journalof

MEMBRANE
SCIENCE
ELSEVIER Joumal of Membrane Science 139 (1998) 201-209

A diffusion model of the pervaporation separation

of ethylene glycol-water mixtures through
crosslinked poly(vinyl alcohol) membrane
ER. Chen*, H.E Chen
Department of Chemical Engineering, Tianjin University, Tianjin 300072, China

Received 28 July 1997; received in revised form 30 September 1997; accepted 30 September 1997

Abstract

The pervaporation separation of ethylene glycol-water mixtures was carried out over the full range of compositions at
temperatures varying from 60 to 80°C, using chemically crosslinked PVA dense membrane which had been developed in our
laboratory. A new thermodynamic diffusion coefficient equation is derived based on the modified Vigne equation. Combining
Lee-Thodos equations, Wilke-Chang equations, Vrentas-Duda's free volume theory, diffusion equations and swelling
equilibrium equations, the permeation fluxes of individual components in ethylene glycol-water mixture through crosslinked
poly(vinyl alcohol) (PVA) dense membrane have been calculated and showed to be in agreement with the experimental values.
© 1998 Elsevier Science B.V.

Keywords: Pervaporation; Diffusion in dense membrane; Crosslinked PVA; Ethylene glycol-water mixtures

1. Introduction meability. Single component permeation through

dense membranes can be satisfactorily described by
The diffusion of small molecules (gases or liquids) Fick's law with a concentration dependent diffusion
in polymeric solids is a subject of which relatively coefficient, as has been done by several authors [2-6].
little interest was shown by the polymer chemists in For multicomponent permeation, many models have
the fifties [ 1]. A number of pervaporation models have been developed by many investigators [7-17]. To date,
been developed. The main researchers focused on how no satisfactory theory exists because the transport
to obtain the diffusion coefficients of penetrants phenomenon is more complicated due to the coupling
through polymeric dense membranes, of fluxes, both in the solution and the diffusion pro-
In a pervaporafion transport model, one should cesses. In the previously described model [18], the
distinguish between single and multicomponent per- coupling effect of the sorption was considered. The
object of this work is to develop a transport model for
the pervaporation separation of binary polar-polar
*Corresponding author. Present address: Department of Chemi- liquid mixtures through a polymer membrane taking
cal Engineering, University of Massachusetts, Amherst, MA 01003, into account the coupling of fluxes, a new thermo-
USA. Fax: +1 413 545 1647. dynamic diffusion coefficient equation is derived

0376-7388/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved.

PII S0376-7388(97)00260-3
202 ER. Chen, H.E Chen/Journal of Membrane Science 139 (1998)201-209

based on the modified Vigne equation. The pervapora- 01n a2 ( 1 V2 V2 V2

tion separation of ethylene glycol (EG)-water mix- 01n~2 -- ~02\ ~/2 -- 1 +V33 - g12 ~11~1 + g13~1 V~
tures by crosslinked poly(vinyl alcohol) (PVA) V2~2(1 _~92)0g12
membrane was used for the experimental study. + g23 (2~2 + q01 -- 2) +
gl (~1 + qo2)20u2
V2 u2t.1 Og12 V2 3 02g12
2. Theoretical +~1 1\ -2u2)-~u2 +VlU2Ul --@-u22j (5)

Thus, knowing the thermodynamic diffusion coeffi-

2.1. Description of the model cients, Di*3, it is possible to calculate the individual
fluxes for permeation of a binary liquid mixture using
The model follows the principle of the solution-
Eqs. (2)-(5). In a pervaporation process, diffusion
diffusion model but the following assumptions have
coefficients of permeating components through dense
been made: membranes in general depend strongly on the con-
e the flow of permeants in the membrane as one-
centrations of penetrants. Therefore, diffusion coeffi-
dimensional steady-state diffusion.
cients will vary considerably across the membrane.
• the interfaces of the membrane are in equilibrium For the determination of the thermodynamic diffusion
with the upstream and downstream phase, coefficients of a polar-polar mixture through dense
• the chemical potential of a component in the
membrane, we suggest that the thermodynamic diffu-
polymeric membrane can be described by sion coefficient of component (index i) through dense
Flory-Huggins thermodynamics [19]. membrane (index 3), Di*3, consists of three parts: the
• during steady-state, the membrane undergoes no self-diffusivity of component i which we define as D*;
structural changes, the basic diffusivity of component i in j at infinite
According to the solution-diffusion model, the flux
dilution of i which we define as Di~; the basic diffu-
of a component i through the membrane can be
sivity of component i in dense membrane (index 3) at
described by infinite dilution of i which we define as Dig. Then we
, 01nai 0~i employ the Vigne relationships [20] to estimate the
Ji = -piDi 3 ~ ~z (1) thermodynamic diffusivity Di*3 from the basic diffu-
sivities (Di*, Di~, D~3). We obtain
Assuming that
3
, 01nai (2) In D*3 = ~iln Di* + Z ~jln Di~ (6)
Di3 ----Di3 01nwi j=l,
j#i
Substitution of Eq. (2) into Eq. (1) gives:
where D~i represents the self-interaction of molecule i;
represents the interaction between molecules i and
Ji=-piDi3~-~z i (3) jD~i~ represents the interaction between molecules i
In Eq. (2), the activity of a component in the mem- and membrane (index 3). The self-diffusivity Di] is
brane can be described by Flory-Huggins thermody- obtained from the Lee equation. The basic diffusivity
namics [19]. For a ternary system, the Olnal/OlnqOl and Di~ can be calculated from the Wilke-Chang equa-
Olna2/Olnqo2 are given by tion. The determination of basic diffusivity Dig is
based on the extension of Vrentas and Duda free
01n a l /" 1 V1 V1 volume theory and steady state pervaporation experi-
01nq01 - ~1 ~ ~11 - 1 + ~ - g12q02 + g23q02~ ments of single components.
--g13(2 2W1 ~92) qa2(1 - - ~ 1 ) O g l 2 For the ternary system water (D-ethylene glycol
- - - (2)-crosslinked PVA (3), Eq. (6) becomes
(~1 q- qtg2)2 0 U 2
+ u~(1 -- 2Ul) Og12 3 02g12 "~ lnD~3 = q°llnD~l + qo21nD~2 + q°31nD~3 (7)
-]- UlU2--~U~) (4) lnD~3 = ~p21nD~2 + wllnD~ + w31nD~3 (8)
 ER. Chen, H.F. Chen/Journal of Membrane Science 139 (1998) 201-209 203

* 00 O0 OG
where the determination ofD~l, D22, D12, D21, D13 and Table 2
D~3 are shown in the following sections. The viscosities of water (1) and EG (2) at various temperatures
T/°C 60 70 80
2.2. Determination of the diffusivities odcP 0.4688 0.4061 0.3565
~2]cP 6.5 4.8 3.4
2.2.1. Self-diffusivities D~I and D~2
According to Lee's self-diffusivity equation [21],
the Di] is defined as the following expressions:
Table 3
Di*° × 105 =- [0.7094G + 0.1916] 2.5 (0 < G < 1.0) The basic diffusivities of water (1) and EG (2) at various
TR temperatures
(9) r/oc 60 70 80
Di*o- _ 0.77 × 10 -5 (G > 1.0) (10) D12/m2 s-I 6.274x10 -1° 8.751x10-l° 1.271x10 9
TR PR -- D21/m2s-1 2.898x 10.9 3.446x10.9 4.040x 10.9

cr= M 1/2/Pc1/2 Vc5/6 (11)

G -- ('OR/T~R'I)* -- (pR/TR)R'I) _ X* - X (12) 2.2.3. Basic diffusivities D~3 and D ~

(pR/TOR'I)* -- (pR/T~R'I)c X* -- 1
2.2.3.1. Derivation of D R and D~ functions. The
X- PR (13) diffusion of small molecules in polymeric
T°RI membranes is usually described by application of
The self-diffusivities of water (1) and EG (2) at the free volume theory of molecular transport. This
various temperatures can be calculated from theory has achieved some degree of success in
Eqs. (9)-(13). The results are shown in Table 1. describing the diffusional behavior of organic
solvents in amorphous polymers [1]. However, the
2.2.2. Basic diffusivities D~2 and D~I1 free volume theory of diffusion, formulated by Fujita
When predicting the basic diffusivities Di~ (i.e., the [1], does not correctly predict the concentration
binary diffusivity of i in j at infinite dilution of i) we dependence of the mutual diffusion coefficient for
employ the Wilke-Chang equation [22]. penetrant molecules of small size in amorphous
T polymers because the key assumption is that the
D i 7 = 7.4 × 10-12(q~jMj)l/2rljV~b/6 (14) molecular weight of the solvent is equal to the
molecular weight of a jumping unit of the polymer
where Vb1=18.9 cm 3 mol - t , Vb2=66.6 cm 3 mo1-1 chain (Mi=Mr). For organic solvents, it can be
[22], ~b1=2.6, and ~b2=1.5 [23]. The viscosities of expected that the molecular weight of the solvent is
water (1) and EG (2) at various temperatures are relatively close to that of the polymeric jumping unit
shown in Table 2. since the polymer is often formed from a monomer
The basic diffusivities of water (1) and EG (2) at which itself is an organic solvent. Hence, it is
various temperatures can be calculated from Eq. (14), reasonable to expect that the diffusional behavior of
and these are shown in Table 3. such solvents can be explained by the theory of Fujita.
On the other hand, for small molecules of low
molecular weight, such as water, the solvent M W
Table 1 will be significantly less than the molecular weight
The self-diffusivities of water (1) and EG (2) at various of the jumping unit of the polymer chain, and the
temperatures predictions of the Fujita version of the free volume
T/°C 60 70 80 theory will no longer be acceptable. But the Vrentas
and Duda version of the free volume theory illustrate
D~l/m2s -1 1.2836x10-9 1.4430x10-9 1.6243x10 9 the effect of solvent size (Mi~Mr). Therefore, the
D~2/m2s i 1.7861x10 9 2.0240x10 9 2.2960x10 9
Vrentas and Duda version of free volume is a more
204 RR. Chen, H.E Chen/Joumal of Membrane Science 139 (1998) 201-209

general adaptation of the theory of Cohen and becomes:

Turnbull [24] to polymer-solvent diffusion than the
theory proposed by Fujita. In this work, the derivation
&i(R + WpMi/Mr)
hi =RTAdiexP - (1 _ wc)f(wi, T) (21)
of 0;“; and 0; functions is based on the development 1
of Vrentas and Duda version of the free volume theory. where w, is the crystallinity of the polymer. We define
According to the Vrentas and Duda [25] version of a new generalized parameter Bi as
the free volume theory, it can be shown that the total
probability P(v*) of finding a free volume exceeding a Bi = ywc) (22)

V*y(Wi
+WPMi/Mr)
cl

given value v* is represented by

VW1
and Eq. (21) becomes:

1
P(v*) = exp n (15) &(W + WpMi/Mr)
[ DTi = RTAdiexp - (23)
f (vi, T)
Here, pr is the specific critical hole free volume of
component i required for displacement of this com- The free volume of the binary system (pure liquid-
ponent; em is the average hole free volume per gram membrane) is given by
of mixture; y is an overlap factor(which should be f(vi, T) =f(O, T) + pi(T)ui (24)
between l/2 and 1). Next, we write the product rp* as
whereflO,T) is the free volume fraction of the polymer
Bdi, thus
itself and p(T) is a proportional constant related to the
Bdi = yv; (16) amount of free volume increased by the diffusing
species i. Substitution of Eq. (24) into Eq. (23) gives:
and v, should be replaced with the average fractional
free volume of the systemf(vi,T). Therefore, Eq. (15) &(Wi + WpMi/Mr)
can be written as DTi= RTAdiexp
- f(o, T) + &f+, (25)
1

1
Bdi(Wi + WrMi/Mr) When Ui=O in Eq. (25), the diffusion coefficient at
P(v*) = exp - (17)
f (vi, T) zero concentration, Dg, is given by

According to the free volume of molecular trans-

0; = RTA,exp[-B$$f] (26)
port, the mobility m& of a penetrant is proportional to
P(v*). Thus
For the ternary system water (l)-EG (2)-crosslinked
Bdi(Wi + WpMi/M,) PVA (3), Eq. (26) can be rewritten as
mdi = Aaexp - (18)
.f(vi, T) I
DE = RTAdiexp[-B$:iy] (27)
where Adi is considered to be dependent primarily
upon the size and shape of the penetrant and hence
may be independent of temperature and concentration. Dg = RTAd~exp[-B$$~] (28)
The usual definition of the mobility, Ittdi, of a
penetrant is
DTi = RTma (19) 2.2.3.2. Determination of 0s and DE. To obtain DE
and 08, the determinations of the free volume
Substitution of Eq. (18) into Eq. (19) gives:

1 (20)
parameters, Aa and Bi, are needed. The free volume

DTi = RTAaexp -
&(K + WpMi/Mr) parameters can be obtained from steady state
f (vi, T) pervaporation experiments of single components.
For each binary system, the flux of component i
As the crystalline regions in polymer are generally can be expressed by
considered to be impenetrable to penetrants, the crys-
Mites are then obstacles and the penetrant molecules dlnai dvi
Ji =-PiDT’~.~ (2%
have to pass around them. Therefore, Eq. (20) 1
 ER. Chen, H.F. Chen/Journal of Membrane Science 139 (1998) 201-209 205

where d In ai/d In vi is calculated using Flory-Huggins Table 4

thermodynamics [19] Free volume parameter ri0,T) of crosslinked PVA membrane at
various temperatures
d In ai
-- 1 - vi - 2Xip/JiVp (30) T/°C 60 70 80
d In vi ri0,T) 0.02211 0.02240 0.02269
Substitution of Eqs. (25) and (30) into Eq. (29)
gives:
Bi(Wi + WpMi/Mr)] When v i = l , Eq. (24) becomes
Ji = -piRTAdiexp f ( 0 , T) ~ J f ( 1 , T) = f ( 0 , T) +/3i(T) (37)

dvi The free volume fraction in a liquid, f(1,T), can be

× (1 - vi - 2XipViVp)dz (31) defined as [26]

where Vp = 1 - vi. The permeability of penetrant i, f ( 1 , T) -- vf - v0 (38)

Pi, gives from Eq. (31) vf
~i(z=0) where vf is the specific volume of the liquid at any
Pi piRTAdi [ (1 - vi)(1 - 2XipVi) temperature and Vo is the specific volume of the liquid
J extrapolated to the temperature T at zero K without a
0 phase change in the form of a density function. The
I Bi(Wi + WpMi/Mr)]dv
x exp . .( 0 , . T) .+/Ti(T)vi
f . ] i (32) liquid density functions of water (1) and EG (2) is
given in Table 5 [27]. Thus, the free volume
assuming parameters, f(1,T) and/3i(T), can be computed with
Eqs. (37) and (38) and data in Tables 4 and 5, and are
,i(z=0)
P summarized in Table 6.
F(Bi, T) = / ( 1 - v i ) ( 1 - 2XipVi) So far, the parameters Xip, A1,T) and /~i(T) in
J0 Eq. (33) have been obtained. The free volume para-
meters Bi and Adi can be calculated from perva-
Bi(Wi ÷ WpMi/Mr)']
x exp . . . . . | dvi (33)
f ( 0 , T) ÷/3i(T)vi J
Substitution of Eq. (33) into Eq. (32) gives: Table 5
Coefficients for the liquid density functionsof water (1) and EG (2)
Pi = RTAdipiF(Bi, T) (34) Equation p = M . A/B [l+(1-r/c)D] (kg m-3)
For two different temperatures, T1, and T2, Eq. (34) Coefficients A B C D
becomes Water 4.6137 0.26214 647.29 0.23072
Pi(T1) _ TlPlF(Bi, T1) (35) EG 1.3352 0.25499 645.00 0.172
Pi (T2) T2P2F(Bi, r2)
In Eq. (33), the interaction parameter Xip (or gi3) Table 6
between i and p is given in the literature [18] and Free volume parameters of water (1) and EG (2) in crosslinked
the free volume fraction of membrane is driven by PVA membrane
[ 16] Penetrants TI°C f(1 ,T) fli(T)
f ( 0 , T) = 0.025 - 0.025(Tg - T)/2Tg (36) Water 60 0.1853 0.1632
70 0.1921 0.1697
The Tg of the crosslinked PVA dense membrane was 80 0.2016 0.1789
found to be 160°C by differential scanning calorimetry
(DSC). Thus, the free volume parameter, ri0,T), of EG 60 0.1483 0.1262
crosslinked PVA membrane at various temperatures is 70 0.1541 0.1316
80 0.1599 0.1372
given in Table 4./Ti(T) can be obtained from Eq. (24).
206 ER. Chen, H.E Chen/Journal of Membrane Science 139 (1998) 201-209

Table 7 lf~o.oo .
Free volume parameters Bi and Adi of water (1) and EG (2) in
crosslinked PVA membrane ~v.eqlma00 l
Penetrants Bi RxA~ -~
Water 0.5333 5.3358 × 10 -9 ~ i " - ~ 0°C
EG 0.1451 6.8969x 10 12 ~

.O 8170.00 -

Table 8
The basic diffusion coefficients D~, D~3 of water (1) and EG (2) at
various temperatures ~ ~ C ~
T/°C D~3/m2s 1 D~3/m2s-I ~ 400.(30 ~ ~ 1

60 9.2305×10 -15 2.2156x10 17 ~o " ~ - ~ ' ~ N

70 1.0803 × 10-14 2.5722 × 10 17
80 1.2575x10 14 2.9712x10-17 a0o ' I ' I ' ' J -'
0.00 0.20 0.40 aro 0.80 1.00
Weight fraction of EG in feed
poration experiments of single components and
Fig. 1. Dependence of total permeation flux on feed composition
Eqs. (33)-(35) by using Roberge numerical inte- for the crosslinked PVA membrane.
gration and Golden Section Search method. The
results are given in Table 7. Thus, the basic diffusion
1600.00--
coefficients D~3 and D~3 of water (1) and EG (2) ~
at various temperatures can be obtained from
Eqs. (27) and (28) and data in Table 7, as shown in
Table 8. "~ ,2oo.0o-
(lJ
O°C [

3. Experimental
O 800.00-
The pervaporation separations of ethylene glycol-
water mixtures were carried out over the full range
of compositions at temperatures of 60, 70 and o~ ~ . ~ "=
80°C, using chemically crosslinked PVA membranes .~ ~0.0o
developed in our laboratory. More details about
the apparatus, the pervaporation cell, and the
experimental procedure have been described else- ~ ~00 ' I ' I ' I ' m
where [28]. o.oo a2o a4o aro ~so ~.00
In Figs. 1-3, the experimental results for per- Weight f r a c t i o n o f EG i n f e e d
vaporation are given as a function of the weight Fig. 2. Dependence of water permeation flux on feed composition
fraction of EG in the feed mixture. The effect of for the crosslinked PVA membrane.
composition of the feed mixture on the total per-
meation flux at different temperatures is shown in
Fig. 1. In the permeation flux of individual compo- permeation flux of ethylene glycol has a maximum
nents (Figs. 2 and 3), the permeation flux of water is point. The results indicate a large synergistic effect of
significantly greater than the permeation flux of ethy- the permeation fluxes of water-ethylene glycol mix-
lene glycol. Especially when the ethylene glycol ture in crosslinked poly(vinyl alcohol) dense mem-
weight fraction in the feed mixture is about 0.7, the brane.
 ER. Chen, H.E Chen/Journal of Membrane Science 139 (1998) 201-209 207

~o0- 4. C o m p a r i s o n b e t w e e n the calculated a n d the

. ~ e x p e r i m e n t a l p e r m e a t i o n fluxes
c.]
1~oo When the equilibrium sorption data, the binary
interaction parameters, the diffusion coefficients and
the concentration profiles are known, it is possible to
o lz00 calculate the individual permeation fluxes in the per-
~ meation of water--ethylene glycol mixtures by using
Eqs. (2)-(5), (7) and (8) and the data in Tables 1, 3
~00 - and 8. In this paper, the solubilities at the feed side are
J calculated using the modified U N I Q U A C - F l o r y - H u g -
gins model which is described in part II [18]. The
4.00- concentrations of penetrants on the surface of a mem-
0J ~ brane on the permeate side, in pervaporation, can be
I approximated as zero [29]. The results of the calcula-
~00 , I ' I ' I ' I ' tions are presented in Table 9 in comparison with
~oo ~ o.4o o.6o ~ 1.oo experimental permeation fluxes in the full range of
Weight fraction of EG in feed
feed compositions at 60, 70 and 80°C, respectively
Fig. 3. Dependence of EG permeation flux on feed composition for (Model A). In order to c h e c k the diffusion coupling
the crosslinked PVA membrane, behavior of fluxes, we calculated the individual per-

Table 9
Comparison of calculated and experimental permeation fluxes of individual components in water-EG mixture through crosslinked PVA
membrane at various temperatures

Temp. in EG content Flux× 103/kg m -2 h-1 Ratio ~

feed (°C) in feed (wt%)
Experimental Model A a Model B b

Water EG Water EG Water EG Water EG

60 10.0 584.8 1.2 658.0 0.7 890.9 0.4 1.125 0.583

30.4 464.7 2.3 592.3 2.5 821.1 1.1 1.274 1.087
50.4 407.2 4.9 475.1 4.6 624.1 1.5 1.167 0.939
70.5 187.7 9.0 350.0 7.0 524.9 2.0 1.865 0.778
82.2 124.9 7.7 203.7 6.7 258.1 1.5 1.631 0.870
90.4 61.4 5.3 88.5 4.9 172.8 1.8 1.444 0.925

70 10.2 768.5 1.5 818.8 0.9 1318.7 0.6 1.065 0.600

30.5 622.4 3.8 688.3 3.1 1078.2 1.5 1.106 0.816
50.2 553.8 9.0 599.9 6.0 861.2 2.0 1.083 0.667
70.6 332.4 16.4 417.9 8.9 698.8 2.7 1.257 0.543
80.6 196.6 12.1 259.2 8.2 396.6 2.1 1.318 0.678
90.6 93.6 10.8 117.4 6.7 182.4 1.8 1.254 0.620

80 10.3 1345.3 2.7 1414.7 1.7 1430.4 0.7 1.052 0.630

30.6 1158.0 7.0 1243.6 5.8 1363.7 1.9 1.074 0.828
49.9 874.0 16.0 883.8 8.9 1148.0 2.7 1.011 0.556
70.7 452.1 19.8 571.7 12.2 832.8 3.2 1.264 0.616
82.5 291.3 18.6 365.1 12.9 519.9 3.0 1.253 0.694
90.8 133.5 17.2 138.4 9.1 232.1 2.3 1.037 0.529

a Model A: It is our modified model which takes into account non-ideal solubility effects and diffusion coupling.
t, Model B: It is our simplified model which takes into account non-ideal solubility effects, but neglects diffusion coupling.
Ratio: calculated permeation flux (Model A)/experimental permeation flux.
208 F.R. Chen, H.F. Chen/Journal of Membrane Science 139 (1998) 201-209

meation fluxes using our simplified model which takes Di* self-diffusivity of component i (m 2 s-1)
into account non-ideal solubility effects, but neglects Di~ basic diffusivity of i in j at infinite dilution
diffusion coupling, and the results are shown in of i (m 2 s -1)
Table 9 (Model B). As can be seen in Table 9, the DT thermodynamic diffusion coefficient for
modified model (Model A) works more successfully single components (m e s -1)
than the simplified model (Model B) does for predict- g binary interaction parameter
ing the permeation of water-ethylene glycol mixtures J permeation flux (kg m -2 h -1)
through crosslinked poly(vinyl alcohol) membrane, M molecular weight
and the enhancement of ethylene glycol permeation by md mobility (m s -1)
the presence of water is more pronounced than that of P permeability (kg m-1 h-1)
water due to the presence of ethylene glycol. On the p pressure (Pa)
whole, the water and ethylene glycol permeation R gas constant (Jmol 1 K - t )
fluxes from the modified model (Model A) are close T operating temperature (K or °C)
to the experimental data. The comparison results Tg glass transition temperature (K)
indicate that, apart from the non-ideal solubility u volume fraction referred to the nonsolvent
effects of the permeating components, the non-ideal part in the ternary system
diffusivity behavior and diffusion coupling contribute V molar volume (m 3 mo1-1)
a major extent to selective permeation. Vb molar volume at the boiling point (m 3 mol)
Z distance along which diffusion takes place
5. Conclusions (m)
6.1. Greek letters
Pervaporation of a polar-polar mixture through
crosslinked PVA membrane is a complex process
which is influenced by several factors. Our model ~ viscosity (Pa s)
# chemical potential (J mo1-1)
takes into account non-ideal solubility effects, non-
ideal diffusivity behavior, and diffusion coupling in /9 density (kg m -3)
order to describe semiquantitatively the performance v volume fraction in the binary system
of a pervaporation membrane for a given separation ~ volume fraction in the temary system
problem. A new thermodynamic diffusion coefficient X Flory-Huggins interaction parameter
equation is derived based on the modified Vigne 6.2. Indices
equation. Combining Lee-Thodos equations,
Wilke-Chang equations, Vrentas-Duda's free volume
1 water
theory, diffusion equations, and swelling equilibrium 2 ethylene glycol
equations, the permeation fluxes of individual com- 3 PVA membrane
ponents in water-ethylene glycol mixtures through
c critical
crosslinked PVA dense membrane have been calcu-
lated and shown to be in good agreement with the i component i
experimental values, j component j
p polymer
R reduced
6. List of symbols r repeat unit of polymer

a activity
Ad, B free volume parameters References
D diffusion coefficient in the polymer fixed
frame of reference (m 2 s-1) [1] H. Fujita, Diffusion in polymer-diluent systems, Adv. Polym.
Sci. 3 (1961) 1.
D* thermodynamic diffusion coefficient for [2] R.C. Binning, R.J. Lee, J.F. Jennings, E.C. Martin, Separation
binary components (m 2 s -1) of liquid mixtures, Ind. Eng. Chem. 53 (1961) 45.
 ER. Chen, H.E Chen/Journal of Membrane Science 139 (1998) 201-209 209

[3] R.B. Long, Liquid permeation through plastic films, Ind. Eng. [16] C.K. Yeom, R.Y.M. Huang, Modelling of the pervaporation
Chem. Fundam. 4 (1965)445. separation of ethanol-water mixtures through crosslinked
[4] R.Y.M. Huang, V.J.C. Lin, Separation of liquid mixtures by poly(vinyl alcohol) membrane, J. Membr. Sci. 67 (1992) 39.
using polymer membranes. 1. Permeation of binary organic [17] A. Heintz, W. Stephan, A generalized solution-diffusion
liquid mixtures through polyethylene, J. Appl. Polym. Sci. 12 model of the pervaporation process through composite
(1968) 2615. membranes. Part II. Concentration polarization, coupled
[5] S.N. Kim, K. Kammermeyer, Actual concentration profiles in diffusion and the influence of the porous support layer, J.
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