Ye Qingguo, Zhong Chongli ^#
(Dept. of Chem. Eng., Qingdao Institute of Chemical Technology, Qingdao 266042; ^#Dept. of Chem. Eng., Beijing University of Chemical Technology, Beijing 100029, China)

Received  Jun. 11, 2000.

Abstract The combinatorial part of the original UNIFAC model is modified to improve its predictive accuracy for asymmetric systems by replacing the volume parameter with an "effective" one. A relation is derived between the "effective" and the original volume parameters. The improved UNIFAC model, the newly proposed combinatorial part coupled with the original residual part, woks well for asymmetric systems such as polymer solutions and those containing small and large molecule than the original UNIFAC model, which is also workable for systems containing only normal fluids. Comparing to those UNIFAC-FV models, the new model has the advantage of not requiring additional information, especially liquid molar volumes, in the calculation, which is very convenient for practical use. Parallel calculations are also carried out for another recently modified UNIFAC model as a comparison.
Keywords Prediction, Activity coefficient, UNIFAC model, Group contribution, Asymmetric system.

1 INTRODUCTION      
The UNIFAC model ^[1] is widely used for phase equilibrium calculations. For normal fluid mixtures, its predictive accuracy is usually good enough to meet the engineering demands, however, when highly asymmetric systems are dealt with, such as polymer solutions, it generally performs poorly. To improve it for such systems, Oishi and Prausnitz ^[2] added a free volume (FV) term to develop the so-called UNIFAC-FV model, which gives greatly improved predictions for solvent activities in polymer solutions. Some modified versions have been developed ^[3,4], however, similar to the UNIFAC-FV model of Oishi and Prausnitz^[2], they all require pure component liquid molar volumes in the calculation, which is troublesome for systems where such volumes are not accurately known, especially for those containing supercritical fluids. Consequently, an alternative method for extending the original UNIFAC model to highly asymmetric systems was proposed whose main idea is not to introduce liquid molar volumes in the modified ones. We, here, call this kind models as liquid volume free (LVF) modified UNIFAC models. Two such models have been developed, one is proposed by Voutsas et al. ^[5], the other one is proposed by us ^[6]. The previous investigation ^[5,6] show that such models work as well as or even better than those UNIFAC-FV type ones ^[2-4] for prediction of activity coefficients of highly asymmetric systems, however, the LVF ones are more convenient for engineering use.
    In this work our previously modified UNIFAC model ^[6] is further improved to make it able to deal with both asymmetric and symmetric systems. A large number of experimental data, including polymer/solvent, small/large molecule and normal/normal fluid systems, were used to test the new model and compared with that proposed by Voutsas et al.^[5] which has not been tested extensively before.

2 THE PROPOSED COMBINATORIAL ACTIVITY COEFFICIENT MODEL
2.1 The proposed combinatorial activity coefficient model   
The activity coefficient for component i form the original UNIFAC model ^[1] is as follows:
ln g_i = ln g_i^C + ln g_i^R                                  (1)
    Where g_i^C and g_i^R are the combinatorial and the residual parts, respectively.
    To improve it for polymer solutions, in our previous work^[6] the residual part was remained unchanged while the combinatorial part was modified as follows :
ln g_i^C = ln(f_i'/x_i)+1-(f_i'/x_i)-0.5zq_i [ln(f_i/q_i)+1-(f_i/q_i)]                             (2)
where f_i =   x_ir_i/Sx_jr_j                                  (3)
q_i =    x_iq_i/Sx_jq_j                                          (4)
and f_i'=    x_ir_i'/Sx_jr_j'                                     (5)
with r_i'=r_i, for small molecule                        (6)
r_i'=(0.6583+0.3709/n)r_i , for polymer         (7)
=0.6583r_i , for polymer                              (8)
    where x_i, r_i and q_i are mole fraction, volume and surface area parameters for component i, respectively. n is the number of monomers constituting the polymer. r_i' is the proposed "effective" volume parameter. Eq.(2) is firstly proposed by Kikic et al.^[7], which reduces to the expression of the original UNIFAC model^[1] if f_i' is set to be identical to f_i.
    Obviously, the key point of our modified combinatorial activity coefficient model is an “effective” volume parameter, r_i', is used in the calculation of f_i', and a relation between the "effective" volume parameter r_i' and the real r_i for polymer was proposed, which stemmed from the theoretical approximate relation between the excluded volume of an n-mer chain molecule and that of the monomer ^[6]. It is evident that our modified UNIFAC model does not need liquid molar volumes in the calculation for polymer/solvent systems, which requires just the same information as that by the original UNIFAC model.
    Though an explicit free volume term is not included in our modified UNIFAC model, the use of an "effective" volume parameter for polymer does a good contribution to the free volume effects. Our previous work shows that the new modified UNIFAC model works better than the UNIFAC-FV model ^[2] for prediction of solvent activities in polymer solutions. As only polymer/solvent systems were dealt in our previous work, the approximate equation r_i'=0.6583r_i was adopted since n is large for a polymer. In this work, we want to extend the model to both asymmetric and symmetric systems containing small, large as well as polydisperse molecules. As a result, eq.(7) is remained. Since the coefficient in eq.(7) is not unit when n is set to one, which is caused by the use of an approximate relation between the excluded volume for an n-mer chain molecule and that for the monomer, eq.(7) was adjusted a little as follows:
r_i'=(0.6583+0.3417/n)r_i , for polymer or large molecule (9)
    Furthermore, n is redefined as follows so that eq.(9)can be used for systems containing both large and small molecules:
n = V_vdW,large/V_vdW,small = r_large/r_small             (10)
where V_vdW is the van der Waals volume. Obviously, eq.(10) means that the "monomer" constituting the large molecule has the same vdW volume as the smaller one for a binary system. Eqs.(1)-(6),(9) and (10) constitute the modified UNIFAC model proposed in this work which is denoted as the UNIFAC-r model for convenience.
    Obviously, eq.(9) reduces to our previous expression, that is eq.(8), for polymer/solvent systems, and for systems with equal molecular sizes, the UNIFAC-r model reduces exactly to the original UNIFAC model. Therefore, it is expected that the proposed UNIFAC-r model can be used for both asymmetric and symmetric systems.
2.2 The R-UNIFAC model^[5]    
Another liquid molar volume free combinatorial activity coefficient model applicable to asymmetric systems is the so-called R-UNIFAC model proposed by Voutsas et al.^[5], where eq.(2) is also adopted with a different expression for f_i' as follows:
f_i' =   x_ir_i^R/Sx_jr_j^R                                                                                   (11)
and R = 0.9[1-(V_vdW,small/V_vdW,large )]=0.9[1- (r_small/r_large)]                    (12)
    From their investigation the R-UNIFAC model works as well as or even better than those free volume ones ^[5] for athermal asymmetric mixtures. We will test and compare systematically the two liquid molar volume free modified UNIFAC models in this work. As only the combinatorial activity coefficient model is given in the work of Voutsas et al. ^[5], the residual part of the original UNIFAC model ^[1] is used, the same as done in our model, as the residual one for the R-UNIFAC model to constitute a complete model applicable to both athermal and thermal mixtures.

3 RESULTS AND DISCUSSION
3.1 Prediction of solvent activities in polymer solutions      
Vapor-liquid equilibrium of polymer solutions are important in the processing of many polymeric materials. As the original UNIFAC model ^[1] does not work well for these systems, many efforts, as mentioned above, have been made to improve it. Our modified UNIFAC model, the UNIFAC-r model, is proposed mainly for this purpose with the advantage of not requiring liquid molar volumes in the calculation.
    In our precious work ^[6], a total of 51 data sets for polymer solutions were adopted to test the proposed model, 36 additional data sets were added in this work. As a result, a large data bank including 87 data sets is available to test the new model and compare with the original UNIFAC as well as the R-UNIFAC models. The predicted results, that is, the average absolute deviation (AAD) of solvent activities, were listed in Table 1. Obviously, the original UNIFAC model gives large AADs, usually about 20%, for most systems, while the UNIFAC-r model shows greatly improved predictions with the AADs within 10% for most systems. The R-UNIFAC model shows improved predictions for many systems, however, it performs fairly poor for some ones. Comparing the two modified UNIFAC models, it is evident that the UNIFAC-r model works better that the R-UNIFAC model for polymer solutions.

Table 1 Predicted results of solvent activities for polymer solutions

*PVA: poly (vinyl acetate); PS: polystyrene; PIB: polyisobutylene; PB: polybutadiene; PEO: poly (ethylene oxide);
PPO: poly(propylene oxide); POD:polyoctadecene; PDD: polydodecene; PD: polydecene; PH polyheptene;
PMA: poly (methyl acrylate); PEA: poly (ethyl acrylate); PBA; poly (n-butyl acrylate); PMMA: poly (methyl methacrylate);
PEMA: poly (ethyl methacrylate); PIP:polyisoprene; PVME: poly(vinyl methyl ether); PDMS: poly(dimethyl siloxane);
PP: polypropylene; PVC: poly (vinyl chloride)
**

3.2 Prediction of activity coefficients for systems containing a small and a large molecules   
In the regression of the original UNIFAC group interaction parameters only the systems containing normal fluids were included. For those systems containing both small and large molecules, due to the high asymmetry, and, of course, also partly due to the exclusion in the regression of the interaction parameters, the original UNIFAC model gives relatively large errors. In this work, we will test the capability of the proposed UNIFAC-r model for this kind of systems.
    A total of 20 systems were collected as shown in Table 2, where the "asymmetry", that is, the ratio of the van der Waals volumes of the constituents, were also listed. Though the activity coefficients for the smaller component are available, those for the larger component are usually scarce. The predicted results reported in Table 2 show that the proposed UNIFAC-r model gives better predictions than the original UNIFAC model for activity coefficients of both components, which are comparable to that from the R-UNIFAC model.

Table 2 Predicted results of activity coefficients for systems containing a small and a large molecule

*n-C₁₁: n-undecane; n-C₁₂: n-dodecane; n-C₁₆: n-hexadecane; n-C₁₈: n-octadecane; n-C₂₀: n-eicosane
**

3.3 Prediction of activity coefficients for systems containing normal fluids  
The above calculations show that the proposed UNIFAC-r model gives improved predictions for highly asymmetric systems where the original UNIFAC model does not work well. It is necessary to further test if it is also workable for those systems containing only normal fluids where the original UNIFAC model usually works very well. 22 systems were collected which were listed in Table 3 according to an increase in the system asymmetry. From the results obtained, it is clear that the UNIFAC-r model gives similar or even better predictive accuracy than that of the original UNIFAC model for systems with an asymmetry smaller than 2.0, which is also the case for the R-UNIFAC models. However, when the asymmetry is larger than 2.0, both the UNIFAC-r and the R-UNIFAC models, as shown in Table 3, give worse results than of the original UNIFAC model, which is caused by the overlarge correction of the combinatorial part of the original one. As a result, it is recommended that the UNIFAC-r model should be used carefully for normal fluid systems with an asymmetry larger than 2.0, which is also the case for the R-UNIFAC model.

4 CONCLUSIONS   
The improved UNIFAC model, the UNIFAC-r model, proposed in this work is applicable to both symmetric and asymmetric systems. Its main advantage is that it does not require liquid molar volumes in the calculation for highly asymmetric systems. The proposed UNIFAC-r model was tested against various kind of systems including polymer/solvent, small/large molecule as well as normal/normal fluid systems, and was compared with the original UNIFAC and the R-UNIFAC models. The results for polymer solutions show that the proposed UNIFAC-r model gives much better predictions than that of the original UNIFAC model. The R-UNIFAC model, however, gives good predictions for most systems but performs poorly for some ones. As to systems containing small and large molecules, the UNIFAC-r model shows comparable accuracy to the R-UNIFAC model, and both of them work better than the original one. Furthermore, it is clear that the UNIFAC-r model, as well as the R-UNIFAC model, gives similar predictions to that of the original UNIFAC model for normal fluid systems when the system asymmetry is smaller than 2.0. However, they should be avoided using for those normal fluid systems with a asymmetry larger than 2.0.
    Comparing to those UNIFAC-FV type models, the proposed UNIFAC-r model does not require additional information excepting those required by the original UNIFAC model in the calculation. Particularly, it does not need liquid molar volumes, so this UNIFAC-r model is very convenient for engineering use especially for those processes containing supercritical fluids.

NOMENCLATURE   
a                activity
n                ratio of can der waals volumes defined by eq.(10)
NP             number of data points
q                 surface area parameter
r                  volume parameter
R                parameter defined by eq.(12)
r'                 effective volume parameter
V_vdW          van der Waals volume
x                 mole fraction
z                coordination number, 10
g                activity coefficient
q                surface area fraction
f', f           volume fraction

Subscripts
i                  component i

Superscripts
C              combinatorial part
cal.           calculated value
exp.          experimental value
R              residual part

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