4656
Ind. Eng. Chem. Res. 1996, 35, 4656-4666
GENERAL RESEARCH Prediction of Vapor-Liquid Equilibria of Associating Mixtures with UNIFAC Models That Include Association Yuan-Hao Fu, Hasan Orbey, and Stanley I. Sandler* Center of Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Two UNIFAC association models are developed using Wertheim’s theory of association rather than a chemical theory. In both models the activity coefficient is the sum of combinatorial, residual, and association contributions. The UNIFAC group-contribution model is used for the combinatorial and residual terms, and two different types of association models are considered. The UNIFAC-AG model uses functional-group-based association, while the UNIFAC-AM model considers association to occur between molecules. For associating mixtures containing acids, alcohols, or water, both activity coefficient models provide better predictions of binary vaporliquid equilibria than the original UNIFAC model. Of those models, the UNIFAC-AM model led to the best predictions. The association term was also added to the more recent, modified UNIFAC model and briefly tested with vapor-liquid equilibrium data for the acetic acid + heptane and acetic acid + butanol mixtures. It was found that, for vapor-liquid equilibria, the modified UNIFAC + association model is only slightly better than the UNIFAC-AM model. However, the modified UNIFAC + association model has four more adjustable parameters than the UNIFAC-AM model. Therefore, use of the UNIFAC-AG or UNIFAC-AM models is recommended. Introduction Hydrogen bonding is frequently modeled as a chemical reaction (see, for example, Kretschmer and Wiebe, 1954; Renon and Prausnitz, 1967). Brandani and Evangelista (1984) developed a UNIQUAC equation for associating mixtures using a chemical reaction model and applied it to vapor-liquid equilibrium. Also, based on chemical theory, Stathis and Tassios (1985) developed a UNIFAC association model for the enthalpy of mixing of mixtures containing alcohols. Recently, we developed a UNIQUAC association model (Fu et al., 1995) based on the statistical mechanical theory of association of Wertheim (1984a,b, 1986a,b). In this model, hydrogen bonding is considered to be the result of a strong attractive force rather than due to chemical reaction. Correlated phase behavior using this model for associating mixtures was found to be better than that with the original UNIQUAC model (Abrams and Prausnitz, 1975). Our purpose here is to build upon this UNIQUAC association model to develop a UNIFAC association model that can be used for the prediction of the phase behavior of hydrogen-bonding mixtures. In this UNIFAC association model, the excess Gibbs free energy is separated into combinatorial, residual, and association contributions. The combinatorial contribution is due to molecular size and shape differences, and the residual contribution is due to differences in the dispersion energies. These two contributions are as in the original UNIFAC equation (Fredenslund et al., 1975), though we also consider the use of a modified UNIFAC model (Weidlich and Gmehling, 1987) later. The association contribution is derived from Wertheim’s theory. * Author to whom correspondence should be addressed.
S0888-5885(95)00545-8 CCC: $12.00
In the UNIFAC model, compounds are treated as a combination of functional groups; however, in the usual form of association theory, association is considered to occur between molecules. For predictions using the group contribution concept, the contribution of association based on functional groups would be useful. Therefore, here we consider two UNIFAC association models. In one the association term is based on association between functional groups, and in the second the association term is taken to be specific to the molecules that hydrogen bond. After developing these models, we fit experimental data to obtain the UNIFAC interaction and association parameters. Finally, we compare the predictions of these two UNIFAC association models with both the original and modified UNIFAC models. UNIFAC Association Models We assume that association occurs in both the vapor and liquid phases; however, we describe these two phases differently. In the liquid phase, the UNIFAC association model is the sum of combinatorial, residual, and association terms assoc ln γi ) ln γcomb + ln γres i i + ln γi
(1)
The combinatorial and residual terms are the same as the original UNIFAC model and are given in the Appendix. We consider the two different assumptions for the association term as discussed below. Molecule-Based Association. First, we consider association to occur between molecules with the association term derived from Wertheim’s equation. Wertheim considered hydrogen bonding to be a strong physical attractive force and, using perturbation theory, derived an equation for the residual Helmholtz free energy due © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4657
to association in a pure compound. Chapman et al. (1990) extended Wertheim’s theory to mixtures and obtained the following expression for the residual Helmholtz free energy of association
[(
assoc
nTa
ln X ∑i xi ∑ A
) nT
RT
X
Ai
2
i
{∑[ ] ∑ ∑[( ) ( ) ] ∑[ )
(2)
2
)
∂ni
j
)
T,P
1
1
xj
+
1 + Mi 2
(
∂nTgE,assoc p
1
) ]
Ai
ln
γassoc i
-
Aj
X
Aj
2
∑k
NT
ν(i) k
ln XAk -
Ak
∂XAj
T,P
+ 2
ln XAk* -
-
∂Nk
XAk
Ak
XAk*
where nT is the total number of moles in the mixture. Ai is an association site in compound i, XAi is the fraction of association sites Ai that are not bonded, xi is the mole fraction of compound i in the liquid phase, and Mi is the total number of association sites in compound i. To derive the contribution to the activity coefficient due to association, we approximate the total excess Gibbs free energy of association to be equal to the difference between the total residual Helmholtz free energy of association in a mixture at ideal solution volume and the temperature T and the sum of the residual Helmholtz free energy of association of the pure compounds at temperature T and pressure P. The details of derivation are given in the appendix. The contribution to the activity coefficient due to association is
ln
i
)
[
∑ A
(
)
∂nTgE,assoc P
γassoc i
ln XAi -
∂ni XAi
]
1 +
2
2
)
[( ) ( ) ] ∑[ ]
T,P
+
∑j ∑ A xj
j
1
1
nT
-
2
XAj
ln XAi* -
∂XAj ∂ni
XAi*
1
+
2
Ai
-
T,P
2
(3)
where * indicates a property of pure compound i. The fraction XAi and its derivative with respect to mole number are also given in the appendix. We refer to the combination of the original UNIFAC model and this association term as the UNIFAC-AM model. Functional-Group-Based Association. If we consider a pure compound to be composed of functional groups, then the molar residual Helmholtz free energy of association of a pure compound is obtained from the following modification of Chapman’s equation:
aassoc i ) RT
∑j
ν(i) j
[( ∑ A
ln XAj* -
j
) ]
XAj*
1
+
2
2
Mj
(4)
where Aj is the association site in group j, XAj* is the fraction of association sites Aj that are not bonded, ν(i) j is the number of group j in the compound i, and Mj is the total number of association sites in group j. For a mixture, we can write the total residual Helmholtz free energy of association as
nTaassoc RT
) nT
∑i ∑j xi
ν(i) j
[∑(
ln XAj -
Aj
) ]
XAj 2
1 + Mj 2
(5)
Using the same analysis as in the derivation of the excess Gibbs free energy of association in the UNIFACAM model, the contribution to the activity coefficient due to association is where NT is the total number of functional groups in the mixture and Nk is the number of group k in the mixture. As before, the details of derivation of eq 6 are
2
]}
1 + 2
(6)
provided in the appendix. We refer to the combination or the original UNIFAC model, and this association term as the UNIFAC-AG model. Vapor-Phase Association We must also consider association in the vapor phase. Here for the vapor phase we use the same association model as in the UNIQUAC association model of Fu et al. (1995), and the details can be found there. Briefly, at low pressure, we assume that any departure from ideal gas behavior in the vapor phase is due to association, and a monomer-dimer association model is used to describe this effect. Values of the dimerization constant in the vapor phase are calculated from PVT data for the pure compounds. Gmehling et al. (1982) have reported values of the dimerization constants for acids. To find the dimerization constants for other associating compounds, we use the reported second virial coefficients for the pure apparent compounds (Dymond and Smith, 1980). For mixtures, the mole fraction of monomer of the associating component is calculated by solving the mass balance equations, and then the fugacity coefficient was computed. Results Before phase behavior using the model developed here can be computed, we have to determine the number of association sites and the type of each site-site association for all components in a mixture. We use the same assumptions for the types of association as in the UNIQUAC association model developed by Fu et al. (1995). UNIFAC Functional Group-Based Association Model. In this model, we use the same parameters as in the original UNIFAC model for interactions among nonassociating groups. However, the interaction parameters need to be recalculated for an associating group such as COOH since the parameters in the original model, regressed from experimental data without explicitly accounting for association, include the effect of association. We assume in this regression process that the association parameter in the liquid phase depends on temperature as follows:
ln R ) AR + BR/T
(7)
Therefore, four parameters (the two UNIFAC interaction parameters and the two association parameters in the equation above) must be determined. What we have done is to correlate isothermal data for acid + alkane binary mixtures to estimate the association parameters of the acid group and recalculated the UNIFAC parameters for the acid-alkane group interactions that then represent dispersive interactions without association. The results for the interaction and association parameters are shown in Tables 1 and 2. Table 3 contains the average errors in correlation using the UNIFAC-
4658 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 1. Interaction Parameters of the UNIFAC-AG Model group 1
group 2
a12
a21
CH2OH COOH H2O COOH COOH CH2OH CH2OH COOH CH2OH CH3CO
CH2 CH2 CH2 CH2OH H2O H2O ACH CCl4 CCl4 CHCl3
251.268 2517.860 553.547 -272.8 -203.977 330.234 233.832 797.761 142.192 -255.481
282.740 358.034 942.311 483.310 -10.987 -65.084 -79.277 159.870 344.975 463.613
Table 2. Association Parameters of the UNIFAC-AG Model group 1
group 2
AR
BR
CH2OH COOH H2O COOH COOH CH2OH CH3CO
CH2OH COOH H 2O CH2OH H2O H 2O CHCl3
-5.301 -5.815 -14.070 -4.656 -10.960 -7.397 -3.149
2989.7 4015.3 4499.0 3211.6 4289.1 3224.3 1049.1
Table 3. Average Errors in Correlation with the UNIFAC-AG and UNIFAC-AM Modelsa UNIFAC mixture acid + alkane alcohol + alkane alcohol + water acid + water acid + acid alcohol + alcohol ketone +chloroform total averageb
UNIFAC-AG UNIFAC-AM
error P (%)
error y
error P (%)
error y
4.225 2.199 3.122 4.408
0.0208 0.0129 0.0226 0.0234
2.931 1.047 1.854 3.801
0.0124 0.0066 0.0155 0.0226
error P (%)
2.689 1.097 1.970 1.798 0.588 0.432 1.450 0.0068 1.246 0.0058 0.424
error y 0.0117 0.0076 0.0104 0.0123 0.0030 0.0056 0.0038
4.221 0.0190 2.439 0.0130 1.798 0.0106
a The errors reported for the UNIFAC model are the results of prediction, not correlation. b The total average error of the UNIFAC-AM model does not include the acid + acid and alcohol + alcohol mixtures.
AG model and prediction using the original UNIFAC model. It should be noted that in the calculation for acids with the original UNIFAC model we use the monomer-dimer association model in the vapor phase but did not explicitly consider association in the liquid phase since the model parameters were determined in that manner. After obtaining the parameters of the UNIFAC-AG model, we also made some predictions for other systems. In Table 4 the predictions using the UNIFAC-AG model are compared with those of the UNIFAC model. Figure 1 is an example of the predictions for an acid + alkane mixture using the UNIFAC and UNIFAC-AG models. From Table 4 and Figure 1 we see that the UNIFAC-AG model results in slightly better predictions for acid + alkane mixtures than the original UNIFAC model. In the original UNIFAC model the volume and area parameters in addition to the group interaction parameters of alcohols were fit to experimental data to improve the predictions. Here we have defined the alcohol functional group to be CH2OH and calculated the volume and area parameters using Bondi’s method. [We also considered the use of OH for the alcohol functional group, as in the original UNIFAC model, but found that the CH2OH group led to better predictions.] The volume and area parameters for the CH2OH group are R ) 1.2044 and Q ) 1.124, respectively, and the association parameters for this group in the liquid phase
are based on correlating isothermal data for alcohol + alkane mixtures; the values are listed in Table 2. We also redetermined the interaction parameters (which now do not include the effects of association) for the alcohol-alkane group, and these are given in Table 1. Table 3 lists the average errors in the correlation of mixture data to obtain the parameters of the UNIFACAG model. We also list the results of the predictions of the UNIFAC model in Table 3. Predictions of the UNIFAC-AG model are given in Table 4 and compared with the original UNIFAC model. It should be noted that, in the original UNIFAC model, we considered the vapor phase to be an ideal gas, while the monomerdimer association model was used for the vapor phase with the UNIFAC-AG model. Figures 2 and 3 are examples of predictions for alcohol + alkane mixtures. In this case, the predictions from the UNIFAC-AG model are clearly better than those from the original UNIFAC model. Because of the insolubility of hydrocarbons in water at low pressures, the interaction parameters between the water and the alkane group in the original UNIFAC model were determined from liquid-liquid equilibrium data. We also used liquid-liquid equilibrium data for water + alkane mixtures here to determine the wateralkane group interaction parameters and the water selfassociation constant. This was done as follows. At equilibrium, we have II xI1γI1 ) xII 1 γ1
(8)
II xI2γI2 ) xII 2 γ2
(9)
and
where the subscripts 1 and 2 represent water and alkane and the superscripts I and II represent the water-rich and alkane-rich phases, respectively. The activity coefficients in eqs 8 and 9 were described by the UNIFAC-AG model, and the liquid-liquid equilibrium data used were from Sorensen and Arlt (1979). We simultaneously fit the two UNIFAC interaction parameters and the two coefficients in eq 7 to experimental data. The results for the parameters obtained in this way are given in Tables 1 and 2. We tested the UNIFAC-AG model for cross-associating mixtures containing acids, alcohols, or water. For acid + acid and alcohol + alcohol mixtures, there is no additional cross-association parameter in the groupbased activity coefficient model being considered here. Table 4 lists the errors in the predictions of pressure and vapor phase mole fractions from both the UNIFACAG and the original UNIFAC models. In these calculations, following Gmehling et al. (1982), the vapor-phase cross-dimerization constant for an acid + acid mixture is assumed to be twice the geometric mean of the selfassociation dimerization constants. For alcohol + alcohol mixtures, we assume an ideal vapor phase in the original UNIFAC model and use the geometric mean of the self-association constants for the vapor-phase crossassociation constant in the UNIFAC-AG model. For the acid + acid mixtures, the UNIFAC-AG model results in a very small improvement over the original UNIFAC model, while the opposite is true for alcohol + alcohol mixtures; however, the difference between the two activity coefficient models for these systems is small and both correctly predict almost ideal mixture behavior.
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4659 Table 4. Errors in Prediction of the Original UNIFAC, UNIFAC-AG, and UNIFAC-AM Models UNIFAC acetic acid + heptane acetic acid + cyclohexane heptane + pentanoic acid ethanol + butane ethanol + hexane ethanol + nonane ethanol + cyclohexane
propanol + hexane propanol + heptane propanol + 2,2,4-trimethylpentane propanol + undecane butanol + pentane butanol + hexane butanol + octane butanol + decane butanol + undecane pentanol + pentane pentanol + hexane acetic acid + propanoic acid ethanol + propanol ethanol + butanol propanol + butanol butanol + pentanol ethanol + pentanol propanol + pentanol ethanol + water acetic acid + water propanoic acid + water acetone + chloroform averageb
UNIFAC-AG
UNIFAC-AM
T (°C)
error P (%)
error y
error P (%)
error y
error P (%)
error y
data sourcea
20 30 25 45 50 75 25 50 70 25 30 35 50 35 50 60 55 75 60 30 60 100 110 100 80 30 25 50 30 40 40 50 70 90 40 40 75 40 40 50 20 50 60 15 45 55
1.595 1.835 5.434 6.081 2.267 4.077 4.753 2.792 2.048 2.323 1.098 1.714 1.432 1.559 3.959 1.693 1.979 1.592 4.574 10.46 2.181 2.419 3.363 2.188 13.62 13.51 5.938 8.105 2.075 2.190 0.126 0.847 1.251 1.010 0.537 0.341 4.242 0.583 2.794 3.984 3.265 5.141 6.158 1.122 0.906 0.473 4.022
0.0268 0.0218 0.0248 0.0250 0.0028 0.0062
2.671 2.220 5.849 5.279 1.431 1.674 0.920 2.085 0.685 0.866 0.619 0.765 0.688 0.821 2.577 2.390 1.247 0.738 0.850 6.808 1.928 1.216 1.141 4.012 4.824 10.17 4.122 6.289 1.821 1.918 0.501 1.239 2.016 1.988 0.683 0.400 4.622 0.666 2.692 3.886 2.032 4.678 4.999 0.471 1.364 0.418 2.813
0.0077 0.0055 0.0262 0.0215 0.0028 0.0061
1.416 1.430 4.924 4.413 1.898 3.739 1.224 1.863 1.399 0.546 1.023 0.828 0.888 0.918 2.199 2.340 1.665 0.806 0.699 6.285 1.605 1.198 1.250 3.987 4.869 9.511 3.637 5.576
0.0086 0.0070 0.0238 0.0193 0.0030 0.0060
0.0134 0.0052 0.0049 0.0137 0.0106
4.532 0.927 2.190 3.461 2.512 2.365 3.722 0.341 0.869 0.421 2.624
0.0074 0.0127 0.0127 0.0134 0.0235 0.0029 0.0041 0.0062 0.0091
1 1 2 2 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.0218 0.0047 0.0179 0.0135 0.0156 0.0147 0.0144 0.0111 0.0129 0.0146 0.0036 0.0112 0.0050 0.0127 0.0209 0.0070 0.0018 0.0144 0.0062 0.0095 0.0090 0.0051 0.0043
0.0136 0.0197 0.0239 0.0279 0.0338 0.0027 0.0040 0.0049 0.0153
0.0155 0.0030 0.0087 0.0117 0.0034 0.0044 0.0079 0.0127 0.0064 0.0063 0.0031 0.0067 0.0041 0.0133 0.0052 0.0051 0.0015 0.0118 0.0060 0.0092
0.0122 0.0047 0.0060 0.0127 0.0045 0.0068 0.0065 0.0129 0.0068 0.0069 0.0030 0.0060 0.0040
0.0103 0.0073 0.0056
0.0091 0.0188 0.0161 0.0224 0.0318 0.0043 0.0032 0.0045 0.0103
a Data source: (1) Lark, B. S.; et al. J. Chem. Eng. Data 1984, 29, 277. (2) Gmehling, J.; et al. DECHEMA Chemistry Data Series, Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt, Germany, 1977. (3) Holderbaum, T.; et al. Fluid Phase Equilib. 1991, 63, 219. b The average errors of the UNIFAC and UNIFAC-AG models do not include the acid + acid and alcohol + alcohol mixtures which are used for correlation in the UNIFAC-AM model.
For alcohol + water, acid + water, and acid + alcohol mixtures, we also need the cross-association and interaction parameters between the two associating functional groups. We fit the binary vapor-liquid phase equilibrium data to obtain the four parameters simultaneously. Tables 1 and 2 list the interaction and crossassociation parameters between each pair of associating functional groups considered. Table 3 lists the average errors in correlation of the UNIFAC-AG model for these cross-associating mixtures. We also show the results of predictions of the original UNIFAC model in Table 3. In calculations with the UNIFAC-AG model, we have used the geometric mean of the two self-association constants for the cross-association constant in the vapor phase for alcohol + water mixtures and assumed that the cross-association in the vapor phase is negligible for the acid + water and acid + alcohol mixtures since the differences in the self-associating constants are so large that acid + acid self-association is dominant. As before, in the UNIFAC model, we have assumed ideal vaporphase behavior except for acids for which we have used
the monomer-dimer model. No vapor-phase crossassociation was assumed to occur in the original UNIFAC model for these mixtures. For ketone + chloroform mixtures that do not selfassociate but can cross-associate, we determined the interaction and cross-association parameters between the ketone and chloroform groups by fitting isothermal vapor-liquid equilibrium data of ketone + chloroform mixtures. The parameters for ketone-chloroform group interactions are shown in Tables 1 and 2. Table 3 lists the average errors in correlation for the mixtures to obtain the parameters of the UNIFAC-AG model. Table 4 contains the errors in pressure and vaporphase mole fractions predicted by the UNIFAC and UNIFAC-AG models for several cross-associating mixtures. Figure 4 is an example of predictions of the UNIFAC and UNIFAC-AG models. From Table 4 and this figure, we find that for most mixtures the UNIFACAG model leads to better predictions than the original UNIFAC model, though the improvement is small. However, for all the mixtures considered here, the
4660 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Figure 1. Vapor-liquid equilibria of the heptane + pentanoic acid mixture at 75 °C. The points are experimental data, the solid line is the UNIFAC predictions, and the dashed line is the UNIFAC-AG predictions.
Figure 2. Vapor-liquid equilibria of the ethanol + nonane mixture at 70 °C. Legend as in Figure 1.
UNIFAC-AG model improves the accuracy of both the pressure predictions and the vapor-phase mole fractions by about 30% compared to the original UNIFAC model. UNIFAC Molecule-Based Association Model. Here we proceed in a similar fashion as above but recognize that hydrogen-bonding and other associations may depend not only on the associating groups but also on the neighboring functional groups. Thus, we consider the association to be specific to the (whole) molecules involved, not only to the hydrogen-bonding functional groups. That is, we consider association to occur between molecules, while the dispersive forces occur between groups which we describe by the UNIFAC group contribution model. To obtain a generalized model for self-associating mixtures, we assume that the molecular association parameter is related to the temperature and molecular weight of either the acids or the alcohols as follows:
ln R ) AR + BR/T + CRMw
(10)
Therefore, for acid + alkane and alcohol + alkane
Figure 3. Vapor-liquid equilibria of the ethanol + cyclohexane mixture at 25 °C. Legend as in Figure 1.
Figure 4. Vapor-liquid equilibria of the ethanol + water mixture at 50 °C. Legend as in Figure 1. Table 5. Interaction Parameters for the UNIFAC-AM Model group 1
group 2
a12
a21
COOH CH2OH H2O COOH COOH CH2OH COOH COOH CH2OH CH3CO
CH2 CH2 CH2 CH2OH H2O H2O CH2OH H2O H2O CHCl3
1109.396 487.237 553.547 299.892 446.674 312.531 507.572 655.043 281.348 -293.267
445.051 -4.924 942.311 -158.036 -133.570 -195.506 -385.855 256.494 81.213 452.894
mixtures, we have five parameters to fit: the two UNIFAC interaction parameters and the three coefficients in the equation above for the association parameter. For acids, we determined these parameters by regressing isothermal vapor-liquid equilibrium data for acid + alkane mixtures. The UNIFAC and association parameters obtained are listed in Tables 5 and 6. Table 3 lists the average errors in correlation for the mixtures in obtaining the parameters of the UNIFACAM model. For comparison we also show the predic-
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4661 Table 6. Association Parameters for the UNIFAC-AM Modela compound 1
compound 2
AR
BR
CR
acid alcohol water acid alcohol acid acid alcohol ketone
inert inert inert acid alcohol alcohol water water CHCl3
-0.0056 -6.881 -14.070 -1.651 -7.616 -10.364 -2.280 -8.345 -13.509
3617.9 3700.8 4499.0 3240.3 3924.3 4766.1 2832.9 3449.0 1500.0
-0.040 64 -0.004 41 -0.003 79 0.034 85 -0.041 90 0.014 22 0.088 51
a ln R ) A + B /T + C M . M R R R w w ) molecular weight of associating compound in self-association ) (Mw1 + Mw2)/2 in crossassociation.
Figure 6. Vapor-liquid equilibria of the propanol + undecane mixture at 60 °C. Legend as in Figure 5.
Figure 5. Vapor-liquid equilibria of the acetic acid + cyclohexane mixture at 25 °C. The points are experimental data, the solid line is the UNIFAC predictions, and the dashed line is the UNIFACAM predictions.
tions of the UNIFAC model in Table 3. In Table 4 we compare the errors in prediction of the UNIFAC-AM model with those of the original UNIFAC and UNIFACAG models. Figure 5 is one example of a prediction using the UNIFAC-AM model. We see from Table 4 that the molecule-based association model results in better predictions than other models. For alcohols, we used isothermal VLE data for the alcohol + alkane mixtures to obtain the parameters listed in Tables 5 and 6. Table 3 lists the average errors in correlation found in obtaining the parameters of the UNIFAC-AM model for alcohol + alkane mixtures. Table 4 gives the errors in prediction for alcohol + alkane mixtures from the UNIFAC-AM, UNIFAC-AG, and original UNIFAC models. In Figures 6 and 7, we present examples of predictions using the UNIFAC and UNIFAC-AM models. From Table 4 and these figures, we see that the predictions of the UNIFAC-AM model are in better agreement with the experimental data than those of the original UNIFAC model and are comparable to those of the UNIFAC-AG model. For cross-associating mixtures, we also assume that for each type of mixture (i.e., acids, alcohols, etc.) the cross-association parameter is related to temperature and molecular weight as
ln R ) AR +
BR + CRMw T
(11)
Figure 7. Vapor-liquid equilibria of the propanol + 2,2,4trimethylpentane mixture at 75 °C. Legend as in Figure 5.
where Mw is now taken to be the average molecular weight of the two associating compounds. For acid + acid and alcohol + alcohol mixtures, we only need to determine the cross-association parameters since the UNIFAC interaction parameters have already been obtained from the analysis of the self-associating mixtures. For acid + acid mixtures, the only available experimental data are for the acetic acid + propanoic acid mixture, so we cannot obtain the dependence of the cross-association parameter on molecular weight. For each mixture, the same vapor-phase association assumptions were used as had been made earlier with the UNIFAC-AG model. Table 3 gives the average errors in correlation for the UNIFAC-AM model to obtain the parameters. Predictions using this model are given in Table 4. The phase diagrams for these systems suggest nearly ideal mixtures, and all the models considered here accurately predict this phase behavior. For alcohol + water, acid + water, and alcohol + acid mixtures that can self-associate and cross-associate, we have to fit five parameters: the two interaction parameters between the associating groups and the three coefficients for the molecule-based cross-association
4662 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
model; however, for those mixtures containing an acid, both association models result in significant improvement in accuracy over the original UNIFAC model. Comparison with the Modified UNIFAC Model A modified UNIFAC model with temperature-dependent interaction parameters has been developed by Weidlich and Gmehling (1987). In this model, the activity coefficient is also a summation of combinatorial and residual terms. Here the combinatorial term is
ln γcomb )1i
(
)
φ′i φ′i z φi φi + ln - qi 1 - + ln xi xi 2 θi θi
(12)
where
xiri3/4 φ′i ) Figure 8. Vapor-liquid equilibria of the propanoic acid + water mixture at 60 °C. Legend as in Figure 5.
parameters. For ketone + chloroform mixtures that can only cross-associate, we determine the two interaction parameters between the ketone and chloroform groups, and the three cross-association parameters between the ketone and chloroform molecules from isothermal vaporliquid equilibrium data. The parameter values we obtained are reported in Tables 5 and 6; the average errors in correlation for all models are given in Table 3. Table 4 and Figure 8 provide an example of a prediction using the UNIFAC-AM model. From this table and figure we see that the predictions for crossassociating systems with the UNIFAC-AM model are better than those with the original UNIFAC model and slightly better than those with the UNIFAC-AG model. Based on the average error, the UNIFAC-AM model is the best of the three group-contribution activity coefficient models considered here. To test the models, we have made the predictions of vapor-liquid equilibria of ternary mixtures using the UNIFAC-AG and UNIFAC-AM models listed in Table 7. We also show the predicted results from the original UNIFAC model for comparison. From the table we see that the UNIFAC-AG and UNIFAC-AM models lead to predictions with error similar to that of the UNIFAC
∑j
(13)
xjrj3/4
The same form for the residual term as in the original UNIFAC model was used, but temperature-dependent parameters were introduced as follows:
Ψnm ) exp(-(anm + bnmT + cnmT2)/T)
(14)
Further, in this modified UNIFAC model, unlike the original UNIFAC model, the volume and area parameters were treated as adjustable parameters. There are six fitted interaction parameters in this model for each group-group interaction. Here we briefly test whether the modified UNIFAC model can be improved by adding the effect of association as we have done with the original UNIFAC model. We chose the isothermal vapor-liquid equilibria data for the acetic acid + heptane and acetic acid + butanol mixtures as test cases. First, we redetermined the six parameters of the acid-alkane group interactions in the modified UNIFAC model and used the same volume and area parameters reported by Gmehling et al. (1993). We then calculated the vapor-liquid equilibria of acetic acid + heptane mixtures. For the acetic acid + butanol mixture, besides the acid-alkane interaction parameters that have been estimated above, we also redetermined the parameters for the modified UNIFAC model
Table 7. Errors in Prediction for Ternary Mixtures Using the UNIFAC, UNIFAC-AG, and UNIFAC-AM Models UNIFAC mixture ethanol + hexane + benzene ethanol + benzene + CCl4 benzene + cyclohexane + ethanol propanol + hexane + heptane benzene + cyclohexane + propane CCl4 + benzene + propanol CCl4 + acetic acid + propanoic acid water + acetic acid + propanoic acid ethanol + water + butanol
T or P 55 °C
error in T or P
UNIFAC-AG error in T or P
error in y1 and y2
UNIFAC-AM error in T or P
error in y1 and y2
15.540 mmHg 0.0197, 0.0296 13.596 mmHg 0.0185, 0.0306 15.374 mmHg 0.0186, 0.0312
35 °C 2.330 mmHg 300 mmHg 0.386 °C 40 °C
error in y1 and y2
5.014 mmHg
2.010 mmHg 0.0087, 0.0122 0.462 °C
2.066 mmHg 0.0050, 0.0115 0.473 °C
3.321 mmHg
2.785 mmHg
0.0054, 0.0117
data sourcea 1 1 1 2
760 mmHg 0.654 °C
0.0117, 0.0186 0.794 °C
0.0143, 0.0175 0.814 °C
0.0147, 0.0175
1
760 mmHg 1.464 °C
0.0219, 0.0168 1.383 °C
0.0167, 0.0197 1.296 °C
0.0167, 0.0198
1
30 °C
0.0484, 0.0449 5.966 mmHg
0.0387, 0.0364 4.204 mmHg
0.0363, 0.0349
3
760 mmHg 4.939 °C
0.0351, 0.0189 0.624 °C
0.0344, 0.0190 0.249 °C
0.0282, 0.0177
1
50 °C
0.0263, 0.0150 3.232 mmHg
0.0117, 0.0133 3.920 mmHg
0.0146, 0.0179
1
9.275 mmHg
3.295 mmHg
a Data source: (a) Gmehling, J.; et al. DECHEMA Chemistry Data Series, Vapor-liquid Equilibrium Data Collection; DECHEMA: Frankfurt, Germany, 1977. (2) Zielkiewicz, J. J. Chem. Thermodyn. 1991, 23, 605. (3) Tamir, A.; et al. Fluid Phase Equilib. 1983, 10, 9.
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4663 Table 8. Average Errors in Prediction for Acetic Acid + Heptane and Butanol + Acetic Acid Mixtures Using the Modified UNIFAC, Modified UNIFAC + Association, and UNIFAC-AM Models modified UNIFAC
modified UNIFAC + association
UNIFAC-AM
error P (%) error y
error P (%)
error y
error P (%) error y
acetic acid + 1.843 0.0140 heptane butanol + 6.221 0.0246 acetic acid
1.716
0.0100
1.802 0.0111
2.714
0.0146
2.877 0.0211
of the alcohol-alkane group interaction from data for the butanol + heptane mixture, and the parameters of the acid-alcohol group interaction from VLE data for the acetic acid + butanol mixture. Table 8 gives the calculated average errors in pressure and vapor phase mole fractions; we also list the average errors of the UNIFAC-AM model for comparison. We see that the UNIFAC-AM model results in smaller errors than the modified UNIFAC model. We then added the molecule-based association term into the modified UNIFAC model and, as before, used the volume and area parameters from Gmehling et al. (1993). There are then eight adjustable parameters in this activity coefficient model (six UNIFAC parameters and only the two association coefficients in eq 10 since here we did not consider the dependence on molecular weight). We fit these eight parameters using data for the acetic acid + heptane mixtures. For acetic acid + butanol mixtures, since we have obtained the interaction parameters between the acid and the alkane groups and the association parameter of acetic acid from above for the modified UNIFAC + association model, we determined the interaction parameters between the alcohol and the alkane groups and the association parameter of butanol from butanol + heptane mixtures. Finally, we determined the parameters for the modified UNIFAC + association model for the acid-alcohol interaction and cross-association parameter between acetic acid and butanol from the acetic acid + butanol mixtures. The average errors in pressure and vapor phase mole fractions resulting from the modified UNIFAC + association model are given in Table 8. We find that the modified UNIFAC + association model results in better agreement with VLE data than the modified UNIFAC model. We also find that the modified UNIFAC + association model, with its additional four adjustable parameters, results in smaller errors than the UNIFAC-AM model; however, the improvement was quite small. Therefore, based on vapor-liquid equilibrium data there is little justification to use the modified UNIFAC + association model over the UNIFAC + association models. Conclusions We have developed two UNIFAC association models based on Wertheim’s statistical mechanical theory of association. Both models use the group-contribution concept for the dispersive forces. The UNIFAC-AG model uses the functional group concept for association, while the UNIFAC-AM model considers association to occur between molecules. For self-associating and crossassociating mixtures, both activity coefficient models provide better predictions of binary vapor-liquid equilibrium than the original UNIFAC model, and the UNIFAC-AM model produces the smallest error. The UNIFAC-AG model has the advantage of using the
group-contribution concept throughout; however, the UNIFAC-AM model is slightly more accurate. Finally, we compared the UNIFAC-AM model with the modified UNIFAC model (Weidlich and Gmehling, 1987) and a modified UNIFAC + association model using vapor-liquid equilibrium data for the acetic acid + heptane and acetic acid + butanol mixtures. We found that, for vapor-liquid equilibrium, the modified UNIFAC + association model is slightly better than the UNIFAC-AM model. However, the modified UNIFAC model has four additional adjustable parameters than the UNIFAC-AM model. Since the improvement is small and the UNIFAC-AM model has fewer parameters, we recommend its use. Acknowledgment This research was supported, in part, by Contract No. DOE-FG02-85ER13436 from the U.S. Department of Energy and Grant No. CTS-9123434 from the U.S. National Science Foundation, both to the University of Delaware. Notation AR ) coefficient for the liquid-phase association parameter aassoc ) molar residual Helmholtz free energy due to association (νip) ) molar residual Helmholtz free energy of comaassoc i pound i due to association at constant temperature and pressure aassoc M (νISE) ) molar residual Helmholtz free energy of mixture due to association at constant temperature and ideal solution volume amn ) UNIFAC interaction parameter BR ) coefficient for the liquid-phase association parameter CR ) coefficient for the liquid-phase association parameter ) molar excess Gibbs free energy of association at gE,assoc P constant temperature and pressure Mi ) number of association sites Ni ) number of moles of functional group i NT ) total number of moles of functional groups in mixture ni ) number of moles of compound i nT ) total number of moles of compounds in mixture P ) total pressure Qi ) area parameter of functional group i qi ) area parameter of compound i Ri ) volume parameter of functional group i ri ) volume parameter of compound i T ) temperature XAi ) mole fraction of association sites Ai that are not bonded Xi ) mole fraction of functional group i xi ) liquid-phase mole fraction of compound i yi ) vapor-phase mole fraction of compound i z ) coordination number Greek Letters RAiBj ) association parameter between association sites Ai and Bj γi ) activity coefficient of compound i φi ) volume fraction of compound i µ ) number of functional group j in compound i Θi ) area fraction of function group i θi ) area fraction of compound i Superscripts * ) property of a pure compound Ai ) association site A in compound i or functional group i assoc ) association comb ) combinatorial E ) excess
4664 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 res ) residual
Θm )
Subscripts M ) mixture P ) constant pressure
Appendix The proposed UNIFAC association model is the sum of combinatorial, residual, and association terms assoc ln γi ) ln γcomb + ln γres i i + ln γi
(A.1)
The combinatorial term is the same as the original UNIFAC model and given by
θi φi z + qi ln + li xi 2 φi xi
φi
) ln ln γcomb i
∑j xjlj
(A.2)
with
z li ) (ri - qi) - (ri - 1) 2
(A.3)
where z is the coordination number (taken to be 10), xi is the mole fraction of compound i in the liquid phase, ri and qi are the volume and surface area parameters of compound i, and ri and qi are the volume and area fractions of compound i, respectively, defined by
ri )
∑j ν(i)j Rj
(A.4)
qi )
∑j ν(i)j Qj
(A.5)
(A.6)
∑j xjrj xiqi
(A.7)
∑j xjqj
where ν(i) j is the number of group j in the compound i, and Ri and Qi are the volume and surface area parameters for group j. The residual term in the original UNIFAC model is given by
ln γres i with
[
(i) ν(i) k [ln Γk - ln Γk ]
ΘmΨkm
(A.8)
]
ΘmΨmk) - ∑ ∑ m m ∑n ΘnΨnm
ln Γk ) Qk 1 - ln(
Γ(i) k
∑k
∑n XnQn
where Xm is the mole fraction of group m in the mixture of groups. Molecule-Based Association. We first consider association to occur between molecules. From Chapman et al. (1990) the residual Helmholtz free energy of association based on molecules is
nTaassoc RT
[(
∑i ∑ A
) nT
xi
ln XAi -
) ]
XAi 2
i
1 ) Mi 2
(A.11)
where nT is the total number of moles in the mixture, Ai is an association site in compound i, XAi is the fraction of association sites Ai that are not bonded, and Mi is the total number of association sites in compound i. In Chapman’s equation, the fraction XAi is obtained from molecular-level information that is difficult to obtain for a real compound. However, based on the assumption in the UNIQUAC association model developed by Fu et al. (1995), we may approximate XAi by
1
XAk ) 1+
X ∑j ∑ B
(A.12)
Bj AkBj BjAk
θ
R
where RAkBj is the association parameter between association site A in compound k and site B in compound j, and θBjAk is the area fraction defined by
( ) ( )
θj exp θBjAk )
ujk T
∑n θn exp -
and
θi )
(A.10)
j
xiri φi )
XmQm
(A.9)
where ln is the residual activity coefficient of group k in pure compound i, Qm is the surface area fraction of group m, and Ψmn ) exp(-amn/T) where amn is the UNIFAC interaction parameter. The surface area fraction is defined to be
unk
(A.13)
T
Here the parameter ujk is an average of the UNIFAC interaction parameters between the functional groups in compound j and the associating functional group in compound k given by
ν(j) ∑ m amk Rm m a
ujk )
∑ m
(A.14)
ν(j) m Rm
where ka is the associating functional group in compound k. To derive the contribution to the activity coefficient due to association, we approximate the total excess Gibbs free energy of association by (see details in Fu et al., 1995)
(νi,p) ∑i xiaassoc i
≈ nTaassoc nTgE,assoc p M (νMISE) - nT
(A.15)
where aassoc M (νM) is the molar Helmholtz free energy of the mixture at ideal solution volume and temperature T, and aassoc (νi,p) is the Helmholtz free energy of asi sociation in pure compound i at temperature T and pressure P. The contribution to the activity coefficient due to association is then
) ∑[ ∑ ∑[( ) ( ) ] ∑[ γassoc i
ln
)
(
∂nTgE,assoc p ∂ni
1
1
xj
-
j
2
Aj
X
Aj
∂XAj
-
∂ni
2
ln XAi* -
2
]
2 (A.16)
where * indicates a property of pure compound i. The fraction XAi is calculated from eq A.12 and
nT
( ) ∂XAj ∂ni
{∑∑[
Aj 2
) -(X )
( ) ( ) ]}
AjBM BMAj
θ
R
m Bm
T,P
XBmRAjBmnT
∂θ
nT
+
T,P
BmAj
∂ni
(A.17)
T
Then using eq A.15 the contribution to the activity coefficient due to association is
γassoc i
]
1 2
+
(
)
∂nTgE,assoc p
)
∂ni
∑j ∑ A Xj
j
)
{[
ln XA ∑k ν(i)k ∑ A
1
1
2
Aj
X
∂XAj
Ak
T,P
A k*
∂θ
)
∂ni ∂θm ∂ni
T,P
umj T
- θBmAj
nT
n
θn exp -
n
∂θn ∂ni
exp -
∂ni
( )
nT
( )
∑n
(
m)i
(
m*i
∑n xnqn)2
(A.19)
assoc
RT
{∑∑[ m Bm
T,P
) nT
[(
∑i ∑j ∑ A xi
ν(i) j
ln XAj -
j
)
XAj 2
1 + 2
]
Mj
(A.20)
We use the same assumptions as those in the UNIQUAC association model of Fu et al. (1995); however, now the model is functional-group based. We may approximate the fraction XAi by
1
XAk ) 1+
∑j ∑ B
(A.21) B j A kB j
X R
( ) ( ) ]} ∂XBm
RAjBMΘmjNT
∂Nk
∂Θmj ∂Nk
+ T,P
(A.25)
T,P
∂Θmj
NT
We refer to the combination of eqs A.1, A.2, A.8, and A.16 as the UNIFAC-AM model. Functional-Group-Based Association. The total residual mixture Helmholtz free energy for association based on functional groups is
nTa
∂Nk
) -(XAj)2
( ) ( ) ( ) [∑ ( ) ( )] [∑ ( )]
NT
-xmqmqi
(A.24)
k
with
xnqn)2
)
∂Nk
∑k ν(i)k ∂N
XBmRAjBmNT
) T,P
∂
)
i
qi(
∂θm ∂ni
∂XAj
NT
and
∑n xnqn) - xiqi2
(A.23)
The fraction XAk is calculated from eq A.21, and the derivative in eq A.23 is equal to
T
(A.18)
∑k ∂n
)
T
unj
∂Nk ∂
∂
unj
T,P
2
where NT is the total number of functional groups in the mixture and Nk is the number of group k in the mixture. In deriving this equation we have used
T,P
exp -
]}
1
X
2
( ) ( ) ( ) [∑ ( ) ( )] [∑ ( )]
2
ln XAk* -
-
∂Nk
+
-
k
NT
-
XAk
k
[( ) ( ) ] ∑[ T,P
+
BmAj
nT
(A.22)
anj
T,P
with nT
T
∑n Θn exp -
ln
∂XBm ∂ni
Θmj )
1 +
2
amj
Θm exp -
+
XAi*
Ai
T,P
]
( ) ( )
1 +
Ai
T,P
nT
XAi
ln XAi -
)
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4665
Θjk
)
∂Nk ∂Θm ∂Nk
T,P
exp -
amj T
T,P
NT
- Θmj
n
Θn exp -
n
∂Θn ∂Nk
exp -
T,P
anj T
anj T
(A.26)
and
NT
( )
Qk(
∂Θm ∂Nk
∑n XnQn) - XkQk2
)
∑n XnQn)
T,P
(
-XmQmQk ) (
∑n XnQn)
m)k
2
(A.27)
m*k
2
We refer to the combination of eqs A.1, A.2, A.8, and A.23 as the UNIFAC-AG model.
j
Literature Cited where RAkBj is the association parameter between association site A in group k and association site B in group j, and Qjk is the local group area fraction defined by
Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116.
4666 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Brandani, V.; Evangelista, F. The UNIQUAC Associated Solution Theory: Vapor-Liquid Equilibria of Binary Systems Containing One Associating and One Inert or Active Component. Fluid Phase Equilib. 1984, 17, 281. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures: a Critical Compilation; Clarendon Press: Oxford, U.K., 1980. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086. Fu, Y.-H.; Sandler, S. I.; Orbey, H. A Modified UNIQUAC Model That Includes Hydrogen Bonding. Ind. Eng. Chem. Res. 1995, 34, 4351. Gmehling, J.; Onken, U.; Grenzheuser, P. DECHEMA Chemistry Data Series, Vapor-Liquid Equilibrium Data Collection. Parts 5; DECHEMA; Frankfurt, Germany, 1982. Gmehling, J.; Li, J.; Schiller, M. Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. Kretschmer, C. B.; Wiebe, R. Thermodynamics of AlcoholHydrocarbon Mixtures. J. Chem. Phys. 1954, 22, 1697. Renon, H.; Prausnitz, J. M. On the Thermodynamics of AlcoholHydrocarbon Solutions. Chem. Eng. Sci. 1967, 22, 299. Stathis, P. J.; Tassios, D. P. Prediction of Enthalpies of Mixing for Systems Containing Alcohols with a UNIFAC/Association Model. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 701.
Sorensen, J. M.; Arlt, W. DECHEMA Chemistry Data Series, Liquid-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt, Germany, 1979. Weidlich, U.; Gmehling, J. A modified UNIFAC Model. 1. Prediction of VLE, hE, and γ∞. Ind. Eng. Chem. Res. 1987, 26, 1372. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984a, 35, 19. Werthein, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamics Perturbation Theory and Integral Equation. J. Stat. Phys. 1984b, 35, 35. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1986a, 45, 459. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 1986b, 42, 477.
Received for review September 1, 1995 Revised manuscript received July 12, 1996 Accepted July 31, 1996X IE950545F
X Abstract published in Advance ACS Abstracts, September 1, 1996.
