Ind. Eng. Chem. Process Des. Dev.
temperature decreased, the production of methanol increases. However, at sufficiently low temperatures or high pressures, supersaturation and subsequent condensation of liquid a t the equilibrium mixture composition is possible. There is thus a limit to maximizing methanol production through the decrease in temperature or increase in pressure. Moreover, the equilibrium conversion cannot be predicted accurately unless the equation of state correctly models the mixture liquid-vapor equilibria. In our computations, we found that the cases of SRK(M) and PR(M) a t 200 "C and 30 MPa did not converge. We attribute this to liquid-phase condensation at these conditions. Conclusions With methanol synthesis reactions, C02 conversion and methanol production are always overestimated by using correction factors calculated from pure-component fugacity coefficients obtained from a generalized chart. Significant errors in the estimation of C 0 2 conversion and CH,OH formation are produced by mistakenly using the liquidphase fugacity coefficients as vapor-phase fugacity coefficients for subcritical compounds. This mistake often has a large effect on calculations involving methanol synthesis reactions at temperatures from 200 to 300 "C and pressures from 5 to 10 MPa. For accurate estimation, these calculations require the use of fugacity coefficients that depend on composition to estimate the nonidealities of the coexisting species in a reaction mixture. This work shows that either the Peng-Robinson or the Soave-Redlich-Kwong
1986,25, 481-486
481
equation of state can be used profitably for this purpose.
Nomenclature K,i = equilibrium correction factor for reaction i Kai = equilibrium constant for reaction i Ni= moles of component i at equilibrium NT = total moles of mixture at equilibrium NIL= initial moles of component i in mixture NIT = total moles of initial mixture X = extent of reaction 1 Y = extent of reaction 2 q$ = vapor-phase fugacity coefficient of i Registry No. CH,OH, 67-56-1;CO, 630-08-0;COP,124-38-9. Literature Cited Bissett, L. Chem. Eng. 1977, 84(21), 155. Chang, T. Ph.D. Thesis, North Carolina State University, Raleigh, 1984. Cherednichenko, V. M., Dissertation, Karpova, Physic0 Chemical Institute, Moscow, U.S.S.R., 1953. Graboski, M. S.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 443. Hougen, 0. A.; Watson, K. M.; Ragatz, R. A. "Chemical Process Principles", 3rd ed.; Wiley: New York, 1964. Peng, D.-Y.; Robinson, D. 8. Ind. Eng. Chem. Fundam. 1976, 15, 59. Smith, W. R.: Missen, R. W. "Chemical Reaction Equilibrium Analysis: Theory and Algorithms"; Wiley: New York, 1982. Soave, G. Chem. Eng. Sci. 1972, 2 7 , 1197. Tarakad, R. R.; Spencer, C. F.;Adler, S. B. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 726. Vonka, P.; Holub, R. Collect. Czech. Chem. Commun. 1975, 4 0 , 931. Wade, L. E.; Gengelbach, R. B.; Taumbley, J. L.; Hallhauer, W. L. "Kirk-0thmer Encyclopedia of Chemical Technology", 3rd ed.; W h y : New York, 1981; Vol. 15, pp 398-415.
Received
f o r review November 26, 1984 Accepted
August 14, 1985
Prediction of Low-Pressure Vapor-Liquid Equilibria of Nan-Hydrocarbon-Containing Systems-ASOG or UNIFAC Parag A. Gupte and Thomas E. Daubert' Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
An extensive comparison of the two group-contribution methods for activity coefficients-UNIFAC and ASOG-is presented. The methods are compared by carrying out bubble point and K-value calculations over a large low-pressure VLE data base. Overall, the two methods yield equivalent results when the methods are compared for various families of systems. However, for certain families, one method may be superior to the other. Prediction results from Raoult's law are also obtained and compared with the more complex models.
Prediction of vapor-liquid equilibria is important for design calculations in the chemical industry. Vapor-liquid equilibria of hydrocarbon-containing systems are commonly predicted by using equations of state. In such cases, the binary interaction parameters kij are either taken as zero or generalized in terms of readily available input parameters like the critical properties. This type of approach yields consistent results for nonpolar systems and for systems where the components are chemically similar. The advantage of this approach is that it can be extended to high pressures. When the system contains non-hydrocarbon compounds and when the components are chemically dissimilar (e.g., hydrocarbon-alcohol systems), the equation of state approach is not applicable. In such cases, the use of zero
* To whom correspondence should be addressed.
interaction constants leads to poor results. Furthermore, interaction constants are not easily generalizable in terms of simple molecular properties. In such cases, the alternative approach is to use GE models for the liquid phase. At low pressures (less than 5 bar), the liquid-phase fugacity is conveniently represented by using activity coefficients. For subcritical components, this relation is f,L
= y.x 1 1 f."L 1
(1)
The activity coefficient yi is related to the excess Gibbs energy of the system by the following well-known relation. n
Ex, In yI
gE = RT
i=l
(2)
Several models for gE are available in the literature. The more popular models are UNIQUAC (Abrams and Prausnitz, 1975), NRTL (Renon and Prausnitz, 1968), and
0196-430518611 125-0481$01.50/0 0 1986 American Chemical Society
482
Ind. Eng. Chem. Process
Des. Dev., Vol. 25,No. 2,
1986
Wilson's equation (Wilson, 1964). These are essentially correlators because they require at least two mixture parameters to describe a binary system. Parameters are commonly regressed by using experimental binary-phase equilibrium data. Once the binary parameters are available, the models may be used to predict ternary and higher-order data. The results from such predictions can be surprisingly accurate for some systems. These models are considered as correlators, inasmuch as they require mixture parameters to predict data. Group-contribution models for activity coefficients have been available in the literature. These models are based on the solution of groups concept. Instead of a mixture of molecules, the solution is considered to be a mixture of groups. The model equations are then applied to the properties of the groups and the interactions between groups. There are thousands of molecules of interest in chemical technology. These molecules can be constructed from a much smaller number of groups. Consequently, the group properties and group energetic interactions can be obtained from a relatively small number of vapor-liquid equilibrium data sets. Once these interactions are available, they may be used to predict the data for a large number of systems for which experimental data are unavailable or are not easy to measure. The best-known group-contribution models for activity coefficients available in the literature are UNIFAC and ASOG. These models are widely used to predict liquidphase fugacities of nonelectrolyte systems. The common approach is to represent the vapor phase by a truncated virial equation. This approach has been recommended by several authors (Prausnitz et al., 1980). The second virial coefficient is obtained from the method of Hayden and O'Connell (1975) or Tsonopoulos (1974, 1975). The combined second virial coefficient-activity coefficient approach is a powerful method to predict VLE of systems containing polar components. Although the above approach is popular and used extensively in the chemical industry, no comparison of the UNIFAC or ASOG method has been made. The present work is the first effort where a comprehensive program was carried out for the two models. The UNIFAC and ASOG models are briefly described in the next section.
UNIFAC The UNIFAC method was developed by Fredenslund et al. (1975). The method was further consolidated, and the group interactions were given in a monograph by Fredenslund et al. (1977). The activity coefficients are represented by In y i = In y I c + In yIR
(3)
The combinatorial terms In :y account for the sizeshape differences between molecules, and the residual term describes the nonidealities due to energetic interactions between groups. In the UNIFAC model, the combinatorial term is given by the Staveri iann-Guggenheim expression. The residual term is a group-contribution version of the residual term in the UNIQUAC model. All the pertinent equations for the UNIFAC model are given in the Appendix section. The parameters needed for the use of UNIFAC are group volumes (&), group surface areas (Qk), and groupinteraction parameters (amnand unm).Tables with revised and updated parameters for 40 groups have been presented by Gmehling et al. (1982). A further revision and introduction of an additional group has been carried out by Macedo et al. (1983). Parameters representing 370 out of 820 possible interactions have been published. These
parameters can be used to predict VLE data for nonelectrolyte binary and multicomponent mixtures in the temperature range 300-425 K. ASOG The ASOG model has been presented by Kojima and Tochiji (1979). The model equations are listed in the Appendix section. Like the UNIFAC method, the activity coefficient in ASOG is represented as a sum of combinatorial and residual contributions. The combinatorial term is calculated by an equation similar to the Flory-Huggins equation. Consequently, only differences in the size of the molecule are taken into account. Wilson's equation (1964) is used to represent the residual contribution, with group fractions instead of mole fractions. The ASOG model has 31 groups. Kojima and Tochiji have estimated 143 out of 465 possible interactions. Unlike the UNIFAC model, the group interaction parameters ski are taken as functions of temperature. In Ukl = mkl + nki/T (4) In eq 4, mkland nklare temperature-independent. The ASOG model thus includes four parameters per pair of groups. The input parameters necessary for ASOG are the group interactions (mkl,nkl,and nlk), the number of nonhydrocarbon atoms in each component, and the number of non-hydrocarbon atoms in each group. For the H20, CH, and C groups, the values for the number of non-hydrogen atoms adopted are 1.6, 0.8, and 0.5, respectively. Tochiji et al. (1981) have modified eq 4 to In
Y1
Ukl
= 1n - - bkl/T
(5)
Yk
where u Land V k are the number of non-hydrogen atoms in groups 1 and K . With this modification, the number of group-interaction parameters is reduced to two per pair (bkl and b l k ) . Group interactions (bkl and b l k ) have been estimated for only 11 groups. Thus, eq 5 was not considered in this work, and the standard ASOG model represented by eq 4 was evaluated. VLE Data Base For evaluation purposes, a large low-pressure data base containing binary, ternary, and quaternary systems was set up. The systems collected were mainly from the published version of the Dortmund data base (Gmehling and Onken, 1977). All the data sets collected were at low pressures (less than 5 bar). Furthermore, all the systems collected included at least one non-hydrocarbon component. The systems were chosen so as to have good representation of various families. The final data base is comprised of 402 separate binary, 208 separate ternary, and 8 quaternary systems. For several systems, there are many isothermal or isobaric data sets. It is estimated that there are about 15000 separate VLE data points included in this data base. The VLE data are "complete"; Le., each data point contains the pressure, temperature, and the compositions in both phases. Pure component data required for evaluation of the various thermodynamic functions were obtained from the AIChE DIPPR-Data Compilation Project (Daubert and Danner, 1985). Evaluation of Methods The distribution coefficient K, can be written as follows for a liquid and vapor phase in equilibrium. y1 yL$IsatPcEat exp[V,(P - P,""t)/RT] K=-= (6) XI
41vp
and +Ftwere obtained by The fugacity coefficients using the truncated virial equation (truncated after the
Ind. Eng. Chem. Process Des. Dev., Val. 25, No. 2, 1986 483 Table I. Prediction Results for Binary Low-Pressure Vapor-Liquid Equilibria for UNIFAC, ASOG, and Raoult’s Law % errors Raoult’s law
% errors UNIFAC
systems 1 2
3 4 5 6 7 8 9 10 11 12 13
14 15 16 17
18 19 20 21 22 23
containing HZO/ alcohol HzO/ aldehyde ketone H20/ester H,O/nitro compds alcohol/ alcohol alcohol/ aldehyde ketone alcohol/ ether alcohol/ ester alcohol/ aliphatic alcohol/ aromatic alcohol/ halogen alcohol/ nitro compd aldehyde-ketone/ aldehydeketone aldehydeketone/ ester aldehydeketone/ aliphatic aldehydeketone/ aromatic aldehydeketone/ halogen ether/ halogens aliphatic/ halogen aliphatic/ nitro compd aromatic/ halogen aromatic/ nitro compd miscellaneous
no. of points 549
no. of systems 12
% errors ASOG
no. of points
no. of systems
597
9
169
4
24
2
113
4
436
21
208
9
181
4
469
14
665
33
619
26
249
10
154
~
KC1 av/bias
KC2 a/! bias
bubble pt
KC1 av/bias
KC2 av/ bias
10.8 -0.3 23.4 16.9
14.7 6.7 26.4 22.1
0.9 -0.6 10.2 9.4
43.1 -42.3 46.5 -45.7
34.3 -29.0 40.8 -36.4
282.0 281.9 13.8 -4.4 6.6 0.4 5.7 1.3
217.0 215.5 14.5 5.2 8.6 1.4 7.3 2.6
70.6 65.0 6.0 -3.8 2.1 0.5 0.6 0.0
52.7 -10.6 39.3 -39.1 7.5 0.2 17.6 -17.3
43.1 -0.1 33.9 -21.9 10.4 2.9 22.2 -20.8
5.3 2.5 5.8 3.1 7.3 0.9 5.2 -0.2 5.6 -0.3 17.4 5.9
0.6 -0.4 1.1 0.6 4.1 -0.1 0.7 -0.1 1.4 0.1 8.5 1.3
27.5 -27.2 27.5 -19.4 40.8 -40.8 29.7 -29.6 33.7 -33.6 32.0 10.1
33.0 -32.9 23.8 -13.1 39.3 -36.2 36.5 -36.2 31.7 -23.8 24.4 9.5
KC2 a/! blas
bubble at
1.1 12.9 -3.1
7.3 0.3 8.2 -3.5
0.5 -0.2 3.4 2.3
11.1 2.5 12.1 -7.5 5.2 0.7 7.6 -6.0
14.1 -9.3 7.3 6.9 5.6 2.4 9.3 -7.8
1.5 -0.9 2.6 -0.0 2.1 -0.0 0.8 -0.1
4
5.9 2.1 10.6 -8.7 7.5 -2.8 5.3 -3.6 9.3 -7.1 7.5 1.1
4.9 -0.7 9.8 -6.2 8.6 -0.6 10.1 -4.8 8.7 -2.5 18.4 9.8
2.4 1.9 5.1 1.0 4.2 -0.8 1.2 1.1 2.4 -0.0 3.7 -1.4
KC1 av/bias
8.8
223
6
24
2
356
11
447
24
244
10
211
4
391
11
717
37
619
26
461
21
190
9
3.2 2.1 4.8 1.9 6.6 -1.0 4.5 -0.5 4.5 -0.5 21.8 1.6
98
6
24.4 22.6
17.1 15.7
5.1 4.1
1.9 5.2
2.7 -0.6
72
4
2.2 -1.0
3.3 0.1
0.1 0.0
88
7
3.2 0.9
4.6 2.1
0.1 -0.0
7.4 -6.4
6.5 -5.2
88
7
5.8 -1.3
6.6 -1.3
0.3 0.2
299
14
13.2 1.4
8.7 5.2
6.8 -2.8
36.9 -36.8
27.8 -25.3
126
5
5.5 -1.8
6.1 2.9
2.4 0.4
175
8
4.3 -1.5
4.9 0.6
1.3 -0.4
9.5 -8.6
10.3 -8.6
115
7
4.5 1.4
3.6 0.6
1.3 1.2
180
7
6.1 3.2
15.0 7.8
2.2 -1.0
21.6 -1.5
28.9 -28.7
143
5
2.1 0.2
4.2 2.6
0.6 0.3
259
9
18.9 11.3 14.0 -13.9 29.0 -28.7
24
17
18.1 13.3 12.6 -12.3 29.6 -28.9
363
298
1.0 0.4 2.5 2.0 4.6 2.0
6
36
7.5 1.6 5.0 -3.6 8.0 -1.7
181
570
6.6 1.0 6.6 3.4 6.7 3.5
162
11
11.5 -10.4 7.6 5.0 8.0 3.0
12.0 -6.0 5.2 0.4 8.6 -0.4
3.0 -1.6 1.6 1.2 3.6 1.7
468
20
1.3 0.4 1.6 -0.6
5.2 1.8 13.3 -10.2
9.8 -4.6 17.4 -14.8
7
17
8.5 -6.3 8.6 -2.9
158
282
3.7 -0.5 3.5 -1.0
187
7
3.0 -1.2 11.9 5.0
3.1 -0.4 13.6 7.2
0.2 0.0 1.5 0.6
578
28
10.1 -4.5
13.4 -5.6
3.5 -1.8
23.0 -18.0
20.1 -17.4
556
28
12.9 1.4
15.9 -7.8
6.1 0.4
second virial coefficient). The second virial coefficient was obtained by using the Hayden-O’Connell method (1975). The Hayden-O’Connell method uses the critical temperature, the critical pressure, the radius of gyration, and the dipole moment as input parameters. Association parameters for @Ftand solvation parameters for div are also required. These parameters have been listed for 92 fluids by Prausnitz et al. (1980). For those systems where the parameters were not available, these were estimated by comparing with chemically similar systems. The modified Rackett equation (Spencer and Danner, 1972) was used to obtain the liquid molar volume V,. The saturation pressure Pisetwas obtained by using the corre-
lation equations prepared in the AIChE DIPPR-Data Compilation Project. The saturation pressure is a significant quantity in eq 6 and is of utmost importance in VLE calculations to represent this quantity accurately. Also, it was observed that the fugacity coefficients GiV and @;set departed significantly from unity for some systems. This shows that for systems containing polar components, vapor-phase nonidealities are not negligible even at low pressures. The evaluations were carried out by using bubble point and K-value calculations. For K-value calculations, all the thermodynamic functions in eq 6 were evaluated at experimental conditions. This K was then compared to the
484
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
91 O ?
se
*I
k
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 485 Table 111. Definitions of Errors deviation, E = [(calculated quantity-experimental
quantity)/experimentalquantity] X 100 av = ( l / n ) I 3 1 4 bias = ( l / n ) E E n = total number of data points KC1 = error in the distribution ratio of component 1, calculated by UNIFAC or ASOG; similarly for KC2, KC3 bubble point = errors in the bubble-point temperature or pressure, calculated by UNIFAC or ASOG
experimental K (Yexp/~e,p). Thus, no equilibrium calculations were carried out in this case. It is felt that for the purpose of model discrimination, this approach is superior to carrying out flash calculations. Flash calculations are more time-consuming, and a significant percentage of data points is lost because the computed V / F ratio is greater than 1 or less than 0. Finally, results for each system were averaged over the several isothermal or isobaric data sets, using the number of data points as weighting factors. The systems were then grouped by family, and composite averages for each binary or ternary family were computed. The UNIFAC and ASOG models were also compared with predictions from Raoult’s law and an ideal vapor-phase assumption. The distribution coefficient K , for this case is given by (neglecting Poynting corrections) pisat
Kiideal
=-
P
(7)
The errors are tabulated by family for both UNIFAC and ASOG in Tables I and 11. The evaluation results are discussed next for binary, ternary, and quaternary systems.
Binary Systems The binary data contained in the VLE data base were comprised of compounds belonging to 12 different chemical families. Thus, there were 66 different binary groupings that can be constructed. The different systems contained in the binary data base were then subdivided into these 66 different groupings. About 22 of these groupings were sufficiently well-represented to be grouped together. The rest of the groupings were sparsely represented and all systems blonging to such groupings were lumped under a “miscellaneous” heading. Table I lists the errors for the UNIFAC model, Raoult’s law, and the ASOG method. The various quantities presented in the tables are defined in Table 111. An examination of the evaluation results reveal that UNIFAC was better for about half of the groupings. The UNIFAC method is more widely applicable and predicted data for a significantly larger number of systems. ASOG lacks binary interaction data for several groups, and often the component groups are not easy to form. Consequently, ASOG is not applicable to the more uncommon systems. The ASOG model was better for all the four groupings containing water. Finally, results from Raoult’s law are comparable to the activity coefficient models for some groupings. This is to be expected for groupings containing chemically similar compounds. Ternary Systems The ternary systems were subdivided into 14 ternary groupings. Such ternary groupings are difficult to form, and the particular choice employed here was obtained by minimizing the number of groupings. The errors are presented in Table 11. The evaluation results indicate that for ternary lowpressure systems, ASOG is a marginally better predictor. The ASOG method appears to be somewhat superior to
UNIFAC for systems containing alcohols and water. The ASOG method is also better than UNIFAC for systems containing chemically similar components. In such systems, however, the use of Raoult’s law can lead to reasonable predictions. Finally, the models were tested for 13 different quaternary systems. The ASOG model was not applicable to five of these systems. The evaluation results are not reproduced here. There was very little difference between the models for the quaternary systems. Usually one model predicted distribution coefficients better for some of the components, whereas the other model predicted better for the remaining components. Conclusions The two group-contribution models for the excess Gibbs energy were compared by using a large low-pressure VLE data base. The models yield comparable results for most families. The UNIFAC method is more widely applicable as it has a larger parameter interaction matrix. For systems containing water and alcohols, the ASOG method was found to yield better results. Predictions from Raoult’s law are comparable to those from the UNIFAC and ASOG models for systems containing chemically similar components. Acknowledgment The Design Institute for Physical Property Data of the American Institute of Chemical Engineers provided financial support of this work. Nomenclature umn,unm = group interactions in the UNIFAC model, K ukl = parameter in the ASOG model bkl, blk = group interactions in the ASOG model, K F = flow rate of feed, mol/h f I L = fugacity of component i in the liquid solution f:L = fugacity coefficient of pure component i at the temferature of the solution G = excess Gibbs energy of mixture, kJ gE = molar excess Gibbs energy of mixture, kJ/mol K , = distribution ratio of component i K1lded= distribution ratio of component i, predicted by Raoult’s law k,, = binary interaction coefficient in an equation of state mkl, mlk = group interactions in the ASOG model nkl,nlk = group interactions in the ASOG model, K n = number of components in the solution P = total pressure, kPa PlSat = saturation pressure of component i, kPa Qk = surface area parameter for group k in the UNIFAC model R = universal gas constant Rk = volume parameter for group h in the UNIFAC model T = temperature, K V = flow rate of vapor product, mol/h V , = liquid molar volume of component z, m3/kmol x, = mole fraction of component i in the liquid xexp= experimental mole fraction in the liquid phase y L = mole fraction of component z in the vapor yexp= experimental mole fraction in the vapor phase Greek Symbols y L = activity coefficient of component i ylc = combinatorial contribution to the activity coefficient y: = residual contribution to the activity coefficient 4: = fugacity coefficient of component i in the vapor mixture 41sat = fugacity coefficient of pure component z at the tem-
perature of the system ul, V k = parameters in the ASOG model
Appendix Model Equations for UNIFAC. In y, = In y: + In y l R
486
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
In yic =
(A-14) J
@i
vjFH = number of non-hydrogen atoms in component j , xi = mole fraction of component j , and M = number of
xiri
=-
components in the solution.
M
C x j r j I
In
(-4-4)
N ?‘i
=
=
-y?
N CVki k
(In
r k
- In r k ( i ) )
(A-15)
V k i = parameter related to the number of non-hydrogen atoms in group k in molecule i, r k = group activity coefficient of group k, and N = number of groups in the solution.
(-4-5)
CVk(i)Rk k
N
4i =
(-4-6)
CVk(i)Qk k
where Rk = volume parameter for group k , Qk = surface area parameter for group k, Vk(i) = number of groups of type k in molecule i, x i = mole fraction of component i, z = coordination number = 10, M = number of components in the solution, and N = number of groups in the solution. In yiR = r k
N &(i) k
(In
rk- In r k ( i ) )
(A-7)
= group activity coefficient of group k
(A-17) i k
In akl =
mkl
+nkl/T
(A-18)
mkl,nkl= group interaction parameters characteristic of groups k and 1 and r k C 0 = r k for solution containing pure 1.
Literature Cited
(A-10) M CV,6)Xj
x,=
(A-11)
M N
C CVn”x
j
I n
amn= group inter ction parameters for the int raction between groups m and n and r k ( i ) = r k for a solution containing only component i. Model Equations for ASOG. (A-12) In yi = In yic + In yiR In yic = In 4 i / x i + 1 - @ i / x i
(A-13)
Abrams, D. S.;Prausnb, J. M. AIChEJ. 1975, 27, 116. Daubert, T. E.;Danner, R. P. “Tables of Physical and Thermodynamic Properties of Pure Components”; American Institute of Chemical Engineers: New York, 1985; Data Compilation Project. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. AIChE J . 1975, 27, 1056. Fredenslund, A.; Grnehllng. J.; Rasmussen, P. “Vapor-Liquid Equilibria Using UNIFAC”; Elsevier Scientlfic: New York, 1977. Gmehllng, J.; Onken, U. DECHEMA -Chem. Data Ser. 1977, No. 7 -8. Gmehling, J.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem, Process Des. Dev. 1982, 27, 118. Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem. Process Des. Dev. 1975, 74, 221. Kojima, K.; Tochiji, K. “Prediction of Vapor-Liquid Equilibria by the ASOG Method”; Elsevier: New York, 1979. Macedeo, E. A.; Weldlch, U.; Gmehling, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 676. Prausnltz, J. M.; Anderson, T. F.; Gens, E. A,: Eckert. C. A,; Hsieh, R.; 0’Connell, J. P. “Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria”, 1st ed.: Prentice Hall: Engiewood Cliffs, NJ, 1980; p 4. Renon, H.; Prausnitz, J. M. AIChE J . 1988, 74, 135. Spencer, C. F.; Danner, R. P. J . Chem. Eng. Data 1972, 77, 236. Tochiji, K.; Lu, B. C.-Y.; Ochi, K.; Kojima, K. AIChE J. 1981, 2 7 , 1022. Tsonopoulos, C. AIChE J. 1974, 2 0 , 263. Tsonopouios, C. AIChE J . 1975, 27, 627. Wilson, G. M. J. Am. Chem. SOC. 1984, 86. 127.
Received for review January 11, 1985 Accepted August 2 2 , 1985
