Znd. Eng. Chem. Res. 1993,32, 1498-1503
1498
Prediction of Vapor-Liquid Equilibria at High Pressures Using Activity Coefficient Parameters Obtained from Low-Pressure Data: A Comparison of Two Equation of State Mixing Rules H a i Huang and S t a n l e y I. Sandler' Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Two recent equation of state mixing rules, one by Dahl and Michelsen (MHV2) and another by Wong and Sandler (W-S), are claimed to be useful for making vapor-liquid equilibrium predictions for nonideal mixtures at high temperatures and pressures using data obtained at low pressures. Here we compare the performance of these two mixing rules in predicting the high-pressure phase behavior of nine binary systems using published activity coefficient parameters. We also compare the predictions for two ternary systems using mixing rule parameters obtained from low-pressure binary mixture data. We find that both mixing rules can be used to make reasonable high-pressure vapor-liquid equilibrium predictions from low-pressure data. However, the errors in the predicted pressure with the W-S mixing rule when used with either the Peng-Robinson or Soave-RedlichKwong equation of state are, on the average, about half or less those obtained when using the MHVB mixing rule. Also, the predictions of the MHVB mixing rule deteriorate significantly as the temperature and pressure range increase, which is not the case with the W-S mixing rule. c1 = 1 - 2112 and c2 = 1 + 2112. For pure substances,
Introduction Since van der Waals presented his equation of state in 1873,more than 100equations and mixing rules have been proposed to describe the phase behavior and thermodynamic properties of pure substances and their mixtures. Here we consider two mixing rules from among the many that have been proposed. The first is by Michelsen (19901, Dahl and Michelsen (19901, and Dahl et al. (1991) (which they refer to as the modified Huron-Vidal second order or MHVB mixing rule), and the second is by Wong and Sandler (1992) and Wong et al. (1992) (the W-S mixing rule). These two mixing rules are the only ones that can use available activity coefficient model parameters obtained from low-pressure data without change to predict phase equilibria at high pressures for complex, nonhydrocarbon mixtures. Other equation of state (EOS)mixing rules, which do not have this feature, have recently been compared by Knudsen et al. (1993). In the past 100 years, several thousand sets of binary vapor-liquid equilibrium data have been measured at or near atmospheric pressure. These data have been correlated using activity coefficientmodels such as the Wilson, NRTL, and UNIQUAC models. I t would be useful, especially for industrial applications, to also be able to predict vapor-liquid equilibrium (VLE) at high pressure by means of an equation of state using these available activity coefficient model parameters. Here we compare the accuracy of the only two mixing rules which claim to allow such predictions.
Pci
and
R T,. bii = n -
In this study, we use the cubic equations of state of Soave, Redlich, and Kwong, SRK (Soave, 1972) and of Peng and Robinson, PR (Peng and Robinson, 1976). Both equations can be written as:
p = - RT a (1) IJ - b (u + c,b)(u + c,b) For the SRK EOS, c1 = 0 and cp = 1, while for PR EOS, t o whom correspondence should be addressed.
(3)
Pci
For the SRK equation, fla = 0.4286 and n b = 0.086 64, while for the PR EOS Q, = 0.457 235 and nb = 0.077 796. For the function f(T,) in the SRK EOS, we follow Dahl and Michelsen (1990) and use f(TJ = 1+ Cly + C g 2 + Cfi3
for T, I 1 (4a)
and
f(T,)= 1+ Cly for T, > 1 (4b) We used the critical constants and vapor pressure data reported in the AICHE DIPPR compilation (Daubert and Danner, 1985) to obtain optimum values of constants 121, CZ,and C3. Table I gives the parameters we obtained and the accuracy of the correlation. For the PR EOS, we obtained good accuracy using the function f(T,) recommended by Stryjek and Vera (1986): f(T,) with
1+ [
+ ~ ~ (T,0.5)(0.7 1 + - T,)1(1- T,O.')
K ~
+
(5)
= 0.378893 + 1.4897153~- 0.17131848~~ 0.0196554~~ (6) where ~1is a constant specific to each fluid. We have used the values of Stryjek and Vera (1986) for this constant. In all of the above, w is the acentric factor, T, is the critical temperature, Pc is the critical pressure, T r = TIT,, and R is the gas constant. While the MHV2 mixing rule was originally used with the SRK EOS and the W-S rule with the PR EOS,for comparison, we have used both mixing rules with both equations here. We first briefly discussed these mixing rules. K~
Equation of State
* Author
R2T2f(TJ2
aii = na
OSS8-5S85/93/2632-~~98$o4.oo/O0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1499 Table I. Parameters of the Temperature Function in the SRK EOS for Pure Substances substance
T(K)
CHI 92-161 Cab 96273 CsH8 112-331 i - C ~ H l o 130-368 n-C&oli 140-378 i-CaHl2 146-424 n-CaHlz 152-422 C& rlI 180-458 C.IH16 190-488 Gale 220-518 Cam 220-538 HzO 280-578 Nz 67-114 coz 220-279 HB 190-338 co 70-119 02 60-139 CzH4 108-268 CHaOH 190-448 CzHsOH 196459 1-CaOH 210-468 2-CsOH 210-468 C2H&O 180-458 CsHaCO 190-488 Cyclo-Ce 280-518 280-518 C7H8 190-518 C1oH22 225-523
Ap/p (%) 0.036 0.120 0.358 0.407 0.245 0.233 0.357 0.178 0.157 0.340 0.315 0.024 0.292 0.016 0.012 0.274 0.023 0.085 0.384 0.902 0.836 0.810 0.067 0.228 0.030 0.122 0.216 0.599
C1 0.518 84 0.643 87 0.722 11 0.766 40 0.746 56 0.855 24 0.862 87 1.013 4 1.078 6 1.059 5 1.113 4 1.063 4 0.534 45 0.813 80 0.530 67 0.783 23 0.521 81 0.618 84 1.439 9 1.397 4 1.243 1 1.339 4 0.973 19 1.030 8 0.864 49 0.823 76 0.90496 1.144 5
c2
cs
-0,247 28 0.153 83 -0.096 42 -0.233 20 0.160 05 -0.281 33 -0.132 34 -0.843 98 -0.751 03 0.104 67 0.278 83 -0.511 46 0.024 44 0.110 90 0.576 33 -1.906 2 -0.127 75 -0.108 11 -0.971 33 0.224 46 1.264 6 0.936 38 -0.254 44 -0.570 3 -0.554 66 -0,309 37 -0.267 78 0.512 06
0.328 57 0.221 54 0.125 42 0.624 96 -0.121 16 0.491 35 0.435 55 1.829 7 1.633 1 -0.316 27 -0.749 60 0.420 13 -0.264 01 -0.505 59 -0.812 10 3.883 4 0.154 44 0.125 70 0.832 34 -1.308 7 -2.379 4 -1.949 6 0.235 65 4.859 8 0.780 92 0.837 89 0.536 94 -1.176 6
MHV2 Mixing Rule. From eq 1, we obtain the following expression for the fugacity, f: ln(h)
+ In b =jpU
(c2 - c1)bRT ln(
-)
=
+ Q (7)
Introducing a = u/(bRT)and u = u/b,the mixture fugacity at zero pressure is h ( k ) + l n b = - l - l n ( u - l ) - L l n ( u + c , ) =u c2
model can be used. In the last form of this equation we have used u=CZiUi
(12)
1
which is a reasonable assumption for liquids. Michelsen (1990) suggests using the P = 0 condition to establish the relationship between u and a through the EOS, i.e.,
and then selecting the smallest, liquid root 1
u = +(a
- c1 - c2) - [a2- 2(c, + c2 + 2)a + (c1- c2)21'/21 (14)
which is valid for a > (cl
+ c2 + 2) + 2(c1 + c2 + 1+ c
~ c ~ ) (15) ~ / ~
Substituting eq 14 into eq 8, we obtain
"(A)+
In b = Q[u(a),al= q(a)
(16)
so that eq 11 becomes
Equation 16 is a transcendental function in a. In order to obtain explicit expression for a from GE and aii, Dahl and Michelsen (1990) suggested the ad hoc procedure of using the second-order polynomial q(a)
-
40
+ 41a + 42a2
(18)
as an approximate expressionfor q(a). Then eq 17becomes
+ c2
- c1 Q(u,a) (8)
which is applicable to both the mixture and the pure components, Le., for component i ln(f.0) RT
+ In bii = Q(uii,aii)
Equations 8 and 9 for each component are combined to yield the following expression for the equation of state excess Gibbs free energy of mixing a t constant temperature and zero pressure, GE:
RT
RT
RT
or
The excess Gibbs free energy in eq 11is based on the pure component standard states, and any activity coefficient
where b = Cribii. Equation 19 is the Dahl-Michelsen or MHV2 mixing rule. It is used as follows. First, the excess Gibbs free energy, GE,is gotten from an empirical activity coefficient model. Then since the aii, bii, and b are known, the quadratic equation above can be solved for a h . A t low equilibrium pressures (C0.5 atm) and low reduced temperatures (T,C 0.61, u varies from 1.1to 1.3, and while a in the SRK EOS can vary from 8 to 20, it is usually between 10 and 14. The relation between a and u approximately agrees with eq 14 in this range. Dahl and Michelsen's recommended values of 41 and 42 are -0.478 and -0.0047, respectively. By a comparison with calculations for similar systems, we find that the corresponding a interval for the PR EOS is a = 11.7-16.7, and in this interval, we obtain the following values q1= -0.4347 and ~2 -0.003 654. While the MHV2 mixing rule leads to predictions of the excess Gibbs free energy equivalent to that of an activity coefficient model at high densities, it violates the low density boundary condition that the second virial coefficient predicted from an equation of state should be quadratic in composition. W-5 Mixing Rule. Wong and Sandler (1992) developed a mixing rule which, while density independent, when
1500 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 Table 11. Sources of High-pressure Experimental Data system n.d.a 56 A 55 B 91 C 39 D 28 E 40 F 11 G H 33 I 33 J 33 108 K a
T (K) 373-523 423-623 298-523 373-473 373-423 393-493 373-413 373-423 373-423 372.7 373-523
P (bar) 1-67 5.6-185 0.07-66.7 3 5-40 0.4-20 3.9-27.3 2.5-11 3.5-25.3 3.6-18 5-8.5 1.5-85 I
source Griswold and Wong (1952) Barr and Barrnett (1959) Griswold and Wong (1952) Griswold and Wong (1952) Campbell et al. (1987) Butcher and Medani (1968) Nieeen et al. (1986) Wilsak et al. (1987) Campbell et al. (1986) Whak et al. (1987) Griswold and Wong (1952)
n.d. is the number of data points.
combined with an equation of state, produces a second virial coefficient which correctly is quadratic in composition and at high density gives an excess free energy that is equivalent to that obtained from an activity coefficient model. The basis for their development was the observation that since there were two equation of state parameters, a and b, to be determined, two boundary conditions could be satisfied. The first is that the second virial coefficient from the equation of state/mixing rule combination should be quadratic in composition, and the second is that the excess Helmholtz free energy at infinite pressure from the equation of state be equal to that from the activity coefficient model. The resulting mixing rule is
yrzizj(bi-RTf ai
bj-&)(?)
were obtained by adjusting the value of kifi In some cases, one of the pure components may not exist as a liquid at the experimental temperature and bubble point pressure of the mixture. In this case, we either do the calculation at a slightly higher pressure to create a liquid phase for the pure component, or take the liquid root of the equation of state to calculate the pure component fugacity coefficient. In our experience the kij obtained from either of these two methods are virtually the same because eq 22 is not sensitive to small changes in pressure. If some activity coefficientmodel parameter information is available, we can also predict the vapor-liquid equilibria for a mixture involving a noncondensable gas with an equation of state and the W-S mixing rule. In this case, to determine kij we use
GE RT = xl[ln y1 - In yl(xl
bm cRT
-
011 + x 2 In y2
biRT
and
where c = -ln(2) for the SRK EOS and - 1)1/2lI2 for the PR EOS. Wong and Sandler (1992) have shown that the excess Helmholtz free energy of mixing is essentially independent of pressure and approximately equal to the excess Gibbs free energy of mixing at low pressure. Therefore, model parameters or low-pressure experimental data for GE are used in place of AE in the equations above. It might appear that the Wong-Sandler mixing rule has one more parameter, kij, than the activity coefficientmodel which it contains. However, this is not the case since ki, is not an independent parameter. Rather, its value is adjusted to ensure that GE computed from the equation of state agrees with that found experimentally at low pressure. This is done by equating
GE = x1 In y1 + x 2 In y2 R~
In this equation x i is the liquid mixture composition, yi is the activity coefficient of component i in the mixture evaluated from the chosen model, and @i,pweand 4i,mk are the fugacity coefficients of component i in the pure liquid and in the mixture evaluated using the equation of state. We ensure that this equation is satisfied only at x1 = xp = 0.5, the temperature T, and the bubble point pressure of the mixture at which the activity coefficient parameters
Thus, in this case also, we can directly use activity coefficient model parameters in the W-S mixing rule. It is important to note that kij is not an independent or freely adjustable parameter. Its value is set so that the equation of statelactivity coefficient model reproduces the low-pressureactivity coefficientbehavior. Therefore, both the MHV2 and W-S mixing rules have the same number of freely adjustable parameters, which equal the number of parameters in the excess free energy model used.
Comparison of the Two Mixing Rules for Binary Systems To compare the MHV2 and W-S mixing rules for predicting vapor-liquid equilibrium over a large range of conditions using activity coefficient model parameters obtained from experimental VLE data at low pressure and temperature, we have considered nine mixtures for which VLE data are available over a large pressure range. These systems are listed in Table 11. In each case we have directly used the activity coefficient model parameters obtained from low-pressuredata for these systems reported in the various parts of Vol. I of the DECHEMA Chemistry Data Series to predict vapor-liquid equilibrium at higher pressures and temperatures using the MHV2 and W-S mixing rules with both the PR and SRK equations of state. In all cases we have used the UNIQUAC activity coefficient model and the parameters reported for a single lowtemperature/low-pressuredata set to make predictions at all higher temperatures and pressures with fixed values of the activity coefficient parameters. Since there are a
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1501 Table 111. Prediction of Vapor-Liquid Equilibria at High Pressure with PR EOS Using Different Parameters Sets Obtained from Low-Pressure Data [System: CHsOH-Ha0 (A)] w-s MHVB parameter Ap/p(%) AyXl00 AP/P(%) AyXl00 I/la, 49 1.99 1.32 1.99 1.11 I/la, 50 2.35 1.61 2.74 1.47 I/la, 51 3.21 1.89 3.92 1.92 I/la, 52 1.96 1.46 2.47 1.39 1.92 I/la, 53 1.46 2.39 1.34 I/la, 54 1.87 1.43 2.02 1.22 I/la, 55 1.25 1.66 1.83 1.14 1.49 2.03 1.40 I/la, 57 2.50 1.43 1.83 2.13 1.26 M a , 58 1.40 2.04 2.29 1.25 I/la, 59 1.44 1.24 I/la, 61 1.86 2.08 average 2.07 1.47 2.39 1.34
number of low-pressure experimental data sets resulting in different activity coefficient parameter sets for each of the binary mixtures we considered, we then repeated the calculation at all temperatures and pressures using separately each of the parameters sets, even though the difference in numerical values among the various parameter sets was quite large. Table I11gives the prediction errors over all data for the CH~OH-HZOmixture using the PR EOS. In this table the column labeled parameter indicates the volume/part and page in the DECHEMA series from which we obtained the UNIQUAC parameters. An important observation to be made from this table is that even though there is a large variation in the values of the activity coefficientparameters among the different low-pressure data sets and they were obtained at different temperatures, any parameter set can be used with either mixing rule to predict the behavior of the methanol-water system with reasonable accuracy over the a temperature range of 150 K and a pressure range of 66 bar. Similar behavior is found for all the other mixtures considered here, though for brevity we do not display this information. A summary comparison for both equations of state and both mixing rules is given in Table IV. The first general observation from this table is that the results we obtain using the SRK and PR equations of state are essentially the same. We also see that, for eight of the mixtures, the W-S mixing rule is equivalent or superior in its pressure predictions to that of the MHVB rule, and for four of these mixtures, CZH&O-HZ~,CH@H-C&, C S H ~ ~ - C H ~ O H , and CSH~&ZH&O, the errors produced by the W-S mixing rule are approximately half those resulting from use of the MHVB mixing rule. Only in one case, for the CSHIZ-C~HSOH mixture, are the errors in the pressure predictions from the MHVB mixing rule slightly less than those with the W-S mixing rule. Also, averaging over all systems we see that the error in pressure is significantly less, and the error in composition is slightly less when the W-S mixing rule is used rather than the MHVB mixing rule. The difference in accuracy of the two mixing rules is clearly shown in Figure 1for the methanol-benzene system and in Figure 2 for the acetone-water system. In these figures we see that while the W-S mixing rule leads to slightly more accurate predictions than the MHV2 model at the conditions at which the activity coefficient model parameters were determined, the difference between the two mixing rules is striking at higher temperatures and pressures. In particular, the W-S mixing rule leads to accurate predictions at all conditions, while the MHVZ predictions are increasingly inaccurate as temperature and
Table IV. Average Deviations in Pressure and Mole Fraction Using the Peng-Robinson and SRK Equations of State with the W-S and MHV2 Mixing Rules
w-s
MHVS
AP/P, AP/P, system (%) AyXl00 (%) Ayxl00 (a) Peng-Robinson Equation of State (over All Parameter Setsp (11) 2.07 1.47 2.39 CHsOH-HzO 1.34 (32) 2.85 1.48 C2HsOH-HzO 2.70 1.03 (12) 2.02 1.08 4.11 1.57 C2&CO-H20 (12) 2.07 2.10 2.75 C2HsCO-CHaOH 2.22 1.93 (4) 5.35 4.24 1.60 CsH12-C2HsOH (28) 3.36 2.02 6.44 3.18 CHsOH-C& 0.40 1.19 0.46 CHsOH42HsOH (17) 1.11 2.31 6.77 1.92 (2) 3.50 ~d+rCHsOH 1.74 6.95 2.56 (4) 3.70 C&In-C2H&O average 2.89 1.61 4.17 1.76 (b) SRK Equation of State
2.10 2.91 1.97 2.07 5.33 2.78 0.88 3.19 3.53 2.75
1.53 1.64 1.09 2.07 2.04 1.83 0.39 2.15 1.68 1.60
2.44 2.92 4.20 2.81 4.95 6.54 1.00 7.38 7.30 4.39
1.39 1.14 1.67 2.18 1.75 3.20 0.46 2.10 2.68 1.84
a The numbers in parentheses indicate the number of low pressure parameter sets used.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole fraction acetone
Figure 1. Predictions of the W-S (solid lines) and MHVB (dashed lines) mixing rules for the acetone-water system with the PengRobinson equation of state and using UNIQUAC parameters reported in DECHEMA Chemistry Data Series, Vol. I, part la, p 168.
pressure increase. Therefore, it appears that temperaturedependent parameters are needed in the MHVS mixing rule to produce results of comparable accuracy to the W-S mixing rule with temperature-independent parameters. Or, put another way, more adjustable parameters are needed with the MHV2 mixing rule to obtain results of equivalent accuracy over a range of temperatures and pressures than are needed with the MHVZ mixing rule. As Michelsen (1990) points out, eq 18 is valid only if a > 10; if a is too small, eq 18 produces quite a large error. If the experimental data were obtained at reduced close to or higher than 0.7,a will temperatures T,= T/T, be less than 10 and using the MHVB mixing rule could produce a larger error. This is not a problem with the W-S mixing rule. As shown in Tables I11 and IV, the behavior of the W-S mixing rule is generally better than that of the MHVS rule, especially for mixtures containing a low critical temperature substance.
1502 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
53K
,
41 3 K
1 OQ
OD
01
02
03
04
05
06
07
08
09
10
Mole traction methanol
Figure 2. Predictions of the W-S (solid lines) and MHVS (dashed lines) mixingrules for the methanol-benzene system with the SoaveRedlich-Kwong equation of state and using UNIQUAC parameters reported in DECHEMA Chemistry Data Series, Vol. I, part 2a, p 212. Table V. Prediction of the VLE for Using the SRK and PR E 0 9 with the W-S and MHVB Mixing Rules with Parameters from Binary VLE Data. w-s MHV2 A p l p (%)
Ay X 100 Aplp (%)
Ay X 100
(a) n-CsHl2-CH3OH-C2H&O (T= 372.7 K and P = 5-8.5 bar) average SRK EOS 2.56 1.46 8.49 1.98 2.89 1.53 7.64 1.85 average PR EOS (b) C2HeCO-CHsOH-H20 (T= 373-523 K and P = 1.5-85 bar) 2.10 4.48 1.76 average SRK EOS 2.44 2.09 4.48 1.75 average PR EOS 2.49 a Results presented are an average over all combinations of the UNIQUAC parameters reported in DECHEMA Chemistry Data Series, Vol. I.
Comparison of the Two Mixing Rules for Ternary Systems We also compare, in Table Va, the errors of the predictions of the two mixing rules for the vapor-liquid equilibrium of the ternary system n-pentane (a)-methanol (b)-acetone (c), using parameters obtained from binary mixture data only. We present only average errors over all possible combinations of binary parameters. In particular we considered two parameter sets (2C, 374 and 375) for the n-pentane + methanol binary, four parameter sets (3 4, 187, 188, 189, and 190) for the n-pentane + acetone binary, and four parameter sets (2C, 72, 73, 74 and 75) for the methanol + acetone binary. [The source of these parameters was the DECHEMA Data Series, with volume number, part number, and page number indicated in parentheses above.] In fact, there was little variation in the accuracy of the predictions with the different parameter sets: 1.68-3.56% in pressure with the W-S mixing rule and 4.19-13.18% in pressure with the MHVZ mixing rule. Thus we see that the predictions, here for a ternary system, are essentially independent of which of the low-pressure binary data sets were used to obtain the activity coefficient model parameters, and that the W-S mixing rule is superior to the MHVZ model regardless of which set of UNIQUAC parameters and which equation of state are used to predict ternary VLE. Indeed, averaging over all parameter sets, the W-S mixing rule produces an error in the pressure which is only about one-third that of the MHV2 mixing rule, and somewhat better composition predictions. Further, and somewhat surprisingly, the magnitude of the prediction errors for this ternary system with the W-S mixing rule is approximately the same as those found for the constituent binary mixtures. Table Vb shows the accuracy of similar predictions for
+
the acetone-methanol-water system over a wider range of temperature and pressure, again using both equations of state, both mixing rules, and various choices of the binary parameters [2C, 74 and 75 for acetone + methanol, la, 190,191,192, and 198 for acetone + water, and la, 49,54, 55, and 58 for methanol + water]. Here we see that the W-S mixing rule produces errors in pressure of only about half those obtained with the MHVB mixing rule. The errors in the predicted composition for this system are small, probably comparable to the experimental uncertainty, though slightly smaller for the MHVB mixing rule for this mixture. Conclusions We have established a number of things in this paper. First, we have presented new parameters for the temperature function in the SRK equation that lead to more accurate vapor pressure predictions. Second, we have shown that either the MHVS or W-S mixing rules can be used to make vapor-liquid predictions for binary mixtures at high temperatures and pressures using published activity coefficient model parameters obtained by regressing low-pressure VLE data. However, the W-S mixing rule leads to more accurate predictions. This is most clearly demonstrated in the figures. Next, we have shown that it is possible to make good Predictions for highpressure, ternary mixture VLE based on parameters obtained from low-pressure binary mixture data. Again, in this case, the W-S mixing rule has a significant advantage in accuracy over the MHVB mixing rule. Finally, we have shown that for such calculations there is little difference between using the Peng-Robinson and Soave-Redlich-Kwong equations, though the best results are obtained with the combination of the Wong-Sandler mixing rule and the Soave-Redlich-Kwong equation of state. Though not previously mentioned, there are several other comments we can make about using the mixing rules discussed here, and especially the W-S rule. First, from either Table I11or V, we see that the errors in the predicted pressures are very similar for most of the parameter sets used, though several parameter sets give larger errors (in excess of 3 % 1. This suggests that the data on which those parameters are based may be less accurate than the other data sets. Thus it is possible that the mixing rules, an equation of state, and high-pressure data may be used to test the quality of a collection of low-pressurevapor-liquid equilibrium data sets. Second, though here we have used activity coefficient or excess Gibbs free energy models which were first fit to experimental data, we need not have done so. The values of the excess Gibbs free energy gotten from experimental data can be used directly in the W-S mixing rule; it is not necessary to first fit these data to an approximate model. Finally, though here we used activity coefficient model parameters which had been regressed from VLE data, any other source of activity coefficient information, such as liquid-liquid or solubility data, could also have been used. Consequently, by using the W-S mixing rule, which does not require temperaturedependent parameters, a great variety of low-pressuredata can be used as the basis for making VLE predictions at high pressures and temperatures. Acknowledgment This research was supported, in part, by Grant No. DEFG02-85ER13436 from the US. Department of Energy
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1503 and Grant No. CTS-9123434 from the U.S. National Science Foundation, both to the University of Delaware.
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Receiued for review November 30, 1992 Accepted April 13, 1993
