Ind. Eng. Chem. Res. 1995,34, 2117-2124
2117
GENERAL RESEARCH Prediction of Vapor-Liquid Equilibria in Polymer Solutions Using an Equation of Statemxcess Gibbs Free Energy Model Nikolaos S. Kalospiros* and Dimitrios Tassios Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Street, Zographou Campus, Zographos, GR-15780, Greece
A new mixing rule has been developed, which is coupled with a translated and modified version of the Peng-Robinson equation of state (EoS) and the entropic-FV activity coefficient model. The resulting EoS/GE model is applied to the correlation and prediction of vapor-liquid equilibria (VLE) in solventipolymer mixtures, after appropriate modification of the cubic EoS to describe the volumetric behavior of the pure polymer. The model yields very good correlation of experimental equilibrium pressures in concentrated polymer solutions, but more importantly, successful predictions of the same data are obtained. A comparison with other EoS-based models applicable to the prediction of VLE in polymer solutions is also presented.
Introduction Phase equilibria in polymer solutions are of fundamental importance in a number of process applications. For example, devolatilization of low molecular weight solvents, like plasticizers, unreacted monomers, or toxic additives, is essential for safety, environmental, and product quality reasons and requires information on the equilibrium pressure of the solvent'polymer mixture as a function of the solvent weight fraction. Consequently, rational design and operation of such processes requires a quantitative description of the vapor-liquid equilibria (VLE) behavior in solventlpolymer systems. A variety of models has been developed for this purpose, which can be classified into two general categories: free-volume activity coefficient or excess Gibbs free energy (GE)models (e.g., Oishi and Prausnitz, 1978; Iwai and Arai, 1985, 1989; Iwai et al., 1990, 1991; Kontogeorgis et al., 1993) and equations of state (EoS) (e.g., Donohue and Prausnitz, 1978; Morris et al., 1987; Sako et al., 1989; High and Danner, 1990; Chen et al., 1990; Kontogeorgis et al., 1994a). The former are based on the Flory-Huggins equation (Flory, 1941, 1942; Huggins, 1941, 1942) and take into account the differences between the free volumes of solvents and polymers. They have been rather successful in the prediction of solvent activities in polymer solutions. However, prediction of equilibrium pressures requires coupling with a model, like a simple EoS, capable of correct description of vapor-phase fugacities. A rather successful activity coefficient model is the predictive entropic-FV model (Elbro et al., 1990; Kontogeorgis et al., 1993). The combinatorial part in this model is a modification of the Flory-Huggins term, where free-volume fractions are used instead of surface area or volume fractions. For the residual part, the entropic-FV model utilizes the UNIFAC term with the linearly temperature-dependent parameter table developed by Hansen et al. (19921, based on low-pressure VLE data of nonpolymeric systems. Contrary to activity coefficient models, EoS have the advantage that they are applicable to both vapor and
liquid phases, and thus prediction of VLE can be performed using only one model for both phases. Most EoS proposed for polymer solutions have their sound theoretical basis on statistical thermodynamics and, thus, are rather complex, noncubic equations. The most important of these EoS are the group-contribution lattice-fluid (GCLF)equation (High and Danner, 1990) and the EoS based on the perturbed hard chain theory of Donohue and Prausnitz (1978, such as the perturbed soft chain theory model (PSCT) and its group-contribution version (GPSCT),developed by Morris et al. (1987). All the above-mentioned EoS models generally yield accurate predictions of VLE in polymeric systems. The group-contribution Flory (GC-Flory) EoS (Chen et al., 19901, on the other hand, although called as such, has been developed as an activity coefficient model and is applicable only to the liquid phase. Consequently, in order to yield VLE predictions, it requires coupling with an appropriate EoS for the description of the vapor phase. Simple cubic EoS have been proposed recently (Sako et al., 1989; Kontogeorgis et al., 1994a; Harismiadis et al., 1994) that perform rather accurately in the correlation of VLE behavior in nearly athermalhonpolar polymer solutions. Specifically, Kontogeorgis et al. (1994a) applied the van der Waals (vdW) EoS to polymeric systems by using the Berthelot combining rule for the evaluation of the mixture attractive-term parameter. Accurate predictions were made possible by linearly correlating the interaction parameter of the Berthelot combing rule with the molecular weight of the solvent (Harismiadiset al., 1994). Extension to systems containing polar solvents has not been examined yet. Recently, the applicability of simple cubic EoS has been extended to complex systems, such as highly nonideal mixtures or mixtures containing supercritical gases, upon coupling with existing excess Gibbs free energy (GE)models, leading to the development of a number of EoS/GE models (Michelsen, 1990a, b; Dah1 and Michelsen, 1990; Holderbaum and Gmehling, 1991; Wong and Sandler, 1992; Orbey et al., 1993; Boukouvalas et al., 1994). Despite the success of these models
0888-5885/95/2634-2117$09.O0/00 1995 American Chemical Society
2118 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 Table 1. Pure-Compound Parameters for the Polar Solvents Considered in This Studs, Used in the t-mPR EoS ~
compound acetone MEK chloroform
Tc (K)
P, (bar)
508.2 535.5 536.4
47.02 41.54 54.72
w 0.306 0.324 0.213
Table 2. Percent Errors between Predicted and Experimental Pure-Polymer Liquid Molar Volumes as a Function of Pressures Using the Method of Kontogeorgis et al. (1994a) and the One Proposed Herea exptl vol % errorb pressure (bar) (cm3/mol) Kontogeorgis et al. this work HDPE (50 000) at 398.15 K 62 600 62 600 62 400 62 100 58 700 56 500
0 10
50 100 1000 2000
0 10 50 100 1000 2000
0.0 -15 -17 -17 -13 -10
PIB (100 000) at 323.15 K 110 700 0.1 110 700 110 400 110 100 105 900 102 700
-11 -12 -12 -8.6 -5.7
286 300 286 200 285 500 284 700 273 400 264 800
0.8 -10 - 14 - 14 -10 -7.5
0.1 -1.3 -2.0 -1.9 1.8 5.0
-0.0 -2.1 -2.5 -2.3 1.6 4.9
a a and b have been evaluated with the latter method using experimental data at 398.25 and 471.15 Kfor HDPE, 317 and 383 K for PIB, and 382 and 469 K for PS. The number-average molecular weight of the polymer is given in parentheses. All experimental data from Rodgers (1993). Error % is defined as [(Vcaled - V e x p d ~ e x pxd l100, where the subscripts calcd and exptl refer to calculated and experimental molar volumes, respectively.
in the prediction of low- and high-pressure VLE in such mixtures, only the mixing rules proposed by Wong and Sandler (1992) have been applied to the correlation of VLE behavior, based on coupling a cubic EoS with the Flory-Huggins GE model (Orbey and Sandler, 1994). Our objective in this work is the development of an EoS/GEmodel, applicable to the prediction, rather than mere correlation, of equilibrium pressures in concentrated polymer solutions. This model is based upon coupling the translated and modified Peng-Robinson (t-mPR) EoS (Magoulas and Tassios, 1990) and the entropic-FV GE model (Kontogeorgis et al., 1993) and is applied successfully in the correlation and prediction of VLE in several solventlpolymer mixtures.
T h e t-mPR EoS In this work, the t-mPR EoS (Magoulas and Tassios, 1990) was used for all nonpolar solvents. This EoS is given by
P=
RT V+ t - b
-
( V + t)2
+
a 2b(V+ t ) - b2
with a = 0.457235-
(RT,)~ a(T,>
PC
c1
99.3 78.7 -52.0
0.81875 0.86676 0.75684
CZ 0.03653 -0.17655 -0.56107
c3
-0.16844 0.23190 1.37066
/
1
0.0 -5.2 -7.1 -7.1 -2.3 1.7
PS (290 000) a t 393.15 K 0 10 50 100 1000 2000
to (cm3/mol)
(1)
- - - Kontogeorgis - This work
300
e t al. (1994a)
400 500 Temperature ( K )
600
Figure 1. Predicted vapor pressure for HDPE with MW equal to 50 000 using the methodology of Kontogeorgis et al. (1994a) and the method proposed in this work. a and b in the methodology of Kontogeorgiset al. are calculated using experimental pure-polymer data a t 398.15 and 471.15 K.
RTC b = 0.077796-
(3)
PC
where T, and P, are the pure-compound critical temperature and pressure, respectively, and m is a function of the acentric factor, w . The translation factor, t , is introduced to improve the prediction of saturated liquid volumes of nonpolymeric compounds and does not affect pure-component and mixture vapor-liquid equilibria calculations. This factor is given as a function of the reduced temperature, T, = TITc,and w :
where to, p, and t c are also functions of w . The exact expressions for the dependence of the parameters m, to, p, and t, on w are given in the Appendix. For polar solvents, we use the Mathias and Copeman (1983) expression for the temperature-dependent correction a:
a(TJ = [l + C,(1 - &)
+ C,(1 -
&)2
+
C&1 - &I3l2
(6)
where the values of CI, CZ, and C3 are evaluated by fitting experimental pure-compound vapor pressure data, obtained from Daubert and Danner (1989). Better prediction of saturated liquid volumes is obtained by adjusting t o to the experimental density value at 25 "C. The values of CI, Cp, C3, and t o for all polar solvents considered in this study are given in Table 1. Critical property and acentric factor values are obtained from Daubert and Danner (19891, except for
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2119 n-alkanes, where the values recommended by Magoulas and Tassios (1990) are used.
Description of Pure Polymers Kontogeorgis et al. (1994a) used a simple scheme for the evaluation of a and b for polymers, using any cubic EoS, assuming that the translation factor is zero. At first, pressure in the EoS is set equal to zero and then a and b are calculated so that the experimental volume of the polymer is exactly reproduced at two different temperatures. Excellent description of the zero pressure volumetric behavior is obtained. A disadvantage of the method is that it occasionally results in large vapor pressures (even of the order of bar), as shown in Figure 1for HDPE with MW equal t o 50 000, which in turn may lead to the prediction of finite polymer solubility in the vapor phase. Additionally, as shown in Table 2, increased errors, up to 15-20%, are also encountered when the EoS is used to predict purepolymer volumes at pressures higher than zero. In this work, we propose a different method for treating pure polymers with a cubic EoS, which leads to much smaller vapor pressures and better prediction of high-pressure volumetric behavior than the methodology of Kontogeorgis et al. (1994a). We also assume a zero translation factor. The new method is based on the free-volume theory for the description of glassy polymers (Williams et al., 1955; Aklonis and MacKnight, 19831, which states that the glassy state is a n iso-free volume one, with the fractional free volume having a universal value equal to 2.5% at all temperatures below the glass transition one, Tg,of the polymer. [Indeed, Williams et al. (1955)have confirmed that the fractional free volume is 2.5% at Tgfor a plethora of polymers.] Since, in the context of a cubic EoS, the free volume is given by V* - b*, the above can be written as -vy: - b*
V*
- 0.025
b* d T
Development of the New Eo$/@ Model The model to be presented in the following is a modification of the EoS/GE model presented by Michelsen (1990a), which is based upon exact matching of the expression for the excess Gibbs free energy, GE, obtained from the EoS, equal t o GE from an existing activity coefficient model at zero pressure, i.e.,
LL GE EoS
GE GEmodel
(12)
= LEI
(7)
where the asterisk denotes quantities per unit mass. Differentiation with respect t o temperature yields
1 db* --_--
Values of the coefficients A,, Ao, and AS, along with values of Tg,V,, and %, are presented in Table 3 for several polymers of industrial importance. Experimental volumes at zero pressure are obtained from Tait’s equation up t o 600 K (Rodgers, 1993). Using this method, the obtained correlation for the zero pressure isobar of pure polymer volume versus temperature is excellent for all polymers studied here, with typical errors less than 0.2% from T, t o 600 K. Compared to the scheme of Kontogeorgis et al. (1994a), the present method has the disadvantage that it achieves correlation of the same, or only slightly better, accuracy with three, rather than two, adjustable parameters. It is characterized, however, by two notable advantages: (1)Very low vapor pressures are obtained for the pure polymer, smaller by at least an order of magnitude than the predictions of the method of Kontogeorgis et al. An example is shown in Figure 1for HDPE with MW equal to 50 000. This ensures the prediction of very low, close to zero, polymer solubilities in the vapor phase. (2) Much better predictions are obtained for pure-polymer volumes at pressures up to 2000 bar, as demonstrated in Table 2 for HDPE, PIB, and PS.
Equation 12 results in the following mixing rule for the reduced attractive-term parameter, a = a/(bRT),of the mixture, which is solved implicitly for a:
1 dV*
V
dT=ag
where a, is the thermal expansion coefficient in the glassy state, a quantity known for a plethora of polymers (Van Krevelen, 1990). Upon integration we obtain
In b* = a,(T - T,)
+ In bg*
(9)
with bg* = b*(T,) = 0.975Vg from eq 7 and V,* the specific volume of the polymer at T , (also known for a variety of polymers using the Tait correlation). We assume that eq 9 also holds at temperatures above T,. The molar quantity b is related t o b* by
b = (MW)b*
(10)
where MW is the number-average molecular weight of the polymer. The attractive term parameter a of the EoS is calculated next, so that the experimental volume at zero pressure is reproduced exactly a t any temperature above Tg.The resulting a vs T curves are fitted to an expression of the form
where for the t-mPR EoS
[
7
a u+l+ 2 qe(a)= -1 - ln(u - 1)- -In 22/2 u + l - . J Z
(14)
and
u = V+t = :[a - 2 -1-4 b 2
(15)
Note that u is the smallest root obtained by solving the EoS at zero pressure and is defined only for a L slim = 4 2& for the t-mPR and PR EoS. Consequently, the model is applicable only for a 2 ali,,,. The mixture covolume parameter, b, is given by
+
b = &bi
(16)
In eqs 13 and 16 the summations are over all components of the system.
2120 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 Table 3. Equation of State Properties for Several Polymers Using the Method Proposed in This Study Tg a, x lo4 V,* polymer (K) (K-l) (cm3/g) A1 A2 HDPE 195 2.69 1.1144 -0.3481 2.3142 x lo5 PIB 1.0826 -0.8299 2.0766 x lo5 198 1.66 PS 373 2.40 0.9778 -0.4760 5.9470 x lo4 PMA 282 2.67 0.8415 -0.3850 2.4183 x lo9 PEA 252 3.25 0.8623 -0.3478 3.2586 x lo9 PMMA 387 0.8678 -0.7603 95901 2.65 PEMA 339 3.09 0.9048 -0.2637 6.9626 x 10" PBD 1.0557 -0.3476 2.9581 x lo9 215 1.89 PVAc 303 2.43 0.8459 -0.4818 6.1352 x lo9 PET 0.7289 -0.3989 1.1100 x 1014 342 3.16 PC 423 3.06 0.8648 -0.2541 1.2478 x lo7 i-PP 1.1527 -0.2196 2.2591 x lo9 263 1.91
A3 2.26 2.04 1.80 3.86 4.02 1.85 4.29 4.10 3.94 5.56 2.68 3.88
0
D
\ n-0.5 U
-0.6 ;
..........................................................
-_
Table 4. Correlation Results Using the EoS@ Model Proposed in This Work
-0.7
0
10
20
30
40
a n-hexanelHDPE (50 000) benzenePIB (45 000) M E W S (290 000) chloroformPS (290 000)
7 10 6 12
353.15 10 313.15 10 343.15 -25 323.15 -5.0
13 36 91. 2.0
7.7 2.8 1.8 2.5
Figure 2. dqe(a)/daand dq(a)/da from eq 18 versus a .
I
0,
a Experimental data for n-hexanelHDPE solution from Wen et al. (1991), benzenePIB from Bawn and Pate1 (1956), chloroform/ PS from Bawn and Wajid (1956), and methyl ethyl ketonePS (MEK) from Bawn et al. (1950). The number-average molecular weight of the polymer is given in parentheses. * Average absolute error % is defined as (100INDP) E / ( P c a i c d - Pexptl)/Pexptl)l, where Pc&d and Pelptl refer to calculated and experimental equilibrium pressures, respectively. NDP denotes the number of experimental data points.
Several modifications of the above model have been proposed, the main feature of which is the replacement of qe(a)with an approximating function q(a),so that the resulting mixing rule for a becomes density independent, simpler to use, since it can be solved explicitly, and applicable to a < alim. For example, fitting of the q e ( dversus a curve to a linear function over the a range between 10 and 14 yields the MHVl EoS/GE model (Michelsen, 1990b) and between 20 and 25 the PSRK model (Holderbaum and Gmehling, 19911, whereas fitting to a quadratic expression in the range 8-18 leads to the MHV2 model (Michelsen, 1990b; Dahl and Michelsen, 1990). Outside the above-mentionedranges, and in particular for large values of a (corresponding to polymeric systems), these approximating functions become much less accurate. In this work, we propose a new, more accurate expression for q(a),based on fitting the dqe(a)/daversus a curve, given by
-301 -
0
4 10
0
20
30
40
1
50
a Figure 3. qe(a) and approximating function q(a) from eq 19 versus a. MHVl and MHV2 approximations are also shown. MHVl approximation: q(a) = 0.394 329 - 0.529 279a; MHV2 approximation: q(a) = -0.389 062 - 0.405 211a - 0.004 4780 3a2.
the correlation however, becomes poorer as approached. Integration of eq 18 yields
slim
is
(19) dqe(a) -=-da
2 4
u+1-&
(17)
with an approximating function of the form dq(d- k --+ s, da a n
(18)
over the range slim -= a < 18. The parameters, is the limit of dqe(a)/daas a approaches infinity, and is equal to -11242 ln[(42 + 11442 - 111 for the t-mPR and PR EoS. The resulting values of K and n are 3.2394 and 1.3708, respectively. As shown in Figure 2, eq 18 provides very good correlation of the dqe(a)/da curve even for a much larger than 18, since it is forced to approach the correct limit at infinite a. The quality of
With a value of q o equal to 5.07278, q(a) and qe(a) coincide essentially for every a (see Figure 3). This is again due to the fact that the slope of q(a) at large values of a is forced to agree with the one of the qe(a) curve. As also shown in Figure 3, linear and quadratic fits of the qe(a)curve in the same a ranges as those used in the development of the MHVl and MHV2 models for the case of the t-mPR EoS provide a poor representation of the exact curve for a's larger than 20. Incorporating q(a)from eq 19 into eq 13 in place of qe(a),we obtain an implicit mixing rule for a. Compared to Michelsen's (1990a) model (eqs 13-16), this new mixing rule is simpler t o use, since it is density independent, and, additionally, is in principle applicable for a < slim.
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2121 Table 5. Prediction Results for HDPE and PIB Solutions with the PSCT, GPSCT, GCLF, vdW EoS, and E o S / P Model Proposed in This Work % AADb
svstemQ
NDP
n-hexanelHDPE (50 000) n-hexanelHDPE (50 000) n-hexanelHDPE (50 000) n-hexanelHDPE (50 000) n-decanelHDPE (50 000) cyclohexanelHDPE (50 000) cyclohexanePIB (100000) cyclohexanePIB (100000) cyclohexanePIB (100000) benzenePIB (45 000) benzenePIB (45000) benzenePIB (45 000)
7 18 6 8 9 6 10 10 9 8 10 11
T (K) 353.15 410.15 416.15 430.15 458.15 373.15 298.15 313.15 338.15 298.15 313.15 338.15
PSCT
GPSCT
GCLF
vdW
this work
18.7 31.5 16.8 12.1 4.8 31.2 12.5 12.4 12.0 24.0 24.9 20.5
15.0 28.0 13.5 8.6 3.5 31.2 12.5 12.4 12.0 24.0 24.9 20.5
24.7 30.6 17.4 11.4 1.2 31.1 11.4 11.4 11.7 25.0 25.9 21.8
11.5 25.9 18.2 15.1 2.2 36.7 20.8 17.1 10.1 26.2 19.3 13.4
22.3 21.3 15.2 10.3 4.1 23.3 7.9 7.5 7.2 5.4 5.3 2.7
Experimental data for HDPE solutions from Wen et al. (1991),and PIB ones from Bawn and Pate1 (1956). The number-average molecular weight of the polymer is given in parentheses. Average absolute error % is defined as (100NDP) c I ( P c d c d - Pexp$PexptI)/, where the P c & d and Pexptl refer to calculated and experimental equilibrium pressures, respectively. NDP denotes the number of expenmental data points. Table 6. Prediction Results for PS Solutions with the PSCT, GPSCT, GCLF, vdW EoS, and E o S / P Model Proposed in This Work %AADb
systema
NDP
T (K)
PSCT
GPSCT
acetonePS (15700) acetonePS (15700) M E W S (290000) M E W S (290000) chloroformPS (90000) chloroformPS (290000) chloroformPS (290000) toluenePS (290000) toluenePS (290000) toluenePS (290000)
8 7 11 6 6 11 12 10 3 3
298.15 323.15 298.15 343.15 298.15-323.15 298.15 323.15 298.15 333.15 353.15
16.3 15.2 12.4 12.5 7.6 4.8 9.9 3.4 8.7 3.0
58.6 64.5 32.6 41.5 nac na na 229. 44.9
GCLF 36.1 36.4 21.9 26.1 na na na 9.1 10.9 5.2
this work 14.0 16.3 10.9 14.4 3.9 5.4 4.8 4.3 8.1 5.4
a Experimental data for acetone and chloroformPS solutions from Bawn and Wajid (1956),and methyl ethyl ketone (MEK) and toluene/ PS ones from Bawn et al. (1950). The number-average molecular weight of the polymer is given in parentheses. Average absolute error % is defined as (100/NDP) z l ( P c & d - Pexptl)/Pexptl/. na denotes unavailability of the chloroform functional group in the respective parameter table.
In this study, the new mixing rule is coupled with the entropic-FV GEmodel (Kontogeorgis et al., 1993),which has been very successful in the prediction of solvent activities in concentrated polymer solutions, and is applied to the correlation and, especially, the prediction of VLE in a number of solventJpolymer mixtures. Results and Discussion At first, the new EoS/GE model developed in the preceding section is applied to the correlation of VLE behavior in polymer solutions, by adopting the UNIQUAC residual term in the entropic-FV @ model and fitting the corresponding interaction parameters, Au12 and Au21, t o the experimental data. Results for a few systems are presented in Table 4. The quality of the correlation is excellent and similar to the one obtained by Orbey and Sandler (1994). The reason for the somewhat poorer results in the n-hexanelHDPE system at 353.15 K is discussed later in this section. The advantage, however, of the new E o S I P model is that it can be easily applied t o the prediction, rather than mere correlation, of VLE in polymer solutions. Such results are presented in Table 5 for hydrocarbod HDPE and PIB systems and in Table 6 for concentrated PS solutions in polar compounds. The interaction parameters in the residual part of the entropic-FV activity coefficient model are set equal to zero for the nearly athermal alkane/HDPE and PIB mixtures in Table 5, while the linearly-temperature-dependent UNIFAC interaction parameters given by Hansen et al.
(1992) are utilized for the benzenePIB system in Table 5 and the PS solutions in Table 6 . In all cases, the model is compared with already existing and widely used noncubic EoS, such as the PSCT EoS and its group-contribution version GPSCT and also the GCLF EoS. The respective results in Tables 5 and 6 were obtained from Kontogeorgis et al. (1994b). The GC-Flory EoS was not included in the comparison, since, as mentioned in the Introduction, it has been developed as a n activity coefficient model. Three graphical examples are shown in Figures 4-6 for the systems n-hexanelHDPE (50000) at 430.15 K, cyclohexane/PIB(100 000) at 298.15 K, and acetonePS (15 700) a t 298.15 K. Additionally, VLE predictions using the vdW EoS in the fashion suggested by Harismiadis et al. (1994) are given for the athermal systems in Table 5 , based on a linear correlation of the binary interaction parameter in the Berthelot combining rule for the mixture attractive term parameter with the MW of the solvent. The EoS is not used to predict VLE in the polar solvent/PS mixtures of Table 6, since the above-mentioned linear correlation was not developed using such data. The following comments summarize our observations: (1)The proposed model generally behaves rather accurately for the HDPE and PIB solutions, shown in Table 5. Specifically, the model yields very good results for the PIB systems, characterized by sigmficantly lower errors-even by 5 times as in the case of benzene/ PIB-than all other models considered.
2122 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995
In
,
I
Experimental Data GC-F EoS A>&.-+ PSCT EoS *2r*2 GPSCT Eo5
4mm.m
Experimental Data ____.... GCLF EoS
Hum..
________
-0.8 L
i-
_ _ Th,s Work
m
U
J2 W
aJ
0.6 :
L
,
3 v,
20.4
,
:
L
a ,
0
0.2f c
0.0 'i, 0."0 0.15 0.20 0.25 0.30 0.35 n-Hexane Weight Fraction Figure 4. Experimental and predicted equilibrium pressures versus weight fraction of solvent for the system n-hexanelHDPE (50 000) at 430.15 K.
4
3 mu...
0.6
Experimental Data - - - vd(N EoS ....--..GCLF EoS .A&.-+ PSCT & GPSCT EoS,
-
/
1
0.2 0.3 0.4 0.0 0.1 Cyclohexane Weight Fraction Figure 5. Experimental and predicted equilibrium pressures versus weight fraction of solvent for the system cyclohexanePIB (100 000) at 298.15 K.
The performance of the model in the prediction of VLE for HDPE solutions is characterized by increased errors, of the same order as with PSCT, GPSCT, GCLF, and vdW EoS. Furthermore, as shown in Table 4, the quality of correlation for the n-hexane/HDPE system is significantly worse than the other systems considered. These problems may be due to the fact that since no HDPE molecular weight value was reported in the reference of the experimental data, an arbitrary value equal to 50 000 was assumed and used with all models in Table 4, which may have led to increased errors in the predicted equilibrium pressures. The specific MW value was chosen, since it has been used previously for the same systems in other studies (Kontogeorgis et al., 1994a, b; Harismiadis et al., 1994). (2)The predictions of the proposed model for the polar solvent4PS solutions in Table 6, are quite accurate and of the same quality as the results of the PSCT EoS. The behavior of the GCLF EoS is consistently poorer than the proposed model, whereas the GPSCT EoS yields unacceptable predictions, due t o its inability t o describe pure-solvent vapor pressures and fugacities.
1
' '
"
''
,=
, I "
I
' ' 1
0
"
'
/.-
-
" " ' 1
" " ' ' I
'
" I
'
I
'
Figure 6. Experimental and predicted equilibrium pressures versus weight fraction of solvent for the system acetonePS (15 700) at 298.15 K.
Finally, due to the fact that the method proposed in this work for the description of pure polymers yields very low vapor pressures, the EoSIGE model predicts extremely small polymer solubilities in the vapor phase (generally less than 10-9. This is in accordance with experiment, since at the temperature and pressure ranges considered here (298-458 K and less than 8 bar, respectively), there is essentially no polymer solubility in the vapor phase. On the other hand, use of the methodology of Kontogeorgis et a1 (1994a) may occasionally lead t o unrealistically high solubilities, due to the prediction of large pure-polymer vapor pressures (see Figure 1). For example, in the case of decane in HDPE at 458.15 K, the resulting solubilities of HDPE or larger. in the vapor phase are in the range of
Conclusion A zero-reference-pressure EoS/GEmodel is developed in this work, based on a new, more accurate approximating expression for the qe(a)function of the model proposed by Michelsen (1990a), which contrary to other approximations, such as MHVl and MHV2 (Michelsen, 1990b), reproduces closely the qe(a)values a t large a's. Since such values are characteristic of polymeric systems, the model is directly applicable to the prediction of VLE in polymer solutions. Specifically, the model is coupled with the successful entropic-FV activity coefficient model (Kontogeorgis et al., 1993) and the t-mPR EoS (Magoulasand Tassios, 1990)to describe the low MW component of the mixture. An appropriate modification of the same EoS is proposed for the description of the volumetric behavior of the pure polymer. Application of the model to several athermal and non-athermal polymer solutions provides excellent correlation of the data, and more importantly, satisfactory predictions. The latter are generally better than the ones obtained with other widely used EoS, such as the PSCT, GPSCT (Morris et al., 1987),and GCLF (High and Danner, 1991) EoS and a recently proposed modification of the vdW EoS (Harismiadis et al., 1994). Acknowledgment The authors wish to thank Mr. G. Kontogeorgis and Dr. I. Economou for providing the results with the PSCT, GPSCT, and GCLF EoS and Mr. N. Spiliotis for
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2123 performing some of the pure-polymer calculations for HDPE and PS.
Appendix Expressions for the Parameters m,to, /?,and t, of the t-mPR EoS. The expressions of these parameters as a function of w are given by (Magoulas and Tassios, 1990)
m = 0.384401
+ 1.52276~- 0.2138080~+ 0.034610~- 0.0019760~
RTC PC
+
0.0674980 - 0.0848520~
to= +-0.014471+
0.0672980~ - 0.0173660~)
/3 = -10.2447 - 28.6312~
RTC tc = -(0.3074 PC
PBD = polybutadiene PC = bisphenol-A polycarbonate PET = poly(ethy1ene terephthalate) PIB = polyisobutylene PEA = poly(ethy1 acrylate) PEMA = poly(ethy1 methacrylate) PMA = poly(methy1 acrylate) PMMA = poly(methy1 methacrylate) PS = polystyrene PVAc = poly(viny1 acetate) t-mPR = translated-modified Peng-Robinson EoS vdW = van der Waals EoS VLE = vapor-liquid equilibria
- zc’) Z:
with
= 0.289 - 0.07010 - 0.02070~
Nomenclature a = attractive-term parameter in EoS ((bar cm6)/mo12) A I ,Az, A3 = pure-polymer parameters in eq 11 b = covolume parameter in EoS (cm3/mol) C1, Cz, C3 = pure-component constants in MathiasCopeman expression m = parameter in eq 4 P = pressure (bar) q(a)= approximating function of qe(a) qe(a)= exact function defined in eq 14 R = ideal gas constant (=83.14 (bar cm3)/(molK) or 8.314 J/(mol K)) s, = infinite-a slope of qe(a) t = translation factor (cm3/mol) to, t, = parameters in eq 5 (cmYmo1) T = temperature (K) u = (V t)/b V = molar volume (cm3/mol) x = molar composition of mixture in liquid phase
+
Greek Symbols a = a/(bRT) a, = thermal expansion coefficient in the glassy state of the pure polymer slim = limiting value of a for which qe(a)is defined a(T,) = attractive-term-parameter temperature correction 8, = parameter in eq 5 Aulz, Auzl = UNIQUAC interaction parameters (J/mol) w = acentric factor Superscripts * = denotes quantity per unit mass Subscripts c = critical property g = at the polymer glass transition i = component in a mixture r = reduced quantity Acronyms
%AAD = percent average absolute deviation EoS = equation of state HDPE = high-density polyethylene i-PP = isotactic polypropylene MEK = methyl ethyl ketone MW = number-average molecular weight
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I39406254 Abstract published in Advance A C S Abstracts, April 15, 1995. @
