Fluid Phase Equilibria 168 Ž2000. 165–182
www.elsevier.nlrlocaterfluid
Vapor–liquid equilibrium in polymer–solvent systems with a cubic
equation of state
Vassiliki Louli, Dimitrios Tassios )
Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical
UniÕersity of Athens, 9, Heroon Polytechniou Str., Zographou Campus, Athens 15780, Greece
Received 8 July 1999; accepted 9 December 1999
Abstract
A cubic equation of state ŽEoS., the Peng–Robinson ŽPR. one, is extended to polymers by using a single set
of energy Ž a. and co-volume Ž b . parameters per polymer fitted to experimental volume data. Excellent results
for the volumetric behavior of the polymer up to 2000 bar pressure are obtained. The EoS is applied to the
correlation of low pressure vapor–liquid equilibrium ŽVLE. data for a variety of polymer solutions by
employing only one interaction parameter in three mixing rules: the van der Waals one fluid ŽvdW1f. — using
the Berthelot combining rule ŽB. for a i j , and the arithmetic mean ŽAM. and Lorentz ŽL. ones for bi j — the
Zhong and Masuoka ŽZM.; and the modified Huron–Vidal 1st order ŽMHV1., by coupling the cubic EoS with
the Flory–Huggins ŽFH. model. Best results are obtained with the ZM mixing rule and the same holds for the
extrapolation with respect to temperature and polymer molecular weight ŽMW.. Low errors in correlation and
extrapolation are also obtained with the other mixing rules, but phase split is observed in certain cases. The
AbbreÕiations: AM, arithmetic mean combining rule; a-PP, atactic polypropylene; B, Berthelot combining rule; cC6,
cyclohexane; EB, ethylbenzene; EoS, equation of state; GC-Flory, group contribution Flory ŽEoS.; GCLF, group contribution lattice fluid ŽEoS.; HDPE, high-density polyethylene; i-PP, isotactic polypropylene; KHFT, method of Kontogeorgis et
al.; LDPE, low-density polyethylene; L, Lorentz combining rule; MHV1, modified Huron–Vidal 1st order mixing rule; MW,
molecular weight; MEK, methyl–ethyl–ketone; nC x, n-alcane with x carbon atoms; NP, number of data points; OS,
method of Orbey and Sandler; OBC, method of Orbey et al.; PBD, cis-1,4-polybutadiene; PcHMA, polyŽcyclohexyl
methacrylate.; PDMS, polyŽdimethylsiloxane.; PEA, polyŽethyl acrylate.; PEMA, polyŽethyl methacrylate.; PEO, polyŽethylene oxide.; PIB, polyisobutylene; PMA, polyŽmethyl acrylate.; PMMA, polyŽmethyl methacrylate.; PMP, polyŽ4-methyl1-pentene.; PoMS, polyŽ o-methylstyrene.; PR, Peng–Robinson; PS, polystyrene; PSCT, perturbed soft chain theory; PVAc,
polyŽvinyl acetate.; PVC, polyŽvinyl chloride.; PVME, polyŽvinyl methyl ether.; SAFT, statistical associating fluid theory;
vdW, van der Waals; vdW1f, van der Waals one fluid mixing rule; VLE, vapor–liquid equilibrium; ZM, Zhong and
Masuoka mixing rule
)
Corresponding author. Tel.: q30-1772-3232; fax: q30-1772-3155.
0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 3 3 9 - 8
166
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
performance of other methods for polymer parameters evaluation is also examined. q 2000 Elsevier Science
B.V. All rights reserved.
Keywords: Cubic equation of state; Mixing rule; Vapor–liquid equilibrium; Polymer solutions
1. Introduction
In recent years, the phase equilibrium in polymer solutions has attracted the interest of the
scientific community due to their fundamental importance in a number of process applications, such
as polymer devolatilization, polymeric membrane–separation processes, vapor phase–photografting,
optimum formulation of paints and coatings, etc. w1x. In order to design such processes, the successful
description of the vapor–liquid equilibrium Ž VLE. behavior in solvent–polymer systems is required.
This need has led to the development of several activity coefficient models, which are mainly
modifications of the Flory–Huggins Ž FH. equation w2,3x Ž e.g., Entropic-FV w4x, UNIFAC-FV w5x, etc.. ,
and various equations of state Ž EoS; e.g., group contribution Flory wGC-Floryx w6x, group contribution
lattice fluid wGCLFx w7x, perturbed soft chain theory wPSCTx w8x, statistical associating fluid theory
wSAFTx w9x, Sanchez–Lacombe w10x, Panayiotou–Vera w11x, etc.. . The latter have the advantage of
being applicable to both vapor and liquid phases in contrast to the former models. However, in the
last few years, cubic EoS have been extensively used as well, because they provide accuracy
comparable to the other models and their simplicity makes them attractive to the practicing engineer.
The first attempt to apply a cubic EoS in polymer solutions has been made by Sako et al. w12x, who
introduced a three-parameter cubic EoS, where apart from the energy Ž a. and co-volume Ž b .
parameters, Prigogine’s external degrees of freedom parameter c is employed for the extension of the
equation to polymers. However, the evaluation of these parameters is quite complicated for both the
solvent and the polymer and thus, this approach does not retain the simplicity of a cubic EoS.
On the other hand, Kontogeorgis, Harismiadis, Fredenslund and Tassios Ž KHFT. w13x proposed in
1994 the use of the van der Waals ŽvdW. EoS with a very simple method for the calculation of the
attractive and repulsive parameters of the EoS. According to this method, the a and b of the polymer
are fitted to two experimental volumetric data at essentially zero pressure. The extension to polymer
solutions was achieved by using the classical van der Waals one fluid mixing rule Ž vdWf1. , the
arithmetic mean ŽAM. combining rule for bi j and the Berthelot one for a i j with only one binary
interaction parameter, l i j . In a series of papers, this approach was applied with satisfactory results to:
Ž1. the correlation and prediction of VLE w13,14x; Ž2. the correlation and prediction of LLE in
polymer solutions and blends w15–17x; and Ž 3. the correlation and prediction of Henry constants w18x.
This simple method, however, has the disadvantage of predicting rather poor volumetric behavior at
high pressures Ž as shown in Fig. 1a and b for polyisobutylene wPIBx and polystyrene wPSx, respectively. and unrealistically high vapor pressures Ž as shown in Fig. 2. for the pure polymer. The latter
causes problems in the correlation of polymer–solvent VLE data involving small molecular weight
ŽMW. polymers, as it leads to the appearance of polymer in the vapor phase. Parameter values with
this method and the Peng–Robinson ŽPR. EoS are presented in Table 1 for five polymers.
Orbey and Sandler ŽOS. w19x, in 1994, fitted the two parameters of the PRSV EoS for the pure
polymer to its volumetric data at the temperature range of interest by assuming that its vapor pressure
is equal to 10y7 MPa. For each MW, a different set of parameters is obtained Ž see Table 1. , which
makes the method time-consuming, while the predicted volumetric behavior is rather poor Ž see Fig. 1a
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
167
Fig. 1. Ža. Percent error in volume Ž DV %. of PIB Ž50,000. vs. pressure at 338.15 K, for all the methods considered in this
study for polymer parameters evaluation ŽOBC is not included, because N DV %N )90%.. Žb. Percent error in volume
Ž DV %. of PS Ž290,000. vs. pressure at 395.15 K, for all methods considered in this study for polymer parameters evaluation
ŽOBC is not included, because N DV %N )90%..
and b. and the vapor pressures are rather high, especially at high temperatures Ž Fig. 2. . In order to
correlate the VLE in polymer solutions, they employed the mixing rule proposed by Wong and
Sandler w20x by coupling the EoS with the FH G E model, and employing two binary interaction
Fig. 2. Predicted vapor pressure of PIB Ž50,000. vs. temperature, for all methods considered in this study for polymer
parameters evaluation.
168
PIB
ŽT range: 326.15–383.15 K.
PS
ŽT range: 388.15–469.15 K.
PEO
ŽT range: 361.15–497.15 K.
arMW
brMW MW
arMW
brMW MW
1170
1350
2700
40,000
45,000
50,000
66,500
100,000
10 6
2.25=10 6
All MWs
848,250
843,750
823,500
748,000
744,750
742,500
733,495
722,900
654,700
629,550
450,893
1.0350
1.0340
1.0320
1.0230
1.0224
1.0221
1.0215
1.0210
1.0160
1.0120
0.9614
870,350
858,005
830,550
823,725
820,752
813,078
784,160
0.9260
0.9220
0.9200
0.9180
0.9175
0.9160
0.9120
600,000 750,000 0.9020
194,000 580,000 0.8420
14,600 770,000 0.8902
454,347 0.8486
351,985 0.7863
287,543 0.7197
376,848 0.8510
a Žcm6 barrmol 2 . and b Žcm3rmol..
brMW MW
arMW
PVME
ŽT range: 303.15–471.15 K.
MW
10,300
15,700
49,000
63,000
70,300
97,200
290,000
arMW
PVAc
ŽT range: 308.15–373.15 K.
brMW MW
arMW
brMW
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
Table 1
PR EoS parameters for the pure polymer calculated with Kontogeorgis et al. Ž italic . and OS methods Žvolumetric data from Tait equation with the Rodgers
parameters w26x.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
169
parameters, k i j and x ŽFlory interaction parameter.. Recently, Zhong and Masuoka w21x simplified
the aforementioned mixing rule by setting A`E equal to zero and obtained satisfactory results by
employing the method of Orbey and Sandler for polymer parameters a and b and only one binary
interaction parameter, i.e., k i j .
Kalospiros and Tassios w22x proposed a method of calculation of polymer parameters a and b
based on the free–volume theory w23,24x, and employed a predictive EoS-G E model, which coupled
the t-mPR EoS with Entropic-FV G E model w4x. But although they were led to good results and low
vapor pressure, P s , for pure polymer, the necessity of properties like the glass transition temperature
of the polymer, its volume at this temperature, and its thermal expansion coefficient in the glassy state
makes the method not very practical.
Finally, Orbey, Bokis and Chen Ž OBC. w25x suggested the use of a common set of critical
properties for all polymers, which are independent of the polymer MW Ž Tc polymer s 1800 K,
Pc polymer s 10 bar.. They employed the modified Huron–Vidal 1st order ŽMHV1. mixing rule by
coupling the SRK EoS with the FH model, and obtained very satisfactory results in the description of
VLE behavior. The predicted volumetric behavior of the polymer is very poor, however, with very
large errors, over 50%, for high MWs Žthis is why it is not included in Fig. 1a and b. . The predicted
P s values are also high, especially at high temperatures, as shown in Fig. 2.
In this study, we evaluate the polymer parameters a and b of a cubic EoS, in this case, the PR one,
by fitting them to pure polymer PVT data and assuming that arMW and brMW are independent of
MW. We proceed then with the correlation of VLE data for a great variety of non-polar and polar
polymer–solvent systems using three different mixing rules, requiring only one adjustable parameter.
Table 2
PR EoS parameters of some representative polymers calculated with the method proposed in this work Žvolumetric data from
Tait equation with the Rodgers parameters w26x.
Polymer
T range ŽK.
P range Žbar.
arMW
brMW
AAE% in V
HDPE
LDPE
PS
PVAc
PMMA
PIB
PEO
PVME
PMA
PEA
PEMA
PVC
PoMS
PcHMA
PBD
PDMS
PMP
i-PP
a-PP
413.15–476.15
394.15–448.15
388.15–469.15
308.15–373.15
387.15–432.15
326.15–383.15
361.15–497.15
303.15–471.15
310.15–493.15
310.15–490.15
386.15–434.15
373.15–423.15
412.15–471.15
396.15–471.15
277.15–328.15
298.15–343.15
514.15–592.15
443.15–570.15
353.15–393.15
0–1960
0–1960
0–2000
0–800
0–2000
0–1000
0–685
0–2000
0–1960
0–1960
0–1960
0–2000
0–1800
0–2000
0–2835
0–1000
0–1960
0–1960
0–1000
1,280,756
1,373,984
1,315,409
1,847,343
1,277,496
2,307,400
2,278,342
1,219,650
1,132,607
1,030,010
1,185,273
998,549
1,414,377
1,234,992
942,530
1,021,986
2,645,412
1,292,409
1,627,639
1.2066
1.1991
0.9549
0.8428
0.8407
1.0882
0.9497
0.9801
0.8664
0.9037
0.9050
0.7154
0.9746
0.9085
1.0549
0.9968
1.3607
1.2444
1.1716
2.77
2.62
2.17
1.38
1.77
1.40
2.48
2.92
3.11
3.34
2.15
1.85
1.95
2.14
2.35
2.10
2.47
3.65
1.77
a Žcm6 barrmol 2 . and b Žcm3rmol..
AAE%sÝabsŽ Vcal yVexp .r Vexp r NP =100.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
170
The possibility of extrapolation with respect to temperature, polymer MW and solvent concentration
is also discussed. Finally, results with the aforementioned methods of polymer parameter estimation
ŽKHFT, OS and OBC. are included for comparison purposes.
2. Evaluation of EoS parameters for polymers
In this study, the evaluation of the energy Ž a. and co-volume Ž b . parameters of the pure polymer in
the PR EoS:
RT
a
Ps
y 2
,
Ž1.
Vyb
V q 2 bV y b 2
is obtained by fitting the available PVT data with a single set of Ž arMW. and Ž brMW. parameters
for all MWs. The source of the required PVT data is the Tait equation, with the parameter values
proposed by Rodgers w26x. Zhong and Masuoka w27x used a similar approach but their parameters were
MW-dependent.
Parameter values for several polymers, along with the percent absolute average error in volume
ŽAAE%. , are presented in Table 2, while typical volume errors Ž DV %. as a function of pressure are
presented graphically in Fig. 1a and b, for PIB and PS. Very satisfactory results are obtained, which
are as expected better than those of the aforementioned methods Ž KHFT, OS, OBC. . Furthermore, the
proposed method predicts reasonably low P s values ŽFig. 2..
3. Correlation of VLE data
3.1. SolÕent parameters
The PR EoS parameters of the solvents examined in this study are determined from:
a s 0.45724
b s 0.0778
Ž RTc .
Pc
RTc
Pc
2
a Ž Tr . ,
Ž2.
,
Ž3.
where Tc and Pc are the pure compound critical temperature and pressure, respectively, obtained from
Daubert and Danner w28x, and Tr is the reduced temperature Ž TrTc . . For the quantity a Ž Tr . , two cases
can be distinguished depending on the polarity of the solvent.
For non-polar compounds, the a Ž Tr . is related to the acentric factor v :
( /
a Ž Tr . s 1 q m 1 y Tr
ž
2
,
Ž4.
m s 0.384401 q 1.52276 v y 0.213808 v 2 q 0.034616 v 3 y 0.001976 v 4 .
Eq. 5 was proposed by Magoulas and Tassios w29x and provides better predictions at low P s.
For polar compounds, the Mathias–Copeman w30x expression is used:
( /
( /
a Ž Tr . s 1 q c1 1 y Tr q c2 1 y Tr
ž
ž
2
( /
q c3 1 y Tr
ž
3 2
,
Ž5.
Ž6.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
171
where the values of c1, c 2 , c 3 are evaluated by fitting the pure compound vapor pressure data
obtained from the correlation of such data by Daubert and Danner w28x. In this study, the c1, c 2 , c 3
values reported by Kalospiros and Tassios w22x were employed.
3.2. Mixing rules
In order to extend the PR EoS to polymer–solvent systems, the following mixing rules are
considered.
3.2.1. ÕdW1f mixing rule
am s Ý
Ý x i x j ai j
i
bm s Ý
j
i
Ý x i x j bi j ,
Ž7.
j
with the combining rules for the cross energy Ž a i j . and co-volume Ž bi j . parameters below:
Berthelot rule ŽB.:
(
ai j s ai a j
bi j
Ž1 y k i j . ,
(b b
Ž 7.1 .
i j
AM rule:
bi j s
bi q bj
,
2
Lorentz Ž 1r3. rule ŽL. :
bi j
žb
s
1r3
q bj1r3
i
Ž 7.2 .
3
/
,
Ž 7.3 .
8
where x iŽ j. is the mole fraction and k i j is the binary interaction parameter. Note that, as Harismiadis
et al. w14x have explained, the Berthelot combining rule is preferable to the geometric mean for this
type of systems.
3.2.2. ZM mixing rule
am
RT
where:
sQ
Qs Ý
i
DD
bm s
1 y DD
Ý xi x j
j
ž
a
by
RT
/
Q
1 y DD
,
ij
,
Ž8.
DD s Ý x i
i
ai
bi RT
,
and:
a
ž b y RT /
1
s
ij
2
ai
aj
ž b y RT / q ž b y RJ / Ž1 y k . .
i
j
ij
Ž 8.1 .
Actually, the only difference between this mixing rule and the Wong–Sandler one is the absence of
excess Helmholtz free energy at infinite pressure, A`E , which was set equal to zero in this case.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
172
Table 3
Overall correlation results for various polymer–solvent systems ŽExperimental data are taken from Wen et al. w32x.
The number of the experimental data is given in the parenthesis.
Polymer
Solvent
T range ŽK. Overall AAD%
vdW1f ŽBrAM. vdW1f ŽBrL. ZM MHV1
PIB Ž4=10 4 –2.25=10 6 . nC4, nC5, nC6, nC8 Ž122.
298–338
Ž4=10 4 –10 5 .
cC6, benzenea , toluenea , EB a Ž129. 298–338
Average error
2.07
3.81
2.94
2.37
3.07
2.72
2.48 3.29
2.00 5.60
2.24 4.45
PIB (1170, 1350, 2700)
nC5 (42)
298 – 328
8.72
35.22
3.92 6.05
PS Ž10,300–2.9=10 5 .
cC6, benzene, toluene, EB Ž111.
acetone, MEK, CHCl3 Ž61.
288–353
298–343
0.74
4.43
2.59
1.00
3.16
2.08
0.90 1.10
3.52 2.84
2.21 1.97
PEO Ž5700–4=10 6 .
Ž6000, 5=10 6b .
Average error
benzene Ž50.
CHCl 3 Ž12.
319–424
298–333
1.88
5.40
3.64
4.33
3.56
3.95
1.41 2.45
6.37 5.94
3.89 4.20
PVAc Ž194,000.
Ž50,500–158,000.
Average error
CHClb3 Ž7.
benzenea Ž33.
333
303–333
4.96
1.98
3.47
5.53
2.22
3.88
5.04 4.91
1.78 1.98
3.41 3.45
PDMS Ž89,000.
nC6 a Ž24.
CHCl 3 Ž8.
298–308
303
0.80
1.00
0.90
8.85
10.8
9.83
0.89 2.48
1.56 0.57
1.23 1.53
PVC Ž34,000.
toluenea Ž8.
316
4.95
3.29
5.65 0.91
PVME Ž14,000.
Ž14,000, 14,600.
Average error
CHCla3 Ž13.
benzenea , toluenea Ž60.
298
298–343
4.28
0.65
2.47
5.67
1.79
3.73
5.97 5.01
0.63 1.18
3.30 3.10
Average error
Average error
a
b
Experimental data taken from Wohlfarth w33x.
Experimental data taken from Gupta et al. w34x.
3.2.3. MHV1 mixing rule
a
bRT
s Ý xi
i
ai
1
q
RTbi
A1
ž
GE
RT
q Ý x i ln
i
b
bi
/
b s Ý x i bi ,
Ž9.
i
where G E is the excess Gibbs free energy and A1 s y0.623 for PR EoS.
In this case, the cubic EoS is coupled with the FH model w2,3,31x, which is:
GE
RT
s x 1 ln
f1
x1
q x 2 ln
f2
x2
q xf 1 f 2 Ž x 1 q x 2 z . ,
Ž 10.
with:
f1 s
x1
x1 q x2 z
and
f2 s
x2 z
x1 q x2 z
,
Ž 11.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
173
where 1 is the solvent, 2 is the polymer, f is the volume fraction and z is the degree of
polymerization, i.e., z s MWpolymerrMWsegment w25x. The only interaction parameter here is the x
found in the FH model.
4. Results
Bubble point pressure calculations are carried out for seven polymers: PIB, PS, polyŽ ethylene
oxide. wPEOx, polyŽvinyl acetate.wPVAcx, polyŽ dimethylsiloxane. wPDMSx, polyŽ vinyl chloride. wPVCx
and polyŽvinyl methyl ether. wPVMEx with a variety of polar and non-polar solvents, using the
aforementioned mixing rules. The VLE data are taken from Wen et al. w32x and Wohlfarth w33x, while
those for PEO Ž5,000,000. –chloroform and PVAc Ž 194,000. –chloroform are from Gupta and
Prausnitz w34x. In each case examined, the average absolute deviation Ž AAD%s ÝabsŽ Pcal y
Pexp .rPexprNP = 100. in bubble point pressure is reported.
4.1. Correlation
Overall correlation results using the EoS polymer parameters proposed here are presented in Table
3, while typical ones are shown graphically in Figs. 3–7. The following comments summarize our
observations on the obtained results.
Ž1. Very satisfactory results are obtained with all mixing and combining rules, with typical errors
for non-polar solvents of 2–3% and a little higher for polar ones Ž Table 3, and Figs. 3–5..
Ž2. Satisfactory results are also obtained for the chloroform Ž CHCl 3 . systems, where hydrogen
bonding exists, as Table 3 and Fig. 6 indicate.
Fig. 3. Correlation of the bubble point pressure for the system PIB Ž50,000. – nC6 Ž k i j vdW1fŽBrAM. s 0.620r0.627r0.640;
k i j vdW1fŽBrL. s 0.541r0.543r0.546; k i j ZM s 0.917r0.917r0.922; x s1.068r1.112r1.083 for T s 298.15r313.15r338.15
K, respectively..
174
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
Fig. 4. Correlation of the bubble point pressure for the system PVAc Ž158,000. –benzene Ž k i j vdW1fŽBrAM. s 0.612r0.619;
k i j vdW1fŽBrL. s 0.542r0.543; k i j ZM s 0.909r0.91; x s 0.667r0.652 for T s 313.15r333.15 K, respectively..
Ž3. Low error in pressure does not, however, guarantee good correlation as suggested by the results
obtained with the vdW1f Ž BrAM. mixing rule for PIB-benzene in Fig. 7. Phase split is predicted at
Fig. 5. Correlation of the bubble point pressure for the system PS Ž290,000. –methyl–ethyl–ketone wMEKx Ž k i j vdW1fŽBrAM.
s 0.521r0.557; k i j vdW1fŽBrL. s 0.449r0.475; k i j ZM s 0.894r0.915; x s 0.853r0.926 for T s 298.15r343.15 K, respectively..
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
175
Fig. 6. Correlation of the bubble point pressure for the systems PEO Ž6000. –chloroform and PVAc Ž194,000. –chloroform
Ž k i j vdW1fŽBrAM. s 0.540r0.610; k i j vdW1fŽBrL. s 0.323r0.512; k i j ZM s 0.705r0.870; x sy0.485r0.026, for each system,
respectively..
313.15 and 338.15 K, even though the corresponding AAD% in bubble point pressure are 4.96% and
4.81%, respectively. For this system, the MHV1 mixing rule also predicts phase split. This is
Fig. 7. Correlation of the bubble point pressure for the system PIB Ž45,000. –benzene Ž k i j vdW1fŽBrAM. s 0.613r0.624r0.634;
k i j vdW1fŽBrL. s 0.557r0.565r0.564; k i j ZM s 0.946r0.949r0.944; x s1.047r1.080r1.034 for T s 298.15r313.15r338.15
K, respectively..
176
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
observed for other systems as well, like PIBŽ 10 6 . –nC5 Ž298.15, 313.15 K. and PIBŽ10 5 . –cyclohexane ŽcC6. Ž338.15 K., where both the vdW1f ŽBrAM. and MHV1 mixing rules lead to phase split.
No such phase split has been observed with the ZM mixing rule.
Ž4. The results for systems involving PIB of very low MW suggest that the vdW1f Ž BrL. mixing
rule fails completely, while the ZM mixing rule gives very satisfactory results.
Ž5. The x values obtained with the MHV1 mixing rule are very often higher than 0.5, suggesting
— in terms of the conventional use of the FH G E model — phase separation. This, of course, is not
necessarily the case as this MHV1 model suffers from the same deficiency as the original MHV1
model, where UNIFAC is the G E model: it fails to reproduce the G E model coupled with it, as
pointed out by Boukouvalas et al. w35x and explained by Kalospiros et al. w36x.
Ž6. The mixing rule of ZM appears to give the best overall results, which is surprising —
especially for polar systems — considering the assumption involved in its development.
4.2. Extrapolation
We examine next the possibility of using the interaction parameter obtained from a given set of
VLE data for extrapolation purposes, since this is of great importance in the case of a process design
when a limited number of experimental data are available.
Satisfactory results for extrapolation with respect to temperature— for reasonable temperature
ranges Žabout 508C. — are obtained, as shown in Table 4 and Fig. 8 for the PEO Ž4 = 10 6 . –benzene
Table 4
Prediction of VLE with respect to temperature and polymer MW using the interaction parameter values of the lowest
available temperature or MW
System
T range ŽK.
vdW1f ŽBrAM. vdW1f ŽBrL.
ki j
AAD%
ki j
AAD%
ki j
AAD%
x
PIB Ž2.25=10 6 . – nC5 Ž21.
PIB Ž10 5 . –cC6 Ž19.
PS Ž49,000. –cC6 Ž24.
PS Ž63,000. –benzene Ž20.
PS Ž2.9=10 5 . –toluene Ž6.
PS Ž2.9=10 5 . –CHCl 3 Ž11.
PS Ž15,700. –acetone Ž7.
PEO Ž4=10 6 . –benzene Ž23.
PVME Ž14,600. –benzene Ž5.
PVME Ž14,600. –toluene Ž11.
Overall error a
308–328
313–338
303–338
303–333
333–353
323.2
323.2
361–423
343.2
343.2
0.669
0.606
0.484
0.488
0.466
0.505
0.554
0.640
0.444
0.413
6.17
6.98
1.46
10.16
16.28
10.84
13.59
18.14
3.75
6.26
9.36
0.618
0.541
0.335
0.404
0.387
0.412
0.488
0.556
0.299
0.285
2.61
1.78
1.21
3.30
4.23
3.42
2.66
4.82
0.85
2.14
2.70
0.962
0.917
0.787
0.828
0.822
0.816
0.933
0.916
0.713
0.682
2.48
1.99
1.28
4.52
5.21
7.02
1.39
4.03
2.47
3.29
3.37
0.997
0.832
0.215
0.217
0.474
0.084
0.980
1.085
0.466
0.636
5.45
3.89
1.13
3.99
3.24
4.56
2.29
5.19
1.56
2.71
3.40
PIB Ž10 6 , 2.25=10 6 . – nC5 Ž17.
PIB Ž5=10 4 , 10 5 . –cC6 Ž21.
PS Ž290,000. –toluene Ž11.
PVAc Ž150,000. –benzene Ž7.
Overall error b
298.2
298.2
298.2
303.2
0.671
0.603
0.457
0.598
6.96
3.02
6.18
3.51
4.92
0.601
0.528
0.372
0.520
4.51
2.82
4.43
3.12
3.72
0.957
0.912
0.817
0.891
5.26
1.85
1.76
3.12
3.00
1.287
0.901
0.344
0.481
11.07
5.25
4.82
1.98
5.78
a
ZM
MHV1
AAD%
For all the systems, the lowest temperature reported is 298 K, apart from PS–benzene Ž288 K., PEO–benzene Ž348 K.
and PVAc systems Ž323 K..
b
For PIB, the lowest MW system is the one of 4=10 4 , while for PS and PVAc, they are those of 49,000 and 5=10 4 ,
respectively.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
177
Fig. 8. Prediction of the bubble point pressure for the system PEO Ž4=10 6 . –benzene at various temperatures using the
interaction parameter values obtained from the lowest temperature data.
system. Best results are given by ZM and vdW1f ŽBrL. mixing rules, since the other two
occasionally lead to a phase split, such as the cases of PIB Ž 2.25 = 10 6 . –nC5 Ž313.15 K. and PIB
Ž10 5 . –cC6.
Fig. 9. Prediction of the bubble point pressure for the system PIB– nC5 at various MWs using the interaction parameter
values obtained from the lowest MW.
178
Table 5
Typical correlation results for some representative systems of this study using all the studied methods for determining EoS polymer parameters
The number of the experimental data is given in the parenthesis.
System
Overall error in pressure
KHFT
vdW1f
ŽBrAM.
PIB Ž1170,1350, 2700. – nC5 Ž42.
PIB Ž50,000. – nC6 Ž24.
PIB Ž45,000. –benzene Ž29.
PS Ž10,300, 49,000, 290,000. –toluene Ž35.
PS Ž290,000. –CHCl 3 Ž22.
PEO Ž6=10 5 . –benzene Ž13.
PVAc Ž194,000. –CHCl 3 Ž7.
PVME Ž14,600. –toluene Ž34.
298–328
298–338
298–338
298–353
298–323
323–343
333
323–343
1.43 a
3.27
4.42
0.73
5.52
0.52
5.03
0.45
OS
vdW1f
ŽBrL.
ZM
MHV1
4.00 a
2.99
1.43
0.92
1.37
2.21
7.65
0.77
1.51a
4.21
1.43
1.12
5.99
0.62
5.03
0.46
3.49 a
2.66
6.46
0.64
4.68
2.32
4.89
1.26
This study
PIB Ž1170, 1350, 2700. – nC5 Ž42.
PIB Ž50,000. – nC6 Ž24.
PIB Ž45,000. –benzene Ž29.
PS Ž10,300, 49,000, 290,000. –toluene Ž35.
PS Ž290,000. –CHCl 3 Ž22.
PEO Ž6=10 5 . –benzene Ž13.
PVAc Ž194,000. –CHCl 3 Ž7.
PVME Ž14,600. –toluene Ž34.
a
298–328
298–338
298–338
298–353
298–323
323–343
333
323–343
8.73
2.99
5.70
0.55
5.04
0.29
4.96
0.46
vdW1f
ŽBrAM.
3.53
3.08
4.93
0.62
5.25
0.36
4.94
0.45
vdW1f
ŽBrL.
ZM
MHV1
14.17
3.02
1.54
1.18
1.59
1.67
5.50
0.78
1.52
4.19
1.44
1.12
5.98
0.62
5.03
0.46
1.55
2.67
6.50
0.62
4.71
2.40
4.91
1.27
14.00
30.38
11.51
25.55
39.57
35.68
110.7
5.25
2.03
4.26
1.42
1.13
5.99
0.62
5.03
0.45
1.41
2.88
4.31
1.17
6.54
1.04
6.33
0.71
OBC
35.22
2.69
2.06
1.55
1.82
1.48
5.53
1.12
3.92
3.98
1.73
1.12
5.99
0.62
5.04
0.45
6.05
2.77
6.44
0.63
4.78
2.44
4.91
1.20
D y vdW1f ŽBrAM. s 4.1Ey4, D y vdW1f ŽBrL. s 2.0Ey4, D y ZM s 3.1Ey4, D y MHV1 s 3.2Ey4.
4.36
24.62
9.98
24.29
32.53
35.57
82.82
4.05
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
T range
ŽK.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
179
The same observations apply to the extrapolation with respect to polymer MW ŽTable 4., when
high values of it are involved Ž) 10,000., where the MW does not affect significantly the VLE Ž Fig.
9..
At last, the extrapolation with respect to solvent concentration from a limited number of data: those
in the lowest and those in the highest solvent concentration range may lead to uncertainties.
4.3. The effect of polymer parameters
Table 5 presents results for some representative systems using the polymer parameters determined
by four methods: the proposed, the KHFT, the OS and the OBC ones, combined with the mixing rules
considered here: vdW1f ŽBrAM., vdW1f ŽBrL., ZM and MHV1.
The first three methods for polymer parameters determination give similar and good results in all
cases, with the exception of the low MW PIB systems. There, the KHFT parameters lead to
significant amounts of polymer in the vapor phase for the system PIB–nC5. This is due to the high
polymer P s values predicted by them, as shown in Fig. 2, and the low solvent mole fractions — due
to the low polymer MW — which make more significant the influence of the polymer EoS
parameters to the VLE correlation. That is why this phenomenon is not observed for the high MW
polymers. For the same low MW systems, the vdW1f Ž BrL. mixing rule gives high errors in
pressure, especially with the parameters of this study.
The OBC parameters give poor results Ž high errors in pressure and unrealistic values for the
interaction parameter. when used with the vdW1f Ž BrAM. and vdW1f ŽBrL. mixing rules, as shown
in Fig. 10 for the system PS Ž290,000. –chloroform. Moreover, their performance deteriorates with
increasing polymer MW due to the unrealistic high value of Pc . On the other hand, they give good
results with the ZM mixing rule, but the interaction parameters assume unrealistically high absolute
Fig. 10. Correlation of the bubble point pressure for the system PS Ž290,000. –chloroform at 298.15 K using the OBC
polymer parameters.
180
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
values ŽFig. 10.. The parameters do provide, however, satisfactory results when used with the MHV1
mixing rule.
5. Conclusions
A cubic EoS, in this case the PR one, is applied to the modeling of VLE of polymer–solvent
systems. Polymer EoS parameters, arMW and brMW, are obtained by fitting polymer PVT data
and provide excellent results in the description of the pure polymer behavior up to very high
pressures.
Correlation of VLE data for a variety of non-polar and polar systems— including hydrogen
bonding ones— is carried out by using three mixing rules: the vdW1f one Ž with the Berthelot
combining rule for a i j and the AM and Lorentz combining rules for bi j . ; the one of ZM and the
MHV1 proposed by Orbey et al., all involving only one interaction parameter. All mixing rules give
low errors in pressure, but phase split may be predicted on occasion with the vdW1f Ž BrAM. and
MHV1 mixing rules, while the vdW1f mixing rule with the Berthelot–Lorentz combining rules for a i j
and bi j , respectively, fails when very small polymer MWs are encountered. Best overall results are
obtained with the ZM mixing rule, especially since no phase split is observed with it.
Extrapolation with respect to temperature and polymer MW Ž) 10,000. is satisfactory, especially
when the ZM and vdW1f Ž BrL. mixing rules are employed, since the vdW1f Ž BrAM. and MHV1
mixing rules lead on occasion to pressure maxima.
Three other methods of evaluating the polymer EoS parameters, those of Kontogeorgis et al.
ŽKHFT., Orbey–Sandler Ž OS. and Orbey et al. Ž OBC. , were also considered. They all give poorer
PVT predictions than the proposed method, especially the OBC one which considers the same
parameters for all polymers, and they also predict substantial polymer P s values. Correlation of
binary VLE data, using the mixing rules considered here with the polymer parameter values of KHFT
and OS methods, gives satisfactory results, except when very small MW polymers are involved,
where the KHFT method may lead to significant polymer amounts in the vapor phase. The OBC
parameters give poor results with the vdW1f mixing rule, unrealistic interaction parameter values with
the ZM mixing rule, but satisfactory results with the MHV1 one.
List of symbols
a
energy parameter of a cubic EoS Žbar cm6rmol 2 .
b
co-volume parameter of a cubic EoS Žcm3rmol.
c1 , c 2 , c 3
pure compound parameters as defined in Mathias–Copeman expression
G
Gibbs free energy
A
Helmholtz free energy
ki j
binary interaction parameter in Eqs. 7.1 and 8.1
m
parameter of Eq. 4
P
pressure Ž bar.
R
universal ideal gas constant Žs 83.14 bar cm3rmol K.
T
absolute temperature ŽK.
V
molar volume Žcm3rmol.
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
x, y
w
181
mole fraction
weight fraction
Greek symbols
a Ž Tr .
energy parameter temperature correction
x
Flory interaction parameter
v
acentric factor
Superscripts
E
excess
s
saturation state
Subscripts
c
cal
exp
iŽ j .
ij
m
r
critical property
calculated value
experimental value
component in a mixture
cross-parameter Ždefined by the combining rules.
mixture
reduced property
References
w1x
w2x
w3x
w4x
w5x
w6x
w7x
w8x
w9x
w10x
w11x
w12x
w13x
w14x
w15x
w16x
w17x
w18x
w19x
w20x
w21x
w22x
w23x
R. Gupta, J. Prausnitz, Ind. Eng. Chem. Res. 35 Ž1996. 1225–1230.
P.J. Flory, J. Chem. Phys. 9 Ž1941. 660–661.
P.J. Flory, J. Chem. Phys. 10 Ž1942. 51–61.
G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, Ind. Eng. Chem. Res. 32 Ž1993. 362–372.
T. Oishi, J.M. Prausnitz, Ind. Eng. Chem., Process. Des. Dev. 17 Ž1978. 333–339.
J. Holten-Andersen, P. Rasmussen, Aa. Fredenslund, Ind. Eng. Chem. Res. 26 Ž1987. 1382.
M.S. High, R.P. Danner, AIChE J. 36 Ž1990. 1625.
W.O. Morris, P. Vimalchand, M.D. Donohue, Fluid Phase Equilib. 32 Ž1987. 103–115.
S.H. Huang, M. Radosz, Ind. Eng. Chem. Res. 29 Ž1990. 2284–2294.
I.C. Sanchez, R.H. Lacombe, J. Phys. Chem. 80 Ž1976. 2528.
C.G. Panayiotou, J.H. Vera, Polymer J. 14 Ž1982. 681–694.
T. Sako, A.H. Wu, J.M. Prausnitz, J. Appl. Polym. Sci. 38 Ž1989. 1839–1858.
G.M. Kontogeorgis, V.I. Harismiadis, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilib. 96 Ž1994. 65–92.
V.I. Harismiadis, G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilib. 96 Ž1994. 93–117.
V.I. Harismiadis, G.M. Kontogeorgis, A. Saraiva, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilib. 100 Ž1994.
63–102.
A. Saraiva, G.M. Kontogeorgis, V.I. Harismiadis, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilib. 115 Ž1994.
73–93.
V.I. Harismiadis, A.R.D. van Bergen, G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, AIChE J. 42 Ž1996.
3170–3180.
S. Bithas, N.S. Kalospiros, G.K. Kontogeorgis, D.P. Tassios, J. Polym. Eng. Sci. 36 Ž1995. 254–261.
N. Orbey, S.I. Sandler, AIChE J. 40 Ž1994. 1203–1209.
D.S.H. Wong, S.I. Sandler, AIChE J. 38 Ž1992. 671.
C. Zhong, H. Masuoka, Fluid Phase Equilib. 123 Ž1996. 59–69.
N.S. Kalospiros, D.P. Tassios, Ind. Eng. Chem. Res. 34 Ž1995. 2117–2124.
M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 Ž1955. 2701.
182
w24x
w25x
w26x
w27x
w28x
w29x
w30x
w31x
w32x
w33x
w34x
w35x
w36x
V. Louli, D. Tassiosr Fluid Phase Equilibria 168 (2000) 165–182
J.J. Aklonis, W.J. MacKnight, in: Introduction to Polymer Viscoelasticity, 2nd edn., Wiley, New York, 1983.
H. Orbey, C.P. Bokis, C. Chen, Ind. Eng. Chem. Res. 37 Ž1998. 1567–1573.
P.A. Rodgers, J. Appl. Polym. Sci. 48 Ž1993. 1061–1080.
C. Zhong, H. Masuoka, Fluid Phase Equilib. 144 Ž1998. 49–57.
T.E. Daubert, R.P. Danner, Physical and Thermodynamic Properties of Pure Compounds: Data Compilation, Hemisphere, New York, 1989.
K. Magoulas, D.P. Tassios, Fluid Phase Equilib. 56 Ž1990. 119–140.
P.M. Mathias, T.W. Copeman, Fluid Phase Equilib. 13 Ž1983. 91–108.
M.L. Huggins, Ann. N.Y. Acad. Sci. 43 Ž1942. 431–443.
H. Wen, H.S. Elbro, P. Alessi, Polymer Solution Data Collection Part 1: Vapor–Liquid Equilibrium, DECHEMA
Chemistry Data Series, DECHEMA, Frankfurt, 1991.
C. Wohlfarth, Vapor–Liquid Equilibrium Data of Binary Polymer Solutions, Elsevier, Amsterdam, 1994.
R.B. Gupta, J.M. Prausnitz, J. Chem. Eng. Data 40 Ž1995. 784–791.
C. Boukouvalas, N. Spiliotis, P. Coutsikos, N. Tzouvaras, D. Tassios, Fluid Phase Equilib. 92 Ž1994. 75–106.
N.S. Kalospiros, N. Tzouvaras, Ph. Coutsikos, D.P. Tassios, AIChE J. 41 Ž1995. 928–937.