant. The head HA at A as meters (feet) of the fluid at that point is hA = hM(ρMⲐρA) − K

(10-8)*

When a gas pressure is measured, unless it is very high, ρA is so much smaller than ρM that the terms involving K in these formulas are negligible. The differential U tube (Fig. 10-2) shows the pressure difference between taps A and B to be pA − pB = [hM(ρM − ρA) + KAρA − KBρB](gⲐgc)

(10-9)*

where hM is the difference in height of the manometric fluid in the U tube; KA and KB are the vertical distances of the upper surface of the *The line leading from the pressure tap to the gauge is assumed to be filled with fluid of the same density as that in the apparatus at the location of the pressure tap; if this is not the case, ρA is the density of the fluid actually filling the gauge line, and the value given for hA must be multiplied by ρAⲐρ where ρ is the density of the fluid whose head is being measured.

FIG. 10-2

Differential U tube.

Closed U tube.

(b) manometric fluid above A and B, respectively; ρA and ρB are the densities of the fluids at A and B, respectively; and ρM is the density of the manometric fluid. If either pressure tap is above the higher level of manometric fluid, the corresponding K is taken to be negative. Valve D, which is kept closed when the gauge is in use, is used to vent off gas which may accumulate at these high points. The inverted differential U tube, in which the manometric fluid may be a gas or a light liquid, can be used to measure liquid pressure differentials, especially for the flow of slurries where solids tend to settle out. Closed U tubes (Fig. 10-3) using mercury as the manometric fluid serve to measure directly the absolute pressure p of a fluid, provided that the space between the closed end and the mercury is substantially a perfect vacuum. The mercury barometer (Fig. 10-4) indicates directly the absolute pressure of the atmosphere in terms of height of the mercury column. Normal (standard) barometric pressure is 101.325 kPa by definition. Equivalents of this pressure in other units are 760 mm mercury (at 0°C), 29.921 inHg (at 0°C), 14.696 lbf/in2, and 1 atm. For cases in which barometer readings, when expressed by the height of a mercury column, must be corrected to standard temperature (usually 0°C), appropriate temperature correction factors are given in ASME PTC, op. cit., pp. 23–26; and Weast, Handbook of Chemistry and Physics, 62d ed., Chemical Rubber, Cleveland, 1984, pp. E36–E37. Tube Size for Manometers To avoid capillary error, tube diameter should be sufficiently large and the manometric fluids of such densities that the effect of capillarity is negligible in comparison with the gauge reading. The effect of capillarity is practically negligible for tubes with inside diameters 12.7 mm (1⁄2 in) or larger (see ASME PTC, op. cit., p. 15). Small diameters are generally permissible for U tubes because the capillary displacement in one leg tends to cancel that in the other. The capillary rise in a small vertical open tube of circular cross section dipping into a pool of liquid is given by 4σgc cos θ h = ᎏᎏ gD(ρ1 − ρ2)

(10-10)

Here σ = surface tension, D = inside diameter, ρ1 and ρ2 are the densities of the liquid and gas (or light liquid) respectively, g = local acceleration due to gravity, gc = dimensional constant, and θ is the contact angle subtended by the heavier fluid. For most organic liquids and water, the contact angle θ is zero against glass, provided the glass is wet with a film of the liquid; for mercury against glass, θ = 140° (International Critical Tables, vol. IV, McGraw-Hill, New York, 1928, pp. 434–435). For further discussion of capillarity, see Schwartz, Ind. Eng. Chem., 61(1), 10–21 (1969). Multiplying Gauges To attain the requisite precision in measurement of small pressure differences by liquid-column manometers,

FIG. 10-4

Mercury barometer.

MEASUREMENT OF FLOW

FIG. 10-5

Inclined U tube.

means must often be devised to magnify the readings. Of the schemes that follow, the second and third may give tenfold multiplication; the fourth, as much as thirtyfold. In general, the greater the multiplication, the more elaborate must be the precautions in the use of the gauge if the gain in precision is not to be illusory. 1. Change of manometric fluid. In open manometers, choose a fluid of lower density. In differential manometers, choose a fluid such that the difference between its density and that of the fluid being measured is as small as possible. 2. Inclined U tube (Fig. 10-5). If the reading R m (ft) is taken as shown and R0 m (ft) is the zero reading, by making the substitution hM = (R − R0) sin θ, the formulas of preceding paragraphs give (pA −pB) when the corresponding upright U tube is replaced by one inclined. For precise work, the gauge should be calibrated because of possible variations in tube diameter and slope. 3. The draft gauge (Fig. 10-6). Commonly used for low gas heads, this gauge has for one leg of the U a reservoir of much larger bore than the tubing that forms the inclined leg. Hence variations of level in the inclined tube produce little change in level in the reservoir. Although hM may be readily computed in terms of reading R and the dimensions of the tube, calibration of the gauge is preferable; often the changes of level in the reservoir are not negligible, and also variations in tube diameter may introduce serious error into the computation. Commercial gauges are often provided with a scale giving hM directly in height of water column, provided a particular liquid (often not water) fills the tube; failure to appreciate that the scale is incorrect unless the gauge is filled with the specified liquid is a frequent source of error. If the scale reads correctly when the density of the gauge liquid is ρ0, then the reading must be multiplied by ρⲐρ0 if the density of the fluid actually in use is ρ. 4. Two-fluid U tube (Fig. 10-7). This is a highly sensitive device for measuring small gas heads. Let A be the cross-sectional area of each of the reservoirs and a that of the tube forming the U; let ρ1 be the density of the lighter fluid and ρ2 that of the heavier fluid; and if R is the reading and R0 its value with zero pressure difference, then the pressure difference is

冣

冢

a g pA − pB = (R − R0) ρ2 − ρ1 + ᎏ ρ1 ᎏ A gc

(10-11)

where g = local acceleration due to gravity and gc = dimensional constant. When A/a is sufficiently large, the term (a/A) ρ1 in Eq. (10-11) becomes negligible in comparison with the difference (ρ2 − ρ1). However, this term should not be omitted without due consideration. In applying Eq. (10-11), the densities of the gauge liquids may not be taken from tables without the possibility of introducing serious error,

FIG. 10-6

Draft gauge.

FIG. 10-7

Two-fluid U tube.

10-9

for each liquid may dissolve appreciable quantities of the other. Before the gauge is filled, the liquids should be shaken together, and the actual densities of the two layers should be measured for the temperature at which the gauge is to be used. When high magnification is being sought, the U tube may have to be enclosed in a constant-temperature bath so that (ρ2 − ρ1) may be accurately known. In general, if highest accuracy is desired, the gauge should be calibrated. Several micromanometers, based on the liquid-column principle and possessing extreme precision and sensitivity, have been developed for measuring minute gas-pressure differences and for calibrating low-range gauges. Some of these micromanometers are available commercially. These micromanometers are free from errors due to capillarity and, aside from checking the micrometer scale, require no calibration. Mechanical Pressure Gauges The Bourdon-tube gauge indicates pressure by the amount of flection under internal pressure of an oval tube bent in an arc of a circle and closed at one end. These gauges are commercially available for all pressures below atmospheric and for pressures up to 700 MPa (about 100,000 lbf/in2) above atmospheric. Details on Bourdon-type gauges are given by Harland [Mach. Des., 40(22), 69–74 (Sept. 19, 1968)]. A diaphragm gauge depends for its indication on the deflection of a diaphragm, usually metallic, when subjected to a difference of pressure between the two faces. These gauges are available for the same general purposes as Bourdon gauges but are not usually employed for high pressures. The aneroid barometer is a type of diaphragm gauge. Small pressure transducers with flush-mounted diaphragms are commercially available for the measurement of either steady or fluctuating pressures up to 100 MPa (about 15,000 lbf/in2). The metallic diaphragms are as small as 4.8 mm (3⁄16 in) in diameter. The transducer is mounted on the apparatus containing the fluid whose pressure is to be measured so that the diaphragm is flush with the inner surface of the apparatus. Deflection of the diaphragm is measured by unbonded strain gauges and recorded electrically. With nonnewtonian fluids the pressure measured at the wall with non-flush-mounted pressure gauges may be in error (see subsection “Static Pressure”). Bourdon and diaphragm gauges that show both pressure and vacuum indications on the same dial are called compound gauges. Conditions of Use Bourdon tubes should not be exposed to temperatures over about 65°C (about 150°F) unless the tubes are specifically designed for such operation. When the pressure of a hotter fluid is to be measured, some type of liquid seal should be used to keep the hot fluid from the tube. In using either a Bourdon or a diaphragm gauge to measure gas pressure, if the gauge is below the pressure tap of the apparatus so that liquid can collect in the lead, the gauge reading will be too high by an amount equal to the hydrostatic head of the accumulated liquid. For measuring pressures of corrosive fluids, slurries, and similar process fluids which may foul Bourdon tubes, a chemical gauge, consisting of a Bourdon gauge equipped with an appropriate flexible diaphragm to seal off the process fluid, may be used. The combined volume of the tube and the connection between the diaphragm and the tube is filled with an inert liquid. These gauges are available commercially. Further details on pressure-measuring devices are found in Sec. 8. Calibration of Gauges Simple liquid-column manometers do not require calibration if they are so constructed as to minimize errors due to capillarity (see subsection “Liquid-Column Manometers”). If the scales used to measure the readings have been checked against a standard, the accuracy of the gauges depends solely upon the precision of determining the position of the liquid surfaces. Hence liquid-column manometers are primary standards used to calibrate other gauges. For high pressures and, with commercial mechanical gauges, even for quite moderate pressures, a deadweight gauge (see ASME PTC, op. cit., pp. 36–41) is commonly used as the primary standard because it is safer and more convenient than use of manometers. When manometers are used as high-pressure standards, an extremely high mercury column may be avoided by connecting a number of the usual U tubes in series. Multiplying gauges are standardized by comparing

10-10

TRANSPORT AND STORAGE OF FLUIDS

L D Simple vane flow straightener (a) FIG. 10-8

Measurement of static pressure.

them with a micromanometer. Procedure in the calibration of a gauge consists merely of connecting it, in parallel with a standard gauge, to a reservoir wherein constant pressure may be maintained. Readings of the unknown gauge are then made for various reservoir pressures as determined by the standard. Calibration of high-vacuum gauges is described by Sellenger [Vacuum, 18(12), 645–650 (1968)].

L Multitube flow straightener

D1

(b)

STATIC PRESSURE Local Static Pressure In a moving fluid, the local static pressure is equal to the pressure on a surface which moves with the fluid or to the normal pressure (for newtonian fluids) on a stationary surface which parallels the flow. The pressure on such a surface is measured by making a small hole perpendicular to the surface and connecting the opening to a pressure-sensing element (Fig. 10-8a). The hole is known as a piezometer opening or pressure tap. Measurement of local static pressure is frequently difficult or impractical. If the channel is so small that introduction of any solid object disturbs the flow pattern and increases the velocity, there will be a reduction and redistribution of the static pressure. If the flow is in straight parallel lines, aside from the fluctuations of normal turbulence, the flat disk (Fig. 10-8b) and the bent tube (Fig. 10-8c) give satisfactory results when properly aligned with the stream. Slight misalignments can cause serious errors. Diameter of the disk should be 20 times its thickness and 40 times the static opening; the face must be flat and smooth, with the knife edges made by beveling the underside. The piezometer tube, such as that in Fig. 10-8c, should have openings with size and spacing as specified for a pitot-static tube (Fig. 10-12). Readings given by open straight tubes (Fig. 10-8d and 10-8e are too low due to flow separation. Readings of closed tubes oriented perpendicularly to the axis of the stream and provided with side openings (Fig. 10-8e) may be low by as much as two velocity heads. Average Static Pressure In most cases, the object of a staticpressure measurement is to obtain a suitable average value for substitution in Bernoulli’s theorem or in an equivalent flow formula. This can be done simply only when the flow is in straight lines parallel to the confining walls, such as in straight ducts at sufficient distance downstream from bends (2 diameters) or other disturbances. For such streams, the sum of static head and gravitational potential head is the same at all points in a cross section taken perpendicularly to the axis of flow. Thus the exact location of a piezometer opening about the periphery of such a cross section is immaterial provided its elevation is known. However, in stating the static pressure, the custom is to give the value at the elevation corresponding to the centerline of the stream. With flow in curved passages or with swirling flow, determination of a true average static pressure is, in general, impractical. In metering, straightening vanes are often placed upstream of the pressure tap to eliminate swirl. Figure 10-9 shows various flow equalizers and straighteners. Specifications for Piezometer Taps The size of a static opening should be small compared with the diameter of the pipe and yet large compared with the scale of surface irregularities. For reliable results, it is essential that (1) the surface in which the hole is made be substantially smooth and parallel to the flow for some distance on either side of the

Equalizer (perforated plate or screen) (c)

D 3 min. 2 Combination equalizer and straightener (d)

b

Q 2D Multiplate type equalizer and straightener (e) FIG. 10-9 Flow equalizers and straighteners [Power Test Code 10, Compressors and Exhausters, Amer. Soc. of Mechanical Engineers, 1997].

opening, and (2) the opening be flush with the surface and possess no “burr” or other irregularity around its edge. Rounding of the edge is often employed to ensure absence of a burr. Pressure readings will be high if the tap is inclined upstream, is rounded excessively on the upstream side, has a burr on the downstream side, or has an excessive countersink or recess. Pressure readings will be low if the tap is inclined downstream, is rounded excessively on the downstream side, has a burr on the upstream side, or protrudes into the flow stream. Errors resulting from these faults can be large. Recommendations for pressure-tap dimensions are summarized in Table 10-3. Data from several references were used in arriving at these composite values. The length of a pressure-tap opening prior to any enlargement in the tap channel should be at least two tap diameters, preferably three or more.

MEASUREMENT OF FLOW TABLE 10-3

Pressure-Tap Holes

Nominal inside pipe diameter, in

Maximum diameter of pressure tap, mm (in)

Radius of hole-edge rounding, mm (in)

1 2 3 4 8 16

3.18 (f) 6.35 (d) 9.53 (r) 12.7 (a) 12.7 (a) 19.1 (e)

<0.40 (1⁄64) 0.40 (1⁄64) 0.40–0.79 (1⁄64–h) 0.79 (h) 0.79–1.59 (h–g) 0.79–1.59 (h–g)

A piezometer ring is a toroidal manifold into which are connected several sidewall static taps located around the perimeter of a common cross section. Its intent is to give an average pressure if differences in pressure other than those due to static head exist around the perimeter. However, there is generally no assurance that a true average is provided thereby. The principal advantage of the ring is that use of several holes in place of a single hole reduces the possibility of completely plugging the static openings. For information on prediction of static-hole error, see Shaw, J. Fluid Mech., 7, 550–564 (1960); Livesey, Jackson, and Southern, Aircr. Eng., 34, 43–47 (February 1962). For nonnewtonian fluids, pressure readings with taps may also be low because of fluid-elasticity effects. This error can be largely eliminated by using flush-mounted diaphragms. For information on the pressure-hole error for nonnewtonian fluids, see Han and Kim, Trans. Soc. Rheol., 17, 151–174 (1973); Novotny and Eckert, Trans. Soc. Rheol., 17, 227–241 (1973); and Higashitani and Lodge, Trans. Soc. Rheol., 19, 307–336 (1975). VELOCITY MEASUREMENTS Measurement of flow can be based on the measurement of velocity in ducts or pipes by using devices such as pitot tubes and hot wire anemometers. The local velocity is measured at various sections of a conduit and then averaged for the area under consideration. w ᎏ =A×V=Q ρ

(10-12)

where w = mass flow rate, lbmⲐs, kgⲐs ρ = density, lbmⲐft3, kgⲐm3 A = area, ftⲐs, mⲐs V = velocity, ftⲐs, mⲐs Q = volumetric flow rate, ft3Ⲑs, m3Ⲑs Equation (10-12) shows that the fluid density directly affects the relationship between mass flow rate and both velocity and volumetric flow rate. Liquid temperature affects liquid density and hence volumetric flow rate at a constant mass flow rate. Liquid density is relatively insensitive to pressure. Both temperature and pressure affect gas density and thus volumetric flow rate. Variables Affecting Measurement Flow measurement methods may sense local fluid velocity, volumetric flow rate, total or cumulative volumetric flow (the integral of volumetric flow rate with respect to elapsed time), mass flow rate, and total mass flow. Velocity Profile Effects Many variables can influence the accuracy of specific flow measurement methods. For example, the velocity profile in a closed conduit affects many types of flow-measuring devices. The velocity of a fluid varies from zero at the wall and at other stationary solid objects in the flow channel to a maximum at a distance from the wall. In the entry region of a conduit, the velocity field may approach plug flow and a constant velocity across the conduit, dropping to zero only at the wall. As a newtonian fluid progresses down a pipe, a velocity profile develops that is parabolic for laminar flow [Eq. (6-41)] and that approaches plug flow for highly turbulent flow. Once a steady flow profile has developed, the flow is said to be fully developed; the length of conduit necessary to achieve fully devel-

10-11

oped flow is called the entrance region. For long cylindrical, horizontal pipe (L < 40D, where D is the inside diameter of the pipe and L is the upstream length of pipe), the velocity profile becomes fully developed. Velocity profiles in flowing fluids are discussed in greater detail in Sec. 6 (p. 6-11). For steady-state, isothermal, single-phase, uniform, fully developed newtonian flow in straight pipes, the velocity is greatest at the center of the channel and symmetric about the axis of the pipe. Of those flowmeters that are dependent on the velocity profile, they are usually calibrated for this type of flow. Thus any disturbances in flow conditions can affect flowmeter readings. Upstream and downstream disturbances in the flow field are caused by valves, elbows, and other types of fittings. Two upstream elbows in two perpendicular planes will impart swirl in the fluid downstream. Swirl, similar to atypical velocity profiles, can lead to erroneous flow measurements. Although the effect is not as great as in upstream flow disturbances, downstream flow disturbances can also lead to erroneous flow measurements. Other Flow Disturbances Other examples of deviations from fully developed, single-phase newtonian flow include nonnewtonian flow, pulsating flow, cavitation, multiphase flow, boundary layer flows, and nonisothermal flows. See Sec. 6. Pitot Tubes The combination of pitot tubes in conjunction with sidewall static taps measures local or point velocities by measuring the difference between the total pressure and the static pressure. The pitot tube shown in Fig. 10-10 consists of an impact tube whose opening faces directly into the stream to measure impact pressure, plus one or more sidewall taps to measure local static pressure. Dynamic pressure may be measured by use of a pitot tube that is a simple impact tube. These tubes measure the pressure at a point where the velocity of the fluid is brought to zero. Pitot tubes must be parallel to the flow. The pitot tube is sensitive to yaw or angle attack. In general angles of attack over 10° should be avoided. In cases where the flow direction is unknown, it is recommended to use a Kiel probe. Figure 10-11 shows a Kiel probe. This probe will read accurately to an angle of about 22° with the flow. The combined pitot-static tube shown in Fig. 10-12 consists of a jacketed impact tube with one or more rows of holes, 0.51 to 1.02 mm (0.02 to 0.04 in) in diameter, in the jacket to measure the static pressure. Velocity V0 m/s (ft/s) at the point where the tip is located is given by V0 = C 兹苶 2gc∆h = C 兹2g 苶 − PS)Ⲑ苶 ρ0 c(PT 苶

(10-13)

where C = coefficient, dimensionless; gc = dimensional constant; ∆h = dynamic pressure (∆hsgⲐgc), expressed in (N⋅m)Ⲑkg[(ft⋅lbf)Ⲑlb or ft of fluid flowing]; ∆hs = differential height of static liquid column corresponding to ∆h; g = local acceleration due to gravity; gc = dimensional constant; pi = impact pressure; p0 = local static pressure; and ρ0 = fluid density measured at pressure p0 and the local temperature. With gases at velocities above 60 m/s (about 200 ft/s), compressibility becomes important, and the following equation should be used: V0 = C

FIG. 10-10

2g k p p −1 ᎏ 莦莦 ᎏ 莦莦 ᎏ莦 冪莦 k − 1 冢 ρ 冣莦冤冢 p 冣 莦莦莦莦莦冥莦 c

0

i

0

0

Pitot tube with sidewall static tap.

(k − 1)/k

(10-14)

10-12

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-11

Kiel probe. Accurate measurements can be made at angles up to 22.5° with the flow stream.

where k is the ratio of specific heat at constant pressure to that at constant volume. (See ASME Research Committee on Fluid Meters Report, op. cit., p. 105.) Coefficient C is usually close to 1.00 (±0.01) for simple pitot tubes (Fig. 10-10) and generally ranges between 0.98 and 1.00 for pitot-static tubes (Fig. 10-12). There are certain limitations on the range of usefulness of pitot tubes. With gases, the differential is very small at low velocities; e.g., at 4.6 m/s (15.1 ft/s) the differential is only about 1.30 mm (0.051 in) of water (20°C) for air at 1 atm (20°C), which represents a lower limit for 1 percent error even when one uses a micromanometer with a precision of 0.0254 mm (0.001 in) of water. Equation does not apply for Mach numbers greater than 0.7 because of the interference of shock waves. For supersonic flow, local Mach numbers can be calculated from a knowledge of the dynamic and true static pressures. The free stream Mach number (M∞) is defined as the ratio of the speed of the stream (V∞) to the speed of sound in the free stream: ∂P ᎏ 冣莦 冪冢莦莦 ∂ρ

(10-15)

V∞ M∞ = ᎏ ∂P ᎏᎏ ∂ρ s = c

(10-16)

A∞ =

s=c

冪冢莦莦莦冣莦

FIG. 10-12

Pitot-static tube.

where S is the entropy. For isentropic flow, this relationship and pressure can be written as: V∞ M∞ = ᎏ 苶T 苶苶s 兹kR

(10-17)

The relationships between total and static temperature and pressure are given by the following relationship: k−1 TT ᎏ = 1 + ᎏ M2 2 TS k−1 PT ᎏ = 1 + ᎏ M2 2 PS

冢

(10-18)

冣

(k − 1)/k

(10-19)

With liquids at low velocities, the effect of the Reynolds number upon the coefficient is important. The coefficients are appreciably less than unity for Reynolds numbers less than 500 for pitot tubes and for Reynolds numbers less than 2300 for pitot-static tubes [see Folsom, Trans. Am. Soc. Mech. Eng., 78, 1447–1460 (1956)]. Reynolds numbers here are based on the probe outside diameter. Operation at low Reynolds numbers requires prior calibration of the probe. The pitot-static tube is also sensitive to yaw or angle of attack than is the simple pitot tube because of the sensitivity of the static taps to orientation. The error involved is strongly dependent upon the exact probe dimensions. In general, angles greater than 10° should be avoided if the velocity error is to be 1 percent or less. Disturbances upstream of the probe can cause large errors, in part because of the turbulence generated and its effect on the static-pressure measurement. A calming section of at least 50 pipe diameters is desirable. If this is not possible, the use of straightening vanes or a honeycomb is advisable. The effect of pulsating flow on pitot-tube accuracy is treated by Ower et al., op. cit., pp. 310–312. For sinusoidal velocity fluctuations, the ratio of indicated velocity to actual mean velocity is given by the factor 兹苶 1 + λ2Ⲑ苶2, where λ is the velocity excursion as a fraction of the mean velocity. Thus, the indicated velocity would be about 6 percent high for velocity fluctuations of ±50 percent, and pulsations greater than ±20 percent should be damped to avoid errors greater than 1 percent. The error increases as the frequency of flow oscillations approaches the natural frequency of the pitot tube and the density of

MEASUREMENT OF FLOW

10-13

FIG. 10-13 Velocity ration versus Reynolds number for smooth circular pipes. [Based on data from Rothfus, Archer, Klimas, and Sikchi, Am. Inst. Chem. Eng. J., 3, 208 (1957).]

the measuring fluid approaches the density of the process fluid [see Horlock and Daneshyar, J. Mech. Eng. Sci., 15, 144–152 (1973)]. Pressures substantially lower than true impact pressures are obtained with pitot tubes in turbulent flow of dilute polymer solutions [see Halliwell and Lewkowicz, Phys. Fluids, 18, 1617–1625 (1975)]. Special Tubes A variety of special forms of the pitot tube have been evolved. Folsom (loc. cit.) gives a description of many of these special types together with a comprehensive bibliography. Included are the impact tube for boundary-layer measurements and shielded total-pressure tubes. The latter are insensitive to angle of attack up to 40°. Chue [Prog. Aerosp. Sci., 16, 147–223 (1975)] reviews the use of the pitot tube and allied pressure probes for impact pressure, static pressure, dynamic pressure, flow direction and local velocity, skin friction, and flow measurements. A reversed pitot tube, also known as a pitometer, has one pressure opening facing upstream and the other facing downstream. Coefficient C for this type is on the order of 0.85. This gives about a 40 percent increase in pressure differential as compared with standard pitot tubes and is an advantage at low velocities. There are commercially available very compact types of pitometers which require relatively small openings for their insertion into a duct. The pitot-venturi flow element is capable of developing a pressure differential 5 to 10 times that of a standard pitot tube. This is accomplished by employing a pair of concentric venturi elements in place of the pitot probe. The low-pressure tap is connected to the throat of the inner venturi, which in turn discharges into the throat of the outer venturi. For a discussion of performance and application of this flow element, see Stoll, Trans. Am. Soc. Mech. Eng., 73, 963–969 (1951). Traversing for Mean Velocity Mean velocity in a duct can be obtained by dividing the cross section into a number of equal areas, finding the local velocity at a representative point in each, and averaging the results. In the case of rectangular passages, the cross section is usually divided into small squares or rectangles and the velocity is found at the center of each. In circular pipes, the cross section is divided into several equal annular areas as shown in Fig. 10-13. Readings of velocity are made at the intersections of a diameter and the set of circles which bisect the annuli and the central circle. For an N-point traverse on a circular cross section, make readings on each side of the cross section at 苶n 苶苶 苶)苶 percent 100 × 兹(2 −苶)/ 1苶N

(n = 1, 2, 3 to N/2)

of the pipe radius from the center. Traversing several diameters spaced at equal angles about the pipe is required if the velocity distri-

bution is unsymmetrical. With a normal velocity distribution in a circular pipe, a 10-point traverse theoretically gives a mean velocity 0.3 percent high; a 20-point traverse, 0.1 percent high. For normal velocity distribution in straight circular pipes at locations preceded by runs of at least 50 diameters without pipe fittings or other obstructions, the graph in Fig. 10-13 shows the ratio of mean velocity V to velocity at the center umax plotted against the Reynolds number, where D = inside pipe diameter, ρ = fluid density, and µ = fluid viscosity, all in consistent units. Mean velocity is readily determined from this graph and a pitot reading at the center of the pipe if the quantity DumaxρⲐµ is less than 2000 or greater than 5000. The method is unreliable at intermediate values of the Reynolds number. Methods for determining mean flow rate from probe measurements under nonideal conditions are described by Mandersloot, Hicks, and Langejan [Chem. Eng. (London), no. 232, CE370-CE380 (1969)]. The hot-wire anemometer consists essentially of an electrically heated fine wire (generally platinum) exposed to the gas stream whose velocity is being measured. An increase in fluid velocity, other things being equal, increases the rate of heat flow from the wire to the gas, thereby tending to cool the wire and alter its electrical resistance. In a constant-current anemometer, gas velocity is determined by measuring the resulting wire resistance; in the constant-resistance type, gas velocity is determined from the current required to maintain the wire temperature, and thus the resistance, constant. The difference in the two types is primarily in the electric circuits and instruments employed. The hot-wire anemometer can, with suitable calibration, accurately measure velocities from about 0.15 m/s (0.5 ft/s) to supersonic velocities and detect velocity fluctuations with frequencies up to 200,000 Hz. Fairly rugged, inexpensive units can be built for the measurement of mean velocities in the range of 0.15 to 30 m/s (about 0.5 to 100 ft/s). More elaborate, compensated units are commercially available for use in unsteady flow and turbulence measurements. In calibrating a hotwire anemometer, it is preferable to use the same gas, temperature, and pressure as will be encountered in the intended application. In this case the quantity I2RwⲐ∆t can be plotted against 兹V 苶, where I = hot-wire current, Rw = hot-wire resistance, ∆t = difference between the wire temperature and the gas bulk temperature, and V = mean local velocity. A procedure is given by Wasan and Baid [Am. Inst. Chem. Eng. J., 17, 729–731 (1971)] for use when it is impractical to calibrate with the same gas composition or conditions of temperature and pressure. Andrews, Bradley, and Hundy [Int. J. Heat Mass Transfer, 15, 1765–1786 (1972)] give a calibration correlation for measurement

10-14

TRANSPORT AND STORAGE OF FLUIDS

of small gas velocities. The hot-wire anemometer is treated in considerable detail in Dean, op. cit., chap. VI; in Ladenburg et al., op. cit., art. F-2; by Grant and Kronauer, Symposium on Measurement in Unsteady Flow, American Society of Mechanical Engineers, New York, 1962, pp. 44–53; ASME Research Committee on Fluid Meters Report, op. cit., pp. 105–107; and by Compte-Bellot, Ann. Rev. Fluid Mech., 8, pp. 209–231 (1976). The hot-wire anemometer can be modified for liquid measurements, although difficulties are encountered because of bubbles and dirt adhering to the wire. See Stevens, Borden, and Strausser, David Taylor Model Basin Rep. 953, December 1956; Middlebrook and Piret, Ind. Eng. Chem., 42, 1511–1513 (1950); and Piret et al., Ind. Eng. Chem., 39, 1098–1103 (1947). The hot-film anemometer has been developed for applications in which use of the hot-wire anemometer presents problems. It consists of a platinum-film sensing element deposited on a glass substrate. Various geometries can be used. The most common involves a wedge with a 30° included angle at the end of a tapered rod. The wedge is commonly 1 mm (0.039 in) long and 0.2 mm (0.0079 in) wide on each face. Compared with the hot wire, it is less susceptible to fouling by bubbles or dirt when used in liquids, has greater mechanical strength when used with gases at high velocities and high temperatures, and can give a higher signal-to-noise ratio. For additional information see Ling and Hubbard, J. Aeronaut. Sci., 23, 890–891 (1956); and Ling, J. Basic Eng., 82, 629–634 (1960). The heated-thermocouple anemometer measures gas velocity from the cooling effect of the gas stream flowing across the hot junctions of a thermopile supplied with constant electrical power input. Alternate junctions are maintained at ambient temperature, thus compensating for the effect of ambient temperature. For details see Bunker, Proc. Instrum. Soc. Am., 9, pap. 54-43-2 (1954). A glass-coated bead thermistor anemometer can be used for the measurement of low fluid velocities, down to 0.001 m/s (0.003 ft/s) in air and 0.0002 m/s (0.0007 ft/s) in water [see Murphy and Sparks, Ind. Eng. Chem. Fundam., 7, 642–645 (1968)]. The laser-Doppler anemometer measures local fluid velocity from the change in frequency of radiation, between a stationary source and a receiver, due to scattering by particles along the wave path. A laser is commonly used as the source of incident illumination. The measurements are essentially independent of local temperature and pressure. This technique can be used in many different flow systems with transparent fluids containing particles whose velocity is actually measured. For a brief review of the laser-Doppler technique see Goldstein, Appl. Mech. Rev., 27, 753–760 (1974). For additional details see Durst, Melling, and Whitelaw, Principles and Practice of Laser-Doppler Anemometry, Academic, New York, 1976. FLOWMETERS In the process industries, flow measurement devices are the largest market in the process instrumentation field. Two web sites for process equipment and instrumentation, www.globalspec.com, and www.thomasnet.com, both list more than 800 companies that offer flow measurement products. There are more than one hundred types of flowmeters commercially available. The aforementioned web sites not only facilitate selection and specification of commercial flowmeters, but also provide electronic access to manufacturers’ technical literature. Devices that measure flow can be categorized in two areas as follows: 1. All types of measuring devices in which the material passes without being divided into isolated quantities. Movement of the material is usually sensed by a primary measuring element which activates a secondary device. The flow rate is then inferred from the response of the secondary device by means of known physical laws or from empirical relationships. 2. A positive-displacement meter, which applies to a device in which the flow is divided into isolated measured volumes. The number of fillings of these known volumes are measured with respect to time. The most common application of flow measurement in process plants is flow in pipes, ducts, and tubing. Table 10-4 lists widely used

flowmeters for these closed conduits as well as the two major classes of open-channel flowmeters. Table 10-4 also lists many other types of flowmeters that are discussed later in this subsection. This subsection summarizes selection and installation of flowmeters, including the measurement of pressure and velocities of fluids when the flow measurement technique requires it. INDUSTRY GUIDELINES AND STANDARDS Because flow measurement is important, many engineering societies and trade organizations have developed flow-related guidelines, standards, and other publications (Table 10-5). The reader should consult the appropriate standards when specifying, installing, and calibrating flow measurement systems. There are also numerous articles in scholarly journals, trade magazines, and manufacturers’ literature related to flow measurement. Different types of flowmeters differ markedly in their degrees of sensitivity to flow disturbances. In the most extreme cases, obtaining highly accurate flow measurements with certain types of flowmeters may require 60D upstream straight pipe and 20D downstream. Valves can be particularly problematic because their effects on a flowmeter vary with valve position. Numerous types of flow straighteners or conditioners, as shown in Fig. 10-9, can significantly reduce the required run of straight pipe upstream of a given flowmeter. CLASSIFICATION OF FLOWMETERS Table 10-4 lists the major classes of flowmeters, along with common examples of each. Brief descriptions are provided in this subsection, followed by more details in subsequent subsections. Differential Pressure Meters Differential pressure meters or head meters measure the change in pressure across a special flow element. The differential pressure increases with increasing flow rate. The pitot tubes described previously work on this principle. Other examples include orifices [see also Eqs. (6-111) and (8-102), and Fig. 10-14], nozzles (Fig. 10-19), targets, venturis (see also Sec. 8 and Fig. 10-17), and elbow meters. Averaging pitot tubes produce a pressure differential that is based on multiple measuring points across the flow path. Differential pressure meters are widely used. Temperature, pressure, and density affect gas density and readings of differential pressure meters. For that reason, many commercial flowmeters that are based on measurement of differential pressure often have integral temperature and absolute pressure measurements in addition to differential pressure. They also frequently have automatic temperature and pressure compensation. Velocity Meters Velocity meters measure fluid velocity. Examples include electromagnetic, propeller, turbine, ultrasonic Doppler, ultrasonic transit time, and vortex meters. Section 8 describes the principles of operation of electromagnetic, turbine, ultrasonic, and vortex flowmeters. Mass Meters Mass flowmeters measure the rate of mass flow through a conduit. Examples include Coriolis flowmeters and thermal mass flowmeters. Coriolis flowmeters can measure fluid density simultaneously with mass flow rate. This permits calculation of volumetric flow rate as well. Section 8 includes brief descriptions of Coriolis and thermal mass flowmeters. Volumetric Meters Volumetric meters (also called positivedisplacement flowmeters) are devices that mechanically divide a fluid stream into discrete, known volumes and count the number of volumes that pass through the device. See Spitzer (2005, op. cit.). Variable-Area Meters Variable-area meters, which are also called rotameters, offer popular and inexpensive flow measurement devices. These meters employ a float inside a tube that has an internal cross-sectional area that increases with distance upward in the flow path through the tube. As the flow rate increases, the float rises in the tube to provide a larger area for the flowing fluid to pass. Open-Channel Flow Measurement Open-channel flow measurements are usually based on measurement of liquid level in a flow channel constructed of a specified geometry. The two most common flow channels used are weirs and flumes. See Spitzer (2005, op. cit.).

MEASUREMENT OF FLOW

10-15

TABLE 10-4 Comparison of Flowmeter Technologies Flowmeter technology

Accuracy* (+/–)

Turndown

Pipe sizes,‡ in

Fluids†

Maximum pressure,‡ psig

Temperature range,‡ o F

Pipe run

Relative pressure loss

Differential Pressure Meters Pitot Averaging pitot

1% R

8 :1

L, G

8 :1

L, G, S

1 to 96

2.76E-4 to 200

−200 to 750

8800

−20 to 2370

0.25 to 72

Orifice

L L Long

0.5 to 1.5% R

4 :1

L, G, S

0.5 to 40

8800

−4 to 2300

M

Eccentric

2% R

4 :1

L, G, S

0.5 to 40

8800

−4 to 2300

M

Segmental

2% R

4 :1

L, G, S, SL

0.5 to 40

8800

−4 to 2300

M

0.5 to 1% R

10 :1

L, G, S

> 0.5

4000

1000

M

0.5 to 1.5% R

10 :1

L, G, S, SL

1 to 120

8800

−4 to 2300

L

0.5 to 2% R

8 :1

L, G, S

2 to 80

>1000

<1000

M

Square-edged

Orifice and multivariable flow transmitter Venturi Flow nozzle V cone

0.5% F

10 :1

L, G, S, SL

0.25 to 120

6000

To 400

M

Wedge

0.5 to 5% R

10 :1

L, G, S, SL

0.5 to 24

>600

‡

M

Velocity Meters Correlation Electromagnetic

0.5% R

10 :1

L, G, SL

0.2 to 2% R

10 :1

L

1 to 60 0.15 to 60

Piping limits 5000

To 600

Long

L

−40 to 350

Short

L

Short

M

Propeller

2% R

15 :1

L

2 to 12

230

0 to 300

Turbine

0.15 to 1% R

10 :1

L, G

0.5 to 30

6000

−450 to 600

L, G, SL

0.5 to 200

6000

−40 to 250

Long

L

1 to 540

6000

−40 to 650

Long

L

0.5 to 16

1500

−330 to 800

Short

M

Ultrasonic Doppler

1 to 30% R

50 :1

Ultrasonic transit time

0.5 to 5% R

Down to zero flow

Vortex

0.5 to 2% R

20 :1

L, G L, G, S

M

Mass Meters Coriolis Thermal (for gases) Thermal (for liquids) Volumetric Variable area

0.1 to 0.3% R

10 :1 to 80 :1

L, G

0.06 to 12

5700

−400 to 800

None

L, M

1% F

50 :1

G

0.125 to 8

4500

32 to 572

Short

L

0.5% F

50 :1

L

0.06 to 0.25

4500

40 to 165

Short

L

0.15 to 2% R

10 :1

L

0.25 to 16

2000

−40 to 600

None

M to H

1 to 5% F

10 :1

L, G

0.125 to 6

6000

<1000

None

M

Open-Channel Flowmeters Weirs

2 to 5% R

25 :1

L

Wide range

NA

NA

L

Flumes

3 to 10% R

40 :1

L

Wide range

NA

NA

L

*F = full scale, R = rate. †L = liquid, G = gas, S = steam, SL = slurry. ‡Dependent on the material selection and application. Readers should consult manufacturers for current capabilities. Adapted from J. Pomroy, Chemical Engineering, pp. 94–102, May 1996; J. W. Dolenc, Chemical Engineering Progress, pp. 22–32, Jan. 1996; R. C. Baker, Introductory Guide to Flow Measurement, American Society of Mechanical Engineers, New York, 2003; R. W. Miller, Flow Measurement Engineering Handbook, 3d ed., McGraw-Hill, New York, 1996; D. W. Spitzer, Industrial Flow Measurement, 3d ed., The Instrumentation, Systems, and Automation Society, Research Triangle Park, N.C., 2005; and manufacturers’ literature at www.globalspec.com.

DIFFERENTIAL PRESSURE FLOWMETERS General Principles If a constriction is placed in a closed channel carrying a stream of fluid, there will be an increase in velocity, and hence an increase in kinetic energy, at the point of constriction. From an energy balance, as given by Bernoulli’s theorem [see Sec. 6, subsection “Energy Balance,” Eq. (6-16)], there must be a corresponding reduction in pressure. Rate of discharge from the constriction can be calculated by knowing this pressure reduction, the area available for flow at the constriction, the density of the fluid, and the coefficient of discharge C. The last-named is defined as the ratio of actual flow to the theoretical flow and makes allowance for stream contraction and frictional effects. The metering characteristics of commonly used differential pressure meters are reviewed and grouped by Halmi [J. Fluids Eng., 95, 127–141 (1973)].

The term static head generally denotes the pressure in a fluid due to the head of fluid above the point in question. Its magnitude is given by the application of Newton’s law (force = mass × acceleration). In the case of liquids (constant density), the static head ph Pa (lbf/ft2) is given by ph = hρgⲐgc

(10-20)

where h = head of liquid above the point, m (ft); ρ = liquid density; g = local acceleration due to gravity; and gc = dimensional constant. The head developed in a compressor or pump is the energy force per unit mass. In the measuring systems it is often misnamed as (ft) while the units are really ft-lb/lbm or kilojoules. For a compressor or turbine, it is represented by the following relationship: E = U1Vθ1 − U2Vθ2

(10-21)

10-16

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-5 Guidelines, Standards, and Other Publications Related to Flow Measurement Number of guidelines and standards*

Technical society American Gas Association (AGA) American Petroleum Institute (API) American Society of Heating, Refrigeration, and Air Conditioning Engineers (ASHRAE) American Society of Mechanical Engineers (ASME) ASTM International (ASTM) British Standards Institution (BSI) Deutsches Institut fur Normung E. V. (DIN) International Electrotechnical Commission (IEC) Instrumentation, Systems, and Automation Society (ISA) International Organization for Standardization (ISO) SAE International (SAE)

2 11 5 18 17 100 48 6 3 212 6

*Number of documents identified by searching for flow measurement on http://global.ihs.com, the web site of a clearinghouse of industry guidelines, codes, and standards.

where U is the blade speed and Vθ is the tangential velocity component of absolute velocity. This equation is known as the Euler equation. Orifice Meters A square-edged or sharp-edged orifice, as shown in Fig. 10-14, is a clean-cut square-edged hole with straight walls perpendicular to the flat upstream face of a thin plate placed crosswise of the channel. The stream issuing from such an orifice attains its minimum cross section (vena contracta) at a distance downstream of the orifice which varies with the ratio β of orifice to pipe diameter (see Fig. 10-15). For a centered circular orifice in a pipe, the pressure differential is customarily measured between one of the following pressure-tap pairs. Except in the case of flange taps, all measurements of distance from the orifice are made from the upstream face of the plate. 1. Corner taps. Static holes drilled one in the upstream and one in the downstream flange, with the openings as close as possible to the orifice plate. 2. Radius taps. Static holes located one pipe diameter upstream and one-half pipe diameter downstream from the plate. 3. Pipe taps. Static holes located 21⁄2 pipe diameters upstream and eight pipe diameters downstream from the plate.

(a)

Coefficient of discharge for square-edged circular orifices for NRe > 30,000 with the upstream tap located between one and two pipe diameters from the orifice plate. [Spitzglass, Trans. Am. Soc. Mech. Eng., 44, 919 (1922).] FIG. 10-15

4. Flange taps. Static holes located 25.4 mm (1 in) upstream and 25.4 mm (1 in) downstream from the plate. 5. Vena-contracta taps. The upstream static hole is one-half to two pipe diameters from the plate. The downstream tap is located at the position of minimum pressure (see Fig. 10-15). Radius taps are best from a practical standpoint; the downstream pressure tap is located at about the mean position of the vena contracta, and the upstream tap is sufficiently far upstream to be unaffected by distortion of the flow in the immediate vicinity of the orifice (in practice, the upstream tap can be as much as two pipe diameters from the plate without affecting the results). Vena-contracta taps give the largest differential head for a given rate of flow but are inconvenient if the orifice size is changed from time to time. Corner taps offer the sometimes great advantage that the pressure taps can be built into the plate carrying the orifice. Thus the entire apparatus can be quickly inserted in a pipe line at any convenient flanged joint without having to drill holes in the pipe. Flange taps are similarly convenient, since by merely replacing standard flanges with special orifice flanges, suitable pressure taps are made available. Pipe taps give the lowest differential pressure, the value obtained being close to the permanent pressure loss. The practical working equation for weight rate of discharge, adopted by the ASME Research Committee on Fluid Meters for use with either gases or liquids, is w = q1ρ1 = CYA2

2g (p − p )ρ ᎏᎏ 冪莦 1−β c

1

2

1

4

苶苶 苶苶 苶苶1 = KYA2 兹2苶苶 gc(p p苶 1 − 2)ρ

(b) Square-edged or sharp-edged orifices. The plate at the orifice opening must not be thicker than one-thirtieth of the pipe diameter, one-eighth of the orifice diameter, or one-fourth of the distance from the pipe wall to the edge of the opening. (a) Pipe-line orifice. (b) Types of plates.

FIG. 10-14

(10-22)

where A2 = cross-sectional area of throat; C = coefficient of discharge, dimensionless; gc = dimensional constant; K = CⲐ兹1苶 − β4, dimensionless; p1, p2 = pressure at upstream and downstream static pressure taps respectively; q1 = volumetric rate of discharge measured at upstream pressure and temperature; w = weight rate of discharge; Y = expansion factor, dimensionless; β = ratio of throat diameter to

MEASUREMENT OF FLOW pipe diameter, dimensionless; and ρ1 = density at upstream pressure and temperature. For the case of subsonic flow of a gas (rc < r < 1.0), the expansion factor Y for orifices is approximated by Y = 1 − [(1 − r)Ⲑk](0.41 + 0.35β4)

(10-23)

where r = ratio of downstream to upstream static pressure (p2/p1), k = ratio of specific heats (cp /cv), and β = diameter ratio. (See also Fig. 10-18.) Values of Y for supercritical flow of a gas (r < rc) through orifices are given by Benedict [J. Basic Eng., 93, 121–137 (1971)]. For the case of liquids, expansion factor Y is unity, and Eq. (10-27) should be used, since it allows for any difference in elevation between the upstream and downstream taps. Coefficient of discharge C for a given orifice type is a function of the Reynolds number NRe (based on orifice diameter and velocity) and diameter ratio β. At Reynolds numbers greater than about 30,000, the coefficients are substantially constant. For square-edged or sharpedged concentric circular orifices, the value will fall between 0.595 and 0.620 for vena-contracta or radius taps for β up to 0.8 and for flange taps for β up to 0.5. Figure 10-15 gives the coefficient of discharge K, including the velocity-of-approach factor (1Ⲑ兹苶 1 − β4), as a function of β and the location of the downstream tap. Precise values of K are given in ASME PTC, op. cit., pp. 20–39, for flange taps, radius taps, vena-contracta taps, and corner taps. Precise values of C are given in the ASME Research Committee on Fluid Meters Report, op. cit., pp. 202–207, for the first three types of taps. The discharge coefficient of sharp-edged orifices was shown by Benedict, Wyler, and Brandt [J. Eng. Power, 97, 576–582 (1975)] to increase with edge roundness. Typical as-purchased orifice plates may exhibit deviations on the order of 1 to 2 percent from ASME values of the discharge coefficient.

10-17

In the transition region (NRe between 50 and 30,000), the coefficients are generally higher than the above values. Although calibration is generally advisable in this region, the curves given in Fig. 10-16 for corner and vena-contracta taps can be used as a guide. In the laminarflow region (NRe < 50), the coefficient C is proportional to 兹苶 N苶 Re. For 1 < NRe < 100, Johansen [Proc. R. Soc. (London), A121, 231–245 (1930)] presents discharge-coefficient data for sharp-edged orifices with corner taps. For NRe < 1, Miller and Nemecek [ASME Paper 58-A-106 (1958)] present correlations giving coefficients for sharpedged orifices and short-pipe orifices (L/D from 2 to 10). For shortpipe orifices (L/D from 1 to 4), Dickerson and Rice [ J. Basic Eng., 91, 546–548 (1969)] give coefficients for the intermediate range (27 < NRe < 7000). See also subsection “Contraction and Entrance Losses.” Permanent pressure loss across a concentric circular orifice with radius or vena-contracta taps can be approximated for turbulent flow by (p1 − p4)Ⲑ(p1 − p2) = 1 − β2

(10-24)

where p1, p2 = upstream and downstream pressure-tap readings respectively, p4 = fully recovered pressure (four to eight pipe diameters downstream of the orifice), and β = diameter ratio. See ASME PTC, op. cit., Fig. 5. See Benedict, J. Fluids Eng., 99, 245–248 (1977), for a general equation for pressure loss for orifices installed in pipes or with plenum inlets. Orifices show higher loss than nozzles or venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to β. See Alvi, Sridharan, and Lakshmana Rao, loc. cit., for details. For the case of critical flow through a square- or sharp-edged concentric circular orifice (where r ≤ rc, as discussed earlier in this subsection), use Eqs. (10-31), (10-32), and (10-33) as given for critical-flow nozzles. However, unlike nozzles, the flow through a sharp-edged orifice continues to increase as the downstream pressure drops below that

FIG. 10-16 Coefficient of discharge for square-edged circular orifices with corner taps. [Tuve and Sprenkle, Instruments, 6, 201 (1933).]

10-18

TRANSPORT AND STORAGE OF FLUIDS

corresponding to the critical pressure ratio rc. This is due to an increase in the cross section of the vena contracta as the downstream pressure is reduced, giving a corresponding increase in the coefficient of discharge. At r = rc, C is about 0.75, while at r ≅ 0, C has increased to about 0.84. See Grace and Lapple, loc. cit.; and Benedict, J. Basic Eng., 93, 99–120 (1971). Measurements by Harris and Magnall [Trans. Inst. Chem. Eng. (London), 50, 61–68 (1972)] with a venturi (β = 0.62) and orifices with radius taps (β = 0.60 − 0.75) indicate that the discharge coefficient for nonnewtonian fluids, in the range N′Re (generalized Reynolds number) 3500 to 100,000, is approximately the same as for newtonian fluids at the same Reynolds number. Quadrant-edge orifices have holes with rounded edges on the upstream side of the plate. The quadrant-edge radius is equal to the thickness of the plate at the orifice location. The advantages claimed for this type versus the square- or sharp-edged orifice are constant-discharge coefficients extending to lower Reynolds numbers and less possibility of significant changes in coefficient because of erosion or other damage to the inlet shape. Values of discharge coefficient C and Reynolds numbers limit for constant C are presented in Table 10-6, based on Ramamoorthy and Seetharamiah [J. Basic Eng., 88, 9–13 (1966)] and Bogema and Monkmeyer (J. Basic Eng., 82, 729–734 (1960)]. At Reynolds numbers above those listed for the upper limits, the coefficients rise abruptly. As Reynolds numbers decrease below those listed for the lower limits, the coefficients pass through a hump and then drop off. According to Bogema, Spring, and Ramamoorthy [J. Basic Eng., 84, 415–418 (1962)], the hump can be eliminated by placing a fine-mesh screen about three pipe diameters upstream of the orifice. This reduces the lower NRe limit to about 500. Permanent pressure loss across quadrant-edge orifices for turbulent flow is somewhat lower than given by Eq. (10-24). See Alvi, Sridharan, and Lakshmana Rao, loc. cit., for values of discharge coefficient and permanent pressure loss in laminar flow. Slotted orifices offer significant advantages over a standard square-edged orifice with an identical open area for homogeneous gases or liquids [G. L. Morrison and K. R. Hall, Hydrocarbon Processing 79, 12, 65–72 (2000)]. The slotted orifice flowmeter only requires compact header configurations with very short upstream pipe lengths and maintains accuracy in the range of 0.25 percent with no flow conditioner. Permanent head loss is less than or equal to that of a standard orifice that has the same β ratio. Discharge coefficients for the slotted orifice are much less sensitive to swirl or to axial velocity profiles. A slotted orifice plate can be a “drop in” replacement for a standard orifice plate. Segmental and eccentric orifices are frequently used for gas metering when there is a possibility that entrained liquids or solids would otherwise accumulate in front of a concentric circular orifice. This can be avoided if the opening is placed on the lower side of the pipe. For liquid flow with entrained gas, the opening is placed on the upper side. The pressure taps should be located on the opposite side of the pipe from the opening. Coefficient C for a square-edged eccentric circular orifice (with opening tangent to pipe wall) varies from about 0.61 to 0.63 for β’s from 0.3 to 0.5, respectively, and pipe Reynolds numbers > 10,000 for either venaTABLE 10-6 Orifices

FIG. 10-17

Herschel-type venturi tube.

contracta or flange taps (where β = diameter ratio). For square-edged segmental orifices, the coefficient C falls generally between 0.63 and 0.64 for 0.3 ≤ β ≤ 0.5 and pipe Reynolds numbers > 10,000, for vena-contracta or flange taps, where β = diameter ratio for an equivalent circular orifice = 兹苶 α (α = ratio of orifice to pipe cross-sectional areas). Values of expansion factor Y are slightly higher than for concentric circular orifices, and the location of the vena contracta is moved farther downstream as compared with concentric circular orifices. For further details, see ASME Research Committee on Fluid Meters Report, op. cit., pp. 210–213. For permanent pressure loss with segmental and eccentric orifices with laminar pipe flow see Lakshmana Rao and Sridharan, Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 98 (HY 11), 2015–2034 (1972). Annular orifices can also be used to advantage for gas metering when there is a possibility of entrained liquids or solids and for liquid metering with entrained gas present in small concentrations. Coefficient K was found by Bell and Bergelin [Trans. Am. Soc. Mech. Eng., 79, 593–601 (1957)] to range from about 0.63 to 0.67 for annulus Reynolds numbers in the range of 100 to 20,000 respectively for values of 2L/(D − d) less than 1 where L = thickness of orifice at outer edge, D = inside pipe diameter, and d = diameter of orifice disk. The annulus Reynolds number is defined as NRe = (D − d)(G/µ)

Discharge Coefficients for Quadrant-Edge * for Limiting N Re constant coefficient†

β

C‡

K‡

Lower

Upper

0.225 0.400 0.500 0.600 0.630

0.770 0.780 0.824 0.856 0.885

0.771 0.790 0.851 0.918 0.964

5,000 5,000 4,000 3,000 3,000

60,000 150,000 200,000 120,000 105,000

*Based on pipe diameter and velocity. †For a precision of about ⫾0.5 percent. ‡Can be used with corner taps, flange taps, or radius taps.

(10-25)

where G = mass velocity pV through orifice opening and µ = fluid viscosity. The above coefficients were determined for β’s (= d/D) in the range of 0.95 to 0.996 and with pressure taps located 19 mm (3⁄4 in) upstream of the disk and 230 mm (9 in) downstream in a 5.25in-diameter pipe. Venturi Meters The standard Herschel-type venturi meter consists of a short length of straight tubing connected at either end to the pipe line by conical sections (see Fig. 10-17). Recommended proportions (ASME PTC, op. cit., p. 17) are entrance cone angle α1 = 21 ± 2°, exit cone angle α2 = 5 to 15°, throat length = one throat diameter, and upstream tap located 0.25 to 0.5 pipe diameter upstream of the entrance cone. The straight and conical sections should be joined by smooth curved surfaces for best results. Rate of discharge of either gases or liquids through a venturi meter is given by Eq. (10-22). For the flow of gases, expansion factor Y, which allows for the change in gas density as it expands adiabatically from p1 to p2, is given by Y=

k 1−r 1−β ᎏ 冣莦冢莦莦 ᎏᎏ 冣莦冢莦莦 ᎏ 冣莦 冪r莦莦冢莦莦 k−1 1−r 1−β r (k − 1)/k

2/k

4

4 2/k

(10-26)

for venturi meters and flow nozzles, where r = p2/p1 and k = specific heat ratio cp/cv. Values of Y computed from Eq. (10-26) are given in Fig. 10-18 as a function of r, k, and β. For the flow of liquids, expansion factor Y is unity. The change in potential energy in the case of an inclined or vertical venturi meter must be allowed for. Equation (10-22) is accordingly modified to give w = q1ρ = CA2

[2g (p − p ) + 2gρ(Z − Z )]ρ ᎏᎏᎏᎏ 冪莦莦 1−β c

1

2

1

4

2

(10-27)

MEASUREMENT OF FLOW where g = local acceleration due to gravity and Z1, Z2 = vertical heights above an arbitrary datum plane corresponding to the centerline pressure-reading locations for p1 and p2 respectively. Value of the discharge coefficient C for a Herschel-type venturi meter depends upon the Reynolds number and to a minor extent upon the size of the venturi, increasing with diameter. A plot of C versus pipe Reynolds number is given in ASME PTC, op. cit., p. 19. A value of 0.984 can be used for pipe Reynolds numbers larger than 200,000. Permanent pressure loss for a Herschel-type venturi tube depends upon diameter ratio β and discharge cone angle α2. It ranges from 10 to 15 percent of the pressure differential (p1 − p2) for small angles (5 to 7°) and from 10 to 30 percent for large angles (15°), with the larger losses occurring at low values of β (see ASME PTC, op. cit., p. 12). See Benedict, J. Fluids Eng., 99, 245–248 (1977), for a general equation for pressure loss for venturis installed in pipes or with plenum inlets. For flow measurement of steam and water mixtures with a Herschel-type venturi in 21⁄2-in- and 3-in-diameter pipes, see Collins and Gacesa, J. Basic Eng., 93, 11–21 (1971). A variety of short-tube venturi meters are available commercially. They require less space for installation and are generally (although not always) characterized by a greater pressure loss than the corresponding Herschel-type venturi meter. Discharge coefficients vary widely for different types, and individual calibration is recommended if the manufacturer’s calibration is not available. Results of tests on the Dall flow tube are given by Miner [Trans. Am. Soc. Mech. Eng., 78, 475–479 (1956)] and Dowdell [Instrum. Control Syst., 33, 1006–1009 (1960)]; and on the Gentile flow tube (also called Beth flow tube or Foster flow tube) by Hooper [Trans. Am. Soc. Mech. Eng., 72, 1099–1110 (1950)]. The use of a multiventuri system (in which an inner venturi discharges into the throat of an outer venturi) to increase both the differential pressure for a given flow rate and the signal-to-loss ratio is described by Klomp and Sovran [J. Basic Eng., 94, 39–45 (1972)]. Flow Nozzles A simple form of flow nozzle is shown in Fig. 1019. It consists essentially of a short cylinder with a flared approach section. The approach cross section is preferably elliptical in shape but may be conical. Recommended contours for long-radius flow nozzles are given in ASME PTC, op. cit., p. 13. In general, the length of the straight portion of the throat is about one-half throat diameter, the upstream pressure tap is located about one pipe diameter from the nozzle inlet face, and the downstream pressure tap about one-half pipe diameter from the inlet face. For subsonic flow, the pressures at points 2 and 3 will be practically identical. If a conical inlet is preferred, the inlet and throat geometry specified for a Herschel-type venturi meter can be used, omitting the expansion section. Rate of discharge through a flow nozzle for subcritical flow can be determined by the equations given for venturi meters, Eq. (10-22) for gases and Eq. (10-27) for liquids. The expansion factor Y for nozzles is the same as that for venturi meters [Eq. (10-26), Fig. 10-18]. The value of the discharge coefficient C depends primarily upon the pipe Reynolds number and to a lesser extent upon the diameter ratio β. Curves of recommended coefficients for long-radius flow nozzles with pressure taps located one pipe diameter upstream and one-half pipe diameter downstream of the inlet face of the nozzle are given in ASME PTC, op. cit., p. 15. In general, coefficients range from 0.95 at a pipe Reynolds number of 10,000 to 0.99 at 1,000,000. The performance characteristics of pipe-wall-tap nozzles (Fig. 10-19) and throat-tap nozzles are reviewed by Wyler and Benedict [J. Eng. Power, 97, 569–575 (1975)]. Permanent pressure loss across a subsonic flow nozzle is approximated by 1 − β2 p1 − p4 = ᎏ2 (p1 − p2) 1+β

FIG. 10-18

Values of expansion factor Y for orifices, nozzles, and venturis.

See Benedict, loc. cit., for a general equation for pressure loss for nozzles installed in pipes or with plenum inlets. Nozzles show higher loss than venturis. Permanent pressure loss for laminar flow depends on the Reynolds number in addition to β. For details, see Alvi, Sridharan, and Lakshamana Rao, J. Fluids Eng., 100, 299–307 (1978). Critical Flow Nozzle For a given set of upstream conditions, the rate of discharge of a gas from a nozzle will increase for a decrease in the absolute pressure ratio p2/p1 until the linear velocity in the throat reaches that of sound in the gas at that location. The value of p2/p1 for which the acoustic velocity is just attained is called the critical pressure ratio rc. The actual pressure in the throat will not fall below p1rc even if a much lower pressure exists downstream. The critical pressure ratio rc can be obtained from the following theoretical equation, which assumes a perfect gas and a frictionless nozzle: k+1 k−1 rc(1 − k)/k + ᎏ β4 rc2/k = ᎏ 2 2

冢

冣

(10-29)

This reduces, for β ≤ 0.2, to

冢

2 rc = ᎏ k+1

k/(k − 1)

冣

(10-30)

where k = ratio of specific heats cp/cv and β = diameter ratio. A table of values of rc as a function of k and β is given in the ASME Research Committee on Fluid Meters Report, op. cit., p. 68. For small values of β, rc = 0.487 for k = 1.667, 0.528 for k = 1.40, 0.546 for k = 1.30, and 0.574 for k = 1.15. Under critical flow conditions, only the upstream conditions p1, v1, and T1 need be known to determine flow rate, which, for β ≤ 0.2, is given by p 2 (k + 1)/(k − 1) wmax = CA2 gck ᎏ1 ᎏ (10-31) v1 k+1

冪莦莦冢莦莦冣莦冢莦莦莦冣 莦莦莦莦莦

For a perfect gas, this corresponds to wmax = CA2p1

M 2 ᎏ 冣莦冢莦莦 ᎏ莦 冪g莦k莦冢莦莦 RT k + 1 冣 莦莦莦莦莦 (k + 1)/(k − 1)

c

1

(10-28)

where p1, p2, p4 = static pressures measured at the locations shown in Fig. 10-19; and β = ratio of nozzle throat diameter to pipe diameter, dimensionless. Equation (10-28) is based on a momentum balance assuming constant fluid density (see Lapple et al., Fluid and Particle Mechanics, University of Delaware, Newark, 1951, p. 13).

10-19

FIG. 10-19

Flow-nozzle assembly.

(10-32)

10-20

TRANSPORT AND STORAGE OF FLUIDS

For air, Eq. (10-31) reduces to 苶1苶 wmax = C1CA2 p1 /兹T

(10-33)

where A2 = cross-sectional area of throat; C = coefficient of discharge, dimensionless; gc = dimensional constant; k = ratio of specific heats, cp/cv; M = molecular weight; p1 = pressure on upstream side of nozzle; R = gas constant; T1 = absolute temperature on upstream side of nozzle; v1 = specific volume on upstream side of nozzle; C1 = dimensional constant, 0.0405 SI units (0.533 U.S. customary units); and wmax = maximumweight flow rate. Discharge coefficients for critical flow nozzles are, in general, the same as those for subsonic nozzles. See Grace and Lapple, Trans. Am. Soc. Mech. Eng., 73, 639–647 (1951); and Szaniszlo, J. Eng. Power, 97, 521–526 (1975). Arnberg, Britton, and Seidl [J. Fluids Eng., 96, 111–123 (1974)] present discharge-coefficient correlations for circular-arc venturi meters at critical flow. For the calculation of the flow of natural gas through nozzles under critical-flow conditions, see Johnson, J. Basic Eng., 92, 580–589 (1970). Elbow Meters A pipe elbow can be used as a flowmeter for liquids if the differential centrifugal head generated between the inner and outer radii of the bend is measured by means of pressure taps located midway around the bend. Equation (10-27) can be used, except that the pressure-difference term (p1 − p2) is now taken to be the differential centrifugal pressure and β is taken as zero if one assumes no change in cross section between the pipe and the bend. The discharge coefficient should preferably be determined by calibration, but as a guide it can be estimated within ±6 percent for circular pipe for Reynolds numbers greater than 105 from C = 0.98 兹苶 RcⲐ2D, where Rc = radius of curvature of the centerline and D = inside pipe diameter in consistent units. See Murdock, Foltz, and Gregory, J. Basic Eng., 86, 498–506 (1964); or the ASME Research Committee on Fluid Meters Report, op. cit., pp. 75–77. Accuracy Square-edged orifices and venturi tubes have been so extensively studied and standardized that reproducibilities within 1 to 2 percent can be expected between standard meters when new and clean. This is therefore the order of reliability to be had, if one assumes (1) accurate measurement of meter differential, (2) selection of the coefficient of discharge from recommended published literature, (3) accurate knowledge of fluid density, (4) accurate measurement of critical meter dimensions, (5) smooth upstream face of orifice, and (6) proper location of the meter with respect to other flow-disturbing elements in the system. Care must also be taken to avoid even slight corrosion or fouling during use. Presence of swirling flow or an abnormal velocity distribution upstream of the metering element can cause serious metering error unless calibration in place is employed or sufficient straight pipe is inserted between the meter and the source of disturbance. Table 10-7 gives the minimum lengths of straight pipe required to avoid appreciable error due to the presence of certain fittings and valves either upstream or downstream of an orifice or nozzle. These values were extracted from plots presented by Sprenkle [Trans. Am. Soc. Mech. Eng., 67, 345–360 (1945)]. Table 10-7 also shows the reduction in spacing made possible by the use of straightening vanes between the fittings and the meter. Entirely adequate straightening vanes can be provided by fitting a bundle of thin-wall tubes within the pipe. The center-to-center distance between tubes should not exceed one-fourth of the pipe diameter, and the bundle length should be at least 8 times this distance. The distances specified in Table 10-7 will be conservative if applied to venturi meters. For specific information on requirements for venturi meters, see a discussion by Pardoe appended to Sprenkle (op. cit.). Extensive data on the effect of installation on the coefficients of venturi meters are given elsewhere by Pardoe [Trans. Am. Soc. Mech. Eng., 65, 337–349 (1943)]. In the presence of flow pulsations, the indications of head meters such as orifices, nozzles, and venturis will often be undependable for several reasons. First, the measured pressure differential will tend to be high, since the pressure differential is proportional to the square of flow rate for a head meter, and the square root of the mean differential pressure is always greater than the mean of the square roots of the differential pressures. Second, there is a phase shift as the wave passes through

TABLE 10-7 Locations of Orifices and Nozzles Relative to Pipe Fittings Distances in pipe diameters, D1 Distance, upstream fitting to orifice

Type of fitting upstream

D2 D1

Without straightening vanes

Single 90° ell, tee, or cross used as ell

0.2 0.4 0.6 0.8

6 6 9 20

9 12

8

2 short-radius 90° ells in form of S

0.2 0.4 0.6 0.8

7 8 13 25

8 10 15

6 11

2 long- or short-radius 90° ells in perpendicular planes

0.2 0.4 0.6 0.8

15 18 25 40

9 10 11 13

5 6 7 9

Contraction or enlargement

0.2 0.4 0.6 0.8

8 9 10 15

Vanes have no advantage

Globe valve or stop check

0.2 0.4 0.6 0.8

9 10 13 21

9 10 10 13

Gate valve, wide open, or plug cocks

0.2 0.4 0.6 0.8

6 6 8 14

With straightening vanes

Distance, vanes to orifice

Distance, nearest downstream fitting from orifice 2 4 2 4 2 4 2 4

5 6 6 9

Same as globe valve

2 4 2 4

the metering restriction which can affect the differential. Third, pulsations can be set up in the manometer leads themselves. Frequency of the pulsation also plays a part. At low frequencies, the meter reading can generally faithfully follow the flow pulsations, but at high frequencies it cannot. This is due to inertia of the fluid in the manometer leads or of the manometric fluid, whereupon the meter would give a reading intermediate between the maximum and minimum flows but having no readily predictable relation to the mean flow. Pressure transducers with flush-mounted diaphragms can be used together with high-speed recording equipment to provide accurate records of the pressure profiles at the upstream and downstream pressure taps, which can then be analyzed and translated into a mean flow rate. The rather general practice of producing a steady differential reading by placing restrictions in the manometer leads can result in a reading which, under a fixed set of conditions, may be useful in control of an operation but which has no readily predictable relation to the actual average flow. If calibration is employed to compensate for the presence of pulsations, complete reproduction of operating conditions, including source of pulsations and waveform, is necessary to ensure reasonable accuracy. According to Head [Trans. Am. Soc. Mech. Eng., 78, 1471–1479 (1956)], a pulsation-intensity limit of Γ = 0.1 is recommended as a practical pulsation threshold below which the performance of all types of flowmeters will differ negligibly from steady-flow performance (an error of less than 1 percent in flow due to pulsation). Γ is the peak-totrough flow variation expressed as a fraction of the average flow rate. According to the ASME Research Committee on Fluid Meters Report (op. cit., pp. 34–35), the fractional metering error E for liquid flow through a head meter is given by (1 + E)2 = 1 + Γ 2/8

(10-34)

When the pulsation amplitude is such as to result in a greater-thanpermissible metering error, consideration should be given to installation of a pulsation damper between the source of pulsations and the

MEASUREMENT OF FLOW flowmeter. References to methods of pulsation-damper design are given in the subsection “Unsteady-State Behavior.” Pulsations are most likely to be encountered in discharge lines from reciprocating pumps or compressors and in lines supplying steam to reciprocating machinery. For gas flow, a combination involving a surge chamber and a constriction in the line can be used to damp out the pulsations to an acceptable level. The surge chamber is generally located as close to the pulsation source as possible, with the constriction between the surge chamber and the metering element. This arrangement can be used for either a suction or a discharge line. For such an arrangement, the metering error has been found to be a function of the Hodgson number NH, which is defined as NH = Qn ∆ps /qps

(10-35)

where Q = volume of surge chamber and pipe between metering element and pulsation source; n = pulsation frequency; ∆ps = permanent pressure drop between metering element and surge chamber; q = average volume flow rate, based on gas density in the surge chamber; and ps = pressure in surge chamber. Herning and Schmid [Z. Ver. Dtsch. Ing., 82, 1107–1114 (1938)] presented charts for a simplex double-acting compressor for the prediction of metering error as a function of the Hodgson number and s, the ratio of piston discharge time to total time per stroke. Table 10-8a gives the minimum Hodgson numbers required to reduce the metering error to 1 percent as given by the charts (for specific heat ratios between 1.28 and 1.37). Schmid [Z. Ver. Dtsch. Ing., 84, 596–598 (1940)] presented similar charts for a duplex double-acting compressor and a triplex double-acting compressor for a specific heat ratio of 1.37. Table 10-8b gives the minimum Hodgson numbers corresponding to a 1 percent metering error for these cases. The value of Q ∆ps can be calculated from the appropriate Hodgson number, and appropriate values of Q and ∆ps selected so as to satisfy this minimum requirement. VELOCITY METERS Anemometers An anemometer may be any instrument for measurement of gas velocity, e.g., a pitot tube, but usually the term refers to one of the following types. The vane anemometer is a delicate revolution counter with jeweled bearings, actuated by a small windmill, usually 75 to 100 mm (about 3 to 4 in) in diameter, constructed of flat or slightly curved radially disposed vanes. Gas velocity is determined by using a stopwatch to find the time interval required to pass a given number of meters (feet) of gas as indicated by the counter. The velocity so obtained is inversely proportional to gas density. If the original calibration was carried out in a gas of density ρ0 and the density of the gas stream being metered is ρ1, the true gas velocity can be found as follows: From the calibration curve for the instrument, find Vt,0 corresponding to the quantity Vm兹苶 ρ苶 ρ苶0, where Vm = measured velocity. Then the actual velocity Vt,1 1/苶 is equal to Vt,0兹苶 ρ苶 ρ1苶. In general, when working with air, the effects of 0/苶 TABLE 10-8a Minimum Hodgson Numbers Simplex double-acting compressor s

NH

s

NH

0.167 0.333 0.50

1.31 1.00 0.80

0.667 0.833 1.00

0.60 0.43 0.34

TABLE 10-8b Minimum Hodgson Numbers Duplex double-acting compressor

Triplex double-acting compressor

s

NH

s

NH

0.167 0.333 0.50 0.667 0.833 1.00

1.00 0.70 0.30 0.10 0.05 0.00

0.167 0.333 0.50 0.667 0.833 1.00

0.85 0.30 0.15 0.06 0.00 0.00

10-21

atmospheric-density changes can be neglected for all velocities above 1.5 m/s (about 5 ft/s). In all cases, care must be taken to hold the anemometer well away from one’s body or from any object not normally present in the stream. Vane anemometers can be used for gas-velocity measurements in the range of 0.3 to 45 m/s (about 1 to 150 ft/s), although a given instrument generally has about a twentyfold velocity range. Bearing friction has to be minimized in instruments designed for accuracy at the low end of the range, while ample rotor and vane rigidity must be provided for measurements at the higher velocities. Vane anemometers are sensitive to shock and cannot be used in corrosive atmospheres. Therefore, accuracy is questionable unless a recent calibration has been made and the history of the instrument subsequent to calibration is known. For additional information, see Ower et al., op. cit., chap. VIII. Turbine Flowmeters They consist of a straight flow tube containing a turbine which is free to rotate on a shaft supported by one or more bearings and located on the centerline of the tube. Means are provided for magnetic detection of the rotational speed, which is proportional to the volumetric flow rate. Its use is generally restricted to clean, noncorrosive fluids. Additional information on construction, operation, range, and accuracy can be obtained from Baker, pp. 215–252, 2000; Miller, op. cit.; and Spitzer, pp. 303–317, 2005. The current meter is generally used for measuring velocities in open channels such as rivers and irrigation channels. There are two types, the cup meter and the propeller meter. The former is more widely used. It consists of six conical cups mounted on a vertical axis pivoted at the ends and free to rotate between the rigid arms of a U-shaped clevis to which a vaned tailpiece is attached. The wheel rotates because of the difference in drag for the two sides of the cup, and a signal proportional to the revolutions of the wheel is generated. The velocity is determined from the count over a period of time. The current meter is generally useful in the range of 0.15 to 4.5 m/s (about 0.5 to 15 ft/s) with an accuracy of ±2 percent. For additional information see Creager and Justin, Hydroelectric Handbook, 2d ed., Wiley, New York, 1950, pp. 42–46. Other important classes of velocity meters include electromagnetic flowmeters and ultrasonic flowmeters. Both are described in Sec. 8. MASS FLOWMETERS General Principles There are two main types of mass flowmeters: (1) the so-called true mass flowmeter, which responds directly to mass flow rate, and (2) the inferential mass flowmeter, which commonly measures volume flow rate and fluid density separately. A variety of types of true mass flowmeters have been developed, including the following: (a) the Magnus-effect mass flowmeter, (b) the axial-flow, transverse-momentum mass flowmeter, (c) the radial-flow, transversemomentum mass flowmeter, (d) the gyroscopic transverse-momentum mass flowmeter, and (e) the thermal mass flowmeter. Type b is the basis for several commercial mass flowmeters, one version of which is briefly described here. Axial-Flow Transverse-Momentum Mass Flowmeter This type is also referred to as an angular-momentum mass flowmeter. One embodiment of its principle involves the use of axial flow through a driven impeller and a turbine in series. The impeller imparts angular momentum to the fluid, which in turn causes a torque to be imparted to the turbine, which is restrained from rotating by a spring. The torque, which can be measured, is proportional to the rotational speed of the impeller and the mass flow rate. Inferential Mass Flowmeter There are several types in this category, including the following: 1. Head meters with density compensation. Head meters such as orifices, venturis, or nozzles can be used with one of a variety of densitometers [e.g., based on (a) buoyant force on a float, (b) hydraulic coupling, (c) voltage output from a piezoelectric crystal, or (d) radiation absorption]. The signal from the head meter, which is proportional to ρV 2 (where ρ = fluid density and V = fluid velocity), is multiplied by ρ given by the densitometer. The square root of the product is proportional to the mass flow rate.

10-22

TRANSPORT AND STORAGE OF FLUIDS

2. Head meters with velocity compensation. The signal from the head meter, which is proportional to ρV 2, is divided by the signal from a velocity meter to give a signal proportional to the mass flow rate. 3. Velocity meters with density compensation. The signal from the velocity meter (e.g., turbine meter, electromagnetic meter, or sonic velocity meter) is multiplied by the signal from a densitometer to give a signal proportional to the mass flow rate. Coriolis Mass Flowmeter This type, described in Sec. 8, offers simultaneous direct measurement of both mass flow rate and fluid density. The coriolis flowmeter is insensitive to upstream and downstream flow disturbances, but its performance is adversely affected by the presence of even a few percent of a gas when measuring a liquid flow. VARIABLE-AREA METERS General Principles The underlying principle of an ideal area meter is the same as that of a head meter of the orifice type (see subsection “Orifice Meters”). The stream to be measured is throttled by a constriction, but instead of observing the variation with flow of the differential head across an orifice of fixed size, the constriction of an area meter is so arranged that its size is varied to accommodate the flow while the differential head is held constant. A simple example of an area meter is a gate valve of the rising-stem type provided with static-pressure taps before and after the gate and a means for measuring the stem position. In most common types of area meters, the variation of the opening is automatically brought about by the motion of a weighted piston or float supported by the fluid. Two different cylinder- and piston-type area meters are described in the ASME Research Committee on Fluid Meters Report, op. cit., pp. 82–83. Rotameters The rotameter, an example of which is shown in Fig. 10-20, has become one of the most popular flowmeters in the chemical-process industries. It consists essentially of a plummet, or “float,” which is free to move up or down in a vertical, slightly tapered tube having its small end down. The fluid enters the lower end of the tube and causes the float to rise until the annular area between the float and the wall of the tube is such that the pressure drop across this constriction is just sufficient to support the float. Typically, the tapered tube is of glass and carries etched upon it a nearly linear scale on which the position of the float may be visually noted as an indication of the flow.

Interchangeable precision-bore glass tubes and metal metering tubes are available. Rotameters have proved satisfactory both for gases and for liquids at high and at low pressures. A single instrument can readily cover a tenfold range of flow, and by providing floats of different densities a two-hundredfold range is practicable. Rotameters are available with pneumatic, electric, and electronic transmitters for actuating remote recorders, integrators, and automatic flow controllers (see Considine, op. cit., pp. 4-35–4-36, and Sec. 8 of this Handbook). Rotameters require no straight runs of pipe before or after the point of installation. Pressure losses are substantially constant over the whole flow range. In experimental work, for greatest precision, a rotameter should be calibrated with the fluid which is to be metered. However, most modern rotameters are precision-made so that their performance closely corresponds to a master calibration plot for the type in question. Such a plot is supplied with the meter upon purchase. According to Head [Trans. Am. Soc. Mech. Eng., 76, 851–862 (1954)], flow rate through a rotameter can be obtained from w = qρ = KDf

W (ρ − ρ)ρ ᎏᎏ 冪莦 ρ

(10-36)

冥

(10-37)

f

f

f

and

冤

Dt µ K = φ ᎏ , ᎏᎏ Df Wf(ρf − ρ)ρ ᎏᎏ ρf

冪莦莦莦莦莦莦莦

where w = weight flow rate; q = volume flow rate; ρ = fluid density; K = flow parameter, m1/2/s (ft1/2/s); Df = float diameter at constriction; Wf = float weight; ρf = float density; Dt = tube diameter at point of constriction; and µ = fluid viscosity. The appropriate value of K is obtained from a composite correlation of K versus the parameters shown in Eq. (10-37) corresponding to the float shape being used. The relation of Dt to the rotameter reading is also required for the tube taper and size being used. The ratio of flow rates for two different fluids A and B at the same rotameter reading is given by wA KA ᎏ=ᎏ wB KB

ᎏᎏ 冪莦 (ρ − ρ )ρ (ρf − ρA)ρA f

B

(10-38)

B

A measure of self-compensation, with respect to weight rate of flow, for fluid-density changes can be introduced through the use of a float with a density twice that of the fluid being metered, in which case an increase of 10 percent in ρ will produce a decrease of only 0.5 percent in w for the same reading. The extent of immunity to changes in fluid viscosity depends upon the shape of the float. According to Baird and Cheema [Can. J. Chem. Eng., 47, 226–232 (1969)], the presence of square-wave pulsations can cause a rotameter to overread by as much as 100 percent. The higher the pulsation frequency, the less the float oscillation, although the error can still be appreciable even when the frequency is high enough so that the float is virtually stationary. Use of a damping chamber between the pulsation source and the rotameter will reduce the error. Additional information on rotameter theory is presented by Fischer [Chem. Eng., 59(6), 180–184 (1952)], Coleman [Trans. Inst. Chem. Eng., 34, 339–350 (1956)], and McCabe, Smith, and Harriott (Unit Operations of Chemical Engineering, 4th ed., McGraw-Hill, New York, 1985, pp. 202–205). TWO-PHASE SYSTEMS

FIG. 10-20

Rotameter.

It is generally preferable to meter each of the individual components of a two-phase mixture separately prior to mixing, since it is difficult to meter such mixtures accurately. Problems arise because of fluctuations in composition with time and variations in composition over the cross section of the channel. Information on metering of such mixtures can be obtained from the following sources.

MEASUREMENT OF FLOW Gas-Solid Mixtures Carlson, Frazier, and Engdahl [Trans. Am. Soc. Mech. Eng., 70, 65–79 (1948)] describe the use of a flow nozzle and a square-edged orifice in series for the measurement of both the gas rate and the solids rate in the flow of a finely divided solid-in-gas mixture. The nozzle differential is sensitive to the flow of both phases, whereas the orifice differential is not influenced by the solids flow. Farbar [Trans. Am. Soc. Mech. Eng., 75, 943–951 (1953)] describes how a venturi meter can be used to measure solids flow rate in a gassolids mixture when the gas rate is held constant. Separate calibration curves (solids flow versus differential) are required for each gas rate of interest. Cheng, Tung, and Soo [J. Eng. Power, 92, 135–149 (1970)] describe the use of an electrostatic probe for measurement of solids flow in a gas-solids mixture. Goldberg and Boothroyd [Br. Chem. Eng., 14, 1705–1708 (1969)] describe several types of solids-in-gas flowmeters and give an extensive bibliography. Gas-Liquid Mixtures An empirical equation was developed by Murdock [J. Basic Eng., 84, 419–433 (1962)] for the measurement of gas-liquid mixtures using sharp-edged orifice plates with either radius, flange, or pipe taps. An equation for use with venturi meters was given by Chisholm [Br. Chem. Eng., 12, 454–457 (1967)]. A procedure for determining steam quality via pressure-drop measurement with upflow through either venturi meters or sharp-edged orifice plates was given by Collins and Gacesa [J. Basic Eng., 93, 11–21 (1971)]. Liquid-Solid Mixtures Liptak [Chem. Eng., 74(4), 151–158 (1967)] discusses a variety of techniques that can be used for the measurement of solids-in-liquid suspensions or slurries. These include metering pumps, weigh tanks, magnetic flowmeter, ultrasonic flowmeter, gyroscope flowmeter, etc. Shirato, Gotoh, Osasa, and Usami [J. Chem. Eng. Japan, 1, 164–167 (January 1968)] present a method for determining the mass flow rate of suspended solids in a liquid stream wherein the liquid velocity is measured by an electromagnetic flowmeter and the flow of solids is calculated from the pressure drops across each of two vertical sections of pipe of different diameter through which the suspension flows in series. FLOWMETER SELECTION Web sites for process equipment and instrumentation, such as www.globalspec.com and www.thomasnet.com, are valuable tools when selecting a flowmeter. These search engines can scan the flowmeters manufactured by more than 800 companies for specific products that meet the user’s specifications. Table 10-4 was based in part on information from these web sites. Note that the accuracies claimed are achieved only under ideal conditions when the flowmeters are properly installed and calibrated for the application. The purpose of this subsection is to summarize the preferred applications as well as the advantages and disadvantages of some of the common flowmeter technologies. Table 10-9 divides flowmeters into four classes. Flowmeters in class I depend on wetted moving parts that can wear, plug, or break. The TABLE 10-9

Flowmeter Classes

Class I: Flowmeters with wetted moving parts Positive displacement Turbine Variable-area

Class III: Obstructionless flowmeters Coriolis mass Electromagnetic Ultrasonic Adapted from Spitzer, op. cit., 2005.

Class II: Flowmeters with no wetted moving parts Differential pressure Vortex Target Thermal Class IV: Flowmeters with sensors mounted external to the pipe Clamp-on ultrasonic Correlation

10-23

potential for catastrophic failure is a disadvantage. However, in clean fluids, class I flowmeters have often proved reliable and stable when properly installed, calibrated, and maintained. Class II flowmeters have no wetted moving parts to break and are thus not subject to catastrophic failure. However, the flow surfaces such as orifice plates may wear, eventually biasing flow measurements. Other disadvantages of some flowmeters in this class include high pressure drop and susceptibility to plugging. Very dirty and abrasive fluids should be avoided. Because class III flowmeters have neither moving parts nor obstructions to flow, they are suitable for dirty and abrasive fluids provided that appropriate materials of construction are available. Class IV flowmeters have sensors mounted external to the pipe, and would thus seem to be ideal, but problems of accuracy and sensitivity have been encountered in early devices. These comparatively new technologies are under development, and these problems may be overcome in the future. Section 8 outlines the following criteria for selection of measurement devices: measurement span, performance, reliability, materials of construction, prior use, potential for releasing process materials to the environment, electrical classification, physical access, invasive or noninvasive, and life-cycle cost. Spitzer, op. cit., 2005, cites four intended end uses of the flowmeter: rate indication, control, totalization, and alarm. Thus high accuracy may be important for rate indication, while control may just need good repeatability. Volumetric flow or mass flow indication is another choice. Baker, op. cit., 2003, identifies the type of fluid (liquid or gas, slurry, multiphase), special fluid constraints (clean or dirty, hygienic, corrosive, abrasive, high flammability, low lubricity, fluids causing scaling). He lists the following flowmeter constraints: accuracy or measurement uncertainty, diameter range, temperature range, pressure range, viscosity range, flow range, pressure loss caused by the flowmeter, sensitivity to installation, sensitivity to pipework supports, sensitivity to pulsation, whether the flowmeter has a clear bore, whether a clamp-on version is available, response time, and ambient conditions. Finally, Baker identifies these environmental considerations: ambient temperature, humidity, exposure to weather, level of electromagnetic radiation, vibration, tamperproof for domestic use, and classification of area requiring explosionproof, intrinsic safety, etc. Note that the accuracies cited in Table 10-4 can be achieved by those flowmeters only under ideal conditions of application, installation, and calibration. This subsection has only given an introduction to issues to consider in the choice of a flowmeter for a given application. See Baker, op. cit., 2003; Miller, op. cit., 1996; and Spitzer, op. cit., 2005, for further guidance and to obtain application-specific data from flowmeter vendors. WEIRS Liquid flow in an open channel may be metered by means of a weir, which consists of a dam over which, or through a notch in which, the liquid flows. The terms “rectangular weir,” “triangular weir,” etc., generally refer to the shape of the notch in a notched weir. All weirs considered here have flat upstream faces that are perpendicular to the bed and walls of the channel. Sharp-edged weirs have edges like those of square or sharpedged orifices (see subsection “Orifice Meters”). Notched weirs are ordinarily sharp-edged. Weirs not in the sharp-edged class are, for the most part, those described as broad-crested weirs. The head h0 on a weir is the liquid-level height above the crest or base of the notch. The head must be measured sufficiently far upstream to avoid the drop in level occasioned by the overfall which begins at a distance about 2h0 upstream from the weir. Surface-level measurements should be made a distance of 3h0 or more upstream, preferably by using a stilling box equipped with a high-precision level gauge, e.g., a hook gauge or float gauge. With sharp-edged weirs, the sheet of discharging liquid, called the “nappe,” contracts as it leaves the opening and free discharge occurs. Rounding the upstream edge will reduce the contraction and increase the flow rate for a given head. A clinging nappe may result

10-24

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-21

Rectangular weir.

FIG. 10-22

if the head is very small, if the edge is well rounded, or if air cannot flow in beneath the nappe. This, in turn, results in an increase in the discharge rate for a given head as compared with that for a free nappe. For further information on the effect of the nappe, see Gibson, Hydraulics and Its Applications, 5th ed., Constable, London, 1952; and Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Flow through a rectangular weir (Fig. 10-21) is given by 苶g苶 q = 0.415(L − 0.2h0)h1.5 0 兹2

(10-39)

where q = volume flow rate, L = crest length, h0 = weir head, and g = local acceleration due to gravity. This is known as the modified Francis formula for a rectangular sharp-edged weir with two end corrections; it applies when the velocity-of-approach correction is small. The Francis formula agrees with experiments within 3 percent if (1) L is greater than 2h0, (2) velocity of approach is 0.6 m/s (2 ft/s) or less, (3) height of crest above bottom of channel is at least 3h0, and (4) h0 is not less than 0.09 m (0.3 ft). Narrow rectangular notches (h0 > L) have been found to give about 93 percent of the discharge predicted by the Francis formula. Thus 苶g苶 q = 0.386Lh1.5 0 兹2

(10-40)

In this case, no end corrections are applied even though the formula applies only for sharp-edged weirs. See Schoder and Dawson, Hydraulics, McGraw-Hill, New York, 1934, p. 175, for further details. The triangular-notch weir has the advantage that a single notch can accommodate a wide range of flow rates, although this in turn reduces its accuracy. The discharge for sharp- or square-edged weirs is given by 苶g苶)/tan φ q = (0.31h2.5 0 兹2

(10-41)

Triangular weir.

See Eq. (10-39) for nomenclature. Angle φ is illustrated in Fig. 10-22. Equations (10-39), (10-40), and (10-41) are applicable only to the flow of water. However, for the case of triangular-notch weirs Lenz [Trans. Am. Soc. Civ. Eng., 108, 759–802 (1943)] has presented correlations predicting the effect of viscosity over the range of 0.001 to 0.15 Pa⋅s (1 to 150 cP) and surface tension over the range of 0.03 to 0.07 N/m (30 to 70 dyn/cm). His equation predicts about an 8 percent increase in flow for a liquid of 0.1-Pa⋅s (100-cP) viscosity compared with water at 0.001 Pa⋅s (1 cP) and about a 1 percent increase for a liquid with one-half of the surface tension of water. For fluids of moderate viscosity, Ranga Raju and Asawa [Proc. Am. Soc. Civ. Eng., J. Hydraul. Div., 103 (HY 10), 1227–1231 (1977)] find that the effect of viscosity and surface tension on the discharge flow rate for rectangular and triangular-notch (φ ⫽ 45°) weirs can be neglected when (NRe)0.2(NWe)0.6 > 900

(10-42)

where NRe (Reynolds number) = 兹苶 gh Ⲑv, g = local acceleration due to gravity, h0 = weir head, ν = kinematic viscosity; NWe (Weber number) 2 = ρgh0Ⲑgcσ, ρ = density, gc = dimensional constant, and σ = surface tension. For the flow of high-viscosity liquids over rectangular weirs, see Slocum, Can. J. Chem. Eng., 42, 196–200 (1964). His correlation is based on data for liquids with viscosities in the range of 2.5 to 500 Pa⋅s (25 to 5000 cP), in which range the discharge decreases markedly for a given head as viscosity is increased. Information on other types of weirs can be obtained from Addison, op. cit.; Gibson, Hydraulics and Its Applications, 5th ed., Constable, London, 1952; Henderson, Open Channel Flow, Macmillan, New York, 1966; Linford, Flow Measurement and Meters, Spon, London, 1949; Lakshmana Rao, “Theory of Weirs,” in Advances in Hydroscience, vol. 10, Academic, New York, 1975; and Merritt, Standard Handbook for Civil Engineers, 2d ed., McGraw-Hill, New York, 1976. 3 0

PUMPS AND COMPRESSORS GENERAL REFERENCES: Meherwan P. Boyce, P.E., Centrifugal Compressors: A Basic Guide, Pennwell Books, Tulsa, Okla., 2002; Royce N. Brown, Compressors: Selection and Sizing, 3d ed., Gulf Professional Publishing, Houston, Tex., 2005; James Corley, “The Vibration Analysis of Pumps: A Tutorial,” Fourth International Pump Symposium, Texas A & M University, Houston, Tex., May 1987; John W. Dufor and William E. Nelson, Centrifugal Pump Sourcebook, McGraw-Hill, New York, 1992; Engineering Data Book, 12th ed., vol. I, Secs. 12 and 13, Gas Processors Suppliers Association, Tulsa, Okla., 2004; Paul N. Garay, P.E., Pump Application Desk Book, Fairmont Press, 1993; Process Pumps, IIT Fluid Technology Corporation, 1992; Igor J. Karassik et al., Pump Handbook, 3d ed., McGraw-Hill, New York, 2001; Val S. Lobanoff and Robert R. Ross, Centrifugal Pumps: Design and Application, 2d ed., Gulf Professional Publishing, Houston, Tex., 1992; A. J. Stephanoff, Centrifugal and Axial Flow Pumps: Theory, Design, and Application, 2d ed., Krieger Publishing, Melbourne, Fla., 1992.

INTRODUCTION The following subsections deal with pumps and compressors. A pump or compressor is a physical contrivance that is used to deliver fluids from one location to another through conduits. The term pump is used when the fluid is a liquid, while the term compressor is used when the fluid is a gas. The basic requirements to define the applica-

tion are suction and delivery pressures, pressure loss in transmission, and flow rate. Special requirements may exist in food, pharmaceutical, nuclear, and other industries that impose material selection requirements of the pump. The primary means of transfer of energy to the fluid that causes flow are gravity, displacement, centrifugal force, electromagnetic force, transfer of momentum, mechanical impulse, and a combination of these energy-transfer mechanisms. Displacement and centrifugal force are the most common energy-transfer mechanisms in use. Pumps and compressors are designed per technical specifications and standards developed over years of operating and maintenance experience. Table 10-10 lists some of these standards for pumps and compressors and for related equipment such as lubrication systems and gearboxes which, if not properly specified, could lead to many operational and maintenance problems with the pumps and compressors. These standards specify design, construction, maintenance, and testing details such as terminology, material selection, shop inspection and tests, drawings, clearances, construction procedures, and so on. There are four (4) major types of pumps: (1) positive displacement, (2) dynamic (kinetic), (3) lift, and (4) electromagnetic. Piston pumps are positive displacement pumps. The most common centrifugal

PUMPS AND COMPRESSORS TABLE 10-10

Standards Governing Pumps and Compressors

ASME Standards, American Society of Mechanical Engineers, New York B73.1-2001, Specification for Horizontal End Suction Centrifugal Pumps for Chemical Process B73.2-2003, Specification for Vertical In-Line Centrifugal Pumps for Chemical Process PTC 10, 1997 Test Code on Compressors and Exhausters PTC 11, 1984 Fans B19.3-1991, Safety Standard for Compressors for Process Industries API Standards, American Petroleum Institute, Washington API Standard 610, Centrifugal Pumps for Petroleum, Petrochemical, and Natural Gas Industries, Adoption of ISO 13709, October 2004 API Standard 613, Special Purpose Gear Units for Petroleum, Chemical and Gas Industry Services, February 2003 API Standard 614, Lubrication, Shaft-Sealing, and Control-Oil Systems and Auxiliaries for Petroleum, Chemical and Gas Industry Services, April 1999 API Standard 616, Gas Turbines for the Petroleum, Chemical, and Gas Industry Services, August 1998 API Standard 617, Axial and Centrifugal Compressors and Expanders— Compressors for Petroleum, Chemical, and Gas Industry Services, June 2003 API Standard 618, Reciprocating Compressors for Petroleum, Chemical, and Gas Industry Services, June 1995 API Standard 619, Rotary-Type Positive Displacement Compressors for Petroleum, Petrochemical, and Natural Gas Industries, December 2004 API Standard 670, Machinery Protection Systems, November 2003 API Standard 671, Special Purpose Couplings for Petroleum, Chemical, and Gas Industry Services, October 1998 API Standard 672, Packaged, Integrally Geared, Centrifugal Air Compressors for Petroleum, Chemical, and Gas Industry Services, March 2004 API Standard 673, Centrifugal Fans for Petroleum, Chemical, and Gas Industry Services, October 2002 API Standard 674, Positive Displacement Pumps—Reciprocating, June 1995 API Standard 675, Positive Displacement Pumps—Controlled Volume, March 2000 API Standard 677, General Purpose Gear Units for Petroleum, Chemical, and Gas Industry Services, April 2006 API Standard 680, Packaged Reciprocating Plant and Instrument Air Compressors for General Refinery Services, October 1987 API Standard 681, Liquid Ring Vacuum Pumps and Compressors for Petroleum, Chemical, and Gas Industry Services, June 2002 API Standard 682, Pumps—Shaft Sealing Systems for Centrifugal and Rotary Pumps, September 2004 API Standard 685, Sealless Centrifugal Pumps for Petroleum, Heavy Duty Chemical, and Gas Industry Services, October 2000 Hydraulic Institute, Parsippany, N.J. (www.pumps.org) ANSI/HI Pump Standards, 2005 (covers centrifugal, vertical, rotary, and reciprocating pumps) National Fire Protection Association, Quincy, Mass. (www.nfpa.org) Standards for pumps used in fire protection systems

pumps are of dynamic type; ancient bucket-type pumps are lift pumps; and electromagnetic pumps use electromagnetic force and are common in modern reactors. Canned pumps are also becoming popular in the petrochemical industry because of the drive to minimize fugitive emissions. Figure 10-23 shows pump classification: TERMINOLOGY Displacement Discharge of a fluid from a vessel by partially or completely displacing its internal volume with a second fluid or by mechanical means is the principle upon which a great many fluidtransport devices operate. Included in this group are reciprocatingpiston and diaphragm machines, rotary-vane and gear types, fluid piston compressors, acid eggs, and air lifts. The large variety of displacement-type fluid-transport devices makes it difficult to list characteristics common to each. However, for most types it is correct to state that (1) they are adaptable to high-pressure operation, (2) the flow rate through the pump is variable (auxiliary damping systems may be employed to reduce the magnitude of pressure pulsation and flow variation), (3) mechanical considerations limit maximum throughputs, and (4) the devices are capable of efficient performance at extremely low-volume throughput rates.

10-25

Centrifugal Force Centrifugal force is applied by means of the centrifugal pump or compressor. Though the physical appearance of the many types of centrifugal pumps and compressors varies greatly, the basic function of each is the same, i.e., to produce kinetic energy by the action of centrifugal force and then to convert this energy into pressure by efficiently reducing the velocity of the flowing fluid. In general, centrifugal fluid-transport devices have these characteristics: (1) discharge is relatively free of pulsation; (2) mechanical design lends itself to high throughputs, capacity limitations are rarely a problem; (3) the devices are capable of efficient performance over a wide range of pressures and capacities even at constant-speed operation; (4) discharge pressure is a function of fluid density; and (5) these are relatively small high-speed devices and less costly. A device which combines the use of centrifugal force with mechanical impulse to produce an increase in pressure is the axial-flow compressor or pump. In this device the fluid travels roughly parallel to the shaft through a series of alternately rotating and stationary radial blades having airfoil cross sections. The fluid is accelerated in the axial direction by mechanical impulses from the rotating blades; concurrently, a positive-pressure gradient in the radial direction is established in each stage by centrifugal force. The net pressure rise per stage results from both effects. Electromagnetic Force When the fluid is an electrical conductor, as is the case with molten metals, it is possible to impress an electromagnetic field around the fluid conduit in such a way that a driving force that will cause flow is created. Such pumps have been developed for the handling of heat-transfer liquids, especially for nuclear reactors. Transfer of Momentum Deceleration of one fluid (motivating fluid) in order to transfer its momentum to a second fluid (pumped fluid) is a principle commonly used in the handling of corrosive materials, in pumping from inaccessible depths, or for evacuation. Jets and eductors are in this category. Absence of moving parts and simplicity of construction have frequently justified the use of jets and eductors. However, they are relatively inefficient devices. When air or steam is the motivating fluid, operating costs may be several times the cost of alternative types of fluid-transport equipment. In addition, environmental considerations in today’s chemical plants often inhibit their use. Mechanical Impulse The principle of mechanical impulse when applied to fluids is usually combined with one of the other means of imparting motion. As mentioned earlier, this is the case in axial-flow compressors and pumps. The turbine or regenerative-type pump is another device which functions partially by mechanical impulse. Measurement of Performance The amount of useful work that any fluid-transport device performs is the product of (1) the mass rate of fluid flow through it and (2) the total pressure differential measured immediately before and after the device, usually expressed in the height of column of fluid equivalent under adiabatic conditions. The first of these quantities is normally referred to as capacity, and the second is known as head. Capacity This quantity is expressed in the following units. In SI units capacity is expressed in cubic meters per hour (m3/h) for both liquids and gases. In U.S. customary units it is expressed in U.S. gallons per minute (gal/min) for liquids and in cubic feet per minute (ft3/min) for gases. Since all these are volume units, the density or specific gravity must be used for conversion to mass rate of flow. When gases are being handled, capacity must be related to a pressure and a temperature, usually the conditions prevailing at the machine inlet. It is important to note that all heads and other terms in the following equations are expressed in height of column of liquid. Total Dynamic Head The total dynamic head H of a pump is the total discharge head hd minus the total suction head hs. Total Suction Head This is the reading hgs of a gauge at the suction flange of a pump (corrected to the pump centerline∗), plus the barometer reading and the velocity head hvs at the point of gauge attachment: hs = hgs + atm + hvs

(10-43)

If the gauge pressure at the suction flange is less than atmospheric, requiring use of a vacuum gauge, this reading is used for hgs in Eq. (10-43) with a negative sign.

10-26

TRANSPORT AND STORAGE OF FLUIDS

Piston Plunger Reciprocating

Double Acting

Simplex Duplex

Steam

Single Acting Double Acting

Simplex Duplex Triplex Multiplex

Power

Simplex Multiplex

Diaphragm

Fluid Operated Mech. Operated

Positive Displacement Single Rotor

Vane Piston Flexible Member Screw

Multiple Rotor

Gear Lobe Circumferential Piston Screw

Blow Cover

Rotary

Fluid Ring Radial Flow Mixed Flow

Single Suction Double Suction Double Valves

Self-priming Non-self-priming Open Impeller Single Stage Semi-open Impeller Multistage Closed Impeller

Axial Flow

Single Suction

Single Stage Multistage

Peripheral

Single Stage Multistage

Self-priming Non-self-priming

Regenerative

Special

Viscous Drag Jet (Ejector-boosted) Gas Lift Hydraulic Ram Electromagnetic Screw Centrifugal Rotating Casing (Pitot)

Pumps Centrifugal

Kinetic

Open Impeller Closed Impeller

Open Screw FIG. 10-23

Classification of pumps. (Courtesty of Hydraulic Institute.)

Before installation it is possible to estimate the total suction head as follows: hs = hss − hfs

(10-44)

where hss = static suction head and hfs = suction friction head. Static Suction Head The static suction head hss is the vertical distance measured from the free surface of the liquid source to the pump centerline plus the absolute pressure at the liquid surface. Total Discharge Head The total discharge head hd is the reading hgd of a gauge at the discharge flange of a pump (corrected to the pump centerline*), plus the barometer reading and the velocity head hvd at the point of gauge attachment: *On vertical pumps, the correction should be made to the eye of the suction impeller.

hd = hgd + atm + hvd

(10-45)

Again, if the discharge gauge pressure is below atmospheric, the vacuum-gauge reading is used for hgd in Eq. (10-45) with a negative sign. Before installation it is possible to estimate the total discharge head from the static discharge head hsd and the discharge friction head hfd as follows: hd = hsd + hfd

(10-46)

Static Discharge Head The static discharge head hsd is the vertical distance measured from the free surface of the liquid in the receiver to the pump centerline,* plus the absolute pressure at the liquid surface. Total static head hts is the difference between discharge and suction static heads.

PUMPS AND COMPRESSORS Velocity Since most liquids are practically incompressible, the relation between the quantity flowing past a given point in a given time and the velocity of flow is expressed as follows: Q = Av

(10-47)

This relationship in SI units is as follows: v (for circular conduits) = 3.54 Q/d 2

(10-48)

where v = average velocity of flow, m/s; Q = quantity of flow, m3/h; and d = inside diameter of conduit, cm. This same relationship in U.S. customary units is v (for circular conduits) = 0.409 Q/d 2

(10-49)

where v = average velocity of flow, ft/s; Q = quantity of flow, gal/min; and d = inside diameter of conduit, in. Velocity Head This is the vertical distance by which a body must fall to acquire the velocity v. hv = v2/2g

(10-50)

Viscosity (See Sec. 6 for further information.) In flowing liquids the existence of internal friction or the internal resistance to relative motion of the fluid particles must be considered. This resistance is called viscosity. The viscosity of liquids usually decreases with rising temperature. Viscous liquids tend to increase the power required by a pump, to reduce pump efficiency, head, and capacity, and to increase friction in pipe lines. Friction Head This is the pressure required to overcome the resistance to flow in pipe and fittings. It is dealt with in detail in Sec. 6. Work Performed in Pumping To cause liquid to flow, work must be expended. A pump may raise the liquid to a higher elevation, force it into a vessel at higher pressure, provide the head to overcome pipe friction, or perform any combination of these. Regardless of the service required of a pump, all energy imparted to the liquid in performing this service must be accounted for; consistent units for all quantities must be employed in arriving at the work or power performed. When arriving at the performance of a pump, it is customary to calculate its power output, which is the product of (1) the total dynamic head and (2) the mass of liquid pumped in a given time. In SI units power is expressed in kilowatts; horsepower is the conventional unit used in the United States. In SI units, (10-51) kW = HQρ/3.670 × 105 where kW is the pump power output, kW; H = total dynamic head, N⋅m/kg (column of liquid); Q = capacity, m3/h; and ρ = liquid density, kg/m3. When the total dynamic head H is expressed in pascals, then kW = HQ/3.599 × 106 (10-52) In U.S. customary units, hp = HQs/3.960 × 103 (10-53) where hp is the pump-power output, hp; H = total dynamic head, lbf⋅ft/lbm (column of liquid); Q = capacity, U.S. gal/min; and s = liquid specific gravity. When the total dynamic head H is expressed in pounds-force per square inch, then

10-27

solids in the liquid and liquid corrosion characteristics requiring special materials of construction. Solids may accelerate erosion and corrosion, have a tendency to agglomerate, or require delicate handling to prevent undesirable degradation. Range of Operation Because of the wide variety of pump types and the number of factors which determine the selection of any one type for a specific installation, the designer must first eliminate all but those types of reasonable possibility. Since range of operation is always an important consideration, Fig. 10-24 should be of assistance. The boundaries shown for each pump type are at best approximate. In most cases, following Fig. 10-24 will select the pump that is best suited for a given application. Low-capacity pumps with high discharge head requirements are best served by positive-displacement pumps. Reciprocating pumps and rotary pumps such as gear and roots rotor-type pumps are examples of positive-displacement pumps. Displacement pumps provide high heads at low capacities which are beyond the capability of centrifugal pumps. Displacement pumps achieve high pressure with low velocities and are thus suited for high-viscosity service and slurry. The centrifugal pump operates over a very wide range of flows and pressures. For low heads but high flows the axial pump is best suited. Both the centrifugal and axial flow pumps impart energy to the fluid by the rotational speed of the impeller and the velocity it imparts to the fluid. NET POSITIVE SUCTION HEAD Net positive suction head available (NPSH)A is the difference between the total absolute suction pressure at the pump suction nozzle when the pump is running and the vapor pressure at the flowing liquid temperature. All pumps require the system to provide adequate (NPSH)A. In a positive-displacement pump the (NPSH)A should be large enough to open the suction valve, to overcome the friction losses within the pump liquid end, and to overcome the liquid acceleration head. Suction Limitations of a Pump Whenever the pressure in a liquid drops below the vapor pressure corresponding to its temperature, the liquid will vaporize. When this happens within an operating pump, the vapor bubbles will be carried along to a point of higher pressure, where they suddenly collapse. This phenomenon is known as cavitation. Cavitation in a pump should be avoided, as it is accompanied by metal removal, vibration, reduced flow, loss in efficiency, and noise. When the absolute suction pressure is low, cavitation may occur in the pump inlet and damage result in the pump suction and on the impeller vanes near the inlet edges. To avoid this phenomenon, it is necessary to maintain a required net positive suction head

hp = HQ/1.714 × 103

(10-54) The power input to a pump is greater than the power output because of internal losses resulting from friction, leakage, etc. The efficiency of a pump is therefore defined as Pump efficiency = (power output)/(power input)

(10-55)

PUMP SELECTION When selecting pumps for any service, it is necessary to know the liquid to be handled, the total dynamic head, the suction and discharge heads, and, in most cases, the temperature, viscosity, vapor pressure, and specific gravity. In the chemical industry, the task of pump selection is frequently further complicated by the presence of

Pump coverage chart based on normal ranges of operation of commercially available types. Solid lines: use left ordinate, head scale. Broken lines: use right ordinate, pressure scale. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert pounds-force per square inch to kilopascals, multiply by 6.895.

FIG. 10-24

10-28

TRANSPORT AND STORAGE OF FLUIDS

(NPSH)R, which is the equivalent total head of liquid at the pump centerline less the vapor pressure p. Each pump manufacturer publishes curves relating (NPSH)R to capacity and speed for each pump. When a pump installation is being designed, the available net positive suction head (NPSH)A must be equal to or greater than the (NPSH)R for the desired capacity. The (NPSH)A can be calculated as follows: (NPSH)A = hss − hfs − p (10-56) If (NPSH)A is to be checked on an existing installation, it can be determined as follows: (NPSH)A = atm + hgs − p + hvs (10-57) Practically, the NPSH required for operation without cavitation and vibration in the pump is somewhat greater than the theoretical. The actual (NPSH)R depends on the characteristics of the liquid, the total head, the pump speed, the capacity, and impeller design. Any suction condition which reduces (NPSH)A below that required to prevent cavitation at the desired capacity will produce an unsatisfactory installation and can lead to mechanical difficulty. The following two equations usually provide an adequate design margin between (NPSH)A and (NPSH)R: (NPSH)A = (NPSH)R + 5 ft

(10-58)

(NPSH)A = 1.35(NPSH)R (10-59) Use the larger value of (NPSH)A calculated with Eqs. (10-58) and (10-59). NPSH Requirements for Other Liquids NPSH values depend on the fluid being pumped. Since water is considered a standard fluid for pumping, various correction methods have been developed to evaluate NPSH when pumping other fluids. The most recent of these corrective methods has been developed by the Hydraulic Institute and is shown in Fig. 10-25. The chart shown in Fig. 10-25 is for pure liquids. Extrapolation of data beyond the ranges indicated in the graph may not produce accurate results. Figure 10-25 shows the variation of vapor pressure and NPSH reductions for various hydrocarbons and hot water as a function of temperature. Certain rules apply while using this chart. When using the chart for hot water, if the NPSH reduction is greater than one-half of the NPSH required for cold water, deduct one-half of cold water NPSH to obtain the corrected NPSH required. On the other

ne

pa

o Pr

e an ut b o

100

Is

50 40 30 20 15 10

fri ge Me thy ran t la lco R-1 ho 1 l Wa ter

ne ta

Bu

5 4 3 2 1.5 1.0

10 8 6 5 4 3 2 1.5 1.0 0.5

0

50

100

NPSH reductions, feet

500 400 300 200 150

Re

Vapor pressure, PSIA

1000

150 200 250 300 400

Temperature, F FIG. 10-25 NPSH reductions for pumps handling hydrocarbon liquids and high-temperature water. This chart has been constructed from test data obtained using the liquids shown (Hydraulic Institute Standards).

hand, if the value read on the chart is less than one-half of cold water NPSH, deduct this chart value from the cold water NPSH to obtain the corrected NPSH. Example 1: NPSH Calculation Suppose a selected pump requires a minimum NPSH of 16 ft (4.9 m) when pumping cold water; What will be the NPSH limitation to pump propane at 55°F (12.8°C) with a vapor pressure of 100 psi? Using the chart in Fig. 10-25, NPSH reduction for propane gives 9.5 ft (2.9 m). This is greater than one-half of cold water NPSH of 16 ft (4.9 m). The corrected NPSH is therefore 8 ft (2.2 m) or one-half of cold water NPSH. PUMP SPECIFICATIONS Pump specifications depend upon numerous factors but mostly on application. Typically, the following factors should be considered while preparing a specification. 1. Application, scope, and type 2. Service conditions 3. Operating conditions 4. Construction application-specific details and special considerations a. Casing and connections b. Impeller details c. Shaft d. Stuffing box details—lubrications, sealing, etc. e. Bearing frame and bearings f. Baseplate and couplings g. Materials h. Special operating conditions and miscellaneous items Table 10-11 is based on the API and ASME codes and illustrates a typical specification for centrifugal pumps. POSITIVE-DISPLACEMENT PUMPS Positive-displacement pumps and those that approach positive displacement will ideally produce whatever head is impressed upon them by the system restrictions to flow. The maximum head attainable is determined by the power available in the drive (slippage neglected) and the strength of the pump parts. A pressure relief valve on the discharge side should be set to open at a safe pressure for the casing and the internal components of the pump such as piston rods, cylinders, crankshafts, and other components which would be pressurized. In the case of a rotary pump, the total dynamic head developed is uniquely determined for any given flow by the speed at which it rotates. In general, overall efficiencies of positive-displacement pumps are higher than those of centrifugal equipment because internal losses are minimized. On the other hand, the flexibility of each piece of equipment in handling a wide range of capacities is somewhat limited. Positive-displacement pumps may be of either the reciprocating or the rotary type. In all positive-displacement pumps, a cavity or cavities are alternately filled and emptied of the pumped fluid by the action of the pump. Reciprocating Pumps There are three classes of reciprocating pumps: piston pumps, plunger pumps, and diaphragm pumps. Basically, the action of the liquid-transferring parts of these pumps is the same, a cylindrical piston, plunger, or bucket or a round diaphragm being caused to pass or flex back and forth in a chamber. The device is equipped with valves for the inlet and discharge of the liquid being pumped, and the operation of these valves is related in a definite manner to the motions of the piston. In all modern-design reciprocating pumps, the suction and discharge valves are operated by pressure difference. That is, when the pump is on its suction stroke and the pump cavity is increasing in volume, the pressure is lowered within the pump cavity, permitting the higher suction pressure to open the suction valve and allowing liquid to flow into the pump. At the same time, the higher discharge-line pressure holds the discharge valve closed. Likewise on the discharge stroke, as the pump cavity is decreasing in volume, the higher pressure developed in the pump cavity holds the suction valve closed and opens the discharge valve to expel liquid from the pump into the discharge line. The overall efficiency of these pumps varies from about 50 percent for the small pumps to about 90 percent or more for the larger sizes.

TABLE 10-11

Typical Pump Specification

Specification

Specification

Description

1.0

Scope: This specification covers horizontal, end suction, vertically split, singlestage centrifugal pumps with top centerline discharge and “back pullout” feature.

2.0

Service Conditions: Pump shall be designed to operate satisfactorily with a reasonable service life when operated either intermittently or continuously in typical process applications.

3.0

Operating Conditions: Capacity U.S. gallons per minute Head ( ft total head) ( psig). Speed ft head) (positive) (lift) ( Suction Pressure ( Liquid to be handled Specific gravity Viscosity ( °F Temperature of liquid at inlet Solids content % Max. size

4.0

rpm psig) )

Pump Construction: 4.1 Casing. Casing shall be vertically split with self-venting top centerline discharge, with an integral foot located directly under the casing for added support. All casings shall be of the “back pullout” design with suction and discharge nozzles cast integrally. Casings shall be provided with bosses in suction and discharge nozzles, and in bottom of casing for gauge taps and drain tap. (Threaded taps with plugs shall be provided for these features.) 4.2 Casing Connections. Connections shall be A.N.S.I. flat-faced flanges. [Cast iron (125) (250) psig rated] [Duron metal, steel, alloy steel (150) (300) psig rated] 4.3 Casing Joint Gasket. A confined-type nonasbestos gasket suitable for corrosive service shall be provided at the casing joint. 4.4 Impeller. Fully-open impeller with front edge having contoured vanes curving into the suction for minimum NPSH requirements and maximum efficiency shall be provided. A hex head shall be cast in the eye of the impeller to facilitate removal, and eliminate need for special impeller removing tool. All impellers shall have radial “pump-out” vanes on the back side to reduce stuffing box pressure and aid in eliminating collection of solids at stuffing box throat. Impellers shall be balanced within A.N.S.I. guidelines to ISO tolerances. 4.4.1 Impeller Clearance Adjustment. All pumps shall have provisions for adjustment of axial clearance between the leading edge of the impeller and casing. This adjustment shall be made by a precision microdial adjustment at the outboard bearing housing, which moves the impeller forward toward the suction wall of the casing. 4.5 Shafts. Shafts shall be suitable for hook-type sleeve. Shaft material shall be (SAE 1045 steel on Duron and 316 stainless steel pumps) or (AISI 316 stainless steel on CD-4MCu pumps and #20 stainless steel pumps). Shaft deflection shall not exceed .005 at the vertical centerline of the impeller. 4.6 Shaft Sleeve. Renewable hook-type shaft sleeve that extends through the stuffing box and gland shall be provided. Shaft sleeve shall be (316 stainless steel), (#20 stainless steel) or (XH-800 Ni-chromeboron coated 316 stainless steel with coated surface hardness of approximately 800 Brinnell). 4.7 Stuffing Box. Stuffing box shall be suitable for packing, single (inside or outside) or double-inside mechanical seal without modifications. Stuffing box shall be accurately centered by machined rabbit fits on case and frame adapter. 4.7.1 Packed Stuffing Box. The standard packed stuffing box shall consist of five rings of graphited nonasbestos packing; a stainless steel packing base ring in the bottom of the box to prevent extrusion of the packing past the throat; a Teflon seal cage, and a two-piece 316 stainless steel packing gland to ensure even pressure on the packing. Ample space shall be provided for repacking the stuffing box. 4.7.1.1 Lubrication-Packed Stuffing Box. A tapped hole shall be provided in the stuffing box directly over the seal cage for lubrication and cooling of the packing. Lubrication liquid shall be supplied (from an external source) (through a by-pass line from the pump discharge nozzle). 4.7.2 Stuffing Box with Mechanical Seal. Mechanical seal shall be of the (single inside) (single outside) (double inside) (cartridge) type and (balanced) (unbalanced). Stuffing box is to be (standard) (oversize) (oversize tapered).

*Omit if not applicable.

4.8

4.9

5.0

6.0

7.0

8.0

Description Suitable space shall be provided in the standard and oversized stuffing box for supplying a (throttle bushing) (dilution control bushing) with single seals. Throttle bushings and dilution control bushings shall be made of (glass-filled Teflon) (a suitable metal material). 4.7.2.1 Lubrication—Stuffing Box with Mechanical Seals. Suitable tapped connections shall be provided to effectively lubricate, cool, flush, quench, etc., as required by the application or recommendations of the mechanical seal manufacturer. Bearing Frame and Bearings: 4.8.1 Bearing Frame. Frames shall be equipped with axial radiating fins extending the length of the frame to aid in heat dissipation. Frame shall be provided with ductile iron outboard bearing housing. Both ends of the frame shall be provided with lip-type oil seals and labyrinth-type deflectors of metallic reinforced synthetic rubber to prevent the entrance of contaminants. 4.8.2 Bearings. Pump bearings shall be heavy-duty, antifriction balltype on both ends. The single row inboard bearing, nearest the impeller, shall be free to float within the frame and shall carry only radial load. The double row outboard bearing (F4-G1 and F4-I1) or duplex angular contact bearing (F4-H1), coupling end, shall be locked in place to carry radial and axial thrust loads. Bearings shall be designed for a minimum life of 20,000 hours in any normal pump operating range. Bearing Lubrication. Ball bearings shall be oil-mist—lubricated by means of a slinger. The oil slinger shall be mounted on the shaft between the bearings to provide equal lubrication to both bearings. Bulls-eye oil-sight glasses shall be provided on both sides of the frame to provide a positive means of checking the proper oil level from either side of the pump. A tapped and plugged hole shall also be provided in both sides of the frame to mount bottle-type constant-level oilers where desired. A tapped and plugged hole shall be provided on both sides for optional straight-through oil cooling device. Baseplate and Coupling: 5.1 Baseplate. Baseplates shall be rigid and suitable for mounting pump and motor. Baseplates shall be of channel steel construction. 5.2 Coupling. Coupling shall be flexible-spacer type. Coupling shall have at least three–and–one-half–inch spacer length for ease of rotating element removal. Both coupling hubs shall be provided with flats 180° apart to facilitate removal of impeller. Coupling shall not require lubrication.* Mechanical Modifications Required for High Temperature: 6.1 Modifications Required, Temperature Range 250–350°F. Pumps for operation in this range shall be provided with a water-jacketed stuffing box. 6.2 Modifications Required, Temperature Range 351–550°F (Maximum). Pumps for operation in this range shall be provided with a water-jacketed stuffing box and a water-cooled bearing frame. Materials: Pump materials shall be selected to suit the particular service requirements. 7.1 Cast Iron—316 SS Fitted. 15″ only; pump shall have cast iron casing and stuffing box cover. 316 SS metal impeller; shaft shall be 1045 steel with 316 SS sleeve. 7.2 All Duron Metal. All pump materials shall be Duron metal. Shaft shall be 1045 steel, with 316 SS sleeve. 316 SS metal impeller optional. 7.3 All AISI 316 Stainless Steel. All pump materials shall be AISI 316 stainless steel. Shaft should be 1045 steel, with 316 SS sleeve. 7.4 All #20 Stainless Steel. All pump materials shall be #20 SS stainless steel. Shaft shall be 316 SS, with #20 SS sleeve. 7.5 All CD-4MCu. All pump materials shall be CD-4MCu. Shaft shall be 316 SS, with #20 SS sleeve. Miscellaneous: 8.1 Nameplates. All nameplates and other data plates shall be stainless steel, suitably secured to the pump. 8.2 Hardware. All machine bolts, stud nuts, and capscrews shall be of the hex-head type. 8.3 Rotation. Pump shall have clockwise rotation viewed from its driven end. 8.4 Parts Numbering. Parts shall be completely identified with a numerical system (no alphabetical letters) to facilitate parts inventory control and stocking. Each part shall be properly identified by a separate number, and those parts that are identical shall have the same number to effect minimum spare parts inventory.

10-30

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-26

Double-acting steam-driven reciprocating pump.

As shown in Fig. 10-26, reciprocating pumps, except when used for metering service, are frequently provided on the discharge side with gas-charged chambers, the purpose of which is to limit pressure pulsation and to provide a more uniform flow in the discharge line. In many installations, surge chambers are required on the suction side as well. Piping layouts should be studied to determine the most effective size and location. If surge chambers are used, provision should be made to keep the chamber charged with gas. A surge chamber filled with liquid is of no value. A liquid-level gauge is desirable to permit checking the amount of gas in the chamber. Reciprocating pumps may be of single-cylinder or multicylinder design. Multicylinder pumps have all cylinders in parallel for increased capacity. Piston-type pumps may be single-acting or doubleacting; i.e., pumping may be accomplished from one or both ends of the piston. Plunger pumps are always single-acting. The tabulation in Table 10-12 provides data on the flow variation of reciprocating pumps of various designs. Piston Pumps There are two ordinary types of piston pumps, simplex double-acting pumps and duplex double-acting pumps. Simplex Double-Acting Pumps These pumps may be directacting (i.e., direct-connected to a steam cylinder) or power-driven (through a crank and flywheel from the crosshead of a steam engine). Figure 10-26 is a direct-acting pump, designed for use at pressures up to 0.690 MPa (100 lbf/in2). In this figure, the piston consists of disks A and B, with packing rings C between them. A bronze liner for the water cylinder is shown at D. Suction valves are E1 and E2. Discharge valves are F1 and F2. Duplex Double-Acting Pumps These pumps differ primarily from those of the simplex type in having two cylinders whose operation is coordinated. They may be direct-acting, steam-driven, or power-driven with crank and flywheel. A duplex outside-end-packed plunger pump with pot valves, of the type used with hydraulic presses and for similar service, is shown in Fig. 10-27. In this drawing, plunger A is direct-connected to rod B, while plunger C is operated from the rod by means of yoke D and tie rods. Plunger pumps differ from piston pumps in that they have one or more constant-diameter plungers reciprocating through packing glands and displacing liquid from cylinders in which there is consider-

TABLE 10-12

Flow Variation of Reciprocating Pumps

Number of cylinders

Single- or doubleacting

Flow variation per stroke from mean, percent

Single Single Duplex Duplex Triplex Quintuplex

Single Double Single Double Single and double Single

+220 to −100 +60 to −100 +24.1 to −100 +6.1 to −21.5 +1.8 to −16.9 +1.8 to −5.2

(a)

(b) FIG. 10-27

Duplex single-acting plunger pump.

able radial clearance. They are always single-acting, in the sense that only one end of the plunger is used in pumping the liquid. Plunger pumps are available with one, two, three, four, five, or even more cylinders. Simplex and duplex units are often built in a horizontal design. Those with three or more cylinders are usually of vertical design. The driver may be an electric motor, a steam or gas engine, or a steam turbine. This is the common type of power pump. An example, arranged for belt drive, is shown in Fig. 10-28 from which the action may be readily traced. Occasionally plunger pumps are constructed with opposed cylinders and plungers connected by yokes and tie rods; this arrangement, in effect, constitutes a double-acting unit. Simplex plunger pumps mounted singly or in gangs with a common drive are quite commonly used as metering or proportioning pumps (Fig. 10-29). Frequently a variable-speed drive or a strokeadjusting mechanism is provided to vary the flow as desired. These pumps are designed to measure or control the flow of liquid within a deviation of ±2 percent with capacities up to 11.35 m3/h (50 gal/min) and pressures as high as 68.9 MPa (10,000 lbf/in2). Diaphragm Pumps These pumps perform similarly to piston and plunger pumps, but the reciprocating driving member is a flexible diaphragm fabricated of metal, rubber, or plastic. The chief advantage of this arrangement is the elimination of all packing and seals exposed to the liquid being pumped. This, of course, is an important asset for equipment required to handle hazardous or toxic liquids. A common type of low-capacity diaphragm pump designed for metering service employs a plunger working in oil to actuate a metallic or plastic diaphragm. Built for pressures in excess of 6.895 MPa (1000 lbf/in2) with flow rates up to about 1.135 m3/h (5 gal/min) per cylinder, such pumps possess all the characteristics of plunger-type metering pumps with the added advantage that the pumping head can be mounted in a remote (even a submerged) location entirely separate from the drive.

PUMPS AND COMPRESSORS

FIG. 10-30

FIG. 10-28

Adrich-Groff variable-stroke power pump. (Courtesy of Ingersoll-

Rand.)

Figure 10-30 shows a high-capacity 22.7-m3/h (100-gal/min) pump with actuation provided by a mechanical linkage. Pneumatically Actuated Diaphragm Pumps (Fig. 10-31) These pumps require no power source other than plant compressed air. They must have a flooded suction, and the pressure is, of course, limited to the available air pressure. Because of their slow speed and large valves, they are well suited to the gentle handling of liquids for which degradation of suspended solids should be avoided. A major consideration in the application of diaphragm pumps is the realization that diaphragm failure will probably occur eventually. The consequences of such failure should be realistically appraised before selection, and maintenance procedures should be established accordingly. Rotary Pumps In rotary pumps the liquid is displaced by rotation of one or more members within a stationary housing. Because internal clearances, although minute, are a necessity in all but a few special types, capacity decreases somewhat with increasing pump

Plunger-type metering pump. (Courtesy of Milton Roy Co.)

Mechanically actuated diaphragm pump.

differential pressure. Therefore, these pumps are not truly positivedisplacement pumps. However, for many other reasons they are considered as such. The selection of materials of construction for rotary pumps is critical. The materials must be corrosion-resistant, compatible when one part is running against another, and capable of some abrasion resistance. Gear Pumps When two or more impellers are used in a rotarypump casing, the impellers will take the form of toothed-gear wheels as in Fig. 10-32, of helical gears, or of lobed cams. In each case, these impellers rotate with extremely small clearance between them and between the surfaces of the impellers and the casing. In Fig. 10-32, the two toothed impellers rotate as indicated by the arrows; the suction connection is at the bottom. The pumped liquid flows into the spaces between the impeller teeth as these cavities pass the suction opening. The liquid is then carried around the casing to the discharge opening, where it is forced out of the impeller teeth mesh. The arrows indicate this flow of liquid. Rotary pumps are available in two general classes, interior-bearing and exterior-bearing. The interior-bearing type is used for handling

Pneumatically actuated diaphragm pump for slurry service. (Courtesy of Dorr-Oliver Inc.)

FIG. 10-31 FIG. 10-29

10-31

10-32

FIG. 10-32

TRANSPORT AND STORAGE OF FLUIDS

Positive-displacement gear-type rotary pump.

liquids of a lubricating nature, and the exterior-bearing type is used with nonlubricating liquids. The interior-bearing pump is lubricated by the liquid being pumped, and the exterior-bearing type is oil-lubricated. The use of spur gears in gear pumps will produce in the discharge pulsations having a frequency equivalent to the number of teeth on both gears multiplied by the speed of rotation. The amplitude of these disturbances is a function of tooth design. The pulsations can be reduced markedly by the use of rotors with helical teeth. This in turn introduces end thrust, which can be eliminated by the use of doublehelical or herringbone teeth. Screw Pumps A modification of the helical gear pump is the screw pump. Both gear and screw pumps are positive-displacement pumps. Figure 10-33 illustrates a two-rotor version in which the liquid is fed to either the center or the ends, depending upon the direction of rotation, and progresses axially in the cavities formed by the meshing threads or teeth. In three-rotor versions, the center rotor is the driving member while the other two are driven. Figure 10-34 shows still another arrangement, in which a metal rotor of unique design rotates without clearance in an elastomeric stationary sleeve. Screw pumps, because of multiple dams that reduce slip, are well adapted for producing higher pressure rises, for example, 6.895 MPa (1000 lbf/in2), especially when handling viscous liquids such as heavy oils. The all-metal pumps are generally subject to the same limitations on handling abrasive solids as conventional gear pumps. In addition, the wide bearing spans usually demand that the liquid have considerable lubricity to prevent metal-to-metal contact. Among the liquids handled by rotary pumps are mineral oils, vegetable oils, animal oils, greases, glucose, viscose, molasses, paints, var-

FIG. 10-33

nish, shellac, lacquers, alcohols, catsup, brine, mayonnaise, sizing, soap, tanning liquors, vinegar, and ink. Some screw-type units are specially designed for the gentle handling of large solids suspended in the liquid. Fluid-Displacement Pumps In addition to pumps that depend on the mechanical action of pistons, plungers, or impellers to move the liquid, other devices for this purpose employ displacement by a secondary fluid. This group includes air lifts and acid eggs. The air lift is a device for raising liquid by means of compressed air. In the past it was widely used for pumping wells, but it has been less widely used since the development of efficient centrifugal pumps. It operates by introducing compressed air into the liquid near the bottom of the well. The air-and-liquid mixture, being lighter than liquid alone, rises in the well casing. The advantage of this system of pumping lies in the fact that there are no moving parts in the well. The pumping equipment is an air compressor, which can be located on the surface. A simplified sketch of an air lift for this purpose is shown in Fig. 10-35. Ingersoll-Rand has developed empirical information on air-lift performance which is available upon request. An important application of the gas-lift principle involves the extraction of oil from wells. There are several references to both practical and theoretical work involving gas lift performance and related problems. Recommended sources are American Petroleum Institute, Drilling and Production Practices, 1952, pp. 257–317, and 1939, p. 266; Trans. Am. Soc. Mining Metall. Eng., 92, 296–313 (1931), 103, 170–186 (1933), 118, 56–70 (1936), 192, 317–326 (1951), 189, 73–82 (1950), and 198, 271–278 (1953); Trans Am. Soc. Mining Metall., and Pet. Eng., 213 (1958), and 207, 17–24 (1956); and Univ. Wisconsin Bull., Eng. Ser., 6, no. 7 (1911, reprinted 1914). An acid egg, or blowcase, consists of an egg-shaped container which can be filled with a charge of liquid that is to be pumped. This container is fitted with an inlet pipe for the charge, an outlet pipe for the discharge, and a pipe for the admission of compressed air or gas, as illustrated in Fig. 10-36. Pressure of air or gas on the surface of the liquid forces it out of the discharge pipe. Such pumps can be handoperated or arranged for semiautomatic or automatic operation. CENTRIFUGAL PUMPS The centrifugal pump is the type most widely used in the chemical industry for transferring liquids of all types—raw materials, materials in manufacture, and finished products—as well as for general services of water supply, boiler feed, condenser circulation, condensate return, etc. These pumps are available through a vast range of sizes, in capacities from 0.5 m3/h to 2 × 104 m3/h (2 gal/min to 105 gal/min), and for discharge heads (pressures) from a few meters to approximately 48 MPa (7000 lbf/in2). The size and type best suited to a particular application can be determined only by an engineering study of the problem. The primary advantages of a centrifugal pump are simplicity, low first cost, uniform (nonpulsating) flow, small floor space, low maintenance expense, quiet operation, and adaptability for use with a motor or a turbine drive.

Two-rotor screw pump. (Courtesy of Warren Quimby Pump Co.)

PUMPS AND COMPRESSORS

FIG. 10-34

10-33

Single-rotor screw pump with an elastomeric lining. (Courtesy of Moyno Pump Division, Robbins & Myers, Inc.)

A centrifugal pump, in its simplest form, consists of an impeller rotating within a casing. The impeller consists of a number of blades, either open or shrouded, mounted on a shaft that projects outside the casing. Its axis of rotation may be either horizontal or vertical, to suit the work to be done. Closed-type, or shrouded, impellers are generally the most efficient. Open- or semiopen-type impellers are used for viscous liquids or for liquids containing solid materials and on many small pumps for general service. Impellers may be of the single-suction or the double-suction type—single if the liquid enters from one side, double if it enters from both sides. Casings There are three general types of casings, but each consists of a chamber in which the impeller rotates, provided with inlet and exit for the liquid being pumped. The simplest form is the circular casing, consisting of an annular chamber around the impeller; no attempt is made to overcome the losses that will arise from eddies and shock when the liquid leaving the impeller at relatively high velocities enters this chamber. Such casings are seldom used. Volute casings take the form of a spiral increasing uniformly in cross-sectional area as the outlet is approached. The volute efficiently converts the velocity energy imparted to the liquid by the impeller into pressure energy. A third type of casing is used in diffuser-type or turbine pumps. In this type, guide vanes or diffusers are interposed between the impeller discharge and the casing chamber. Losses are kept to a minimum in a well-designed pump of this type, and improved efficiency is obtained over a wider range of capacities. This construction is often used in multistage high-head pumps. Action of a Centrifugal Pump Briefly, the action of a centrifugal pump may be shown by Fig. 10-37. Power from an outside source is applied to shaft A, rotating the impeller B within the stationary casing C. The blades of the impeller in revolving produce a reduction in

pressure at the entrance or eye of the impeller. This causes liquid to flow into the impeller from the suction pipe D. This liquid is forced outward along the blades at increasing tangential velocity. The velocity head it has acquired when it leaves the blade tips is changed to pressure head as the liquid passes into the volute chamber and thence out the discharge E. Centrifugal Pump Characteristics Figure 10-38 shows a typical characteristic curve of a centrifugal pump. It is important to note that at any fixed speed the pump will operate along this curve and at no other points. For instance, on the curve shown, at 45.5 m3/h (200 gal/min) the pump will generate 26.5-m (87-ft) head. If the head is increased to 30.48 m (100 ft), 27.25 m3/h (120 gal/min) will be delivered. It is not possible to reduce the capacity to 27.25 m3/h (120 gal/min) at 26.5-m (87-ft) head unless the discharge is throttled so that 30.48 m (100 ft) is actually generated within the pump. On pumps with variable-speed drivers such as steam turbines, it is possible to change the characteristic curve, as shown by Fig. 10-39. As shown in Eq. (10-50), the head depends upon the velocity of the fluid, which in turn depends upon the capability of the impeller

A form of acid egg. External controls required for automatic operation are not shown.

FIG. 10-36

FIG. 10-35

head.

Simplified sketch of an air lift, showing submergence and total FIG. 10-37

A simple centrifugal pump.

10-34

TRANSPORT AND STORAGE OF FLUIDS estimate of the off-design performance of pumps can be obtained through similarity relationship or the affinity laws. These are: 1. Capacity (Q) is proportional to impeller rotational speed (N). 2. Head (h) varies as square of the impeller rotational speed. 3. Brake horsepower (BHP) varies as the cube of the impeller rotational speed. These equations can be expressed mathematically and appear in Table 10-13. System Curves In addition to the pump design, the operational performance of a pump depends upon factors such as the downstream load characteristics, pipe friction, and valve performance. Typically, head and flow follow the following relationship: (Q2 )2 h2 ᎏ2 = ᎏ (Q1) h1

Characteristic curve of a centrifugal pump operating at a constant speed of 3450 r/min. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; to convert horsepower to kilowatts, multiply by 0.746; and to convert inches to centimeters, multiply by 2.54.

FIG. 10-38

to transfer energy to the fluid. This is a function of the fluid viscosity and the impeller design. It is important to remember that the head produced will be the same for any liquid of the same viscosity. The pressure rise, however, will vary in proportion to the specific gravity. For quick pump selection, manufacturers often give the most essential performance details for a whole range of pump sizes. Figure 10-40 shows typical performance data for a range of process pumps based on suction and discharge pipes and impeller diameters. The performance data consists of pump flow rate and head. Once a pump meets a required specification, then a more detailed performance data for the particular pump can be easily found based on the curve reference number. Figure 10-41 shows a more detailed pump performance curve that includes, in addition to pump head and flow, the brake horsepower required, NPSH required, number of vanes, and pump efficiency for a range of impeller diameters. If detailed manufacturer-specified performance curves are not available for a different size of the pump or operating condition, a best

where the subscript 1 refers to the design condition and 2 to the actual conditions. The above equation indicates that head will change as a square of the water flow rate. Figure 10-42 shows the schematic of a pump, moving a fluid from tank A to tank B, both of which are at the same level. The only force that the pump has to overcome in this case is the pipe friction, variation of which with fluid flow rate is also shown in the figure. On the other for the use shown in Fig. 10-43, the pump in addition to pipe friction should overcome head due to difference in elevation between tanks A and B. In this case, elevation head is constant, whereas the head required to overcome friction depends on the flow rate. Figure 10-44 shows the pump performance requirement of a valve opening and closing. Pump Selection One of the parameters that is extremely useful in selecting a pump for a particular application is specific speed Ns. Specific speed of a pump can be evaluated based on its design speed, flow, and head: NQ1/2 Ns = ᎏ (10-61) H3/4 where N = rpm, Q is flow rate in gpm, and H is head in ft⋅lbf /lbm. Specific speed is a parameter that defines the speed at which impellers of geometrically similar design have to be run to discharge one gallon per minute against a one-foot head. In general, pumps with a low specific speed have a low capacity and high specific speed, high capacity. Specific speeds of different types of pumps are shown in Table 10-14 for comparison. Another parameter that helps in evaluating the pump suction limitations, such as cavitation, is suction-specific speed. NQ1/2 S = ᎏᎏ (NPSH)3/4

Characteristic curve of a centrifugal pump at various speeds. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; to convert horsepower to kilowatts, multiply by 0.746; and to convert inches to centimeters, multiply by 2.54.

FIG. 10-39

(10-60)

(10-62)

Typically, for single-suction pumps, suction-specific speed above 11,000 is considered excellent. Below 7000 is poor and 7000–9000 is of an average design. Similarly, for double-suction pumps, suctionspecific speed above 14,000 is considered excellent, below 7000 is poor, and 9000–11,000 is average. Figure 10-45 shows the schematic of specific-speed variation for different types of pumps. The figure clearly indicates that, as the specific speed increases, the ratio of the impeller outer diameter D1 to inlet or eye diameter D2 decreases, tending to become unity for pumps of axial-flow type. Typically, axial flow pumps are of high flow and low head type and have a high specific speed. On the other hand, purely radial pumps are of high head and low flow rate capability and have a low specific speed. Obviously, a pump with a moderate flow and head has an average specific speed. A typical pump selection chart such as shown in Fig. 10-46 calculates the specific speed for a given flow, head, and speed requirements. Based on the calculated specific speed, the optimal pump design is indicated. Process Pumps This term is usually applied to single-stage pedestal-mounted units with single-suction overhung impellers and with a single packing box. These pumps are ruggedly designed for

PUMPS AND COMPRESSORS

800

10-35

3500 16

15

600

12

400

13

11

14 5 7

8

Selection and curves

6

200

10

Feet

9 100 80

1

2

4 3

60 40

20

20

40

60 80 100

200

400

600 800 1000

2000

GPM Range Pump No. 1.5 × 6 E 731 Plus 1 3 × 1.5 × 6 731 Plus 2 3×2×6 731 Plus 3 4×3×6 731 Plus 4 1.5 × 1 × 8 731 Plus 5 3 × 1.5 × 8 731 Plus 6 3 × 1.5 × 8.5 E 731 Plus 7 3 × 2 × 8.5 E 731 Plus 8 FIG. 10-40

Curve A-8475 A-6982 A-8159 A-8551 A-8153 A-8155 A-8529 A-8506

Range Pump No. 4 × 3 × 8.5 731 Plus 9 6 × 4 × 8.5 731 Plus 10 2 × 1 × 10 E 731 Plus 11 3 × 1.5 × 11 E 731 Plus 12 3 × 2 × 11 731 Plus 13 4 × 3 × 11 731 Plus 14 3 × 1.5 × 13 E 731 Plus 15 3 × 2 × 13 731 Plus 16

Curve A-8969 A-8547 A-8496 A-8543 A-8456 A-7342 A-8492 A-7338

Performance curves for a range of open impeller pumps.

ease in dismantling and accessibility, with mechanical seals or packing arrangements, and are built especially to handle corrosive or otherwise difficult-to-handle liquids. Specifically but not exclusively for the chemical industry, most pump manufacturers now build to national standards horizontal and vertical process pumps. ASME Standards B73.1—2001 and B73.2—2003 apply to the horizontal (Fig. 10-47) and vertical in-line (Fig. 10-48) pumps, respectively. The horizontal pumps are available for capacities up to 900 m3/h (4000 gal/min); the vertical in-line pumps, for capacities up to 320 m3/h (1400 gal/min). Both horizontal and vertical in-line pumps are available for heads up to 120 m (400 ft). The intent of each ANSI specification is that pumps from all vendors for a given nominal capacity and total dynamic head at a given rotative speed shall be dimensionally interchangeable with respect to mounting, size, and location of suction and discharge nozzles, input shaft, base plate, and foundation bolts. The vertical in-line pumps, although relatively new additions, are finding considerable use in chemical and petrochemical plants in the United States. An inspection of the two designs will make clear the relative advantages and disadvantages of each. Chemical pumps are available in a variety of materials. Metal pumps are the most widely used. Although they may be obtained in iron, bronze, and iron with bronze fittings, an increasing number of pumps of ductile-iron, steel, and nickel alloys are being used. Pumps are also available in glass, glass-lined iron, carbon, rubber, rubberlined metal, ceramics, and a variety of plastics, such units usually being employed for special purposes. Sealing the Centrifugal Chemical Pump Although detailed treatment of shaft seals is presented in the subsection “Sealing of

Rotating Shafts,” it is appropriate to mention here the special problems of sealing centrifugal chemical pumps. Current practice demands that packing boxes be designed to accommodate both packing and mechanical seals. With either type of seal, one consideration is of paramount importance in chemical service: the liquid present at the sealing surfaces must be free of solids. Consequently, it is necessary to provide a secondary compatible liquid to flush the seal or packing whenever the process liquid is not absolutely clean. The use of packing requires the continuous escape of liquid past the seal to minimize and to carry away the frictional heat developed. If the effluent is toxic or corrosive, quench glands or catch pans are usually employed. Although packing can be adjusted with the pump operating, leaking mechanical seals require shutting down the pump to correct the leak. Properly applied and maintained mechanical seals usually show no visible leakage. In general, owing to the more effective performance of mechanical seals, they have gained almost universal acceptance. Double-Suction Single-Stage Pumps These pumps are used for general water-supply and circulating service and for chemical service when liquids that are noncorrosive to iron or bronze are being handled. They are available for capacities from about 5.7 m3/h (25 gal/min) up to as high as 1.136 × 104 m3/h (50,000 gal/min) and heads up to 304 m (1000 ft). Such units are available in iron, bronze, and iron with bronze fittings. Other materials increase the cost; when they are required, a standard chemical pump is usually more economical. Close-Coupled Pumps (Fig. 10-49) Pumps equipped with a builtin electric motor or sometimes steam-turbine-driven (i.e., with pump

10-36

TRANSPORT AND STORAGE OF FLUIDS Pump 1.5 × 1 × 6 E 731 Plus

Curve A-8475-1

Speed 3525 RPM

Impeller data Imp. no.

175

6.0′′

Total head in feet

150

45

5.0′′

100

4.5′′

75

4.0′′

50

55

58

61

63

Max. Inlet sphere 0.28′′ area

63

6

2.7σ

65 68

55 50

3.5′′

50

No. of vanes

Max. dia. 6.0′′ Min. dia. 3.5′′

5.5′′

125

P-3708

A

B C

25

E

F

3.0 4.0

10

6.0

8.0

10.0

D

NPSI Req’d (ft)

BHP

A B

5

C F

0

0

20

40

60

E

D

120 100 80 U.S. gallons per minute

140

160

FIG. 10-41 Typical pump performance curve. The curve is shown for water at 85°F. If the specific gravity of the fluid is other than unity, BHP must be corrected.

Capacity

Q1 N1 ᎏ=ᎏ Q2 N2

Q1 D1 ᎏ=ᎏ Q2 D2

Head

H1 (N1)2 ᎏ = ᎏ2 H2 (N2)

h1 (D1)2 ᎏ = ᎏ2 h2 (D2)

Brake horsepower

BHP1 (N1)3 ᎏ = ᎏ3 BHP2 (N2)

BHP1 (P1)3 ᎏ = ᎏ3 BHP2 (P2)

n ti o fri c

Constant impeller speed

m

Constant impeller diameter

B Flow

ste

The Affinity Laws

A

Sy

TABLE 10-13

difficult-to-handle liquids. They have the following advantages: the liquid level is above the impeller, and the pump is thus self-priming; and the shaft seal is above the liquid level and is not wetted by the pumped liquid, which simplifies the sealing task. When no bottom connections are permitted on the tank (a safety consideration for highly corrosive or toxic liquid), the vertical wet-pit pump may be the only logical choice. These pumps have the following disadvantages: intermediate or line bearings are generally required when the shaft length exceeds about 3 m (10 ft) in order to avoid shaft resonance problems; these

Total head, feet

impeller and driver on the same shaft) are known as close-coupled pumps. Such units are extremely compact and are suitable for a variety of services for which standard iron and bronze materials are satisfactory. They are available in capacities up to about 450 m3/h (2000 gal/min) for heads up to about 73 m (240 ft). Two-stage units in the smaller sizes are available for heads to around 150 m (500 ft). Canned-Motor Pumps (Fig. 10-50) These pumps command considerable attention in the chemical industry. They are close-coupled units in which the cavity housing the motor rotor and the pump casing are interconnected. As a result, the motor bearings run in the process liquid and all seals are eliminated. Because the process liquid is the bearing lubricant, abrasive solids cannot be tolerated. Standard single-stage canned-motor pumps are available for flows up to 160 m3/h (700 gal/min) and heads up to 76 m (250 ft). Two-stage units are available for heads up to 183 m (600 ft). Canned-motor pumps are being widely used for handling organic solvents, organic heat-transfer liquids, and light oils as well as many clean toxic or hazardous liquids or for installations in which leakage is an economic problem. Vertical Pumps In the chemical industry, the term vertical process pump (Fig. 10-51) generally applies to a pump with a vertical shaft having a length from drive end to impeller of approximately 1 m (3.1 ft) minimum to 20 m (66 ft) or more. Vertical pumps are used as either wet-pit pumps (immersed) or dry-pit pumps (externally mounted) in conjunction with stationary or mobile tanks containing

Capacity, GPM FIG. 10-42

Variation of total head versus flow rate to overcome friction.

PUMPS AND COMPRESSORS TABLE 10-14

B

10-37

Specific Speeds of Different Types of Pumps

Pump type

Specific speed range

Below 2,000 2,000–5,000 4,000–10,000 9,000–15,000

Process pumps and feed pumps Turbine pumps Mixed-flow pumps Axial-flow pumps

A Flow

he

ad

cu

Total head, feet

rv e

bearings must be lubricated whenever the shaft is rotating. Since all wetted parts must be corrosion-resistant, low-cost materials may not be suitable for the shaft, column, etc. Maintenance is more costly since the pumps are larger and more difficult to handle. For abrasive service, vertical cantilever designs requiring no line or foot bearings are available. Generally, these pumps are limited to about a 1-m (3.1-ft) maximum shaft length. Vertical pumps are also used to pump waters to reservoirs. One such application in the Los Angeles water basin has fourteen 4-stage pumps, each pump requiring 80,000 hp to drive them. Sump Pumps These are small single-stage vertical pumps used to drain shallow pits or sumps. They are of the same general construction as vertical process pumps but are not designed for severe operating conditions. Multistage Centrifugal Pumps These pumps are used for services requiring heads (pressures) higher than can be generated by a single impeller. All impellers are in series, the liquid passing from one impeller to the next and finally to the pump discharge. The total head then is the summation of the heads of the individual impellers. Deepwell pumps, high-pressure water-supply pumps, boiler-feed pumps, fire pumps, and charge pumps for refinery processes are examples of multistage pumps required for various services. Multistage pumps may be of the volute type (Fig. 10-52), with single- or double-suction impellers (Fig. 10-53), or of the diffuser type (Fig. 10-54). They may have horizontally split casings or, for extremely high pressures, 20 to 40 MPa (3000 to 6000 lbf/in2), vertically split barrel-type exterior casings with inner casings containing diffusers, interstage passages, etc.

s Sy

te

System friction

m

Static difference FIG. 10-43 Variation of total head as a function of flow rate to overcome both friction and static head.

Valve partially closed Valve open

Head

Pump head capacity curve

2

hea S y ste m

d cu

1

rv e

S1 Q2

Q1

Frictional resistance

PROPELLER AND TURBINE PUMPS Axial-Flow (Propeller) Pumps (Fig. 10-55) These pumps are essentially very-high-capacity low-head units. Normally they are designed for flows in excess of 450 m3/h (2000 gal/min) against heads of 15 m (50 ft) or less. They are used to great advantage in closed-loop circulation systems in which the pump casing becomes merely an

Total static head

Capacity

Typical steady-state response of a pump system with a valve fully and partially open.

FIG. 10-44

20000

15000

9000 10000

8000

7000

6000

5000

4000

3000

2000

1500

1000

900

800

700

600

500

Values of specific speed, Ns

D2

D2 Radial-vane area D2 >2 D1

Francis-flow area

Mixed-flow area

Axial-flow area

D2 = 1.5 to 2 D1

D2 < 1.5 D1

D2 <1 D1

FIG. 10-45 Specific speed variations of different types of pump.

Axis of rotation

10-38

TRANSPORT AND STORAGE OF FLUIDS

50 60 80 100 400

150 200

Capacity, GPM 300 400 600 800 1000

300

17 14 70 r 50 p 11 rp m m 97 50 r p 0r pm m

Head, ft of liquid

200 150

2000 4000 35 29 50 r p 50 rp m m

100 80 60 50 40 30 Specific speed

400

600 800 1000

2000

4000

Typical impeller proportion as a function of specific speed above

Relationships between specific speed, rotative speed, and impeller proportions (Worthington Pump Inc., Pump World, vol. 4, no. 2, 1978).

FIG. 10-46

elbow in the line. A common installation is for calandria circulation. A characteristic curve of an axial-flow pump is given in Fig. 10-56. Turbine Pumps The term “turbine pump” is applied to units with mixed-flow (part axial and part centrifugal) impellers. Such units are available in capacities from 20 m3/h (100 gal/min) upward for heads up to about 30 m (100 ft) per stage. Turbine pumps are usually vertical. A common form of turbine pump is the vertical pump, which has the pump element mounted at the bottom of a column that serves as the discharge pipe (see Fig. 10-57). Such units are immersed in the liquid to be pumped and are commonly used for wells, condenser circulating water, large-volume drainage, etc. Another form of the pump has a shell surrounding the pumping element which is connected to the intake pipe. In this form, the pump is used on condensate service in power plants and for process work in oil refineries.

FIG. 10-47 Horizontal process pump conforming to American Society of Mechanical Engineers Standards B73.1—2001.

FIG. 10-48 Vertical in-line process pump conforming to ASME standard B73.2—2003. The pump shown is driven by a motor through flexible coupling. Not shown but also conforming to ASME Standard B73.2—2003 are vertical inline pumps with rigid couplings and with no coupling (impeller-mounted on an extended motor shaft).

Regenerative Pumps Also referred to as turbine pumps because of the shape of the impeller, regenerative pumps employ a combination of mechanical impulse and centrifugal force to produce heads of several hundred meters (feet) at low volumes, usually less than 20 m3/h (100 gal/min). The impeller, which rotates at high speed with small clearances, has many short radial passages milled on each side at the periphery. Similar channels are milled in the mating surfaces of the casing. Upon entering, the liquid is directed into the impeller passages and proceeds in a spiral pattern around the periphery, passing alternately from the impeller to the casing and receiving successive impulses as it does so. Figure 10-58 illustrates a typical performance-characteristic curve.

FIG. 10-49

Close-coupled pump.

FIG. 10-50

Canned-motor pump. (Courtesy of Chempump Division, Crane Co.)

PUMPS AND COMPRESSORS

10-39

Ejectors and injectors are the two types of jet pumps of interest to chemical engineers. The ejector, also called the siphon, exhauster, or eductor, is designed for use in operations in which the head pumped against is low and is less than the head of the fluid used for pumping. The injector is a special type of jet pump, operated by steam and used for boiler feed and similar services, in which the fluid being pumped is discharged into a space under the same pressure as that of the steam being used to operate the injector. Figure 10-59 shows a simple design for a jet pump of the ejector type. The pumping fluid enters through the nozzle at the left and passes through the venturi nozzle at the center and out of the discharge opening at the right. As it passes into the venturi nozzle, it develops a suction that causes some of the fluid in the suction chamber to be entrained with the stream and delivered through this discharge. The efficiency of an ejector or jet pump is low, being only a few percent. The head developed by the ejector is also low except in special types. The device has the disadvantage of diluting the fluid pumped by mixing it with the pumping fluid. In steam injectors for boiler feed and similar services in which the heat of the steam is recovered, efficiency is close to 100 percent. The simple ejector or siphon is widely used, in spite of its low efficiency, for transferring liquids from one tank to another, for lifting acids, alkalies, or solid-containing liquids of an abrasive nature, and for emptying sumps. ELECTROMAGNETIC PUMPS Vertical process pump for dry-pit mounting. (Courtesy of Lawrence Pumps, Inc.) FIG. 10-51

These pumps are particularly useful when low volumes of lowviscosity liquids must be handled at higher pressures than are normally available with centrifugal pumps. Close clearances limit their use to clean liquids. For very high heads, multistage units are available. JET PUMPS Jet pumps are a class of liquid-handling device that makes use of the momentum of one fluid to move another.

FIG. 10-52

Six-stage volute-type pump.

The necessity of circulating liquid-metal heat-transfer media in nuclear-reactor systems has led to development of electromagnetic pumps. All electromagnetic pumps utilize the motor principle: a conductor in a magnetic field, carrying a current which flows at right angles to the direction of the field, has a force exerted on it, the force being mutually perpendicular to both the field and the current. In all electromagnetic pumps, the fluid is the conductor. This force, suitably directed in the fluid, manifests itself as a pressure if the fluid is suitably contained. The field and current can be produced in a number of different ways and the force utilized variously. Both alternating- and direct-current units are available. While dc pumps (Fig. 10-60) are simpler, their high-current requirement is a definite limitation; ac pumps can readily obtain high currents by making use of transformers. Multipole induction ac pumps have been

10-40

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-53

Two-stage pump having double-suction impellers.

built in helical and linear configurations. Helical units are effective for relatively high heads and low flows, while linear induction pumps are best suited to large flows at moderate heads. Electromagnetic pumps are available for flow rates up to 2.271 × 103 m3/h (10,000 gal/min), and pressures up to 2 MPa (300 lbf/in2) are practical. Performance characteristics resemble those of centrifugal pumps. PUMP DIAGNOSTICS Pump problems vary over a large range depending on the type of pumps and the usage of the pumps. They can be classified in the following manner by the pump type and the service: 1. Positive-displacement pumps—reciprocating pumps problems can be classified into the following categories: a. Compressor valve problems: plate valves, feather valves, concentric disk valves, relief valves b. Piston and rod assembly: piston rings, cylinder chatter, cylinder cooling, piston-rod packing c. Lubrication system 2. Positive displacement pumps—gear-type and roots-type problems can be classified into the following categories: a. Rotor dynamic problems: vibration problems, gear problems or roots rotor problems, bearing and seal problems b. Lubrication systems

FIG. 10-54

Seven-stage diffuser-type pump.

3. Continuous flow pumps such as centrifugal pumps problems can be classified into the following categories: a. Cavitation b. Capacity flow c. Motor overload d. Impeller e. Bearings and seals f. Lubrication systems Table 10-15 classifies different types of centrifugal pump-related problems, their possible causes, and corrective actions that can be taken to solve some of the more common issues. These problems in the table are classified into three major categories for these type of pumps: 1. Cavitation 2. Capacity flow 3. Motor overload The use of vibration monitoring to diagnose pump and compressor problems is discussed at the end of the subsection on compressor problems. COMPRESSORS A compressor is a device which pressurizes a working fluid. One of the basic purposes of using a compressor is to compress the fluid and to

PUMPS AND COMPRESSORS

FIG. 10-55

10-41

Axial-flow elbow-type propeller pump. (Courtesy of Lawrence

Pumps, Inc.)

deliver it at a pressure higher than its original pressure. Compression is required for a variety of purposes, some of which are listed below: 1. To provide air for combustion 2. To transport process fluid through pipelines 3. To provide compressed air for driving pneumatic tools 4. To circulate process fluid within a process Different types of compressors are shown in Fig. 10-61. Positivedisplacement compressors are used for intermittent flow in which successive volumes of fluid are confined in a closed space to increase their pressures. Rotary compressors provide continuous flow. In rotary compressors, rapidly rotating parts (impellers) accelerate fluid to a high speed; this velocity is then converted to additional pressure by gradual deceleration in the diffuser or volute which surrounds the impeller. Positive-displacement compressors can be further classified as either reciprocating or rotary type, as shown in Fig. 10-61. The reciprocating compressor has a piston having a reciprocating motion within a cylinder. The rotary positive-displacement compressors have rotating elements whose positive action results in compression and displacement. The rotary positive-displacement compressors can be further subdivided into sliding vane, liquid piston, straight lobe, and helical lobe compressors. The continuous flow compressors (Fig. 10-61) can be classified as either dynamic compressors or ejectors. Ejectors entrain the in-flowing fluid by using a high-velocity gas or steam jet and then convert the velocity of the mixture to pressure in a diffuser. The dynamic compressors have rotating elements, which accelerate the inflowing fluid, and convert the velocity head to pressure head, partially in the rotating elements and partially in the stationary diffusers or blade. The dynamic compressors can be further subdivided into centrifugal, axial-flow, and mixed-flow compressors. The main flow of gas in the centrifugal compressor is radial. The flow of gas in an axial com-

Characteristic curve of an axial-flow pump. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert horsepower to kilowatts, multiply by 0.746.

FIG. 10-56

FIG. 10-57

Vertical multistage turbine, or mixed-flow, pump.

pressor is axial, and the mixed-flow compressor combines some characteristics of both centrifugal and axial compressors. It is not always obvious what type of compressor is needed for an application. Of the many types of compressors used in the process industries, some of the more significant are the centrifugal, axial, rotary, and reciprocating compressors. They fall into three categories, as shown in Fig. 10-62. For very high flows and low pressure ratios, an axial-flow compressor would be best. Axial-flow compressors usually have a higher efficiency, as seen in Fig. 10-63, but a smaller operating region than does

FIG. 10-58 Characteristic curves of a regenerative pump. To convert gallons per minute to cubic meters per hour, multiply by 0.2271; to convert feet to meters, multiply by 0.3048; and to convert horsepower to kilowatts, multiply by 0.746.

10-42

FIG. 10-59

TRANSPORT AND STORAGE OF FLUIDS

Simple ejector using a liquid-motivating fluid.

a centrifugal machine. Centrifugal compressors operate most efficiently at medium flow rates and high pressure ratios. Rotary and reciprocating compressors (positive-displacement machines) are best used for low flow rates and high pressure ratios. The positivedisplacement compressors and, in particular, reciprocating compressors were the most widely used in the process and pipeline industries up to and through the 1960s. In turbomachinery the centrifugal flow and the axial-flow compressors are the ones used for compressing gases. Positive-displacement compressors such as reciprocating, gear-type, or lobe-type are widely used in the industry for many other applications such as slurry pumping. The performance characteristics of a single stage of the three main types of compressors are given in Table 10-16. The pressure ratios of the axial and centrifugal compressors have been classified into three groups: industrial, aerospace, and research. The industrial pressure ratio is low because the operating range needs to be large. The operating range is defined as the range between the surge point and the choke point. The surge point is the point at which the flow is reversed in the compressor. The choke point is the point at which the flow has reached Mach = 1.0, the point where no more flow can get through the unit, a “stone wall.” When surge occurs, the flow is reversed, and so are all the forces acting on the compressor, especially the thrust forces. Surge can lead to total destruction of the compressor. Thus surge is a region that must be avoided. Choke conditions cause a large drop in efficiency, but do not lead to destruction of the unit. Note that with the increase in pressure ratio and the number of stages, the operating range is narrowed in axial-flow and centrifugal compressors.

Compressor Selection To select the most satisfactory compression equipment, engineers must consider a wide variety of types, each of which offers peculiar advantages for particular applications. Among the major factors to be considered are flow rate, head or pressure, temperature limitations, method of sealing, method of lubrication, power consumption, serviceability, and cost. To be able to decide which compressor best fits the job, the engineer must analyze the flow characteristics of the units. The following dimensionless numbers describe the flow characteristics. Reynolds number is the ratio of the inertia forces to the viscous forces ρVD NRe = ᎏ (10-63) µ where ρ is the density of the gas, V is the velocity of the gas, D is the diameter of the impeller, and µ is the viscosity of the gas. The specific speed compares the adiabatic head and flow rate in geometrically similar machines at various speeds. 苶 N兹Q Ns = ᎏ (10-64) 3/4 Had where N is the speed of rotation of the compressor, Q is the volume flow rate, and H is the adiabatic head. The specific diameter compares head and flow rates in geometrically similar machines at various diameters DH1/4 DS = ᎏ (10-65) 苶 兹Q The flow coefficient is the capacity of the flow rate of the machine Q1 φ = ᎏ3 (10-66) ND The pressure coefficient is the pressure or the pressure rise of the machine H Ψ=ᎏ (10-67) N 2D2 In selecting the machines of choice, the use of specific speed and diameter best describe the flow. Figure 10-64 shows the characteristics of the three types of compressors. Other considerations in chemical plant service such as problems with gases which may be corrosive or have abrasive solids in suspension must be dealt with. Gases at elevated temperatures may create a potential explosion hazard, while air at the same temperatures may be handled quite normally; minute amounts of lubricating oil or water may contaminate the process gas and so may not be permissible, and for continuous-process use, a high degree of equipment reliability is required, since frequent shutdowns for inspection or maintenance cannot be tolerated. COMPRESSION OF GASES Theory of Compression In any continuous compression process the relation of absolute pressure p to volume V is expressed by the formula pV n = C = constant (10-68) The plot of pressure versus volume for each value of exponent n is known as the polytropic curve. Since the work W performed in proceeding from p1 to p2 along any polytropic curve (Fig. 10-65) is

冕 p dV 2

W=

FIG. 10-60

pump.

Simplified diagram of a direct-current-operated electromagnetic

(10-69)

1

it follows that the amount of work required is dependent upon the polytropic curve involved and increases with increasing values of n. The path requiring the least amount of input work is n = 1, which is equivalent to isothermal compression. For adiabatic compression (i.e., no heat is being added or taken away during the process), n = k = ratio of specific heat at constant pressure to that at constant volume.

PUMPS AND COMPRESSORS TABLE 10-15

10-43

Pump Problems Possible causes

Corrective action Cavitating-Type Problems

Plugged suction screen. Piping gaskets with undersized IDs installed, a very common problem in small pumps. Column tray parts or ceramic packing lodged in the impeller eye. Deteriorated impeller eye due to corrosion. Flow rate is high enough above design that NPSH for flow rate has increased above NPSH. Lined pipe collapsed at gasket area or ID due to buildup of corrosion products between liner and carbon-steel pipe. Poor suction piping layout, too many elbows in too many planes, a tee branch almost directly feeding the suction of the other pump, or not enough straight run before the suction flange of the pump.

Check for indications of the presence of screen. Remove and clean screen. Install proper-sized gaskets.

Vertical pumps experience a vortex formation due to loss of submergence required by the pump. Observe the suction surface while the pump is in operation, if possible. Spare pump begins to cavitate when attempt is made to switch it with the running pump. The spare is “backed off” by the running pump because its shutoff head is less than the head produced by the running pump. This is a frequent problem when one pump is turbine-driven and one is motor-driven. Suction piping configuration causes adverse fluid rotation when approaching impeller. Velocity of the liquid is too high as it approaches the impeller eye. Pump is operating at a low-flow-producing suction recirculation in the impeller eye. This results in a cavitationlike sound.

Review causes of vortexing. Consider installation of a vortex breaker such as a bell mouth umbrella or changes to sump design.

Remove suction piping and debris. Replace impeller and overhaul pump. Reduce flow rate to that of design. Replace deteriorated piping. Redesign piping layout, using fewer elbows and laterals for tees, and have five or more straight pipe diameters before suction flange.

Throttle discharge of running pump until spare can get in system. Slow down running pump if it is a turbine or variable-speed motor.

Install sufficient straight run of suction piping, or install vanes in piping to break up prerotation. Install larger suction piping or reduce flow through pump. Install bypass piping back to suction vessel to increase flow through pump. Remember bypass flow may have to be as high as 50 percent of design flow.

Capacity-Type Problems Check the discharge block valve opening first. It may be partially closed and thus the problem. Wear-ring clearances are excessive (closed impeller design). Impeller-to-case or head clearances are excessive (open impeller design). Air leaks into the system if the pump suction is below atmospheric pressure. Increase in piping friction to the discharge vessel due to the following: 1. Gate has fallen off the discharge valve stem. 2. Spring is broken in the spring-type check valve. 3. Check valve flapper pin is worn, and flapper will not swing open. 4. Lined pipe collapsing. 5. Control valve stroke improperly set, causing too much pressure drop. Suction and/or discharge vessel levels are not correct, a problem mostly seen in lower-speed pumps. Motor running backward or impeller of double suction design is mounted backward. Discharge pressure developed in both cases is about one-half design value. Entrained gas from the process lowering NPSH available. Polymer or scale buildup in discharge nozzle areas. Mechanical seal in suction system under vacuum is leaking air into system, causing pump curve to drop. The pump may have formed a vortex at high flow rates or low liquid level. Does the vessel have a vortex breaker? Does the incoming flow cause the surface to swirl or be agitated? Variable-speed motor running too slowly. Bypassing is occurring between volute channels in a double-volute pump casing due to a casting defect or extreme erosion. The positions of impellers are not centered with diffuser vanes. Several impellers will cause vibration and lower head output. When the suction system is under vacuum, the spare pump has difficulty getting into system. Certain pump designs use an internal bypass orifice port to alter head-flow curve. High liquid velocities often erode the orifice, causing the pump to go farther out on the pump curve. The system head curve increase corrects the flow back up the curve. Replacement impeller is not correct casting pattern; therefore NPSH required is different. Volute and cutwater area of casing is severely eroded.

Open block valve completely. Overhaul pump. Renew wear rings if clearance is about twice design value for energy and performance reasons. Reposition impeller to obtain correct clearance. Take actions as needed to eliminate air leaks. Take the following actions: 1. Repair or replace gate valve. 2. Repair valve by replacing spring. 3. Overhaul check valve; restore proper clearance to pin and flapper bore. 4. Replace damaged pipe. 5. Adjust control valve stroke as necessary. Calibrate level controllers as necessary. Check for proper rotation and mounting of impeller. Reverse motor leads if necessary. Reduce entrained gas in liquid by process changes as needed. Shut down pump and remove scale or deposits. Change percentage balance of seal faces or increase spring tension. Reduce flow to design rates. Raise liquid level in suction vessel. Install vortex breaker in suction vessel. Adjust motor speed as needed. Overhaul pump; repair eroded area. Overhaul pump; reposition individual impellers as needed. Reposition whole rotor by changing thrust collar locator spacer. Install a positive-pressure steam (from running pump) to fill the suction line from the block valve through the check valve. Overhaul pump, restore orifice to correct size.

Overhaul pump, replace impeller with correct pattern. Overhaul pump; replace casing or repair by welding. Stress-relieve after welding as needed.

10-44

TRANSPORT AND STORAGE OF FLUIDS

TABLE 10-15

Pump Problems (Concluded) Possible causes

Corrective action Overload Problems

Polymer buildup between wear surfaces (rings or vanes). Excessive wear ring (closed impeller) or cover-case clearance (open impeller). Pump circulating excessive liquid back to suction through a breakdown bushing or a diffuser gasket area. Minimum-flow loop left open at normal rates, or bypass around control valve is open. Discharge piping leaking under liquid level in sump-type design. Electrical switch gear problems cause one phase to have low amperage. Specific gravity is higher than design specification.

Pump motor not sized for end of curve operation. Open impeller has slight rub on casing. Most often occurs in operations from 250 to 400°F due to piping strain and differential growth in the pump. A replacement impeller was not trimmed to the correct diameter.

Remove buildup to restore clearances. Replace wear rings or adjust axial clearance of open impeller. In severe cases, cover or case must be replaced. Overhaul pump, replacing parts as needed. Close minimum-flow loop or control valve bypass valve. Inspect piping for leakage. Replace as needed. Check out switch gear and repair as necessary. Change process to adjust specific gravity to design value, or throttle pump to reduce horsepower requirements. This will not correct problem with some vertical turbine pumps that have a flat horsepower-required curve. Replace motor with one of larger size, or reduce flow rate. Increase clearance of impeller to casing. Remove impeller from pump and turn to correct diameter.

COMPRESSOR

POSITIVE DISPLACEMENT Intermittent Flow RECIPROCATING

SLIDING VANE FIG. 10-61

CONTINUOUS FLOW

ROTARY

LIQUID PISTON

EJECTOR

STRAIGHT LOBE

HELICAL

DYNAMIC

CENTRIFUGAL

AXIAL

MIXED FLOW

Principal types of compressors.

Since most compressors operate along a polytropic path approaching the adiabatic, compressor calculations are generally based on the adiabatic curve. Some formulas based upon the adiabatic equation and useful in compressor work are as follows: Pressure, volume, and temperature relations for perfect gases: p2 /p1 = (V1 /V2)k (10-70) T2 /T1 = (V1 /V2)

(10-71)

p2 /p1 = (T2 /T1)k / (k − 1)

(10-72)

Adiabatic Calculations Adiabatic head is expressed as follows: In SI units, k × ZRT p2 (k − 1) / k Had = ᎏ1 −1 (10-73a) ᎏ k−1 p1 where Had = adiabatic head, N⋅m/kg; R = gas constant, J/(kg⋅K) = 8314/ molecular weight; T1 = inlet gas temperature, K; p1 = absolute inlet pressure, kPa; and p2 = absolute discharge pressure, kPa. In U.S. customary units,

冤冢 冣

k Had = ᎏ ZRT1 k−1

冥

(k − 1) / k

冤冢ᎏpp 冣 2 1

−1

冥

(10-73b)

where Had = adiabatic head, ft⋅lbf/lbm; R = gas constant, (ft⋅lbf)/ (lbm⋅°R) = 1545/molecular weight; T1 = inlet gas temperature, °R;

Positive Displacement Centrifugal Compressor

20 Pressure Ratio

k−1

30

10

Axial Flow Compressor

1 10

103

104

105

106

Flow, ft3/m FIG. 10-62

Performance characteristics of different types of compressors.

PUMPS AND COMPRESSORS Adiabatic efficiency is given by the following relationship:

1

ideal work ηad = ᎏᎏ actual work

Adiabatic Efficiency

0.9 0.8

T2 − T1 ηad = ᎏ T2a − T1

Centrifugal Rotary Positive Displacement

0.6 0.5

10

100

1000

ηad

0.5 0.75 Specific Speed Ns = NQ /H

p1 = absolute inlet pressure, lbf/in2; and p2 = absolute discharge pressure, lbf/in2. The work expended on the gas during compression is equal to the product of the adiabatic head and the mass flow of gas handled. Therefore, the adiabatic power is as follows: In SI units, WHad k × ZWRT1 kWad = ᎏ = ᎏᎏ 103 k−1

(k − 1) / k

冤冢 冣

k kWad = 2.78 × 10−4 ᎏ Q1p1 k−1

p2 ᎏ p1

−1

(k − 1) / k

冤冢 冣 p2 ᎏ p1

冥 冥

(10-75a)

where kWad = power, kW; W = mass flow, kg/s; and Q1 = volume rate of gas flow, m3/h, at compressor inlet conditions. In U.S. Customary units, WHad k ZWRT1 hpad = ᎏ = ᎏ ᎏ 550 k − 1 550 Q1P1 k hpad = ᎏ ᎏ 3600 k−1

1

(k − 1) / k

冤冢 冣 p2 ᎏ p1

(k − 1) / k

冤冢ᎏp 冣 p2

冥

−1

−1

冥

(10-74b)

(10-79)

(n − 1 )/ n

Polytropic head can be expressed by the following relationship: n Hpoly = ᎏ ZRT1 n−1

(n − 1 )/ n

冦冤冢ᎏPP 冣 2

−1

1

(10-75b)

(k − 1)/k η pc = ᎏ (n − 1)/n

(10-80)

(10-81)

Polytropic efficiency is the limited value of the isentropic efficiency as the pressure ratio approaches 1.0, and the value of the polytropic efficiency is higher than the corresponding adiabatic efficiency. A characteristic of polytropic efficiency is that the polytropic efficiency of a multistage unit is equal to the stage efficiency if each stage has the same efficiency. If the compression cycle approaches the isothermal condition, pV = constant, as is the case when several stages with intercoolers are used, a simple approximation of the power is obtained from the following formula: In SI units, (10-82a)

In U.S. customary units, hp = 4.4 × 10−3Q1p1 ln p2 /p1

(10-82b)

Reciprocating Compressors Reciprocating compressors are used mainly when high-pressure head is required at a low flow. Reciprocating compressors are furnished in either single-stage or multistage types. The number of stages is determined by the required compressor ratio p2/p1. The compression ratio per stage is generally limited to 4, although low-capacity units are furnished with compression ratios of 8 and even higher. Generally, the maximum compression ratio is determined by the maximum allowable discharge-gas temperature.

Performance Characteristics of Compressors Pressure ratio per stage

Types of compressors

Industrial

Positive displacement

Up to 30

—

—

75–82

—

Centrifugal

1.2–1.9

2.0–7.0

13

75–87 25

Large

Axial

1.05–1.3

1.1–1.45

2.1

80–91

Narrow 3–10

Aerospace

冥冧

Likewise for polytropic efficiency, which is often considered as the small stage efficiency, or the hydraulic efficiency:

kW = 2.78 × 10−4Q1p1 ln p2 /p1

where hpad = power, hp; W = mass flow, lb/s; and Q1 = volume rate of gas flow, ft3/min. Adiabatic discharge temperature is T2 = T1(p2 /p1)(k − 1) / k (10-76) The work in a compressor under ideal conditions as previously shown occurs at constant entropy. The actual process is a polytropic process as shown in Fig. 10-65 and given by the equation of state Pvn = constant.

TABLE 10-16

1

1

(10-74a)

−1

(k − 1 )/ k

冢ᎏPPᎏ冣 − 1 = ᎏᎏ 冢ᎏPPᎏ冣 − 1 2

FIG. 10-63 Variation of adiabatic efficiency with specific speed for the three types of compressors.

or

(10-78)

where T2a is the total actual discharge temperature of the gas. The adiabatic efficiency can be represented in terms of total pressure change: 2

or

(10-77)

In terms of the change in total temperatures the relationship can be written as:

Axial

0.7

0.4

10-45

Research

Efficiency, %

Operating range surge − choke, %

10-46

TRANSPORT AND STORAGE OF FLUIDS

Compressor coverage chart based on the normal range of operation of commercially available types shown. Solid lines: use left ordinate, head. Broken lines: use right ordinate, pressure. To convert cubic feet per minute to cubic meters per hour, multiply by 1.699; to convert feet to meters, multiply by 0.3048; and to convert pounds-force per square inch to kilopascals, multiply by 6.895; (°F − 32)5⁄9 = °C. FIG. 10-64

Single-acting air-cooled and water-cooled air compressors are available in sizes up to about 75 kW (100 hp). Such units are available in one, two, three, or four stages for pressure as high as 24 MPa (3500 lbf/in2). These machines are seldom used for gas compression because of the difficulty of preventing gas leakage and contamination of the lubricating oil. The compressors most commonly used for compressing gases have a crosshead to which the connecting rod and piston rod are connected. This provides a straight-line motion for the piston rod and permits simple packing to be used. Figure 10-66 illustrates a simple single-stage machine of this type having a double-acting piston. Either single-acting (Fig. 10-67) or double-acting pistons (Fig. 10-68) may be used, depending on the size of the machine and the number of stages. In some machines double-acting pistons are used in the first stages and single-acting in the later stages. On multistage machines, intercoolers are provided between stages. These heat exchangers remove the heat of compression from the gas and reduce its temperature to approximately the temperature existing at the compressor intake. Such cooling reduces the volume of gas going to the high-pressure cylinders, reduces the power required for compression, and keeps the temperature within safe operating limits. Figure 10-69 illustrates a two-stage compressor end such as might be used on the compressor illustrated in Fig. 10-66. Compressors with horizontal cylinders such as illustrated in Figs. 10-66 to 10-68 are most commonly used because of their accessibility. However, machines are also built with vertical cylinders and other arrangements such as right-angle (one horizontal and one vertical cylinder) and V-angle.

Fig. 10-65

Polytropic compression curves.

Compressors up to around 75 kW (100 hp) usually have a single center-throw crank, as illustrated in Fig. 10-66. In larger sizes compressors are commonly of duplex construction with cranks on each end of the shaft (see Fig. 10-70). Some large synchronous motordriven units are of four-corner construction; i.e., they are of doubleduplex construction with two connecting rods from each of the two crank throws (see Fig. 10-71). Steam-driven compressors have one or more steam cylinders connected directly by piston rod or tie rods to the gas-cylinder piston or crosshead. Valve Losses Above piston speeds of 2.5 m/s (500 ft/min), suction and discharge valve losses begin to exert significant effects on the actual internal compression ratio of most compressors, depending on the valve port area available. The obvious results are high temperature rise and higher power requirements than might be expected. These effects become more pronounced with higher-molecular-weight gases. Valve problems can be a very major contributor to down time experienced by these machines. Control Devices In many installations the use of gas is intermittent, and some means of controlling the output of the compressor is therefore necessary. In other cases constant output is required despite variations in discharge pressure, and the control device must operate to maintain a constant compressor speed. Compressor capacity, speed, or pressure may be varied in accordance with requirements. The nature of the control device will depend on the function to be regulated. Regulation of pressure, volume, temperature, or some other factor determines the type of regulation required and the type of the compressor driver. The most common control requirement is regulation of capacity. Many capacity controls, or unloading devices, as they are usually termed, are actuated by the pressure on the discharge side of the compressor. A falling pressure indicates that gas is being used faster than it is being compressed and that more gas is required. A rising pressure indicates that more gas is being compressed than is being used and that less gas is required. An obvious method of controlling the capacity of a compressor is to vary the speed. This method is applicable to units driven by variable-speed drivers such as steam pistons, steam turbines, gas engines, diesel engines, etc. In these cases the regulator actuates the steamadmission or fuel-admission valve on the compressor driver and thus controls the speed. Motor-driven compressors usually operate at constant speed, and other methods of controlling the capacity are necessary. On reciprocating compressors discharging into receivers, up to about 75 kW (100 hp), two types of control are usually available. These are automaticstart-and-stop control and constant-speed control. Automatic-start-and-stop control, as its name implies, stops or starts the compressor by means of a pressure-actuated switch as the

PUMPS AND COMPRESSORS

FIG. 10-66

Typical single-stage, double-acting water-cooled compressor.

gas demand varies. It should be used only when the demand for gas will be intermittent. Constant-speed control should be used when gas demand is fairly constant. With this type of control, the compressor runs continuously but compresses only when gas is needed. Three methods of unloading the compressor with this type of control are in common use: (1) closed suction unloaders, (2) open inlet-valve unloaders, and (3) clearance unloaders. The closed suction unloader consists of a pressure-actuated valve which shuts off the compressor intake. Open inlet-valve unloaders (see Fig. 10-72) operate to hold the compressor inlet valves open and thereby prevent compression. Clearance unloaders (see Fig. 10-73) consist of pockets or small reservoirs which are opened when unloading is desired. The gas is compressed into them on the compression stroke and reexpands into the cylinder on the return stroke, thus preventing the compression of additional gas. It is sometimes desirable to have a compressor equipped with both constant-speed and automatic-start-and-stop control. When this is done, a switch allows immediate selection of either type. Motor-driven reciprocating compressors above about 75 kW (100 hp) in size are usually equipped with a step control. This is in reality a variation of constant-speed control in which unloading is accomplished in a series of steps, varying from full load down to no load.

Three-step control (full load, one-half load, and no load) is usually accomplished with inlet-valve unloaders. Five-step control (full load, three-fourths load, one-half load, one-fourth load, and no load) is accomplished by means of clearance pockets (see Fig. 10-74). On some machines, inlet-valve and clearance-control unloading are used in combination. Although such control devices are usually automatically operated, manual operation is satisfactory for some services. When manual operation is provided, it often consists of a valve or valves to open and close clearance pockets. In some cases, a movable cylinder head is provided for variable clearance in the cylinder (see Fig. 10-75). When no capacity control or unloading device is provided, it is necessary to provide bypasses between the inlet and discharge in order that the compressor can be started against no load (see Fig. 10-76). Nonlubricated Cylinders Most compressors use oil to lubricate the cylinder. In some processes, however, the slightest oil contamination

FIG. 10-67 Two-stage single-acting opposed piston in a single step-type cylin-

der.

10-47

FIG. 10-68

Typical double-acting compressor piston and cylinder.

10-48

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-69 Two-stage double-acting compressor cylinders with intercooler.

is objectionable. For such cases a number of manufacturers furnish a “nonlubricated” cylinder (see Fig. 10-77). The piston on these cylinders is equipped with piston rings of graphitic carbon or Teflon* as well as pads or rings of the same material to maintain proper clearance between the piston and the cylinder. Plastic packing of a type that requires no lubricant is used on the stuffing box. Although oil-wiper rings are used on the piston rod where it leaves the compressor frame, minute quantities of oil might conceivably enter the cylinder on the rod. If even such small amounts of oil are objectionable, an extended cylinder connecting piece can be furnished. This simply lengthens the piston rod enough so that no portion of the rod can alternately enter the frame and the cylinder. In many cases, a small amount of gas leaking through the packing is objectionable. Special connecting pieces are furnished between the cylinder and the frame, which may be either single-compartment or double-compartment. These may be furnished gastight and *®Du Pont tetrafluoroethylene fluorocarbon resin.

FIG. 10-70

Duplex two-stage compressor (plan view).

vented back to the suction or filled with a sealing gas or fluid and held under a slight pressure. High-Pressure Compressors There is a definite trend in the chemical industry toward the use of high-pressure compressors with discharge pressures of from 34.5 to 172 MPa (5000 to 25,000 lbf/in2) and with capacities from 8.5 × 103 to 42.5 × 103 m3/h (5000 to 25,000 ft3/min). These require special design, and a complete knowledge of the characteristics of the gas is necessary. In most cases, these types of applications use the barrel-type centrifugal compressor. The gas usually deviates considerably from the perfect-gas laws, and in many cases temperature or other limitations necessitate a thorough engineering study of the problem. These compressors usually have five, six, seven, or eight stages, and the cylinders must be properly proportioned to meet the various limitations involved and also to balance the load among the various stages. In many cases, scrubbing or other processing is carried on between stages. High-pressure cylinders are steel forgings with single-acting plungers (see Fig. 10-78). The compressors are usually designed so that the pressure load against the plunger is opposed by one or more single-acting pistons of the lower pressure stages. Piston-rod packing is usually of the segmental-ring metallic type. Accurate fitting and correct lubrication are very important. High-pressure compressor valves are designed for the conditions involved. Extremely high-grade engineering and skill are necessary. Piston-Rod Packing Proper piston-rod packing is important. Many types are available, and the most suitable is determined by the gas handled and the operating conditions for a particular unit. There are many types and compositions of soft packing, semimetallic packing, and metallic packing. In many cases, metallic packing is to be recommended. A typical low-pressure packing arrangement is shown in Fig. 10-79. A high-pressure packing arrangement is shown in Fig. 10-80. When wet, volatile, or hazardous gases are handled or when the service is intermittent, an auxiliary packing gland and soft packing are usually employed (see Fig. 10-81). Metallic Diaphragm Compressors (Fig. 10-82) These are available for small quantities [up to about 17 m3/h (10 ft3/min)] for compression ratios as high as 10:1 per stage. Temperature rise is not a serious problem, as the large wall area relative to the gas volume permits sufficient heat transfer to approach isothermal compression. These compressors possess the advantage of having no seals for the process gas. The diaphragm is actuated hydraulically by a plunger pump.

PUMPS AND COMPRESSORS

10-49

FIG. 10-71 Four-corner four-stage compressor (plan view).

FANS AND BLOWERS Fans are used for low pressures where generally the delivery pressure is less than 3.447 kPa (0.5 lb/in2), and blowers are used for higher pressures. However, they are usually below delivery pressures of 10.32 kPa (1.5 lbf/in2). These units can either be centrifugal or the axial-flow type. Fans and blowers are used for many types of ventilating work such as air-conditioning systems. In large buildings, blowers are often used due to the high delivery pressures needed to overcome the pressure drop in the ventilation system. Most of these blowers are of the centrifugal type. Blowers are also used to supply draft air to boilers and furnaces. Fans are used to move large volumes of air or gas through ducts, supplying air for drying, conveying material suspended in the gas stream, removing fumes, condensing towers and other highflow, low-pressure applications. Axial-Flow Fans These are designed to handle very high flow rates and low pressure. The disc-type fans are similar to those of a household fan. They are usually for general circulation or exhaust work without ducts. The so-called propeller-type fans with blades that are aerodynamically designed (as seen in Fig. 10-83) can consist of two or more stages. The air in these fans enters in an axial direction and leaves in an axial direction. The fans usually have inlet guide vanes followed by a rotating blade, followed by a stationary (stator) blade. Centrifugal Blowers These blowers have air or gases entering in the axial direction and being discharged in the radial direction. These blowers have three types of blades: radial or straight blades, forward-curved blades, and backward-curved blades (Figs. 10-84 to 10-86). Radial blade blowers as seen in Fig. 10-84 are usually used in largediameter or high-temperature applications. The blades being radial in direction have very low stresses as compared to the backward or forward

FIG. 10-72

Inlet-valve unloader.

FIG. 10-73

Clearance-control cylinder. (Courtesy of Ingersoll-Rand.)

FIG. 10-74 Actual indicator diagram of a two-stage compressor showing the operation of clearance control at five load points.

10-50

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-75 Sectional view of a cylinder equipped with a hand-operated valve lifter on one end and a variable-volume clearance pocket at other end.

curve blades. The rotors have anywhere between 4 to 12 blades and usually operate at low speeds. These fans are used in exhaust work especially for gases at high temperature and with suspensions in the flow stream. Forward-Curved Blade Blowers These blowers discharge the gas at a very high velocity. The pressure supplied by this blower is lower than that produced in the other two blade characteristics. The number of blades in such a rotor can be large—up to 50 blades—and the speed is high—usually 3600–1800 rpm in 60-cycle countries and 3000–1500 rpm in 50-cycle countries. Backward-Curved Blade Blowers These blowers are used when a higher discharge pressure is needed. It is used over a wide range of applications. Both the forward and backward curved blades do have much higher stresses than the radial bladed blower.

Piston equipped with carbon piston and wearing rings for a nonlubricated cylinder.

FIG. 10-77

Bypass arrangement for a single-stage compressor. On multistage machines, each stage is bypassed in a similar manner. Such an arrangement is necessary for no-load starting.

FIG. 10-76

FIG. 10-78

Forged-steel single-acting high-pressure cylinder.

PUMPS AND COMPRESSORS

FIG. 10-79

10-51

Typical packing arrangements for low-pressure cylinders.

High-pressure, low-capacity compressor having a hydraulically actuated diaphragm. (Pressure Products Industries.)

FIG. 10-82

FIG. 10-80 Typical packing arrangement, using metallic packing, for highpressure cylinders.

FIG. 10-83

FIG. 10-81

Two-stage axial-flow fan.

Soft packing in an auxiliary stuffing box for handling gases.

The centrifugal blower produces energy in the air stream by the centrifugal force and imparts a velocity to the gas by the blades. Forward curved blades impart the most velocity to the gas. The scroll-shaped volute diffuses the air and creates an increase in the static pressure by reducing the gas velocity. The change in total pressure occurs in the impeller—this is usually a small change. The static pressure is increased in both the impeller and the diffuser section. Operating efficiencies of the fan range from 40 to 80 percent. The discharge total pressure is the summation of the static pressure and the velocity head. The power needed to drive the fan can be computed as follows. Power (kW) = 2.72 × 10−5QP

(10-83)

where Q is the fan volume (m3/h) and P is the total discharge pressure

FIG. 10-84

Straight-blade, or steel-plate, fan.

in cm of water column. In U.S. customary units, hp = 1.57 × 10−4Qp

(10-84)

where hp is the fan power output, hp; Q is the fan volume, ft3/min; and p is the fan-operating pressure, inches water column. air power output Efficiency = ᎏᎏ (10-85) shaft power input

10-52

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-85

Forward-curved blade, or “scirocco”-type, fan.

FIG. 10-87

Approximate characteristic curves of various types of fans.

CONTINUOUS-FLOW COMPRESSORS

Centrifugal compressors in general are used for higher pressure ratios and lower flow rates compared to lower-stage pressure ratios and higher flow rates in axial compressors. The pressure ratio in a singlestage centrifugal compressor varies depending on the industry and application. In the petrochemical industry the single stage pressure ratio is about 1.2:1. Centrifugal compressors used in the aerospace industry, usually as a compressor of a gas turbine, have pressure ratios between 3:1 to as high as 9:1 per stage. In the petrochemical industry, the centrifugal compressors consist mainly of casings with multiple stages. In many instances, multiple casings are also used, and to reduce the power required to drive these multiple casings, there are intercoolers between them. Each casing can have up to 9 stages. In some cases, intercoolers are also used between single stages of compressor to reduce the power required for compression. These compressors are usually driven by gas turbines, steam turbines, and electric motors. Speed-increasing gears may be used in conjunction with these drivers to obtain the high speeds at which many of these units operate. Rotative speeds of as high as 50,000 rpm are not uncommon. Most of the petrochemical units run between 9,000 and 15,000 rpm. The compressor’s operating range is between two major regions as seen in Fig. 10-89, which is a performance map of a centrifugal compressor. These two regions are surge, which is the lower flow limit of stable operation, and choke or stonewall, which is the maximum flow through the compressor at a given operating speed. The centrifugal compressor’s operating range between surge and choke is reduced as the pressure ratio per stage is increased or the number of stages are added.

Continuous-flow compressors are machines where the flow is continuous, unlike positive-displacement machines where the flow is fluctuating. Continuous-flow compressors are also classified as turbomachines. These types of machines are widely used in the chemical and petroleum industry for many services. They are also used extensively in many other industries such as the iron and steel industry, pipeline boosters, and on offshore platforms for reinjection compressors. Continuous-flow machines are usually much smaller in size and produce much less vibration than their counterpart, positive-displacement units. Centrifugal Compressors The flow in a centrifugal compressor enters the impeller in an axial direction and exits in a radial direction. In a typical centrifugal compressor, the fluid is forced through the impeller by rapidly rotating impeller blades. The velocity of the fluid is converted to pressure, partially in the impeller and partially in the stationary diffusers. Most of the velocity leaving the impeller is converted into pressure energy in the diffuser as shown in Fig. 10-88. It is normal practice to design the compressor so that half the pressure rise takes place in the impeller and the other half in the diffuser. The diffuser consists of a vaneless space, a vane that is tangential to the impeller, or a combination of both. These vane passages diverge to convert the velocity head into pressure energy.

FIG. 10-88

FIG. 10-86

Backward-curved-blade fan.

Fan Performance The performance of a centrifugal fan varies with changes in conditions such as temperature, speed, and density of the gas being handled. It is important to keep this in mind in using the catalog data of various fan manufacturers, since such data are usually based on stated standard conditions. Corrections must be made for variations from these standards. The usual variations are as follows: When speed varies, (1) capacity varies directly as the speed ratio, (2) pressure varies as the square of the speed ratio, and (3) horsepower varies as the cube of the speed ratio. When the temperature of air or gas varies, horsepower and pressure vary inversely as the absolute temperature, speed and capacity being constant. See Fig. 10-87. When the density of air or gas varies, horsepower and pressure vary directly as the density, speed and capacity being constant.

Pressure and velocity through a centrifugal compressor.

PUMPS AND COMPRESSORS 10

ηc denotes efficiency related

Re10 = 106

to total exhaust pressure and total inlet pressure

6

Ns =

3

ηc

.30 .6.60 ηc .700 .80 .85

Ds 1

10-53

Ds =

N Q1 3/4 Had 1/4 DHad Q1

N = rpm Q = ft3/sec Had = ft • lbf/lbm D = ft

φ ad

.6

= 02 .0 04 .0

.3

1 .0

2 .0

4 .0

.1

.2

.4

1

φad =

V2abs u2

.1 10

30

60

100

300 Ns

600

1000

3000

6000 10000

Centrifugal compressor map. (Balje, O. E., “A Study of Reynolds Number Effects in Turbomachinery,” Journal of Engineering for Power, ASME Trans., vol. 86, series A, p. 227.)

FIG. 10-89

A compressor is said to be in surge when the main flow through the compressor reverses its direction. Surge is often symptomized by excessive vibration and a large audible sound. This flow reversal is accompanied with a very violent change in energy, which causes a reversal of the thrust force. The surge process is cyclic in nature and if allowed to cycle for some time, irreparable damage can occur to the compressor. In a centrifugal compressor, surge is usually initiated at the exit of the impeller or at the diffuser entrance for impellers producing a pressure ratio of less than 3:1. For higher pressure ratios, the initiation of surge can occur in the inducer. A centrifugal compressor impeller can have three types of blades at the exit of the impeller. These are forward-curved, backward-curved, and radial blades. Forward-curved blades are not often used in a centrifugal compressor’s impeller because of the very high-velocity discharge at the compressor that would require conversion of the high velocity to a pressure head in the diffuser, which is accompanied by high losses. Radial blades are used in impellers of high pressure ratio since the stress levels are minimal. Backward-curved blades give the highest efficiency and the largest operating margin of any of the various types of blades in an impeller. Most centrifugal compressors in the petrochemical industry use backward-curved impellers because of the higher efficiency and larger operating range. Process compressors have impellers with very low pressure ratio impellers and thus large surge-to-choke margins. The common method of classifying process-type centrifugal compressors is based on the number of impellers and the casing design. Sectionalized casing types have impellers that are usually mounted on the extended motor shaft, and similar sections are bolted together to obtain the desired number of stages. Casing material is either steel or cast iron. These machines require minimum supervision and maintenance and are quite economic in their operating range. The sectionalized casing design is used extensively in supplying air for combustion in ovens and furnaces. The horizontally split type have casings split horizontally at the midsection and the top. The bottom halves are bolted and doweled together. This design type is preferred for large multistage units. The internal parts such as shaft, impellers, bearings, and seals are readily accessible for inspection and repairs by removing the top half. The casing material is cast iron or cast steel.

Barrel casings are used for high pressures in which the horizontally split joint is inadequate. This type of compressor consists of a barrel into which a compressor bundle of multiple stages is inserted. The bundle is itself a horizontally split casing compressor. Compressor Configuration To properly design a centrifugal compressor, one must know the operating conditions—the type of gas, its pressure, temperature, and molecular weight. One must also know the corrosive properties of the gas so that proper metallurgical selection can be made. Gas fluctuations due to process instabilities must be pinpointed so that the compressor can operate without surging. Centrifugal compressors for industrial applications have relatively low pressure ratios per stage. This condition is necessary so that the compressors can have a wide operating range while stress levels are kept at a minimum. Because of the low pressure ratios for each stage, a single machine may have a number of stages in one “barrel” to achieve the desired overall pressure ratio. Figure 10-90 shows some of the many configurations. Some factors to be considered when selecting a configuration to meet plant needs are: 1. Intercooling between stages can considerably reduce the power consumed. 2. Back-to-back impellers allow for a balanced rotor thrust and minimize overloading the thrust bearings. 3. Cold inlet or hot discharge at the middle of the case reduces oil-seal and lubrication problems. 4. Single inlet or single discharge reduces external piping problems. 5. Balance planes that are easily accessible in the field can appreciably reduce field-balancing times. 6. Balance piston with no external leakage will greatly reduce wear on the thrust bearings. 7. Hot and cold sections of the case that are adjacent to each other will reduce thermal gradients and thus reduce case distortion. 8. Horizontally split casings are easier to open for inspection than vertically split ones, reducing maintenance time. 9. Overhung rotors present an easier alignment problem because shaft-end alignment is necessary only at the coupling between the compressor and driver. 10. Smaller, high-pressure compressors that do the same job will reduce foundation problems but will have greatly reduced operational range.

10-54

TRANSPORT AND STORAGE OF FLUIDS

(a) Fillet weld

(b) Full penetration fillet weld

(c) Butt weld

(d) Slot weld

(e) Electron beam weld

(f) Machined

Four post, three cooling points Series flow, cooling point

Parallel flow, suction in ends

Series flow (basic compressor)

(g) Riveted FIG. 10-91

Parallel flow, suction in center

Series flow, one cooling point

Series flow, two cooling points

Series flow, one cooling point suction on ends, cool ends

Series flow, one cooling point, suction in center, warm ends

Series flow, with double flow inlet and side stream

FIG. 10-90

Various configurations of centrifugal compressors.

Impeller Fabrication Centrifugal-compressor impellers are either shrouded or unshrouded. Open, shrouded impellers that are mainly used in single-stage applications are made by investmentcasting techniques or by three-dimensional milling. Such impellers are used, in most cases, for the high-pressure-ratio stages. The

(h) Riveted

(i) Riveted

(j) Riveted

Several fabrication techniques for centrifugal impellers.

shrouded impeller is commonly used in the process compressor because of its low pressure ratio stages. The low tip stresses in this application make it a feasible design. Figure 10-91 shows several fabrication techniques. The most common type of construction is seen in A and B where the blades are fillet-welded to the hub and shroud. In B, the welds are full penetration. The disadvantage in this type of construction is the obstruction of the aerodynamic passage. In C, the blades are partially machined with the covers and then butt-welded down the middle. For backward lean-angled blades, this technique has not been very successful, and there has been difficulty in achieving a smooth contour around the leading edge. D illustrates a slot-welding technique and is used where bladepassage height is too small (or the backward lean-angle too high) to permit conventional fillet welding. In E, an electron-beam technique is shown. Its major disadvantage is that electron-beam welds should preferably be stressed in tension but, for the configuration of E, they are in shear. The configurations of G through J use rivets. Where the rivet heads protrude into the passage aerodynamic performance is reduced. Riveted impellers were used in the 1960s—they are very rarely used now. Elongation of these rivets occurs at certain critical surge conditions and can lead to major failures. Materials for fabricating these impellers are usually low-alloy steels, such as AISI 4140 or AISI 4340. AISI 4140 is satisfactory for most applications; AISI 4340 is used for large impellers requiring higher strengths. For corrosive gases, AISI 410 stainless steel (about 12 percent chromium) is used. Monel K-500 is employed in halogen gas atmospheres and oxygen compressors because of its resistance to sparking. Titanium impellers have been applied to chlorine service. Aluminumalloy impellers have been used in great numbers, especially at lower temperatures (below 300°F). With new developments in aluminum alloys, this range is increasing. Aluminum and titanium are sometimes selected because of their low density. This low density can cause a shift in the critical speed of the rotor, which may be advantageous. Axial-Flow Compressors Axial-flow compressors are used mainly as compressors for gas turbines. They are also used in the steel industry as blast furnace blowers and in the chemical industry for large nitric acid plants. They are mainly used for applications where the head required is low and the flow large. Figure 10-92 shows a typical axial-flow compressor. The rotating element consists of a single drum to which are attached several rows

PUMPS AND COMPRESSORS

FIG. 10-92

10-55

Axial-flow compressor. (Courtesy of Allis-Chalmers Corporation.)

of decreasing-height blades having airfoil cross sections. Between each rotating blade row is a stationary blade row. All blade angles and areas are designed precisely for a given performance and high efficiency. The use of multiple stages permits overall pressure increases up to 30:1. The efficiency in an axial flow compressor is higher than the centrifugal compressor. Pressure ratio per casing can be comparable with those of centrifugal equipment, although flow rates are considerably higher for a given casing diameter because of the greater area of the flow path. The pressure ratio per stage is less than in a centrifugal compressor. The pressure ratio per stage in industrial compressors is between 1:05 and 1:15, and for aeroturbines, 1.1 and 1.2. The axial-flow compressors used in gas turbines vary depending on the type of turbines. The industrial-type gas turbine has an axial-flow compressor of a rugged construction. These units have blades that have low aspect ratio (R = blade height/blade chord) with minimum streamline curvation, and the shafts are support on sleeve-type bearings. The industrial gas turbine compressor has also a lower pressure ratio per stage (stage = rotor + stationary blade), giving a low blade loading. This also gives a larger operating range than its counterpart the aero axial gas turbine compressor but considerably less than the centrifugal compressor. The axial-flow compressors in aero gas turbines are heavily loaded. The aspect ratio of the blades, especially the first few stages, can be as high as 4.0, and the effect of streamline curvature is substantial. The streamline configuration is a function of the annular passage area, the camber and thickness distribution of the blade, and the flow angles at the inlet and outlet of the blades. The

shafts on these units are supported on antifriction bearings (roller or ball bearings). The operation of the axial-flow compressor is a function of the rotational speed of the blades and the turning of the flow in the rotor. The stationary blades (stator) are used to diffuse the flow and convert the velocity increased in the rotor to a pressure increase. One rotor and one stator make up a stage in a compressor. One additional row of fixed blades (inlet guide vanes) is frequently used at the compressor inlet to ensure that air enters the first stage rotors at the desired angle. In addition to the stators, another diffuser at the exit of the compressor further diffuses the gas and, in the case of gas turbines, controls its velocity entering the combustor. The axial-flow compressor has a much smaller operating range “Surge to Choke” than its counterpart in the centrifugal compressor. Because of the steep characteristics of the head/flow capacity curve, the surge point is usually within 10 percent of the design point. The axial-flow compressor has three distinct stall phenomena. Rotating stall and individual blade stall are aerodynamic phenomena. Stall flutter is an aeroelastic phenomenon. Rotating stall (propagating stall) consists of large stall zones covering several blade passages and propagates in the direction of the rotor and at some fraction of rotor speed. The number of stall zones and the propagating rates vary considerably. Rotating stall is the most prevalent type of stall phenomenon. Individual blade stall occurs when all the blades around the compressor annulus stall simultaneously without the occurrence of the stall propagation mechanism. The phenomena of stall flutter is caused by self-excitation of the blade and is aeroelastic. It must be distinguished from classic flutter, since classic flutter is a coupled

10-56

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-95

Approximate performance curves for a rotary positive-displacement compressor. The safety valve in discharge line or bypass must be set to operate at a safe value determined by construction.

Fig. 10-93

torsional-flexural vibration that occurs when the freestream velocity over an airfoil section reaches a certain critical velocity. Stall flutter, on the other hand, is a phenomenon that occurs due to the stalling of the flow around a blade. Blade stall causes Karman vortices in the airfoil wake. Whenever the frequency of the vortices coincides with the natural frequency of airfoil, flutter will occur. Stall flutter is a major cause of compressor-blade failure. Positive-Displacement Compressors Positive-displacement compressors are machines that are essentially constant-volume machines with variable discharge pressures. These machines can be divided into two types: 1. Rotary compressors 2. Reciprocating compressors Many users consider rotary compressors, such as the “Rootes”-type blower, as turbomachines because their behavior in terms of the rotor dynamics is very close to centrifugal and axial flow machinery. Unlike the reciprocating machines, the rotary machines do not have a very high vibration problem but, like the reciprocating machines, they are positive-displacement machines. Rotary Compressors Rotary compressors are machines of the positive-displacement type. Such units are essentially constant-volume machines with variable discharge pressure. The volume can be varied only by changing the speed or by bypassing or wasting some of the capacity of the machine. The discharge pressure will vary with the resistance on the discharge side of the system. A characteristic curve typical of the form produced by these rotary units is shown in Fig. 10-93.

FIG. 10-94

Two-impeller type of rotary positive-displacement blower.

Screw-type rotary compressor.

Rotary compressors are generally classified as of the straight-lobe type, screw type, sliding-vane type, and liquid-piston type. Straight-Lobe Type This type is illustrated in Fig. 10-94. Such units are available for pressure differentials up to about 83 kPa (12 lbf/in2) and capacities up to 2.549 × 104 m3/h (15,000 ft3/min). Sometimes multiple units are operated in series to produce higher pressures; individual-stage pressure differentials are limited by the shaft deflection, which must necessarily be kept small to maintain rotor and casing clearance. Screw Type This type of rotary compressor, as shown in Fig. 10-95, is capable of handling capacities up to about 4.248 × 104 m3/h (25,000 3 ft /min) at pressure ratios of 4:1 and higher. Relatively small-diameter rotors allow rotative speeds of several thousand rev/min. Unlike the straight-lobe rotary machine, it has male and female rotors whose rotation causes the axial progression of successive sealed cavities. These machines are staged with intercoolers when such an arrangement is advisable. Their high-speed operation usually necessitates the use of suction- and discharge-noise suppressors. The bearings used are sleeve-type bearings. Due to the side pressures experienced, tilting pad bearings are highly recommended. Sliding-Vane Type This type is illustrated in Fig. 10-96. These units are offered for operating pressures up to 0.86 MPa (125 lbf/in2) and in capacities up to 3.4 × 103 m3/h (2000 ft3/min). Generally, pressure ratios per stage are limited to 4:1. Lubrication of the vanes is required, and the air or gas stream therefore contains lubricating oil. Liquid-Piston Type This type is illustrated in Fig. 10-97. These compressors are offered as single-stage units for pressure differentials up to about 0.52 MPa (75 lbf/in2) in the smaller sizes and capacities up to 6.8 × 103 m3/h (4000 ft3/min) when used with a lower pressure differential. Staging is employed for higher pressure differentials. These units have found wide application as vacuum pumps on wet-vacuum service. Inlet and discharge ports are located in the impeller hub. As the vaned impeller rotates, centrifugal force drives the sealing liquid

FIG. 10-96

Sliding-vane type of rotary compressor.

PUMPS AND COMPRESSORS

FIG. 10-99

FIG. 10-97

Liquid-piston type of rotary compressor.

against the walls of the elliptical housing, causing the air to be successively drawn into the vane cavities and expelled against discharge pressure. The sealing liquid must be externally cooled unless it is used in a once-through system. A separator is usually employed in the discharge line to minimize carryover of entrained liquid. Compressor capacity can be considerably reduced if the gas is highly soluble in the sealing liquid. The liquid-piston type of compressor has been of particular advantage when hazardous gases are being handled. Because of the gas-liquid contact and because of the much greater liquid specific heat, the gas-temperature rise is very small. EJECTORS An ejector is a simplified type of vacuum pump or compressor which has no pistons, valves, rotors, or other moving parts. Figure 10-98 illustrates a steam-jet ejector. It consists essentially of a nozzle which discharges a high-velocity jet across a suction chamber that is con-

10-57

Booster ejector with multiple steam nozzles.

nected to the equipment to be evacuated. The gas is entrained by the steam and carried into a venturi-shaped diffuser which converts the velocity energy into pressure energy. Figure 10-99 shows a large-sized ejector, sometimes called a booster ejector, with multiple nozzles. Nozzles are devices in subsonic flow that have a decreasing area and accelerate the flow. They convert pressure energy to velocity energy. A minimum area is reached when velocity reaches sonic flow. In supersonic flow, the nozzle is an increasing area device. A diffuser in subsonic flow has an increasing area and converts velocity energy into pressure energy. A diffuser in supersonic flow has a decreasing area. Two or more ejectors may be connected in series or stages. Also, a number of ejectors may be connected in parallel to handle larger quantities of gas or vapor. Liquid- or air-cooled condensers are usually used between stages. Liquid-cooled condensers may be of either the direct-contact (barometric) or the surface type. By condensing vapor the load on the following stage is reduced, thus minimizing its size and reducing consumption of motive gas. Likewise, a precondenser installed ahead of an ejector reduces its size and consumption if the suction gas contains vapors that are condensable at the temperature condition available. An aftercondenser is frequently used to condense vapors from the final stage, although this does not affect ejector performance. Ejector Performance The performance of any ejector is a function of the area of the motive-gas nozzle and venturi throat, pressure of the motive gas, suction and discharge pressures, and ratios of specific heats, molecular weights, and temperatures. Figure 10-100, based on the assumption of constant-area mixing, is useful in evaluating single-stage-ejector performance for compression ratios up to 10 and area ratios up to 100 (see Fig. 10-101 for notation). For example,* assume that it is desired to evacuate air at 2.94 lbf/in2 with a steam ejector discharging to 14.7 lbf/in2 with available steam pressure of 100 lbf/in2. Entering the chart at p03/p0b = 5.0, at p0b/p0a = 2.94/100 = 0.0294 the optimum area ratio is 12. Proceeding horizontally to the left, wb/wa is approximately 0.15 lb of air per 1 lb of steam. This value must be corrected for the temperature and molecularweight differences of the two fluids by Eq. (10-86). 苶苶M w/wa = wb/wa兹T 0a苶苶T b /苶苶M 0b苶苶a

(10-86)

In addition, there are empirical correction factors which should be applied. Laboratory tests show that for ejectors with constant-area mixing the actual entrainment and compression ratios will be approximately 90 percent of the calculated values and even less at very small values of p0b/p0a. This compensates for ignoring wall friction in the mixing section and irreversibilities in the nozzle and diffuser. In theory, each point on a given design curve of Fig. 10-100 is associated with an optimum ejector for prevailing operating conditions. Adjacent points on the same curve represent theoretically different ejectors for the new conditions, the difference being that for each ratio of p0b/p0a there is an optimum area for the exit of the motive-gas nozzle. In practice, however, a segment of a given curve for constant A2/At represents the

FIG. 10-98

Typical steam-jet ejector.

*All data are given in U.S. customary units since the charts are in these units. Conversion factors to SI units are given on the charts.

10-58

FIG. 10-100

TRANSPORT AND STORAGE OF FLUIDS

Design curves for optimum single-stage ejectors. [DeFrate and Hoerl, Chem. Eng. Prog., 55, Symp. Ser. 21, 46 (1959).]

performance of a single ejector satisfactorily for estimating purposes, provided that the suction pressure lies within 20 to 130 percent of the design suction pressure and the motive pressure within 80 to 120 percent of design motive pressure. Thus the curves can be used to select an optimum ejector for the design point and to estimate its performance at off-design conditions within the limits noted. Final ejector selection should, of course, be made with the assistance of a manufacturer of such equipment. Uses of Ejectors For the operating range of steam-jet ejectors in vacuum applications, see the subsection “Vacuum Systems.” The choice of the most suitable type of ejector for a given application depends upon the following factors: 1. Steam pressure. Ejector selection should be based upon the minimum pressure in the supply line selected to serve the unit. 2. Water temperature. Selection is based on the maximum water temperature. 3. Suction pressure and temperature. Overall process requirements should be considered. Selection is usually governed by the minimum suction pressure required (the highest vacuum). 4. Capacity required. Again overall process requirements should be considered, but selection is usually governed by the capacity required at the minimum process pressure.

FIG. 10-101

Notation for Fig. 10-100.

Ejectors are easy to operate and require little maintenance. Installation costs are low. Since they have no moving parts, they have long life, sustained efficiency, and low maintenance cost. Ejectors are suitable for handling practically any type of gas or vapor. They are also suitable for handling wet or dry mixtures or gases containing sticky or solid matter such as chaff or dust. Ejectors are available in many materials of construction to suit process requirements. If the gases or vapors are not corrosive, the diffuser is usually constructed of cast iron and the steam nozzle of stainless steel. For more corrosive gases and vapors, many combinations of materials such as bronze, various stainless-steel alloys, and other corrosion-resistant metals, carbon, and glass can be used.

VACUUM SYSTEMS Figure 10-102 illustrates the level of vacuum normally required to perform many of the common manufacturing processes. The attainment of various levels is related to available equipment in Fig. 10-103. Vacuum Equipment The equipment shown in Fig. 10-103 has been discussed elsewhere in this section with the exception of the diffusion pump. Figure 10-104 depicts a typical design. A liquid of low absolute vapor pressure is boiled in the reservoir. The vapor is ejected at high velocity in a downward direction through multiple jets and is condensed on the walls, which are cooled by the surrounding coils. Molecules of the gas being pumped enter the vapor stream and are driven downward by collisions with the vapor molecules. The gas molecules are removed through the discharge line by a backing pump such as a rotary oil-sealed unit. Diffusion pumps operate at very low pressures. The ultimate vacuum attainable depends somewhat upon the vapor pressure of the pump liquid at the temperature of the condensing surfaces. By providing a cold trap between the diffusion pump and the region being evacuated, pressures as low as 10−7 mmHg absolute are achieved in this manner. Liquids used for diffusion pumps are mercury and oils of low vapor pressure. Silicone oils have excellent characteristics for this service.

PUMPS AND COMPRESSORS

FIG. 10-102

10-59

Vacuum levels normally required to perform common manufacturing processes. (Courtesy of Compressed Air

magazine.)

SEALING OF ROTATING SHAFTS Seals are very important and often critical components in large rotating machinery especially on high-pressure and high-speed equipment. The principal sealing systems used between the rotor and stationary elements fall into two main categories: (1) noncontacting seals and (2) face seals. These seals are an integral part of the rotating system, they affect the dynamic operating characteristics of the machine. The stiffness and damping factors will be changed by the seal geometry and pressures. In operation the rotating shafts have both radial and axial movement. Therefore any seal must be flexible and compact to ensure maximum sealing minimum effect on rotor dynamics.

Noncontact Seals Noncontact seals are used extensively in gas service in high speed rotating equipment. These seals have good mechanical reliability and minimum impact on the rotor dynamics of the system. They are not positive sealing. There are two types of noncontact seals: (1) labyrinth seals and (2) ring seals. Labyrinth Seals The labyrinth is one of the simplest of the many sealing devices. It consists of a series of circumferential strips of metal extending from the shaft or from the bore of the shaft housing to form a cascade of annular orifices. Labyrinth seal leakage is greater than that of clearance bushings, contact seals, or filmriding seals. The major advantages of labyrinth seals are their simplicity, reliability, tolerance to dirt, system adaptability, very low shaft power

10-60

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-103

Vacuum levels attainable with various types of equipment. (Courtesy of Compressed Air magazine.)

consumption, material selection flexibility, minimal effect on rotor dynamics, back diffusion reduction, integration of pressure, lack of pressure limitations, and tolerance to gross thermal variations. The major disadvantages are the high leakage, loss of machine efficiency, increased buffering costs, tolerance to ingestion of particulates with

FIG. 10-104

Typical diffusion pump. (Courtesy of Compressed Air magazine.)

resulting damage to other critical items such as bearings, the possibility of the cavity clogging due to low gas velocities or back diffusion, and the inability to provide a simple seal system that meets OSHA or EPA standards. Because of some of the foregoing disadvantages, many machines are being converted to other types of seals. Labyrinth seals are simple to manufacture and can be made from conventional materials. Early designs of labyrinth seals used knifeedge seals and relatively large chambers or pockets between the knives. These relatively long knives are easily subject to damage. The modern, more functional, and more reliable labyrinth seals consist of sturdy, closely spaced lands. Some labyrinth seals are shown in Fig. 10-105. Figure 10-105a is the simplest form of the seal. Figure 10-105b shows a grooved seal; it is more difficult to manufacture but produces a tighter seal. Figure 10-105c and 10-105d is rotating labyrinth-type seals. Figure 10-105e shows a simple labyrinth seal with a buffered gas for which pressure must be maintained above the process gas pressure and the outlet pressure (which can be greater than or less than the atmospheric pressure). The buffered gas produces a fluid barrier to the process gas. The eductor sucks gas from the vent near the atmospheric end. Figure 10-105f shows a buffered, stepped labyrinth. The step labyrinth gives a tighter seal. The matching stationary seal is usually manufactured from soft materials such as babbitt or bronze, while the stationary or rotating labyrinth lands are made from steel. This composition enables the seal to be assembled with minimal clearance. The lands can therefore cut into the softer materials to provide the necessary running clearances for adjusting to the dynamic excursions of the rotor. To maintain maximum sealing efficiency, it is essential that the labyrinth lands maintain sharp edges in the direction of the flow.

PUMPS AND COMPRESSORS

Low pressure

Stationary

Rotating surface (steel)

High pressure Leakage path

Low press ure

Rotating surface (steel)

(a) Simplest design. (Labyrinth materials: aluminum, bronze, babbitt or steel)

Low pressure

Low pressure

High pressure Stationary sleeve

(c) Rotating labyrinth type, before operation. (Sleeve material: babbitt, aluminum, nonmetallic or other soft materials)

Inlet Pi

Leakage path

Rotating surface

Rotating surface (steel)

(d) Rotating labyrinth, after operation. Radial and axial movement of rotor cuts grooves in sleeve material to simulate staggered type shown in b. Buffered (or barrier) inlet Pb

Outlet Po Atmosphere Pa

Process Pp

High pressure Leakage path

(b) More difficult to manufacture but produces a tighter seal. (Same material as in a.)

High pressure

Stationary sleeve

Stationary

10-61

Process Pp

Atmosphere Pa

Straight labyrinth

Stepped labyrinth

Shaft (f) Buffered combination labyrinth

(e) Buffered-vented straight labyrinth FIG. 10-105

Various configurations of labyrinth seals.

Leakage past these labyrinths is approximately inversely proportional to the square root of the number of labyrinth lands. This translates into the following relationship if leakage is to be cut in half in a four labyrinth seal: The number of labyrinths would have to be increased to 16. The Elgi leakage formula can be modified and written as: (g/Vo)(Po − Pn) mᐉ = AK ᎏᎏ n + ln (Pn /Po)

冤

where mᐉ = leakage A = leakage area of single throttling

冥

1/2

(10-87)

K = labyrinth constant (K = .9 for straight labyrinths, K = .75 for staggered labyrinths) Po = absolute pressure before the labyrinth Pn = absolute pressure after the last labyrinth Vo = specific volume before the labyrinth n = number of lands The leakage of a labyrinth seal can be kept to a minimum by providing: (1) minimum clearance between the seal lands and the seal sleeve, (2) sharp edges on the lands to reduce the flow discharge coefficient, and (3) grooves or steps in the flow path for reducing dynamic head carryover from stage to stage.

10-62

TRANSPORT AND STORAGE OF FLUIDS Seal housing

Flange gasket

Atmosphere (outlet) Po

Process gas (inlet) PL

End plate Metal chamber

Circumferential groove

Radial vent

Outlet Bearing

Small radial clearance

Process Pp

Po

Segmented carbon seal

(a ) FIG. 10-106

Sensing pressure Ps

Segmented metal retainer Key

Moderate clearance

Buffer inlet Pa

Extension spring

(b)

Floating-type restrictive ring seal.

The labyrinth sleeve can be flexibly mounted to permit radial motion for self-aligning effects. In practice, a radial clearance of under 0.008 is difficult to achieve. Ring Seals The restrictive ring seal is essentially a series of sleeves in which the bores form a small clearance around the shaft. Thus, the leakage is limited by the flow resistance in the restricted area and controlled by the laminar or turbulent friction. There are two types of ring seals: (1) fixed seal rings and (2) floating seal rings. The floating rings permit a much smaller leakage; they can be either the segmented type as shown in Fig. 10-106a or the rigid type as shown in Fig. 10-106b. Fixed Seal Rings The fixed-seal ring consists of a long sleeve affixed to a housing in which the shaft rotates with small clearances. Long assemblies must be used to keep leakage within a reasonable limit. Long seal assemblies aggravate alignment and rubbing problems, thus requiring shafts to operate below their capacity. The fixed bushing seal operates with appreciable eccentricity and, combined with large clearances, produces large leakages, thus making this kind of seal impractical where leakage is undesirable. Floating Seal Rings Clearance seals that are free to move in a radial direction are known as floating seals. The floating characteristics permit them to move freely, thus avoiding severe rubs. Due to differential thermal expansion between the shaft and bushing, the bushings should be made of material with a higher coefficient of thermal expansion. This is achieved by shrinking the carbon into a metallic retaining ring with a coefficient of expansion that equals or exceeds that of the shaft material. It is advisable in high shearing applications to lock the bushings against rotation. Buildup of dirt and other foreign material lodged between the seal ring and seat will create an excessive spin and damage on the floating seal ring unit. It is therefore improper to use soft material such as babbitt and silver as seal rings. Packing Seal A common type of rotating shaft seal consists of packing composed of fibers which are first woven, twisted, or braided into strands and then formed into coils, spirals, or rings. To ensure initial lubrication and to facilitate installation, the basic materials are often impregnated. Common materials are braided and twisted rubber and duck, flax, jute, and metallic braids. The so-called plastic packings can be made up with varying amounts of fiber combined with a binder and lubricant for high-speed applications. Maximum temperatures that base materials of packings withstand and still give good service are as follows:

Flax Cotton Duck and rubber Rubber Metallic (lead-based) Metallic (aluminum-based) Metallic (copper-based)

°C

°F

38 93 149 177 218 552 829

100 200 300 350 425 1025 1525

Packing may not provide a completely leak-free seal. With shaft surface speeds less than approximately 2.5 m/s (500 ft/min), the packing may be adjusted to seal completely. However, for higher speeds some leakage is required for lubrication, friction reduction, and cooling. Application of Packing Coils and spirals are cut to form closed or nearly closed rings in the stuffing box. Clearance between ends should be sufficient to allow for fitting and possible expansion due to increased temperature or liquid absorption of the packing while in operation. The correct form of the ring joint depends on materials and service requirements. Braided and flexible metallic packings usually have butt or square joints (Fig. 10-107a). With other packing material, service experience indicates that rings cut with bevel or skive joints (Fig. 10-107b) are more satisfactory. A slight advantage of the bevel joint over the butt joint is that the bevel permits a certain amount of sliding action, thus absorbing a portion of ring expansion.

(a)

(b) FIG. 10-107

Butt (a) and skive (b) joints for compression packing rings.

PUMPS AND COMPRESSORS

FIG. 10-108

Seal cage or lantern ring. (Courtesy of Crane Packing Co.)

In the manufacture of packings, the proper grade and type of lubricant is usually impregnated for each service for which the packing is recommended. However, it may be desirable to replenish the lubricant during the normal life of the packing. Lack of lubrication causes packing to become hard and lose its resiliency, thus increasing friction, shortening packing life, and increasing operating costs. An effective auxiliary device frequently used with packing and rotary shafts is the seal cage (or lantern ring), shown in Fig. 10-108. The seal cage provides an annulus around the shaft for the introduction of a lubricant, oil, grease, etc. The seal cage is also used to introduce liquid for cooling, to prevent the entrance of atmospheric air, or to prevent the infiltration of abrasives from the process liquid. The chief advantage of packing over other types of seals is the ease with which it can be adjusted or replaced. Most equipment is designed so that disassembly of major components is not required to remove or add packing rings. The major disadvantages of a packing-type seal are (1) short life, (2) requirement for frequent adjustment, and (3) need for some leakage to provide lubrication and cooling. Mechanical Face Seals This type of seal forms a running seal between flat precision-finished surfaces. It is an excellent seal against leakages. The sealing surfaces are planes perpendicular to the rotating shaft, and the forces that hold the contact faces are parallel to the shaft axis. For a seal to function properly, there are four sealing points: 1. Stuffing box face 2. Leakage down the shaft 3. Mating ring in the gland plate 4. Dynamic faces Mechanical Seal Selection There are many factors that govern the selection of seals. These factors apply to any type of seal: 1. Product 2. Seal environment 3. Seal arrangement 4. Equipment 5. Secondary packing 6. Seal face combinations 7. Seal gland plate 8. Main seal body

10-63

Product Physical and chemical properties of the liquid or gas being sealed places constraints on the type of material, design, and arrangement of the seal. Pressure. Pressure affects the choice of material and whether balanced or unbalanced seal design can be used. Most unbalanced seals are good up to 100 psig stuffing box pressure. Over 100 psig, balanced seals should be used. Temperature. The temperature of the liquid being pumped is important because it affects the seal face material selection as well as the wear life of the seal face. Lubricity. In any mechanical seal design, there is rubbing motion between the dynamic seal faces. This rubbing motion is often lubricated by the fluid being pumped. Most seal manufacturers limit the speed of their seals to 90 ft/s (30 m/s). This is primarily due to centrifugal forces acting on the seal, which tends to restrict the seal’s axial flexibility. Abrasion. If there are entrained solids in the liquid, it is desirable to have a flushed single inside type with a face combination of very hard material. Corrosion. This affects the type of seal body: what spring material, what face material, and what type of elastomer or gasket material. The corrosion rate will affect the decision of whether to use a single or multiple spring design because the spring can usually tolerate a greater amount of corrosion without weakening it appreciably. Seal Environment The design of the seal environment is based on the product and the four general parameters that regulate it: 1. Pressure control 2. Temperature control 3. Fluid replacement 4. Atmospheric air elimination Seal Arrangement There are four types of seal arrangements: 1. Double seals are standard with toxic and lethal products, but maintenance problems and seal design contribute to poor reliability. The double face-to-face seal may be a better solution. 2. Do not use a double seal in dirty service—the inside seal will hang up. 3. API standards for balanced and unbalanced seals are good guidelines; too low a pressure for a balanced seal may encourage face lift-off. 4. Arrangement of the seal will determine its success more than the vendor. Over 100 arrangements are available. Equipment The geometry of the pump or compressor is very important in seal effectiveness. Different pumps with the same shaft diameter and the total differential head can present different sealing problems. Secondary Packing Much more emphasis should be placed on secondary packing especially if Teflon is used. A wide variation in performance is seen between various seal vendors, depending on seal arrangement there can be differences in mating ring packing. Seal Face Combinations The dynamics of seal faces is better understood today. Seal-face combinations have come a long way in the past 8–10 years. Stellite is being phased out of the petroleum and petrochemical applications. Better grades of ceramic are available, cost of tungsten has come down, and relapping of tungsten are available near most industrial areas. Silicon carbide is being used in abrasive service. Seal Gland Plate The seal gland plate is caught in between the pump vendor and the seal vendor. Special glands should be furnished by seal vendors, especially if they require heating, quenching, and drain with a floating-throat bushing. Gland designs are complex and may have to be revisited, especially if seals are changed. Main Seal Body The term seal body makes reference to all rotating parts on a pusher seal, excluding shaft packing and seal ring. In many cases it is the chief reason to avoid a particular design for a particular service. Basically, most mechanical seals have the following components as seen in Fig. 10-109. 1. Rotating seal ring 2. Stationary seal ring 3. Spring devices to provide pressure 4. Static seals A loading device such as a spring is needed to ensure that in the event of loss or hydraulic pressure the sealing surfaces are kept closed. The amount of the load on the sealing area is determined by the

10-64

TRANSPORT AND STORAGE OF FLUIDS

FIG. 10-112

FIG. 10-109

Mechanical-seal components.

degree of “seal balance.” Figure 10-110 shows what seal balance means. A completely balanced seal is when the only force exerted on the sealing surfaces is the spring force; i.e., hydraulic pressure does not act on the sealing surface. The type of spring depends on the space available, loading characteristics, and the seal environment. Based on these considerations, either a single or multiple spring can be used. In small axial space, belleville springs, finger washers, or curved washers can be used. Shaft-sealing elements can be split up into two groups. The first type may be called pusher-type seals and includes the O-ring, V-ring, U-cup, and wedge configurations. Figure 10-111 shows some typical pusher-type seals. The second type is the bellow-type seals, which differ from the pusher-type seals in that they form a static seal between themselves and the shaft.

FIG. 10-110

Internal mechanical seal.

Internal and External Seals Mechanical seals are classified broadly as internal or external. Internal seals (Fig. 10-112) are installed with all seal components exposed to the fluid sealed. The advantages of this arrangement are (1) the ability to seal against high pressure, since the hydrostatic force is normally in the same direction as the spring force; (2) protection of seal parts from external mechanical damage; and (3) reduction in the shaft length required. For high-pressure installations, it is possible to balance partially or fully the hydrostatic force on the rotating member of an internal seal by using a stepped shaft or shaft sleeve (Fig. 10-110). This method of relieving face pressure is an effective way of decreasing power consumption and extending seal life. When abrasive solids are present and it is not permissible to introduce appreciable quantities of a secondary flushing fluid into the process, double internal seals are sometimes used (Fig. 10-113). Both sealing faces are protected by the flushing fluid injected between them even though the inward flow is negligible. External seals (Fig. 10-114) are installed with all seal components protected from the process fluid. The advantages of this arrangement are that (1) fewer critical materials of construction are required, (2) installation and setting are somewhat simpler because of the exposed position of the parts, and (3) stuffing-box size is not a limiting factor. Hydraulic balancing is accomplished by proper proportioning of the seal face and secondary seal diameters. Throttle Bushings These bushings (Fig. 10-115) are commonly used with single internal or external seals when solids are present in the fluid and the inflow of a flushing fluid is not objectionable. These close-clearance bushings are intended to serve as flow restrictions through which the maintenance of a small inward flow of flushing fluid prevents the entrance of a process fluid into the stuffing box. A typical complex seal utilizes both the noncontact and mechanical aspects of sealing. Figure 10-116 shows such a seal with its two major elements. This type of seal will normally have buffering via a labyrinth

Balanced internal mechanical seal.

FIG. 10-111 Various types of shaft-sealing elements.

FIG. 10-113

Internal bellows-type double mechanical seal.

PUMPS AND COMPRESSORS

10-65

Seal oil

11

Suction gas chamber

4

Process gas

5

8

FIG. 10-114

7

2

1

External mechanical seal. 9

seal and a positive shutdown device. For shutdown, the carbon ring is tightly sandwiched between the rotating seal ring and the stationary sleeve with gas pressure to prevent gas from leaking out when no oil pressure is available. In operation seal oil pressure is about 30–50 psi over the process gas pressure. The high-pressure oil enters the top and completely fills the seal cavity. A small percentage is forced across the carbon ring seal faces. The rotative speed of the carbon ring can be anywhere between zero and full rotational speed. Oil crossing the seal faces contacts the process gas and therefore is “contaminated oil.” The contaminated oil leaves through the contaminated oil drain to a degassifier for purification. The majority of the oil flows through the uncontaminated seal oil drain. Materials Springs and other metallic components are available in a wide variety of alloys and are usually selected on the basis of temperature and corrosion conditions. The use of a particular mechanical seal is frequently restricted by the temperature limitations of the organic materials used in the static seals. Most elastomers are limited to about 121°C (250°F). Teflon will withstand temperatures of 260°C (500°F) but softens appreciably above 204°C (400°F). Glass-filled Teflon is dimensionally stable up to 232 to 260°C (450 to 500°F). One of the most common elements used for seal faces is carbon. Although compatible with most process media, carbon is affected by strong oxidizing agents, including fuming nitric acid, hydrogen chloride, and high-temperature air [above 316°C (600°F)]. Normal mating-face materials for carbon are tungsten or chromium carbide, hard steel, stainless steel, or one of the cast irons. Other sealing-face combinations that have been satisfactory in corrosive service are carbide against carbide, ceramic against ceramic, ceramic against carbon, and carbon against glass. The ceramics have also been mated with the various hard-facing alloys. When selecting seal materials the possibility of galvanic corrosion must also be considered.

FIG. 10-115

3

External mechanical seal and throttle bushing.

6

1. Rotating carbon ring 2. Rotating seal ring 3. Stationary sleeve 4. Spring retainer 5. Spring 6. Gas and contaminated oil drain FIG. 10-116

10

7. Floating babbitt-faced steel ring 8. Seal wiper ring 9. Seal oil drain line 10. Buffer gas injection port 11. Bypass orifice

Mechanical contact shaft seal.

BEARINGS Many factors enter into the selection of the proper design for bearings. Some of these factors are: 1. Shaft speed range. 2. Maximum shaft misalignment that can be tolerated. 3. Critical speed analysis and the influence of bearing stiffness on this analysis 4. Loading of the compressor impellers 5. Oil temperatures and viscosity 6. Foundation stiffness 7. Axial movement that can be tolerated 8. Type of lubrication system and its contamination 9. Maximum vibration levels that can be tolerated Types of Bearings Figure 10-117 shows a number of different types of journal bearings. A description of a few of the pertinent types of journal bearings is given here: 1. Plain journal. Bearing is bored with equal amounts of clearance (on the order of one and one-half to two thousands of an inch per inch of journal diameter) between the journal and bearing. 2. Circumferential grooved bearing. Normally the oil groove is half the bearing length. This configuration provides better cooling but reduces load capacity by dividing the bearing into two parts. 3. Cylindrical bore bearings. Another common bearing type used in turbines. It has a split construction with two axial oil-feed grooves at the split. 4. Pressure or pressure dam. Used in many places where bearing stability is required, this bearing is a plain journal bearing with a pressure pocket cut in the unloaded half. This pocket is approximately 1⁄32 of an inch deep with a width 50 percent of the bearing length. This groove or channel covers an arc of 135° and terminates abruptly in a sharp-edge-edge dam. The direction or rotation is such that the oil is pumped down the channel toward the sharp edge. Pressure dam bearings are for one direction of rotation. They can be used in conjunction with cylindrical bore bearings as shown in Fig. 10-117.

10-66

TRANSPORT AND STORAGE OF FLUIDS

Bearing type

Load capacity

Suitable direction of rotation

Resistance to half-speed whirl

Stiffness and damping

Worst

Moderate

Cylindrical bore Good

Cylindrical bore with dammed groove Good

Moderate

Good

Moderate

Lemon bore

Increasing Three lobe Moderate

Good

Good

Excellent

Offset halves

Tilting pad Moderate

FIG. 10-117

Best

Good

Comparison of general bearing types.

5. Lemon bore or elliptical. This bearing is bored with shims split line, which are removed before installation. The resulting shape approximates an ellipse with the major axis clearance approximately twice the minor axis clearance. Elliptical bearings are for both directions of rotation. 6. Three-lobe bearing. The three-lobe bearing is not commonly used in turbomachines. It has a moderate load-carrying capacity and can be operated in both directions. 7. Offset halves. In principle, this bearing acts very similar to a pressure dam bearing. Its load-carrying capacity is good. It is restricted to one direction of rotation. 8. Tilt-pad bearings. This bearing is the most common bearing type in today’s machines. It consists of several bearing pads posed around the circumference of the shaft. Each pad is able to tilt to assume the most effective working position. This bearing also offers the greatest increase in fatigue life because of the following advantages: • Thermal conductive backing material to dissipate heat developed in oil film. • A thin babbitt layer can be centrifugally cast with a uniform thickness of about 0.005 inch. Thick babbitts greatly reduce bearing life. Babbitt thickness in the neighborhood of .01 reduce the bearing life by more than half. • Oil film thickness is critical in bearing stiffness calculations. In a tilting-pad bearing, one can change this thickness in a number of ways:

(a) change the number of pads; (b) direct the load on or in between the pads; (c) change the axial length of pad. The previous list contains some of the most common types of journal bearings. They are listed in the order of growing stability. All of the bearings designed for increased stability are obtained at higher manufacturing costs and reduced efficiency. The antiwhirl bearings all impose a parasitic load on the journal, which causes higher-power losses to the bearings and in turn requires higher oil flow to cool the bearing. Thrust Bearings The most important function of a thrust bearing is to resist the unbalanced force in a machine’s working fluid and to maintain the rotor in its position (within prescribed limits). A complete analysis of the thrust load must be conducted. As mentioned earlier, compressors with back-to-back rotors reduce this load greatly on thrust bearings. Figure 10-118 shows a number of thrust-bearing types. Plain, grooved thrust washers are rarely used with any continuous load, and their use tends to be confined to cases where the thrust load is very short duration or possibly occurs at standstill or low speed only. Occasionally, this type of bearing is used for light loads (less than 50 lb/in2), and in these circumstances the operation is probably hydrodynamic due to small distortions present in the nominally flat bearing surface. When significant continuous loads have to be taken on a thrust washer, it is necessary to machine into the bearing surface a profile to generate a fluid film. This profile can be either a tapered wedge or occasionally a small step.

PUMPS AND COMPRES