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Model equations for interactions of hydrated species in transmembrane transport
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DESALINATION Desalination 163 (2004) 177-192
ELSEVIER
Model equations for interactions of hydrated species in transmembrane transport Andrzej
SlCzak " a*, Slawomir Grzegorczyn b, Jacek W~sik c
aDepartment of Biomedical Fundamentals of Sport, CzOstochowa Technical University, 19B Armia Krajowa Al, 42-200 Czestochowa Tel. & Fax: +48 (34) 361-3876; email: [email protected];[email protected] bDepartment of Biophysics, Silesian Medical Academy, 19 Henryk Jordan Str, 41-808 Zabrze Clnstitute of Physics, Pedagogical University, 13/15 Armia Krajowa Al, 42-200 Cz~stochowa Received 17 July 2003; accepted 16 October 2003
Abstract
The relationship between friction coefficients and hydration numbers of permeating solutes through membranes is presented in the framework of generalised Spiegler-Kedem-Katchaisky frictional model equations (SKI(model). The membrane system contained multicomponent, non-ionic and homogeneous hydrated solutions. In the framework of this model the expressions linking the permeability parameters of the membrane (Lp, o, coik) with the friction coefficients (f~,)for n-component non-ionic hydrated solutions were presented. The matrix of transformation from phenomenological onto frictional coefficients was worked out. Moreover, the method of calculation of the frictional and phenomenological coefficients change with the hydration number of solutes for binary and ternary non-ionic solutions was also presented. The friction coefficients and their dependencies on hydration number of glucose for aqueous glucose solution (binary solution) and glucose solution in aqueous ethanol solution (ternary solution), permeating through flat polymer membranes are also presented. Keywords: Membrane transport; Spiegler-Kedem-Katchalskyfrictional model equation; Multicomponent solutions; Hydratation number
1. I n t r o d u c t i o n
Transport of substances through membranes performs an essential part in biological, physico*Corresponding author.
chemical and technical systems [1]. This transport depends on membrane structure, its thickness and environmental influence, e.g., solution mixing near membranes [2,3]. It also depends on existence and direction of external fields:
Presented at the PERMEA 2003, Membrane Science and Technology Conference of Visegrad Countries (Czech Republic, Hungary, Poland and Slovakia), September 7-11, 2003, Tatranskd Matliare, Slovakia. 0011-9164/04/$- See front matter © 2004 Elsevier B.V. All rights reserved PII: S0011-9164(04)00099-2
A. Slfzak et aL ~Desalination 163 (2004) 177-192
178
electrical or gravitational [4-6]. The most direct approach for transport phenomena in membranes is based on specific mathematical models. The oldest are the Fick's diffusion model equations [7]. Among the most recent are those that have been elaborated on the grounds of statistical mechanics based on the Nemst-Planck [8], Stefan-Maxwell [9,10], and Gibbs-Duhem equations [11] and linear non-equilibrium thermodynamics [12-14]. The phenomenological "frictional" coefficients in equations of the Onsager were used by Spiegler [15]. Then Mackay and Meares [16] and Ginzburg and Katchalsky [17] applied Spiegler's model to the analysis of experimental data. Spiegler's frictional model equations, based on the principle of a balance between gross thermodynamic forces acting on the system and frictional interactions between the components of the system, are one of the fundamental treatments of transmembrane transport processes [9]. The appearance of friction forces (Fk) acting on the k-th component of a solution from other solution components (F k = ~ ] F~j; J*k
j = 1, ..., n, m, w) is taken into consideration in the frictional model. In stationary flows through the membrane the frictional force (Fk) equilibrates thermodynamic forces (X¥) acting on a moving solution component through the membrane ( ~ ] Xkj+¥ k --0). The direct frictional j,k
force (Fk) depends on interactions between k and j solution components (/÷k andj = 1, 2, ..., n; w, m; w, for water, m, for membrane) and can be presented as
(l) where f~ is the frictional coefficient of mutual interactions between one mole ofk and one mole o f j 0÷k) solution components; or between one mole of k component and the membrane m, v~ is the velocity of the i-th component of the solution (for membrane vm = 0). The friction coefficients
depends on the kind of solutes and on the solutes concentration in the membrane. Spiegler's frictional model equations were used by Kedem and Katchalsky [3] to give a physical interpretation of the phenomenological coefficients of membrane (Lp, o, co) and by Meares and co-workers [5] to give a mechanical interpretation of the molecular interaction by means of the macroscopic friction coefficients (R~). The generalisation of the Spiegler-KedemKatchalsky frictional model equations for multicomponent non-ionic solutions was proposed by ~l~zak [11,12]. The diffusive permeability and reflection coefficients for membrane transport of electrolyte and non-electrolyte were derived by Vaidhyanathan [13,14] on the basis of the statistical mechanical expression for the mean force acting on a molecule in a multicomponent system. In this paper the non-hydrated species was treated as one component and the total water present as another. The frictional model is used to describe the transmembrane transport of electrolyte [5-7] and non-electrolyte [1] solutions. As is well known [8], the hydration of ions (or other solutes) can occur, thus, the values of resistance and friction coefficients for solutes without hydration may differ from the values of coefficients describing interactions between hydrated species and water - - real components of a system. Sch6nert [8] has shown how the thermodynamics of irreversible processes can be applied to describe the inter- (mutual) diffusion, intra(tracer) diffusion, transference, electrical conductivity in a binary non-electrolyte and electrolyte solution in which the solute is hydrated in several steps. The Kedem-Katchalsky model [12] is based on the non-equilibrium transport description in a stationary state. The thermodynamic forces - gradient of mechanical pressure, gradient of osmotic pressure for solute and thermodynamic fluxes - - volume and solute fluxes are the essential components in this model. The dependencies between thermodynamic fluxes and forces are
A. Sl~zak et al. / Desalination 163 (2004) 177-I92
described by Spiegler-Kedem model equations [181: n
J,-- J L+E J,L i=l (2)
LP
179
2. Model equations for the interaction of the unhydrated species
In a steady state Je(x) is dependent on x mad equal to the macroscopic flow. An equation joining the thermodynamic forces (Xw, X 1 , . . . , X , ) and fluxes (Jw,J1,..., 3,,) by means of the friction matrix R can be expressed as
i=1
n J*=C'(1-°*)J~-Ei,
d=, °~t' dx
c. sl
(3)
=
where dv = S - 1 " dV/dt, Jw = S -1 • d1%]dt, d e = S q • dn]d t are, respectively, the volume, water and solute fluxe s at point x; S is the surface area of the membrane; n~ and n, are the number of moles for water and solute transported through the membrane in time dt; vk is the velocity of the k-th component; Lp is the hydraulic permeability coefficient of membrane for water; g w and F t are the partial molar volume of water and of the i-th substance; Vis the volume of solute; p is the mechanical pressure; o~ is the membrane reflection coefficient for the i-th solute, c% is the permeability coefficient of the k-th solute under the influence of thermodynamic force d~z~/dx;and C, is the average solute concentration on the membrane at point x, s = 1, 2, ..., n. The aim of this theoretical study is to show the relationship between the frictional coeffi-cients and the hydration numbers of permeating species through membranes in the framework of generalised Spiegler-Kedem-Katchalsky frictional model equations for a single membrane system containing multicomponent, non-ionic and homogeneous solutions. The method of calculation of the frictional coefficients dependencies on the hydration number of solutes for binary and ternary non-electrolyte solutions is also presented. Moreover, the determining of matrix transformation between phenomenological and frictional coefficients for the membrane was also a goal of this work.
=R. C1
(4)
s. The friction matrix R is defined by the following expression: n
r..÷Ez
-4,
.-.
-i.
...
-A.
n
-A.
R=
A.+IEA, f#l
n
... f . +
f. f~n
(5) where i=w, 1, 2, ..., n; J~ is the water flux; J, is the molar flux of the s-th solute; and s = 1, 2,...,n. Rewriting the friction matrix R in a form which allows us to introduce the CE concentration and using Onsager's reciprocal relation, we obtain
q
fki = ~-~kfjk
(6)
A consequence of the above equation is a relationship between minors of the friction
180
A. Sl~zak et al. / Desalination 163 (2004) 177-192
coefficients matrix detRkj = Cjdet Rjk
kl
(7)
where k, j =w, 1, 2, ..., n. In this approach the entire amount of every species of the system is treated as one component. Consequently, the friction coefficients fkj from Eq. (5) do not describe real interactions between species if hydration in the system occurs. Below we present the matrix method transformation of phenomenological into frictional coefficients. The thermodynamic forces occurring in Eq. (4) are determined by
(11)
c.
The vectors of thermodynamic fluxes defined by Eq. (11) can be transformed mutually on each other in accordance with Eq. (2), defining volume flux by
Jv =P'J
(12)
where
(8)
vw l...L
p= (9) --
m
where g W and Vs are the molar volumes for water and solute s, and cs is the average concentration of solute on the membrane. This concentration is connected with solute concentration inside the membrane (Cs) by means of the distribution coefficient Ks, c s = C s / K s. Let's define the vectors of thermodynamic forces on the membrane
[ i] 0
1 ...
0
0 ---
(12a)
Taking into account Eqs. (8) and (9), the relationship between vectors of thermodynamic forces defined by Eq. (10) can be presented in the form
X = B.X.
(13)
where °..
<
d~ x
x~=
~
(lO)
d~ n dx and vectors of thermodynamic fluxes through the membrane
-v, B=
-v.
K,
cl
0
o (13a)
c~
A. ~lczak et al. / Desalination 163 (2004) 177-192
Next, taking into account the vectors defined by Eqs. (10) and (11), the equations of the Spiegler-Kedem model [Eqs. (2) and (3)], which connect thermodynamic forces and fluxes, can be
%
zpol -o.+Cx0-O )L.o
i, =
presented in the form J, = I".X~
°°.
where
-(0nl + C10-O1)LpO n
X = R'J c
(16)
The relationship between the vectors of thermodynamic fluxes Jc and J, defined by Eq. (11), can be presented as
$c = K-1
Taking into consideration Eqs. (12)-(14), (16) and (17), the matrix of friction coefficients R can be expressed as R = B "I"-1 "P "K -1
(18)
Eq. (18) shows the dependence between matrixes R and r in a local form. In order to express the dependence of friction on phenomenological coefficients in global form, we have to integrate over the thickness of the membrane. Assuming that the difference of concentration on the membrane is small (the dependence of concentration in the membrane on direction perpendicular to the membrane surface is linear) and small thickness of the membrane, we obtain after integration the dependence in global form:
(17) a = !. ~.}-l.p.K-1
where
Ax i
..- 0
where 1
0
(17a)
CI
0
(15)
+c.(1-o.)L.o.
is the matrix determined by phenomenological coefficients of the membrane, which allows to transform vector of forces (X~) onto the vector of fluxes (J~). The matrix 1" depends on the transport properties of the membrane. Next the matrix of friction coefficients R, defined by Eq. (5), transforms the flux vector Jc defined by Eq. (11), onto the vector of thermodynamic forces X, in accordance with the equation
K=
(14)
LpOn
-c.(1-o.)L.
0
181
0
1
c.
(19)
182
A. ~lfzak et al. /Desalination 163 (2004) 177-192
-LL -E
..L
1
=-
ic, --~-"
g=
"'"
0
¢1
(19a)
J
0
...
0
C w
0
1 ....
0
(19b)
=~
Pwg
-L
0
/q
0
7_,p~"
=
•..
Zp~'.
1
0
] (19c)
- ('~11+'C1(1-0'1) Zp ~'1 .... (~nl +"Cl(1 -~'1) Zp a'n [ -al,, .... g=+~'(1--')Zva' ]
= (&c)/(A In c) is the mean concentration on the membrane and Ax is the thickness of the membrane. We used the dependence
Table 1 Values of the parameters of Nephrophane membranes for aqueous glucose (index 1) and/or ethanol (index 2) solutions
f c.(x),i~ = ~
Coefficient
Value
Lp'1012 [m3"(Ns)<] o1"102 o2"102
5.0 4- 0.2 6.8 4- 0.3 2.5 4- 0.1
on'10 l° [mol'(Ns) -1] o~'10 t° [mol'(Ns) -1] o2fl013 [mol"(Ns)-1]
8.0 4- 0.4 2.00 4- 0.05 8.1 4- 2.5
o12"1012[mol'(Ns) <] ~x'104 [m] q0w K1 /('2 el [mol'm-3] c2 [mol'm-3] ~'w"106 [m3"(m°l)-l] Vl "106 [m3"(m°1)-1] V2"106 [m3"(mol)-1]
2.0 4- 0.9 2.00 + 0.02 0.71 4- 0.06 0.53 4- 0.12 0.564-0.13 12.5 + 100 25 + 200 18.05 120.2 58.3
0
.~
which performed the above assumptions for r = ±1, ±2, ±3. The global phenomenological coefficients, I'v' ° , ' ~ , used forthe calculations arc listed in Table 1. Next, m accordance with form (5) o f matrix R, the friction coefficients, which determine interactions between transported solutes ~.), solutes and water (f~) and interactions between transported solutes and the membrane ~m,f,~,,), Can be calculated from the following equations: f m = -Rm,1
(20)
f.q = -Rld+l
(21) 11
f , ~ = RI,t + ~ a=l
R,,,+ 1
(22)
183
A. Slfzak et al. /Desalination 163 (2004) 177-192
L . = R.1. +1 +
(23)
=-},+1,j+1 +
s~=i
fu = -R~+1,,+1
(25)
The dependence of Eq. (25) is in a local form, so in order to obtain the dependence in global form, we have to integrate over the thickness of the membrane, using the above-mentioned assumptions, we get the dependence in global form: 1 . p.R_:t . R -a. ~
(26)
where ~c is the thickness of the membrane, and matrixes [', K and B are presented, respectively, by Eqs. (19c), (19b) and (19a). In the case where matrix R is known, the matrix ~' can be calculated by means of Eq. (26), and next taking into consideration the form of matrix 1", defined in Eq. (19c), the global phenomenologieal coefficients of the membrane can be calculated from equations: Lt, : -I'1,1 ~" = _ ~ l , n + l n
~1,1
"?1,,+1
(29)
(24)
The index w is for water, indices i and s for solutes and m for membrane. When phenomenological coefficients for the membrane are known, the matrix r can be built [Eq. (15)], and on the basis of Eq. (19), the matrix R can be calculated. Next Eqs. (20)-(24) allow calculation of the friction coefficients for this membrane. The inverse transformation, from friction coefficients to phenomenological coefficients, in the ease when the friction coefficients are known (known matrix R) can be obtained from Eq. (18) and expressed by the equation F =P-K-I.R-B
1-
(27)
(28)
This method allows the transformation of the phenomenological coefficients into the friction coefficients [Eqs. (19) and (20)-(24)] orthe friction coefficients into the phenomenological coefficients [Eqs. (26) and (27)-(29)].
3. Model equations for interactions of hydrated species
When the solute molecules can hydrate, the solute transport through the membrane will depend on hydration number. Therefore, the hydrated molecules are subjected to more complex interactions with other solution components and the membrane. The influence of hydration number, that is the number of water molecules connected with the solute molecule, on friction coefficients was introduced by Koter [25]. Let's consider the system consisted of the membrane and watery solutions of n solutes, which can be hydrated. If h~ is defined as the hydration number of solute s, then the flux of free water can be presented by the relation [25,29] n
J:
-" J,,,
- E h,4
(30)
s=l
The solute fluxes do not depend on hydration numbers, so the dependenciesbetween the vector of fluxes without hydration J, defined by Eq. (11) and the vector of fluxes with hydration J~, can be expressed by
A
.~, = J*1 =A"
A •j
[J,,J
(31)
184
A. Sl¢zak et al./ Desalination 163 (2004) 177-192
where s =1, ..., n and
connects the vectors ofthermoc~namic forces X* and thermodynamic fluxes Jc for hydrated solutes. The matrix R* in Eq. (34) is the matrix of friction coefficients for hydrated solutes:
! -h 1 . . . . h, A=
1
...
0
•
"'"
0
0
...
1
(32) /I
-£1
..-
,I=I /I A
Next, the relation betweeAathe fluxes Jc (solutes without hydration) and J c (solutes with hydration), also defined by Eq. (11), can be expressed by A
-I-
-.-
J i n
S,=
N
...
A
J~ = A c "Jc
E
(33)
(37) where Taking into consideration Eqs. (33), (35) and (36), the matrix of friction coefficients R" can be connected with matrix R (for unhydrated solutes) by equation
1 - CI h 1 . . . . C n hn
cw Ac=
l0
cw ...
0
(34) R" = ~kT) -1" R" A -I C
0
0
-.-
1
The relationship between vectors of thermodynamic forces X (without hydration) and X" (with hydration), defined by Eq. (10) and cor~nected with suitable thermodynamic fluxes J (without hydration) and J* (with hydration), can be expressed by
Expression (38) takes into consideration the influence of the solute hydration number on friction coefficients. Taking into consideration Eq. (38) and the form of matrixes (32) and (34), the following expression for coefficients of matrix R* were obtained:
a=l
X* =
.
= ( A T ) -1.
:j
= (ilk T)-I " X
(38)
÷E n,.:,, a=l I=I
(39)
f,,
(4o)
(35)
:.J
:L,-
+ s=l
The relation x * = R*- Jc
(36)
f~,,:f~-h,
+
(41)
185
A. gl¢zak et al. ~Desalination 163 (2004) 177-192 n
gfl
(42) ht
-~IJa,,-h,f,v,.)]
+ k=l
/~=& + %f.,+n, (:L-h~f.. - % a--1 E" f.,) 43)
The thermodynamic forces existing on the membrane system are the result of the definite thermodynamicforces acting on a water molecule (or on a solute molecule) at point x of the membrane, expressed by Eqs. (8) and (9). Taking into consideration expressions (8) and (9) in (46), we obtain ,
~
dp
(47)
w h e r e / ¢ . -- (c,)/(G)'h,.
In addition, Eq. (36) has a solution in the form of the Kramer formulas:
+h, L E k dx
d,*
1
C~
det R*
{x.
1
C~
detR*
(48)
Ga,
(44) Now taking into consideration expressions (47) and (48) in Eq. (46) we obtain
' , +E R.(-,)
s;
K s d~ s
j$-
! detR*
(45)
{-[A*-Vw+~tB[~t+h'~rw)]-~
(49)
1
l,s = 1, 2, ...,n.
where Taking into consideration Eqs. (2), (44) and (45), we obtain (46)
+
v.
Ck
+ L E B;h, t
Let us write the generalised KedemKatchalsky equation for volume flux in a local form [10]
where
A':GL~RL+E
* C~-~ detRw,(-1 )'
(50)
$
B; -- c. v. ~t R,. (-1) 8
where Lp* and o I are the local hydraulic permeability and reflection coefficients, respectively, l = 1, 2, ..., n. The coefficients L~* and o t are defined by the following expressions:
A. Slqzaket aL / Desafination 163 (2004) 177-192
186
(51)
J;=C,J*
1-o; -
6o~ dx
(56)
where
(52) o~ + Pa~
detR*
Combining Eqs. (49) and (50) we obtain expressions for the local hydraulic permeability and reflection coefficients, respectively:
is the diffusive permeability coefficient of the membrane for the s-th solute under the influence of thermodynamic force dnJdx, and
L;*-
o;=1-
1
[ ~ * v w + El Bl'~l+hl-Vw)]
o; =
(54)
k
J~*[
is the reflection coefficient for the s-th solute. After integrating Eqs. (50) and (56) over the thickness of the membrane, we obtain
J; =7.. ~ - ~
~'A~t
(57)
Combining Eqs. (44), (47) and (48), we obtain
( - !-E: detR* k
Ca
d~ k
-
(55)
J,• = c ( 1-o a")-C a +
n
~k," A~k
(58)
~-1
where
where
= v.~R~a(-1)
z;=
+E ~1 +hl~rw)detR~(-1) ,+a
(59)
l
~ " detR*
(55a)
l
_,) ~.
Ps*k= -~'w detR~a ( - 1 ) a - E hl~'w detRt~
k
K~ ct
(60)
°i=
l
(55b)
k
c~ N~
Using in Eq. (55) the differential pressure (dP/dx) calculated from Eq. (50), we find the following expression:
(61) Ax.detR*
and
A. ~l¢zak et al. I DesalinatT"on 163 (2004) 177-192
)' + ~ A* :~'w~'w d e t R ~ + ' ~ ~'sT-r: detRws(-1)', B; =c~ W-I/"w detR/w(-1 * &
] 87
'~',~', detR; (-1) ,+'
$
p,,:_~, detR~,(il),_E ht _Vw detRt,, (- 1,+ --* ) , + =K, d e t R ~ ( -, 1 ) ' * * , A ~ = / ¢ o - ~ ~ l
and Ap =Po-P~, c', is the mean concentration on the membrane and Ax the thickness of the membrane. 4. Results and discussion As a model membrane, the flat dialyse membrane called Nephrophane was used for the theoretical study. Suitable values of the permeability parameters for this membrane are listed in Table 1, and the method of the measurement of the data was presented in previous papers [3,24]. The values of friction coefficients (f.e) and their dependencies on the hydration number (hi) for binary and ternary solutions were calculated by means of Mathcad 2000 professional on the basis of data presented in Table 1. The matrices P, B and r were defined on the basis ofdatapresented in Table 1 and Eqs. (12a), (19a) and (19c). The matrix of friction coefficients R was calculated on the basis o f Eqs. (19) and (20)-(24). Next the matrix of friction coefficients of the hydrated molecules R* was calculated on the basis of Eqs. (39)-(43). As a binary solution, the aqueous glucose solution was used (c a = 60 mol.m -3) and as a ternary - - glucose in c 2 = 25 mol.m -3 aqueous ethanol solution. The concentration ca refers to the glucose, and c 2 to the ethanol. The hydration number h a refers to the glucose, and h 2 to the ethanol. The friction coefficients fro, f2=, f~m, f ~ , f2~, f~a, f~2, f 2 and f21 describe the interactions of glucose-membrane, ethanolmembrane, water-membrane, glucose-water, ethanol-water, water-glucose, water-ethanol, g l u c o s e - e t h a n o l and e t h a n o l - glucose, respectively. The dependencies o f the abovementioned friction coefficients on hydration number for glucose (h~) are presented in Figs.
Ck
1-5. On the basis of dependencies o f f~k(hO, presented in Figs. 1-5; and in Eqs. (59), (60) and (61) the values of hydration permeability coefficients Lp, reflection coefficient o 1 and diffusion permea-bility coefficients were calculated. From these calculations the result is that Lp, ol, 0)11 and 0)22 do not depend on the hydration number o f glucose, h x. Moreover, the diffusion permeability coeffi-eients o)21 and %2 do depend on the hydration number o f glucose. The dependencies of 0)2a and %2on the hydration number o f glucose are presented in Figs. 6 and 7. Fig. 1 shows the dependencies of the f m coefficient on hydration number h a for binary (plot 1) and ternary (plots 2 and 3) solutions, under fixed values of: c a = 60 mol.m -3 and c z = 25 mol.m -~. Plots 2 and 3 were received suitably 1AO-E 2_ O E
1,35.
z e~
"
1.110-
i
"7
E ,,_~ 126-
1,20-
0,00
4.00
8.00
12,00
16,00
hi Fig. 1. The dependencies off~m coeffieient on hydration number h 1. Graph 1 was obtained for the binary solution (c 1 = 60 mol.m -3, c2 = 0 tool.m-3), and graphs 2 and 3 for
ternary solutions (c 1 = 60 rnol.m-3, c2 = 25 mol,m-3). Graph 2 was obtained for h2 = 0 and graph 3 for h~ = 10. Index 1 refers to glucose and index 2 to ethanol.
A. Slezak et al. / Desalination 163 (2004) 177-192
188
12.00 - -
1 __,.---...----m-
,._.. E
4.30-
E -5
E zZ
--
11.00 -
0
03
1&l~) -
4.20-
"7
E
E
o,... 04
~
~.~-
4.10
8.00-
4.60
' 0.00
I 4,00
'
I 8.00
'
I 12.~0
'
I 16,00
hi Fig. 2. The dependencies of thef~mcoefficient on hydration number h I for the ternary solution with a fixed glucose concentration c 1= 60 mol-m-a and ethanol concentration c2 = 25 mol'm -3. Graph 1 was obtained for h2 = 0 and graph 2 for h2 = 10. Index 1 refers to glueose and index 2 to ethanol. f o r h 2 = 0 and h 2 = 10. From Fig. 1 shows that an increase in the hydration number o f glucose hi causes a small increase ofthef~m coefficient but is slightly higher for the binary than for the temary solution. The change o f hydration number of ethanol h 2 causes smaller changes offl~ than for h~. An increase o f h 2 causes a very small decrease ofthef~m coefficient. Fig. I also shows that an increase o f h 1 has a mild influence on the interaction o f glucose with the membrane, resulting in a slight increase for this interaction. An increase o f h2 causes a very small decrease o f the glucose-membrane interaction. The results in Fig. 1 also indicate that the addition o f ethanol to the glucose-water solution causes a very small increase o f the glucose-membrane interaction, higher for a greater value o f the glucose hydration number. Fig. 2 shows the dependencies o f the f ~ coefficient on hydration number ha for ternary solutions under fixed values of: c 1 = 60 mol'm -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received
7.00
' 0.00
I 4.00
t
I 8.00
'
I 12.00
'
I 16.00
hi Fig. 3. The dependencies of the fw= coefficient on hydration number h 1. Graph 1 was obtained for binary solutions (c1= 60 mol.m-3, % = 0 mol'rn-3), and graphs 2 and 3 for ternary solutions (c1 = 60 mol-m-3, c2 = 25 tool.m-3). Graph 2 was obtained for h2 = 0 and graph 3 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol. suitably forh 2 = 0 and h 2 = 10. From the results in Fig. 2, an increase o f the hydration number o f glucose causes a small decrease o f the f~, coefficient (ethanol-membrane interaction) while an increase o f the ethanol hydration number causes a small increase o f the f~, coefficient. From the calculated results, it is seen that the value o f f ~ , is greater than the f m coefficient, which means a higher interaction o f ethanol than glucose with the Nephrophane membrane. Fig. 3 shows the dependencies o f the f~= coefficient on hydration number h 1 for binary (plot 1) and temary (plots 2 and 3) solutions under fixed values of: ca = 60 mol-m -~ and c2 = 25 mol.m -3. Plots 2 and 3 were received suitably for h 2 = 0 and h 2 = 10. The results in Fig. 3 indicate that an increase o f the glucose hydration number ha causes an increase o f the f ~ coefficient (interaction o f water with the Nephrophane membrane) for binary and ternary solutions. The increase o f the hydration number o f glucose h 1
189
A. ~Iczak et al. / Desalination 163 (2004) 177-192 2.10 - -
2/
0.20--
g--
'.,-.. E "B
2.00 -
E
g
0.16 -
g
O3
Z,~
1.90 -
0.12 -
"7
1.80
-
0.08-
,.¢...
1.70 0.13o,-
1,60
'
0.00
I 4.00
'
I 8.C0
'
I 12.00
'
I 16.00
h~
' 0.~
I 4.00
'
I 8.00
'
1 12.00
'
] 16.00
h~
Fig. 4. The dependencies of the f ~ coefficient on the hydration number hv Graph 1 was obtained for the binary solution (c 1 = 60 mol-m-3, c2 = 0 tool.m-3), and graphs 2 and 3 for ternary solutions (c~ = 60 mol.m-3, c2 = 25 tool" m-3). Graph 2 was obtained for h 2 = 0 and graph 3 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Fig. 5. The dependencies of the f12 coefficient on the hydration number h 1 for a ternary solution with fixed glucose concentration c~ = 60 rnol.m -3 and ethanol concentration c2 = 25 mol-m-3. Graph 1 was obtained for h 2 = 0 and graph 2 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
and/or ethanol hydration number h 2 is very small. The addition o f ethanol to the binary glucosewater solution causes decrease off~m coefficient. Fig. 4 shows the dependencies o f the f ~ coefficient on hydration number h~ for binary (plot 1) and ternary (plots 2 and 3) solutions, under fixed values of: ca = 60 mol'm -3 and c: = 25 m o l ' m -3. Plots 2 and 3 were received suitably for h 2 = 0 and h 2 = 10, From Fig. 4 it can be see that the f w coefficient, representing the interaction o f glucose with water, decreases with an increase o f the glucose hydration number h a and does not depend on the ethanol hydration number. The addition o f ethanol to the binary glucose-water solution causes a small decrease o f the f ~ coefficient, which is higher when the hydration number o f glucose is greater. From the calculated results, thef2~ coefficient does not depend on the hydration number o f glucose h a and amounts to 9.5"1012 N ' s ' m -~ for h 2 = 0 and 9.25.1012 N's-m -1 for h2 = 10, c a =
60 tool.m-3 and c2 = 25 mol.m -3. An increase o f h 2 causes a small decrease o f t h e f ~ value. Fig. 5 shows the dependencies o f the f 2 coefficient on hydration number ha for ternary solutions under fixed values of: Cl = 60 m o l ' m -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h2 = 0 and h2 = 10. From Fig. 5 it is seen that an increase o f the hydration number o f glucose hj causes a considerable increase o f the f 2 coefficient, which is connected with the interaction o f ethanol with glucose. An increase o f the ethanol hydration number h 2 also causes an increase o f the f 2 coefficient value but smaller than for an analogical change o f h 1. Fig. 6 shows the dependencies o f the permeability coefficient COzaon the hydration number h a for ternary solutions under fixed values of: ca = 60 mol-m -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h2 = 0 and h2 = 10. From Fig. 6 it is seen that the co21 coefficient depends strongly on the h a hydration number.
190
A. SI¢zak et aL /Desalination 163 (2004) 177-192 5.00
-
2,01
-
4.00
--
at
2.00-
"5
1
.k
.t
,1
-5 2.oo J
3,00
E) T-
$
ZOO
1.99
1.99
1,00
0.00
J 0.00
I 4.00
'
I 8.03
~
I i2.00
I
'
16.00
hi
1.96
' 0.00
I 4.00
'
[ 8.00
'
I 12.00
'
I 18,00
hi
Fig. 6. The dependencies of the permeability coefficient c%~ on hydration number h I for the ternary solution with fixed glucose concentration c~ = 60 mol.m-3 and ethanol concentration c2 = 25 mol.m-3. Graph 1 was obtained for h2 = 0 and graph 2 for h 2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Fig. 7. The dependencies of the permeability coefficient w]2 on hydration number hi for the ternary solution with fixed glucose concentration cl = 60 mol.m-3 and ethanol concentration c2= 25 mol.m-3. Graph 1 was obtained for h 2 = 0 and graph 2 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Increase o f ha causes a considerable increase o f coza, while an increase o f h2 causes a very small decrease o f the 6% coefficient, which is connected with mutual interactions o f glucose and ethanol in the membrane. Fig. 7 shows the dependencies o f the permeability coefficient 6012on the hydration number h a for ternary solutions under fixed values of. ca = 60 m o l ' m -3 and c2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h 2 = 0 and h2 = 10. From Fig. 7 it is seen that for h2 = 0 an increase o f the glucose hydration number h a causes a very small increase o f f.D12, while for h~_ = 10 an increase o f h 1 causes a decrease o f the 6012 coefficient.
depend on hydration numbers for glucose. However, an increase o f the hydration number o f glucose causes a significant increase o f the o321 coefficient and a small decrease o f c % when h2÷0. 2. The values o f frictional coefficients - f~m, fl*, and f2*.calculated on the basis o f Eqs. (39) and (42) depend on the hydration numbers for both glucose and ethanol. The increase in the glucose and/or ethanol hydration number causes a decrease off2*m and an increase o f fl*~ and f~m values. 3. The frictional coefficient fl*w, calculated on the basis o f Eq. (41), depends on the glucose hydration number but does not depend on the ethanol hydration number, while the frictional coefficient f2*w depends on the ethanol hydration number but does not depend on the glucose hydration number. The increase o f a suitable hydration number causes a small decrease o f these coefficients. 4. The frictional coefficient f12, calculated on
5. Conclusions 1. The hydraulic permeability coefficient Lp, the reflection coefficient for glucose o 1 and the diffusion permeability coefficients co]~ and ¢o22 calculated on the basis o f Eqs. (59)-(61) do not
A. Slgzak et al. / Desalinan'on 163 (2004) 177-192
the basis o f Eq. (43), strongly depends on the hydration numbers for both glucose and ethanol. The increase o f the glucose and/or ethanol hydration number causes an increase of this coefficient value.
6. Symbols C
--
Cs
D
F• g
-
A
f~ h
s
^
J,J J
%
?7 s
A
^
,Jc - -
,Jc,Jv
-
-
-
Solute concentration inside the membrane Solute concentration outside the membrane Matrix o f membrane phenomenological coefficients for hydrated solute molecules Friction force Frictional coefficients of mutual interactions with hydration Friction coefficient for k and i components Friction coefficient which determines interactions between transported solutes (i), water (w) and membrane (m) Friction coefficient which determines interactions between transported solutes and water Frictional coefficients of mutual interactions Hydration number for solute s Solute flux Volume flux Water flux Vectors o f the thermodynamic fluxes Vector o f fluxes with hydration Distribution coefficient Local hydraulic permeability coefficient Global hydraulic permeability, reflection and diffusive permeability coefficients Number o f moles for solute
n~ p R R0 S t V Vj
---------
v~
---
X,X~
--
XkT. X~, Xk X , X£
----
~c
--
W
191
Number o f moles for water Mechanical pressure matrix o f friction coefficients Minor o f matrix R Membrane surface area Time Volume o f solute Partial molar volume ofi-th substance Partial molar volume o f water Velocity o f solution i-th component Vectors o f the thermodynamics force Thermodynamic force Thermodynamic force Thermodynamic forces for hydrated solutes Thickness of membrane
Greek
r
0i 60lr
Matrix determined by phenomenological coefficients of the membrane, which allows the transformation ofvectorX~ into 3~ Osmotic pressure Local reflection coefficient Local diffusive permeability coefficient
References [1] H.K. Lonsdale, The growth of membrane technology. J. Membr. ane Sci., 10 (1982) 81-181. [2] PM. Barry and J.M. Diamond, Effects of unstirred layers on membrane phenomena. Physiol. Rev., 64 (1984) 763-872. [31 A. ~lqzak, Irreversiblethermodynamic model equations of transport across a horizontally mounted membrane.Biophys. Chem., 34 (1989) 91-102. [4] A. ~lqzak, K. Dworecki and J.E. Anderson, Gravitational effectsontransmembraneflux: the RayleighTaylor convective instability. J. Membr. Sci., 23 (1985) 71-81.
192
A. $l~zak et al. / Desalination 163 (2004) 177-192
[5] K. Dworeeki and S. W~lsik, The investigation of time-dependent solute transport through horizontaUy situated membrane: the effect of configurationmembrane system. J. Biol. Phys., 23 (1997) 181-195. [6] W. Push, Measurement techniques of transport through membranes. Desalination, 59 (1986) 105108. [7] A. Fiek, Ober diffusion. Anrr Phys. Chem., 94 (1985) 59-86. [8] G.D. Mehta, T.F. Morse, E.A. Mason and M.H. Deneshpajooh, Generalized Nemst-Planek and Stefan-Maxwell equations for membrane transport. J. Chem. Phys., 64 (1976) 3917-3923. [9] L.F. Del Castillo and E.A. Mason, Generalization of membrane reflection coefficients for non-ideal, nonisothermal, multicomponent systems with external forces and viscous flow. J. Membr. Sei., 28 (1986) 229-267. [10] P.J.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes: the binary friction model. Chem. Eng. J., 64 (1996) 319343. [11] B. Tomieki, The Gibbs-Duhem equation in membrane transport. Eur. Biophys. J., 17 (1989) 137-142. [12] A. Katehalsky and P.F. Curran, Non-equilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge, MA, 1965. [13] B. Tomieki, Mechanics and thermodynamics in membrane permeation, J. Membr. Sei., 39 (1988) 11-24.
[14] A. NarCbska and S. Koter, Irreversible thermodynamics of transport across charged membranes. A comparative treatment. Polish J. Chem., 71 (1997) 1707-1717. [15] K.S. Spiegler, Transport processes in ionic membrane. Trans. Faraday Soe., 54 (1958) 1408-1428. [16] D. Mackay and P. Meares, The electrical conductivity and eleetroosmotie permeability of a cationexchange resign, Trans. Faraday Sot., 55 (1959) 1221-1238. [17] B.Z. Ginzburg and A. Katchalsky, The frictional coefficients of the flows of non-electrolytes through artificial membranes. J. Gen. Physiol., 47 (1963) 403-418. [18] K.S. Spiegler and O. Kedem, Thermodynamics of
hypertiltration (reverse osmosis): criteria for efficient membranes. Desalination, 1 (1966) 311-326. [ 19] Lord Rayleigh, Investigation of the character of an incompressible fluid of variable density, Proc. Math. Soc. (London), 4 (1873) 363. [20] R. Zwanzing, J.G. Kirkwood, I. Oppenheim and B.J. Adler, Statistical mechanical theory of transport processes. VII. The coefficient of thermal conductivity monatomic liquid. J. Chem. Phys., 22 (1954) 783790. [21] O. Kedem and A. Katchalsky, A physical interpretation of the phenomenological coefficients of membrane permeability. J. Gen. Physiol., 45 (1961) 143-179.
[22] P. Meares, J.F. Thain and D.G. Dowson, Transport across ion-exchange resign membranes:the frictional model of transport, in: G. Eisman, ed., Membranes. A series of Advances, Vol. 1, Marcel Dekker, New York, 1972, pp. 55-123. [23] A. Narebska, S. Koter and W. Kujawski, Ions and water Iransport across charged Nation membranes. Desalination, 51 (1984) 3-17. [24] V.S. Vaidhyanatan, On the permeability of electrolytes through biological membranes, L Theoret. Biol., 7 (1964) 334-338. [25] S. Koter, Interactions ofhydrated species in transport across membranes. Z. Phys. Chem. Neue Folge, 148 (1986) 247-253. [26] V.S. Vaidhyanatan and W.H. Perkins, On the permeability of non-electrolytes through biological membranes. J. Theoret. Biol., 7 (1964) 329-333. [27] A.~ l~zak and B. Turczyfiski, Generalization of the Spiegler-Kedem-Katehalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions. Biophys. Chem., 44 (1992) 139-142. [28] A. $1¢zak, A frictional interpretation of the phenomenological coefficients of membrane permeability for multieomponent solutions. J. Biol. Phys., 23 (1997) 239-250. [29] H. Sch6nert, Transport processes and tracer-diffusion in solutions with hydrated species. I. Binary nonelectrolyte solutions. Z. Phys. Chem. Neue Folge, 138 (1983) 1-16.
iiii
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DESALINATION Desalination 163 (2004) 177-192
ELSEVIER
Model equations for interactions of hydrated species in transmembrane transport Andrzej
SlCzak " a*, Slawomir Grzegorczyn b, Jacek W~sik c
aDepartment of Biomedical Fundamentals of Sport, CzOstochowa Technical University, 19B Armia Krajowa Al, 42-200 Czestochowa Tel. & Fax: +48 (34) 361-3876; email: [email protected];[email protected] bDepartment of Biophysics, Silesian Medical Academy, 19 Henryk Jordan Str, 41-808 Zabrze Clnstitute of Physics, Pedagogical University, 13/15 Armia Krajowa Al, 42-200 Cz~stochowa Received 17 July 2003; accepted 16 October 2003
Abstract
The relationship between friction coefficients and hydration numbers of permeating solutes through membranes is presented in the framework of generalised Spiegler-Kedem-Katchaisky frictional model equations (SKI(model). The membrane system contained multicomponent, non-ionic and homogeneous hydrated solutions. In the framework of this model the expressions linking the permeability parameters of the membrane (Lp, o, coik) with the friction coefficients (f~,)for n-component non-ionic hydrated solutions were presented. The matrix of transformation from phenomenological onto frictional coefficients was worked out. Moreover, the method of calculation of the frictional and phenomenological coefficients change with the hydration number of solutes for binary and ternary non-ionic solutions was also presented. The friction coefficients and their dependencies on hydration number of glucose for aqueous glucose solution (binary solution) and glucose solution in aqueous ethanol solution (ternary solution), permeating through flat polymer membranes are also presented. Keywords: Membrane transport; Spiegler-Kedem-Katchalskyfrictional model equation; Multicomponent solutions; Hydratation number
1. I n t r o d u c t i o n
Transport of substances through membranes performs an essential part in biological, physico*Corresponding author.
chemical and technical systems [1]. This transport depends on membrane structure, its thickness and environmental influence, e.g., solution mixing near membranes [2,3]. It also depends on existence and direction of external fields:
Presented at the PERMEA 2003, Membrane Science and Technology Conference of Visegrad Countries (Czech Republic, Hungary, Poland and Slovakia), September 7-11, 2003, Tatranskd Matliare, Slovakia. 0011-9164/04/$- See front matter © 2004 Elsevier B.V. All rights reserved PII: S0011-9164(04)00099-2
A. Slfzak et aL ~Desalination 163 (2004) 177-192
178
electrical or gravitational [4-6]. The most direct approach for transport phenomena in membranes is based on specific mathematical models. The oldest are the Fick's diffusion model equations [7]. Among the most recent are those that have been elaborated on the grounds of statistical mechanics based on the Nemst-Planck [8], Stefan-Maxwell [9,10], and Gibbs-Duhem equations [11] and linear non-equilibrium thermodynamics [12-14]. The phenomenological "frictional" coefficients in equations of the Onsager were used by Spiegler [15]. Then Mackay and Meares [16] and Ginzburg and Katchalsky [17] applied Spiegler's model to the analysis of experimental data. Spiegler's frictional model equations, based on the principle of a balance between gross thermodynamic forces acting on the system and frictional interactions between the components of the system, are one of the fundamental treatments of transmembrane transport processes [9]. The appearance of friction forces (Fk) acting on the k-th component of a solution from other solution components (F k = ~ ] F~j; J*k
j = 1, ..., n, m, w) is taken into consideration in the frictional model. In stationary flows through the membrane the frictional force (Fk) equilibrates thermodynamic forces (X¥) acting on a moving solution component through the membrane ( ~ ] Xkj+¥ k --0). The direct frictional j,k
force (Fk) depends on interactions between k and j solution components (/÷k andj = 1, 2, ..., n; w, m; w, for water, m, for membrane) and can be presented as
(l) where f~ is the frictional coefficient of mutual interactions between one mole ofk and one mole o f j 0÷k) solution components; or between one mole of k component and the membrane m, v~ is the velocity of the i-th component of the solution (for membrane vm = 0). The friction coefficients
depends on the kind of solutes and on the solutes concentration in the membrane. Spiegler's frictional model equations were used by Kedem and Katchalsky [3] to give a physical interpretation of the phenomenological coefficients of membrane (Lp, o, co) and by Meares and co-workers [5] to give a mechanical interpretation of the molecular interaction by means of the macroscopic friction coefficients (R~). The generalisation of the Spiegler-KedemKatchalsky frictional model equations for multicomponent non-ionic solutions was proposed by ~l~zak [11,12]. The diffusive permeability and reflection coefficients for membrane transport of electrolyte and non-electrolyte were derived by Vaidhyanathan [13,14] on the basis of the statistical mechanical expression for the mean force acting on a molecule in a multicomponent system. In this paper the non-hydrated species was treated as one component and the total water present as another. The frictional model is used to describe the transmembrane transport of electrolyte [5-7] and non-electrolyte [1] solutions. As is well known [8], the hydration of ions (or other solutes) can occur, thus, the values of resistance and friction coefficients for solutes without hydration may differ from the values of coefficients describing interactions between hydrated species and water - - real components of a system. Sch6nert [8] has shown how the thermodynamics of irreversible processes can be applied to describe the inter- (mutual) diffusion, intra(tracer) diffusion, transference, electrical conductivity in a binary non-electrolyte and electrolyte solution in which the solute is hydrated in several steps. The Kedem-Katchalsky model [12] is based on the non-equilibrium transport description in a stationary state. The thermodynamic forces - gradient of mechanical pressure, gradient of osmotic pressure for solute and thermodynamic fluxes - - volume and solute fluxes are the essential components in this model. The dependencies between thermodynamic fluxes and forces are
A. Sl~zak et al. / Desalination 163 (2004) 177-I92
described by Spiegler-Kedem model equations [181: n
J,-- J L+E J,L i=l (2)
LP
179
2. Model equations for the interaction of the unhydrated species
In a steady state Je(x) is dependent on x mad equal to the macroscopic flow. An equation joining the thermodynamic forces (Xw, X 1 , . . . , X , ) and fluxes (Jw,J1,..., 3,,) by means of the friction matrix R can be expressed as
i=1
n J*=C'(1-°*)J~-Ei,
d=, °~t' dx
c. sl
(3)
=
where dv = S - 1 " dV/dt, Jw = S -1 • d1%]dt, d e = S q • dn]d t are, respectively, the volume, water and solute fluxe s at point x; S is the surface area of the membrane; n~ and n, are the number of moles for water and solute transported through the membrane in time dt; vk is the velocity of the k-th component; Lp is the hydraulic permeability coefficient of membrane for water; g w and F t are the partial molar volume of water and of the i-th substance; Vis the volume of solute; p is the mechanical pressure; o~ is the membrane reflection coefficient for the i-th solute, c% is the permeability coefficient of the k-th solute under the influence of thermodynamic force d~z~/dx;and C, is the average solute concentration on the membrane at point x, s = 1, 2, ..., n. The aim of this theoretical study is to show the relationship between the frictional coeffi-cients and the hydration numbers of permeating species through membranes in the framework of generalised Spiegler-Kedem-Katchalsky frictional model equations for a single membrane system containing multicomponent, non-ionic and homogeneous solutions. The method of calculation of the frictional coefficients dependencies on the hydration number of solutes for binary and ternary non-electrolyte solutions is also presented. Moreover, the determining of matrix transformation between phenomenological and frictional coefficients for the membrane was also a goal of this work.
=R. C1
(4)
s. The friction matrix R is defined by the following expression: n
r..÷Ez
-4,
.-.
-i.
...
-A.
n
-A.
R=
A.+IEA, f#l
n
... f . +
f. f~n
(5) where i=w, 1, 2, ..., n; J~ is the water flux; J, is the molar flux of the s-th solute; and s = 1, 2,...,n. Rewriting the friction matrix R in a form which allows us to introduce the CE concentration and using Onsager's reciprocal relation, we obtain
q
fki = ~-~kfjk
(6)
A consequence of the above equation is a relationship between minors of the friction
180
A. Sl~zak et al. / Desalination 163 (2004) 177-192
coefficients matrix detRkj = Cjdet Rjk
kl
(7)
where k, j =w, 1, 2, ..., n. In this approach the entire amount of every species of the system is treated as one component. Consequently, the friction coefficients fkj from Eq. (5) do not describe real interactions between species if hydration in the system occurs. Below we present the matrix method transformation of phenomenological into frictional coefficients. The thermodynamic forces occurring in Eq. (4) are determined by
(11)
c.
The vectors of thermodynamic fluxes defined by Eq. (11) can be transformed mutually on each other in accordance with Eq. (2), defining volume flux by
Jv =P'J
(12)
where
(8)
vw l...L
p= (9) --
m
where g W and Vs are the molar volumes for water and solute s, and cs is the average concentration of solute on the membrane. This concentration is connected with solute concentration inside the membrane (Cs) by means of the distribution coefficient Ks, c s = C s / K s. Let's define the vectors of thermodynamic forces on the membrane
[ i] 0
1 ...
0
0 ---
(12a)
Taking into account Eqs. (8) and (9), the relationship between vectors of thermodynamic forces defined by Eq. (10) can be presented in the form
X = B.X.
(13)
where °..
<
d~ x
x~=
~
(lO)
d~ n dx and vectors of thermodynamic fluxes through the membrane
-v, B=
-v.
K,
cl
0
o (13a)
c~
A. ~lczak et al. / Desalination 163 (2004) 177-192
Next, taking into account the vectors defined by Eqs. (10) and (11), the equations of the Spiegler-Kedem model [Eqs. (2) and (3)], which connect thermodynamic forces and fluxes, can be
%
zpol -o.+Cx0-O )L.o
i, =
presented in the form J, = I".X~
°°.
where
-(0nl + C10-O1)LpO n
X = R'J c
(16)
The relationship between the vectors of thermodynamic fluxes Jc and J, defined by Eq. (11), can be presented as
$c = K-1
Taking into consideration Eqs. (12)-(14), (16) and (17), the matrix of friction coefficients R can be expressed as R = B "I"-1 "P "K -1
(18)
Eq. (18) shows the dependence between matrixes R and r in a local form. In order to express the dependence of friction on phenomenological coefficients in global form, we have to integrate over the thickness of the membrane. Assuming that the difference of concentration on the membrane is small (the dependence of concentration in the membrane on direction perpendicular to the membrane surface is linear) and small thickness of the membrane, we obtain after integration the dependence in global form:
(17) a = !. ~.}-l.p.K-1
where
Ax i
..- 0
where 1
0
(17a)
CI
0
(15)
+c.(1-o.)L.o.
is the matrix determined by phenomenological coefficients of the membrane, which allows to transform vector of forces (X~) onto the vector of fluxes (J~). The matrix 1" depends on the transport properties of the membrane. Next the matrix of friction coefficients R, defined by Eq. (5), transforms the flux vector Jc defined by Eq. (11), onto the vector of thermodynamic forces X, in accordance with the equation
K=
(14)
LpOn
-c.(1-o.)L.
0
181
0
1
c.
(19)
182
A. ~lfzak et al. /Desalination 163 (2004) 177-192
-LL -E
..L
1
=-
ic, --~-"
g=
"'"
0
¢1
(19a)
J
0
...
0
C w
0
1 ....
0
(19b)
=~
Pwg
-L
0
/q
0
7_,p~"
=
•..
Zp~'.
1
0
] (19c)
- ('~11+'C1(1-0'1) Zp ~'1 .... (~nl +"Cl(1 -~'1) Zp a'n [ -al,, .... g=+~'(1--')Zva' ]
= (&c)/(A In c) is the mean concentration on the membrane and Ax is the thickness of the membrane. We used the dependence
Table 1 Values of the parameters of Nephrophane membranes for aqueous glucose (index 1) and/or ethanol (index 2) solutions
f c.(x),i~ = ~
Coefficient
Value
Lp'1012 [m3"(Ns)<] o1"102 o2"102
5.0 4- 0.2 6.8 4- 0.3 2.5 4- 0.1
on'10 l° [mol'(Ns) -1] o~'10 t° [mol'(Ns) -1] o2fl013 [mol"(Ns)-1]
8.0 4- 0.4 2.00 4- 0.05 8.1 4- 2.5
o12"1012[mol'(Ns) <] ~x'104 [m] q0w K1 /('2 el [mol'm-3] c2 [mol'm-3] ~'w"106 [m3"(m°l)-l] Vl "106 [m3"(m°1)-1] V2"106 [m3"(mol)-1]
2.0 4- 0.9 2.00 + 0.02 0.71 4- 0.06 0.53 4- 0.12 0.564-0.13 12.5 + 100 25 + 200 18.05 120.2 58.3
0
.~
which performed the above assumptions for r = ±1, ±2, ±3. The global phenomenological coefficients, I'v' ° , ' ~ , used forthe calculations arc listed in Table 1. Next, m accordance with form (5) o f matrix R, the friction coefficients, which determine interactions between transported solutes ~.), solutes and water (f~) and interactions between transported solutes and the membrane ~m,f,~,,), Can be calculated from the following equations: f m = -Rm,1
(20)
f.q = -Rld+l
(21) 11
f , ~ = RI,t + ~ a=l
R,,,+ 1
(22)
183
A. Slfzak et al. /Desalination 163 (2004) 177-192
L . = R.1. +1 +
(23)
=-},+1,j+1 +
s~=i
fu = -R~+1,,+1
(25)
The dependence of Eq. (25) is in a local form, so in order to obtain the dependence in global form, we have to integrate over the thickness of the membrane, using the above-mentioned assumptions, we get the dependence in global form: 1 . p.R_:t . R -a. ~
(26)
where ~c is the thickness of the membrane, and matrixes [', K and B are presented, respectively, by Eqs. (19c), (19b) and (19a). In the case where matrix R is known, the matrix ~' can be calculated by means of Eq. (26), and next taking into consideration the form of matrix 1", defined in Eq. (19c), the global phenomenologieal coefficients of the membrane can be calculated from equations: Lt, : -I'1,1 ~" = _ ~ l , n + l n
~1,1
"?1,,+1
(29)
(24)
The index w is for water, indices i and s for solutes and m for membrane. When phenomenological coefficients for the membrane are known, the matrix r can be built [Eq. (15)], and on the basis of Eq. (19), the matrix R can be calculated. Next Eqs. (20)-(24) allow calculation of the friction coefficients for this membrane. The inverse transformation, from friction coefficients to phenomenological coefficients, in the ease when the friction coefficients are known (known matrix R) can be obtained from Eq. (18) and expressed by the equation F =P-K-I.R-B
1-
(27)
(28)
This method allows the transformation of the phenomenological coefficients into the friction coefficients [Eqs. (19) and (20)-(24)] orthe friction coefficients into the phenomenological coefficients [Eqs. (26) and (27)-(29)].
3. Model equations for interactions of hydrated species
When the solute molecules can hydrate, the solute transport through the membrane will depend on hydration number. Therefore, the hydrated molecules are subjected to more complex interactions with other solution components and the membrane. The influence of hydration number, that is the number of water molecules connected with the solute molecule, on friction coefficients was introduced by Koter [25]. Let's consider the system consisted of the membrane and watery solutions of n solutes, which can be hydrated. If h~ is defined as the hydration number of solute s, then the flux of free water can be presented by the relation [25,29] n
J:
-" J,,,
- E h,4
(30)
s=l
The solute fluxes do not depend on hydration numbers, so the dependenciesbetween the vector of fluxes without hydration J, defined by Eq. (11) and the vector of fluxes with hydration J~, can be expressed by
A
.~, = J*1 =A"
A •j
[J,,J
(31)
184
A. Sl¢zak et al./ Desalination 163 (2004) 177-192
where s =1, ..., n and
connects the vectors ofthermoc~namic forces X* and thermodynamic fluxes Jc for hydrated solutes. The matrix R* in Eq. (34) is the matrix of friction coefficients for hydrated solutes:
! -h 1 . . . . h, A=
1
...
0
•
"'"
0
0
...
1
(32) /I
-£1
..-
,I=I /I A
Next, the relation betweeAathe fluxes Jc (solutes without hydration) and J c (solutes with hydration), also defined by Eq. (11), can be expressed by A
-I-
-.-
J i n
S,=
N
...
A
J~ = A c "Jc
E
(33)
(37) where Taking into consideration Eqs. (33), (35) and (36), the matrix of friction coefficients R" can be connected with matrix R (for unhydrated solutes) by equation
1 - CI h 1 . . . . C n hn
cw Ac=
l0
cw ...
0
(34) R" = ~kT) -1" R" A -I C
0
0
-.-
1
The relationship between vectors of thermodynamic forces X (without hydration) and X" (with hydration), defined by Eq. (10) and cor~nected with suitable thermodynamic fluxes J (without hydration) and J* (with hydration), can be expressed by
Expression (38) takes into consideration the influence of the solute hydration number on friction coefficients. Taking into consideration Eq. (38) and the form of matrixes (32) and (34), the following expression for coefficients of matrix R* were obtained:
a=l
X* =
.
= ( A T ) -1.
:j
= (ilk T)-I " X
(38)
÷E n,.:,, a=l I=I
(39)
f,,
(4o)
(35)
:.J
:L,-
+ s=l
The relation x * = R*- Jc
(36)
f~,,:f~-h,
+
(41)
185
A. gl¢zak et al. ~Desalination 163 (2004) 177-192 n
gfl
(42) ht
-~IJa,,-h,f,v,.)]
+ k=l
/~=& + %f.,+n, (:L-h~f.. - % a--1 E" f.,) 43)
The thermodynamic forces existing on the membrane system are the result of the definite thermodynamicforces acting on a water molecule (or on a solute molecule) at point x of the membrane, expressed by Eqs. (8) and (9). Taking into consideration expressions (8) and (9) in (46), we obtain ,
~
dp
(47)
w h e r e / ¢ . -- (c,)/(G)'h,.
In addition, Eq. (36) has a solution in the form of the Kramer formulas:
+h, L E k dx
d,*
1
C~
det R*
{x.
1
C~
detR*
(48)
Ga,
(44) Now taking into consideration expressions (47) and (48) in Eq. (46) we obtain
' , +E R.(-,)
s;
K s d~ s
j$-
! detR*
(45)
{-[A*-Vw+~tB[~t+h'~rw)]-~
(49)
1
l,s = 1, 2, ...,n.
where Taking into consideration Eqs. (2), (44) and (45), we obtain (46)
+
v.
Ck
+ L E B;h, t
Let us write the generalised KedemKatchalsky equation for volume flux in a local form [10]
where
A':GL~RL+E
* C~-~ detRw,(-1 )'
(50)
$
B; -- c. v. ~t R,. (-1) 8
where Lp* and o I are the local hydraulic permeability and reflection coefficients, respectively, l = 1, 2, ..., n. The coefficients L~* and o t are defined by the following expressions:
A. Slqzaket aL / Desafination 163 (2004) 177-192
186
(51)
J;=C,J*
1-o; -
6o~ dx
(56)
where
(52) o~ + Pa~
detR*
Combining Eqs. (49) and (50) we obtain expressions for the local hydraulic permeability and reflection coefficients, respectively:
is the diffusive permeability coefficient of the membrane for the s-th solute under the influence of thermodynamic force dnJdx, and
L;*-
o;=1-
1
[ ~ * v w + El Bl'~l+hl-Vw)]
o; =
(54)
k
J~*[
is the reflection coefficient for the s-th solute. After integrating Eqs. (50) and (56) over the thickness of the membrane, we obtain
J; =7.. ~ - ~
~'A~t
(57)
Combining Eqs. (44), (47) and (48), we obtain
( - !-E: detR* k
Ca
d~ k
-
(55)
J,• = c ( 1-o a")-C a +
n
~k," A~k
(58)
~-1
where
where
= v.~R~a(-1)
z;=
+E ~1 +hl~rw)detR~(-1) ,+a
(59)
l
~ " detR*
(55a)
l
_,) ~.
Ps*k= -~'w detR~a ( - 1 ) a - E hl~'w detRt~
k
K~ ct
(60)
°i=
l
(55b)
k
c~ N~
Using in Eq. (55) the differential pressure (dP/dx) calculated from Eq. (50), we find the following expression:
(61) Ax.detR*
and
A. ~l¢zak et al. I DesalinatT"on 163 (2004) 177-192
)' + ~ A* :~'w~'w d e t R ~ + ' ~ ~'sT-r: detRws(-1)', B; =c~ W-I/"w detR/w(-1 * &
] 87
'~',~', detR; (-1) ,+'
$
p,,:_~, detR~,(il),_E ht _Vw detRt,, (- 1,+ --* ) , + =K, d e t R ~ ( -, 1 ) ' * * , A ~ = / ¢ o - ~ ~ l
and Ap =Po-P~, c', is the mean concentration on the membrane and Ax the thickness of the membrane. 4. Results and discussion As a model membrane, the flat dialyse membrane called Nephrophane was used for the theoretical study. Suitable values of the permeability parameters for this membrane are listed in Table 1, and the method of the measurement of the data was presented in previous papers [3,24]. The values of friction coefficients (f.e) and their dependencies on the hydration number (hi) for binary and ternary solutions were calculated by means of Mathcad 2000 professional on the basis of data presented in Table 1. The matrices P, B and r were defined on the basis ofdatapresented in Table 1 and Eqs. (12a), (19a) and (19c). The matrix of friction coefficients R was calculated on the basis o f Eqs. (19) and (20)-(24). Next the matrix of friction coefficients of the hydrated molecules R* was calculated on the basis of Eqs. (39)-(43). As a binary solution, the aqueous glucose solution was used (c a = 60 mol.m -3) and as a ternary - - glucose in c 2 = 25 mol.m -3 aqueous ethanol solution. The concentration ca refers to the glucose, and c 2 to the ethanol. The hydration number h a refers to the glucose, and h 2 to the ethanol. The friction coefficients fro, f2=, f~m, f ~ , f2~, f~a, f~2, f 2 and f21 describe the interactions of glucose-membrane, ethanolmembrane, water-membrane, glucose-water, ethanol-water, water-glucose, water-ethanol, g l u c o s e - e t h a n o l and e t h a n o l - glucose, respectively. The dependencies o f the abovementioned friction coefficients on hydration number for glucose (h~) are presented in Figs.
Ck
1-5. On the basis of dependencies o f f~k(hO, presented in Figs. 1-5; and in Eqs. (59), (60) and (61) the values of hydration permeability coefficients Lp, reflection coefficient o 1 and diffusion permea-bility coefficients were calculated. From these calculations the result is that Lp, ol, 0)11 and 0)22 do not depend on the hydration number o f glucose, h x. Moreover, the diffusion permeability coeffi-eients o)21 and %2 do depend on the hydration number o f glucose. The dependencies of 0)2a and %2on the hydration number o f glucose are presented in Figs. 6 and 7. Fig. 1 shows the dependencies of the f m coefficient on hydration number h a for binary (plot 1) and ternary (plots 2 and 3) solutions, under fixed values of: c a = 60 mol.m -3 and c z = 25 mol.m -~. Plots 2 and 3 were received suitably 1AO-E 2_ O E
1,35.
z e~
"
1.110-
i
"7
E ,,_~ 126-
1,20-
0,00
4.00
8.00
12,00
16,00
hi Fig. 1. The dependencies off~m coeffieient on hydration number h 1. Graph 1 was obtained for the binary solution (c 1 = 60 mol.m -3, c2 = 0 tool.m-3), and graphs 2 and 3 for
ternary solutions (c 1 = 60 rnol.m-3, c2 = 25 mol,m-3). Graph 2 was obtained for h2 = 0 and graph 3 for h~ = 10. Index 1 refers to glucose and index 2 to ethanol.
A. Slezak et al. / Desalination 163 (2004) 177-192
188
12.00 - -
1 __,.---...----m-
,._.. E
4.30-
E -5
E zZ
--
11.00 -
0
03
1&l~) -
4.20-
"7
E
E
o,... 04
~
~.~-
4.10
8.00-
4.60
' 0.00
I 4,00
'
I 8.00
'
I 12.~0
'
I 16,00
hi Fig. 2. The dependencies of thef~mcoefficient on hydration number h I for the ternary solution with a fixed glucose concentration c 1= 60 mol-m-a and ethanol concentration c2 = 25 mol'm -3. Graph 1 was obtained for h2 = 0 and graph 2 for h2 = 10. Index 1 refers to glueose and index 2 to ethanol. f o r h 2 = 0 and h 2 = 10. From Fig. 1 shows that an increase in the hydration number o f glucose hi causes a small increase ofthef~m coefficient but is slightly higher for the binary than for the temary solution. The change o f hydration number of ethanol h 2 causes smaller changes offl~ than for h~. An increase o f h 2 causes a very small decrease ofthef~m coefficient. Fig. I also shows that an increase o f h 1 has a mild influence on the interaction o f glucose with the membrane, resulting in a slight increase for this interaction. An increase o f h2 causes a very small decrease o f the glucose-membrane interaction. The results in Fig. 1 also indicate that the addition o f ethanol to the glucose-water solution causes a very small increase o f the glucose-membrane interaction, higher for a greater value o f the glucose hydration number. Fig. 2 shows the dependencies o f the f ~ coefficient on hydration number ha for ternary solutions under fixed values of: c 1 = 60 mol'm -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received
7.00
' 0.00
I 4.00
t
I 8.00
'
I 12.00
'
I 16.00
hi Fig. 3. The dependencies of the fw= coefficient on hydration number h 1. Graph 1 was obtained for binary solutions (c1= 60 mol.m-3, % = 0 mol'rn-3), and graphs 2 and 3 for ternary solutions (c1 = 60 mol-m-3, c2 = 25 tool.m-3). Graph 2 was obtained for h2 = 0 and graph 3 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol. suitably forh 2 = 0 and h 2 = 10. From the results in Fig. 2, an increase o f the hydration number o f glucose causes a small decrease o f the f~, coefficient (ethanol-membrane interaction) while an increase o f the ethanol hydration number causes a small increase o f the f~, coefficient. From the calculated results, it is seen that the value o f f ~ , is greater than the f m coefficient, which means a higher interaction o f ethanol than glucose with the Nephrophane membrane. Fig. 3 shows the dependencies o f the f~= coefficient on hydration number h 1 for binary (plot 1) and temary (plots 2 and 3) solutions under fixed values of: ca = 60 mol-m -~ and c2 = 25 mol.m -3. Plots 2 and 3 were received suitably for h 2 = 0 and h 2 = 10. The results in Fig. 3 indicate that an increase o f the glucose hydration number ha causes an increase o f the f ~ coefficient (interaction o f water with the Nephrophane membrane) for binary and ternary solutions. The increase o f the hydration number o f glucose h 1
189
A. ~Iczak et al. / Desalination 163 (2004) 177-192 2.10 - -
2/
0.20--
g--
'.,-.. E "B
2.00 -
E
g
0.16 -
g
O3
Z,~
1.90 -
0.12 -
"7
1.80
-
0.08-
,.¢...
1.70 0.13o,-
1,60
'
0.00
I 4.00
'
I 8.C0
'
I 12.00
'
I 16.00
h~
' 0.~
I 4.00
'
I 8.00
'
1 12.00
'
] 16.00
h~
Fig. 4. The dependencies of the f ~ coefficient on the hydration number hv Graph 1 was obtained for the binary solution (c 1 = 60 mol-m-3, c2 = 0 tool.m-3), and graphs 2 and 3 for ternary solutions (c~ = 60 mol.m-3, c2 = 25 tool" m-3). Graph 2 was obtained for h 2 = 0 and graph 3 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Fig. 5. The dependencies of the f12 coefficient on the hydration number h 1 for a ternary solution with fixed glucose concentration c~ = 60 rnol.m -3 and ethanol concentration c2 = 25 mol-m-3. Graph 1 was obtained for h 2 = 0 and graph 2 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
and/or ethanol hydration number h 2 is very small. The addition o f ethanol to the binary glucosewater solution causes decrease off~m coefficient. Fig. 4 shows the dependencies o f the f ~ coefficient on hydration number h~ for binary (plot 1) and ternary (plots 2 and 3) solutions, under fixed values of: ca = 60 mol'm -3 and c: = 25 m o l ' m -3. Plots 2 and 3 were received suitably for h 2 = 0 and h 2 = 10, From Fig. 4 it can be see that the f w coefficient, representing the interaction o f glucose with water, decreases with an increase o f the glucose hydration number h a and does not depend on the ethanol hydration number. The addition o f ethanol to the binary glucose-water solution causes a small decrease o f the f ~ coefficient, which is higher when the hydration number o f glucose is greater. From the calculated results, thef2~ coefficient does not depend on the hydration number o f glucose h a and amounts to 9.5"1012 N ' s ' m -~ for h 2 = 0 and 9.25.1012 N's-m -1 for h2 = 10, c a =
60 tool.m-3 and c2 = 25 mol.m -3. An increase o f h 2 causes a small decrease o f t h e f ~ value. Fig. 5 shows the dependencies o f the f 2 coefficient on hydration number ha for ternary solutions under fixed values of: Cl = 60 m o l ' m -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h2 = 0 and h2 = 10. From Fig. 5 it is seen that an increase o f the hydration number o f glucose hj causes a considerable increase o f the f 2 coefficient, which is connected with the interaction o f ethanol with glucose. An increase o f the ethanol hydration number h 2 also causes an increase o f the f 2 coefficient value but smaller than for an analogical change o f h 1. Fig. 6 shows the dependencies o f the permeability coefficient COzaon the hydration number h a for ternary solutions under fixed values of: ca = 60 mol-m -3 and c 2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h2 = 0 and h2 = 10. From Fig. 6 it is seen that the co21 coefficient depends strongly on the h a hydration number.
190
A. SI¢zak et aL /Desalination 163 (2004) 177-192 5.00
-
2,01
-
4.00
--
at
2.00-
"5
1
.k
.t
,1
-5 2.oo J
3,00
E) T-
$
ZOO
1.99
1.99
1,00
0.00
J 0.00
I 4.00
'
I 8.03
~
I i2.00
I
'
16.00
hi
1.96
' 0.00
I 4.00
'
[ 8.00
'
I 12.00
'
I 18,00
hi
Fig. 6. The dependencies of the permeability coefficient c%~ on hydration number h I for the ternary solution with fixed glucose concentration c~ = 60 mol.m-3 and ethanol concentration c2 = 25 mol.m-3. Graph 1 was obtained for h2 = 0 and graph 2 for h 2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Fig. 7. The dependencies of the permeability coefficient w]2 on hydration number hi for the ternary solution with fixed glucose concentration cl = 60 mol.m-3 and ethanol concentration c2= 25 mol.m-3. Graph 1 was obtained for h 2 = 0 and graph 2 for h2 = 10. Index 1 refers to glucose and index 2 to ethanol.
Increase o f ha causes a considerable increase o f coza, while an increase o f h2 causes a very small decrease o f the 6% coefficient, which is connected with mutual interactions o f glucose and ethanol in the membrane. Fig. 7 shows the dependencies o f the permeability coefficient 6012on the hydration number h a for ternary solutions under fixed values of. ca = 60 m o l ' m -3 and c2 = 25 mol.m -3. Plots 1 and 2 were received suitably for h 2 = 0 and h2 = 10. From Fig. 7 it is seen that for h2 = 0 an increase o f the glucose hydration number h a causes a very small increase o f f.D12, while for h~_ = 10 an increase o f h 1 causes a decrease o f the 6012 coefficient.
depend on hydration numbers for glucose. However, an increase o f the hydration number o f glucose causes a significant increase o f the o321 coefficient and a small decrease o f c % when h2÷0. 2. The values o f frictional coefficients - f~m, fl*, and f2*.calculated on the basis o f Eqs. (39) and (42) depend on the hydration numbers for both glucose and ethanol. The increase in the glucose and/or ethanol hydration number causes a decrease off2*m and an increase o f fl*~ and f~m values. 3. The frictional coefficient fl*w, calculated on the basis o f Eq. (41), depends on the glucose hydration number but does not depend on the ethanol hydration number, while the frictional coefficient f2*w depends on the ethanol hydration number but does not depend on the glucose hydration number. The increase o f a suitable hydration number causes a small decrease o f these coefficients. 4. The frictional coefficient f12, calculated on
5. Conclusions 1. The hydraulic permeability coefficient Lp, the reflection coefficient for glucose o 1 and the diffusion permeability coefficients co]~ and ¢o22 calculated on the basis o f Eqs. (59)-(61) do not
A. Slgzak et al. / Desalinan'on 163 (2004) 177-192
the basis o f Eq. (43), strongly depends on the hydration numbers for both glucose and ethanol. The increase o f the glucose and/or ethanol hydration number causes an increase of this coefficient value.
6. Symbols C
--
Cs
D
F• g
-
A
f~ h
s
^
J,J J
%
?7 s
A
^
,Jc - -
,Jc,Jv
-
-
-
Solute concentration inside the membrane Solute concentration outside the membrane Matrix o f membrane phenomenological coefficients for hydrated solute molecules Friction force Frictional coefficients of mutual interactions with hydration Friction coefficient for k and i components Friction coefficient which determines interactions between transported solutes (i), water (w) and membrane (m) Friction coefficient which determines interactions between transported solutes and water Frictional coefficients of mutual interactions Hydration number for solute s Solute flux Volume flux Water flux Vectors o f the thermodynamic fluxes Vector o f fluxes with hydration Distribution coefficient Local hydraulic permeability coefficient Global hydraulic permeability, reflection and diffusive permeability coefficients Number o f moles for solute
n~ p R R0 S t V Vj
---------
v~
---
X,X~
--
XkT. X~, Xk X , X£
----
~c
--
W
191
Number o f moles for water Mechanical pressure matrix o f friction coefficients Minor o f matrix R Membrane surface area Time Volume o f solute Partial molar volume ofi-th substance Partial molar volume o f water Velocity o f solution i-th component Vectors o f the thermodynamics force Thermodynamic force Thermodynamic force Thermodynamic forces for hydrated solutes Thickness of membrane
Greek
r
0i 60lr
Matrix determined by phenomenological coefficients of the membrane, which allows the transformation ofvectorX~ into 3~ Osmotic pressure Local reflection coefficient Local diffusive permeability coefficient
References [1] H.K. Lonsdale, The growth of membrane technology. J. Membr. ane Sci., 10 (1982) 81-181. [2] PM. Barry and J.M. Diamond, Effects of unstirred layers on membrane phenomena. Physiol. Rev., 64 (1984) 763-872. [31 A. ~lqzak, Irreversiblethermodynamic model equations of transport across a horizontally mounted membrane.Biophys. Chem., 34 (1989) 91-102. [4] A. ~lqzak, K. Dworecki and J.E. Anderson, Gravitational effectsontransmembraneflux: the RayleighTaylor convective instability. J. Membr. Sci., 23 (1985) 71-81.
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[5] K. Dworeeki and S. W~lsik, The investigation of time-dependent solute transport through horizontaUy situated membrane: the effect of configurationmembrane system. J. Biol. Phys., 23 (1997) 181-195. [6] W. Push, Measurement techniques of transport through membranes. Desalination, 59 (1986) 105108. [7] A. Fiek, Ober diffusion. Anrr Phys. Chem., 94 (1985) 59-86. [8] G.D. Mehta, T.F. Morse, E.A. Mason and M.H. Deneshpajooh, Generalized Nemst-Planek and Stefan-Maxwell equations for membrane transport. J. Chem. Phys., 64 (1976) 3917-3923. [9] L.F. Del Castillo and E.A. Mason, Generalization of membrane reflection coefficients for non-ideal, nonisothermal, multicomponent systems with external forces and viscous flow. J. Membr. Sei., 28 (1986) 229-267. [10] P.J.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes: the binary friction model. Chem. Eng. J., 64 (1996) 319343. [11] B. Tomieki, The Gibbs-Duhem equation in membrane transport. Eur. Biophys. J., 17 (1989) 137-142. [12] A. Katehalsky and P.F. Curran, Non-equilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge, MA, 1965. [13] B. Tomieki, Mechanics and thermodynamics in membrane permeation, J. Membr. Sei., 39 (1988) 11-24.
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[22] P. Meares, J.F. Thain and D.G. Dowson, Transport across ion-exchange resign membranes:the frictional model of transport, in: G. Eisman, ed., Membranes. A series of Advances, Vol. 1, Marcel Dekker, New York, 1972, pp. 55-123. [23] A. Narebska, S. Koter and W. Kujawski, Ions and water Iransport across charged Nation membranes. Desalination, 51 (1984) 3-17. [24] V.S. Vaidhyanatan, On the permeability of electrolytes through biological membranes, L Theoret. Biol., 7 (1964) 334-338. [25] S. Koter, Interactions ofhydrated species in transport across membranes. Z. Phys. Chem. Neue Folge, 148 (1986) 247-253. [26] V.S. Vaidhyanatan and W.H. Perkins, On the permeability of non-electrolytes through biological membranes. J. Theoret. Biol., 7 (1964) 329-333. [27] A.~ l~zak and B. Turczyfiski, Generalization of the Spiegler-Kedem-Katehalsky frictional model equations of the transmembrane transport for multicomponent non-electrolyte solutions. Biophys. Chem., 44 (1992) 139-142. [28] A. $1¢zak, A frictional interpretation of the phenomenological coefficients of membrane permeability for multieomponent solutions. J. Biol. Phys., 23 (1997) 239-250. [29] H. Sch6nert, Transport processes and tracer-diffusion in solutions with hydrated species. I. Binary nonelectrolyte solutions. Z. Phys. Chem. Neue Folge, 138 (1983) 1-16.