Development Of A Universal Group Contribution Equation Of State

Ind. Eng. Chem. Res. 2003, 42, 6205-6211

6205

GENERAL RESEARCH Development of a Universal Group Contribution Equation of State. 4. Prediction of Vapor-Liquid Equilibria of Polymer Solutions with the Volume Translated Group Contribution Equation of State Li-Sheng Wang,‡ Jens Ahlers,† and Ju 1 rgen Gmehling*,† Universita¨ t Oldenburg, Technische Chemie, D-26111 Oldenburg, Germany, and Department of Chemical Engineering, Beijing Institute of Technology, 100081, China

The simplified mixing rule for the Peng-Robinson EOS parameters proposed by Ahlers and Gmehling has been extended to calculate the VLE behavior of polymer solutions. To avoid the calculation of the molar volumes in the mixing rule as required in the UNIFAC-FV model, the quadratic mixing rule for parameter b has been modified to take into account the free volume contribution to the excess Gibbs energy for the solvent-polymer solutions. The group interaction parameters and relative van der Waals surface area parameters of the original UNIFAC model are used for the calculations. Satisfactory predictions are obtained for 18 solvent-polymer systems, which cover a wide range of molar mass of the polymers. Additionally, activity coefficients at infinite dilution based on weight fraction for 36 systems have been calculated. The results of this paper show that the presented mixing rule results in a simple, predictive model with satisfying accuracy for the application in industrial process design. 1. Introduction The prediction of the phase equilibrium behavior of polymer solutions with conventional models has received much attention for designing numerous separation processes, such as polymer devolatilization, pervaporation, the selection of the optimum membrane for a given separation process, consideration of safety aspects (e.g., most suitable material for safety gloves), and so forth, where a reliable description for the vaporliquid equilibrium (VLE) of solvent-polymer systems is required. It has been demonstrated that the group contribution method UNIFAC in its original form cannot be used directly for polymer solutions since it generally leads to an underestimation of solvent activity. The Modified UNIFAC model cannot be used either since its combinatorial term with an empirical exponent leads to an overestimation of solvent activities. To extend the UNIFAC model for the prediction of the VLE behavior of polymer solutions, the UNIFAC-FV group contribution activity model1 has been developed, where the Oishi-Prausnitz term is added to calculate the free volume effects. The origin of this term is based on the Flory equation of state with the simplification that Flory’s potential energy parameter is equal to zero. For mixtures with modest molar mass, this term provides only a small contribution that is usually negligible in the UNIFAC model. There are other activity models * To whom correspondence should be addressed. Tel.: +49441-798-3831. Fax: +49-441-798-3330. E-mail: gmehling@ tech.chem.uni-oldenburg.de. Web site: http://www.unioldenburg.de/tchemie. † Universita¨t Oldenburg. ‡ Beijing Institute of Technology.

applicable to polymer solutions, such as the simple activity model ENTROPIC-FV proposed by Kontogeorgis et al.,2 with a combined combinatorial-free volume term and retaining of the residual term of UNIFAC. Phase equilibria of systems containing nonpolar and supercritical components can be calculated with the help of cubic equations of state (EOS). The application range of cubic equations of state has been extended to polar systems by the development of gE mixing rules. When the attractive term a of the equation of state is linked to the excess Gibbs energy of a gE model or a group contribution method (EOS-gE model), predictions of high accuracy from atmospheric to high pressures can be obtained. Group contribution methods such as UNIFAC,3 Modified UNIFAC,4 and ASOG5 are widely used in EOS-gE models for systems with modest molar mass, for example, in PSRK6 and PRASOG,7 because of their ability to predict high-pressure VLE using parameters derived from low-pressure data. The PRASOG model combines the PR EOS with the group contribution method ASOG using a zero-pressure gE mixing rule consistent with the second virial coefficient condition. The PSRK model, combining the SRK EOS with the group contribution method UNIFAC, allows employing the large parameter matrix of UNIFAC for the reliable prediction of VLE and gas solubilities for a large variety of systems. To improve the prediction ability of the PSRK model for asymmetric systems, Li et al.8 introduced an empirical correction into the model. So-called effective van der Waals volumes and surface areas for the subgroups in alkanes allow a reliable prediction of asymmetric systems up to pentatetracontane. However, it was pointed out by Li et al. that by simply canceling the combinatorial part of the PSRK mixing rule (∑xi ln(b/bi)) with the Flory-Huggins term in the UNIFAC

10.1021/ie0210356 CCC: $25.00 © 2003 American Chemical Society Published on Web 10/25/2003

6206 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 Table 1. Peng-Robinson Twu r-Function Parameters L, M, and N for Some Solvents component

Pc [MPa]

vc [cm3/mol]

Tc [K]

ω

L

M

N

butane pentane hexane heptane octane decane dodecane cyclohexane 1-pentene benzene toluene ethylbenzene ethanol 1-propanol acetone 2-butanone 4-methyl-2-pentanone chloroform carbon tetrachloride 1,4-dioxane vinyl acetate ethyl actetate

3.800 3.369 3.014 2.734 2.495 2.100 1.810 4.073 3.526 4.894 4.114 3.607 6.384 5.168 4.702 4.154 3.273 5.472 4.560 5.208 4.357 3.830

255.00 304.00 370.00 432.00 492.00 603.00 713.00 308.00 300.00 259.00 316.00 374.00 167.00 218.50 209.00 267.00 371.00 239.00 276.00 238.00 265.00 286.00

425.20 469.70 507.40 540.30 568.80 617.90 658.80 553.40 464.70 562.10 591.70 617.10 516.20 536.70 508.10 535.60 571.00 536.40 556.40 587.00 525.00 523.20

0.193 0.251 0.296 0.351 0.394 0.490 0.562 0.213 0.245 0.212 0.257 0.301 0.635 0.624 0.309 0.329 0.400 0.216 0.194 0.288 0.340 0.363

0.176 56 0.752 33 0.965 06 0.939 55 0.832 91 0.655 06 0.458 39 0.537 89 0.353 95 0.884 02 0.650 71 0.805 93 1.130 43 1.279 39 0.606 55 0.643 97 0.0726 72 0.843 99 0.080 81 0.963 59 0.949 74 1.186 84

0.852 13 0.897 51 0.942 50 0.909 56 0.871 60 0.825 30 0.794 47 -1.944 7 0.841 77 0.875 28 0.853 04 0.893 10 0.986 59 0.855 56 0.892 91 0.873 00 0.950 89 0.960 46 0.885 69 0.945 60 0.926 16 0.9737 2

2.272 41 0.934 40 0.816 37 0.901 35 1.080 34 1.465 50 2.037 81 0.334 47 1.575 45 0.723 70 1.051 81 0.965 63 1.13010 0.947 26 1.284 68 1.220 77 12.117 9 0.809 27 3.663 51 0.787 34 0.892 08 0.730 18

model, better prediction results could be achieved for asymmetric systems without the correction. Following this idea and suggestions of Chen,9 Ahlers and Gmehling10 proposed the model VTPR, which contains a more simplified mixing rule for Peng-Robinson EOS parameter a (see eq 6 of this paper). Together with a nonlinear quadratic mixing rule for parameter b (see eqs 9 and 10), the VTPR model leads to satisfactory descriptions of asymmetric systems containing longchain alkanes. In recent years a few models for the calculation of polymer solutions have been proposed, which are based on a van der Waals type cubic equation of state (EOS).7,11,12 When models such as UNIFAC-FV or ENTROPIC-FV are applied in a EOS-gE mixing rule to predict the VLE and PVT properties of polymer solutions, volumetric properties at the temperature of the solution must be calculated at first in the mixing rule. To avoid calculating the molar volumes in the mixing rule for polymer solutions, some authors introduced a binary interaction parameter for achieving better agreement with the experimental data.11,12 However, this parameter is usually system- and temperature-dependent. As one part of the development of a predictive equation of state based on a large database, the objective of this work is to apply the new VTPR model to the VLE prediction for polymer solutions. 2. Peng-Robinson Equation of State and gE Mixing Rule for Polymer Solution 2.1. Equation of State. The equation of state used in this work is the volume translated Peng-Robinson EOS

P)

a(T) RT (1) v + c - b (v + c)(v + c + b) + b(v + c - b)

where a and b are mixture energy and size parameters, respectively. For solvent species, the pure component parameters aii and bii can be obtained from critical data

Table 2. Peng-Robinson Equation of State Parameters of Some Polymers and Mean Relative Deviation in Volume (Extract of the Table Taken from Louli and Tassios12)

polymer

temp range [K]

LDPE PS PVAc PMMA PIB PEO PDB PVC PP

394-476 388-469 308-373 387-432 326-383 361-497 277-328 412-471 443-570

pressure range a/M [bar] [dm6 bar/mol g] 0-1960 0-2000 0-800 0-1000 0-685 0-2835 0-1960 0-1800 0-1000

1.373984 1.315409 1.847343 1.277496 2.307400 2.278342 0.942530 0.998549 1.627639

b/M [dm3/mol g]

AAD [%]

1.1991 × 10-3 0.9549 × 10-3 0.8428 × 10-3 0.8407 × 10-3 1.0882 × 10-3 0.9497 × 10-3 1.0549 × 10-3 0.7154 × 10-3 1.1716 × 10-3

2.62 2.17 1.38 1.77 1.40 2.48 2.35 1.85 1.77

Tc and Pc:

aii(T) ) (0.457235R2Tci2/Pci)R(T)

(2)

bii ) 0.077796RTci/Pci

(3)

In eq 2, the R-function proposed by Twu et al.13 is used:

R(T) ) TrN(M-1) exp[L(1 - TrNM)]

(4)

The R-function guarantees an accurate description of the pure component vapor pressures. Table 1 contains a list of the Twu parameter N, M, L of different solvents. If no Twu parameters are available, the R(T)-value can be calculated as a function of the acentric factor ω, which is described in detail by Ahlers and Gmehling.10 The Peng-Robinson EOS parameters aii and bii for the polymers investigated in this work are taken from the paper of Louli and Tassios.12 They provide specified aii and bii values for different kinds of polymers and assume that a/M and b/M are independent of molecular weight MW. Table 2 shows a selection of these data and the mean relative deviations in volume. When a mixing rule for calculating the equation of state parameters a and b of the mixture is employed, the properties of various polymers with different solvents can be obtained.

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The volume translation improves significantly the description of liquid densities. In the VTPR model the translation parameter ci is defined as the difference between experimental and predicted volume at the reduced temperature Tr ) 0.7. If no experimental liquid densities are available, ci can be obtained from critical data:14

RTc,i (1.5448zc,i - 0.4024) Pc,i

ci ) 0.252

(5)

A linear mixing rule is applied for calculating the parameter c of the mixture. However, since the volume translation has no influence on the VLE calculation, it is also possible to use the original Peng-Robinson equation of state for the following VLE calculations of polymer solutions. 2.2. gE Mixing Rule. In the VTPR model the following mixing rule for the attractive parameter a is used:10

a ) b

∑

xi

aii gEres + bii A

(6)

Besides the parameters of the pure components aii and bii (solvents: eq 2-4; polymers: Table 2), this gE mixing rule contains the residual part of the original UNIFAC method gEres (section 2.4). The parameter A only depends on the chosen EOS and reference state. According to Ahlers and Gmehling,10 the liquid volume at 1 atm was chosen as the reference state, and a constant packing fraction 1/u is assumed: L vL vi ) u ) ui ) b bii

(7)

Fischer and Gmehling have shown15 that the assumption of a constant packing fraction (1/u) for the PSRK model is justified. The calculated v/b values (respectively u values) for about 80 components scatter around a value of u ) 1.1, which is in agreement with the value empirically fitted to experimental phase equilibrium data. The same way Ahlers and Gmehling10 obtained a constant value of u ) 1.22498 for the VTPR model. As a result, the parameter A

A)

u + (1 - x2) 1 ln 2x2 u + (1 + x2)

(8)

amounts to -0.53087. 2.3. b Mixing Rule. The parameter b of the mixture is calculated using a quadratic mixing rule:

b)

∑i ∑j xixjbij

(9)

In the VTPR model the binary parameter bij is obtained by applying the following combining rule:9 3/4

bij

bii3/4 + bjj3/4 ) 2

(10)

The exponent 0.75 was adopted from the group contribution method Modified UNIFAC (Dortmund),16 where the same exponent is used for the relative van der Waals

Table 3. Calculated Results for the Cross Co-volume bij Based on Different Molecular Size and Exponents E (Eq 10a) system: hexane (1) + alkane (2) butane (symmetric) butane (symmetric) butane (symmetric) hexatriacontane (asymmetric) hexatriacontane (asymmetric) hexatriacontane (asymmetric) LDPE (M ) 300 kg/mol) (polymer) LDPE (M ) 300 kg/mol) (polymer) LDPE (M ) 300 kg/mol) (polymer)

b1 b2 [L/mol] [L/mol] 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11

0.07 0.07 0.07 1.18 1.18 1.18 359.73 359.73 359.73

E

b12 [L/mol]

0.5 0.09 0.75 0.09 1 0.09 0.5 0.5 0.75 0.57 1 0.64 0.5 93.09 0.75 143.20 1 179.92

volume parameter ri in the combinatorial part. The exponent of 0.75 provides a better description of the free volume effect of asymmetric systems, which has always a positive contribution to the activity coefficients. Therefore, when an exponent E ) 1 (smaller free volume contribution) instead of 0.75 is used, the vapor pressures will be predicted too low for highly asymmetric systems, as already shown by Ahlers and Gmehling for the system ethane + decane.10 The free-volume effect increases with increasing volume differences of the compounds considered. To take into account the free volume contribution to the excess Gibbs energy in polymer solutions, a suitable value for the exponent E of the cross co-volume bij between component i and j was determined empirically. The influence of the exponent E, which seems to represent a sensitive parameter for the description of the free volume effect, can be discussed with the help of the following generalized form of eq 10:

bEij

bEii + bEjj ) 2

(10a)

Table 3 lists some results for the cross co-volume based on different molecular size and the effect of the exponent E on the values of the co-volume. For systems with compounds of similar molecular sizes, for example, hexane + butane, the different values of E deliver similar results. For systems with compounds of different molecular sizes, so-called asymmetric systems such as hexane + hexatriacontane, the value of the cross covolume is reduced from 0.64 to 0.5 L/mol if the exponent E is reduced from 1 to 0.5. As can be seen in Table 3, for polymer solutions the cross co-volume strongly depends on the exponent E. The VTPR model provides too low bubble point predictions for polymer-solvent systems (Figure 1 A-F). With a smaller cross co-volume bij, the predicted saturation pressure is increased. Since a lower exponent E of the cross-volume bij leads to smaller bij values, better vapor pressure predictions should be obtained, choosing smaller E values. Empirically for the VTPR model an optimal exponent E ) 0.5 was found. When a polymer is dissolved in a solvent, the chain of the polymer will become more flexible than it is in the polymer state (in fact, a lot of pure polymers are in a crystallized state). Therefore, the cross covolume of a polymer with a solvent will be much smaller. That is the reason for us to choose a value E ) 0.5. 2.4. UNIFAC Model for Polymer Solution. In eq 6 (gE mixing rule) only the residual part of the UNIFAC model is required (gEres) for the calculation of the pa-

6208 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003

Figure 1. Experimental and with the VTPR model predicted values for different polymer solutions. Comparison of different values of the exponent E of the binary parameter bij (eq 10a). (s) E ) 0.5, (‚‚‚) E ) 0.6, (- - -) E ) 0.75. (A) Toluene (1) + polystyrene [M ) 290 kg/mol] (2) at 298.15 K (b) and 333.15 K (2); (B) 1,4-dioxane (1) + polystyrene [M ) 10.92 kg/mol] (2) at 323.15 K (2); (C) benzene (1) + poly(ethylene oxide) [M ) 10 kg/mol] (2) at 343.15 K (b); (D) 2-butanone (1) + polystyrene [M ) 290 kg/mol] (2) at 343.15 K (9), 2-butanone (1) + polystyrene [M ) 290 kg/mol] (2) at 298.15 K (2), 2-butanone (1) + polystyrene [M ) 10.92 kg/mol] (2) at 321.65 K (b); (E) benzene (1) + polystyrene [M ) 63 kg/mol] (2) at 333.15 K (9), 318.15 K (b), and 288.15 K (2); (F) cyclohexane (1) + polyisobutylene [M ) 50 kg/mol] (2) at 311.15 K (b).

rameter a in the mixture. To calculate the activity coefficient γi , the following equations are used:

∑i xi ln γres i

gE ) gEres ) RT

ln γi ) ln γres i and

(11) (12)

ln γres i )

∑k v(i)k (ln Γk - ln Γ(i)k )

(13)

in which vk(i) is the number of group k in component i, Γk is the group residual activity coefficient, and Γk(i) is the group residual activity coefficient in a reference solution only containing molecules of type i. The group activity coefficient can be calculated from the original

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6209

Figure 2. Experimental and predicted values for different polymer solutions. (s) VTPR model with the exponent E ) 0.5 for the binary parameter bij (eq 10a). (A) 2-Butanone (1) + poly(methyl methylacrylate) [M ) 19.77 kg/mol] (2) at T ) 321.65 K (2), toluene (1) + poly(methyl methylacrylate) [M ) 19.77 g/mol] (2) at T ) 321.65 K (b); (B) vinyl acetate (1) + poly(vinyl acetate) [M ) 143 kg/mol] (2) at 303.15 K (b); (C) ethyl benzene (1) + polybutadiene [M ) 250 kg/mol] (2) at 353.15 K (b); (D) n-butane (1) + polyisobutylene [M ) 1000 kg/mol] (2) at 319.65 K (b), at 308.15 K (2), and at 298.15 K (9), (E) tetrachloromethane (1) + polypropylene [M ) 15 kg/mol] (2) at 298.15 K (b); (F) chloroform (1) + polystyrene [M ) 1.040 kg/mol] (2) at 298.15 K (b); (G) cyclopentane (1) + low-density polyethylene [76 kg/mol] (2) at 474.15 K (b), pentane (1) + low-density polyethylene [76 kg/mol] (2) at 423.65 K (2), cyclopentane (1) + low-density polyethylene [76 kg/mol] (2) at 425.65 K (9); (H) toluene (1) + low-density polyethylene [76 kg/mol] (2) at 393.15 K (b), 2-propylamine (1) + low-density polyethylene [76 kg/mol] (2) at 427.15 K (9), 1-pentene (1) + low-density polyethylene [76 kg/mol] (2) at 423.15 K (2).

6210 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 Table 4. Binary Solvent-Polymer Solutions Investigated in This Work

no. polymer 1

PS

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

molar mass of figure no. temp polymer shown in data [K] [g/mol] this work source

system solvent (1) toluene

333.15 290000 298.15 290000 PS 1,4-dioxane 323.15 10920 PEO benzene 343.15 10000 PS 2-butanone 343.15 290000 298.15 290000 321.65 10920 PS benzene 333.15 63000 318.15 63000 288.15 63000 PIB cyclohexane 311.15 50000 PMMA 2-butanone 321.65 19770 PMMA toluene 321.65 19770 PVAc vinyl acetate 303.15 143000 PBD ethyl benzene 353.15 250000 PIB n-butane 319.65 1000000 308.15 1000000 298.15 1000000 PP carbon tetrachloride 298.15 15000 PS chloroform 298.15 1040 LDPE cyclopentane 474.15 76000 425.65 76000 LDPE pentane 423.65 76000 LDPE toluene 393.15 76000 LDPE 2-propylamine 427.15 76000 LDPE 1-pentene 423.15 76000

1A 1A 1B 1C 1D 1D 1D 1E 1E 1E 1F 2A 2A 2B 2C 2D 2D 2D 2E 2F 2G 2G 2G 2H 2H 2H

17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18

expression of the UNIFAC model:

ln Γk ) Qk[1 - ln(

θmψmk) - ∑(θmψkm/∑θnψnm)] ∑ m m n

(14)

θm )

QmXm

∑n

(15)

Q nX n

where the group interaction parameter ψnm is defined as in the UNIFAC model3:

(

ψnm ) exp -

)

anm T

(16)

The group interaction parameters and relative van der Waals surface areas of the UNIFAC model are used for all calculations in this work. To apply the above model for polymer-containing solutions, the group fraction Xm can be calculated from

Xm )

∑j djv′m,jxj ∑j ∑n djv′n,jxj

(17)

where d j is the degree of polymerization of molecule j (for the solvent species, d ) 1), v′m,j is the number of group m in a solvent species j or in a repeat unit of polymer j, and xj is the mole fraction of component j in the mixture. 3. Results and Discussions Bubble point pressure calculations are carried out for 18 binary systems of solvent-polymer solutions in which 8 polymers including LDPE (low-density polyethylene), PS (polystyrene), PEO (poly(ethylene oxide)), PMMA (poly(methyl methacrylate)), PVAc (poly(vinyl acetate)), PBD (polybutadiene), PIB (polyisobutylene),

Table 5. Experimental and Calculated Solvent Activity Coefficients at Infinite Dilution Based on Weight Fraction for Polymer Solutions polymer (molar mass M) (2) + solvent (1) PEO (300000) + ethanol PEO (4000000) + carbon tetrachloride PEO (300000) + benzene PEO (300000) + 1-propanol PEO (4000000) + acetone PEO (300000) + 2-butanone PEO (300000) + ethyl acetate PBD (23956) + carbon tetrachloride PBD (23956) + pentane PBD (23956) + ethyl acetate PBD (23956) + 4-methyl-2-pentanone PBD (23956) + cyclohexane PS (102800) + 1,4-dioxane PS (97600) + 2-butanone PS (97600) + ethanol PS (2145000) + benzene PS (102800) + ethylbenzene LDPE (235000) + hexane LDPE (235000) + heptane LDPE (235000) + octane LDPE (235000) + cyclohexane LDPE (235000) + benzene LDPE (235000) + toluene LDPE (82000) + octane LDPE (82000) + 3-methylhexane PVC (101500) + acetone PVC (101500) + 2-butanone PVC (101500) + acetaldehyde PVAC (331400) + vinyl acetate PVAC (331400) + ethyl acetate PVAC (331400) + toluene PVAC (500000) + acetone PMMA (200000) + benzene PMMA (200000) + ethylbenzene PMMA (200000) + butylbenzene PMMA (200000) + methyl acetate

AAD T [K] Ω1∞exp.a Ω1∞calc. [%]b 373.2 375.1 423.7 393.2 393.2 423.2 373.2 373.0 353.0 353.0 363.2 373.0 423.2 420.0 423.2 420.9 423.2 423.2 423.2 423.2 423.2 423.2 423.2 418.3 418.3 398.2 413.2 413.2 423.2 423.2 448.2 417.1 383.2 383.2 383.2 398.2

a Experimental data of Hao et al.17 Ω1,exp∞|/Ω1,exp∞) × 100.

b

9.82 3.65 4.61 8.29 7.74 5.5 5.67 1.82 5.51 6.02 6.89 4.18 5.17 7.13 17.5 5.45 4.96 5.91 5.26 4.85 3.95 4.25 3.94 4.73 5.04 10.8 10.2 11.4 5.28 5.97 6.57 5.84 5.65 7.5 10.07 6.25

10.95 3.31 4.80 9.71 7.34 6.7 5.85 2.28 5.62 7.17 8.02 4.97 5.34 6.03 14.31 4.79 5.31 4.45 4.85 5.09 3.96 4.69 5.11 5.15 5.54 9.43 8.11 7.19 4.89 5.5 5.53 7.79 5.36 6.74 8.67 5.55

11.5 9.3 4.1 17.1 5.2 21.8 3.2 25.2 2.0 19.1 16.4 18.9 3.3 15.4 18.2 12.1 7.1 24.7 7.8 4.9 0.3 10.4 29.7 8.9 9.9 12.7 20.5 36.9 7.4 7.9 15.8 33.4 5.1 10.2 13.8 11.2

AAD [% ] ) (|Ω1,calc∞ -

and PP (polypropylene) are included with a variety of polar and nonpolar solvents. The experimental data, that is, temperature and molecular weight of these polymers with different solvents, are listed in Table 4. The results obtained are shown in Figures 1 and 2. Figure 1A-F shows the calculated and experimental bubble point pressures in different polymer solutions using different values of E in the combining rule for the co-volume bij. For UNIFAC and Modified UNIFAC the solvent activities of polymer solutions are overestimated. When the combinatorial part of the group contribution method UNIFAC in the VTPR model is skipped, this effect can be avoided. In all cases the predicted bubble point pressures are too low compared with experimental values when E ) 0.75 is used, as defined in the original VTPR mixing rule. Much better results are obtained by using an E value of 0.5. Agreement with experiment is usually within 10%. With the use of an exponent of E ) 0.5, calculations have been performed with the new model for further systems. A comparison of the experimental and the predicted results is shown in Figure 2A-H. It can be seen that also for these systems good agreement is obtained, although again systems measured at different temperatures (298-474 K) and with very different molar mass of the polymer (1000-1000000 kg/mol) were considered. Additionally, experimental17 and calculated activity coefficients at infinite dilution based on weight fraction for 36 polymer-solvent systems are given in Table 5. These data are mainly measured with the help of inverse gas chromatography. In this case a temperature

Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6211

range between 353 and 448 K is covered, a temperature range which is typical for this technique. The mean relative deviations are approximately 14%. When compared with the experimental accuracy of this technique, the predicted results can be regarded as satisfactory.

i ) component i ij ) interaction between components i and j k ) structural group res ) residual part

4. Conclusions A simplified mixing rule for Peng-Robinson EOS parameters is presented in this paper for polymer solutions. A quadratic mixing rule for parameter b has been modified to take into account the free volume contribution to the excess Gibbs energy of solventpolymer solutions. The original group interaction parameters and relative van der Waals surface areas of the UNIFAC model are used in the new model. Satisfactory prediction results are obtained for vapor-liquid equilibria and activity coefficients at infinite dilution of solvent-polymer systems, which cover a large range of molar mass of the polymers. The results of this paper show that the presented mixing rules provide a simple, reliable, purely predictive model with satisfying accuracy for application in process design.

(i) ) component i E ) excess property E ) exponent in the combining rule res ) residual part ∞ ) infinite dilution

Acknowledgment L.-S. Wang thanks DAAD (Deutscher Akademischer Austauschdienst) for the financial support of his research stay in Oldenburg. Nomenclature a ) cohesive energy parameter of the PR equation of state a1 ) solvent activity A ) constant in the gE mixing rule anm ) temperature-independent group interaction parameter b ) volumetric parameter of the PR equation of state c ) temperature-independent volume correction for the VTPR equation [m3/mol] d ) number of repeat units EOS ) equation of state g ) Gibbs energy L, M, N ) Twu-Bluck-Cunningham-Coon R function parameters M ) molar mass of the polymer [g/mol] P ) pressure [bar] PR ) Peng-Robinson equation of state PSRK ) predictive Soave-Redlich-Kwong R ) general gas constant [J/mol K] Q ) relative van der Waals surface area parameter of the structural group k T ) absolute temperature [K] u ) inverse packing fraction v ) molar volume [m3/mol] VLE ) vapor-liquid equilibrium VTPR ) volume translated Peng-Robinson equation of state w ) weight fraction x ) liquid mole fraction X ) group mole fraction Greek Letters R ) temperature-dependent function of a(T) γ ) activity coefficient θ ) surface area fraction ψ ) temperature-dependent function in the residual part v′m,j ) number of group m in a solvent species j or in a repeat unit of the polymer j Ω ) activity coefficient based on weight fraction Subscripts c ) at the critical point

Superscripts

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Received for review December 17, 2002 Revised manuscript received August 26, 2003 Accepted August 26, 2003 IE0210356