3142
Ind. Eng. Chem. Res. 1998, 37, 3142-3150
Vapor-Liquid Equilibrium of Polymer + Solvent Mixtures by the Chain-of-Rotators Equation of State Carlos R. Novenario,† James M. Caruthers, and Kwang-Chu Chao* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
Vapor-liquid equilibrium (VLE) of polymer + solvent mixtures is described by the chain-ofrotators (COR) equation of state (EOS) using segment parameters that are applicable to polymers of varying molecular weight. Segment parameters a and b are obtained from small molecules of the same chemical structure as the polymer segment. The segment parameter c is then obtained by fitting polymer pressure-volume-temperature (PVT) data. van der Waals and free energy matching mixing rules are investigated to describe mixtures of various types. Excellent correlation of VLE data is obtained for all mixtures studied by using only one constant interaction coefficient for a, k12, in the van der Waals mixing rule independent of temperature or polymer molecular weight. VLE predictions from the COR EOS by using van der Waals mixing rules with zero k12 as well as by using UNIFAC free energy matched mixing rules are presented. For the majority of systems studied, both predictive methods gave better or equal results compared to the UNIFAC-FV model (FV ) free volume). Introduction Equations of state are widely used in phase equilibrium calculations as both vapor and liquid phases are described over wide ranges of conditions by the same equation. Whereas most EOS theories are developed to represent the behavior of volatile fluids, extension of these theories to systems containing large-chain molecules is necessary in a variety of industrial applications. A number of equations of state have been proposed to address this need. The capabilities and limitations of several EOSs are reviewed elsewhere (Danner and High, 1993; Kontogeorgis et al., 1994a,b). Overall, both review articles recommended the UNIFAC-FV (UNIFAC-free volume) solution model (Oishi and Prausnitz, 1978) over any of the equations of state studied in terms of accuracy and ease of use. In the past few years, cubic EOSs have been proposed for polymer solutions (Sako et al., 1989; Kontogeorgis et al., 1994b; Harismiadis et al., 1994). To date, these equations have only been demonstrated to represent VLE of nonpolar or slightly polar solutions. The perturbedhard-sphere-chain (PHSC) EOS developed by Song et al. (1994a,b) was used by Gupta and Prausnity (1995, 1996) to correlate VLE of various polymer + solvent mixtures giving excellent representation of the data with one binary interaction parameter. Two additional parameters are required in the PHSC EOS to describe hydrogen-bonding systems. Recently, a mixing rule based on a solution model was proposed by Novenario, Caruthers, and Chao (1996) to apply to cubic and noncubic EOSs. The free energy from the EOS is matched with the solution model at a molar volume equal to a constant multiple of the hard-core volume. The matching condition simulates the low pressures of the experimental data used to regress solution model parameters. The ability of the EOS to represent phase behavior beyond the low reference * Author to whom correspondence is addressed. † New address: Seagate Technology, 7801 Computer Avenue South, Bloomington, MN 55435-5489.
pressure extends the useful range of the solution model. A predictive method is obtained when a parametrized model (e.g., UNIFAC) is incorporated into the EOS. Solution model-based mixing rules have been scarcely used for polymer solutions. Orbey and Sandler (1994) applied the Wong-Sandler (1992) method to incorporate the Flory-Huggins (F-H) solution model into the Stryjek-Vera (1986) modified Peng-Robinson equation to correlate VLE data. The mass-specific polymer parameters reported in their work were dependent on molecular weight. Two binary parameters, the F-H χ and kij, were fitted to polymer + solvent VLE data. Kalospiros and Tassios (1995) proposed an incorporation method at zero reference pressure using the entropicFV model (Elbro et al., 1990; Kontogeorgis et al., 1993) with a modified Peng-Robinson EOS (Magoulas and Tassios, 1990). Satisfactory predictions were presented for a few nonpolar and slightly polar solutions. In this work, we investigate the description of vaporliquid equilibrium of polymer + solvent mixtures by the chain-of-rotators (COR) equation of state using segment parameters applicable to polymers of varying molecular weight. Segment parameters a and b are obtained from small molecules of the same chemical structure as the polymer segment. The segmental c is then obtained from fitting the polymer pressure-volume-temperature (PVT) data. Using the a, b, and c thus obtained, van der Waals mixing rules accurately correlate VLE data for various nonpolar and polar polymer + solvent binaries using one interaction parameter. Predictive methods using van der Waals- and UNIFAC-based mixing rules are compared with the UNIFAC-FV solution model. New COR Parameters Chien, Greenkorn, and Chao (1983) combined the rotational and translational configurational partition functions to obtain the repulsive pressure for the COR EOS for polyatomic molecules. The Redlich-Kwong
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Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3143 Table 1. Chain-of-Rotators EOS Segmental Parameters for Polymers (r ) 1.5)
polymers
surrogate molecule for parameters a and b of polymer segment
polystyrene (PS) polyisobutylene (PIB) poly(ethylene glycol) (PEG) poly(methyl methacrylate) (PMMA) poly(vinyl chloride) (PVC) poly(vinyl acetate) (PVAC) poly(R-methylstyrene) (PAMS) poly(ethyl acrylate) (PEA)
ethylbenzene 2,2,4-trimethylpentane 1,2-dimethoxyethane methyl isobutyrate 1,3-dichlorobutane ethyl acetate cumene ethyl propionate
a1 ((bar cm6)/mol2) a2 × 103 (K-1) b1 (cm3/mol) b2 × 105 (K-1.5) 1.112 62 × 108 1.282 25 × 108 8.448 63 × 107 9.693 00 × 107 1.037 19 × 108 7.650 76 × 107 1.334 72 × 108 9.926 46 × 107
1.973 41 2.298 18 2.420 44 2.400 71 1.741 42 2.397 31 1.937 11 2.340 67
209.187 257.150 166.801 184.375 176.368 150.310 232.791 176.376
c
4.437 15 5.813 31 5.546 04 5.394 68 3.804 66 5.439 30 4.446 70 5.305 68
5.213 5.741 5.546 4.870 8.580 5.196 6.235 9.626
Table 2. Chain-of-Rotators EOS Parameters for Some Solvents (r ) 1.5) solvents
a1 ((bar cm6)/mol2)
a2 × 103 (K-1)
b1 (cm3/mol)
b2 × 105 (K-1.5)
c
toluene 2-butanone cyclohexane acetone benzene m-xylene n-pentane water chloroform dichloromethane carbon tetrachloride ethylbenzene tetrahydrofuran cumene
8.653 85 × 6.669 23 × 107 7.662 24 × 107 4.957 02 × 107 6.510 42 × 107 1.139 42 × 108 6.776 12 × 107 1.799 67 × 107 5.423 30 × 107 4.021 70 × 107 6.589 68 × 107 1.112 62 × 108 5.528 99 × 107 1.334 72 × 108
2.014 45 2.151 44 2.112 72 2.221 22 2.130 21 2.005 76 2.540 06 1.933 82 2.188 52 2.274 65 2.029 37 1.973 41 2.100 69 1.937 11
180.973 141.547 165.381 118.047 155.233 206.085 172.716 37.110 131.553 102.502 149.872 209.187 121.816 232.791
4.787 26 4.748 61 5.515 66 5.043 80 4.943 08 4.529 17 7.048 59 3.775 06 5.658 99 5.906 86 5.250 84 4.437 15 5.740 97 4.446 70
2.593 3.030 4.175 2.209 1.453 3.482 3.574 0.000 2.491 2.407 3.453 3.018 4.515 3.923
107
form of the attractive pressure was later adopted by Pults, Greenkorn, and Chao (1989a,b) to simplify the equation to the following:
PV 1 + y + y2 - y3 ) + RT (1 - y)3 a(T) 3y + 3Ry2 - (R + 1)y3 c (1) (R - 1) 2 RT[V + b(T)] (1 - y)3 where the packing fraction y ) b(T)/4V is a reduced density; b is the excluded volume parameter; c is the rotational degrees of freedom; R is the nonsphericity parameter of the model rotators; a is the attractive parameter. Pults et al. (1989a) presented COR parameters for several nonpolar and slightly polar groups and some molecules. In extending eq 1 to polar and associating substances, Novenario et al. (1998) changed the temperature dependence for a and b to
a ) a1 exp(-a2T)
(2)
b ) b1 exp(-b2T1.5)
(3)
The five constants a1, a2, b1, b2, and c fitted from pure fluid saturated properties were reported for 106 nonpolar and polar molecules (Novenario et al., 1998). These parameters were obtained by minimizing the error between the experimental and calculated vapor pressure and saturated liquid volume. The average deviations were found to be around 1% for the vapor pressure and well-below 1% for the saturated liquid volume (Novenario et al., 1998). Sy-Siong-Kiao et al. (1996) extended eq 1 to polymers and obtained excellent representation of PVT data of polymer melts. In Sy-Siong-Kiao’s work and in this work, the COR parameters a, b, and c of a polymer are
expressed in terms of segment parameters according to
a ) asegr2
(4)
b ) bsegr
(5)
c ) csegr
(6)
where r is the number of segments in a polymer molecule of a given number average molecular weight. By eqs 4-6, the same segment parameters are used for a polymer irrespective of the molecular weight. The use of such segment parameters is true to the chemical nature of the polymer and advantageous in being able to describe polymers of varying molecular weight with one set of parameters. The parameters reported by Sy-Siong-Kiao (1996), being fitted only to liquid density data, do not represent vapor-liquid equilibrium phenomena of polymer mixtures. For an equation of state to describe VLE, it is essential that the attractive parameter a be fitted to the vapor pressure. Since polymers do not exert any vapor pressure, we turn to the method of Elbro et al. (1990) to find the a and b of a surrogate molecule that has the same chemical structure as the polymer segment of interest and identify these parameters as being of the segment. The COR EOS parameters of the surrogate molecule are determined by fitting vapor pressure and saturated liquid density as reported by Novenario et al. (1998). The rotational modes of the surrogate molecule are perceived to be different from those of the corresponding polymer segment due to the bonds that incorporate the segments in a chain. With parameters a and b of the polymer segment taken to be the same as those of the surrogate molecule, c of a polymer segment is determined by fitting pure polymer PVT data. Segmental parameters for the polymers studied in this work and their surrogate molecules are given in Table 1.
3144 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 3. van der Waals Binary Interaction Constants for the Attractive Parameters a polymer
solvent
T (K)
toluene 2-butanone cyclohexane carbon tetrachloride chloroform acetone benzene
321.65 321.65 307.15 293.15 323.15 323.15 318.15 333.15 293.15
10 300 10 300 440 000 500 000 290 000 15 700 900 000 900 000 500 000
0.018 41 0.026 71 0.020 35 0.005 31 0.000 64 0.047 83 0.022 50 0.022 50 0.014 82
Danner and High (1993) Danner and High (1993) Danner and High (1993) Danner and High (1993) Danner and High (1993) Danner and High (1993) Wohlfarth (1994) Wohlfarth (1994) Danner and High (1993)
338.15 353.20 313.15 313.15 318.15 328.15
100 000 50 000 50 000 1170 1170 1170
0.037 63 0.037 63 0.002 42 0.037 48 0.037 48 0.037 48
Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Danner and High (1993) Danner and High (1993) Danner and High (1993)
ethylbenzene
343.15 333.15 293.10 313.10 333.10 313.10 333.10 298.15 298.15 343.75
600 000 5 000 000 600 600 600 6000 6000 12 000 200 3272
0.012 30 -0.029 76 -0.231 12 -0.231 12 -0.231 12 -0.231 12 -0.231 12 -0.231 12 -0.231 12 0.016 52
PMMA
2-butanone toluene
321.65 321.65
19 770 19 770
0.023 98 0.021 15
Danner and High (1993) Danner and High (1993)
PVC
toluene tetrahydrofuran
316.35 315.65
34 000 34 000
0.015 98 0.006 39
Danner and High (1993) Danner and High (1993)
PVAC
benzene toluene
48 200 158 000 158 000 194 000
0.011 79 0.016 79 0.016 79 -0.005 27
PS
m-xylene PIB
benzene cyclohexane pentane
PEG
benzene chloroform water
polymer Mn
k12
VLE data source
Danner and High (1993) Gupta and Prausnitz (1995) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994)
chloroform
303.15 313.15 333.15 333.15
Danner and High (1993) Wohlfarth (1994) Wohlfarth (1994) Gupta and Prausnitz (1995)
PAMS
cumene
338.15
17 000
0.021 91
Wohlfarth (1994)
PEA
toluene chloroform dichloromethane carbon tetrachloride benzene
296.65 296.65 296.65 296.65 296.65
38 600 38 600 38 600 38 600 38 600
0.003 38 -0.032 55 -0.021 51 0.019 70 0.001 74
Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994) Wohlfarth (1994)
The nonsphericity parameter R that characterizes the model rotator of the COR equation is assigned a value of 1.5 in this work. This value of R corresponds to a dumbbell model rotator of two contacting spheres. In this work, COR parameters for both polymers and solvents are obtained based on R ) 1.5. Average absolute deviations from pure polymer PVT data using the parameters given in Table 1 range from 2% to 5%. Parameters for the solvents in this work are reported in Table 2. The values of the parameters are changed slightly from those reported previously using R ) 1.078 (Novenario et al., 1998), but the fitting of the puresolvent-saturated properties remains scarcely changed.
where xi is the mole fraction of component i; bi and ci are the parameters of pure i.
Mixing Rules for VLE Correlation and Prediction
while the fugacity coefficient of component i in a mixture, φi, at T and P is
Two classes of mixing rules for the attractive parameter a are investigated in this work to correlate and predict VLE of polymer + solvent systems: (1) van der Waals- and (2) solution model-based. In both cases, b and c for the mixture are expressed as follows:
ln φi )
b) c)
∑i xibi
(7)
∑i xici
(8)
From eq 1, the fugacity coefficient of pure i, φ0i , at temperature T and pressure P is
ln φ0i )
yi(4 - 3yi) (1 - yi)2
[
(R + 4)yi - 3yi2 ci + (R - 1) + 2 (1 - yi)2
]
(R + 1) ln(1 - yi) -
[]
ln(1 + 4yi) ai + (zi - 1) - ln zi RT bi
[
(9)
ci (R + 4)y - 3y2 + + (R 1) 2 (1 - y)2 (1 - y)2 n2a ∂ bi ln(1 + 4y) nb (R + 1) ln(1 - y) + (z RT ∂ni b y(4 - 3y)
]
[ ] () ( )
1) - ln z (10)
where yi ) bi/4Vi, y ) b/4V, ai, bi, ci are parameters of
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3145
Figure 1. Vapor-liquid equilibrium of pentane + PIB (1170). Data taken from Danner and High (1993).
Figure 3. Vapor-liquid equilibrium of cyclohexane + PS (440 000). Data taken from Danner and High (1993).
Figure 2. Vapor-liquid equilibrium of cyclohexane + PIB (50 000). Data taken from Wohlfarth (1994).
Figure 4. Vapor-liquid equilibrium of toluene + PS (10 300). Data taken from Danner and High (1993).
pure i; a and b are parameters of the mixture; n is the number of moles; zi ) PVi/RT and z ) PV/RT. Using the van der Waals mixing rule,
a)
∑i ∑j
When a solution model is incorporated into the COR EOS, the expressions analogous to eqs 11 and 12 are
a)b xixjaij
(11)
[(
RT
ln 1 +
()
-
1
GEsm
+
RT
) ∑ ( ) ]
∑i xi ln bi - ln b
0.6
xi
and
∂
( ) n2a nb
∂ni
i
()
) -bi
a
b
2
+
2
∑i xiaij
b
∂ (12)
where aii ) ai; aij ) (1 - kij)(aiaj)1/2; kij is a binary interaction parameter for unlike molecules i and j.
( )
n2a ai nb ) + ∂ni bi
RT
[)
1 ln 1 + 0.6
(
()
-ln γi + ln
+
ai
(13)
bi
]
bi bi +1b b
(14)
where GEsm and γi are the excess Gibbs free energy and
3146 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 5. Vapor-liquid equilibrium of cumene + PAMS (17 000). Data taken from Wohlfarth (1994).
Figure 7. Vapor-liquid equilibrium of benzene + PEG (600 000). Data taken from Danner and High (1993).
Figure 6. Vapor-liquid equilibrium of acetone + PS (15 700). Data taken from Danner and High (1993).
Figure 8. Vapor-liquid equilibrium of toluene + PVAC (158 000). Data taken from Wohlfarth (1994).
activity coefficient from a solution model. The constant 0.6 is the ratio V/b for the COR EOS at a low-pressure state of matching with the model. The derivation of eqs 13 and 14 has been given by Novenario et al. (1996).
Both correlative and predictive methods are important in process modeling. Whereas it is necessary to predict the phase behavior in the absence of experimental data, correlation of existing data is useful when a high degree of accuracy is required. The challenge is to adequately correlate available data with a minimum of parameters. The UNIFAC-FV method is chosen to represent methods of current use as it is the recommended method by Danner and High (1993) and Kontogeorgis et al. (1994) as a result of extensive studies. A predictive method for all mixtures studied here is obtained when UNIFAC, with the large number of reported parameters, is incorporated into COR. The parameter values reported by Hansen et al. (1991) are used in this work. Although the COR EOS with van der Waals mixing rules falls into distinctly different methods for VLE representation, being correlative when using best-fit k12 or predictive if k12 ) 0, the distinction is not necessarily
Results and Discussion Vapor-liquid equilibrium of binary polymer solutions obtained by the following methods are compared with experimental data: (1) Correlation by the COR EOS using van der Waals mixing rule with the best-fit k12. The k12 values for all the mixtures considered in this work are given in Table 3. (2) Prediction by the COR EOS using van der Waals mixing rule with k12 ) 0. (3) Prediction by the COR EOS using mixing rule based on the UNIFAC solution model. (4) Prediction by the UNIFAC-FV solution model.
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3147
Figure 9. Vapor-liquid equilibrium of 2-butanone + PMMA (19 770). Data taken from Danner and High (1993).
Figure 11. Vapor-liquid equilibrium of carbon tetrachloride + PS (500 000). Data taken from Danner and High (1993).
Figure 10. Vapor-liquid equilibrium of chloroform + PS (500 000). Data taken from Danner and High (1993).
Figure 12. Vapor-liquid equilibrium of chloroform + PEA (38 600). Data taken from Wohlfarth (1994).
permanent, but depends on usage. As best-fit k12 is determined for more mixtures with use in the future, correlation for the binary interaction parameter may be developed which can make the method predictive. In light of this potential, the results reported here using best-fit k12 is of value as an indication of the limit of future predictive methods with correlated k12. In Figures 1-19, VLE correlations and predictions are presented with experimental data in the same form that the data were reported. The solvent concentration in the liquid solution is shown as weight fraction w1. The equilibrium solvent property is given as a weight fraction activity coefficient Ω1 ()φ1/w1φ01), activity a1 ()φ1/φ01), or partial pressure of the solvent P1 ()the equilibrium pressure of the system as the solvent is taken to consist only of the volatile solvent). Figures 1-3 show calculated VLE for nonpolar mixtures pentane + PIB, cyclohexane + PIB, and cyclohexane + PS. Excellent fit of the data is obtained in all cases and the prediction is almost as good as the
correlation for cyclohexane + PIB when k12 is set to zero. UNIFAC incorporated into COR gave better predictions than UNIFAC-FV for all these mixtures. Two examples of mixtures where the polymer segment and the solvent are of comparable structures are given in Figures 4 (toluene + PS) and 5 (cumene + PAMS). The data are likewise well-correlated. UNIFAC + COR gave better predictions than UNIFACFV, but van der Waals predictions are about the same as those of UNIFAC-FV. Polar mixtures are given in Figures 6-9. Correlation using the van der Waals mixing rule best represents the data in all these cases except at low solvent concentrations (w1 < 0.1) for 2-butanone + PMMA (Figure 9) where UNIFAC-FV gave a more accurate description. Except for that mixture, UNIFAC + COR again yielded better (for acetone + PS in Figure 6) or comparable (for benzene + PEG in Figure 7 and toluene + PVAC in Figure 8) results versus UNIFAC-FV.
3148 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 13. Vapor-liquid equilibrium of carbon tetrachloride + PEA (38 600). Data taken from Wohlfarth (1994).
Figure 15. Vapor-liquid equilibrium of tetrahydrofuran + PVC (34 000). Data taken from Danner and High (1993).
Figure 14. Vapor-liquid equilibrium of toluene + PVC (34 000). Data taken from Danner and High (1993).
Figure 16. Vapor-liquid equilibrium of water + PEG (200). Data taken from Wohlfarth (1994).
Figures 10 and 11 present the calculated VLE for chloroform and carbon tetrachloride with PS, respectively. The data are accurately correlated in both solutions while significant deviations especially in the low solvent region is observed for the UNIFAC-FV model. The van der Waals predictions for chloroform + PS are excellent which is not surprising as the bestfit binary constant is of the order of 10-4. Mixtures of PEA with the same solvents are given in Figures 12 and 13. Satisfactory fit of the data is achieved in these cases, although the van der Waals predictions show marked deviation from the data. Mixtures of PVC with toluene and tetrahydrofuran are presented in Figures 14 and 15. Again, the van der Waals mixing rule accurately fit the data. However, predictions from UNIFAC-FV gave large deviations (especially in the low-solvent range for tetrahydrofuran + PVC), and incorporation of UNIFAC into COR indicates no significant improvement.
Figures 16-19 show VLE correlations for four water + PEG mixtures. The best-fit binary parameter has a high negative value to account for the large attractive potential between the unlike pair. The UNIFAC-FV was found to be inapplicable for these systems. All the other predictive methods studied (UNIFAC, UNIFAC incorporated into COR, and van der Waals mixing rule with k12 ) 0) gave unacceptable results (a1 > 1) and were not included in the plots. Data at 313.10 K for PEG(600) in Figure 17 and PEG(6000) in Figure 18 are well-correlated with k12 ) -0.231 12. The same binary parameter gave absolute deviations up to 3.5% and 6% for water + PEG(200) in Figure 16 and water + PEG(12 000) in Figure 19, respectively. Conclusion A surrogate molecule approach to parameter estimation for polymers in the chain-of-rotators equation of state allows a good description of vapor-liquid equilib-
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3149
Figure 17. Vapor-liquid equilibrium of water + PEG (600). Data taken from Wohlfarth (1994).
Figure 19. Vapor-liquid equilibrium of water + PEG (12 000). Data taken from Wohlfarth (1994).
Nomenclature
Figure 18. Vapor-liquid equilibrium of water + PEG (6000). Data taken from Wohlfarth (1994).
rium of polymer solutions. The examples presented in this work show generally excellent correlation of the data using one binary interaction parameter in the van der Waals mixing rule that is independent of temperature or polymer molecular weight. Adequate correlation is obtained even for the strongly interacting mixture of water + PEG which cannot be described by the other methods studied. The predictive methods in this work, through incorporation of UNIFAC (using model parameters available in the literature) into COR by the method previously developed by the same authors (Novenario et al., 1996) and without the use of any adjustable parameter in the van der Waals mixing rule, give generally good VLE representation. In most cases, incorporation of UNIFAC into the COR EOS improves the performance of the solution model better than the addition of a free volume term as in the UNIFAC-FV method. Both predictive methods proposed give better or comparable results versus the widely used UNIFACFV model for the majority of mixtures studied.
y ) packing fraction, )b/4V R ) nonspherecity parameter for the elemental rotator in the COR EOS a ) EOS attractive parameter b ) EOS excluded volume parameter c ) COR rotational degrees of freedom parameter ai ) EOS attractive parameter for pure i bi ) EOS excluded volume parameter for pure i ci ) COR rotational degrees of freedom parameter for pure i aseg ) segmental attractive parameter for the polymer bseg ) segmental excluded volume parameter for the polymer cseg ) segmental rotational degrees of freedom parameter for the polymer r ) number of segments in the polymer chain n ) number of moles ni ) number of moles of component i z ) compressibility factor ) PV/RT k12 ) binary interaction coefficient for a in the van der Waals mixing rule GEsm ) excess Gibbs free energy from a solution model γi ) activity coefficient of component i from a solution model w1 ) weight fraction of the solvent in a binary polymer solution f1 ) fugacity of the solvent in a binary polymer solution f 01 ) fugacity of pure solvent Ω1 ) solvent weight fraction activity coefficient in a binary polymer solution a1 ) solvent activity in a binary polymer solution P1 ) partial pressure of the solvent (also the equilibrium pressure) in a binary polymer solution
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Received for review December 1, 1997 Revised manuscript received March 23, 1998 Accepted April 14, 1998 IE970906M
