Ind. Eng. Chem. Res. 1993, 32, 362-372
362
Vaaelenak, J. A.; Croeemann, I. E.; Westerberg, A. W. Optimal Retrofit D e s i of Multiproduct Batch Plants. Ind. Eng. Chem. Res. 1987,26,718-726. Viswanathan, J.; Crossmann, I. E. A Combined Penalty Function and Outer Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990,14,769-782. Yeh, N. C.; Reklaitis, G. V. Synthesis and Sizing of Batch/Semi-
continuous Processes. Presented at the AIChE Annual Meeting, Chicago, IL, 1985; paper 35a. Received for review May 28, 1992 Revised manuscript received October 19, 1992 Accepted November 12, 1992
GENERALRESEARCH Simple Activity Coefficient Model for the Prediction of Solvent Activities in Polymer Solutions Georgios M. Kontogeorgis and Aage Fredenslund* Znstitut for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
Dimitrios P . Tassios Department of Chemical Engineering, National Technical University of Athens, Zographos, 15773, Greece
A new simple activity coefficient model for the prediction of solvent activities in polymer solutions is presented. The model consists of two terms, a recently proposed "combined" combinatorialfree-volume term by Elbro et al. and the residual term of UNIFAC. The linear temperature-dependent parameter table for UNIFAC, recently developed by Hansen e t al., is used. The ability of the new model to predict solvent activity coefficients a t infinite dilution for many different polymer-solvent systems is shown. Considering the significant errors often found in the experimental measurements of these coefficients and the scatter of the experimental data, the obtained results are very good: they compare favorably with the UNIFAC-FV model by Oishi and Prausnitz and they are better than the two rather complicated and recently developed equations of state proposed for polymer solutions by High and Danner and Chen et al.
Introduction Since Flory (1970)developed his well-known equation of state for polymer solutions, much work has been carried out in order to establish a model for accurate predictions-and not simply correlations-of solvent activities in polymer solutions. It has been demonstrated that UNIFAC in its original form (Fredenslund et al., 1977)cannot be used for this purpose, since it generally leads to underestimation of solvent activities. This is due to the Staverman-Guggenheim combinatorial term used in W A C , which does not account for the significant free-volume differences between solvents and polymers that occur in most polymemolvent solutions. Free-volume differences explain the frequent occurrence of partial miscibility at high temperatures for polymer-solvent solutions and the existence of a lower critical solution temperature. The modified UNIFAC model of Larsen et al. (1987)cannot be used either, since it always leads to a large overestimation of solvent activities. Thia is a general deficiency of the models that have a combinatorial term of the empirical "exponential" form: di di In 7;= In - + 1- Xi
xi
where diis a modified segment fraction of component i:
Both activity coefficient models and equations of state have been used for the thermodynamic modeling of polymer solutions. Two somewhat different kinds of activity coefficient models have been proposed: (1)the Flory-Huggins (FH) model, i.e., the FH(x) approach (Flory, 1953); (2) the UNIFAC-FV models introduced by Oishi and Prausnitz (1978)and Iwai and Ami (1989). A similar but simpler and theoretically more sound free-volume expression has been recently proposed by Elbro et al. (19901,and it will be the basis of the model presented here. 1. The Flory-Huggins Model. From the FloryHuggins theory the following expression for the solvent activity in a binary polymer solution can be derived: dV0l
In al = In dyol+ 1 - - + x(d~"1)~ 1
X1
(3)
where x is the FH parameter. Although this parameter was initially introduced to account for the energetic interactions between the polymer and the solvent alone, it was recognized early that acceptable resulta are obtained only if we assume that it can be regarded as a free-energy term with an enthalpic and an entropic part (denoted by the subscripts h and s, respectively):
x = Xh + x s
(4)
The Flory-Huggins approach can qualitatively describe a number of phenoma occurring in polymer solutions, but 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 363 it has some serious deficiencies: (a) The combinatorial term doee not account quantitatively for the free-volume differences between polymers and solvents in most polymer solutions. For the case of 'athermal" polymer solutions this can be seen in the work of Elbro et al. (1990). (b) The FH parameter was initially assumed to be constant, but it was soon realized that is exhibits considerable variation with both temperature and composition. One of the moet successful expressions that has been proposed in order to describe this dependency is the KoningsveldKleintjens (1971) equation:
where the energy parameter B is a function of temperature:
B = Bo + B
l P
(6)
Equations 5 and 6 reveal the rather complicated dependency of the FH parameter with temperature and volume fraction of the polymer. Although eq 5 seems to represent this dependency better than a polynomial volume expansion of the FH parameter (Danner, 1991), it is implied that the entropic contribution is constant, independent of temperature and concentration, which is not always correct. Despite these problems, eq 3 is still used in industry. One of the reasons is the direct and simple relationship between the FH parameter and the solubility parameter of solvent (1) and polymer (2): (7)
Equation 7 provides a qualitative criterion for investigating whether a solvent is soluble in a polymer and vice versa: the closer the two solubility parameters are, the more compatible the components will be. 2. The UNIFAC-FV Models. Following Flov (1970), Oishi and Prausnitz added a free-volume correction term to the original UNIFAC expression. T w o parameters of this term (b and cl) have been adjusted to solvent-polymer experimental equilibrium data. The value of c1 was set equal to 1.1. However, since this parameter reflects the external degrees of freedom per solvent molecule, this low value is highly unrealistic, especially for high molecular weight solvents. Furthermore, Soremen et al. (1990) have recently demonstrated some of the shortcomings of Oishi's model when they tried to predict their own experimental data for the infinite dilution activity coefficients of several hydrocarbons and aromatic solvents in poly(ethy1ene glycol). Yet, despite these problems, UNIFAC-FV is a very succeaeful model for estimating solvent activities in polymer solutions. Iwai and Arai (1989) proposed a somewhat different free-volume expression based on a partition function. All the parameters are estimated through group-contribution equations. For several relatively simple homopolymer and copolymer solutions their model performs well, but no thorough evaluation has been reported. Aa far as the equation-of-state (EOS) approach to polymer solutions is concerned, two categories of models exist: (1) the van der Waals-type EOS, like that of Holten-Andersen et al. (1987) and the group-contribution Flory equation (GCFLORY EOS) by Chen et al. (1990); (2) the lattice-fluid EOS, like the group-contribution lattice
fluid equation (GCLF EOS) propoaed by High and Danner (1990).
Despite their obvious and important differences, the aforementioned equations exhibit several common characteristic features: (a) They are derived from a partition function and have, therefore, some theoretical foundation. (b) They are based on a group-contribution concept. (c) They are rather complex and sometimes not fully consistent. The purpose of this work is to develop a new simple activity coefficient model based on a previously proposed free-volume term (Elbro et al., 1990). Since our aim is to develop a purely predictive model, this will be based on the group-contribution concept and the group energy interaction parameters will be obtained from mixtures of low molecular weight compounds only. New Simple Activity Coefficient Model
The new activity coefficient model is based on a simple definition of free volume (FV). Free volume, the available volume to the center of mass of a molecule, is a difficult concept to describe. Bondi (1968) mentions that "every author defines this volume as what he wants it to mean". The free volume may be defined as
v, = v-
(8)
vi
where V is the liquid molar volume and Vi is the so-called "inaccessible volume". In this work, we make the assumption that the inaccessible volume is equal to the van der Waals volume (V,), as calculated by the method of Bondi, i.e.,
vi = v,
(9)
The van der Waals volume of a molecule is defined (van Krevelen, 1990) as the space occupied by this molecule, which is impenetrable to other molecules with normal thermal energies (Le., corresponding with ordinary temperatures). Equation 9 is the obvious choice and many investigations have proved that other expressions for the inaccessible volume (for instance, the expansion volume at 0 K) do not lead to acceptable results (Elbro et al., 1990, Elbro, 1992). However, eq 9 is not, by any means, the only choice. Attempts to include the difference of shape between the two components into the free-volume expression have been reported. Combining two equations presented by Bondi (1968), the following somewhat different approximate expression for the free volume can be derived:
Vf,, = Aw(V1/' - V,'/')
(10)
It can be seen that, in eq 10, the free volume also depends on van der Waals surface area A,. The new simple activity coefficient model we propose (ELBRO-FV) is based on the FV definitions given by eqs 8 and 9 and on the residual term of UNIFAC. The expression for the activity coefficient of component i is In yi = In r F + In p
(11)
where the free-volume (FV) part is given by (Elbro et al., 1990)
$F 4F In y f ' = In - + 1 - xi
xi
364 Ind. Eng. Chem. Res., Vol. 32, No.2, 1993 Table I. Experimental and Predicted Activity Coefficients at Infinite Dilution for Athermal Polymer Solutions0 oolvmer solution T (K) exptl value ELBRO-FV UNIFAC-FV GCFLORY GCLF 5.65 5.07 4.26 6.06 4.48 LDPE (35000) + nC6 383.20 10.3% 24.5% 7.3% 20.7% 6.21 5.73 5.19 4.27 398.20 4.57 8.4% 9.2% 25.5% 20.2% 5.91 6.54 4.74 423.20 5.51 4.21 10.7% 6.7% 19.8% 27.7% 7.03 5.97 4.96 448.20 5.98 4.26 17.8% 27.8% 0.390 28.6% 7.96 5.24 473.20 6.10 6.79 4.19 30.5% 11.5% 14.1% 31.3% 5.45 4.17 383.20 5.08 4.64 3.94 LDPE (35000) + nC7 7.4% 17.8% 8.7% 22.4% 6.47 5.40 4.71 473.20 5.66 3.95 4.9% 19.8% 12.8% 26.9% 4.91 4.56 393.15 LDPE (82000) + 3MeC6 7.1% 4.72 418.25 5.04 6.2% HDPE (105000) + 3MeC6 5.48 4.78 418.60 12.6% 5.71 4.85 425.80 15.0% 4.46 4.50 4.15 4.36 PIB (24500) + cC6 3.19 298.20 2.3% 3.2% 4.7% 26.8% 4.62 4.81 4.59 3.22 3.62 313.1 PIB (860000) + cC6 4.6% 3.9% 24.7% 33.0% 7.51 6.15 7.33 6.55 4.32 313.1 PIB (860000) + C5 2.5% 10.6% 16.1% 41.1% 1.20 7.54 6.63 6.28 323.1 4.38 4.7% 12.6% 7.9% 39.2% 4.61 4.56 PIB (53000) + cC6 4.64 4.33 323.2 3.23 1.1% 5.1% 1.8% 29.4% 4.45 4.88 5.42 4.95 423.2 3.46 8.8% 11.1% 1.5% 29.1%
..
a
Percentage values represent AAD.
and & are the FV fractions associated with component
i:
The residual part of the ELBRO-FV model is given by UNIFAC (Fredenslund et al., 1977). The interaction parameters are assumed to be temperature dependent through a simple linear relationship: amn = amn,l + a m n , A T - 7'0) (14) where Tois a reference temperature, equal to 298.15 K. The interaction parameter matrix used in this work has been recently developed by Hansen et al. (1992),and it is based exclusively on experimental vapor-liquid equilibrium (VLE) data of low molecular weight compounds. It should be noted that the parameters of this new extended parameter matrix have not been estimated using the new free-volume term; instead the Staverman-Guggenheim combinatorial term of UNIFAC has been used. The Staverman-Guggenheim combinatorial term has been succeeefullyused in UNIFAC for the modeling of VLE for mixtures of low molecular weight compounds. As a consequence, the approach followed in this work is similar to the one originally introduced by Oishi and Prausnitz (19781, with three important differences: (a) The new simple equation (12) for the combinatorial-FV part of the activity coefficient is used instead of the rather complicated term proposed by Oishi and Prausnitz. Their term has two separate contributions,a combinatorial one given by UNIFAC (Fredenslund et al., 1977) and a free-volume one derived by Flory (1970). (b) No adjustable parameters for the free-volume part are used in the new model. On the other hand, Oishi and
Prausnitz term contains two empirical constants ( b and cl) that have been optimized based on VLE data of approximately 30 polymer-solvent solutions. (c) A new linear temperature-dependent parameter matrix is used in the new model instead of the temperature-independent parameters of original UNIFAC that were adopted by Oishi and Prausnitz model. Since both parameter matrices were derived from VLE data of low molecular weight systems, we may consider both UNIFAC-FV and ELBRO-FV as purely predictive models. The theoretical derivation of the 'combined" combinatorial-FV term used in this work is presented elsewhere (Elbro et al., 1990). In the same reference it is explained why this term accounts for both combinatorial and freevolume contributions.
Athermal Polymer Solutions The solutions which are assumed to have zero enthalpy of mixing and, as a consequence, canceling of energetic interactions are called atherrnal. It is apparent that solutions of hydrocarbons in polyethylene (PE) and polyiaobutylene (PIB) are treated by the ELBRO-FV model as athermal, since the only main group present in these solutions is the hydrocarbon one. As a consequence, in these polymer solutions the residual part does not contribute to the activity coefficient. Solvent activity coefficients at infinite dilution for several PE and PIB athermal solutions-at different temperatures and polymer molecular weights-have been eatimated using ELBRO-FV model, and the results are presented in Tables I and 11. The predictions obtained by three other models (UNIFAC-FV, GCFLORY, and GCLF) are also preaented. It can be observed that the new model can predict the solvent activities with rather good accuracy and as accurately as the more complicated
Ind. Eng. Chem. Res., Vol. 32, No.2, 1993 365 Table 11. Mean Percent Deviations between Calculated and Experimental Activity Coefficients at Infinite Dilution for Athermal Polymer Solutions ELBRO-FV UNIFAC-FV GCFLORY GCLF polymer solution temp range (K) PIB (53000) + nC6 323.2-423.2 6.0 6.2 5.7 31.8 PIB (53000) + cC6 323.2-423.2 2.4 3.5 5.6 29.6 LDPE (35000) + nC6 383.2-473.2 7.6 27.5 14.9 18.3 LDPE (35000) + cC6 383.2-473.2 4.1 17.3 36.4 15.8 LDPE (35 000) + nC7 383.2-473.2 5.9 24.6 11.8 16.0 LDPE (35000) + nC8 383.2-473.2 5.7 22.9 10.2 14.4 ~~
models. It should be emphasized that the combinatorial and the free-volume contributions are here combined into one simple term, while all the other models have two different expressions for these two contributions, leading to rather complicated equations. No difference in the predictions of activities of branched hydrocarbons in the two polymers is observed, while the results are somewhat worse at very high molecular weights. The experimental data presented in this work have been obtained from the Polymer Solution Data Collection of Wen et al. (1991). The experimental errors of activity coefficients at infinite dilution are often rather high for polymer solutions and no final conclusions can be easily drawn. There is, however, a tendency for the ELBRO-FV model to underestimate the solvent activities at low temperatures. This could be explained by the fact that exor shape effects are not included in the cess-volume (P) FV expreasion of the model or that these systems-though treated as athermal-do exhibit some energetic interactions. Another reason could be the dependency of free volume on the length and flexibility of polymer’s chain, Le., how the segments are connected to each other into chains (Patterson, 1969). It has been shown by Hildebrand and Scott (1964) and Elbro et al. (1990) that excess-volume effects-although existing in both “athermal” and real polymer solutions (Mtiller and Raemussen, 1991; Flory et al., 1968)-have little effect on GEcalculations. An attempt to incorporate the differences in shape between the solvent and the repeat unit of the polymer has been made in this work by adding an empirical Stauerman-Guggenheirn (SG) correction term to eq 12. The resulting expreasion for the free-volume activity coefficient of component i is
4P 4P In $ = In + 1- - i q i ( In xi
Xi
2 + 2) 1-
6.50
I
~
< -:-::::-----I/ P
i
3.50 380.00
400.00
420.00
460 00
440.00
48
00
Temperature ( K )
-
EXPERIKENTAL DATA ELBRO-FV UNIFAC-FV GCFLORY
Figure 1. Experimental and predicted infinite dilution activity coefficienta for n-heptane in LDPE (35000). 7’50
h
(15)
Using eq 15 (GK-FV model), we get somewhat better predictions only when the ELBRO-FV model leads to predictions lower than the experimental ones. This happens because the SG correction term is always positive. Furthermore, as can be seen from Table III,the magnitude of this term is rather small for these systems, although-as will be discussed later-its influence can be much greater for polymer solutions with groups having very different volume and surface area (Rand Q)values. The SG term is also important for very polar mixtures where the reaidual term has a significant value, and therefore, even a small increase in the FV term might lead to rather different resulta for the activity coefficient. The result of this discussion is that even with the inclusion of the possible influence of VE and shape effects in the free volume, the solvent activities for “athermal polymer solutions” are underestimated by the ELBRO-FV model because the small but existing energetic interactions are not taken into account. This conclusion is supported by Figure 1, which shows the experimental and predicted activity coefficients of n-heptane in low-density polyethylene (LDPE). If this polymer solution were athermal,
453
1 5
6
7
E
9
Number of Carbon Atoms in Solvent
-
EXPERIMENTAL DATA
ntbs ELBRO-FV
UNIFAC-FV
pb%~o
Figure 2. Experimental and predicted variation of infinite dilution activity coefficient with number of carbon atom in the solvent molecule. (The polymer considered ie PIB (24 500) and the temperature is 298.15 K.)
the experimental activity coefficients would be temperature independent, but they are not. Moreover, at higher temperatures the activity coefficienta are not underestimated. This could be expected, since the energy interactions between the solvent and the polymer reduce with increasing temperature. At these conditions the freevolume part is predominant. There is a general belief (Rasmussen and Rasmussen, 1989; Patterson, 1969; Somcynsky, 1982)that the free-volume dissimilarities between polymer and solvent increase with increasing temperature,
366 Ind. Eng. Chem. Res., Vol. 32, No.2, 1993 Table Exmrimental and Predicted Activity Coefficients at Infinite Dilution for Athermal Polymer Solutions Using~ _111. . ~ . ELBRO-FVaGd GK-FV Models: Importance of the Staverman-Guggenheim ’Shape” Correction Term T (K) exptl value ELBRO-FV GK-FV .Dolvmer - solution LDPE (35000) + hexane 383.20 5.65 5.07 6.24 10.3% 7.2% AAD 5.75 5.91 5.51 423.20 6.7% 2.6% AAD 6.79 473.20 6.10 7.09 11.5% 16,2% AAD 5.08 4.64 4.82 383.20 LDPE (35000) + heptane 8.7% 5.2% AAD 4.14 418.25 4.32 4.03 LDPE (82000) + decane 6.6 % 4.0% AAD 3.73 4.02 3.81 393.15 LDPE (82000) + dodecane 7.3% 5.1% AAD 3.77 4.10 418.25 3.86 5.7% 7.9% AAD 4.73 4.90 418.25 5.04 LDPE (82000) + 3-methylhexane 6.2% 2.7% AAD 4.72 418.25 4.97 4.99 LDPE (82 000) + 2,2,4-trimethyl-C5 4.9% 0.5% AAD 4.79 5.48 4.97 418.60 HDPE (105000) + 3-methylhexane 12.6% 9.3% AAD 5.64 323.20 6.29 5.69 PIB (53000) + hexane 9.4% 10.4% AAD 4.15 298.20 4.36 4.19 PIB (24500) + cyclohexane 3.7% 4.1% AAD 4.a 4.82 5.98 298.2 PIB (40000) + octane 19% 19.3% AAD 9% 7% overall AAD ~
~
5 50
Table IV. Experimental and Predicted Activity Coefficients at Infinite Dilution with the ELBRO-FV Model Using Two Different Values for the Polymer Density: Study of the Sensitivity of the ELBRO-FV Model with the Poiymer Density” exptl ELBRO- ELBROsvstam T (.K,) value FV/Taitb FV/GCVc nC6-PIB 323.20 6.29 5.64 4.75 (M.= 53000) 348.20 6.44 5.88 4.92 373.20 6.58 6.14 5.14 398.20 6.72 5.42 6.44 6.80 423.20 6.86 5.79 323.20 4.56 cC6-PIB 4.33 3.65 423.20 4.88 4.22 4.95 (M”= 53000) nC&PIB (M,= 4C,000) 298.20 5.98 4.82 4.14 nC12-LDPE 3.73 3.71 393.15 4.02 3.77 3.77 418.25 4.10 (M.= 82000) nC6-LPDE 5.02 5.07 383.20 5.65 6.79 6.78 (M”= 35000) 473.20 6.10 4.12 393.15 4.44 nC9-LDP E 4.14 418.25 4.51 4.24 4.24 (M,= 82000)
4 5.00
c
e,
.H
0
.H
4
50
e, 0
u h4.00
5 3
.H
4
0 3.50
4
3 00 i 300 00
400 00
500 00
Temperature ( K )
-
EXPERIMENTAL DATA ELBRO-FV UNIFAC-FV GCFLORY s+++d GCLF
Figure 3. Experimental and predicted infinite dilution activity coefficients for cyclohexane in PIB (53 000).
and indeed the ELBRO-FV model seems capable of predicting this behavior. Figures 2 and 3 present experimental and calculated activity coefficients at infinite dilution for several hydrocarbons (with increasing molecular weight) in PIB and the temperature dependence of activity coefficients for cyclohexane in PIB. In both cases, the ELBRO-FV model seems capable of predicting the observed behavior. The predictions of the ELBRO-FV model are rather sensitive to the values used for the polymer density (as shown in Table IV);therefore, experimental densities (for the polymer) should be used whenever possible. In this work the Tait equation has been used for the estimation of the polymer density as a function of temperature i f the temperature of the system was included in the temper-
“The mean difference between the two values used for the polymer density is 0.5% for PE and 6.5% for PIB. bELBRO-FV/Tait means that the experimental density of the polymer ia used BB calculated from the Tait correlation. ELBRO-FV/GCV means that the polymer density used has been predicted using the GCVOL method.
ature range used for the estimution of the Tait equation constants. If this is not the case or if the parameters are not available, a recently developed groupcontribution model for the prediction of polymer density is used (GCVOL model (Elbro et al., 1991)). A rather different, simpler and somewhat theoretical groupcontribution expression has been proposed by van Krevelen (van Krevelen and Hoftyser, 1972;van Krevelen, 1990).
Polymer Systems with Energy Interactions For polymer solutions with energy interactionsespecially for thoee having stsong polar forcea-the reaidual term can be significantly larger than 1. Table V provides a list of values of the residual term for several polymer systems calculated by UNIFAC using the new temperature-dependent parameter matrix. The maximumdegree of nonideality for solvent-polymer solutions is observed-for VLE calculations-at infinite
, Ind. Eng. Chem. Res., Vol. 32,No. 2, 1993 367
9.00
4
I
Table V. Residual Term of the ELBRO-FV Model for Polymer Solution6 with Energy Interactionr T (K) residual term polymer solution benzenePIB (53000) 323.2 1.47 423.2 1.16 298.2 CHClS-PIB (40000) 1.65 383.2 1.16 p-xylene-LDPE (35000) 413.0 BuOAePVC (40000) 1.50 iPrOH-PIP (10800) 328.2 9.82 ethylene glycol-PIP (10800) 328.2 438.13 328.2 methanol-PIP (10800) 20.38 acetonitrile-PIP (10800) 328.2 11.72 328.2 12.30 acetic acid-PIP (10800) 328.2 acetone-PIP (10800) 3.84 dioxane-PIP (10800) 2.21 328.2 1.30 473.2 chlorobenzene-PS (97000) 423.2 acetonitrile-PS (97000) 2.14 MEK-PBD (22600) 2.37 353.2 2.21 373.2 butanol-PBD (22600) 353.2 4.88 373.2 3.78 ~~
-
-EXPERIMENTAL ELBRO-N UNIFAC-N GCFLORY
DATA
CCCH
Figure 4. Experimental and predicted infinite dilution activity coefficients for benzene in PE (35000). 4
I
5.00
-
4
c:
0, .w
0
;r:
rr 4.60 Q, 0
u
h
5
*?4.20 -
ccr Q, 28.00
4
0
0
u
4
L7 Temperature ( K ) -
3.80 400.00
I
420.00
I "
1 '
440.00
" ' 1 '
I
" I "
460.00
" ' 3
I '
" " I
480.00
'
fl
50( 00
EXPERINENTAL DATA ELBRO-FV
" I UNIFAC-FV
Figure 6. Experimental and predicted infinite dilution activity coefficients for 1,2-dichloroethane in PS (97000). ctm WPERIMENTAL DATA
Hm ELBRO-FV
UNIFAC-FY
Figure 5. Experimental and predicted infinite dilution activity coefficients for acetonitrile in PBD (22600).
dilution (very small amount of the solvent). We have, therefore, tested the ELBRO-FV model for the prediction of infiiite dilution activity coefficients. Polymer solutions comprising 7 different polymers and more than 30 Merent solvents have been considered. The resulta have been compared with both the experimental valuea (Wen et al., 1991)and the predictions obtained by the GCFLORY, GCLF, and UNIFAC-FV models and are presented in Tables VI-VIII. Table WI,in particular, presents the predictions of the ELBRO-FV model for solutions of various solventa in polyisoprene (PIP), compared only with the UNIFAC-FV method. This has been done because the GCLF equation does not include the C H 4 group (existing in the repeat unit of PIP) and we have found that the GCFLORY EOS does not yield reliable results for PIP systems. The same problem holds for polybutadiene (PBD) systems, even for the eimplest ones (see also Tables VI and VII). It seems that the energy parameters between the hydrocarbon and the double-bond group are not very accurate, and there-
E3 24.00
is
2 u 22.00 E c4 GI
d 20.00
u
E z
18.00
I
Figure 7. Influence of the physical R and Q values of OH group in the predictions of infinite dilution activity coefficienta for isopropyl alcohol in PBD (22600).
368 Ind. Eng. Chem. Res.,Vol. 32,No. 2, 1993 Table VI. Experimental and Predicted Activity Coefficients at Infinite Dilution for Polymer Solutions with Energy Interactions" polymer solution T (K) exptl value ELBRO-FV UNIFAC-FV CCFLORY CCLF PIB (53000) + toluene 323.2 5.30 5.28 6.19 7.01 3.07 42.0% 16.9% 32.3% 0.4% 4.91 5.24 7.88 4.98 3.03 423.2 38.3% 6.6% 60.6% 1.4% 2.93 2.10 ne 2.53 2.74 298.2 PIB (40000) + CC1, 15.8% 16.9% 8.4% 14.54 3.18 7.69 7.67 6.17 443.0 HDPE (15000) + MIBK 24.6% 48.3% 24.3% 7.75 3.62 2.71 3.73 3.59 383.2 LDPE (35000) + p-xylene 27.2% 2.7% 3.7% 10.67 3.49 2.80 3.67 3.71 473.2 5.9% 24.3% 1.1% 5.43 4.72 5.04 4.68 5.60 353.2 PVAc (141420) + EtOAc 3.1% 15.7% 16.4% 9.9% 5.54 4.65 4.82 5.15 5.70 373.2 18.4% 2.9% 15.3% 9.6% 10.17 3.64 7.24 9.98 11.8 PVC (41000) + acetone 393.2 69.1% 13.9% 38.6% 15.5% 10.86 3.57 7.37 10.13 10.8 PVC (101500) + acetone 398.2 66.9% 31.8% 6.1% 6.8% 10.14 14.63 15.59 15.65 45.0 383.2 PVC (41000) + heptane 77.5% 67.5% 65.3% 65.2% 3.53 7.64 7.95 8.80 8.69 413.2 PVC (101500) + MEK 59.4% 8.5% 12.1% 1.3% 6.48 12.4 6.68 7.73 10.04 PVC (41000) + EtOAc 393.2 46.1% 47.8% 37.6% 19.4% 5.41 12.2 12.95 8.46 10.29 413.0 PVC (40000) + BuOAc 55.6% 6.10% 30.7 % 15.6% 12.0 ns ns 10.93 10.93 PVC (101500) + acetaldehyde 398.2 8.9% 8.8% 10.46 11.4 11-01 ns ns 413.2 8.2% 3.4% 5.36 8.74 16.1 8.96 10.96 413.2 PVC (101500) + cyclohexane 44.3% 66.7% 45.7% 31.0% 18.21 72.35 20.2 17.42 353.2 3.34 PBD (22600) + 1-butanol 9.8% 83.5% 13.8% 17.07 18.2 54.96 3.37 15.47 363.2 81.4% 15.0% 6.2% 16.1 16.09 42.36 3.43 13.79 373.2 0.1 % 14.3% 78.7% 76.9 86.3 4.11 59.46 339.2 PBD (93000) + methanol 12.3% 94.6% 22.7% 68.2 49.1 252.01 4.03 32.27 369.2 38.9% 91.8% 34.3% 5.57 5.98 12.83 3.21 5.90 353.2 PBD (22600) + EtOAc 6.7% 1.3% 46.3% 10.4 9.49 373.2 33.63 PBD (22600) + acetone 3.63 11.36 8.7 % 65.1% 9.2% 10.31 11.6 32.81 3.49 12.35 339.2 PBD (93000) + acetone 11.1% 69.9% 6.5% 5.58 6.89 18.13 3.34 7.02 363.2 PBD (22600) + MIBK 19.0% 51.5% 1.9% 3.61 4.12 6.82 3.56 4.125 353.2 PBD (22600) + cC6 12.5% 65.6% 13.5% 0.13% 4.97 9.89 PBD (22600) + hexane 6.36 5.79 373.2 5.44 21.9% 55.7% 14.4% 8.9% 4.27 400.2 PS (3524) + benzene 3.98 5.78 4.20 4.76 6.8% 35.4% 11.6% 1.50% 4.43 5.69 5.12 423.2 PS (86700) + benzene 6.78 4.54 22.2% 19.3% 20.2% 10.0% PS (1780000) + benzene 6.78 5.45 5.10 4.44 420.9 4.53 24.4% 6.4% 18.6% 16.9% 423.2 PS (97000) + EtCeHS 4.96 7.02 4.23 4.47 4.63 14.7% 41.5% 6.6% 9.7% 5.17 PS (97000) + dioxane 423.2 5.38 4.65 5.69 4.35 15.9% 4.2% 9.9% 10.1% 473.2 9.12 PS (97000) + MEK 7.95 7.87 6.27 12.8% 13.7% 31.2% Percentage values represent AAD.
fore, the GCFLORY EOS should not be used for polymer solutions where the polymer repeat unit has a double-bond group (like PIP and PBD). The reaulta presented in Tables VI-VIII prove that the two activity coefficient models (ELBRO-FV and UNI-
FAC-FV) are of comparable accuracy and significantly better than the two EOS. Since they are also simpler and can be used for many different polymer solutions (due to the large UNIFAC parameter table), they are recommended for low-prewure VLE calculations. The new ac-
Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 369 Table VII. Mean Percent Deviations between Experimental and Predicted Activity Coefficients at Infinite Dilution for Polvmer Solutiona with Energy -. Interactions polymer solution temp range ELBRO-FV UNIFAC-FV GCFLORY GCLF PIB (53000) + benzene 323.2-423.2 3.3 8.2 22.5 46.6 PIB (53000) + toluene 323.2-423.2 1.9 12.1 44.5 40.6 12.4 391.2-413.2 2.9 80.5 27.1 PE (16600) + benzene 1.2 PE (35000) + benzene 383.2-413.2 7.6 69.4 31.8 3.6 PE (35000)+ p-xylene 383.2-413.2 4.6 136 26.2 2.2 383.2-413.2 4.3 97.9 PE (35000) + toluene 28.4 391.2-423.2 PE (16600) + toluene 5.5 0.5 90.6 26.8 22.8 15 PVAc (500000)+ EtOAc 31.9 354-411.1 6.3 PVAc (1500000)+ EtOAc 13.1 8.6 21.2 374.8-417.3 7.8 27 26.4 339.2-369.2 PBD (93000) + methanol 10.6 353.2-373.2 PBD (22800) + iPrOH 9.0 84.1 11.3 7.5 353.2-373.2 86.0 PBD (22600) + l-propanol PBD (22600) + acetone 1.4 10.0 353.2-313.2 65.1 10.0 339.2-369.2 7.6 PBD (93000) + acetone 67.8 353.2-313.2 17.9 PBD (22600) + MIBK 2.5 50.9 10.2 PBD (22600) + MEK 8.3 353.2-373.2 58.7 31.7 PBD (22600) + acetonitrile 1.6 353.2-313.2 ne ns 7.5 2.4 353.2-313.2 PBD (22600) + EtOAc 45.5 PBD (22600) + cyclohexane 1.9 353.2-313.2 12.5 70.0 13.8 52.1 20.9 PBD (22600) + hexane 10.0 353.2-313.2 14.5 PBD (22600) + benzene 4.6 6.5 353.2-313.2 15.5 5.5 1.7 353.2-313.2 7.1 ns PBD (22600) + CC14 16 395.2-444.8 PS (3524) + benzene 6.1 41.4 3.9 PS (86700)+ benzene 7.1 21.5 24.5 423.2-448.2 17.8 8.8 11.0 423.2-473.2 63.9 ne PS (97000) + 1,2-C12Et Table VIII. Experimental and Predicted Solvent Activity Coefficients for Several PIP Polymer Solutionr PIP systems exptl value ELBRO-FV UNIFAC-FV IT .- = 328.2 - ~ -K) + ~SODIODYIalcohol 50.4 38.53 31.53 23.6% 31.4% Ap;D - 21.66 36.2 29.28 + l-butanol ~
~~
I
AAD
+ methanol AAD + ethylene glycol AAD + acetonitrile AAD + acetic acid AAD + cC6one AAD + acetone AAD + MEK AAD + benzene AAD + hexane AAD + chloroform AAD + 1,2-C12Et AAD + CC14 AAD + 1,4-dioxane AAD + THF AAD + ethyl acetate
AAD
overall AAD
105 1145 68.6 37.9 1.32 11.3 11.4 4.37 6.36 2.13 4.25 1.71 6.08 4.38 7.41
~
19.1% 81.95 21.9% 908.35 20.1% 41.68 30.5% 33.45 11.7% 5.37 26.6% 15.92 1.9% 12.13 6.4% 4.47 2.5% 5.07 20.2% 2.99 40.8% 5.48 29 % 2.13 20.4% 6.29 3.5% 4.98 13.1% 7.30 2.2% 11.7%
40.2% 89.98 14.3%
52.30 23.8% 17.66 53.4% 4.57 31.6% 13.38 22.6% 10.01 11.6% 4.36 0.2% 4.62 21.3% 2.55 19.1% 6.53 53.6% 1.71 0% 5.96 1.9% 3.93 10.3% 6.63 11.2% 22.8%
tivity coefficient model-the simpler of the two-is based on a recently developed parameter table, the energy parameters are temperature dependent, and it generally yields better predictions than UNIFAC-FV for systems with significant energy interactions. It should be mentioned, however, that all four models have a number of deficiencies, and some of them are
common. These are discussed in some detail in the Appendix. Figures 4-6 show that ELBRO-FV model can follow the observed trends of solvent activity coefficients at infinite dilution with the temperature and the molecular weight of the polymer. Table M and Figure 7 deal with the 'alcohol case". It is obvious that the best results are obtained if the "physical" R and Q values for OH, derived by Bondi (R = 0.52999, Q = 0.5841, are used in the free-volume part of the model. Since, however, the energy interaction parameters were obtained wing "fitted" R and Q values for this group (R = 1.00, Q = 1.20), theae artificial values must be retained in the residual part. Methanol is an exception to this rule, since ita 'physical" R and Q values (R = 1.4311, Q = 1.432) have been originally used for the estimation of the energy parameters. Notice,finally, the resulta obtained for the system ethylene glyml-PIP, where two OH group are present in the solvent molecule. Use of the artificial values for the R and Q parameters leads to a nonphysical low value for the free-volume contribution to the activity coefficient. The fact that the new FV term should be always used with "physical" R and Q values justifies-in some way-ita theoretical nature. Note (Table X)the importance of the "shape" correction SG factor (GK-FV model) for polymer solutions with energy interactions. Use of the "shape" factor often leads to improved predictions. If, however, the predictions of solvent activities with the ELBRO-FV model are not underestimated, the GK-FV term leads to worse resulta. These remarks point out the need of some correction, particularly for highly polar systems. Reeathation of the group energy parameters of the residual term using the new FV term instead of the combinatorial term of UNIFAC leads, in some cases, to better predictions, but preliminary investigations have ehown that the improvement is rather small (Kontogeorgis, 1991). Conclusion A new activity coefficient model for the prediction of solvent activities in polymer solutions has been introduced. The new model (ELBRO-FV) combines an approximate free-volume expression and the residual term of W A C .
370 Ind. Eng. Chem. Res., Vol. 32,No. 2, 1993 Table IX. Exwrimental and Predicted Solvent Activity Coefficients at Infinite Dilution for Alcohol Polymer Solutions? Importance of Physical R , 8 values for the OH Group; Importance of the Staverman-Guggenheim Correction (GK-FV Term) ELBRO-FV model GK-FV model' R, Q values for OH group R, Q values for OH group polymer solutionb physical fitted Dhveid fitted 38.53 iPrOH-PIP; T = 328.2 K (50.4) 31.26 40.22 36.75 AAD 23.6% 37.9% 20.2% 27.1% l-butanol-PIP; T 328.2 K (36.2) 29.28 24.41 27.66 30.16 19.1% AAD 32.6% 16.7% 23.6% ethylene glycol-PIP; T = 328.2 K (1145) 908.35 275.63 978.98 415.16 AAD 20.7% 75.9% 14.5% 63.7% 81.95 81.95 methanol-PIP; T = 328.2 K (105) 92.22 92.22 21.9% 21.9% AAD 12.2% 12.2% 25.43 20.47 28.37 1-propanol-PBDa; T = 339.2 K (32.5) 27.28 AAD 21.7% 37.0% 12.7% 16.1% 1-propanol-PBDa; T = 369.2 K (20.8) 17.57 14.42 19.21 19.59 AAD 15.5% 30.7% 5.8% 7.6% 59.47 methanol-PBDa; T = 339.2 K (76.9) 71.60 71.60 59.47 22.7% AAD 6.9% 22.7% 6.9% 17.42 1-butanol-PBDb; T = 353.2 K (20.2) 14.69 19.10 18.79 AAD 13.8% 27.3% 5.4% 6.9% For PBDa M, = 93000,for PBDb M, = 22 600,and for PIP M, = 10800. The experimental values are in parentheses. StavermanGuggenheim correction term.
Table X. Experimental and Predicted Activity Coefficients at Infinite Dilution Using ELBRO-FV and GK-FV Models:" Importance of the Staverman-Guggenheim Correction for Slightly and Strongly Polar Polymer Solutions exptl ELBRO- GKFV FV polymer solution T (K) value PIB (53000)+ benzene 323.2 5.93 5.88 6.24 0.7% 5.2% 423.2 5.56 5.33 5.65 4.1% 1.6% 423.2 PIB (53000)+ toluene 4.98 5.31 4.91 1.4% 8.2% PVC (41000)+ ethyl 10.04 11.57 393.2 12.4 19.4% 6.7% acetate PVC (41000)+ acetone 393.2 11.80 9.97 11.3 15.5% 3.9% PIP (10800)+ 47.68 49.94 328.2 68.6 acetonitrile 30.5% 27.2% PIP (10880)+ acetic acid 328.2 37.9 33.45 36.18 11.7% 4.5% PIP (10800)+ acetone 15.92 16.66 328.2 17.3 7.9% 3.7% PIP (10800)+ MEK 11.4 12.12 12.49 328.2 6.4% 9.6% PBD (22600)+ 353.2 23.5 26.28 28.99 acetonitrile 11.8% 23.4% 22.55 24.87 373.2 23.8 5.3% 4.5% Percentage values represent AAD.
The energy parameters are assumed to be temperature dependent and have been previously estimated from low molecular weight VLE data using the original Staverman-Guggenheim combinatorial term of UNIFAC. We have presented an extensive evaluation of the new model for a large variety of polymer solutions, ranging from athermal to strongly polar and hydrogen-bonding solutions. Despite some problems observed mainly in hydrogenbonding polymer solutions, the predicted solvent activity coefficienta at infinite dilution are in very good agreement with the experimental data. Furthermore, the predictions of ELBRO-FV model compare favorably with the empirically based UNIFAC-FV activity coefficient model and are significantly better than the two rather complicated EOS, the GCLF and the GCFLORY models. It should be mentioned that the new activity coefficient model representa a successful compromise between simplicity and accuracy. Considering the extensive UNIFAC parameter matrix available, the proposed new model is
" 1 4 40
1 350.00 360.00 370.00 38( 00
3.20 340 00
TEMPERATURE ( K ) ETHYLACETATE - PED (22600)
-
ACETONE - PED (22600)
Figure 8. Variation of the combinatorial-FV part of the ELBROFV model with temperature for two polar polymer solutions. Table XI. Temperature Dependency of the Combinatorial and the Residual Parts of the ELBRO-FV Model for Polymer Solutions with Energy Interactions (All Calculations Made at Infinite Dilution) combinatorial residual polymer solution T(K) art Dart methanol-PBD (93000) 339.2 3.905 15.228 349.2 3.974 12.436 359.2 4.035 9.965 369.2 4.091 7.890 acetone-PBD (93OOO) 339.2 4.076 3.029 349.2 4.177 2.886 359.2 4.271 2.755 369.2 4.359 2.636 ethyl acetate-PVAC (1500000) 374.8 5.034 1.026 388.4 5.136 1.022 401.8 5.250 1.019 417.3 5.403 1.015 3.495 benzene-LDPE (35OOO) 383.2 1.263 3.542 1.222 298.2 423.2 3.656 1.160 3.824 448.2 1.107 473.2 4.066 1.059
cr. 3.20
Ind. Eng. Chem. Res., Vol. 32, No.2, 1993 371
,
SG = Staverman-Guggenheim'shape-correction" term THF = tetrahydrofuran VLE = vaporliquid equilibria Symbokr A = surface area (cm*/g-mol) am,,= energy parameter between groups m and n am,,,l first adjustable energy parameter (eq 14) = second adjustable energy parameter (eq 14) 2.20 b 3: constant in the free-volume term of UNIFAC-FV model c1 = external degree of freedom parameter used in UNIFAC-FV model G = Gibbs energy f!1 70 M, = number-average molecular weight of the polymer P = pressure R = universal gas constant r = dimensionless segment/volume parameter 120 I Q = dimensionless surface area parameter 34000 35000 36000 37000 38000 T = absolute temperature (K) TEMPERATURE (K) To= reference temperature in eq 14, equal to 298.15 K t = temperature ("C) ETHYLACETATE - PBD 22600) ACETONE - PBD (22606) V = molar volume Vi = inaccessible volume Figure 9. Variation of the reeidual part of the ELBRO-FVmodel n = molar fraction with temperature for two polar polymer solutione (the aame as in z = coordination number Figure 8). Greek Letters both a reliable and widely applicable tool for VLE calcua = entropy parameter in eq 5 lations in polymer solutions. a = activity p = energy parameter in eq 5 Acknowledgment Bo = adjustable parameter in eq 6 p1 = adjustable parameter in eq 6 The authors wish to thank professor R. P. Danner for y = parameter in eq 5;it is a function of z providing us with the POLYPROG program with which y = molar activity coefficient many of the calculations concerning the UNIFAC-FV, 6 = solubility parameter GCLF, and GCGLORY models were made. We also want B = surface area fraction to thank Profmeor Peter Rasmuesen for his constructive p = polymer's density (g/cm3) comments and H.S.Elbro for many inspiring discussions. 6 = modified segment/volume or free-volume fraction x = FH free-energy parameter Nomenclature Superscripts A bbreuicrt ions c = combinatorial AAD average absolute deviation E = excess property BuOAc = butyl acetate fv = free volume cC6 = cyclohexane vol = volume cC6one = cyclohexanone res = residual CCl, = carbon tetrachloride CHC1, = chloroform Subscripts 1,P-Cl2Et= 1,Edichloroethane 1 = component 1 (solvent) EOS = equation of state 2 = component 2 (polymer) exptl = experimental (value) f = free volume EtOAc = ethyl acetate fn = free volume (only in eq 10) EtC6H5= ethylbenzene i = component i FH = Flory-Huggins (model/parameter) j = component j FV = free volume h = enthalpic GC = group contribution s = entropic GCLF = groupcontribution lattice fluid EOS w = van der Waals GCFLORY = groupcontribution Flory EOS HDPE = high-density polyethylene Appendix. Deficiencies of Activity Coefficient iPrOH = isopropyl alcohol Models and Equations of State Used for the LDPE = low-density polyethylene Prediction of Solvent Activities in Polymer MEK = methyl ethyl ketone Solutions 3MeC6 = 3-methylhexane We have found in this work that the equations of state MIBK = methyl isobutyl ketone (GCLF, GCFLORY) and the activity coefficient models nCx = normal-hydrocarbon with x carbon atoms (UNIFAC-FV, ELBRO-FV, GK-FV) that have been prons = no solvent (wed in several tables, meaning that the posed and used for the prediction of phase equilibria in particular model cannot be used with a specific solvent) polymer solutions reveal a number of deficiencies. PBD = polybutadiene One case where such a problem occurs is the hydroPE = polyethylene gen-bonding (alcohol) systems. All models usually unPIB = polyieobutylene derestimate the solvent activities in these systems, thus PIP = polyisoprene indicating that the 'hydrogen-bonding effect" has not been PS = polystyrene taken properly into account. A second problem appears PVAC = poly(viny1 acetate) when we consider mixtures of normal hydrocarbons in PVC = poly(viny1 chloride)
9 1
-
-
372 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
poly(viny1 chloride). Their experimental infinite dilution activity coefficients have very high values (of the order 30 or a), higher than the one8 of even the most polar solvents in the same polymer. Again, the predicted activity coefficients are much lower than the experimental ones. The residual term has a very small value. Therefore, a need of insight into the actual physics of these solutions is essential. In addition to the aforementioned difficulties,a number of particular problems regarding specific polymel-solvent solutions and predictive models have been observed. Since the knowledge of these problems provides guidelines for the correct use of the availablemodels, the most important of them are mentioned here: 1. The GCLF EOS cannot be used quantitatively for PIB solutions (probably due to the unfortunate parameters concerning the quaternary carbon atom). 2. The GCFLORY EOS cannot be used for solutions containing PIP and PBD, probably due to the inaccurate energy parameters between the hydrocarbon and the double-bond groups. 3. The two equations of state often lead to very erroneous predictions for alcohol solutions. The GCFLORY EOS gives very high values, while GCLF gives very low ones. It seems, finally, that the ELBRO-FV model leads to somewhat inferior predictions at very high temperatures (above 130 "C), especiallyfor strongly polar solutions. As we observe from Table XI and Figures 8 and 9, the combinatorial-free volume contribution always increases with temperature, while the residual (energetic) contribution decreases as the temperature increases and approaches unity at very high temperatures. These trends are in agreement with both the observed behavior and theoretical considerations. It seems, however, that at high temperatures the decrease of the residual term-as predicted by the ELBRO-FV model-is quite significant and cannot be always counterbalanced by the increase of the free-volume part of the activity coefficient model. It should, however, be emphasized that a large amount of data concerning infinite dilution activity coefficientsand most (over 80%) of the experimental polymer-solvent VLE data at intermediate concentrations are in the temperature range 273.15-323.15 K (Wen et al., 1991),where ELBRO-FV can be safely used.
Literature Cited Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; Wiley: New York, 1968; p 256. Chen, F.; Fredenslund, Aa.; Rasmusaen, P. Group-Contribution Flory Equation of State for Vapor-Liquid Equilibria in Mixtures with Polymers. Znd. Eng. Chem. Res. 1990, 29 (5), 875. Danner, R. P. Personal communication, 1991. Elbro, H. S. Phase Equilibria of polymer solutions-with special emphasis on free volumes. Ph.D. Dissertation, The Technical University of Denmark, Lyngby, 1992; p E4. Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23 (21), 4707.
Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. Group Contribution Method for the Prediction of Liquid Densities as a Function of Temperature for Solvents, Oligomers, and Polymers. Znd. Eng. Chem. Res. 1991,30 (12), 2576. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: London, 1953. Flory, P. J. Discuss. Faraday SOC.1970,49, 7. Flory, P. J.; Ellenson, J. L.; Eichinger, B. E. Thermodynamics of Mixing of n-Alkanes with Polyisobutylene. Macromolecules 1968, I (3), 279. Fredenslund, Aa.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNZFAC; Elsevier Scientific: New York, 1977. Hansen, H. K.; Cob, B.; Kuhlmann, B. 'UNIFAC with Lineary Temperature-Dependent Group-Interaction Parameters"; Technical report (No. 9212); IVC-SER Research Engineering Center, Institut for Kemiteknik, The Technical University of Denmark Lyngby, 1992. High, M. S.; Danner, R. P. Application of the Group Contribution Lattice-Fluid EOS to Polymer Solutions. AIChE J. 1 9 9 0 , s (11). 1625.
Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes; Dover: New York, 1964. Holten-Andersen, J.; Rasmuasen, P.; Fredenslund, Aa. Phase Equilibria of Polymer Solutions by Group Contribution. 1. VaporLiquid Equilibria. Znd. Eng. Chem. Res. 1987, 26 (7), 1382. Iwai, Y.; Arai, Y. Measurement and Prediction of Solubilities of Hydrocarbon Vapors in Molten Polymers. 3. Chem. Eng. JPR. 1989,22 (2), 155.
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Received for review May 11, 1992 Revised manuscript received October 19, 1992 Accepted November 9, 1992
