A theoretical analysis of donnan dialysis across charged porous membranes

Journal of Membrane

Science,

Elsevier Science Publishers

48 (1990) 155-179 B.V., Amsterdam - Printed

155 in The Netherlands

A THEORETICAL ANALYSIS OF DONNAN DIALYSIS ACROSS CHARGED POROUS MEMBRANES

ELEANOR

H. CWIRKO and RUBEN G. CARBONELL

Department

of Chemical Engineering,

North Carolina State University,

Raleigh, NC 27695

(U.S.A.)

(Received November

22,1988; accepted in revised form June 19,1989)

Summary A previously developed model [ 1 ] for the transport of ions in charged capillary tubes is used to determine the effect of membrane properties (average pore radius and surface charge density) and solution concentration on steady-state ion and solution fluxes in two types of Donnan dialysis separations. Specifically, the concentration of a dilute metal ion solution into an acid stripping solution by ion exchange across a cation exchange membrane and the removal of an acid contaminant from a metal ion solution using an anion exchange membrane are studied. The model is able to predict the changes in metal ion flux caused by changes in acid and metal ion concentrations in the strip and feed solutions in the Donnan dialysis experiments performed by Lake and Melsheimer [ 21. These effects could not be explained by the mass-transfer resistance model used by Lake and Melsheimer [2] and by others [3,4] to describe ion transport across a charged membrane in Donnan dialysis separations. The time-dependent ion concentrations in well-stirred solutions undergoing Donnan dialysis are also modeled. The results are shown to match the experimental data of Ng and Snyder [ 41 when the boundary layer resistance are negligible and reasonable values for the membrane pore parameters are assumed.

Introduction Donnan dialysis is a membrane separation process that uses ion-selective membranes to prevent the flow of certain ions from one solution to another. The ions which are permeable to the membrane will “equilibrate” between the two solutions until the Donnan equilibrium equation [ 51 is satisfied. Writing aiI and aiIIas the activities of ion i in the solutions I and II on opposite sides of a charged membrane, the Donnan equation can be written as \ l/z; /

=K, where Zi is the ion valence, and K is a constant. Equation (1) applies to any ion that can move through the membrane; as a result, the constant K serves to equate the indicated ratios of activities of any pair of ions. For example, when

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156

a dilute solution of a NiSO, (feed solution) is separated from a solution of sulfuric acid (strip solution) by a cation exchange membrane (negative fixed charges in the membrane matrix), the sulfate ions are excluded from the membrane, and the H+ and Ni2+ ions redistribute across the membrane until $$

= ( ;;1:“,:;:I)“2,

(2)

assuming activity coefficients equal to 1 for both ions. In effect, the high concentration gradient of the H+ ions causes a flux of H+ from the acid solution to the dilute metal solution. In order to maintain electroneutrality in both solutions, the metal ions must flow from the dilute metal solution into the acid solution. The metal ions will continue to flow in this direction even when the flux is against the concentration gradient of metal ions. This type of Donnan dialysis will be referred to as Donnan ion exchange. A schematic of this process is shown in Fig. la. A similar process can be used to remove multi-valent anions, such as phosphate ions, from a dilute solution using a positively charged membrane and a basic strip solution. A different application of Donnan dialysis is in the removal of acids from streams containing multi-valent metal ions. For this separation, a positively charged membrane is used in order to prevent the flow of the positively charged metal ions from the feed stream to the pure water or dilute acid receiving stream. However, the H+ ions, which are much smaller and have a lower valence than the metal ions, will diffuse across the membrane into the receiving solution. The hydrogen ions will redistribute until the mobile ions, in this case H+ and the negative ions in solution, meet the Donnan equilibrium condition given in eqn. (1). If the metal ions are sufficiently excluded from the membrane, the acid can be transported against its concentration gradient in this process. This separation will be referred to as diffusion dialysis. A schematic of the diffusion dialysis process is shown in Fig. lb. Donnan dialysis separations are especially attractive since the separation is driven completely by electrochemical potential gradients across the membrane, and the only energy costs are those required to pump the solutions through the dialysis cell. The major drawback, however, is that the ion fluxes tend to be rather small compared to membrane processes, such as electrodialysis, in which an additional external driving force is applied. The fairly recent development of very thin ion exchange membranes which are chemically and mechanically durable has made Donnan dialysis more economically feasible. However, it would be extremely valuable if one could predict the membrane properties, such as pore size and fixed charge density, which would optimize the ion fluxes in a particular Donnan dialysis separation. Lake and Melsheimer [ 21, Wen and Hamil [ 61, and Sudoh et al. [ 71 have conducted experimental studies of the Donnan ion exchange process for the

157

strip feed Solution (dilute

solution (strong acid)

solution

of metal salt) negatively

charged

visualization

of a single charged pore

visualization

of a single charged pore

membrane

positively charged membrane recovered metal solution

recovered acid

T

Ii-

I

“I <<“II

+

Nj-+

b

W feed solution mixed acid and

y:.

I

:,

“II

H”f+

dilute acid solution

metal solution

Fig. 1. Schematic of Donnan dialysis separations: (a) ion exchange using a cation exchange membrane; (b) diffusion dialysis using an anion exchange membrane.

concentration of dilute solutions of metal ions using flat-plate dialyzers. In addition, Ng and Snyder [3,4] have studied Donnan ion exchange of copper and nickel ions in a shell-and-tube dialyzer. These studies have shown that either the membrane itself or the boundary layer on the feed stream side can be the dominant resistance to transport, depending on the magnitude of the Reynolds number and the metal concentration in the feed stream. In modeling

158

the metal ion transport across the membrane in these systems, Lake and Melsheimer [2] and Ng and Snyder [3,4] have used a simple mass-transfer resistance model, N, =Kn,,(&l),

(3)

where N, is the flux of metal ions, dc, is the difference in metal ion concentration across the interior of the membrane, and K,,, is the membrane mass transfer coefficient. The form of eqn. (3 ) implies that the flux of metal ions is due solely to diffusion caused by the concentration gradient of the metal ions. However, ions in charged membranes can also be transported by the osmotic flow through the pores, and by migration in the electrical field which arises in the pores [ 8,9]. Since the acid and metal ion concentration levels in the external solutions affect the osmotic flow and the electrical potential gradient in the membrane pores, K,,, in eqn. (3) will vary not only with membrane type, but also with the metal ion and acid concentration levels in the solutions. Lake and Melsheimer did find in their experiments [ 21 that K,,, depended on the external electrolyte conditions. We will show that by including the effects of the electrical potential and the osmotic flow on the ion fluxes, Lake and Melsheimer’s results for K,,, can be predicted. Wen and Hamil [6] and Ng and Snyder [ 3,4] also noticed that the metal ion concentration in the feed stream tends to decrease exponentially with time. This observation allowed them to fit the metal ion concentration disappearance in the feed stream using a firstorder rate law. This is a purely empirical approach, however, with no explanation as to why this behavior is seen or why deviations from the first order rate expressions occur (see Ref. [ 41). Also, the rate constants in these expressions must be determined experimentally for each different membrane and for each different initial composition of the strip solution. Neither one of these two previous approaches to modeling ion transport in Donnan dialysis allows one to predict the dependence of the ion and solvent fluxes on membrane properties such as the average pore size or average pore surface charge density or on the acid and metal concentrations in the external solutions. In recent years, it has been shown that ion and solution transport through charged porous membranes can be predicted quite well by modeling the membrane as a bank of parallel capillary tubes with uniformly charged surfaces. While the numerical solution of the transport equations for ion and solvent flow through charged capillaries containing a single binary electrolyte has become almost routine [8-l 21 f solutions for systems containing three or more ions are almost non-existent.. Recently, Cwirko and Carbonell [l] used the method of spatial averaging to reduce the ion and solvent transport equations in charged capillaries containing three or more ions to a set of ordinary differential equations which can be easily solved by finite difference methods. The purpose of this paper is to show how these equations can be used to predict ion and solution fluxes in Donnan dialysis membrane separations. The effects of

159

ion valences, metal ion and acid concentration differences across the membrane, and membrane properties on ion and solution fluxes are examined. Mass transfer resistances in the liquid films on either side of the membrane will be neglected, since we are primarily interested in the effect of the membrane on the separation. In the following section, we present a brief outline of the model equations which are derived in more detail in an earlier paper [ 11. The succeeding sections contain the predictions of this model for the steady-state Donnan ion exchange process using a cation exchange membrane, and the Donnan diffusion process using an anion exchange membrane. The model predictions for metal ion fluxes will be compared to the experimental results of Lake and Melsheimer for Donnan ion exchange across a cation exchange membrane with well-stirred solutions. The model is shown to predict the dependence of the metal ion flux on the strip solution acid concentration and the feed solution metal concentration that was seen by Lake and Melsheimer, but could not be explained by the simple mass transfer resistance model. Finally, the time-dependent ion concentration profiles in well-stirred solutions undergoing typical Donnan dialysis separations are predicted assuming that the membrane is the only significant resistance to mass-transfer. These results are compared to the experimental data of Ng and Snyder [ 41, and shown to match quite well when boundary layer resistances are negligible. Model equations for ion transport in charged capillaries

Consider a long, narrow, charged cylindrical capillary, such as the one shown in Fig. 1, separating solutions I and II. The solutions have different concentrations of ion i, denoted C,(i), and Cn (i). The capillary has a radius r,, length L, an a uniform surface charge density, q, on its interior surface. This surface charge causes radial gradients in the ion concentrations, in the electrical potential and in the hydrostatic pressure in the pore fluid [91.As explained in many previous papers [s-12], the relevant equations describing ion and solution transport in this situation are the radial and axial components of the Nernst-Planck equations for ion fluxes, the radial and axial components of the Navier-Stokes equation, ignoring inertial terms and including an electrical body force term, the continuity equation for each ion and for the whole solution, and Poisson’s equation to describe the dependence of the electrical potential on the ion concentrations in the pore. It is possible to reduce this set of partial differential equations to a set of ordinary differential equations using the techniques of area averaging. The area average of a function f(r,z) inside the pore is denoted as (f) , and defined as

(f(r,z) >

=A [f(r,z)rdr. ro J

0

(4)

160

The radial deviation of a function from its average value is denoted as f(r,z), and defined as

f=f-

*

(5)

Using these definitions, the ion and solution transport equations can be reduced to the dimensionless ordinary differential equations listed in Table 1 when there are N different ions in solutions I and II [ 11. The dimensionless variables used in Table 1 are defined in the notation section. Equations (I.l)1.6) are the area-averaged form of the basic transport equations for ion and solution flow in capillary tubes. Equation (I.1 ) is the area-averaged axial component of the Nernst-Planck equation for ion fluxes, valid for dilute solutions of electrolytes with constant ion diffusivities. Equations (I.2 ) and (I.3 ) are the area-averaged continuity equation for ion i, and the solution, respectively, when the solution is assumed to have a constant density. Equation (1.4) results from taking the area-average of the axial component of the Navier-Stokes equation. This equation gives an expression for the Peclet number, Pe= ( u,)L/D,, which is a direct measure of the solution flow rate through the pore. Equation (1.5) is simply a statement of overall electroneutrality in the pore, where C, is the number of fixed pore surface charges per pore volume. This condition is obtained from the area average of Poisson’s equation using the assumption that the pore radius is much smaller than the pore length. Finally, equation (1.6) gives the definition of the area-averaged current through a pore, which is zero in Donnan dialysis applications since there is no applied potential difference across the membrane. In the area-averaged transport equations given in Table 1, the effects of the radial profiles of the ion concentration and electrical potential fields are contained in the “dispersion terms”, ( EiGi), (E&D/&), ( Ai), and (&) . If these terms are expressed as functions of the area-averaged ion concentrations, velocity, potential, and pressure, eqns. (I.1 )- (1.6) become a set of ordinary differential equations for the area-averaged ion fluxes, concentrations, solution velocity, pressure and potential in the pore. The closed-form expressions for these dispersion terms w_hich are listed in Table 1 were derived in Ref. [l] under the res$iction, .z$<< 1, but were also shown to give accurate results even when zi a26 2, a condition which is satisfied as long as aiQ= ZiFqr, /CRT is less than 10. Many experimental and commercial charged membranes satisfy this criterion, at least for 1:lelectrolytes [11. In Donnan dialysis separations, the area-averaged current through the pores is constant and equal to zero, and the total pressure drop across the pores is also equal to zero. When the area-averaged current is known, eqns. (I.1 )- (1.6) require 2N+ 1 boundary conditions for their solution. Since the pore is long and narrow, one can show that the following equilibrium conditions between the pore mouths and the external solutions are valid [ 1,9,12]:

161 TABLE 1 Transport equations for area-averaged quantities in pore Dimensionless

area-averaged transport equations

(N,>=(Qpe+(~)-D’

d(N)=0

dz

I z

i=1,2,...,N

i= 1,2,...,N- 1

dPe -_=o dF Pe= _d(U?

(1.1) (1.2) (I.31

-
(1.4) (I.51 (1.6)

Closure expressions Ei= _*i(&)fjT

(I.71 (&,,I~ are modified Bessel functions of the first kind)

(I.81

(I.9 1 (1.10) (1.11) (1.12) (1.13) Dimensionless

parameters

(1.14) (1.15) mC =p 2q C,(l) rJ’G(1)

(1.16)

162

(6)

These equations are the result of assuming a balance between electrical and diffusive fluxes of ions in the bulk solution adjacent to the membrane. Equation (6) can be integrated across the transition region between the external solutions and the pore mouths. This result for i= 1 can be combined with the result for i= 2,3,...,N in order to eliminate the electrical potential from the expression for the ion concentrations. The area-average of the result is [ 1] i=2,3,...,N

(8)

at z= 0 (pore mouth adjacent to solution I) and

at z= L (pore mouth adjacent to solution II), where ~i=(eXp(-_i~))/(eXp(-_z,~))z"zl.

(10)

The quantity @i is a measure of the radial curvature of the electrical potential in the pore, and it depends on the area-averaged ion concentrations in the pore through the 8 function. If the electrical potential is essentially constant across the radius of the pore, Si will be equal to 1. When the electrical potential is significantly non-uniform in the radial direction, Oi will be greater than one. Assuming that the 8is on both sides of the pore are equal to one, the relationships represented in eqns. (8) and (10) can be combined with the electroneutrality condition in the pore,

(11) in order to solve for the area-averaged concentration of ions at each end of the pore. Here Cm= Zq/r$ is the effective concentration of fixed charges in the pore. The IMSL routine DZBREN [131 was used to solve these non-linear algebraic equations for ( C,) at z = 0 and z = L. The area-averaged concentrations thus obtained can then be corrected when the Ois do not equal 1 by the iterative method described by Cwirko and Carbonell [ 11. These values for the areaaveraged ion concentrations at each end of the pore provide 2N-2 independent boundary conditions. Area-averaging the results obtained from the integration of eqn. (6) across the transition regions leads to the conditions [ 11,

163

(@-@+ln((c’>+a)

at z=O

(12)

and (G)-@ii=--+

(

(e,)%+a II

>

at z=L,

(13)

where cu= -ln(exp(-zz,S)). By integrating eqn. (7) across the pore mouths and area-averaging the results, the following conditions are obtained, (P)-(n)=&I=&

at 2~0

(14)

(P) - =pn-I&,

at z=L.

(15)

When @i and 4 are set equal to zero as reference conditions, and pn is known (pii is then zero in most Donnan dialysis separations), eqns. (12), (14), and (15) provide the final three boundary conditions required for the solution of eqns. (I.l)-(1.6). Once the differential equations have been solved, the condition given in eqn. (13) can be used to determine Qn =A@ which is often called the membrane potential. In this investigation, the differential equations were solved by a finite difference method using the BAND (J)subroutine developed by Newman [ 141. It is important to note a few things about the model equations and boundary conditions presented above. First, there are 10 independent dimensionless groups which appear in the boundary conditions and in the differential equations when 3 ions are present in the external solutions. Seven of these parameters, C&/C,(l), zJzl, zS/zl, 02/D1, OS/&, 9; and j3, arise in the transport equations listed in Table 1, and the other three Cn (1) /C, (1) , Cn (3) /C, (3)) and Ci (3 ) /C, ( 1) arise in the boundary conditions given by eqns. (8) and (9 ) . Three of these groups, C&/C, (1)) q, and j?, depend on the pore characteristics, q and r,, while the others depend on the external electrolyte composition and the physical and electrical properties of the ions. When the ion concentration ratios, C,,(l)/C,(l), Cn(3)/C,(3), C,(3)/C,(l), are held constant, the value of C,/Ci ( 1) gives an indication of how completely the coions are excluded from the pore. This is the ratio of the surface charges per unit volume of fluid in the pore, to the charge per unit volume of ion 1 in the bulk solution. The larger this ratio, the greater the capacity of the membrane to include oppositely charged ions and exclude ions of like charge. Put another way, when q and the external ion concentration ratios are constant, C,/Ci (1) is a measure of the ratio of the pore radius to the Debye screening length and hence is a measure of the ability of the pore to exclude coions. The quantity S is a measure of the

164

non-uniformity of the radial electrical potential profile inside the pore. This parameter is nothing more than the dimensionless potential gradient in the fluid evaluated at the pore wall,

a6 _

ar r=l _

=-

qy

where r% r/r,,. As 4 increases, the gradient of the electrical potential rises. As 4 approaches zero, the equations listed in Table 1 and the boundary conditions given by eqns. (8)-( 15) reduce to those which apply when the radial profiles of the electrical potential and ion concentrations are uniform [ 15,161. The quantity /3appears only in the equations involving the convective flow through the pore. This dimensionless parameter affects the ion fluxes through the convective term, (&) Pe, and the dispersive term (&G) , in the expression for the total flux, ( Ni). It is important to note that when the solution viscosity and the diffusivity of ion 1 are held constant, as they will be for all of the situations investigated in this paper, /I is not an independent parameter; instead, it is directly proportional to the ratio of 4 to C&/C, (1). In order to predict ion fluxes through a charged porous membrane using this capillary tube model, it is necessary to know the average pore radius, the average pore length, the average pore surface charge density, and the pore area per unit of total membrane area. This last quantity can be equated to the membrane porosity divided by the pore tortuosity (pore length divided by membrane thickness). According to this model, the ion fluxes across a membrane decrease directly with increasing pore length and decreasing pore area. The pore characteristics of a membrane can be estimated from the porosity, the ion exchange capacity, and the thickness of the membrane, combined with measurements of the hydraulic permeability, and either the electrical resistance or salt permeability (salt flux divided by the salt concentration difference across the membrane) of the membrane at high concentrations of a single binary electrolyte [lo]. Results and discussion Donnan ion exchange of a dilute metal ion stream using a cation exchange membrane In a Donnan ion exchange process such as the one pictured in Fig. la, the equilibrium condition given in eqn. (1) for the mobile cations, along with the condition that there is no flux of anions across the membrane dictate the highest possible ratio of metal ion concentration in the strip solution to that in the feed stream that can be achieved in this process. However, leakage of the anions into the feed solution and osmotic flow of the solvent into the strip solution will both tend to reduce the equilibrium metal ion concentration ratio.

165

Ideally, one would like to maximize both the metal ion flux and the equilibrium metal ion concentration ratio. Using the model outlined in the previous section, it is possible to investigate the effects of some of the important dimensionless parameters in the model on the metal ion flux through a charged pore. In the following discussion, H+ will be referred to as ion 1, the metal ion will be ion 3, and the common anion will be ion 2. Consider first the separation of metal ions from a feed stream (solution II), which may have varying concentrations of hydrogen and metal ions, through a negatively charged pore into a strip solution (solution I) containing fixed concentrations of metal and hydrogen ions. In particular, we wish to investigate the effects of the dimensionless parameters, C,/C,( l), 4; Cri(1 )/C,( l), Cn(3)/C1(3), z2/ z1 and zS/zl on the metal ion flux. Figure 2 shows a graph of the dimensionless metal ion flux vs. C,/C, (1) for several different valences of the metal ion and anion in solution, i.e., for several different combinations of zg/zl and zg/zl. The following diffusivities of the ions in free solution were taken from those listed by Newman [ 141:DNiz+=0.72x 10m5 cm’/sec, D Na+= 1.32 x 10m5 cm2/sec, DH+ = 9.33 x 10e5 cm2/sec, Dsop = 1.06 x lop5 cm2/sec, and DC,_ = 2.0x 10-' cm2/sec. The negative fluxes shown in Fig. 2 indicate that the metal ion is moving from the feed stream (II) into the strip solution (I ) . From Fig. 2 it is apparent that when other conditions are equal, the flux of multi-valent metal ions is significantly higher than the flux of univalent metal ions in these ion exchange separations. However, the valence of the anion which is present in the solutions has very little effect on the metal ion flux. Figure 2 also shows very clearly that the highest metal ion fluxes are obtained when 1Cm/C1 (1) ( is as high as possible. This is to be expected since high values 1Cm/C, (1) ( indicate good coion (the anion in this case) exclusion

Fig. 2. Metal ion flux vs. - C,/C,( 1) for different metal ion/anion combinations in Donnan ion exchange. Curve A represents the NiCl,-HCl and NiSO,-H,SO, systems, while curve B represents the NaCl-HCl and Na,SO,-H,SO, systems. In both curves, q=l.l, C,,(l)/C,(l) =10e3, C,,(3)/ C,(3)=10-2,andC,(1)/C,(3)=200.

166 TABLE

2

Dependence

of Ni2+ ion flux and solution

velocity

on 4 when C,/C,

( 1) = 10 and on CJC, (1)

when~=llinDonnanionexchange.Inbothcases,C~,(1)/C~(1)=O.O1,C~,(:3)/C,(3)=l,C,(3)/ C, (1) = 5 X 10m3 and the common anion is SOi-

10

I$V) -Pe

1o-4 1.226 0.22 x 1o-5

0.11 1.230 0.0022

10

1.253 1.1 0.018

10

11.0 1.383 0.064

0.1

1.0

11.0 0.022 - 0.056

11.0 0.139 0.037

10

11.0 1.383 0.064

100

11.0 11.81 0.041

from the pore. As the anion flux is reduced because of exclusion, the metal ion flux must increase in order to balance the charge flow caused by the H+ ion flux into the feed solution. In Fig. 2, s was held constant as C,/Cr (1) was changing. In Table 2, the metal ion fluxes and solution Peclet numbers are shown for constant 4 while varying C,/Ci (1 ), and for constant C,/Ci (1) while varying q. It is evident from these results that changes in C,/C, (1) have a very large effect on the ion fluxes in Donnan ion exchange, but that the effect of q is rather small. This indicates that the uniform potential model (Q= 0) should give reasonably accurate predictions of metal ion fluxes in Donnan ion exchange even when q is as high as 11. However, the predictions of the solution velocity are greatly dependent on @. Since q= CJ *r,*/eRT, one can vary @ by changing the pore radius, r. while keeping C,/C, ( 1) constant. The pore radius is expected to have a significant effect on the solution flow, hence the dependence of the Peclet number on q. The negative values of the Peclet number indicate that the solution is flowing from the low ionic strength feed solution (II) to the high ionic strength strip solution (I), which is the normal direction for osmotic flow across neutral semi-permeable membranes. When a charged membrane separates a solution of high ionic strength from one of low ionic strength, this osmotic flow in the absence of a pressure may actually occur in either direction, due to the effects of the electrical potential gradient which arises across the charged membranes. A more detailed study of the physical causes of these effects is given by Sasidhar and Ruckenstein [ 9,101. In Donnan ion exchange applications, it is important to maximize the metal ion flux while minimizing the osmotic flow into the strip solution, since the osmotic flow in this direction tends to dilute the strip solution and hence reduce the final concentration of the recovered metal ions. Figure 2, along with the data in Table 2 indicates that this can best be accomplished by keeping C,/Ci (1) as high as possible to get the highest possible ion flux but also keeping 4 as low as possible to reduce the osmotic flow into the strip solution. If the concentrations of the ions in the two solutions are given, C,/C, (1) is proportional to q/r,, while 4 is proportion to qr,. Therefore, with other characteristics

167

-3.0

-2.0

-1.0

'4-&J

0.0

1.0

2.0

2.0

1.0

3.0

log(E)

Fig. 3. Metal ion flux vs. - C,,,/C,( 1) for different Ni2+ and H+ concentration ratios in Donnan ion exchange. Curves A, B, C, and D represent the Ni‘2+ flux across the pore when Cn( 1 )/ C,( 1) ~0.001, 0.01, 0.1, and 0.5, respectively, while C&(3)/C,(3) is held constant at 1.0. Curves E, F, and G give the Ni2+ flux across the pore when &(3)/C,(3) =O.l, 0.01, and 0.005, respectively, while C,,(l)/C,(l), is held constant at 0.001. In all curves, gzl.1 and Ci(l)/Cr(3) =200. IonlisH+,ion2isSOz-,andion3isNi2+. Fig. 4. Relationship between Ni.‘+ flux and the equilibrium displacement factor in Donnan dialysis. For allpoints, C,/C,(l) = 10,4= 1.1, and Cr(l)C,(3) =200. Ion 1 is H+, ion 2 is SO:-, and ion 3 is Ni’+.

being equal (pore length and pore area per unit membrane area), the best membranes for Donnan ion exchange separations are those with very small pore radii, but a very high surface charge. Given the best possible membrane for a particular separation, it is also important to investigate the effects of the ion concentrations in the feed and the strip solutions on the metal ion flux. Figure 3 illustrates the effect of the hydrogen ion concentration ratio, Cn ( 1 )/C, (1) , and the metal ion concentration ratio, C,, (3) /C, (3)) on the metal ion flux when the common anion is SOSand the metal ion is Ni’+, and q= 1.1. These results indicate that the metal ion flux is affected more by the hydrogen ion concentration ratio across the pore or membrane than by the metal ion concentration ratio. However, it is im;ortant to note that in both of these figures, the effect of the concentration ratios appears to become greater as the system comes closer to equilibrium, i.e., as Cn ( 1 )/C, (1) approaches ( Cn (3 ) /C, (3 ) ) ljz3.One can define an equilibrium displacement factor, E, as

168

E=

C,(l) G,(l)

(1’3)

( >

C,(3) lh

G,(3)

so that E gives a measure of the net driving force for metal ion flux across the charged pore or membrane. In fact, for a given value of &/C,(l), all of the metal ion fluxes calculated for various C, (1) /Cn ( 1) and Ci (3 ) /Cn (3 ) ratios seem to depend monotonically on logE, as demonstrated in Fig. 4. It is evident from Fig. 4 that when E is greater than 100, there is little to be gained by adjusting the ion concentration ratios so as to increase E. However, when E is between 1 and 100, appropriate adjustments in the hydrogen ion concentrations on either side of the membrane can cause a significant increase in the metal ion flux. Previous investigators [3,4,6] have correlated the metal ion flux in Donnan dialysis to the metal ion concentration in the feed solution, or to the metal ion concentration gradient across the membrane. However, as noted in the previous discussion, the flux of metal ions across the membrane can be controlled by the adjustment of the hydrogen ion concentration ratio alone. If the acid which diffuses into the feed solution is continuously removed, perhaps by a diffusion dialysis process similar to the one illustrated in Fig. lb, the metal ion flux can be kept at its initial level much longer, and the final equilibrium concentration of metal ions in the strip solution will be higher than would have been possible if the acid had not been removed from the feed solution. Although there have been several experimental studies of Donnan ion exchange across cation exchange membranes [ 2-71, a quantitative comparison of these experimental results for ion fluxes and predictions of the model presented here is difficult for two reasons. First, none of the previous investigators have performed the necessary measurements to determine and average pore radius, surface charge, and pore length of the membrane pores. These are necessary input parameters for the model. Also, in many of the experimental studies [ 2,3,4,7], the boundary layer resistances to ion transport were significant, so the membrane model alone can not be used to predict ion fluxes in these cases. Lake and Melsheimer [ 21, however, did several stirred-cell experiments with Nafion 120*, a perfluorosulfonic acid membrane with an equivalent weight of 1200 (1200 g polymer per mole of sulfonic acid groups) and a thickness of 10 mil., or 0.025 cm. In these experiments, the metal ion flux was determined for various external acid and metal ion concentrations when only the membrane resistance was controlling. Considerable work has been done elsewhere to characterize the pore structure of Nafion. From the work of Gierke and Hsu *Nafion is a registered trademark of E. I. duPont and Co.

169

[ 181, the characteristic pore radius in the membrane used by Lake and Melsheimer was estimated from hydraulic permeability data as 8x lOBa cm, and its

porosity was given as 33%. However, Gierke and Hsu used X-ray spectroscopy techniques to show that the pore structure of Nafion, rather than being a bank of uniformly sized pores, is closer to a network of clusters, 40-50 A in diameter, connected by narrow channels about 10 A in diameter. We used 16 A for the pore diameter (r,, =8 x 10e8 cm) determined by hydraulic permeability measurements for the uniform pore size needed in the model since this is the characteristic radius for transport of solvent through the membrane pores. Two parameters, the pore length and the surface charge, remain to be specified. We assumed the pore length was 1.4 times the membrane thickness, or 0.035 cm. The pore surface charge was chosen as 6.6~ lop6 C/cm2 so that the model prediction for the metal ion flux would match the experimental result when the feed solution contained a 0.268 Msolution of NaNO,, and the strip solution contained a 1.95 A4 solution of nitric acid. This value for q is about half the value calculated using the equivalent weight of the membrane, assuming all sulfonic acid groups dissociate and distribute evenly along the pore surfaces. This difference could be due to counterion adsorption in an immobile Stern layer at the pore surface which would effectively reduce the pore surface charge [lo]. It could also be due to the non-uniformity of the actual Nafion pores described above. The larger clusters will be less effective in excluding anions than the smaller uniform pores assumed in the model, so a lower surface charge is required on the “theoretical” pores. A comparison of the model predictions using the above pore parameters with Lake and Melsheimer’s experimental results for the metal ion fluxes are shown in Table 3 for three different metal ions and several different electrolyte concentrations in the feed and strip solutions. The model predictions match the experimental results extremely well, especially considering our rather crude estimates of the membrane pore characteristics. The largest errors occur when the divalent calcium ion is in the feed solution. The model predictions for the Ca2+ ion fluxes are 35-50% higher than the fluxes determined in the experiments. These results indicate that despite our highly idealized model of the Nafion membrane, all of the changes in metal ion flux due to changes in electrolyte type and concentration can be predicted at least qualitatively by this model. This was impossible with the mass transfer resistance model used by Lake and Melsheimer (see eqn. 3 ) . Diffusion dialysis across positively charged membranes The model is now used to determine the effects of many of the dimensionless variables on the hydrogen ion flux in the diffusion dialysis of an acid from a metal salt solution using anion exchange membranes. The feed solution from which the acid is to be removed will be solution I, and the dilute acid receiving solution will be referred to as solution II. As in the previous section, ion 1 is H+, ion 2 is the common anion and ion 3 is the metal ion. First, the effects of

170 TABLE

3

Comparison of Lake and Melsheimer experimental results with model predictions for metal ion fluxes in Donnan ion exchange. The metal ion is ion 3, the common negative ion, NO,, is ion 2, andH+ isionl.Inallcases,Cr(3)=Cn(l)=0 Metal ion

C,(l)

G,(3)

(ion 3)

-
-
cm2mem.-set Na+ Na+ Na+ K+ K+

1.95 3.03 1.0 0.95

Kf Ca2+ Ca*+

1.95 3.04 0.99 1.99

Ca2+

3.0

0.268 0.073 0.073 0.071 0.066 0.065 0.012 0.010 0.010

4.27 x 10W7 4.64 x 10W7 3.89 x 10F7 4.97 x 10-7 5.55 x 1o-7 6.05 x 10V7 1.75x 10-7 1.95x 10-7 2.15x

1O-7

4.25 x 1O-7 5.11 x 3.53 x 4.88X 5.72x 7.14x

1o-7 1o-7 10-7

1O-7 1o-7

1.18X 1o-7 1.43x 1o-7 1.58X 1o-7

the ion valences and the fixed charge concentration in the membrane will be examined. In diffusion dialysis separations of an acid from a dilute metal ion solution, the membrane must allow the acid to flow through, while rejecting the flow of the metal ions from the feed solution. The selectivity of the membrane for the hydrogen ion over the metal ion in the solution is at least as important as the acid flux through the membrane in determining the efficacy of the diffusion dialysis separation. In Fig. 5, the pore selectivity for the H+ ion, defined as the ratio of the hydrogen ion flux to the metal ion flux, and the dimensionless acid flux are plotted against the C,/Ci (1) value for several different combinations for metal ions and anions in solution. From Fig. 5 it is apparent that the pore selectivity is essentially independent of C,/Ci (1) when the metal ion is univalent. Since both H+ and the metal ion have the same valency in this case, the electrical fields in the pore and across the pore-solution interface affect both ions equally. Hence, the selectivity of the pore for H+ ions over univalent metal ions is almost exactly equal to the ratio of the ionfluxesduetopurediffusion,orD,[CII(1)-C1(1)]/{D,[CII(3)-CI(3)]}. For this reason, diffusion dialysis is not an effective method of selectively removing acid from a solution of univalent metal ions. On the other hand, when the metal ion is divalent instead of univalent, the pore selectivity for H+ is much higher, due to the electrical field effects both inside the membrane and at the membrane-solution interfaces. This selectivity can be increased by changing C,/Ci ( 1) as shown in Fig. 5. By comparing the hydrogen ion flux results and the selectivity results in Fig. 5, it appears that one could combine the highest possible acid flux with a pore selectivity greater than 50 by using a

-2.0

-1.0

0.0

1.0

2.0

Fig. 5. Selectivity and H+ flux through diffusion dialysis membrane as a function of C,/C, (1) for different combinations of metal and coion in solution. The A, B, C, and D curves represent the NiSO,-H,SO,, Na2S04-H2S04, NiC&-HCl, and NaCl-HCl systems, respectively. In all curves, Q=1.1,C~~(1)/C~(1)=O.O1,CII(3)/CI(3)=0.0,andC~(1)/C,(3)=2.0. Fig. 6. Selectivity and H+ flux through diffusion dialysis membrane as a function of C,/Ci (1) for different hydrogen ion concentration ratios. The A, B, and C curves represent the acid flux and selectivity when C,,(l)/C,(l) =O.Ol, 0.1, and 1.0, respectively. In all curves, S= 1.1, C,,(3)/ C1(3)=0.0,andC1(1)/Cr(3)=2.0.10n lisH+,ion2isSOz-,andion3isNi2+.

membrane with pore characteristics such that C,/Ci (1) is less than 1. However, because of the complex coupling of the mass, charge and momentum transport in this problem, the selectivity is very much dependent on the hydrogen ion concentration gradient across the membrane. (For a more detailed discussion of these coupling effects, see Ref. [ 191. ) At low values of C&/C, ( 1 ), where the selectivity of the pore is due almost exclusively to the electrical potential gradient which arises to prevent a net flow of charge, the H+ flux and selectivity are reduced dramatically as the H+ ion concentration ratio is increased from 0.01 to 1. This effect is shown in Fig. 6. As the hydrogen ion concentration ratio is increased, the concentration gradient of the hydrogen ion is decreased, while the concentration gradient of the metal ion remains essentially unchanged. Because of the reduced hydrogen ion concentration gradient, the potential gradient required to maintain a zero net current is also reduced. Since it is this diffusion potential which gives rise to the selectivity of the pore for acid when C,/Ci (1) < 1, the selectivity of the pore also decreases dramatically with the increasing hydrogen ion concentration ratio. In fact, when the acid concentration ratio is equal to 1 for C,/C,( 1) < 1, the selectivity of the pore is less than one. This indicates that the membrane is permitting metal

172

ions to be removed from the feed solution at a faster rate than the acid is being removed. In contrast, when C,/Ci (1) is greater than 1, the acid flux from the feed solution is significant, and the selectivity is greater than 1, even when the acid concentration is equal on both sides of the membrane. This selectivity is due to the exclusion of the positive ions from the pore due to the large positive charge on the pore walls. The divalent metal ion is excluded more completely than the H+ ion, consequently, the metal ion flux is reduced more than the acid flux. From this analysis we can conclude that although a weakly charged membrane, with C,/Ci (1) << 1 will permit a faster acid flux with good selectivity when the acid concentration difference between the two solutions is high, a more highly charged membrane, with C,/C, (1) >> 1 is required in order to selectively remove acid when the acid concentration in the feed solution begins to approach that of the receiving solution. This suggests that a two-stage or multi-stage approach might be best for most diffusion dialysis separations. In the fist stage, a weakly charged or neutral porous membrane could be used to reduce the acid concentration in the feed to about ten times that in the receiving solution. In the second stage, a positively charged membrane with C,/ C, (1)2 1 would be used in order to maintain good selectivity and acid flux as the feed solution acid concentration approaches that in the receiving solution. It should be emphasized, however, that the boundary layer effects have been neglected in this analysis. Martinez et al. have indicated that in very thin weakly charged track-etched polycarbonate membranes, these boundary layer resistances are difficult to overcome without actually forcing fluid through the pores [ 20,211. It has also been demonstrated that the surface charge density in weakly charged membranes is not independent of ion concentrations as assumed here, but that the negative charge on the pore surface tends to increase when the electrolyte concentration is increased [ 11,22-241. Nevertheless, the use of these weakly charged membranes as a preliminary stage in diffusion dialysis separations seems to have some merit and warrants further investigation. So far, we have looked at the effects of the pore characteristics on the ion fluxes using only the dimensionless parameter, Cm/C1(1) . The parameter 4 has been held constant at 1.1. In Table 4, the acid flux and selectivity of a pore are TABLE 4

Dependence of ionfluxes and selectivity on 9 for a diffusion dialysis separation keeping C,/C,( I) constantat30,withC,,(l)/C,(l)=O.Ol,C,,(3)/C,(3) =0, C1(3)/C,(1)=0.5,where the common anion is SOi-c

(N,) l

0.011 0.252 102

1.1

3.3

0.257 98

0.290 83.5

11 0.443 49.8

173

shown for several values of q, while holding C,/C, ( 1) constant at 30 when the metal ion is Ni2+, and the anion is SO:-. As q is increased from 0.011 to 11, the acid flux increases by about 80 percent, but the selectivity decreases by about 50 percent. A similar percentage increase is seen in the acid flux when the anion is Cl-. The effect of Qon the acid flux and selectivity decreases with decreasing C,/Ci (1) so that when Cm/C1(1) = 0.01, the acid flux increases only 6 percent as q is increased from 0.011 to 11. The effects of C,/Ci (1) and S on the osmotic flow rate of solution in a diffusion dialysis separation are shown in Fig. 7. The positive Peclet numbers indicate that the solution usually flows from the feed stream (high ionic strength) to the dilute acid receiving stream (low ionic strength) in these separations, opposite to the direction of normal osmosis. In most diffusion dialysis separations, osmotic flow from the feed solution to the receiving solution is desirable, since it tends to increase the metal ion concentration in the feed solution and increase the total amount of acid which can be removed from the feed solution. From Fig. 7, it is apparent that large positive Peclet numbers can be obtained only when C&/C, (1) < 10, and Q> 10. However, these conditions are not favorable for good selectivity of the membrane as shown in Fig. 6. Consequently, the advantages of the large osmotic flow from the feed are usually more than negated by the disadvantage of losing metal ions to the receiving stream. Until this point, we have considered only the flux of ions when the external solutions are maintained at constant concentrations. In the actual separation processes, the ion concentrations in the external solutions will change as the ions move across the membrane. In the following section, the transient behavior of the external solution concentrations is modeled, and results are presented for both diffusion dialysis and Donnan ion exchange separations.

Fig. 7. Peclet number dependence on C,/C,( 1) and Q in diffusion dialysis of H,SO, from NiSO,. Curves A, B, and C represent q=ll, 3.3, and 1.1, respectively. In all curves, C,,(l)/C,(l) =O.Ol, C,,(3)/C,(3)=0.0,andCI(l)/C,(3)=2.0.Ion1isH+,ion2isSOq-,andion3isNi2+.

174

Transient concentration profiles in batch donnan dialysis separation Consider a batch Donnan dialysis process in which a membrane separates two well-stirred aqueous solutions that both contain the same 3 ionic components. The macroscopic mass balances for the ions in solutions I and II for the separation process are

d[VICI(i) 1 dt

=-(NL)A

dWd$i)l=(Ni)A

i=123 , , ,...

(17)

i=123

(18)

3

9

,...

dV, ,,=-(uz)A

(19)

v, + v,, = v; + v;1

(20)

In these equations, A is the total pore area available on the face of the membrane, and Vi and Vg are the initial volumes of solution I and II, respectively. The electroneutrality conditions, Xi”, ,Ci (i).q = 0 and CL 1Cn (i)zi = 0 in each solution can be used to eliminate the component mole balances for ion 3 given by eqns. (17) and (18). These mass balances are also applicable to the batch separation system used by Ng and Snyder [ 3,4], in which the feed and strip solutions are rapidly recirculated through a shell and tube dialyzer such that the ion concentrations do not change appreciably in any one pass through the dialyzer. The assumptions made in formulating equations (17 )- (20) are: (1) the characteristic time for concentration changes in the external solutions is much larger than the characteristic time for transport in the membrane (the membrane fluxes are quasi-steady), and (2 ) the densities of the two solutions are approximately equal and do not change appreciably as the ion concentrations change. Assumption (1) is valid as long as the solution volumes are much larger than the pore volume of the membrane, and assumption (2) is valid as long as the solutions are dilute. Euler’s one-step method [ 171 has been used to solve eqns. (17)-(20) with the obvious initial conditions. The ionic fluxes and solution velocity at each time-step were obtained from the subroutine used to solve the equations in Table 1. Figure 8 shows a comparison of the model predictions with the experimental results of Ng and Snyder [ 41 for the Donnan dialysis of a copper sulfate solution across a perfluorosulfonic acid membrane into a 0.5 M sulfuric acid strip solution. Both solutions were 500 ml in volume. The Reynolds number in the feed compartment during the experiments was high enough so that the boundary layer resistance was negligible at the initial conditions. The average pore radius and porosity of this membrane were estimated to be 8 x 10W8 cm, and 0.33, respectively, from the results reported by Gierke and Hsu [ 181 for an 1100 equivalent weight (g polymer/ionic group) perfluorosulfonic mem-

175

7

x

NGAND

SNYDER

EXP.

(1983)

k? :, c’ z 1 :K e z E “9 s

_

x0

I 60

Fig. 8. Comparison

I

I

120 180 TIME(MIN)

I 240

of model predictions

300

with experimental

results for copper concentration

in a

CuSo, feed solution during Donnan ion exchange using a 0.5 M sulfuric acid strip solution.

brane. The pore length was estimated to be 1.5 times the membrane thickness, or 5.55 X 10m6cm. The value of 4 was then chosen to be 5 x lO-‘j C/cm2 so that the model results for feed stream copper concentration would match the experimental results at 60 minutes. This value of 9 is about 25% lower than that used to model the Nafion 120 membrane in the Lake and Melsheimer experiments. Ng and Snyder do not specify the equivalent weight (EW) of the perfluorosulfonic acid membrane that they used, so it could have been different from 1200 EW membrane used by Lake and Melsheimer. In Fig. 8, the model predictions for the copper concentration in the feed as a function of time are shown to match the experimental results quite well through a three order of magnitude drop in the copper ion concentration. In fact, the change in the slope of the semilog plot of copper concentration vs. time which Ng and Snyder attribute to boundary layer effects at low feed stream copper concentrations is predicted by this membrane model alone. As demonstrated by Ng and Snyder [4] the boundary layer resistance in the feed stream does increase as the copper concentration decreases, however these investigators neglect the fact that the membrane resistance also increases dramatically as the acid and copper equilibrate across the membrane. Our model results indicate that the copper flux through the membrane after 240 minutes is less than one tenth of its initial flux. Therefore, it is likely that the membrane resistance remains controlling throughout the entire experiment. Conclusions

The method of area-averaging has been used to model ion and solution fluxes through charged capillary pores in typical Donnan dialysis separations. This model takes into account the effects of concentration, electrical potential and pressure driving forces on ion fluxes across charged cylindrical pores. Com-

176

parison with experiments done by Lake and Melsheimer [ 21 on Donnan ion exchange across Nation 120 with well-stirred solutions show that the dependence of the metal ion flux on the acid concentration in the strip solution, on the metal ion concentration in the feed, and on the metal ion type could all be predicted by this model. These effects could not be predicted by the mass transfer resistance model which has been used in the past [2-41. This resistance model, described by eqn. (3)) neglects the coupling of the ion fluxes through the electrical potential gradient and the osmotic flow in the membrane pores. Equation (3 ) differs significantly from the equation for the area-averaged ion fluxes used in the capillary pore model (eqn. I.1 ) . This expression for the areaaveraged ion fluxes includes contributions due to migration in the electric field, due to convection, and due to coupling of the radial profiles of the ion concentrations and the electrical potential. Consequently, the effect of one ion’s concentration in the external solutions on another ion’s flux can be predicted by using the model equations in Table 1, but not by using eqn. (3). The use of the capillary pore model for transport in charged membranes also allows one to predict the optimum pore characteristics for a particular separation. In ion exchange separations, the model results indicate that given the same total pore area and external electrolyte concentrations, the rate of ion exchange is maximized when the pore radii and surface charge density in a membrane are as small and large as possible, respectively. However, in the diffusion dialysis of an acid from a multi-valent metal salt, the pore radius and surface charge which optimize the acid flux and selectivity in a membrane depend very much on the external electrolyte concentrations. When the acid concentration driving force is large, a weakly charged or neutral membrane with wide pores tends to optimize the rate of the separation and the selectivity. As the system moves closer to equilibrium, however, a more highly charged, smaller pore is required to maintain good selectivity of the membrane for acid over the metal, although this trend in pore size and surface charge will tend to decrease the acid flux. The time-dependent ion concentrations in well-stirred solutions undergoing Donnan dialysis were also modeled. These results are compared to the experimental data of Ng and Snyder [ 41, and are shown to match quite well when boundary layer resistances are negligible. List of symbols activity, mol/cm3 ai concentration of ion i in pore, mol/cm3 ci concentration of ion i in solutions I and II, respectively, mol/cm” G(i), C,,(i)

Zi

ci/Ci ( 1)) dimensionless concentration in pore diffusivity of ion i in the pore, taken to be that in dilute, free solution, cm’/sec

177

E F

W I K L

N Ni Ni P

P Pe Q

(7 r

r. R T (4)

Greek letters a

P t

equilibrium displacement constant for Donnan ion exchange, defined by eqn. (16) Faraday’s constant, 96,487 C/eq. area-averaged current flux through pore, C/cm’-set dimensionless ionic strength in pore, 5 CE=, 24 ( & ) Donnan equilibrium constant, dimensionless length of pore, cm total number of ions in solution flux of ion i in pore, mol/cm’-set NiLIDly dimensionless flux of ion i hydrostatic pressure in the pore, dyn/cm2 dimensionless hydrostatic pressure = Pri/8pD, (v,)L/D,, Peclet number for convective flow in the pore, dimensionless surface charge density on the pore wall, C/cm2 dimensionless surface charge density on pore wall, Fqr,/cRT radial coordinate in pore, cm pore radius, cm gas constant, 8.314 C-V/mol-K absolute temperature, K area-averaged axial velocity of the center of mass of solution in pore, cm/set axial coordinate in pore, cm z/L, dimensionless axial coordinate valence of ion i

correction to electrical potential boundary condition due to nonuniform radial potential profile in pore, defined in eqn. (13), dimensionless mensionless group affecting the osmotic flow ro2G(1)RT/8,G,di through the pore permittivity of solution, taken to be that of water, 6.906 x lo-l2 C/V-cm dimensionless radial coordinate, scaled r[cRT/2F2CI(1)I]-1’2, by the Debye length ‘I2 dimensionless pore radius ro[cRT/2F2CI(1)T]factor in ion concentratfon boundary conditions which corrects for radial curvature of the electrical potential in the pores. Defined by eqn. (10) viscosity of solution in the pore, Pa-set IO( qo) /II ( qo), where IO and 1, are modified Bessel functions of the first kind

dimensionless osmotic pressure = (rg RT/8,uD, ) C E I ( Ci ) electrical potential in the pore, V electrical potenal in solution I and solution II, respectively, V F@/RT, dimensionless electrical potential Superscripts _

dimensionless quantity deviation of the field from its area-averaged value in the pore

Brackets

area-averaged over the pore cross-sectional area

0

References 1

E. H. Cwirko and R. G. Carbonell, Transport of electrolytes in charged pores: analysis using the method of spatial averaging, J. Colloid Interface Sci., 129 (1989) 513.

2

M. A. Lake and S. S. Melsheimer,

3

J., 24 (1978) 130. P. K. Ng and D. D. Snyder,

4 5 6 7 8 9 10 11

12

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17

Mass transfer

Mass transfer

characterization

characterization

of Donnan

of Donnan

dialysis, AIChE

dialysis:

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the nickel dialysis,

J.

C. P. Wen and H. F. Hamil, Metal counterion transport in Donnan dialysis. J. Membrane Sci., 8 (1981) 51. M. Sudoh, H. Kamei and S. Nakamura, Donnan dialysis concentration of cupric ions, J. Chem. Eng. Jpn., 20 (1987) 34. J. C. Fair and J. F. Osterle, Reverse electrodialysis in charged capillary membranes, J. Chem. Phys., 54 (1971) 3307. V. Sasidhar and E. Ruckenstein, Electrolyte osmosis through capillaries, J. Colloid Interface Sci., 82 (1981) 439. V. Sasidhar and E. Ruckenstein,

Anomalous

effects during electrolyte

osmosis across charged

porous membranes, J. Colloid Interface Sci., 85 (1982) 332. G. B. Westermann-Clark and J. L. Anderson, Experimental verification of the space-charge model for electrokinetics in charged microporous membranes, .J. Electrochem. Sot., 130 (1983) 839. H. J. M. Hijnen, J. Van Daalen and J. A. M. Smit, The application of the space-charge model to the permeability properties of charged microporous membranes, J. Colloid Interface Sci.. 107 (1985) 525. IMSL User’s Manual-Math/Library, version 1.0, softcover edition, 1987, p. 770. J. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, NJ, 1973, p. 414. R. Z. Schkigl, Der Theorie der anomalen Osmose, Z. Phys. Chem. N. F., 3 (1955) 73. A. A. Sonin, Osmosis and ion transport in charged porous membranes: a macroscopic, mechanistic model, in: E. Selegny (Ed.), Charged Gels and Membranes - Part I. Reidel, Dordrecht, 1976, p. 256. B. Carnahan and J. 0. Wilkes, Sons, New York, NY, 1973.

Digital

Computing

and Numerical

Methods.

John Wiley and

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19 20 21

22

23

24

T.D. Gierke and W. Y. Hsu, The cluster-network model of ion clustering in perfluorosulfonated membranes, in: A. Eisenberg and H. L. Yeager (Eds.), Perfluorinated Ionomer Membranes, ACS Symposium Series No. 180, American Chemical Society, Washington, DC, 1982, p. 283. E. H. Cwirko, Ph.D. Thesis, North Carolina State University, in preparation. L. Martinez, A. F. Tejerina and J. I. Arribas, Diffusion of LiCl through Nuclepore membranes of polycarbonate, J. Non-Equilib. Thermodyn., 12(3) (1986) 245. L. Martinez, A. Hernandez and A. F. Tejerina, Cation transport numbers in polycarbonate microporous membranes. Effect of unstirred diffusion layers, Sep. Sci. Technol., 22 (1) (1987) 85. L. Martinez, A. Hernandez and A. F. Tejerina, Concentration dependence of some electrochemical properties of polycarbonate microporous membranes and evaluation of their electrokinetic charge, Sep. Sci. Technol., 22 (6) (1987) 1625. J. H. Petropoulos, D. G. Tsimboukis and K. Kouzeli, Non-equipotential volume membrane models. Relation between the Gluekauf and equipotential surface models, J. Membrane Sci., 16 (1983) 379. J. H. Petropoulos, Y. Kimura and T. Iijima, Ionic partition equilibria of a simple organic electrolyte in chemically modified charged cellulosic membranes, J. Membrane Sci., 38 (1988) 39.