Ind. Eng. Chem. Res. 1998, 37, 1173-1180
1173
RESEARCH NOTES Prediction of Vapor-Liquid Equilibria at Low and High Pressures from UNIFAC Activity Coefficients at Infinite Dilution Dan Geana˜ * and Viorel Feroiu† Department of Applied Physical Chemistry, University “Politechnica” Bucharest, Spl. Independentei 313, Bucharest, Romania
A modified procedure of using the Huron-Vidal mixing rule based on UNIFAC activity coefficients at infinite dilution (HVID) at low pressures is shown to be useful for making vaporliquid equilibrium (VLE) predictions for polar and nonpolar systems at low and high pressures. Four cases were considered: (a) symmetric polar systems; (b) systems containing alkanols and hydrocarbons; (c) systems containing acetone and hydrocarbons; (d) hydrocarbon mixtures. Prediction of both low- and high-pressure VLE for binary systems is good. The HVID model, using UNIFAC ‘93 activity coefficients at infinite dilution, performs satisfactorily in the VLE predictions of ternary systems also. Introduction Equation of state (EOS) mixing rules, based on local composition concepts for excess Gibbs energy, were introduced by Huron and Vidal (1979). The original Huron-Vidal mixing rule at infinite pressure was not largely used because the model parameters (in GE at infinite pressure) must be adjusted and not related to available parameters at low pressure. The past few years have seen a rapid growth in the number of ideas for the direct incorporation of existing model parameters of excess Gibbs energy models in equations of state, for example, Heidemann and Kokal (1990), Dahl and Michelsen (1990, MHV2), Dahl et al. (1991, MHV2-UNIFAC), Holderbaum and Gmehling (1991, PSRK), Wong and Sandler (1992, WS), Orbey et al. (1993, WS-UNIFAC), Boukouvalas et al. (1994, LCVM), Voutsas et al. (1995, LCVM-UNIFAC), Crisciu and Geana˜ (1995, COMB). These so-called GE-EOS mixing rules use available activity coefficient model parameters from low-pressure data, without change, for predicting phase equilibria at high pressures and temperatures. Michelsen and Heidemann (1996) analyzed the reproduction of the GE base model at correlating temperature and the nature of the extrapolation to higher temperatures by the above mixing rules. The mixing rule models MHV2, PSRK, WS, and LCVM have been used in connection with group contribution methods such as UNIFAC, as these allow EOS to become predictive tools. Recently, Feroiu and Geana˜ (1996), following the approach of Soave et al. (1994), suggested the coupling of the Huron-Vidal mixing rule with infinite-dilution activity coefficients (called the HVID model). The procedure is based on the reduced UNIQUAC model, suitable for infinite-pressure conditions, coupled to SRK * Author to whom correspondence is addressed. E-mail:
[email protected]. † E-mail:
[email protected].
equation of state. Infinite-pressure activity coefficients at infinite dilution were obtained from low-pressure activity coefficients at infinite dilution at several temperatures, using GE model parameters from DECHEMA tables. The parameters of the reduced UNIQUAC model at infinite pressure were calculated at each temperature. A temperature-dependent function of the parameters was determined and extrapolated at higher temperatures. In this work, following the previous paper (Feroiu and Geana˜, 1996), we propose a procedure of using UNIFAC ‘93 (Gmehling et al. 1993) activity coefficients at infinite dilution and low pressure in the Huron-Vidal mixing rule. It is shown that this method can be useful for making vapor-liquid equilibrium (VLE) predictions for polar and nonpolar systems at low and high pressures. More specifically, four cases were considered: (a) symmetric polar systems at both low and high temperatures and pressures; (b) systems containing alkanols and hydrocarbons; (c) systems containing acetone and hydrocarbons; (d) hydrocarbon mixtures. In case a, six binary systems were studied, and we found that the errors in the predicted pressure and vapor phase composition with the HVID-UNIFAC ‘93 mixing rule are, on the average, less than those obtained when using the MHV2 model coupled to UNIFAC ‘87 (Larsen et al., 1987). In case b, a modification of the original values of the reduced UNIQUAC component surface areas (q f q′) for alkanols was needed. Good results were obtained for systems containing acetone and hydrocarbons, in case c. In case d, the predictions for the systems containing hydrocarbons at high pressures were based only on UNIFAC ‘93 pure-component group parameters (Rk and Qk) in the estimation of the infinite-dilution activity coefficients at P ) 0. The results of VLE predictions in ternary systems acetonemethanol-water, methanol-benzene-cyclohexane, and ethane-butane-pentane were reasonably good, for both low and high pressures.
S0888-5885(97)00472-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/28/1998
1174 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 Table 1. Constants of the Temperature Function (Eq 4), Obtained from UNIFAC ’93 Activity Coefficients at Infinite Dilution at Low Temperatures system
u012
u112
u021
u121
2-propanol-water methanol-water acetone-water ethanol-water methanol-benzene acetone-methanol
308.2 -38.5 324.3 213.6 200.7 146.3
-26794 19502 31286 -21405 -49223 29859
996.2 936.1 896.2 999.5 202.7 -40.2
-195745 -227607 -194112 -216537 154645 0
HVID Mixing Rule Model Based on UNIFAC In this study, we use the cubic equation of state of Soave, Redlich, and Kwong, SRK (Soave, 1972):
P)
a(Tr) RT V - b V(V + b)
(1)
For the function a(Tr) we used the form proposed by Mathias and Copeman (1983). The critical constants of pure substances were taken from Reid et al. (1987), and the values of constants C1, C2, C3 in a(Tr) function, from Dahl et al. (1991). For mixtures, the Huron-Vidal mixing rules were used:
b) a
)
bRT
(
∑i Xibi
ai
∑i Xi b RT -
(2)
)
ln γi(Pf∞) ln 2
i
(3)
where γi(Pf∞) is the activity coefficient of the component in the mixture at infinite pressure. As in the previous work (Feroiu and Geana˜, 1996), a reduced UNIQUAC model (to its residual part only), suitable for infinite-pressure conditions is used. The interaction parameters of the reduced UNIQUAC model are considered temperature-dependent:
1 uj ) u0ij + u1ij T
(4)
Neglecting the combinatorial part, only component surface areas, qi, are used for the calculation of activity coefficients. For a binary system, the interaction parameters are related to the infinite dilution activity coefficients at ∞ (Pf∞), calculated from its value infinite pressure γi(j) ∞ at zero pressure γi(j)(Pf0). The corresponding equations (11-13) are given in the paper of Feroiu and Geana˜ (1996), where the volumes (Vi, Vj) are the solutions of eq 1 at P f 0. At higher temperatures we used the extrapolation procedure proposed by Heidemann and Kokal (1990):
ξi )
() bi Vi
P)0
)1+β
( ) ( ) ai biRT
2
+δ
ai biRT
3
(5)
with β ) -0.041 404 612 711 and δ ) 0.004 861 102 317 for SRK EOS. Equation 5 is applied whenever
( )
ai ai < biRT biRT
lim
) 6.7027
(6)
Figure 1. Ethanol-water binary system. The temperature dependence of reduced UNIQUAC (HVID) parameters obtained from UNIFAC ‘93 activity coefficients at infinite dilution. Comparison with direct best-fitted (optimum) parameters to experimental VLE data.
The procedure for using activity coefficient at infinite dilution at low temperature/low pressure, from UNIFAC ‘93 parameter tables, in prediction of high-pressure VLE, is that involving steps 1-5, given by Feroiu and Geana˜ (1996). Two modifications are made: in step 1, ∞ (Pf0) is the activity coefficient at infinite dilution γi(j) calculated from UNIFAC ‘93 parameter tables (Gmehling et al., 1993), and step 4 is not involved if the calculation is at temperatures in the range of applicability of UNIFAC parameters. Compared to the MHV2-UNIFAC method of Dahl and Michelsen (1990), the HVID-UNIFAC model uses only infinite-dilution activity coefficients. In comparison with the procedure of Orbey et al. (1993), based on the WS-UNIFAC mixing rule, our model does not need the third binary interaction parameter (kij). Moreover, our mixing rule is more simple than MHV2 and WS, in the VLE calculations (regarding the mixing rule equations, the equations for fugacity coefficients and the GE model which is only “half” of the classical UNIQUAC). Results and Discussion (a) Symmetric Polar Systems. The ability of the HVID mixing rules coupled with UNIFAC ‘93 (Gmehling et al., 1993), for predicting vapor-liquid equilibrium, over a large range of conditions, was tested on six binary mixtures for which VLE data are available over large temperature and pressure ranges. These systems are listed in Table 1 and have been used in the literature in the study of the MHV2 (Dahl and Michelsen 1990) and WS (Wong and Sandler, 1992) mixing rules, as well as in our work on the HVID model (Feroiu and Geana˜, 1996). This fact allows also a comparison of all these mixing rules.
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1175 Table 2. VLE Results for the HVID-UNIFAC ‘93 Model and the MHV2 Model Coupled to UNIFAC ‘87 (Dahl and Michelsen, 1990) HVID-UNIFAC ‘93
MHV2-UNIFAC ‘87
system
T [K]
P [bar]
No. Dpt.
∆P/P × 100
∆Y × 100
∆P/P × 100
∆Y × 100
data ref
2-propanol-water methanol-water acetone-water ethanol-water methanol-benzene acetone-methanol
423-573 373-523 373-523 423-573 393-493 373-473
5.2-123.5 1.0-70.6 1.1-67.6 5.6-128.9 2.9-56.8 3.5-39.9
76 103 75 74 72 60
2.7 2.7 2.1 1.4 2.9 1.3
2.8 1.8 1.0 0.5 1.3 1.7
7.6 2.6 3.3 3.3 1.8
2.7 1.5 1.9 2.1 2.0
a b b a c b
2.2
1.5
3.7
2.0
average a
b
c
Barr-David and Dodge, 1959. Griswold and Wong, 1952. Butcher and Medani, 1968.
Figure 2. VLE prediction for the ethanol-water binary system by the HVID-UNIFAC ‘93 model.
The choice of the UNIFAC ‘93 version (Gmehling et al., 1993) is motivated by the fact that a large database was used to fit temperature-dependent group interaction parameters simultaneously to VLE, LLE, HE, and γ∞ data. Therefore, more reliable values of infinite-dilution activity coefficients are predicted. The group interaction parameters are temperature-dependent, with the recommended range of temperature being 20-125 °C (Gmehling et al., 1993). First we studied the capability of the HVID model, based on UNIFAC ‘93 activity coefficients at infinite dilution, to represent the low-temperature data sets available in the literature (i.e., DECHEMA tables; Gmehling and Onken, 1977). The errors in pressure and vapor composition of the direct calculation with the UNIFAC ‘93 model were reasonable. In our previous work (Feroiu and Geana˜, 1996), the results suggested that temperature-dependent parameters are needed in the HVID mixing rule to produce better predictions. We tried to calculate the parameters of the reduced UNIQUAC model, also called HVID parameters, directly from UNIFAC ‘93 activity coefficients at infinite dilution, for each temperature. Figure 1 shows, for the ethanol-water system, the tem-
Figure 3. VLE prediction for the ethanol-hexane binary system at T ) 323 K by the HVID-UNIFAC ‘93 model: - - -, qethanol ) 1.97; s, q′ethanol ) 1.2.
perature dependence of the HVID parameters obtained from UNIFAC ‘93 (dashed curves), in comparison to the values of the direct best-fitted (optimum) parameters at each high temperature. Also shown are the lines of extrapolated values of the HVID parameters obtained from UNIFAC ‘93 in the low-temperature range. As can be observed, at high temperatures, the HVID parameters obtained directly from UNIFAC ‘93 deviate from their optimum values. Contrarily, the linear extrapolation of HVID parameters obtained from UNIFAC ‘93 at low temperatures follows reasonably the temperature dependence of optimum parameters. Therefore, as in the previous work (Feroiu and Geana˜, 1996), linear temperature dependencies of the parameters were adopted as given in eq 4. The values of the constants in the temperature function (eq 4), obtained from UNIFAC ‘93 activity coefficients at infinite dilution at low temperatures are given in Table 1 for the six studied systems. These functions were extrapolated to predict vapor-liquid equilibrium at higher temperatures and pressures using the HVID mixing rules with the SRK equation of state coupled to the reduced UNIQUAC model.
1176 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 Table 3. High-Pressure VLE Predictions for the Ternary System Acetone-Methanol-Water: Experimental Data of Griswold and Wong (1952) HVID-UNIFAC ’93
MHV2-UNIFAC ‘87
HVID-DECHEMA
MHV2-DECHEMA
T [K]
P [bar]
No. Dpt.
∆P/P × 100
∆Y × 100
∆P/P × 100
∆Y × 100
∆P/P × 100
∆Y × 100
∆P/P × 100
∆Y × 100
423 573
1.2-3.9 47.5-82
50 55
3.2 2.3
2.3 0.8
7.6 8.5
3.0 2.1
4.7 3.1
2.1 1.1
2.8 7.0
2.3 1.6
Table 4. Prediction of VLE by HVID Mixing Rules with Reduced UNIQUAC Model and SRK EOS, from UNIFAC ‘93 Activity Coefficients at Infinite Dilution: Experimental Data of Maczynski et al. (1982) system
T [K]
P [bar]
No. Dpt.
∆P/P × 100
∆Y × 100
methanol-pentane methanol-hexane methanol-cyclohexane ethanol-hexane ethanol-toluene 1-propanol-hexane 1-butanol-hexane 1-pentanol-hexane 1-hexanol-hexane
372.7 318-343 323-328 293-333 313-333 318-348 313-333 303-323 333-363
3.5-8.5 0.40-2.12 0.50-1.00 0.18-1.08 0.08-0.51 0.09-1.40 0.03-0.78 0.04-0.54 0.01-1.9
11 89 18 81 43 45 55 43 69
1.5 1.8 4.0 2.3 1.0 2.6 3.4 3.6 4.7
1.0 1.0 2.6 1.7 0.5 1.2 0.4
The results for VLE predictions at high pressure are presented in Table 2 for the six investigated systems. The absolute average deviations in bubble pressure and vapor phase composition (mole fraction) are given for the HVID-UNIFAC ‘93 model and compared with the corresponding values reported in the literature for the MHV2 model coupled to UNIFAC ‘87 (Dahl and Michelsen, 1990). The general observation from Table 2 is that the results we obtain using the HVID model with parameters based on UNIFAC ‘93 activity coefficients at infinite dilution are generally good. One example is shown in the diagrams of Figure 2 for the system ethanol-water. As observed in Table 2, the HVID mixing rule is superior in its pressure and vapor phase composition predictions to that of the MHV2 model. The predictions of the MHV2 model are increasingly inaccurate as temperature increases, suggesting that temperature-dependent parameters are also needed in this model. Table 3 shows equilibrium calculation errors for the ternary mixture acetone-methanol-water, based on UNIFAC ‘93 activity coefficients at infinite dilution. Binary parameters given in Table 1 were used in VLE prediction of the ternary system, as well as the previously determined parameters based on DECHEMA tables. It is noted that both sets of parameters give good predictions in the ternary mixture, with the HVIDUNIFAC ‘93 parameters leading to better results. As compared with the MHV2-UNIFAC ‘87 mixing rule (Dahl and Michelsen, 1990), our model is superior in its pressure and vapor phase composition predictions of the ternary mixture. (b) Systems Containing Alkanols and Hydrocarbons. It is well-known that such mixtures are not satisfactorily correlated even with the usual GE models. It is expected that the prediction of VLE in these systems will be a severe test for any kind of mixing rule model. Figure 3 shows an example for hexane-ethanol at T ) 323 K, also presented by Feroiu and Geanaˇ (1989), and by Soave et al. (1994) in their Figure 3. As can be seen, the predictions of bubble and dew pressures (dashed curves) are wrong; the bubble-point curve shows the maxima and minima that are associated with liquid-liquid instability. It was found that the value of UNIQUAC component surface areas of ethanol strongly affects the calculation of VLE. Therefore, the
Figure 4. VLE prediction for the methanol-pentane binary system by the HVID-UNIFAC ‘93 model.
original value of UNIQUAC component surface areas, qethanol ) 1.97, must be changed to obtain better results in VLE calculations. After some trials, a value q′ethanol ) 1.2 is chosen, which leads to a good agreement of predicted VLE with experimental data (solid curves in Figure 3). Similar behavior is found with other alkanol-hydrocarbon systems (Table 4). For methanol, a value of q′methanol ) 1.15 was estimated as appropriate (compared with the original value of 1.43). For all the other calculated alkanols (propanol to hexanol), q′alkanol ) 1.2 was used. It can be mentioned that Prausnitz et al. (1980) have modified the original formulation of UNIQUAC by using a pure-component surface areas parameter q′ for alcohols, in the residual term. The surface of interaction q′ is smaller than the geometric external surface q, indicating that “for alcohols, intermolecular attraction is determined primarily by the OH group”.
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1177 Table 5. VLE Predictions for the Ternary System Methanol-Benzene-Cyclohexane: Experimental Data of Kogan et al. (1966) T [K]
P [bar]
328.2 0.3-1.2
No. Dpt. ∆P/P × 100 ∆Y × 100 28
2.3
obs q′methanol ) 1.15
2.1
Table 6. Prediction of VLE by HVID Mixing Rules with the Reduced UNIQUAC Model and SRK EOS, from UNIFAC ‘93 Activity Coefficients at Infinite Dilution: Experimental Data of Maczynski et al. (1979) system
T [K]
P [bar]
acetone-hexane acetone-heptane acetone-decane acetone-benzene acetone-toluene acetone-cyclohexane
293-328 323-338 321-338 313-323 308-328 298-328
0.16-0.92 0.18-1.40 0.23-1.34 0.18-0.79 0.15-0.90 0.30-1.07
No. ∆P/P ∆Y Dpt. × 100 × 100 38 17 32 35 44 45
1.2 2.0 4.8 2.0 0.7 3.1
0.7 0.8 0.1 0.5 0.3 0.5
One example is displayed in Figure 4 for a methanolpentane system at T ) 372.7 K. Table 5 shows equilibrium calculation errors for the ternary mixture methanol-benzene-cyclohexane. Binary parameters based on UNIFAC ‘93 activity coefficients at infinite dilution, leading to results reported in Table 4 at temperature 328.2 K, were used in VLE prediction of the ternary system. For the symmetric polar systems presented in section a, we used the geometric external surface qalcohol, because calculations performed with q′ values lead to similar results in VLE predictions. Surely, for the ternary system predictions, a value of q′methanol ) 1.15 was used, due to the presence of the pair methanolcyclohexane which needs, as discussed above, such values of the reduced UNIQUAC component surface areas. (c) Systems Containing Acetone and Hydrocarbons. Soave et al. (1994) found that for the systems containing ketones, it is not possible to reproduce experimental data, with their procedure, unless original Qk values are multiplied by a factor greater than 1. A value of 2.5 for the multiplier has been chosen for the >CO group by Soave et al. (1994). We made VLE predictions with our HVID procedure in six binary acetone-hydrocarbon systems. It is noteworthy that a change of the original value of UNIQUAC component surface areas, qacetone ) 2.34, was not necessary, and the original value of UNIFAC ‘93, Qk ) 1.67 for the CH3CO group, was not modified. The results for VLE predictions are presented in Table 6 for the investigated systems. The absolute average deviations in bubble pressure and vapor phase composition (mole fraction) are given for the HVID-UNIFAC ‘93 model. (d) Hydrocarbon Systems. The study of systems involving gases at normal temperatures is limited by
the available groups in UNIFAC ‘93. To overcome this limitation, Dahl et al. (1991), Holderbaum and Gmehling (1991), and Voutsas et al. (1995) introduced new groups for gases and corresponding interaction parameters, in a previous UNIFAC version. For example, Dahl et al. (1991) added 13 gases (as hydrocarbon groups: CH4, C2H2, C2H4, C2H6, C3H6, C3H8, and C4H10) to the Larsen et al. (1987) UNIFAC. Here we must note that such added group parameters may be used only with the corresponding EOS mixing rule model. In this work we used the group parameters as given in the original UNIFAC ‘93: for alkanes the group “CH2” with four subgroups “CH3”, “CH2”, “CH”, “C”, and the group “ACCH2” with three subgroups “ACCH3”, “ACCH2”, and “ACCH”. These available groups allow the study of high-pressure binary hydrocarbon mixtures given in Table 7. For the calculations of these hydrocarbon systems, the extrapolation formula (eq 5) was applied at temperatures near and above the critical point of a pure component. It must be mentioned that in alkane-alkane systems only the combinatorial part of the UNIFAC model is used, since no interaction coefficients are involved. As a consequence, the infinite-dilution activity coefficients at P ) 0, calculated from UNIFAC ‘93, are not temperature-dependent. However, using these values in our procedure, the infinite-dilution activity coefficients at P ) ∞ and consequently the parameters of reduced UNIQUAC model will be temperature-dependent. Table 7 shows the results of VLE predictions for the studied systems. The results are good, taking into account that only UNIFAC pure-component group parameters (Rk and Qk) are involved in the estimation of the infinite-dilution activity coefficients at P ) 0. One example of VLE prediction is displayed in Figure 5 for the ethane-butane system. An inconsistency may be pointed out for alkanealkane systems: according to UNIFAC there is no residual term, yet our procedure will lead to u12 and u21 parameter values in the reduced (residual part) of the UNIQUAC model. This inconsistency may be accepted, considering the reduced UNIQUAC equation only as a tool for generation of activity coefficient values on the whole scale of mole fraction. Table 8 shows equilibrium calculation errors for the ternary mixture ethane-butane-pentane. Binary parameters based on UNIFAC ‘93 activity coefficients at infinite dilution, at temperature 338 K, were used in VLE prediction of the ternary system. A limitation of the HVID-UNIFAC ‘93 model is determined by unavailable group contribution parameters for gases as CH4, CO2, H2, etc. However, it is probably of interest to see the capability of the mixing rule model based on the reduced UNIQUAC equation
Table 7. Prediction of VLE at High Pressures by HVID Mixing Rules with the Reduced UNIQUAC Model and SRK EOS, from UNIFAC ‘93 Activity Coefficients at Infinite Dilution system
T [K]
P [bar]
No. Dpt.
∆P/P × 100
DY × 100
data ref
ethane-propane ethane-butane propane-butane propane-pentane propane-cyclohexane propane-decane propane-propylbenzene butane-decane
273-343 303-363 303-393 336-373 313-393 377.6 313-393 344-411
4.70-51.0 4.40-53.3 3.50-42.0 3.30-43.8 2.50-40.6 3.50-46.8 3.50-43.9 3.50-27.5
52 43 37 37 18 6 17 13
3.1 5.5 1.3 2.5 4.3 3.9 7.1 3.2
1.3 2.4 0.8 0.5 1.8 1.5 0.9 0.4
a b c d e f e g
a Matschke and Thodos, 1962. b Lhotak and Wichterle, 1981. c Beranek and Wichterle, 1981. et al., 1985. f Reamer and Sage, 1966. g Reamer and Sage, 1964.
d
Vejorsta and Wichterle, 1974. e Laugier
1178 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998
Figure 5. VLE prediction for the ethane-butane binary system by the HVID-UNIFAC ‘93 model.
Figure 6. VLE correlation for the methane-CO2 binary system by the HVID model.
Table 8. VLE Predictions for the Ternary System Ethane-Butane-Pentane: Experimental Data of Kogan et al. (1966) T [K]
P [bar]
No. Dpt.
∆P/P × 100
∆Y × 100
338.75
34-60
21
4.4
0.9
in correlating binary systems containing such gases. Figures 6 and 7 show two examples of phase diagrams for methane-CO2 and hydrogen-methane binary systems, calculated by fitting the parameters of the reduced UNIQUAC model to experimental VLE data. As can be seen, the correlation of the data is very good, and many similar good correlations were obtained for other gas-containing systems. Based on the values of the correlated parameters, it will be possible to add new groups for gases in UNIFAC ‘93 tables. A relatively large VLE database on high-pressure gas-containing binary systems is needed. The compromise solution, to correlate the binary parameters directly from experimental data for systems with gases, seems to be reasonable at this stage. Conclusions The procedure previously proposed (Feroiu and Geana˜, 1996) is extended to predict vapor-liquid equilibria at high pressure using the HVID mixing rule model and the infinite-dilution activity coefficients from UNIFAC ‘93. The procedure is based on the reduced UNIQUAC model, suitable for infinite-pressure conditions, coupled to SRK equation of state. Infinite-pressure activity coefficients at infinite dilution were obtained from low-pressure activity coefficients at infinite dilution at several temperatures, using UNIFAC ‘93 tables. The parameters of the reduced UNIQUAC model at infinite pressure were calculated at each temperature. A temperature-de-
Figure 7. VLE correlation for the hydrogen-methane binary system by the HVID model.
pendent function of the parameters was determined and extrapolated at higher temperatures. The method was applied to predict the high-pressure VLE for symmetric polar binary systems and one ternary mixture, for which data are available over large temperature and pressure ranges. VLE predictions were made for alkanol-hydrocarbon and acetone-
Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1179
hydrocarbon systems. In the first case, a modification of the original values of the reduced UNIQUAC component surface areas (q f q′) for alkanols was needed. Good results were obtained for systems containing hydrocarbons at high pressures, based only on UNIFAC ‘93 pure-component group parameters (Rk and Qk) in the estimation of the infinite-dilution activity coefficients at P ) 0. For all cases, the results of VLE ternary system predictions were reasonably good. A more extensive study of the model predictions in multicomponent high-pressure VLE would evidently be desirable. The infinite-dilution activity coefficients from UNIFAC ‘93, at low pressures, are converted in high-pressure formal parameters, with a minor calculation effort. In compensation, our mixing rule is more simple than the others, in the VLE calculations (regarding the mixing rules, the equations of fugacity coefficients and the GE model which is only “half” of classical UNIQUAC). The results obtained using the HVID-UNIFAC ‘93 model were generally good. Consequently, low-pressure activity coefficients at infinite dilution obtained from UNIFAC ‘93 parameter tables can be used as the basis for VLE predictions at low and high pressures and temperatures. Acknowledgment The authors are grateful to Ministry of Education and Ministry of Research and Technology in Bucharest, for grants supporting this work. Notations a, b ) equation of state parameters G ) Gibbs free energy P ) pressure R ) gas constant T ) temperature V ) molar volume u ) reduced UNIQUAC parameter, K Greek Letters γ ) activity coefficient ξ, β, δ ) notations in eq 15 Subscripts i, j ) components c ) critical r ) reduced Superscripts E ) excess ∞ ) infinite dilution Abbreviations COMB ) combination of mixing rules with reference state at P ) 0 and P ) ∞ EOS ) equation of state HVID ) Huron-Vidal mixing rule coupled with infinitedilution activity coefficients LCVM ) linear combination of Vidal and Michelsen mixing rules MHV2 ) modified Huron-Vidal mixing rule of second order No. Dpt. ) number of data points PSRK ) predictive Soave-Redlich-Kwong SRK ) Soave-Redlich-Kwong equation of state UNIQUAC ) universal quasi-chemical UNIFAC ) UNIQUAC functional group activity coefficients
VLE ) vapor-liquid equilibrium WS ) Wong-Sandler mixing rule
Literature Cited Barr-David, F.; Dodge, B. F. Vapor-liquid equilibrium at high pressures. The system ethanol-water and 2-propanol-water. J. Chem. Eng. Data 1959, 4, 107. Beranek, P.; Wichterle, I. Vapor-liquid equilibria in the propanen-butane system at high pressures. Fluid Phase Equilib. 1981, 6, 279. Boukouvalas, C.; Spiliotis, N.; Coutsikos, P.; 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 at elevated temperatures. J. Appl. Chem. 1968, 18, 100. Crisciu, A.; Geana˜, D. A New Mixing Rule for Cubic Equations of State. Proceedings of the Conference on Chemistry and Chemical Engineering, Bucharest, Oct 20-21, 1995; Univ. “Politechnica”: Bucharest, Romania, 1995; Vol. 3, p 44. Dahl, S.; Michelsen, M. L. High-pressure vapor-liquid equilibrium with a UNIFAC based equation of state. AIChE J. 1990, 36, 1829. Dahl, S.; Fredenslund, A.; 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 Pressure. Ind. Eng. Chem. Res. 1991, 30, 1936. Feroiu, V.; Geana˜, D. Prediction of Vapor-Liquid Equilibria at Low Pressure by a General Equation of State. Rev. Roum. Chim. 1989, 34, 1329. Feroiu, V.; Geana˜, D. Prediction of vapor-liquid equilibria at high pressures using activity coefficients at infinite dilution. Fluid Phase Equilib. 1996, 120, 1. Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Compilation: DECHEMA Data Series DECHEMA: Frankfurt, Germany, 1977. Gmehling, J.; Li, J.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. 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. Heidemann, R. A.; Kokal, S. L. Combined excess free energy models and equations of state. Fluid Phase Equilib. 1990, 56, 17. Holderbaum, T.; Gmehling, J. PRSK: a group contribution equation of state based on UNIFAC. Fluid Phase Equilib. 1991, 70, 251. Huron, M.-J.; Vidal, J. New mixing rules in simple equation of state for representing vapor-liquid equilibria of strongly nonideal mixtures. Fluid Phase Equilib. 1979, 3, 255. Kogan, V. B.; Fridman, V. M.; Kafarov, V. V. Vapor-liquid equilibria (Russ.); Nauka: Moskva, Russia, 1966; Vol. 2. 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. Laugier, S.; Valtz, A.; Chareton, A.; Richnon, D.; Renon, H. Vaporliquid equilibria measurements on the systems ethane-npropylcyclohexane, propane-cyclohexane, propane-n-propylcyclohexane, propane-n-propylbenzene and propane-mesitylene. GPA RR-88, 1985. Lhotak, V.; Wichterle, I. Vapor-liquid equilibria in the ethanen-butane system at high pressures. Fluid Phase Equilib. 1981, 6, 229. Maczynski, A.; Maczynska, Z.; Treszczanowicz, T.; Dunajska, K. Verified vapor-liquid equilibrium data. Binary systems of hydrocarbons and oxygen compounds without alcohols and acids; Polish Academy of Sciences: Warszawa, Poland, 1979; Vol 4. Maczynski, A.; Bilinski, A.; Oracz, P.; Treszczanowicz, T. Verified vapor-liquid equilibrium data. Binary systems of hydrocarbons and alcohols; Polish Academy of Sciences: Warszawa, Poland, 1982; Vol. 6. Mathias, P. M.; Copeman, T. W. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Evaluation of the
1180 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91. Matschke, D. E.; Thodos, G. Vapor-liquid equilibria for the ethane-propane system. J. Chem. Eng. Data 1962, 7, 232. Michelsen, M. L.; Heidemann, R. A. Some Properties of Equation of State Mixing Rules Derived from Excess Gibbs Energy Expressions. Ind. Eng. Chem. Res. 1996, 35 (5), 278. 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. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A.; Hsieh, R.; O’Connel, J. P. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1980. Reamer, H. H.; Sage, B. H. Phase equilibria in hydrocarbon systems: Phase behavior in the butane-n-decane system. J. Chem. Eng. Data 1964, 9, 24. Reamer, H. H.; Sage, B. H. Phase equilibria in hydrocarbon systems: Volumetric and phase behavior of the propane-ndecane system. J. Chem. Eng. Data 1966, 11, 17. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.
Soave, G. Equilibrium Constant from a modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. Soave, G.; Bertucco, A.; Vecchiato, L. Equation-of-State Group Contributions from Infinite-Dilution Activity Coefficients. Ind. Eng. Chem. Res. 1994, 33, 975. Vejorsta, J.; Wichterle, I. The propane-pentane system at high pressures. Collect. Czech., Chem.Commun. 1974, 39, 1246. Voutsas, E. C.; Spiliotis, N.; Kalospiros, N. S.; Tassios, D. Prediction of Vapor-Liquid Equilibria at Low and High Pressures Using UNIFAC-Based Models. Ind. Eng. Chem. Res. 1995, 34, 681. Wong, D.; Sandler, S. Theoretically Correct Mixing Rule for Cubic Equations of State. AIChE J. 1992, 38, 671.
Received for review July 7, 1997 Revised manuscript received November 18, 1997 Accepted November 26, 1997 IE970472V
