681
Ind. Eng. Chem. Res. 1995,34, 681-687
Prediction of Vapor-Liquid Equilibria at Low and High Pressures Using UNIFAC-BasedModels Epaminondas C. Voutsas, Nikolaos Spiliotis, Nikolaos S. Kalospiros, and Dimitrios Tassios* Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Street, Zographou Campus, 15780 Athens, Greece
Four recently developed models involving equations of state combined with the UNIFAC GE expression: MHV2, PSRK, LCVM,and the Wong-Sandler models are compared with respect to their performance in the prediction of vapor-liquid equilibria (VLE). The following cases are considered: (a) polar mixtures with components of similar size at low and high pressures; (b) asymmetric systems containing supercritical gases and n-alkanes; (c) infinite dilution activity coefficients in asymmetric hydrocarbon mixtures. All models perform satisfactorily i n the VLE prediction of symmetric systems (case a), but only the LCVM model for asymmetric ones (cases b and c). The conventional y-4 approach has also been included in the comparison when applicable; i.e., low-pressure VLE.
Introduction Recently, attempts to incorporate excess Gibbs free energy (GE)models into the mixing rule expressions for the attractive term of cubic equations of state (EOS) have led to the development of equation of state models applicable to low- and high-pressure vapor-liquid equilibrium (VLE) of polar systems. Of special interest is the use of the group contribution UNIFAC class GE models, as these allow EOS to become predictive tools. These models can be classified into the two following categories: those stemming from Vidal (1978) and Michelsen (1990a,b) mixing rules and the Wong and Sandler model (Orbey et al., 1993) using the mixing rules presented by Wong and Sandler (1992). In order to develop models of the first category, one obtains the mixing rule expression for the attractiveterm parameter, a, of any EOS by setting the GE expression obtained by the EOS equal to that of a GE model-hence the name EOS/GE models. Although such a matching was originally postulated at infinite pressure (Vidal, 1978), it is a matching at the zero-pressure limit that has led t o two successful zero reference pressure (ZRP) EOS@ models (MHV2, Dahl and Michelsen (1991); PSRK, Holderbaum and Gmehling (1991). Boukouvalas et al. (1994) have developed a mixing rule, leading to the LCVM model, that is a linear combination of the mixing rules of Vidal (1978) and Michelsen (1990b) and, as such, has no reference pressure. In all the above-mentioned models, a linear mixing rule is assumed for the covolume parameter, b, of the EOS; i.e., b = Zqbi. As a consequence of these mixing rules for a and b, all ZRP models and the LCVM model do not satisfy the theoretical requirement that the second virial coefficient be a quadratic function of composition and, therefore, are inconsistent with statistical mechanical theory. In order to overcome the above deficiency, Wong and Sandler (1992) developed a set of mixing rules, in which the attractive parameter, a, and the covolume parameter, b, of an equation of state are determined to give the correct quadratic composition dependence of the second virial coefficient. However, in contrast to the models of the first category, an extra parameter is introduced in this model, a second virial
* To whom correspondence should be addressed. 0888-5885/95/2634-0681$09.00/0
coefficient binary interaction parameter, K - . The value of Ku is determined by equating the calculated through the GE model t o the corresponding value calculated from the EOS, at a specific composition. Finally, a useful tool for the prediction of VLE is the conventional y-C#I approach with the UNIFAC group contribution GE model, the major advantage of which is the existence of extensive parameter tables for the interaction of a plethora of groups. However, this approach is limited by the use of the virial EOS up t o about 10-15 atm and by the use of UNIFAC versions to temperatures approximately between 275 and 425 K (Fredenslund, 1989). The objective of this study is the comparison of these EOS/UNIFAC models with respect to their accuracy in the prediction of VLE for polar and nonpolar systems as well as for systems of varying degree of asymmetry with respect to size. More specifically, we consider three cases: (a) symmetric systems a t both low and high temperatures and pressures; (b) asymmetric systems; (c) infinite dilution activity coefficients in hydrocarbon mixtures. In addition, the y-4 approach is also considered in the prediction of low-pressure VLE using three UNIFAC versions. The remainder of the paper is organized as follows. The models considered are briefly presented in the next section. The data base used in the comparison follows. The results of the models are compared and discussed next. We close with our conclusions.
&
The Models The EOS/"IFAC models considered in this study are the following: (a) MHV2 (Dahl and Michelsen, 1990; Dahl et al., 1991) is a model based on the second-order approximationof the modified Huron-Vidal mixing rule (Michelsen, 1990b). It combines the SRK EOS with the modified UNIFAC GEmodel (Larsen et al., 1987). (b) PSRK (Holderbaum and Gmehling, 1991) is a model based on the first-order approximation of the modified Huron-Vidal mixing rule (Michelsen, 1990a,b). It combines the SRK EOS with the original UNIFAC GE model (Hansen et al., 1991). (c) LCVM (Boukouvalas et al., 1994) is based on a linear combination of the mixing rules of Vidal(l978) and Michelsen (1990b).This model combines the t-mPR EOS (Magoulasand Tassios,
0 1995 American Chemical Society
682 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 Table 1. VLE Results Obtained by Using the Original and the Modified WS Models system
rep
N
Trange(K)
Prange(bar)
methanollbenzene methanovwater acetone/water acetoneimethanol
1 2 2 2
40 55 63 35
373-493 373-523 373-523 373-473
3.1-57.6 1-83 1.1-67.6 3.5-39.9
K" 0.278 0.085 0.295 0.100
original WSb AP% Ay(x100) 2.4 2.7 2.8 2.8
Ki 0.281 0.087 0.283 0.105
1.5 1.7 1.7 1.8
modified WSb AP% Ay(x100) 1.7 1.8 2.6 2.1
1.7 1.4 1.5 2.0
1,Butcher and Medani (1968); 2, Griswold and Wong (1952). A€% is the average absolute percent error in bubble point pressure. Ay is the average absolute deviation in vapor phase mole fraction: Ay = (1/2)C&(1/NDP)Z1~lyj,fal - yj,iewl.
Table 2. Results for Polar Systems at Low Pressures with Different UNIFAC-Type Models original UNIFAC rep
T(K)
Prange (bar)
methanolhenzene
1
methanoywater
2
n-pentanelmethanol
3
373 413 373 423 372.7 397.7 422.6
3.1-4.2 6.7-11.8 1.0-3.4 5.1-13.7 4.5-7.9 10.4-15 17-25.3
system
a
AP% 4.9 12.3 2.1 1.1 3.9 6.2 12.6
new UNIFAC
modified UNIFAC
Ay (x100)
AP%
Ay(x1GO)
AP%
Ay (x100)
2.1 4.2 1.0 1.4 2.3 2.6 4.8
1.5 4.9 3.1 0.8 6.5 8.9 17.1
0.7 1.5 1.4 1.2 3.8 4.6 6.8
0.8 0.9 4.0 3.1 3.9 5.3 8.2
0.4 2.3 1.7 2.0 1.6 2.2 3.4
1, Butcher and Medani (1968); 2, Griswold and Wong (1952); 3, Wilsak et al. (1987).
1990) with the original UNIFAC (Hansen et al., 1991). As mentioned above this model has no reference pressure. (d) Wong and Sandler (WS). Orbey et al. (1993), using the mixing rules of Wong and Sandler (19921, developed a predictive UNIFAC-based model that combines the PRSV EOS (Stryjek and Vera, 1986) with the UNIQUAC GE model (Abrams and Prausnitz, 1975). In order for the model to be fully predictive, the interaction parameters of UNIQUAC are estimated by use of infinite dilution activity coefficients predicted from the original UNIFAC (Hansen et al., 1991). In the present work, we have used a modified approach in applying the Wong and Sandler model. Here we must note that the above models have been used in the same fashion as suggested in the original publications (Dahl and Michelsen (19901, Dahl et al. (1991) for MHV2, Holderbaum and Gmehling (1991) for PSRK, Boukouvalas et al. (1994) for LCVM, and Orbey et al. (1993) for WS), i.e., the same EOS, pure component EOS parameters, and the same UNIFAC version. Thus, details on the development and the use of the models can be found in the original publications. Finally, the original UNIFAC version proposed by Hansen et al. (1991)has been used in the y-4 approach, after comparison with other UNIFAC versions such as the modified UNIFAC (Larsen et al., 1987) and the original UNIFAC with linearly temperature-dependent parameters (Hansen et al., 1992), hereafter referred t o as new UNIFAC. The virial EOS truncated to the second term with second virial coefficients from Tsonopoulos' correlation (Tsonopoulos, 1974, 1975) is used t o describe the vapor phase. Modified Method of Using the Wong and Sandler Model. In order to avoid the cumbersome method proposed by Orbey et al. (1993), i.e., t o estimate UNIQUAC parameters from infinite dilution activity coefficients predicted from UNIFAC, we have developed a much simpler method. In this method there is no need for using the UNIQUAC GE model. The unique GE model that has been used is the original UNIFAC. Specifically, the value of the second virial coefficient binary interaction parameter, K,, is estimated directly by solving the equation
a t x z = 0.5 and T = 25 "C, where G E m 1 ~ is * ~the excess Gibbs free energy from original UNIFAC and the one from the PRSV EOS using the WS mixing rules. In Table 1 some indicative VLE results for polar systems are demonstrated at both low and high pressures for the Wong and Sandler model. Both approaches, for the binary interaction parameter, Kg, have been applied, i.e., the one of Orbey et al. (1993) and the one proposed in this study. A comparison between the two suggests that our modification performs at least as satisfactorily as the original model.
os
Data Base The data base used for the comparison of the models considered in this study consists of the following cases. Case A. Symmetric polar systems: (a) binary polar mixtures at low pressures where all models can be applied to the prediction of VLE, including the y-4 approach; (b) binary polar mixtures at high pressures where the y-4 approach is inapplicable. Only the EOSl GE models have been compared in the prediction of VLE. Case B. Asymmetric systems containing light gases: (a) methaneln-alkanes (propane to n-eicosane); (b) propaneln-pentane and propaneln-eicosane, both for VLE predictions. Case C. Asymmetric hydrocarbon systems: (a) npentaneln-alkane (n-hexadecane to n-octacosane); (b) n-alkane (n-hexadecane to n-dotriacontane)/n-hexane, for comparison of the models in the prediction of infinite dilution activity coefficients. Activity coefficients from the original and the modified UNIFAC are also compared.
Results and Discussion Case A. Symmetric Polar Systems. In Table 2 we present VLE prediction results for binary polar systems a t low pressures obtained by using the aforementioned UNIFAC versions. It appears from the limited data base in this table that none of these versions has a consistent advantage over the others for low-pressure VLE predictions. This is in agreement with the study of Hansen et al. (19921, who, using an extensive data base, demonstrated that the three UNIFAC versions considered here have only marginal differences with
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 683 Table 3. Results for Polar Systems at Low Pressures MHV2 system
rep
N
T(K)
acetone/chloroform n-Dentanelacetone methanoybenzene methanovwater ethanovwater 2-propanoVwater n-pentanelmethanol acetonelwater acetone/methanol pentanovwater
1 2 3 4 5 5 6 4 4 7
137 27 20 30 17 19 18 39 25 6
298-333 373-423 373-413 373-423 423 423 373-398 373-423 373-423 369-393
P range
(bar)
AP%
0.2-1.013 0.2-6.7 3.1-11.8 1-13.7 5.6-9.9 5.2-9.2 4.5-15 1.1-11 3.5-14 1.013
4.7 1.7 0.8 1.2 3.3 4.4 1.2 3.2 1.7 9.5
ws
PSRK
AY ( ~ 1 0 0 ) AP%
0.6 1.0 0.6 1.3 0.8 2.0 0.9 2.0 1.4 1.9
1.2 1.9 2.1 1.6 2.3 1.8 8.0 2.7 1.8 4.1
AY ( ~ 1 0 0 ) AP%
0.8 0.9 1.2 1.2 0.8 1.6 4.0 1.1 1.5 2.4
1.9 2.3 1.0 1.7 3.3 5.4 6.8 2.7 1.7 4.2
Y-4
AY ( ~ 1 0 0 ) AP%
0.9 1.4 0.8 1.2 1.0 2.1 3.3 1.7 1.4 3.0
LCVM
AY (x100) AP%
1.2 7.6 8.6 1.6 3.9 4.8 5.3 5.0 3.3 6.7
0.9 3.3 3.1 1.2 2.6 1.9 2.4 2.2 1.5 3.1
1.1 1.5 2.2 2.0 2.1 4.0 8.5 3.0 1.8 8.8
AY (~100)
0.8 1.0 1.2 1.2 1.4 3.0 3.9 1.5 1.4 4.4
1,Hala et al. (1968); 2, Campbell et al. (1986);3, Butcher and Medani (1968);4, Griswold and Wong (1952); 5 , Barr-David and Dodge (1979); 6, Wilsak et al. (1987); 7, Gmehling et al. (1981). (1
14.00
4.50
12.00
4.00
3.50 10.00
‘23.00
A i
0
,d’
O
n
I) 8.00 v
v
a
a 2.50
__ LCVM ___
6.00
2.00 4.00
4
5
; ’
2~000.0
0.2
0.6
0.4
0.8
1 .o
’~000.0 .jO
0.2
0.4
0.6
0.8
Figure 1. 1. Prediction of the P-x-y diagram for the system methanovwater at 423 K. Experimental data from Griswold and Wong (1952).
Xlr Yl Figure 2. Prediction of the P-x-y diagram for the system methanolhenzene at 373 K. Experimental data from Butcher and Medani (1968).
respect to their performance in VLE predictions for polar systems at low pressures and provided general guidelines for the use of the most successful UNIFAC version per system. In the following we use the original UNIFAC of Hansen et al. (1991) to predict low-pressure VLE through the y-4 approach. VLE predictions for binary polar systems at low pressures are presented in Table 3, and typical examples are shown graphically in Figures 1 and 2, and in Table 4 and Figure 3, for binary systems a t high pressures. The y-4 approach is not included in the latter case since it is inapplicable at such high pressures. The following comments summarize our observations: 1. All four EOS/UNIFAC models and the y-4 approach with the original UNIFAC (Hansen et al., 19911, for the low-pressure systems, give similar results (see Tables 3 and 4 and Figures 1and 2). As expected, the results obtained with the y-q5 approach generally become poorer at elevated pressures (see, for example, results for methanolhenzene and n-pentanelmethanol in Table 2), due t o uncertainties mainly in the estimation of the cross second virial coefficients. 2. Although all models perform similarly in the prediction of VLE, there may be differences from case to case. For example, as shown in Figure 3 with the results for the system 2-propanollwater at 523 K, the MHV2, PSRK, and WS models predict the critical point
at a lower composition than the experimentally observed value. The LCVM model on the other hand, while not giving the best average percent error in predicted bubble point pressures and vapor phase compositions, predicts the critical point closely. 3. Considering the degree of nonideality and the high pressures-especially in Table 4-involved, the performance of the four EOS/UNIFAC models can be considered very satisfactory. Case B. Asymmetric Systems Containing Light Gases and n-Alkanes. Results for the methanelnbutane system at 377.6 K and methaneln-eicosane at 373.3 K are shown in Figures 4 and 5, respectively, while the average absolute percent error in bubble point pressure prediction for methaneln-alkane systems is plotted versus the carbon number in the n-alkane in Figure 6. The model of Wong and Sandler is not applicable in this case, because there are no specifically estimated UNIFAC interaction parameters for the methane group in this model, as in the case of the M W 2 , PSRK, and LCVM models. The y-q5 approach is also inapplicable in such systems, since they contain supercritical methane. Bubble point pressure predictions for propanelnpentane at 344.3 K and propaneln-eicosane a t 333.4 K are presented in Figures 7 and 8, respectively. For these systems, zero interaction parameters for UNIFAC
x13
Yl
684 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 Table 4. Results for Polar Systems at High Pressures system ethanoliwater 2-propanollwater methanofienzene methanoliwater acetone/water acetone/methanol a
N 61 55 20 25 41 10
rep 1 1 2 3 3 3
T(K) 473-623 473-573 453-493 473-523 473-523 473
Prange (bar) AP% 2.1 17.9-185.5 18.5-123.5 4.9 4.5 16.3-57.6 17-83 4.8 1.6 16-67.6 2.1 29.5-39.9
ws
MHV2 PSRK Ay(x100) AP% Ay(x100) AP% 1.6 1.5 0.5 4.7 1.3 3.0 1.3 6.7 2.7 2.8 2.4 5.3 1.8 1.1 0.8 1.9 1.3 3.3 1.2 2.7 3.1 3.2 2.8 3.6
LCVM
Ay (x100) AP% 2.2 1.0 3.5 2.7 2.5 6.3 1.2 1.5 2.0 1.6 3.6 3.1
Ay ( ~ 1 0 0 ) 1.0 2.2 3.4 0.8 1.1 3.2
1, Barr-David and Dodge (1959); 2, Butcher and Medani (1968); 3, Griswold and Wong (1952).
70
j I
/
,, ,
40
,
I
I
I
~
I
I
I
)
,
,
I
,
'
)
,
,
t
~ I
I
1 1
I I I
I~ I \ I I I , ,
I
~
~
~
~
\
0.3
~
02
0.1
0.3
0.5
0.4
X1
Figure 5. Prediction of the bubble point pressures for the system methaneln-eicosane at 373.3 K. Experimental data from Huang et al. (1988). 100
;
0 00
00000
Exp
01
Pts
/
02
x1,
0 3
0 4
0.5
C6
Y1
Figure 4. Prediction of the P-x-y diagram for the system methaneln-butane at 377.6 K. Experimental data from Knapp et al. (1982).
have been used with the LCVM, PSRK, and WS models and the y-q5 approach, while for MHV2 the nonzero interaction parameters proposed by Dah1 et al. (1991) were used. For alkanes with carbon numbers greater than 18, where no experimental T,and Pc, and consequently w , values exist, those recommended by Magoulas and Tassios (1990) have been used. The following comments summarize our observations:
0
- -
5. ,
0
I
# ,
!
8
4
1
8
'
/
I
10
~
I
1 Il
I
I
I
,
20
/
J
1
1
~
1
l
~
30
I I , 8 I1
1
I
I
1
8
4C
Carbon N u m b e r Figure 6. Average absolute percent error in bubble point pressure prediction for methaneln-alkane systems as a function of carbon atoms in the n-alkane. The data base for these systems is the one used by Boukouvalas et al. (1994).
1. For systems containing components with no large size differences, all EOSAJNIFAC models perform similarly (see Figures 4 and 6-for low carbon numbers-and Figure 7). This is in agreement with the results presented above for symmetric polar systems (case A). 2. However, as the size difference between the components increases, the results for the MHV2 and
Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 686 1.1 PSRK
1 .o t
0.9 1
00000
0
Or
0.2
0.0
0.6
0.4
0.8
1
x1, y1
oonoc
i i N FAC
.o
Figure 7. Prediction of the P-x-y diagram for the system propaneln-pentane at 344.26 K. Experimental data from Knapp et al. (1982). 25
E x p , Pts
2
Figure 9. I&te dilution activity coefficient of n-pentane in large n-alkane (2' = 303.15-343.15 K and P = 1 atm) as a function of the number of carbon atoms in the n-alkane. Experimental data from Kniaz (1991).
Exp. Pts
- LCVM
00000 Exp
1
Pts
ws
Y ._ c 1.0 L? C W 0
5 0.8 I C
L
< 0.6 L
a, U
x
.-
: ,
0.4
e
U 0
c
; . 0.2
.--J u
L
0.2
0.4
0.6
0.9
1 .o
X1
,g 0.0 3
Corban a t o m s in n-alkanes
Figure 8. Prediction of the bubble point pressures for the system propaneln-eicosane at 333.4 K. Experimental data from Gregorowicz et al. (1992).
Figure 10. Infinite dilution activity coefficient of large n-alkane in n-hexane (2' = 250.8-288.3 K and P = 1 atm) as a function of the number of carbon atoms in the n-alkane. Experimental data from Kniaz (1991).
PSRK models become progressively poorer and fail at high asymmetries (see Figures 5 and 6). This is more pronounced with the MHV2 model, which gives error in bubble point pressure greater than 20% for the methaneln-alkane systems for carbon numbers greater than 7, where the ratio bn-heptandbmethane is equal to 4.8, while for the PSRK model this occurs at carbon number equal to 10 where the ratio of b's is 7.1. This also explains the better performance of the latter over the former in the case of n-propaneln-eicosane (Figure 81, where the b ratio is 7.9, even though no interaction parameters are used with the PSRK as compared to the MHV2 model. Increased errors with increased asymmetry are also observed with the WS model as suggested with the system propaneln-eicosane at 333.4 K (Figure 81, where the obtained results are much worse than those obtained for the propaneln-pentane mixture at 344.3 K (Figure 7). On the other hand, LCVM gives
satisfactory results even for highly asymmetric systems, such as methaneln-dotriacontane, methaneln-hexatriacontane, etc. (Figure 6). Case C. Infinite Dilution Activity Coefficients in Hydrocarbon Mixtures. Experimental and predicted infinite dilution activity coefficients are presented in Figure 9 for small n-alkanes in large ones, and in Figure 10, for large n-alkanes in small ones. Only the combinatorial part of the UNIFAC model is used here, since no interaction coefficients are involved. Results with the combinatorial expressions of UNIFAC, that of the original and that of the modified, are presented. Original UNIFAC underestimates substantially the infinite dilution activity coefficient values in agreement with the findings of Kniaz (1991) and of Kontogeorgis et al. (1994). The MHV2, PSRK, and WS models give very poor results. Actually, the PSRK model gives infinite dilu-
686 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995
tion activity coefficients greater than 1for small alkanes in large ones (Figure 91, and the WS model gives infinite dilution activity coefficients that increase with increasing asymmetry in the case for larger alkanes in small ones (Figure 10). Finally, the MHV2 model strongly underestimates infinite dilution activity coefficients in both cases. Best results are obtained with the modified UNIFAC and the LCVM models. The former is better with infinite dilution activity coefficients of large alkanes in small ones (Figure 101, and the latter with infinite dilution activity coefficients of small in large (Figure 9). A point worth mentioning is that no model reproduces the infinite dilution activity coefficientpredictions of the GE model it is combined with, e.g., MHV2 and modified UNIFAC. Furthermore, the predictions of WS, L C W , and PSRK vastly differ although they are all combined with the same GE model, the original UNIFAC. This paradoxical behavior, whereby the models do not reproduce the GE model they are associated with, has been theoretically explained elsewhere (Kalospiros et al., 1994; Coutsikos et al., 1994). No further discussion will be offered here, since it is outside the scope of the present work.
Conclusions Four recently developed EOS/UNIFAC models have been compared with respect to their performance in the prediction of low- and high-pressure VLE behavior of symmetric and asymmetric systems as well as infinite dilution activity coefficients in asymmetric n-alkane mixtures. No significant difference is observed in the performance of these models when applied to systems with components of similar size regarding the prediction of VLE behavior. When systems containing components with significant size difference are involved, however, such as mixtures of light gases (methane, propane) with large alkanes or asymmetric n-alkane mixtures, only LCVM car. generate satisfactory predictions in both VLE behavior and infinite dilution activity coefficients, while the other three fail. The conventional y-q5 approach with the original UNIFAC model performs very satisfactorily in the VLE predictions for polar systems a t low pressures.
Acknowledgment E.C.V. acknowledges the National Scholarship Foundation of Greece for their financial support.
Nomenclature Latin Symbols a: EOS attractive term (cohesion) parameter (m6 bar/ kmol2)
b: EOS covolume parameter (m3/kmol)
P:excess Gibbs free energy (Jkmol) Kij: second virial coefficient binary interaction parameter P : pressure (bar)
T: temperature (K) mole fraction
x:
Greek Symbols w:
acentric factor
Subscripts c: critical value
i: component z
Superscripts
cal: calculated value exp: experimental value Table Symbols
AP%: average absolute percent error in bubble pressure Ay: average absolute error in vapor phase composition NDP: total number of data points per system Acronyms
EOS: equation of state
VLE: vapor-liquid equilibrium ZRP: zero reference pressure
Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly and Completely Miscible Systems. AZChE J. 1975,21,116. Barr-David, F.; Dodge, B. F. Vapor-Liquid Equilibrium at High Pressures. The Systems Ethanol-Water and 2-PropanolWater. J. Chem. Eng. Data 1969,4,107. Boukouvalas, C.; Spiliotis, N.; Coutsikos, Ph.; Tzouvaras, N.; Tassios, D. Prediction of Vapor-Liquid Equilibrium with the LCVM Model a Linear Combination of the Vidal and Michelsen Mixing Rules coupled with the original UNIFAC and the t-mPR Equation of State. Fluid Phase Equilib. 1994,92,75. Butcher, K. L.; Medani, M. S. Thermodynamic Properties of Methanol-Benzene Mixtures a t Elevated Temperatures. J. Appl. Chem. 1968,18,100. Campbell, S.W.; Wilsak, R. A.; Thodos, G. Isothermal VaporLiquid Equilibrium Measurements for the n-Pentane Acetone System at 372.7, 397.7 and 422.6 K. J. Chem. Eng. Data 1988, 31, 424. Coutsikos, Ph.; Kalospiros, N. S.; Tassios, D. Capabilities and Limitations of Wong-Sandler Mixing Rules., in preparation. Dahl, S.; Michelsen, M. L. High-Pressure Vapor-Liquid Equilibrium with a UNIFAC-Based Equation of State. MChE J. 1990, 36, 1829. Dahl, S.; Fredenslund, Aa.; Rasmussen, P. The MHV2 Model: A UNIFAC-Based Equation of State Model for Prediction of Gas Solubility and Vapor-Liquid Equilibria at Low and High Pressures. Znd. Eng. Chem. Res. 1991,30, 1936. Fredenslund, Aa. Unifac and Related Group-Contribution Models for Phase Equilibria. Fluid Phase Equilib. 1989,52,135. Fredenslund, Aa.; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AZChE J. 1975,21,1086. Gmehling, J.;Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection.Aqueous-Organic Systems (Supplement 1);DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1981; Vol. I, Part la. Gregorowicz, J.; de Loos, T. W.; de Swaan Arons, J. The System Propane+Eicosane: P, T, x Measurements in the Temperature Range 288-358 K. J. Chem. Eng. Data 1992,37,356. Griswold, J.; Wong, S. Y. Phase-Equilibria of the AcetoneMethanol-Water System from 100 "C into the Critical Region. Chem. Eng. Prog., Symp. Ser. 1952,48,18. Hala, E.;Wichterle, I.; Polak, J.; Boublik, T. Vupour-Liquid Equilibrium Data at Normal Pressures; Pergamon Press: Oxford, U.K., 1968. Hansen, H. K.; Rasmussen, P.; Fredenslund, Aa.; Schiller, M.; Gmehling, J. Vapor-Liquid Equilibria by UNIFAC Group Contribution. 5 . Revision and Extension. Z n d . Eng. Chem. Res. 1991, 30,2352. Hansen, H. K.; Coto, B.; Kuhlmann, B. UNZFAC with Linearly Temperature-Dependent Group-ZnteractionParameters; Institut for Kemiteknik, DTH: Lyngby, Denmark, 1992; SEP 9212. Holderbaum, T.; Gmehling, J. PSRK. A Group Contribution Equation of State Based on UNIFAC. Fluid Phase Equilib. 1991, 70,251. Huang, S.; Lin, H.-M.; Chao, K-C. Solubility of Carbon Dioxide, Methane and Ethane in n-Eicosane. J. Chem. Eng. Data 1988, 33, 145. Huron, M.-J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibria of Strongly Nonideal Mixtures. Fluid Phase Equilib. 1979,3,255. Kalospiros, N. S.; Tzouvaras, N.; Coutsikos, Ph.; Tassios, D. Analysis of Zero-Reference-Pressure EOSI@ Models. MChE J. in press.
Ind. Eng. Chem. Res., Vol. 34,No.2, 1995 687 Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M. Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1982;Vol. VI. Kniaz, K. Influence of size and shape effects on the solubility of hydrocarbons: the role of the combinatorial entropy. Fluid Phase Equilib. 1991,68,35. Kontogeorgis, G. M.; Coutsikos, Ph.; Tassios, D.; Fredenslund, Aa. Improved models for the prediction of activity coefficients in nearly athermal mixtures. Part I. Empirical modifications of free-volume models. Fluid Phase Equilib. 1994,92, 35. Larsen, B. L.;Rasmussen, P.; Fredenslund, Aa. A Modified UNIFAC Group-Contribution Model for Prediction of Phase Equilibria and Heats of Mixing. Ind. Eng. Chem. Res. 1987,26,
2274.
Orbey, H.; Sandler, S. I.; Wong, D. S. H. Accurate equation of state predictions at high temperatures and pressures using the existing UNIFAC model. Fluid Phase Equilib. 1993,85,41. Tsonopoulos, C. An empirical correlation of second virial coefficients. AIChE J. 1974,20, 263. Tsonopoulos, C. Second virial coefficients of Polar Haloalkanes. AIChE J. 1976,21,827. Vidal, J. Mixing Rules and Excess Properties in Cubic Equations of State. Chem. Eng. Sci. 1978,33,787. Wilsak, R. A.; Campbell, S. W.; Thodos, G. Vapor-Liquid Equilibrium Measurements for the n-Pentane-Methanol system at 372.7,397.7and 422.6K. Fluid Phase Equilib. 1987,33,157. Wong, D. S. H.; Sandler, S. I. A Theoretically Correct Mixing Rule for Cubic Equations of State. AIChE J. 1992,38,671.
Magoulas, K.; Tassios, D. Thermophysical Properties of n-Alkanes from C1 to C20 and Their Prediction for Higher Ones. Fluid Phase Equilib. 1990,56,119. Michelsen, M. L.A Method for Incorporating Excess Gibbs Energy Models in Equations of State. Fluid Phase Equilib. 1990a,60,
Received for review J u n e 16, 1994 Accepted October 12, 1994 @
IE940377B
47. Michelsen, M. L. A modified Huron-Vidal Mixing Rule for Cubic Equations of State. Fluid Phase Equilib. 1990b,60,213.
@
Abstract published in Advance ACS Abstracts, December
15, 1994.
