Practical equation of state and activity coefficient models based on the exp-6 fluid

Fluid Phase Equilibria, 69 (1991) 99-139

99

Elsevier Science Publishers B.V., Amsterdam

Practical equation of state and activity coefficient models based on the exp-6 fluid Lawrence R. Dodd 1 and Stanley I. Sandler * Department of Chemical Engineering, University of Delaware, Newark, DE 19716 (USA)

(Received January 11, 1991; accepted in final form May 28, 1991)

ABSTRACT Dodd, L.R. and Sandler, S.I., 1991. Practical equation of state and activity coefficient models based on the expd fluid. Fluid Phase Equilibria, 69: 99-139. In this paper we test a partition function model recently developed for the expd fluid mixture against experimental data for real fluids. This model leads to a unique size and density-dependent mixing rule providing non-quadratic density-dependence which satisfies the low-density and pure fluid boundary conditions. The equation of state calculations with this model gave good results, especially for highly asymmetric systems. Applications of a similar mixing rule to cubic engineering equations of state for real fluids yielded better predictions than the van der Waals one-fluid mixing rule. However, except in the near critical region, any clear advantage of the new density-dependent mixing rule for real fluids is lost when adjustable parameters are added. We also show that an activity coefficient model based on this expd model leads to good correlations for low pressure vapor-liquid equilibria of both simple and very complex mixtures, indeed mixtures that could not easily be described by other models with the same number of parameters.

INTRODUCTION

In our previous work using the modified expd fluid, we were able to obtain a model for the partition function for the exp-6 fluid using generalized van der Waals theory. In this work we consider the extension of this partition function to real fluid mixtures based on the insights obtained from our computer simulation studies. We then compare the predictions

’ Present address: Department of Chemical Engineering, University of California, Berkeley, CA 94720, USA. * Author to whom correspondence should be addressed. 0378-3812/91/$03.50

0 1991 Elsevier Science Publishers B.V. All rights reserved

100

and correlations of these models, including the use of densi~-dependent mixing rules, with those of commonly used equations of state using high pressure vapor-liquid equilibrium data, and with the results of a number of excess Gibbs free energy (activity coefficient) models for low pressure VLE data. In our previous study we found that the partition function for N modi~ed exp-6 molecules in a volume V at temperature 7’ is

with 17= 7rpa3/6 = 7rN~$~/(6V) where the effective hard-sphere dHS is

diameter

and LX(T)= 0.1045832 [/$-

- ;]

-6.445300&

The equation of state that results from this partition function is p=-

RT u3 + (b/4)u2 + (b/4)*u - (b/4) V

[u-(b/4)13

where u = ~~/~)~~ is R is the gas constant, and a = -axr3Na. We also found that, tion for exp-6 mixtures

RT a --G

(3)

the molar volume, P is pressure, T is temperature, A?, is Avogadro’s number, b = (2?~/3>[d~~(T)]~lV,, to a reasonable is

approximation,

the partition func-

101

where a3 = Cx,uj and the &PM are defined in Appendix A. The functions cr and p for mixtures are aij

=

a( Eij/kT) = Cl

pi = p( EJkT)

=

2 +c* [pp]

c3 g

+

cq

(5)

[&g]

with c1 = - 6.3212, c2 = 0.80082, c3 = -0.56667 and cq = - 0.59098 being parameters fitted to mixture simulation data. The resulting equation of state is P _ RT u3 + (~t;~/4)b,~v~ V

+ (c~;~/16)b&p 1V -

knk/4)13

- (c~f~/64)b;~,

---

RT amix

v v (6)

The repulsive term is the Boublik-Mansoori equation (see Appendix A for the fugacity expression and the aBM definitions). Also amix= CCxixjaij, where aij = aTyW(T) + a$(T, p’, x), has both density-independent (vdW) and density-dependent (dd) parts defined as aij = [ -"ijUi~Na] 0;;”

-1 +

-3(

~03N,)2( XiPi

+

Xjpj)

(yqimj

(7)

add 11

and b,, = Cxibi and bi = 2’rr[d~~‘(Eii/kT)13N,/3 where d,yS is the effective hard-core diameter as defined by Barker and Henderson (1967) and p’ = l/v. Note that th e aij, and therefore amix, are functions of temperature, density and composition, and only at low density is amix quadratic in composition. This equation of state mixing rule was found to describe the exp-6 model fluid quite well. In the generalized van der Waals description of the pressure @/2kT represents the attractive and soft-core repulsive forces of the fluid. This ratio, by definition, goes to zero as temperature increases. Thus, the “attractive” pressure Pat* over kT goes to zero at infinite temperature as well. However, the attractive pressure Pat* need not go to zero at infinite temperature, though it does in many empirical equations of state (e.g. Redlich-Kwong, 1949). In fact, the attractive pressure need only be less than linear in temperature in the limit of infinite temperature for Patt/kT to have the correct infinite temperature limit. The temperature dependence of our model for the attractive pressure obeys this limit.

102

APPLICATION

TO REAL FLUID MIXTURES

We now consider the application of eqn. (6) to real fluid mixtures. The first system studied was the binary mixture of argon and methane, Both these species are nearly spherical and aE = 14 based on virial coefficient data (Hirschfelder et al., 1964) the mixture should therefore be conformal. One set of E and cr parameters for this equation was obtained by fitting the saturated vapor pressures and volumes of each pure fluid from T/T, = 0.5 to the critical point. A second set was found by matching experimental critical properties to those of the exp-6 fluid given earlier (Dodd and Sandler, 1989). The cross parameters for all calculations were found from Ejj

=

fi(l

aij = &(l

-k,)

(8)

- Zij)

where kij and fij are binary interaction constants. We also used eqn. (6) with standard one-fluid van der Waals mixing rules obtained by omitting the densi~-dependent term, a$$, of eqn. (7); i.e. CL,~= CCxixjaij and bmiv= Shown in Table 1 are the results of bubble-point pressure calculations using eqn. (6) together with the results of the Peng-Robinson equation of CCx,Xjbjj.

TABLE I Bubble-point calculations results for the expd equation of state (eqn. (6)) with van der Waals one-fluid mixing rules (eqn. (6)-vdW) and with density-dependent mixing rules (eqn. (6)-dd), and for the Peng-Robinson equation of state with van der Waals one-fluid mixing rules (PR-vdW). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

argon / methane

T range (Kelvin) 115.77 to 132.97 P range (bar) 3.040 to 22.291 Number of data points = 33 Equation

t - pure component $ - pure component

P% ~1% N

parameters parameters

P% ~1%

obtained obtained

kzz N

P% ~1%

from critical point. from binodal curve.

Jqa

G,

N

103

state (Peng and Robinson, 19761, using the van der Waals one-fluid mixing rules and the following two combining rules with interaction parameters

aij = b,j

JG(l bii + bjj

= y(l

kij)

-

‘ij)

Both equations of state were considered with: (1) all binary interaction parameters set to zero (kij I=Zij= 0); (2) k,j optimized and lij = 0; and (3) k,, and Zij optimized. Here by optimized we mean chosen to minimize errors in the bubble-point pressure. We see from the results in Table 1 that the perfo~an~e of eqn, (6) is strongly dependent on the way in which the pure component parameters are obtained, and on the number of binary interaction parameters used. When using pure fluid parameters obtained from critical point data, the predictions of eqn. (6) are not very good, though some of this error can be reduced by using one or two binary constants. However, using interaction parameters with pure ffuid parameters obtained from vapor pressure and density data results in correlation of the mixture data that is comparable to the Peng-Robinson equation of state. Consequently, in all subsequent calculations we used E and cr, fit only to vapor pressure and density data. We also see in Table 1 that the predictions using eqn. (6) for this simple system are slightly better with the densi~-dependent mixing rules than with van der Waals one-fluid mixing rules when both are used without binary interaction parameters (i.e. kij = lij = 0). However, this difference disappears when optimized binary interaction parameters are used. The density-dependent mixing rule does lead to smaller values of binary interaction parameters, though this is only a minor advantage. Similar conclusions for other density-dependent mixing rules were obtained by Deiters (1987) and Lee et al. (1989). Next we applied the exp-6 equation of state proposed here, eqn. (61, to mixtures of non-spherical molecules. We tested this equation both with and without the densi~-dependent correction term, and compared the results with those of the Peng-Robinson equation using a comparable number of adjustable parameters. The mixtures studied were largely hydrocarbons with supercritical nitrogen, carbon dioxide, or hydrogen. We performed calculations with zero, one and two binary interaction parameters fitted to minimize bubble-point pressure errors. The results for bubble point pressures and vapor compositions are presented in Tables Bl-B5 of Appendix B, and comparisons with experimental two-phase volumes are given in Tables B6-139 for those cases in which such data were available. (All data were taken from the DECHEMA Chemistry Data Series (Knapp et al.,

104 250

200

;: 2

150

: 2 g

100

L 50

0 0.0

0.2

Mole

0.4

0.6

Fraction

0.8

0

Argon

Fig. 1. Binary phase diagram of argon/n-butane at 340.0 K. o, experimental data. The predictions of eon. (6) are shown for the density-dependent mixing rule with cs and cq optimized (solid curve), for the density-dependent mixing rule with kij and lij optimized (small-dashed curve - - - ), and for van der Waals one-fluid mixing rules with k,j and lij optimized (long-dashed curve - - - 1. Also shown are the predictions of the Peng-Robinson equation with k,j, and l,, optimized (dot-dashed curve -a-).

1982j.j We see in these tables that the performance of eqn. (6) is comparable to the Peng-Robinson equation, and that, in general, any advantage of the density-dependent mixing rule diminishes when binary interaction parameters are introduced. However, for highly asymmetric systems, such as the hydrogen-containing mixtures, the predictions of eqn. (6) are superior to the Peng-Robinson equation of state which, with zero values of the binary interaction parameters, converged at only a few experimental points. The binary phase diagrams for n-butane with argon and nitrogen respectively, are presented in Figs. 1 and 2, and for hydrogen with n-hexane in Fig. 3. All these diagrams show the predictions of eqn. (6) with and without density-dependent mixing rules, and of the Peng-Robinson equation with van der Waals one-fluid mixing rules. For both equations the binary interaction parameters (kij and Iii) were fit to minimize bubble-point pressure errors. Finally, we also examined the density-dependent mixing rule of eqn. (7) by fitting cg and c4 (instead of k,, and Zij) of the pi temperature functions of eqn. (5) to minimize bubble-point pressure errors. There are two conclusions to be drawn from these figures. First, for eqn. (6), the density-dependent mixing rule with two binary interaction parameters does not perform as well as the van der Waals one-fluid mixing rule in predicting the critical point of the mixture, although on the average it results in better bubble-point pressure predictions. Second, if c3 and c4 are

105

300

,\ /

\

I,/

250-

~1

,';',

‘,

,,/ ,'.' 5

‘.\ '\

I

‘1 ':

zoo-

e : 7

150-

D 2 a

loo-

50-

Oo10 Mole

I.11

Fraction

Nitrogen

Fig. 2. Binary phase diagram of nitrogen/n-butane at 339.4 K. o, experimental data. The predictions of eqn. (6) are shown for the density-dependent mixing rule with c3 and cq optimized (solid curve), for the density-dependent mixing rule with k,j and I,, optimized (small-dashed curve - - - ), and for van der Waals one-fluid mixing rules with k,j and Ii, optimized (long-dashed curve - - - ). Also shown are the predictions of the Peng-Robinson equation with k,, and li, optimized (dot-dashed curve -.-).

,’ ; ’

750

’ ’ ’ ’

z

600

’

e :

I

’ I

450

rz 6 L

300

150

0 0.0

0.2

Mole

0.4

Fraction

0.6

0.3

1.1D

Hydrogen

Fig. 3. Binary phase diagram of hydrogen/n-hexane at 477.5 K. o, experimental data. The predictions of eqn. (6) are shown for the density-dependent mixing rule with ca and c4 optimized (solid curve), for the density-dependent mixing rule with kij and lij optimized (small-dashed curve - - - 1, and for van der Waals one-fluid mixing rules with k,, and I,, optimized (long-dashed curve - - - ). Also shown are the predictions of the Peng-Robinson equation with k,, and I,,, optimized (dot-dashed curve -a-).

106

optnmized rather than kij and li,, the best overall prediction of the mixture critical point is obtained. This is most evident in Fig. 3 where we see the Peng-Robinson equation of state underpredicts the mixture critical point, while eqn. (6) with the density-dependent mixing rule overpredicts the critical point if the binary interaction parameters are optimized, but fits the two-phase region almost exactly if we optimize cf and c4 instead. Thus, for eqn. (6), the densi~-dependent mixing rule constants c3 and c4 for the 6 parameter have a greater effect on the predictions than the binary interaction parameters. Based on the systems studied, the density-dependent mixing rule developed for the expd mixture improves predictions in asymmetric mixtures because of the explicit size dependence of the densi~-dependent correction, Us. Moreover, even without the density-dependent mixing rule, eqn. (6) occasionally performs better than the Peng-Robinson equation for mixtures in the two-phase region. A disadvantage of the density-dependent mixing rule developed here is that the correction to the van der Waals one-fluid mixing rule adds an additional term in the expression for the attractive pressure of the equation of state, the @: of eqn. (71, which requires additional fitting to mixture data for the best performance (e.g. optimizing c3 and c,). Consequently, for application to real mixtures below, we develop a density-dependent correction that has the essential features found necessary for the description of the expd mixture, but is in terms of only the pure fhrid equation of state parameters and thus does not require fitting any additional mixture constants.

EMPIRICAL MODIFICATION OF THE DENSITY-DEPENDENT

MIXING RULE

Based on our observations in the previous section and the mean density approximation (MDA) of Leland and co-workers (Mansoori and Leland, 1972; Leland, 1976; Gonsalves and Leland, 1978), we now develop a new equation of state mixing rule, which is consistent with, and simpler than, our previous work. The mean density appro~mation for the radial distribution function is gij(r/oij;

pl? P2,“.9

T) = g ’ (r/c

; pb:,

Eij/kT)

(10)

where g, and g ’ are the radial distribution functions for the mixture and a pure fluid respectively, p02 is the density reduced with a composition-dependent diameter u-,. Using this in the compressibili~ expression

(11)

107

which relates the radial distribution functions of the components in a mixture, g,,, to the mixture compressibility, 2, results in the following expression (Huber and Ely, 1987, 1989)

’ - l = C

( 1‘(Z$

Cxi”j

i

l)lpm~,e,,/kT

:

j

(12)

x

Here 2: is the pure fluid compressibility evaluated at the reduced density pa: and reduced temperature l ij/kT. As pointed out by Khan et al. (1987), when mixing rules of the form of eqn. (12) are used for both the repulsive and attractive terms they result in multiple hard-core singularities, e.g. three for the l-l, 2-2 and l-2 interactions in a binary mixture. To avoid divergence in the equation of state, only the largest value can be the hard-core volume for the mixture at all compositions. This can result in an unrealistically large hard-core volume, particularly for dilute mixtures of highly asymmetric components. Instead we propose to use eqn. (12) for only the attractive term, i.e. Zatt =

C

CXiXj

i

1

3z;4”l~$,~~,,,,

~

i

j

x

(13)

We refer to the above mixing rule as the mean density approximation for the attractive term (MDAAT). The mixing rule of Lee et al. (1989) which was proposed earlier is a special case of eqn. (13) with Uii/Ux = 1 for all ij. This modification of the mean-density approximation is general and can be used with any equation of state. This is in contrast with the versions of Ely (1986) and Dieters (1986) which use pertubative approaches about hard spheres in developing a mixture equation of state. Applying eqn. (13) to the Peng-Robinson equation of state, we obtain P=

RT v-bmi,

- ~ ~x’x’

(v + Sbij/8i,~yV + tbij/Sij)

(14)

where s = 1 + \/z and t = 1 - fi, and we use the combining rules of eqn. (9). The only unknowns in eqn. (14) are each of the aij = (qj/oXj3, which are dependent on composition. An important feature of eqn. (14) is the explicit composition dependence of the attractive term which changes with density as did our simulation data for exponential-6 mixtures. Equation (14) can be rewritten as p=

RT v-bb,i,

--

amix v2

(15)

108

where a mix =

c cxixj i

j

‘ij

(1 +

sp’bij/~ij)(

1+

(16) t/Sbij/8ij)

with p’ = l/v equal to the molar density of the mixture. Equation (16) meets the low-density boundary condition in that it reduces to the van der Waals one-fluid mixing rule, amix= CCxixjaij as p’ + 0. Because of the added density dependence of the MDAAT, and therefore amix, the order of the mixture equation of state with respect to volume is increased above that for the pure fluid. In practice, this adds little complexity to the calculations if one first computes the volume by solving eqn. (15) using amix= CCxixjUij, recalculates amix using this volume in eqn. (16), and then resolves eqn. (15), repeats the procedure until the value of amix (and volume) converges (see Lee et al., 1989). In the original derivation of the MDA (Mansoori and Leland, 1972) a: was set equal to XXnixj~$ However, Huber and Ely (1989) point out that there is no theoretical justification for using this, and Gonsalves and Leland (1978) found that 0: = Cx,ai was best for mixtures of components with large size differences. Therefore, we have used the latter with the arithmetic-average combining rule for oij = i(ali + ojj). The unknown parameter ajj/aii must be found from the ratio of some measure of the molecular sizes of the components. Two alternatives are the cube root of the ratio of the critical volumes and the ratio of UNIFAC group volume parameters, r (Gmehling et al., 1982; Sander et al., 1983). In Tables BlO-B13 of Appendix B we compare the bubble-point correlations using the Peng-Robinson equation of state with the van der Waals one fluid, the MDAAT, and the n-fluid mixing rule of Lee et al. (1989). (Though not presented here, similar results have been found for the Soave (1972) version of the Redlich-Kwong equation of state.) For systems for which experimental densities are available, we show the errors in these predictions in Tables B14 and B15. For all these systems, oi//Uii was determined from critical volumes as this resulted in slightly better predictions than using UNIFAC r parameters. We see that in most cases the MDAAT and Lee’s n-fluid mixing rule result in better predictions (all binary interaction parameters set equal to zero) than the van der Waals one-fluid mixing rule. Furthermore, MDAAT of eqns. (15) and (16) is best for high pressure (and hence high density) systems. Most striking, as seen in Table 2, is that the MDAAT mixing rule predicts reasonably accurate phase behavior for the CO,/water system even without the use of a binary parameter, while the usual van der Waals one-fluid and Lee’s mixing rules converge for only a few points. We also show the predictions of the expd mixture equation of state of the previous

109 TABLE 2 Bubble-point calculation results for the expd equation of state (eqn. (6)) with the van der Waals one-fluid (eqn. (6)-vdW) and density-dependent mixing rules (eqn. (6)-dd), and the Peng-Robinson equation of state with van der Waals one-fluid mixing rules (PR-vdW), Lee’s n-fluid mixing rule (PR-Lee), and the MDAAT (PR-MDAAT). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

carbon dioxide/water T (Kelvin) 541.15 P range (bar) 200 to 1500 Number of data ooints = 14 Equation eqn. (6)-vdW

eqn. (B)-dd PR-vdW PR-Lee PR-MDAAT

P%

~1%

N

P% ~1% kl2

N

13.4 20.9 29.3 36.3 24.4

19.3 12.9 12.3 19.2 15.8

11 13 2 1 12

4.6 20.7 8.7 9.8 7.5

11 13 13 13 13

I-

18.7 13.0 12.6 9.0 15.2

-0.01 0.00 0.05 0.69 0.02

4.6 4.5 8.6 9.4 6.7

18.7 13.2 8.5 7.2 13.6

-0.01 -0.07 0.09 0.11 0.05

0.00 0.03 0.02 0.01 0.02

11 13 13 13 14

section both with and without the density-dependent mixing rule, and note that overall this equation performs best. However, once a binary interaction parameter, k,,, is fit to the VLE data, the results for all equations and mixing rules are almost indistinguishable, though as might be expected based on the previous discussion, the binary interaction parameters for the density-dependent mixing rules are again smaller than those for the van der Waals mixing rules. The only real advantage of the density-dependent mixing rule that remains when a binary interaction parameter is used is that this mixing rule is better able to predict and/or correlate the phase behavior in the vicinity of the mixture critical point.

ACTIVITY

COEFFICIENT

MODEL

Using the partition function of eqn. (4), we can also derive an activity coefficient model. The excess Helmholtz free energy for a mixture at constant temperature and volume is given by AE=A(Nr,

A$,...,

V, T) - CAi(N;:, i

c::, T) -kTCN, i

In xi

(17)

where A and Ai are the Helmholtz free energies of the mixture and pure component i respectively, while V and F are their volumes. Using the

110

definition for the Helmholtz free energy (A = -kT In Q) and of the generalized van der Waals partition function for the mixture and for the pure fluids, eqn. (17) can be written as

where @ and V, are the mean potential and free volume respectively, and the subscript i denotes a pure component property. To obtain the excess Gibbs free energy for mixing at constant pressure, and consequently activity coefficients, from the expression for the Helmholtz free energy of eqn. (X3), the mixing process must be considered carefully. As discussed by Scatchard (1937), and elaborated upon by Sandler et al. (1989), the excess Gibbs free energy for mixing at constant pressure can be represented very accurately by the Helmholtz free energy for mixing at ideal volume. That is, mixing in which the pure fluids are at temperature T and pressure P, and the mixture is at the same temperature and a pressure so that V= XV;:. Therefore, in what follows, we take the mixture volume to be the sum of the pure liquid volumes. Thus, we can write eqn. (18) as

(19) combinatorial =[C~b+[&~

residual (20)

where gE is the molar excess Gibbs free energy for mixing at constant i‘ and P. The combinatorial part of the free energy arises from entropic contributions, while the residual part is the enthalpic contribution due to the attractive intermolecular forces. The partial molar excess Gibbs free energy of component k is

TJ’,n,+nk

= ln ryrnb + In ypd

(21)

where nT = Cn, is the total number of moles. Note that as a result of the redefinition of the partition functions for pure fluids and mixtures introduced in our previous work (Dodd and Sandler, 1990, the free volumes and combinatorial terms now include the

111

effects of soft-core repulsive forces, and are temperature dependent. Using the expressions for the pure fluid and mixture partition function (eqns. (1) and (4)) we obtain

-1 raid

gE

RT

i

-

i

j

j

~xiaipiai

(22)

where pi = Ni/F. For the combinatorial term we use the Boublik-Mansoori free volume for the mixture and the Carnahan-Starling (Carnahan and Starling, 1969) free volume for the pure fluid, both containing the BarkerHenderson temperature-dependent, hard-sphere diameter, and write comb =

-p,“”

ln(1

- 77)+ 7j

P,“”

-

4-37&

P3B”rl ExiTi

(1 - 17)2

i

(1 -

%I2

(23) Here the reduced densities are defined as

(24)

and rli(T) = ;;(d;s)3 I

and the p,““, /3,“” and p3BM, defined composition and temperature-dependent

in Appendix A, are functions of effective hard-core diameters,

dHS.

From the pure-fluid equation of state (see eqn. (3)) we find that the liquid volume of the expd fluid can be approximated as K = A&( dFs)3

(25)

where Ni is the number of type i atoms, and d&,/kT) *’ = d, + (q/kT)

(26)

with d, = 0.7519587 and d, = -0.5589981. We have found this simple equation to be valid for l j/kT = 1.0 to 2.0 (for (Ye= 14) where the average percent deviation is only 0.108%; the value of Ai changes from 1.04 to 1.69

112

in this temperature range. Following Scatchard, we write for the volume of the mixture V= C,V, = C,A,N,(dys)3. However, in analogy with eqn. (25), we also have I/=

ANCX,(d;s)3

(27)

I

for the mixture volume. Equating these two expressions and solving for the mixture A in terms of composition, temperature-dependent hard-sphere diameters, and the pure fluid hi values, we have

Replacing the volumes appearing in eqns. (22) and (23) with eqns. (25) and (28), the excess Gibbs free energy can be written analytically as a function of only temperature and composition, and then be differentiated with respect to mole number (eqn. (21)) to obtain the activity coefficient model shown in Appendix C.

TEST OF THE NEW ACTIVITY COEFFICIENT MODEL

The exp-6 activity coefficient model developed here was tested with low pressure, isothermal vapor-liquid equilibrium data for a number of real binary mixtures. The goal was to show that good correlations are possible using a model developed from theory. The exp-6 activity coefficient model has three parameters: the pure fluid energy parameters, lii and cZ2, and the size ratio, (+&(~ii. The geometric combining rules of eqn. (8) were used for cl2 and c12. The results of calculations for the n-pentane/propionaldehyde system, and the aqueous systems of pyrrolidine/water morpholine/water are presented in Table 3. We compare these results with those of four other activity coefficient models: the three-parameter NRTL (non-random two liquid) model of Renon and Prausnitz (19681, and the two-constant (threesuffix) Margules (1895), van Laar (1910) and Wilson (1964) models. For each isotherm the parameters for all models were fit to minimize the sum of the squared relative deviations in the activity coefficients in a bubblepoint pressure calculation. Pressures in all calculations were less than 1.5 bar allowing us to neglect vapor-phase non-idealities. Calculations with the exp-6 model were also done with the size ratio set equal to the ratio of the UNIQUAC r parameters, (a22/a11)3 = y2/r1 (taken from Gmehling and Onken, 1977ff) thus reducing the number of adjusted parameters to two. We see in Table 3 that all models fit the n-pentane/propionaldehyde system equally well. The correlation of the data for this system for the

113 TABLE 3 Test of expd activity coefficient model with low-pressure bubble-point calculations. Shown are the average pressure deviations (API, in units of mm Hg, and average vapor mole fraction deviations for component one (Ay,) for the two- and three-parameter exp-6 model (exp-6 [2] and expd [3] respectively), and the Margules, van Laar, Wilson and NRTL models (N is the number of data points). All parameters were fit to the sum of the squared, relative deviations in activity coefficients

T = 333.15 (Kelvin)

T = 353.15 (Kelvin

0.0058

0.0046

0.0087

0.0099 0.0076

t - (Eng and Sandier, 1984); $ - (Wu et al., 1990)

two-parameter expd and NRTL models are shown in Fig. 4, where we see that the results for two-parameter exp-6 model are equivalent to those of the three-parameter NRTL model. More interesting phase behavior is

114

1000 G X

i

900

800 5 P

700

L a 600

500 0.0

0.2

Mole

0.4

Fraction

0.6

0.3

1.0

N-Pentane

Fig. 4. Low-pressure binary diagram for n-pentane in propionic aldehyde at 313.15 K. Shown are the experimental data of Eng and Sandler (1984), o, and the predictions of the two-parameter expd model (solid curve> and the three-parameter NRTL model (dashed curve). The curves are indistinguishable.

found in the pyrrolidine/water and morpholine/water systems (Wu et al., 1990). The two-phase regions in these systems widen, narrow and then widen again as a function of composition, as is seen in Figs. 5-8. This change in curvature is best predicted with the exp-6 models. As we see in Figs. 5 and 6 the three-parameter exp-6 model is superior to the threeparameter NRTL model in following the shape of the two-phase region for

150

’

0.0

Mole

Fraction

Pyrrolidine

Fig. 5. Low-pressure binary phase diagram for pyrrolidine in water at 333.15 K. Shown are the experimental data of Wu et al. (19891, o, and the predictions of the three-parameter expd model (solid curve) and the three-parameter NRTL model (dashed curve).

115

Joow 0.0

0.2

J.0

0.4

Mole Fraction

Morpholtne

Fig. 6. Low-pressure binary phase diagram for morpholine in water at 348.35 K, Shown are the experimental data of Wu et al. (19891, o, and the predictions of the three-parameter exp-6 model (solid curve) and the three-parameter NRTL model (dashed curve).

these systems. Likewise, in Figs. 7 and 8 we see that among the two-parameter models, the exp-6 model does as well as, or better, than the Margules, van Laar and Wilson models. A more extensive test of the three-parameter version of the exp-model was made with fifty-eight binary systems over 138 isotherms. All data were

600

550

h

s”

1

500

z 4w ?J

D

I

a

400

350

0.0

8/ 0.2

z

, 0.4

Mole Fraction

b

I

0.6

’

/

0.8

5

I

Pyrroiidine

Fig. 7. Low-pressure binary phase diagram for pyrrolidine in water at 353.15 K. Shown are the experimental data of Wu et al. (19891, 0, and the predictions of the two-parameter exp-6 model (solid curve) and the two-parameter Ma&es model (dashed curve).

116

)

250 0.0

0.2

0.4

Mole Fraction

0.6

0.8

0

Morpholine

Fig. 8. Low-pressure binary diagram for morpholine in water at 368.35 K. Shown are the experimental data of Wu et al. (1989), o, and the predictions of the two-parameter expd model (solid curve) and the two-parameter van Laar model (dashed curve).

taken from the DECHEMA Chemistry Data Series (Gmehling and Onken, 1977) and included only systems that passed both the point-wise and integral consistency tests. The systems studied are separated into the five classes of mixtures devised by Rarey-Nies (1987). He classified binary mixtures of alkanes and halogenated alkanes with esters, ethers, ketones and aldehydes as type I mixtures. Type II mixtures are binaries of water and alcohols; alkanes with alcohols are type III mixtures. Type IV mixtures are acetone/chloroform mixtures, and binary aromatic mixtures are type V. Thus, type I mixtures are simple non-polar systems, type II mixtures are those that associate through hydrogen bonding, type III mixtures are hydrogen-bonding components with non-polar alkanes, type IV mixtures combine Lewis base components with components containing acidic protons and exhibit strong negative deviations from Raoult’s law, and finally type V mixtures are binary systems of aromatic components. The specific mixtures considered are shown in Tables 4 and 5. The average deviation in bubble-point pressures and vapor-phase mole fractions for each class of mixture with the three-parameter exp-6 and NRTL models and the two-parameter Margules, van Laar, Wilson and UNIQUAC (Abrams and Prausnitz, 1975) models are shown in Table 6. We see in this table that the exp-6 model is as good as, or better than, all the other activity coefficient models for all but the type IV and V mixtures. Further, we see that by neglecting the density-dependent correction of the exp-6 mixing rule in the development of the activity coefficient model (i.e.,

117 TABLES 4 and 5 Binary systems used in the test of the three-parameter exp-6 activity coefficient model (N is the number of isotherms). Data taken from the DECHEMA Chemistry Data Series (Gmehling and Onken, 1977~ TAF3LE 4

TABLE 5 TYD~

7

I

Binary System acetone/n-hexane acetone/cyclohexsne n-pentane/acetone acetone/n-heptane acetone/n-decane acetone/tetrachloromethane acetone/ croton aldehyde diethyl ether/acetone hexafluorobenzene/cyclohexane diethyl ether/chloroform 2,2-dimethylbutane/n-hexane n-pentane/n-hexane ethyl iodide/n-heptane cyclopentane/tetrachloro~thane ~tr~hloro~th~e/cyclohex~e methylcyclohexane/toluene

j_

5 5 3 1 2 1 1 1 5 1 1 I 2 I 4 4

methanol/water ethanol/water 1-propanol/water Z-propanol/water 1-butanol/water acetaldehyde/water water/thiazole acetone/water methyl acetate/water Zbutanonefwater tetrahydrofuran/water 1,4_dioxane/water diethylamine/water water/pyridine water/&methylpyridine water/cycIohexanone water/butyl acetate methyl diethylaminelwater

n-pent~e/l-but~ol cyclohexane/l-buknol cyclohexane/ethanol ethanol/cyclohexane n-hexane/ethanol methanol/cyclohexane n-hexane/meth~oi cyclohexane/2-propanoi 2-propanol/methyicyclohexane Zpropanol/n-heptane 2-propanol/n-decane n-hexanell-propanol I-propanol/n-decane eth~ol/toluene toluene/I-butano1 toluene/l-pentanol

N 1 1 2 5 2 3 1 1 1 1 1 I I 1 2

Type IV

Type II Binary System

Type III Binary System

;$_ z 3 6 4 1 6 1 1 2 I 3 2 3 4 2 1 1 2 -

Type V Binary System benzene/chIorobenzene benzene/hexafluorobenzene hexafluorobenzene/toluene hexafluorobenzenefp-xylene ethylbenzene/nitrobenzene propylbenzene/nitroben~ne butylbenzene/nitrobenzene

N 2 5 3 2 1 1 1

118

TABLE 6 Test of the three-parameter expd activity coefficient model with isothermal, low-pressure, vapor-liquid equilibrium bubble-point calculations. Shown below are the average pressure deviations (API, in units of mm Hg, average vapor mole fraction deviations for component one (Ay,) for the three-parameter expd model (expd), and for the Margules, van Laar, Wilson, NRTL, and UNIQUAC (UNIQ) models. All parameters were fit to the sum of the squared, relative deviations in activity coefficients. Also shown are results for the expd activity coefficient model developed without density-dependent correction ($)

Type 1 Model +

exp-6i

AP (mm Rg) AYE

12.15 2.92 3.22 3.13 0.0043 0.0055 0.0055 0.0175

Model 4

exp-6t

AP (mm Hg) AYE

7.36 9.31 10.52 27.14 0.0074 0.0111 0.0137 0.0294

Model -

exp-6t

AP (mm Hg) AI1

6.08 7.54 7.16 22.25 0.0095 0.0146 0.0145 0.0345

exp-6t exp-6f

exp-6’(

Margules van Laar Wilson 3.29 0.0058

3.87 0.0058

NRTL

UNIQ

3.47 3.31 3.58 0.0043 0.0044 0.0054

Type II exg6$

exp-6s exp-61

Margules van Laar Wilson 12.16 0.0167

9.04 0.0109

NRTL

UNIQ

8.99 5.14 9.09 0.0089 0.0061 0.0102

Type III exp-6$ exp-6f

exp-61

Margules van Laar Wilson 8.26 0.0157

7.73 0.0147

NRTL

UNIQ

4.94 3.99 6.17 0.0062 0.0051 0.0117

Type IV Model d

exp-6t

AP (mm Hg) AYE

3.09 2.21 15.33 8.49 0.0253 0.0057 0.0026 0.0382

Model +

exp-6t

AP (mm Hg) AYE

6.78 1.65 1.88 0.83 0.0082 0.0082 0.0045 0.0222

exp-6:

exp-69 exp-6(

Margules van Laar Wilson 2.18 0.0034

2.22 0.0036

NRTL

UNIQ

2.29 2.22 3.55 0.0039 0.0036 0.0048

Type V

t - ~11, 6~2, and

UZZ/UI~

exp-6t exp-65 exp-61

Margules van Laar Wilson 0.54 0.0026

fit.

density-dependent correction neglected with ~11, 622, and 022/u~~ fit.

0-

(612 -

f22),

(~1

-

~2)

and

(CZI -

UNIQ

1.25 1.67 1.16 1.30 0.0048 0.0053 0.0066 0.0053

$ -

7 - (~12 -

NRTL

~111, and CII)

UZZ/U~I

fit.

fit.

neglecting a? and using a van der Waals one-fluid mixing rule) we obtain a simpler equation, but a correlation which is not as good. This is to be expected since this correction term is important at the liquid densities considered here. We also see in Table 6 the density-dependent correction gives slightly better results for all but the type IV mixtures.

119

In the above analysis of the exp-6 model we fitted the pure fluid energy parameters lii and l22 to the low-pressure vapor-liquid equilibrium data, and then used the geometric combining rule, ei2 = l2i = \IEll,E22, for the cross parameters. An alternative is to fix lii/k and e22/k using the critical temperature and the known value of l/kT, for the exp-6 pure fluid (Dodd and Sandler, 1989) and then to fit (cl2 - e22)/k and (eZ1 - lll)/k to the mixture data. Parameters in the Wilson, NRTL and UNIQUAC models are all such energy differences. Using these energy differences and the size ratio as the adjustable parameters in the expd model, the average deviations for the type IV and type V mixtures drop significantly as is seen in Table 6, while the deviations for other mixtures increase slightly. Finally, we also show in this table the results for the exp-6 model with the energy differences optimized and the size ratio set equal to the ratio of the UNIQUAC r parameters, (~~~/(~ii)~ = r2/r1. The results for this twoparameter model are worse than for any other model considered. Consequently, for the model developed here, treating ll2 - l22, l21- lii and a,,/a,, as adjustable parameters is best for general use.

CONCLUSIONS

We find that the application of the exp-6 equation of state with densitydependent mixing rules to experimental data leads to better predictions for highly asymmetric mixtures, especially in the near critical region, than those using a van der Waals one-fluid mixing rule. In particular the density-dependent mixing rule we have developed is better able to account for the non-quadratic composition dependence in asymmetric mixtures, particularly for the smaller components in the mixture, and also satisfies the low density quadratic composition boundary condition. As our densitydependent mixing rule reduces to the standard quadratic mixing rule for equal-sized mixtures, it is applicable to simple symmetric mixtures. Thus, a shortcoming of previous density-dependent mixing rules, their failure for relatively simple systems, is avoided with the new model proposed here. On the basis of the performance of the exp-6 mixture model for non-spherical, non-exp-6, fluids, we decided to extend the mixing rule to engineering equations of state known to be useful in describing highly non-spherical systems. From the study of our density-dependent mixing rule with empirical equations of state we find that when no binary interaction parameters are used, the density-dependent mixing rule performs better than the standard van der Waals one-fluid mixing rules. However, the clear superiority of the density-dependent mixing rule vanishes when one or more of the binary

120

interaction parameters are optimized to bubble-point pressures, except in the vicinity of the mixture critical point. The generalized van der Waals partition function was also used to develop a new activity coefficient model. The two and three parameter versions of this new, theoretically-based activity coefficient model were found to be as good as, and frequently much better than, two- and three-parameter models now in use by chemical engineers. Among the three parameter versions of this new model, the one in which the energy differences (ark - +) and (eZ1 - err) and the size ratio c~~~/cr~rare fit is, in general, the best for a large variety of mixtures with different chemical functionality.

REFERENCES Abrams, D.S. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs free energy of partly or completely miscible systems. AIChE J., 21: 116-128. Barker, J.A. and Henderson, D,, 1967. Perturbation Theory and Equation of State for Fluids. II. A Successfuf Theory of Liquids. J. Chem. Phys., 47: 4714-4721. Boubhk, T., 1970. Hard-sphere equation of state. J. Chem. Phys., 53: 471-472. Carnahan, N.F. and Starling, K.E., 1969. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys., 51: 635-636. Deiters, U.K., 1986. Calculation of fluid-fluid and solid-fluid equilibria in cryogenic mixtures at high pressures. In K.C. Chao and R. Robinson (Ed.), Equations of State: Theory and Applications, ACS Symp. Ser. 300: 331-350. Deiters, U.K., 1987. Density-dependent mixing rules for the calculation of fluid phase equilibria at high pressures. Fluid Phase Equilibria, 33: 267-293. Dodd, L.R. and Sandler, S.I., 1989. Monte Carlo Study of the Buckingham Exponential-Six Fluid. Mol. Simul., 2: 15-22. Dodd, L.R. and Sandler, S.I., 1991. Study of the Buckingham exponential-six fluid mixtures. Fluid Phase ~uilib~a, 63: 279-315. Ely, J., 1986. Improved mixing rules for one-fluid conformal solution calculations. In KC. Chao and R. Robinson (Ed.), Equations of State: Theory and Applications, ACS Symp. Ser. 300: 331-350. Eng, R. and Sandler, S.I., 1984. Vapor-liquid equilibria for three aldehyde/hydrocarbon mixtures. J. Chem. Eng. Data, 29: 156-161. Gmehling, J. and Onken, U. (Eds.), 1977ff. Vapor-Liquid Equilibria Data Collection. DECHEMA Chemistry Data Series, Vol. I, DECHEMA, Frankfurt. Gmehling, J., Rasmussen, P. and Fredenslund, A., 1982. Vapor-liquid equilibria by UNIFAC group contribution: revision and extension. 2. Ind. Eng. Chem. Process Des. Dev., 21: 118-127. Consalves, J.B. and Leland, T.W., 1978. A modification of regular theory to incorporate an improved approximation for pair distribution functions in mixtures. AIChE J., 24: 279-285. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., 1964 (corrected printing). Molecular Theory of Gases and Liquids. Wiley, New York.

121 Huber, M.L. and Ely, J.F., 1987. Improved conformal solution theory for mixtures with large size ratios. Fluid Phase Equilibria, 37: 105-121. Huber, M.L. and Ely, J.F., 1989. Properties of Lennard-Jones Mixtures at Various Temperatures and Energy Ratios with a Size Ratio of Two. National Institute of Standards and Technology Technical Note 1331 (Thermophysics Division), Boulder, CO. Khan, M.A., Li, M.H., Lee, L.L. and Starling, K.E., 1987. Equation of state composition dependence. Fluid Phase Equilibria, 37: 141-151. Knapp, H., Ddring, R., Oellrich, L., Plocker, U. and Prausnitz, J.M., 1982. Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances. DECHEMA Chemistry Data Series, Vol. VI, DECHEMA, Frankfurt. Lee, K.H., Dodd, L.R. and Sandler, S.I., 1989. The generalized van der Waals partition function. V. Mixture of square-well fluids of different sizes and energies. Fluid Phase Equilibria, 50: 53-77. I-eland, T.W., 1976. Recent Developments in the Theory of Fluid Mixtures. Adv. Cryog. Eng., 21: 466-484. Mansoori, G.A. and Leland, Jr., T.W., 1972. Statistical thermodynamics of mixtures-a new version for the theory of conformal solution. J. Chem. Sot., Faraday Trans. II, 68: 320-344. Mansoori, G.A., Carnahan, N.F., Starling, ICE. and Leland, T.W., 1971. Equilibrium thermodynamic properties of the mixture of hard spheres. J. Chem. Phys., 54: 1523-1525. Margules, M., 1895. S.-B. Akad. Wiss. Wien, Math.-Naturwiss. K. II., 104: 104. Peng, D.Y. and Robinson, D.B., 1976. A new two constant equation of state. Ind. Eng. Chem. Fundam., 15: 59-64. Rarey-Nies, J., 1987. Test of New Activity Models with Experimental Data. Personal Communication. Redlich, 0. and Kwong, J.N.S., 1949. On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solution. Chem. Rev., 44: 233-244. Renon, H. and Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135-144. Sander, B., Skjold-Jorgensen, S. and Rasmussen, P., 1983. Gas solubility calculations. I. UNIFAC. Fluid Phase Equilibria, 11: 105-126. Sandler, S.I., Fischer, J. and Reschke, F., 1989. Free energies of mixing. Fluid Phase Equilibria, 45: 251-264. Scatchard, G., 1937. Change of volume of mixing and the equations for non-electrolyte mixtures. Trans. Faraday Sot., 33: 160-166. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 1197-1203. van Laar, J.J., 1910. Uber dampfspannungen von Binken gemischen. Z. Phys. Chem., 72: 723-751. Wilson, G.M., 1964. Vapor-liquid equilibrium. XI. A new expression for the excess Gibbs free energy of mixing. J. Am. Chem. Sot., 86: 127-130. Wu, H.S., Locke III, W.E. and Sandier, S.I., 1990. Vapor-liquid equilibrium of binary mixtures containing pyrrolidine. J. Chem. Eng. Data, 35: 169-172.

122

Appendix A

FUGACITY EXPRESSION FOR THE BOUBLIK-MANSOORI

EQUATION

The fugacity coefficient for the Boublik-Mansoori (Boublik, 1970; Mansoori et al., 1971) is given as

equation

+(ZBM - 1) - In (ZBM)

_ 1

of state

(Al)

where

!?&) i

E

(cq”+2)

q

(1-4

Here, A’ is the reduced residual Helmholtz free energy given by B”-AIG)

=(4M+2)

’

NkT

+cgM

P-4 +

(ctFM- 1) ln Cl- 77)

q

(I-

VI2

w

where AIG is the Helmholtz free energy of a mixture of ideal gases at the same density, temperature and composition, where TN ‘7ETV -F ED BMs3--2 F E3 ED a~M~3--33+1=3a~M-cy,BM-1 F2

Ql

(A41

123

with D = &dii E = zxjd; F = &di W) i i i where d,, is the hard-sphere diameter of pure 1. Also, p,“” = 1 - c$~, p,“” = (u:~ + 2 and p,“” = p,“” - PpM + 1. When the Boublik-Mansoori equation is used with an attractive term, as it is in this work, the fugacity coefficient for the mixture is found by adding the fugacity for the attractive term to eqn. (Al) and replacing In ZBM by In 2, the compressibility of the mixture.

Appendix B

BUBBLE-POINT CALCULATION RESULTS

See Tables Bl-B15.

Appendix C

ACTIVITY COEFFICIENT MODEL

The activity coefficient of component

k in the exp-6 model is given by

In yk = In yrid + In $Omb The residual contribution

(Cl)

is

1+

ln YFid=

ln

where rkdd is the contribution of the density-dependent activity coefficient model and is defined as In rzd = 43 C

j

•t 63[2

-

[

’

Cxtxjcijl__

i

-

‘kk

CXiCik

-

i

IJ

3(6kk

-

~F~A)I

C i

-

P)

mixing rule to the

C~jckj j

CXixjcij j

rid

I

(9

124

also aij = -aijSij Cij = 8,

6ij

=

(XiPi +

1-

XjPj)(

Sij)/Sij

h,/h

=

Ui:/

C

w

XlUi

I

s, = ( dg3/ I

c x1(d;s)3= ( d,H,S)3/F 1

Note that for molecules of equal size Sij = 1, cij = 0, and In yfd goes to zero. The combinatorial contribution to the activity coefficient comes from Boublik-Mansoori and Carnahan-Starling free volume terms and is written

lnypmb=

-[1-~]‘[1+2~][ln(l-1))+ (191)2] -3[I-$][l-$]&+ln

6,+(I-6,)+RcOmb

(C5)

where Rcomb arises from energy differences and is given as comb_ 17(4- 377) 17k(4- 3%) + ln 6 + W - %)

R

-

(1-?J)*

-

(1 -77k)2

*

I--)7

P)

W7)

6, =

( dk”k”)3/cx/( d;‘)” = ( d,H,S)3/F

I temperature-dependent, In these equations dHS is the Barker-Henderson hard-sphere diameter and (Y,/3, A and A, are defined in the text.

125 TABLE Bl Bubble-point calculation results for expd equation of state (eqn. (6)) with van der Waals l-fluid (eqn. (6)-vdW) and density-dependent mixing rule (eqn. (6)-dd), and the PengRobinson equation of state (PR-vdW). Shown are average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

22.3

6.5

14

P range

3.7 3.3

4.2 5.3

0.10 0.12

14 14

T (Kelvin) 310.90 (bar) 10.970 to 285.160

3.7 3.0

4.2 5.3

0.09 0.18

0.00 -0.03

14 14

126 TABLE B2 Bubble-point calculation results for exp-6 equation of state (eqn. (6)) with van der Waals l-fluid (eqn. (6)-vdW) and density-dependent mixing rule (eqn. (6)-dd), and the PengRobinson equation of state (PR-vdW). Shown are average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

5.9 5.5

12.1 12.7

0.04 0.05

11 11

T (Kelvin) 310.90 P range (bar) 6.980 to 73.960

5.9 3.8

12.1 13.8

0.05 0.42

0.00 -0.19

11 11

127

TABLE B3 Bubble-point calculation results for expd equation of state (eqn. (6)) with van der Waals l-fluid (eqn. (6)-vdW) and density-dependent mixing rule (eqn. (6&dd), and the PengRobinson equation of state (PR-vdW). Shown are average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

3.6 3.7

36.3

16.6 16.5

30.4

8 8

11

2.4 2.4

2.5

15.2 0.04 15.1 0.04

8 8

2.4 2.3

7.239

to 55.916

21.649

to 65.913

4.0

0.12

11

15.2 17.3

0.00 -0.25

0.01 0.10

8 8

4.0 9.7

0.01 0.75

0.04 -0.65

11 11

128

TABLE B4 Bubble-point calculation results for expd equation of state (eqn. (6)) with van der Waals l-fluid (eqn. (6)-vdW) and density-dependent mixing rule (eqn. (6)-dd), and the PengRobinson equation of state (PR-vdW). Shown are average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

I

Equation

eqn. (6)-vdW eqn. (6)-dd PR-vdW

hydrogen sulfide / nitrogen T (Kelvin) 321.87 P range (bar) 40.678 to 206.839 Number of data points = 11 1 P% 24.9 24.9 22.6

y,%

N

1 P%

~1%

LIZ

N

11.5 11.5 10.8

11 11 11

2.1 2.0 2.8

2.7 2.6 3.4

0.12 0.12 0.18

11 11 11

1 P%

~1%

k12

!I,

N

2.7 0.9 7.2

0.00 0.65 -0.11

0.04 -0.43 -0.18

11 11 11

P%

~1%

hz

(11

N

5.0 2.2 15.6

5.2 6.1 22.2

0.01 0.45 0.07

0.01 -0.18 -0.09

~1%

hz

f,z

2.1 1.3 0.8

hydrogen / n-hexane T (Kelvin) 477.59 P range (bar) 34.473 to 344.733 Number of data points = 10 Equation

P%

y,%

eqn. (6)-vdW eqn. (6)-dd PR-vdW

6.4 7.2 17.0

4.6 7.6 33.5

N 9 9 1

PI

~1%

k1z

5.0 6.9 13.1

5.2 7.8 26.4

0.04 0.02 0.34

N 9 9 1

9 9 2

hydrogen / n-hexane T (Kelvin) 444.26 P range (bar) 34.473 to 482.625 Number of data noints = 14 Equation

P%

eqn. (6)-vdW eqn. (6)-dd PR-vdW

17.5 21.0 -

y,%

N

3.5 14 5.1 14 --

P% 12.4 14.7 11.2

~1%

ku

N

14 3.6 0.12 5.4 0.15 14 3.4 0.52 12

P% 12.4 8.2 -

3.6 4.2 -

N

0.01 0.04 14 0.49 -0.16 14 -

hydrogen / n-heptane T (Kelvin) 498.85 P range (bar) 24.516 to 343.225 Number of data points = 8 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

IP% 5.4 7.0 14.5

~1%

N

6.6 9.5 32.3

8 8 2

I PI

3.9 6.2 11.5

~1%

hz

6.5 9.6 29.1

0.07 0.04 0.18

N

8 8 2

J’%

3.9 2.6 2.7

~1%

hz

(11

6.5 7.3 25.0

0.00 0.35 -0.91

0.02 -0.11 -0.36

N

8 8 2

129

TABLE B5 Bubble-point calculation results for expd equation of state (eqn. (6)) with van der Waals l-fluid (eqn. (6)-vdW) and density-dependent mixing rule (eqn. (6)-dd), and the PengRobinson equation of state (PR-vdW). Shown are average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

17.9

7.3 12 2.2 27.7 -4.11 -1.40

2

130

TABLE B6 Bubble-point calculation results for expd equation of l-fluid (eqn. (6)-vdW) and density-dependent mixing Robinson equation of state (PR-vdW). Shown are the in liquid and vapor volumes (N = number of converged

P range (bar) 13.930 to 184.850

T (Kelvin) 380.13 P range (bar) 24.340 to 122.860 Number of data points = 9

T (Kelvin) 310.90

state (eqn. (6)) with van der Waals rule (eqn. (6)-dd), and the Pengaverage absolute percent deviations data points)

131

TABLE B7 Bubble-point calculation results for exp-6 equation of l-fluid (eqn. (6)-vdW) and density-dependent mixing Robinson equation of state (PR-vdW). Shown are the in liquid and vapor volumes (N = number of converged

state (eqn. (6)) with van der Waals rule (eqn. (6)-dd), and the Pengaverage absolute percent deviations data points)

nitrogen / n-butane T (Kelvin) 380.19 P range (bar) 24.270 to 135.760 Number of data points = 11 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

VI%

u.%

N

u,%

v,%

kll

N

V,%

I+%

kll

3.8 3.8 3.0

17.7 18.8 10.4

11 11 11

4.3 4.4 3.7

12.0 11.9 6.7

0.04 0.05 0.07

11 11 11

4.3 2.3 2.1

12.0 7.6 2.1

0.05 0.42 -0.07

v,%

vu%

5.9 3.2 9.9

h

N

0.00 -0.19 -0.13

11 11 11

Cl2

ela

N

20.8 16.5 1.9

0.00 0.68 -0.29

0.00 -0.57 -0.33

10 10 9

nitrogen / n-butane T (Kelvin) 410.85 P range (bar) 34.950 to 69.460 Number of data points = 10 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

v,%

I+%

N

~1%

vv%

klz

N

5.9 5.9 6.6

20.8 20.9 11.4

10 10 10

5.9 5.9 6.3

20.8 20.8 10.8

0.00 0.00 0.22

10 10 10

carbon dioxide / n-butane T (Kelvin) 310.90 P range (bar) 6.980 to 73.960 Number of data points = 12 Equation

v,%

vu%

N

eqn. (6)-vdW eqn. (6)-dd PR-vdW

7.9 10.2 5.8

71.1 79.2 34.5

12 12 12

~1% v.% 4.3 4.2 7.7

23.0 25.0 3.4

klz

N

v,%

v,%

k,z

h2

N

0.12 0.13 0.12

12 12 12

4.3 4.2 8.4

23.0 22.3 4.2

0.01 -0.01 0.10

0.04 0.06 -0.04

12 12 11

carbon dioxide / n-butane T (Kelvin) 344.30 P range (bar) 10.500 to 80.580 Number of data points = 13 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

v,%

vu%

N

V,%

V”%

klz

N

~1%

vu’%

klz

(12

N

6.1 7.9 2.8

58.2 64.8 31.3

13 13 13

2.2 2.9 4.3

13.7 14.3 3.2

0.11 0.12 0.13

13 13 13

2.2 2.9 6.8

13.7 14.2 1.6

0.01 0.06 0.07

0.04 0.03 -0.07

13 13 11

132

TABLE B8 Bubble-point calculation results for exp-6 equation of l-fluid (eqn. (6)-vdW) and density-dependent mixing Robinson equation of state (PR-vdW). Shown are the in liquid and vapor volumes (N = number of converged

4.1

62.4

8

5.5

58.3

7

state (eqn. (6)) with van der Waals rule (eqn. (6)-dd), and the Pengaverage absolute percent deviations data points)

0.2

5.8

-0.02

1.1 4.6

10.0 2.8

0.13 0.13

7 7

1.9 3.8

11.7 -0.05 2.6 0.14

0.07 0.01

7 7

1.5 6.8

2.6 3.4

0.12 0.18

11 11

0.8 4.7

3.3 0.75 10.8 -0.19

-0.65 -0.22

11 11

133

TABLE B9 Bubble-point calculation results for expd equation of l-fluid (eqn. (6)-vdW) and density-dependent mixing Robinson equation of state (PR-vdW). Shown are the in liquid and vapor volumes (N = number of converged

state (eqn. 6)) with van der Waals rule (eqn. (6)-dd), and the Pengaverage absolute percent deviations data points)

hydrogen sulfide / nitrogen T (Kelvin) 321.87 P range (bar) 40.678 to 206.839 Number of data points = 11 Equation

'

eqn. (6)-vdW eqn. (B)-dd PR-vdW

u,% uv% N 0.5 34.0 0.5 33.8 3.2 22.9

11 11 11

VI% I$% hz 1.4 1.4 5.4

3.2 3.1 2.1

0.12 0.12 0.18

N 11 11 11

4%~ ~"96 42 1.4 0.6 3.2

112

3.2 0.00 0.04 1.9 0.65 -0.43 7.3 -0.11 -0.18

N

11 11 11

hydrogen / n-hexane T (Kelvin) 477.59 P range (bar) 34.473 to 344.733 Number of data points = 10 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

IV,% 3.0 6.1 14.0

uv% N 1 ~1% vv% 8.4 8.8 0.0

6 6 1

3.9 6.4 13.3

8.3 8.9 0.0

hz 0.04 0.02 0.34

N 1 ~1% vu% 6 6 1

3.9 3.6 14.3

8.3 6.4 0.0

hz

Cn

0.01 0.01 0.45 -0.18 0.07 -0.09

N 6 6 2

hydrogen / n-hexane T (Kelvin) 444.26 P range (bar) 34.473 to 482.625 Number of data points = 10 Equation eqn. (6)-vdW eqn. (6)-dd PR-vdW

u,%

uv%

N

1.0 18.7 14 3.3 21.5 14 -_

VI% U”% 3.7 6.2 3.7

klz

11.6 0.12 12.6 0.15 9.4 0.52

N 14 14 12

~196 uv% 3.7 4.6 -

11.6 8.1 -

k1z

f12

0.01 0.04 0.49 -0.16 -

h’

14 14 -

134 TABLE BlO Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

11.0 23.3

6.9 7.6

51 51

7.4 5.2

6.8 5.2

0.07 -0.13

51 51

3.6 4.5

T range (Kelvin) 310.90 to 410.90

4.1 5.1

0.00 -0.18

-0.06 -0.04

49 51

135

TABLE Bll Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

34.0 26.0 54 31.6

23.6

54

5.2 5.2

T range

4.0 2.2

4.8 5.2

28 30

4.1 3.9

0.17 0.15

54 54

4.1 4.2

8.5 7.6

0.01 0.02

-0.09 -0.10

54 54

5.1 5.0

-0.02 0.00

0.03 0.02

31 31

(Kelvin) 277.65 to 377.59 2.275 to 96.318

1.5 1.6

5.7 5.5

-0.03 -0.01

30 30

1.4 1.6

136 TABLE B12 Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

benzene / methane T range (Kelvin) 338.71 to 501.15 P range (bar) 6.895 to 330.943 Number of data points = 35 Equation

P%

~1%

N

P%

~1%

Cl2

N

P%

ye%

kn

e12

N

vdW Lee MDAAT

12.6 33.7 7.7

19.4 25.4 17.7

35 34 35

6.0 4.4 6.0

16.7 17.2 16.9

0.94 -0.09 0.02

35 35 35

4.6 2.5 4.6

16.3 17.1 16.8

0.02 -0.06 0.00

-0.01 0.03 -0.02

35 35 35

N

chlorotrifluoromethane / dichlorodifluoromethane T range (Kelvin) 255.00 to 290.00 P range (bar) 2.145 to 27.831 Number of data points = 17 Equation vdW Lee MDAAT

P%

~1%

N

6.2 5.4 6.1

3.8 3.0 3.7

17 17 17

P%

~1%

k12

N

1.3 1.2 1.3

2.0 2.0 2.0

0.03 0.03 0.03

17 17 17

P%

~1%

kn,

t-12

1.0 1.0 1.0

3.1 3.1 3.1

0.00 -0.01 -0.01

-0.04 -0.05 -0.05

17 17 17

~1%

klz

e12

N

15.0 17.7 16.0

0.00 -0.05 -0.01

-0.02 -0.02 -0.03

9 9 9

benzene / ethylene T range (Kelvin) 348.15 to 346.15 P range (bar) 15.199 to 91.191 Number of data points = 9 Equation

PI

~1%

vdW Lee MDAAT

8.9 12.7 6.6

14.3 27.8 13.6

N 9 9 9

P%

~1%

112

3.9 2.4 3.9

15.2 17.8 16.0

0.03 -0.03 0.02

N 9 9 9

P% 1.6 1.8 1.6

carbon dioxide / water T range (Kelvin) 541.15 to 541.15 P range (bar) 199.996 to 1499.967 Number of data points = 14 Equation

P%

~1%

N

P%

~1%

k12

N

P%

~1%

kn

C12

N

vdW Lee MDAAT

29.3 36.3 24.4

12.3 19.2 15.8

2 1 12

8.7 9.8 7.5

12.6 9.0 15.2

0.65 0.09 0.02

13 13 13

8.6 9.4 6.7

8.5 7.2 13.6

0.09 0.11 0.05

0.02 0.01 0.02

13 13 14

137 TABLE B13 Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in pressure and vapor mole fraction of component one (N = number of converged data points)

15.5

3.5

37

7.1

3.3

0.04

37

5.9 9.8

1.013 to 1.013

T range (Kelvin) 348.15 to 398.15

3.1 0.01 -0.02 3.8 -0.15 -0.07

37 35

138 TABLE B14 Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in liquid and vapor volumes (N = number of converged data points)

nitrogen / n-butane T range (Kelvin) 310.90 to 410.85 P range (bar) 10.970 to 285.160 Number of data points = 51 Equation

VI%

v,%

N

VI%

V”%

klz

N

vr%

vv%

klz

ha

N

1 Equation

1~1%

vu%

N

1 vl%

vv%

klz

N 1 vl%

v,%

kls

4,

N I

vdW Lee MDAAT

1.9 3.5 2.0

10.9 14.0 7.0

23 23 23

2.6 2.3 2.5

3.6 4.0 3.6

0.09 -0.10 0.06

23 23 23

1.9 2.4 1.8

3.8 3.7 3.9

0.06 -0.10 0.03

-0.03 -1 23 0.01 23 -0.04 23

carbon dioxide / n-butane T range (Kelvin) 310.90 to 410.90 P range (bar) 6.980 to 80.580 Number of data points = 33 Equation

v,%

v,%

N

v,%

v,%

k12

N

v,%

v,%

k,z

e12

N

carbon dioxide / iso-butane T range (Kelvin) 310.93 to 394.26 P range (bar) 7.239 to 71.842 Number of data points = 29 Equation

VI%

vv%

N

VI%

V”%

klz

N

w%

vv%

kll

e12

N

vdW Lee MDAAT

5.6 6.2 5.5

24.2 10.7 19.6

29 28 29

8.7 9.8 8.1

5.0 5.0 7.0

0.10 0.07 0.07

29 28 29

14.3 7.2 12.7

5.3 4.6 5.6

0.07 0.10 0.07

-0.08 0.05 -0.06

26 28 25

139 TABLE B15 Bubble-point calculation results for the Peng-Robinson equation of state with van der Waals l-fluid (vdW), Lee’s mixing rule (Lee) and the mean density approximation for the attractive term (MDAAT) (eqn. (13)). Shown are the average absolute percent deviations in liquid and vapor volumes (N = number of converged data points)

vdW Lee MDAAT

6.6

5.2

0.15

54

5.5

4.7

0.02

-0.10

54

3.3

12.8

41

3.8

5.3 3.2

0.12 0.04

41 41

4.5 3.8

4.7 3.4

0.11 0.05

-0.02 0.01

41 41

2.1 2.4 2.1

42.9 12.9 33.7

48 48 48

3.7 3.0 4.0

5.8 3.5 5.6

0.13 0.04 0.10

48 48 48

4.9 3.0 5.5

4.4 3.5 4.2

0.10 0.04 0.07

-0.03 0.00 -0.04

48 48 48

ethane / iswbutane T range (Kelvin) 311.26 to 394.04 P range (bar) 10.687 to 53.709 Number of data points = 32 Equation

VI%

u,%

N

u,%

U”%

klz

N

VI%

vu%

klz

e12

N

vdW Lee MDAAT

7.5 8.4 8.2

3.0 6.6 3.7

30 28 30

7.0 7.1 7.2

2.8 2.4 2.7

-0.01 -0.03 -0.01

30 30 30

8.3 6.2 6.4

2.3 2.7 3.0

-0.03 -0.02 0.00

-0.03 0.03 0.02

30 31 31