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Preface to the Fifth Edition The data given in the previous edition has been thoroughly checked. Errors occurring in some pages have been corrected. This little book has been the companion volume of the author’s book—Fundamentals of Compressible Flow. They together will continue to serve the needs of students, scientists and engineers in areas of aerospace, mechanical, chemical and gas engineering. It is hoped that the new format of this edition will be more convenient to use.
New Delhi July, 2006
S.M. YAHYA
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Preface to the First Edition There was a long-felt need for gas tables to support the courses in Gas Dynamics, Turbomachinery and Aerodynamics taught at different levels. These tables have been prepared for compressible fluids of common use. Long and detailed tables in a wide range covering a number of values of γ = cp /cv besides being expensive are not fully used by a majority of students. Various formulas which form the basis of the tables are also included which will be useful for ready reference. With numerical work of this magnitude it is highly probable that some errors may have crept in; reference to these errors and suggestions for improvement will be highly appreciated. The ranges of various tables have been chosen according to their importance and frequency in use.
New Delhi July, 1978
S.M. YAHYA
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Contents Preface to the Fifth Edition
v
Preface to the First Edition
vii
Notation
1
Introduction
2
1.
Gas Dynamic Properties of Some Gases and Vapours 1.1 1.2 1.3 1.4 1.5
Air Hydrogen Carbon Dioxide Steam Freon–21
2.
Properties of Standard Atmosphere
3.
Isentropic Flow of Perfect Gases 3.1 γ = 1.3 3.2 γ = 1.4 3.3 γ = 1.67
4.
47 52 57
62 71
Flow of Perfect Gases with Friction (Fanno Flow) 6.1 γ = 1.3 6.2 γ = 1.4 6.3 γ = 1.67
7.
21 28 40
Oblique Shocks in Perfect Gases 5.1 Wave Angles for Given Deflections, γ = 1.4 5.2 Upstream and Downstream Mach Numbers and Deflections from Normal Shock Relations, γ = 1.4
6.
19
Normal Shocks in Perfect Gases 4.1 γ = 1.3 4.2 γ = 1.4 4.3 γ = 1.67
5.
14 15 16 17 18
75 81 87
Isothermal Flow of Perfect Gases with Friction 7.1 γ = 1.3 7.2 γ = 1.4
93 99
Contents
x
8.
Flow of Perfect Gases with Heat Transfer (Rayleigh Flow) 8.1 8.2 8.3 8.4
9.
γ = 1.3 γ = 1.4 γ = 1.67 Maximum Heat Transfer in Perfect Gases
Prandtl-Meyer Function with Mach Angles 9.1 α and ω (M) as functions of M 9.2 M and α as functions of ω (M)
10.
105 111 117 123
125 130
Approximate Velocities and Mach Numbers of Air Corresponding to Dynamic Heads, and Pressure Ratios 10.1 Air velocities and Mach numbers as functions of dynamic heads in mm W.G. 10.2 Air velocities and Mach numbers as functions of dynamic heads in mm Hg. 10.3 Mach Numbers as functions of pressure ratios (γ = 1.4)
135 136
Manometer Inclinations Required for the Desired Magnification of the Deflection
138
12.
Static Pressure Drop in Gas Pipe Lines
139
13.
Isentropic Compressor Power
141
14.
Pumping Power for Fans and Blowers
143
15.
S.I, Metric and British Units
11.
15.1 Principal dimensions and units 15.2 Conversion of Units
134
147 148
Introduction
1
Notation a c cp cv
Velocity of sound fluid velocity specific heat at constant pressure specific heat at constant volume
A
area of cross-section
D
hydraulic mean diameter
f h L 1 Lmax
mean coefficient of friction
F
impulse function
enthalpy, actual deflection k constant length of the pipe line magnified value of manometer deflection maximum pipe length required for choking conditions
m P ∆p Qmax R t v Z α γ δ ρ η
mass flow rate power pressure rise or drop maximum heat transfer Gas constant temperature in °C specific volume altitude Mach angle, manometer inclination = cp /cv deflection or half wedge angle density efficiency
Subscripts o Stagnation state y downstream of normal shock 2 downstream of oblique shock
corresponding to M =
1 γ
(isothermal flow)
1
Mach number pressure pressure ratio volume flow rate entropy temperature in K mass flow rate change of entropy
σ wave angle µ coefficient of viscosity ω(M) Prandtl-Meyer function
x l s
Superscript * corresponding to choking conditions (M = 1) *t
M p pr Q s T w ∆s
upstream of normal shock upstream of oblique shock isentropic
Gas Tables for Compressible Flow Calculations
2
Introduction This section is designed to help the user to make the best use of the data given in the tables. Detailed explanation of the various theories behind the formulae mentioned in this section are not included; the user is expected to be aware of these while using the tables. However, some explanatory notes and figures have been included. 1. Gas dynamic properties of some vapours and gases It is well known that the specific heats (cp and cv) of gases and vapours vary with temperature. Therefore, the values of the ratio of specific heats, Gas constant and the maximum mass flow parameter also vary with temperature. These have been tabulated for a range of temperatures at given values of pressure. Only some of the commonly used gases and vapours have been chosen for these tables.
Formulae γ = cp cv
...(1.1)
R = cp − cv
...(1.2)
a = γRT
...(1.3)
w T0 A * P0
Lγ F 2 I =M G MN R H γ + 1JK
γ +1 γ −1
OP PQ
12
...(1.4)
2. Properties of the standard atmosphere Values of temperature, velocity of sound, pressure, density and dynamic viscosity of air are tabulated against altitude in the range of 0 to 20,000 meters. The lapse rate in temperature is 0.65°C for each 100 meters of altitude upto an altitude of 11000 metres. Kinematic viscosity (µ/ρ) can be determined with the help of the last two columns. 3. Isentropic flow of perfect gases Subsonic and supersonic isentropic flows in nozzles and diffusers are shown in figures 1 and 2 respectively. Pressures corresponding to the choking and stagnation states in both the cases are also shown.
Assumptions (a) One dimensional isentropic flow (b) Perfect gas with constant specific heats and molecular weight.
Introduction
3
h
p0 M=0
p p*
M<1 M = M* = 1
p
M>1 S
Fig. 1. Isentropic Accelerating Flow
h p0
M=0 p
M<1 p*
M = M* = 1 p
M>1 S
Fig. 2. Isentropic Decelerating Flow
Gas Tables for Compressible Flow Calculations
4
Formulas T0 = T0* = constant
...(3.1)
T = T0
...(3.2)
1 γ −1 2 1+ M 2
FG IJ H K
γ γ −1
F γ − 1 M IJ = G1 + H 2 K
...(3.3)
FG IJ H K
1 γ −1
F γ − 1 M IJ = G1 + H 2 K
...(3.4)
p T = p0 T0 ρ T = ρ0 T0
γ 1 − γ 2
1 2 1− γ
γ +1 2 M 2 M* = * = γ −1 2 c 1+ M 2 c
A
1 = * M A
F F*
A A
*
γ +1 2 γ −1
LMF 2 I FG1 + γ − 1 M IJ OP b NGH γ + 1JK H 2 K Q 2
g
1 + γM 2
= M
×
...(3.5)
F γ − 1 M IJ 2 b γ + 1g G1 + H 2 K
p = p0
...(3.7)
2
FG 2 IJ H γ + 1K M1+
γ +1 2 γ −1
b g
γ −1 2 M 2
4. Normal shocks in perfect gases A normal shock from state X to state Y is shown in figure 3.
Assumptions (a) (b) (c) (d)
...(3.6)
One dimensional flow with normal shocks Constant area duct Perfect gas with constant specific heats and molecular weight Adiabatic flow
...(3.8)
Introduction
5
Formulas T0 x = T0 y
...(4.1)
LM 2 + M OP γ −1 =M MM 2 γ M − 1PPP Q N γ −1
12
2 x
My
py px
...(4.2)
2 x
=
1 + γM 2x 1+
γM 2y
=
γ −1 2γ M 2x − γ +1 γ +1
FG H
IJ FG KH b g b g
...(4.3)
IJ K
γ −1 2 2γ γ −1 2 Mx M 2x − 1 1+ M x Ty − γ 2 1 2 = = 2 γ −1 2 Tx γ +1 My 1+ M 2x 2 2 γ −1 1+
ay ax
ρy ρx
FT I =G J HT K
12
y
...(4.5)
x
=
p y Tx p x Ty
...(4.6)
IJ K
γ 1− γ
F γ − 1 M IJ = G1 + H 2 K
γ γ −1
FG H
px γ −1 2 Mx = 1+ p0x 2
p0y py p0y p0x
2 y
=
p0y p y p x p y p x p0x
p ∆s = ln 0 x R p0y p0y px
...(4.4)
=
...(4.7)
...(4.8)
...(4.9)
...(4.10)
p0y p y py px
...(4.11)
Gas Tables for Compressible Flow Calculations
6
h
M = 1/ g
py Y
Fanno line
M<1
p*
M<1
Norma l shock
M=1 p*
p = constant
M=1
Rayleigh line
M>1 M>1
px X s
Fig. 3. Normal Shock, Fanno and Rayleigh Lines
5. Oblique shocks in perfect gases An oblique shock wave is inclined at an angle to the direction of upstream flow. Figure 4 shows an oblique shock wave at the corner of a wall. Supersonic flow passing through an oblique shock (wave angle = σ) is deflected through an angle (δ). The same picture is obtained when a supersonic flow is deflected through an angle (δ) either over a wedge or a wall corner.
Oblique shock
My
M1 = c1/a1 M2 = c2 /a2
Mx
s d
Fig. 4. Flow Through Oblique Shock Wave
Figure 4 shows the upstream and downstream Mach numbers corresponding to a normal shock.
Introduction
7
Assumptions (a) (b) (c) (d)
Perfect gas, with constant specific heats and molecular weight, adiabatic flow, absence of body forces, absence of external work.
Formulas tan δ =
M12 sin 2 σ − 2 cot σ
b
2 + M12 γ + cos 2 σ
M x = M1 sin σ
g
...(5.1) ...(5.2)
b
M y = M 2 sin σ − δ
g
...(5.3)
Table 5.1 is used to determine the wave angle for given values of upstream Mach numbers and deflection. Table 5.2 enables calculations of properties downstream of the oblique shock through normal shock tables. The gas property ratios corresponding to Mach number Mx can be found from Table 4.2. 6. Flow of perfect gases with friction (Fanno Flow) A Fanno line is shown on h-s co-ordinates in figure 3. Friction in the passage leads to increase in entropy and takes the process towards the choking conditions (M = M* = 1).
Assumptions (a) (b) (c) (d)
One dimensional flow with friction, Constant area duct, perfect gas with constant specific heats and molecular weight, adiabatic flow.
Formulas T0 = T0* = constant p *
p
c c
*
=
=
1 M
γ +1 γ −1 2 2 1+ M 2
FG H
ρ* =M ρ
...(6.1)
IJ K
γ +1 γ −1 2 2 1+ M 2
FG H
...(6.2)
IJ K
...(6.3)
Gas Tables for Compressible Flow Calculations
8
T T
*
=
γ +1 γ −1 2 2 1+ M 2
FG H
IJ K
...(6.4)
2
p0 p *0
F F*
4f
γ +1 2 γ −1
LM 2 FG1 + γ − 1 M IJ OP b H 2 KP 1 = M MM MN bγ + 1g PPQ =
g ...(6.5)
1 + γM 2
LM b g FG H N
M 2 γ +1 1 +
γ −1 2 M 2
L max 1 − M 2 γ + 1 = + ln 2γ D γM 2
IJ OP KQ
FG H
12
...(6.6)
bγ + 1g M
2 1+
2
γ −1 2 M 2
IJ K
...(6.7)
γ+1
s − s* γ +1 ⎡ ⎤ 2γ = ln M2 ⎢ γ −1 2 ⎞⎥ cp ⎛ ⎢ 2 M2 ⎜ 1 + M ⎟⎥ 2 ⎝ ⎠ ⎦⎥ ⎣⎢
...(6.8)
7. Isothermal flow of perfect gases with friction The combined effect of friction and heat transfer in long ducts can be considered to produce an isothermal flow model.
e
j
*t *t *t *t *t Here the reference values of the properties p , ρ , c , T0 , p 0 and L max are taken at the
Mach number
M*t =
1 γ
Assumptions (a) (b) (c) (d)
One dimensional flow with friction and heat transfer, Constant area duct, Perfect gas with constant specific heats and molecular weight, Isothermal flow.
Introduction
9
Formulas p p
*t
c c *t
T0 T0* t p0 p *0t 4f
1
=
M
...(7.1)
γ
=
ρ*t M = γ ρ
=
2γ γ −1 2 M 1+ 3γ − 1 2
=
...(7.2)
FG H
1 M
γ
IJ K
...(7.3)
LM 2 γ FG1 + γ − 1 M IJ OP N 3γ − 1 H 2 K Q 2
γ γ −1
L max 1 − γM 2 = + ln γM 2 D γM 2
e j
...(7.4)
...(7.5)
8. Flow of perfect gases with heat transfer (Rayleigh flow) A Rayleight line is shown in figure 3. Heat addition increases the entropy and takes the process towards the choking conditions, (M = M* = 1). This behaviour imposes a limit to heat addition; therefore for a given initial Mach Number there is a fixed value of the maximum possible heat transfer (Qmax). This has been tabulated in Table 8.4.
Assumptions (a) One dimensional flow with heat transfer, (b) Constant area frictionless duct, (c) Perfect gas with constant specific heats and molecular weight.
Formulas p p
*
=
1+ γ
...(8.1)
1 + γM 2 γ
⎡ ⎛ γ − 1 2 ⎞ ⎤ γ−1 2 ⎜1 + M ⎟⎥ ⎢ p0 1+ γ 2 ⎝ ⎠⎥ ⎢ = * 2 γ +1 p0 1 + γM ⎣⎢ ⎦⎥
T T
*
=
F 1+ γ I GH 1 + γM JK 2
...(8.2)
2
M2
...(8.3)
Gas Tables for Compressible Flow Calculations
10
T0 T0*
ρ ρ*
b g FGH γ 2− 1 M IJK e1 + γM j
2 1 + γ M2 1 + =
=
2
2
...(8.4)
2
c* 1 + γM 2 = c 1 + γ M2
b g F GH
s − s* 1+ γ = ln M 2 cp 1 + γM 2
...(8.5)
I JK
γ +1 γ
...(8.6) ...(8.7)
F = F * = constant
e j b g
2
M2 − 1 Q max = cpT 2 γ + 1 M2
...(8.8)
9. Prandtl-Meyer functions with Mach angles Supersonic flow of a gas along a wall turning in the clock-wise direction is shown in figure 5. Mach wave and the Mach angle (α) are also shown.
Mach wave
M>1 a
M
Wall
Fig. 5. Prandtl-Meyer Expansion
Assumptions (a) Perfect gas with constant specific heats and molecular weight, (b) Isentropic flow.
+d M
Introduction
11
Formulas sin α =
1 M
b g
...(9.1) γ +1 γ −1 2 M − 1 − tan −1 M 2 − 1 tan −1 γ −1 γ +1
e
ω M =
j
...(9.2)
10. Approximate velocities and Mach numbers of air At low velocities (M < 0.30) correct values of the velocity of a gas can be determined by measur-
FG H
IJ K
1 2 ρc . Conversely the value of the dynamic head required for 2 obtaining a given velocity can also be determined. Knowledge of the temperature of the gas along with the dynamic head gives the value of Mach Number.
ing the dynamic head p 0 − p =
b
2 RT p 0 − p ρ
c= M=
g
...(10.1)
c
bγRTg
...(10.2)
12
For lower velocities dynamic head is measured in millimetres of water (mmW.G) on a vertical or inclined tube manometer. At higher velocities dynamic head is more conveniently measured in millimeters of mercury (mm Hg). Tables 10.1 and 10.2 give approximate velocities and Mach numbers at the prescribed values of pressure and temperature; the approximation is less realistic at higher Mach numbers on account of increasing deviation of the actual flow from incompressible flow conditions. Therefore, at higher Mach numbers Table 10.2 gives only rough estimates of the velocity and Mach number. On account of this discrepancy the use of Table 10.3 is recommended at higher Mach numbers. 11. Measurement of low pressures on inclined manometers. Low pressures and velocities of gases can be measured by magnifying the actual manometer deflection (h) many times. This is done by inclining the manometer tube/tubes by angles shown in table 11. Actual (h) and magnified (A) values of the deflection are shown in figure 6. Plane of the manometer tubes
l h a
Horizontal plane
Fig. 6. Actual (h) & Magnified (A) Values of Deflections in Inclined Tube Manometer
Gas Tables for Compressible Flow Calculations
12
12. Static pressure drop (pr = outlet pressure/inlet pressure) in gas pipe lines Compressors are required to raise the pressure of gases to compensate for the pressure drop in long pipe lines. In these tables values of static pressure drop in the pipe lines have been tabulated with the assumption of isothermal flow. The flow has been further assumed to start with a low Mach number of M1 = 0.02. Pressure drop has been expressed as pressure ratio p r = p 2 p1 .
Following relations have been used to compute the values of the pressure ratio: p r = φ M1 , γ , f , L D
...(12.1)
p r = 0.02 2500 − 4 γ f L D
...(12.2)
Following values of the parameters have been covered: γ = 1.3, 1, 4
f = 0.002, 0.0025, 0.0030, 0.0035, 0.0040, 0.0050 L D = 10 − 2 × 105 These values of the pressure drop can be employed to obtain the magnitudes of the compressor power. 13. Isentropic compressor power The ideal power required to drive a compressor is given by:
c
h
c ′p T2 s − T1′ = Isentropic power P′ = m
b γ −1g γ − 1
c ′p T1′ p r P′ = m
...(13.1)
The pressure ratio is given by
b
p r = exit pressure entry pressure p 2 p1 = T2 s T1
g
b g
γ γ −1
...(13.2)
For the sake of generality following values have been used in the calculations: c ′p = 1.0 kJ kg K
T1′ = 300 K The actual power (for a given pressure ratio) can be calculated by multiplying the values in the table by the factor k given by: k = c p × T1 η0 T1′ = c p × T1 300 η0
where T1 = actual temperature of the gas at entry η0 = overall efficiency of the compressor and motor (prime mover) together.
...(13.3)
Introduction
13
14. Approximate values of pumping power for fans and blowers Fans and blower of various sizes are required to raise the pressure of a gas from a few millimetres of W.G. to large values. Table 14 presents approximate values of the power requirement of fans and blowers in a wide range. Power requirement for a specific application corresponding to values of Q and ∆p which do not appear in the tables can also be found with the help of available data. Following formulas have been used: ν Q=m
...(14.1)
P = Q ⋅ ∆p η 0
...(14.2)
Overall all efficiency η0 = 0.65 (assumed) P = Q ⋅ ∆p 66.261 kW
Q is in
m3/s
...(14.3)
; ∆p is in mm W.G.
15. Dimensions and units Principal dimensions and units occurring in gas dynamics are given in Table 15.1. Table 15.2 gives the conversion of some units.
Gas Tables for Compressible Flow Calculations
14
TABLE 1.1 Gas dynamic properties of air at various temperatures T
cp
cv
K
kJ/kg K
kJ/kg K
273 373 473 573 673
1.004 1.010 1.025 1.045 1.069
0.716 0.723 0.737 0.758 0.782
773 873 973 1073 1173
1.092 1.115 1.136 1.154 1.170
1273 1373 1473 1573 1673
1.185 1.197 1.208 1.218 1.227
γ
w T0 A∗ p0
R
a
kJ/kg K
m/s
1.402 1.397 1.390 1.379 1.367
0.288 0.288 0.288 0.287 0.287
332.0 387.4 435.3 476.2 513.8
0.0404 0.0403 0.0402 0.0400 0.0400
0.806 0.828 0.849 0.867 0.884
1.355 1.347 1.338 1.331 1.324
0.286 0.287 0.287 0.287 0.286
547.3 580.9 611.3 640.2 666.3
0.0400 0.0398 0.0396 0.0396 0.0397
0.898 0.910 0.920 0.930 0.940
1.320 1.315 1.313 1.309 1.305
0.287 0.287 0.288 0.288 0.287
694.3 719.9 746.3 770.3 791.7
0.0395 0.0395 0.0394 0.0393 0.0394
Gas Dynamic Properties of Some Gases and Vapours
15
TABLE 1.2 Gas dynamic properties of hydrogen at various temperatures T
cp
cv
K
kJ/kg K
kJ/kg K
273 373 473 573
14.195 14.448 14.503 14.534
10.071 10.324 10.380 10.409
673 773 873 973
14.581 14.663 14.779 14.930
1073 1173 1273 1373 1473 1573 1673 1773
γ
w T0 A∗ p0
R
a
kJ/kg K
m/s
1.409 1.399 1.397 1.396
4.124 4.124 4.123 4.125
1260.0 1467.2 1650.7 1816.7
0.01068 0.01066 0.1065 0.01065
10.456 10.538 10.654 10.805
1.395 1.391 1.387 1.381
4.125 4.125 4.125 4.125
1967.9 2106.4 2235.0 2355.0
0.01065 0.01064 0.01059 0.0106
15.115 15.313 15.518 15.736
10.991 11.188 11.393 11.611
1.375 1.368 1.362 1.355
4.124 4.125 4.125 4.125
2466.8 2573.4 2674.4 2770.5
0.01059 0.01058 0.01055 0.01053
15.950 16.166 16.370 16.565
11.825 12.040 12.245 12.440
1.349 1.342 1.337 1.331
4.125 4.126 4.125 4.125
2862.8 2952.0 3037.4 3120.7
0.01052 0.0105 0.0105 0.01047
Gas Tables for Compressible Flow Calculations
16
TABLE 1.3 Gas dynamic properties of carbon dioxide at various temperatures T
cp
cv
K
kJ/kg K
kJ/kg K
273 373 473 573
0.815 0.914 0.993 1.057
0.626 0.725 0.804 0.868
673 773 873 973
1.110 1.155 1.192 1.223
1073 1173 1273 1373
γ
w T0 A∗ p0
R
a
kJ/kg K
m/s
1.302 1.261 1.237 1.218
0.189 0.189 0.189 0.189
259.2 298.1 332.6 363.1
0.0485 0.0481 0.0476 0.0473
0.921 0.966 1.003 1.034
1.205 1.195 1.188 1.182
0.189 0.189 0.189 0.189
391.5 417.9 442.8 466.4
0.0472 0.0470 0.0469 0.0469
1.249 1.272 1.290 1.306
1.060 1.082 1.101 1.117
1.178 1.176 1.172 1.169
0.189 0.190 0.189 0.189
488.8 511.8 530.9 550.8
0.0467 0.0466 0.0467 0.0466
1473 1573 1673 1773
1.319 1.332 1.342 1.350
1.130 1.142 1.152 1.161
1.167 1.166 1.164 1.162
0.189 0.190 0.190 0.189
570.1 590.4 608.5 624.2
0.0466 0.0465 0.0465 0.0465
1873 1973 2073 2173
1.357 1.364 1.369 1.374
1.168 1.175 1.181 1.185
1.162 1.161 1.159 1.159
0.189 0.189 0.188 0.189
641.3 657.9 672.1 690.0
0.0465 0.0466 0.0465 0.0464
Gas Dynamic Properties of Some Gases and Vapours
17
TABLE 1.4 Gas dynamic properties of superheated steam at p = 0.981 bar T
cp
cv
K
kJ/kg K
kJ/kg K
413 433 453 473 493
1.997 1.985 1.975 1.980 1.985
1.482 1.485 1.490 1.500 1.505
513 533 553 573 593
1.990 1.997 2.000 2.013 2.025
613 633 653 673 693
2.040 2.045 2.050 2.071 2.085
γ
w T0 A∗ p0
R
a
kJ/kg K
m/s
1.347 1.336 1.325 1.320 1.318
0.515 0.500 0.485 0.480 0.480
535.0 537.8 539.6 547.4 558.5
0.0297 0.0300 0.0305 0.0306 0.0306
1.515 1.525 1.536 1.548 1.556
1.313 1.310 1.302 1.300 1.300
0.475 0.472 0.464 0.465 0.469
565.6 574.0 578.0 588.6 601.6
0.0307 ‘0.0308 0.0311 0.0310 0.0308
1.571 1.584 1.594 1.607 1.619
1.298 1.291 1.286 1.288 1.287
0.469 0.461 0.456 0.464 0.466
611.0 613.8 618.8 634.4 644.9
0.0308 0.0310 0.0311 0.0308 0.0308
At p = 9.81 bar 453 493 533 573 613
2.500 2.278 2.190 2.152 2.139
1.738 1.658 1.624 1.616 1.620
1.438 1.374 1.348 1.332 1.320
0.762 0.620 0.566 0.536 0.519
704.6 648.0 637.8 639.5 648.1
0.0250 0.0273 0.0283 0.0291 0.0294
653 693 733 773 813
2.135 2.144 2.156 2.173 2.194
1.633 1.650 1.671 1.691 1.717
1.307 1.299 1.290 1.285 1.278
0.502 0.494 0.485 0.482 0.477
654.6 667.0 677.2 691.9 703.9
0.0298 0.0299 0.0302 0.0302 0.0303
853 893 933
2.215 2.240 2.265
1.742 1.767 1.792
1.271 1.267 1.264
0.473 0.473 0.473
716.3 731.7 746.8
0.0304 0.0304 0.0304
Gas Tables for Compressible Flow Calculations
18
TABLE 1.5 Gas dynamic properties of superheated Freon-21 at various temperatures T
cp
cv
K
kJ/kg K
kJ/kg K
γ
R
a
kJ/kg K
m/s
w T0 A∗ p0
At p = 0.10 bar 243 283 323 363 403
0.531 0.578 0.622 0.661 0.695
0.447 0.496 0.540 0.579 0.615
1.187 1.165 1.152 1.142 1.130
0.084 0.082 0.082 0.082 0.080
155.7 164.4 174.7 184.3 190.9
0.0705 0.0713 0.0733 0.0698 0.0705
At p = 1.0 bar 293 0.616 313 0.630 333 0.646 353 0.662 373 0.678
0.521 0.539 0.557 0.575 0.593
1.182 1.168 1.159 1.151 1.143
0.095 0.091 0.089 0.087 0.085
181.4 182.5 185.4 188.0 190.4
0.0662 0.0673 0.0679 0.0685 0.0691
At p = 10.0 bar 333 0.812 383 0.799 393 0.792 403 0.786 413 0.783 423 0.782 433 0.782 443 0.782 453 0.783 463 0.785 473 0.786
0.644 0.645 0.647 0.651 0.655 0.658 0.663 0.667 0.672 0.676 0.681
1.261 1.239 1.224 1.207 1.195 1.188 1.179 1.172 1.165 1.161 1.154
0.168 0.154 0.145 0.135 0.128 0.124 0.119 0.115 0.111 0.109 0.105
265.6 270.3 264.1 256.3 251.4 249.7 246.5 244.4 242.1 242.1 239.4
0.0508 0.0527 0.0541 0.0558 0.0566 0.0580 0.0590 0.0599 0.0608 0.0614 0.0624
At p = 20.0 bar 403 0.984 423 0.909 443 0.872 463 0.852 473 0.846
0.703 0.697 0.696 0.699 0.701
1.396 1.304 1.252 1.219 1.207
0.281 0.212 0.176 0.153 0.145
397.6 342.0 312.5 293.8 287.7
0.0408 0.0458 0.0459 0.0527 0.0538
Properties of Standard Atmosphere
19
TABLE 2 Properties of standard atmosphere Z
t
T
a
p
ρ
m
°C
K
m/s
bar
kg/m3
µ
Ns m
2
× 10 6
0 100 200 300 400 500
15.00 14.35 13.70 13.05 12.40 11.75
288.15 287.50 286.85 286.20 285.55 284.90
340.5 340.0 339.7 339.5 338.9 338.5
1.0133 1.000 0.989 0.978 0.966 0.955
1.225 1.213 1.202 1.190 1.178 1.167
17.93 17.90 17.88 17.84 17.80 17.77
600 700 800 900 1000
11.10 10.45 9.80 9.15 8.50
284.25 283.60 282.95 282.30 281.65
338.0 337.6 337.3 337.0 336.6
0.943 0.932 0.921 0.909 0.898
1.156 1.145 1.134 1.123 1.111
17.74 17.70 17.66 17.64 17.61
1100 1200 1300 1400 1500
7.85 7.20 6.55 5.90 5.25
281.00 280.35 279.70 279.05 278.40
336.1 335.7 335.4 335.0 334.5
0.888 0.877 0.866 0.856 0.845
1.100 1.090 1.079 1.070 1.058
17.58 17.55 17.50 17.48 17.45
1600 1700 1800 1900 2000
4.60 3.95 3.30 2.65 2.00
277.75 277.10 276.45 275.80 275.15
334.1 333.7 333.4 333.0 332.6
0.835 0.825 0.815 0.805 0.795
1.048 1.037 1.027 1.017 1.006
17.40 17.38 17.35 17.31 17.28
2100 2200 2300 2400 2500
1.35 0.70 0.05 −0.60 −1.25
274.50 273.85 273.20 272.55 271.90
332.2 332.0 331.5 331.0 330.6
0.785 0.775 0.766 0.756 0.747
0.996 0.986 0.976 0.967 0.957
17.25 17.22 17.17 17.14 17.12
Gas Tables for Compressible Flow Calculations
20
TABLE 2 Properties of standard atmosphere Z
t
T
a
p
ρ
m
°C
K
m/s
bar
kg/m3
µ
Ns m
2
× 10 6
2600 2700 2800 2900 3000
-1.90 -2.55 -3.20 -3.85 -4.50
271.25 270.60 269.95 269.30 268.65
330.2 329.7 329.4 329.0 328.7
0.737 0.727 0.719 0.710 0.701
0.948 0.938 0.929 0.920 0.909
17.09 17.05 17.01 16.98 16.95
3500 4000 4500 5000
-7.75 -11.00 -14.25 -17.50
265.40 262.15 258.90 255.65
326.7 324.7 322.7 320.7
0.657 0.616 0.577 0.540
0.863 0.819 0.777 0.736
16.80 16.62 16.45 16.27
5500 6000 6500 7000
-20.75 -24.00 -27.25 -30.50
252.40 249.15 245.90 242.65
318.7 316.5 314.5 312.4
0.505 0.472 0.440 0.411
0.697 0.660 0.624 0.589
16.10 15.94 15.76 15.60
7500 8000 8500 9000
-33.75 -37.00 -40.25 -43.50
239.40 236.15 232.90 229.65
310.03 308.2 306.0 304.0
0.383 0.356 0.331 0.307
0.557 0.525 0.495 0.466
15.42 15.24 15.06 14.89
9500 10000 11000 12000
-46.75 -50.00 -56.50 -56.50
226.40 223.15 216.65 216.65
301.8 299.6 295.2 295.2
0.285 0.264 0.226 0.193
0.439 0.413 0.364 0.311
14.72 14.54 14.17 14.17
13000 14000 15000 16000
-56.50 -56.50 -56.50 -56.50
216.65 216.65 216.65 216.65
295.2 295.2 295.2 295.2
0.165 0.141 0.120 0.103
0.266 0.227 0.194 0.165
14.17 14.17 14.17 14.17
17000 18000 19000 20000
-56.50 -56.50 -56.50 -56.60
216.65 216.65 216.65 216.65
295.2 295.2 295.2 295.2
0.0878 0.0751 0.0641 0.0548
0.141 0.121 0.103 0.088
14.17 14.17 14.17 14.17
Isentropic Flow of Perfect Gases
21
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A M
M*
T/T0
p/p0
A/A*
F/F*
A
*
p p0
0.00 0.02 0.04 0.06 0.08 0.10
0.000 0.0214 0.0429 0.064 0.086 0.107
1.000 0.999 0.999 0.999 0.999 0.998
1.000 0.999 0.998 0.997 0.995 0.994
∞ 29.268 14.644 9.774 7.342 5.885
∞ 23.324 11.679 7.805 5.874 4.720
∞ 29.261 14.629 9.751 7.312 5.848
0.12 0.14 0.16 0.18 0.20
0.1285 0.150 0.171 0.193 0.214
0.998 0.997 0.996 0.995 0.994
0.991 0.987 0.984 0.979 0.975
4.917 4.228 3.712 3.312 2.994
3.954 3.410 3.005 2.693 2.445
4.872 4.174 3.651 3.243 2.917
0.22 0.24 0.26 0.28 0.30
0.235 0.256 0.277 0.298 0.319
0.993 0.991 0.989 0.988 0.987
0.969 0.963 0.957 0.951 0.944
2.735 2.520 2.340 2.186 2.054
2.245 2.079 1.941 1.824 1.724
2.651 2.428 2.239 2.078 1.938
0.32 0.34 0.36 0.38 0.40
0.341 0.362 0.383 0.403 0.424
0.985 0.983 0.981 0.979 0.977
0.936 0.928 0.920 0.911 0.902
1.939 1.838 1.750 1.672 1.602
1.638 1.564 1.498 1.442 1.391
1.874 1.759 1.610 1.523 1.446
0.42 0.44 0.46 0.48 0.50
0.444 0.465 0.486 0.506 0.526
0.974 0.972 0.969 0.967 0.964
0.893 0.883 0.873 0.863 0.853
1.540 1.484 1.434 1.388 1.348
1.347 1.308 1.273 1.241 1.213
1.343 1.281 1.225 1.173 1.149
0.52 0.54 0.56 0.58 0.60
0.547 0.567 0.587 0.607 0.627
0.961 0.958 0.955 0.952 0.948
0.842 0.831 0.819 0.807 0.796
1.310 1.277 1.246 1.218 1.193
1.188 1.165 1.145 1.127 1.111
1.103 1.061 1.021 0.984 0.950
Gas Tables for Compressible Flow Calculations
22
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
0.62 0.64 0.66 0.68 0.70
0.646 0.666 0.686 0.705 0.725
0.945 0.942 0.938 0.935 0.932
0.784 0.772 0.760 0.748 0.735
1.170 1.149 1.130 1.113 1.097
1.097 1.084 1.072 1.062 1.052
0.918 0.887 0.859 0.832 0.807
0.72 0.74 0.76 0.78 0.80
0.744 0.763 0.782 0.800 0.820
0.928 0.924 0.920 0.916 0.912
0.723 0.710 0.697 0.685 0.673
1.083 1.070 1.059 1.048 1.039
1.044 1.037 1.030 1.025 1.020
0.783 0.760 0.738 0.718 0.698
0.82 0.84 0.86 0.88 0.90
0.838 0.857 0.875 0.893 0.911
0.908 0.904 0.900 0.896 0.892
0.659 0.647 0.634 0.621 0.608
1.032 1.025 1.018 1.013 1.009
1.015 1.012 1.009 1.006 1.004
0.680 0.663 0.645 0.629 0.614
0.92 0.94 0.96 0.98 1.00
0.929 0.947 0.965 0.983 1.000
0.887 0.883 0.878 0.874 0.870
0.596 0.583 0.571 0.558 0.546
1.006 1.003 1.001 1.0004 1.000
1.003 1.002 1.0006 1.0002 1.000
0.599 0.585 0.571 0.558 0.546
1.02 1.04 1.06 1.08 1.10
1.017 1.035 1.052 1.068 1.085
0.865 0.860 0.856 0.851 0.846
0.533 0.521 0.509 0.497 0.485
1.0003 1.001 1.003 1.005 1.008
1.0001 1.0006 1.001 1.002 1.003
0.534 0.522 0.511 0.500 0.489
1.12 1.14 1.16 1.18 1.20
1.102 1.118 1.135 1.151 1.167
0.842 0.837 0.832 0.827 0.822
0.474 0.462 0.451 0.439 0.428
1.012 1.016 1.021 1.026 1.032
1.005 1.006 1.008 1.010 1.012
0.479 0.469 0.460 0.450 0.442
A
*
Isentropic Flow of Perfect Gases
23
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
1.22 1.24 1.26 1.28 1.30
1.183 1.198 1.214 1.230 1.245
0.817 0.813 0.807 0.803 0.798
0.418 0.407 0.396 0.386 0.376
1.038 1.045 1.053 1.062 1.070
1.014 1.016 1.019 1.022 1.024
0.434 0.425 0.417 0.409 0.402
1.32 1.34 1.36 1.38 1.40
1.260 1.276 1.291 1.305 1.320
0.793 0.788 0.783 0.778 0.773
0.365 0.356 0.346 0.337 0.327
1.080 1.089 1.100 1.111 1.123
1.027 1.029 1.033 1.036 1.039
0.398 0.387 0.381 0.374 0.367
1.42 1.44 1.46 1.48 1.50
1.334 1.348 1.363 1.377 1.391
0.768 0.763 0.758 0.752 0.748
0.318 0.309 0.300 0.292 0.284
1.135 1.148 1.161 1.175 1.189
1.042 1.045 1.048 1.051 1.055
0.361 0.352 0.346 0.340 0.337
1.52 1.54 1.56 1.58 1.60
1.405 1.418 1.432 1.445 1.458
0.743 0.737 0.732 0.727 0.723
0.275 0.267 0.259 0.252 0.245
1.205 1.220 1.237 1.254 1.271
1.058 1.062 1.065 1.068 1.072
0.332 0.326 0.321 0.316 0.310
1.62 1.64 1.66 1.68 1.70
1.472 1.485 1.497 1.510 1.523
0.717 0.712 0.707 0.703 0.698
0.237 0.230 0.223 0.216 0.210
1.289 1.308 1.328 1.348 1.369
1.075 1.079 1.083 1.086 1.089
0.306 0.301 0.296 0.292 0.288
1.72 1.74 1.76 1.78 1.80
1.535 1.547 1.559 1.572 1.584
0.693 0.687 0.683 0.678 0.673
0.203 0.197 0.191 0.185 0.180
1.391 1.413 1.436 1.459 1.484
1.093 1.097 1.100 1.104 1.108
0.283 0.279 0.275 0.271 0.267
A
*
Gas Tables for Compressible Flow Calculations
24
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
1.82 1.84 1.86 1.88 1.90
1.595 1.607 1.618 1.629 1.641
0.688 0.663 0.658 0.654 0.648
0.174 0.168 0.163 0.158 0.153
1.509 1.535 1.562 1.589 1.618
1.111 1.115 1.118 1.122 1.125
0.263 0.259 0.255 0.252 0.248
1.92 1.94 1.96 1.98 2.00
1.652 1.663 1.674 1.685 1.696
0.644 0.639 0.634 0.629 0.625
0.148 0.144 0.139 0.135 0.131
1.647 1.678 1.708 1.740 1.773
1.128 1.132 1.136 1.139 1.143
0.245 0.241 0.238 0.235 0.231
2.02 2.04 2.06 2.08 2.10
1.706 1.717 1.727 1.737 1.747
0.620 0.616 0.611 0.606 0.602
0.127 0.122 0.118 0.114 0.111
1.807 1.842 1.877 1.914 1.951
1.146 1.150 1.153 1.156 1.160
0.228 0.225 0.222 0.219 0.216
2.12 2.14 2.16 2.18 2.20
1.757 1.767 1.776 1.786 1.796
0.597 0.593 0.588 0.583 0.579
0.107 0.104 0.100 0.097 0.094
1.990 2.029 2.071 2.133 2.156
1.163 1.166 1.169 1.173 1.175
0.213 0.211 0.208 0.205 0.203
2.22 2.24 2.26 2.28 2.30
1.805 1.815 1.824 1.833 1.842
0.575 0.571 0.566 0.562 0.558
0.091 0.088 0.085 0.082 0.080
2.199 2.245 2.292 2.339 2.388
1.179 1.183 1.186 1.189 1.192
0.200 0.197 0.195 0.192 0.190
2.32 2.34 2.36 2.38 2.40
1.851 1.859 1.868 1.877 1.885
0.553 0.549 0.545 0.541 0.537
0.077 0.074 0.072 0.069 0.067
2.439 2.490 2.543 2.598 2.654
1.196 1.198 1.202 1.205 1.208
0.187 0.185 0.183 0.180 0.178
A
*
Isentropic Flow of Perfect Gases
25
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
2.42 2.44 2.46 2.48 2.50
1.894 1.902 1.910 1.918 1.926
0.532 0.528 0.524 0.520 0.516
0.065 0.063 0.061 0.059 0.057
2.711 2.769 2.829 2.891 2.954
1.211 1.214 1.217 1.219 1.223
0.176 0.174 0.172 0.170 0.168
2.52 2.54 2.56 2.58 2.60
1.934 1.942 1.950 1.957 1.965
0.512 0.508 0.504 0.500 0.496
0.055 0.053 0.051 0.049 0.048
3.019 3.086 3.154 3.224 3.295
1.226 1.228 1.231 1.234 1.237
0.166 0.164 0.162 0.160 0.158
2.62 2.64 2.66 2.68 2.70
1.972 1.980 1.987 1.994 2.001
0.493 0.489 0.485 0.481 0.478
0.046 0.045 0.043 0.042 0.041
3.370 3.444 3.521 3.600 3.681
1.240 1.242 1.245 1.248 1.250
0.157 0.155 0.153 0.151 0.150
2.72 2.74 2.76 2.78 2.80
2.008 2.015 2.022 2.029 2.036
0.474 0.470 0.467 0.463 0.460
0.039 0.038 0.037 0.036 0.034
3.764 3.849 3.936 4.025 4.116
1.253 1.256 1.258 1.261 1.264
0.148 0.146 0.145 0.143 0.142
2.82 2.84 2.86 2.88 2.90
2.042 2.048 2.055 2.062 2.068
0.456 0.453 0.449 0.446 0.442
0.033 0.032 0.031 0.030 0.029
4.210 4.306 4.404 4.504 4.607
1.266 1.268 1.271 1.273 1.275
0.140 0.138 0.137 0.135 0.134
2.92 2.94 2.96 2.98 3.00
2.074 2.081 2.087 2.093 2.099
0.439 0.435 0.432 0.429 0.426
0.028 0.027 0.026 0.0256 0.025
4.713 4.821 4.931 5.044 5.160
1.278 1.281 1.283 1.285 1.288
0.133 0.131 0.129 0.128 0.127
A
*
Gas Tables for Compressible Flow Calculations
26
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
3.05 3.10 3.15 3.20 3.25
2.113 2.127 2.141 2.155 2.168
0.417 0.409 0.402 0.394 0.387
0.0227 0.0209 0.0193 0.0177 0.0163
5.461 5.781 6.119 6.477 6.857
1.293 1.298 1.304 1.309 1.315
0.124 0.121 0.118 0.115 0.112
3.30 3.35 3.40 3.45 3.50
2.181 2.193 2.205 2.217 2.228
0.379 0.373 0.366 0.359 0.352
0.0151 0.0139 0.0128 0.0118 0.011
7.258 7.684 8.133 8.608 9.110
1.319 1.325 1.329 1.334 1.338
0.109 0.107 0.104 0.101 0.099
3.55 3.60 3.65 3.70 3.75
2.239 2.250 2.261 2.271 2.281
0.346 0.339 0.334 0.327 0.322
0.010 0.0093 0.0086 0.0079 0.0073
9.640 10.200 10.792 11.416 12.075
1.343 1.347 1.351 1.356 1.360
0.097 0.095 0.093 0.092 0.088
3.80 3.85 3.90 3.95 4.00
2.290 2.299 2.308 2.318 2.326
0.316 0.310 0.305 0.299 0.294
0.0068 0.0063 0.0058 0.0054 0.005
12.769 13.502 14.274 15.087 15.940
1.363 1.367 1.371 1.375 1.378
0.086 0.085 0.083 0.081 0.079
4.05 4.10 4.15 4.20 4.25
2.335 2.343 2.351 2.359 2.367
0.289 0.284 0.279 0.274 0.270
4.6 × 10−3 4.27 × 10−3 3.9 × 10−3 3.68 × 10−3 3.41 × 10−3
16.846 17.796 18.796 19.850 20.953
1.382 1.385 1.388 1.391 1.395
0.077 0.076 0.075 0.073 0.072
4.30 4.35 4.40 4.45 4.50
2.374 2.381 2.383 2.395 2.402
0.265 0.261 0.256 0.252 0.248
3.17 × 10−3 2.94 × 10−3 2.73 × 10−3 2.54 × 10−3 2.36 × 10−3
22.116 23.338 24.622 25.971 27.390
1.397 1.400 1.403 1.406 1.408
0.070 0.068 0.067 0.066 0.065
A
*
Isentropic Flow of Perfect Gases
27
TABLE 3.1 Isentropic flow of perfect gases (γγ = 1.3) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
4.55 4.60 4.65 4.70 4.75
2.408 2.415 2.421 2.427 2.433
0.243 0.240 0.235 0.232 0.228
2.2 × 10−3 2.05 × 10−3 1.91 × 10−3 1.77 × 10−3 1.65 × 10−3
28.874 30.434 32.070 33.786 35.585
1.412 1.414 1.417 1.420 1.422
0.063 0.062 0.061 0.060 0.059
4.80 4.85 4.90 4.95 5.00
2.438 2.444 2.450 2.455 2.460
0.224 0.221 0.217 0.214 0.211
1.54 × 10−3 1.44 × 10−3 1.34 × 10−3 1.25 × 10−3 1.17 × 10−3
37.472 39.448 41.518 43.687 45.950
1.424 1.427 1.429 1.431 1.433
0.058 0.057 0.055 0.0547 0.054
6.00 7.00 8.00 9.00 10.00 ∞
2.543 2.598 2.635 2.662 2.680 2.769
0.156 0.119 0.094 0.076 0.063 0
120.100 285.300 625.000 1275.000 2438.000 ∞
1.468 1.491 1.508 1.520 1.527 1.565
0.038 0.029 0.023 0.018 0.015 0
32.0 × 10−5 10.0 × 10−5 3.6 × 10−5 1.4 × 10−5 0.6 × 10−5 0
Gas Tables for Compressible Flow Calculations
28
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
0.00 0.01 0.02 0.03 0.04 0.05
0.000 0.011 0.022 0.033 0.044 0.0548
1.000 0.999 0.999 0.999 0.999 0.999
1.000 0.999 0.999 0.999 0.998 0.998
∞ 57.874 28.942 19.300 14.482 11.592
∞ 45.65 22.834 15.232 11.435 9.158
∞ 57.869 28.934 19.288 14.465 11.571
0.06 0.07 0.08 0.09 0.10
0.0657 0.0766 0.0876 0.0985 0.1094
0.999 0.999 0.998 0.998 0.998
0.997 0.996 0.995 0.994 0.993
9.666 8.292 7.262 6.461 5.822
7.643 6.562 5.753 5.125 4.624
9.642 8.263 7.229 6.425 5.781
0.11 0.12 0.13 0.14 0.15
0.1204 0.1313 0.1422 0.1531 0.1640
0.9976 0.9971 0.9966 0.996 0.995
0.992 0.989 0.988 0.986 0.984
5.299 4.864 4.497 4.182 3.910
4.215 3.875 3.588 3.343 3.132
5.255 4.815 4.445 4.126 3.849
0.16 0.17 0.18 0.19 0.20
0.1750 0.186 0.1965 0.207 0.218
0.9949 0.9943 0.994 0.993 0.992
0.982 0.980 0.978 0.975 0.973
3.673 3.464 3.278 3.112 2.964
2.947 2.786 2.642 2.515 2.400
3.607 3.394 3.205 3.035 2.882
0.21 0.22 0.23 0.24 0.25
0.229 0.239 0.251 0.261 0.272
0.991 0.990 0.989 0.988 0.987
0.9697 0.967 0.964 0.961 0.957
2.829 2.707 2.597 2.495 2.403
2.297 2.205 2.120 2.043 1.973
2.744 2.618 2.503 2.398 2.300
0.26 0.27 0.28 0.29 0.30
0.283 0.294 0.304 0.315 0.325
0.986 0.9855 0.985 0.983 0.982
0.954 0.951 0.947 0.943 0.939
2.317 2.238 2.166 2.098 2.035
1.909 1.849 1.795 1.745 1.698
2.211 2.128 2.051 1.979 1.912
Isentropic Flow of Perfect Gases
29
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
0.31
0.336
0.981
0.936
1.976
1.655
1.849
0.32 0.33 0.34 0.35
0.347 0.357 0.368 0.378
0.979 0.978 0.977 0.976
0.932 0.927 0.923 0.918
1.922 1.871 1.823 1.778
1.614 1.577 1.542 1.509
1.790 1.735 1.683 1.634
0.36 0.37 0.38 0.39 0.40
0.389 0.400 0.410 0.421 0.431
0.975 0.973 0.972 0.970 0.969
0.914 0.909 0.905 0.900 0.895
1.735 1.696 1.658 1.623 1.590
1.479 1.450 1.424 1.398 1.375
1.587 1.543 1.501 1.462 1.424
0.41 0.42 0.43 0.44 0.45
0.442 0.452 0.463 0.473 0.483
0.967 0.966 0.964 0.963 0.961
0.891 0.886 0.881 0.876 0.870
1.558 1.529 1.500 1.474 1.448
1.353 1.332 1.312 1.294 1.276
1.388 1.354 1.322 1.290 1.261
0.46 0.47 0.48 0.49 0.50
0.494 0.504 0.514 0.524 0.534
0.959 0.958 0.956 0.954 0.952
0.865 0.859 0.854 0.848 0.843
1.425 1.402 1.380 1.360 1.340
1.259 1.244 1.230 1.216 1.203
1.232 1.205 1.178 1.154 1.129
0.51 0.52 0.53 0.54 0.55
0.545 0.555 0.565 0.575 0.585
0.951 0.948 0.947 0.945 0.943
0.837 0.832 0.826 0.820 0.814
1.321 1.303 1.287 1.270 1.255
1.190 1.178 1.167 1.157 1.147
1.106 1.084 1.062 1.042 1.022
0.56 0.57 0.58 0.59 0.60
0.595 0.605 0.615 0.625 0.635
0.941 0.939 0.937 0.935 0.933
0.808 0.802 0.796 0.790 0.784
1.240 1.226 1.213 1.200 1.188
1.138 1.129 1.121 1.113 1.105
1.002 0.984 0.966 0.948 0.932
A
*
Gas Tables for Compressible Flow Calculations
30
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
0.61 0.62 0.63 0.64 0.65
0.645 0.655 0.664 0.674 0.684
0.931 0.928 0.926 0.924 0.922
0.778 0.772 0.765 0.759 0.753
1.177 1.166 1.155 1.145 1.136
1.098 1.091 1.085 1.078 1.073
0.915 0.900 0.885 0.869 0.855
0.66 0.67 0.68 0.69 0.70
0.693 0.703 0.713 0.722 0.732
0.919 0.917 0.915 0.913 0.911
0.746 0.740 0.734 0.727 0.721
1.126 1.118 1.109 1.102 1.094
1.068 1.063 1.058 1.053 1.049
0.841 0.827 0.814 0.802 0.789
0.71 0.72 0.73 0.74 0.75
0.741 0.751 0.760 0.769 0.779
0.908 0.906 0.904 0.901 0.899
0.714 0.708 0.701 0.695 0.688
1.087 1.081 1.074 1.068 1.062
1.045 1.041 1.038 1.035 1.031
0.777 0.765 0.753 0.742 0.732
0.76 0.77 0.78 0.79 0.80
0.788 0.797 0.807 0.816 0.825
0.896 0.894 0.892 0.889 0.886
0.682 0.676 0.669 0.663 0.656
1.057 1.052 1.047 1.043 1.038
1.028 1.026 1.023 1.021 1.018
0.721 0.711 0.700 0.691 0.681
0.81 0.82 0.83 0.84 0.85
0.834 0.843 0.852 0.861 0.870
0.884 0.881 0.879 0.876 0.874
0.649 0.643 0.636 0.630 0.624
1.034 1.031 1.027 1.024 1.021
1.017 1.015 1.013 1.011 1.009
0.672 0.663 0.654 0.645 0.636
0.86 0.87 0.88 0.89 0.90
0.879 0.888 0.897 0.906 0.915
0.871 0.868 0.869 0.863 0.861
0.617 0.610 0.604 0.598 0.591
1.018 1.015 1.013 1.011 1.009
1.008 1.007 1.006 1.005 1.004
0.628 0.620 0.612 0.604 0.596
Isentropic Flow of Perfect Gases
31
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
0.91 0.92 0.93 0.94 0.95
0.923 0.932 0.941 0.949 0.958
0.858 0.855 0.853 0.849 0.847
0.585 0.578 0.572 0.566 0.559
1.007 1.006 1.004 1.003 1.002
1.003 1.0025 1.002 1.0014 1.0009
0.589 0.582 0.575 0.567 0.561
0.96 0.97 0.98 0.99 1.00
0.966 0.975 0.983 0.992 1.000
0.844 0.842 0.839 0.836 0.834
0.553 0.547 0.541 0.534 0.528
1.0014 1.0008 1.0003 1.0001 1.000
1.0006 1.0003 1.0001 1.000 1.000
0.554 0.547 0.541 0.535 0.528
1.01 1.02 1.03 1.04 1.05
1.0083 1.017 1.025 1.033 1.041
0.831 0.828 0.825 0.822 0.819
0.522 0.516 0.510 0.504 0.498
1.0001 1.0003 1.0007 1.0013 1.002
1.000 1.0001 1.0003 1.0005 1.0008
0.522 0.516 0.510 0.505 0.498
1.06 1.07 1.08 1.09 1.10
1.049 1.057 1.065 1.073 1.081
0.816 0.814 0.811 0.808 0.805
0.492 0.486 0.480 0.474 6.468
1.003 1.004 1.005 1.006 1.008
1.0012 1.0016 1.002 1.0025 1.003
0.493 0.488 0.483 0.477 0.472
1.11 1.12 1.13 1.14 1.15
1.089 1.097 1.105 1.113 1.120
0.802 0.799 0.796 0.794 0.791
0.463 0.457 0.451 0.445 0.439
1.009 1.011 1.013 1.015 1.017
1.0036 1.004 1.005 1.0057 1.006
0.467 0.462 0.457 0.452 0.447
1.16 1.17 1.18 1.19 1.20
1.128 1.136 1.143 1.151 1.158
0.788 0.785 0.782 0.779 0.776
0.434 0.428 0.423 0.418 0.412
1.019 1.022 1.025 1.028 1.030
1.007 1.008 1.009 1.010 1.011
0.443 0.438 0.434 0.429 0.425
A
*
Gas Tables for Compressible Flow Calculations
32
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
1.21 1.22 1.23 1.24 1.25
1.166 1.173 1.181 1.188 1.195
0.774 0.771 0.768 0.765 0.762
0.407 0.402 0.396 0.391 0.386
1.034 1.037 1.039 1.043 1.047
1.012 1.013 1.014 1.015 1.016
0.421 0.416 0.412 0.408 0.404
1.26 1.27 1.28 1.29 1.30
1.203 1.210 1.217 1.224 1.231
0.759 0.756 0.753 0.750 0.747
0.381 0.376 0.371 0.366 0.361
1.050 1.054 1.058 1.062 1.066
1.017 1.018 1.019 1.021 1.022
0.400 0.396 0.392 0.388 0.385
1.31 1.32 1.33 1.34 1.35
1.238 1.245 1.252 1.259 1.266
0.745 0.742 0.738 0.736 0.733
0.356 0.351 0.346 0.342 0.337
1.071 1.075 1.079 1.084 1.089
1.023 1.024 1.025 1.027 1.028
0.381 0.378 0.374 0.370 0.367
1.36 1.37 1.38 1.39 1.40
1.273 1.280 1.286 1.293 1.300
0.730 0.727 0.724 0.721 0.718
0.332 0.328 0.323 0.318 0.314
1.094 1.099 1.104 1.110 1.115
1.029 1.031 1.032 1.033 1.035
0.364 0.360 0.357 0.354 0.350
1.41 1.42 1.43 1.44 1.45
1.306 1.313 1.319 1.326 1.333
0.716 0.713 0.709 0.706 0.704
0.309 0.305 0.301 0.297 0.293
1.121 1.126 1.132 1.138 1.144
1.036 1.037 1.039 1.040 1.042
0.347 0.344 0.341 0.338 0.335
1.46 1.47 1.48 1.49 1.50
1.339 1.345 1.352 1.358 1.365
0.701 0.698 0.695 0.693 0.689
0.288 0.285 0.280 0.276 0.272
1.150 1.156 1.163 1.170 1.176
1.043 1.044 1.046 1.047 1.048
0.332 0.329 0.326 0.323 0.320
A
*
Isentropic Flow of Perfect Gases
33
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
1.51 1.52 1.53 1.54 1.55
1.371 1.377 1.383 1.389 1.395
0.687 0.684 0.681 0.678 0.675
0.268 0.265 0.261 0.257 0.253
1.183 1.189 1.197 1.204 1.211
1.050 1.052 1.053 1.055 1.056
0.317 0.315 0.312 0.309 0.307
1.56 1.57 1.58 1.59 1.60
1.402 1.408 1.414 1.419 1.425
0.673 0.670 0.667 0.664 0.661
0.249 0.246 0.242 0.238 0.235
1.219 1.227 1.234 1.242 1.250
1.057 1.059 1.061 1.062 1.063
0.304 0.302 0.300 0.297 0.294
1.61 1.62 1.63 1.64 1.65
1.431 1.437 1.443 1.448 1.454
0.658 0.656 0.653 0.650 0.647
0.232 0.228 0.225 0.222 0.218
1.258 1.267 1.275 1.284 1.292
1.065 1.067 1.068 1.069 1.071
0.292 0.289 0.287 0.285 0.282
1.66 1.67 1.68 1.69 1.70
1.460 1.466 1.471 1.477 1.483
0.645 0.642 0.639 0.636 0.634
0.215 0.212 0.208 0.206 0.203
1.301 1.310 1.319 1.328 1.337
1.073 1.074 1.076 l.077 1.078
0.280 0.278 0.275 0.273 0.271
1.71 1.72 1.73 1.74 1.75
1.485 1.494 1.499 1.504 1.509
0.631 0.628 0.626 0.623 0.620
0.200 0.196 0.194 0.191 0.187
1.347 1.357 1.367 1.376 1.387
1.080 1.082 1.083 1.085 1.086
0.269 0.267 0.265 0.262 0.260
1.76 1.77 1.78 1.79 1.80
1.515 1.520 1.526 1.531 1.536
0.617 0.615 0.612 0.609 0.607
0.185 0.182 0.179 0.177 0.174
1.397 1.407 1.418 1.428 1.439
1.087 1.089 1.091 1.092 1.094
0.258 0.256 0.254 0.252 0.250
Gas Tables for Compressible Flow Calculations
34
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
1.81 1.82 1.83 1.84 1.85
1.541 1.546 1.551 1.556 1.561
0.604 0.602 0.598 0.596 0.594
0.171 0.168 0.166 0.164 0.161
1.450 1.461 1.472 1.484 1.495
1.095 1.096 1.098 1.099 1.100
0.248 0.247 0.245 0.243 0.241
1.86 1.87 1.88 1.89 1.90
1.566 1.571 1.576 1.581 1.586
0.591 0.588 0.585 0.583 0.581
0.158 0.156 0.154 0.152 0.149
1.507 1.518 1.531 1.543 1.555
1.102 1.104 1.105 1.107 1.108
0.239 0.237 0.235 0.234 0.232
1.91 1.92 1.93 1.94 1.95
1.591 1.596 1.600 1.605 1.609
0.578 0.575 0.573 0.570 0.568
0.147 0.145 0.142 0.140 0.138
1.568 1.580 1.593 1.606 1.619
1.110 1.111 1.113 1.114 1.116
0.230 0.228 0.227 0.225 0.223
1.96 1.97 1.98 1.99 2.00
1.615 1.619 1.624 1.628 1.633
0.566 0.563 0.561 0.558 0.555
0.136 0.134 0.132 0.129 0.128
1.633 1.646 1.659 1.674 1.687
1.117 1.118 1.120 1.121 1.123
0.222 0.220 0.219 0.217 0.216
2.01 2.02 2.03 2.04 2.05
1.638 1.642 1.646 1.651 1.655
0.553 0.551 0.548 0.546 0.543
0.126 0.114 0.122 0.120 0.118
1.702 1.716 1.731 1.745 1.760
1.124 1.126 1.127 1.128 1.130
0.214 0.213 0.211 0.209 0.208
2.06 2.07 2.08 2.09 2.10
1.660 1.664 1.668 1.673 1.677
0.541 0.538 0.536 0.534 0.531
0.116 0.115 0.113 0.111 0.109
1.775 1.790 1.805 1.821 1.837
1.131 1.133 1.134 1.135 1.137
0.207 0.205 0.204 0.202 0.200
A
*
Isentropic Flow of Perfect Gases
35
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
2.11 2.12 2.13 2.14 2.15
1.681 1.685 1.689 1.694 1.698
0.529 0.527 0.524 0.522 0.519
0.108 0.106 0.1043 0.1027 0.1010
1.853 1.869 1.885 1.902 1.918
1.138 1.139 1.141 1.142 1.143
0.199 0.198 0.197 0.195 0.194
2.16 2.17 2.18 2.19 2.20
1.702 1.706 1.710 1.714 1.718
0.517 0.515 0.513 0.510 0.508
0.0996 0.0980 0.0965 0.095 0.0935
1.935 1.953 1.969 1.987 2.005
1.145 1.146 1.147 1.148 1.150
0.193 0.191 0.190 0.188 0.187
2.21 2.22 2.23 2.24 2.25
1.722 1.726 1.730 1.734 1.737
0.506 0.504 0.501 0.499 0.497
0.0921 0.0906 0.0892 0.0878 0.0865
2.023 2.041 2.059 2.078 2.096
1.151 1.153 1.154 1.155 1.156
0.186 0.185 0.184 0.183 0.181
2.26 2.27 2.28 2.29 2.30
1.741 1.745 1.749 1.753 1.756
0.495 0.492 0.490 0.488 0.486
0.0851 0.0838 0.0825 0.0812 0.0800
2.115 2.135 2.154 2.173 2.193
1.158 1.159 1.160 1.162 1.163
0.180 0.179 0.178 0.1765 0.175
2.31 2.32 2.33 2.34 2.35
1.760 1.764 1.767 1.771 1.775
0.484 0.482 0.479 0.477 0.475
0.0787 0.0775 0.0763 0.0751 0.0740
2.213 2.233 2.254 2.274 2.295
1.164 1.165 1.167 1.168 1.169
0.174 0.173 0.172 0.171 0.169
2.36 2.37 2.38 2.39 2.40
1.778 1.782 1.785 1.789 1.792
0.473 0.471 0.469 0.467 0.464
0.0728 0.0717 0.0706 0.0695 0.0684
2.316 2.338 2.359 2.381 2.403
1.170 1.172 1.173 1.174 1.175
0.168 0.167 0.166 0.165 0.164
A
*
Gas Tables for Compressible Flow Calculations
36
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
2.41 2.42 2.43 2.44 2.45
1.796 1.799 1.803 1.806 1.809
0.463 0.461 0.458 0.456 0.454
0.0673 0.0663 0.0653 0.0643 0.0633
2.425 2.448 2.471 2.494 2.517
1.176 1.178 1.179 1.180 1.181
0.163 0.162 0.161 0.160 0.159
2.46 2.47 2.48 2.49 2.50
1.813 1.816 1.819 1.823 1.826
0.452 0.450 0.448 0.446 0.444
0.0623 0.0613 0.0604 0.0595 0.0585
2.540 2.564 2.588 2.612 2.637
1.182 1.183 1.184 1.186 1.187
0.158 0.157 0.156 0.155 0.154
2.51 2.52 2.53 2.54 2.55
1.829 1.832 1.835 1.838 1.842
0.442 0.441 0.438 0.436 0.434
0.0576 0.0567 0.0558 0.0550 0.0542
2.662 2.686 2.712 2.737 2.763
1.188 1.189 1.190 1.191 1.192
0.153 0.152 0.151 0.150 0.149
2.56 2.57 2.58 2.59 2.60
1.845 1.848 1.851 1.854 1.857
0.433 0.431 0.429 0.427 0.425
0.0533 0.0525 0.0517 0.0509 0.0501
2.789 2.815 2.842 2.869 2.896
1.193 1.195 1.196 1.197 1.198
0.1487 0.1475 0.147 0.146 0.145
2.61 2.62 2.63 2.64 2.65
1.860 1.863 1.866 1.869 1.872
0.423 0.421 0.419 0.417 0.415
0.0494 0.0486 0.0478 0.0471 0.0464
2.923 2.951 2.980 3.007 3.306
1.199 1.200 1.201 1.202 1.203
0.144 0.143 0.1425 0.142 0.141
2.66 2.67 2.68 2.69 2.70
1.875 1.878 1.881 1.884 1.886
0.414 0.412 0.410 0.408 0.407
0.0457 0.0449 0.0443 0.0436 0.043
3.065 3.094 3.123 3.153 3.183
1.204 1.205 1.206 1.207 1.208
0.140 0.139 0.138 0.1375 0.137
A
*
Isentropic Flow of Perfect Gases
37
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
2.71 2.72 2.73 2.74 2.75
1.889 1.892 1.895 1.898 1.900
0.405 0.403 0.402 0.399 0.398
0.0423 0.0417 0.0410 0.0404 0.0398
3.213 3.244 3.275 3.306 3.337
1.209 1.210 1.211 1.212 1.213
0.136 0.135 0.134 0.1335 0.133
2.76 2.77 2.78 2.79 2.80
1.903 1.906 1.908 1.911 1.914
0.396 0.395 0.393 0.391 0.389
0.0392 0.0386 0.0379 0.0374 0.0369
3.369 3.402 3.434 3.467 3.500
1.214 1.215 1.216 1.217 1.218
0.132 0.131 0.130 0.1297 0.129
2.81 2.82 2.83 2.84 2.85
1.915 1.919 1.922 1.925 1.927
0.388 0.386 0.384 0.382 0.381
0.0363 0.0357 0.0352 0.0347 0.0342
3.534 3.567 3.602 3.636 3.670
1.219 1.220 1.221 1.222 1.223
0.128 0.1275 0.127 0.126 0.125
2.86 2.87 2.88 2.89 2.90
1.929 1.932 1.935 1.937 1.940
0.379 0.378 0.376 0.375 0.373
0.0336 0.0331 0.0326 0.0321 0.0317
3.706 3.741 3.777 3.813 3.850
1.224 1.225 1.226 1.227 1.228
0.1245 0.124 0.123 0.1225 0.122
2.91 2.92 2.93 2.94 2.95
1.942 1.945 1.947 1.950 1.952
0.371 0.369 0.368 0.367 0.365
0.0312 0.0307 0.0303 0.0298 0.0294
3.887 3.924 3.961 4.000 4.037
1.229 1.230 1.2305 1.231 1.232
0.121 0.120 0.1198 0.119 0.1185
2.96 2.97 2.98 2.99 3.00
1.955 1.957 1.959 1.962 1.964
0.363 0.362 0.360 0.358 0.357
0.0289 0.0285 0.0281 0.0276 0.0272
4.076 4.115 4.155 4.195 4.235
1.233 1.234 1.235 1.236 1.237
0.118 0.117 0.117 0.116 0.115
Gas Tables for Compressible Flow Calculations
38
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) A
p p0
M
M*
T/T0
p/p0
A/A*
F/F*
3.05 3.10 3.15 3.20 3.25
1.976 1.987 1.997 2.008 2.018
0.349 0.342 0.335 0.328 0.321
0.0253 0.0235 0.0218 0.0202 0.0188
4.441 4.657 4.884 5.121 5.369
1.241 1.246 1.249 1.253 1.257
0.112 0.109 0.106 0.104 0.101
3.30 3.35 3.40 3.45 3.50
2.028 2.037 2.047 2.056 2.064
0.315 0.308 0.302 0.296 0.289
0.0175 0.0163 0.0151 0.0141 0.0131
5.628 5.900 6.184 6.480 6.789
1.260 1.264 1.268 1.272 1.274
0.098 0.096 0.0935 0.091 0.089
3.55 3.60 3.65 3.70 3.75
2.073 2.081 2.089 2.095 2.104
0.284 0.278 0.273 0.268 0.262
7.113 7.450 7.802 8.169 8.552
1.278 1.281 1.284 1.287 1.289
0.087 0.085 0.083 0.081 0.079
3.80 3.85 3.90 3.95 4.00
2.111 2.118 2.125 2.132 2.138
0.257 0.252 0.247 0.243 0.238
8.63 × 10−3 8.06 × 10−3 7.53 × 10−3 7.04 × 10−3 6.592 × 10−3
8.950 9.366 9.800 10.249 10.719
1.292 1.295 1.298 1.300 1.303
0.077 0.075 0.0738 0.072 0.071
4.05 4.10 4.15 4.20 4.25
2.145 2.151 2.156 2.162 2.168
0.234 0.229 0.235 0.221 0.217
6.16 × 10−3 5.77 × 10−3 5.4 × 10−3 5.06 × 10−3 4.74 × 10−3
11.207 11.715 12.243 12.792 13.362
1.305 1.308 1.310 1.312 1.315
0.069 0.0676 0.066 0.0648 0.0635
4.30 4.35 4.40 4.45 4.50 5.00
2.173 2.178 2.184 2.188 2.194 2.236
0.213 0.209 0.205 0.202 0.198 0.167
4.45 × 10−3 4.17 × 10−3 3.92 × 10−3 3.68 × 10−3 3.46 × 10−3 1.89 × 10−3
13.955 14.571 15.210 15.872 16.562 25.000
1.317 1.319 1.321 1.323 1.325 1.341
0.062 0.0608 0.060 0.058 0.057 0.047
12.2 × 10−3 11.38 × 10−3 10.62 × 10−3 9.9 × 10−3 9.24 × 10−3
A
*
Isentropic Flow of Perfect Gases
39
TABLE 3.2 Isentropic flow of perfect gases (γγ = 1.4) M
M*
T/T0
6.00 7.00 8.00 9.00 10.00
2.295 2.333 2.359 2.377 2.390
0.122 0.093 0.072 0.058 0.047
∞
2.45
0
p/p0 63.3 × 10-5 24.2 × 10−5 10.2 × 10−5 4.74 × 10−5 2.36 × 10−5 0
A
p p0
A/A*
F/F*
53.180 104.143 190.110 327.189 536.00
1.365 1.381 1.392 1.399 1.404
0.033 0.025 0.0195 0.0156 0.0126
1.429
0
∞
A
*
Gas Tables for Compressible Flow Calculations
40
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) M
M*
T/T0
p/p0
A/A*
F/F*
A A
*
p p0
0.00 0.02 0.04 0.06 0.08 0.10
0.000 0.023 0.046 0.069 0.092 0.115
1.000 0.999 0.999 0.998 0.997 0.997
1.000 0.999 0.998 0.997 0.995 0.992
∞ 28.132 14.078 9.397 7.061 5.661
∞ 21.662 10.851 7.255 5.464 4.392
∞ 28.113 14.054 9.366 7.021 5.615
0.12 0.14 0.16 0.18 0.20
0.138 0.161 0.184 0.207 0.229
0.995 0.994 0.992 0.989 0.987
0.988 0.984 0.979 0.973 0.967
4.733 4.070 3.576 3.193 2.886
3.686 3.184 2.809 2.522 2.293
4.675 4.003 3.499 3.107 2.793
0.22 0.24 0.26 0.28 0.30
0.252 0.275 0.297 0.319 0.342
0.984 0.981 0.978 0.975 0.971
0.961 0.954 0.945 0.937 0.928
2.640 2.435 2.262 2.115 1.989
2.110 1.958 1.833 1.725 1.635
2.535 2.320 2.138 1.982 1.848
0.32 0.34 0.36 0.38 0.40
0.363 0.385 0.407 0.428 0.450
0.967 0.963 0.958 0.954 0.949
0.919 0.909 0.899 0.889 0.878
1.880 1.784 1.700 1.626 1.560
1.558 1.491 1.432 1.381 1.336
1.728 1.623 1.529 1.445 1.370
0.42 0.44 0.46 0.48 0.50
0.471 0.493 0.513 0.534 0.555
0.945 0.939 0.934 0.928 0.923
0.867 0.855 0.843 0.831 0.818
1.502 1.448 1.401 1.358 1.320
1.297 1.262 1.231 1.203 1.178
1.300 1.238 1.181 1.128 1.080
0.52 0.54 0.56 0.58 0.60
0.575 0.595 0.615 0.635 0.655
0.917 0.911 0.905 0.899 0.893
0.806 0.793 0.780 0.767 0.753
1.286 1.254 1.225 1.200 1.176
1.157 1.137 1.120 1.105 1.091
1.035 0.994 0.955 0.919 0.885
Isentropic Flow of Perfect Gases
41
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) F/F*
A
p p0
M
M*
T/T0
p/p0
A/A*
0.62 0.64 0.66 0.68 0.70
0.674 0.693 0.712 0.731 0.750
0.886 0.879 0.873 0.866 0.859
0.739 0.726 0.712 0.698 0.685
1.154 1.135 1.118 1.102 1.087
1.085 1.068 1.058 1.049 1.042
0.854 0.824 0.796 0.769 0.745
0.72 0.74 0.76 0.78 0.80
0.768 0.786 0.804 0.821 0.839
0.853 0.845 0.838 0.831 0.823
0.671 0.657 0.644 0.630 0.616
1.075 1.063 1.053 1.043 1.035
1.035 1.029 1.024 1.019 1.016
0.721 0.698 0.677 0.657 0.638
0.82 0.84 0.86 0.88 0.90
0.856 0.873 0.889 0.906 0.922
0.817 0.809 0.802 0.795 0.787
0.603 0.589 0.570 0.563 0.550
1.028 1.022 1.016 1.011 1.008
1.012 1.009 1.007 1.005 1.003
0.619 0.602 0.585 0.569 0.555
0.92 0.94 0.96 0.98 1.00
0.938 0.954 0.969 0.985 1.000
0.779 0.772 0.765 0.757 0.749
0.537 0.524 0.511 0.499 0.487
1.005 1.003 1.001 1.0003 1.000
1.002 1.001 1.0005 1.0001 1.000
0.539 0.525 0.512 0.499 0.487
1.02 1.04 1.06 1.08 1.10
1.015 1.029 1.044 1.058 1.072
0.742 0.735 0.727 0.720 0.712
0.475 0.463 0.451 0.440 0.428
1.0003 1.001 1.003 1.005 1.007
1.0001 1.0004 1.001 1.0015 1.002
0.475 0.463 0.452 0.442 0.432
1.12 1.14 1.16 1.18 1.20
1.086 1.100 1.113 1.126 1.139
0.705 0.698 0.690 0.682 0.675
0.418 0.407 0.396 0.385 0.375
1.010 1.014 1.018 1.022 1.027
1.003 1.005 1.006 1.007 1.008
0.421 0.412 0.402 0.393 0.385
A
*
Gas Tables for Compressible Flow Calculations
42
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) F/F*
A
p p0
M
M*
T/T0
p/p0
A/A*
1.22 1.24 1.26 1.28 1.30
1.152 1.164 1.176 1.188 1.200
0.668 0.661 0.654 0.647 0.638
0.365 0.355 0.346 0.336 0.327
1.032 1.038 1.044 1.051 1.057
1.010 1.011 1.013 1.015 1.017
0.376 0.368 0.361 0.353 0.346
1.32 1.24 1.36 1.38 1.40
1.212 1.224 1.235 1.246 1.257
0.632 0.625 0.618 0.612 0.604
0.318 0.309 0.300 0.293 0.284
1.065 1.073 1.081 1.089 1.098
1.018 1.021 1.022 1.024 1.026
0.338 0.332 0.325 0.318 0.312
1.42 1.44 1.46 1.48 1.50
1.268 1.278 1.289 1.299 1.308
0.598 0.591 0.585 0.578 0.570
0.276 0.268 0.261 0.253 0.246
1.108 1.117 1.127 1.138 1.148
1.028 1.030 1.032 1.035 1.036
0.306 0.300 0.294 0.289 0.283
1.52 1.54 1.56 1.58 1.60
1.319 1.329 1.338 1.348 1.356
0.565 0.558 0.552 0.545 0.538
0.239 0.233 0.226 0.220 0.214
1.160 1.171 1.183 1.195 1.207
1.038 1.041 1.043 1.045 1.047
0.278 0.273 0.267 0.263 0.258
1.62 1.64 1.66 1.68 1.70
1.365 1.375 1.384 1.393 1.400
0.533 0.527 0.521 0.515 0.508
0.208 0.202 0.196 0.191 0.185
1.220 1.234 1.247 1.261 1.275
1.049 1.051 1.053 1.055 1.057
0.253 0.248 0.244 0.239 0.236
1.72 1.74 1.76 1.78 1.80
1.409 1.417 1.425 1.433 1.440
0.503 0.498 0.492 0.486 0.480
0.179 0.175 0.169 0.165 0.160
1.290 1.305 1.320 1.336 1.351
1.060 1.062 1.064 1.065 1.067
0.232 0.228 0.224 0.220 0.216
A
*
Isentropic Flow of Perfect Gases
43
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) F/F*
A
p p0
M
M*
T/T0
p/p0
A/A*
1.82 1.84 1.86 1.88 1.90
1.448 1.456 1.464 1.471 1.477
0.475 0.469 0.464 0.459 0.453
0.155 0.151 0.147 0.143 0.138
1.368 1.385 1.402 1.419 1.435
1.069 1.072 1.073 1.075 1.077
0.213 0.209 0.206 0.202 0.199
1.92 1.94 1.96 1.98 2.00
1.485 1.492 1.498 1.505 1.511
0.448 0.443 0.438 0.433 0.428
0.135 0.131 0.127 0.124 0.120
1.455 1.474 1.493 1.511 1.530
1.079 1.081 1.083 1.085 1.086
0.196 0.193 0.189 0.187 0.184
2.02 2.04 2.06 2.08 2.10
1.518 1.525 1.531 1.537 1.542
0.424 0.418 0.414 0.405 0.404
0.117 0.114 0.110 0.107 0.104
1.551 1.571 1.592 1.613 1.632
1.088 1.090 1.092 1.094 1.095
0.181 0.178 0.175 0.173 0.170
2.12 2.14 2.16 2.18 2.20
1.548 1.554 1.560 1.566 1.570
0.400 0.396 0.391 0.387 0.380
0.101 0.098 0.096 0.093 0.091
1.656 1.678 1.700 1.723 1.745
1.097 1.099 1.101 1.102 1.104
0.168 0.165 0.163 0.160 0.158
2.22 2.24 2.26 2.28 2.30
1.577 1.582 1.587 1.592 1.596
0.378 0.374 0.370 0.366 0.361
0.088 0.085 0.083 0.081 0.078
1.769 1.793 1.817 1.842 1.865
1.106 1.107 1.108 1.110 1.111
0.156 0.153 0.151 0.149 0.147
2.32 2.34 2.36 2.38 2.40
1.603 1.607 1.612 1.617 1.620
0.358 0.354 0.350 0.346 0.341
0.077 0.075 0.073 0.071 0.068
1.893 1.918 1.945 1.971 1.995
1.113 1.115 1.116 1.118 1.120
0.145 0.143 0.141 0.139 0.137
A
*
Gas Tables for Compressible Flow Calculations
44
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) F/F*
A
p p0
M
M*
T/T0
p/p0
A/A*
2.42 2.44 2.46 2.48 2.50
1.626 1.631 1.635 1.639 1.642
0.338 0.335 0.331 0.328 0.323
0.067 0.065 0.063 0.062 0.060
2.025 2.053 2.081 2.109 2.135
1.121 1.122 1.123 1.124 1.125
0.135 0.133 0.131 0.129 0.128
2.52 2.54 2.56 2.58 2.60
1.648 1.652 1.656 1.660 1.663
0.321 0.317 0.314 0.311 0.306
0.058 0.057 0.055 0.053 0.052
2.168 2.197 2.228 2.258 2.285
1.127 1.128 1.130 1.131 1.132
0.126 0.124 0.123 0.121 0.120
2.62 2.64 2.66 2.68 2.70
1.668 1.672 1.676 1.680 1.682
0.304 0.301 0.298 0.295 0.291
0.051 0.049 0.048 0.047 0.046
2.321 2.353 2.385 2.417 2.445
1.134 1.135 1.136 1.137 1.138
0.118 0.117 0.115 0.114 0.112
2.72 2.74 2.76 2.78 2.80
1.687 1.690 1.694 1.697 1.700
0.288 0.285 0.282 0.279 0.276
0.045 0.044 0.042 0.041 0.040
2.484 2.518 2.552 2.587 2.615
1.139 1.141 1.142 1.143 1.145
0.111 0.109 0.108 0.107 0.106
2.82 2.84 2.86 2.88 2.90
1.704 1.707 1.710 1.713 1.714
0.274 0.271 0.268 0.265 0.262
0.039 0.038 0.037 0.036 0.035
2.660 2.694 2.731 2.768 2.800
1.146 1.147 1.148 1.149 1.150
0.104 0.103 0.101 0.100 0.099
2.92 2.94 2.96 2.98 3.00
1.720 1.723 1.726 1.728 1.730
0.260 0.258 0.225 0.252 0.250
0.034 0.0337 0.033 0.032 0.031
2.843 2.881 2.920 2.960 2.990
1.151 1.152 1.153 1.154 1.155
0.098 0.097 0.096 0.095 0.094
A
*
Isentropic Flow of Perfect Gases
45
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) A/A*
F/F*
A
p p0
M
M*
T/T0
p/p0
3.05 3.10 3.15 3.20 3.25
1.739 1.746 1.752 1.760 1.765
0.244 0.238 0.232 0.226 0.221
0.0294 0.0276 0.0259 0.0245 0.0230
3.100 3.205 3.312 3.423 3.536
1.157 1.159 1.162 1.164 1.166
0.0908 0.088 0.086 0.083 0.081
3.30 3.35 .3.40 3.45 3.50
1.771 1.776 1.782 1.787 1.790
0.216 0.211 0.206 0.201 0.196
0.0217 0.0204 0.0193 0.0182 0.0170
3.653 3.772 3.895 4.022 4.135
1.168 1.169 1.172 1.173 1.175
0.079 0.077 0.075 0.073 0.071
3.55 3.60 3.65 3.70 3.75
1.797 1.802 1.807 1.811 1.815
0.192 0.188 0.184 0.179 0.176
16.21 × 10−3 15.32 × 10−3 14.5 × 10−3 13.7 × 10−3 12.97 × 10−3
4.284 4.421 4.559 4.704 4.849
1.177 1.178 1.180 1.182 1.183
0.0693 0.0676 0.0659 0.0643 0.063
3.80 3.85 3.90 3.95 4.00
1.819 1.824 1.827 1.832 1.834
0.172 0.168 0.165 0.161 0.157
12.28 × 10−3 11.6 × 10−3 11.02 × 10−3 10.45 × 10−3 9.94 × 10−3
5.000 5.154 5.311 5.472 5.607
1.185 1.186 1.187 1.188 1.189
0.061 0.0598 0.0584 0.057 0.056
4.05 4.10 4.15 4.20 4.25
1.838 1.842 1.845 1.848 1.852
0.154 0.151 0.148 0.145 0.142
9.4 × 10−3 8.93 × 10−3 8.48 × 10−3 8.06 × 10−3 7.66 × 10−3
5.806 5.978 6.155 6.336 6.519
1.191 1.193 1.194 1.195 1.196
0.054 0.053 0.052 0.051 0.049
4.30 4.35 4.40 4.45 4.50
1.855 1.858 1.861 1.864 1.865
0.139 0.137 0.134 0.132 0.128
7.28 × 10−3 6.93 × 10−3 6.6 × 10−3 6.28 × 10−3 6.00 × 10−3
6.708 6.900 7.097 7.298 7.455
1.197 1.198 1.199 1.200 1.201
0.0487 0.0477 0.0467 0.0457 0.045
A
*
Gas Tables for Compressible Flow Calculations
46
TABLE 3.3 Isentropic flow of perfect gases (γγ = 1.67) F/F*
p p0
M*
T/T0
p/p0
4.55 4.60 4.65 4.70 4.75
1.869 1.871 1.874 1.876 1.878
0.126 0.124 0.122 0.119 0.117
5.7 × 10−3 5.4 × 10−3 5.2 × 10−3 4.9 × 10−3 4.7 × 10−3
7.712 7.925 8.143 8.365 8.591
1.202 1.203 1.204 1.205 1.206
0.044 0.043 0.042 0.041 0.040
4.80 4.85 4.90 4.95 5.00
1.881 1.883 1.885 1.887 1.888
0.115 0.113 0.111 0.109 0.107
4.5 × 10−3 4.3 × 10−3 4.1 × 10−3 3.9 × 10−3 3.7 × 10−3
8.822 9.058 9.298 9.543 9.720
1.206 1.207 1.208 1.208 1.209
0.039 0.0389 0.0381 0.0374 0.037
6.00 7.00 8.00 9.00 10.00
1.918 1.937 1.950 1.960 1.965
0.076 0.057 0.045 0.036 0.029
165.0 × 10−5 81.0 × 10−5 43.0 × 10−5 24.5 × 10−5 14.5 × 10−5
15.680 23.850 34.600 48.250 65.200
1.220 1.228 1.232 1.234 1.238
0.026 0.019 0.015 0.012 0.009
∞
1.995
0
∞
1.248
0
0
A/A*
A
M
A
*
Normal Shocks in Perfect Gases
47
TABLE 4.1 Normal shocks in perfect gases (γγ = 1.3) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.00 1.02 1.04 1.06 1.08 1.10
1.000 0.981 0.962 0.944 0.927 0.911
1.000 1.046 1.092 1.139 1.185 1.237
1.000 1.010 1.021 1.031 1.041 1.051
1.000 0.999 0.999 0.999 0.999 0.999
1.832 1.875 1.918 1.963 2.010 2.058
1.12 1.14 1.16 1.18 1.20
0.896 0.881 0.867 0.853 0.840
1.287 1.338 1.391 1.444 1.497
1.061 1.070 1.080 1.089 1.099
0.998 0.997 0.995 0.994 0.993
2.107 2.157 2.209 2.262 2.316
1.22 1.24 1.26 1.28 1.30
0.828 0.816 0.804 0.793 0.783
1.552 1.608 1.664 1.722 1.780
1.109 1.119 1.128 1.138 1.148
0.990 0.987 0.985 0.982 0.978
2.372 2.428 2.486 2.545 2.605
1.32 1.34 1.36 1.38 1.40
0.772 0.762 0.753 0.744 0.735
1.839 1.899 1.960 2.022 2.085
1.158 1.167 1.177 1.187 1.197
0.975 0.971 0.966 0.961 0.956
2.667 2.728 2.792 2.856 2.922
1.42 1.44 1.46 1.48 1.50
0.726 0.714 0.709 0.702 0.694
2.149 2.214 2.279 2.346 2.413
1.207 1.217 1.227 1.237 1.247
0.951 0.945 0.939 0.933 0.926
2.988 3.056 3.125 3.195 3.265
1.52 1.54 1.56 1.58 1.60
0.687 0.680 0.673 0.666 0.660
2.481 2.551 2.621 2.692 2.764
1.257 1.268 1.278 1.288 1.300
0.919 0.912 0.905 0.897 0.889
3.337 3.410 3.484 3.559 3.635
Gas Tables for Compressible Flow Calculations
48
TABLE 4.1 Normal shocks in perfect gases (γγ = 1.3) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.62 1.64 1.66 1.68 1.70
0.654 0.647 0.642 0.636 0.630
2.836 2.910 2.985 3.060 3.137
1.310 1.320 1.331 1.342 1.353
0.881 0.873 0.864 0.856 0.847
3.712 3.791 3.870 3.991 4.031
1.72 1.74 1.76 1.78 1.80
0.625 0.620 0.615 0.609 0.605
3.214 3.292 3.371 3.451 3.532
1.364 1.375 1.386 1.397 1.407
0.837 0.828 0.819 0.810 0.800
4.155 4.196 4.281 4.366 4.452
1.82 1.84 1.86 1.88 1.90
0.600 0.596 0.591 0.587 0.583
3.614 3.697 3.780 3.865 3.950
1.420 1.432 1.443 1.455 1.466
0.790 0.781 0.771 0.761 0.751
4.539 4.628 4.717 4.807 4.898
1.92 1.94 1.96 1.98 2.00
0.578 0.574 0.570 0.567 0.563
4.037 4.124 4.212 4.301 4.390
1.479 1.491 1.503 1.515 1.527
0.741 0.731 0.721 0.711 0.700
4.991 5.084 5.178 5.274 5.370
2.02 2.04 2.06 2.08 2.10
0.559 0.556 0.552 0.548 0.546
4.482 4.574 4.667 4.760 4.856
1.539 1.552 1.565 1.578 1.590
0.690 0.680 0.670 0.660 0.650
5.467 5.565 5.665 5.765 5.867
2.12 2.14 2.16 2.18 2.20
0.542 0.539 0.536 0.533 0.530
4.950 5.046 5.144 5.242 5.341
1.600 1.617 1.630 1.643 1.656
0.639 0.629 0.619 0.609 0.600
5.968 6.071 6.176 6.281 6.387
Normal Shocks in Perfect Gases
49
TABLE 4.1 Normal shocks in perfect gases (γγ = 1.3) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.22 2.24 2.26 2.28 2.30
0.527 0.524 0.522 0.519 0.516
5.441 5.542 5.643 5.746 5.849
1.669 1.683 1.697 1.711 1.725
0.590 0.580 0.571 0.561 0.552
6.494 6.603 6.712 6.822 6.934
2.32 2.34 2.36 2.38 2.40
0.514 0.511 0.508 0.506 0.505
5.954 6.059 6.166 6.273 6.380
1.738 1.753 1.767 1.781 1.797
0.542 0.533 0.523 0.515 0.505
7.045 7.158 7.272 7.387 7.504
2.42 2.44 2.46 2.48 2.50
0.502 0.499 0.497 0.495 0.493
6.489 6.599 6.711 6.822 6.935
1.810 1.825 1.840 1.854 1.869
0.496 0.487 0.478 0.469 0.460
7.621 7.740 7.857 7.977 8.098
2.52 2.54 2.56 2.58 2.60
0.491 0.489 0.487 0.485 0.483
7.050 7.163 7.278 7.394 7.512
1.885 1.899 1.915 1.930 1.945
0.452 0.444 0.436 0.427 0.420
8.220 8.343 8.467 8.592 8.715
2.62 2.64 2.66 2.68 2.70
0.481 0.479 0.477 0.475 0.474
7.629 7.748 7.865 7.988 8.770
1.962 1.977 1.993 2.009 2.025
0.411 0.404 0.396 0.388 0.381
8.845 8.973 9.101 9.231 9.361
2.72 2.74 2.76 2.78 2.80
0.472 0.470 0.468 0.467 0.466
8.233 8.351 8.481 8.606 8.733
2.042 2.058 2.074 2.091 2.107
0.374 0.366 0.359 0.352 0.345
9.494 9.626 9.760 9.895 10.031
Gas Tables for Compressible Flow Calculations
50
TABLE 4.1 Normal shocks in perfect gases (γγ = 1.3) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.82 2.84 2.86 2.88 2.90
0.464 0.462 0.461 0.459 0.458
8.859 8.987 9.116 9.246 9.380
2.124 2.141 2.158 2.175 2.194
0.338 0.332 0.325 0.318 0.313
10.167 10.305 10.444 10.583 10.725
2.92 2.94 2.96 2.98 3.00
0.456 0.455 0.454 0.452 0.451
9.508 9.641 9.774 9.908 10.041
2.210 2.227 2.245 2.263 2.280
0.306 0.300 0.294 0.288 0.282
16.865 11.008 11.151 11.295 11.450
3.05 3.10 3.15 3.20 3.25
0.448 0.445 0.442 0.439 0.436
10.385 10.733 11.086 11.445 11.809
2.325 2.371 2.418 2.465 2.513
0.268 0.255 0.242 0.229 0.218
11.735 12.183 12.563 13.027 13.421
3.30 3.35 3.40 3.45 3.50
0.434 0.431 0.428 0.426 0.424
12.180 12.556 12.937 13.325 13.720
2.561 2.611 2.661 2.711 2.762
0.207 0.196 0.186 0.178 0.168
13.740 14.145 14.556 14.972 15.391
3.55 3.60 3.65 3.70 3.75
0.422 0.420 0.418 0.416 0.414
14.116 14.520 14.930 15.345 15.766
2.815 2.868 2.922 2.976 3.031
0.159 0.151 0.143 0.136 0.129
15.823 16.259 16.700 17.147 17.600
3.80 3.85 3.90 3.95 4.00
0.412 0.411 0.409 0.407 0.406
16.193 16.625 17.063 17.507 17.960
3.087 3.144 3.201 3.259 3.317
0.122 0.116 0.110 0.105 0.099
18.060 18.526 18.997 19.475 19.961
Normal Shocks in Perfect Gases
51
TABLE 4.1 Normal shocks in perfect gases (γγ = 1.3) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
4.05 4.10 4.15 4.20 4.25
0.404 0.403 0.401 0.400 0.398
18.411 18.872 19.338 19.810 20.288
3.378 3.438 3.498 3.561 3.623
0.094 0.089 0.085 0.081 0.077
20.449 20.945 21.447 21.956 22.470
4.30 4.35 4.40 4.45 4.50
0.397 0.396 0.395 0.394 0.393
20.771 21.260 21.755 22.255 22.760
3.686 3.750 3.815 3.880 3.945
0.073 0.069 0.066 0.062 0.0594
22.999 23.516 24.049 24.588 25.181
4.55 4.60 4.65 4.70 4.75
0.392 0.391 0.389 0.388 0.387
23.272 23.789 24.312 24.841 25.375
4.013 4.081 4.149 4.218 4.287
0.056 0.054 0.051 0.048 0.046
25.683 26.241 26.804 27.373 27.948
4.80 4.85 4.90 4.95 5.00
0.387 0.386 0.385 0.384 0.383
25.915 26.460 27.011 27.568 28.130
4.358 4.429 4.502 4.574 4.646
0.044 0.042 0.039 0.038 0.036
28.529 29.117 29.710 30.309 30.901
6.00 7.00 8.00 9.00 10.00
0.370 0.363 0.357 0.354 0.351
40.570 55.261 72.220 91.430 112.911
6.271 8.189 10.401 12.907 15.710
0.014 0.0061 0.00283 0.0014 0.00074
∞
0.340
∞
0
∞
44.320 60.140 78.400 99.101 122.25 ∞
Gas Tables for Compressible Flow Calculations
52
TABLE 4.2 Normal shocks in perfect gases (γγ = 1.4) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.00 1.02 1.04 1.06 1.08 1.10
1.000 0.981 0.962 0.945 0.928 0.911
1.000 1.047 1.095 1.144 1.194 1.245
1.000 1.013 1.026 1.039 1.052 1.065
1.000 0.999 0.999 0.999 0.999 0.998
1.893 1.938 1.984 2.032 2.082 2.133
1.12 1.14 1.16 1.18 1.20
0.897 0.882 0.868 0.855 0.842
1.297 1.349 1.403 1.458 1.513
1.078 1.090 1.103 1.115 1.128
0.998 0..997 0.996 0.994 0.993
2.185 2.238 2.294 2.350 2.407
1.22 1.24 1.26 1.28 1.30
0.830 0.818 0.807 0.796 0.786
1.570 1.627 1.685 1.745 1.805
1.141 1.153 1.166 1.178 1.191
0.991 0.988 0.985 0.983 0.979
2.466 2.526 2.587 2.650 2.714
1.32 1.34 1.36 1.38 1.40
0.776 0.766 0.757 0.748 0.740
1.866 1.928 1.991 2.055 2.120
1.204 1.216 1.229 1.242 1.255
0.976 0.972 0.967 0.963 0.958
2.778 2.845 2.911 2.979 3.049
1.42 1.44 1.46 1.48 1.50
0.731 0.723 0.716 0.708 0.701
2.186 2.253 2.320 2.389 2.458
1.267 1.281 1.294 1.307 1.320
0.953 0.947 0.942 0.936 0.930
3.120 3.192 3.265 3.338 3.413
1.52 1.54 1.56 1.58 1.60
0.694 0.687 0.681 0.675 0.668
2.529 2.600 2.673 2.746 2.820
1.334 1.347 1.361 1.374 1.388
0.923 0.917 0.909 0.903 0.895
3.489 3.567 3.645 3.725 3.805
Normal Shocks in Perfect Gases
53
TABLE 4.2 Normal shocks in perfect gases (γγ = 1.4) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.62 1.64 1.66 1.68 1.70
0.663 0.657 0.651 0.646 0.641
2.895 2.971 3.048 3.126 3.205
1.402 1.416 1.430 1.444 1.458
0.887 0.880 0.872 0.864 0.856
3.886 3.969 4.053 4.138 4.224
1.72 1.74 1.76 1.78 1.80
0.635 0.631 0.626 0.621 0.616
3.285 3.336 3.447 3.530 3.613
1.473 1.487 1.502 1.517 1.532
0.847 0.839 0.830 0.822 0.813
4.315 4.443 4.533 4.578 4.670
1.82 1.84 1.86 1.88 1.90
0.612 0.608 0.604 0.599 0.596
3.698 3.781 3.870 3.957 4.045
1.547 1.562 1.578 1.593 1.608
0.804 0.795 0.786 0.776 0.767
4.762 4.855 4.947 5.043 5.142
1.92 1.94 1.96 1.98 2.00
0.592 0.588 0.584 0.581 0.577
4.134 4.224 4.315 4.407 4.500
1.624 1.639 1.656 1.671 1.687
0.758 0.749 0.739 0.730 0.721
5.239 5.338 5.438 5.538 5.641
2.02 2.04 2.06 2.08 2.10
0.574 0.571 0.567 0.564 0.561
4.594 4.688 4.784 4.881 4.978
1.704 1.720 1.737 1.754 1.770
0.712 0.702 0.693 0.684 0.674
5.743 5.847 5.953 6.058 6.166
2.12 2.14 2.16 2.18 2.20
0.558 0.556 0.553 0.550 0.547
5.077 5.176 5.276 5.378 5.480
1.787 1.805 1.822 1.839 1.857
0.665 0.656 0.646 0.637 0.628
6.274 6.383 6.493 6.604 6.716
Gas Tables for Compressible Flow Calculations
54
TABLE 4.2 Normal shocks in perfect gases (γγ = 1.4) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.22 2.24 2.26 2.28 2.30
0.545 0.542 0.539 0.537 0.534
5.583 5.687 5.792 5.898 6.005
1.875 1.892 1.910 1.928 1.947
0.619 0.610 0.601 0.592 0.583
6.829 6.945 7.060 7.176 7.294
2.32 2.34 2.36 2.38 2.40
0.532 0.529 0.527 0.525 0.523
6.113 6.222 6.331 6.442 6.553
1.965 1.984 2.003 2.021 2.040
0.574 0.566 0.557 0.548 0.540
7.412 7.532 7.653 7.775 7.897
2.42 2.44 2.46 2.48 2.50
0.521 0.519 0.517 0.515 0.513
6.666 6.780 6.894 7.008 7.125
2.059 2.079 2.098 2.118 2.138
0.532 0.523 0.515 0.507 0.499
8.027 8.145 8.271 8.398 8.526
2.52 2.54 2.56 2.58 2.60
0.511 0.509 0.507 0.506 0.504
7.242 7.360 7.479 7.599 7.720
2.157 2.177 2.197 2.218 2.238
0.491 0.483 0.475 0.467 0.460
8.655 8.785 8.916 9.145 9.181
2.62 2.64 2.66 2.68 2.70
0.502 0.500 0.498 0.497 0.496
7.842 7.965 8.088 8.213 8.338
2.259 2.279 2.300 2.322 2.343
0.453 0.445 0.438 0.431 0.423
9.315 9.451 9.587 9.724 9.863
2.72 2.74 2.76 2.78 2.80
0.494 0.493 0.491 0.490 0.488
8.465 8.592 8.721 8.849 8.980
2.364 2.386 2.407 2.429 2.451
0.416 0.409 0.403 0.396 0.389
10.002 10.142 10.283 10.426 10.569
Normal Shocks in Perfect Gases
55
TABLE 4.2 Normal shocks in perfect gases (γγ = 1.4) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.82 2.84 2.86 2.88 2.90
0.487 0.485 0.484 0.483 0.481
9.111 9.243 9.376 9.510 9.645
2.473 2.495 2.518 2.540 2.563
0.383 0.376 0.370 0.364 0.358
10.714 10.859 11.006 11.154 11.302
2.92 2.94 2.96 2.98 3.00
0.480 0.478 0.477 0.476 0.475
9.781 9.917 10.055 10.194 10.333
2.586 2.609 2.632 2.656 2.679
0.352 0.346 0.339 0.334 0.328
11.452 11.603 11.754 11.907 12.061
3.05 3.10 3.15 3.20 3.25
0.472 0.469 0.467 0.464 0.462
10.686 11.045 11.409 11.780 12.156
2.738 2.798 2.860 2.922 2.985
0.315 0.301 0.288 0.276 0.264
12.450 12.846 13.247 13.656 14.071
3.30 3.35 3.40 3.54 3.50
0.459 0.457 0.455 0.453 0.451
12.538 12.926 13.320 13.791 14.125
3.049 3.114 3.180 3.247 3.315
0.253 0.243 0.232 0.222 0.213
14.492 14.920 15.354 15.795 16.242
3.55 3.60 3.65 3.70 3.75
0.449 0.447 0.445 0.444 0.442
14.536 14.953 15.376 15.805 16.239
3.384 3.454 3.525 3.596 3.669
0.204 0.195 0.187 0.179 0.172
16.700 17.156 17.715 18.095 18.575
3.80 3.85 3.90 3.95 4.00
0.441 0.439 0.438 0.436 0.435
16.680 17.126 17.578 18.036 18.500
3.743 3.817 3.893 3.969 4.047
0.164 0.157 0.151 0.145 0.138
19.060 19.553 20.051 20.556 21.068
Gas Tables for Compressible Flow Calculations
56
TABLE 4.2 Normal shocks in perfect gases (γγ = 1.4) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
4.05 4.10 4.15 4.20 4.25
0.434 0.432 0.431 0.430 0.429
18.969 19.445 19.926 20.413 20.906
4.125 4.205 4.285 4.367 4.449
0.133 0.127 0.122 0.117 0.113
21.586 22.111 22.642 23.179 23.723
4.30 4.35 4.40 4.45 4.50
0.428 0.427 0.426 0.425 0.424
21.405 21.909 22.420 22.936 23.458
4.532 4.616 4.702 4.788 4.875
0.108 0.104 0.099 0.095 0.0917
24.273 24.829 25.393 26.078 26.539
4.55 4.60 4.65 4.70 4.75
0.423 0.422 0.421 0.420 0.419
23.986 24.520 25.059 25.605 26.156
4.963 5.052 5.142 5.233 5.325
0.088 0.085 0.081 0.078 0.075
27.120 27.710 28.305 28.907 29.516
4.80 4.85 4.90 4.95 5.00
0.418 0.417 0.4167 0.416 0.415
26.713 27.276 27.845 28.419 29.000
5.418 5.512 5.607 5.703 5.800
0.072 0.069 0.067 0.0642 0.0617
30.130 30.751 31.379 32.013 32.654
6.00 7.00 8.00 9.00 10.000
0.404 0.397 0.393 0.389 0.388
41.833 57.000 74.500 94.333 116.500
7.941 10.469 13.387 16.693 20.388
0.0296 0.0154 0.0085 0.005 0.003
46.815 63.552 82.865 104.753 129.217
∞
0.378
∞
∞
0
∞
Normal Shocks in Perfect Gases
57
TABLE 4.3 Normal shocks in perfect gases (γγ = 1.67) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.00 1.02 1.04 1.06 1.08 1.10
1.000 0.981 0.962 0.945 0.928 0.913
1.000 1.051 1.102 1.155 1.208 1.262
1.000 1.020 1.040 1.059 l.078 1.098
1.000 0.999 0.999 0.999 0.999 0.999
2.055 2.107 2.161 2.217 2.274 2.333
1.12 1.14 1.16 1.18 1.20
0.898 0.884 0.871 0.858 0.846
1.318 1.375 1.432 1.490 1.551
1.118 1.137 1.157 1.176 1.195
0.998 0.997 0.996 0.995 0.993
2.393 2.455 2.518 2.584 2.649
1.22 1.24 1.26 1.28 1.30
0.835 0.824 0.813 0.805 0.794
1.611 1.672 1.735 1.798 1.863
1.215 1.234 1.253 1.273 1.293
0.991 0.989 0.987 0.984 0.981
2.718 2.787 2.857 2.929 3.002
1.32 1.34 1.36 1.38 1.40
0.784 0.775 0.767 0.758 0.751
1.928 1.995 2.062 2.131 2.201
1.312 1.332 1.352 1.372 1.394
0.978 0.975 0.971 0.967 0.963
3.076 3.152 3.229 3.308 3.388
1.42 1.44 1.46 1.48 1.50
0.743 0.736 0.729 0.723 0.716
2.271 2.342 2.415 2.488 2.562
1.412 1.433 1.453 1.475 1.497
0.958 0.953 0.948 0.943 0.938
3.468 3.551 3.634 3.719 3.805
1.52 1.54 1.56 1.58 1.60
0.709 0.703 0.697 0.692 0.687
2.638 2.715 2.792 2.871 2.950
1.516 1.537 1.560 1.580 1.605
0.933 0.927 0.921 0.915 0.908
3.892 3.981 4.071 4.162 4.255
Gas Tables for Compressible Flow Calculations
58
TABLE 4.3 Normal shocks in perfect gases (γγ = 1.67) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
1.62 1.64 1.66 1.68 1.70
0.681 0.676 0.671 0.667 0.662
3.031 3.112 3.195 3.278 3.365
1.624 1.646 1.668 1.691 1.715
0.902 0.896 0.889 0.882 0.875
4.347 4.442 4.538 4.635 4.734
1.72 1.74 1.76 1.78 1.80
0.657 0.653 0.649 0.645 0.641
3.448 3.535 3.622 3.711 3.800
1.736 1.760 1.783 1.806 1.832
0.868 0.861 0.854 0.847 0.840
4.833 4.933 5.036 5.139 5.244
1.82 1.84 1.86 1.88 1.90
0.637 0.633 0.630 0.626 0.623
3.891 3.982 4.075 4.168 4.266
1.853 1.878 1.903 1.927 1.954
0.832 0.824 0.817 0.809 0.802
5.349 5.456 5.564 5.673 5.785
1.92 1.94 1.96 1.98 2.00
0.619 0.616 0.613 0.610 0.607
4.358 4.454 4.552 4.651 4.753
1.976 2.002 2.027 2.053 2.080
0.794 0.786 0.778 0.771 0.763
5.895 6.008 6.122 6.237 6.355
2.02 2.-04 2.06 2.08 2.10
0.604 0.601 0.598 0.596 0.594
4.851 4.952 5.055 5.158 5.266
2.105 2.131 2.157 2.184 2.213
0.756 0.747 0.740 0.732 0.725
6.472 6.500 6.711 6.832 6.953
2.12 2.14 2.16 2.18 2.20
0.591 0.588 0.586 0.584 0.582
5.368 5.475 5.582 5.891 5.805
2.238 2.266 2.293 2.321 2.354
0.717 0.709 0.701 0.694 0.687
7.078 7.203 7.328 7.456 7.586
Normal Shocks in Perfect Gases
59
TABLE 4.3 Normal shocks in perfect gases (γγ = 1.67) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.22 2.24 2.26 2.28 2.30
0.579 0.577 0.575 0.573 0.571
5.911 6.022 6.135 6.248 6.367
2.378 2.405 2.435 2.464 2.500
0.678 0.671 0.664 0.656 0.649
7.714 7.845 7.977 8.110 8.245
2.32 2.34 2.36 2.38 2.40
0.569 0.567 0.565 0.563 0.562
6.478 6.595 6.713 6.831 6.953
2.523 2.553 2.583 2.613 2.651
0.641 0.634 0.627 0.619 0.614
8.379 8.517 8.654 8.794 8.935
2.42 2.44 2.46 2.48 2.50
0.560 0.558 0.556 0.555 0.554
7.071 7.193 7.315 7.438 7.567
2.674 2.705 2.736 2.767 2.805
0.605 0.598 0.591 0.584 0.578
9.004 9.218 9.362 9.507 9.651
2.52 2.54 2.56 2.58 2.60
0.552 0.550 0.548 0.547 0.546
7.689 7.815 7.943 8.071 8.205
2.831 2.863 2.895 2.928 2.968
0.571 0.564 0.557 0.551 0.545
9.800 9.949 10.098 10.249 10.401
2.62 2.64 2.66 2.68 2.70
0.544 0.543 0.541 0.540 0.539
8.333 8.463 8.595 8.729 8.870
2.994 3.027 3.061 3.094 3.138
0.538 0.531 0.525 0.518 0.514
10.556 10.709 10.865 11.022 11.180
2.72 2.74 2.76 2.78 2.80
0.538 0.536 0.535 0.534 0.533
8.995 9.135 9.273 9.411 9.557
3.163 3.197 3.232 3.267 3.311
0.506 0.500 0.494 0.488 0.483
11.339 11.500 11.662 11.824 12.000
Gas Tables for Compressible Flow Calculations
60
TABLE 4.3 Normal shocks in perfect gases (γγ = 1.67) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
2.82 2.84 2.86 2.88 2.90
0.532 0.531 0.529 0.528 0.527
9.691 9.833 9.976 10.118 10.270
3.337 3.373 3.409 3.445 3.492
0.476 0.471 0.465 0.459 0.455
12.153 12.319 12.487 12.656 12.831
2.92 2.94 2.96 2.98 3.00
0.526 0.525 0.524 0.523 0.523
10.408 10.555 10.703 10.851 11.010
3.518 3.555 3.593 3.630 3.680
0.448 0.443 0.438 0.433 0.428
12.997 13.169 13.342 13.517 13.690
3.05 3.10 3.15 3.20 3.25
0.520 0.518 0.516 0.514 0.511
11.379 11.763 12.154 12.551 12.954
3.763 3.860 3.958 4.053 4.160
0.415 0.402 0.390 0.378 0.367
14.137 14.589 15.048 15.515 15.989
3.30 3.35 3.40 3.45 3.50
0.509 0.508 0.506 0.505 0.504
13.363 13.779 14.201 14.620 15.075
4.262 4.367 4.473 4.581 4.705
0.357 0.346 0.336 0.326 0.318
16.471 16.960 17.456 17.959 18.471
3.55 3.60 3.65 3.70 3.75
0.502 0.500 0.498 0.497 0.496
15.504 15.951 16.404 16.864 17.330
4.800 4.912 5.026 5.141 5.258
0.309 0.298 0.290 0.282 0.274
18.988 19.514 20.047 20.588 21.135
3.80 3.85 3.90 3.95 4.00
0.495 0.494 0.493 0.492 0.491
17.801 18.279 18.764 19.255 19.761
5.376 5.496 5.618 5.741 5.886
0.266 0.258 0.251 0.244 0.239
21.691 22.253 22.823 23.400 23.990
Normal Shocks in Perfect Gases
61
TABLE 4.3 Normal shocks in perfect gases (γγ = 1.67) Mx
My
py /px
Ty /Tx
p0y /p0x
p0y /px
4.05 4.10 4.15 4.20 4.25
0.489 0.488 0.487 0.486 0.485
20.255 20.764 21.280 24.802 22.330
5.992 6.119 6.249 6.379 6.511
0.231 0.225 0.218 0.213 0.207
24.577 25.175 25.783 26.397 27.018
4.30 4.35 4.40 4.45 4.50
0.485 0.484 0.483 0.482 0.482
22.864 23.405 23.952 24.505 25.081
6.645 6.781 6.918 7.057 7.222
0.201 0.196 0.191 0.186 0.181
27.647 28.283 28.926 29.578 30.250
4.55 4.60 4.65 4.70 4.75
0.481 0.480 0.479 0.479 0.478
25.630 26.202 26.780 27.365 27.955
7.338 7.482 7.626 7.773 7.921
0.176 0.171 0.167 0.163 0.158
30.901 31.574 32.254 32.942 33.637
4.80 4.85 4.90 4.95 5.00
0.478 0.477 0.476 0.476 0.476
28.552 29.155 29.765 30.380 31.021
8.070 8.221 8.373 8.528 8.700
0.155 0.151 0.147 0.143 0.140
34.340 35.049 35.766 36.491 37.230
6.00 7.00 8.00 9.00 10.00
0.467 0.462 0.459 0.457 0.456
44.780 61.050 79.810 101.080 124.85
12.165 16.250 20.960 26.300 32.260
0.089 0.058 0.041 0.029 0.022
53.400 72.530 94.600 119.600 147.50
∞
0.448
∞
∞
0
∞
Gas Tables for Compressible Flow Calculations
62
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 2°) M1
σ
M1
σ
M1
σ
1.02 1.04 1.06 1.08 1.10
83.49 80.93 79.05 77.56 76.30
1.62 1.64 1.66 1.68 1.70
40.15 39.50 39.00 38.45 37.93
2.22 2.24 2.26 2.28 2.30
28.32 28.05 27.80 27.54 27.30
1.12 1.14 1.16 1.18 1.20
75.22 68.71 65.55 63.10 61.10
1.72 1.74 1.76 1.78 1.80
37.45 36.93 36.45 36.00 35.50
2.32 2.34 2.36 2.38 2.40
27.05 26.81 26.51 26.35 26.12
1.22 1.24 1.26 1.28 1.30
59.24 57.50 56.15 54.75 53.50
1.82 1.84 1.86 1.88 1.90
35.10 34.70 34.25 33.85 33.45
2.42 2.44 2.46 2.48 2.50
25.90 25.70 25.50 25.25 25.05
1.32 1.34 1.36 1.38 1.40
52.30 51.15 50.10 49.12 48.15
1.92 1.94 1.96 1.98 2.00
33.05 32.70 32.35 32.00 31.65
2.52 2.54 2.56 2.58 2.60
24.85 24.65 24.45 24.25 24.06
1.42 1.44 1.46 1.48 1.50
47.27 46.42 45.60 44.82 44.05
2.02 2.04 2.06 2.08 2.10
31.30 30.95 30.65 30.35 30.03
2.62 2.64 2.66 2.68 2.70
23.90 23.70 23.52 23.35 23.15
1.52 1.54 1.56 1.58 1.60
43.35 42.65 42.00 41.35 40.75
2.12 2.14 2.16 2.18 2.20
29.74 29.40 29.15 28.85 28.59
2.72 2.74 2.76 2.78 2.80
23.00 22.83 22.65 22.50 22.35
Oblique Shocks in Perfect Gases
63
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 2°) M1
σ
M1
σ
2.82 2.84 2.86 2.88 2.90
22.20 22.05 21.88 21.74 21.58
4.05 4.10 4.15 4.20 4.25
15.63 15.45 15.25 15.10 14.94
2.92 2.94 2.96 2.98 3.00
21.45 21.30 21.15 21.00 20.85
4.30 4.35 4.40 4.45 4.50
14.75 14.62 14.45 14.30 14.15
3.05 3.10 3.15 3.20 3.25
20.53 20.20 19.90 19.60 19.29
4.55 4.60 4.65 4.70 4.75
14.03 13.88 13.75 13.60 13.48
3.30 3.35 3.40 3.45 3.50
19.00 18.75 18.45 18.21 17.95
4.80 4.85 4.90 4.95 5.00
13.35 13.23 13.10 12.98 12.85
3.55 3.60 3.65 3.70 3.75
17.70 17.50 17.25 17.05 16.81
5.50 6.00 6.50 7.00 7.50
11.80 10.91 10.16 9.55 9.00
3.80 3.85 3.90 3.95 4.00
16.60 16.40 16.20 16.00 15.81
8.00 8.50 9.00 9.50 10.00
8.50 8.08 7.70 7.35 7.00
M1
σ
Gas Tables for Compressible Flow Calculations
64
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 6°) M1
σ
M1
σ
1.02 1.04 1.06 1.08 1.10
1.62 1.64 1.66 1.68 1.70
44.65 44.00 43.36 42.75 42.15
2.22 2.24 2.26 2.28 2.30
31.70 31.42 31.15 30.85 30.60
1.12 1.14 1.16 1.18 1.20
1.72 1.74 1.76 1.78 1.80
41.56 41.02 40.50 40.00 39.47
2.32 2.34 2.36 2.38 2.40
30.35 30.10 29.84 29.60 29.38
M1
σ
1.22 1.24 1.26 1.28 1.30
71.35 70.80 70.30 67.38 63.50
1.82 1.84 1.86 1.88 1.90
39.00 38.55 38.05 37.65 37.22
2.42 2.44 2.46 2.48 2.50
29.15 28.92 28.70 28.47 28.25
1.32 1.34 1.36 1.38 1.40
61.05 59.08 57.45 55.96 54.65
1.92 1.94 1.96 1.98 2.00
36.80 36.40 36.00 35.62 35.24
2.52 2.54 2.56 2.58 2.60
28.05 27.85 27.64 27.45 27.25
1.42 1.44 1.46 1.48 1.50
53.42 52.30 51.24 50.25 49.35
2.02 2.04 2.06 2.08 2.10
34.90 34.50 34.16 33.83 33.50
2.62 2.64 2.66 2.68 2.70
27.05 26.85 26.66 26.50 26.30
1.52 1.54 1.56 1.58 1.60
48.45 47.62 46.81 46.05 45.35
2.12 2.14 2.16 2.18 2.20
33.20 32.88 32.60 32.28 31.97
2.72 2.74 2.76 2.78 2.80
26.13 25.95 25.80 25.61 25.46
Oblique Shocks in Perfect Gases
65
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 6°) M1
σ
M1
σ
2.82 2.84 2.86 2.88 2.90
25.30 25.15 25.00 24.81 24.67
4.05 4.10 4.15 4.20 4.25
18.60 18.45 18.27 18.10 17.95
2.92 2.94 2.96 2.98 3.00
24.32 24.38 24.21 24.05 23.95
4.30 4.35 4.40 4.45 4.50
17.78 17.65 17.45 17.30 17.17
3.05 3.10 3.15 3.20 3.25
23.60 23.30 22.95 22.63 22.32
4.55 4.60 4.65 4.70 4.75
17.00 16.90 16.75 16.62 16.50
3.30 3.35 3.40 3.45 3.50
22.05 21.76 21.50 21.23 21.00
4.80 4.85 4.90 4.95 5.00
16.36 16.25 16.10 16.00 15.89
3.55 3.60 3.65 3.70 3.75
20.73 20.50 20.25 20.00 19.80
5.50 6.00 6.50 7.00 7.50
14.84 14.00 13.27 12.70 12.16
3.80 3.85 3.90 3.95 4.00
19.60 19.40 19.20 19.00 18.80
8.00 8.50 9.00 9.50 10.00
11.70 11.34 11.00 10.70 10.45
M1
σ
Gas Tables for Compressible Flow Calculations
66
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 10°) M1
σ
M1
σ
1.02 1.04 1.06 1.08 1.10
1.62 1.64 1.66 1.68 1.70
50.20 49.40 48.63 47.90 47.15
2.22 2.24 2.26 2.28 2.30
35.45 35.15 34.90 34.60 34.34
1.12 1.14 1.16 1.18 1.20
1.72 1.74 1.76 1.78 1.80
46.50 45.83 45.22 44.65 44.06
2.32 2.34 2.36 2.38 2.40
34.00 33.80 33.53 33.30 33.00
1.22 1.24 1..26 1.28 1.30
1.82 1.84 1.86 1.88 1.90
43.50 43.00 42.47 42.00 41.50
2.42 2.44 2.46 2.48 2.50
32.80 32.54 32.30 32.00 31.85
1.32 1.34 1.36 1.38 1.40
68.30 68.00 67.75
1.92 1.94 1.96 1.98 2.00
41.00 40.60 40.15 39.73 39.30
2.52 2.54 2.56 2.58 2.60
31.65 31.40 31.20 31.00 30.80
1.42 1.44 1.46 1.48 1.50
67.50 62.30 60.00 58.20 56.71
2.02 2.04 2.06 2.08 2.10
38.91 38.50 38.15 37.80 37.43
2.62 2.64 2.66 2.68 2.70
30.60 30.40 30.20 30.00 29.82
1.52 1.54 1.56 1.58 1.60
55.34 54.16 53.05 52.10 51.10
2.12 2.14 2.16 2.18 2.20
37.10 36.76 36.43 36.10 35.80
2.72 2.74 2.76 2.78 2.80
29.65 29.45 29.27 29.11 28.95
M1
σ
Oblique Shocks in Perfect Gases
67
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 10°) M1
σ
M1
σ
2.82 2.84 2.86 2.88 2.90
28.78 28.60 28.44 28.30 28.15
4.05 4.10 4.15 4.20 4.25
22.04 21.90 21.70 21.54 21.36
2.92 2.94 2.96 2.98 3.00
28.00 27.81 27.70 27.55 27.35
4.30 4.35 4.40 4.45 4.50
21.25 21.05 20.90 20.78 20.61
3.05 3.10 3.15 3.20 3.25
27.00 26.70 26.36 26.10 25.80
4.55 4.60 4.65 4.70 4.75
20.50 20.35 20.21 20.10 19.95
3.30 3.35 3.40 3.45 3.50
25.45 25.16 24.90 24.65 24.38
4.80 4.85 4.90 4.95 5.00
19.85 19.72 19.60 19.50 19.40
3.55 3.60 3.65 3.70 3.75
24.14 23.90 23.66 23.45 23.25
5.50 6.00 6.50 7.00 7.50
18.40 17.60 16.95 16.40 15.93
3.80 3.85 3.90 3.95 4.00
23.00 22.80 22.60 22.42 22.25
8.00 8.50 9.00 9.50 10.00
15.53 15.20 14.90 14.65 14.45
M1
σ
Gas Tables for Compressible Flow Calculations
68
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 20°) M1
σ
M1
σ
1.62 1.64 1.66 1.68 1.70
2.22 2.24 2.26 2.28 2.30
47.50 47.14 46.74 46.37 46.00
2.82 2.84 2.86 2.88 2.90
39.30 39.12 38.95 38.76 38.57
1.72 1.74 1.76 1.78 1.80
65.08 65.00 64.95
2.32 2.34 2.36 2.38 2.40
45.65 45.30 44.98 44.65 44.35
2.92 2.94 2.96 2.98 3.00
38.40 38.25 38.10 37.90 37.75
1.82 1.84 1.86 1.88 1.90
64.94 64.90 60.90 59.20 57.90
2.42 2.44 2.46 2.48 2.50
44.04 43.75 43.45 43.16 42.90
3.05 3.10 3.15 3.20 3.25
37.37 37.00 36.70 36.35 36.00
1.92 1.94 1.96 1.98 2.00
56.80 55.83 54.95 54.15 53.40
2.52 2.54 2.56 2.58 2.60
42.60 42.35 42.10 41.85 41.60
3.30 3.35 3.40 3.45 3.50
35.70 35.41 35.15 34.86 34.60
2.02 2.04 2.06 2.08 2.10
52.75 52.10 51.50 50.90 50.35
2.62 2.64 2.66 2.68 2.70
41.40 41.15 40.93 40.70 40.50
3.55 3.60 3.65 3.70 3.75
34.35 34.10 33.90 33.65 33.45
2.12 2.14 2.16 2.18 2.20
49.84 49.35 48.88 48.40 48.00
2.72 2.74 2.76 2.78 2.80
40.30 40.10 39.90 39.70 39.50
3.80 3.85 3.90 3.95 4.00
33.25 33.05 32.85 32.65 32.45
M1
σ
Oblique Shocks in Perfect Gases
69
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 20°) M1
σ
4.05 4.10 4.15 4.20 4.25
32.30 32.12 31.96 31.80 31.64
4.30 4.35 4.40 4.45 4.50
31.50 31.35 31.20 31.05 30.95
4.55 4.60 4.65 4.70 4.75
30.80 30.70 30.56 30.45 30.33
4.80 4.85 4.90 4.95 5.00
30.22 30.11 30.00 29.90 29.80
5.50 6.00 6.50 7.00 7.50
28.93 28.25 27.70 27.29 26.92
8.00 8.50 9.00 9.50 10.00
26.62 26.36 26.15 25.98 25.80
M1
σ
M1
σ
Gas Tables for Compressible Flow Calculations
70
TABLE 5.1 Oblique shocks in perfect gases Wave angles for given deflection (γγ = 1.4, δ = 30°) M1
σ
M1
σ
2.22 2.24 2.26 2.28 2.30
2.82 2.84 2.86 2.88 2.90
54.50 54.15 53.85 53.55 53.30
4.05 4.10 4.15 4.20 4.25
45.05 44.85 44.64 44.46 44.30
2.32 2.34 2.36 2.38 2.40
64.69 64.70 64.71
2.92 2.94 2.96 2.98 3.00
53.00 52.73 52.50 52.25 52.00
4.30 4.35 4.40 4.45 4.50
44.13 44.00 43.83 43.70 43.55
2.42 2.44 2.46 2.48 2.50
64.72 64.73 64.75 64.77 64.78
3.05 3.10 3.15 3.20 3.25
51.50 50.80 50.50 50.00 49.60
4.55 4.60 4.65 4.70 4.75
43.40 43.30 43.15 43.00 42.90
2.52 2.54 2.56 2.58 2.60
64.30 62.00 60.95 60.10 59.35
3.30 3.35 3.40 3.45 3.50
49.15 48.80 48.40 48.10 47.75
4.80 4.85 4.90 4.95 5.00
42.75 42.67 42.50 42.45 42.35
2.62 2.64 2.66 2.68 2.70
58.70 58.15 57.62 57.15 56.70
3.55 3.60 3.65 3.70 3.75
47.45 47.15 46.90 46.60 46.35
5.50 6.00 6.50 7.00 7.50
41.47 40.80 40.30 39.85 39.50
2.72 2.74 2.76 2.78 2.80
56.26 55.88 55.50 55.10 54.80
3.80 3.85 3.90 3.95 4.00
46.10 45.87 45.65 45.40 45.21
8.00 8.50 9.00 9.50 10.00
39.23 39.00 38.80 38.67 38.50
M1
σ
Oblique Shocks in Perfect Gases
71
TABLE 5.2 Oblique shocks in perfect gases (γγ = 1.4) Upstream and downstream Mach numbers and deflection from normal shock relations Mx
σ
M1
δ
My
M2
1.05 1.10 1.15 1.20 1.25
60 60 60 60 60
1.212 1.270 1.328 1.385 1.443
2.039 4.018 5.927 7.761 9.515
0.953 0.912 0.875 0.842 0.812
1.124 1.100 1.081 1.065 1.053
1.30 1.35 1.40 1.45 1.50
60 60 60 60 60
1.501 1.559 1.616 1.674 1.732
11.189 12.780 14.290 15.720 17.072
0.786 0.762 0.739 0.719 0.701
1.044 1.038 1.033 1.031 1.029
1.55 1.60 1.65 1.70 1.75
60 60 60 60 60
1.789 1.847 1.905 1.963 2.021
18.348 19.553 20.688 21.758 22.766
0.684 0.668 0.654 0.641 0.628
1.0293 1.030 1.032 1.035 1.038
1.80 1.85 1.90 1.95 2.00
60 60 60 60 60
2.078 2.136 2.194 2.252 2.309
23.715 24.609 25.453 26.246 26.995
0.616 0.605 0.596 0.586 0.577
1.042 1.045 1.050 1.055 1.060
2.10 2.20 2.30 2.40 2.50
60 60 60 60 60
2.425 2.540 2.656 2.771 2.887
28.368 29.591 30.685 31.664 32.543
0.561 0.547 0.534 0.523 0.513
1.070 1.081 1.091 1.102 1.112
2.60 2.70 2.80 2.90 3.00
60 60 60 60 60
3.002 3.117 3.233 3.348 3.464
33.334 34.049 34.696 35.283 35.817
0.504 0.495 0.488 0.481 0.475
1.123 1.133 1.142 1.151 1.160
Gas Tables for Compressible Flow Calculations
72
TABLE 5.2 Oblique shocks in perfect gases (γγ = 1.4) Mx
σ
M1
3.10 3.20 3.30 3.40 3.50
60 60 60 60 60
3.579 3.695 3.811 3.926 4.041
3.60 3.70 3.80 3.90 4.00
60 60 60 60 60
1.05 1.10 1.15 1.20 1.25
δ
My
M2
36.305 36.750 37.158 37.533 37.878
0.469 0.464 0.459 0.455 0.451
1.168 1.176 1.184 1.191 1.198
4.157 4.272 4.387 4.503 4.618
38.196 38.490 38.762 39.015 39.249
0.447 0.444 0.441 0.438 0.435
1.205 1.211 1.217 1.222 1.227
70 70 70 70 70
1.117 1.171 1.224 1.277 1.330
1.531 3.050 4.551 6.027 7.473
0.953 0.912 0.875 0.842 0.813
1.025 0.991 0.962 0.937 0.916
1.30 1.35 1.40 1.45 1.50
70 70 70 70 70
1.383 1.437 1.489 1.543 1.596
8.884 10.258 11.591 12.881 14.127
0.786 0.762 0.739 0.719 0.701
0.897 0.882 0.868 0.857 0.847
1.55 1.60 1.65 1.70 1.75
70 70 70 70 70
1.649 1.703 1.756 1.809 1.862
15.328 16.483 17.592 18.656 19.676
0.684 0.668 0.654 0.641 0.628
0.838 0.831 0.825 0.820 0.816
1.80 1.82 1.90 1.95 2.00
70 70 70 70 70
1.916 1.968 2.022 2.075 2.128
20.652 21.586 22.478 23.331 24.145
0.616 0.606 0.596 0.586 0.577
0.813 0.809 0.807 0.806 0.805
Oblique Shocks in Perfect Gases
73
TABLE 5.2 Oblique shocks in perfect gases (γγ = 1.4) Mx
σ
M1
2.10 2.20 2.30 2.40 2.50
70 70 70 70 70
2.235 2.341 2.447 2.554 2.660
2.60 2.70 2.80 2.90 3.00
70 70 70 70 70
3.10 3.20 3.30 3.40 3.50
δ
My
M2
25.664 27.047 28.308 29.456 30.503
0.561 0.547 0.534 0.523 0.513
0.8031 0.8028 0.8035 0.8047 0.806
2.767 2.873 2.979 3.086 3.193
31.459 32.332 33.132 33.865 34.537
0.504 0.496 0.488 0.481 0.475
0.8087 0.811 0.814 0.816 0.819
70 70 70 70 70
3.299 3.405 3.512 3.618 3.725
35.156 35.725 36.251 36.736 37.185
0.469 0.464 0.459 0.455 0.451
0.822 0.825 0.827 0.830 0.833
3.60 3.70 3.80 3.90 4.00
70 70 70 70 70
3.831 3.937 4.044 4.150 4.257
37.602 37.988 38.347 38.682 38.993
0.447 0.444 0.441 0.438 0.435
0.835 0.837 0.839 0.842 0.844
1.05 1.10 1.15 1.20 1.25
80 80 80 80 80
1.066 1.117 1.168 1.218 1.269
0.821 1.648 2.478 3.309 4.138
0.953 0.912 0.875 0.842 0.812
0.970 0.931 0.896 0.865 0.838
1.30 1.35 1.40 1.45 1.50
80 80 80 80 80
1.320 1.371 1.422 1.472 1.523
4.963 5.781 6.591 7.390 8.177
0.786 0.762 0.739 0.719 0.701
0.813 0.791 0.772 0.754 0.738
Gas Tables for Compressible Flow Calculations
74
TABLE 5.2 Oblique shocks in perfect gases (γγ = 1.4) Mx
σ
M1
δ
1.55 1.60 1.65 1.70 1.75
80 80 80 80 80
1.574 1.625 1.675 1.726 1.777
8.950 9.710 10.454 11.182 11.893
0.684 0.668 0.654 0.641 0.628
0.723 0.710 0.698 0.687 0.677
1.80 1.85 1.90 1.95 2.00
80 80 80 80 80
1.828 1.878 1.929 1.980 2.031
12.587 13.263 13.921 14.561 15.183
0.616 0.605 0.595 0.586 0.577
0.668 0.659 0.652 0.644 0.638
2.10 2.20 2.30 2.40 2.50
80 80 80 80 80
2.132 2.234 2.335 2.437 2.538
16.373 17.491 18.541 19.525 20.445
0.561 0.547 0.534 0.523 0.513
0.626 0.617 0.608 0.601 0.595
2.60 2.70 2.80 2.90 3.00
80 80 80 80 80
2.640 2.741 2.843 2.945 3.046
21.306 22.110 22.862 23.564 24.220
0.504 0.496 0.488 0.481 0.475
0.589 0.585 0.581 0.578 0.575
3.10 3.20 3.30 3.40 3.50
80 80 80 80 80
3.147 3.249 3.351 3.452 3.554
24.834 25.408 25.945 26.447 26.918
0.469 0.464 0.459 0.455 0.451
0.572 0.569 0.567 0.565 0.564
3.60 3.70 3.80 3.90 4.00
80 80 80 80 80
3.656 3.757 3.858 3.960 4.062
27.359 27.773 28.162 28.527 28.871
0.447 0.444 0.441 0.438 0.435
0.563 0.561 0.560 0.559 0.558
My
M2
Flow of Perfect Gases with Friction
75
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
0.02 0.04 0.06 0.08 0.10
53.617 26.806 17.868 13.398 10.716
0.021 0.043 0.064 0.086 0.107
1.150 1.150 1.150 1.149 1.148
29.268 14.644 9.775 7.342 5.885
23.324 11.679 7.805 5.874 4.720
1915.510 474.428 208.052 115.078 72.200
0.12 0.14 0.16 0.18 0.20
8.927 7.648 6.690 5.943 5.346
0.128 0.150 0.171 0.193 0.214
1.147 1.146 1.145 1.144 1.143
4.917 4.228 3.712 3.312 2.994
3.954 3.410 3.005 2.693 2.445
49.020 35.120 26.157 20.058 15.730
0.22 0.24 0.26 0.28 0.30
4.856 4.450 4.104 3.807 3.551
0.235 0.256 0.277 0.298 0.319
1.142 1.140 1.138 1.136 1.134
2.735 2.520 2.340 2.186 2.054
2.245 2.071 1.941 1.825 1.725
12.562 10.176 8.341 6.904 5.759
0.32 0.34 0.36 0.38 0.40
3.326 3.127 2.950 2.792 2.649
0.341 0.362 0.382 0.403 0.424
1.132 1.130 1.128 1.126 1.123
1.939 1.839 1.750 1.672 1.602
1.639 1.564 1.499 1.442 1.391
4.837 4.085 3.465 2.950 2.520
0.42 0.44 0.46 0.48 0.50
2.520 2.403 2.295 2.196 2.106
0.445 0.465 0.485 0.506 0.526
1.120 1.118 1.115 1.111 1.108
1.540 1.484 1.434 1.389 1.348
1.347 1.308 1.272 1.241 1.213
2.157 1.850 1.588 1.364 1.172
0.52 0.54 0.56 0.58 0.60
2.022 1.944 1.872 1.804 1.741
0.547 0.567 0.587 0.607 0.627
1.105 1.102 1.098 1.095 1.091
1.311 1.277 1.246 1.219 1.193
1.188 1.166 1.145 1.127 1.111
1.007 0.864 0.741 0.634 0.541
Gas Tables for Compressible Flow Calculations
76
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
0.62 0.64 0.66 0.68 0.70
1.682 1.626 1.574 1.525 1.479
0.646 0.667 0.686 0.705 0.725
1.087 1.083 1.079 1.075 1.071
1.170 1.149 1.130 1.113 1.097
1.097 1.084 1.072 1.062 1.052
0.460 0.390 0.329 0.276 0.231
0.72 0.74 0.76 0.78 0.80
1.435 1.393 1.354 1.316 1.280
0.744 0.763 0.782 0.800 0.820
1.067 1.063 1.058 1.054 1.049
1.083 1.070 1.059 1.048 1.039
1.044 1.037 1.030 1.025 1.020
0.191 0.157 0.127 0.102 0.081
0.82 0.84 0.86 0.88 0.90
1.246 1.214 1.183 1.153 1.125
0.838 0.857 0.875 0.893 0.911
1.045 1.040 1.035 1.030 1.025
1.032 1.025 1.019 1.013 1.009
1.016 1.012 1.009 1.006 1.004
0.062 0.047 0.035 0.024 0.016
0.92 0.94 0.96 0.98 1.00
1.098 1.072 1.047 1.023 1.000
0.929 0.947 0.965 0.983 1.000
1.020 1.015 1.010 1.005 1.000
1.006 1.003 1.001 1.0004 1.000
1.003 1.002 1.001 1.0002 1.000
0.010 0.005 0.0023 0.0005 0.000
1.02 1.04 1.06 1.08 1.10
0.978 0.956 0.936 0.916 0.897
1.017 1.035 1.051 1.068 1.085
0.995 0.989 0.984 0.978 0.973
1.0003 1.0014 1.003 1.005 1.008
1.0001 1.0006 1.001 1.002 1.003
0.0005 0.0020 0.0043 0.0074 0.0112
1.12 1.14 1.16 1.18 1.20
0.878 0.861 0.843 0.827 0.810
1.102 1.118 1.135 1.151 1.167
0.968 0.962 0.957 0.951 0.946
1.012 1.016 1.021 1.026 1.032
1.005 1.006 1.008 1.010 1.012
0.015 0.020 0.026 0.032 0.0382
Flow of Perfect Gases with Friction
77
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
1.22 1.24 1.26 1.28 1.30
0.795 0.780 0.765 0.751 0.737
1.183 1.198 1.214 1.230 1.245
0.940 0.934 0.928 0.923 0.917
1.038 1.046 1.053 1.062 1.070
1.014 1.017 1.019 1.022 1.024
0.045 0.052 0.059 0.066 0.074
1.32 1.34 1.36 1.38 1.40
0.723 0.710 0.697 0.685 0.673
1.260 1.275 1.290 1.305 1.320
0.912 0.906 0.900 0.894 0.889
1.080 1.089 1.100 1.111 1.123
1.027 1.030 1.033 1.036 1.039
0.082 0.089 0.098 0.106 0.114
1.42 1.44 1.46 1.48 1.50
0.662 0.650 0.639 0.629 0.618
1.334 1.349 1.363 1.377 1.391
0.883 0.877 0.871 0.866 0.860
1.135 1.147 1.160 1.175 1.189
1.042 1.045 1.048 1.052 1.055
0.123 0.131 0.139 0.148 0.156
1.52 1.54 1.56 1.58 1.60
0.608 0.598 0.588 0.579 0.569
1.404 1.418 1.432 1.445 1.458
0.854 0.848 0.842 0.837 0.831
1.205 1.220 1.237 1.254 1.271
1.058 1.062 1.065 1.068 1.072
0.165 0.173 0.182 0.190 0.199
1.62 1.64 1.66 1.68 1.70
0.561 0.552 0.543 0.535 0.527
1.472 1.485 1.497 1.510 1.523
0.825 0.819 0.814 0.808 0.802
1.290 1.308 1.328 1.348 1.369
1.076 1.079 1.083 1.086 1.089
0.207 0.216 0.224 0.233 0.241
1.72 1.74 1.76 1.78 1.80
0.519 0.511 0.503 0.496 0.488
1.535 1.547 1.559 1.572 1.584
0.796 0.791 0.785 0.780 0.774
1.391 1.413 1.436 1.459 l.484
1.093 1.097 1.100 1.104 1.108
0.249 0.257 0.265 0.273 0.281
Gas Tables for Compressible Flow Calculations
78
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
1.82 1.84 1.86 1.88 1.90
0.482 0.475 0.467 0.461 0.455
1.595 1.607 1.618 1.630 1.641
0.768 0.763 0.757 0.752 0.746
1.509 1.535 1.562 1.590 1.618
1.111 1.115 1.118 1.122 1.125
0.289 0.297 0.307 0.313 0.320
1.92 1.94 1.96 1.98 2.00
0.448 0.442 0.436 0.429 0.424
1.652 1.663 1.674 1.684 1.696
0.741 0.735 0.729 0.724 0.719
1.647 1.678 1.708 1.740 1.773
1.129 1.132 1.136 1.139 1.143
0.328 0.335 0.343 0.350 0.357
2.02 2.04 2.06 2.08 2.10
0.418 0.412 0.407 0.402 0.396
1.706 1.716 1.727 1.737 1.747
0.713 0.708 0.703 0.697 0.692
1.807 1.842 1.878 1.914 1.951
1.146 1.150 1.153 1.156 1.160
0.364 0.372 0.378 0.386 0.393
2.12 2.14 2.16 2.18 2.20
0.391 0.386 0.381 0.376 0.371
1.757 1.767 1.777 1.786 1.796
0.687 0.682 0.676 0.671 0.667
1.990 2.030 2.070 2.113 2.156
1.163 1.167 1.169 1.173 1.176
0.399 0.406 0.412 0.419 0.426
2.22 2.24 2.26 2.28 2.30
0.366 0.362 0.357 0.353 0.348
1.805 1.815 1.824 1.833 1.842
0.661 0.656 0.651 0.646 0.641
2.200 2.245 2.292 2.339 2.388
1.179 1.183 1.186 1.189 1.192
0.432 0.438 0.444 0.451 0.457
2.32 2.34 2.36 2.38 2.40
0.344 0.340 0.335 0.331 0.327
1.851 1.859 1.868 1.877 1.885
0.636 0.631 0.627 0.622 0.617
2.438 2.490 2.543 2.598 2.654
1.196 1.199 1.202 1.205 1.209
0.463 0.469 0.474 0.480 0.486
Flow of Perfect Gases with Friction
79
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
2.42 2.44 2.46 2.48 2.50
0.323 0.319 0.316 0.312 0.308
1.893 1.902 1.910 1.918 1.926
0.612 0.608 0.603 0.598 0.594
2.710 2.770 2.829 2.891 2.954
1.211 1.214 1.217 1.220 1.223
0.492 0.497 0.503 0.508 0.514
2.52 2.54 2.56 2.58 2.60
0.305 0.300 0.297 0.294 0.291
1.934 1.942 1.950 1.957 1.965
0.589 0.585 0.580 0.575 0.571
3.019 3.086 3.154 3.224 3.295
1.226 1.228 1.231 1.234 1.237
0.519 0.524 0.529 0.534 0.539
2.62 2.64 2.66 2.68 2.70
0.287 0.284 0.280 0.278 0.275
1.972 1.980 1.987 1.994 2.001
0.567 0.562 0.558 0.554 0.549
3.369 3.443 3.521 3.601 3.681
1.240 1.243 1.245 1.248 1.250
0.544 0.549 0.554 0.559 0.564
2.72 2.74 2.76 2.78 2.80
0.271 0.268 0.265 0.263 0.260
2.008 2.015 2.022 2.029 2.036
0.545 0.541 0.537 0.533 0.529
3.765 3.849 3.936 4.025 4.116
1.253 1.256 1.258 1.261 1.263
0.568 0.573 0.578 0.582 0.586
2.82 2.84 2.86 2.88 2.90
0.257 0.254 0.251 0.248 0.246
2.042 2.048 2.055 2.062 2.068
0.525 0.520 0.516 0.512 0.508
4.210 4.306 4.404 4.505 4.607
1.266 1.268 1.271 1.273 1.276
0.591 0.595 0.599 0.604 0.608
2.90 2.94 2.96 2.98 3.00
0.243 0.241 0.238 0.235 0.233
2.074 2.080 2.097 2.093 2.099
0.505 0.500 0.497 0.493 0.489
4.713 4.820 4.931 5.045 5.160
1.278 1.281 1.283 1.285 1.288
0.611 0.616 0.619 0.624 0.628
Gas Tables for Compressible Flow Calculations
80
TABLE 6.1 Flow of perfect gases with friction (Fanno flow, γ = 1.3)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
3.10 3.20 3.30 3.40 3.50
0.221 0.210 0.200 0.191 0.182
2.127 2.155 2.181 2.205 2.228
0.471 0.453 0.437 0.421 0.405
5.781 6.477 7.259 8.133 9.110
1.299 1.310 1.319 1.330 1.338
0.647 0.664 0.681 0.697 0.711
3.60 3.70 3.80 3.90 4.00
0.174 0.166 0.159 0.152 0.145
2.250 2.271 2.290 2.308 2.326
0.391 0.377 0.363 0.351 0.338
10.200 11.420 12.790 14.274 15.940
1.347 1.356 1.363 1.371 1.378
0.725 0.738 0.751 0.762 0.773
4.10 4.20 4.30 4.40 4.50
0.139 0.134 0.128 0.123 0.118
2.343 2.358 2.374 2.388 2.402
0.327 0.315 0.305 0.294 0.285
17.796 19.848 22.116 24.622 27.390
1.385 1.391 1.398 1.403 1.409
0.783 0.792 0.802 0.811 0.819
4.60 4.70 4.80 4.90 5.00
0.114 0.110 0.105 0.102 0.098
2.415 2.427 2.438 2.449 2.460
0.276 0.267 0.258 0.250 0.242
30.433 33.786 37.472 41.520 45.950
1.414 1.420 1.424 1.429 1.433
0.827 0.834 0.841 0.848 0.854
6.00 7.00 8.00 9.00 10.00
0.071 0.053 0.041 0.033 0.0277
2.543 2.598 2.635 2.662 2.681
0.179 0.138 0.108 0.088 0.072
1.468 1.491 1.507 1.519 1.527
0.904 0.936 0.957 0.972 0.983
120.1 285.3 625.2 1275.0 2438.0
Flow of Perfect Gases with Friction
81
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
0.02 0.04 0.06 0.08 0.10
54.770 27.382 18.251 13.684 10.943
0.022 0.044 0.066 0.087 0.109
1.199 1.199 1.199 1.198 1.197
28.942 14.482 9.666 7.262 5.822
22.834 11.435 7.643 5.753 4.624
1778.450 440.352 193.031 106.718 66.922
0.12 0.14 0.16 0.18 0.20
9.116 7.809 6.830 6.067 5.455
0.131 0.153 0.175 0.196 0.218
1.196 1.195 1.194 1.192 1.190
4.864 4.183 3.673 3.278 2.964
3.875 3.343 2.950 2.643 2.400
45.407 32.511 24.197 18.543 14.533
0.22 0.24 0.26 0.28 0.30
4.955 4.538 4.185 3.882 3.619
0.239 0.261 0.283 0.304 0.326
1.188 1.186 1.184 1.181 1.178
2.708 2.496 2.317 2.166 2.035
2.205 2.043 1.909 1.795 1.698
11.596 9.387 7.687 6.357 5.299
0.32 0.34 0.36 0.38 0.40
3.388 3.185 3.004 2.842 2.695
0.347 0.368 0.389 0.410 0.431
1.176 1.172 1.169 1.166 1.163
1.922 1.823 1.736 1.658 1.590
1.615 1.542 1.479 1.423 1.375
4.447 3.752 3.180 2.705 2.308
0.42 0.44 0.46 0.48 0.50
2.563 2.442 2.333 2.231 2.138
0.452 0.473 0.494 0.514 0.534
1.159 1.155 1.151 1.147 1.143
1.529 1.474 1.425 1.380 1.340
1.332 1.294 1.260 1.230 1.203
1.974 1.692 1.451 1.245 1.069
0.52 0.54 0.56 0.58 0.60
2.052 1.972 1.897 1.828 1.763
0.555 0.575 0.595 0.615 0.635
1.138 1.134 1.129 1.124 1.119
1.303 1.270 1.240 1.213 1.188
1.179 1.157 1.138 1.120 1.105
0.917 0.787 0.674 0.575 0.491
4 f Lmax D
Gas Tables for Compressible Flow Calculations
82
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
0.62 0.64 0.66 0.68 0.70
1.703 1.645 1.592 1.541 1.493
0.654 0.674 0.693 0.713 0.732
1.114 1.109 1.104 1.098 1.093
1.166 1.145 1.127 1.109 1.094
1.092 1.079 1.068 1.058 1.049
0.417 0.353 0.298 0.249 0.208
0.72 0.74 0.76 0.78 0.80
1.448 1.405 1.365 1.326 1.289
0.751 0.770 0.788 0.807 0.825
1.087 1.082 1.076 1.069 1.064
1.081 1.068 1.057 1.047 1.038
1.041 1.035 1.028 1.023 1.018
0.172 0.141 0.114 0.092 0.073
0.82 0.84 0.86 0.88 0.90
1.254 1.221 1.189 1.158 1.129
0.843 0.861 0.879 0.897 0.915
1.058 1.051 1.045 1.039 1.003
1.031 1.024 1.018 1.013 1.009
1.015 1.011 1.008 1.006 1.004
0.056 0.042 0.031 0.022 0.015
0.92 0.94 0.96 0.98 1.00
1.101 1.074 1.048 1.024 1.000
0.932 0.949 0.966 0.983 1.000
1.026 1.020 1.013 1.006 1.000
1.006 1.003 1.001 1.0003 1.000
1.003 1.001 1.0006 1.0001 1.000
0.0089 0.005 0.0020 0.0005 0.000
1.02 1.04 1.06 1.08 1.10
0.977 0.955 0.933 0.913 0.894
1.016 1.033 1.049 1.065 1.081
0.993 0.986 0.979 0.973 0.966
1.0003 1.0013 1.003 1.005 1.008
1.0001 1.0005 1.0012 1.002 1.003
0.00046 0.0018 0.004 0.0066 0.0099
1.12 1.14 1.16 1.18 1.20
0.875 0.856 0.838 0.821 0.804
1.097 1.113 1.128 1.143 1.158
0.959 0.952 0.945 0.938 0.932
1.011 1.015 1.020 1.025 1.030
1.004 1.006 1.007 1.009 1.011
0.0138 0.0182 0.023 0.028 0.034
Flow of Perfect Gases with Friction
83
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0/p0*
F/F*
1.22 1.24 1.26 1.28 1.30
0.788 0.773 0.757 0.743 0.728
1.173 1.188 1.202 1.217 1.231
0.925 0.918 0.911 0.903 0.897
1.037 1.043 1.050 1.058 1.066
1.013 1.015 1.017 1.019 1.022
0.039 0.045 0.0517 0.0582 0.065
1.32 1.34 1.36 1.38 1.40
0.715 0.701 0.688 0.676 0.663
1.245 1.259 1.273 1.286 1.300
0.889 0.883 0.876 0.869 0.862
1.075 1.084 1.094 1.104 1.115
1.024 1.027 1.029 1.032 1.035
0.0716 0.0785 0.0855 0.093 0.099
1.42 1.44 1.46 1.48 1.50
0.651 0.640 0.628 0.617 0.606
1.313 1.326 1.339 1.352 1.364
0.855 0.848 0.841 0.834 0.828
1.126 1.138 1.150 1.163 1.176
1.037 1.040 1.043 1.046 1.048
0.107 0.114 0.121 0.128 0.136
1.52 1.54 1.56 1.58 1.60
0.596 0.586 0.576 0.566 0.557
1.377 1.389 1.401 1.413 1.425
0.821 0.814 0.807 0.800 0.794
1.189 1.204 1.219 1.234 1.250
1.052 1.055 1.057 1.061 1.063
0.143 0.151 0.158 0.165 0.172
1.62 1.64 1.66 1.68 1.70
0.547 0.538 0.529 0.521 0.513
1.437 1.448 1.460 1.471 1.482
0.787 0.780 0.774 0.767 0.760
1.267 1.284 1.301 1.319 1.340
1.067 1.069 1.073 1.075 1.078
0.179 0.187 0.194 0.201 0.208
1.72 1.74 1.76 1.78 1.80
0.505 0.496 0.489 0.481 0.474
1.493 1.504 1.515 1.525 1.536
0.754 0.747 0.741 0.735 0.728
1.357 1.376 1.397 1.418 1.439
1.082 1.085 1.088 1.091 1.093
0.215 0.222 0.228 0.235 0.242
4 f Lmax D
Gas Tables for Compressible Flow Calculations
84
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
1.82 1.84 1.86 1.88 1.90
0.467 0.459 0.453 0.446 0.439
1.546 1.556 1.566 1.576 1.586
0.722 0.716 0.709 0.703 0.697
1.461 1.484 1.507 1.530 1.555
1.096 1.099 1.102 1.105 1.108
0.248 0.255 0.262 0.268 0.274
1.92 1.94 1.96 1.99 2.00
0.433 0.426 0.420 0.414 0.408
1.596 1.605 1.615 1.624 1.633
0.691 0.685 0.678 0.673 0.667
1.580 1.610 1.633 1.660 1.687
1.111 1.114 1.117 1.120 1.123
0.281 0.287 0.293 0.299 0.305
2.02 2.04 2.06 2.08 2.10
0.402 0.397 0.391 0.386 0.380
1.642 1.651 1.659 1.668 1.677
0.661 0.655 0.649 0.643 0.637
1.716 1.747 1.775 1.806 1.837
1.126 1.128 1.131 1.134 1.137
0.311 0.317 0.323 0.328 0.334
2.12 2.14 2.16 2.18 2.20
0.375 0.369 0.365 0.359 0.355
1.685 1.694 1.702 1.710 1.718
0.632 0.623 0.621 0.615 0.609
1.870 1.902 1.935 1.970 2.005
1.139 1.142 1.145 1.147 1.150
0.339 0.345 0.350 0.356 0.361
2.22 2.24 2.26 2.28 2.30
0.350 0.346 0.341 0.336 0.332
1.726 1.733 1.741 1.748 1.756
0.604 0.598 0.594 0.588 0.583
2.041 2.078 2.115 2.154 2.193
1.153 1.155 1.158 1.160 1.163
0.366 0.371 0.376 0.381 0.386
2.32 2.34 2.36 2.38 2.40
0.327 0.323 0.319 0.315 0.311
1.764 1.771 1.778 1.785 1.792
0.578 0.573 0.567 0.563 0.557
2.233 2.274 2.316 2.360 2.403
1.165 1.168 1.170 1.173 1.175
0.391 0.396 0.400 0.405 0.409
Flow of Perfect Gases with Friction
85
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
2.42 2.44 2.46 2.48 2.50
0.307 0.303 0.299 0.296 0.292
1.799 1.806 1.813 1.819 1.826
0.553 0.548 0.543 0.538 0.533
2.448 2.494 2.540 2.588 2.637
1.178 1.179 1.182 1.185 1.187
0.414 0.419 0.423 0.427 0.432
2.52 2.54 2.56 2.58 2.60
0.289 0.285 0.281 0.278 0.275
1.832 1.838 1.845 1.851 1.857
0.528 0.524 0.519 0.515 0.510
2.687 2.737 2.789 2.842 2.896
1.189 1.191 1.193 1.195 1.198
0.436 0.440 0.445 0.448 0.453
2.62 2.64 2.66 2.68 2.70
0.271 0.268 0.265 0.262 0..258
1.863 1.869 1.875 1.881 1.886
0.506 0.501 0.497 0.493 0.488
2.951 3.007 3.065 3.123 3.183
1.200 1.202 1.204 1.206 1.208
0.457 0.460 0.464 0.468 0.472
2.72 2.74 2.76 2.79 2.80
0.256 0.253 0.250 0.247 0.244
1.892 1.898 1.903 1.908 1.914
0.484 0.479 0.476 0.471 0.467
3.244 3.306 3.370 3.435 3.500
1.210 1.212 1.214 1.216 1.218
0.476 0.479 0.483 0.486 0.489
2.82 2.84 2.86 2.88 2.90
0.241 0.238 0.236 0.233 0.231
1.919 1.925 1.929 1.935 1.940
0.463 0.459 0.455 0.451 0.447
3.567 3.636 3.706 3.780 3.850
1.220 1.222 1.224 1.226 1.228
0.493 0.497 0.499 0.503 0.506
2.92 2.94 2.96 2.98 3.00
0.228 0.226 0.223 0.220 0.218
1.945 1.949 1.954 1.959 1.964
0.443 0.439 0.436 0.432 0.428
3.924 4.000 4.076 4.155 4.235
1.229 1.231 1.233 1.235 1.237
0.509 0.513 0.516 0.519 0.522
Gas Tables for Compressible Flow Calculations
86
TABLE 6.2 Flow of perfect gases with friction (Fanno flow, γ = 1.4)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
3.10 3.20 3.30 3.40 3.50
0.207 0.196 0.186 0.177 0.168
1.987 2.008 2.028 2.046 2.064
0.411 0.394 0.377 0.362 0.348
4.657 5.121 5.628 6.184 6.789
1.245 1.253 1.260 1.267 1..274
0.537 0.550 0.563 0.575 0.586
3.60 3.70 3.80 3.90 4.00
0.160 0.153 0.146 0.139 0.134
2.081 2.096 2.111 2.125 2.138
0.334 0.321 0.308 0.297 0.286
7.450 8.169 8.951 9.800 10.720
1.281 1.287 1.293 1.298 1.303
0.597 0.607 0.616 0.625 0.633
4.10 4.20 4.30 4.40 4.50
0.128 0.123 0.117 0.113 0.108
2.150 2.162 2.173 2.184 2.194
0.275 0.265 0.255 0.246 0.237
11.715 12.792 13.955 15.210 16.562
1.308 1.312 1.317 1.321 1.325
0.641 0.648 0.655 0.662 0.668
4.60 4.70 4.80 4.90 5.00
0.104 0.100 0.096 0.093 0.089
2.203 2.212 2.220 2.228 2.236
0.229 0.221 0.214 0.207 0.200
18.018 19.583 21.264 23.070 25.000
1.330 1.332 1.335 1.339 1.342
0.673 0.678 0.684 0.689 0.694
6.00 7.00 8.00 9.00 10.00
0.064 0.047 0.037 0.029 0.024
2.295 2.333 2.359 2.377 2.391
0.146 0.111 0.087 0.069 0.057
53.180 104.14 190.11 327.19 535.94
1.366 1.381 1.392 1.399 1.404
0.730 0.753 0.768 0.779 0.787
Flow of Perfect Gases with Friction
87
TABLE 6.3 Flow of perfect gases with friction (Fanno flow, γ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
0.02 0.04 0.06 0.08 0.10
57.753 28.862 19.235 14.420 11.535
0.023 0.046 0.069 0.092 0.115
1.333 1.332 1.332 1.331 1.330
28.132 14.078 9.397 7.061 5.661
21.662 10.851 7.256 5.465 4.392
1493.072 369.405 161.762 89.320 55.830
0.12 0.14 0.16 0.18 0.20
9.600 8.220 7.187 6.381 5.739
0.138 0.161 0.184 0.207 0.229
1.327 1.325 1.322 1.319 1.317
4.733 4.071 3.576 3.193 2.887
3.686 3.183 2.810 2.532 2.293
37.892 27.086 20.125 15.393 12.110
0.22 0.24 0.26 0.28 0.30
5.207 4.766 4.392 4.071 3.795
0.252 0.275 0.297 0.319 0.342
1.313 1.308 1.304 1.300 1.296
2.640 2.435 2.262 2.115 1.989
2.110 1.959 1.833 1.726 1.635
9.589 7.747 6.332 5.225 4.337
0.32 0.34 0.36 0.38 0.40
3.549 3.333 3.141 2.968 2.814
0.364 0.385 0.407 0.429 0.450
1.289 1.284 1.278 1.273 1.267
1.880 1.785 1.700 1.626 1.560
1.558 1.490 1.432 1.381 1.336
3.639 3.063 2.591 2.200 1.868
0.42 0.44 0.46 0.48 0.50
2.672 2.545 2.427 2.318 2.220
0.472 0.493 0.514 0.534 0.555
1.260 1.253 1.245 1.238 1.232
1.501 1.450 1.402 1.359 1.320
1.297 1.262 1.231 1.203 1.178
1.597 1.365 1.168 1.0005 0.855
0.52 0.54 0.56 0.58 0.60
2.127 2.042 1.962 1.888 1.819
0.575 0.596 0.616 0.635 0.655
1.223 1.215 1.807 1.200 1.191
1.286 1.255 1.225 1.200 1.176
1.157 1.138 1.120 1.105 1.091
0.734 0.628 0.536 0.457 0.388
4 f Lmax D
Gas Tables for Compressible Flow Calculations
88
TABLE 6.3 Flow of perfect gases with friction (Fanno flow, γ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
0.62 0.64 0.66 0.68 0.70
1.754 1.693 1.635 1.581 1.530
0.674 0.693 0.712 0.731 0.750
1.182 1.173 1.165 1.155 1.146
1.155 1.135 1.118 1.102 1.087
1.079 1.068 1.058 1.050 1.042
0.329 0.278 0.234 0.196 0.163
0.72 0.74 0.76 0.78 0.80
1.481 1.435 1.391 1.350 1.311
0.768 0.786 0.804 0.821 0.839
1.137 1.128 1.118 1.108 1.099
1.075 1.063 1.053 1.043 1.035
1.035 1.029 1.024 1.019 1.016
0.134 0.110 0.089 0.071 0.056
0.82 0.84 0.86 0.88 0.90
1.273 1.237 1.203 1.170 1.139
0.856 0.873 0.889 0.906 0.922
1.089 1.079 1.070 1.060 1.050
1.028 1.022 1.016 1.012 1.008
1.012 1.010 1.007 1.005 1.003
0.043 0.0325 0.0238 0.0167 0.011
0.92 0.94 0.96 0.98 1.00
1.108 1.080 1.052 1.026 1.000
0.938 0.954 0.969 0.985 1.000
1.040 1.030 1.020 1.010 1.000
1.005 1.003 1.0012 1.0003 1.000
1.002 1.001 1.0005 1.0001 1.000
0.007 0.0037 0.00156 0.00037 0.000
1.02 1.04 1.06 1.08 1.10
0.975 0.952 0.929 0.907 0.886
1.015 1.029 1.044 1.058 1.072
0.990 0.980 0.970 0.960 0.950
1.0003 1.0012 1.0026 1.005 1.007
1.0001 1.0004 1.001 1.0016 1.002
0.00035 0.0013 0.0029 0.005 0.0074
1.12 1.14 1.16 1.18 1.20
0.866 0.846 0.827 0.809 0.791
1.086 1.099 1.113 1.126 1.139
0.940 0.930 0.920 0.911 0.900
1.010 1.014 1.018 1.022 1.027
1.0034 1.0045 1.006 1.007 1.008
0.010 0.0135 0.017 0.021 0.025
Flow of Perfect Gases with Friction
89
TABLE 6.3 Flow of perfect gases with friction (Fanno flow, γ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
1.22 1.24 1.26 1.28 1.30
0.774 0.757 0.741 0.726 0.710
1.151 1.164 1.176 1.188 1.200
0.891 0.882 0.872 0.862 0.852
1.032 1.038 1.044 1.051 1.057
1.010 1.012 1.013 1.015 1.017
0.0291 0.0335 0.038 0.043 0.047
J.32 1.34 1.36 1.38 1.40
0.696 0.682 0.668 0.654 0.641
1.211 1.223 1.235 1.246 1.257
0.843 0.834 0.825 0.816 0.806
1.065 1.073 1.081 1.089 1.098
1.019 1.021 1.022 1.024 1.026
0.052 0.057 0.062 0.067 0.072
1.42 1.44 1.46 1.48 1.50
0.629 0.616 0.605 0.593 0.582
1.268 1.278 1.288 1.298 1.309
0.797 0.788 0.780 0.771 0.761
1.108 1.117 1.127 1.138 1.148
1.028 1.030 1.032 1.035 1.036
0.078 0.083 0.088 0.093 0.0975
1.52 1.54 1.56 1.58 1.60
0.571 0.560 0.550 0.540 0.530
1.318 1.328 1.338 1.347 1.356
0.753 0.745 0.736 0.728 0.718
1.160 1.171 1.183 1.195 1.207
1.039 1.041 1.043 1.045 1.047
0.103 0.108 0.113 0.118 0.123
1.62 1.64 1.66 1.68 1.70
0.521 0.511 0.502 0.493 0.485
1.365 1.374 1.383 1.391 1.400
0.711 0.703 0.695 0.687 0.678
1.220 1.234 1.250 1.261 1.275
1.049 1.051 1.053 1.056 1.057
0.128 0.133 0.138 0.142 0.146
1.72 1.74 1.76 1.78 1.80
0.476 0.468 0.460 0.452 0.445
1.408 1.416 1.425 1.432 1.440
0.671 0.664 0.656 0.648 0.640
1.290 1.305 1.320 1.336 1.351
1.060 1.0615 1.0635 1.065 1.067
0.152 0.156 0.161 0.165 0.169
4 f Lmax D
Gas Tables for Compressible Flow Calculations
90
TABLE 6.3 Flow of perfect gases with friction (Fanno flow, γ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
1.82 1.84 1.86 1.88 1.90
0.437 0.430 0.423 0.416 0.409
1.448 1.455 1.463 1.469 1.477
0.634 0.626 0.619 0.612 0.604
1.370 1.385 1.402 1.419 1.436
1.070 1.0715 1.073 1.075 1.077
0.174 0.178 0.183 0.187 0.191
1.92 1.94 1.96 1.98 2.00
0.403 0.396 0.390 0.384 0.378
1.484 1.491 1.497 1.504 1.510
0.598 0.591 0.585 0.578 0.571
1.455 1.474 1.492 1.511 1.530
1.079 1.081 1.083 1.085 1.086
0.195 0.199 0.203 0.207 0.211
2.02 2.04 2.06 2.08 2.10
0.372 0.366 0.361 0.355 0.350
1.517 1.523 1.530 1.535 1.541
0.565 0.558 0.552 0.545 0.539
1.551 1.571 1.592 1.613 1.632
1.088 1.090 1.092 1.094 1.095
0.215 0.219 0.222 0.226 0.229
2.12 2.14 2.16 2.18 2.20
0.345 0.339 0.335 0.329 0.324
1.547 1.553 1.559 1.565 1.570
0.534 0.527 0.522 0.516 0.509
1.656 1.678 1.700 1.723 1.744
1.097 1.099 1.101 1.102 1.104
0.233 0.237 0.240 0.244 0.246
2.22 2.24 2.26 2.28 2.30
0.320 0.315 0.311 0.306 0.302
1.575 1.581 1.586 1.591 1.596
0.505 0.499 0.493 0.487 0.482
1.770 1.793 1.818 1.842 1.865
1.106 1.107 1.108 1.110 1.111
0.250 0.254 0.257 0.259 0.262
2.32 2.34 2.36 2.38 2.40
0.298 0.294 0.289 0.285 0.281
1.601 1.606 1.611 1.615 1.620
0.477 0.472 0.467 0.461 0.456
1.893 1.920 1.945 1.971 1.995
1.113 1.115 1.116 1.118 1.119
0.265 0.269 0.272 0.275 0.277
4 f Lmax D
Flow of Perfect Gases with Friction
91
TABLE 6.3 Flow of perfect gases with friction (Fanno flow, γ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
2.42 2.44 2.46 2.48 2.50
0.278 0.274 0.270 0.267 0.263
1.625 1.629 1.634 1.638 1.642
0.452 0.446 0.442 0.437 0.432
2.025 2.053 2.081 2.110 2.135
1.121 1.122 1.123 1.125 1.126
0.280 0.283 0.286 0.288 0.290
2.50 2.54 2.56 2.58 2.60
0.260 0.256 0.253 0.249 0.246
1.646 1.651 1.655 1.658 1.663
0.428 0.423 0.418 0.414 0.409
2.168 2.198 2.228 2.260 2.285
1.127 1.129 1.130 1.131 1.132
0.294 0.296 0.299 0.301 0.303
2.62 2.64 2.66 2.68 2.70
0.243 0.240 0.236 0.234 0.231
1.667 1.670 1.674 1.678 1.682
0.405 0.401 0.397 0.393 0.388
2.321 2.352 2.385 2.417 2.445
1.134 1.135 1.136 1.1375 1.138
0.306 0.308 0.311 0.313 0.314
2.72 2.74 2.76 2.78 2.80
0.228 0.225 0.222 0.219 0.217
1.685 1.688 1.692 1.694 1.699
0.385 0.381 0.377 0.373 0.368
2.484 2.518 2.552 2.587 2.616
1.139 1.141 1.142 1.143 1.144
0.317 0.319 0.322 0.324 0.325
2.82 2.84 2.86 2.88 2.90
0.214 0.211 0.209 0.207 0.204
1.702 1.705 1.709 1.712 1.715
0.365 0.361 0.358 0.354 0.350
2.658 2.694 2.730 2.767 2.797
1.145 1.146 1.148 1.1485 1.149
0.328 0.330 0.332 0.334 0.335
2.92 2.94 2.96 2.98 3.00
0.202 0.199 0.197 0.195 0.192
1.718 1.721 1.724 1.727 1.730
0.347 0.343 0.340 0.337 0.333
2.843 2.881 2.920 2.960 2.990
1.150 1.152 1.153 1.1537 1.154
0.338 0.340 0.342 0.343 0.344
Gas Tables for Compressible Flow Calculations
92
TABLE 6.3 Flow of perfect gases with friction (γγ = 1.67)
M
p/p*
c/c* = ρ*/ρ
T/T*
p0 /p0*
F/F*
4 f Lmax D
3.10 3.20 3.30 3.40 3.50
0.182 0.172 0.163 0.154 0.146
1.744 1.756 1.768 1.780 1.790
0.317 0.302 0.288 0.275 0.262
3.205 3.423 3.653 3.895 4.134
1.159 1.164 1.168 1.172 1.174
0.354 0.362 0.369 0.376 0.381
3.60 3.70 3.80 3.90 4.00
0.139 0.132 0.126 0.120 0.1145
1.800 1.808 1.817 1.825 1.833
0.251 0.239 0.229 0.219 0.210
4.420 4.703 5.000 5.311 5.608
1.178 1.182 1.185 1.187 1.189
0.389 0.394 0.400 0.404 0.407
4.10 4.20 4.30 4.40 4.50
0.109 0.105 0.100 0.096 0.092
1.840 1.846 1.852 1.858 1.864
0.202 0.194 0.186 0.178 0.172
5.979 6.336 6.708 7.098 7.456
1.193 1.195 1.197 1.199 1.200
0.413 0.417 0.421 0.4246 0.426
4.60 4.70 4.80 4.90 5.00
0.088 0.085 0.082 0.079 0.075
1.868 1.873 1.878 1.883 1.887
0.166 0.159 0.154 0.150 0.142
7.925 8.365 8.827 9.300 9.721
1.203 1.205 1.207 1.208 1.209
0.431 0.434 0.437 0.439 0.440
6.00 7.00 8.00 9.00 10.00
0.053 0.040 0.030 0.024 0.0199
1.918 1.938 1.951 1.960 1.967
0.102 0.077 0.059 0.047 0.038
15.680 23.850 34.580 48.240 65.180
1.220 1.227 1.232 1.235 1.238
0.459 0.472 0.479 0.485 0.489
Isothermal Flow of Perfect Gases with Friction
93
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3) p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
4 f Lmax D
0.02 0.04 0.06 0.08 0.10
43.853 21.926 14.617 10.963 8.770
0.0228 0.0456 0.0684 0.0912 0.1140
0.897 0.897 0.897 0.897 0.897
27.328 13.674 9.128 6.858 5.499
1914.515 473.593 207.311 114.403 71.580
0.12 0.14 0.16 0.18 0.20
7.308 6.264 5.481 4.872 4.385
0.1368 0.1596 0.1824 0.2052 0.2280
0.898 0.899 0.900 0.900 0.902
4.596 3.953 3.472 3.100 2.804
48.440 34.576 25.645 19.574 15.274
0.22 0.24 0.26 0.28 0.30
3.986 3.654 3.373 3.132 2.923
0.250 0.273 0.296 0.319 0.342
0.903 0.904 0.905 0.907 0.908
2.563 2.363 2.195 2.053 1.930
12.127 9.762 7.947 6.528 5.401
0.32 0.34 0.36 0.39 0.40
2.741 2.579 2.436 2.308 2.192
0.365 0.387 0.410 0.433 0.456
0.910 0.912 0.914 0.916 0.918
1.824 1.731 1.650 1.578 1.514
4.495 3.759 3.154 2.654 2.237
0.42 0.44 0.46 0.48 0.50
2.088 1.993 1.907 1.827 1.754
0.478 0.501 0.524 0.547 0.570
0.920 0.923 0.925 0.927 0.930
1.457 1.405 1.360 1.319 1.282
1.888 1.593 1.344 1.133 0.953
0.52 0.54 0.56 0.58 0.60
1.686 1.624 1.566 1.512 1.461
0.593 0.615 0.638 0.661 0.684
0.933 0.936 0.938 0.942 0.945
1.248 1.218 1.190 1.166 1.144
0.799 0.668 0.555 0.459 0.377
M
Gas Tables for Compressible Flow Calculations
94
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
0.62 0.64 0.66 0.68 0.70
1.415 1.370 1.329 1.289 1.253
0.707 0.729 0.753 0.775 0.798
0.948 0.952 0.955 0.958 0.962
1.124 1.105 1.089 1.074 1.061
0.307 0.248 0.197 0.154 0.119
0.72 0.74 0.76 0.78 0.80
1.218 1.185 1.154 1.124 1.096
0.821 0.844 0.866 0.889 0.912
0.966 0.970 0.974 0.978 0.983
1.049 1.039 1.031 1.023 1.016
0.089 0.065 0.045 0.029 0.018
0.82 0.84 0.86 0.88 0.90
1.069 1.044 1.019 0.997 0.974
0.935 0.958 0.980 1.003 1.026
0.987 0.991 0.996 1.0007 1.005
1.010 1.006 1.002 0.999 0.998
0.0095 0.0038 0.0078 0.22 × 10−4 0.0013
0.92 0.94 0.96 0.98 1.00
0.953 0.933 0.913 0.895 0.877
1.049 1.072 1.095 1.117 1.140
1.010 1.015 1.020 1.025 1.031
0.997 0.997 0.9975 0.999 1.001
0.0044 0.0092 0.015 0.023 0.0316
1.02 1.04 1.06 1.08 1.10
0.859 0.843 0.827 0.812 0.797
1.163 1.185 1.208 1.231 1.254
1.036 1.042 1.047 1.053 1.059
1.004 1.008 1.012 1.017 1.023
0.0413 0.052 0.0635 0.076 0.088
1.12 1.14 1.16 1.18 1.20
0.783 0.769 0.756 0.743 0.731
1.277 1.299 1.322 1.345 1.368
1.065 1.071 1.078 1.083 1.090
1.029 1.037 1.045 1.053 1.063
0.102 0.116 0.131 0.146 0.161
4 f Lmax D
Isothermal Flow of Perfect Gases with Friction
95
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
1.22 1.24 1.26 1.28 1.30
0.718 0.707 0.696 0.685 0.674
1.391 1.414 1.437 1.459 1.482
1.097 1.103 1.110 1.116 1.124
1.073 1.083 1.094 1.106 1.119
0.177 0.193 0.209 0.226 0.242
1.32 1.34 1.36 1.38 1.40
0.664 0.654 0.645 0.635 0.626
1.505 1.528 1.550 1.573 1.596
1.131 1.138 1.145 1.152 1.160
1.132 1.146 1.161 1.176 1.192
0.259 0.276 0.293 0.310 0.328
1.42 1.44 1.46 1.48 1.50
0.617 0.609 0.601 0.592 0.594
1.619 1.642 1.664 1.687 1.710
1.168 1.175 1.183 1.191 1.199
1.209 1.227 1.245 1.264 1.284
0.345 0.362 0.380 0.397 0.415
1.52 1.54 1.56 1.58 1.60
0.577 0.569 0.562 0.555 0.548
1.733 1.756 1.778 1.801 1.824
1.207 1.215 1.224 1.232 1.241
1.305 1.327 1.349 1.372 1.396
0.433 0.450 0.468 0.485 0.503
1.62 1.64 1.66 1.68 1.70
0.541 0.534 0.528 0.522 0.516
1.847 1.869 1.893 1.915 1.938
1.249 1.258 1.267 1.276 1.285
1.421 1.447 1.474 1.502 1.530
0.520 0.538 0.555 0.572 0.589
1.72 1.74 1.76 1.78 1.80
0.510 0.504 0.498 0.493 0.487
1.961 1.984 2.007 2.029 2.052
1.294 1.303 1.313 1.323 1.132
1.560 1.591 1.622 1.655 1.689
0.607 0.624 0.641 0.658 0.675
4 f Lmax D
Gas Tables for Compressible Flow Calculations
96
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
1.82 1.84 1.86 1.88 1.90
0.482 0.476 0.471 0.466 0.461
2.075 2.098 2.121 2.143 2.166
1.342 1.352 1.362 1.372 1.382
1.724 1.760 1.797 1.836 1.876
0.692 0.709 0.726 0.742 0.759
1.92 1.94 1.96 1.98 2.00
0.457 0.452 0.447 0.443 0.438
2.189 2.212 2.235 2.257 2.280
1.392 1.403 1.413 1.424 1.434
1.917 1.959 2.003 2.048 2.094
0.775 0.792 0.808 0.825 0.841
2.02 2.04 2.06 2.08 2.10
0.434 0.430 0.426 0.422 0.417
2.303 2.326 2.348 2.372 2.394
1.445 1.456 1.467 1.478 1.489
2.142 2.191 2.242 2.294 2.348
0.857 0.873 0.889 0.905 0.921
2.12 2.14 2.16 2.18 2.20
0.414 0.409 0.406 0.402 0.398
2.417 2.440 2.463 2.485 2.508
1.501 1.512 1.524 1.536 1.547
2.404 2.462 2.521 2.581 2.644
0.936 0.952 0.967 0.983 0.998
2.22 2.24 2.26 2.28 2.30
0.395 0.392 0.388 0.385 0.381
2.531 2.554 2.577 2.599 2.622
1.559 1.571 1.583 1.596 1.608
2.708 2.775 2.843 2.914 2.986
1.013 1.028 1.043 1.058 1.073
2.32 2.34 2.36 2.38 2.40
0.378 0.375 0.372 0.368 0.365
2.645 2.668 2.691 2.714 2.736
1.620 1.633 1.645 1.658 1.671
3.061 3.138 3.217 3.298 3.383
1.088 1.103 1.117 1.132 1.147
4 f Lmax D
Isothermal Flow of Perfect Gases with Friction
97
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
2.42 2.44 2.46 2.48 2.50
0.362 0.359 0.356 0.354 0.351
2.759 2.782 2.805 2.827 2.850
1.684 1.697 1.710 1.724 1.737
3.469 3.557 3.649 3.743 3.839
1.161 1.175 1.189 1.204 1.218
2.52 2.54 2.56 2.58 2.60
0.348 0.345 0.343 0.340 0.337
2.873 2.896 2.919 2.942 2.964
1.750 1.764 1.778 1.792 1.805
3.939 4.042 4.147 4.255 4.366
1.232 1.246 1.259 1.273 1.287
2.62 2.64 2.66 2.68 2.70
0.334 0.332 0.329 0.327 0.325
2.987 3.010 3.033 3.055 3.078
1.819 1.834 1.848 1.862 1.877
4.481 4.599 4.720 4.845 4.973
1.301 1.314 1.328 1.341 1.354
2.72 2.74 2.76 2.78 2.80
0.322 0.320 0.318 0.315 0.313
3.101 3.124 3.147 3.169 3.192
1.891 1.906 1.921 1.936 1.951
5.105 5.240 5.379 5.522 5.669
1.367 1.381 1.394 1.407 1.419
2.82 2.84 2.86 2.88 2.90
0.311 0.309 0.307 0.304 0.302
3.215 3.238 3.261 3.283 3.306
1.966 1.981 1.996 2.012 2.027
5.821 5.976 6.136 6.300 6.469
1.432 1.445 1.458 1.471 1.483
2.92 2.94 2.96 2.98 3.00
0.300 0.298 0.296 0.294 0.292
3.329 3.352 3.375 3.398 3.421
2.043 2.059 2.075 2.091 2.107
6.642 6.821 7.004 7.192 7.385
1.495 1.508 1.521 1.533 1.545
4 f Lmax D
Gas Tables for Compressible Flow Calculations
98
TABLE 7.1 Isothermal flow of perfect gases with friction (γγ = 1.3)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
3.10 3.20 3.30 3.40 3.50
0.283 0.274 0.266 0.258 0.251
3.534 3.648 3.763 3.876 3.991
2.189 2.274 2.361 2.451 2.544
8.433 9.631 10.998 12.555 14.327
1.605 1.664 1.721 1.776 1.831
3.60 3.70 3.80 3.90 4.00
0.243 0.237 0.231 0.225 0.219
4.105 4.218 4.332 4.447 4.561
2.639 2.737 2.838 2.942 3.048
16.341 18.626 21.214 24.142 27.449
1.883 1.935 1.985 2.035 2.083
4.10 4.20 4.30 4.40 4.50
0.214 0.209 0.204 0.199 0.195
4.675 4.789 4.903 5.017 5.131
3.157 3.269 3.383 3.500 3.619
31.181 35.384 40.112 45.423 51.380
2.130 2.176 2.221 2.265 2.308
4.60 4.70 4.80 4.90 5.00
0.191 0.187 0.183 0.179 0.175
5.245 5.359 5.473 5.587 5.701
3.742 3.867 3.995 4.125 4.258
58.053 65.517 73.854 83.155 93.517
2.351 2.392 2.433 2.472 2.512
5.50 6.00 6.50 7.00 7.50
0.159 0.146 0.135 0.125 0.117
6.271 6.841 7.411 7.981 8.551
4.965 5.738 6.578 7.486 8.461
165.267 283.672 473.496 769.836 1221.350
2.697 2.867 3.024 3.169 3.306
8.00 8.50 9.00 9.50 10.00
0.109 0.103 0.097 0.092 0.0877
9.121 9.691 10.262 10.831 11.402
9.503 10.613 11.789 13.033 14.344
1894.190 2876.708 4285.063 6269.795 9020.438
3.433 3.553 3.667 3.773 3.875
4 f Lmax D
Isothermal Flow of Perfect Gases with Friction
99
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4) p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
4 f Lmax D
0.02 0.04 0.06 0.08 0.10
42.257 21.128 14.086 10.564 8.451
0.0237 0.0473 0.0710 0.0946 0.1183
0.8751 0.8752 0.8756 0.8761 0.8770
26.488 13.255 8.849 6.650 5.333
1777.227 439.327 192.122 105.892 66.159
0.12 0.14 0.16 0.18 0.20
7.043 6.037 5.282 4.695 4.226
0.142 0.165 0.189 0.213 0.237
0.8775 0.8784 0.8795 0.881 0.882
4.458 3.835 3.369 3.010 2.723
44.699 31.847 23.573 17.952 13.975
0.22 0.24 0.26 0.28 0.30
3.841 3.521 3.251 3.018 2.817
0.260 0.284 0.307 0.331 0.355
0.883 0.885 0.887 0.888 0.891
2.490 2.297 2.135 1.997 1.879
11.066 8.883 7.208 5.901 4.865
0.32 0.34 0.36 0.38 0.40
2.641 2.486 2.347 2.224 2.113
0.378 0.402 0.426 0.449 0.473
0.893 0.895 0.897 0.900 0.903
1.777 1.687 1.609 1.539 1.478
4.033 3.358 2.805 2.348 1.968
0.42 0.44 0.46 0.48 0.50
2.012 1.921 1.837 1.761 1.690
0.497 0.521 0.544 0.568 0.592
0.906 0.909 0.912 0.915 0.918
1.424 1.375 1.331 1.292 1.256
1.651 1.384 1.159 0.968 0.807
0.52 0.54 0.56 0.58 0.60
1.625 1.565 1.509 1.457 1.408
0.615 0.639 0.663 0.686 0.710
0.922 0.926 0.929 0.934 0.938
1.225 1.196 1.170 1.147 1.126
0.670 0.553 0.454 0.370 0.299
M
Gas Tables for Compressible Flow Calculations
100
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
4 f Lmax D 0.238 0.188 0.145 0.1098 0.0808
0.62 0.64 0.66 0.68 0.70
1.363 1.320 1.280 1.243 1.207
0.733 0.757 0.781 0.805 0.828
0.942 0.947 0.951 0.956 0.961
1.107 1.090 1.075 1.061 1.049
0.72 0.74 0.76 0.78 0.80
1.174 1.142 1.112 1.083 1.056
0.852 0.876 0.899 0.923 0.946
0.966 0.971 0.976 0.981 0.987
1.039 1.029 1.022 1.015 1.009
0.0573 0.0386 0.0242 0.0135 62.5 × 10−4
0.82 0.84 0.86 0.88 0.90
1.031 1.006 0.983 0.960 0.939
0.970 0.994 1.017 1.041 1.065
0.993 0.998 1.004 1.010 1.017
1.004 1.001 0.998 0.996 0.995
18.6 × 10−4 0.75 × 10−4 6.0 × 10−4 31.8 × 10−4 75.8 × 10−4
0.92 0.94 0.96 0.98 1.00
0.918 0.899 0.880 0.962 0.845
1.088 1.112 1.135 1.159 1.183
1.023 1.029 1.036 1.043 1.050
0.995 0.9958 0.997 0.999 1.0025
1.36 × 10−2 2.11 × 10−2 2.98 × 10−2 3.98 × 10−2 5.07 × 10−2
1.02 1.04 1.06 1.08 1.10
0.828 0.813 0.797 0.783 0.768
1.207 1.230 1.254 1.278 1.302
1.057 1.064 1.072 1.079 1.087
1.0062 1.0106 1.0157 1.0215 1.028
0.0626 0.0753 0.0887 0.1028 0.1174
1.12 1.14 1.16 1.18 1.20
0.755 0.741 0.728 0.716 0.704
1.325 1.349 1.372 1.396 1.419
1.095 1.102 1.110 1.118 1.127
1.035 1.043 1.051 1.061 l.070
0.1325 0.1481 0.164 0.180 0.197
Isothermal Flow of Perfect Gases with Friction
101
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4)
M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
4 f Lmax D
1.22 1.24 1.26 1.28 1.30
0.693 0.681 0.671 0.660 0.650
1.443 1.467 1.491 1.515 1.538
1.135 1.144 1.153 1.162 1.171
1.081 1.092 1.103 1.116 1.129
0.214 0.231 0.248 0.266 0.284
1.32 1.34 1.36 1.38 1.40
0.640 0.631 0.621 0.612 0.604
1.562 1.585 1.609 1.633 1.656
1.180 1.189 1.198 1.208 1.218
1.142 1.157 1.172 1.187 1.204
0.302 0.319 0.337 0.356 0.374
1.42 1.44 1.46 1.48 1.50
0.595 0.587 0.578 0.571 0.563
1.680 1.704 1.727 1.751 1.775
1.228 1.238 1.248 1.258 1.268
1.221 1.238 1.257 1.276 1.296
0.392 0.410 0.428 0.447 0.465
1.52 1.54 1.56 1.58 1.60
0.556 0.549 0.542 0.535 0.528
1.798 1.822 1.846 1.869 1.893
1.279 1.290 1.301 1.312 1.323
1.317 1.338 1.360 1.383 1.407
0.483 0.501 0.519 0.537 0.555
1.62 1.64 1.66 1.69 1.70
0.522 0.515 0.509 0.503 0.497
1.917 1.940 1.964 1.987 2.011
1.334 1.345 1.357 1.368 1.381
1.431 1.457 1.483 1.510 1.538
0.573 0.591 0.609 0.627 0.645
1.72 1.74 1.76 1.78 1.80
0.491 0.486 0.480 0.475 0.469
2.035 2.058 2.082 2.106 2.129
1.393 1.405 1.417 1.429 1.442
1.567 1.596 1.627 1.658 1.691
0.663 0.680 0.697 0.715 0.733
Gas Tables for Compressible Flow Calculations
102
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4) M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
1.82 1.84 1.86 1.88 1.90
0.464 0.459 0.454 0.449 0.445
2.153 2.177 2.200 2.224 2.248
1.455 1.467 1.480 1.493 1.507
1.724 1.758 1.793 1.830 1.868
0.749 0.767 0.784 0.801 0.818
1.92 1.94 1.96 1.98 2.00
0.440 0.435 0.431 0.427 0.423
2.272 2.295 2.319 2.343 2.366
1.520 1.534 1.547 1.561 1.575
1.906 1.946 1.986 2.029 2.072
0.835 0.852 0.868 0.885 0.901
2.02 2.04 2.06 2.08 2.10
0.418 0.414 0.410 0.406 0.402
2.390 2.414 2.437 2.461 2.484
1.589 1.603 1.617 1.632 1.647
2.116 2.162 2.208 2.257 2.306
0.918 0.934 0.950 0.966 0.982
2.12 2.14 2.16 2.18 2.20
0.398 0.395 0.391 0.387 0.384
2.508 2.532 2.556 2.579 2.603
1.662 1.676 1.691 1.707 1.722
2.357 2.409 2.463 2.517 2.574
0.998 1.014 1.029 1.045 1.061
2.22 2.24 2.26 2.28 2.30
0.381 0.377 0.374 0.370 0.367
2.627 2.650 2.674 2.697 2.721
1.737 1.753 1.769 1.785 1.800
2.632 2.691 2.752 2.815 2.879
1.076 1.092 1.107 1.122 1.137
2.32 2.34 2.36 2.38 2.40
0.364 0.361 0.358 0.355 0.352
2.745 2.768 2.792 2.816 2.839
1.817 1.833 1.849 1.866 1.883
2.945 3.013 3.082 3.153 3.226
1.152 1.167 1.182 1.197 1.211
4 f Lmax D
Isothermal Flow of Perfect Gases with Friction
103
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4) M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
2.42 2.44 2.46 2.48 2.50
0.349 0.346 0.343 0.341 0.338
2.863 2.887 2.911 2.934 2.958
1.899 1.917 1.934 1.951 1.968
3.301 3.378 3.456 3.537 3.619
1.226 1.240 1.255 1.269 1.283
2.52 2.54 2.56 2.58 2.60
0.335 0.333 0.330 0.327 0.325
2.982 3.005 3.029 3.053 3.076
1.986 2.004 2.022 2.039 2.058
3.704 3.791 3.880 3.971 4.064
1.297 1.311 1.325 1.339 1.353
2.62 2.64 2.66 2.68 2.70
0.322 0.320 0.318 0.315 0.313
3.100 3.123 3.147 3.171 3.195
2.076 2.094 2.113 2.132 2.151
4.160 4.258 4.359 4.462 4.567
1.367 1.380 1.394 1.407 1.421
2.72 2.74 2.76 2.78 2.80
0.311 0.308 0.306 0.304 0.302
3.218 3.242 3.265 3.289 3.313
2.169 2.189 2.208 2.227 2.247
4.675 4.785 4.898 5.015 5.133
1.434 1.447 1.461 1.473 1.487
2.82 2.84 2.86 2.88 2.90
0.299 0.297 0.295 0.293 0.291
3.337 3.360 3.384 3.407 3.431
2.267 2.286 2.306 2.327 2.346
5.255 5.379 5.506 5.637 5.770
1.499 1.512 1.525 1.538 1.551
2.92 2.94 2.96 2.98 3.00
0.289 0.287 0.285 0.283 0.282
3.455 3.478 3.502 3.526 3.549
2.367 2.387 2.408 2.429 2.450
5.906 6.046 6.189 6.335 6.485
1.563 1.575 1.588 1.601 1.613
4 f Lmax D
Gas Tables for Compressible Flow Calculations
104
TABLE 7.2 Isothermal flow of perfect gases with friction (γγ = 1.4) M
p/p*t
c/c*t = ρ*t/ρ
T0 /T0*t
p0 /p0*t
4 f Lmax D
3.10 3.20 3.30 3.40 3.50
0.273 0.264 0.256 0.248 0.241
3.668 3.786 3.905 4.023 4.141
2.557 2.667 2.781 2.898 3.018
7.286 8.182 9.183 10.229 11.541
1.673 1.732 1.790 1.846 1.900
3.60 3.70 3.80 3.90 4.00
0.235 0.228 0.222 0.217 0.211
4.259 4.378 4.496 4.614 4.733
3.143 3.271 3.402 3.537 3.675
12.922 14.454 16.152 18.029 20.104
1.953 2.005 2.056 2.105 2.153
4.10 4.20 4.30 4.40 4.50
0.206 0.201 0.196 0.192 0.188
4.851 4.969 5.088 5.206 5.324
3.817 3.962 4.110 4.263 4.418
22.391 24.911 27.682 30.725 34.062
2.201 2.247 2.292 2.336 2.380
4.60 4.70 4.80 4.90 5.00
0.184 0.179 0.176 0.172 0.169
5.443 5.561 5.679 5.798 5.916
4.578 4.741 4.907 5.077 5.250
37.718 41.716 46.084 50.850 56.043
2.422 2.464 2.505 2.544 2.584
5.50 6.00 6.50 7.00 7.50
0.154 0.141 0.130 0.121 0.112
6.507 7.099 7.691 8.282 8.874
6.168 7.175 8.268 9.450 10.718
89.591 139.367 211.379 313.219 454.338
2.769 2.939 3.097 3.243 3.379
8.00 8.50 9.00 9.50 10.00
0.1056 0.0994 0.0939 0.0889 0.0845
9.466 10.057 10.649 11.241 11.832
12.075 13.518 15.050 16.668 18.375
646.325 903.222 1241.856 1682.182 2247.66
3.506 3.626 3.739 3.847 3.948
Flow of Perfect Gases with Heat Transfer
105
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) T0 /T0*
c/c* = ρ*/ρ
M
p/p*
p0 /p0*
T/T*
0.02 0.04 0.06 0.08 0.10
2.299 2.295 2.289 2.281 2.270
1.255 1.254 1.252 1.250 1.247
0.002 0.009 0.0189 0.0333 0.0516
0.00184 0.0073 0.0164 0.029 0.0449
0.002 0.0037 0.008 0.015 0.0227
0.12 0.14 0.16 0.18 0.20
2.258 2.243 2.226 2.207 2.186
1.244 1.239 1.235 1.230 1.224
0.073 0.098 0.127 0.158 0.191
0.064 0.086 0.110 0.138 0.167
0.0325 0.044 0.057 0.072 0.0875
0.22 0.24 0.26 0.28 0.30
2.164 2.140 2.114 2.087 2.059
1.128 1.212 1.205 1.198 1.191
0.277 0.264 0.302 0.342 0.382
0.198 0.231 0.265 0.300 0.336
1.105 0.123 0.143 0.163 0.185
0.32 0.34 0.36 0.38 0.40
2.030 1.999 1.968 1.936 1.904
1.183 1.176 1.167 1.159 1.152
0.422 0.462 0.502 0.542 0.580
0.373 0.408 0.445 0.481 0.516
0.208 0.231 0.255 0.279 0.305
0.42 0.44 0.46 0.48 0.50
1.871 1.837 1.804 1.769 1.736
1.143 1.135 1.127 1.119 1.111
0.617 0.654 0.688 0.722 0.753
0.551 0.585 0.618 0.649 0.680
0.330 0.356 0.382 0.408 0.434
0.52 0.54 0.56 0.58 0.60
1.702 1.670 1.634 1.600 1.567
1.103 1.096 1.088 1.081 1.074
0.783 0.811 0.837 0.861 0.884
0.708 0.736 0.762 0.787 0.810
0.460 0.486 0.512 0.538 0.564
Gas Tables for Compressible Flow Calculations
106
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
0.62 0.64 0.66 0.68 0.70
1.534 1.500 1.468 1.436 1.405
1.067 1.061 1.054 1.048 1.042
0.904 0.923 0.939 0.954 0.967
0.832 0.852 0.870 0.887 0.903
0.589 0.615 0.639 0.664 0.688
0.72 0.74 0.76 0.78 0.80
1.374 1.344 1.314 1.284 1.255
1.037 1.032 1.028 1.023 1.019
0.978 0.989 0.997 1.003 1.009
0.917 0.930 0.942 0.952 0.961
0.712 0.736 0.758 0.781 0.804
0.82 0.84 0.86 0.88 0.90
1.227 1.199 1.173 1.146 1.120
1.016 1.012 1.010 1.007 1.005
1.013 1.015 1.0172 1.0173 1.017
0.969 0.976 0.983 0.987 0.991
0.825 0.846 0.867 0.887 0.907
0.92 0.94 0.96 0.98 1.00
1.095 1.070 1.046 1.023 1.000
1.003 1.002 1.0007 1.0002 1.000
1.015 1.012 1.009 1.005 1.000
0.995 0.997 0.998 0.999 1.000
0.927 0.946 0.964 0.982 1.000
1.02 1.04 1.06 1.08 1.10
0.977 0.956 0.935 0.914 0.894
1.0002 1.0008 1.002 1.003 1.005
0.994 0.988 0.982 0.975 0.967
0.999 0.998 0.997 0.995 0.993
1.017 1.034 1.050 1.066 1.082
1.12 1.14 1.16 1.18 1.20
0.874 0.855 0.836 0.818 0.800
1.007 1.009 1.0127 1.016 1.020
0.958 0.950 0.942 0.933 0.924
0.990 0.987 0.984 0.981 0.976
1.097 1.111 1.126 1.139 1.153
Flow of Perfect Gases with Heat Transfer
107
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
1.22 1.24 1.26 1.28 1.30
0.784 0.767 0.751 0.735 0.719
1.024 1.028 1.034 1.039 1.045
0.914 0.904 0.894 0.885 0.875
0.972 0.967 0.963 0.958 0.953
1.166 1.179 1.192 1.204 1.216
1.32 1.34 1.36 1.38 1.40
0.705 0.689 0.676 0.662 0.648
1.051 1.058 1.065 1.073 1.081
0.865 0.854 0.844 0.834 0.824
0.948 0.943 0.938 0.932 0.927
1.227 1.238 1.249 1.260 1.270
1.42 1.44 1.46 1.48 1.50
0.635 0.622 0.609 0.598 0.586
1.089 1.098 1.108 1.117 1.128
0.813 0.803 0.793 0.783 0.773
0.921 0.915 0.910 0.904 0.898
1.281 1.291 1.300 1.309 1.318
1.52 1.54 1.56 1.58 1.60
0.575 0.563 0.552 0.542 0.531
1.138 1.149 1.161 1.173 1.185
0.763 0.753 0.743 0.733 0.723
0.893 0.887 0.882 0.876 0.870
1.327 1.336 1.344 1.352 1.360
1.62 1.64 1.66 1.68 1.70
0.521 0.512 0.502 0.492 0.484
1.195 1.213 1.227 1.241 1.256
0.713 0.704 0.694 0.685 0.676
0.865 0.859 0.853 0.847 0.842
1.368 1.376 1.383 1.390 1.397
1.72 1.74 1.76 1.78 1.80
0.475 0.465 0.457 0.449 0.441
1.272 1.288 1.305 1.322 1.340
0.670 0.657 0.648 0.639 0.631
0.837 0.831 0.825 0.821 0.815
1.404 1.411 1.417 1.423 1.430
Gas Tables for Compressible Flow Calculations
108
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
1.82 1.84 1.86 1.88 1.90
0.433 0.426 0.418 0.411 0.404
1.358 1.377 1.397 1.417 1.438
0.622 0.614 0.606 0.597 0.589
0.810 0.805 0.799 0.795 0.790
1.436 1.442 1.447 1.453 1.459
1.92 1.94 1.96 1.98 2.00
0.397 0.390 0.383 0.377 0.371
1.451 1.482 1.505 1.528 1.552
0.581 0.573 0.565 0.557 0.551
0.785 0.780 0.775 0.771 0.766
1.464 1.469 1.474 1.479 1.484
2.02 2.04 2.06 2.08 2.10
0.365 0.358 0.353 0.347 0.341
1.576 1.602 1.628 1.655 1.683
0.543 0.536 0.528 0.522 0.515
0.761 0.757 0.752 0.748 0.744
1.488 1.493 1.498 1.502 1.506
2.12 2.14 2.16 2.18 2.20
0.336 0.331 0.325 0.320 0.315
1.711 1.740 1.770 1.800 1.832
0.508 0.501 0.495 0.487 0.482
0.739 0.735 0.731 0.727 0.723
1.511 1.515 1.518 1.523 1.527
2.22 2.24 2.26 2.28 2.30
0.310 0.306 0.301 0.296 0.292
1.865 1.898 1.932 1.967 2.003
0.475 0.469 0.463 0.457 0.451
0.718 0.715 0.711 0.707 0.703
1.530 1.534 1.537 1.541 1.545
2.32 2.34 2.36 2.38 2.40
0.287 0.283 0.279 0.275 0.271
2.040 2.078 2.117 2.156 2.197
0.445 0.440 0.434 0.428 0.423
0.699 0.696 0.693 0.688 0.686
1.548 1.551 1.555 1.557 1.561
Flow of Perfect Gases with Heat Transfer
109
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
2.42 2.44 2.46 2.48 2.50
0.267 0.263 0.259 0.255 0.252
2.238 2.281 2.325 2.370 2.416
0.417 0.412 0.407 0.402 0.397
0.682 0.678 0.675 0.672 0.670
1.564 1.567 1.569 1.573 1.575
2.52 2.54 2.56 2.58 2.60
0.248 0.245 0.242 0.238 0.235
2.464 2.512 2.562 2.612 2.664
0.392 0.387 0.382 0.378 0.373
0.666 0.663 0.659 0.657 0.654
1.578 1.580 1.583 1.586 1.588
2.62 2.64 2.66 2.68 2.70
0.232 0.228 0.225 0.222 0.219
2.718 2.772 2.828 2.885 2.945
0.368 0.364 0.360 0.356 0.351
0.650 0.648 0.645 0.642 0.640
1.591 1.593 1.596 1.598 1.600
2.72 2.74 2.76 2.78 2.80
0.217 0.214 0.211 0.208 0.206
3.004 3.065 3.128 3.192 3.258
0.347 0.343 0.339 0.335 0.331
0.636 0.634 0.631 0.629 0.627
1.603 1.605 1.607 1.609 1.612
2.82 2.84 2.86 2.88 2.90
0.203 0.200 0.198 0.195 0.193
3.326 3.395 3.465 3.537 3.611
0.327 0.323 0.319 0.316 0.312
0.624 0.622 0.619 0.617 0.614
1.613 1.615 1.617 1.619 1.620
2.92 2.94 2.96 2.98 3.00
0.190 0.188 0.185 0.183 0.181
3.687 3.765 3.844 3.925 4.007
0.309 0.305 0.302 0.298 0.295
0.612 0.609 0.607 0.605 0.603
1.623 1.625 1.626 1.628 1.630
Gas Tables for Compressible Flow Calculations
110
TABLE 8.1 Flow of perfect gases with heat transfer (γγ = 1.3) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
3.10 3.20 3.30 3.40 3.50
0.170 0.161 0.152 0.143 0.136
4.451 4.947 5.500 6.118 6.806
0.279 0.264 0.251 0.238 0.226
0.593 0.583 0.574 0.566 0.558
1.638 1.645 1.653 1.659 1.665
3.60 3.70 3.80 3.90 4.00
0.128 0.122 0.116 0.110 0.106
7.571 8.422 9.365 10.411 11.570
0.215 0.205 0.195 0.186 0.178
0.551 0.544 0.538 0.532 0.527
1.670 1.675 1.680 1.684 1.688
4.10 4.20 4.30 4.40 4.50
0.100 0.096 0.092 0.088 0.084
12.850 14.265 15.825 17.545 19.440
0.170 0.163 0.156 0.149 0.143
0.521 0.516 0.512 0.508 0.504
1.692 1.6951.698 1.701 1.705
4.60 4.70 4.80 4.90 5.00
0.081 0.077 0.074 0.071 0.068
21.518 23.803 26.309 29.056 32.060
0.138 0.132 0.127 0.122 0.118
0.499 0.496 0.493 0.489 0.487
1.707 1.709 1.713 1.714 1.715
6.00 7.00 8.00 9.00 10.00
0.048 0.035 0.027 0.0216 0.0175
76.970 191.300 413.400 833.400 1582.00
0.083 0.062 0.048 0.038 0.031
0.464 0.450 0.440 0.434 0.429
1.732 1.742 1.748 1.753 1.756
Flow of Perfect Gases with Heat Transfer
111
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) T0 /T0*
c/c* = ρ*/ρ
M
p/p*
p0 /p0*
T/T*
0.02 0.04 0.06 0.08 0.10
2.398 2.394 2.388 2.378 2.367
1.268 1.267 1.265 1.262 1.259
0.0023 0.0092 0.021 0.036 0.056
0.00192 0.0077 0.017 0.030 0.0468
0.00096 0.0038 0.0086 0.015 0.0237
0.12 0.14 0.16 0.18 0.20
2.353 2.336 2.317 2.295 2.273
1.255 1.251 1.246 1.241 1.235
0.079 0.107 0.137 0.171 0.207
0.066 0.089 0.115 0.143 0.174
0.034 0.0457 0.059 0.074 0.091
0.22 0.24 0.26 0.28 0.30
2.248 2.221 2.193 2.163 2.131
1.228 1.221 1.214 2.206 1.198
0.245 0.284 0.325 0.367 0.409
0.206 0.239 0.274 0.310 0.347
0.108 0.128 0.148 0.169 0.192
0.32 0.34 0.36 0.38 0.40
2.099 2.066 2.031 1.996 1.961
1.190 1.182 1.174 1.165 1.157
0.451 0.493 0.535 0.576 0.615
0.384 0.421 0.457 0.493 0.529
0.215 0.239 0.263 0.288 0.314
0.42 0.44 0.46 0.48 0.50
1.925 1.888 1.852 1.815 1.778
1.148 1.139 1.131 1.122 1.114
0.653 0.690 0.725 0.758 0.790
0.564 0.597 0.630 0.661 0.691
0.339 0.365 0.392 0.418 0.444
0.52 0.54 0.56 0.58 0.60
1.741 1.704 1.668 1.632 1.596
1.106 1.098 1.090 1.082 1.075
0.819 0.847 0.872 0.896 0.917
0.720 0.747 0.773 0.796 0.819
0.471 0.497 0.523 0.548 0.574
Gas Tables for Compressible Flow Calculations
112
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) M
p/p*
p0/p0*
T/T*
T0/T0*
c/c* = ρ*/ρ
0.62 0.64 0.66 0.68 0.70
1.560 1.525 1.490 1.457 1.423
1.068 1.061 1.055 1.049 1.043
0.935 0.953 0.968 0.981 0.993
0.839 0.859 0.877 0.893 0.908
0.599 0.625 0.650 0.673 0.697
0.72 0.74 0.76 0.78 0.80
1.391 1.358 1.327 1.296 1.266
1.037 1.032 1.028 1.023 1.019
1.002 1.011 1.017 1.022 1.025
0.922 0.934 0.945 0.955 0.964
0.721 0.744 0.766 0.788 0.810
0.82 0.84 0.86 0.88 0.90
1.236 1.207 1.179 1.151 1.125
1.015 1.012 1.010 1.007 1.0045
1.027 1.028 1.028 1.027 1.025
0.972 0.978 0.984 0.998 0.992
0.831 0.852 0.872 0.892 0.911
0.92 0.94 0.96 0.98 1.00
1.098 1.073 1.048 1.023 1.000
1.003 1.002 1.001 1.0002 1.000
1.021 1.017 1.012 1.006 1.000
0.995 0.997 0.998 0.999 1.000
0.930 0.950 0.966 0.983 1.000
1.02 1.04 1.06 1.08 1.10
0.977 0.955 0.933 0.911 0.891
1.0002 1.0007 1.002 1.0033 1.005
0.993 0.985 0.978 0.969 0.960
0.999 0.998 0.997 0.996 0.994
1.016 1.032 1.048 1.063 1.078
1.12 1.14 1.16 1.18 1.20
0.871 0.851 0.832 0.814 0.796
1.007 1.009 1.012 1.016 1.019
0.951 0.942 0.932 0.922 0.912
0.991 0.988 0.985 0.982 0.978
1.092 1.106 1.120 1.133 1.146
Flow of Perfect Gases with Heat Transfer
113
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) M
p/p*
p0/p0*
T/T*
T0/T0*
c/c* = ρ*/ρ
1.22 1.24 1.26 1.28 1.30
0.778 0.761 0.745 0.728 0.713
1.023 1.028 1.033 1.038 1.044
0.902 0.891 0.881 0.869 0.859
0.975 0.971 0.967 0.962 0.958
1.158 1.171 1.182 1.194 1.205
1.32 1.34 1.36 1.38 1.40
0.698 0.683 0.668 0.654 0.641
1.049 1.056 1.063 1.070 1.078
0.848 0.837 0.827 0.816 0.805
0.953 0.948 0.944 0.939 0.934
1.216 1.226 1.237 1.247 1.256
1.42 1.44 1.46 1.48 1.50
0.628 0.615 0.602 0.590 0.578
1.086 1.094 1.103 1.111 1.122
0.795 0.784 0.773 0.763 0.753
0.929 0.924 0.919 0.914 0.909
1.266 1.275 1.284 1.293 1.301
1.52 1.54 1.56 1.58 1.60
0.567 0.556 0.544 0.534 0.524
1.132 1.142 1.153 1.164 1.176
0.742 0.732 0.722 0.711 0.702
0.904 0.899 0.894 0.889 0.884
1.309 1.317 1.325 1.333 1.340
1.62 1.64 1.66 1.68 1.70
0.513 0.503 0.494 0.484 0.476
1.187 1.200 1.213 1.226 1.240
0.692 0.682 0.673 0.663 0.654
0.879 0.874 0.869 0.865 0.860
1.347 1.354 1.362 1.368 1.375
1.72 1.74 1.76 1.78 1.80
0.467 0.458 0.449 0.442 0.434
1.254 1.269 1.284 1.299 1.316
0.644 0.635 0.626 0.617 0.609
0.855 0.850 0.845 0.841 0.836
1.381 1.387 1.393 1.399 1.405
Gas Tables for Compressible Flow Calculations
114
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
1.82 1.84 1.86 1.88 1.90
0.425 0.418 0.411 0.403 0.396
1.332 1.349 1.367 1.385 1.403
0.600 0.592 0.584 0.575 0.567
0.832 0.827 0.823 0.818 0.814
1.410 1.415 1.421 1.426 1.431
1.92 1.94 1.96 1.98 2.00
0.389 0.383 0.376 0.369 0.364
1.422 1.442 1.462 1.482 1.503
0.559 0.551 0.544 0.536 0.529
0.810 0.806 0.801 0.797 0.793
1.436 1.441 1.446 1.450 1.455
2.02 2.04 2.06 2.08 2.10
0.357 0.352 0.345 0.340 0.334
1.525 1.547 1.569 1.592 1.616
0.522 0.514 0.507 0.500 0.494
0.789 0.785 0.782 0.778 0.774
1.459 1.463 1.467 1.471 1.475
2.12 2.14 2.16 2.18 2.20
0.329 0.324 0.318 0.314 0.308
1.640 1.665 1.690 1.717 1.743
0.487 0.480 0.474 0.467 0.461
0.770 0.767 0.763 0.759 0.756
1.480 1.483 1.487 1.490 1.494
2.22 2.24 2.26 2.28 2.30
0.304 0.299 0.294 0.290 6.285
1.771 1.798 1.827 1.856 1.886
0.455 0.448 0.443 0.437 0.431
0.750 0.749 0.746 0.743 0.739
1.497 1.500 1.504 1.507 1.510
2.32 2.34 2.36 2.38 2.40
0.281 0.277 0.273 0.268 0.265
1.916 1.947 1.980 2.012 2.045
0.426 0.420 0.415 0.409 0.404
0.736 0.733 0.730 0.727 0.724
1.513 1.516 1.519 1.522 1.525
Flow of Perfect Gases with Heat Transfer
115
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
2.42 2.44 2.46 2.48 2.50
0.261 0.257 0.253 0.249 0.246
2.078 2.114 2.149 2.185 2.222
0.398 0.393 0.388 0.383 0.379
0.721 0.718 0.716 0.713 0.710
1.528 1.530 1.533 1.536 1.540
2.52 2.54 2.56 2.58 2.60
0.242 0.239 0.235 0.232 0.229
2.260 2.298 2.337 2.377 2.418
0.374 0.369 0.365 0.360 0.356
0.707 0.704 0.702 0.699 0.697
1.542 1.544 1.546 1.548 1.551
2.62 2.64 2.66 2.68 2.70
0.226 0.223 0.220 0.217 0.214
2.459 2.502 2.545 2.589 2.634
0.351 0.347 0.343 0.338 0.334
0.695 0.692 0.689 0.687 0.685
1.553 1.555 1.557 1.559 1.561
2.72 2.74 2.76 2.78 2.80
0.211 0.208 0.206 0.203 0.200
2.680 2.727 2.775 2.823 2.873
0.330 0.326 0.322 0.318 0.315
0.683 0.680 0.678 0.676 0.674
1.563 1.565 1.567 1.569 1.571
2.82 2.84 2.86 2.88 2.90
0.198 0.195 0.193 0.190 0.188
2.923 2.975 3.028 3.081 3.136
0.311 0.307 0.304 0.300 0.297
0.672 0.669 0.667 0.665 0.663
1.572 1.574 1.576 1.578 1.580
2.92 2.94 2.96 2.98 3.00
0.185 0.183 0.181 0.178 0.176
3.191 3.248 3.306 3.365 3.424
0.293 0.290 0.286 0.284 0.281
0.661 0.659 0.658 0.655 0.654
1.582 1.583 1.585 1.586 1.588
Gas Tables for Compressible Flow Calculations
116
TABLE 8.2 Flow of perfect gases with heat transfer (γγ = 1.4) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
3.10 3.20 3.30 3.40 3.50
0.166 0.156 0.148 0.130 0.1275
3.741 4.097 4.465 4.878 5.328
0.265 0.251 0.237 0.225 0.214
0.645 0.637 0.629 0.622 0.616
1.595 1.603 1.609 1.615 1.620
3.60 3.70 3.80 3.90 4.00
0.125 0.119 0.113 0.107 0.1026
5.817 6.349 6.925 7.551 8.227
0.204 0.193 0.185 0.176 0.168
0.609 0.604 0.598 0.594 0.589
1.625 1.630 1.633 1.637 1.641
4.10 4.20 4.30 4.40 4.50
0.098 0.093 0.089 0.085 0.0818
8.958 9.747 10.598 11.515 12.502
0.161 0.154 0.147 0.141 0.135
0.585 0.581 0.577 0.573 0.569
1.645 1.647 1.653 1.654 1.656
4.60 4.70 4.80 4.90 5.00
0.078 0.075 0.072 0.069 0.0667
13.563 14.702 15.923 17.232 15.634
0.130 0.125 0.120 0.115 0.111
0.567 0.564 0.561 0.558 0.556
1.658 1.661 1.663 1.665 1.667
6.00 7.00 8.00 9.00 10.00
0.0467 0.0345 0.0265 0.021 0.017
38.946 75.415 136.62 233.88 381.62
0.0785 0.058 0.045 0.036 0.029
0.536 0.524 0.516 0.511 0.507
1.681 1.690 1.695 1.699 1.702
Flow of Perfect Gases with Heat Transfer
117
TABLE 8.3 Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) T0 /T0*
c/c* = ρ*/ρ
M
p/p*
p0 /p0*
T/T*
0.02 0.04 0.06 0.08 0.10
2.665 2.659 2.651 2.638 2.626
1.298 1.297 1.295 1.292 1.289
0.0025 0.0113 0.025 0.044 0.069
0.0021 0.0085 0.0190 0.033 0.052
0.001 0.004 0.009 0.017 0.026
0.12 0.14 0.16 0.18 0.20
2.605 2.583 2.558 2.530 2.503
1.284 1.278 1.273 1.266 1.259
0.097 0.131 0.167 0.207 0.251
0.074 0.098 0.127 0.157 0.190
0.037 0.051 0.065 0.082 0.100
0.22 0.24 0.26 0.28 0.30
2.468 2.433 2.397 2.358 2.320
1.251 1.243 1.234 1.225 1.215
0.294 0.341 0.388 0.436 0.485
0.225 0.261 0.298 0.335 0.374
0.119 0.140 0.162 0.185 0.209
0.32 0.34 0.36 0.38 0.40
2.278 2.236 2.193 2.149 2.107
1.207 1.197 1.187 1.178 1.168
0.531 0.578 0.623 0.667 0.710
0.412 0.450 0.487 0.525 0.561
0.233 0.258 0.284 0.310 0.337
0.42 0.44 0.46 0.48 0.50
2.061 2.016 1.971 1.927 1.884
1.158 1.148 1.138 1.129 1.120
0.749 0.787 0.822 0.855 0.887
0.595 0.628 0.660 0.691 0.720
0.364 0.390 0.417 0.445 0.471
0.52 0.54 0.56 0.58 0.60
1.838 1.795 1.751 1.708 1.667
1.111 1.102 1.094 1.086 1.078
0.914 0.939 0.962 0.982 1.001
0.747 0.773 0.797 0.819 0.840
0.497 0.524 0.549 0.575 0.600
Gas Tables for Compressible Flow Calculations
118
TABLE 8.3 Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
0.62 0.64 0.66 0.68 0.70
1.625 1.585 1.545 1.506 1.468
1.070 1.063 1.056 1.049 1.044
1.015 1.028 1.039 1.049 1.056
0.859 0.877 0.893 0.907 0.921
0.625 0.649 0.673 0.697 0.719
0.72 0.74 0.76 0.78 0.80
1.431 1.394 1.358 1.324 1.291
1.038 1.033 1.028 1.023 1.019
1.061 1.065 1.067 1.066 1.066
0.933 0.944 0.954 0.962 0.970
0.742 0.764 0.785 0.806 0.826
0.82 0.84 0.86 0.88 0.90
1.257 1.226 1.194 1.164 1.135
1.016 1.012 1.010 1.007 1.005
1.063 1.059 1.055 1.049 1.043
0.976 0.982 0.986 0.990 0.994
0.845 0.865 0.883 0.902 0.919
0.92 0.94 0.96 0.98 1.00
1.106 1.078 1.052 1.025 1.000
1.003 1.0017 1.0008 1.0002 1.000
1.036 1.028 1.019 1.009 1.000
0.996 0.998 0.999 0.999 1.000
0.936 0.953 0.969 0.985 1.000
1.02 1.04 1.06 1.08 1.10
0.975 0.951 0.928 0.906 0.884
1.0002 1.0007 1.0017 1.003 1.0046
0.989 0.979 0.968 0.957 0.945
0.999 0.999 0.998 0.997 0.995
1.015 1.029 1.043 1.056 1.070
1.12 1.14 1.16 1.18 1.20
0.863 0.842 0.822 0.803 0.784
1.0066 1.009 1.011 1.015 1.018
0.934 0.922 0.910 0.898 0.886
0.993 0.991 0.988 0.986 0.983
1.082 1.095 1.106 1.118 1.130
Flow of Perfect Gases with Heat Transfer
119
TABLE 8.3 Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) c/c* = ρ*/ρ
M
p/p*
p0 /p0*
T/T*
T0 /T0*
1.22 1.24 1.26 1.28 1.30
0.766 0.748 0.731 0.715 0.698
1.022 1.026 1.030 1.035 1.040
0.874 0.861 0.849 0.837 0.825
0.960 0.978 0.974 0.971 0.967
1.140 1.150 1.161 1.171 1.181
1.32 1.34 1.36 1.38 1.40
0.683 0.668 0.653 0.639 0.625
1.045 1.051 1.057 1.063 1.070
0.813 0.801 0.789 0.778 0.765
0.964 0.960 0.957 0.953 0.950
1.189 1.198 1.208 1.216 1.225
1.42 1.44 1.46 1.48 1.50
0.611 0.598 0.586 0.573 0.561
1.077 1.084 1.092 1.099 1.108
0.754 0.743 0.731 0.720 0.708
0.946 0.942 0.938 0.935 0.931
1.233 1.240 1.248 1.256 1.263
1.52 1.54 1.56 1.58 1.60
0.549 0.538 0.527 0.517 0.506
1.116 1.125 1.134 1.144 1.154
0.698 0.688 0.677 0.667 0.656
0.927 0.923 0.920 0.916 0.913
1.269 1.276 1.283 1.289 1.296
1.62 1.64 1.66 1.68 1.70
0.496 0.486 0.476 0.467 0.458
1.163 1.174 1.184 1.195 1.206
0.646 0.636 0.626 0.617 0.607
0.908 0.905 0.901 0.898 0.895
1.302 1.307 1.313 1.319 1.325
1.72 1.74 1.76 1.78 1.80
0.449 0.441 0.432 0.425 0.416
1.217 1.229 1.241 1.254 1.266
0.598 0.589 0.580 0.571 0.562
0.891 0.887 0.884 0.881 0.878
1.329 1.335 1.340 1.345 1.350
Gas Tables for Compressible Flow Calculations
120
TABLE 8.3 Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
1.82 1.84 1.86 1.88 1.90
0.409 0.401 0.394 0.387 0.380
1.280 1.293 1.306 1.320 1.334
0.554 0.546 0.537 0.529 0.521
0.875 0.871 0.868 0.865 0.862
1.354 1.358 1.362 1.367 1.371
1.92 1.94 1.96 1.98 2.00
0.373 0.367 0.360 0.354 0.348
1.348 1.363 1.378 1.393 1.408
0.514 0.506 0.498 0.491 0.483
0.858 0.856 0.853 0.850 0.848
1.375 1.379 1.383 1.387 1.391
2.02 2.04 2.06 2.08 2.10
0.342 0.336 0.330 0.325 0.319
1.425 1.441 1.457 1.475 1.490
0.477 0.470 0.463 0.456 0.450
0.844 0.841 0.838 0.836 0.834
1.394 1.397 1.401 1.404 1.408
2.12 2.14 2.16 2.18 2.20
0.314 0.309 0.304 0.299 0.294
1.509 1.527 1.545 1.563 1.580
0.443 0.437 0.431 0.425 0.418
0.831 0.828 0.826 0.823 0.821
1.411 1.414 1.417 1.419 1.423
2.22 2.24 2.26 2.28 2.30
0.289 0.285 0.280 0.276 0.272
1.600 1.620 1.640 1.660 1.678
0.413 0.407 0.401 0.396 0.390
0.818 0.816 0.814 0.811 0.810
1.426 1.428 1.431 1.433 1.435
2.32 2.34 2.36 2.38 2.40
0.267 0.263 0.259 0.255 0.251
1.700 1.721 1.742 1.764 1.783
0.385 0.380 0.375 0.369 0.364
0.807 0.805 0.803 0.800 0.800
1.438 1.441 1.444 1.446 1.450
Flow of Perfect Gases with Heat Transfer
121
TABLE 8.3 Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
2.42 2.44 2.46 2.48 2.50
0.248 0.244 0.241 0.237 0.233
1.807 1.830 1.853 1.876 1.897
0.359 0.355 0.350 0.345 0.341
0.796 0.795 0.793 0.791 0.789
1.450 1.453 1.455 1.457 1.459
2.52 2.54 2.56 2.58 2.60
0.230 0.227 0.223 0.220 0.217
1.923 1.947 1.971 1.996 2.018
0.336 0.332 0.328 0.324 0.319
0.786 0.785 0.783 0.781 0.780
1.461 1.460 1.465 1.467 1.469
2.62 2.64 2.66 2.68 2.70
0.214 0.211 0.208 0.206 0.203
2.050 2.073 2.098 2.125 2.148
0.315 0.311 0.307 0.304 0.299
0.778 0.776 0.774 0.773 0.772
1.470 1.472 1.474 1.476 1.477
2.72 2.74 2.76 2.78 2.80
0.200 0.197 0.195 0.192 0.189
2.179 2.207 2.234 2.263 2.287
0.296 0.292 0.288 0.285 0.281
0.769 0.768 0.767 0.765 0.764
1.479 1.481 1.482 1.484 1.485
2.82 2.84 2.86 2.88 2.90
0.187 0.185 0.182 0.180 0.178
2.320 2.350 2.379 2.409 2.435
0.278 0.275 0.272 0.268 0.265
0.762 0.761 0.759 0.758 0.757
1.487 1.488 1.489 1.491 1.492
2.92 2.94 2.96 2.98 3.00
0.175 0.173 0.171 0.168 0.167
2.471 2.502 2.533 2.565 2.587
0.262 0.259 0.256 0.253 0.250
0.755 0.754 0.753 0.751 0.750
1.494 1.495 1.496 1.498 1.500
Gas Tables for Compressible Flow Calculations
122
TABLE 8.3
Flow of perfect gases with heat transfer (γγ = 1.67) (Rayleigh flow) M
p/p*
p0 /p0*
T/T*
T0 /T0*
c/c* = ρ*/ρ
3.10 3.20 3.30 3.40 3.50
0.157 0.148 0.139 0.132 0.124
2.765 2.941 3.127 3.324 3.521
0.236 0.223 0.211 0.200 0.189
0.745 0.738 0.733 0.728 0.725
1.505 1.511 1.515 1.520 1.524
3.60 3.70 3.80 3.90 4.00
0.118 0.112 0.106 0.101 0.096
3.750 3.980 4.221 4.473 4.715
0.180 0.172 0.163 0.156 0.148
0.720 0.716 0.713 0.709 0.707
1.528 1.532 1.535 1.538 1.542
4.10 4.20 4.30 4.40 4.50
0.092 0.087 0.084 0.080 0.077
5.014 5.303 5.605 5.918 6.213
0.142 0.136 0.129 0.124 0.119
0.703 0.700 0.698 0.695 0.694
1.544 1.546 1.548 1.550 1.553
4.60 4.70 4.80 4.90 5.00
0.074 0.070 0.067 0.065 0.062
6.588 6.944 7.314 7.698 8.045
0.114 0.109 0.105 0.101 0.098
0.691 0.689 0.687 0.685 0.685
1.555 1.556 1.558 1.559 1.560
6.00 7.00 8.00 9.00 10.00
0.044 0.032 0.0248 0.0190 0.0159
12.860 19.440 28.070 39.051 52.660
0.068 0.051 0.039 0.031 0.025
0.672 0.664 0.660 0.656 0.653
1.573 1.580 1.585 1.587 1.589
Flow of Perfect Gases with Heat Transfer
123
TABLE 8.4
Maximum heat transfer in perfect gases M
Qmax /cpT γ = 1.3
γ = 1.4
γ = 1.67
0.00
∞
∞
∞
0.10 0.20 0.30 0.40 0.50
21.300 5.000 2.000 0.960 0.490
20.410 4.800 1.920 0.920 0.468
18.400 4.307 1.722 0.825 0.421
0.60 0.70 0.80 0.90 1.00
0.2475 0.116 0.044 0.0097 0.000
0.2368 0.111 0.0421 0.0093 0.000
0.2129 0.0994 0.0378 0.00836 0.000
1.10 1.20 1.30 1.40 1.50
0.00791 0.0292 0.0615 0.102 0.1511
0.0076 0.0280 0.0588 0.0980 0.1450
0.00682 0.0252 0.0528 0.0880 0.130
1.60 1.70 1.80 1.90 2.00
0.2065 0.268 0.337 0.411 0.489
0.1980 0.257 0.323 0.394 0.468
0.178 0.231 0.290 0.354 0.421
2.10 2.20 2.30 2.40 2.50
0.574 0.661 0.756 0.854 0.958
0.550 0.633 0.725 0.818 0.918
0.494 0.569 0.652 0.736 0.826
Gas Tables for Compressible Flow Calculations
124
TABLE 8.4
Maximum heat transfer in perfect gases M
Qmax /cpT γ = 1.3
γ = 1.4
γ = 1.67
2.60 2.70 2.80 2.90 3.00
1.067 1.179 1.298 1.419 1.546
1.023 1.130 1.243 1.360 1.481
0.920 1.016 1.118 1.223 1.331
3.50 4.00 4.50 5.00 6.00
2.245 3.060 3.978 5.000 7.397
2.152 2.930 3.812 4.800 7.089
1.934 2.633 3.427 4.314 6.373
7.00 8.00 9.00 10.00 ∞
10.220 13.480 17.174 21.306 ∞
9.796 12.919 16.458 20.418 ∞
8.805 11.613 14.794 18.354 ∞
Prandtl–Meyer Functions and Mach Angles
125
TABLE 9.1
Prandtl–Meyer functions and Mach angles M
α
ω (M) γ = 1.3
γ = 1.4
γ = 1.67
1.00
90.00
0.00
0.00
0.00
1.02 1.04 1.06 1.08 1.10
78.63 74.06 70.63 67.81 65.38
0.131 0.367 0.670 1.014 1.400
0.126 0.351 0.637 0.968 1.336
0.113 0.315 0.570 0.865 1.191
1.12 1.14 1.16 1.18 1.20
63.23 61.30 59.55 57.94 56.44
1.820 2.270 2.740 3.233 3.750
1.735 2.160 2.607 3.075 3.558
1.543 1.917 2.310 2.718 3.140
1.22 1.24 1.26 1.28 1.30
55.05 53.75 52.53 51.38 50.29
4.275 4.820 5.375 5.944 6.523
4.057 4.569 5.093 5.627 6.170
3.573 4.017 4.469 4.928 5.393
1.32 1.34 1.36 1.38 1.40
49.25 48.27 47.33 46.44 45.59
7.112 7.710 8.311 8.923 9.542
6.721 7.280 4.844 8.413 8.987
5.864 6.340 6.820 7.300 7.784
1.42 1.44 1.46 1.48 1.50
44.77 43.98 43.23 42.51 41.81
10.164 10.791 11.421 12.056 12.693
9.565 10.147 10.730 11.317 11.905
8.270 8.756 9.244 9.732 10.220
1.52 1.54 1.56 1.58 1.60
41.14 40.50 39.87 39.27 38.68
13.333 13.975 14.620 15.263 15.910
12.495 13.086 13.677 14.268 14.860
10.708 11.195 11.681 12.167 12.650
Gas Tables for Compressible Flow Calculations
126
TABLE 9.1
Prandtl–Meyer functions and Mach angles M
α
ω (M) γ = 1.3
γ = 1.4
γ = 1.67
1.62 1.64 1.66 1.68 1.70
38.12 37.57 37.04 36.53 36.03
16.556 17.203 17.850 18.500 19.144
15.452 16.043 16.633 17.222 17.810
13.131 13.611 14.089 14.564 15.037
1.72 1.74 1.76 1.78 1.80
35.55 35.08 34.62 34.18 33.75
19.790 20.435 21.080 21.723 22.360
18.397 18.982 19.565 20.146 20.725
15.508 15.976 16.441 16.903 17.363
1.82 1.84 1.86 1.88 1.90
33.33 32.92 32.53 32.13 31.75
23.004 23.644 24.281 24.916 25.550
21.302 21.877 22.450 23.020 23.586
17.820 18.273 18.723 19.170 19.614
1.92 1.94 1.96 1.98 2.00
31.39 31.03 30.68 30.34 30.00
26.181 26.805 27.435 28.060 28.681
24.151 24.712 25.271 25.827 26.380
20.054 20.491 20.925 21.355 21.782
2.02 2.04 2.06 2.08 2.10
29.67 29.35 29.04 28.74 28.44
29.300 29.915 30.530 31.120 31.750
26.930 27.476 28.020 28.561 29.097
22.206 22.626 23.042 23.455 23.865
2.12 2.14 2.16 2.18 2.20
28.15 27.86 27.58 27.31 27.04
32.350 32.952 33.561 34.150 34.750
29.631 30.161 30.690 31.212 31.733
24.271 24.674 25.073 25.468 25.860
Prandtl–Meyer Functions and Mach Angles
127
TABLE 9.1
Prandtl–Meyer functions and Mach angles M
α
ω (M) γ = 1.3
γ = 1.4
γ = 1.67
2.22 2.24 2.26 2.28 2.30
26.77 26.52 26.26 26.01 25.77
35.332 35.920 36.503 37.080 37.660
32.250 32.763 33.273 33.780 34.283
26.249 26.364 27.017 27.395 27.771
2.32 2.34 2.36 2.38 2.40
25.54 25.30 25.07 24.85 24.63
38.233 38.803 39.372 39.936 40.495
34.783 35.279 35.771 36.261 36.746
28.141 28.510 28.875 29.236 29.594
2.42 2.44 2.46 2.48 2.50
24.41 24.20 23.99 23.78 23.58
41.052 41.607 42.158 42.705 43.250
37.229 37.708 38.183 32.655 39.124
29.501 30.301 30.650 31.000 31.337
2.52 2.54 2.56 2.58 2.60
23.38 23.19 22.99 22.81 22.62
43.790 44.325 44.860 45.390 45.920
39.589 40.051 40.509 40.963 41.415
31.507 32.012 32.344 32.674 33.000
2.62 2.64 2.66 2.68 2.70
22.44 22.26 22.08 21.91 21.74
46.440 46.960 47.477 47.990 48.500
41.863 42.307 42.749 43.187 43.622
33.324 33.645 33.961 34.277 34.588
2.72 2.74 2.76 2.78 2.80
21.57 21.41 21.25 21.08 20.93
49.008 49.511 50.011 50.508 51.002
44.053 44.481 44.906 45.328 45.746
34.897 35.203 35.505 35.806 36.104
Gas Tables for Compressible Flow Calculations
128
TABLE 9.1
Prandtl–Meyer functions and Mach angles M
α
ω (M) γ = 1.3
γ = 1.4
γ = 1.67
2.82 2.84 2.86 2.88 2.90
20.77 20.62 20.46 20.32 20.17
51.490 51.980 52.463 52.943 53.420
46.161 46.573 46.982 47.388 47.790
36.398 36.691 36.980 37.267 37.551
2.92 2.94 2.96 2.98 3.00
20.03 19.89 19.75 19.61 19.47
53.895 54.365 54.831 55.300 55.758
48.190 48.586 48.980 49.370 49.757
37.832 38.111 38.387 38.661 38.932
3.05 3.10 3.15 3.20 3.25
19.14 18.82 18.51 18.21 17.92
56.900 58.020 59.120 60.202 61.266
50.713 51.650 52.569 53.470 54.355
39.600 40.252 40.890 41.513 42.122
3.30 3.35 3.40 3.45 3.50
17.63 17.37 17.10 16.85 16.60
62.311 63.341 64.350 65.341 66.318
55.222 56.073 56.907 57.726 58.530
42.720 43.301 43.871 44.430 44.975
3.55 3.60 3.65 3.70 3.75
16.36 16.13 15.91 15.68 15.47
67.277 68.220 69.150 70.060 70.954
59.316 60.092 60.850 61.595 62.327
45.510 46.032 46.544 47.045 47.535
3.80 3.85 3.90 3.95 4.00
15.26 15.05 14.86 14.67 14.48
71.833 72.700 73.551 74.388 75.210
63.044 63.748 64.440 65.118 65.785
48.015 48.485 48.946 49.397 49.840
Prandtl–Meyer Functions and Mach Angles
129
TABLE 9.1
Prandtl–Meyer functions and Mach angles M
α
ω (M) γ = 1.3
γ = 1.4
γ = 1.67
4.05 4.10 4.15 4.20 4.25
14.30 14.11 13.94 13.78 13.61
76.020 76.815 77.600 78.367 79.123
66.439 67.081 67.713 68.333 68.943
50.273 50.698 51.114 51.522 51.923
4.30 4.35 4.40 4.45 4.50
13.45 13.29 13.14 12.98 12.84
79.868 80.600 81.321 82.031 82.730
69.541 70.130 70.706 71.274 71.832
52.315 52.700 53.078 53.449 53.812
4.55 4.60 4.65 4.70 4.75
12.69 12.56 12.42 12.28 12.15
83.413 84.088 84.751 85.408 86.051
72.380 72.920 73.450 73.970 74.480
54.169 54.520 54.863 55.201 55.532
4.80 4.85 4.90 4.95 5.00
12.03 11.89 11.77 11.66 11.54
86.685 87.309 87.923 88.528 89.123
74.990 75.482 75.971 76.449 76.920
55.861 56.178 56.492 56.800 57.104
Gas Tables for Compressible Flow Calculations
130
TABLE 9.2
Mach numbers and Mach angles as functions of ω (M) ω (M)
γ = 1.3
γ = 1.4
γ = 1.67
M
α
M
α
M
α
0
1.000
90.000
1.000
90.000
1.000
90.000
0.5 1.0 1.5 2.0
1.049 1.079 1.105 1.128
72.358 67.905 64.833 62.422
1.051 1.082 1.108 1.133
72.098 67.574 64.451 62.000
1.055 1.089 1.117 1.144
71.437 66.732 63.476 60.915
2.5 3.0 3.5 4.0 4.5
1.150 1.171 1.191 1.209 1.228
60.411 58.675 57.138 55.755 54.500
1.155 1.177 1.198 1.218 1.238
59.950 58.180 56.615 55.205 53.920
1.169 1.194 1.217 1.240 1.262
58.774 56.920 55.278 53.800 52.455
5.0 5.5 6.0 6.5 7.0
1.247 1.265 1.282 1.300 1.316
53.340 52.270 51.265 50.327 49.442
1.257 1.275 1.294 1.312 1.330
52.738 51.642 50.619 49.658 48.753
1.283 1.305 1.326 1.347 1.368
51.200 50.050 48.964 47.950 46.990
7.5 8.0 8.5 9.0 9.5
1.333 1.349 1.366 1.383 1.398
48.605 47.810 47.054 46.332 45.641
1.348 1.366 1.383 1.400 1.418
47.896 47.082 46.305 45.566 44.857
1.389 1.409 1.430 1.450 1.471
46.081 45.216 44.392 43.605 42.848
10.0 11.0 12.0 13.0 14.0
1.415 1.447 1.478 1.510 1.541
44.978 43.730 42.569 41.485 40.468
1.435 1.469 1.503 1.537 1.571
44.177 42.894 41.700 40.585 39.537
1.490 1.532 1.573 1.615 1.656
42.120 40.750 39.470 38.270 37.140
Prandtl–Meyer Functions and Mach Angles
131
TABLE 9.2
Mach numbers and Mach angles as functions of ω (M) ω (M)
γ = 1.3
γ = 1.4
γ = 1.67
M
α
M
α
M
α
15.0 16.0 17.0 18.0 19.0
1.572 1.603 1.634 1.665 1.695
39.508 38.602 37.740 36.923 36.140
1.605 1.638 1.673 1.707 1.741
38.547 37.611 36.721 35.874 35.065
1.696 1.740 1.784 1.828 1.873
36.071 35.060 34.089 33.165 32.281
20.0 21.0 22.0 23.0 24.0 25.0
1.727 1.758 1.789 1.820 1.851 1.883
35.394 34.679 33.993 33.333 32.700 32.084
1.775 1.810 1.845 1.879 1.915 1.950
34.290 33.548 32.835 32.148 31.486 30.847
1.918 1.964 2.010 2.058 2.107 2.157
31.433 30.615 29.830 29.073 28.339 27.629
26.0 27.0 28.0 29.0 30.0
1.915 1.945 1.978 2.010 2.043
31.493 30.921 30.367 29.831 29.310
1.986 2.023 2.059 2.096 2.134
30.229 29.632 29.053 28.491 27.945
2.207 2.259 2.313 2.366 2.423
26.941 26.273 25.624 25.000 24.377
31.0 32.0 33.0 34.0 35.0
2.075 2.108 2.142 2.175 2.209
28.805 28.315 27.838 27.373 26.921
2.172 2.210 2.249 2.289 2.328
27.415 26.900 26.397 25.908 25.430
2.480 2.540 2.600 2.663 2.727
23.777 23.192 22.620 22.061 21.515
36.0 37.0 38.0 39.0 40.0
2.243 2.278 2.312 2.347 2.383
26.480 26.050 25.630 25.222 24.820
2.369 2.411 2.452 2.495 2.538
24.965 24.510 24.066 23.631 23.206
2.793 2.861 2.932 3.005 3.081
20.980 20.456 19.942 19.437 18.942
Gas Tables for Compressible Flow Calculations
132
TABLE 9.2
Mach numbers and Mach angles as functions of ω (M) ω (M)
γ = 1.3
γ = 1.4
γ = 1.67
M
α
M
α
M
α
41.0
2.418
24.428
2.584
22.790
3.159
18.456
42.0 43.0
2.454 2.491
24.045 23.670
2.626 2.672
22.382 21.983
3.240 3.325
17.980 17.510
44.0
2.528
23.305
2.718
21.591
3.411
17.045
45.0
2.565
22.943
2.765
21.207
3.502
16.590
46.0
2.603
22.591
2.812
20.830
3.597
16.142
47.0 48.0
2.642 2.680
22.245 21.906
2.861 2.911
20.459 20.096
3.696 3.800
15.700 15.262
49.0
2.720
21.573
2.961
19.735
3.906
14.832
50.0
2.760
21.246
3.013
19.387
4.018
14.410
51.0
2.800
20.925
3.013
19.387
4.136
14.000
52.0 53.0
2.841 2.883
20.610 20.300
3.065 3.119
19.041 18.700
4.260 4.390
13.580 13.168
54.0
2.925
20.000
3.174
18.366
4.526
12.764
55.0
2.967
19.695
3.229
18.036
4.670
12.364
56.0
3.010
19.400
3.287
17.711
4.822
11.968
57.0 58.0
3.055 3.100
19.110 18.825
3.346 3.406
17.391 17.076
4.983 5.153
11.578 11.190
59.0
3.145
18.543
3.470
16.765
5.334
10.806
60.0
3.191
18.265
3.592
16.155
5.525
10.424
61.0
3.237
18.000
3.660
15.857
5.731
10.048
62.0 63.0
3.285 3.334
17.723 17.457
3.727 3.797
15.561 15.270
5.950 6.185
9.675 9.304
64.0
3.383
17.195
3.868
14.983
6.438
8.935
65.0
3.433
16.937
3.941
14.698
6.710
8.571
Prandtl–Meyer Functions and Mach Angles
133
TABLE 9.2
Mach numbers and Mach angles as functions of ω (M) ω (M)
γ = 1.3
γ = 1.4
γ = 1.67
M
α
M
α
M
α
66.0 67.0 68.0 69.0 70.0
3.484 3.536 3.588 3.642 3.697
16.682 16.430 16.182 15.937 15.694
4.016 4.094 4.173 4.255 4.340
14.417 14.140 13.965 13.593 13.325
7.000 7.323 7.670 8.050 8.468
8.209 7.850 7.490 7.135 6.782
71.0 72.0 73.0 74.0 75.0
3.753 3.810 3.868 3.927 3.987
15.455 15.219 14.985 14.754 14.525
4.426 4.515 4.608 4.703 4.801
13.059 12.800 12.535 12.277 12.021
8.928 9.440 10.010 10.651 11.378
6.430 6.081 5.733 5.387 5.043
Gas Tables for Compressible Flow Calculations
134
TABLE 10.1
Approximate velocities and Mach numbers of air at p = 1.0133 bar t = 25°C (T = 298.15 K and ρ = 1.183 kg/m3 ; a = 347.07 m/s) Dynamic head (mm W.G.)
Velocity (c) m/s
Mach number (M)
0 1 2 3 4 5
0.00 4.07 5.76 7.05 8.14 9.10
0.000 0.0117 0.0166 0.0203 0.0235 0.0262
10 15 20 25 30
12.87 15.76 18.20 20.35 22.29
0.0371 0.0454 0.0525 0.0586 0.0643
35 40 45 50 55
24.08 25.74 27.30 28.78 30.18
0.0694 0.0742 0.0787 0.0829 0.0270
60 65 70 75 80
31.53 32.81 34.05 35.25 36.40
0.0908 0.0945 0.0981 0.1015 0.1049
85 90 95 100
37.52 38.61 39.67 40.70
0.1081 0.1113 0.1143 0.1173
200 300 400 500 600
57.56 70.49 81.40 91.00 99.69
0.1658 0.2032 0.2346 0.2623 0.2873
Approximate Velocities and Mach Numbers of Air
135
TABLE 10.2
Approximate velocities and Mach numbers of air at p = 1.0133 bar t = 25°C (T = 298.15 K and ρ = 1.183 kg/m3 ; a = 347.07 m/s) Dynamic head (mm Hg)
Velocity (c) m/s
Mach number (M)
50 60 70 80
106.06 116.19 125.50 134.16
0.306 0.335 0.362 0.387
90 100 110 120
142.30 150.00 157.30 164.32
0.410 0.432 0.453 0.474
130 140 150 160
171.03 177.48 183.71 189.74
0.493 0.511 0.529 0.547
170 180 190 200
195.50 201.25 206.76 212.13
0.564 0.580 0.596 0.611
250 300 350 400 450 500
237.17 259.81 280.62 300.00 318.17 335.47
0.683 0.748 0.808 6.865 0.917 0.966
Gas Tables for Compressible Flow Calculations
136
TABLE 10.3
Mach numbers as functions of pressure ratios (γγ = 1.4) p/p0
M
p/p0
M
1.00 0.99 0.98 0.97 0.96 0.95
0.000 0.120 0.170 0.209 0.242 0.272
0.74 0.73 0.72 0.71 0.70
0.670 0.686 0.702 0.717 0.732
0.94 0.93 0.92 0.91 0.90
0.298 0.324 0.347 0.370 0.391
0.69 0.68 0.67 0.66 0.65
0.748 0.763 0.778 0.794 0.809
0.89 0.88 0.87 0.86 0.85
0.411 0.431 0.451 0.469 0.488
0.64 0.63 0.62 0.61 0.60
0.825 0.840 0.855 0.871 0.886
0.84 0.83 0.82 0.81 0.80
0.505 0.523 0.540 0.557 0.574
0.59 0.58 0.57 0.56 0.55
0.902 0.918 0.933 0.949 0.965
0.79 0.78 0.77 0.76 0.75
0.590 0.607 0.623 0.639 0.655
0.54 0.53 0.52 0.51 0.50
0.981 0.997 1.014 1.030 1.046
Pressure Ratios and Mach Numbers of Air
137
TABLE 10.3
Mach numbers as functions of pressure ratios (γγ = 1.4) p/p0
M
p/p0
M
0.49 0.48 0.47 0.46 0.45
1.063 1.080 1.097 1.114 1.132
0.24 0.23 0.22 0.21 0.20
1.587 1.615 1.645 1.676 1.709
0.44 0.43 0.42 0.41 0.40
1.150 1.168 1.186 1.204 1.223
0.19 0.18 0.17 0.16 0.15
1.742 1.778 1.815 1.855 1.897
0.39 0.38 0.37 0.36 0.35
1.242 1.262 1.282 1.302 1.322
0.14 0.13 0.12 0.11 0.10
1.941 1.989 2.040 2.096 2.157
0.34 0.33 0.32 0.31 0.30
1.344 1.365 1.387 1.410 1.433
0.09 0.08 0.07 0.06 0.05
2.225 2.300 2.385 2.484 2.601
0.29 0.28 0.27 0.26 0.25
1.457 1.481 1.506 1.532 1.560
0.04 0.03 0.02 0.01 0.00
2.746 2.935 3.208 3.693 ∞
Gas Tables for Compressible Flow Calculations
138
TABLE 11
Manometer inclinations required for the desired magnification of the deflection Magnification required (1/h)
Inclination from the horizontal α = (sin−1 h/1) degrees
2 3 4 5 6 7 8 9 10
30 19.47 14.47 11.53 9.59 8.21 7.18 6.38 5.74
Static Pressure Drop in Gas Pipe Lines
139
TABLE 12.1
Static pressure drop (pr = outlet pressure/inlet pressure) in gas pipe lines (Isothermal flow, f = 0.002 − 0.005, γ = 1.3, M1 = 0.02) L/D
Static pressure ratio, pr = p2 /p1 f = 0.0020
f = 0.0025
f = 0.0030
f = 0.0035
f = 0.0040
f = 0.0050
10 50 100 250 400
0.9999 0.9998 0.9997 0.9995 0.9992
0.9999 0.9998 0.9997 0.9993 0.9989
0.9999 0.9998 0.9996 0.9992 0.9987
0.9999 0.9998 0.9996 0.9991 0.9985
0.9999 0.9997 0.9995 0.9989 0.9983
0.9999 0.9997 0.9995 0.9987 0.9979
500 600 700 800 900
0.9989 0.9987 0.9985 0.9983 0.9981
0.9986 0.9984 0.9981 0.9979 0.9976
0.9984 0.9981 0.9978 0.9975 0.9971
0.9982 0.9978 0.9975 0.9970 0.9967
0.9979 0.9975 0.9970 0.9967 0.9962
0.9974 0.9968 0.9963 0.9958 0.9953
1000 2000 3000 4000 5000
0.9979 0.9958 0.9937 0.9916 0.9895
0.9973 0.9947 0.9921 0.9895 0.9869
0.9968 0.9937 0.9905 0.9874 0.9842
0.9963 0.9927 0.9890 0.9853 0.9816
0.9958 0.9916 0.9874 0.9832 0.9789
0.9947 0.9895 0.9842 0.9789 0.9736
6000 7000 8000 8500 9000
0.9874 0.9853 0.9832 0.9821 0.9811
0.9842 0.9816 0.9789 0.9776 0.9763
0.9811 0.9779 0.9747 0.9731 0.9715
0.9779 0.9742 0.9704 0.9685 0.9667
0.9747 0.9704 0.9661 0.9639 0.9618
0.9683 0.9629 0.9575 0.9548 0.9520
1 × 104 1.5 × 104 2.0 × 104 2.5 × 104 3 × 104
0.9789 0.9683 0.9575 0.9465 0.9355
0.9736 0.9602 0.9465 0.9327 0.9186
0.9683 0.9520 0.9355 0.9187 0.9015
0.9629 0.9438 0.9243 0.9044 0.8841
0.9575 0.9355 0.9130 0.8899 0.8662
0.9466 0.9187 0.8899 0.8602 0.8294
4 × 104 5 × 104 6 × 104 7 × 104 8 × 104
0.9130 0.8899 0.8663 0.8419 0.8168
0.8899 0.8602 0.8295 0.7975 0.7642
0.8663 0.8295 0.7909 0.7505 0.7077
0.8419 0.7975 0.7505 0.7003 0.6462
0.8168 0.7642 0.7077 0.6462 0.5783
0.7642 0.6928 0.6132 0.5215 0.4099
1 × 105 2 × 105
0.7642 0.4098
0.6928 –
0.6132 –
0.5215 –
0.4099 –
– –
Gas Tables for Compressible Flow Calculations
140
TABLE 12.2
Static pressure drop (pr = outlet pressure/linlet pressure) in gas pipe lines (Isothermal flow, f = 0.002 − 0.005, γ = 1.4, M1 = 0.02) L/D
Static pressure ratio, pr = p2 /p1 f = 0.0020
f = 0.0025
f = 0.0030
f = 0.0035
f = 0.0040
f = 0.0050
10 50 100 250 400
0.9999 0.9998 0.9997 0.9994 0.9991
0.9999 0.9998 0.9997 0.9993 0.9988
0.9999 0.9998 0.9996 0.9991 0.9986
0.9999 0.9998 0.9996 0.9990 0.9984
0.9999 0.9997 0.9995 0.9988 0.9982
0.9999 0.9997 0.9994 0.9986 0.9977
500 600 700 800 900
0.9988 0.9986 0.9984 0.9982 0.9979
0.9986 0.9983 0.9980 0.9977 0.9975
0.9983 0.9979 0.9976 0.9973 0.9969
0.9980 0.9976 0.9972 0.9968 0.9964
0.9977 0.9973 0.9968 0.9964 0.9959
0.9972 0.9966 0.9960 0.9955 0.9949
1000 2000 3000 4000 5000
0.9977 0.9955 0.9932 0.9910 0.9887
0.9972 0.9943 0.9915 0.9887 0.9859
0.9966 0.9932 0.9898 0.9864 0.9830
0.9961 0.9921 0.9882 0.9842 0.9802
0.9955 0.9910 0.9864 0.9819 0.9773
0.9944 0.9887 0.9830 0.9773 0.9716
6000 7000 8000 8500 9000
0.9864 0.9842 0.9819 0.9808 0.9796
0.9830 0.9802 0.9773 0.9759 0.9744
0.9796 0.9762 0.9727 0.9710 0.9692
0.9762 0.9721 0.9681 0.9661 0.9640
0.9727 0.9681 0.9635 0.9611 0.9588
0.9658 0.9600 0.9541 0.9512 0.9482
1 × 104 1.5 × 104 2 × 104 2.5 × 104 3 × 104
0.9773 0.9658 0.9541 0.9423 0.9304
0.9716 0.9571 0.9423 0.9273 0.9121
0.9658 0.9482 0.9303 0.9121 0.8935
0.9600 0.9393 0.9182 0.8967 0.8745
0.9541 0.9303 0.9059 0.8809 0.8551
0.9423 0.9121 0.8809 0.8485 0.8150
4 × 104 5 × 104 6 × 104 7 × 104 8 × 104
0.9060 0.8809 0.8551 0.8285 0.8010
0.8809 0.8485 0.8149 0.7797 0.7429
0.8551 0.8149 0.7725 0.7277 0.6800
0.8285 0.7797 0.7277 0.6717 0.6106
0.8010 0.7430 0.6800 0.6106 0.5322
0.7430 0.6633 0.5727 0.4648 0.3225
1 × 105 2 × 105
0.7429 0.3225
0.6633 –
0.5727 –
0.4648 –
0.3225 –
– –
Isentropic Compressor Power
141
TABLE 13.1
Isentropic compressor power in kW (cp =1.0 kJ/kg K, T1 = 300 K, γ = 1.3) m = mass flow rate in kg/s
pr = p2 /p1 1.0
1.5
2.0
3.0
4.0
5.0
1.1 1.2 1.3 1.4 1.5
6.671 12.891 18.724 24.221 29.425
10.007 19.337 28.086 36.332 44.137
13.342 25.782 37.448 48.443 58.850
20.014 38.673 56.172 72.664 88.274
26.685 51.564 74.896 96.886 117.700
33.356 64.455 93.620 121.107 147.123
1.6 1.7 1.8 1.9 2.0
34.367 39.078 43.580 47.894 52.036
51.551 58.617 65.370 71.840 78.054
68.735 78.156 87.160 95.787 104.072
103.102 117.234 130.740 143.681 156.108
137.470 156.312 174.320 191.574 208.144
171.837 195.390 217.900 239.468 260.180
2.1 2.2 2.3 2.4 2.5
56.022 59.864 63.575 67.163 70.638
84.033 89.796 95.362 100.744 105.957
112.044 119.729 127.149 134.326 141.276
168.065 179.593 190.724 201.489 211.914
224.087 239.457 254.299 268.652 282.552
280.109 299.321 317.873 335.815 353.190
2.6 2.7 2.8 2.9 3.0
74.008 77.279 80.459 83.552 86.564
111.011 115.919 120.688 125.328 129.846
148.015 154.558 160.917 167.104 173.128
222.023 231.837 241.376 250.656 259.693
296.031 309.116 321.835 334.208 346.257
370.038 386.396 402.293 417.760 432.822
3.5 4.0 4.5 5.0 5.5
100.563 113.098 124.479 134.926 144.598
150.844 169.646 186.719 202.389 216.897
201.125 226.195 248.959 269.852 289.196
301.688 339.293 373.438 404.779 433.794
402.250 452.390 497.917 539.705 578.391
502.813 565.488 622.397 674.631 722.990
6.0 7.0 8.0 9.0 10.0
153.615 170.041 184.751 198.107 210.365
230.423 255.062 277.126 297.160 315.548
307.230 340.083 369.501 396.213 420.731
460.845 510.124 554.252 594.320 631.096
614.460 680.165 739.002 792.426 841.462
768.075 850.207 923.753 990.533 1051.827
Gas Tables for Compressible Flow Calculations
142
TABLE 13.2
Isentropic compressor power in kW (cp =1.0 kJ/kg K, T1 = 300 K, γ = 1.4) m = mass flow rate in kg/s
pr = p2 /p1 1.0
1.5
2.0
3.0
4.0
5.0
1.1 1.2 1.3 1.4 1.5
8.281 16.041 23.351 30.271 36.845
12.422 24.061 35.027 45.406 55.268
16.563 32.082 46.703 60.541 73.690
24.844 48.123 70.054 90.812 110.536
33.124 64.164 93.406 121.083 147.381
41.406 80.205 116.757 151.353 184.227
1.6 1.7 1.8 1.9 2.0
43.114 49.109 54.856 60.380 65.700
64.671 73.663 82.285 90.571 98.551
86.228 98.217 109.713 120.761 131.401
129.342 147.326 164.569 181.141 197.101
172.456 196.434 219.425 241.522 262.802
215.570 245.543 274.282 301.902 328.502
2.1 2.2 2.3 2.4 2.5
70.834 75.795 80.598 85.254 89.774
106.251 113.693 120.898 127.882 134.661
141.668 151.591 161.197 170.509 179.548
212.501 227.386 241.795 255.763 269.322
283.335 303.181 322.393 341.018 359.095
354.169 378.977 402.992 426.272 448.869
2.6 2.7 2.8 2.9 3.0
94.166 98.439 102.600 106.657 110.615
141.249 147.659 153.901 159.986 165.922
188.332 196.878 205.201 213.314 221.230
282.498 295.317 307.801 319.971 331.845
376.664 393.756 410.402 426.628 442.460
470.830 492.195 513.002 533.286 553.075
3.5 4.0 4.5 5.0 5.5
129.103 145.789 161.046 175.135 188.251
193.654 218.684 241.569 262.702 282.376
258.206 291.579 322.092 350.270 376.501
387.309 437.368 483.137 525.405 564.7521
516.412 583.158 644.183 700.540 753.003
645.515 728.947 805.229 875.675 941.253
6.0 7.0 8.0 9.0 10.0
200.540 223.077 243.418 262.016 279.190
300.810 334.616 365.127 393.023 418.785
401.081 446.154 486.836 524.031 558.381
601.621 669.232 730.254 786.047 837.570
802.161 892.309 973.672 1048.062 1116.761
1002.702 1115.386 1217.090 1310.078 1395.951
Pumping Power for Fans and Blowers
143
TABLE 14
Approximate values of pumping power (kW) for fans and blowers ∆p = Pressure rise in millimeters W.G.
Q (m 3 /s ) 1
2
3
4
5
0.1 0.2 0.3 0.4 0.5
0.0015 0.0030 0.0045 0.0060 0.0075
0.0030 0.0060 0.0091 0.0121 0.0151
0.0045 0.0091 0.0136 0.0181 0.0226
0.0060 0.0121 0.0181 0.0241 0.0302
0.0075 0.0151 0.0226 0.0302 0.0377
1.0 1.2 1.4 1.6 1.8
0.0151 0.0181 0.0211 0.0241 0.0272
0.0302 0.0362 0.0423 0.0483 0.0543
0.0453 0.0543 0.0634 0.0724 0.0815
0.0604 0.0724 0.0845 0.0966 0.1087
0.0755 0.0906 0.1056 0.1207 0.1358
2.0 2.5 3.0 3.5 4.0
0.0302 0.0377 0.0453 0.0528 0.0604
0.0604 0.0755 0.0906 0.1056 0.1207
0.0906 0.1132 0.1358 0.1585 0.1811
0.1207 0.1509 0.1811 0.2111 0.2415
0.1509 0.1886 0.2264 0.2641 0.3018
5.0 10.0 20.0 50.0 100
0.0755 0.1509 0.3018 0.7546 1.5092
0.1509 0.3018 0.6037 1.5092 3.0184
0.2264 0.4528 0.9055 2.2638 4.5276
0.3018 0.6037 1.2073 3.0184 6.0367
0.3773 0.7546 1.5092 3.7730 7.5460
150 200 400 600 800
2.2638 3.0184 6.0367 9.0551 12.0735
4.5276 6.0367 12.0735 18.1102 24.1469
6.7913 9.0551 18.1102 27.1653 36.2204
9.0551 12.0735 24.1469 36.2204 48.2939
11.3189 15.0918 30.1837 45.2755 60.3673
1000 1500 2000 2500 3000
15.0918 22.6378 30.1837 37.7296 45.2755
30.1837 45.2755 60.3673 75.4592 90.5510
45.2755 67.9133 90.5510 113.1888 135.8265
60.3673 90.5510 120.7347 150.9183 181.1020
75.4592 113.1888 150.9183 188.6479 226.3775
Gas Tables for Compressible Flow Calculations
144
TABLE 14
Approximate values of pumping power (kW) for fans and blowers ∆p = Pressure rise in millimeters W.G.
Q (m 3 /s ) 10
20
30
40
50
0.1 0.2 0.3 0.4 0.5
0.0151 0.0302 0.0453 0.0604 0.0755
0.0302 0.0604 0.0906 0.1207 0.1509
0.0453 0.0906 0.1358 0.1811 0.2264
0.0604 0.1207 0.1811 0.2415 0.3018
0.0755 0.1509 0.2264 0.3018 0.3773
1.0 1.2 1.4 1.6 1.8
0.1509 0.1811 0.2113 0.2415 0.2717
0.3018 0.3622 0.4226 0.4829 0.5433
0.4528 0.5433 0.6339 0.7244 0.8150
0.6037 0.7244 0.8451 0.9659 1.0866
0.7546 0.9055 1.0564 1.2073 1.3583
2.0 2.5 3.0 3.5 4.0
0.3018 0.3773 0.4528 0.5282 0.6037
0.6037 0.7546 0.9055 1.0564 1.2073
0.9055 1.1319 1.3583 1.5846 1.8110
1.2073 1.5092 1.8110 2.1129 2.4147
1.5092 1.8865 2.2638 2.6411 3.0184
5.0 10.0 20.0 50.0 100
0.7546 1.5092 3.0184 7.5459 15.0918
1.5092 3.0184 6.0367 15.0918 30.1837
2.2638 4.5276 9.0551 22.6378 45.2755
3.0184 6.0367 12.0735 30.1837 60.3673
3.7730 7.5459 15.0918 37.7296 75.4592
150 200 400 600 800
22.6378 30.1837 60.3673 90.5510 120.7347
45.2755 60.3673 120.7347 181.1020 241.4693
67.9133 90.5510 181.1020 271.6530 362.2040
90.5510 120.7347 241.4693 362.2040 482.9387
113.1888 150.9183 301.8367 452.7550 603.6733
1000 1500 2000 2500 3000
150.9183 226.3775 301.8367 377.2958 452.7550
301.8367 452.7550 603.6733 754.5917 905.5100
452.7550 679.1325 905.5100 1131.8876 1358.2650
603.6733 905.5100 1207.3467 1509.1833 1811.0200
754.5917 1131.8876 1509.1833 1886.4792 2263.7751
Pumping Power for Fans and Blowers
145
TABLE 14
Approximate values of pumping power (kW) for fans and blowers ∆p = Pressure rise in millimeters W.G.
Q (m 3 /s ) 100
200
300
400
500
0.1 0.2 0.3 0.4 0.5
0.1509 0.3018 0.4528 0.6037 0.7546
0.3018 0.6037 0.9055 1.2073 1.5092
0.4528 0.9055 1.3583 1.8110 2.2638
0.6037 1.2073 1.8110 2.4147 3.0184
0.7546 1.5092 2.2638 3.0184 3.7730
1.0 1.2 1.4 1.6 1.8
1.5092 1.8110 2.1129 2.4147 2.7165
3.0184 3.6220 4.2257 4.8294 5.4331
4.5276 5.4331 6.3386 7.2441 8.1496
6.0367 7.2441 8.4514 9.6588 10.8661
7.5459 9.0551 10.5643 12.0735 13.5827
2.0 2.5 3.0 3.5 4.0
3.0184 3.7730 4.5276 5.2821 6.0367
6.0367 7.5459 9.0551 10.5643 12.0735
9.0551 11.3189 13.5827 15.8464 18.1102
12.0735 15.0918 18.1102 21.1286 24.1469
15.0918 18.8648 22.6378 26.4107 30.1837
5.0 10.0 20.0 50.0 100
7.5459 15.0918 30.1837 75.4592 150.9183
15.0918 30.1837 60.3673 150.9183 301.8367
22.6378 45.2755 90.5510 226.3775 452.7550
30.1837 60.3673 120.7347 301.8367 603.6733
37.7296 75.4592 150.9183 377.2958 754.5917
150 200 400 600 800
226.3775 301.8367 603.6733 905.5100 1207.3467
452.7550 603.6733 1207.3467 1811.0200 2414.6934
679.1325 905.5100 1811.0200 2716.5300 3622.0400
905.5100 1207.3467 2414.6934 3622.0400 4829.3867
1131.8876 1509.1833 3018.3667 4527.5503 6036.7334
1000 1500 2000 2500 3000
1509.1833 2263.7751 3018.3667 3772.9585 4527.5503
3018.3667 4527.5503 6036.7334 7545.9170 9055.1006
4527.5503 6791.3252 9055.1006 11318.8750 13582.6504
6036.7334 9055.1006 12073.4668 15091.8340 18110.2012
7.546 MW 11.319 MW 15.092 MW 18.865 MW 22.638 MW
Gas Tables for Compressible Flow Calculations
146
TABLE 14
Approximate values of pumping power (kW) for fans and blowers ∆p = Pressure rise in millimeters W.G.
Q (m 3 /s ) 600
700
800
1000
1500
0.1 0.2 0.3 0.4 0.5
0.9055 1.8110 2.7165 3.6220 4.5276
1.0564 2.1129 3.1693 4.2257 5.2821
1.2073 2.4147 3.6220 4.8294 6.0367
1.5092 3.0184 4.5276 6.0367 7.5459
2.2638 4.5276 6.7913 9.0551 11.3189
1.0 1.2 1.4 1.6 1.8
9.0551 10.8661 12.6771 14.4882 16.2992
10.5643 12.6771 14.7900 16.9029 19.0157
12.0735 14.4882 16.9029 19.3175 21.7322
15.0918 18.1102 21.1286 24.1469 27.1653
22.6380 27.1653 31.6930 36.2204 40.7480
2.0 2.5 3.0 3.5 4.0
18.1102 22.6378 27.1653 31.6929 36.2204
21.1286 26.4107 31.6929 36.9750 42.2571
24.1469 30.1837 36.2204 42.2571 48.2939
30.1837 37.7296 45.2755 52.8214 60.3673
45.2755 56.5944 67.9133 79.2321 90.5510
5.0 10.0 20.0 50.0 100
45.2755 90.5510 181.1020 452.7550 905.5100
52.8214 105.6428 211.2857 528.2142 1056.4283
60.3673 120.7347 241.4693 603.6733 1207.3467
75.4592 150.9183 301.8367 754.5917 1509.1833
113.1888 226.3780 452.7550 1131.8876 2263.7751
150 200 400 600 800
1358.2650 1811.0200 3622.0400 5433.0601 7244.0801
1584.6426 2112.8567 4225.7134 6338.5703 8451.4268
1811.0200 2414.6934 4829.3867 7244.0801 9658.7734
2263.7751 3018.3667 6036.7334 9055.1006 12073.4668
3395.6626 4527.5503 9055.1006 13582.6504 18110.2012
1000 1500 2000 2500 3000
9055.1006 13582.6504 18110.2012 22637.7500 27165.3008
10564.2832 15846.4258 21128.5664 26410.7090 31692.8516
12073.4668 18110.2012 24146.9336 30183.6680 36220.4023
15.091 MW 22.637 MW 30.184 MW 37.729 MW 45.275 MW
22.637 MW 33.957 MW 45.275 MW 56.594 MW 67.913 MW
S.I. Metric and British Units
147
TABLE 15.1
S.I, Metric and British units Principal dimensions and units Quantity
Dimensions
S.I.
Metric
British
Length Mass Time Temperature Area Volume Density Force Torque Velocity Acceleration Pressure Work and energy Heat Specific heat Power Specific entropy Dynamic viscosity
L M T θ L2 L3 M/L3 ML/T2 ML2/T2 L/T L/T2 M/LT2 ML2/T2 ML2/T2 L2/T2θ ML2/T3 L2/T2θ M/LT
m kgm sec °K m2 m3 kgm/m3 kgf m kgf m/sec m/s2 kgf/m2 kgf m cal cal/kgm °K kgf m/s cal/kgm °K kgf-sec/m2
ft lbm sec °R ft2 ft3 Ibm/ft3 lbf ft lbf ft/sec ft/sec2 lbf/ft2 ft lbf Btu Btu/lbm °R ft lbf/sec Btu/lbm °R lbf-sec/ft2
Kinematic viscosity Weight
L2/T ML/T2
m kg s K m2 m3 kg/m3 N = kg m/s2 mN m/s m/s2 Pa = N/m2 J = Nm Joule = J J/kg K Nm/s, J/s J/kg K Ns/m2 kg/ms m2/s N = kg m/s2
m2/sec kgf
ft2/sec lbf
Gas Tables for Compressible Flow Calculations
148
TABLE 15.2
Conversion of units Length 1m 1 cm 1 km 1 ft 1 in 1 mile 1 nautical mile
= = = = = = =
3.28 ft 0.3937 in 0.6215 mile 30.48 cm 2.54 cm 1.609 km 6076 ft = 1.853 km
Area 1 m2 1 cm2 1 ft2 1 in2
= = = =
10.765 ft2 0.155 in2 0.0929 m2 6.452 cm2
Volume 1 m3 = 1000 litres = 35.315 ft2 1 litre = 1000 cc = 0.001 m3 = 0.0353 ft3 1 cc = 0.061 in3 3 1 ft = 0.0283 m3 = 28.328 litres 3 1 in = 16.387 cm3 1 gallon (British) = 8 pints = 4.546 litres = 0.160 ft3 = 277.3 in3 1 pint = 568.25 cc Mass 1 kgm = 2.204 lbm 1 lbm = 0.4537 kgm 1 metric ton (tonne) = 0.984 ton Temperature F − 32 1.8 = 1.8 °C + 32 = 273.15 + °C = Kelvins (in S.I. Units) = 459.4 + °F = 1.8 °K
Pressure 1 bar
= = = 1 m bar = = 1 mm W.G. = 1 torr = = 1 atmosphere = = = Density 1 kg/m3 1 lbm/ft3
°F °K °R
=
Force 1N = 105 dynes = 0.102 kgf = 0.2248 lbf 1 kgf = 9.807 N = 2.204 lbf 1 metric ton = 1000 kgf 1 lbf = 4.448 N = 0.4537 kgf Velocity 1 m/s 1 km/hr 1 mile/hr 1 ft/sec 1 knot
= = = = =
3.28 ft/sec 0.621 mile/hr 1.609 km/hr 0.305 m/s 1.689 ft/sec = 0.515 m/s
= 0.0624 lbm/ft3 = 16.025 kg/m3
Work and Energy 1 Nm = 1 Joule = 107 ergs = 0.102 kgf m = 0.7374 ft lbf 1 kgf m = 9.807 Nm = 7.229 ft lbf 1 ft lbf = 0.1383 kgf m = 1.356 Nm = 1.356 J Heat 1 kJ 1 k cal 1 Btu Power 1 Nm/s
o
°C
105 N/m2 = 105 Pa = 100 kN/m2 0.1 MN/m2 = 1.0197 kgf/cm2 14.504 lbf/in2 10-3 bar = 100 N/m2 = 100 Pa 10.2 mm W.G. 1 kgf/m2 = 9.807 N/m2 = 0.098 m bar 1 mm Hg = 13.6 mm W.G. = 133.38N/m2 1.334 m bar 1.013 bar = 1.033 kgf/cm2 14.696 lbf/in2 = 29.921 in Hg 760 mm Hg = 10.336 m W.G.
1 kW
= 1000 J = 0.2388 k cal = 0.9478 Btu = 427 kgf m = 4.187 kJ = 3.968 Btu = 778 ft lbf = 1.055 kJ = 0.252 k cal
= 1 J/s = 1 watt = 0.102 kgf m/sec = 0.7375 ft lb/sec = 1000 J/s = 102 kgf m/sec = 737.5 ft lbf/sec
= 1.36 metric H.P. = 1.34 H.P. (FPS) 1 metric H.P. = 75 kgf m/sec = 0.735 kW = 0.985 H.P. 1 H.P. (FPS) = 550 ft lb/sec = 1.015 metric H.P. = 0.746 kW Dynamic Viscosity 1 Ns/m2 = 1 kg/ms = 0.102 kgf/m2 = 0.0209 lbf sec/ft2 = 10 poise 1 lbf sec/ft = 47.847 Ns/m2 Kinematic Viscosity 1 m2/s = 10.76 ft2/sec = 104 cm2/sec = 104 Stokes 1 ft2/sec = 0.0929 m2/s
