Modeling Vapor-Liquid Equilibria: Cubic Equations of State and their Mixing Rules (Cambridge Series in Chemical Engineering)

(2.4. 1 6)

which results in

where

r: ; is the molar volume of pure liquid i ,

x, _, V

(2.4. 1 7)

-]

8;

is the solubility parameter defined by the expression 8, -

1

and t:. {[""P is the internal energy change upon vaporization at the normal boiling point. Among the advantages of the regular solution model are that it requires only pure com­ ponent property information and is easi ly extendable to multicomponent mixtures. One disadvantage of this model is that it is only applicable to mixtures of hydrocarbons and other nonpolar components and frequently is not very accurate even for those mi x ­ tures. Also, the regular solution model is not accurate for polar or hydrogen-bonding components and by its form can only lead to positive deviations from Raoult's law. The most accurate prediction methods are based on the mixture group contribution concept. The idea behind such models is that each molecule is considered to be a IS

Modeling Vaporo-Liquid Eq u i l ibria

collection of functional groups and that thus the behavior of a mixture can be pre­ dicted based on known functional group-functional group i nteractions (or interaction parameters) . When one regresses avai lable experimental data containing these func­ tional groups, a matrix of functional group-functional group interaction parameters is obtained that is then used for predictions involving mixtures for which data are not available. The UNIFAC (Fredenslund, Gmehling, and Rasmussen 1 977) and ASOG (Koj ima and Tochigi 1 979) models are the two most important group contribution models for mixtures, but the UNIFAC model, being applicable to the largest number of compounds, is the most commonly used. The UNIFAC model is the group contri­ bution version of the UNIQUAC model, whereas the ASOG model is based upon the Wilson equation. The UNIFAC model is presented briefly in the next paragraph. When using the UNIFAC model one needs to identify the functional subgroups present in each molecule by means of the UNIFAC group table. Next, similar to the UNIQUAC model, the activity coefficient for each species is written as eqn. (2.4. 1 4 ) except for the the residual term, which is evaluated by a group contribution method in UNIFAC . The residual contribution of the logarithm of the activity coefficient of group k in the mixture, In rk , is obtained from ,

(2.4. 1 8)

where

em

Xm Q m . . = (surface area fractiOn of group m) = '"""' � X" Q "

(2.4. 1 9)

n

in which Xm is the mole fraction of group m i n the mixture, and

Vtm n

= exp( - Qmn / T )

m

(2.4.20)

where Qm n is a measure of interaction energy between groups and n and the sums are over all groups in the mixture. The residual contribution to the activity coefficient of species i is then computed from In y; (residual)

= L vii )[ In rk - In rY)]

vii

(2.4.2 1 )

k

In eqn. (2.4.2 1 ) l is the number of k groups present i n species i , and In ry> i s the residual contribution to the activity coefficient of group k in a pure fluid of species i molecules. The purpose of the last term is to ensure that, in the limit of pure species i (which is still a mixture of groups unless of course the molecules of species i consist of a single functional group), the residual term is zero. It needs to be stressed that any extrapolation method, and more seriously any pre­ diction method, can be of limited and unproven accuracy. Therefore, it is always 16

Thermodynamics of Ph ase Eq u i l i b ri u m

desirable to base engineering design work on experimental data. It is only in the ab­ sence of experimental data, and then only for preliminary design work, that predictive methods should be used. An advantage of the y -¢ method is that very nonideal mixtures can be described because an activity coefficient model, with suitable values of its parameters, can give very large excess Gibbs free energies of mixing. However, there are also important disadvantages of the y -¢ method. In particular, because a different model is being used for the vapor and liquid phases, this method is incapable of properly describing critical region behavior. Indeed, a critical point will not be predicted, as the mathemat­ ical conditions for its occurrence can not be satisfied. Also, excess Gibbs free-energy (or activity coefficient) model s are based on the mixing of pure liquids at a specified temperature and pressure (the standard state) to form a liquid mixture at these same conditions. This poses a problem when one (or more) of the components in the mix­ ture is not a liquid at the standard-state temperature and pressure, and especially when the mixture temperature is above the critical temperature of one or more components. Therefore, the y -¢ method is not useful for the description of, for example, supercrit­ ical extraction and other mixtures containing supercritical components. Finally, the y -¢ method can be used for the calculation of VLE, but in contrast to equations of state, other thermodynamic properties such as densities, enthalpies, and heat capaci­ ties can not also be computed from the same model unless the Gibbs free-energy of the mixture is known as a function of temperature and pressure, as well as composition, which is not generally the case. In this monograph we use activity coefficient models in two ways. First we use them in the traditional y -¢ method to correlate and predict VLE behavior at low to moderate pressures. Second, we also incorporate these models into equations of state for the description of the VLE of nonideal mixtures at high pressures, as wi ll be discussed in Chapter 4. Programs for binary VLE calculations with activity coefficient models are provided on the diskette incl uded with this monograph. The programming detai ls are given in Appendix D . 2.5.

Equation o f State M o d e l s for Vapor-Li q u i d Phase Eq u i l ibrium Calculations

If an equation of state is used to describe the liquid phase, the fugacity of a species in a liquid mixture is computed from

[ (TX;P, P. x;)] -L

In

/;

= In ¢;

R T jv [RT ( oN;oP ) l 1

= -

V =x

v

-

-

-

T. V . NJ ,;

d V - ln Z L

(2.5 . 1 ) 17

Modeling Vapor-Li q u i d Eq u i l i bria

where the compressibility factor Z is computed from an EOS . This equation differs from eqn. (2.3. 1 ) only in that the liquid phase (smallest volume) solution to the EOS is used in calculating the fugacity. Most equations of state used in engineering are either extended forms of the virial equation [eqn. (2.3.2)], or variations of the classic van der Waals equation, such as the Peng-Robinson, eqn. (2. 3 . 3 ) . There are also lesser-used empirical equations fitted to experimental data and equations derived from theory such as the equations from the perturbed hard chain theory (PHCT), the statistical associating fluid theory (SAFf), the chain of rotators model (COR), and others. The central theme of this monograph is phase equilibrium calculations and predictions with two-parameter cubic equations of state coupled with novel mixing and combining rules; consequently, we will limit our discussion to this subject in the following chapter. A more thorough discussion of these and other equations may be found in the recent review of Sandler et al. (Sandler 1 994, Chap. 2).

18

3 Vapor-Liquid Equilibri um Mod eli n g with Two- Parameter Cubic Equation s of State an d th e van d er Waals Mixin g Rules

WO-PARAMETER cubic equations of state coupled with the classical van der T Waals mixing rules are probably the most extensively used modeling tool for the VLE of hydrocarbon mixtures and of hydrocarbons with organic gases. In this chapter, after a brief review of recent modifications of cubic EOS for pure compounds (Section 3 . 1 ), we discuss the capabilities and limitations of the van der Waals mixing rules and their modifications.

3. 1 .

Cubic Equations of State and Their Mod ifications for Phase Eq u i l ibrium Calculations of Nonideal M ixtu res Many cubic equations of state are available in the literature; some recent and com­ prehensive reviews are available (Anderko 1 990; Sandler 1994). For the purposes of illustration, here we use the Peng-Robinson ( 1 976) equation of state

p=

a(T) + l:: - b 1::< 1:: b ) + b ( l:: - h)

__!!!____

_

(3 . 1 . 1 )

though the principles and models discussed are general and thus applicable to all other two-parameter cubic equations of state. The form of this equation (and others in this category) was originally chosen to give a reasonable representation of the volumetric behavior of hydrocarbons in the gasoline carbon-number range. The parameters in the equation were then determined. First, to ensure that the equation of state predicts the correct critical temperature, Tc , and pressure, Pc o of the mixture, the following conditions were i nvoked at the critical point:

( � �) ( ;�) = T,

=

T.

0

(3 .1.2)

For the Peng-Robinson equation this leads to (3 . 1 . 3 )

19

Mode l i ng Vapor-Li q u i d Equ i l i bria

and

b = 0.077796 RPTcc

(3 .1.4)

These relations ensure that a critical point i s obtained from the EOS . The term a(T) in eqn. ( 3 . 1 . 3 ) i s temperature dependent, is unity at the critical temperature, and has been chosen to ensure that the vapor pressure calculated from the EOS at other temperatures is acceptably accurate. One such representation is ( 3 . 1 .5 ) which i s applicable t o hydrocarbons and organic gases with the following form for K : K = 0.37464 + 1 . 54226w - 0.26992w

2

( 3 . 1 .6)

where w is the Pitzer acentric factor defined as w = - 1 .0 - log 10

[

P""P(T,. = 0. 7 ) _

P,.

J

( 3 . 1 .7)

where T,. = T / Tc is the reduced temperature. In this form, the equation i s completely predictive once the three constants (critical temperature, critical pressure, and acentric factor) are given. Consequently, this equation is a two-parameter EOS (a and that depends upon the three constants (Tc , P"' and w). The same temperature dependence of the a term can be used for nonhydrocarbon fluids ; however, as we discuss below, the completely generalized form of the a func­ tion given in eqn. (3. 1 .5 ) does not give accurate vapor pressures in this case. Other expressions have to be used for nonhydrocarbons for the dependence of a on temper­ ature, usually with one or more parameters that are specific to the fluid of interest, rather than completely generalized in terms of the critical properties and the acentric factor. Several investigators (Mathias and Copeman 1983 ; Stryjek and Vera 1 986a,b) have introduced ways of adding further species-specific constants to provide accurate vapor pressure correlations, especially at lower temperatures and for nonhydrocarbon fluids, that are needed for a better description of vapor-liquid equilibrium. Here we use the temperature dependence of the a term proposed by Stryjek and Vera ( 1 986a) . In their approach, eqn. ( 3 . 1 .6) is replaced by the relation

b)

(3. 1 .8) with K0

= 0.378893

+ 1 .4897 1 5 3w - 0. 1 7 1 3 1 848w 2 + 0. 0 1 96554w 3

( 3 . 1 .9)

and where the constant K 1 is specific for each pure compound and i s used to fit low-temperature saturation pressures accurately. This version of the Peng-Robinson EOS is referred to as the PRSV equation. The pure component constants of the 20

Vapor-Liq u i d Eq u i l i brium M o d e l i ng

Table 3.1 . 1 . Pure component parameters for PRSV equation of state Pc , bar

(1)

K)

Compound

Tc . K

Acetone

508 . 10

46.96

0 . 3 0667

- .00888

B enzene

562.16

48 .98

0 .20929

0.07019

Carbon dioxide

304.21

73.82

0. 22500

0 .04285

Cyclohexane

553 .64

40.75

0.20877

0.07023

Ethanol

513 .92

61.48

0.64439

- .03374

Methane

190 . 55

45.95

0.01045

- . 00159

Methanol

512 .58

80.96

0.565 3 3

- . 16816

Methyl acetate

506.85

46.9 1

0 . 3 2027

0.05791

n - Butanol

562 . 98

44.13

0 . 59022

0 . 3 343 1

n -Decane

617. 50

21.03

0.49052

0.04510

n -Heptane

540.10

27.36

0. 35022

0.04648

n - Hexane

507. 30

30.12

0. 30075

0.05104

n - P entanc

469.70

3 3 .69

0. 25143

0.03946 0.03136

P ropane

369.82

42.50

0. 15416

2-P ropanol

508 .40

47 .64

0.66372

0. 23264

Water

647 .29

220.90

0 . 34380

- 0 .06635

PRSV equation for the substances considered here are given in Table 3 . 1 . 1 ; K 1 of the PRS V equation is obtained by fitting pure component saturation pressure ( P v"P ) versus temperature data. A computer program to optimize K 1 for a set of versus p vap data is provided on the diskette accompanying this monograph, and the program details are presented in Appendix D. The effect of thi s parameter on the accuracy of vapor pressure correlations for several fluids is shown in Figure 3 . 1 . 1 . Before leaving this subj ect we should note several other i mportant points. The Stryjek-Vera modification of a takes care of the inaccuracies in temperature depen­ dence of the a term at low temperatures. However, since the a term is based on vapor pressure, it is not well defined at temperatures above the critical temperature of a component. The behavior of this function for the PRSV EOS used here is shown in Figure 3 . 1 .2 for various values of its parameters. For fluids for which both w and K 1 have nonzero values, the value of the a function extrapolated from subcritical condi­ tions tends to increase very rapidly with temperature at supercritical conditions. This is a problem when dealing with a fluid whose critical temperature is very low, such as hydrogen. In these cases the use of other a functions becomes necessary. Twu et al . ( 1 99 1 ; 1 995a,b) have made a thorough analysis of this problem and have proposed a new a function that avoids extrema in the supercritical region and smoothly goes to zero at infinite temperature. In this monograph the fu nction given in eqn . (3. 1 .8) will be used, but readers interested in applications to mixtures that contain fluids in the highly supercritical state, such as hydrogen-containing mixtures, may wish to consider alternative forms such as those presented by Twu et al.

T

21

Modeling Vapor-Li q u i d Eq u i l i bria

1 00

�

1 0 ·1

::I C/l C/l CD

2

2

.0

ai ... ...

a.

1 o -2

2

1 o-3 260

lOj

280

300

320

340

360

380

400

temperature, K

Figure 3.1 . 1 . Effect of the Kt parameter on the pure component saturation pressure calculated with the PRSV equation of state . Points denote experimental saturation pressures of methanol ( 0) and butanol (D) (Vargaftik 1 975). Dashed lines represent results calculated with Kt 0, and solid lines are results calculated with K t values reported in Table 3 . 1 . 1 . =

1 .6 1 .4 1 .2 s:::: 0

1 .0

s::::

0.8

�

.2 d

I

1

I PRSV '" =0.35. K1 =-0.06

0.6

0.4 PRSV m =O, K1 =0

0.2 0.0 0

2

3

4

5

6

7

8

9

1 0

reduced temperature Figure 3.1.2. The parameter a (see eqn. 3 . 1 . 3 ) as a function of reduced temperature ( T / T,_. ) . Points rep­ resent a values required to reproduce experimental ly reported compressibilities for various fluids, and lines signify calculated a values from the PRSV equation of state with different values of K t and acentric factor (w ) .

22

Vapor-Liquid Eq u i l i b n u m Modeling

Mathias and Klotz (1994) have shown that utilizing multiproperty fitting (that is, simultaneously fitting the parameters of the a function to data such as the enthalpy of vaporization and heat capacity in addition to vapor pressure) greatly improves the overall performance of an EOS . This should be remembered when saturation pressure versus temperature information is not sufficiently accurate for good parameter estimation and when the EOS is intended for calculation of other properties, such as excess enthalpies, along with phase equilibrium. One shortcoming of two-parameter cubic equations of state is their inaccuracy in

predicting liquid density ; there is typically a 5 - to 10-percent error, and it is greater as the critical point is approached. Further, when the two parameters in a cubic EOS are adj usted to give the correct critical pressure and critical temperature, the critical volume (or equivalently the critical compressibility Zc = Pc'Lcl R Tc) will be in error. One indication of this is that for all fluids, the Soave-Redlich-Kwong (SRK) (Soave 1 972) equation gives Zc = 1/3 , and the PR equation gives Zc = 0. 3074, whereas the critical compressibi lities for common fluids range from 0.12 for hydrogen fluoride, 0. 229 for water, around 0.27 for many hydrocarbons, and from 0.286 to 0. 3 1 1 for the noble gases. One way to improve the poor liquid density predictions of cubic equations of state is to allow the hard core parameter to be temperature dependent. Xu and Sandler ( 1 987) did this with the original PR equation using fluid-specific a and parameters to obtain good accuracy in both densities and vapor pressures. Others have introduced various temperature dependencies into two-parameter cubic equations ( Fuller 1 976; Heyen 1980) . A problem with such approaches is that if the temperature dependencies of the a and parameters are not carefully coordinated, the isotherms may cross over in certain regions of the pressure-volume-temperature and pressure-enthalpy-temperature space, leading to negative heat capacities and other anomalies, as shown by Trebble and B ishnoi ( 1 986). A clever method of improving the saturated liquid molar volume predictions of a cubic EOS was introduced by Peneloux, Rauzy, and Freze (1982) by translating the calculated volumes without changing the predicted phase equilibria. For example, the volume translations '!_ --+ '!_ + c and --+ + c applied to the PR equation give the correct liquid saturation volume at some temperature depending upon the choice of the parameter c and lead to an improvement in the liquid densities overall while, because of their small magnitude, having a negligible effect on the vapor densities. For the Peng-Robinson EOS used here, a more detailed volume translation was introduced by Mathias, Naheiri, and Oh ( 1 989).

b

b

b

b b

3.2.

General Characteristics of M ixing and Co m b i n i ng Ru les In order to use equations of state for the correlation and prediction of the phase behavior of mixtures, a compositional dependence has to be introduced. This is done by devising mixing and combining rules for the EOS parameters. All extensions of 23

Modeling Vapor-Li q u i d Eq u i l i bria

equations of state to mixtures are, at least partially, empirical in nature, because there is no exact statistical mechanical solution relating the properties of dense fluids to their intermolecular potentials, nor is detailed information available on such intermolecular potential functions. One of the exact results we do have from statistical mechanics is the virial equation of state

B

C

Z = --= = 1 + ··· 1::: + 1:::2 + RT

PV

(3.2. 1 )

where Z is the compressibility factor, P is the pressure, T is the temperature, 1::: i s the molar volume, and B and C are second and third virial coefficients, respectively. The virial coefficients are related to the intermolecular potential between molecules, and for pure fluids they are functions of temperature only. Also, for mixtures the only composition dependence of the virial coefficients is given by

B = L L X ; XjBij ( T) j c = L L L X ; X j Xk Cijk ( T) , j k

(3 .2.2) and so forth.

(3.2.3)

Since eqns. ( 3 . 2 . 2 and 3 .2 . 3 ) are exact, these equations can b e considered low-density boundary conditions that should be satisfied for mixtures by other equations of state when expanded into the virial form. At present, there is no exact high-density boundary condition for mixture equations of state. However, there is the observation that, at liquid densities, the empirical activity coefficient models (such as those of van Laar, Wilson, NRTL, UNIQUAC, etc.) discussed earlier provide a good representation of the excess or nonideal part of the free energy of mixing. Therefore, another boundary condition that could be imposed is as follows: excess free energy of mixing

excess free energy of mixing

calculated from an EOS

calculated from an activity

( 3 . 2 .4)

coefficient model Equations (3.2.2) and (3 . 2.4) have been used to develop EOS mixing rules in recent years, and they will be considered here. B ecause so many such models have been proposed over the years, we discuss only those that have stood the test of time or are relatively new and appear to be important advances for EOS modeling of industrial mixtures. We start here with the traditional van der Waals mixing rules that are based on eqn. (3.2.2). These have been used for decades with success to describe the high-pressure VLE of hydrocarbon mixtures and of hydrocarbons with inorganic gases. They are, however, not adequate for mixtures involving organic chemicals. The capabilities and limitations of the van der Waals mixing rules are discussed in Sections 3 . 3 and 3 .4 together with the reasons for their failure when applied to highly 24

Vapor-Li q u i d Eq u i l i brium Modeling

nonideal mixtures. The next step in the development of mixing rules for the cubic equations was the introduction of more than one binary interaction parameter in the conventional van der Waals mixing rules. Section 3 . 5 is devoted to the analysis of such multiparameter van der Waals mixing rules. In recent years equation of state mixing rules have been developed that combine excess free-energy models with a cubic EOS (Dahl and Michelsen 1 990 ; Michelsen 1 990a, b; Holderbaum and Gmehling 1 99 1 ; Wong and S andler 1 992; B oukouvalas et al. 1 994; Orbey and Sandler 1 995c). Various methods based on this concept will be considered in detail in Chapters 4 and 5 . 3.3.

Conventional van der Waals M ixing Rules with a Single Binary I nteraction Param ete r ( I PVDW Model) The first successful method of general izing a pure fluid EOS to mixtures was the one-fluid model proposed by van der Waals. The underlying assumption of thi s model is that the same EOS used for pure fluids can be used for mixtures if a satisfactory way is found of obtaining the mixture EOS parameters. The common method for doing this is based on expanding the EOS in virial form, that is, in powers of ( 1 I 1::) . For the Peng-Robinson equation one obtains (3.3. 1 ) To satisfy the boundary condition of eqn. (3.2.2), the composition dependence of two-parameter cubic equations of state of the van der Waals family must conform to the relation " x I- x J· B I}· · (T) = B(x I • T) = " �� I

j

b - __!!_ RT _

( 3 . 3 .2)

It should also be noted that (3 . 3 . 3 )

Clearly, it is not possible to set a composition dependence of the h parameter to satisfy eqns. ( 3 . 3 . 2 and 3 . 3 . 3 ) simultaneously for two-parameter cubic equations of state. Unti l recently, the most common way of choosing mixture parameters was to satisfy only eqn. ( 3 . 3 .2) with the van der Waals one-fluid mixing rules as follows:

j

b

=

L L X; Xj hij j

(3.3.4) (3 . 3 . 5 )

25

M o d e l i ng Vapor-Liq u i d E q u i l i bria

The following combining rules are frequently used to obtain the cross coefficients aij and bij from the corresponding pure component parameters :

aij

= Ja;; ajj ( l - kij )

bij =

( 3 . 3 .6)

4Cb ; ; + bjj )( I - lij )

(3.3.7)

where kij and lij are the binary interaction parameters obtained b y fitting EOS predic­ tions to measured phase equilibrium and volumetric data. Generally lij is set to zero, in which case

b=

L: x;h;

(3.3.8)

We will refer t o this one-binary-interaction-parameter-per-pair version o f the van der Waals mixing rules (eqns. 3 . 3 .4, 3 . 3 .6, and 3 . 3 . 8 ) as the l PVDW model. There is only a qualitative j ustification for eqns. ( 3 . 3 . 6 and 3 . 3 .7). The a parameter is related to attractive forces, and, from intermolecular potential theory, the parameter in the attractive part of the intermolecular potential for a mixed interaction is given by a relation like eqn . ( 3 . 3 .6). Similarly, the excluded volume parameter b would be given by eqn. ( 3 . 3 . 7) if the molecules were hard spheres. However, there is no direct relation between the attractive part of the i ntermolecul ar potential and the a parameter in a cubic EOS , and real molecules are not hard spheres. The fugacity coefficient of species i in a one-phase mixture is obtained from the PR-EOS using l PVDW as follows : In

-

¢;(T, P, x;) =

b; b ( Z - I ) - I n (Z - B) _ _A_

2./2B with A =

B=

(2L::X ) j= l

a·

1 '1

a

_

b; b

In

[z

+ ( 1 + ./2)B

Z + ( 1 - ./2)B

-­ ­

J

( 3 . 3 .9)

aP

R2 T2

(3. 3 . 1 0)

bP

RT

Also for future reference, note that the pure component fugacity coefficient from the PR equation is In

26

¢;(T, P) = (Z -

I ) - I n ( Z - B) -

-A

2./2B

In

[z

+ (1 + ./2)B

Z + ( 1 - ./2)B

]

(3.3. 1 1 )

Vapor-Li q u i d Eq u i l i bri u m Modeling

3 .4 .

Vapor-Li q u i d Phase Eq u i l i bri u m Calc u l ations with the I PVDW Model The performance of two-parameter cubic equations of state with the conventional van der Waals mixing rules ( 1 PVDW model) is relatively well known and is presented here mainly for reference, but also to indicate certain misconceptions abqut this method. The results presented in this section were obtained using the computer program VDW provided on the accompanying diskette. The program details are presented in Sec­ tion D.3 of Appendix D. When fitting VLE data with the l PVDW model, it is found that the binary interac­ tion parameter kij is approximately zero for relatively simple mixtures, such as alkane mixtures, whereas for some other mixtures such as hydrocarbons with industrial gases like carbon dioxide and organic solvents, it is not only nonzero but will also change in value with temperature. For highly nonideal mixtures, which are our main concern here, accurate correlation of VLE is not possible by this method. In Figure 3 .4. 1 the results for the methane and n -pentane (Knapp et al . 1 982) binary system are presented. This is a typical mixture for which the van der Waals one-fluid mixing rules with a single constant binary interaction parameter performs very well

1 80 1 60

o

1ii

oj .... ::I Ill Ill Cl)

....

c..

,.

VLE data at 377 K

0 + VLE data at 444 K

1 40

.£J

e VLE data at 31 0 K

�:; 'V

1 20 1 00 80 60 40 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of methane

Figure 3.4. 1 . VLE correlation of the methane and n ­

pentane binary system a t 3 1 0, 3 7 7 , and 444 K with the I PVDW mixing rule and the PRSV equation of state. The lines represent VLE results calculated with the binary interaction parameter k 1 2 = 0.02 1 5 . (Data are from the DECHEMA Chemistry Series, Gmehling . and Onken 1 977, Vol . 6, p. 445 ; data files for this system on the accompanying disk are C 1 C5 3 1 0.DAT, C 1 C5377.DAT, and C l C5444.DAT. )

27

Model i ng Vapor-Liquid Eq u i l i bria

70

0 e V L E data a t 2 7 7 K

'V .II. VLE data at 31 0 K

60

(;;

50

ai' ....

40

Q)

30

•

0 + V L E data at 3 4 4 K • 0 • •

..c :::::1 rn rn

a.

20

10 0 ������----�--�� 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of carbon dioxide

Figure 3.4.2. VLE correlation of the carbon dioxide

=

and propane binary system at 277, 3 1 0, and 344 K with the l PVDW mixing rule and the PRSV equa­ tion of state. Solid lines represent VLE results cal­ culated with the binary interaction parameter k 1 2 0. 1 2 1 , and the dashed lines denote results calcu­ lated with k t 2 0.0. (Data are from DECHEMA Chemistry Series, Gmehling and Onken 1 977, Vol . 6, p. 589; data files for this system on the accompany­ ing disk are C02C3277 .OAT, C02C3 3 1 O. DAT, and C02C3344.DAT. )

=

(solid lines in the figure) over a wide range of temperature (from 310 to 444 K). Moreover, the binary interaction parameter is small (k;j = 0.02 1 5), and, even if it were set to zero, a good description of the VLE behavior is possible (dashed lines in the figure). Another application of this mixing rule is for the description of mixtures of inor­ ganic gases with hydrocarbons. An example is shown in Figure 3 .4.2 for the carbon dioxide and propane (Knapp et al. 1 982) mixture. Here, setting the binary interaction parameter to zero leads to unsatisfactory results, which are shown as dashed lines in the figure. However, the use of a nonzero but constant binary interaction parameter (k;i = 0. 1 2 1 0 in this case) leads to very good correlation of VLE at all temperatures considered. Consequently, for such cases some experimental VLE data are neces­ sary to fit thi s parameter. The binary interaction parameter obtained at temperatures and pressures away from the design conditions may be used to extrapolate phase equilibrium information to the actual design case. Our main concern here is with mixtures significantly more nonideal than those discussed above. Many industrial mixtures fall into this category, and for these systems the 1 PVDW model is not adequate. To see this we first consider the n-pentane and ethanol (Gmehling a� d Onken 1 977) binary mixture (Figure 3 .4.3). Alkane and alcohol 28

Vapor-Li q u i d Eq u i l i brium Model i ng

20 18 16 ... CIS .c

a) ... ::s Ill Ill CD

... a.

14 12 10 8 6 4 2

�������--��

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of n -pentane

Figure 3.4.3. VLE correlation of the n -pentane and ethanol binary system at 373, 398, and 423 K with the I PVDW mixing rule and the PRSV equation of state. Solid lines are the VLE result s calculated with a differ­ ent binary interaction parameter, for each temper­ ature. (The points are the data of Campbell , Wil sak, and Thodos 1 987; data files for this system on the accompanying disk are PE373. DAT, PE398. DAT, and PE423 .DAT. )

k12,

mixtures are industrially important and are also a stringent test of EOS models. There are many studies in the literature for which the van der Waals one-fluid mixing rules lead to predictions of false liquid-liquid splits. The example selected for study here is of a mixture that is not very asymmetric in the size or vapor pressure of its components and therefore should not be very difficult to model . Still, in the temperature range of 373 to 423 K, there is not a single isotherm that can be accurately correlated with the I PVDW mixing rule, even when using a different binary interaction parameter for each isotherm . For example, the azeotropic point pressure is always underestimated. The results for the more asymmetric propane and methanol (Galivel-Solastiuk, Laugier, and Richon 1986) mixture at 3 1 3 K is shown in Figure 3 .4.4. In this case the correlation with the I PVDW model is poor, giving a fal se liquid-liquid split and underpredicting the pressure over the whole concentration range. Similar results are obtained at other temperatures for this system with this mixing rule. Difficulties are also encountered when water and alcohol mixtures are considered. The correlation of the 2-propanol and water binary system at 353 K is shown in Figure 3 .4.5 . Here we see that at a temperature of 353 K, and also at lower temperatures, the l PVDW mixing rule gives a false liquid split and poorly represents the VLE data. At higher temperatures, the results for this system improve somewhat, as shown in Figure 3.4.6 for 523 K, but the correlation is still not acceptable for industrial design. 29

Modeling Vapor- L i q u i d Eq u i l ibria

1 .4

1 .2

«< c..

:::!: Cli 5

0..

en en Ql

•

.. .

•

1 .0 e

VLE data at 31 3 K

0.8

0.6

0.4 0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of propane

Figure 3.4.4. VLE correlation of the propane and

methanol binary system at 3 1 3 K with the 1 PVDW mixing rule and the PRSV equation of state ; symbols are the experimental data. A binary i nteraction param­ eter k 1 2 0.045 1 was used. (Points are the data of Galivel-Solastiouk et a! . 1 986; the data file for this sys­ tem on the accompanying disk is PM3 1 3 .DAT.) =

1 .2 .A. VLE data at 353 K

1 .1

iu

.c

1 .0

0 .8

...

0.7

Q.

LI.A.

0.9

Cli ... :I en en Ql

Ll� •

•

0.6 0.5 0.4

L......I...__L.._.._.__.__._-...�.__._...�.._._.J.__J

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 3.4.5. VLE correlation of the 2-propanol and water binary system at 353 K with the I PVDW mixing rule and the PRSV equation of state. A binary i nterac­ tion parameter k 1 2 -0. 1 62 1 was used. (Points are the data of Wu, Hagewiessche, and Sandler 1 98 8 ; the data file for this system on the accompanying disk is 2PW80. DAT. ) =

30

Vapor-Liq u i d Equ i l i br i u m Modeling

't'

a;

v

60

.c

oi ....

:J C/l C/l Q)

c.

50

't' VLE data at 523 K

40

30

������--���

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

mole fraction of 2-propanol

Figure 3.4.6. VLE correlation of the 2-propanol and

water binary system at 523 K with the l PYDW mix­ ing rule and the PRSV equation of state; • and "V are the experimental data. A binary i nteraction parameter k 1 2 = -0. 1 1 20 was used. (Points are the data of B arr­ David and Dodge 1 959; the data file for this system on the accompanying disk is 2PW250. DAT. )

Similarly poor results are obtained with the l PVDW mixing rule for the acetone and water (Griswold and Wong 1 952) system. The correlation of the experimental data for this binary system is shown in Figure 3 .4.7 and 3 .4.8. The correlation results are very poor at low (Figure 3 .4.7) and at high (Figure 3 .4.8) temperatures. The overall conclusion is that even though the conventional van der Waals mixing rules are simple to use and conform to the second virial coefficient boundary condition, they are very limited in their application and are not useful for either the correlation or the prediction of the VLE of complex mixtures . Finally, we would like to point out an important but overlooked point about the van der Waals one-fluid mixing rules . It is frequently assumed that when the binary interaction parameters of some of the pairs in a multicomponent mixture are not available as a result of insufficient data, it is acceptable to set the binary interaction coefficient of these pairs to zero (usually the binary pairs for which data are missing are the more nonideal pairs) . The results of doing this are presented in Figure 3 .4.9 for the acetone and water binary system at 423 K, which shows that such an assumption may cause problems . For this case, when k 1 2 is set to zero, there is a range of concentration in the water-rich region for which the EOS model leads to unrealistic phase equilibrium predictions. Indeed, a computer program using this model may not even converge unless the programmer has taken many precautions. Why the van der Waals one-fluid mixing rules cannot describe highly nonideal mixtures can be understood by starting with the relation between the molar excess 31

Modeling Vapor-Liquid Eq u i l i bria

e VLE data at 298

0.3

(ij

.c

ai ....

• •

0.2

•

K

•

•

•

::I Ill Ill CD

.... a.

0.1

--

0.0

L...--'... -'---'---'-....--'-....L. .. --... ... -...1. ..L .... ... _.__.l_.__J

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 mole fraction of acetone

Figure 3.4.7. VLE correlation of the acetone and water binary system at 298 K with the I PVDW mixing rule and the PRSV equation of state. A binary interac­ tion parameter k 1 2 -0.256 was used. (Points are the data of Griswold and Wong 1 95 2 ; the data file for this system on the accompanying disk is AW25 .DAT.) =

70

65

(ij

60

.c

ai ....

55

.... a.

50

::I Ill Ill CD

£ VLE data at

45

40

523 K

�����--�--�-�--��

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

mole fraction of acetone

Figure 3.4.8. VLE correlation of the acetone and

water binary system at 523 K with the I PVDW mixing rule and the PRSV equation of state. A binary inter­ action parameter k 1 2 -0.093 was used. (Points are the data of Griswold and Wong, 1 95 2 ; data file for thi s system on the accompanying disk is AW250. DAT. ) =

32

Vapor-Li q u i d Equ i l i brium M o d e l i ng

1 4.5

• no convergence region

1 3.0

«;

.D

1 1 .5

oi ...

1 0.0

en Q)

8.5

:::::1 en

... Q.

•

•

•

0

•

7.0

0

• VLE data at 423 K

0

•

5.5 4.0

•

• VLE predictions at 423 K

0 L..... ..o. .o... .J.... __._....L. ...J .... .. ....,. �..J_....�-'--.... ....:. .. �.J.----'

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

Figure 3.4.9. VLE prediction of the acetone and water

binary system at 423 K with the I PYDW mixing rule and the PRSV equation of state . The binary interaction parameter k 1 2 was set to zero. In this figure D and • de­ note experimental data, and 0 and • signify calculated values. There is a region where convergence is not ob­ tained for water-rich mi xtures. (Data are from Griswold and Wong, 1 95 2 ; the data file for this system on the accompanying disk is AW 1 50.DAT. )

Gibbs free energy of mixing, gex , and fugacity coefficients obtained from an EOS :

gex

=

RT[ ln ¢ - I>i ln ¢i ] i

(3 .4.1)

where ¢ and ¢i are the fugacity coefficients of the mixture and of the pure component

i calculated from eqn . (2.3 .9) [or eqn . ( 3 . 3 . 1 1 ) for the PR EOSJ using the mixture and pure component parameters, respectively. For the van der Waals cubic EOS, this leads to the following expression for the excess Gibbs free energy of a binary mixture:

(3 .4.2) This equation shows that the excess Gibbs free energy computed from a cubic EOS of the van der Waals type and the one-fluid mixing rules contains three contributions. The first, which is the Flory free-volume term, comes from the hard core repulsion terms and is completely entropic in nature. The second term is very similar to the excess free-energy term in the regular solution theory, and the third term is similar to a term that appears in augmented regular solution theory. Consequently, one is led 33

M o d e l i ng Va p o r-Li q u i d Eq u i l i bria

to expect that the combination of a cubic EOS with the van der Waals mixing rules can only represent those mixtures that are describable by augmented regular solution theory. This excludes polar and hydrogen-bonding fluids. Equation (3.4.2) also shows that setting the binary interaction parameter k1 2 equal to zero does not result in an ideal solution, for even with k 1 2 = 0, Qe x is not equal to zero ; indeed, none of the three terms in the equation vanish, nor do the terms cancel. 3.5.

Nonq uad ratic Co m b i n i ng Ru les for the van der Waals One-Fluid Model (2PVDW Model) An empirical approach t o overcome the shortcomings o f the van der Waals one-fluid model for a cubic EOS has simply been to add an additional composition dependence and parameters to the combining rule for the a parameter, generally leaving the b parameter rule unchanged. Some examples are the combining rules of Panagiotopoulos and Reid ( 1 986) ( 3 .5 . 1 ) Adachi and Sugie ( 1 986) ( 3 . 5 . 2) Sandoval, Wilseck-Vera, and Vera ( 1 989) (3.5.3) and Schwartzentruber and coworkers ( 1 986, 1 989) (3. 5 .4) where i n the last equation K;; = K ; ; , l; ; = -I;; , m ; ; = 1 m ;; , and K;; = I;; = 0. It should be pointed out that these combining rules do not satisfy the boundary condition of eqn. (3.2.2). B y an appropriate choice of their parameters, these combining rules reduce to one another and to eqn. ( 3 . 3 .6). For the binary systems considered here, all these models reduce to -

(3.5.5) The combinations o f eqns. ( 3 . 3 .4, 3 . 3 .6, 3 . 3 . 8 , and 3 .5 . 5 ) will b e referred t o a s the 2PVDW model. This model has been shown to provide a good correlation of VLE data of highly nonideal systems that previously could be correlated only with activity coefficient models. 34

Vapor-Li q u i d Eq u i l i br i u m Mode l i ng

Note that with this modification to the van der Waals one-fluid model, the fugac­ ity coefficient expression for species i given in eqn. (3. 3 .9) will change because an additional compositional dependence has been introduced to the term of the EOS. For the PR EOS, with the van der Waals one-fluid model, a more general form of the fugacity coefficient expression of species i in a mixture is

a

ln ¢; ( T, P, x;) = b ( Z - I ) - ln( Z - B )

-

b;

A_((BNa jaN;h. a v. NH,

__ 2 J2 B

+ I _

b; h

) [z In

+ ( 1 + J2) B

Z + ( l - J2)B

]

(3.5.6)

where the derivative term depends on the form chosen for the composition dependence of kij . For example, for eqn. (3.5.5 ) used here the derivative term for a binary system becomes (3.5.7) For more details see Stryjek and Vera ( 1 986b). Some binary VLE correlations with the 2PVDW model are presented in the remain­ der of this section. The calculations with this model were performed with the program VOW, and the computational details are presented in Section 0.3 of Appendix D. The results for the n -pentane and ethanol binary system with this model are given in Fig­ ure 3.5. 1 , where we show the correlation of the individual isotherms with the 2PVDW model (solid lines) and also the prediction of VLE behavior at 423 and at 373 K with the parameters obtained from fitting VLE data at 398 K (short dashed lines). Also included in the figure is the correlation of each isotherm with the I PVDW model (medium dashed lines). Several conclusions can be drawn from these results. First, for this binary system, the VLE at any temperature can accurately be correlated with the two-parameter 2PVDW model. Second, the predictions with this model are generally not very good when one attempts to use the parameters obtained at one temperature to predict the VLE at other temperatures, even over a small range of temperature. We will see this in other examples which follow. Finally it is interesting to note that these less-than-accurate predictions are still better than the correlations of the individual isotherms of thi s system with the 1 PVDW model. The more asymmetric alkane-alcohol systems, however, can not successfully be represented by the 2PVDW model. The results for the propane and methanol system are shown in Figure 3.5.2. There the 2PVDW model (solid line) is more accurate than the l PVDW model (dashed lines); however, the 2PVDW mixing rule also predicts a false liquid-liquid split. Alcohol-hydrocarbon systems are generally difficult to describe, and more complex mixing rules that we will discuss later can overcome this

35

Modeling Vapor-Li q u i d Eq u i l i bria

21 19 17 15 ...

ell .J:l

13

ai ... ::I Ill Ill ell ...

Q.

9 7 5 3 1 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of n -pentane

Figure 3.5.1. VLE correlation of the n-pentane and ethanol binary system with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are the results of correlation with k 1 2 / k21 0. 1 95/0.049 at 373 K, 0.2056/0.073 at 398 K, and 0. 207/0.096 at 423 K. Short dashed lines are the results of VLE predic­ tions with k 1 2 f k2 1 0.200/0.073 a t all temperatures, and the medium dashed li nes are I PVDW model corre­ lations presented earlier in Figure 3 .4 . 3 . (Data are from Campbell et al . , 1 987; data files for this system on the accompanying disk are PE3 7 3 .DAT, PE398.DAT and PE423 .DAT. ) =

=

problem, but only with a constraint fit of their parameters . However, with the 2PDVW mixing rule it is not possible to obtain a sati sfactory fit even in thi s manner. The results of correlation and prediction with the 2PVDW model of the 2-propanol and water system are presented in Figure 3 . 5 . 3 and 3 . 5 .4, respectively. In Figure.3 . 5 . 3 , the VLE correlation with the 2PVDW model (solid line) a t 353 K i s shown. For com­ parison we have also included the correlated results for this system with the l PVDW model . The two-parameter model can correlate the data almost within experimental accuracy in contrast to the 1 PVDW model that fails seriously. The results at 523 K are even more interesting. Here the correlation with the 2PVDW mixing rule (solid line) is very accurate and much better than correlation with the 1 PVDW mixing rule (medium dashed lines), but the prediction with the 2PVDW model with parameters obtained at 353 K (short dashed lines) is the least accurate. This shows that the use of temperature-i ndependent parameters is not good for the extrapolation of VLE in­ formation with the 2PVDW model. That is, the model is good for correlation but not extrapolation, and then for only moderately nonideal systems. The results for the acetone and water system are similar. In Figure 3 . 5 . 5 we present the results of correlating VLE data for this binary mixture at 298 K. The use of the 36

1 .4

•

1 .2

al a.

::::E

ai � ::II Ill Ill Gl

:--- - - - - -

---

1 .0 •

---

VLE data at 31 3 K

0.8 0.6

�

0..

0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 0.8

0.9 1 .0

mole fraction of propane

Figure 3.5.2. VLE correlation of the propane and

methanol binary system at 3 1 3 K with the 2PVDW mixing rule and the PRS V equation of state . Solid lines denote correlation results with 1 2 / 0.0953/0.0249 . Dashed lines show l PVDW model correlations presented earlier in Figure 3 .4.4. (Points are the data of Galivel-Solastiouk et al. , 1 986; data file for this system on the accompanying disk is PM3 1 3 .DAT.)

k kzt

1 .2 1 .1 1 .0

....

al .0

0.9

ai

....

0.8

....

0.7

::::J Ill Ill Gl

0..

.t.

0.6

VLE data at 353 K

0.5

0.4

..._. ...._.__.___._.... ...

_._

....... �...._-.......�.

_

0.0

0.1

0.2

0.3

0.4

0.5

_

0.6

0.7

0.8

_._�

0.9

1 .0

mole fraction of 2-propanol

Figure 3.5.3. VLE correlation of the 2-propanol and

water binary system at 353 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with = 0.0953/0.0249 . Dashed lines show l PVDW model correlations pre­ sented earlier in Figure 3 .4.5 . (Points are the data of Wu et al . , 1 98 8 ; data file for this system on the accom­ panying disk is 2PW80. DAT. )

k tzl kzt

1ii

60

:::1 en en Q)

50

.0

a) ....

Q.

T

VLE data at 523 K

40

30

....___,_ _. __.__._�...J.� ..�--'----'---'-�...J..... .....

0.0

0.1

0.2

0.3

0. 4

0.5

0.6

0.7

0.8

mole fraction of 2-propanol

Figure 3.5.4. VLE correlation of the 2-propanol and water binary system at 523 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k 1 2 l k2 1 -0.02391-0. 1 378. Short dashed lines show results of predictions with binary interaction parameters k 1 2 l k21 = -0.09 1 1 I -0. 1 766 obtained at 353 K , medium dashed lines represent I PVDW model corre­ lations presented earlier in Figure 3 .4.6. ( Points are the data of Barr-David and Dodge, 1 959; data file for this system on the accompanying disk is 2PW250.DAT.)

0.3

e

VLE data at 298 K

0.2

0.1

.. .... .__ -'-.... ..._ .. �-'-.o...-'-...J. ...I. -... ..�-'--' ...J.. ---' 0.0 ....__._.... 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of acetone

Figure 3.5.5. VLE correlation of acetone and water

binary system at 298 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k 1 2 l k 2 1 -0. 1 4 1 61 - 0. 2822, and the medium dashed lines represent I PVDW model correlations presented earlier in Figure 3 .4.7. (Points are the experimental data of Gri swold and Wong 1 95 2 ; data file for this system o n the accompanying disk i s AW25.DAT. ) =

Vapor-Li q u i d E q u i l ibrium M o d e l i ng

65

(ii

60

_ _ _ _ .....

.c

ai ...

::::J en en G) ... a.

55

50

45

•

VLE data at 523 K

..._ ..___ ._ ....___.... ..._ ..___.... ._ ..___.... ._ ..___ ._ ....... 4 0 ..._....___.... 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

mole fraction of acetone

Figure 3.5.6. VLE correlation of acetone and water binary system at 523 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k 1 2 / k2 1 = 0.0445 / - 0. 1 52 1 . Short dashed lines reflect predictions with k 1 2 / k2 1 = -0. 1 4 1 6/ - 0.2822 obtained at 298 K, and the medium dashed lines represent I PVDW model correlations pre­ sented earl ier in Figure 3 .4 . 8 . (Points are the data of Griswold and Wong, 1 95 2 ; data file for this system on the accompanying disk is AW250.DAT. )

two-binary-interaction-parameter 2PVDW model provides a significant improvement in correlation over the one-parameter model . The difference between the results from the two models is less at 523 K, as shown in Figure 3.5.6. Moreover the predic­ tions of the 2PVDW model with parameters from the fit of the 298 K (short dashed lines) experimental data are very poor, again indicating the need for the temperature­ dependent parameters in the 2PVDW model. The two-binary-interaction-per-pair (2PVDW) approach allows one to use activity coefficient data for the prediction of VLE of moderately nonideal mixtures. The relationship between an activity coefficient and an EOS is Yi =

[r/J; (T, P , X; )/ ¢; (T. P)]

(3.5.8)

-;jy ;( T,

where P, x;) is the fugacity coefficient of species i in a mixture, and r/J; (T, P) is the pure component fugacity coefficient, both obtained from the EOS at the tem­ perature and pressure of the mixture. Consequently, the two interaction parameters per binary in this class of mixing rules can be related to the activity coefficients over the whole composition range or to the values at a specified composition, such as the two infinite dilution activity coefficients of a mixture (Torress-Marchal, Catalino, and De Brito 1 989; Pividal et al . 1 992), thus eliminating the need for VLE data over a 39

Modeling Vapor-Li q u i d Eq u i l i bria

range of compositions. In this latter case one can write eqn. (3.5.8) for a cubic EOS for the two infinite dilution limits, obtaining two equations for the two parameters in eqn. (3.5.5), in terms of the pure component EOS parameters and infinite dilution activity coefficients. For the PR equation we have

(3.5.9) with A.; =

and

I;;

=

�� ( Z; - 1 ) - 1n [ Z; - (b; P;vap j R T ) ]

a;

[

vap R T / 1n vap 2../iR Tb; Z; - (../i - 1 ) b; P; I R T r;:;

Z ; + (v 2 + 1 )b; P;

(3.5. 1 0)

]

(3.5. 1 1 )

where Y]':' is infinite dilution activity coefficient of species j in i , Z; is the pure component liquid phase compressibility at saturation, and ¢j(T, P) is the fugacity coefficient of species j in a mixture at infinite dilution in species i , with Kj; being obtained by index rotation. In this case it is possible to predict the phase behavior of some moderately nonideal systems successfully using only infinite dilution activity coefficient information at the same temperature. Experimentally measured infinite dilution activity coefficients of the binary pairs in one another are used in eqn. (3.5 .9). However, this kind of data may not always be avai lable, so that a predictive group con­ tribution method such as UNIFAC may have to be used to obtain the necessary infinite dilution activity coefficients, and thus the method becomes completely predictive. An example of the use of this method is given in Figure 3.5.7 for the methyl acetate( ! ) and cyclohexane(2) binary system (Pividal et al. 1 992) at 3 1 3 K. The infinite dilution activity coefficient of each component in the other is available for this binary pair, the mixture is nearly symmetric and deviates only moderately from ideal solution behavior (y;"" /yj"" = 4.8 1 /4.54). The solutions of eqns. (3.5.9 to 3.5 . 1 1 ) give values of the binary interaction coefficients of k 1 2 0.0905 and k2 1 = 0. 1 1 67, and predictions with these parameters are shown in the figure as dashed lines. Direct correlation of VLE with the 2PVDW model gives the binary interaction parameters as k 1 2 = 0.0944, and k2 1 = 0. 1 027, and the results obtained with these parameters are shown as the solid line in the same figure. For this case, predictions based on infinite dilution activity coefficients are good, and a reasonable representation of the VLE data over the whole concentration range is possible with this information. However, this approach becomes less dependable as the nonideality of the mixture increases. For example, in Figure 3.5.8, the results for the ethanol and heptane binary system are shown. This mixture is more nonideal, (y/"0 IY]"' = 1 6.27 I 14.2 1 ), and the predictions obtained (long dashed lines) with parameters k12 = 0.0565 and k 2 1 = 0. 1 62 1 based on only infinite dilution activity coefficients are much worse than those obtained by =

40

Vapor-Li q u i d Eq u i l i bri u m Modeling

0.6

0.5

0.4

0.3

I I I I

I

I

I

I

I

I

I

/

/

/

/

,..

,

""'

,...

....

e VLE data at 31 3 K

... .__.__._.... _ .._; �_.___.__._...._.... ... .... .._ .. 0.2 ...._._..._ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of methylacetate

Figure 3.5.7. VLE correlation of the methyl acetate

and cyclohexane binary system at 3 1 3 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are model predictions obtained by direct correlation of VLE data, and the dashed lines are predictions using infinite dilution activity coefficient data. See text for details. (Points are experimental data from the DECHEMA Chemistry Series, Gmehling and Onken 1 977, Vol . I , Pt. 5, p. 392; the data file for this system on the accompanying disk is MAC640.DAT. )

correlating data over the whole composition range (solid line) to obtain the parameters = 0.020 1 and k2 1 = 0. 1 376. There is another, albeit more cumbersome, way of using activity coefficient infor­ mation. This i s described by Pividal et al. ( 1 992) as follows: First, a two-parameter excess free-energy model is selected, and its parameters are obtained from the mea­ sured activity coefficients at infinite dilution. Next, by use of these parameters the ac­ tivity coefficients are computed from the Gibbs free-energy model at a selected mole fraction (Pi vidal et al. used x = 0.5). Finally the two EOS mixing rule parameters are determined by equating these midconcentration activity coefficient values to the activ­ ity coefficient expressions from the EOS . This procedure requires no optimization and leads to predictions shown in Figure 3 . 5 . 8 as short dashed lines for the ethanol and hep­ tane binary system. These results are considerably better than the predictions obtained by solving eqns. (3.5.9 to 3.5. 1 1 ) (long dashed lines) but still inferior to the direct fit of the data (solid lines). Moreover, a liquid two-phase split is erroneously predicted. There are several problems associated with these multiparameter combining rules that limit their use in process design for mixtures containing many compounds, or to mixtures which contain some species with similar characteri stics (such as with mix­ tures of isomers, etc.). The first of these problems is the so-called dilution effect in

k12

41

Modeling Vapor- L i q u i d Eq u i l i bria

0.7

�

gj ....

::::J C/1 C/1 Gl

0.6

0.5

....

a.

0.4 e VLE data at 333 K

0.3

0.2 L..... ...o. ..o-J .J.... -....1."'---'�-'--.... ... .. -...I.o... ....o �.J... ..J.. .. ---' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of ethanol

Figure 3.5.8. VLE correlation of the ethanol and n ­

heptane binary system a t 3 3 3 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are model predictions obtained by direct corre­ lation of VLE data, and the dashed lines are predic­ tions using infinite dilution activity coefficient data. See text for details. (Points are experimental data from the DECHEMA Chemistry Series, Gmehling and Onken 1 977, Vol . I , Pt. 5, p. 392; the data file for this system on the accompanying disk for this system is ETC7 3 3 3 . DAT. )

which, as the number of components in a mixture increases, the mole fraction of any one component in the system becomes smaller. This leads to smaller contributions from the terms with the higher-order composition dependence and the added param­ eter(s), thus effectively reducing the mixing rule to the quadratic one-fluid mixing rule of van der Waals ( l PVDW model) as the number of components in the mixture increases. The second difficulty with mixing rules in this category, as pointed out by Michelsen and Kistenmacher ( 1 990), is that they do not result in the correct treatment of mul­ ticomponent mixtures containing two or more identical subcomponents. Consider a mixture of three components, and for the sake of demonstration allow two of these components, such as 2 and 3, to become identical. (This is not a trivial thought experiment, for there are multicomponent mixtures of industrial interest in which two or more components have very similar EOS parameters such as the isomers ] -methyl­ naphthalene and 2-methyl-napthalene.) In this case, to be internally consistent, the mixing rule for the parameter a should reduce to that for a binary mixture with the composition (x2 + x3 ) for the new component 2 + 3. The one-fluid combining rule of van der Waals satisfies this criterion, whereas the combining rules being considered in this section do not. 42

Vapor-Li q u i d Eq u i l i brium M o d e l i ng

Finally, a difficulty with these mixing rules is that, because of the added compo­ sition dependence of the a parameter, they fail to satisfy the theoretical quadratic composition dependence of the second virial coefficient given in eqn. (3.2.2). This fact is usually ignored on the grounds that pure component second virial coefficients calculated from a cubic EOS are in poor agreement with experimental data. However, the expression for the fugacity coefficient of a species in a mixture (eqn. 2.3. 1 ) shows that its evaluation from the EOS involves an integral of a term that includes a partial derivative with respect to composition from zero density to the density of interest. Therefore, an error in the composition dependence of the EOS at low densities will affect the fugacity coefficient calculated from eqn. (2.3. 1 ) at all densities. It should be stressed that the correct composition dependence of the second virial coefficient and correct numerical values of the pure component second virial coefficients are two different problems, and both will affect the computed fugacity coefficient. Models that have the correct second virial composition dependence eliminate one of these sources of error.

43

4 Mixing Rules that Combine an Equation of State with an Activity Coeffi cient Mod el

ANY mixtures of interest in the chemical industry exhibit strong nonideality,

M greater than that describable by regular solution theory, and have traditional ly

been described by activity coefficient (or free-energy) models for the liquid phase and an equation of state for the vapor phase. However, as discussed earlier, there are numerous problems with the activity coefficient description. For example, there are difficulties in defining standard states (especially for supercritical components), the parameters in these models are very temperature dependent, and critical phenomena are not predicted because a different model is used for the vapor and liquid phases. Also, other thermodynamic properties (densities, enthalpies, entropies, etc.) can not usually be obtained from the same model because the Gibbs energy is rarely known as a function of temperature and pressure. Therefore, interest exists in mixture EOS models that are capable of describing greater degrees of nonideality than is possible with the van der Waals one-fluid model and its variations. A very attractive route for developing better mixing rules is to combine an EOS with activity coefficient models, and this approach is the subject of this chapter.

4. 1 .

The Combination of Equation of State Models with Excess Free- Energy (EOS-G ex) Models: An Overview

It is possible to obtain the activity coefficient of a species i in a mixture from an EOS using the relation given in eqn. (3 .5.8), y; = ([>; I¢; , where ([>; is the fugacity coefficient of component i in the mixture and ¢; is the pure component fugacity, both of which are computed from an EOS at the temperature and pressure of the mixture. The molar excess Gibbs free-energy is ccx

RT 44

=

I:>; In Y; I

(4. 1 . 1 )

M ix i ng Ru les that Combine an Eq uati on of State with an Activity Coefficient Model

The combination of eqns. (3.5.8) and (4. 1 . 1 ) leads to

G'os(T,RT x;) ln ¢(T, xJ - �" x; ln ¢;(T, ; [ Z(T, x;) - �x;Z;(T, ] [ Z(T, x;) - � x; Z;(T, ] (1Y.(T, P.x; ) Z 1 L:x; 1Y.;( T. I') -) A�0s ( T, x; ) - [" x; ---� Z(T,Z;(T, x;)] RT (1Y.Cf.l',x; ) -Z - L:x; 1Y.(T. P; ) Z; ) ( £ ( Z ] ) [1 -RT z - "x z "x "x Z I � � . � __!!!_ _ ( ) ) "x (c(V) I bRT - � b;RT c (V· ) z) ( 1 - £Y. ] ( - ) C(V) [ z "x "x-RT � . � - hRT _ ) C(V ) - "x ( __!!!_ b;RT C(¥_) C(¥_) 2:n [ �:�:���: ] b h h P,

__ _

---=.::..::...._

=

P,

=

P)

P)

P,

- In -

P)

In

P,

v- d y_ -

- d y_ v ] Z;-1

-

X

i

X

(4. 1 .2)

or, equivalently, for the excess Helmholtz free energy of mixing

P,

=

--=:;:_:::__ __ __

i

-

P, P)

In

y_

00

1

d¥_ -

-- d y_ y_j

1

j

X

(4. 1 .3)

For the two-parameter cubic equations considered here, eqns. 4. 1 .2 and 4. 1 .3) become cex E

os

=

I

l

In

I

l

l

I

l

In _____x_ 1 !!;_ .!::'i

a

+

-I

I

(4. 1 .4)

and

Aex -� - =

l

I

� I I

In

l

I

l

_

,

In

I

!!L !::'i

+

a

(4. 1 .5)

Here is a molar-volume-dependent function specific to the EOS chosen. For = example, for the Peng-Robinson equation In . Equations (4. 1 .4 or 4. 1 .5) use one degree of freedom in establishing a relation between the EOS parameters a and and the mixture composition with activity coefficient or excess Gibbs free-energy models. Coupled with another independent piece of information that connects a or (or both) to composition, one can solve for the EOS parameters a and in terms of the mole fractions. There are several alternative ways of implementing such a procedure, and they are reviewed here and in the following sections. 45

Modeling Vapor-Li q u i d Eq u i l i bria

For later reference we note that in the limit of infinite pressure, �; ---+ b ; and �mi x ---+ bmi x so that C( �mix = bmi x ) = C( �; = b; ) = C* ; for the Peng-Robinson equation C* = ln( ../2 - 1 )/ ../2 = -0.62323 . Then eqns. (4. 1 .2 and 4. 1 .3) become

A.�os(T, P

---+

C* R T

oo ,

x; )

oo,

x; )

=

and

Q�0s(T, P ---+ C* R T

=

_I_ [� _

"" a ; x � b; I

]

a [ RT

"" a ; x; � b;

]

RT I

b

b-

I

I

(4. 1 .6)

+

P "" b - b; ) R T � x; (

(4. 1 .7)

I

IfA.�os or Q�os is equated to those from an activity coefficient model , then eqns. (4. 1 .6 and 4. 1 .7) are a mathematically rigorous combination of an EOS and an activity coefficient model ; they have been obtained in the limit of infinite pressure. From the definition of an excess property change upon mixing, it is necessary that the pure components and the mixture be in the same state of aggregation; therefore, to use eqns. (4. 1 .6 and 4. 1 . 7) the same must be true. It is obvious from these equations that the excess Gibbs and Helmholtz free energies of mixing computed from an EOS are a function of pressure, whereas activity coefficient models are independent of pressure or density. Therefore, the equality between Qcx (or A.e' ) from an EOS and from an activity coefficient model can be made at only a single pressure. Models that combine equations of state and activity coefficient models can be categorized into two groups: those that make this link at infinite pressure (Huron and Vidal 1 979; Wong and Sandler 1 992) and those that make this link at low or zero pressure (Dahl and Michelsen 1 990; Holderbaum and Gmehling 1 99 1 ; Michelsen 1 990b, among others). However, in the zero pressure limit there is no mathematically rigorous solution applicable to all phase equilibrium problems . This is because, above some temperature, the cubic EOS will not have the necessary real root for the liquid phase at zero pressure. Such a case is schematically shown in Figure 4. I . 1 in which several isotherms calculated from the reduced PR EOS [eqn. (4. 1 . 8)]

P,. =

3 .2573 T,. -,---V,. - 0.2534

--

4.85 1 4a V} + 0.5068 Vr - 0.0642

( 4. 1 .8)

are shown. In eqn. ( 4. 1 .8) the reduced pressure, temperature, and volume are defined as P,. = P / Pc , T,. = T/ Tc, and V,. = � Pc/(0.307 R T,.) respectively. The PR EOS with w set to zero can give a real positive root for the liquid molar volume at the limit of zero pressure at a reduced temperature of 0.8 ( P = 0 in the figure) and yet fail to give a root at a reduced temperature of 0.9. The limiting reduced temperature for obtaining a real root for liquid molar volume is about 0.85 for this EOS with w = 0. Consequently, an approxi mation must be introduced for the temperature range T,. > 0. 85 to obtain a pseudoliquid root. In spite of this shortcoming, these 46

Mixing Rules that Combine an Equation of State with an Activity Coeffic ient Model

.

2 0

1 .5 Q)

:;

II) II)

c.

I'

1 .0

\

Q)

"'0

I

0.5

u

"'0 Q) ...

/

/

- � - - - -

\ /1 ,,

Q)

:;,

Tr =0.9 ,

0.0

-

Tr =0.8

v

-0.5

-1 .0

2

0

3

5

4

Figure 4.1.1. Reduced pressure ( P I Pc ) versus re­ duced volume [J:::. Pc /(0. 307 R Tc)] as a function of re­ duced temperature (T j T,. ) from the Peng-Robinson equation of state with acentric factor set to zero. See text for details. reduced volume

approximate models are successful in many cases and are therefore considered in this monograph. Most of these approximate zero pressure models take the form Q�xos

--

RT

=

-

" � x ; ln

;

( ) + qn ( b b;

-

a; a " X; --- - � bRT

b; R T

;

)

(4. 1 .9)

where qn is a model-dependent approximate parameter (or function). When using a zero pressure model, the approximate eqn. (4. 1 .9) is used instead of the exact relation given by eqn. ( 4. 1 . 7), which was obtained at the limit of infinite pressure. By relaxing mathematical rigor in establishing the connection between excess free­ energy models and EOS , several successful approximate models have been developed in the limit of infinite pressure. One such model that uses excess Helmholtz free energy was introduced by Orbey and Sandler ( 1 995c) and is as follows: A.�xos

--

RT

=

-

" x; ln �

;

( )+C ( b h;

-

•

a -- - " � x; bRT

;

a;

--

b; R T

)

(4. 1 . 1 0)

Equation ( 4. 1 . 1 0) is algebraically very similar to eqn. ( 4. 1 . 9) except that the approx­ imate parameter qn is now replaced with the parameter C* , which, as pointed out earlier, is dependent on the EOS used. This seemingly small difference, however, has a nontrivial effect on the results obtained, as will be shown later. In this approach, the approximate eqn. (4. 1 . 1 0) replaces the exact expression given by eqn. (4. 1 .6). 47

Modeling Vapor-Liq u i d Eq u i l i bria

4.2.

The H u ron-Vidal (HVO) Model

Vidal ( 1 978) and later Huron and Vidal ( 1 979) proposed the first successful combi­ nation of an EOS and activity coefficient models by requiring that the mixture EOS at liquid densities should behave like an activity coefficient model. To ensure a liquid density at all temperatures, this equality was made using eqn. ( 4. 1 . 7) and the relation

(Z,/(T,

P = oo ,

x)

=

Q�"o5(T, P =

x)

oo ,

x)

x)

(4.2. 1 )

where Q�x(T, P = oo , and Q�05(T, P = oo , are the excess Gibbs free energies at infinite pressure (i.e., at liquid-like densities) calculated from an activity coefficient model and the EOS, respectively. Because Qex = Aex P J:::ex, with Aex in the liquid state being almost independent of pressure, to use eqn. ( 4. 1 . 7) it is necessary that

vex

_

= V- � -

"X

l

V1 =

-

b - "x- b · �

l

I

=

0

+

(4.2.2)

since at infinite pressure from an equation of state 1:: ; = h; . That is, to keep Q�xos finite in order to use eqn. (4. 1 . 7), eqn. ( 4.2.2) must also be used. Equations (4. 1 .7 and 4.2.2) provide the two equations necessary to determine the two EOS constants. The resulting mixing rule for the a parameter is

a b [ LX; (a·)b: + C* ] =

cex

(4.2.3)

where Qex is an excess free energy of mixing expression appropriate for the mixture of interest. Equations (4.2.3 and 3 . 3 . 8 ) constitute the original Huron-Vidal mixing rule. We will refer to this mixing rule as the HVO model in this monograph. Some results obtained with the HVO mixing rule are presented in the following paragraphs. These results were obtained with the computer program HV provided on the accompanying diskette; the computational details are presented in Section D.4 of Appendix D. The correlation of data for the methane and pentane binary system is shown in Figure 4.2. 1 . In this case the van Laar excess Gibbs free-energy model has been used in the HVO model; the two model parameters were fitted to VLE data on the 277 K isotherm, and the vapor-liquid equilibria at higher temperatures were predicted with the same temperature-independent parameters. The results are very good in this case and similar to those obtained with the l PVDW and 2PVDW models. The results for the carbon dioxide and propane binary system, shown as dashed lines in Figure 4.2.2, on the other hand are not as good. When compared with the performance of the l PVDW model (solid lines in Figure 4.2.2), the use of the same parameters for all isotherms leads to inferior results at higher temperatures despite the use of an extra parameter in the Huron-Vidal model. This indicates that, for the mixtures containing supercritical components, the HVO mixing rule, when combined 48

Mixing Ru les that Com b i n e an Equation of State with an Activity Coefficient Model

1 80 t:. A

1 60 1 40

...

r:u .c

ai ... :::1

Ill Ill

Cll ...

a.

VLE data at 31 0 K data at 377 K VLE data at 444 K

A

o • VLE

o

e

1 20 1 00 80 60 40 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of methane Figure 4.2. 1 . VLE correlation of the methane and

pentane binary system at 3 1 0, 377, and 444 K with the Huron-Vidal original ( HVO) mixing rule with the van Laar excess free-energy model and the PRSV equa­ tion of state. The van Laar model parameters used are �/ f3 = A 1 2 / A 2 1 = 0. 1 20 1 /0. 1 430. Points are experi­ mental data from the DECHEMA Chemistry Data Se­ ries, Gmehling and Onken 1 977, Vol. 6, p. 445 ; data files for this system on the accompanying disk are C l C5 3 l O.DAT, C l C5377 .DAT and C l C5444.DAT.

with the conventional excess free-energy models (such as that of van Laar) is inferior to the van der Waals one-fluid model . This was observed earlier by Shibata and Sandler ( 1 989). It should be noted, however, that by choosing specific algebraic forms for the excess free-energy term it is possible to reduce the HVO model to the I PVDW model. See, for example, Huron and Vidal ( 1 979) and Orbey and Sandler ( 1 995a,c) for the details of such modified excess free-energy models. For more nonideal mixtures, the HVO model shows good correlation capabi lities but is not satisfactory for extrapolation over a range of temperatures. See, for example, the results presented in Figures 4.2.3 and 4.2.4 for the acetone and water binary system and those in Figures 4.2.5 and 4.2.6 for the 2-propanol and water binary system. In these figures, the dashed lines are direct correlations of the isothermal VLE data with the HVO mixing rule. The solid lines are predictions with model parameters obtained from the DECHEMA Chemistry Series at or near room temperature for the excess free-energy model. In each case the model was observed to be superior to both the one-parameter ( I PVDW) and the two-parameter (2PVDW) van der Waals models for the correlation of VLE data. On the other hand, poor predictive performance observed in these figures indicates that, even though the Huron-Vidal approach allows the use of cex models with an EOS, one cannot use excess Gibbs free-energy model parameters obtained from the y -¢ method at low pressure (for example those in the 49

Model i ng Vapor-Liquid Eq u i l ibna

70

60

.... CIS .c

gj .... ::I C/1 C/1

(I) .... a.

<) + VLE data at 344 K

1:!. & VLE data at 31 0 K

0 e VLE data at 277 K

50

40

30

20

10 0 ������� 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of carbon dioxide

Figure 4.2.2. VLE correlation (dashed lines) of

the carbon dioxide and propane binary system at 277, 3 1 0, and 344 K with the Huron-Vidal original (HVO) mixing rule with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are � / {3 = A 1 z / A 2 1 = 1 . 1 8 1 6/ 1 .690 1 . Solid lines repre­ sent the I PVDW model correlations presented earlier in Figure 3 .4.2. (Points are experimental data from the DECHEMA Chemistry Data Series, Gmehling and Onken 1 977, Vol . 6, p. 589; data fi les for this system on the accompanying disk are C02C3277. DAT, C02C33 1 O.DAT and C02C3344. DAT. )

DECHEMA Chemistry Data Series) with this EOS model. This is because the excess Gibbs free energy of mixing from experiment and as calculated from an EOS is very pressure dependent; therefore, the excess Gibbs free energies at infinite pressure and at the pressure at which experimental data were obtained can be very different. Consequently, a fundamental shortcoming of the Huron-Vidal approach is the use of the pressure-dependent Gibbs excess free-energy in the EOS rather than Helmholtz excess free-energy, which is much less pressure dependent. This shortcoming was corrected by Wong and Sandler ( 1 992), and their work is discussed next. 4.3.

T h e Wong-Sandier (WS) Model

Wong and Sandler ( 1 992) have developed a mixing rule that combines an EOS with a free-energy model but produces the desired EOS behavior at both low and high densities without being density dependent, uses the existing table of cex parameters, allows extrapolation over wide ranges of temperature and pressure, and provides a 50

M ixing Ru les that Com b i n e an Equation of State with an Activity Coefficient Model

o e V L E data a t 2 9 8 K

0.3

0.2

0.1

0 .0

....__,_ ._. ...._........ ...._ .... ..._ _._...J_ .._. ...... _._...I-.o---1.....1.-.i... . ..- ...J

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

Figure 4.2.3. VLE correlation of the acetone and wa­

ter binary system at 298 K with the Huron-Vidal orig­ inal ( H VO) mixing rule with the van Laar excess free­ energy model and the PRSV equation of state. The dashed lines denote results calculated with van Laar model parameters � / tl A 1 2/A21 3 . 5 1 2 1 /2. 2227 obtained from fitting the experimental data, and the solid lines represent the results obtained with model parameters �I tl A !2/ A21 1 .9399/ 1 . 8022 ob­ tained at the same temperature from the DECHEMA Chemistry Series (Gmehling and Onken 1 977, Vol . I , Pt. I , p. 238). =

=

=

=

conceptually simple method of accurately extending UNIFAC or other low-pressure VLE prediction methods to high temperature and pressure. This mixing rule is based on several observations. The first is that, although eqns. ( 3 . 3 .4 and 3.3.5) are sufficient conditions to ensure the proper composition dependence of the second virial coef­ ficient, they are not necessary conditions. In particular, the van der Waals one-fluid mixing rules of eqns. (3.3.4 and 3 . 3 . 5 ) place constraints on the two functions a and to satisfy the single relation

B(x .. T)

=

�. xI xJ B I} ( T ) �. L... L... I

J

=

�. x I x J � L.... L...

·

I

J

(

h I} - !!.!!__ RT

)

b

=

h - _!!____ RT

(4.3 . 1 )

The original version of the Wong-Sandier mixing rule (Wong and Sandler 1 992) uses the last equality of eqn . ( 4.3. 1 ) as one of the restrictions on the parameters together with the combining rule

(

h - _!!____ RT

) � [ (b . . I}

=

2

I -

!!!_ R T __

) + ( - _!!j_ ) J o hJ

RT

(4.3 .2)

kI ) )

b

which introduces a second virial coefficient binary interaction parameter kij . Note that eqn. (4.3 . 1 ) does not provide relations for the parameters a and separately but only 51

Modeling Vapor-Liquid Eq u i l i bria

._o .!

...

«< .c

60

Cli ... ::I rn rn G)

...

a.

,.

I

50

I I I I I I

�I

/

I

<

I

/

//

I

i

/

" / e

....- -'6if /,-/ / ,19 /0 I

/

/

/

/

I

I

/

I

I

/

/o

e

o VLE data at 523 K ���-L�-L�-L�

40 �� 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 mole fraction of acetone

Figure 4.2.4. VLE correlation of the acetone and wa­ ter binary system at 523 K with the Huron-Vidal orig­ inal (HVO) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with van Laar model parameters � / f3 A 12/A21 4 . 2206/ I . 7264 obtained from fitting the experimental data, and the solid lines denote the results obtained with model parameters � /{3 A t z/Azt 2 . 1 700/ 1 . 7264 from the DECHEMA Chemistry Series at 353 K (Gmehling and Onken 1 977, Vol . 1 , Pt. 1 , p. 3 34). =

=

=

=

for the sum (b - a/ R T) , and thus an additional equation is needed. Also, by using eqn. (4. 3 . 1 ) as one of the relations to determine the EOS parameters, the proper com­ position dependence of the second virial coefficient is assured, regardless of which additional equation is used. The second equation in their mixing rule is based on the observation that the excess Helmholtz free-energy of mixing calculated from a cubic EOS is much less sensitive to pressure than the Gibbs free-energy, as can be seen in Figure 4.3. 1 . Consequently, to an excellent approximation:

Qe' ( T , P =

1 bar, x )

=

.Ae' ( T , P =

1 bar, x )

=

.Ae' (T .

high pressure, x )

(4.3.3)

The first of these equalities results from the relation Q.ex Aex + P l':ex and that the P l':cx term at low pressures. The second of these equalities is a result of the essential pressure independence of .Aex at liquid densities. Therefore, the second equation for the a and b parameters comes from eqn. (4.3 .3 ) in the form of =

A�'05 (T, P = oo , x ) = .A�'(T, P = oo , x ) = .A�' (T , = Q�' ( T, low P , x ) 52

low P, x ) (4. 3 .4)

M ixing Rules that Com b i n e an Equation of State with an Activity Coeffi c ient Model

1 .1 1 .0 0.9 0.8

0.6

oe VLE

data at 353 K

0.5 0. 4

L...-l. ... .......... ._; ....I-.... .._ .. _...._.... _ ��--'-�J... ... .. ... ..J

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction 2-propanol

Figure 4.2.5. VLE correlation of the 2-propanol and water binary system at 353 K with the Huron-Vidal original (HVO) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines represent results calcu­ lated with a = 0.2893 and <1 2 / <2 1 = 0.7882/3 . 9479 obtained from fitting the experimental data, and the solid lines denote results calculated with a = 0. 2893 and r1 2 / rz 1 = 0. 1 509/ 1 . 805 1 obtained from the DECHEMA Chemi stry Series at 303 K (Gmehling and Onken 1 977, Vol. 1, Pt. 1 , p. 325).

Combining eqns. (4.3 . 1 , 4.3 .4, and 4. 1 .6) gives the following mixing rules: (4.3.5) and (4.3 .6) where the cross term in eqn. (4. 3.5) is obtained from eqn. (4.3 .2). Any excess Gibbs free-energy model may be used in eqn. ( 4.3 .6). The model parameters are the parameter k ij in eqn . (4.3.2) and the parameters of the excess Gibbs free-energy model used in eqn. (4. 3 .6). This mixing rule is referred to as the WS mixing rule in the remainder of this monograph. The WS mixing rule satisfies the low-density boundary condition that the second virial coefficient be quadratic in composition and the high-density condition that ex­ cess free energy be produced like that of currently used activity coefficient models, whereas the mixing rule itself is independent of density. This model provides a cor­ rect alternative to the earlier ad hoc density-dependent mixing rules (Copeman and 53

Mode l i ng Vapor-Li q u i d Eq u i l ibria

a;

65

0

Gi ... :::J en en CD ...

55

c.

45 oe

VLE data at 523 K

35 ...._._.. .. .... ..._ _._.. .. .... ..._ .. _._.. .1-...o ..._ --'--'--'---'--"-..... 0.0 O.t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 mole fraction of 2-propanol

Figure 4.2.6. VLE correlation of the 2-propanol and

water binary system at 523 K with the Huron-Vidal original (HVO) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines represent results calcu­ lated with a = 0. 2893 and r u / r2 1 = 0. 3952/4. 1 5 1 8 obtained by fitting the experimental data, and the solid lines denote results calculated with a = 0. 2893 and r 1 2 / r2 1 = 0. 1 0 1 9/ 1 . 2 1 85 obtained from the DECHEMA Chemistry Series (Gmehling and Onken 1 977, Vol. I , Pt. I , p. 325). 0 . 4 5 0 .----...,....---,...,

� � ..._ X CD

0 � � ..._ ""

G •x a t 1 bar, A ex at 1 &1 0 0 0 bar 0.350

0.250

0.1 50

X CD

(!)

0.050

'I

I

1

1

/

/

/

/ - - - --- --

G•x at 1 000 bar

..___

....__

"- , ' '--

"

"

""

- 0 . 0 5 0 L...----�---;__...._---L._ . _____J 0.8 0.4 0.6 0.2 0.0 1 .0 mole fraction of methanol

Figure 4.3. 1 . The excess Gibbs and Helmholtz energies

of mixing for the methanol and benzene binary system at 373 K calculated with the Wong-Sandier (WS) mixing rule and the PRSV equation of state at I and 1 000 bar.

54

M ix i ng Rules that Com b i n e an Equation of State with an Activity Coefficient Model

Mathias 1 986; Michel, Hooper, and Prausnitz 1 989; Sandler et al. 1 986) and retains an important feature of the one-fluid model: that the EOS for the pure fluids and the mixture have the same density dependence. This mixing rule has been successful in several ways. First, when combined with any cubic EOS that gives the correct vapor pressure and an appropriate activity coefficient model for the (r term, it has been shown to lead to very good correlations of vapor­ liquid, liquid-liquid, and vapor-liquid-liquid equilibria, indeed generally comparable to those obtained when the same activity coefficient models are used directly in the y -

Model i ng Vapor-Li q u i d Eq u i l i bria

1 60 6 .a. VLE data at 444 K

1 40

�

1 20

::I Ill Ill Q)

80

.c

ai .... ....

Q.

D • VLE data at 377 K

1 00

60 40 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of methane

Figure 4.3.2. VLE correlation of the methane and n ­

pentane binary system a t 3 1 0, 3 7 7 , and 444 K with the Wong-Sandier (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are � / {3 A 12/A21 0. 1 924/0.67 1 9, and the Wong­ Sandler mixing rule parameter k 1 2 0 . 5 2 1 6. (Points are experimental data from the DECHEMA Chem­ istry Data Series, Gmehling and Onken Vol . 6, p . 445 ; data files for this system on the accompanying disk are C 1 C53 1 0.DAT, C 1 C5377 . DAT and C I C5444.DAT. ) =

=

=

in Figures 4.3.6 to 4.3.8, are observed for the acetone and water binary system. This illustrates the good accuracy of extrapolations of vapor-liquid phase behavior to high temperatures that can be obtained with the WS mixing rule and a simple EOS . To demonstrate the differences between the WS and the HVO models, the results of VLE predictions for the 2-propanol and water binary system at 353 K with the parameters obtained from the DECHEMA tables at 303 K are shown in Figure 4.3.9 in which the solid line is the prediction with the WS mixing rule and the dashed line describes the results of the HVO model . The significant advantage of the WS model over the HVO model in predictions is clearly visible in this figure. Experience has shown that the choice of activity coefficient model to be coupled with the EOS has some effect on the predictive performance of both the WS and HVO models. For example, as the complexity of the mixture increases, the NRTL and the UNIQUAC models, which have a temperature dependence of their parameters, usually give better results over a range of temperatures than the simple van Laar model. A desirable characteristic of an excess free-energy-based mixing rule is that it goes smoothly to the conventional van der Waals one-fluid mixing rule for some values of its parameters. This is useful because in multicomponent mixtures only some of the binary pairs may form highly nonideal mixtures requiring mixing rules such as

56

M ix i ng Ru les that Com b i n e an Eq uation of State with an Activity Coefficient Model

80

...... ...

0+ VLE data a t 3 4 4 K

70 60

Cll .D

50

a:i

40

:::::1 Ill Ill Cl)

30

a.

20 10

Q L.......J...L..JL..o..J..I.�-'-...L�.J..o___J._�J..J 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of carbon dioxide

Figure 4.3.3. VLE correlation of the carbon dioxide

and propane binary system at 277, 3 1 0, and 344 K with the Wong-Sandier (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are � I f3 A 1 2 / A21 0.7897/0.792 8 , and the 0.3565. Wong-Sandier mixing rule parameter k 1 2 (Points are experimental data from the DECHEMA Chemistry Data Series, Gmehling and Onken , Vol . 6, p. 589; data files for this system on the accompany­ ing disk are C02C3277.DAT, C02C33 1 0. DAT and C02C3344.DAT. ) =

=

=

those described in this section, whereas other binary pairs in the same mixture can be adequately described using the van der Waals mixing rule of eqns. ( 3 . 3 .4 and 3 . 3 . 8). Orbey and Sandler ( 1 995a) have proposed a slightly reformulated version of the WS mixing rule to accomplish this. They retained the mixing rule of eqns. (4.3 . 5 and 4.3 .6) but replaced the combining rule of eqn. (4.3 .2) by

(b - a ) . RT

lj

=

1

l (b; + bi ) -

,JZfiZij

1 ki ) �( - j

(4.3 .7)

This equation introduces the binary interaction parameter in a manner similar to that of eqn . (3.3 .6) of the van der Waals one-fluid mixing rule. Next, the following modified form of the NRTL equation was used for the excess free-energy term : ex Q

RT

with

=

'""'

� X;

(� ) x/ Gj; Tj i

=-X_k_G_k_i----=L k

(4.3.8)

.

(4. 3 9 ) 57

Model i ng Vapor-Li q u i d E q u i l i bria

1 .1 1 .0

1ii

..0

gj .... :::::1 Ill Ill Cl)

.... Q.

0.9 0.8 0.7 0.6

e

VLE data at 353 K

0.5 0.4

0...--'.. ""----'--...._...._.... .__. _.... ...._ .___ __J_�...._, ...

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1 .0

mole fraction 2-propanol

Figure 4.3.4. VLE correlation (solid lines) of the

2-propanol and water binary system at 353 K with the Wong-Sandier (WS) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines are cal­ culated with a = 0 . 2529, r1 2 / r2 1 = 0. 1 562/2.7548, and with the Wong-Sandier mixing rule parame­ ter = 0.2529 obtained by fitting the experimen­ tal data. The solid lines represents results calcu­ lated with a = 0. 2893 and r 1 2 / r2 1 = 0. 1 509/ 1 . 805 1 obtained from the DECHEMA Chemistry Series at 303 K (Gmehling and Onken 1 977, Vol . l , Pt. I , p. 325) and with the Wong-Sandier mixing rule param­ eter k 1 2 = 0. 3659 obtained by matching the excess Gibbs free-energy from the equation of state and from the NRTL model at 303 K. Experimental data ( e , 0 ) are from Wu , Hagewiessche, and Sandler 1 98 8 .

k1 2

where b J is the volume parameter in the EOS for species j. This modified NRTL form was suggested earlier by Huron and Vidal ( 1 979) for use in their model. There are four dimensionless parameters in eqns. (4.3.7 to 4.3.9): a , TiJ , r1 ; , and k iJ . This version of the WS mixing rule can be used as a four-parameter model to cor­ relate the behavior of complex mixtures or in several ways that have fewer adjustable parameters. For example, one can solve the two relations obtained from eqn. (4. 3 . 8 ) in the infinite dilution limit r1 ;

=

b;

ln yiJX - TiJ [; exp( - a riJ )

(4. 3 . 1 0)

J

for selected values of a and kiJ , where the Y/j is the infinite dilution activity coefficient of species i in j . Orbey and Sandler ( 1 995a) found that setting a 0. 1 and kiJ 0 worked quite well for several nonideal mixtures of organic chemicals, and thus one could use the model to predict the complete ph
58

=

M ix i ng Rules that Com b i n e an Equation of State with an Activity Coeffic ient Model

70

1ii

.&J

60

ai ... ::I Ul Ul Q) ...

a.

e VLE data at

50

523 K

40

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 mole fraction of 2-propanol

Figure 4.3.5. VLE correlation (solid lines) of the 2propanol and water binary system at 523 K with the Wong-Sandier (WS) mixing rule combined with the NRTL excess free-energy model and the PRSV equa­ tion of state. The dashed lines represent results calcu­ lated with a = 0.2893, T J 2 / T2 I = 0.4302/ 2 . 5 280, and with the Wong-Sandier mixing rule parame­ ter k 1 2 = 0 . 3 1 59 obtained by fitting the experimental data. The solid lines denote results calculated with a = 0. 2893 , r1 2 / T2 1 0. 1 0 1 9/ 1 . 2 1 85 obtained from the DEC HEM A Chemi stry Series at 303 K (Gmehling and Onken 1 977, Vol . 1 , Pt. 1 , p. 325) with the Wong­ Sandier mixing rule parameter k 1 2 = 0 . 3 659 obtained by matching the excess Gibbs free energy from the equation of state and from the NRTL model at 303 K. Points are the experimental data from B arr-David and Dodge 1 959. -

=

temperature and pressure from the two infinite dilution activity coefficients at a single temperature. This model could also be used in a completely predictive fashion, in the absence of any experimental data, with infinite dilution activity coefficients obtained from the UNIFAC model . Also, by setting a = 0 and solving for r1 2 and r2 1 from ; = Tj

[ RT C*

2 ,;a;a; b;

+ bj (

1 - k ij ) -

a; ] b;

(4 . 3. 1 1 )

one recovers the van der Waals one-fluid mixing rule with a single binary interaction parameter kij that can be used for only slightly nonideal binary pairs in a multi­ component mixture. It should be noted that eqn. (4.3. 1 1 ) is not unique, and that other expressions also lead to the van der Waals mixing rule. Further details of this approach are available elsewhere (Orbey and Sandler 1 995a,c ) . 59

Modeling Vapor-Liquid Eq u i l i bria

e

0.3

1ii

.0

Gi ...

VLE data at 298 K

0.2

::::J en en CD ...

a.

0.1

0.0

L...o.--L...o....J�--'-"'-oo.l.----...l---L�.L..-'-�"'--'

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

Figure 4.3.6. VLE correlation of the acetone and wa­

ter binary system at 298 K with the Wong-Sandier (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with van Laar model parameters �I fJ = A 1 2 l /\ 2 1 = 1 . 772412 . 029 1 and the Wong-Sandier mi xing rule parameter k 1 2 = 0.2529 obtained by fitting the experimental data. The solid lines denote the results using model parame­ ters � I fJ = A 1 2 l /\ 2 1 = 1 . 93991 1 . 8022 obtained at the same temperature from the DECHEMA Chemistry Data Series (Gmehling and Onken 1 977, Vol. 1 , Pt. I , p. 238, which also lists the experimental data shown as points) and with the Wong-S andier mixing rule pa­ rameter k 1 2 = 0.24 1 7 obtained by matching the excess Gibbs free-energy from the equation of state and from the van Laar model at 298 K.

Because the WS mixing rule uses VLE information only at low pressure, it can also be used to make predictions at high pressure based on low-pressure prediction techniques such as UNIFAC and other group contribution methods (Orbey, Sandler, and Wong 1 993). This completely predictive method using the WS and other excess free-energy-based EOS models is discussed in Chapter 5 . The W S mixing rule has also been applied to some asymmetric mixtures such as hydrogen and hydrocarbon binaries (Huang, Sandler, and Orbey 1 994 ). However, in this case there is a potential problem, depending on the temperature, if the usual temperature dependence of the EOS a parameter is used. The problem arises because the temperature-dependent a term in equations of state is usually obtained by fitting vapor pressure and other subcritical data, as pointed out earlier, and is poorly defined at supercritical, high reduced temperatures (such as can be encountered with hydrogen owing to its low critical temperature or with other asymmetric mixtures containing 60

Mixing Ru les that Com b i n e an Equation of State with an Activity Coeffi c ient Model

4.0

3.5

a;

3.0

... ...

.c

oj

2.5

a.

2.0

::I en en Q)

1 .5

e

VLE data at 373 K

1 .0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1 .0

mole fraction of acetone

Figure 4.3.7. VLE correlation of the acetone and wa­ ter binary system at 373 K with the Wong-Sandier (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with van Laar model parameters � /fJ = A 1 z / A 2 1 = 2.0287/ 1 . 6009 and the Wong-Sandier mixing rule parameter k ! 2 0. 2779 obtained by fitting the experimental data. The solid lines denote the results using model parame­ ters � I fJ = A 1 2 / A 2 1 1 .9399/ 1 . 8022 obtained at the same temperature from the DECHEMA Chemistry Se­ ries (Gmehling and Onken 1 977, Vol . I , Pt. 1 , p. 238) and with the Wong-Sandier mixing rule parameter k 1 z = 0 . 24 1 7 obtained by matching the excess Gibbs free-energy from the equation of state and from the van Laar model at 298 K. Points are the experimental data of Griswold and Wong 1 95 2 . =

=

very light components) . The mixture b parameter in the WS model can be written as

b

=

( .. - !!.!!_ _) RT

RT � "" "" � X I X }. bI) I

RT -

[

}

"" a ; � x 1I

b;

+

)]

_ Qe'y' (- T , x_ ; __ C*

(4.3. 1 2)

and the denominator of this equation contains three terms. The excess free-energy term can be negative or positive and should vanish at high temperatures. For simplicity this term is neglected in the discussion here. In this case, unless the (a; /b; R T ) term is larger than unity for all the components of the mixture, there will be a composition of the (liquid or vapor) mixture at which the denominator becomes zero and then changes sign from negative to positive. The first of these possibilities is theoretically not allowed because of the discontinuity that results in the value of b of the mixture, 61

M o d e l i ng Vapor-Li q u i d Eq u i l ibria

g;

60

...

.c

ai ::J UJ UJ

�

a.

50 e

VLE data at 523 K

��

40 �� 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 mole fraction of acetone

Figure 4.3.8. VLE correlation of the acetone and water binary system at 523 K with the Wong­ Sandier (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state . The dashed lines represent re­ sults calculated with the van Laar model pa­ rameters Uf3 = A 1 2 / A 2 J = 1 . 9520/ 1 . 3 8 1 2 and the Wong-Sandier mixing rule parameter k 1 2 = 0. 264 1 obtained by fitting the experimental data. The solid lines denote the results using model parame­ ters �I f3 = A 1 2 / A 2 1 = 1 . 9399/ 1 . 8022 obtained at the same temperature from the DECHEMA Chemistry Se­ ries (Gmehling and Onken 1 977, Vol . I , Pt. I , p. 238) and with the Wong-Sandier mixing rule parameter k 1 2 = 0.24 1 7 obtai ned by matching the excess Gibbs free-energy from the equation of state and from the van Laar model at 298 K. Points are the experimental data of Griswold and Wong 1 95 2 .

and the second case is not allowed because it leads to meaningless negative values for the b parameter. In order to avoid these possibilities it is necessary at all temperatures to have

(4.3 . 1 3)

a;

b -RT < 0 I

for all the components of the mixture, which will ensure that the (a; I b; R T ) term will always be larger than unity. For cubic equations of state this term can be written in a generalized form; for example, for the Peng-Robinson equation of state we have

. 45 7 24 ) a ( T ) = S .8 77 1 2 [ a ( T) _bRTa_ = ( 00.077 8 T, J T,.

62

(4.3. 1 4)

M ixing Rules that Com b i n e an Equation of State with an Activity Coeffici ent Model

1 .1 1 .0

1;;

.c

:; Ill

oj

a. Ill

Cll

0.9 0.8 0.7 0.6

e

VLE data at 353 K

0.5 0.4

...._._ .. ...._.__.____._._ ... ........ ... _ ... ...... _.__.___,

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 4.3.9. Comparison of VLE predictions of the

2-propanol and water binary system at 353 K from the Wong-Sandier (solid lines) and Huron-Vidal original (dashed lines) models with both model parameters ob­ tained by fitting the experimental data at 303 K. Points are the experimental data of Barr-Davi d and Dodge 1 959.

Thus, the requirement of eqn. (4.3 . 1 3) becomes a ( T) :::: T, /5 . 877 1 2 for the Peng­ Robinson EOS . For most cubic equations of state at reduced temperatures below 2 the requirement of eqn. (4.3 . 1 3) is met, but, depending upon the temperature de­ pendence of a (T ), it may not be satisfied at higher reduced temperatures. A simple solution for mixtures with one or more highly supercritical components is to use the equation a ( T) = T,. I M for the highly supercritical component, with M being an EOS­ dependent constant, for example, 5 . 877 1 2 for the Peng-Robinson EOS . This removes the singularity and does not cause any numerical problems in the calculation of phase behavior or thermodynamic properties; however, in this case, for highly asymmetric mixtures such as those containing hydrogen with heavy hydrocarbons, the VLE corre­ lations and predictions may not be as accurate as those that have been discussed here. 4.4.

Approximate M ethods of Co m b i n i ng Free-Energy Models and Eq uations of State: The M HV I , M HV2, LCVM, and HVOS Models

The problem with the HVO model resulting from the pressure dependence of the excess Gibbs free-energy of mixing led a number of investigators to propose EOS mixing rules based on the idea of combining activity coefficient models and equations of state at low (or zero) pressures, as originally suggested by Mollerup ( 1 986) . This approach requires solving for the liquid density at zero (or low) pressure from the 63

Modeling Vapor-Li q u i d Eq u i l i bria

volumetric EOS for each species in the mixture. Consequently, the complication that arises in this category of methods is that there may not be a liquid density solution for one or more of the pure components at the temperature of the mixture and at zero (or low) pressure (see Figure 4. 1 . 1 ). Avoiding this complication requires some sort of approximate extrapolation technique. Several approximate models of this kind are reviewed below. The modified Huron-Vidal (MHV) mixing rule of Michelsen ( 1 990b) is one of the most used of this class. The idea behind this mixing rule is to use eqn. ( 4. 1 .4) at P = 0 to obtain

Q.�xos< P

RT

=

0)

a- ()t) = - L xi In ( 'fJy_o /hih -- I1 ) - L xi In ( !?.._bi ) + ( )c bRT i i / - � xi ( bi�T ) c ( � ) (4.4. 1 )

where superscript 0 denotes the value of the volume at P more like the Michelsen form when rewritten as

=

0. Equation ( 4.4. 1 ) looks

0 X·) p b q{s} L: xiq{sd + -cex(T y ' RT ' ' + I: xi In ( - ) � i =

=

(4.4.2)

i

s = ajbRT and q{s} is the function V0 a q{s) = - In ( =- 1 ) + - C( V0) b bRT -

where

( 4.4.3)

These equations use one of the degrees of freedom in choosing the mixture EOS parameters. The second equation that is used to define the mixing rule completely is eqn. ( 3 . 3 . 8 ) . However, note that, because it is no longer necessary to be concerned with divergence of the excess Gibbs free energy at infinite pressure, it is not necessary to impose eqn. (3 . 3 . 8 ) ; other choices, such as the last equality of eqn. (4. 3 . 1 ), could be used as well. At temperatures at which there is no liquid root of the EOS to use in the right-hand side of eqn. ( 4.4.3 ) Michelsen ( 1 990b) chose an approximate linear extrapolation

,

(4.4.4)

s, resulting in b " a " ai Q.�x - � = x In ( - ) + q1 ( -- - � xi -- ) i RT bi bRT i bi RT i

that is used for all values of

(4.4.5)

This relation, combined with eqn. (3.3.8), is known as the MHY l model and will be referred to as such in this monograph. It is interesting to note that an alternate derivation : i mi = ..!:::7: / of the MHV I model is to start with eqn. ( 4.4. 1 ) and assume that .!:::�

x

64

b

x

bi =

M i x i ng Rules that Combine an Equation of State with an Activity Coefficient Model

=

constant, so that C(�,i x ) C(ljl ) q 1 , which immediately gives eqn. (4.4.5). In this derivation it is seen that an assumption inherent in the MHV l model is that the ratio of the zero-pressure liquid molar volume to the close-packing parameter is the same for the mixture and for each of its pure components. Later, for better accuracy, a quadratic extrapolation was proposed for q { & } =

b

(4.4.6) and parameters were chosen to ensure continuity of the function q and its derivatives. The relation between the excess Gibbs free energy from an EOS and from an activity coefficient model takes the following quadratic form in this approach: q1

(& - '"'7 x; &; ) + ( - '"'7 x; &;2 ) Q.�xRT( T, + �'"' x; ( b ) q2

&

2

=

X; )

b;

In

(4.4.7)

This equation, together with eqn. (3.3.8), is known as MHV2 model (Dahl and Michelsen 1 990). The MHV 1 and MHV2 models are considered here for correla­ tion and prediction of the VLE of various mixtures, and computer programs that use these methods are provided in the accompanying disk. Further details are given later in this section. As indicated above, eqn. (3.3.8) is not the only option that can be used with eqn. (4.4. 1 ) . Indeed Tochigi et al. ( 1 994) have proposed a MHV mixing rule consistent with the second viria1 coefficient boundary condition by combining eqns. (4. 3 . 1 and 4.3.2 and 4.4.5). In their implementation they have eliminated the binary interaction parameter in eqn. (4.3 .2), leading to the mixing rule

-RT - b I '"'�; x; --RT + - [ Q.�xRT( T, + �'"'; x; In ( -b;b)] l ( '"' � x·I b·I - RT ) I b = '"' ) x;) '"' � x (T, [G + � X; ( b ] -RT ; b; ; b;RT a

b;

_

a;

X ) t

1

(4.4.8)

q1

and

X � ;

1

2

]

a;

(4.4.9)

In -

q1

This mixing rule model has been tested for five binary and three ternary systems by the authors and shown to be approximately equivalent in accuracy to the MHV I model . Therefore, we do not consider it further. Boukouvalas et al . ( 1 994) proposed a new mixing rule by forming the following linear combination of the HVO and Michelsen models (LCVM)

bRT ( -. + --- ) -RTQ.�x + --- �; x; In ( -b;b ) + '"'�; x; -h;RT a

--

=

A

C

A

1

q1

A '"'

1

q1

a;

(4.4. 1 0)

65

M o d e l i ng Vapor-Li q u i d E q u i l i bria

This is a hybrid model and lacks a firm theoretical basis because it combines Q�x of the HVO model, which is evaluated at infinite pressure, and Q�x of the Michelsen model , which is evaluated at zero pressure. Nevertheless the authors have shown that this model can be used to obtain successful correlation and prediction of the VLE of various nonideal systems. As we will discuss in Section 5 . 1 , part of this success is due to a cancellation of errors . This model is also included in the computer programs on the disk accompanying this monograph and is tested below in thi s section for VLE correlations and predictions. The EOS-Gex models that are based on the zero pressure limit are mathematically approximate because of the lack of liquid density roots of the EOS at zero pressure and some temperatures. If mathematical rigor in establishing the tie between excess free­ energy models and EOS is to be sacrificed, successful approximate models can also be developed at the limit of infinite pressure. One such model was introduced by Orbey and Sandler ( 1 995c) and is also tested in this monograph. In this model it is assumed that there is a universal linear algebraic function that relates the liquid molar volumes to their hard core volumes, such as � = where is a positive constant larger than unity. This assumption is similar to the concept of constant packing fraction introduced by Peneloux, Abdou!, and Rauzy ( 1 989) and also to the assumption inherent in the MHV 1 model discussed earlier. Using the Helmholtz free-energy function for the reasons explained earlier, and assuming that at infinite pressure, both for mixtures and for the pure components, approaches a unique value, one obtains

ub,

RT

xos A� --

u

u - u ) ( -- ) C{ub} ub ) - " x; ( -" x; ( - u bRT ; ub; ;

= -�

�

ln

ln

I

1

+

a

L:x; ; [ !?._b; ] ( bRT )c{ub} - L:x;(__!!!_ ; b;RT ) c{ub;} u u C { b} C*) cex b ) + C ( - - L:x;­ ) Ae x I: ( RT RT ; b; bRT ; b; = -

In

+

a -

(4.4. 1 1 )

Further, since approaches unity at both infinite pressure and at very low temperatures, for simplicity, as an approximation, one can take = 1 at al l conditions (so = and obtain EOS =--

-y = = -

x; ln

-

*

a

a;

(4.4. 1 2)

In this model, eqns. ( 4.4. 1 2 and 3 . 3 .8) are used to obtain the EOS parameters and h. This model is referred to as Huron-Vidal as modified by Orbey-Sandler (HVOS) model in this monograph and is also included in the programs supplied on the accompanying disk. It is an approximate model but is in agreement with the spirit of the van der Waals hard core concept, and it is algebraically very similar to several of the commonly used zero-pressure models mentioned in this section. Yet it does not a

66

M ixing Ru les that Com bine an Equation of State with an Activity Coeffi cient Model

1 80 ,.......,--,..,...,...'I'""T"""r-r-"'"'!'""1 • 1 60

1 80 r-r'"'I""T...,....,.."T"T"-.,.,r-r"T.T'""""'" •

1 80

1 40

1 40

1 20

1 20

1 00

1 00

80

80

60

80

40

40

20

20

.

o- �.-.._..a

0 0.0

0.2

0.4

0.8

0.8

1 .0

0.0

mole fraction of methane

02

0.4

0.8

0.8

1 .0

mole fraction of methane

1 80 .---,.....,..,...,"T"!"TT"T""T""I 1 80 1 40

1 40

1 20

1 20

1 00

1 00

80

80

80

eo

40

40

20

20

0

��� 0.0

02

0.4

0.6

0.8

mole fraction of methane

1 .0

0

· �.,..���

-=::::e:�==--��� 0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of methane

Figure 4.4. 1 . VLE correlations of the methane and n -pentane binary system

with various approximate EOS-Gex models. Clockwi se from top left : HVOS , MHV2, MHV 1 , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines are model predictions. The points are measured VLE data at 444 K ( & , 6 ) , 377 K (., D) and 3 1 0 K ( e , O ) from the DECHEMA Chemistry Data Series (Gmehling and Onken 1 977, Vol . 6, p. 445 ) .

contain any arbitrary constant o r function because no hypothetical liquid volumes are needed. It is simpler than those zero-pressure models, and, as wi l l be shown below, it is as successful. Some comparisons of VLE correlations with the approximate EOS-Gex models discussed in this section (MHV I , MHV2, LCVM and HVOS) follow. These results have been calculated using the computer program HV, which is described in Section 0.4 of Appendix D and included on the disk accompanying this monograph . To make the comparison a fair one, the same activity coefficient model was used in all cases. For the methane and n-pentane binary system shown in Figure 4.4. 1 , the lowest­ temperature isotherm data were fitted with two van Laar parameters in each model, and the vapor-liquid equilibria at other temperatures were predicted. The parameters are reported in Table 4.4. 1 . All the models perform almost identically, albeit with different parameters, as seen in the table, and each provides an excellent description of the system. However, one must also keep in mind that, for this binary mixture, the I PVDW model performs comparably with only one parameter, whereas all the excess free-energy-based models are multiparameter models. 67

Modeling Vapor-Liq u i d Eq u i l i bria

Table 4.4. 1. Van Laar model parameters (A 1 2 / A 2 1 ) of some binary mi xtures for various approximate EOS-Gex m i x in g rul es EOS - G ex Model

B i nary

M i xture

Methane + n -pentane

MHV I

MHV2

LCVM

HVOS

-0.432/-0.677

- 0.154/ - 1.066

- 0.268/-0. 350

-0.428/-0. 632

0.611/1.01

0.552/0.906

0.728/1.227

0. 77611 .284

1.979/1.245

2.253/1 . 3 00

2.614/1.524

2. 652/1 . 577

( 310 K)

C0 2 +

propane

(278 K)

Acetone + water (373 K)

70

70

�

.a

eli ...

::I Ul Ul CD

...

a.

60 50

Ill .a

30

::I Ul Ul CD

...

eli ...

40

...

a.

20

50

40

30

10

10 0

60

0.0

0.2

0.4

0.6

0.6

0.2

1 .0

oi ... ::I Ul Ul CD

...

a.

...

50

Ill .a

30

::I Ul Ul CD

oi ...

40

...

a.

20

1 .0

60 50

40 30

20

10

10 0

0.8

70

70

�

0.6

mole fraction of carbon dioxide

mole fraction of carbon dioxide

.c

0.4

0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of carbon dioxide

0

0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of carbon dioxide

Figure 4.4.2. VLE correlations of the carbon dioxide and propane binary system with various

approximate EOS-Gex models. Clockwise from top left: HVOS , MHV2, MHV I , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines are model predictions. The points are measured VLE data at 343 K ( 0, + ) , 3 1 0 K ( L'I. , &) and 277 K ( 0 , •) from the DECH EMA Chemi stry Data Series (Gmehling and Onken 1 977, Vo l . 6, p . 589).

68

M ixing Ru les that Com b i n e an Equation of State with an Activity Coefficient Model

The results for the carbon dioxide and propane binary system are shown in Fig­ ure 4.4.2. Again the lowest-isotherm data were used to obtain the model param­ eters, and the behavior at other temperatures is predicted. In this case again, all the approximate EOS-cex models are comparable, and there is a slight deterio­ ration of the performance in all models along the highest-temperature isotherm. Similar results were obtained with the HVO model (see Figure 4.2.2). Only the l PVDW (Figure 3 .4.2) and the WS (Figure 4.3.3) models can correlate all isotherms with a single set of parameters to the desired high accuracy. However, the 1 PVDW model accomplishes this with only one parameter. The overall conclusion we reach from these results is that the real value of EOS-cex models is in the correlation and prediction of the vapor-liquid phase behavior of highly nonideal mixtures; for mixtures of hydrocarbons and hydrocarbons with inorganic gases, the results of the simpler I PVDW model are very good, and more complicated models are not needed. The performance of the approximate EOS-cex models for the more non ideal ace­ tone and water system is shown in Figures 4.4.3 to 4.4. 5 . In these figures the results were obtained in two ways. First, each isotherm has been separately correlated with each model (solid lines) . The results indicate that all these models are very good for correlation, though each with a different set of parameters (see Table 4.4. 1 ) . Next, the VLE behavior was predicted with the parameters of the excess free-energy term (van Laar expression in this case) obtained from the DECHEMA tables at 298 K (dashed lines). Now we see that the performance of the different models in predicting the VLE behavior with the parameters from the DECHEMA tables is significantly different. At 298 K, among the approximate models considered in this section, all except the MHV l model gave very good predictions (see Figure 4.4. 3). The MHV I model , however, predicted an incorrect liquid-liquid split. At 373 K, (Figure 4.4.4), the best predictions are obtained with the MHV2 model, followed by the MHV I model. The HVOS and LCVM models performed comparably, and they were infe­ rior to MHV-type models because they underestimated the saturation pressure. Most importantly, if one compares these results with the predictions obtained with the WS model (Figure 4.3.7), it can be seen that the WS model performs better than all of the approximate models considered here. For the 523 K isotherm of this sys­ tem, shown in Figure 4.4.5, the predictions of the HVOS and the LCVM models, which are comparable, are superior to those obtained using the MHV 1 and MHV2 models (the reverse of the previous case), which overpredict the saturation pressure. However, looking back at Figure 4 . 3 . 8 , we see that the WS model again leads to pre­ dictions for this isotherm that are superior to all of the approximate models considered here. Another nonideal binary mixture investigated here is the 2-propanol and water system, the results for which are presented in Figures 4.4.6 and 4.4.7. Again, two types of calculations were carried out. First, at each temperature the model param-

69

Modeling Vapor-Li q u i d Eq u i l ibria

0.3

0.3

:;;

� :::1

.t:l

:;;

.t:l

�

0.2

., ., CD a

en en

�

0.1

0.0

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1 .0

mole fraction of acetone

0.3

0.4

0.6

0.8

1 .0

0.3

:;;

:;;

� :::1

� :II

.t:l

.t:l

en en CD

en en

a

0.2

mole fraction of acetone

!!!

Q.

0.1

0.0

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

mole fraction of acetone

1 .0

0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of acetone

Figure 4.4.3. VLE correlations of the acetone and water binary system at 298 K with various approximate EOS-Gex models. Clockwise from top left: HYOS, MHV2, MHY I , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRS V equation of state. The solid l i nes represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters obtained from the DECHEMA tables at 298 K. The points are measured VLE data at 298 K from the DECHEMA Chemistry Data Series, Gmehling and Onken 1 977, Vol . I , Pt. 1 , p. 238.

eters were separately fi tted to the experimental data (dashed lines). Second, pre­ dictions were made at the higher temperatures with the parameters of the excess free-energy model (NRTL in this case) obtained from the DECHEMA correlation at 303 K. For this system, all four models successfully correlated the data on the 353 K isotherm with parameters reported in Table 4.4.2. However, the predictive performance of the approximate models was different. Best results were obtained with the MHV2 model . The MHV 1 model overpredicted the saturation pressure, whereas the HVOS and LCVM models again behaved very similarly and under­ predicted the pressure. None of these approximate models, however, was able to predict the phase behavior as accurately as the WS model (see Figure 4 . 3 . 5 ) at 353 K. At 523 K (Figure 4.4. 7), the correlations were less accurate than those achieved at 353 K, and the predictions with all models were rather poor, especially when compared with the good predictions from the WS model (Figure 4 . 3 .4) at this temperature. 70

M ixing Rules that Combine an Equation of State with an Activity Coefficient Model

Table 4.4.2. NRTL model parameters (in cal/mol ) of 2-propanol and water binary mixture for various approx imate EOS-Gex mixing rules EOS - G cx

Temperature ( K )

MHV I

353 523

:;;

! ::J

LCVM

HVOS

- 1 92/ 1 8 1 5

- 1 26/ 1 967

7/2079

33/3 1 1 8

- 665/3020

3021772

- 94/2523

-449/363 1

4.0

4.0

3.5

3.5

.Q

3.0

.. ..

2.5

�

Model

MHV2

:;;

3.0

.. .. !s.

2.0

! ::J

.Q

Gl

2.0 1 .5

1 .5

1 .0

1 .0 0.0

0.2

0.4

0.6

0.8

0.0

1 .0

:;;

! ::J

"' "' �

D.

0.4

o.e

0.8

1 .0

mole fraction of acetone

mole fraction of acetone

.Q

0.2

4.0

4.0

3.5

3.5

:;;

! ::J

3.0

2.0

.Q

3.0

"' "' �

2.5

D.

2.0 1 .5

1 .5

1 .0

1 .0 0.0

0.2

0.4

o.e

0.8

mole fraction of acetone

1 .0

0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of acetone

Figure 4.4.4. VLE correlations of the acetone and water binary system at 373 K with various approximate EOS-Gex models. Clockwise from top left: HVOS , MHV2, MHV I , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters reported in the DECHEMA tables for 298 K. Points are VLE data at 373 K of Griswold and Wong 1 95 2 .

4.5.

General Comments on t h e Correlative and Predictive Capabi l ities of Various M ixing Rules with Cubic Eq uations of State In this and the preceding chapter we examined the correlative and semipredictive capabilities of various mixing rules. We use the term semi predictive in the sense that 71

M o d e l i n g Vapor-Li q u i d Eq u i l i bria

80

80

:a

� :::J

J:J rn rn Cll

c.

70

:a

J:J

�

60

70

60

rn rn

;.

50

50

40

40 0.0 0.1

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7

80

80

� :::J

70

;;;

� :::J

J:J

J:J

60

rn rn Cl>

rn ..

!!!

Cl.

0.2 0.3 0.4 0.5 0.6 0.7

mole fraction of acetone

mole fraction of acetone

:a

-

c.

50

70

60

50

40

40 0.0 0.1

0.2 0.3 0.4 0.5

0.6 0.7

mole fraction of acetone

0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7

mole fraction of acetone

Figure 4.4.5. VLE correlations of the acetone and water binary system at 523 K with various approximate EOS- Gex models. Clockwise from top left: HVOS, MHV2, MHV 1 , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSY equation of state. Solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters reported in the DECHEMA tables for 298 K. Points are VLE data at 373 K of Griswold and Wong 1 95 2 .

a minimum amount of experimental binary VLE data, such as those used to obtain the parameters reported in the DECHEMA tables, are used in one way or another to obtain the model parameters. In Chapter 5, we investigate completely predictive procedures that are based on group contribution methods. However, before that, we present a summary of the correlative performance we have observed with models so far considered. In general, the behavior of a model in the correlation (and extrapolation with respect to temperature) of VLE information is an indicator of its predictive potential . The mixing rules considered are the I PVDW model, the 2PVDW model , two rigorous EOS-cex models (namely, the HVO and the WS mixing rules), and four approximate EOS-cex models (the modified MHV l and MHV2 models based on the zero pressure limit, the HVOS model, and the hybrid LCVM model, which combines the MHV I model developed at zero pressure with the HVO model developed at infinite pressure). For moderately asymmetric mixtures of nonpolar components, such as the mix­ ture of methane with n-pentane, the conventional van der Waals mixing rules with no interaction parameter provide a sufficient description, and the EOS-ccx models 72

M i x i ng Rules that Combine an Equation of State with an Activity Coefficient Model

Iii

1 .1

1 .1

1 .0

1 .0

!" :::1

0.7

<>.

0.6

.Q 01 01

!!!

Iii

!" :::1

0.9

.Q

0.8

01 01 Gl

ii

0.9 0.8 0.7 0.6

0.5 0.4

0.4 0.0

0.2

0.4

0.6

0.8

0.0

1 .0

mole fraction of 2-propanol

Ia

.Q

�

01 01 Gl

ii

1 .1

1 .1

1 .0

1 .0

Iii

0.9

.Q

;�

0.8

01 01

0.7

!!!

<>.

0.6

0.8

mole fraction of 2-propanol

1 .0

0.8 0.5

0.6

0.8

0.7

0.4 0.4

0.6

0.8

0.4 0.2

0.4

0.9

0.5

0.0

0.2

mole fraction of 2-propanol

1 .0

0.0

0.2

0.4

0.6

0.8

1 .0

mole fraction of 2-propanol

Figure 4.4.6. VLE correlations of the 2-propanol and water binary system

at 353 K with various approximate EOS-Gex models. Clockwi se from top left: HVOS, MHY2, MHV I , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines represent correlations with van Laar parameters fit to experimental data, the dashed lines represent predictions with the van Laar parameters obtained from the DECHEMA tables for data at 303 K. Points are data from Wu. Hagewiesche, and Sandler 1 98 8 .

introduce an unnecessary number of parameters and complexity. Furthermore, they do not necessarily lead to better results. However, as the asymmetry and difference in chemical functionality of the components of a mixture increase, and when gases such as carbon dioxide are involved, then the l PVDW and 2PVDW methods require binary interaction parameters and lose their predictive capability. In such cases, as we discuss in Chapter 5 , the EOS-Gcx methods are useful. If such mixtures do not contain highly supercritical components, the WS model is the most accurate of the EOS- Gcx models for extrapolations in temperature and pressure. Some of the EOS-Gex mod­ els discussed have been combined with the UNIFAC group contribution methods for asymmetric systems, resulting in completely predictive VLE models for mixtures, and these will be discussed in Chapter 5 . For highly nonideal and polar mixtures o f organic compounds, the I PVDW model is inadequate, and the various forms of the 2PVDW model have limitations as a correla­ tive method and suffer from computational problems such as the dilution effect and the Michelsen-Kistenmacher syndrome mentioned in Section 3 . 5 . Such models should only be used with caution as semipredictive methods, and they have little utility a s 73

M o d e l i ng Vapor-Liq u i d Eq u i l ibria

75 70

t;l

J:J Gi

!'i

., .,

!

<>.

•

65

. � -. . .

55

Ia

60

::II "' .,

50

!

!

<>.

45

•• •

J:J

40 35 0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.1

mole fraction of 2-propanol

70

70

... . .

... . .

J:J

t;l

60

J:J

t;l

60

., .,

50

!

., .,

50

! ::II

�

0.2 0.3 0.4 0.5 0.6 0.7 0.8

mole fraction of 2-propanol

! :::1 <>.

40

40

0.0

0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

mole fraction of 2-propanol

0.0 0.1

0.2 0.3

0.4 0.5 0.8 0.7 0.8

mole fraction of 2-propanol

Figure 4.4.7. VLE correlations of the 2-propanol and water binary system at 523 K with various approximate EOS-cex models. Clockwise from top left: HVOS , MHV2, MHV 1 , and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines are predictions using van Laar parameters obtained from the DECH EMA tables at 303 K. Points are the VLE data at 523 K from Barr-David and Dodge 1 959.

completely predictive methods. On the other hand, the combined EOS-Gex models are of real value in these cases. Among such models analyzed here, only the HVO and the WS models are mathematical ly rigorous, and, as shown earlier, of the two, only the WS model has predictive capabilities. All of the approximate methods (MHV 1 , MHV2, HVOS, and LCVM) demonstrate good correlative capabilities, and some predictive capabilities, though they are generally less accurate than the WS method for extrapola­ tion . This is especially obvious when extended ranges of temperature are considered . Although the quality of predictions of the WS mixing rule remains about the same over wide temperature ranges, predictions from the other more approximate methods can deteriorate considerably. Among the approximate models considered here, not one is consistently superior to the others. Overall , the behavior of the MHV 1 and MHV2 models were similar, and the performance of the LCVM and HVOS methods were also comparable in most cases. All of the EOS- Gex models considered here can be used in a completely predictive mode if the excess free-energy term in the mixing rule is obtained from a predictive model like UNIFAC . This is considered in Chapter 5. 74

5 Com pletely Pred ictive

EOS-G ex Mod els

N the last several years mixing rules that combine predictive excess free-energy methods, such as the UNIFAC model and equations of state, have been a subject of intense interest. Several mixing-combining rules have been proposed (Dahl and Michelsen 1 990; Holderbaum and Gmehling 1 99 1 ; Orbey et al. 1 993 ; Boukouvalas et al . 1 994; Kalospiros et al. 1 995 ; Voutsas et al . 1 995 ) and used to predict the vapor­ liquid phase behavior of highly nonideal mixtures. A brief discussion of the nature of these models and their capabilities is presented in this chapter. More details on this subject can be found elsewhere (Orbey and Sandler 1 995c ) . In this monograph we will use the original UNIFAC group contribution method. The principles involved, however, are general and are applicable to any group contribution method. 5. 1 .

Completely Predictive EOS-G ex Models fo r M ixtu res of Condensable Compounds Here we consider the prediction of the VLE behavior o f mixtures, the constituents of which exist as pure liquids at the temperature and pressure at which the UN/FAC model parameters are evaluated. The reason for this restriction is that the excess free energy in most models is defined with respect to the pure liquid state. However, it is also possible to treat mixtures of noncondensable gases with condensable compounds by means of such predictive models, and this is the subject of Section 5 . 3 . The goal o f predictive phase equilibrium models is t o provide reliable and accurate predictions of the phase behavior of mixtures in the absence of experimental data. For low and moderate pressures, this has been accomplished to a considerable extent by using the group contribution activity coefficient methods, such as the UNIFAC or ASOG models, for the activity coefficient term in eqn. (2.3.8). The combination of such group contribution methods with equations of state is very attractive because it makes the EOS approach completely predictive and the group contribution method

75

Model i ng Vapor-Liq u i d Eq u i l i bria

for activity coefficients suitable for use at high pressures and temperatures at which the y -ep method is not applicable. Selecting a proper EOS mixing rule to be used with a group contribution excess free-energy model i s an interesting problem for the design engineer because in such a combination the characteristics of the two types of models are superimposed, and the overall accuracy of the combined method is a result of that choice. If the temperature and pressure are near ambient and the constituents of a mixture exist as pure liquid over the whole range of temperature and pressure, it may be most convenient to use the group contribution activity coefficient method directly in the conventional way (that is, in a y -ep model), for there would be little advantage to coupling it with an EOS . (Of course if one needs to compute other properties such as excess volumes in addition to phase equilibrium properties from the same model simultaneously, then an EOS should be used.) The real potential of the predictive EOS-G•x methods is realized when the temperature and pressure of application are considerably higher than those of the ambient, thus rendering one or more of the constituents of the mixture supercritical . In such circumstances, application of the conventional activity coeffi cient approach becomes difficult owing to lack of a pure component vapor pressure, or the reliability of extrapolating the group contribution method parameters is questionable. Most group contribution methods, such as the original UNIFAC model, were designed to be used at near room temperature, and their extrapolation capabi lity over a range of temperature is more reliable when they are used in an EOS-Gex mixing rule than in a y -ep model, as we will show in the following paragraphs . I n the remainder o f this section w e examine several EOS-G•• models using three prototype binary mixtures that form strongly nonideal solutions. For compari son, we also include the predictions of the UNIFAC model used directly in the y -ep method wherever applicable. The systems considered are the methanol and benzene (Butcher and Medani 1 968), the acetone and water (Gmehling and Onken 1 977), and the 2-propanol and water (Barr-David and Dodge 1 959) binary mixtures. Note that there are many systems with small to moderate solution nonideality for which all or most of the methods mentioned above work reasonably well . We are not concerned with such systems here because the method selection would not be a problem in such cases. Rather we consider only those systems that are more nonideal and for which the differences between the models discussed here are clearly evident. We use the fol lowing mixing rules combined with the original UNIFAC excess free­ energy model: the rigorous HVO and WS models and four approximate models (the HVOS, MHV l , MHV2, and LCVM mixing rules) . The programs used for the VLE predictions, named WSUNF and HVUNF, are provided on the diskette accompanying this monograph, and some of the computational details are described in Sections D.6 and D.7 of Appendix D, respectively. Other computational details may be found in Appendixes B and C. Before proceeding, it is necessary to stress once again a characteristic difference between the WS model and the other five model s used here. All the EOS-Gex models, 76

Completely Predictive EOS-G ex Models

0.20 0.1 9

�

.c

ai ...

;::) rn rn Cll ... Q.

0.1 8 0.1 7 0.1 6 0.1 5 0.1 4 0.1 3 0.1 2 e

0.1 1

VLE

data at 293 K

0.4

0.5

0.1 0 0.0

0.1

0.2

0.3

0.6

0.7

0.8

0.9

1 .0

mole fraction of methanol

Figure 5 . 1 . 1 . VLE prediction for the methanol and ben­

zene binary system at 293 K by various methods. Circles represent experimental data, the solid line with crosses shows the UNIFAC predictions, and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS , H VO, and MHV I models, respectively ; the dotted line is from the MHV2 model ; and the dot-dash line reflects the results of the LCVM model. (Points are VLE data from the DECHEMA Chemistry Series, Gmehling and Onken 1 977, Vol . I , Pt. 2a, p. 220; the data file name on the accompanying disk for this system is MB20.DAT. )

including the WS mixing rule, require as input the excess free-energy model (that is, the G �x term and the two activity coefficients calculated from it). (See Appendixes B and C for a summary of the relations. ) Only the WS model also requires a value for the additional parameter kij ; however, this parameter is chosen so that the G �x curve from the UNIFAC model is matched as closely as possible by A�'bs calculated from the EOS at the low pressure (and temperature) at which the UNIFAC parameters were obtained (see Section 4.3). Thus, the parameter kij does not contain any further information than that already included in the G �x term from the UNIFAC model ; indeed, it merely provides the additional flexibility to better match the excess free-energy curve from the UNIFAC model with that from EOS(WS)-Gex (UNIFAC) model. As we discuss later, this additional flexibility, unique to the WS model, is useful in some circumstances. The first system we consider is the methanol and benzene binary system . The results of predictions at 293 K are shown in Figure 5 . 1 . 1 . (In the case of the WS mixing rule, k;; was set equal to 0.2808 using the procedure described above. ) Although none of the models is highly accurate, there are considerable differences among the performance of the various methods. The original UNIFAC model used in the y -¢ approach with an ideal vapor phase predicts the VLE behavior reasonably ; however, it is not the best 77

M o d e l i ng Vapor-Li q u i d Eq u i l i bria

0.6

0.5

0.4

1ct -.. >< "'

(!)

0.3

0.2

0.1

0.0 0.0 0.1

0.2

0.3

0.4 0.5 0.6 0.7

0.8

0.9 1 .0

mole fraction of methanol

Figure 5 . 1 .2. Excess Gibbs free-energy predictions

for the methanol and benzene binary system at 293 K. The circles represent the excess Gibbs energy calcu­ lated using experimental data (see text), the line - · · · ­ reflects the UNIFAC predictions, and the smooth solid line denotes the results of the WS model . The large, medium, and short dashed lines are from the HVOS , HVO, and MHV I models, respectively; the dotted line is from the MHV2 model ; and the dot-dash line repre­ sents the results of the LCVM model.

model. The WS model that uses the UNIFAC model in the EOS- cex is more accurate than the original UNIFAC model for this mixture. All the other EOS-cex models show deviations from the experimental behavior to different extents, and the MHV I model even shows a false l iquid-liquid split. Some insight into the behavior of various EOS-cex models can be obtained if one examines the excess free-energy curves derived from these models and compares them with the excess Gibbs free energy computed from the measured data (we used the measured P-x-y data and the ideal gas assumption to obtain activity coefficients and hence the "experimental" excess Gibbs free-energy information through eqns. 2.3 .6, 8 , and 1 0) . The results are shown in Figure 5 . 1 .2. In the figure the points represent the excess Gibbs free energy computed from experimental data, and the lines are the model predictions. None of the models can represent the experimental excess Gibbs free-energy behavior exactly, and consequently their VLE predictions deviate from the experimental behavior. Of course there is a direct correlation between the capability of a model in representing the experimental excess Gibbs free energy and its prediction of VLE behavior. Models that overpredict the excess free energy also overpredict the equilibrium total pressure and conversely for models that underpredict the cex . 78

Completely Pred ictive EOS-G ex Models

30 28 26 ....

as .c

ai .... :::1 Ill Ill Q)

....

Q.

24 22 20 18 16 14 e VLE data at

12

453 K

10 0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0

mole fraction of methanol

Figure 5.1 .3. VLE prediction for the methanol and benzene binary system at 453 K by various methods. The circles denote experimental data, the solid line with crosses represents the UNIFAC predictions, and the smooth solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS , HVO, and MHV I model s, respec­ tively ; the dotted line is from the MHV2 model; and the dot-dash line denotes the results of the LCVM model. (Points ( 0 , e) are VLE data from B utcher and Medani 1 968; the data file name on the accompanying disk for this system is MB I 80.DAT.)

In Figure 5 . 1 .3 the prediction of the VLE behavior of the various models for the methanol and benzene binary system at 453 K is shown. The direct use of the UNIFAC activity coefficient model in the y -
79

Modeling Vapor-Liq u i d Eq u i l ibria

e

0.3

VLE data at 298 K

0.2

0.1

0.0

L..... .. .-... .J.... � ...J -'-��...J..... ._ �...I. ... '---'-�.L... ... _J

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1 .0

mole fraction of acetone

Figure 5 . 1 .4. VLE predictions for the acetone and wa­ ter binary system at 298 K by various methods. Solid line with crosses represents the UNIFAC predictions, and the smooth solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS , HVO, and MHV I models, respec­ tively; the dotted line is from the MHV2 model ; and the dot-dash line denotes the results of the LCVM model . (The points ( 0 , • ) are VLE data for this system at 298 K from Griswold and Wong 1 95 2 ; the data file for this system on the accompanying disk is AW25.DAT. )

values calculated from measured VLE data is shown in Figure 5 . 1 .5 . Among the EOS-Gex models, the WS model agrees most closely with the UNIFAC curve, as a result of the matching procedure described earlier. (That there is scatter in the experimental values in the dilute acetone region indicates some inaccuracy in the measurements . ) A comparison o f the V L E predictions for the acetone and water system a t 4 7 3 K using the 298 K parameters is revealing (Figure 5 . 1 .6) The most accurate results are obtained with the WS mixing rule. The extrapolations using the HVOS and LCVM models are reasonably good, whereas the MHV2 model, which produced results similar to those of the HVOS and LCVM models at 298 K, now shows the largest overprediction of the saturation pressure. The MHV 1 model also overpredicts the saturation pressure, whereas the HVO model fails in representing the azeotrope and by underpredicting saturation pressure over the whole concentration range. As before, the LCVM model is a combination of the worst two models for extrapolating VLE information, but because of a cancellation of errors is better than both. In this case the direct use of the UNIFAC activity coefficient model gave poor results, indicating that the extrapolation of VLE predictions in the y -¢ formalism is less reliable than using the UNIFAC model in the EOS-Gex formalism. 80

0.5

0.4

.... cc: -.. )( "' Q)

0.3

-------

0.2

-

:...... ...... .....

'

'

'

0.1

'

'

\

\

\

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

\

\

' 1 .0

mole fraction of acetone

Figure 5.1.5. Excess Gibbs free-energy predictions

for the acetone and water binary system at 298 K. The circles are calculated from experimental data (see text) , the solid line with crosses reflects the UNIFAC predictions, and the smooth solid line denotes the re­ sults of the WS model . The large, medium, and short dashed lines are from the HVOS , HVO, and MHV l models, respectively ; the dotted line is from the MHV2 model ; and the dot-dash line represents the results of the LCVM model. 35 33

iii

.J:J

Cli ....

::I (/) (/) Cl)

....

0.

31 29 27 25 23 21 19 e

17

VLE data at 473 K

15 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

Figure 5.1.6. VLE prediction for the acetone and wa­

ter binary system at 473 K by various methods . The solid line with crosses represents the UNIFAC predic­ tions, and the smooth solid line reflects the results of the WS model . The large, medium, and short dashed lines are from the HVOS , HVO, and MHV l models, respectively ; the dotted line is from the MHV2 model ; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 , • ) are the experimental data of Gri swold and Wong 1 95 2 ; the data file for this system on the accompanying disk is AW200.DAT. )

Modeling Vapor-Liquid Eq u i l ibria

0.5

0.4

1ct ...... )( Gl

(!)

0.3

0.2

0.1

0.0 0.0 0.1

0.2 0.3 0.4 0.5 0.6 0.7

0.8

0.9

1 .0

mole fraction of 2 -propanol

Figure 5.1.7. Excess Gibbs free-energy predictions for the 2-propanol and water binary system at 298 K. The circles represent the energy calculated using ex­ perimental data (see text), the solid line with crosses reflects the UNIFAC predictions, and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and M H Y I models, respectively; the dotted line is from the MHY2 model; and the dot-dash line repre­ sents the results of the LCVM model .

The last example we consider i s the 2-propanol and water system, which was selected to show how inaccuracies in the activity coefficient model used affect the performance of the EOS-Gex model. In Figure 5. I . 7, the excess free-energy predic­ tions for thi s system are shown. By setting kij of the WS model to 0.3834, we can represent the UNIFAC behavior very closely, though not exactly. All of the other EOS-Gex models deviate from the UNIFAC behavior to varying extents, especially in the midconcentration range. Note also that the excess Gibbs free-energy behavior derived from the experimental data can not be represented by any one of the model s over the whole concentration range. Examining the VLE behavior of this binary sys­ tem at 298 K shown in Figure 5 . 1 . 8 , we see that not one of the models is satisfactory. The direct use of the UNIFAC model does not provide an accurate description of this system and even gives a false liquid-liquid split in the water-rich region. An EOS-Gex model such as the WS model that can reproduce the UNIFAC excess free-energy be­ havior closely also inherits its problems, and therefore also shows an erroneous phase split. This poor low-temperature description also causes a problem when EOS-Gex models are used to predict phase behavior at elevated temperatures and pressures. In Figure 5 . 1 .9, the VLE predictions for the 2-propanol and water binary system at 523 K are shown. The MHV2 model gives the largest overprediction of pressure, 82

Completely Pre d i ctive EOS- G ex Models

0.07

a;

0.06

�

0 . 05

.c

3

ai Ill Ill

c.

0.04 •

0.03

VLE data at 298 K

�_L������_L�

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 5.1.8. VLE prediction for the 2-propanol and

water binary system at 298 K. The solid line with crosses represents the UNIFAC predictions, and the solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHV l models, respectively; the dotted line is from the MHV2 model ; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 , e) are measured data reported in the DECHEMA Chemi stry Data Series, Gmehling and Onken 1 977, Vol . 1, Pt. 1 b, p. 220; the data file for this system on the accompanying disk is 2PW2 5 . DAT. )

followed by the MHV I and WS models, whereas the HVO model underpredicts the pressure. The best results are obtained with the HVOS and LCVM models, which behave almost identically. However, because these two models underpredicted the experimental saturation pressures at 298 K, the agreement here must be viewed as fortuitous. The 2-propanol and water system is considered again now using the UNIQUAC model, which is the correlative model closest to the UNIFAC model, to examine the effect of activity coefficient model choice. B ecause the UNIQUAC model is correla­ tive, it is possible to fit the parameters of each EOS-cex model to VLE data at 298 K. The fitted parameters are given in Table 5 . 1 . 1 . The VLE correlations at 298 K and the predictions at 523 K are shown in Figures. 5 . 1 . 1 0 and 5 . 1 . 1 1 , respectively. In this case all of the model s are able to provide a very accurate correlation of the low-pressure data, as shown in Figure 5 . 1 . 1 0. However, when the same parameters are used to predict VLE behavior of this system at 523 K, the performance of various models dif­ fers, as shown in Figure 5 . 1 . 1 1 . The WS model once again gives the test prediction, followed by the HVOS and LCVM models, both of which somewhat underpredict the saturation pressure. The HVO model underpredicts the pressure significantly, and 83

Modeling Vapor-Li q u i d Eq u i l i bria

Table 5.1.1. UNIQUAC model parameters (cal/mol) for the 2propanol and water system" at 298 K for various mixi n g rules

fl. u 1 2 fl. u 2 1

HVO

M HV 1

MHV2

LCVM

HVOS

wsb

646.29

745 . 5 7

706.23

666 . 5 7

724.34

8 3 7 . 65

2 1 4.79

- 240.76

- 1 2 8 . 26

- 9 3 .44

- 1 08 .02

-28.38

" U N IQUAC

2-propanol ; b

kij

=

pure component parameters are

r =

0. 1 5

0 . 92 , q

=

q

'

=

r

=

1 .4 for water.

2.78, q

=

q'

=

2.5 1

for

70

iii

.0

60

::I Ill Ul Ql

50

oi ....

c.

40 e

30

VLE data at 523 K ����--��

��

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 5 . 1 .9. VLE prediction for the 2-propanol and

water binary system at 523 K . The solid line reflects the results of the WS model . The large, medi um, and short dashed lines are from the HVOS , HVO, and MHV l models, respectively ; the dotted line is from the M H V 2 model; and the dot-dash line denotes the results of the LCVM model . (The points ( 0 , e ) are the measured VLE data of B arr-David and Dodge 1 959; the data file for thi s system on the accompanying disk is 2PW250.DAT. )

both the MHV 1 and MHV2 models overpredict the pressure, the MHV2 model being more seriously in error. From the analysis of various mixing rules for the prediction of mixtures of con­ densable compounds, we reach the following conclusions: 1 . At low pressures, for which the UNIFAC method was developed, the Helmholtz free energy and Gibbs free energy calculated from an EOS are virtually indistinguishable 84

Completely Pred i ctive EOS-G ex Models

a;

0.06

..0

� :::1

0.05

Ill Ill CD ...

Q.

0.04

e

VLE data at 298 K

0.03 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 5 . 1 . 10. VLE correlations for the 2-propanol and

water binary system at 298 K using UNIQUAC method with the EOS- Gex models . The solid line reflects the results of the WS model . The large, medium, and short dashed lines which almost coincide are from the HVOS , HVO, and MHV 1 models, respectively ; the dotted line is from the MHV2 model ; and the dot-dash line denotes the results of the LCVM model. (The points ( 0, e) are the measured VLE data reported in the DECHEMA Chem­ istry Data Series, Gmehling and Onken 1 977, Vol . 1 , Pt. 1 b, p . 220.)

and can be used interchangeably. With this in mind, there are two conditions that must be sati sfied for successful VLE predictions by such models: a. The excess free energy function obtained from the direct use of the UNIFAC model should be in agreement with the experimental excess Gibbs free energy at low pressure. b. The excess free energy computed from the EOS-Gex (UNIFAC) model at low pressure should match the excess free energy obtained from the direct use of the UNIFAC model in the y -¢ method as closely as possible. If either one of these conditions is not satisfied, the combined EOS-Gex model may fail to represent the low-pressure equilibria or lead to inaccurate extrapo­ lations at high pressures and temperatures with the parameters obtained at low pressures, or both. 2. Among the mixing rules tested here in the predictive EOS-Gex formalism, only the WS model can be made to match the excess free energy from a conventional activity coefficient model closely by varying the model parameter kij . This flexibility can also be used to incorporate infinite dilution activity coefficient information into this model, as discussed in the next section . 85

M o d e l i ng Vapor-Li q u i d Eq u i l ibria

70 .... cU .c

•

60

ai .... ::I Ill Ill CD

a.

50 e

VLE data at 523 K

40

30

....--'-""'""... '---'---'-.o... -' ---'-_._ ....I ...... _.__._��..J....... ... ...

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of 2-propanol

Figure 5.1 . 1 1 . VLE predictions for the 2-propanol

and water binary system at 523 K using the U NIQUAC method with the EOS-Gex models, and parameters ob­ tained at 298 K. The solid line refl ects the results of the WS model . The large, medium, and short dashed lines are from the HVOS, HVO, and MHV l models, respectively ; the dotted line i s from the MHV2 model ; and the dot -dash line denotes the results of the LCVM model. (The points ( 0 , e) are measured VLE data from B arr-Davi d and Dodge 1 959.)

5.2.

Prediction o f I nfi n ite D i l ution Activity Coefficients with the EOS-G ex App roach An extension of completely predictive EOS- c e x models for modeling mixtures of con­ densable compounds is to use them to obtain infinite dilution activity coefficients . The capability of EOS models in predicting infinite dilution activity coefficients is impor­ tant for describing dilute solution behavior. Among the EOS mixing rules discussed in this monograph, the conventional l PVDW model is not suitable for the correlation of the infinite dilution activity coefficient behavior because it is a one-parameter model, whereas there are two infinite dilution activity coefficients per pair of components to estimate. Consequently, this model can be made to fit one, but not both components. The two-parameter 2PVDW model can be made to fit both infinite dilution activity coefficients in a binary mi xture but is then not capable of also replicating the excess free energy over the whole composition range (Pi vidal et al. 1 992). The use of infinite dilution activity coefficient information in obtaining parameters in the 2PVDW model was presented earlier in Section 3 . 5 .

86

Completely Pred i ctive EOS-G ex Models

There are several recent studies analyzing the utility ofEOS-Gcx models (Kalospiros et al. 1 994; Orbey and Sandler 1 996a) for the prediction of infinite dilution activity coefficients . We present a brief analysis of these studies in the following paragraphs. Among the EOS-Gex models considered in this monograph the WS mixing rule can exactly reproduce the infinite dilution activity coefficient of one species in the other if one selects an appropriate value for its kiJ parameter, but not both infinite dilution activity coefficients. All of the other EOS-Gcx models can give only approximate estimates because they have no additional degrees of freedom once the predictive model (here UNIFAC) is specified. As an example, we consider the predictions of the infinite dilution activity coefficients of some heavy alkanes in n -hexane. The results are presented in Figure 5 . 2 . 1 . The dots are data taken from Kniaz ( 1 99 1 ), and the lines are predictions using various EOS-Gex models . As has been explained above, with the WS model a value of kiJ can be selected for each binary to represent one of the infinite dilution activity coefficients exactly, and thus the results for the WS model are not shown because they are coincident with the experimental data. All other EOS-Gex models show various degrees of deviation from the measured infinite dilution data. Of the other models, the best predictions are obtained with the HVOS model (long dashed line) . Only the MHV 1 model (short dashed line) generates positive deviations (activity coefficients larger than unity) ; all the other models, including the use of the UNIFAC model in the y -¢ formalism (solid line) , give negative deviations from ideal behav­ ior. The HVOS model slightly overpredicts the activity coefficients; the remaining

G�'

•

1 .00

0.75

0.50

0.25

0.00 8

10

12

14

16

18

20

alkane number

Figure 5.2. 1 . Infinite dilution activity coefficients of

n -alkanes in hexane. The dots denote experimental data ( Kniaz 1 99 1 ) ; the solid line represents the UNIFAC predictions; the large, medium, and short dashed lines are from HVOS , HVO, and MHV 1 models, respec­ tively; the dotted line is from MHV2 model ; and the dot-dash line reflects results from the LCVM model.

87

Model i ng Vapor- L i q u i d Eq u i l i bria

EOS-cex models underpredict the experimental activity coefficients, and the largest error comes from the MHV2 model (dotted line) . The LCVM (dot-and-dash line) model, which is a linear combination of the HVO (medium dashed line) and the MHV 1 models, gives better results than both again as a result of the cancellation of errors. 5.3.

Completely Predictive EOS-G e x Models for M ixtu res o f Condensable Compounds with Supercritical Gases In many industrially important processes, mixtures of supercritical gases with pure or mixed solvents are encountered. These are highly asymmetric systems that are usually difficult to measure with high accuracy at the desired design conditions, and which are generally above (or below) room temperature (See, for example, the compilation of Fogg and Gerard, 1 99 1 for details of the experimental accuracy in gas solubility measurements in liquids, etc.). Therefore, it is important to have reliable predictive models for the vapor-liquid phase equilibrium and other properties such as the Henry's constant, activity coefficient, and so forth, of gases in liquids for such systems. There are many empirical methods for the prediction of the dilute solution phase behavior for such systems (Catte et al. 1 993); however, the best candidates for this type of modeling are equations of state. An EOS can simultaneously provide the phase behavior, activity coefficients, the Henry ' s constants, and volumetric and calorimetric properties. The prediction of the phase behavior of mixtures of solvents with supercritical gases using an EOS is an area of ongoing research, and there are several emerging methods, such as those recently proposed by Dahl, Fredenslund, and Rasmussen ( 1 99 1 ) , Apostolou, Kalospiros, and Tassios ( 1 995), and Fischer and Gmehling ( 1 995) that are all based on the EOS-cex combination and use a predictive excess free-energy model such as UNIFAC for the G �x term. Here we use two such EOS-Gex models to demonstrate the capabilities of these models and to address several issues for future development in this area. As mentioned earlier, because the present excess free-energy models used in the EOS-Gex formalism were developed for mixtures of pure liquids, in principle the EOS­ cex approach is applicable to mixtures in which all constituents exist as pure liquids at the temperature and pressure at which temperature-independent model parameters of the excess free-energy model (here UNIFAC) have been evaluated. Using such models to describe the mixture of a supercritical gas dissolved in a liquid is an ap­ proximation that has certain consequences, as we discuss later in this section. The EOS- cex models used so far to describe gas solubility are the approximate mixing rules such as the MHV2, HVOS, or LCVM models, and we will restrict our discussions to these models. When the UNIFAC group contribution method is used for the prediction of phase equilibrium properties of a mixture, two types of input parameters are used in the prediction. One set is made up of the volume ( R;) and shape or surface area ( Q; )

88

Completely Pre d i ctive EOS- G ex Models

parameters of the groups that compose the pure constituents of the mixture, and the other set consists of two group interaction parameters per each pair of groups in the mixture. Thus, the first problem to be addressed is to choose size and shape parameters for the groups that form the gases involved. Some of the gases treated so far by the methods discussed here are light industri al gases, C02 , CO, H2 S , N2 and the like, and light paraffins from methane to butane(s). All of the present models treat these gases as individual groups, for each is relatively small, and define the volume and surface area parameters for each gas. In some cases these structural parameters are estimated using semi theoretical methods like those of Bondi ( 1 968) and Apostolou et al. ( 1 995), but their values are essentiall y arbitrary ; for example Dahl et al . ( 1 99 1 ) u se R; = 1 .7640 and Q; = 1 . 9 1 00 for oxygen, whereas Apostolou et al. ( 1 995) report 0. 8570 and 0.940 for these parameters; a similar trend is observed for other gases. Consequently, the gas parameters in Dahl ' s correlation are about twice the size of the gases of Apostolou et al. There has not yet been a thorough analysis of which values are the best. ln addition, these values must be set before the optimum values of the group interaction parameters of the UNIFAC model can be found; thus, any deficiency that exi sts in the selection of these structural parameters will affect the group interaction parameters. With the treatment of gases as individual groups, some binary (or multicomponent) gas-liquid mixtures are reduced to mixtures of only two groups. For example, the carbon dioxide and methanol mixture considered at the conclusion of this section is actually a molecular mixture because both molecules are treated as groups by the UNIFAC approach. Similarly, mixtures of carbon dioxide with benzene or with paraffinic hydrocarbon liquids contain only two groups. The results for such systems are remarkably successful, as will be discussed in this section. The description of mixtures with more than two groups is possible for some of the present models, and the results look promising (Apostolou et al. 1 995). Here we consider only two of the approximate models discussed in this monograph coupled with the PRSV EOS to estimate the VLE of gas-solvent mixtures. These are the HVOS (Orbey and S andler 1 995c) and LCVM models (Boukouvalas et al . 1 994 ) , each with parameters for gases reported b y Apostolou e t a l . ( 1 995 ) i n both cases. To test the capabilities of these models, we used the original UNIFAC model with two temperature-independent parameters per pair of groups (instead of four in the modified UNIFAC models) for simplicity. We optimized the binary group interaction parameters of the UNIFAC model in the EOS formalism at a selected temperature for various mixtures and then estimated the vapor-liquid phase behavior at other temperatures. We first investigated the behavior of mixtures of the normal paraffinic solvents pentane, heptane, and decane with gaseous methane. These mixtures consist of two main UNIFAC groups, methane and the main methyl group CH2 ; thus, there are only two binary interaction parameters to evaluate. We used the VLE data for the 377 K isotherm of the methane and n -heptane mixture to obtain these parameters for both the HVOS and LCVM models; the parameter values are reported in Table 5 . 3 . 1 . We then estimated the VLE at all other temperatures of the three mixtures. The results 89

Modeling Vapor-Liquid Eq u i l i bria

Table 5.3.1. UNIFAC interaction parameters

(u 1 2 /u 2 1 ) for various groups used in VLE cal­ culations with the LCVM and HVOS EOS-Gex models Group Pair

HVOS Model

CH4 + CH 2 C02 + CH30H C02 + -CH=

25 1 . 3 9 1 1 00.0 I

L C VM M o del

- 1 00.8 1 / 1 09 . 09

- 32 . 2 8/97 . 1 7 209 .64/94 .94

1 40.96/- 1 1 . 1 6

30.62/98.62

250

iii

.c

•

200

e

VLE data at 377 K VLE data at 477 K

1 50

ai

...

::I Ill Ill Q)

...

c.

1 00

50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

5.3.1. VLE predictions for the methane and heptane binary system using the H VOS (solid lines) and LCYM (dashed lines) models. See text for de­ tails. (Measured data ( 0 , e and D , •) reported in the DECHEMA Chemistry Data Series, Knapp et al . 1 982, pp. 4 7 1 -4 72. The data files for this system on the accompanying disk are C 1 C7377 .DAT and C 1 C7477.DAT. ) mole fraction of n-heptane

Figure

are presented in Figures 5 . 3 . 1 t o 5 .3. 3. I n these figures the solid lines result from the HVOS model, and the dashed lines are from the LCVM model. The performance of the two models is essentially comparable, and both are very good considering the change in size of the paraffinic hydrocarbon and the ranges of temperature and pressure involved. Undoubtedly even better results would be obtained if all the data were used in obtaining the binary parameters instead of only the data for one binary mixture at one temperature, as we have done here. Another mixture containing only two UNIFAC groups i s the carbon dioxide and benzene binary system. For this system we fitted the binary group parameters of carbon dioxide with the arvmatic - CH = group at 3 1 3 K and predicted the behavior at 393 and 273 K. These results are presented in Figure 5 . 3 .4. The results are very good for 90

Com p l ete ly Predictive EOS-G ex Models

1 80 e

1 60

•

1 40

/ij

..c

::; en

ai

c.

en Q)

•

VLE data at 31 0 K VLE data at 41 0 K

1 20 1 00 80 60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of n-pentane

Figure 5.3.2. VLE predictions for the methane and n-pentane binary system using the HVOS (solid lines) and LCVM (dashed lines) models . See text for de­ tails. (The points (0, e for 3 1 0 K, and D, • for 4 1 0 K ) are from the DECHEMA Chemistry Data Series, Knapp et al . 1 982, p. 445 . The data fi les for thi s sys­ tem on the accompanying disk are C 1 C53 1 O.DAT and C I C54 1 0.DAT. )

e

•

/ij

VLE data at 377 K VLE data at 542 K

300

..c

� :J

c.

200

en en Q)

· =

\

1 00

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of n-decane

Figure 5.3.3. VLE predictions for the methane and n-decane binary system using the HVOS (solid lines) and LCVM (dashed lines) models . See text for details. (The points ( 0 , e for 377 K, and D, • at 542 K) are measured VLE data reported in DECHEMA Chemi stry Data Series Knapp et a! . 1 982, p. 486 and 489. The data fi les for this system on the accompanying disk are C 1 C 1 0377. DAT and C 1 C l 0542.DAT. )

91

M o d e l i ng Vapor-Liquid Eq u i l i bria

2 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of carbon dioxide

Figure 5.3.4. V LE predictions for the carbon diox­

ide and benzene binary system using the HVOS (solid li nes) and LCYM (dashed lines) models . See text for details. (Data are from Gupta et a! . 1 982 and Kaminishi et a! . 1 987; data file names on the accompanying disk for this system are C02BZ27 3 .DAT, C02BZ3 1 3 .DAT and C02BZ393 .DAT. )

the 273 and 3 1 3 K isotherms and, in spite of some overprediction of pressure, are still acceptable at 393 K. The last binary system we considered was the carbon dioxide and methanol binary mixture, the results for which are presented in Figure 5 . 3 . 5 . In this case the model could be considered a molecular one because both carbon dioxide and methanol are treated as groups; nevertheless, the mixture represents a stringent test in as much as most EOS models predict false liquid-liquid splits at lower temperatures for this mixture even when correlation rather than prediction is used (Schwartzentruber et al. 1 986; Orbey and S andler 1 995b ) . In this case we fitted the group parameters for the carbon dioxide and methanol system to data at 394 K and then predicted the behavior at 273 and 477 K. The results at 394 and 273 K are reasonably good except in the vicinity of the critical point for the mixture at 394 K, and they deteriorate noticeably around the critical point at 477 K. One noteworthy finding, however, is that the typical false liquid-liquid split that has been observed with several EOS-rnixing rule combinations at the 273 K isotherm is not found with the models considered here. Moreover, some of the previous EOS models were very sensitive to the parameter values (that is, with some sets of parameters they gave false phase splits, whereas with other sets they did not). The models used here were much less sensitive because, even with group parameters other than those that give the most accurate VLE predictions, false liquid-liquid splits were still avoided. Overall, we consider the approximate group contribution-EOS-based models con­ sidered here to be promising for the prediction of gas-liquid phase equilibrium. Future 92

Completely Predictive EOS-G ex Models

18 16 14 al 11..

12

::!E ai

10

a_

8

.....

::I

VJ VJ Q)

6 4 2 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of carbon dioxide

Figure 5.3.5. VLE predictions for the carbon diox­

ide and methanol binary system using the HYOS (solid lines) and LCVM (dashed lines) models. See text for details. (Data are from Hong and Kobayashi 1 988 and Kaminishi, Yokoyama and Takahashi 1 987. Data file names on the accompanying disk for this system are C02ME27 3 . DAT, C02ME394. DAT, and C02ME477 .DAT.)

efforts should be concentrated on selecting the best alternative among these models and obtaining the best pure component structural parameters, as well as on developing a more inclusive group interaction parameter base. It is also important to determine whether the simultaneous representation of both dilute solution (Henry ' s coefficients) and bulk solution (VLE at finite composition ranges) behavior is possible with a single set of parameters.

93

6 Ep ilogue

T rules for cubic equations of state to the calculation of VLE in mixtures. In principle,

HE main theme in this monograph has been the application of various recent mixing

these models are not restricted to VLE and are applicable to all other phase equilibrium problems. Vapor-liquid phase equilibrium has been considered here because only VLE has thus far been studied in detail with these mixing rules. Indeed, the significant improvement obtained with these models in such a mature area as VLE correlation is an encouraging indicator for their extension to other types of phase equilibrium. We have tried to present the models currently used in systematic, critical, but unprej udiced manner. Consequently, we have not declared any one of these models to be better than all others for every mixture. Indeed, several EOS-Gcx models perform quite well in some circumstances. Further, it can be difficult to determine in those cases in which the EOS-Gex description is not accurate whether the problem is with the mixing rule, the EOS , or the excess free-energy model used. More importantly, we hope that by providing programs for all the models presented, the reader will be encouraged to examine some or all of the models to determine which is best for the system he or she is studying. In closing, we would like to present our thoughts about future areas for study and possible application for these EOS-G ex mixing rules. However, before proceeding, it might be worthwhile to comment on the argument that cubic equations of state are too simple to be useful for phase equilibrium calculations more complex than basic vapor-liquid equilibria. Van Konynenburg and Scott ( 1 980) have shown in a comprehensive fashion that the van der Waals cubic EOS, combined with the simple van der Waals one-fluid mixing rule, is capable of describing a wide range of complex phase behavior, albeit qualitatively. Indeed, the use of simple models to represent complex behavior in thermodynamics is not new ; for example many simple, two­ parameter liquid solution models are capable of describing complex liquid solution behavior, such as liquid-liquid phase separation. Cubic equations of state, in spite of their simplicity, have been shown to be very versatile thermodynamic models. They are capable of representing the continuity

94

E p i l ogue

between the fluid phases and consequently can be used to represent liquid-liquid equilibrium (LLE) and vapor-liquid-liquid equilibrium (VLLE), as well as VLE. Many such examples have appeared in the literature. See, for example, the application of PRSV EOS to the LLE of ternary mixtures using various mixing rules considered here (Ohta 1 989; Wong and Sandler 1 992). We consider the areas discussed in the sections below in no particular order to be promising for future investigation with the models considered in this monograph. These choices are subj ective, and in some cases even speculative. However, by men­ tioning them here we hope to stimulate research into these challenging areas of ther­ modynamic modeling. In some of these areas significant progress has been made, and in these cases we cite some of the recent work. 6. 1 .

Systematic I nvestigation of EOS and M ixing Ru le Combi nations for the Thermodynam ic Model i ng of M ixtu re Behavior at H igh D i l ution The evaluation of Henry 's constants of gases in pure and mixed solvents for gas sol­ ubility calculations and of the infinite dilution activity coefficients of solutes in pure and mixed solvents are in this category. Some of the models considered here have already been used in this problem area (see Section 5 . 2 for examples) . It is useful to point out that similar models, such as the LCVM and HVOS models, which perform comparably for VLE calculations in the finite concentration range, give significantly different predictions for dilute solution behavior. Such a case is shown in Figure 5 . 2. 1 . A thorough explanation and comprehension of these differences is essential for the development of predictive EOS models that can be applied with a single set of pa­ rameters over the whole concentration range.

6.2.

Simu ltan eous Correlation and Prediction of VLE and Other M ixtu re Properties such as Enthal py, Entropy, H eat Capacity, etc. The accurate representation of phase behavior by any model requires an accurate correlation of the excess Gibbs free energy of the solution, whereas the representation of calorimetric properties of solutions requires an accurate correlation of the excess enthalpy. For most liquid mixtures, the composition dependence of enthalpy is a more complicated function than that of the Gibbs free energy. See, for example, Figure 6.2. 1 in which the excess Gibbs free energy and excess enthalpy of the acetone and water mixture at ambient temperature obtained from measured data are shown. The EOS-Gex models considered in this monograph make use of analytic expressions that were developed solely for the representation of excess Gibbs (or Helmholtz) free energy of liquid mixtures and have not been very successful for the simultaneous representation of the excess enthalpy of solutions with parameters that have been tuned 95

Modeling Vapor-Li q u i d Eq u i l i bria

0.5

0.4

.....

0.3

ct

--

X G>


0.2

!

I

0.1

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

300

\

200 �

0 E --

1 00 0

3

-1 00

a;

-200

>a.

.r::

"E

-300

CD

Ill Ill CD 0 X CD

-400

\

-500 -600 -700 - 800

\ \_

L...-'... �l...-l..-.... ..o. ._j -'-���..._._._ . �......

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of acetone

Figure 6.2. 1 . Excess Gibbs free energy and excess en­ thalpy of the acetone and water binary mixture at 293 K. The excess Gibbs free energy was calculated from VLE data as described in Section 5 . 1 . The excess enthalpy data are as reported in the DECHEMA Chemistry Data Series Heat of Mixing Collection, Christiansen et a! . 1 984, Vol . 1 , Pt. 1 b, pp. 1 48-9.

to the Gibbs free energy. See, for example, the representation of excess enthalpy of the benzene and cyclohexane binary mixture at 393 K (Figure 6.2.2) with temperature­ independent model parameters fitted to room temperature VLE and excess enthalpy data (Orbey and Sandler 1 996b ). In this case all models fit the excess enthalpy at 293 K within experimental accuracy ; however, the EOS-Gex models using the van Laar excess free-energy expression predicted an incorrect trend for the excess enthalpy at 96

E p i l ogue

1 400 1 200

0 E

3

>a.

(ij

.r::

'E

1 000 800 600

Q)

II) II)

Q) 0 X Q)

400 200

• excess enthalpy data at 293 K

.a. excess enthalpy data at 393 K

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 .0

mole fraction of benzene

Figure 6.2.2. Excess enthalpy for the benzene and

cyclohexane system at 293 K (dots) and at 393 K (tri­ angles). The lines denote correlations at 293 K and predictions at 393 K using various models . The solid line reflects predictions using the 2PVDW model, the dotted line represents the predictions using the van Laar activity coefficient model, the short dashed lines signify predictions using the HYOS model, and the long dashed line denotes predictions made with the WS model. Data are from the DECHEMA Chem­ istry Series (Gmehling and Onken 1 977, Vol 3, Pt. 2, p. 992) . .

393 K as did the van Laar activity coefficient model . Only the 2PVDW model , which is not an EOS-Gex model, predicted a qualitatively correct trend. Consequently, the development of better liquid excess property models that can simultaneously account for enthalpy and phase equilibrium behavior is needed, not on ly for general use but also for use with the EOS-Gex mixing rules. Indeed, a general comment that can be made is that in some cases, such as this one, the EOS-Gex model inherits the shortcomings of the excess free-energy model that it includes. Consequently, it is the inadequacy of the underlying cex model that is at fault when one tries to obtain the simultaneous representation of both the excess free energy and the excess enthalpy. 6.3.

Representation o f Polymer-Solvent and Polymer-Superc ritical F l u i d VLE and LLE with the EOS Models Polymeric materials, both as end products and intermediates, are an ever-increasing segment of the chemical industry. Representation of polymer mixtures by equations of state especially developed for this task is a fairly mature area (see the review of 97

Modeling Vapor-Liquid Eq u i l i bria

Sanchez and Panayiotou in Sandler 1 994, Chap. 3 ) . Recently the phase equilibria of mixtures of polymers in organic liquid solvents and in supercritical fl uids have became very important (Folie and Radosz 1 995 ) . The EOS -cex models considered here have the potential of describing such mixtures, and some work has been done on EOs-c•x models that are applicable to polymer-solvent systems (Harismaidis et al. 1 994; Kontogeorgis et al. 1 994a,b ; Orbey and Sandler 1 994; Kalaspiros and Tassios 1 995 ; Xiong and Kiran 1 995) . However, the results so far indicate that there is a need for more work, especially in developing accurate predictive, rather than correlative, models. One point to stress is that the results of these early studies seem to indicate that the EOS parameters used for the pure polymers are not very critical to the success of these models, but how the solvent is described appears to be more important to the final results. However, this needs to be investigated further. 6.4.

S i m u ltaneous Representation of Chem ical Reaction and Phase Eq u i l ibrium and the Evaluation of Phase Enve lopes of Reactive M ixtu res In some cases of industrial interest, chemical reaction may occur together with phase equilibrium. Reactive distillation is a good example. In such cases it is important to be able to predict the phase envelope as a function of temperature and pressure so that the design engineer will know whether a second liquid phase will form during the progress of the reaction, which may affect the reaction kinetics and other design factors (Wu et al. 1 99 1 a,b ). It has already been shown that a cubic EOS with the simple van der Waals mixing rules can be used with acceptable accuracy in predicting the phase envelope for many cases in which only phase equilibria of simple mixtures are encountered. See, for example, the recent review of Sad u s ( 1 994 ) . Reactive systems usually consist of mixtures of molecules with very different functional groups, which consequently are very nonideal, and the types of mixing rules discussed here may be necessary for their representation. A systematic investigation of recent mixing rules with cubic equations of state in representing the phase behavior of such reactive mixtures would be very u seful.

6.5.

Co rrelation o f Phase Eq u i l i bri um for M ixtu res that Form M i c rostructu red M icellar Sol utions Liquid-liquid emulsions and other microstructured fluids have been the subj ect of much academic and industrial interest, for they offer a new area for scientific research, and their behavior influences many engineering technologies ; indeed they represent the backbone of several emerging chemical and biochemical processes. Colloids are in a transition domain between macroscopic and microscopic regimes of matter and

98

E p i l ogue

are not well understood. One can classify the types of phase behavior observed in such liquid-liquid emulsions into two general categories, depending on the scale involved. The first type is conventional, macroscopic phase equilibrium. The second type is the formation of microstructures in some (and sometimes in each) of the separated phases. Many important characteristics of emulsions are dictated by their microstructured phases. However, it is not possible to investigate the microstructure of such systems without an understanding of the macroscopic phase behavior. A knowledge of the macroscopic phase behavior is a prerequisite for identifying the phase boundaries in systems within which the microstructures are formed. Only a few recent attempts have been made to describe the macroscopic phase behavior of such systems quantitatively ( Kahlweit et al. 1 98 8 ; S assen et al. 1 992; Kao et al . 1 993 ; Knudsen, Stenby, and Andersen 1 994 ), but all have been limited in their scope and success. Kao et al. used the Peng-Robinson EOS and a phenomenological multiparameter mixing rule to de­ scribe ternary phase behavior of the water, carbon dioxide and C4E 1 (2-butoxyethanol) system . For the C4E 1 and water system, Kao et al . were able to correlate isothermal VLE data successfully with two binary parameters, but those parameters could not be used to represent the LLE between these two species at higher pressures. The closed­ loop LLE exhibited by these two species at higher pressures could be correlated with their model but only with a set of two binary interaction parameters that were a func­ tion of temperature. This study supported the findings of van Pelt, Peters, and de Swaan Arons ( 1 99 1 ), which showed that, when coupled with nonquadratic mixing rules, equations of state can represent closed-loop liquid-liquid mi scibi lity gaps char­ acteristic of so-called type VI systems that cannot be represented by the conventional van der Waals mixing rules. However, it was also shown that the parameters of such phenomenological mixing rules provide little or no extrapolation capability. Knudsen et al. ( 1 994) studied surfactant systems using the MHV2 model con­ sidered in this monograph with a modified Soave-Redlich-Kwong equation of state. They investigated the same surfactant and water binary system previously investigated by Kao et al. and found that a reasonably successful correlation could be obtained with the MHV2 equation coupled with the UNIQUAC model by fitting two strongly temperature-dependent parameters per binary pair to the data. Even though not very successful, the results of that work, and that of Kao et al ., are somewhat encouraging. First, they show that equations of state can correlate the phase behavior of the binary pairs in a ternary micellar system. Second, they were able to predict, albeit only qualitatively, ternary phase behavior on the basis of these correlations, which is an important goal in modeling such systems. However, it was clear from the results that there is still much to be done to develop accurate extrapolations with respect to temperature and pressure with these EOS models for such systems . The challenge of quantitatively predicting ternary phase behavior using only data on binary systems remains for these systems, and indeed more generally. It should be noted that even when using activity coefficient models directly, temper­ ature dependent parameters are needed. There is no excess Gibbs free energy model 99

Modeling Vapor-Li q u i d Eq u i l i bria

with temperature-independent parameters that can describe such behavior. The devel­ opment of such a model would be an important contribution to applied engineering thermodynamics. 6.6.

Systematic I nvestigation of LLE and VLLE for Nonelectrolyte M ixtu res with an EOS A thorough investigation of the use of cubic equations of state in the EOS-Gex for­ malism for the description of LLE and VLLE needs to be undertaken . As indicated above, the prediction of phase transitions from VLE to VLLE and to LLE is smooth with an EOS . Thi s is a significant advantage in computer simulations because no a priori knowledge of the number of phases present may be available, and, consequently, the applicability of a single model to all possible situations would be an important advantage. Among the systems that should be considered in such an analysis are fluid mixtures near the solvent critical point. S upercritical extraction, the production of liquefied natural gas or gas condensates, and enhanced recovery of hydrocarbon resources with carbon dioxide and methane are a few examples of such systems . It is in the vicinity of their critical points that supercritical solvents have their largest extractive powers, and such mixtures can exhibit transitions from VLE to LLE and VLLE. Although experimental data for the analysis of such systems are available (see, for example, recent works of Patton and Luks 1 995 and Peters et al. 1 995), most EOS models are not sati sfactory for quantitative description of such systems. Computational tools are also needed for the analysis of azeotropic separations (see, for example, the work of Bossen, Jorgensen, and Gani 1 993 and Coats, Mullins, and Thies 1 99 1 ) . The recent review of Dohrn and Brunner ( 1 995) contains much information on additional systems that can be studied with the models presented here. The computational aspects of EOS modeling of systems that exhibit LLE and VLLE behavior are also somewhat more complicated than for VLE; the works of Michelsen and his colleagues (Heidemann and Michelsen 1 995; Michelsen 1 986, 1 987, 1 993, 1 994; Mollerup and Michelsen 1 992) provide an excellent discussion of algorithms that can be used.

1 00

A P P E N D IX A

Bibl iography of G eneral Thermod ynamics and Phase Equilibria References

VARIETY of good reference sources are available for those who wish to learn

Amore about phase equilibrium calculations and the recent advances in the subject.

A partial list of source books is given below. Some of them are recent and provide up-to-date developments, and some dated sources introduce the basic principles in a coherent and easy-to-understand fashion.

Malanowski, S . , and Anderko, A. Modeling Phase Equilibria. J. Wiley and Sons, New York, 1 992. Null, R. H . Phase Equilibrium in Process Design. Wi ley-Interscience, New York, 1 970. Prausnitz, J. M., Lichtenthaler, R. N . , and de Azevedo, E. G. Molecular Thermody­ namics of Fluid Phase Equilibria . 2d ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1 986. Reid, R. C., Prausnitz, J. M., and Poling, B. E. The Properties of Gases and Liquids. 4th ed. McGraw-Hill, New York, 1 987. Sandler, S . l. Models for Thermodynamic and Phase Equilibria Calculations. Marcel­ Dekker, New York, 1 994. Sandler, S. I. Chemical and Engineering Thermodynamics, 2d ed. J . Wiley and Sons, New York, 1 98 8 . Van Ness, H. C . Classical Thermodynamics of Non-Electrolyte Solutions. Pergamon Press, Oxford, 1 964. Walas, S. M. Phase Equilibria in Chemical Engineering. Butterworth-Heinemann, B oston, 1 98 5 .

101

APPEN DIX B

Summary of the A l gebraic D etail s for the Various Mixing Rul es and Computational Method s U sing These Mixing Rul es

N this appendix we present the algebraic expressions for the EOS parameters a and b and for the fugacity coefficient expressions for mixtures for each of the various mixing rules considered in this monograph. These are the basic relations needed to do VLE calculation s . Comments concerning the activity coefficient models and programming details are also included . Although some of the material in this appendix has been presented in the preceding chapters, it is repeated here for the purposes of clarity and completeness. This appendix is not intended to provide exhaustive mathematical or thermodynamic details; for those the interested reader should refer to the various books and papers given in the reference section of this monograph. It should also be noted that by following the derivations provided here one can develop the expressions that can be used to modify the programs included with thi s monograph to accommodate new mixing rules that are proposed in the future. I.

Activity Coefficient Models The general constraint for VLE is

-L

-v

!; (T , P, X; ) = !; (T, P, y;)

(B . I . l )

where f; is the fugacity of species i in a homogeneous liquid or vapor mixture. We use the overbar to indicate a property of a species in a mixture, and the superscripts L and V represent the liquid and vapor phases, respectively. Also T and P are absolute temperature and pressure and x and y are mole fractions (of species i ) in the liquid and vapor, respectively. In the y -¢ method, the equilibrium constraint in eqn. ( B . I . l ) is rewritten as

-

(B.I.2) X; Y;(T , P, X; ) /;L (T, P) = y;
homogeneous vapor mixture, and J;L is the fugacity of pure component i as a liquid. 1 02

Appendix B: S u m mary of the Algebraic Details for the Various M ixing Rules

{ VL[p _ p vap (T)] }

The pure component fugacity can be computed from

J/(T, P) = p vaP (T )¢; [T, p vaP ( T) ] exp

-i

(B.l.3)

RT

where ¢; i s the fugacity coefficient for the pure component, and the exponential term, written here assuming that the pure liquid molar volume, l:: �, is independent of pressure, is the Poynting correction. In general the p ure component fugacity is obtained from an EOS using the relation In

[ J; ( : P) ] =

I n ¢; (T, P )

= -1R-T fv [ RVT _ _!_N; J d V - ln Z + ( Z - 1) V=x

V

Z V

(B.I.4)

in which N; i s the number of moles of species i, is the total volume, = P 1 N; R T is the compressibility factor, and R i s the gas constant. For a pure liquid at its saturation vapor pressure, p vap , as an approximation we have that

jj (T, P) = p vaP (T)

( B .l . 5 )

provided the saturation pressure is low. If the saturation pressure is low but the liquid i s a t a pressure higher than its vapor pressure, w e need t o add the Poynting correction in eqn. (B .l.3 . ) In most cases, however, this term is usually small enough to be neglected unless the total pressure is very high. With these in mind eqn. (B.I.2) becomes

x; y; (T, P, x;)P;vap (T) - y;¢; (T, P, y;)P _

,

-

,

(B . l .6)

Furthermore, for most vapor mixtures at low pressure, ¢; is very close to unity (there are exceptions to this assumption; for example, associating gases such as hydrogen fluoride or acetic acid), and that leads to the equilibrium relation we used in this monograph to calculate the vapor-liquid phase equilibrium by the direct use of activity coefficient methods :

x; y; (T, P , x; ) P/ P ( T ) = y; P "

( B . L7)

In eqn. (B.l.7), the activity coefficient i s obtained from an excess Gibbs free-energy model, which provides an expression for the molar excess Gibbs free energy of a mixture, Q�x , u sing the rigorous thermodynamic relation (B.l . 8 ) In eqn. ( B . I . 8 ) , the activity coefficient is obtained b y multiplying the molar excess Gibbs free-energy expression with total number of moles of the mixture, N, and then differentiating the resultant total excess Gibbs free-energy term with respect to the 1 03

Model i ng Vapor-Liq u i d Eq u i l ibria

T

mole number of species i, keeping all other mole numbers and and P constant. Note that all conventional excess free-energy models are pressure independent with temperature-dependent parameters. Thus the model parameters are constant when the derivative with respect to mole number in eqn. (B .l.8) i s taken. II.

Eq uation o f State Models In the EOS approach the equilibrium constraint of eqn . (B.I. I ) is again used, except in this case the same EOS is used for both phases: -L

X;¢ ;

(T, P , X; )P

=

-v

Y;¢ ; ( T , P, y;)P

or -L

x; ¢ ;

(T,

P,

-v

x; ) = y;¢ ;

(T,

( B . II. I )

y; )

P,

Here the same expression i s used for the fugacity coefficients of species i i n both homogeneous liquid and vapor phases : P , Z; )

[J; (T,z; P ] v [RT ( - ) ] i RT

ln (f; = In

1 = -

V = oo

v

- -

ap

o N;

T. V . Nj#i

dV

ln Z

(B .ll.2)

where here Z; is used as a generic mole fraction term ; when eqn. (B . II.2) is applied to the liquid phase, x ; is substituted; for the vapor phase y; is used instead. The pressure and the compressibility factor terms appearing in this equation must be obtained using the liquid and vapor molar volumes as appropriate, as will be explained next. In the EOS approach, an equation, like the Peng-Robinson equation below, is used to obtain l:: or, equivalently, Z in eqn. (B.II.2). p =

or P

v-

_!!!___ =

a(T)

(B.II.3)

b l::( l:: + b) + b(l:: - b) N [Na ( T)] NRT ---V - ( Nb) V [ V + (Nb)] + (Nb) [ V - ( Nb)]

T

When a homogeneous liquid phase is in equilibrium with a vapor, and P are the same in both phases. In such a case when the EOS is solved at a selected T, and P, and composition, three volume roots are obtained at temperatures less than the critical temperature. For the liquid phase, the liquid composition must be used, and the smallest of the volume roots is taken as the solution. The compressibility factor Z is then calculated with that root. If the vapor composition is used, the largest root for the volume is used. 1 04

Appendix B: Sum mary of the Alge b raic Deta i l s for the Various Mixing Rules

(B .l .2 B.I .3)

The eqns. and are general and applicable to any mixing and com­ bining rule included in thi s monograph. Different mixing rules, however, result in different composition dependencies for the EOS parameters and b in eqn. and because of this, each model i s different. For the mixing-combining rules dis­ cussed in this monograph, the expressions for the mixture and b parameters and for the fugacity coefficient of species i in a one-phase mixture are given next.

(B . I . 3 ) ,

a

a

I I .A.

One- Parameter van der Waals One -Fl u i d M o del (I PVDW) In this case the mixing rules for the two parameters of the PR EOS are

a = L L Z; Zja;,;

and

j

=

b

l:z;b;

(B . I .4) (B.I .5) I

with the combining rule

a;,;

=

.,;a;a;c 1 - k ;j )

With these mixing and combining rules, the fugacity coefficient of species i in a homogeneous mixture obtained from eqn. is

-

ln ; (T, P, Z;)

=

( B . I l . 2) B) ( 2Lz;a;.i )

b; b ( Z - 1 ) - ln(Z _

A_

.i = I

_

2v'2B

a B = hP/RT, A = aP/(RT?,

.

_ b; b

In

[z

(1 + Z + (1 +

v'l)B] v'l)B (B I ) . I.6

= PY_/RT.

where and Z Note that throughout this appendix in the double index notation m;; = m ; , where m is any indexed variable. In this and all cases that follow, the compressibility factor Z is computed from the EOS .

I I . B.

Two- Parameter van der Waals One -Fl u i d M o del (2PVDW)

(B.IlI.l4.A

In this case eqns. employed in Section

to B . II . 6) are used with the exception that the binary parameter is now a composition-dependent two-parameter term (B . II.7)

and the fugacity expression then takes the form In

-

; (T, P, Z;) =

b;

b - In ( Z -

A --

2v'2B

B)

((aNajaN;a h_ N1"'

b; + 1 - -

b

) [z In

+ (1 +

v'l) B Z + ( 1 - v'l) B

]

(B . l . 8 ) l OS

Modeling Vapor-Li q u i d E q u i l i bria

with

for binary mixtures.

II.C.

Wong -San dier M ixing Ru le (WS) In the Wong-Sandier mixing rule the EOS parameters for a homogeneous liquid or vapor mixture is computed from

b =

RT L. L. Z ; Zj (b - �) RT 11

( B .II. l O )

[""z a; + _Q,_e/ (C*___ T_,z_;)] RT I

J

� ·1 b; I

with

� [( ( �) RT . b-

.

I}

=

2

bI -

!!.!.__

RT

) ( 1 - .!!J. R T .._ ) Jo + b.

-

kI } )

(B.II. l l )

·

or, alternatively,

( B . II. l 2) and

, Z; ) a = bRT [ Q�x(T C* RT

+

L. I

I -b; R T

z·

a;

]

(B .ll. l 3 )

In these equations, Q�x, the molar excess Gibbs free-energy obtained from any ex­ cess free-energy model, is a function of temperature and composition only. Even though the Wong-Sandier derivation involves the Helmholtz free-energy of the mix­ ture, this substitution is due to the assumption that Q�x (T, Z; ) A��5 ( T, P. Z ; ) . See Section 4.3 for details. The C* term i s the EOS-dependent constant, as explained in Section 4. 1 . For the PR EOS C* = [ln( J2 - l )J/J2 = -0.62323. Either of the combining rules of eqns . (B .II. l l or B .II. l 2) can be used, yielding slightly different results; again, see Section 4.3 for details.

=

1 06

Appendix B: S u m m ary of the Algebraic Deta i l s for the Various Mixing Rules

The fugacity coefficient expression for species i in a mixture for the Wong-Sandier mixing rule i s

(B.II. l 4)

( aNb ) = ( a N Q ) - Q 2 ( aN D ) aN 2 - D N aN - D ) - aN ( aNh ) RTb ( a N D ) 2_ ( aN a ) = RTD

The partial derivative terms are 1

I

N

T. N;#i

2

1

(I

I

1

aN;

+

aN;

I

I

aN;

(B . l l . 1 5 ) (B . II. 1 6)

with

Q = L L ZiZj (b - _!!_ RT ) . .

(B.II. 1 7)

D-

( B . II . l 8)

j

i

and

"'

I}

____!!!____

Q x (T, � - C*RTZ; ) + L.. z1 h; RT I

( 2. aN2 Q )

.

(h - _!!_ )

N aM aND ln y; a; --+ aN; h; RT 1 Z; ) ln y; = RT aN; =

2 L: z; .I

[aNQ�x(T,C* J

I I . D.

RT

(B.II. 1 9)

0

(B.II.20) (B .II.2 1 )

T. N , ., ,

H u ron -Vi da l (Origi nal) M ixing Ru l e ( HVO) In the Huron-Vidal mixing rule the mixture EOS parameters are given as

b = L: z;b;

( B . II . 22)

which is identical to the mixing rule for the rules, and

[

a = b L z - + --'--; c• .

I

I

a

h

Q�'(T. Z ; )

;

]

b parameter in

I PVDW and 2PVDW

(B.ll.23) 1 07

Modeling Vapor-Li q u i d Eq u i l i bria

In this case the fugacity coefficient of species i in a mixture becomes

- = bb; ( Z - (Z - B) B] - 2J2 ( b;RT + ) [ ZZ ++ (( l +- J2) J2)B 1 ) - ln

ln ¢;

1

a;

ln y;

C*

1

ln

(B.II.24)

where again the compressibility factor i s computed from the EOS .

I I . E.

Mo di fie d H u ron -Vi dal First -O r der Mixing Ru le (MHV I )

b

In this mixing rule the expression for the parameter is the same as the HVO (eqn. B . II . 22) 1 PVDW and 2PVDW mixing rules in Section TI.D, and for the parameter we have

"' a = (b R T) I 7 Z; b;RT + q; [ Gcx(RTT, + "'""7 Z; ( b;b ) ] l q1 - -b;b ( Z - ( Z - B) ( -b ) + - ( b; ) ] [ --l n + + - -; b;RT 2J2 q q1 q1 b [z + ( + J2)B J Z + J2)B 1

a;

-

Z; )

y

ln

a

(B . II.25)

where is an empirical parameter obtained by fitting pure component information. The fugacity coefficient of species i in a homogeneous solution is In ¢;

=

I ) - ln

1

x ln

I I . F.

a;

ln y;

1

1

h;

- - 1

1

(1 -

(B .ll.26)

Mo difie d Hu ron-Vi dal Secon d- O r der Mix i ng Ru le ( MHV2) In this mixing rule eqn. (B . II.22) again L ; Z ; h; ) is used for the parameter. The parameter is obtained by solving the following quadratic expression for =

b c: ajbRT (b = a q2 t: + q1 c: + [ -q, "'""7 z; t:; - q2 "'""7 Z;E;2 - Q�xRT(T, z; ) - "'""7 z; ( bb; ) ] b. t: 1q q2 2

ln

=0

(B . II. 27)

In and choosing the larger o f the two real roots for t o obtain a in terms of eqn. ( B . Il.27), and are empirical constants obtained by fitting pure component properties. See Section 4.4 and Dahl and Michelsen ( 1 990) for details. The fugacity expression for i in a homogeneous mixture is _

ln ¢;

1 08

h;

= -b ( Z -

1) - In

J2)B J aNaNc:); [zZ ++ +- J2)B ( Z - B) - 2J2 ( 1

ln

(1 (1

( B . Il.28)

Appendix B: S u m m ary of the Algebraic Deta i l s for the Various M ixing Rules

where

o Ne

q , B; + q2 (s2 + sf) + ln y; + I n ( b j b; ) + (b; /b) q , + 2q2 s

a N;

I I .G.

I

(B.11.29)

Li near Com bi nation of Hu ron-Vi dal an d Miche ls en Mo dels (LCV M) As the name implies, this model is a combination of HVO and MHV l models, and once again eqn.(B .ll.22) ( b = L ; Z; b; ) is used for the b parameter. The a parameter is obtained from a =

bRT

[ ( -. A

1 -A

+ -C q1

) Q�x( T, z; ) RT

()

1 -A b a; + -- " L.. Z ; I n - + " L.. Z; -q1 ; h; ; b; R T

]

( B . II. 30)

where A is an arbitrary parameter that has been selected to give the best results for the particular system under consideration once the excess free-energy model is chosen. Note that A = I gives the HVO model, and A = 0 gives the MHV 1 model. The fugacity coefficient of species i in a homogeneous mixture is

-

=

ln ¢;

b;

b ( Z - 1 ) - I n (Z - B )

- _../2I { � + �+� b; R T ( C* q1 ) 2

x ln

II.H.

+ ( 1 + .Ji) B Z + ( I - ..fi) B

[z

I n y; +

.!.....=._!: q1

J

[ ( !?._ ) tn

b;

b

+ ; h

-

1

]}

(B . II.3 I )

O r bey-San dle r Mo dification of the H u ron-Vi da l Mixing Ru le (HVOS) Again

�

a =

=

L ; Z; b;

bRT

IL

a

� [ Q�xR( TT, Z; ) + L; Z; In ( !?._b, )]

is used, and the

parameter relation is

Z; � + ; b, R T C

The fugacity coefficient of species i i n a mixture becomes

-

I n ¢;

=

b ( Z - 1 ) - In (Z - B)

J

(B.II.32)

b;

-

I

2

[ ../2 [z

x ln

a

b; � T

+

In y 1 + • In c .' c

+ ( I + ..fi) B Z + ( 1 - ..fi) B

]

() b b;

+

I c•

( � )J b

-

1

(B.II.33)

1 09

Modeling Vapor-Li q u i d E q u i l i bria

IH .

T h e Program m i ng Detai ls fo r t h e V L E Calcu lations The VLE calculations presented here were done using an isothermal bubble point algorithm . A flow diagram of the algorithm is presented in Figure B . l for the EOS methods. When optimizing the model parameters, we used the objective function

F

=

L I P;.exp - P;.cal l

(B .III. l )

for the minimization in a simplex formalism.

Calculate

�L ( T, P,x,) for all

components using

Calculate

t-

1;v (T,P,y,) for all

components using zv

� = y,· for all

components

Computed bubble-point pressure and vapor compositions are correct

Figure B . l .

1 10

Appendix B: S u m m ary of the Algebraic Details for the Various M ixing Rules

When the WS model i s used in the predictive mode, the excess Helmholtz free energy calculated from the EOS at each liquid composition in a given data set was matched, as closely as possible, with the excess Gibbs free energy calculated from the UNIFAC model at the same liquid compositions by adj usting the value of the kij pa­ rameter in the model. The kij parameter obtained this way was used in eqn . (B.II. l l ) . The UNIFAC model was also used t o calculate the Q�x term in eqns. (B .Il. l 0, B .Il. l 3 , B .II. l 8 , and B . II . 2 1 ) . For all other predictive EOS models, the only input require­ ment for the mixing rule was the Q.�x term (and the activity coefficients calculated from it using eqn. [B.l.8J), which was obtained from the UNIFAC model. Thus, in the predictive mode, no experimental VLE data are needed in any of the excess free-energy-based EOS models for VLE calculations. Note, however, that the pro­ grams provided on the accompanying disk require the experimental P-x-y data only if one wants to calculate the deviations from those data.

I l l

APPEN DIX C

D erivation of H el m hol tz and G ibbs Free- E n ergy D epartu re Fu nctions from the Peng-Robin son Equ ation of State at Infi nite Pressure

T

HE Helmholtz free-energy departure function (from ideal gas behavior) for the Peng-Robinson equation at a given temperature, pressure, and composition is (Wong and S andler 1992) CA -

A'G ) RT

= - n[ I

P (Y. -

b)

RT

]+

The first logarithmic term in eqn. for the PR EOS

P( Y. - b)

=1

a

ln

2./2b R T

[

Y. +

y_ + ( 1

(I

- ./2)b + ./2)b

]

(C. 1 )

(C. l ) is zero in the limit of infinite pressure because

a (T)

( ¥_ - b) (C.2) R T ( ¥_2 + 2b ¥_ - b2) RT and as P b; thus, the right-hand side in eqn. (C.2) becomes unity, and its oo , Y. logarithm is zero. In the second term in eqn. (C. 1 ), the argument of the logarithmic

�

. [y_

_

�

] [

]

expression at the infinite pressure limit becomes hm

P-"> x

+ ( 1 ./2)b ¥_ + ( I + ./2)b -

2

;b R T

=

b + ( I - ./2)b b + ( 1 + ./2)b

ln ( h - 1 )

2

=

�

(v 2

-

?

= _a_ [ln ( h - 1 )/ h] bRT

(C.3)

1 )-

=

a __ C*

bRT

(C.4)

Because the excess Helmholtz free energy is closely related to this departure term (see Wong and Sandler 1 992 for details) the same C* term appears in the excess Helmholtz free-energy term obtained from the PR EOS . If a similar analysis is done for the Gibbs free energy, the analogue to eqn. (C. I ) is ( Q. - Q.'G )

RT

1 12

=

- In

[ P(Y.R T- b) ] +

a

2./2R T

ln

[

Y. + ( I -

y_ +

]

./2)b Y. 1+ R T ( 1 + ./2)b (C.5)

p

Appendix C: Derivation of H e l m h o ltz and G i bbs Free-Energy Departu re Functions

At infinite pressure the last term is infinite; however, when the excess Gibbs free energy of the mixture is evaluated, this term cancels out between the pure component and mixture terms, but only if one assumes b = L i xi bi for mixtures. In that case, the relation between excess Helmholtz and Gibbs free energy becomes

RT

RT

(C.6)

See the work of Huron and Vidal ( 1 979) and Fischer and Gmehling ( 1 995) for further details. Note that for other cubic equations of state a similar analysis holds, but the numerical value of the C* term is different, a point that has sometimes been overlooked in literature.

I 13

APPEN DIX D

Comp u ter Programs for B i n ary Mi xtu res

HE disk that accompanies this monograph contains the programs and sample data

T files that can be used to correlate and predict vapor-liquid equilibria using various

equations of state and activity coefficient models. All programs are coded in FORTRAN using MICROSOFT FORTRAN Version 5 . 1 and are also supplied as stand­ alone executable modules (EXE files) that run on DOS or WINDOWS-based personal computers . The files are compressed and must be decompressed, preferably i nto a di­ rectory in the hard disk of your personal computer. There are four compressed files on the disk: EXEFILES .ZIP, DATFILES .ZIP, FORFILES.ZIP, and MAKFILES .ZIP. EXEFILES contains the ten executable programs, seven of which are described be­ low (the remaining three are for prediction of the VLE of multicomponent mixtures, and they are described in Appendix E). DATFILES contains all the data files (a total of fifty-six files: forty with the DAT extension ; two with the DTA extension; four with the ACT extension; two with the VDW extension; two with the HVN extension; two with the WSN extension ; and one each with the WSU, HVW, HVU , and WSW extensions [see also Appendix E] ) used in this monograph. FORFILES contains the FORTRAN subroutine programs (a total of seventy-nine), and MAKFILES contains nine MAKE files that are used to build nine multimodule FORTRAN executable programs, the subroutines for which are provided. The tenth FORTRAN executable program, VDWMIX, is a single-module program and does not need a MAK file, (see Appendix E). All that follows applies to operating in DOS or in a DOS window of a computer using any version of the WINDOWS operating system . If you are i nterested in only using stand-alone executable modules, only EXEFILES .ZIP and DATFILES .ZIP need to be decompressed. To do this, insert the accompanying disk into the floppy drive designated drive A. Next create a subdi­ rectory in your root (C : >) directory (for example, a directory called TEST). To create a subdirectory called 'test' type the following command: • At C : \>type MD TEST and press RETURN. (This results in creation of a subdirectory named TEST under the root directory C. ) Type the fol lowing commands :

1 14

Appendix D: Computer Programs for Bi nary M i xtu res

•

At C : \> type CD TEST and press RETURN. At C : \ TEST> type A: and press RETURN. • At A : \> type PKUNZIP EXEFILES C : \ TEST and press RETURN. (This results in the decompression of the ten executable files and one auxiliary file [PKZIP.EXE] into the subdirectory TEST). • At A:\> type PKUNZIP DATFILES C : \ TEST and press RETURN to decompress the data files and place them in the TEST subdirectory. (However, in this case an overwrite warning message will appear for the auxiliary file PKZIP.EXE. Type "n" to proceed. ) To decompress a l l the files the following commands are used : • At C : \> type CD TEST and press RETURN. • At C : \ TEST> type A: and press RETURN. • At A: \> type PKUNZIP * C : \ TEST and press RETURN. This results in decompression of all the files into the subdirectory TEST. In thi s case you can use FORTRAN and MAKE files with the Microsoft FORTRAN package to change or rebuild the executable modules, or both. For further details on this mode, refer to the Microsoft FORTRAN manuals. The EXE files can be run directly from the DOS prompt. To do this, the directory where the EXE files reside (for example TEST directory) is selected, and the name of the EXE file is typed at the prompt. Each program is separately described in the following sections, and a tutorial section is included to facilitate the use of each program. In these tutorials, the output that will appear on the screen is indicated in bold and in a smaller font. The information the user is to supply is shown in the normal font. •

D. I .

Program AC: VLE by D i rect Use of Activity Coefficient Models The Program AC can be used to correlate or predict VLE using activity coefficients model s directly, without an EOS , that is, using the y -ep method. There are five activ­ ity coefficient models available in this program: UNIQUAC, the Non-Random Two Liquid (NRTL), the van Laar, UNIFAC, and the Wilson models. The gas phase is assumed to be ideal in this program. The instructions that appear on the screen must be followed to execute the program. See the tutorial given later in this section. The program can be used in two ways. If no experimental T - P -x-y information is avail­ able, the user only needs to supply the temperature and saturation pressure of each compound at the temperature of interest as input. These data are entered fol l owing the commands that appear on the screen . In this mode the program will return isothermal x - y - P predictions at the temperature entered in the composition range x1 = 0 to 1 at intervals 0 . 1 , 0.2, 0.3, and so on. In the second mode, available isothermal VLE data can be correlated. The data needed are the temperature, the measured mole fractions (of species I ) in the liquid and 1 15

Model i ng Vapor-Liq u i d Eq u i l ibria

vapor phases, and the pressure. The program reads previously stored data or accepts new data entered from the keyboard. Again, the activity coefficient models require ex­ perimental pure component saturation pressures as input information. Consequently, if new data are entered from the keyboard, the first data point must be Xex p = 0. Yex p = 0, p;'ap, where P2vap i s the pure component vapor pressure of the second com­ and Pexr I , Yexp = 1 , and Pex p = P,vap, where ponent, and the last data point must be X exp vap P, is the pure component vapor pressure of the first component (see Example D. l .B in the tutorial) . On the accompanying disk, the extension ACT, such as MW25 .ACT, has been used for the sample data files employed with this program. The UNIFAC model is predictive; hence, its use leads to the direct prediction of VLE without any parameter optimization. In this option, however, the user must supply information as to the groups constituting the molecules required in the UNIFAC model. For the other activity coefficient models in the correlative mode, the program uses a simplex optimization routine to optimize the activity coefficient model parameters, thus minimizing the absolute error between the experimental and calculated pressures. Owing to the nature of the simplex optimization routine, a local minimum, rather than a global minimum, may be obtained. Therefore, the fi nal results of the optimization may depend on the the initi al guess for the parameters . Also, an inappropriate choice of values for the initial parameters may result in a divergence, in which case calculations with new initial guesses should be attempted. The results from the program AC can be sent to a printer, to a disk file, or both . To make thi s choice, the commands that appear on the screen upon the completion of calculations should be followed. Please see the following tutorial for further de­ tails. =

=

Tutorial on the Use of AC . EXE Exam ple 0, I A Fitting Activity Coefficient Model Parameters to VLE Data • •

Change to the directory containing AC . EXE (e.g., A>, or C>, etc . ) . Start the program by typing AC at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN) . The following appears :

AC : VLE CALCULATIONS WITH VARIOUS ACTIVITY COEFFICIENT MODELS YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA ARE AVAILABLE TO COMPARE RESULTS WITH

YOU MUST SUPPLY THE TEMPERATURE , COMPOUND AT THAT TEMPERATURE .

1 16

AND SATURATI ON PRESSURE OF EACH

Appendix D: Computer Program s for Bi nary M i xtures

IN THI S MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED ,

IN THE COMPOS ITION RANGE X l = O TO 1

AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I SOTHERMAL x - y - P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

TO FIT THE MODEL

PARAMETERS TO THE VLE DATA . ALTERNATIVELY ,

YOU CAN CALCULATE

VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA .

•

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) 1 OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (With this entry it will be possible either to use a previously stored data file or enter data avai lable and store it in a file . ) A t " DO YOU WANT T O USE AN EXISTING DATA FILE ( Y/ N ) ? " type Y (or y ) and press RETURN. At " INPUT EXISTING DATA FILE NAME ( f or examp l e , a : mw2 5 . ac t ) : " type A: MW25 .ACT. (This selection results in the use of an existing data file stored in the disk in drive A with the name MW25 .ACT. ) At " SELECT AN ACTIVITY COEFFIC IENT MODEL

•

type 2 and press RETURN . (Thi s selection results in the use of the van Laar equation as the activity coefficient model. ) A t " INPUT INITIAL GUESSES FOR VAN LAAR PARAMETERS P 1 2 , P2 1

•

LAAR MODEL ) : " type 1 , I and press RETURN. At "DO YOU WANT TO FIT THE PARAMETERS TO VLE DATA ( 1 )

•

•

•

O = EXIT 1 =NRTL 2 =VAN

LAAR

3 =UNIFAC 4 =WILSON S =UNIQUAC"

( PIJ ARE DIMENS IONLESS KAPPA PARAMETERS OF THE VAN

THE PARAMETERS JUST ENTERED ( 2 ) ?" type I and press

OR DO YOU WANT TO DO A CALCULATION OF VLE WITH

RETURN.

(At this point the program starts the optimization . When the calculations are completed, the final results appear on the screen as shown below. ) AC -VLE FROM ACTIVITY COEFFICIENT MODELS THE VAN LAAR MODEL PARAMETERS P 1 2 , P2 1 : .

5853

.

3458

METHANOL WATER TEST DATA FOR PROGRAM AC 2 5 C TEMPERATURE ( K )

: 2 9 8 . 15

1 17

Mode l i ng Vapor-Li q u i d Eq u i l i bria

PRESSURE IS IN THE UNITS OF THE DATA . YCAL

ACT 1

ACT2

SUM

23 . 7024

1. 7991

1 . 0000

1 . 0000

39 . 0018

1 . 5451

1. 0068

1 . 0000

53 . 0000

52 . 5250

1 . 3511

1 . 0284

1 . 0000

69 . 8000

68 . 3688

1 . 1811

1. 0789

1 . 0000

82 . 3000

82 . 3242

1. 0869

1 . 1443

1. 0000

98 . 5000

9 9 . 7 12 2

1. 0258

1 . 2430

1 . 0000

PEXP

XEXP

YEXP

. 0000

. 0000

. 0001

23 . 7000

. 0873

. 4187

. 4416

37 . 5000

. 19 0 0

. 6 187

. 6241

. 3 4 17

. 7350

. 7538

. 4943

. 7 934

. 8334

. 6919

. 8822

. 9090

PCAL

. 84 9 2

. 9384

. 9 583

112 . 0 0 0 0

113 . 7 67

1 . 0053

1 . 3288

1 . 0000

1. 0000

1 . 0000

1 . 0000

127 . 7 000

127 . 69 8

1 . 0000

1 . 4148

1. 0000

fractions of species I and the total pressure, respectively, and columns three, five, (The first, second and fourth columns are the experimental liquid and vapor mole six, and seven are the calcul ated vapor mole fractions of species I , pressure, and

activity coefficients of species I and 2 , respectively. Column eight lists the sum of vapor phase mole fractions that are calcu l ated separately and printed as a check; values should be unity or very close to unity. ) •

At "DO YOU WANT A PRINT-OUT { Y/ N ) ? " type Y (or y) and press RETURN .

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A:MW25 .0UT and press

(This command sends the results on the screen to your printer. ) ( YI N ) ? " type Y (or y) and press RETURN. RETURN. (Thi s command saves the results above in your disk in drive A under the name MW25 . 0UT in ASCII file code. ) •

At "DO YOU WANT TO DO ANOTHER VLE CALCULATION { Y/ N ) ? " type N (or n) and press RETURN.

Exam ple D. I . B: Use of U N I FAC to Predict VLE Data • •

Change to the directory containing AC . EXE (e. g . , A> or C>, etc . ) . Start the program by typing AC at the DOS prompt. Press ENTER (or press RETURN). The program introduction message appears on the screen. Press ENTER (or press RETURN) . The following appears :

AC : VLE CALCULATIONS WITH VARIOUS ACTIVITY COEFFICIENT MODELS YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA ARE AVAILABLE TO COMPARE RESULTS WITH

YOU MUST SUPPLY THE TEMPERATURE ,

AND SATURATION PRESSURE OF EACH

COMPOUND AT THAT TEMPERATURE . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERE D , AT INTERVALS OF 0 . 1 .

I 18

IN THE COMPOS ITION RANGE X l = O TO 1

Append1x D: Computer Programs for Bi nary M i xtures

MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) PARAMETERS TO THE VLE DATA .

TO FIT THE MODEL

ALTERNATIVELY ,

YOU CAN CALCULATE

VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA .

•

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN .

•

At "DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type n

•

At " INPUT NEW DATA FILE NAME : " type A:TEMP.ACT and press RETURN .

(or N ) and press RETURN. (The preceding command will lead to saving a data fi l e to the disk in the A drive under the name TEMP.ACT. A disk must be in that drive. ) •

At " INPUT A TITLE FOR THE NEW FILE : " type 'temporary data fi l e for methanol-water at 25"C' and press RETURN. (You can enter any title composed of up to forty alphanumeric characters for the title statement given above to describe your file for later reference . )

•

At " INPUT NUMBER OF DATA POINTS : " type 4 and press RETURN.

•

At " INPUT TEMPERATURE in K : " type 298. 1 5 and press RETURN.

•

At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVI SION" ( ex : i f original data in mm Hg , type 7 5 0

i f original data i n p s i a , type 1 4 . 5 etc . ) : " type 750 and press RETURN. •

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( XEXP ) OF SPECIES 1 , VAPOR MOLE FRACTION ( XEXP ) OF SPEC IES 1 , AND BUBBLE POINT PRESSURE ( PEXP ) IN THE UNITS OF THE ORIGINAL DATA" ( three in a row, separated by commas ) REMINDER : FIRST DATA POINT MUST BE X= O AND P =SATURATION P OF PURE SPEC IES 2 LAST DATA POINT MUST BE X = 1 AND P =SATURATION P OF PURE SPECIES 1 •

At " INPUT XEXP , YEXP , PEXP : " type 0, 0, 23. 7, and press RETURN .

•

At " INPUT XEXP , YEXP , PEXP : " type 0. 1 9, 0.6 1 87 , 5 3 , and press RETURN.

•

At " INPUT XEXP , YEXP , PEXP : " type 0. 849, 0.93 84, 1 1 2 , and press RETURN.

•

At " reminder : thi s ent ry is the last INPUT XEXP , YEXP , PEXP : " type I , I , 1 27 . 7 , and press RETURN.

enter X=Y= 1 and P=Psat . of pure spec ies 1

(When the number of sets of data specified by NP, here four, is entered, the program writes the data to the file under the name TEMP.ACT specified above and then continues. Thi s data file now is an existing data fi l e and can be used if 1 19

Mode l i ng Vapor-Liq u i d Eq u i l ibria

the program is run again. The data tile appears as shown below if called by an editor program. ) temporary data f i l e for methanol water system a t 2 5C 4 2 9 8 . 15 750 0

0

23 . 7

. 19

. 6187

53

. 849

. 9384

112

1

1

127 . 7

•

At " SELECT AN ACTIVITY COEFFICIENT MODEL O =EXIT l=NRTL 2 =VAN LAAR 3 =UNIFAC 4 =WILSON S =UNIQUAC" type 3 and press RETURN . (This choice results in the use of UNIFAC for the activity coeffi cient mode l . )

•

At

"THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK . THESE DATA ARE STORED IN TWO FILES NAMED UNF i l . DTA AND UNF I 2 . DTA . UNF i l . DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION . UNFI 2 . DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION . IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY , THEN ENTER 1 , OTHERWI SE ENTER 2 " type 2 and press RETURN. (The data files UNFI l .DTA and UNFI2.DTA are provided on the disk that disk directory that is u sed to run the programs. In this case I should be entered .

accompanies thi s monograph. It is better if these data files are copied to the hard

An entry of 2, as above, indicates that these tiles are not present in the current directory. In thi s case the user must provide the directory and file names as below. ) •

At

"TYPE THE DIRECTORY & THE NAME OF THE FILE WHERE UNIFAC GROUP PARAMETER INFORMATION IS STORED ( default = a : UNFi l . DTA ) " type a: UNFi l .DTA and press RETURN. •

At

"TYPE THE DIRECTORY & THE NAME OF THE FILE WHERE UNIFAC BINARY INTERACTION PARAMETER INFORMATION I S STORED ( default = a : UNFI 2 . DTA ) " type a: UNFI2.DTA and press RETURN. •

At

"ENTER COMPONENT INFORMATION ENTER COMPONENT NAME ( max . 1 20

12 Characters ) FOR COMPONENT 1

Appendix D: Computer Program s for Binary M i xtures

OR ENTER <pre s s RETURN> TO TERMINATE ENTRIES" type METHANOL and press RETURN . (Fol lowing the preceding comment, a group selection table will appear on the screen . The user must follow the instructions at the top of the table to choose one CH3 0H group for methanol and enter press RETURN . ) •

At

"ENTER COMPONENT INFORMATION ENTER COMPONENT NAME (max .

12 Charac ters ) FOR COMPONENT 2

OR ENTER <pre s s RETURN> TO TERMINATE ENTRIES" type WATER and press RETURN. Following the preceding comment the group selection table will again appear on the screen. The user should fol low the i nstructions at the top of the table to choose one H2 0 group for water and then press RETURN. After the last entry, a summary of the parameter input appears on the screen . Press RETURN to continue. The fol lowing results will appear on the screen: AC -VLE FROM ACTIVITY COEFFICIENT MODELS THE UNIFAC MODEL temporary data f i l e f o r methanol wat e r , TEMPERATURE

25

c

(K) : 298 . 15

PRESSURE I S IN T HE UNITS OF THE DATA . YEXP

XEXP

PCAL

PEXP

YCAL

ACT 1

ACT2

SUM

. 0000

. 0000

. 0000

23 . 7000

23 . 7000

2 . 2446

1 . 0000

1 . 0000

. 19 0 0

. 6188

. 6446

53 . 0000

5 6 . 1636

1 . 4921

1 . 0398

1 . 0000

. 8490

. 9384

. 9542

112 . 0 0 0 0

114 . 4 8 8 6

1. 0076

1. 4659

1 . 0000

1 . 0000

1 . 000

1 . 000

127 . 7 0 0 0

127 . 7 00

1. 0000

1 . 6046

1 . 0000

(This i s a part o f the methanol-water binary system data used in Example D . I .A. As before, in this table the first, second, and fourth columns are the measured liquid and vapor mole fractions of species 1 and the pressure, respectively, and columns three, five, six, and seven are the calculated vapor mole fractions of species

1 , pressure, and the activity coefficients of species

1 and 2, respectively.

Column eight is the calculation confirmation line described earli er. ) •

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type Y (or y) and press RETURN. (This command sends the results on the screen to printer. )

•

At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type Y and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A :TEMP.OUT and press RETURN . (This command saves the results given above o n the disk i n drive A with the name TEMP.OUT as an ASCII file . ) 121

Model i ng Vapor-Liquid Eq u i l i bna

•

At "DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y/ N ) ? " type N (or n) and press RETURN .

Exam ple D. l .C: Di rect Use of Activity Coefficient Model to Pred ict VLE • •

Change to the directory containing AC . EXE ( e . g . , A> or C>, etc . ) Start the program by typing AC at the DOS prompt. The program introduction message appears on the screen . Press RETURN. The following appears :

AC : VLE CALCULATIONS WITH VARIOUS ACTIVITY COEFFICIENT MODELS YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA ARE AVAILABLE TO COMPARE RE SULTS WITH

YOU MUST SUPPLY THE TEMPERATURE ,

AND SATURATION PRESSURE OF EACH

COMPOUND AT THAT TEMPERATURE . IN THI S MODE THE PROGRAM WILL RETURN I S OTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED ,

IN THE COMPOS ITION RANGE X 1 = 0 TO 1

AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I S OTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) PARAMETERS TO THE VLE DATA .

TO FIT THE MODEL

ALTERNATIVELY ,

YOU CAN CALCULATE

VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA .

•

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE

THE PROGRAM" type I and press RETURN.

(This results in the selection of the predictive mode of the program. In this mode no experimental VLE data can be entered to, or accessed from, the di sk. The user must supply a temperature and the pure component vapor pressures following the commands on the screen. In addition, the user must select a model and provide the model parameters. The program returns temperature entered in the liquid mole fraction range

x1

x-y- P predictions at the =

0 to 1 at intervals 0. 1 ,

0.2, 0 . 3 , etc . ) •

At "YOU MAY ENTER A TITLE ( 2 5 CHARACTERS MAX . ) FOR THE MIXTURE TO BE PREDICTED ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) : " enter methanol-water 25 C and press RETURN . (The title entry is optional . )

•

At " INPUT TEMPERATURE in K : " enter 298 . 1 5 and press RETURN.

•

At " INPUT VAPOR PRESSURE OF COMPONENT 1 ( IN

1 27 .698 and press RETURN . 1 22

ANY

UNIT ) : " enter

Appendix D: Computer Programs fo r Binary M i xtures

•

At " INPUT VAPOR PRESSURE OF COMPONENT 2 ( IN

ANY

UNIT ) : " enter

23.70 and press RETURN . •

At " INPUT FACTOR TO CONVERT PRESSURE INTO BAR BY DIVISION" ( type 1 i f you entered vapor pres sures in bar type 7 50 if you entered them in mm Hg . etc . ) : " enter 750 and press RETURN

•

At " SELECT AN ACTIVITY COEFFICIENT MODEL O =EXIT l=NRTL 2 =VAN LAAR 3 =UNIFAC 4 =WILSON S =UNIQUAC" type

I • •

and press RETURN.

At " INPUT ALPHA OF THE NRTL MODEL : " enter 0.35 and press RETURN. At " INPUT REDUCED NRTL PARAMETERS P 1 2 AND P2 1 [ PIJ=AIJ/ ( RT ) , WHERE AIJ IS IN CAL /MOL . 1 : " enter 1 , 1 and press RETURN. The following results appear on the screen:

AC -VLE FROM ACTIVITY COEFFIC IENT MODELS THE NRTL MODEL PARAMETERS P 1 2 , P2 1 1 . 0000 1 . 0000 ALPHA= . 3 5 0 methanol - water 2 5C TEMPERATURE ( K ) : 2 9 8 . 1 5 PRESSURE I S IN T HE UNITS OF THE DATA . SUM

ACT 1

ACT2

5 . 4997

1 . 0000

1. 0000

7 1 . 0950

3 . 8660

1 . 0185

1 . 0000

93 . 6455

2 . 8698

1. 0733

1. 0000

. 8153

104 . 8255

2 . 2308

1 . 1672

1. 0000

. 8320

110 . 7753

1 . 8044

1. 3083

1. 0000

. 8434

114 . 4 5 2 9

1 . 5119

1 . 5119

1. 0000

. 8542

117 . 3457

1 . 3083

1 . 8044

1 . 0000

. 7 000

. 8680

12 0 . 1 9 4 6

1 . 1672

2 . 2308

1 . 0000

. 8000

. 8896

123 . 2530

1 . 07 3 3

2 . 8698

1 . 0000

. 9000

. 9274

12 6 . 2 19 3

1. 0185

3 . 8660

1. 0000

1 . 0000

1 . 0000

127 . 69 8 0

1. 0000

5 . 4997

1. 0000

YEXP

XEXP

YCAL

PEXP

PCAL

. 0000

. 0000

23 . 7024

. 1000

. 6944

. 2000

. 7827

. 3000 . 4000 . 5000 . 6000

• •

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type n (or N) and press RETURN . At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " enter N (or n) and press RETURN.

•

At "DO YOU WANT TO DO ANOTHER CALCULATION ( Y/ N ) ? " type N (or n) and press RETURN

1 23

Mode l i ng Vapor-Li q u i d Eq u i l ibria

0. 2.

Program KO PT: Eval uation of the {K i ) Param eter fo r the PRSV Equation of State The program KOPT is used for the evaluation of the K1 constant of pure fluids in the PRSV equation (see Section 3 . 1 ). The data required for thi s program are critical temperature (in Kelvin), critical pressure (in bar), and acentric factor of the fluid as well as data for the temperature (in Kelvin) versus vapor pressure (in any units ) . The program returns the K1 value, which minimizes the average difference between the estimated and experimental vapor pressures. A si mplex optimization routi ne is used in the calculations. The program reads previously stored data or accepts new data entered from the keyboard. The extension OAT, such as ACETONE. OAT (one of the sample data sets included on the accompanying disk), was used for the data files for this program on the accompanying disk. A tutorial i s provided below to demonstrate the use of the KOPT program. As a requirement of the simplex mini mization procedure, an initial guess for K 1 must be provided. The initial guess can be a positive or a negative number, usually in the range from zero to one. The results from KOPT can be sent to a printer, to a disk file, or both. To make this choice, the commands that appear on the screen at the completion of calculations must be followed. Please see the following tutorial for further details.

Tuto rial on the Use of KO PT. EXE Example D.2.A: Determi nation of O pti m u m K1 i n the PRSV Equation of State with Existi ng Data • •

Change to the directory containing KOPT.EXE (e.g . , A> or C>, etc . ) . Start the program typing KOPT at the DOS prompt. Press RETURN (or ENTER) . An introductory message appears on the screen. Press RETURN.

•

At

"KOPT : OPTIMI ZES PURE COMPONENT KAPPA- 1 PARAMETER IN THE PRSV EOS DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type Y (or y) and press RETURN. •

At " INPUT THE NAME OF THE EXISTING DATA FILE

•

At " INPUT AN INITIAL GUESS FOR THE KAPPA- 1 PARAMETER : " type 0. 1

( for example : a : acetone . dat ) : " type a: acetone .dat. and press RETURN. (Following the preceding command, the results of the intermediate iterations are graphically shown in the form of an error bar on the screen so that the user can follow the convergence of the calculations. Next, a message showing the results of the optimi zation appears on the screen. To proceed, press RETURN . )

1 24

Appendix D: Computer Programs fo r Bi nary M ixtures

•

At "DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type y (or Y) and press RETURN. (With this command the results shown below are sent to the printer. )

KOPT : KAPPA- 1 OPTIMI ZATION FOR THE PRSV EQUATION acetone . dat ACETONE VARGAFTIK 4 2 7 - DECHEMA COMBINED KAPPA- 1 = - . 0 1 0 0 PEXP ( BAR )

T (K)

PCAL

AAD

273 . 1500

. 0935

. 0937

. 1563

283 . 1500

. 1552

. 1552

293 . 1500

. 2473

295 . 8500

vv

VL ( CM3 /MOL ) 8 1 . 5939

241384 . 2

. 0001

82 . 4861

150681 . 4

. 24 7 1

. 0660

8 3 . 4442

97672 . 6

. 2666

. 27 8 4

4 . 4392

83 . 7 150

87394 . 8

303 . 1500

. 3801

. 3798

. 0823

84 . 4 7 5 1

65452 . 8

312 . 6500

. 5333

. 5548

4 . 02 9 6

85 . 5295

45984 . 8

313 . 1500

. 5660

. 5656

. 07 5 6

85 . 5872

45168 . 9

329 . 6500

1 . 0133

1 . 0270

1 . 3477

87 . 6269

25885 . 4

323 . 1500

. 8190

. 8 187

. 0372

8 6 . 7905

31990 . 6

351. 7500

2 . 0200

2 . 0677

2 . 3600

90 . 8483

13415 . 4

386 . 1500

5 . 0600

5 . 1506

1. 7907

97 . 4385

5593 . 1

4 17 . 6 5 0 0

10 . 1300

10 . 3 107

1. 7 8 4 0

106 . 2131

2786 . 3

454 . 1500

20 . 2600

20 . 3683

. 5343

123 . 1467

1316 . 7

47 8 . 1500

30 . 3900

3 0 . 1622

. 7495

144 . 0444

792 . 8

487 . 6500

40 . 52 0 0

• .

no convergence for thi s data point . .

PERCENT AAD ( OVERALL ) , SUM ( ABS ( PEXP - PCAL ) / PEXP ) * 1 0 0 /NP : •

1 . 497

At "DO YOU WANT T O SAVE THE RESULTS T O AN OUTPUT FILE ( YI N ) ? " type y and press RETURN .

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type a: acetone. out and press RETURN. (The last two commands save the results above in the disk in the A drive under the name acetone.out in ASCII code. )

•

A t " DO YOU WANT T O DO ANOTHER KAPPA- 1 CALCULATION ( Y/ N ) ? " type n (or N ) and press RETURN .

Exam ple D. 2.B: Determ i nation of O pti m u m of State Entering N ew Data

Ki

i n the PRSV Equation

Change to the directory containing KOPT.EXE (e. g . , A> or C>, etc . ) •

Start the program typing KOPT at the DOS prompt. Press RETURN (or ENTER) .

1 25

Model i ng Vapor-Liquid Eq u i l i b na

The program introduction message appears on the screen. Press RETURN . •

At

"KOPT : OPTIMI ZES PURE COMPONENT KAPPA- 1 PARAMETER IN THE PRSV EOS DO

YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type N (or n) and press RETURN. •

At " INPUT NAME OF THE DATA FILE TO BE CREATED : " type a: temp l .dat and press RETURN . (The preceding command will lead to saving a data fi le named temp l .dat on the disk in drive A. If you choose to do this, a disk must be present in the A drive. )

•

A t " INPUT A TITLE FOR THE NEW DATA FILE : " type 'T V S P DATA FOR PURE ACETONE' and press RETURN. (For the title statement above you can enter any title of up to forty alphanumeric characters to describe your file for later reference. )

•

A t " INPUT T C ( K ) , P C ( BAR ) , ACENTRIC FACTOR W : " type

•

At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN.

•

At " INPUT FACTOR TO CONVERT PRESSURE INTO BAR BY DIVI SION"

508 . 1 ,46.96,0.30667 , and press RETURN.

( ex : i f source data are in mm Hg , type 7 5 0 i f source data are i n p s i a type 14 . 5 etc . ) : " type 1 and press RETURN. •

At " INPUT T ( K ) , PSAT : " type 2 83 . 1 5 , 0 . 1 55 1 89 , and press RETURN.

•

At " INPUT T ( K ) , PSAT : " type 3 1 3 . 1 5 , 0.56598, and press RETURN.

•

At " INPUT T ( K ) , PSAT : " type 478 . 1 5 , 30.39, and press RETURN. (When the specified number of sets of data, here three, has been entered, the program writes the data to the file u nder the name temp l .pur and continues. Thi s data file becomes an existing data file and can be used when the program is run again. The data file appears as shown below if called by an editor program . )

T VS P DATA FOR PURE ACETONE 46 . 96

508 . 1

. 3 0667

3 1 . 0000 2 8 3 . 15

. 155189

3 1 3 . 15

•

478 . 15

56598

30 . 39 •

At " INPUT AN INITIAL GUESS FOR KAPPA- 1 PARAMETER : " type 0. 1 and press RETURN. (Fol lowing the preceding command, the results of intermediate iterations are displayed on the screen as an error bar for the user to follow the convergence of the calculations. Next, an i ntermediate message summarizing the results appears on the screen. Press RETURN to continue. )

1 26

Appen dix D: Computer Programs for Bi nary M ixtures

•

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type y (or Y) and press RETURN. (With this command the results, like those shown below, are sent to the printer. )

KOPT : KAPPA- 1 OPTIMI ZATION FOR THE PRSV EQUATION teJIIP . OUt T VS P DATA FOR PURE ACETONE KAPPA- 1 = - . 0 1 0 0 T (K)

PEXP ( BAR )

283 . 1500

. 1552

313 . 1500

•

47 8 . 1500

5660

30 . 3900

PCAL

AAD

VL ( CM3 /MOL )

VV

. 1552

•

0001

82 . 4861

15068 1 . 4

5656

•

0756

85 . 5872

45168 . 9

. 7495

144 . 0444

792 . 8

•

3 0 . 1622

PERCENT AAD ( OVERALL ) , SUM (ABS ( PEXP - PCAL ) / PEXP ) * 1 0 0 / NP : •

. 275

A t "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type a: temp ) .out and press RETURN. (With these commands the results cited above are saved on the disk i n drive A under the name temp l .out in ASCII code . )

•

At " DO YOU WANT T O DO ANOTHER KAPPA- 1 CALCULATION ( Y / N ) ? " type n ( o r N) and press RETURN.

D. J .

Program VDW: Bi nary VLE with the van der Waals One- F l u i d Mixing Ru les ( I PVDW and 2PVDW) The program VDW can be used to calculate binary VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either l PVDW or 2PVDW; see S ections 3 . 3 t o 3 . 5 ) . The program can b e used in two way s . If experimental isothermal VLE data are available, the program can be used with u ser-provided binary interaction parameter(s) to calculate VLE at measured liquid mole fractions, and the calculated and experimen­ tal bubble pressures and vapor mole fractions can be then compared. Alternatively, the program can be used to optimize the values of the binary i nteraction parameter(s) by fitting them to experimental composition versus total pressure data using a si mplex algorithm. In this mode, the program reads previously stored data or accepts new VLE data entered from the keyboard. The data needed are critical temperatures (K), pressures (bars) , acentric factors, the K 1 constants of the PRSV EOS for both pure com­ ponents, isothermal VLE data in the form of measured liquid and vapor mole fractions of the first component (that is, x and y i n the liquid and vapor phases, respectively), and

The datafile structure of this and all the remaining programs that use an EOS to calculate VLE is the same, and the the total pressure, P, (in any units) at a given temperature.

1 27

Modeling Vapor-Li q u i d Eq u i l i bria

VDW2 can also be used with the other EOS programs that are de­

scribed in Sections D.4 to D. 7. To help convergence, it is advantageous to designate the data file createdfor

component with the lower critical temperature as the first component in these data files.

The sample data files on the accompanying disk are identified with the OAT extension, such as MW25 .DAT, etc. A tutorial is provided below (see Examples D . 3 . A and D . 3 . B ) . During program execution, a s a requirement o f the simplex approach, initial guess(es) for the binary interaction parameter(s), (k 12 for l PVDW or k 1 2 and k2 1 for 2PVDW model) must be provided by the user. The initial guess(es) can be positive or negative number(s ) . Depending upon the nonideality of the system, an initial guess may have to be significantly different from zero (such as - 0 . 1 5 for the acetone-water binary system, as shown in Example D . 3 . B below) to achieve convergence. If con­ vergence cannot be obtained with a (set of) initial guess(es), the user should try again with different choices. When no experimental VLE information is avai lable, the user only needs to supply the critical temperature, the critical pressure, acentric factor, PRSV K 1 parameter for each compound, and a temperature as input following the directions that appear on the screen. In this mode the program will return isothermal x-y- P predictions at the temperature entered in the composition range x1 = 0 to 1 at intervals of 0. 1 . Several temperature values can be selected successively. A tutorial is provided below (see Example D . 3 .C ) . The results from the program VOW c a n be sent t o a printer, t o a d i s k fi le, o r both. The commands that appear on the screen upon the completion of calculations must be followed to make this choice. Please see the following tutorial for further details.

Tutorial on the Use of VDW. EXE Example DJ.A: Fitting Binary VLE Data with the van der Waals One-Fl u i d M ixing Ru l e •

Change to the directory containing VDW.EXE (e. g . , A> or C>, etc . ) .

•

Start the program by typing VOW at the DOS prompt and then press RETURN (or ENTER) . A program introduction message appears o n the screen . Press RETURN to continue.

•

At

"VDW : BINARY VLE WITH VAN DER WAALS ONE - FLUID MIXING RULES 1 : CONVENTIONAL ( 1PVDW ) 2 : 2 - PARAMETER COMPOSITION DEPENDENT MIXING RULE ( 2 PVDW ) type I and press RETURN.

DO YOU WANT TO USE 1 - PARAMETER VDW MODEL OR 2 - PARAMETER VDW MODEL ( 1 / 2 ) ? "

1 28

Appendix D: Com p uter Programs for Bi nary M i xtu res

(This results in the selection of l PVDW model for the VLE calculations. The message below appears on the scree n . ) VDW : BINARY VL E CALCULATIONS WITH VAN DER WAALS ONE - FLUID MIXING RULES YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RESULTS WITH ARE AVAI LABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , ALONG WITH A

( PAIR OF )

ACENTRIC FACTOR ,

AND A TEMPERATURE

PREVIOUSLY SELECTED MODEL PARAMETER ( S ) .

IN THI S MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED,

IN THE COMPOS ITION RANGE X 1 = 0 TO 1

AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

TO FIT MODEL PARAMETERS

TO THAT VLE DATA .

YOU CAN CALCULATE VLE WITH PREVIOUSLY

ALTERNATIVELY ,

SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA . •

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" enter 2 and press

RETURN.

(With this selection, the user can enter new VLE data from the keyboard, or use At "DO YOU WANT T O USE AN EXISTING DATA FILE ( Y/ N ) ? " type Y

previously entered VLE data for correlation with l PVDW or 2PVDW models . )

•

(or y) and press RETURN.

•

At

" INPUT NAME OF THE EXISTING DATA FILE ( for example , a : pe 3 7 3 . dat ) : " type a : pe37 3 . dat and press RETURN . (Thi s results in the use of an existing data file pe373 .dat on the disk in your A drive. ) •

At " INPUT INITIAL GUESS FOR BINARY INTERACTION PARAMETER K12 : " type 0. 1 and press RETURN.

•

At

" DO YOU WANT TO FIT Kij TO VLE DATA ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH VALUE OF Kij ENTERED ( 2 ) ? " type 1 and press RETURN. (At this stage the program is run to optimize the kiJ parameter and intermediate results will be displayed on the screen as an error bar. Next a message summarizing the results of parameter optimization appears on the screen . Press RETURN to continue after inspecting the results . The information below appears on the screen. ) VDW : VAN DER WAALS MODEL ( S )

+

PRSV EOS VL E PROGRAM

PENTANE ETHANOL 3 7 2 . 7 K

1 29

Modeling Vapor-Liquid Eq u i l i bria

K12 = . 1 2 2 7 T (it ) = 3 7 2 . 7 0 PHASE VOLUMES ARE IN C C / MOL, PRESSURE I S IN UNITS OF THE DATA . VL-CAL

VV-CAL

. 00003

69 . 10

13444 . 1

. 59494

74 . 4 1

5037 . 3

. 64 5 6 4

80 . 23

4365 . 9

. 64764

89 . 23

4338 . 9 4330 . 2

Y-EXP

Y-CAL

220 . 876

. 00000

544 . 8 6 8

. 49100

537 . 400

615 . 4 62

. 62 9 0 0

618 . 800

618 . 682

. 69000

P - EXP

P - CAL

. 0000

220 . 000

. 0830

422 . 600

. 17 1 0 . 3030

X-EXP

. 4410

654 . 300

619 . 692

. 72400

. 64948

98 . 7 3

. 62 6 0

67 8 . 1 0 0

633 . 974

. 74700

. 68772

111 . 11

4206 . 9 4156 . 0

. 7360

684 . 300

6 3 8 . 9 13

. 76800

. 73800

118 . 02

. 8390

682 . 600

633 . 474

. 80300

. 81050

124 . 0 8

4178 . 2

. 9370

658 . 100

613 . 884

. 86000

. 9 1147

129 . 43

4305 . 0

. 9999

591 . 000

591 . 57 8

. 99990

. 99984

132 . 65

4466 . 8

pres s return to cont inue .

species 1 , total pressure, and vapor mole fraction of species I , respectively.

(The first, second, and third columns are the measured liquid mole fraction of

The third, fifth, sixth, and seventh columns are total pressure, vapor mole

fraction, and liquid and vapor saturated phase volumes, respectively, calculated at the experimental liquid mole fraction s . ) Press RETURN t o continue. •

At "DO YOU WANT A PRINT- OUT ( YI N ) ? " type y (or Y) and press RETURN. (Thi s command sends the results, similar to those shown above, to the printer. )

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? "

•

At

type y (or Y ) and press RETU RN II

INPUT A NAME FOR THE OUTPUT FILE :

II

type A : TEMP2.0UT and press RETURN. (With this command the results shown above are saved in the disk in drive A under the name TEMP2 .0UT in ASCII code . ) •

A t " DO YOU WANT T O DO ANOTHER VL E CALCULATION ( Y / N ) ? " enter n (or N) and press RETURN .

Exam ple D. 3 . B: Fitting B i n ary VLE Data with the Two-parameter van der Waals One-fl u i d M ixing Ru l e •

Change to the directory containing VDW.EXE (e.g . , A> or C>, etc . ) .

•

Start the program by typing VOW at the DOS prompt and then press RETURN (or ENTER). The program introduction message appears on the screen. Press RETURN to continue.

1 30

Appendix D: Computer Programs for B i n ary M 1 xtures

•

At

"VDW : BINARY VLE WITH VAN DER WAALS ONE - FLUID MIXING RULES 1 : CONVENTIONAL ( 1PVDW ) 2 : 2 - PARAMETER COMPOSITION DEPENDENT MIXING RULE ( 2 PVDW ) DO YOU WANT TO USE 1 - PARAMETER VDW MODEL OR 2 - PARAMETER VDW MODEL ( 1 / 2 ) ? " type 2 and press RETURN . (This results in the selection of 2PVDW model for the VLE calculations. Next the message below w i l l appear on the screen.) VDW :

BINARY VL E CALCULATIONS WITH VAN DER WAALS ONE - FLUID MIXING RULES

YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAI LABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , ALONG WITH A

( PAIR OF )

AND

ACENTRIC FACTOR,

A TEMPERATURE

PREVIOUSLY SELECTED MODEL PARAMETER ( S ) .

IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

TO FIT MODEL PARAMETERS

TO THAT VLE DATA .

YOU CAN CALCULATE VLE WITH PREVIOUSLY

ALTERNATIVELY ,

SELECTED PARAMETERS •

AND

COMPARE THE RESULTS WITH THE VLE DATA .

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" enter 2 and press RETURN.

•

At "DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type n (or N) and press RETURN .

•

At " INPUT NEW DATA FILE NAME : " type a : temp 3 .dat and press RETURN. (The preceding command will lead to saving a data file named temp3 . dat on the disk in the A drive. You must have a disk in the A drive, or choose another directory, by typing c : temp3 .out, for example, to save the fi l e on the hard drive . )

•

At " INPUT A TITLE FOR THE NEW DATA FILE : " type ' acetone-water temporary file at 1 00 C' and press RETURN . (You can enter any title up to forty alphanumeric characters to describe your file for l ater reference. )

•

At " CRITICAL PARAMETERS : TC=CRITICAL TEMP , K PC=CRITICAL PRESSURE , BAR W= ACENTRIC FACTOR KAP= PRSV KAPPA- 1 PARAMETER INPUT TC 1 , PC1 , W1 , KAP 1 : " type 508. 1 , 46.96, 0 . 3 0667 , -0.008 8 8 , and press RETURN.

1 3 1

M o d e l i ng Vapor-Liq u i d Equ i l i bria

(These are the pure component constants of acetone for the PRSV EOS from Table 3 . 1 . 1 . ) •

At " INPUT TC2 , PC2 , W2 , KAP2 : " type 647 .286, 220.90, 0 . 343 8 , - 0.06635, and press RETURN . (These are the pure component constants of water for the PRSV EOS from Table 3 . 1 . 1 . )

•

At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN.

•

At " INPUT TEMPERATURE in K : " type 373 . 1 5 and press RETURN .

•

At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVI SION ( e . g . i f original data in mm Hg , type 7 5 0 i f original data i n p s i a , type 14 . 5 etc . ) : " type 750 and press •

RETURN.

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( X1EXP ) , 'VAPOR MOLE FRACTION ( Y lEXP ) OF SPECIES 1 , BUBBLE POINT PRESSURE ( PEXP ) ( three in a row, separated with commas ) " " INPUT X1EXP , Y1EXP , PEXP : " type 0. 1 08 , 0.632, 2089 . 2 8 , and then press RETURN . " INPUT X1EXP , Y1EXP , PEXP : " type 0.480, 0.747, 2606.43 , and then press RETURN . " INPUT X1EXP , YlEXP , PEXP : " type 0.77 1 , 0 . 8 3 7 , 276 1 .5 8 , a n d then press

RETURN. When the number of sets o f data, specified by NP, here three i s entered, the program writes the data to the file with the name temp3.dat and continues. These data now become an existing data file and can be used when thi s program is run again or when using the other programs described below. This data file looks as shown below if called by an editor program. acetone -water trial file at 100 C 508 . 1

46 . 96

. 3 0667

-0 . 0089

647 . 2 8 6

220 . 8975

. 3438

-0 . 0664

. 108

. 632

2089 . 2 8

. 48

. 747

2606 .43

. 77 1

. 837

2761 . 58

3 3 7 3 . 15 750

The program then continues a s shown below. •

At

" INPUT INITIAL GUESSES FOR BINARY INTERACTION PARAMETERS K 1 2 , K2 1 : " , type 0. 1 ,0. 1 and press RETURN . 1 32

Appendix D: Computer Programs for Bi nary M ixtures

(Because the 2PDW model was selected at an earlier stage, here values of two parameters are needed. ) •

A t " DO YOU WANT T O F I T Kij T O VLE DATA ( 1 ) OR DO YOU WANT CALCULATION OF VLE WITH Kij ENTERED ( 2 ) ? " type 1 and press RETURN. (With the entry given above, the data fit process is initiated. However, the initial guesses 0. 1 /0. 1 are not suitable for the acetone-water binary system; therefore, the message below appears on the screen . )

INITIAL GUESS ( ES ) YOU SELECTED . 100

. 100

LEADS T O NEGATIVE LOG VALUES ENTER 1 TO SELECT ANOTHER ( SET OF ) Kij VALUE ( S ) ENTER 2 FOR ANOTHER VLE CALCULATION :

Type 1 and press RETURN. (The preceding entry will give the user the opportunity to try new i nitial guesses, as shown below. ) •

At

" INPUT INITIAL GUESSES FOR BINARY INTERACTION PARAMETERS K12 , K2 1 : " , type - 0 . 1 5 , - 0 . 1 5 and press RETURN. •

At

" DO YOU WANT TO FIT Kij TO VLE DATA ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH VALUE OF Kij ENTERED ( 2 ) ? " type 1 and press RETURN. (At this stage the program runs to optimize kij . Intermediate results are displayed on the screen in the form of an error bar. Next a summary of optimization results appears on the screen. Press RETURN to continue.) The following results appear on the screen: VDW : VAN DER WAALS MODEL ( S )

+

PRSV EOS VLE PROGRAM

acetone -water t emporary f i l e at 1 0 0 C K12= - . 07 2 8 K2 1 = - . 2 3 5 1 T (K) = 3 7 3 . 15 PHASE VOLUMES ARE IN CC /MOL , PRESSURE IS IN UNITS OF THE DATA . X- EXP

P - EXP

P - CAL

Y- EXP

Y-CAL

VL - CAL

VV-CAL

. 65198

. 1080

2089 . 2 80

2090 . 522

. 63200

29 . 60

10604 . 7

•

4800

2606 . 430

2601 . 658

•

74700

•

74949

55 . 7 1

8377 . 3

•

7710

2761 . 580

2788 . 980

•

83700

•

83700

77 . 54

7744 . 4

pre s s return to cont inue .

1 33

M o d e l i ng Vapor-Liquid Eq u i l ibria

Press RETURN to continue. • •

At "DO YOU WANT A PRINT -OUT ( Y/ N ) ? " type n (or N) and press RETURN . At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y ) and press RETURN .

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A:TEMP3 .0UT and press RETURN. (With this command the results similar to those shown above are saved on the disk in drive A under the name TEMP3 . 0 UT as an ASCII file.)

•

At "DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y/ N ) ? " , enter n (or N ) and press RETURN .

Exam p l e D. 3.C: Binary VLE Predictions Using the van der Waals One-Fl u i d Model •

Change to the directory containing VDW.EXE ( e . g . , A> or C>, etc . ) .

•

Start the program by typing VDW at the DOS prompt. Press RETURN (or ENTER). The program introduction message appears on the screen. Press RETURN to continue.

•

At

"VDW : BINARY VLE WITH VAN DER WAALS ONE - FLUID MIXING RULES 1 : CONVENTIONAL ( 1PVDW ) 2 : 2 - PARAMETER COMPOSITION DEPENDENT MIXING RULE ( 2 PVDW ) DO YOU WANT TO USE 1 - PARAMETER VDW MODEL OR 2 - PARAMETER VDW MODEL ( 1 / 2 ) ? " type 2 and press RETURN. The fol l owing message will appear on the screen: VDW : BINARY VL E CALCULATIONS WITH VAN DER WAALS ONE - FLUID MIXING RULES

YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , ALONG WITH A

( PAIR OF )

ACENTRIC FACTOR,

AND A TEMPERATURE

PREVIOUSLY SELECTED MODEL PARAMETER ( S ) .

IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED ,

IN THE COMPOS ITION RANGE X 1 = 0 TO 1

AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I SOTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

1 34

TO FIT MODEL PARAMETERS

Appendix D: Compute r Programs for Bi nary M i xtu res

TO THAT VLE DATA .

ALTERNATIVELY ,

YOU CAN CALCULATE VLE WITH PREVIOUSLY

SELECTED PARAMETERS AND COMPARE THE RE SULTS WITH THE VLE DATA .

•

At "ENTER 1 FOR MODE ( 1 )

I

2 FOR MODE ( 2 )

I

OR 0 TO TERMINATE

THE PROGRAM" enter 1 and press RETURN. (Thi s example is presented to demonstrate a case for which no experimental

VLE

data are available, so that no data are entered to, or accessed from, the disk. The user should provide, following the commands that appear on the screen,

Tc , P"

compound in addition to a temperature, and the mixing rule parameter(s) kiJ . The the acentric factor and

K1

parameter of the PRSV equation of state for each

program returns i sothermal

x-y-P

predictions at the temperature selected.

Repeated temperature entries are al lowed . ) •

At

"YOU MAY ENTER A TITLE ( 3 0 CHARACTERS MAX . ) FOR THE MIXTURE T O B E PREDICTED ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) :

11

enter acetone-water 1 00 C

and press RETURN . •

At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPA1 PARAMETER OF THE PRSV EOS INPUT TC 1 , PC 1 , W1 , KAPPA- 1 : " enter 508. 1 , 46.96, .30667 , -0.008 8 8 , and press RETURN.

•

At " INPUT TC2 , PC2 , W2 , KAPPA- 2 : " enter 647 . 286, 220.8975 , 0. 343 8 ,

•

At " INPUT TEMPERATURE in K : " type 373 . 1 5 , and press

•

At " INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE

- .06635 , and press

RETURN.

DEFAULT IS BAR

,

RETURN.

TYPE 1 IF YOU WANT PRESSURE IN BAR

.

( type 7 5 0 i f you want pres sure in IIDII Hg , etc . ) : " enter 750 and press RETURN. •

At " INPUT BINARY INTERACTION PARAMETERS K12 , K2 1 : " enter - 0.07 1 6, -0.2356, and press

RETURN.

(At this stage, the program runs and a summary of results appears on the screen. In this case percent error in total pressure i s not reported because there i s no experimental information. Press RETURN to continue.) The following results appear on the screen : VDW : VAN DER WAALS MODEL ( S ) acetone -wate� 1 0 0

+

PRSV EOS VLE PROGRAM

c

K12= - . 07 1 6 K2 1 = - . 2 3 5 6 T (K) = 373 . 15

1 35

Modeling Vapor-Liq u i d Eq u i l ibria

PHASE VOLUMES ARE IN CC /MOL . FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR I S : 7 5 0 . 0 0 P - EXP

X- EXP

P - CAL

Y - CAL

VL - CAL

VV-CAL

. 00004

22 . 51

30295 . 7

Y-EXP

761 . 381

. 0000 . 1000

2 0 5 3 . 37 8

. 64 4 9 7

2 9 . 07

10809 . 0

. 2000

233 1 . 920

. 69616

35 . 81

9434 . 4

. 3000

2461 . 554

. 7 1942

42 . 7 5

8899 . 2

. 4000

2548 . 672

. 7 3674

49 . 88

8568 . 7

. 5000

2620 . 572

•

75374

57 . 18

8311 . 1

. 6000

2687 . 615

. 77401

64 . 63

8081 . 3

. 7000

2751 . 583

•

80135

7 2 . 17

7868 . 8

. 8000

2806 . 006

. 84 1 0 3

79 . 74

7688 . 7

. 9000

2833 . 493

. 90169

87 . 27

7583 . 4

1 . 0000

2799 . 59 6

1. 0000

94 . 67

7640 . 9

pre s s return to cont inue .

Press RETURN to continue. • •

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type y (or Y) and press RETURN . At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y ) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type c : temp4.out and press RETURN. (With this command results similar to those reported above are written to a file named temp4.out in the C directory. )

•

At type Y (or y ) and press RETURN.

"DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y / N } ? " (This entry allows the user to calculate VLE at another temperature for the binary

mixture under consideration.) •

At " INPUT TEMPERATURE in K : " type 425 . 1 5 K and press RETURN.

•

At " INPUT BINARY INTERACTION PARAMETERS Kl2 , K2 1 : " type -0.07 1 6, -0.2356 and press RETURN. (A summary of results appear on the screen . Press RETURN to continue . ) The following results appear on t h e screen:

VDW - VAN DER WAALS MODEL ( S ) acetone

+

wat er 1 0 0

K12 = - . 0 7 1 6 K2 1 = - . 2 3 5 6 T ( K) =425 . 15

1 36

c

+

PRSV E O S VL E PROGRAM

Appendix D: Computer Programs for Binary M ixtures

PHASE VOLUMES ARE IN C C / MOL FACTOR YOU ENTERED TO CONVERT PRESSURE FROM P - EXP

X- EXP

P - CAL

Y- EXP

Y-CAL

BAR

IS : 7 5 0 . 00

VL - CAL

VV-CAL

. 0000

3768 . 422

. 00001

23 . 7 6

6 8 18 . 6

. 1000

6212 . 298

. 42057

3 0 . 94

3903 . 6

. 2000

7 14 5 . 4 9 3

. 51822

38 . 45

3 3 13 . 4

. 3000

7734 . 740

. 57 6 8 2

46 . 32

3 0 10 . 2

. 4000

8 17 1 . 3 9 2

. 62304

54 . 5 6

2810 . 6

. 5000

8516 . 83 8

. 66528

63 . 16

2663 . 8

. 6000

87 9 6 . 6 0 0

. 70862

72 . 09

2549 . 2

. 7000

9 0 14 . 0 7 9

. 75749

81.26

2459 . 0

. 8000

9151 . 503

. 81693

90 . 58

2393 . 6

. 9000

9167 . 150

. 89395

99 . 85

2361 . 1

1. 0000

8989 . 156

1. 00000

108 . 91

2378 . 9

pre s s return to cont inue .

Press RETURN to continue. •

At "DO YOU WANT A PRINT- OUT ( Y / N ) ? " type N (or n) and press RETURN .

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? "

•

type y (or Y) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type c : temp4.out and press RETURN. earlier to save the results of previous VLE predictions at 373 . 1 5 K.)

(With this command the results above are appended to the file temp4.out opened

•

At

" DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y I N ) ? " type N (or n ) and press RETURN. •

At "DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y/ N ) ? " , type n (or N) and press RETURN .

0.4 .

Program HV: Binary VLE with the H u ron-Vi dal M ixing Rule (HVO) and Its Modifications ( M H V I , M HV2, LCVM, and HVOS) the following Gibbs excess free-energy-based mixing rules : HVO, MHV I ,

The program HV can be used to calcul ate VLE using the PRSV EOS and one of

MHV2,

LCVM, or HVOS . This program allows the NRTL, van Laar, Wilson, or the UNI­ QUAC excess free-energy model s to be used in the EOS formalism. Any mixing rule and excess free-energy model combination can be chosen during the program execu­ tion following the directions that appear on the screen. A tutorial is provided in thi s

1 37

M o d e l i ng Vapor-Liquid Eq u i l i bria

section . The program can be used in two ways. When isothermal VLE data are avail­ able, the program can be used to calculate VLE with model parameters provided by the user at measured liquid mole fractions and to compare the calculated bubble pressures and vapor mole fractions with the measured values. Alternatively, the program can be used to obtain parameters of a selected model by fi tting them to measured liquid composition versus bubble pressure data with a simplex algorithm. In this mode the program reads previously stored data or accepts new data entered from the keyboard. The data structure is identical to that described in Section D . 3 for the program VDW, and details concerning the data input requirements can be found there. The data fi les that can be used by this program are those on the disk with the DAT extension. In this mode, initial guesses for model parameters must be provided by the user for the parameter optimization by the simplex method. The initial guesses can be positive or negative numbers . The input parameters required are in reduced form, and a value be­ tween zero and one for each is usually satisfactory. If convergence is not achieved with a set of initial guesses, the user should try again with a different choice of parameters. In the absence of any experimental VLE data, the program can be used to calcu­ late VLE at a given temperature using internally generated liquid mole fractions of component 1 from 0 to 1 at intervals of 0 . 1 . In this case the user only needs to supply the critical temperature, critical pressure, acentric factor, and PRSV

K1

parameter for

each compound, and a temperature as input following the directions that appear on the screen. In this mode the program will return isothermal x-y- P predictions at the temperature entered in the composition range x1

=

0 to I at intervals of 0. 1 . Several

temperature values can be selected successively. A tutorial is provided below (see Example D.4.C). The results from the program HV can be sent to a printer, to a disk file, or both . To make this choice, the commands that appear on the screen upon the comple­ tion of calculations must be followed. Please see the following tutorial for further details.

Tutorial on the Use of HV. EXE Example D.4.A: Use of the HVO Model to Correlate Experi mental Data •

Change the directory containing HY.EXE ( e . g . , A> or C>, etc . ) .

•

Start the program by typing HV at the DOS prompt. Press RETURN (or ENTER). A program introduction message appears . Press RETURN to continue. The following appears on the screen:

HV : BINARY VL E CALCULATIONS WITH THE HURON-VIDAL MODEL AND I T S VARIATIONS YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

1 38

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAI LABLE ,

Appendix D: Computer Programs for Bi nary M i xtu res

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRES SURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND ,

AND

ACENTRIC FACTOR ,

A TEMPERATURE

ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED ,

IN THE COMPOSITION RANGE X 1 = 0 TO 1

AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I S OTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) TO THE VLE DATA .

ALTERNATIVELY ,

SELECTED PARAMETERS

•

AND

TO FIT MODEL PARAMETERS YOU CAN CALCULATE VLE WITH PREVIOUSLY

COMPARE THE RESULTS WITH THE VLE DATA .

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered VLE data. ) •

A t " SELECT A MIXING RULE MODEL : HV-0= HURON-VIDAL ORIGINAL MHVl= MODIFIED HURON-VIDAL 1ST ORDER MHV2 = MODIFIED HURON-VIDAL 2ND ORDER LCVM= LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN O =EXIT l=HV-0 2 =MHV1 3 =MHV2 4 =LCVM S =HVOS" type 5 and press

HVOS = HURON-VIDAL MODIFIED BY ORBEY AND SANDLER RETURN .

(This results in the selection of the HVOS model for the mixing rule . ) •

A t " SELECT AN EXCESS FREE ENERGY MODEL : l =NRTL 2 =VAN LAAR 3 =UNIQUAC 4 =WILSON" type 3 and press RETURN . (This results i n the selection of the UNIQUAC model for the excess free-energy term in the HVOS mixing rule . )

•

At

"THE UNIQUAC MODEL WILL REQUIRE PURE COMPONENT PARAMETERS R,

Q, Q ' .

IF YOU DO NOT HAVE THEM PLEASE SELECT ANOTHER MODEL . " •

At

"DO YOU WANT TO SELECT ANOTHER ACTIVITY COEFFIC IENT MODEL ( Y/ N ) ? " type n (or N ) and press RETURN. •

At " DO YOU WANT TO USE AN EXISTING DATA FILE ( Y I N ) ? " type Y

•

At

(or y) and press RETURN . " INPUT THE NAME OF EXISTING DATA FILE ( for example a : pe 3 7 3 . dat ) " (Both pe3 7 3 . dat and aw I OO.dat are data files provided on the accompanying disk.

type a:aw l OO . dat.

1 39

Mode l i ng Vapor-Liq u i d Eq u i l i bria

The aw l OO.dat file contain s isothermal VLE data for the acetone-water binary system at 1 00° C . ) •

At

"ENTER UNIQUAC PURE COMPONENT SURFACE AND VOLUME PARAMETERS INPUT UNIQUAC PARAMETERS Rl , Q l , Q l ' : " type 2 . 5 7 , 2 . 34, 2 . 34, and press RETURN. (These are the surface and volume parameters for component 1 , acetone, obtained from Prausnitz et a!. 1 980, p. l 45 . ) •

At " INPUT UNIQUAC PARAMETERS R2 , Q2 , Q2 ' : " type 0.92, 1 .4, 1 .0, and press RETURN . (These are the surface and volume parameters for component 2, water. )

•

At

[ PIJ=EXP ( -AIJ/ ( RT ) > , AIJ IN CAL /MOL ] : " type l , I and press RETURN .

" INPUT INITIAL GUESSES FOR P 1 2 AND P2 1 OF THE UNIQUAC MODEL •

At "DO YOU WANT TO FIT THE PARAMETERS TO VLE DATA ( 1 )

OR DO YOU WANT TO CALCULATE VLE WITH THE VALUES OF THE PARAMETERS JUST ENTERED ( 2 ) ? " type l and press RETURN. (At this stage the program attempts to optimize the two model parameters of the UNIQUAC model, and intermediate results wi l l be continuously displayed on the screen as an error bar. When the optimization is completed, a summary of the results appears on the screen . Press RETURN to continue. The results given bel ow appear on the screen.)

HV : BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VARIATIONS ACETONE WATER 1 0 0 C EOS MODEL = HVOS ; EXCESS ENERGY MODEL UNIQUAC P 1 2 ( =EXP { -A12 / RT} )

. 2900

UNIQUAC P2 1 ( =EXP { -A2 1 / RT } )

. 9 947

UNIQUAC

TEMPERATURE i n K : 3 7 3 . 1 5 PHASE VOLUMES ARE IN CC /MOL, PRESSURE IS IN UNITS OF THE DATA VL - CAL

VV-CAL

P - CAL

Y-EXP

Y-CAL

832 . 610

845 . 123

. 09020

. 10260

22 . 72

27219 . 5

. 0 040

848 . 120

862 . 565

. 10 9 0 0

. 12 1 1 1

22 . 77

26654 . 6

. 0045

879 . 150

874 . 893

. 11800

. 13376

22 . 80

26269 . 0

. 0080

977 . 4 10

9 5 8 . 2 67

. 20700

. 21077

23 . 03

23922 . 8

X-EXP

P - EXP

. 0033

. 0480

1680 . 7 30

1630 . 24

. 54500

. 54615

2 5 . 63

13786 . 0

. 0820

18 3 5 . 8 8 0

1938 . 15

. 61300

. 62416

27 . 87

11488 . 2

. 0980

2001 . 37 0

2 0 3 4 . 17

. 63 7 0 0

. 64413

28 . 93

10913 . 5

. 1080

2089 . 280

2083 . 01

. 63200

. 65372

29 . 59

1 0 64 1 . 3

. 2200

2301 . 310

2 3 3 4 . 17

. 70500

. 69929

3 7 . 18

9420 . 4

1 40

Appendix D: Computer Programs for Binary M ixtures

3080

2399 . 570

2400 . 25

•

7 1 17 9

43 . 30

9140 . 6

. 3160

2404 . 740

2405 . 08

. 7 1900

. 7 12 7 7

43 . 86

9 12 0 . 8

. 3970

2503 . 000

2453 . 13

. 72700

. 72335

49 . 63

892 6 . 9

•

71500

•

pre s s return to see more resu l t s on the screen .

Press RETURN to continue. . 4800

2606 . 430

2506 .75

. 74700

. 7 3729

55 . 65

8718 . 3

. 52 6 0

2570 . 230

2 5 3 9 . 14

. 74600

. 74701

59 . 02

8595 . 9

. 6950

2678 . 830

2 6 6 5 . 47

. 80100

. 79889

7 1 . 64

8140 . 7

. 7 150

2 62 1 . 950

2680 . 01

. 8 14 0 0

. 80707

7 3 . 14

8090 . 0

. 7420

2699 . 52 0

2699 . 05

. 82 3 0 0

. 81893

7 5 . 18

8024 . 0

. 7710

2720 . 200

2718 . 53

. 83700

. 83279

77 . 38

7956 . 7

. 8540

2756 . 400

2 7 6 5 . 94

. 87 8 0 0

. 87 9 9 2

83 . 66

7791 . 1

. 9440

2766 . 7 50

2795 . 44

. 94600

. 94694

90 . 4 6

7 67 5 . 2

. 97 1 0

2761 . 580

2797 . 90

•

97200

. 9 7 12 4

92 . 49

7 6 57 . 9

. 97 7 0

2766 . 750

2797 . 93

. 97800

. 97 6 9 6

92 . 9 5

7655 . 5

pre s s return to cont inue .

Press RETURN to continue. • •

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type y (or Y) and press RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y ) and press RETURN.

•

At " INPUT A FILE NAME FOR THE OUTPUT FILE : " type A :TEMP4. 0UT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP4.0UT in ASCII code . )

•

A t " DO YOU WANT T O DO ANOTHER VL E CALCULATION ( Y / N ) ? " type n (or N) and press RETURN.

Exam p l e D.4.B: Use of the HV Model to Correlate N ew Data • •

Change the directory containing HV.EXE (e.g., A> or C>, etc . ) . Start the program by typing

HV

at the DOS prompt. Press RETURN (or ENTER).

A program introduction message appears on the screen. Press RETURN. The following message appears on the screen: HV : BINARY VL E CALC�LATIONS WITH THE HURON-VIDAL MODEL AND I T S VARIATIONS

YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RESULTS WITH ARE AVAI LABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND ,

AND

ACENTRIC FACTOR,

A TEMPERATURE

141

Modeling Vapor-Li q u i d Eq u i l i bria

ALONG W�TH A SET OF PREV�OUSLY SELECTED MODEL PARAMETERS . �N TH� S MODE THE PROGRAM W�LL RETURN � SOTHERMAL x-y-P PRED�CT�ONS AT THE TEMPERATURE ENTERED �N THE COMPOS�T�ON RANGE X 1 = 0 TO 1 AT �NTERVALS OF 0 . 1 . MODE

(2) :

� F YOU HAVE � SOTHERMAL x - y - P DATA,

YOU CAN ENTER THESE DATA

FOLLOW�NG COMMANDS THAT W�LL APPEAR ON THE SCREEN ( OR USE PREV�OUSLY ENTERED DATA ) TO THE VLE DATA .

ALTERNAT�VELY ,

TO F�T MODEL PARAMETERS YOU CAN CALCULATE VLE W�TH PREV�OUSLY

SELECTED PARAMETERS AND COMPARE THE RESULTS W�TH THE VLE DATA . •

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered VLE data.) •

At " SELECT A MIXING RULE MODEL : HV-0= HURON-VIDAL ORIGINAL MHV1= MODIFIED HURON-VIDAL 1ST ORDER MHV2 = MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS = HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O =EXIT l=HV-0 2 =MHV1 3 =MHV2 4 = LCVM S =HVOS " type 1 and press

RETURN . (This results in the selection o f the original Huron-Vidal model, HVO, for the mixing rule model . ) •

A t " SELECT AN EXCESS FREE ENERGY MODEL : l =NRTL 2 =VAN LAAR 3 =UNIQUAC 4 =WILSON? " type 2 and press RETURN . (This results in the selection o f the van Laar model for the excess free-energy term in the HVO mixing rule.)

•

At "DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type

•

At " INPUT NEW DATA FILE NAME : " type a: temp4. dat and press

N

(or n) and press RETURN . RETURN. (The preceding command results in saving a data file named temp4.dat on the disk in drive A. You must have a disk in the A drive, or select another drive by typing, for example, c :temp4.out, to save the results in the hard drive . ) •

A t " INPUT A TITLE FOR THE NEW DATA FILE : " type ' methanol - water trial data at 1 00 C' and press RETURN . (You can enter any title composed of up to forty alphanumeric characters to describe your fi l e for l ater reference. )

1 42

Appendix D: Computer Programs for Bi nary M i xtures

•

At "CRITICAL PARAMETERS : TC=CRITICAL TEMP , K PC=CRITICAL PRESSURE , BAR W=ACENTRIC FACTOR KAP=THE PRSV EOS KAPPA1 PARAMETER INPUT TC 1 , PC 1 , W1 , KAP 1 : " type 5 1 2. 5 8 , 80.9579, 0 . 5 65 3 3 , -0. 1 68 1 6, and press RETURN. (These are constants of methanol for the PRSV EOS from Table 3 . 1 . 1 . )

•

At " INPUT TC2 , PC2 , W2 , KAP2 : " type 647 .286, 220 8975 , 0 . 3438 .

,

-0.06635 , and press RETURN. (These are constants of water for the PRS V EOS from Table 3 . 1 . 1 . ) •

At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN.

•

At " INPUT TEMPERATURE in K : " type 3 7 3 . 1 5 and press RETURN .

•

At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION ( e . g . if original data in mm Hg , type 7 5 0 i f original data i n p s i a , type 14 . 5 etc . ) " type 750 and press RETURN . •

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( X1EXP ) OF SPECIES 1 , VAPOR MOLE FRACTION ( Y1EXP ) OF SPEC IES 1 , BUBBLE POINT PRESSURE ( PEXP ) ( three in a row, s eparated by commas ) " . " INPUT X1EXP , Y1EXP , PEXP : " type 0.03 5 , 0 . 1 9 1 , 93 1 , and press RETURN. " INPUT X1EXP , Y1EXP , PEXP : " type 0.28 1 , 0.6 1 9, 1 5 35 .96, and press RETURN. " INPUT X1EXP , Y1EXP , PEXP : " type 0 . 8 26, 0.9 1 1 , 2337.76, and press RETURN. (When the number of items of data, speci fied by NP, here three, is entered, the program writes the data to a file under the name temp4.dat as spec ified above and continues. Thi s data fi le becomes an existing data file and can be used when this program or other EOS programs arc run again. This data file appears as shown below if called by an editor program. )

methanol -water trial data at 1 0 0

c

512 . 58

80 . 9579

. 56533

- . 16816

647 . 2 8 6

220 . 8975

. 34 3 8

- 6 . 6 3 5 0 0 0 0 0 0 0 0 0 0 0 1D - 0 2

3

3 7 3 . 15

750 . 035

. 19 1

931

. 281

. 6 19

1535 . 96

. 826

. 9 11

2337 . 7 6

1 43

M o d e l i ng Vapor-Li q u i d Eq u i l i bria

(Note that this format i s exactly the same as that of the input data created following the tutorial given in Example D . 3 . B as all EOS programs use the same data structure. When the data entry process is complete, the program continues as below. ) •

At " INPUT INITIAL GUESSES FOR VAN LAAR PARAMETERS P 1 2 , P2 1 type I , 1 and press RETURN.

( PIJ ARE DIMENS IONLESS PARAMETERS OF THE VAN LAAR MODEL ) "

•

At

" DO YOU WANT TO FIT THE PARAMETERS TO VLE DATA ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH THE VALUES OF THE PARAMETERS JUST ENTERED ( 2 ) ? " type 1 and press RETURN. (At thi s stage the program attempts to optimize the two model parameters of the van Laar model, and the intermediate results are continuously displayed on the screen in the form of an error bar. When the optimization is complete, a message displaying summary of results appears on the screen for inspection . Press RETURN to continue. The results given below appear on the screen. )

HV : BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VARIATIONS methanol -water trial data at 1 0 0 C EOS MODEL = HVO ; EXCESS ENERGY MODEL

VAN LAAR

=

P 1 2 ( =DIMENS IONLESS KAPPA12 OF VAN LAAR )

1 . 44 6 8

P2 1 ( =DIMENSIONLESS KAPPA2 1 O F VAN LAAR )

. 67 0 5

TEMPERATURE T ( K )

373 . 15

PHASE VOLUMES ARE IN CC/MOL , PRESSURE IS IN UNITS OF THE DATA . X- EXP

P - EXP

P - CAL

Y- EXP

Y - CAL

VL-CAL

VV-CAL

. 0350

931 . 000

933 . 475

. 19100

. 2 1068

23 . 50

24622 . 4

. 2810

1535 . 960

1535 . 978

. 61900

. 60611

30 . 58

147 7 8 . 0

. 8260

2337 . 760

2368 . 4 82

. 91100

•

92059

47 . 2 6

9405 . 2

pre s s return to cont inue .

•

At "DO YOU WANT A PRINT-OUT { Y / N ) ? " type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer. )

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE { Y/ N ) ? "

Press RETURN to continue.

•

type y (or Y) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A : TEMP4.0UT and press RETURN. (With these commands the results shown above are saved on a d i s k in drive A

under the name TEMP4.0UT in ASCII code. ) •

A t "DO YOU WANT T O D O ANOTHER VLE CALCULATION ( YI N ) ? " type n (or N) and press RETURN.

1 44

Appendix D: Computer Programs for Bi nary M i xtures

Exam ple D.4.C: Binary VLE Predicti ons Using the H u ron-Vidal Model •

Change to the directory containin g HV.EXE (e. g . , A> or C>, etc . ) .

•

S tart the program by typing HV at the DOS prompt. Press RETURN (or ENTER) . The program i ntroduction message appears on the screen. Press RETURN . The following message appears on the screen :

HV : BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VARIATIONS

YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

ARE

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND ,

AND

AVAILABLE , ACENTRIC FACTOR,

A TEMPERATURE

ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) TO THE VLE DATA .

ALTERNATIVELY ,

SELECTED PARAMETERS •

AND

TO FIT MODEL PARAMETERS YOU CAN CALCULATE VLE WITH PREVIOUSLY

COMPARE THE RE SULTS WITH THE VLE DATA .

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" (This results in the program being used in the predictive mode. This example i s type 1 and press RETURN .

presented to demonstrate a case for which n o experi mental VLE data are

must provide, following the commands that appear on the screen, T,. P" the

avai lable. In this case no data are entered to, or accessed from, the disk. The user ,

acentric factor, and the K1 parameter of the PRS V equation of state for each

compound in addition to a temperature and model parameter(s) for the selected mode l . The program returns i sothermal x - y - P predictions at the temperature selected . Repeated temperature entries are allowed . ) •

At " SELECT A MIXING RULE MODEL : HV-0= HURON-VIDAL ORIGINAL MHVl= MODIFIED HURON-VIDAL 1ST ORDER MHV2 = MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS = HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O =EXIT l=HV-0 2 =MHV1 3 =MHV2 4 = LCVM S =HVOS " type 5 and press RETURN. (This commands results i n the use of the HVOS model for the mixing rule model . ) 1 45

Modeling Vapor-Li q u i d Eq u i l ibria

•

At " SELECT AN EXCESS FREE ENERGY MODEL : 1 =NRTL 2 =VAN LAAR 3 =UNIQUAC 4 =WILSON" type 1 and press RETURN. (This command results i n the selection of the NRTL model to be used as the excess free-energy term in the HVOS mixing rule.)

•

At

"YOU MAY ENTER A TITLE ( 3 0 CHARACTERS MAX . ) FOR THE MIXTURE TO BE PREDICTED ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) : " enter methanol-water 1 00 C and press RETURN. •

At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAP=KAPPA1 PARAMETER OF THE PRSV EOS INPUT TC 1 , PC 1 , W1 , KAP - 1 : " enter 5 1 2 . 5 8 , 80.9579, 0.56533 , -0. 1 68 1 6, and press RETURN .

•

At " INPUT TC2 , PC2 , W2 , KAP - 2 : " enter 647 .286,220. 8975 , 0 . 343 8 , - .06635 , and press RETURN.

•

At " INPUT TEMPERATURE in K : " type 373 . 1 5 and press RETURN.

•

At

" INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE DEFAULT IS BAR TYPE 1 IF YOU WANT PRESSURE IN BAR

,

.

( type 7 5 0 i f you want pres sure in mm . Hg , etc . ) " enter 750 and press RETURN. •

At " INPUT ALPHA OF THE NRTL MODEL : " type 0.35 and press RETURN .

•

At " INPUT REDUCED NRTL PARAMETERS P 1 2 , P2 1 : [ PIJ =AIJ/ ( RT ) AND AIJ IN CAL /MOL ] " enter 0 . 5 , 0. 5 and press RETURN. (The program then runs, and a summary of the results appears on the screen. In this case the percent error in total pressure is not reported because there is no experimental information . Press RETURN to continue . ) The following results appear o n the screen:

HV : BINARY VL E CALCULATIONS WITH THE HURON-VIDAL MODEL AND I T S VARIATIONS methanol -water 1 0 0 C EOS MODEL = HVOS ; EXCESS ENERGY MODEL= NRTL ALPHA= . 3 5 0 0 NRTL P 1 2 ( =Al2 / RT )

. 5000

NRTL P 2 1 ( =A2 1 / RT )

. 5000

TEMPERATURE in K : 3 7 3 . 1 5 PHASE VOLUMES ARE IN CC /MOL . FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR I S : 7 5 0 . 0 0

1 46

AppendiX 0: Computer Programs for B i nary M i xtures

X - EXP

P - EXP

P - CAL

Y-EXP

760 . 002

. 0000

Y - CAL

VL - CAL

VV-CAL

. 00001

22 . 51

30351 . 2

. 10 0 0

1184 . 10 8

. 41416

25 . 35

19313 . 7

. 2000

14 9 1 . 7 3 9

•

57447

2 8 . 24

1523 6 . 4

. 3000

172 2 . 962

. 6 6 3 17

3 1 . 18

13129 . 2

. 4000

1904 . 907

. 72 3 10

3 4 . 17

1182 9 . 8

. 5000

2055 . 883

. 7 6994

37 . 21

1092 5 . 2

. 6000

2 1 8 8 . 113

•

81120

40 . 29

10234 . 6

. 7000

2309 . 505

. 85137

43 . 42

9669 . 0

. 8000

2424 . 550

. 89389

46 . 58

9 184 . 0

. 9000

2534 . 854

. 94210

49 . 76

8758 . 4

1 . 0000

2638 . 801

. 99999

52 . 94

8387 . 2

pres s return to cont inue .

Press RETURN to continue. •

At "DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type N (or n) and press RETURN.

•

At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ?" type n (or N ) and press RETURN.

•

At

"DO YOU WANT TO DO VLE CALCULATION AT ANOTHER TEMPERATURE ( Y / N ) ? " type N (or n ) and press RETURN. •

At " DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y/ N ) ? " type n (or N) and press RETURN .

D. S .

Program WS: Bi nary VLE from Wong-San dier M ixing Rule The program WS i s used t o calculate VLE using the PRSV EOS and the Wong­ Sandier Mixing rule. One of four (UNIQUAC, van Laar, Wilson, or NRTL) excess free-energy models can be used with this mixing rule following the i nstructions that appear on the screen during program execution. Thi s program can be used in two ways, as shown in the tutorial that follows. If measured isothermal VLE data are avai l able, the program can be used to calcu­ late VLE at the measured liquid mole fractions with user-provided model parame­ ters and to compare the calculated bubble pressures and vapor mole fractions with the experimental ones. Alternatively, the program can be used to opti mize param­ eters of a selected model by fitting parameters to measured l iquid mole fraction versus bubble pressure data. Parameter optimization is done using a simplex algo­ rithm. In thi s mode the program reads previously stored data or accepts new data entered from the keyboard. The i nput data structure is identical to that described in

1 47

Modeling Va p or-Liq u i d Eq u i l ibria

Section 0.3 for the program VOW, and the details concerning the input data can be

found there.

During parameter optimization, as a requirement of the simplex method, an initial guess must be provided for each parameter. The initial guesses may be positive or negative numbers ; they are in reduced form and thus a value of between zero and one is a u seful choice in many cases . However, depending on the nonideality of the mixture, an i nitial guess may need to be significantly different from unity in order for the program to converge. If convergence cannot be achieved with a set of initial guesses, the user should try again with different initial guesses (see Examples D . 5 . A and D.5.B). If no experimental V L E data are avai lable, the program can b e used for predictions using i nternally generated liquid mole fractions of species 1 in the range from 0 to 1 at intervals of 0. 1 . In this case the user must provide all model parameters and temperature in addition to pure component critical temperature and pressure, acentric factor, and the

K1

parameter of the PRSV equation of state for each compound. An

example is given below (Example D . 5 .C) for this mode of operation of the program. The results from the program can be sent to a pri nter, to a disk file, or both . Thi s choice is made following the commands that appear on the screen upon completion of the calculations. Please see the following tutorial for further details.

Tutorial on the Use of WS.EXE Exam ple D.S .A: Use of the WS Model to Correlate Data •

Change the directory containing W S . EXE (e. g . , A> or C>, ) .

•

Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER).

•

A message introducing the program appears on the screen . Press RETURN to continue. The following appears on the screen :

WS : BINARY VLE CALCULATIONS WITH THE WONG- SANDLER MIXING RULE YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND,

AND

ACENTRIC FACTOR,

A TEMPERATURE

ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I SOTHERMAL x-y-P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN

1 48

Append ix D: Computer Programs for Bi nary M i xtures

( OR USE PREVIOUSLY ENTERED DATA ) TO THE VLE DATA .

ALTERNATIVELY ,

TO FIT MODEL PARAMETERS YOU CAN CALCULATE VLE

WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA . •

•

At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered data.) At

"DO YOU WANT TO DO A PARAMETER FIT ( ENTER A) OR CALCULATION WITH PREVIOUSLY FITTED PARAMETERS ( ENTER B ) ? " type A and press RETU RN . •

At

"ENTER NUMBER OF PARAMETERS TO BE FIT ( 2 OR 3 ) ( 2 ) : TWO PARAMETERS OF EXCESS FREE ENERGY MODEL ARE FIT ( 3 ) : IN ADDITION K12 PARAMETER OF THE WS MODEL I S FIT ( other parameters such as alpha of the NRTL mode l , or UNIQUAC pure component parameters must be supp l i ed by user . ) : "

•

•

type 3 and press RETURN. (With this command, all three parameters in the WS mixing rule, the two excess free-energy model parameters, and the binary interaction parameter, kiJ , are optimized . ) A t "DO YOU WANT T O USE AN EXISTING DATA FILE ( Y / N ) ? " type y (or Y) and press RETURN. At

" INPUT NAME OF THE EXISTING DATA FILE ( for example a : pe 3 7 3 . dat ) : " •

type a:pe373 .dat. At " SELECT AN EXCESS FREE ENERGY MODEL : O =EXIT 1 =UNIQUAC 2 =VAN LAAR 3 =WILSON 4 =NRTL" type 2 and press

•

RETURN. (This results in selection of the van Laar model for the excess-energy term i n the WS mixing rule . ) At

" PARAMETERS P 1 2 AND P2 1 ARE REDUCED AS DESCRIBED BELOW . AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES . FOR UNIQUAC , P IJ=EXP ( -AIJ / RT ) . FOR VAN LAAR , PIJ=AIJ . FOR WILSON , PIJ= ( VLPJ/VLPI ) * EXP ( -AI J / RT ) . FOR NRTL , PIJ=AIJ/RT . WITH THI S REDUCTION ,

IT I S POSSIBLE TO USE INITIAL GUESSES

IN THE RANGE OF ZERO TO ONE . INITIAL VALUES RECOMMENDED FOR P 1 2 AND P2 1 ARE 0 . 1 . INPUT INITIAL GUESSES FOR P 1 2 , P2 1 : " type 0. 1 , 0. 1 and press RETURN . 1 49

Modeling Vapor-Li q u i d Eq u i l i bria

•

At " INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12 : "

type 0 . 3 and press RETURN. (At thi s stage the program i s run to optimize the two parameters of the NRTL model and the binary interaction parameter, kij . Intermediate results will be continuously displayed on the screen in the form of an error bar. When the optimization is complete, a message summarizing the results appears on the screen for inspection . Press RETURN to continue.) The following results appear on the screen : WS : THE WONG- SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS PENTANE ETHANOL 3 7 2 . 7 K EXCESS ENERGY MODEL = VAN LAAR K12 = . 3 0 8 4 P 1 2 ( =DIMENS IONLESS KAPPA1 2 OF VAN LAAR )

1 . 2172

P2 1 ( =DIMENS IONLESS KAPPA2 1 OF VAN LAAR ) 2 . 9 7 0 9 TEMPERATURE i n K : 3 7 2 . 7 0 PHASE VOLUMES ARE IN C C / MOL, PRESSURE I S IN UNITS OF THE DATA . X- EXP

P - EXP

P - CAL

Y-EXP

Y-CAL

. 0000

220 . 000

220 . 608

. 00000

. 00011

69 . 10

13461 . 1

. 0830

422 . 600

392 . 650

. 49100

. 46706

70 . 22

7335 . 6

. 17 1 0

537 . 400

5 17 . 3 2 6

. 62900

. 6 1874

72 . 21

5396 . 2

VL - CAL

VV-CAL

. 3030

618 . 800

618 . 967

. 69000

. 7 0420

77 . 00

4376 . 9

. 4410

654 . 3 00

660 . 413

. 72400

. 73733

84 . 65

4045 . 8

. 62 6 0

678 . 100

67 8 . 3 9 9

. 74700

. 7 57 9 6

99 . 44

3 9 11 . 1

7360

684 . 3 00

683 . 632

. 7 6800

. 77313

110 . 22

3869 . 2

•

. 8390

682 . 600

682 . 601

. 80300

•

8 0 117

12 0 . 5 3

3864 . 0

. 9370

658 . 100

657 . 64 8

. 86000

. 86940

128 . 94

4006 . 8

. 9999

591 . 000

5 9 1 . 174

. 99990

. 99964

132 . 65

4470 . 6

press return to cont inue .

•

•

•

I SO

Press RETURN to continue. At "DO YOU WANT A PRINT- OUT ( Y / N ) ? " type y (or Y) and press RETURN . (With thi s command the results, shown above, are sent to the pri nter. ) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y ) and press RETURN. At " INPUT A NAME FOR THE OUTPUT FILE" : type A:TEMP5 .0UT and press RETURN . (With this command the results shown above are saved on the disk in drive A under the name TEMPS .OUT in ASCII code . )

Appendix D: Computer Programs for Bi nary M i xtures

•

At " DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y / N ) ?" type n (or N) and press RETURN.

Exam ple D.S . B: Use of the WS Model to Correlate Data • • •

Change the directory containing WS.EXE (e.g., A> or C>, etc . ) . Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER) . The message introducing the WS program appears on the screen. Press RETURN to continue. The following appears on the screen:

WS : BINARY VL E CALCULATIONS WITH THE WONG- SANDLER MIXING RULE YOU MODE

CAN

USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS .

(1) :

IF NO T - P -x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND ,

AND

ACENTRIC FACTOR,

A TEMPERATURE

ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x - y - P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) TO THE VLE DATA .

ALTERNATIVELY ,

TO FIT MODEL PARAMETERS YOU CAN CALCULATE VLE

WITH PREVIOUSLY SELECTED PARAMETERS •

AND

COMPARE THE RE SULTS WITH THE VLE DATA .

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM"

•

type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered data. ) At

" DO YOU WANT TO DO A PARAMETER FIT ( ENTER A ) OR CALCULATION WITH PREVIOUSLY FITTED PARAMETERS ( ENTER B ) ? " type A and press RETURN. •

At "ENTER NUMBER OF PARAMETERS TO BE FIT ( 2 OR 3 ) (2)

: TWO PARAMETERS OF EXCESS FREE ENERGY MODEL ARE FIT

(3)

: IN ADDITION K12 PARAMETER OF THE WS MODEL I S FIT

( other parameters such as alpha of the NRTL mode l , or UNIQUAC pure component parameters must be supp l i ed by user . ) : " type 2 and press RETURN.

(With this command, the excess free-energy parameters in the WS mix ing rule are optimized. The initial value of the binary interaction parameter, kij , is used in computations. ) lSI

Mode l i ng Vapor-Liq u i d E q u i l i bria

•

At " DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type n (or N) and press RETURN .

•

At

" PROVIDE THE FOLLOWING INPUT INFORMATION INPUT NEW DATA FILE NAME : " type tempo6.DAT and press RETURN. •

At " INPUT A TITLE FOR THE NEW DATA FILE : " type methanol-water at 373 K and press RETURN .

•

At "CRITICAL PARAMETERS : TC=CRITICAL TEMPERATURE , K PC=CRITICAL PRESSURE , BAR W=ACENTRIC FACTOR KAP=THE PRSV EOS KAPPA- 1 PARAMETER INPUT TC 1 , PC 1 , W1 , KAP 1 : " type 5 1 2. 5 8 , 80.9579, 0 .565 3 3 , -0. 1 68 1 6, and press RETURN. (These are parameters of methanol for the PRSV EOS from Table 3 . 1 . 1 . )

•

At " INPUT TC2 , PC2 , W2 , KAP2 : " type 647 .286, 220 . 8975 , 0.343 8 , -0.06635 , and press RETURN.

(These are parameters of water for the PRSV EOS from Table 3 . 1 . 1 . ) •

At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN.

•

At " INPUT TEMPERATURE T in K : " type 373 . 1 5 and press RETURN.

•

At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVI SION (e.g.

i f ori ginal data in mm Hg ,

i f origina l data i n p s i a , •

type

type

750

14 . 5 e t c . ) : " type 750 and press RETURN .

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( X1EXP ) OF SPEC IES 1 , VAPOR MOLE FRACTION ( Y1EXP ) OF SPEC IES 1 , AND BUBBLE POINT PRESSURE ( PEXP ) ( three in a row ,

s eparated by comma s )

INPUT X1EXP , Y 1EXP , PEXP : " type 0.035, 0. 1 9 1 , 93 1 . •

At " INPUT X1EXP , Y1EXP , PEXP : " type 0 . 2 8 1 , 0 . 6 1 9, 1 5 35 .96.

•

At " INPUT X1EXP , Y1EXP , PEXP : " type 0 . 8 26, 0.9 1 J , 2 3 3 7 . 7 6 . (When the number o f items o f data, specified b y NP, here three, i s entered, the program writes the data to a file under the name tempo6.dat as specified above and continues. Thi s data file becomes an

existing data file and can be used when

this program or other EOS programs are run again . ) •

A t " SELECT AN EXCESS FREE ENERGY MODEL : O =EXIT 1 =UNIQUAC 2 =VAN LAAR 3 =WILSON 4 =NRTL" type 2 and press RETURN . (This results in selection of the van Laar model for the excess energy term in the

WS mixing rule. ) •

At

" PARAMETERS P 1 2 AND P2 1 ARE REDUCED AS DESCRIBED BELOW . AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES . 1 52

Appendix D: Computer Programs for Binary M i xtures

FOR UNIQUAC , PIJ=EXP ( -AIJ / RT ) . FOR VAN LAAR , P I J=AIJ . FOR WILSON , P I J= ( VLPJ/VLPI ) *EXP ( -AI J / RT ) . FOR NRTL , PIJ=AIJ / RT . WITH THI S REDUCTION, IT I S POSSIBLE TO USE INITIAL GUESSES IN THE RANGE OF ZERO TO ONE . INITIAL VALUES RECOMMENDED FOR P 1 2 AND P2 1 ARE 0 . 1 . INPUT INITAL GUESSES FOR P 1 2 , P 2 1 : " type 0. 1 , 0 . 1 and press RETURN . •

At " INPUT THE WS MIXING-RULE PARAMETER K12 : " type 0 . 2 and press RETURN. (At this stage the program optimizes the two parameters of the van Laar model. Intermediate results will continuously be displayed on the screen in the form of an error bar. When the optimization is completed, a message summarizing the results appears on the screen . Press RETURN to continue.) The following results appear on the screen :

WS : THE WONG- SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS methanol - water at 3 7 3 K EXCESS ENERGY MODEL = VAN LAAR K12= . 2 0 0 1 P 1 2 ( =DIMENS IONLES S KAPPA12 O F VAN LAAR )

. 6358

P2 1 ( =DIMENS IONLESS KAPPA2 1 OF VAN LAAR )

. 1095

TEMPERATURE in K : 3 7 3 . 1 5 PHASE VOLUMES ARE IN CC/MOL , PRESSURE IS IN UNITS OF THE DATA . X- EXP

P - EXP

. 0350

93 1 . 000

P - CAL

Y- EXP

Y-CAL

VL - CAL

VV-CAL

931 . 024

. 19100

. 2 0 8 17

22 . 90

24717 . 6

•

2810

1535 . 960

1535 . 929

. 61900

. 61062

26 . 45

1482 0 . 6

•

8260

2337 . 7 60

2375 . 031

. 91100

. 92047

43 . 50

9391 . 6

pre s s return to cont inue .

•

•

•

Press RETURN to continue. At " DO YOU WANT A PRINT-OUT ( Y / N ) ? " type y (or Y) and press RETURN . (With this command the results, shown above, are sent to the printer. ) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y /N ) ? " type n (or N ) and press RETURN . At "DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y / N ) ? " type n (or N) and press RETURN.

Exam ple D. S .C: Use of the WS M ixing Ru le in the Pred ictive Mode • •

Change to the directory containing WS.EXE (e. g. , A> or C>, etc . ) Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER). 1 53

Mode l i ng Vapor-Liquid Eq u i l i bria

•

the message introducing the program appears on the screen . Press RETURN to continue. The following appears on the screen :

WS : BINARY VLE CALCULATIONS WITH THE WONG- SANDLER MIXING RULE

YOU CAN USE THI S PROGRAM FOR VLE CALCULATION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND ,

ACENTRIC FACTOR ,

AND A TEMPERATURE

ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I SOTHERMAL x-y-P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA ) TO THE VLE DATA . ALTERNATIVELY ,

TO FIT MODEL PARAMETERS YOU CAN CALCULATE VLE WITH PREVIOUSLY

SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA . •

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM"

•

type 1 and press RETURN. (This example serves to demonstrate the predictive mode of the program WS, which is selected with the preceding entry. This mode i s used in the absence of VLE data, and therefore no data are entered to, or can be accessed from the disk in thi s mode. Instead, the user provides the critical temperature, critical presssure, acentric factor, and the PRSV k1 parameter for each pure component, selects an excess free-energy model; provides model parameters and a temperature. The program will return isothermal x - y - P predictions at the temperature entered, in the composition range x 1 = 0 to I , at intervals of 0. 1 .) At

"YOU MAY ENTER A TITLE ( 2 5 CHARACTERS MAX . ) FOR THE MIXTURE TO BE PREDICTED ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) : " enter 'meoh-water binary system ' •

and press RETURN . At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAP=KAPPA- 1 PARAMETER OF THE PRSV EOS INPUT TC 1 , PC 1 , W1 , KAP 1 : " enter 5 1 2.58, 80.9579, 0.56533,

•

1 54

-0. 1 68 1 6, and press RETURN . At " INPUT TC2 , PC2 , W2 , KAP2 : " enter 647 .286, 220. 8975, 0.343 8, -0.06635, and press RETURN.

Appendix D: Computer Programs for Bi nary M i xtures

• •

At " INPUT TEMPERATURE in K : " type 373. 1 5 and press RETURN. At

" INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE . DEFAULT IS BAR , TYPE 1 IF YOU WANT PRESSURE IN BAR .

( type 7 5 0 i f you want pres sure

•

in nun Hg , etc . ) : " enter 750 and press RETURN. At " SELECT AN EXCESS FREE ENERGY MODEL :

•

O =EXIT l =UNIQUAC 2 =VAN LAAR 3 =WILSON 4 =NRTL" type 2 and press RETURN. (This results in selection of the van Laar model for the excess energy term in the WS mixing rule.) At

" INPUT REDUCED PARAMETERS P 1 2 AND P2 1 . BELOW AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES . FOR UNIQUAC , P IJ=EXP ( -AIJ / RT ) . FOR VAN LAAR , P I J=AIJ . FOR WILSON , P I J= ( VLPJ/VLPI ) *EXP ( -AI J / RT ) . FOR NRTL , P I J=AI J / RT . WITH THI S REDUCTION ,

IT IS POSSIBLE TO USE INITIAL GUESSES IN THE

RANGE OF ZERO TO ONE . INPUT PARAMETERS P 1 2 , P2 1 : " type 0. 7727, 0.3088, and press RETURN. •

At " INPUT THE ws MIXING-RULE PARAMETER K12 : " type 0. 1 and press RETURN. (At this stage the program runs, and a message summarizing results appears on the screen. No average absolute deviation in bubble pressure is reported in that message because measured bubble pressure information is not available. Press RETURN to continue. ) The following results appear o n the screen:

WS : THE WONG- SANDLER MIXING RULE FOR BINARY VL E CALCULATIONS meoh-water binary system EXCESS ENERGY MODEL = VAN LAAR K12= . 1 0 0 0 P 1 2 ( =DIMENS IONLESS KAPPA12 O F VAN LAAR )

. 7727

P2 1 ( =DIMENS IONLESS KAPPA2 1 O F VAN LAAR )

. 3088

TEMPERATURE in K : 3 7 3 . 1 5 PHASE VOLUMES ARE IN C C / MOL . FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR I S : 7 5 0 . 0 0 X- EXP

P - EXP

P - CAL

Y- EXP

VL-CAL

VV-CAL

22 . 51

30349 . 3

. 3847 1

24 . 29

20242 . 6

. 53 2 0 2

26 . 25

16663 . 5

. 62 4 2 0

28 . 43

14 6 1 0 . 4

Y-CAL

. 0000

760 . 051

. 00008

. 1000

1132 . 2 00

. 2000

1369 . 104

. 3000

1555 . 544

1 55

Model i ng Vapor-Li q u i d Eq u i l i bna

. 4000

17 2 0 . 619

. 69525

30 . 86

13161 . 7

. 5000

187 6 . 244

. 75594

33 . 59

12027 . 8

. 6000

2 027 . 9 84

. 81065

3 6 . 63

11088 . 2

. 7000

2178 . 725

. 86148

40 . 04

10283 . 0

. 8000

2 3 3 0 . 114

. 90957

43 . 86

9577 . 9

. 9000

2483 . 203

. 95559

4 8 . 14

8951 . 0

1 . 0000

2638 . 7 82

1 . 00000

52 . 94

8 3 87 . 3

pre s s return to cont inue .

Press RETURN to continue. • At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type n (or N) and press RETURN. • At"DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y ) and press RETURN. • At " INPUT A NAME FOR THE OUTPUT FILE : " type A:TEMP8 .0UT and press RETURN . • At " DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y / N ) ? " type y (or Y) and press RETURN. • At " INPUT TEMPERATURE T in K : " enter 393. 1 5 and press RETURN . • At " SELECT AN EXCESS FREE ENERGY MODEL : O =EXIT l =UNIQUAC 2 =VAN LAAR 3 =WILSON 4 =NRTL" type 2 and press RETURN. (This command results in the selection of the van Laar model to be used as the excess free-energy term in the WS mixing rule.) • At " INPUT REDUCED PARAMETERS P 1 2 AND P2 1 . BELOW AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES . FOR UNIQUAC , P IJ=EXP ( -AIJ / RT ) . FOR VAN LAAR , PIJ=AIJ . FOR WILSON , P I J= ( VLPJ/VLP I ) * EXP ( -AI J / RT ) . FOR NRTL , PIJ=AIJ/RT . WITH THIS REDUCTION , IT IS POSSIBLE TO USE INITIAL GUESSES IN THE RANGE OF ZERO TO ONE . INPUT PARAMETERS P 1 2 , P2 1 : " type 0.7727, 0.3088, and press RETURN. •

At " INPUT THE WS MIXING-RULE PARAMETER Kl2 : " type 0. 1 and press RETURN. (At this stage the program runs, and the message summarizing the results appears again on the screen. Press RETURN to continue.) The fol lowing results appear on the screen:

WS : THE WONG- SANDLER MIXING RULE FOR BINARY VL E CALCULATIONS meoh-water binary system

1 56

Appe n d i x D: Computer Programs for Bi nary M i xtu res

EXCESS ENERGY MODEL = VAN LAAR K12= . 10 0 0 P 1 2 ( =DIMENS IONLESS KAPPA12 O F VAN LAAR )

. 7727

P2 1 ( =DIMENS IONLESS KAPPA2 1 O F VAN LAAR )

. 3088

TEMPERATURE i n K : 3 9 3 . 1 5 PHASE VOLUMES ARE IN CC/MOL . FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR I S : 7 5 0 . 0 0 X- EXP

P - EXP

P - CAL

Y- EXP

Y-CAL

VL - CAL

VV-CAL

. 0000

1488 . 88 1

. 00007

22 . 95

1622 1 . 6

. 1000

2145 . 238

. 3 6245

24 . 7 8

11147 . 8

. 2000

2564 . 061

. 50879

26 . 81

9261 . 0

. 3000

2892 . 341

. 60225

2 9 . 07

8161 . 2

. 4000

3181 . 187

. 67 5 1 9

31 . 60

7379 . 1

. 5000

3451 . 7 65

. 7 3 8 14

34 . 45

6763 . 4

. 6000

3 7 14 . 1 0 4

. 79549

37 . 65

6250 . 5

. 7000

3973 . 501

. 84938

41 . 25

5808 . 5

. 8000

4232 . 937

. 90089

45 . 32

5429 . 2

. 9000

4494 . 9 06

. 9 5102

49 . 92

5070 . 1

1 . 0000

4760 . 618

1 . 00001

5 5 . 12

4753 . 9

pres s return to cont inue .

Press RETURN to continue • At "DO YOU WANT A PRINT - OUT ( Y/ N ) ? " type n (or N) and press RETURN . • At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT F:ILE ( Y/ N ) ? " type y (or Y ) and press RETURN . • A t " :INPUT A NAME FOR THE OUTPUT F:ILE : " type A:TEMP8.0UT and press RETURN. (With this entry the results above are appended to the file TEMP8.0UT, which already contains the predictions for this binary system at 3 7 3 . 1 5 K) . • At " DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y/ N ) ? " type n (or N ) and press RETURN . • At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (YIN)?" type n (or N) and press RETURN. 0.6 .

Program WSU N F: Binary VLE Predictions Using the Wo ng-Sandier M ixing Rule Com b i n ed with the U N I FAC Excess Free-Energy Model The program WSUNF is used to predict VLE by means of the PRSV EOS coupled with the Wong-Sandier mi xing rule and the UNIFAC group contribution method 1 57

Modeling Vapor-Liquid Eq u i l i b ria

without using any measured VLE data. To use the Wong-Sandier mixing rule this way, it is necessary to determine the value of the kij parameter of this mixing rule to match the excess Gibbs free energy from the EOS with the excess free energy of the UNIFAC activity coefficient model as closely as possible at or near 25°C (see Section 5 . 1 for details). Thus, one task of the WSUNF program is the evaluation of the optimum kij parameter of the WS model by matching the excess free-energy functions mentioned above. This is accompli shed by entering the appropriate commands during execution (a tutorial is provided below). The program can be used in two ways. If measured isothermal VLE data are avai l­ able, the program can be run to predict VLE at the measured liquid mole fractions; then the calculated and measured bubble pressures and vapor mole fractions are com­ pared. In this mode the program reads previously stored data or accepts new data entered from the keyboard. The input data structure is identical to that used for all other EOS mixture programs, and the details of the input data have been given in Section 0 . 3 . I f n o experi mental data arc available, bubble pressures and vapor mole fractions are calculated at liquid mole fractions x1 0 to I at intervals of 0. 1 . In this mode no data are entered to, or accessed from, the disk. Instead, the user provides critical tem­ perature, critical pressure, acentric factor, and the PRSV k1 parameter for each pure component in addition the Wong-Sandier mixing-rule parameter k 1 2 and a tempera­ ture. The program then returns i sothermal bubble pressure and vapor mole fraction predictions at the temperature entered . In either mode, during the matching of excess energy functions from the equation of state and from UNIFAC, the k 1 2 parameter is varied to minimize the obj ective function F = L I G��s G��IFAc I using a simplex algorithm. As a requirement of the simplex approach, an initial guess must be provided for the k 1 2 . The initial guess may be a positive or a negative number; usually between zero and one. If convergence cannot be achieved with the selected initial guess, the user should try again with different choices. The results from WSUNF can be sent to a printer, to a disk file, or both. This selection is made from the commands that appear on the screen at the completion of the calculations . See the following tutorial for further details. =

-

Tutorial on the Use of WSU N F. EXE Example D. 6.A: Use of the Wong-San d i e r M ixing Ru le and U N I FAC for Binary VLE Pred icti ons Using an Existi ng Data File • •

1 58

Change to the directory containing WSUNF.EXE (e.g. , A>). Start the program by typing WSUNF at the DOS prompt. Press RETURN (or ENTER) .

Appendix D: Computer Programs for B i nary M i xtures

•

A program introduction message appears on the screen. Press RETURN to continue. At

"THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK . THESE DATA ARE STORED IN TWO FILES NAMED UNF I 1 . DTA AND UNFI2 . DTA . UNF I 1 . DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION UNF I 2 . DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION . IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY THEN ENTER 1 , OTHERWISE ENTER 2 : " type 2 and press RETURN.

•

(The data files UNFI I .DTA and UNFI2.DTA are provided on the disk that accompanies thi s monograph. The program is easier to use if these data files are copied to the hard disk directory used to run the programs . In this case, I must be entered . An entry of 2, as above, i ndicates that these files are not present in the current directory. In this case the user must provide the directory and file names as indicated below. ) At

"TYPE THE DIRECTORY AND NAME OF THE FILE WHERE UNIFAC GROUP PARAMETER INFORMATION I S STORED ( default = a : UNFI 1 . DTA ) " •

type a: UNFI I .DTA and press RETURN . At "TYPE THE DIRECTORY AND NAME OF THE FILE WHERE UNIFAC BINARY INTERACTION PARAMETER INFORMATION I S STORED ( defau l t = a : UNF I 2 . DTA ) " type a:UNFI2.DTA and press RETURN . The following message appears on the screen :

WSUNF : BINARY VLE CALCULATIONS WITH THE WONG- SANDLER MIXING RULE AND THE UNIFAC MODEL YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

AND

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND

AND

ACENTRIC FACTOR

A TEMPERATURE .

IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS I T ION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I SOTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

AND

TO PREDICT VLE BEHAVIOR

COMPARE RESULTS WITH THE VLE DATA . •

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM : "

enter 2 and press RETURN. 1 59

M o d e l i n g Vapor-Li q u i d Eq u i l i b ri a

•

•

At "DO YOU WANT TO USE AN EXISTING DATA FILE ( Y/ N ) ? " type y (or Y) and press RETURN. At

" INPUT THE NAME OF EXISTING DATA FILE ( for example : a : am2 5 . dat ) : "

•

type a:am25 .dat and press RETURN . (This results in the us e o f am25 .dat, the existing isothermal V LE data for the acetone plus methanol binary system at 25"C. ) At "ENTER UNIFAC GROUP PARAMETER INFORMATION

•

type 'acetone' and press RETURN . (Following the preceding comment a group selection table wi l l appear o n the screen. The user must follow the instructions at the top of the table to choose one CH3 and one CH3CO group for acetone and press RETURN.) At "ENTER UNIFAC GROUP PARAMETER INFORMATION

•

type ' methanol ' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose one CH3 0H group for methanol and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue. ) At

ENTER NAME OF THE COMPONENT 1 "

ENTER NAME OF THE COMPONENT 2 "

" INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12 : " •

type 0. 1 and press RETURN. At " DO YOU WANT TO FIT THE K12 TO Gex OF UNIFAC ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH K12 ENTERED ( 2 ) ? " type 1 and press RETURN. (At this point the program calculates a value of k;1 that matches the excess Gibbs free-energy values from the EOS and from the UNIFAC model. Intermediate results will continuously be displayed on the screen in the form of an error bar. When the optimization is completed a message summarizing the results appears on the screen . Press RETURN to continue. ) Calculated results are then displayed o n the screen a s shown below :

WSUNF : THE WONG- SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS WITH THE UNIFAC MODEL ACETONE METHANOL 2 5 C K12 = . 1 0 2 0 TEMPERATURE ( K )

=

2 9 8 . 15

PHASE VOLUMES ARE IN C C / MOL , PRESSURE IS IN UNITS OF THE DATA .

1 60

Append ix D: Computer Programs for Bi nary M i xtures

X-EXP

P - EXP

P - CAL

Y- EXP

Y - CAL

VL-CAL

VV-CAL

. 0001

127 . 7 00

12 6 . 8 4 7

. 00010

. 00037

47 . 72

14 5 9 3 5 . 7

. 0610

146 . 2 00

145 . 47 8

. 21600

. 17 8 5 2

49 . 13

127134 . 2

. 0860

153 . 200

152 . 004

. 26800

. 23233

49 . 74

121637 . 0

. 0940

156 . 000

153 . 97 2

. 29000

. 24 7 8 8

49 . 94

120069 . 9

. 2040

178 . 600

176 . 2 59

. 43500

. 40774

52 . 85

104762 . 7

. 2360

183 . 400

181 . 400

. 46800

. 4 4 17 8

53 . 7 6

101763 . 7

. 4020

205 . 200

201 . 794

. 59 8 0 0

. 57 6 2 8

58 . 95

91364 . 5

. 4600

211 . 200

207 . 152

. 62 3 0 0

. 61433

60 . 95

88968 . 7

. 5820

220 . 800

2 1 6 . 4 17

. 69500

. 68933

6 5 . 47

85099 . 6

. 6610

224 . 700

221 . 213

. 74100

. 73771

6 8 . 62

83219 . 0

. 7860

231 . 000

226 . 992

. 80700

. 81983

7 3 . 94

81046 . 8

. 8120

231. 200

227 . 897

. 81000

. 83841

7 5 . 10

8 0 7 14 . 1

pre s s return to see more resu l t s on the screen .

(This message appears when the number of data points exceeds twelve. Press RETURN to continue.) . 87 8 0

230 . 780

229 . 669

. 85500

. 88891

78 . 11

80063 . 3

. 9999

230 . 911

230 . 553

. 99990

•

99990

8 3 . 94

79703 . 0

pre s s return to cont inue .

•

•

•

•

Press RETURN to continue. A t " DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer. ) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y ) and press RETURN. At " INPUT A NAME FOR THE OUTPUT FILE : " type A :TEMP6.0UT and press RETURN . (With this command the results shown above are saved on the disk in drive A under the name TEMP6.0UT in ASCII code. ) A t " DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y / N ) ? " type n (or N) and press RETURN.

Exam p l e D. 6.B: Use of the WS M ixing Ru le and U N I FAC for Bi nary VLE Predictions Without an Existi ng Data File • •

Change to the directory containing WSUNF.EXE (e.g., A>). Start the program by typing WSUNF at the DOS prompt. Press RETURN (or ENTER) .

161

M o d e l i ng Vapor-Liq u i d Eq u i l i bria

•

A program introduction message appears on the screen . Press RETURN to continue. At

"THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK . THESE DATA ARE STORED IN TWO FILES NAMED UNFI 1 . DTA AND UNFI2 . DTA . UNF I 1 . DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION . UNFI 2 . DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION . THEN ENTER 1 ; OTHERWISE ENTER 2 : " type I and press RETU RN .

IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY ,

(The data files UNFI l .DTA and UNFI2.DTA are provided on the disk that accompanies thi s monograph. The program is easier to use if these data files are copied to the hard disk directory used to run the programs . In this case, an entry of I must be used. An entry of 2, as discussed earlier in Example D.6.A, indicates that these files are not present in the current directory. In that case the user must provide the directory and file names . ) The fol lowing message appears on the screen :

WSUNF : BINARY VLE CALCULATIONS WITH THE WONG - SANDLER MIXING RULE AND THE UNIFAC MODEL YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

AND

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND

AND

ACENTRIC FACTOR

A TEMPERATURE .

IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE I S OTHERMAL x-y-P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

AND

TO PREDICT VLE BEHAVIOR

COMPARE RESULTS WITH THE VLE DATA . •

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" •

•

1 62

enter 2 and press RETURN. At "DO YOU WANT TO USE AN EXISTING DATA FILE ( Y / N ) ? " type n (or N) and press RETURN . At " PROVIDE THE FOLLOWING INPUT INFORMATION : INPUT NEW DATA FILE NAME : " type a: tempo8 .dat and press RETURN . (The preceding command will lead to saving a data file named tempo8.dat on the disk in drive A. You must have a disk in the A drive or select another directory by typing c: tempo8.dat, for example, to save the file on the hard drive. )

Appendix D: Computer Programs for Bi nary M i xtures

•

•

At " INPUT A TITLE FOR THE NEW DATA FILE : " type methanol water 25C and press RETURN. At "CRITICAL PARAMETERS : TC=CRITICAL TEMP K I

PC=CRITICAL PRESSURE , BAR W=ACENTRIC FACTOR KAPPA=THE PRSV EOS KAPPA- 1 PARAMETER

•

• • •

INPUT TC 1 , PC 1 , W1 , KAPPA- 1 : " type 5 1 2. 80, 80.9579, 0.565 3 3 , -0. 1 68 1 6, and press RETURN. (These are EOS constants for methanol . ) A t " INPUT TC2 , PC2 , W2 , KAPPA- 2 : " type 647 .286, 220. 8975 , 0. 3438, -0.06635. (These are EOS constants for water. ) At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN. At " INPUT TEMPERATURE in K : " type 298. 1 5 and press RETURN. At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVIS IO:N ( e . g . if original data in mm Hg , type 7 5 0 i f original data i n psi a , type 1 4 . 5 etc . ) : " enter 750 and press RETURN. •

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( X1EXP ) OF SPECIES 1 , VAPOR MOLE FRACTION ( Y1EXP ) OF SPECIES 1 , AND BUBBLE POINT PRESSURE (l· r�: . ( three in a row, separated by commas ) INPUT X1EXP , Y1EXP , PEXP : " type 0. 1 9, 0.6 1 87, 5 3 , and press RETURN. •

•

•

At " INPUT X1EXP , Y1EXP , PEXP : " type 0.4943 , 0.7934, 82. 3, and press RETURN . At " INPUT X 1EXP , Y 1EXP , PEXP : " type 0. 8492, 0.93 84, 1 1 2, and press RETURN. (When the number of items of data specified by NP, here three, has been entered, the program writes the data to a file with the name tempo8. dat as specified above and continues . This data file becomes an existing data file and can be used when this program or other EOS programs are run again.) At "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1 "

type ' methanol ' and press RETURN.

•

(Following the preceding comment a group selection table will appear on the screen. The user must fol low the instructions at the top of the table to choose one CH30H for methanol and press RETURN . ) A t "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2 "

type 'water' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose one H2 0 group for water and then press RETURN. Following 1 63

'

)

Modeling Vapor- L i q u i d E q u i l i bria

•

this a summary of group selections will appear on the screen. Press RETURN to continue. ) At " INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12 : "

•

type 0. 1 and press RETURN. At "DO YOU WANT TO FIT THE K12 TO Gex OF UNIFAC ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH Kl2 ENTERED ( 2 ) ? " type I and press RETURN. (At this stage the program obtains a value of kij that matches the excess Gibbs free-energy values from the EOS and from the UNIFAC model. Intermediate results will continuously be displayed on the screen in the form of an error bar. When the optimization is completed a message summarizing the results appears on the screen. Press RETURN to continue. ) Calculated results are displayed on the screen a s shown below.

WSUNF : THE WONG- SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS WITH THE UNI FAC MODEL METHANOL WATER 2 5C K12 = . 0 8 7 7 TEMPERATURE ( K )

= 298 . 15

PHASE VOLUMES ARE IN C C / MOL , PRESSURE IS IN UNITS OF THE DATA X- EXP

P - EXP

P - CAL

Y- EXP

. 1900

53 . 000

55 . 682

•

61870

Y-CAL •

64050

VL - CAL

VV-CAL

24 . 66

333363 . 3

•

4943

82 . 3 00

83 . 919

•

79340

•

82829

31 . 50

220964 . 8

•

8492

112 . 000

112 . 9 63

•

93840

•

95350

42 . 16

163964 . 9

pres s return to cont inue .

•

•

•

•

1 64

Press RETURN to continue. At "DO YOU WANT A PRINT- OUT { Y / N ) ? " type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer. ) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE { Y/ N ) ? " type y (or Y ) and press RETURN. At " INPUT A NAME FOR THE OUTPUT FILE : " type A:TEMP8 .0UT and press RETURN. (With thi s command the results shown above are saved on the disk in drive A under the name TEMP8 . 0UT in ASCII code. ) A t "DO YOU WANT T O DO ANOTHER VLE CALCULATION { Y/ N ) ? " type y (or Y) and press RETURN. In the first part of this example, we matched excess Gibbs energy from the PRSV equation of state with excess Gibbs energy from UNIFAC at 25"C and obtained

Appe ndix D: Computer Programs for Binary M i xtures

the Wong-Sandler mixing rule binary interaction parameter, k 1 2 , as 0.0869. Also we compared predictions at 25oC with k 1 2 0. 0869 to experimental data entered from the keyboard. In the second part of this example, shown below, we use the same k 1 2 value to predict isothermal VLE data at 1 00°C, this time using internal ly generated liquid mole fractions x 1 0, 0. 1 , 0.2, etc . This mode is implemented as described below. The following message reappears on the screen: =

=

WSUNF : BINARY VL E CALCULATIONS WITH T HE WONG- SANDLER MIXING RULE AND THE UNI FAC MODEL

YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

{1) :

IF NO T - P -x-y DATA TO COMPARE RESULTS WITH ARE AVAI LABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

ACENTRIC FACTOR

AND PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND AND A TEMPERATURE . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

{2) :

IF YOU HAVE ISOTHERMAL x-y-P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN { OR USE PREVIOUSLY ENTERED DATA )

TO PREDICT VLE BEHAVIOR

AND COMPARE THE RE SULTS WITH THE VLE DATA .

•

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM : " •

enter 1 and press RETURN. At

"YOU MAY ENTER A TITLE ( 2 5 CHARACTERS MAX . ) FOR THE MIXTURE TO BE PREDICTED ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) : " enter meoh-water I 00 C and •

press RETURN. At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPA- 1 PARAMETER OF THE PRSV EOS

•

• •

INPUT TC l , PC 1 , W1 , KAPPA- 1 : " enter 5 1 2. 5 8 , 80.9579, 0.565 3 3 , -0. 1 68 1 6, and press RETURN. At " INPUT TC2 , PC2 , w2 , KAPPA- 2 : " enter 647 .286, 220. 897 5 , 0. 343 8, -0.06635 , and press RETURN. At " INPUT TEMPERATURE in K : " type 3 7 3 . 1 5 and press RETURN. At " INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE DEFAULT IS BAR , TYPE 1 IF YOU WANT PRESSURE IN

BAR .

1 65

Model1ng Vapor-Liquid Eq u i l i bria

( type 7 5 0 if you want pres sure in mm Hg , etc . ) : " enter 750 and press RETURN. •

At "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1 " type 'methanol ' and press RETURN . (Following the preceding comment a group selection table will appear o n the screen. The user must fol l ow the instructions at the top of the table to choose one CH30H for methanol and press RETURN . )

•

A t "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2 " type 'water' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose one H20 group for water and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue. )

•

At " INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12 : " type 0.086 and press RETURN.

•

At " DO YOU WANT TO FIT THE K 1 2 TO Gex OF UNIFAC ( 1 ) OR DO YOU WANT TO CALCULATE VLE WITH K12 ENTERED ( 2 ) ? " type 2 and press RETURN . (At this stage the program runs with k 1 2 = 0.086, and a summary of intermediate results appears on the screen for inspection. Because no experimental data are entered in this case, no average absolute deviation in pressure is reported. Press RETURN to continue . ) The following results appear on the screen :

WSUNF : THE WONG- SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS WITH THE UNIFAC MODEL METHANOL WATER 1 0 0 C K12= . 0 8 6 0 TEMPERATURE ( K ) = 3 7 3 . 1 5 PHASE VOLUMES ARE IN CC /MOL . FACTOR YOU ENTERED TO CONVERT PRESSURE FROM X-EXP

P - EXP

P - CAL

Y - EXP

Y - CAL

BAR

IS : 7 5 0 . 0 0

VL - CAL

VV-CAL

. 0000

760 . 510

. 00076

22 . 51

30330 . 7

. 1000

1160 . 189

. 40078

24 . 40

19742 . 1

. 2000

1424 . 83 7

. 55177

26 . 49

15992 . 9

. 3000

1624 . 02 0

. 64 0 0 5

28 . 79

13974 . 3

. 4000

1790 . 43 0

. 70473

31 . 33

12 6 3 0 . 4

. 5000

1940 . 430

. 7 59 0 1

34 . 13

1 1 6 14 . 9

. 6000

2082 . 555

. 80856

37 . 2 1

10786 . 2

1 66

Appendix D: Computer Programs for Bi nary M i xtures

. 7000

222 1 . 469

•

85619

40 . 61

10077 . 2

. 8000

2359 . 695

. 90342

44 . 34

9453 . 0

. 9000

2 4 9 8 . 57 9

. 9 5117

48 . 44

8893 . 8

1 . 0000

2638 . 77 8

1 . 00000

52 . 94

8387 . 3

pre s s return t o cont inue .

At " DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type y (or Y) and press RETURN .

Press RETURN to continue. •

(With this command the results shown above are sent t o the printer.)

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y ) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A : TEMP8 .0UT and press RETURN . (With this command the results shown above are appended to the file temp8 .out, which already exists on the disk in drive A . )

•

At

"DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y/ N ) ? " type n (or N ) and press RETURN. •

At " DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y/ N ) ? " type n (or

D. 7 .

N) and press RETURN.

Program HVU N F: Bi nary V L E Pred ictions from t h e H u ron-Vidal M ixing Ru le (HVO) and Its Mod ifications (M HV I , M HV2, LCVM, and HVOS) The program HVUNF can be used to predict VLE using the PRSV EOS and one of the Gibbs excess-energy-based mixing rules, HVO, MHV I , MHV2, LCVM , and HVOS coupled with the UNIFAC group contribution method without the use of any measured VLE data. The program can be used in two ways. If experimental VLE data (isothermal) are available, the program can be run to calculate VLE at the measured liquid mole fractions ; then, the calculated and measured bubble pressures and vapor mole fractions are compared. In thi s mode the program reads previously stored data or accepts new data entered from the keyboard. The input data structure is identical to that used for all other EOS mixture programs, and the detail s of the input data have been descri bed in Section 0 . 3 . If no experimental data are available, bubble pressures and vapor mole fractions are calculated over the liquid mole fraction range of

x1

=

0 to l at intervals of

0. 1 . In this mode no data are entered to, or accessed from , the disk. Instead, the user provides critical temperature, critical presssure, the acentric factor and PRSV K1

parameter for each pure component, and temperature. The program then returns 1 67

Model 1ng Vapor- L1quid Eq u i l ibria

i sothermal bubble pressure and vapor mole fraction predictions at the temperature entered in the composition range

x1

=

0 to 1 , at intervals of 0. 1 .

The results from the program HVUNF can be sent to a printer, to a disk file, or both. To make this choice, fol low the commands that appear on the screen upon the completion of calculations. Please see the following tutori al for further details.

Tuto rial on the Use of HVU N F. EXE Exam ple D.7.A: Use of the H u ron-Vidal Class of M i xi ng Ru l e , Here HVOS, with U N I FAC t o Predict B i n ary V L E Data • •

Change to the directory containing HVUNF.EXE (e.g . , A> or C>, etc . ) . Start the program by typing HVUNF at the DOS prompt. Press RETURN (or ENTER) . A program introduction message appears . Press RETURN t o continue.

•

At

'"'rHE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK . THESE DATA ARE STORED IN TWO FILES NAMED UNF i l . DTA AND UNFI 2 . DTA . UNF i l . DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION UNFI 2 . DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION . IF

THEN ENTER 1 ; OTHERWISE ENTER 2 : " type I and press RETURN.

YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY ; (The data fi les UNFl l .DTA and UNFI2.DTA are provided on the disk that accompanies this monograph. The program is easier to run if these data files are copied to the hard disk directory used to run the programs . In thi s case 1 , as above, must be entered. An entry of 2, as shown earlier in Example D.6.A, indicates that these files are not present in the current directory. In that case the u ser must provide the directory and file names.) •

"HVUNF :

At BINARY VLE CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES

AND THE UNIFAC EXCESS FREE ENERGY MODEL SELECT A MIXING RULE MODEL HV-O=HURON-VIDAL ORIGINAL MHVl=MODIFIED HURON-VIDAL 1ST ORDER MHV2 =MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS=HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O =EXIT l =HV-0 2 =MHV1 3 =MHV2 4 =LCVM S =HVOS " type 5 and press RETURN. (This results in using the HVOS model for the mixing rule model.) 1 68

Appendix D: Computer Programs for Bi nary M ixtu res

The following message appears on the screen : HVUNF : BINARY VL E CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES AND THE UNIFAC MODEL

YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAI LABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

AND

CRITICAL PRESSURE ,

PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND

AND

ACENTRIC FACTOR

A TEMPERATURE .

IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

AND

TO PREDICT VLE BEHAVIOR

COMPARE RESULTS WITH THE VLE DATA .

•

At

" ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" enter 2 and press RETURN. (or Y) and press RETURN.

•

At " DO YOU WANT TO USE AN EXI STING DATA FILE ( Y/ N ) ? " type y

•

At

" INPUT THE NAME OF EXISTING DATA FILE ( for example , a : ab2 S . dat ) : " type a : ab25 .dat and press RETURN. •

At "ENTER UNIFAC GROUP PARAMETER INFORMATION NAME OF COMPONENT 1 " type ' acetone' and press RETURN . (Following the preceding comment a group selection table will appear on the screen . The user must follow the instructions at the top of the table and for this example choose one CH3 and one CH3CO group for acetone and enter press RETURN . )

•

A t "ENTER UNIFAC GROUP PARAMETER INFORMATION NAME OF COMPONENT 2 " type 'benzene' and press RETURN. (The group selection table will again appear on the screen. For this example the user must choose six ACH groups for benzene and then press RETURN. Following this, a summary of group selections will appear on the screen for inspection of the entries. After inspection you can press any key to continue. At thi s stage the program runs , and, when point-to-point calculations for each data point in the ab25 .dat data file are completed, the results are displayed on the screen as shown below.) 1 69

Modeling Vapor-Li q u i d Eq u i l i bria

HVUNF : VLE CALCULATIONS WITH HURON-VIDAL TYPE MODELS AND UNIFAC ab2 5 . dat ACETONE -BENZENE AT 2 5C FROM DECHEMA- 1 - 3 B - 1 6 3 TEMPERATURE ( K ) = 2 9 8 . 1 5 MIXING RULE : HVOS PHASE VOLUMES ARE IN C C / MOL , PRESSURE IS IN UNITS OF THE DATA . P - EXP

X-EXP

P - CAL

Y-EXP

Y-CAL

VL-CAL

VV-CAL

. 0001

95 . 600

9 5 . 040

. 00010

. 00035

87 . 09

194516 . 0

. 0500

106 . 400

106 . 678

. 14 6 0 0

. 15 2 5 7

86 . 96

17 3 2 0 8 . 4

. 1000

116 . 600

117 . 52 5

. 2 6000

. 26900

86 . 83

1 5 7 14 6 . 8

. 1500

126 . 300

127 . 618

. 35300

. 36118

86 . 7 0

144 6 5 3 . 1

. 2000

135 . 400

137 . 0 18

. 42900

. 43632

86 . 57

134673 . 1

. 2500

144 . 0 0 0

145 . 7 80

. 49400

. 49911

86 . 44

126528 . 6

. 3000

152 . 100

153 . 958

. 54900

. 55269

86 . 3 0

119762 . 6

. 3500

159 . 900

161 . 606

. 59800

. 59928

86 . 16

114055 . 6

. 4000

167 . 2 00

168 . 7 68

. 64100

. 64051

86 . 02

109180 . 0

. 4500

174 . 2 00

17 5 . 4 9 3

. 68000

. 67757

8 5 . 87

104963 . 8

. 5000

180 . 800

181 . 824

. 7 1500

•

7 1 14 1

85 . 72

1 0 127 9 . 9

. 5500

187 . 000

187 . 802

. 74700

. 74278

85 . 57

9802 9 . 7

pre s s return to see more resu l t s on the screen .

(Thi s message appears when the number of data points exceeds twelve. Press RETURN to continue . ) . 6000

193 . 000

193 . 463

. 77800

•

77231

85 . 41

9 5 13 6 . 4

. 6500

198 . 600

198 . 844

. 80600

. 80051

85 . 25

92539 . 6

. 7000

203 . 800

203 . 975

. 83300

. 82787

85 . 08

90191 . 1

. 7500

208 . 800

208 . 883

. 85900

. 85484

84 . 9 1

88052 . 7

. 8000

213 . 300

213 . 590

. 88400

. 88184

84 . 7 3

86094 . 6

. 8500

217 . 500

218 . 113

. 90900

. 90932

84 . 55

84292 . 7

. 9000

221 . 400

222 . 454

. 99600

. 93779

84 . 3 5

82632 . 3

. 9500

224 . 800

2 2 6 . 612

. 96500

. 96779

84 . 15

81102 . 2

. 9999

227 . 700

230 . 556

. 99990

. 99993

83 . 95

79702 . 0

pre s s return to cont inue .

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type y (or Y) and press RETURN.

Press RETURN to continue. •

(The results above will be sent to the printer. ) •

1 70

type y (or Y) and press RETURN.

A t "DO YOU WANT T O SAVE THE RESULTS T O AN OUTPUT FILE ( Y/ N ) ? "

Appendix D: Computer Programs for B1nary M i xtures

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A:TEMP9. 0UT and press RETURN. (With this command the results shown above are saved i n the disk in drive A under the name TEMP9 .0UT in ASCII code . )

•

At "DO YOU WANT T O DO ANOTHER VL E CALCULATION ( Y/ N ) ? " type n (or N) and press RETURN.

Exam ple 0.7 . 8: Use of the H u ron-Vidal Class of M ixing Ru l e , Here HVOS, with U N I FAC t o Predict Binary VLE • •

Change to the directory containing HVUNF. EXE (e.g . , A> or C>, etc . ) . Start the program by typing HVUNF at the DOS prompt. Press RETURN (or ENTER). The program introduction message appears . Press RETURN to continue.

•

At

"THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK . THESE DATA ARE STORED IN TWO FILES NAMED UNF i l . DTA AND UNFI 2 . DTA . UNFi l . DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION UNFI2 . DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION . IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY , THEN ENTER 1 ; OTHERWISE ENTER 2 : " type

1

and press RETURN .

(The data files UNFI I .DTA and UNFI2.DTA are provided on the disk that copied to the hard disk directory u sed to run the programs. In this case an entry of accompanies this monograph. The program is easier to run if these data files are

1 i s required. An entry of 2, as shown earlier in Example D.6.A, indicates that

these files are not present in the current directory. In that case the user must provide the directory and file names . ) •

"HVUNF :

At BINARY VLE CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES

AND THE UNIFAC EXCESS FREE ENERGY MODEL SELECT A MIXING RULE MODEL HV-O=HURON-VIDAL ORIGINAL MHVl =MODIFIED HURON-VIDAL 1ST ORDER MHV2 =MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS= HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O =EXIT l=HV-0 2 =MHV1 3 =MHV2 4 =LCVM S =HVOS " type 5 and press RETURN. (This results in using the HVOS model for the mixing rule mode l . ) The following message appears on the screen : HVUNF : BINARY VLE CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES AND THE UNIFAC MODEL

171

Modeling Vapor- Liquid Eq u i l i bria

YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRESSURE ,

ACENTRIC FACTOR

AND PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND AND A TEMPERATURE . IN THI S MODE THE PROGRAM WILL RETURN I S OTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x - y - P DATA,

YOU

CAN

ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

TO PREDICT VLE BEHAVIOR ,

AND COMPARE RE SULTS WITH THE VLE DATA .

•

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" enter 2 and press RETURN. •

At " DO YOU WANT TO USE AN EXISTING DATA FILE ( Y / N ) ? " type n (or N) and press RETU RN .

•

A t "PROVIDE FOLLOWING INPUT INFORMATION : INPUT NEW DATA FILE NAME : " type a: temp09.dat and press RETURN. (The preceding command will lead to saving a data fi l e named temp09 .dat on the disk in drive A. You must have a disk in the A drive, or select another directory, by typing c : temp09 .dat, for example, to save the fi le on the hard drive. )

•

A t " INPUT A TITLE FOR THE NEW DATA FILE : " type acetone-benzene 25°C.

•

At

"CRITICAL PARAMETERS : TC=CRITICAL TEMP , K PC=CRITICAL PRESSURE ,

BAR

W=ACENTRIC FACTOR KAPPA=KAPPA- 1 PARAMETER OF THE PRSV EOS INPUT TC 1 , PC 1 , W1 , KAPPA- 1 : " type 508. 1 , 46.96, 0.30667 , -0.008 8 8 , and press RETURN. (These are EOS constants for acetone . ) •

A t " INPUT TC2 , PC2 , W2 , KAPPA- 2 : " type 562. 1 6, 48.98, 0. 20929, 0.070 1 9, and press RETURN . (These are EOS constants for benzen e . )

•

At " INPUT NUMBER OF DATA POINTS : " type 3 and press RETURN .

•

At " INPUT TEMPERATURE in K : " type 298. 1 5 and press RETURN.

•

At

" INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION ( e . g . i f original data in mm Hg , type 7 5 0 i f original data i n p s i a , type 14 . 5 etc . ) : " enter 750 and press RETURN. 1 72

Appendix D: Com p uter Programs for Bi nary M i xtures

•

At

" INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION ( X1EXP ) SPECIES 1 , VAPOR MOLE FRACTION ( Y1EXP ) OF SPEC IES 1 , AND BUBBLE POINT PRESSURE ( PE:XP ( three in a row , separated by commas ) INPUT X1EXP , Y 1EXP , PEXP : " type 0 . 1 , 0.26, 1 1 6.6, and press RETURN. •

At " INPUT X1EXP , Y1EXP , PEXP : " type 0 . 5 , 0.7 1 50, 1 80 . 8 , and press RETURN.

•

At " INPUT X1EXP , Y1EXP , PEXP : " type 0.9, 0.965 , 224 . 8 , and press RETURN . (When the number of items of data specified by NP, here three, is entered, the program writes the data to a file under the name temp09 .dat as specified above and continues. This data fi l e becomes an

existing data file

and can be used when

the program is run again. The data used here are part of those from the data fi le ab2 5 .dat used in the previous example.) •

At "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1 " type ' acetone' and press RETURN . (Following the preceding comment a group selection table will appear on the screen . The user must fol low the instructions at the top of the table to choose one CH3 and one CH3CO for acetone and press RETURN. )

•

A t " ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2 " type 'benzene' and press RETURN. (The group selection table will agai n appear on the screen. For this example the u ser should choose six ACH groups for benzene and then press RETURN . Following this a summary o f group selections will appear o n the screen. Press RETURN to continue . )

HVUNF : VLE CALCULATIONS WITH HURON-VIDAL TYPE MODELS AND UNIFAC a : t emp 0 9 . dat ACETONE BENZENE 2 5C TEMPERATURE ( K ) = 2 9 8 . 1 5 MIXING RULE : HVOS PHASE VOLUMES ARE IN CC /MOL , PRESSURE IS IN UNITS OF THE DATA . X-EXP

P - EXP

P - CAL

Y-EXP

Y-CAL

VL - CAL

VV- CAL

. 1000

116 . 600

117 . 5 15

. 26000

. 2 6899

86 . 83

157160 . 1

7 1140

85 . 72

1 0 12 9 0 . 3

. 93779

84 . 35

82642 . 5

5000

180 . 800

181 . 805

•

. 9000

224 . 800

222 . 427

. 96500

•

71500

•

pre ss return to cont inue .

At "DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type y (or Y) and press RETURN.

Press RETURN to continue . •

(The results above will be sent to the printer. )

1 73

Modeling Vapor- L i q u i d Eq u i l i b ri a

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y) and press RETURN.

•

At " INPUT A NAME FOR THE OUTPUT FILE : " type A : TEMP l O.OUT and press RETURN. (With thi s command the results shown above are saved on the disk in drive A under the name TEMP I O.OUT in ASCII code . )

•

At "DO YOU WANT T O DO ANOTHER VL E CALCULATION ( Y/ N ) ? " type y (or Y) and press RETURN .

•

At " SELECT A MIXING RULE MODEL HV-O=HURON-VIDAL ORIGINAL MHVl=MODIFIED HURON-VIDAL 1ST ORDER MHV2 =MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS=HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O =EXIT l=HV-0 2 =MHV1 3 =MHV2 4 =LCVM S =HVOS " type 5 and press RETURN. The following message appears on the screen:

HVUNF : B I NARY VL E CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES AND THE UNIFAC MODEL

YOU CAN USE THI S PROGRAM FOR VLE PREDICTION IN TWO WAYS . MODE

(1) :

IF NO T - P -x-y DATA TO COMPARE RE SULTS WITH ARE AVAILABLE ,

YOU MUST SUPPLY CRITICAL TEMPERATURE ,

CRITICAL PRES SURE ,

ACENTRIC FACTOR

AND PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND AND A TEMPERATURE . IN THI S MODE THE PROGRAM WILL RETURN I SOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOS ITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0 . 1 . MODE

(2) :

I F YOU HAVE ISOTHERMAL x-y-P DATA ,

YOU CAN ENTER THESE DATA

FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN ( OR USE PREVIOUSLY ENTERED DATA )

TO PREDICT VLE BEHAVIOR,

AND COMPARE RESULTS WITH THE VLE DATA .

•

At

"ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" enter 1 and press RETURN. (In the first part of this example, we compared VLE predictions for the acetone-benzene binary mixture at 25oC with experimental data entered from the keyboard. In the second part, shown below, we use the same model ( HVOS ) to predict isothermal VLE data at 1 00° C , this time using internally generated liquid 1 74

Appendix D: Computer Programs for Bi nary M ixtu res

mole fractions

x1

=

0, 0 . 1 , 0 . 2 , etc . This mode is implemented by entering

1

above. ) •

At

"YOU MAY ENTER A TITLE ( 2 5 CHARACTERS MAX . ) FOR THE MIXTURE TO BE PREI ( OR YOU MAY PRESS RETURN TO SKIP THE TITLE ) : " enter acetone-water ] 00 C and press RETURN.

•

At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPA- 1 PARAMETER OF THE PRSV EOS INPUT TC 1 , PC 1 , W1 , KAPPA- 1 : " enter 508. 1 , 46.96, 0.3067 , -0.0089, and press

•

RETURN.

At " INPUT TC2 , PC2 , W2 , KAPPA- 2 : " enter 562. 1 6, 48.98, 0 . 2093 , 0.0702, and press RETURN.

•

At " INPUT TEMPERATURE in K : " type 3 7 3 . 1 5 and press

•

At " INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE

RETURN.

DEFAULT I S BAR, TYPE 1 I F YOU WANT PRESSURE IN BAR .

( type 7 5 0 i f you want pres sure in mm Hg , etc . ) : " enter 750

and press RETURN. •

At "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1" type acetone' and press RETURN . (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table to choose one CH3 and one CH3CO for acetone and press RETURN . )

•

A t "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2 " type 'benzene' and press RETURN. (The group selection table will agai n appear on the screen . In thi s example the user should choose six ACH group for benzene and then press RETURN . Fol lowing thi s a summary of group selections will appear on the screen. Press RETURN to conti nue . ) The following results appear on the screen :

HVUNF : VLE CALCULATIONS WITH HURON-VIDAL TYPE MODELS AND UNIFAC acetone-benzene l O O C TEMPERATURE ( K ) = 3 7 3 . 1 5 REMINDER : XEXP VALUES

ARE

INTERNALLY GENERATED

NO ACTUAL EXPERIMENTAL DATA

ARE

AVAI LABLE

MIXING RULE : HVOS PHASE VOLUMES

ARE

IN CC /MOL .

FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS 7 5 0 . 0 0

1 75

Model i ng Vapor-Li q u i d Eq u i l i bria

P - CAL

Y-EXP

Y-CAL

VL-CAL

VV-CAL

. 0000

1345 . 7 99

. 00000

95 . 50

16499 . 6

. 1000

1569 . 039

. 21950

9 5 . 52

1407 5 . 6

. 2000

1767 . 497

. 37449

9 5 . 54

12432 . 7

. 3000

1944 . 52 6

. 49211

9 5 . 54

11249 . 1

. 4000

2103 . 222

. 58669

95 . 53

10357 . 0

. 5000

2246 . 3 62

. 66667

95 . 49

9660 . 3

. 6000

2 37 6 . 377

. 73752

95 . 42

9100 . 3

. 7000

2495 . 281

. 80316

95 . 32

8639 . 5

. 8000

2604 . 568

. 86678

95 . 16

8253 . 4

. 9000

2705 . 021

. 93133

94 . 9 5

7926 . 6

1 . 0000

2796 . 386

1. 00000

94 . 67

7650 . 5

P - EXP

X-EXP

pre s s return to cont inue .

At "DO YOU WANT A PRINT-OUT ( Y / N ) ? " type y (or Y) and press RETURN. Press RETURN to continue.

•

(The results above will be sent to the printer.) •

•

type y (or Y) and press RETURN .

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " At " INPUT A NAME FOR THE OUTPUT FILE ? " type A : TEMP I O.OUT and (With thi s command the results shown above are appended to the file temp I O.out

press RETURN.

in the disk in drive A in ASCII code. ) •

At

"DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y/ N } ? " type n (or N ) and press RETURN . •

At "DO YOU WANT TO DO ANOTHER VLE CALCULATION ( Y / N ) ? " type n (or N) and press RETURN .

1 76

APPEN DIX E

Com puter Program s for Multicom ponent Mixtures

The accompanying disk contains the programs and sample data files that can be used to predict vapor-liquid equilibria of multicomponent mixtures using the EOS models discussed in thi s monograph. All the programs coded in FORTRAN using MICROSOFT FORTRAN Version 5 . 1 and are also supplied as stand-alone executable modules (EXE files) that run on DOS or WINDOWS-based personal computers. For more details, see the introduction section of Appendix D . Each program is separately described in the following sections, and tutorials are included to facilitate the use of each program. In these tutorials, the output that will appear on the screen is indicated in bold and in a smaller font. The information the user is to supply is shown here in the normal font.

E. I .

Program VDWM IX: M u lticom ponent VLE Calculations with van der Waals One-F l u i d M ixing Rules The program VDWMIX i s used to calculate multicomponent VLE using the PRSV EOS and the van der Waal s one-fluid mixing rules (either l PVDW or 2PVDW; see Sections 3 . 3 to 3 . 5 and Appendix D . 3 ) . The program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten) , the l iquid mole fractions, the temperatures at which the calculations are to be done (for as many sets of calculations as the user wishes, up to a maximum of fi fty) , critical temperatures, pressures (bar), acentric factors, the

K1

constants of the

PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and the vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply binary interaction parameter(s) for each pair of components in the multicomponent mixture. These interaction parameters can be 1 77

Model i ng Vapor-Li q u i d Eq u i l i bria

obtained using the program VDW (see Appendix D . 3 ) if experimental data are avai l­ able for each of the binary pairs. Alternatively, the user can select an already existing data fi le (we use extension VDW for these fi les, and some examples of such data files are provided on the accompanying disk) to calculate multicomponent VLE for the mixture of that input file. The results from the program VDWMIX can be sent to a printer, to a disk file, or both. The commands that appear on the screen upon the completion of the cal­ culations must be followed to make this choice. Please sec the following tutorial for further details.

Tutorial on the Use of VDWM IX.EXE Exam p l e E. I .A: Creati ng a N ew I n put File and Calc u l ation of M u lticomponent VLE •

Change to the directory containing the program VDWMIX.EXE (e. g . , A> or C>, etc . ) .

•

Start the program by typing VDWMIX at the DOS prompt. A program introduction message appears on the screen . Press ENTER (or press RETURN) . The following appears :

•

At

"VDWMIX : MtiLTICOMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE - FLUID MIXING RULES . THIS PROGRAM CAN BE USED FOR I SOTHERMAL BUBBLE POINT CALCULATIONS CREATING A NEW INPUT FILE , OR USING A PREVIOUSLY STORED INPUT FILE . YOU MUST SUPPLY NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , TEMPERATURE , AND MODEL PARAMETER ( S ) FOR EACH PAIR OF COMPONENTS . ENTER 1

TO

CREATE A NEW INPUT FILE , 2 TO USE A PREVIOUSLY

STORED INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?"

type 1 and RETURN. (With this selection a new i nput fi l e will be created.) •

At "ENTER A NAME FOR THE NEW INPUT FILE ( * * * * * * * VDW ) : " •

enter a name for the new fi l e (such as A:TEST.VDW) and press RETURN . •

At "ENTER A TITLE FOR THE NEW INPUT FILE : " enter a descriptive title for the file (for example "ACETONE-METHANOL-WATER AT 523 K") and press RETURN.

1 78

Appendix E: Computer Programs for M u lticomponent M i xtu res

•

At " INPUT NUMBER OF COMPONENTS : " enter 3 and press RETURN.

•

At

"HOW MANY SETS OF I SOTHERMAL BUBBLE POINT CALCULATIONS DO YOU WANT TO DO? ( FOR EACH SET YOU MUST PROVIDE A NEW LIQUID COMPOSITION AND TEMPERATURE ) : " enter 3 and press RETURN . •

At " INPUT PURE COMPONENT PARAMETERS : TC=CRITICAL TEMPERATURE , K PC=CRITICAL PRESSURE , BAR W=PITZER ' S ACENTRIC FACTOR KAP=KAPPA- 1 PARAMETER OF THR PRSV EOS INPUT TC , PC , W, KAP OF COMPONENT 1 : " type 5 0 8 . 1 , 46.96, 0. 30667, - 0.008 8 , and press RETURN.

•

At " INPUT TC , PC , W, KAP OF COMPONENT 2 : " type 5 1 2 . 5 8 , 80.96, 0.565 3 3 , - 0 . 1 68 1 6, and press RETURN.

•

At " INPUT TC , PC , W, KAP 1 OF COMPONENT 3 : " type 647 .29, 220.90, 0. 343 8 , -0.0663 5 , and press RETURN.

•

At " INPUT TEMPERATURE ( K ) OF SET NO . 1 : " enter 5 2 3 . 1 5 and press RETURN .

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 1 : " enter 0 .05 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 1 : " enter 0.05 and press RETURN .

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 1 : " enter 0.90 and press RETURN .

•

At " INPUT TEMPERATURE ( K ) OF SET NO . 2 : " enter 5 2 3 . 1 5 and press RETURN .

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 2 : " enter 0. 1 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 2 : "

•

A t " INPUT LIQUID MOLE FRACTION O F COMPONENT 3 I N SET 2 : "

enter 0. 1 and press RETURN . enter 0 . 8 and press RETURN . •

At " INPUT TEMPERATURE ( K ) OF SET NO . 3 : " enter 5 2 3 . 1 5 and press RETURN .

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 3 : " enter 0. 1 5 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 3 : " enter 0. 1 5 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 3 : " enter 0.7 and press RETURN .

1 79

Model 1 ng Vapor-Li q u i d Eq u i l i bria

•

At

"DO YOU WANT TO INPUT EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION AND PRESSURE FOR COMPARI SON WITH THE CALCULATED VALUES ( Y / N ) ? " type n (or N ) and RETURN. (The entry of experimental vapor mole fractions and bubble point pressures i s optional. In thi s example n o entry i s made, because no experimental data were available.) •

At

"PROVIDE BINARY INTERACTION PARAMETER ( S ) FOR EACH PAIR OF COMPONENTS IN THE MIXTURE . THERE ARE TWO OPTIONS : 1=0NE PARAMETER VDW MODEL ( 1PVDW ) 2 = TWO PARAMETER VDW MODEL ( 2 PVDW ) ENTER 1 FOR THE 1PVDW MODEL , OR 2 FOR THE 2 PVDW MODEL . 1/2?" type 2 and press RETURN. (With thi s selection the user will be prompted to provi de two binary interaction parameters for each pair in the mixture . ) •

A t " 2 PVDW OPTION : TWO PARAMETERS PER PAIR ARE REQUIRED . INPUT INTERACTION PARAMETER FOR THE PAIR 1 2 : " type 0.026 and RETURN.

•

At " INPUT INTERACTION PARAMETER FOR THE PAIR 1 3 : " type 0.046 1

•

At " INPUT INTERACTION PARAMETER FOR THE PAIR 2 1 : " type 0.0076

and RETURN. and RETURN . •

A t " INPUT INTERACTION PARAMETER FOR THE PAIR 2 3 : " type -0.0429 and RETURN .

•

At " INPUT INTERACTION PARAMETER FOR THE PAIR 3 1 : " type - . 1 56 and RETURN .

•

At " INPUT INTERACTION PARAMETER FOR THE PAIR 3 2 : " type -0.0845 and RETURN . (These binary interaction parameters were obtained using the program VDW. EXE described in Appendix 0 . 3 , and the data fi les am200.dat, mw250.dat, and aw250.dat, respectively, for acetone-methanol , methanol-water, and acetone-water binary pairs . ) After the last o f the binary interaction parameters is entered, the program calculates the VLE and the following appears on the screen:

VDWMIX : MULTICOMPONENT VL E CALCULATIONS WITH THE VAN DER WAALS ONE FLUID MIXING RULES .

1 80

Append i x E: Computer Programs for M u lticomponent M i xtures

INPUT FILE

NAME :

TEST . VDW

ACETONE -METHANOL-WATER AT 5 2 3 K SET NO .

TEMP ( K )

PEXP ( BAR )

PCAL

VLIQ ( CM3 /MOL ) .

VVAP

1

523 . 15

53 . 182

35 . 382

600 . 4

2

523 . 15

59 . 462

43 . 976

492 . 6

3

523 . 15

63 . 939

54 . 115

424 . 5

pre s s return for phase compos it ions .

Press RETURN to continue. The following appears : PHASE COMPOSITIONS ( IN MOLE FRACTION ) SET NO .

COMPONENT

1

2

3

XEXP

YEXP

YCAL

1

. 0500

. 1578

2

. 0500

. 1041

3

. 9000

. 7381

1

. 1000

. 1998

2

. 1000

. 1610

3

. 8000

. 6392

1

. 1500

. 2261

2

. 1500

. 2063

3

. 7000

. 5676

pre s s return for the binary parameter matrix .

Press RETURN to continue. The binary parameter matrix for the VOW mixing rule appears : THE BINARY PARAMETER MATRIX FOR THE VDW MIXING RULE 2

1 1

. 0000

•

3

0260

. 0461

2

. 0076

. 0000

-. 0429

3

- . 1560

- . 0845

. 0000

• •

At " DO YOU WANT A PRINT- OUT ( Y / N ) ? " type n (or N) and press RETURN. At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y /N) ? " type n (or N ) and press RETURN.

•

At "DO YOU WANT TO START A NEW CALCULATION ( Y/ N ) ? " type n (or N) and press RETURN .

181

Modeling Vapor-Liquid Eq u i l i bria

Exam ple E. I . B: Calculation of M u lticomponent VLE Using an Existing I n put File •

Change to the directory containing the program VDWMIX .EXE (e. g . , A> or C>, etc . ) .

•

Start the program by typing VDWMIX at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN ) . The following appears :

•

At

"VDWMIX : MULTI COMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE - FLUID MIXING RULES . THI S PROGRAM CAN BE USED FOR ISOTHERMAL BUBBLE POINT CALCULATIONS CREATING A NEW INPUT FILE , OR USING A PREVIOUSLY STORED INPUT FILE . YOU MUST SUPPLY NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , TEMPERATURE , AND MODEL PARAMETER ( S ) FOR EACH PAIR OF COMPONENTS . ENTER 1

TO

CREATE A NEW INPUT FILE , 2 TO USE A PREVIOUSLY

STORED INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?"

type 2 and RETURN. (This results in the selection of an already existing i nput file . ) •

A t "ENTER THE NAME O F EXI STING INPUT FILE ( for example , a : AMW2 5 0 . VDW ) : " enter a: AMW250. VDW and RETURN. The following appears on the screen :

VDWMIX : MULTI COMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE FLUID MIXING RULES . INPUT FILE

NAME :

amw2 5 0 . vdw

ACETONE -METHANOL-WATER 2 5 0 C SET NO .

TEMP ( K )

PEXP ( BAR )

PCAL

VLIQ ( CM3 /MOL )

VVAP

1

523 . 15

62 . 060

61 . 015

4 8 . 4 17

462 . 8

2

523 . 15

58 . 480

58 . 023

4 1 . 567

515 . 3

3

523 . 15

52 . 890

52 . 3 7 8

34 . 57 6

6 16 . 9

pre s s return for phase compos it ions .

Press RETURN to continue. 1 82

Appendix E: Computer Programs for M u lticomponent M i xtures

PHASE COMPOSITIONS { IN MOLE FRACTION ) SET NO .

COMPONENT

XEXP

YEXP

YCAL

1

1

. 13 7 0

. 2370

. 2390

2

. 0940

. 14 8 0

. 14 0 7

3

. 7690

. 6150

. 62 0 3

2

3

1

. 0880

. 1920

. 1941

2

. 0840

. 14 5 0

. 14 3 3

3

. 82 8 0

. 6630

. 6626

1

. 04 3 0

. 14 3 0

. 14 4 2

2

. 04 9 0

. 0950

. 10 6 1

3

. 9080

. 7620

. 7497

pre s s return for the binary parameter matrix .

Press RETURN to continue. The binary parameter matrix for the VOW mixing rule appears : BINARY PARAMETER MATRIX FOR THE 1

2

MIXING RULE

3

1

. 0000

. 0260

. 0460

2

. 0070

. 0000

- . 04 2 0

3

- . 1560

- . 0840

. 0000

•

VDW

At "DO YOU WANT A PRINT- OUT ( Y / N ) ? " enter y (or Y) and RETURN. (This command sends the results, simi lar to those shown above, to the printer. )

•

At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " enter y (or Y ) and press RETURN.

•

At " ENTER A NAME FOR THE OUTPUT FILE : " type a file name of your choice (for example A: OUTPUT l .OUT) and press RETURN. (With this command the results shown above are saved on the disk in drive A with the name OUTPUT I .OUT as an ASCII file.)

•

At " DO YOU WANT TO START A NEW CALCULATION ( Y/ N ) ? " type n (or N) and press RETURN to terminate the program .

E.2.

Program WSM MAI N : M u ltico m ponent VLE Calcu lations with Wong-San d i e r M ixing Rules The program WSMMAIN can be used to calculate multicomponent VLE using the PRSV EOS and the Wong-Sandier mixing rule. One of the three (the UNIQUAC, Wilson, or NRTL) excess free-energy model s can be used with this mixing rule by following the instructions that appear on the screen during program execution. 1 83

Modeling Vapor-Liquid Eq u i l i b ria

This program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for thi s mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which calculations are to be done (for the number of sets of calculations, as the the user wishes, up to a maximum of fifty) , critical temperatures, pressures (bar), acentric factors, the

Kt

constants of the PRSV equation for each

compound in the mixture, and, if available, the experimental bubble point pressure and vapor phase compositions (these last entries are optional, and are used for a comparison between the experimental and calculated results) . In addition, the user is requested to supply model parameters for each pair of components in the multicomponent mixture. These model parameters can be obtained using the program WS (see Appendi x

0.5)

if experimental data are available for each of the binary pairs . Alternatively, the user can select an already existing fi l e (for these files we usc the extensions WSN, WSW, and WSU, respectively, for the WS - NRTL , WS-WILSON, and WS - UNIQUAC options, and some examples are provided on the accompanying disk) and calculate the multicomponent VLE for the mixture of that input file. The results from the program WSMMAIN can be sent to a pri nter, to a disk file, or both. The commands that appear on the screen upon the completion of the calculations must be followed to make this choice. Please see the following tutorial for further details.

Exam ple E.2.A: Creating a N ew I n put File and Calcu l ation of M u lticomponent VLE •

Change to the directory containing the program WSMMAIN.EXE ( e . g . , A> or C>, etc . ) .

•

Start the program by typing WSMMAIN at the DOS prompt. A program introduction message appears on the screen . Press ENTER (or press RETURN) . The following appears:

WSM : MULTICOMPONENT VLE CALCULATIONS WITH THE WONG- SANDLER MIXING RULE . YOU HAVE TO SELECT AN EXCESS ENERGY MODEL THE SELECTIONS

TO

BE USED IN THE MIXING RULE .

ARE :

l =NRTL 2 =WILSON 3 =UNIQUAC 1/2/3?

Type 1 and press RETURN. (This results in the selection of the NRTL model . ) •

At

"WSM : MULTICOMPONENT VLE CALCULATIONS WITH T HE WONG- SANDLER-NRTL MIXING RULE . THIS PROGRAM CAN BE USED FOR ISOTHERMAL BUBBLE POINT CALCULATIONS CREATING A NEW INPUT FILE , OR USING A PREVIOUSLY STORED INPUT FILE .

1 84

Append ix E: Computer Programs for M u lticomponent M i xtu res

YOU MUST SUPPLY THE NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , TEMPERATURE , AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS . ENTER 1 TO CREATE A NEW INPUT FILE ,

2 TO SELECT A PREVIOUSLY

STORED INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?"

Type 1 and press RETURN . (With thi s selection the user is prompted to create a new input fi le . ) •

At

"ENTER A NAME FOR THE NEW INPUT FILE ( format : * * * * * * * * . WSN ) : " type a: test l . wsn and press RETURN. (This will result in the creation of an i nput file named TEST l .WSN that will be stored on the disk on drive A . ) •

A t "ENTER A TITLE FOR THE NEW INPUT FILE : " type ' acetone-methanol-water at 250 C by WS+NRTL model ' and press RETURN. (The title i s a descriptive statement, with a maximum 60 characters, about the input file to be created.)

•

At " INPUT NUMBER OF COMPONENTS : " type 3 and press RETU RN .

•

At

"HOW MANY SETS OF I SOTHERMAL BUBBLE POINT CALCULATIONS DO YOU WANT TO DO? ( FOR EACH SET YOU PROVIDE A NEW LIQUID COMPOS ITION AND TEMPERATURE ) : " type 2 and press RETURN . •

At "ENTER PURE COMPONENT PARAMETERS : TC=CRITICAL TEMPERATURE , K PC=CRITICAL PRESSURE , BAR W=PITZER ' S ACENTRIC FACTOR KAP=THE KAPPA- 1 PARAMETER OF THE PRSV EOS INPUT TC , PC , W, KAP OF COMPONENT 1 : " type 508. 1 , 46.96, 0.30667 , - 0.008 8 , and press RETURN.

•

At " INPUT TC , PC , W, KAP OF COMPONENT 2 : " type 5 1 2 . 5 8 , 80.96, 0.565 3 3 , - 0 . 1 68 1 6, and press RETURN.

•

At " INPUT TC , PC , W , KAP 1 OF COMPONENT 3 : " type 647 .29, 220.90, 0 . 343 8 , -0.0663 5 , and press RETURN.

•

At " INPUT TEMPERATURE ( K ) OF SET 1 : " enter 5 2 3 . 1 5 and press

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 1 : "

RETURN. enter 0.05 and press RETURN. •

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 1 : " enter 0.05 and press RETURN. 1 85

Model i ng Vapor-Li q u i d Eq u i l i bria

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 1 : " enter 0.90 and press RETURN.

•

At " INPUT TEMPERATURE ( K ) OF SET 2 : " enter 5 23 . 1 5 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 2 : " enter 0. 1 5 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 2 : " enter 0. 1 5 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 2 : " enter 0 . 7 and press RETURN.

•

At

"DO YOU WANT TO INPUT EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION AND PRESSURE FOR COMPARISON WITH CALCULATED VALUES ( Y/ N ) ? " type n (or N ) and press RETURN . •

At

" INPUT MODEL PARAMETERS . THEY ARE : KIJ=THE WONG- SANDLER MODEL BINARY INTERACTION PARAMETER . ALPHAIJ=THE NRTL MODEL ALPHA PARAMETER . AIJ=THE NRTL MODEL ENERGY PARAMETERS , TWO FOR EACH PAIR OF COMPONENTS , IN CAL /MOLE .

[AIJ=TAUIJ*RT ]

INPUT KIJ, ALPHAIJ FOR THE PAIR 1 2 : " type 0.05, 0 . 3 5 , and press RETURN . •

At " INPUT KIJ AND ALPHAIJ FOR THE PAIR 1 3 : " type 0 . 3 5 , 0 . 3 5 , and

•

At " INPUT KIJ AND ALPHAIJ FOR THE PAIR 2 3 : " type 0.05, 0.35, and

press RETURN. press RETURN . •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 2 : " type 45 1 .5 8 and press RETURN. •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 3 : " type 452.77 and press RETURN . •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 1 : " type 95 .0 and press RETURN . •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 3 : " type 1 97 . 5 2 and press RETURN. •

At type 1 042 . 8 8 and press RETURN.

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 1 : "

1 86

Appendix E: Computer Programs for M u lticompo nent M ixtures

•

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 2 : " type 520.60 and press RETURN. (These binary interaction parameters were obtained using the program WS.EXE described in Appendix . D . 5 , and the data files am200.dat, mw250.dat, and aw250.dat for acetone-methanol, methanol-water, and acetone-water binary mixtures, respectively. ) Following these entries of the model parameters, the information is written to the disk in drive A, and the program then calculates the multicomponent VLE. The following results appear on the screen : WSM : MULTICOMPONENT VLE WITH THE WONG - - SANDLER MIXING RULE INPUT FILE

NAME :

t e st l . wsn

acetone -methano l -water at 2 5 0 C by WS +NRTL model PCAL

VLIQ ( CM3 /MOL )

VVAP

523 . 15

52 . 527

3 1 . 481

643 . 3

523 . 15

65 . 370

42 . 124

443 . 4

SET NO .

TEMP ( K )

1 2

PEXP ( BAR )

pre s s return f o r the phase compos i t i ons .

Press RETURN to see phase compositions. The following appears : PHASE COMPOS I TIONS ( IN MOLE FRACTION ) SET NO .

COMPONENT

XEXP

1

2

YEXP

YCAL

1

. 0500

. 1611

2

. 0500

. 1133

3

. 9000

. 7256

1

. 1500

. 2536

2

. 1500

. 22 3 0

3

. 7000

. 52 3 4

pres s return for parameter matrices .

Press RETURN to continue. The following parameter matrix for the k;J parameter of the Wong-Sandier mixing rule appears : PARAMETER MATRIX FOR THE KIJ PARAMETER 1

2

3

1 I

. 0000

. 0500

. 3500

2 I

. 0500

. 0000

. 0500

3 I

. 3500

. 0500

. 0000

pre s s return for the alpha parameter matrix .

Press RETURN to continue. The following parameter matrix for the NRTL model parameter appears : 1 87

Modeling Vapor-Li q u i d Eq u i l ibria

PARAMETER MATRIX FOR THE ALPHA PARAMETER 1

2

3

1 I

. 0000

. 3500

. 3 500

2 I

. 3500

. 0000

. 3500

3 I

. 3500

. 3500

. 0000

pre s s return for the NRTL mode l energy parameter matrix .

Press RETURN to continue. The fol l owing parameter matrix for the NRTL model energy parameters appears : PARAMETER MATRIX FOR THE NRTL ENERGY PARAMETER , AIJ 2

1 . 00

451 . 58

452 . 77

95 . oo

. oo

197 . 52

1042 . 88

520 . 60

. oo

1 2 1

3 1

3

•

At "DO YOU WANT A PRINT -OUT ( Y/ N ) ? " type y (or Y) and press RETURN .

•

At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y ) and RETURN .

•

At "ENTER A NAME FOR THE OUTPUT FILE : type a:output l .out.

•

At

"DO YOU WANT TO START A NEW CALCULATION WITH THE NRTL MODEL ( Y/ N ) ? " type n (or N ) and press RETURN. •

At " DO YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL (Y /N) ? " type n (or N ) and press RETURN.

E.2.B. Calculation of M u lticomponent VLE Using an Existi ng I n put Fi le •

Change to the directory containing the program WSMMAIN.EXE (e. g . , A> or

•

Start the program by typing WSMMAIN at the DOS prompt. A program

C>, etc . ) . introduction message appears on the screen. Press ENTER (or press RETURN) . The following appears : WSM : MULTICOMPONENT VLE CALCULATIONS WITH THE WONG- SANDLER MIXING RULE . YOU HAVE TO SELECT AN EXCESS ENERGY MODEL TO BE USED IN THE MIXING RULE . THE SELECTIONS ARE : l=NRTL 2 =WILSON 3 =UNIQUAC 1/2/3?"

1 88

Appen d i x E: Computer Programs for M u lt1component M ixtures

Type 1 and press RETURN. (This results i n the selection of the NRTL mode l . ) •

At

"WSM : MULTICOMPONENT VLE CALCULATIONS WITH WONG- SANDLER-NRTL MIXING RULE . THI S PROGRAM CAN BE USED FOR I SOTHERMAL BUBBLE POINT CALCULATIONS , CREATING A NEW INPUT FILE , OR USING A PREVIOUSLY STORED INPUT FILE . YOU MUST SUPPLY THE NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND , TEMPERATURE , AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS . ENTER 1 TO CREATE A NEW INPUT FILE ,

2 TO SELECT A PREVIOUSLY

STORED INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?"

type 2 and press RETURN. (This results in the selection of an already existing input file.) •

At "ENTER THE NAME OF EXISTING INPUT FILE ( for example , a : PE4 2 3 . WSN ) : " type a : amw250.wsn and press RETURN. The following appears on the screen :

WSM : MULTICOMPONENT VLE WITH THE WONG- SANDLER MIXING RULE INPUT FILE

NAME :

AMW2 5 0 . WSN

INPUT FILE : ACETONE -METHANOL -WATER 2 5 0 C SET NO .

TEMP ( K )

PEXP ( BAR )

PCAL

VLIQ ( CM3 /MOL )

VVAP

1

523 . 15

62 . 060

62 . 22 6

38 . 088

485 . 0

2

523 . 15

58 . 480

58 . 552

34 . 702

542 . 3

3

523 . 15

52 . 890

5 1 . 57 0

3 1 . 14 0

661 . 1

pre s s return for the phase compos i t ions .

Press RETURN to see phase composition s . The fol lowing appears : PHASE COMPOS ITIONS S E T NO . 1

2

COMPONENT

( IN MOLE FRACTION ) XEXP

YEXP

YCAL

1

. 13 7 0

. 2370

. 2 7 17

2

. 0940

. 14 8 0

. 1548

3

. 7690

. 6150

. 57 3 5

1

. 0880

. 19 2 0

. 2 154

2

. 0840

. 14 5 0

. 1574

3

. 82 8 0

. 6630

. 6272

1 89

Modeling Vapor-Li q u i d Eq u i l i bria

3

1

. 04 3 0

. 14 3 0

. 14 4 2

2

. 0490

•

0950

. 1142

3

. 9080

. 7620

•

7416

press return for parameter matrices .

Press RETURN to continue. The following parameter matrix for the

kij

parameter of the Wong-Sandier mixing rule appears : PARAMETER MATRIX FOR THE KIJ PARAMETER

1 I

2 I

3 I

3

1

2

. 0000

. 0500

. 3500

. 0500

. 0000

. 0500

. 3500

. 0500

. 0000

pre s s return for the alpha parameter matrix .

Press RETURN to continue. The following parameter matrix for the NRTL model

a

parameter appears :

PARAMETER MATRIX FOR THE ALPHA PARAMETER 1

2

3

1 I

. 0000

. 3500

. 3500

2 I

. 3500

. 0000

. 3500

3

. 3500

. 3500

. 0000

pre s s return for the NRTL mode l energy parameter matrix .

Press RETURN to continue. The fol lowing parameter matrix for the NRTL model-energy parameters appears : PARAMETER MATRIX FOR THE NRTL ENERGY PARAMETER, AIJ 2

1

451 . 58

452 . 77

95 . 00

. 00

197 . 52

1042 . 88

520 . 60

. 00

1 I

. 00

2 I 3 I

3

• •

At " DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type n (or N) and press RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type n (or N) and RETURN.

•

At

"DO YOU WANT TO START A NEW CALCULATION WITH THE NRTL MODEL ( Y/ N ) ? " type n (or N ) and press RETURN. 1 90

Appendix E: Computer Programs for M u lticomponent M i xtures

•

At " DO YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL (Y /N) ? " type n (or N ) and press RETURN.

E.J.

Program HVM MAI N : M u lticomponent V L E Cal cu lations with Modified H u ron-Vidal (HVOS) M ixing Rule The program HVMMAIN c a n be used t o calculate multicomponent VLE using the PRS V EOS and the HVOS mixing rule ( see Section D.4). One of the three (the UNIQUAC , Wilson, or NRTL) excess free-energy models is selected for use with thi s mixing rule by following the instructions that appear on the screen during execution of the program. This program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the i sothermal bubble point pressure and the compo­ sition of the coexisting vapor phase for this mixture. In this mode the information needed is number of components (up to a maximum of ten) , the liquid mole fractions, the temperatures at which calculations are to be done (for the number of sets of cal­ culations the user wishes, up to a maximum of fifty) , critical temperatures, pressures (bar), acentric factors, the K1 constants of the PRS V equation for each compound in the mixture, and, if available, the experimental bubble point pressure and vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results) . In addition, the user is requested to supply model parameters for each pair of components in the multicomponent mixture. These model parameters can be obtained using the program HV (see Section D.4) if ex­ perimental data are available for each of the binary pairs . Alternatively, the user can select an already existing file (for these files we use the extensions HVN, HVW, and HVU, respectively, for the HVOS-NRTL, HVOS-WILSON, and HVOS-UNIQUAC options, and some examples are provided on the accompanying disk) and ·calculate the multicomponent VLE for the mixture of that input file. The results from the program HVMMAIN can be sent to a printer, to a disk file, or both. The commands that appear on the screen upon the completion of the calculations must be followed to make this choice. Please see the following tutorial for further details.

Tuto rial on the Use of HVM MAI N . EXE Exam ple E . 3 .A: Calculation of M u lticomponent VLE Creating a N ew I n put F i l e •

Change to the directory containing the program HVMMAIN .EXE (e.g., A> or C>, etc . ) .

•

Start the program by typing HVMMAIN at the DOS prompt. A program introduction message appears on the screen . Press ENTER (or press RETURN) . 191

Modeling Vapor-Li q u i d Eq u i l i bria

The following appears : " HVM :

MULTICOMPONENT VLE CALCULATIONS WITH THE HVOS MIXING RULE . YOU HAVE TO

SELECT AN EXCESS ENERGY MODEL TO BE USED IN THE HVOS MIXING RULE . THE SELECTIONS

ARE :

1 =NRTL 2 =WI LSON 3 =UNIQUAC 1/2/3?"

Type 1 and press RETURN (This results in selection of the NRTL model . ) • HVM:

At

MULTICOMPONENT VLE CALCULATIONS WITH THE HVOS

+

NRTL MODEL .

THI S PROGRAM CAN BE USED FOR ISOTHERMAL BUBBLE POINT CALCULATIONS , CREATING A NEW INPUT FILE , OR USING A PREVIOUSLY STORED INPUT FILE . YOU MUST SUPPLY THE NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND, TEMPERATURE , AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS . ENTER 1 TO CREATE A NEW INPUT FILE , 2 TO SELECT A PREVIOUSLY STORED INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?"

type 1 and press RETURN. (With this selection the user is prompted to create a new input fi l e . ) •

At

"ENTER A NAME FOR THE NEW INPUT FILE ( format : * * * * * * * * . HVN ) : " type a:test l .hvn and press RETURN. (This results in the creation of an input file named test l .hvn that will be stored on the disk in drive A. ) •

At "ENTER A TITLE FOR THE NEW INPUT FILE : " type ACETONE-METHANOL-WATER AT 523 K and press RETURN. (The title is a descriptive statement, maximum 60 characters, for the input fi le to be created. )

•

At " INPUT NUMBER OF COMPONENTS : " type 3 and press RETURN .

•

At

"HOW MANY SETS OF I SOTHERMAL BUBBLE POINT CALCULATIONS DO YOU WANT TO DO? ( FOR EACH SET YOU PROVIDE A NEW LIQUID COMPOSITION AND TEMPERATURE ) : " type 2 and press RETURN. •

At "ENTER PURE COMPONENT PARAMETERS : TC=CRITICAL TEMPERATURE , K PC=CRITICAL PRESSURE , BAR W=PITZER ' S ACENTRIC FACTOR

1 92

Appendix E: Computer Programs for M u lticomponent M ixtures

KAP=THE KAPPA- 1 PARAMETER OF THE PRSV EOS INPUT TC , PC , W, KAP OF COMPONENT 1 : " type 508 . 1 , 46.96, 0.30667, -0.008 8 , and press RETURN . •

At " INPUT TC , PC , W, KAP OF COMPONENT 2 : " type 5 1 2. 5 8 , 80.96, 0.565 3 3 , -0. 1 68 1 6, and press RETURN.

•

At " INPUT TC , PC , W , KAP OF COMPONENT 3 : " type 647 .29, 220.90, 0 . 343 8, - 0.0663 5 , and press RETURN.

•

At " INPUT TEMPERATURE ( K ) OF SET 1 : " enter 5 2 3 . 1 5 and press

•

A t " INPUT LIQUID MOLE FRACTION O F COMPONENT 1 I N SET 1 : "

RETURN . enter 0.05 and press RETURN. •

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 1 : " enter 0.05 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 1 : " enter 0.90 and press RETURN.

• •

At INPUT TEMPERATURE ( K ) OF SET 2 : " enter 5 23 . 1 5 and press RETURN . At " INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 2 : " enter 0. 1 0 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 2 : " enter 0. 1 0 and press RETURN.

•

At " INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 2 : " enter 0.80 and press RETURN .

•

At

" DO YOU WANT TO ENTER EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION AND PRESSURE FOR COMPARI SON WITH THE CALCULATED VALUES ( Y/ N ) ? " type n (or N ) and press RETURN. •

At

"ENTER EXCESS GIBBS ENERGY MODEL PARAMETERS . THEY ARE : ALPHAIJ= THE NRTL MODEL ALPHA PARAMETER . AIJ= THE NRTL MODEL ENERGY PARAMETERS , TWO FOR EACH PAIR OF COMPONENTS . IN CAL MOLE .

[AIJ=TAUIJ*RT ]

INPUT ALPHAIJ FOR THE PAIR 1 2 : " type 0.35 and press RETURN. •

At " INPUT ALPHAIJ FOR THE PAIR 1 3 : " type 0.35 and press RETURN.

•

At " INPUT ALPHAIJ FOR THE PAIR 2 3 : " type 0.35 and press RETURN.

•

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 2 : " type - 1 03 . 0 and press RETURN. •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 3 : " type 278.86 and press RETURN. 1 93

M o d e l i ng Vapor-Li q u i d Eq u i l i b n a

•

At

" lNPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 1 : " type 476.29 and press RETURN. •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 3 : " type - 1 1 5 . 5 8 and press RETURN . •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 1 : " type 2322.80 and press RETURN. •

At

" INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 2 : " type 1 0 1 9 .48 and press RETURN. These binary interaction parameters were obtained using the program HV.EXE described in Appendix. 0.4, and the data files am200.dat, mw250.dat, and aw250.dat for acetone-methanol, methanol-water, and acetone-water binary pairs, respectively. Following the entry of the model parameters, the information is written to a disk in drive A, and the program calculates the multicomponent VLE. The following results appear on the screen: HVM :

MULTICOMPONENT VL E WITH T HE HVOS MIXING RULE

INPUT FILE

NAME :

tes t l . hvn

ACETONE -METHANOL -WATER AT 5 2 3 K PCAL

VVAP

VLIQ ( CM3 /MOL )

SET NO .

TEMP ( K )

1

523 . 15

53 . 045

35 . 352

603 . 2

2

523 . 15

60 . 077

43 . 964

483 . 8

PEXP ( BAR )

pres s return for phase compo s i t ions .

Press RETURN to see phase compositions. The following appears: PHASE COMPOS ITIONS ( IN MOLE FRACTION ) S E T NO . 1

2

COMPONENT

XEXP

YEXP

YCAL

1

. 0500

. 1563

2

. 0500

. 1072

3

. 9000

. 7365

1

. 1000

. 2050

2

. 10 0 0

. 1648

3

. 8000

. 6302

pre s s return for the NRTL mode l alpha parameter matrix .

Press RETURN to continue. The following parameter matrix for the NRTL model 1 94

a

parameter appears:

Appendix E: Computer Programs for M u lticomponent M i xtures

PARAMETER MATRIX FOR THE ALPHA PARAMETER 2

1 1 I

2 I 3 I

3

. 0000

. 3500

. 3500

. 3 500

. 0000

. 3500

3500

. 3500

. 0000

.

pre s s return for the NRTL model energy parameter matrix .

Press RETURN to continue. The following parameter matrix for the NRTL model energy parameters appears : PARAMETER MATRIX FOR THE NRTL MODEL ENERGY PARAMETER AIJ ( CAL/MOLE )

. oo

-103 . 00

278 . 86

47 6 . 2 9

. 00

-115 . 58

2322 . 80

1019 . 48

. 00

1 I 2 3 1

3

2

1

• •

At "DO YOU WANT A PRINT- OUT ( Y/ N ) ? " type y (or Y) and press RETURN. At " DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type y (or Y ) and RETURN .

•

A t "ENTER A NAME FOR THE OUTPUT FILE : " type a: output l .out.

•

At

" DO YOU WANT TO START A

NEW

CALCULATION WITH THE NRTL MODEL ( Y/ N ) ? "

type n (or N) and press RETURN. •

At "DO YOU WANT TO SELECT A

NEW

EXCESS ENERGY MODEL (Y / N ) ? "

type n (or N ) and press RETURN.

Exam ple E J . B: Calculation of M u lticomponent VLE Using an Existing I n put File •

Change to the directory containing the program HVMMAIN .EXE (e. g . , A> or C>, etc . )

•

Start the program by typing HVMMAIN at the DOS prompt . A program introduction message appears on the screen . Press ENTER (or press RETURN). The following appears:

" HVM :

MULTICOMPONENT VL E CALCULATIONS WITH THE HVOS MIXING RULE . YOU HAVE T O SELECT

AN EXCESS ENERGY MODEL TO BE USED IN THE HVOS MIXING RULE . THE SELECTIONS ARE : 1 =NRTL 2 =WILSON 3 =UNIQUAC 1/2/3?"

1 95

Modeling Vapor-Liq u i d E q u i l i bria

Type 1 and press RETURN. (This results i n the selection of the NRTL mode l . ) • " HVM :

At

MULT:ICOMPONENT VL E CALCULATIONS WITH THE HVOS

+

NRTL MODEL .

THI S PROGRAM CAN BE USED FOR :I SOTHERMAL BUBBLE POINT CALCULATIONS , CREATING A NEW :INPUT FILE , OR USING A PREVIOUSLY STORED :INPUT FILE . YOU MUST SUPPLY THE NUMBER OF COMPONENTS , LIQUID MOLE FRACTION, CRITICAL TEMPERATURE , CRITICAL PRESSURE , ACENTRIC FACTOR, PRSV KAPPA- 1 PARAMETER FOR EACH COMPOUND, TEMPERATURE , AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS . ENTER 1 TO CREATE A NEW :INPUT FILE , 2 TO SELECT A PREVIOUSLY STORED :INPUT FILE , OR 0 TO TERMINATE THE PROGRAM . 0/1/2?

type 2 and press RETURN. (This results in the use of an existing input file . ) •

A t "ENTER NAME O F THE EXI STING INPUT FILE ( for example , a : PE4 2 3 . HVN ) : " type AMW250.HYN and press RETURN. The following appears on the screen :

HVM :

MULT:ICOMPONENT VLE WITH THE HVOS MIXING RULE

:INPUT FILE

NAME :

AMW2 5 0 . HVN

ACETONE -METHANOL-WATER 2 5 0 C SET NO .

TEMP ( K )

PEXP ( BAR )

PCAL

VL:IQ ( CM3 /MOL )

VVAP

1

523 . 15

62 . 060

61 . 734

48 . 421

4 52 . 7

2

523 . 15

58 . 480

58 . 477

4 1 . 539

508 . 4

3

523 . 15

52 . 890

5 2 . 14 1

34 . 546

621 . 6

pre s s return for phase compos it ions .

Press RETURN to see phase compositions . The following appears : PHASE COMPOSITIONS SET NO . 1

2

3

( IN MOLE FRACTION)

COMPONENT

XEXP

YEXP

YCAL

1

. 13 7 0

. 2370

. 2455

2

. 0940

. 14 8 0

. 14 3 6

3

. 7 690

. 6150

. 6109

1

. 0880

. 1920

. 1988

2

. 0840

. 14 5 0

. 14 6 8

3

. 82 8 0

. 6630

. 6 544

1

. 04 3 0

. 14 3 0

. 14 1 6

2

. 0490

. 0950

. 1087

3

. 9080

. 7620

. 7496

pre s s return for the NRTL model a lpha parameter matrix .

1 96

Appen dix E: Computer Programs fo r M u lticomponent M i xtures

Press RETURN to continue. The following parameter matrix for the NRTL model

a

parameter appears:

PARAMETER MATRIX FOR THE ALPHA PARAMETER 1

2

3

11

. 0000

. 3 500

. 35001

21

. 3500

. 0000

. 35001

31

. 3500

. 3500

. 00001

pre s s return for the NRTL model energy parameter matrix .

Press RETURN to continue. The fol lowing parameter matrix for the NRTL model energy parameters appears : PARAMETER MATRIX FOR THE NRTL ENERGY PARAMETER AIJ ( CAL/MOLE ) 2

1 11

. 00

21

47 6 . 29

31

2322 . 80

3

-103 . 00

278 . 861

00

-115 . 58 1

•

1019 . 48

•

001

•

At "DO YOU WANT A PRINT-OUT ( Y/ N ) ? " type n (or N) and press RETURN.

•

At

" DO YOU WANT TO SAVE THESE RESULTS TO AN OUTPUT FILE ( Y/ N ) ? " type n (or N ) and RETURN. •

At

" DO YOU WANT TO START A NEW CALCULATION WITH THE NRTL MODEL (Y / N ) ? " type n (or N ) and press RETURN . •

At "DO YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL ( Y/ N ) ? " type n (or N ) and press RETURN.

1 97

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References

Abrams, D. S . , and Prausnitz, J. M . , 1 975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems.

A/ChE J. , 2 1 : 1 1 6-- 1 28 . Adachi, Y. , and Sugie, H., 1 986. A new mixing rule- modified conventional mixing rule. Fluid Phase Eq. , 2 3 : 1 03- 1 1 8 . Anderko, A . , 1 990. Equation of state models for modeling of phase equilibria. Fluid Phase Eq. , 6 1 : 1 45-225 . Apostolou, D. A., Kalospiros, N. S . , and Tassios, D. P. , 1 995 . Prediction o f gas solubilities using the LCVM equation of state/excess Gibbs free energy model. Ind. Eng. Chern. Res. , 34:948-957 . Barr-David, F. , and Dodge, B . F. , 1 959. Vapor liquid equilibrium a t high pressures, the systems ethanol + water and 2-propanol + water. J. Chern. Eng. Data, 4: 1 07- 1 2 1 .

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204

Ind ex

1 PVDW model, 26-27, 35, 1 05 2PVDW model, 34, 1 05 Acentric factor, 20, 2 1 Acetone

Computer program(s) for binary mixtures, 1 1 4 for multicomponent mixtures, 1 77 Critical

PRSV parameters, 2 1

compressibility, 23

VLE with water, 3 1 , 36, 49, 56, 69, 79-8 1

temperature, 1 9, 2 1

Activity coefficient, 6, 1 1 , I 03

at infinite dilution, 86 from an equation of state, 39

Activity coefficient models, 8, 1 1 - 1 7 , 1 02 Margules, 1 3

pressure, 1 9, 2 1 Cubic equations o f state, 1 9 Cyclohexane PRSV parameters, 2 1 VLE with methyl acetate, 4 1

NRTL, 1 3 UNIFAC, 1 6 UNIQUAC, 1 3 Wilson , 1 3

(a) parameter, Peng-Robinson equation, 20

van Laar, 1 2 Alpha

Antoine equation, 9 ASOG model, 1 6 Benzene

PRSV parameters, 2 1 VLE with carbon dioxide, 92-93 VLE with methanol, 77-79 Binary interaction parameters, 26, 34, 40, 5 1 , 57 Carbon dioxide PRSV parameters, 2 1 V LE with benzene, 90-92 VLE with methanol, 92-93 VLE with propane, 28, 48, 69

DECHEMA, 9 , 1 4 Density dependent mixing ru le, 53 Enthalpy, 95 Entropy, 95 Ethanol PRSV parameters, 2 1 VLE with n-heptane, 28, 35, 40-42 Equation of state, 7 Models, 1 7, 1 04 Excess free energy Gibbs, of mixing, 33 Helmholtz, mixing, 44 Molar, Gibbs, 44 FORTRAN, 1 1 4

Fugacity, 6, 1 7 , I 03

Fugacity coefficient, 6, 7, 8, 1 04

Chemical potential, 5 Chemical reaction, 98

Gamma-Phi method, 7

Combining rule(s), 23

Gibbs

Compressibility factor, 7, 1 8 Computational methods

excess energy, of mixing, 44

free energy departure function, 1 1 2

205

I ndex

Gibbs (continued) free energy in ideal mixture, 5 partial molar free energy, 5

PRSV parameters, 2 1 VLE with methane, 89-9 1 n -Heptane PRSV parameters, 2 1

Heat capacity, 95 Henry 's constants, 95 Helmholtz excess free energy, of mixing, 44 free energy departure function, 1 1 2

see Huron-Vidal model

Huron-Vidal (HVO) model , 48, 1 07 HVO model ,

HVOS model, 63, 66 Ideal gas equation, 8

VLE with ethanol, 40-42 VLE with methane, 89-9 1 n-Hexane, PRSV parameters, 2 1 Nonelectrolyte mixtures, I 00

Non-quadratic combi ning rules, 34 n -Pentane PRSV parameters, 2 1

VLE with ethanol, 28, 35

VLE with methane, 27, 48, 67, 89-90

Infinite dilution activity coefficient, 8 6, 95

NRTL model, 1 3 Modified form of, 57

Kappa (K) Parameter

One-fluid model, 25, 1 05

Infinite pressure, limit, 46

Peng-Robinson, 20 PRSV, 20, 2 1

Partial molar Gibbs free energy, 5

Peng-Robinson equation of state, 7, 1 9, 1 04

LCVM model, 63, 65, 1 09

virial form for the, 25

Liquid-liquid equilibrium (LLE), 95, 97, 1 00

reduced form, 44

LLE,

see

liquid-liquid equilibrium

Pitzer's acentric factor, 20, 2 1 Polymer, 97

Margules equation, 1 2

Poynting correction, 9

Methane

Predictive models, 75

PRSV parameters, 2 1 VLE with n -decane, 89-9 1 VLE with n -heptane, 89-90 VLE with n -pentane, 27, 48, 67, 89-9 1 Methanol PRSV parameters, 2 1 VLE with benzene, 77-79 VLE with propane, 29, 35 Methyl acetate PRSV parameters, 2 1 VLE with cyclohexane, 4 1 MHV l model, 63, 64, 1 08 MHV2 mode l , 63, 65, 1 08 Micellar solutions, 9 8 Michelsen-Ki stenmacher (syndrome) , 4 2

Mixing rule(s), 23, 25, 44 HVOS, 63, 66, I 09

LCVM, 63, 65, 1 09 MHV l , 63, 64, 1 08

for mixtures of condensable compounds, 75

for mixtures with supercritical gases, 88

Propane

PRSV parameters, 2 1 VLE with carbon dioxide, 28, 48, 69 VLE with methanol, 29, 35 Propanol, 2PRSV parameters, 2 1 VLE with water, 29, 36, 49, 56, 69, 82-84 PRSV equation, 20 Raoult's law, I I

Redlich-Kister equation, I I

Reduced

Peng-Robi nson equation of state, 46 temperature, 22, 46 pressure, 46 Regular solution model, 1 5 Saturation pressure, pure component, 9, 2 1

van der Waals, 26, 34

Simplex formalism, I I 0 Solubility parameter, 1 5

Wong and Sandler, 50, 1 06

Supercritical

MHV2, 63, 65, 1 08

Multicomponent, computer programs for, mixtures, 1 77

fluid, 97 gases. 88

n - B utanol, PRSV parameters, 2 1

UNIFAC model, 1 6

n- Decane

UNIQUAC model, 1 3

206

I ndex

van der Waals mixing rules, 26, 34 van Laar equation, 1 2 Vapor-liquid equilibrium, 6 , 7 , 1 9 of, acetone with water, 3 1 , 36, 49, 56, 69, 79-8 1 of, benzene with carbon dioxide, 90--9 2

of, water with acetone, 3 1 , 36, 49, 56, 69, 79-8 1 of, water with 2-propanol, 29, 36, 49, 56, 69, 82-84

programming, I I 0

Vapor-l iquid-liquid equilibrium, 95 , I 00

of, benzene with methanol, 77-79

Vapor pressure, pure liquid, 9

of, carbon dioxide with benzene, 90--92

Virial

of, carbon dioxide with methanol, 92-93

equation of state, 7, 24

of, carbon dioxide with propane, 28, 48, 69

second coefficient, 7 , 24

of, cyclohexane with methyl acetate, 4 1

third coefficient, 7, 24 see

of, ethanol with n -pentane, 2 8 , 3 5

VLE,

of, ethanol with n-heptane, 40--4 2

VLLE,

of, methane with n -decane, 89-9 1

Vol ume, reduced, 46

of, methane with n -heptane, 89-90 of, methane with n-pentane, 27, 48, 67, 89-9 1 of, methanol with benzene, 77-79 of, methanol with propane, 29, 35 of, methyl acetate with cyclohexane, 4 1 of, n-decane with methane, 89-9 1 of, n-heptane with ethanol, 40--4 2 of, n-heptane with methane, 89-90 of, n-pentane with ethanol, 28, 35 of, n-pentane with methane, 27, 48, 67, 89-9 1 of, propane with carbon dioxide, 28, 48, 69 of, propane with methanol, 29, 35

vapor-liquid equilibrium

see

vapor-liquid-liquid equilibrium

Water PRSV parameters, 2 1 VLE with acetone, 3 1 , 36, 49, 56, 69, 79-8 1 VLE with 2-propanol, 30, 36, 49, 56, 69, 82-84 Wilson equation, 1 3 Wohl expansion, 1 2

Wong-Sandier model, 50, I 06 WS model,

see

Wong-Sandier model

of, 2-propanol with water, 29, 36, 49, 56, 69, 82-84

Zero pressure, limit, 46

207