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Concentration Polarization

Chapter 7

7.1. CURRENT–VOLTAGE RELATIONSHIP

Concentration polarization has been widely studied based on the current(I)–voltage (V) curve. The following investigation is its pioneering work presentedby Peer (1956). The I–V curve of a cation exchange membrane (Permutit C-10)indicated in Fig. 7.1 has an S-type shape including three characteristic regions:Region I shows Ohmic behavior, and it transforms as voltage is increased intoRegion II in which the current varies very slowly with voltage and exhibits aplateau presenting the limiting current density ilim. Region III is the overlimitingcurrent sphere in which current increases gradually. The limiting current densityequation is given by

ilim ¼FDC

ðt̄� tÞd(7.1)

t̄ and t are the transport number of counter-ions in an ion exchange membraneand in a solution. C is the electrolyte concentration in the solution,D the diffusionconstant of electrolytes dissolving in the solution, F the Faraday constant and dthe boundary layer thickness. Peer suggested that the current in excess of ilim wasfound to be carried partly by hydrogen ions but mostly by chloride ions.

Spiegler calculated the current–voltage relationship for an anion exchangemembrane placed in a 1,1 valent electrolyte solution based on the Nernst–Planckmodel, and introduced the potential drop across the membrane DE in an elect-rodialysis system in Fig. 7.2 as follows (Spiegler, 1971):

DE ¼ �ð2RS þ RmÞi þal

� �þ b� g

h iln1þ ði=ilimÞ

1� ði=ilimÞ(7.2)

where

a ¼ �FD

t̄þ � tþ¼

FD

t̄� � t�

b ¼RT

F

� �ðt̄� � t̄þÞ

g ¼RT

F

� �ðt� � tþÞ

Here, i is the current density, ilim limiting current density, Rs electric resistanceof a solution, Rm electric resistance of a membrane, l equivalent conductanceof a solution, t̄�; t̄þ transport number of anions and cations in a membrane,

DOI: 10.1016/S0927-5193(07)12007-6

dx.doi.org/10.1016/S0927-5193(07)12007-6.3d

200

180

160

140

120

100

80

60

40

20

0 1 2 3

V=(Vi−iR i=0) volts

i lim

II

III

Membrane Area = 0.126 cm2

i am

ps ×

10

6

4 5

Figure 7.1 Current–voltage curve (Membrane area 0.126 cm2) (Peer, 1956).

Concentratingside

Ion exchangemembrane

Cm

C C

C

C

Desaltingside

Figure 7.2 Concentration polarization.

Ion Exchange Membranes: Fundamentals and Applications98

t–, t+ transport number of anions and cations in a solution, R Gas constant andT Absolute temperature. DE ( ¼ DE (total)) was plotted against i, obtainingFig. 7.3 in which 1st term and 2nd term correspond to the terms in Eq. (7.2).Fig. 7.3 shows that upon increasing the voltage across the membrane, a current

Figure 7.3 Current–voltage curve (Calculation) (Spiegler, 1971).

Concentration Polarization 99

plateau is reached. The limiting current is realized when the electrolyte concen-tration reaches zero (Cm ¼ 0) in Fig. 7.2, however an overlimiting current seenin Fig. 7.1 does not appear in Fig. 7.3.

Cowan and Brown expressed the current–voltage relationship (Cowanplot) by the following equation (Cowan and Brown, 1959).

V

I¼ Rþ

ðV e þ V c þ VpÞ

I(7.3)

where Ve is the electrode potential, Vc concentration potential, Vp polarizationpotential and IR the Ohmic voltage. Equation (7.3) indicates that the interceptof a plot of V/I against I�1 presents the resistance of the cell and the slope ofthe plot presents the cell voltage plus their derivatives. Polarization voltagemanifest itself as a rapid change of slope as shown in Fig. 7.4 obtained for ananion exchange membrane. The pH of diluting stream begins to change at acurrent very near the value at which the resistance slope change and continue todecrease as current density increases. The pH change is attributed to the occur-rence of water dissociation on the desalting surface of the anion exchangemembrane. The point at which negative slope cuts positive slope shows thelimiting current Ilim.

Kooistra reported from the measurement of a I–V curve of a cell pair thatthe limiting current density of a cation exchange membrane ilim cation is less thanthat of an anion exchange membrane ilim anion placed in a 0.02M NaCl solution(Fig. 7.5) (Kooistra, 1967). This is due to the transport number of Na+ ions tobe less than that of Cl– ions in a NaCl solution.

Figure 7.4 Voltage/current–reciprocal current curve (Cowan and Brown, 1959).


7.2. CONCENTRATION POLARIZATION POTENTIAL

An ion exchange membrane is assumed to be immersed in a 1,1 valentelectrolyte solution of electrolyte concentration C. Then, an electrodialysissystem in Fig. 7.2 is formed by passing an electric current through the membrane.Next, the electric current is interrupted, and the potential across the membrane Ejust after the electric current interruption is measured. E is the concentrationpolarization potential, expressed by the following equation.

E ¼ ED þ Em ¼ 2RT

F

� �ðt̄� tÞ ln

ðC þ DCÞg1ðC � DCÞg2

(7.4)

Diffusion potential : ED ¼RT

F

� �ð2t� 1Þ ln

ðC � DCÞg2ðC þ DCÞg1

(7.5)

Membrane potential : Em ¼RT

F

� �ð2t̄� 1Þ ln

ðC þ DCÞg1ðC � DCÞg2

(7.6)

g1 and g2 are the activity coefficients of ions in solutions of the concentrationC +DC and C–DC, respectively. DC is computed by substituting E observed

3

2

1

00 5

i (mA/cm2)

E(V

)

i lim, cation

i lim, anion

10

Figure 7.5 Current–voltage curve (Kooistra, 1967).


experimentally into Eq. (7.4). Further, we have the following equation by sub-stituting i and DC into ilim and C in Eq. (7.1).

DDCd

¼i

F

� �ðt̄� tÞ (7.7)

d is obtained by substituting DC calculated above into Eq. (7.7).Onoue (1962) and Cooke (1961a) observed DC and d. Table 7.1 shows the

data measured by Onoue for m-phenol sulfonic acid cation exchange membrane.

7.3. CHRONOPOTENTIOMETRY

In order to obtain a clear picture of the mechanism of polarization anddepolarization, Forgacs measured the potential differences between an ionexchange membrane during constant current transfer and after electric currentinterruption (Forgacs, 1962). The results are summarized in Fig. 7.6. In the firstinstant of the application of a direct current through the solution–membrane

Table 7.1 Concentration polarization of a cation exchange membrane

CNaCl (M) i (A dm–2) E (mV) DC (M) d� 102 (cm)

0.1 0.25 21.3 0.037 3.00.50 38.0 0.060 2.40.75 70.0 0.080 1.91.00 95.0 0.095 1.9

0.5 1 8.5 0.075 1.42 15.2 0.13 1.34 28.5 0.24 1.16 40.0 0.31 1.0

1.5 1 3.2 0.10 2.12 6.0 0.18 1.93 8.4 0.25 1.86.67 15.0 0.44 1.4

2.5 2 2.9 0.13 1.44 5.1 0.23 1.26 9.5 0.40 1.0

Source: Onoue (1962).


system, a potential drop E0 is obtained. This potential increases nonlinearly,achieves a maximum value Em and decreases again to a steady value Es aftertime ts such that Es>E0. This was observed for all conditions of solution con-centration, current densities and flow conditions (static, stirred and continuouslyflowing liquid). After the interruption of the current, the potential suddenlydrops to a residual value Er which slowly disappeared with time. The value of Er

depends upon the time of current application, but remains constant for times ofcurrent transfer greater than ts.

When, instead of interrupting the current, the current is suddenly reversed,the following effects are obtained (Fig. 7.7).

(1)
With the reversal of the current, there is a potential drop of reversed signto a value which is a function of the time of current application, butachieves a constant value at time ts.
(2)
The change in potential drop with continued application of the reversedcurrent is similar to that occurring during the application of the direct(nonreversed) current.
(3)
After the interruption of the reversed current, two types of depolari-zation curves may develop. If the duration of the reversed current issufficiently long, the depolarization curve is the same as previously de-scribed (Fig. 7.6). But if the time of reversed current application isshorter, when this is stopped, the residual potential changes its sign,arrives at a maximum value, and disappears with time (Fig. 7.7). It isimportant to note that this maximum is achieved after a specific time lagand not immediately.

+

−

0

+

−

0

Time

Eo

Em

Es

Er

ts

E (

Pot

entio

n dr

opon

mem

bran

e)I (

App

lied

curr

ent)

Figure 7.6 Potential–time curve (Forgacs, 1962).


The information provides profitable aspects in electrodialysis reversal(DER) system (cf. Chapter 2 in Applications).

Sistat and Pourcelly (1997) investigated the chronopotentiometric responseof a cation exchange membrane as a function of the applied current density belowthe limiting one. Taking into account classical assumption such as the variation ofconcentration associated to the Teorell–Meyer–Sievers segmentation model, theconcept of the Nernst layer model, the quasi-electro-neutrality condition withinthe diffusion boundary layer and the validity of the Donnan equation at thesolution–membrane boundaries, theoretical values of the trans-membrane poten-tial difference were obtained which fit very well the experimental V–t responses.Moreover, this method allows the calculation of the diffusion boundary layerthickness. In addition, a decrease of the diffusion boundary layer thickness wasobserved when the current density increases. This result was attributed to theoccurrence of convection effects inside the diffusion boundary layer.

7.4. REFRACTIVE INDEX

When a ray of light shines on the interface between two solutions thathave different refractive indexes, the light is refracted and the refractive indexincreases with the electrolyte concentration of the solutions. This phenomenon is

0I

Time

+

−

0E

+

−

Figure 7.7 Potential–time curve (Forgacs, 1962).


the principle of the Schlieren-diagonal method, which was applied to measurethe concentration polarization on the surface of an ion exchange membraneby Takemoto (1972). In this experiment, an Aciplex CK-2 cation exchangemembrane or a CA-2 anion exchange membrane was integrated in the three-compartment optical glass cell. The effective membrane area and the distancebetween the membranes were maintained at 0.3 cm2 (1 cm height, 0.3 cm width)and at 0.2 cm, respectively. A NaCl solution (0.1M or 0.05M) was put in thecentral desalting cell, and in the electrode cells placed on both outsides of thecentral cell, and an electric current was passed through Ag–AgCl electrodes.The NaCl concentration changes were observed by the Schlieren diagonalmethod, changing the current density incrementally.

Figs. 7.8 and 7.9 show the concentration distribution in a boundary layerformed on the cation exchange membrane and anion exchange membrane,respectively. In the 0.1MNaCl solution, the limiting current density for the cationexchange membrane (2.93A dm–2) was less than that for the anion exchangemembrane (3.99A dm–2). This difference is due to the lower mobility of Na+ ions(Na+ ion mobility uNa, zNauNaF ¼ 50S cm2 mol–1) compared to Cl– ions (Cl– ionmobility uCl, zCluClF ¼ 76S cm2 mol–1) and lower transport number of Na+ ions(Na+ ion transport number tNa ¼ 0.40) compared to Cl– ions (Cl– ion transportnumber tCl ¼ 0.60) in the NaCl solution. The boundary layer thickness d on thecation exchange membrane (0.0366 cm) was also less than that on the anion

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.00

0.02

0.04

0.06

0.08

0.10

0.12

=0.0580 cm=0.0366 cm

C(M

)

x (cm)

0.05MNaCl

0.1MNaCl

2.28(LCD)4.30(over LCD)

2.93A/dm2(LCD)

2.10

1.13A/dm2

Figure 7.8 NaCl concentration distribution in a boundary layer. Cation exchangemembrane (Aciplex CK-2) (Takemoto, 1972).


exchange membrane (0.0402 cm), and these are not affected by the currentdensity. In the 0.05M NaCl solution, however, the limiting current density for thecation exchange membrane (2.28A dm–2) was larger than that for the anionexchange membrane (1.81A dm–2). The thickness of the boundary layer on thecation exchange membrane (0.058 cm) was greatly increased compared to that onthe anion exchange membrane (0.0362 cm).

It was apparent in Fig. 7.8 that the NaCl concentration fluctuated on thesurface of the cation exchange membrane placed in the 0.05 M NaCl solution atthe limiting current density within the range of the distance from the membranesurface xo0.01 cm. The concentration fluctuation was found to propagate fromthe membrane surface toward the inside of the solution adjacent to the membrane.The velocity of the propagation was unstable, but it was estimated to be about0.1 cm s–1. So, the frequency of fluctuation was estimated to be 10 times s�1.The overlimiting current phenomena seen in the cation exchange membranes aresupposedly related the concentration fluctuation. The concentration fluctuationwas not observed on the anion exchange membrane.

Shaposhnik et al. used the three-frequency laser interferometry method forstudying the concentration polarization of ion exchange membranes in widerange of current densities (Shaposhnik et al., 2000). The concentration and

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.00

0.02

0.04

0.06

0.08

0.10

0.12

=0.0580 cm=0.0366 cm

C(M

)

x(cm)

0.05MNaCl

0.1MNaCl

2.28(LCD)4.30(over LCD)

2.93A/dm2(LCD)

2.10

1.13A/dm2

Figure 7.9 NaCl concentration distribution in a boundary layer. Anion exchange mem-brane (Aciplex CA-2) (Takemoto, 1972).


temperature fields of a solution involved in electrodialysis of sodium chloride aremeasured simultaneously. The proposed method makes it possible to measureabsolute values of local concentration of acids and bases that form during waterdissociation at the solution–membrane interfaces following an increase in thecurrent density above the limiting value. According to an analysis of the con-centration and temperature distribution in an electrodialyzer channel, maximumvariations in the measured quantities occur near the interface.

7.5. NATURAL CONVECTION

Under an applied electric current, a solution in a boundary layer flowsupward on the surface of the membrane because of the decrease in electrolyteconcentration, and flows downward because of the increase in electrolyte concen-tration. These phenomena (natural convection) occur owing to the decrease andincrease in solution density in the boundary layer. Frilette observed the naturalconvection using the electrodialysis cell (Fig. 7.10) incorporated with two Permutitcation exchange membranes and equilibrated with a 0.0996M NaCl solution(Frilette, 1957). A constant electric current was passed and was turned off after thepredetermined time had elapsed by keeping total coulombs passed to be constant.

Figure 7.10 Apparatus for measuring natural convection: (A) luicite end block; (B)graphite plate; (C) spacer; (D) membrane (Frilette, 1957).


After the completion of the experiment, the contents of the central chamber weredrained off slowly and collected in two equal portions; the portion first collectedcorresponded to the solution in the lower half of the central chamber. The twoportions were analyzed for chloride ions. In all experiments it was found that a netmovement of NaCl into the lower half of the chamber had occurred, with nosignificant change in total electrolyte content of the chamber. The results of thisexperiment are shown in Table 7.2, which is characterized by a relatively largetransfer of salt from the upper half into the lower half of the chamber.

In order to determine the ascending flow velocity vy caused by the naturalconvection on the desalting surface of an ion exchange membrane, Tanakasuspended fine cellulose fibers in a 0.5M NaCl solution in a transparentpolyethylene cell incorporated with a cation exchange membrane (Aciplex CK-2)and an anion exchange membrane (Aciplex CA-2) (Tanaka, 2004). Effectivemembrane area was adjusted to 4 cm2 (2 cm height, 2 cm width). The movement ofthe cellulose fibers was observed with a microscope while applying an electriccurrent through Ag–AgCl electrodes. The flow velocity was plotted against cur-rent density and is presented in Fig. 7.11, which shows that vy on the cationexchange membrane is larger than vy on the anion exchange membrane. Thisevent means that concentration polarization occurs more easily on the cationexchange membrane than on the anion exchange membrane. The ascending flowsupposedly produces the overlimiting current.

Table 7.2 Transfer of electrolytes in a cell owing to natural convection (Constant Current�Timea)

Current (mA) Current Density(A cm–2)

Time (min) Lost, UpperLayer (meq)

Gained, LowerLayer (meq)

Transferred

Upper Layer(eq F–1)

Lower Layer(eq F–1)

46 1.87 40 0.303 0.310 –0.264 +0.27061.3 2.49 30 0.345 0.310 –0.301 +0.27092 3.74 20 0.367 0.365 –0.320 +0.318138 5.61 13 1/3 0.465 0.457 –0.406 +0.400184 7.48 10 0.483 0.488 –0.422 +0.426230 9.35 8 0.506 0.493 –0.442 +0.430368 14.96 5 0.504 0.511 –0.440 +0.446

Source: Frilette (1957).aTotal of 1.840 A min for all runs or 1.147� 10–3 F.

IonExchangeMem

branes:

FundamentalsandApplica

tions

108

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

v y(1

0-2cm

/s)

Anion membr.Catio

n membr.

i(A/dm2)

Figure 7.11 Ascending flow velocity in a boundary layer (0.1M NaCl, Aciplex CK-2/CA-3) (Tanaka, 2004).


The natural convection occurring in the boundary layer had beendiscussed near a vertical electrode incorporated with an electrolyzer (Wagner,1949; Tobias et al., 1952; Ibl and Muller, 1958).

7.6. FLUCTUATION

Occurrence of fluctuation of electric currents through ion exchangemembranes is attributed to the depletion of salt on one side of the membrane,which creates a thin layer of high resistance. Joule heating in this depletion layerand ensuring temperature gradient, as well as the concentration gradient, giverise to buoyant forces, which may create a turbulent convection current. Theturbulence mixes the depletion layer so that the electric resistance fluctuates, andconsequently the current flickers (Lifson et al., 1978). Lifson suggests that thefollowing experimental results support the above mentioned conjecture.

(1)
Noise is coincident with the increase of the electric resistance by thedepletion process.


(2)
When the current density is reduced, it reaches a critical value, belowwhich the convection current changes from turbulent to laminar, and thefluctuation disappears.
(3)
Fluctuation reduces with temperature, because the expansion coefficientof water decreases with temperature and its viscosity increases.
(4)
A nonionic water-soluble polymer added to the compartment on the sideof the depletion layer reduces the fluctuation, by increasing the bulkviscosity of the solution.
(5)
Fluctuation depends on the membrane’s orientation in the gravitationalfield.
(6)
The convection current in the depletion layer can be observed directly,using a laser-beam, by adding latex particles, which create optical noiseas they drift with the convection current across the beam. The opticalnoise is observed only coincidently with the current noise.
Krol et al. (1999) studied the overlimiting ion transport through aNeocepta CMX cation and AMX anion exchange membrane. This techniqueis used to characterize the fluctuations in membrane voltage drop observed inthe overlimiting current region. Above the limiting current the measurementsshow large voltage drop fluctuation in time indicating hydrodynamic insta-bility. The amplitude of this fluctuation is increasing with increasing appliedcurrent density. The fluctuation occurs when a set up is used where thereis no forced convection and the depleted diffusion layer is stabilized bygravitation.

Tanaka observed the voltage fluctuation and water dissociation at overlimiting current using an electrodialysis cell integrated with a cation exchangemembrane (Aciplex K-172) and an anion exchange membrane (Aciplex A-172)(Tanaka, 2004). Effective membrane area was adjusted to 0.383 cm2 and a 5mMNaCl solution was supplied into a desalting cell. An electric current was passedthrough Ag–AgCl electrodes, and the voltage fluctuation was measured using avoltmeter. Current efficiencies for water dissociation arising on a cation exchangemembrane ZH and those arising on an anion exchange membrane ZOH werecalculated from the changes of pH and volume of a solution in the concentratingcell. Voltage fluctuation on a cation and anion exchange membrane obtained bysetting current density at 0.131A cm–2 is shown in Fig. 7.12, which indicates thatvoltage fluctuates on the cation exchange membrane but does not fluctuate on theanion exchange membrane. ZH and ZOH were calculated as 0.041 and 0.336,respectively. Accordingly, water dissociation on the cation exchange membranewas confirmed to be strongly suppressed compared to that on the anion exchangemembrane. The mechanism of the fluctuation is discussed in Section 7.8.5 in thischapter.

Li et al. (1983) measured light scattering spectra from polystyrene latexnear an Ionics Inc., #61-CZL-386 cation exchange membrane in 0.020M HCl

Figure 7.12 Voltage fluctuation at overlimiting current density (5mM NaCl, 0.131Acm–2) (Tanaka, 2004).


and 0.020M NaCl solution. For HCl, spectra on the depleted side showedevidence for turbulent flow at and overlimiting current density. Spectra on theconcentrate side showed similar spectra above approximately twice the limitingcurrent density, although of lower intensity. For NaCl, depleted side spectrabegan at currents approximately twice the limiting value. On the concentratedside, at least six to seven times the limiting current density was needed fornonzero difference spectra. From these experimental results, they suggest thatthe electrical noise spectra correspond principally to fluctuations induced byturbulent flow on the depleted side of the membrane.

7.7. OVERLIMITING CURRENT

In many investigations devoted so far to the limiting current density,considerable works are proceeded toward the elucidation of the mechanism ofthe overlimiting current. However, in spite of many discussions on the origin ofthe overlimiting current, a quantitative theory accounting for Region III inFig. 7.1 observed in the current density–voltage curve for the cation exchangemembrane is still lacking. Concentration polarization occurs easily on a cationexchange membrane than on an anion exchange membrane. This is becausetransport number of counter-ions in a solution is larger on a cation exchange


membrane than on an anion exchange membrane. From this, it is estimated thatwater dissociation occurs easily on a cation exchange membrane than on ananion exchange membrane. At overlimiting current density, the deficit of ions ata cation exchange membrane–solution interface is estimated to be compensatedby generation of H+ ions and OH– ions caused by water dissociation in aboundary layer. However, contrary to this estimation, Rosenberg and Tirrell,and Cooke show that the water dissociation is rather difficult to occur on acation exchange membrane than on an anion exchange membrane (Rosenbergand Tirrell, 1957; Cooke, 1961b). Accordingly, water dissociation never preventsthe deficit of ions at the cation exchange membrane–solution interface andaccordingly never contributes to generate ions at overlimiting current on thesurface on the cation exchange membrane. Natural convection, voltage fluc-tuation, concentration fluctuation, optical fluctuation described above areestimated to relate to the overlimiting current for the cation exchange mem-brane. However, the discussion on the mechanism of these phenomena isinsufficient. Here, we try to inspect another aspect concerning the overlimitingcurrent in this section.

7.7.1 Co–Ion Leakage

This idea is that the ionic transport is compensated by passing co-ionsthrough an ion exchange membrane at overlimiting current. However, thepermselectivity of counter-ions for commercially available membranes placedin a dilute salt solution was observed to be maintained at around 90–100%, andco-ion leakage was not arisen (Block et al., 1966).

7.7.2 Electro–Osmotic Convection

Most studies on polarization are based on the Nernst diffusion model forthe boundary layer, and little attention has been paid to the role of electro-osmotic convection. In practice, electro-osmotic convection term is contained inthe flux equations. Frilette suggested that electro-osmotic streaming occur at thesurface of the membrane, the Nernst layer would become thinner due to shortrange turbulence, and the electro-osmotic streaming would support conductionby transporting ions toward the membrane surface (Frilette, 1957). However,experimental and numerical simulation studies show that the electro-osmoticwater transport does not support the overlimiting current (Mazanares et al.,1991).

7.7.3 Space Charge

Rubinstein and Shtilman confirmed that the water dissociation on theSelemion CMV cation exchange membrane at overlimiting currents is verysmall. At the same time the membranes remained highly permselective. Hence,


an electric current many times greater than the limiting one was transferredexclusively by the salt cations. The computation in this study was proceededbased on the following Nernst–Planck–Poisson equation for a 1,1 valent elec-trolyte (Rubinstein and Shtilman, 1979; Rubinstein, 1981).

@p

@t¼ divðgrad pþ p gradcÞ (7.8)

@n

@t¼ divðgrad n� n gradcÞ (7.9)

�Dc ¼ n� p (7.10)

p and n are dimensionless counter-ion concentration and dimensionless co-ionconcentration respectively. e is a square ratio of the bulk Debye radius tothe thickness of the unstirred layer. t is dimensionless time. c is dimensionlesselectric potential.

The computed ion concentration profile in an unstirred layer on acation exchange membrane is illustrated in Fig. 7.13 which is distinguished bythree distinct regions, Region I, II and III. Solid lines and dotted lines show

C+ < CR

−−−−−−−−

++++++++

p

n

x

I

II III

Figure 7.13 Concentration profiles of cations and anions in an unstirred layer formedon the desalting surface of a cation exchange membrane (Rubinstein and Shtilman, 1979,Rubinstein, 1981).


cation concentration and anion concentration respectively. In Region I, localelectro-neutrality is preserved with high accuracy. Region II is a chargedregion developed in the course of concentration polarization. Its dimensionscan exceed by orders of magnitude the equilibrium Debye radius. In thisregion the counter-ion (cation) concentration is much higher than that ofco-ions, which is very low. The electric field at the end of Region II is veryhigh. Ion transport in this region is dominated by migration as opposed todiffusion. Region III is a boundary layer for counter-ions in which their con-centration rapidly approaches its high interface value C̄þ within the cationexchange membrane. Region III is comparable in thickness (though smaller)with the equilibrium Debye radius. If we assume here that the interface con-centration of exchange groups within the cation exchange membrane inFig. 7.13 is C̄R; in order to maintain the electric neutrality in a total system,C̄R4C̄þ must hold and potential difference must be generated between asolution and a membrane.

7.7.4 Hydrodynamic Convection

The Nernst model assumes that there is no hydrodynamic convection, ormixing in the Nernst layer. However, Gavish and Lifson argued that thisassumption cannot be strictly valid and suggested that hydrodynamic convectionaffects the current–voltage relationship in a way which is sufficient to explain theoverlimiting current behavior as follows (Gavish and Lifson, 1979).

Turbulent convection in the depleted layer is the source of the current noise.The convection is attributed to the force of gravity acting on the depleted layer,where concentration gradients create density gradients, resulting in hydrodynamicconvection current, which at high enough current densities becomes turbulent.The turbulent convection injects salt from the bulk of the solution into thedepleted layer, thus reducing its resistance significantly. The current separates theinjected positive and negative ions, thus tending to rebuild the depleted layeragainst its destruction by the turbulent convection. The dynamic balance betweenthese two opposing trends sets up a stationary state in which the depleted layer isonly partially depleted.

The phenomena mentioned above were discussed on the basis of the modelillustrated in Fig. 7.14, consisting of two sublayers. One sublayer, in the range of0oxod1, namely between the membrane surface and d1, is characterized by afixed concentration C̄1: The other layer, like the Nernst diffusion layer, is char-acterized by a fixed concentration gradient, C 0. Thus, the concentration in thetotal depletion layer varies according to

CðxÞ ¼ C̄1; for 0oxod1 (7.11)

CðxÞ ¼ C0 þ C0ðx� dÞ; C0 ¼ðC0 � C̄1Þ

ðd� d1Þ; for d1oxod (7.12)

Figure 7.14 Schematic description of the Nernst model for high current densities. (A)The concentration profile. (B) The specific resistance profile. DR is the net rise ofresistance due to the formation of the depleted and enriched layer. (Gavish and Lifson,1979).oComp: Change figure parts to uppercase ‘A’, ‘B’ in artwork.>


The width of the enriched layer, as well as that of depleted layer, remains d forall values of the current densities, as in the Nernst model. The surface con-centration in the enriched layer continues to grow the same way as in theNernst layer, namely its local concentration varies in the range –doxo0according to

CðxÞ ¼ C0 þ C0ðxþ dÞ; C0 ¼ðC2 � C0Þ

d¼

ðC0 � C̄1Þ

ðd� d1Þ(7.13)

The voltage difference DV created solely by the formation of the Nernst layer isgiven by

DV ¼ DVP þ JDR (7.14)

DVP is the polarization potential established across the membrane.

DVP ¼RT

F

� �2Dtþ ln

C2

C̄1

� �(7.15)

R is the gas constant, T the absolute temperature, Dt+ difference of the trans-port numbers of the permeable cations between in the membrane and in thesolution.

J is electric current density defined by

J ¼ Jþ þ J� ¼�DFC0

Dtþ(7.16)


D is the diffusion constant of salt, F the Faraday constant.

DR is the Ohmic resistance of the cell by an amount

DR ¼

Z d

�drðxÞ � r0

� �dx ¼ L�1

Z d

�d

1

CðxÞ�

1

C0

� �dx (7.17)

where r(x) is the specific resistance at point x in the Nernst layer, and L is theequivalent conductance. They are related by r –1

¼ LC.Using Eqs. (7.11)–(7.17) inclusive, the current–voltage relationship of the

Nernst model may be transformed to dimensionless scale potential E and scaledcurrent density I, defined by

E ¼FDVRT

; I ¼J

JL¼

C0dC0

(7.18)

JL is the limiting current density. The result is shown as follows puttinge ¼ C̄1/C0.

E ¼ 2Dtþ þ ð2DtþÞ�1

� �ln

1þ I

1� I�

I

Dtþ; for Io1� � (7.19)

E ¼ 2Dtþ þ ð2DtþÞ�1

� �ln

1þ I

�þ

I � 1þ �

2tþ��

I

Dtþ,

for I41� � ð7:20Þ

The transition between the two regimes of low and high current densities issmooth, in the sense that Eqs. (7.19) and (7.20) yield the same value of E atI ¼ 1 – e in the common boundary. At this point C1 ¼ C̄1 and d1 ¼ 0.

Gavish and Lifson compared the current–voltage relationship of thepresent model with the corresponding experimental results obtained by Cooke(1961a) in Fig. 7.15.

7.8. MASS TRANSPORT IN A BOUNDARY LAYER (Tanaka, 2004)

Concentration polarization occurs because of the difference betweenthe ion transport number in a solution and in an ion exchange membrane.The main subject to be studied in this field must be to make clear ‘‘themechanism of ionic transport in a depleted boundary layer’’. Concentrationdecrease in the depleted layer gives rise to the natural convection owing to thedecrease in solution density as suggested by Gavish (Gavish and Lifson,1979) (cf. Section 7.7.4). Accordingly, the mass transport in this region isanalyzed based on the extended Nernst–Planck equation including convec-tion term. In this section, we discuss the phenomena in the boundary layerformed on the desalting surface of an ion exchange membrane placed in

0 0.5

1 2 340

30

20

E

I

10

0

40

30

20

slope = 1/ε

1-ε

10

00 0.5 1.0 1.5

1.0 1.5

Figure 7.15 Voltage plotted against current. (1) Nernst model; (2) Modified Nernstmodel (Eqs. (7.19) and (7.20)) with e ¼ 1/175; (3) Experimental curve according Cooke.(Note that all three curves coincide for 1o1–e) (Gavish and Lifson, 1979)


a NaCl solution as illustrated in Fig. 7.16, based on the investigationproceeded by Tanaka (2004).

7.8.1 The Equation of Material Balance, the Extended Nernst–Planck Equation,

the Reduced Diffusion Coefficient, the Reduced Transport-Diffusion Coefficient

Mass transport with natural convention in a boundary layer near thesurface of a vertical membrane is a three-dimensional process shown inFig. 7.17. The theory of three-dimensional free convection in a liquid is under-standable from the equation of continuity and the equation of motion asexpressed below.

Equation of continuity:

@

@xrvx þ

@

@yrvy þ

@

@zrvz ¼ 0 (7.21)

Cation exchange membrane

Anion exchangemembrane

C' C'

JNa<0

JCl>0

i<0

− +VK<0

d /dx>0

dC/dx>0

Co

JNa>0

JCl<0

i>0

VA<0

d /dx<0

dC/dx>0

Co

x x

Boundary layer Boundary layerBulk

Desalting cell

Figure 7.16 Boundary layer formed on a desalting surface of an ion exchange mem-brane.


where r is the solution density. vx, vy and vz are x-, y- and z-components ofsolution velocity.

Equation of motion:

vx@vx@x

þ vy@vx@y

þ vz@vx@z

¼ 0 (7.22)

vx@vy@x

þ vy@vy@y

þ vz@vy@z

¼ gr� r0

r(7.23)

vx@vz@x

þ vy@vz@y

þ vz@vz@z

¼ 0 (7.24)

where r0 is the solution density in bulk and g the gravitational acceleration.Here, we separate the flux of ions J in a boundary layer into x-component

Jx, a y-component Jy and a z-component Jz, as shown in Fig. 7.17, and expressthe material balance of three components:

Jx dj þ Jy;in x�dj þ Jz;in x�dj ¼ Jx xj þ Jy;out x�dj þ Jz;out x�dj (7.25)

Jx xj þ Jy;in 0�xj þ Jz;in 0�xj ¼ Jx 0j þ Jy;out 0�xj þ Jz;out 0�xj (7.26)

y

Jy,out | 0-x Jy,out | x-��

l

C C'

Jz,out | 0-x Jz,out | x-�

Jx,m Jx | 0 Jx | x Jx | �

Jz Jz,in | x-�

Jz,in | 0-x Jy

Jx

z

Co

b

x

x �Jy,in | 0-x Jy,in | x-�

Ion exchange Boundary layer Bulk

membrane

0

Figure 7.17 Ion flux in a boundary layer formed on a desalting surface of an ionexchange membrane.


From the material balance of ionic fluxes in the x-component:

Jx dj ¼ Jx xj ¼ Jx 0j ¼ Jx mj (7.27)

Each term in Eq. (7.27) concerns with an electric current, and Eq. (7.27)represents indirectly electric current balance in Fig. 7.17. Jx9m is ionic flux acrossan ion exchange membrane, which corresponds to an electric current acrossan ion exchange membrane. The material balance in the y-component and thez-component is expressed by the following equations:

Jy;in x�dj ¼ Jy;out x�dj (7.28)

Jy;in 0�xj ¼ Jy;out 0�xj (7.29)

0.0 0.1 0.2 0.3 0.4 0.5-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(eq/

A •

cm

2 )(1

05 s/c

m2 )

C(10-3mol/cm3)

A

k

Figure 7.18 NaCl concentration vs. reduced diffusion coefficient and reduced transport-diffusion coefficient.


Jz;in x�dj ¼ Jz;out x�dj (7.30)

Jz;in 0�xj ¼ Jz;out 0�xj (7.31)

Equations (7.28)–(7.31) do not concern them with an electric current and areexcluded from the electric current balance. Eqs. (7.27)–(7.31) are one-dimensionalexpressions of three-dimensional mass transport phenomena in the boundarylayer (Bird et al., 1960). They are equivalent to the equation of continuity(Eq. (7.21)) and the equation of motion (Eqs. (7.22)–(7.24)) and can be termed theequation of material balance.

Based on Eq. (7.27), the transport of ion i at x in the boundary layer isexpressed by Ji,x using the extended Nernst–Planck equation including thediffusion, migration and convection as follows:

Ji;x ¼ �Di

dCi

dx�

FDiziCi

RT

dcdx

þ Civx (7.32)

where Ci is the concentration of ions i, c the electric potential, Di the diffusionconstant of ions i, zi the ionic charge number of ions i, F the Faraday constant,R the gas constant and T the absolute temperature. In a NaCl solutionEq. (7.32) is expressed by Eq. (7.33).

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

�'(1

05 s/c

m2 )

�'

(eq/

A •

cm

2 )

C(10-3mol/cm3)

�'K

�'A

�'

Figure 7.19 NaCl concentration vs. reduced diffusion coefficient and reduced transport-diffusion coefficient.


JNa ¼ �DNadCNa

dx�

FDNaCNa

RT

dcdx

þ CNav

JCl ¼ �DCldCCl

dxþ

FDClCCl

RT

dcdx

þ CClv(7.33)

In the electrodialysis of a NaCl solution, the electrical neutrality expressed byEq. (7.34) is assumed to be satisfied in the boundary layer.

CNa ¼ CCl ¼ C (7.34)

The material balance of Na+ ions and Cl– ions in an ion exchange membraneand in a boundary layer is expressed by Eq. (7.35) introduced from Eq. (7.27).

JNa ¼ J̄Na ¼i

F

� �t̄Na

JCl ¼ J̄Cl ¼ �i

F

� �t̄Cl

(7.35)

where t̄Na and t̄Cl are the transport number of Na+ ions and Cl– ions in an ionexchange membrane.

0.00 0.01 0.02 0.03 0.04 0.05-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Jdiff

, Jm

igr,

Jco

nv(1

0-7m

ol/c

m2

• s)

x(cm)

JCl=JCl,diff+JCl,migr+JCl,conv

JNa=JNa,diff+JNa,migr+JNa,conv

JNa,conv=JCl,conv

JCl,migr

JNa,migrJ Cl,d

iff

J Na,diff

Figure 7.20 Diffusion flux, migration flux and convection flux of Na+ ions and Cl– ionsin a boundary layer (cation exchange membrane).


Using Eqs. (7.33)–(7.35), we obtain Eqs. (7.36) and (7.37).Equation of concentration gradient:

dC

dx¼ avC � bi (7.36)

Equation of potential gradient:

dcdx

¼RT

F

� �a0v� b0

i

C

� �(7.37)

where

a ¼1

2

1

DNaþ

1

DCl

� �(7.38)

b ¼1

2F

t̄Na

DNa�

t̄Cl

DCl

� �(7.39)

a0 ¼1

2

1

DNa�

1

DCl

� �(7.40)

0.00 0.01 0.02 0.03 0.04 0.05-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

J diff

, Jm

igr,

J con

v(10

-7m

ol/c

m2

• s)

x(cm)

JCl = JCl,diff + JCl,migr + JCl,conv

JNa=JNa,diff+JNa,migr+JNa,conv

JNa,conv = JCl,conv

JCl,migr

JNa,migr

J Cl,diff

J Na,diff

Figure 7.21 Diffusion flux, migration flux and convection flux of Na+ ions and Cl– ionsin a boundary layer (anion exchange membrane).


b0 ¼1

2F

t̄Na

DNaþ

t̄Cl

DCl

� �(7.41)

a and a0 are termed the reduced diffusion coefficient of NaCl, because they areanother expression of DNa and DCl. b and b0 are termed the reduced transport-diffusion coefficient, because they relate t̄Na; t̄Cl; DNa and DCl. C vs. a, bK andbA, and C vs. a0, b0K and b0A in a NaCl solution electrodialysis system arepresented in Figs. 7.18 and 7.19 respectively. These parameters facilitate thecomputation in this chapter.

7.8.2 Ionic Flux in a Boundary Layer

Substituting the equation of concentration gradient Eq. (7.36) and theequation of potential gradient Eq. (7.37) into the extended Nernst–Planckequation Eq. (7.33), the flux of Na+ ions JNa and Cl– ions JCl are divided intothe terms of diffusion Jdiff, electro-migration Jmigr and convection Jconv:

JNa ¼ JNa;diff þ JNa;migr þ JNa;conv

JCl ¼ JCl;diff þ JCl;migr þ JCl;conv(7.42)

0.00 0.01 0.02 0.03 0.04 0.05-6

-5

-4

-3

-2

-1

0

1

2

i diff

, i m

igr(

A/d

m2 )

x(cm)

4.30(overLCD)

2.93(LCD)

2.10

1.13A/dm2

i=idiff+imigr

idiff

imigr

i=idiff+imigr

imigr

i=idiff+imigr

i=idiff+imigr

imigr

imigr

Figure 7.22 Diffusion current density and migration current density in a boundary layer(cation exchange membrane).


Here, on a cation exchange membrane:

JNa;diff ¼ �DNaðavC � bKiÞ

JCl;diff ¼ �DClðavC � bKiÞ(7.43)

JNa;migr ¼ �DNaða0vC � b0KiÞ

JCl;migr ¼ DClða0vC � b0KiÞ(7.44)

On an anion exchange membrane:

JNa;diff ¼ �DNaðavC � bAiÞ

JCl;diff ¼ �DClðavC � bAiÞ(7.45)

JNa;migr ¼ �DNaða0vC � b0AiÞ

JCl;migr ¼ DClða0vC � b0AiÞ(7.46)

and on a cation exchange membrane and an anion exchange membrane:

JNa;conv ¼ JCl;conv (7.47)

0.00 0.01 0.02 0.03 0.04 0.050

1

2

3

4

5

i diff, i

mig

r(A

/dm

2 )

x(cm)

3.99(LCD)3.75

2.27

1.19A/dm2

idiff

imigr

i migr

i migr

imigr

i=idiff + imigr

i=idiff+ imigr

i=idiff + imigr

i=idiff + imigr

Figure 7.23 Diffusion current density and migration current density in a boundary layer(anion exchange membrane).


Applying Eqs. (7.38)–(7.41) and Eqs. (7.43)–(7.47) to the concentration distri-bution in a boundary layer observed by means of the Schlieren-diagonal method(Figs. 7.8 and 7.9), ionic fluxes in a boundary layer at the limiting current densityare calculated. The ionic fluxes on the cation exchange membrane placed in a0.1M NaCl solution are shown in Fig. 7.20, in which the absolute values of Jconvis recognized to be decreased, and the absolute values of Jdiff and Jmigr areincreased. It is estimated that Jconv which does not carry an electric currentconverts to Jdiff and Jmigr, which carry an electric current instead of Jconv in theboundary layer.

Gavish expressed the phenomenon mentioned above as ‘‘The turbulentconvection injects salt from the bulk of the solution into the depleted layer, andthe current separates the injected positive and negative ions, thus tending torebuild the depleted layer against its destruction by the turbulent convection’’(Gavish and Lifson, 1979) (cf. Section 7.7.4).

Fig. 7.21 shows the ionic fluxes on the anion exchange membrane, indicatingthat the absolute values of Jconv and Jmigr are decreased and Jdiff is increased.In this situation, Jconv converts to Jdiff but does not convert to Jmigr.

0 2 4 60

2

4

6

8

10

12

14

16

18

-vx(

10-3

cm/s

)

x(10-2cm)

2.28(LCD)4.30(overLCD)

2.93(LCD)2.101.13A/dm2

1 3 5

Figure 7.24 Solution velocity in a boundary layer (cation exchange membrane).


7.8.3 Current Density in a Boundary Layer

Current density i is divided into the terms of diffusion current idiff,migration current imigr and convection current iconv:

i ¼ idiff þ imigr þ iconv (7.48)

idiff ¼ iNa;diff þ iCl;diff ¼ F ðJNa;diff � JCl;diff Þ (7.49)

imigr ¼ iNa;migr þ iCl;migr ¼ F ðJNa;migr � JCl;migrÞ (7.50)

iconv ¼ F ðJNa;conv � JCl;convÞ ¼ 0 (7.51)

Substituting Eqs. (7.43)–(7.46) into Eqs. (7.48)–(7.50), idiff and imigr are calcu-lated for the Schlieren-diagonal experiment. The result for the cation exchangemembrane placed in a 0.1M NaCl solution is shown in Fig. 7.22. The sign of idiff(>0) on the cation exchange membrane is in the opposite to that of i (o0).Namely, idiff decreases i, and this is because 9JNa,diff9o9JCl,diff9 holds inFig. 7.20. The absolute values of imigr on the cation exchange membrane areincreased due to the conversion of Jconv to imigr and support an electric current.The result for the anion exchange membrane is shown in Fig. 7.23.

0 43210

1

2

3

-vx(

10-3

cm/s

)

x(10−2cm)

1.81(LCD)

3.99(LCD)3.75

2.27A/dm2

1.19

5

Figure 7.25 Solution velocity in a boundary layer (anion exchange membrane).

0

0

vy,out

v+dxx

vy,in

−vx+dx−vx−vx,0

y

l

z

b

x

Figure 7.26 Convection of a solution in a boundary layer formed on an ion exchangemembrane.


0 2 4 51 3 6-2

0

2

4

6

8

10

12

-dv x

/dx

= (v

y,ou

t -

v y,in

)/l (

s-1)

x (10-2cm)

2.28 A/dm2(LCD)0.05 M NaCl

4.30 A/dm2 (overLCD) 2.93 A/dm2(LCD) 2.10 A/dm2 1.13A/dm2

0.1 M NaCl

Figure 7.27 Solution velocity gradient in a boundary layer (cation exchange mem-brane).


7.8.4 Solution Velocity in a Boundary Layer

(1)
x-component of convection velocity vx
The x-component of solution velocity vx is introduced from the equation
of concentration gradient Eq. (7.36) as follows:
vx ¼1

aCdC

dxþ bi

� �(7.52)

The NaCl concentration distribution observed by the Schlieren-diagonalmethod (Figs. 7.8 and 7.9), a and b are substituted into Eq. (7.52) and nx iscalculated (Figs. 7.24 and 7.25), showing vx on the cation exchange membraneto be larger than that on the anion exchange membrane. vx at x ¼ 0, vx,x ¼ 0

corresponds to the sum of electro-osmosis velosmo and concentration-osmosisvconosmo passing through an ion exchange membrane:

vx;x¼0 ¼ velosmo þ vconosmo (7.53)

0 1 2 3 4 5 6-1

0

1

2

3

4

5

6

-dv x

/dx

= (v

y,ou

t-v y

,in)/

l (s-1

)

x (10-2 cm)

2.28 A/dm2(LCD)0.05 M NaCl

3.99 A/dm2(LCD) 3.75 A/dm2 2.27 A/dm2 1.19 A/dm2

0.1 M NaCl

Figure 7.28 Solution velocity gradient in a boundary layer (anion exchange membrane).


(2)
y-component of convection velocity vy
In Fig. 7.26, the convection velocity on y-axis in the range of x ¼ x to
x+dx on x-axis is assumed to be vy,in at the bottom and vy,out at the top. Takingaccount of the material balance, we have
�blvxþdx þ bvy;indx ¼ �blvx þ bvy;outdx (7.54)

from Eq. (7.54)

�dvx

dx¼

vy;out � vy;in

l(7.55)

vx in Figs. 7.24 and 7.25 are differentiated and substituted into Eq. (7.55) toobtain (vy,out – vy,in)/l, which is shown in Fig. 7.27 (cation exchange membrane)and Fig. 7.28 (anion exchange membrane).

Inspecting these figures, –dvx/dx ¼ (vy,out – vy,in)/l ¼ 0, that is vy,out ¼ vy,inis realized in almost whole region in the boundary layer. On the other hand,vy,out>vy,in (–dVx/dx>0) is seen near x ¼ 0 particularly at high current density.This phenomenon is attributed to that vx,0 is so little than vx that the velocity

Boundary layer Bulk

v

vx

vy,out

Ion-exchangemembrane

vy,inv

vx

Figure 7.29 Convection velocity near an ion exchange membrane in a boundary layer(illustration).


excepting vx,0 is rejected to pass through the membrane and converted to vy,out.Convection velocity in this situation is illustrated in Fig. 7.29.

When limiting current density is applied, vy,outovy,in (–dVx/dxo0) isrecognized as shown in Figs. 7.27 and 7.28 at the point distant from themembrane to some extent. Convection velocity in this situation is illustrated inFig. 7.30.

7.8.5 Fluctuation Phenomena in a Boundary Layer

When the limiting current density is applied across the cation exchangemembrane placed in a 0.05M NaCl solution, the NaCl concentration fluctuateson the membrane surface (Fig. 7.8). When the overlimiting current is appliedacross the cation exchange membrane placed in a 5mM NaCl solution, thevoltage fluctuates (Fig. 7.12). These phenomena are supposedly due to the eventsrepresented in Fig. 7.31 consisting of the following steps:

(1)
Concentration velocity v near the membrane indicated in Fig. 7.30 isdecreased. This event generates the optical noise (fluctuation).
(2)
Ionic flux by the convection Jconv ¼ Cv is decreased.

v vy,out

vx

C

C

C0

vvy,in

C0 xvx

Cation exchangemembrane

Boundary layer Bulk

Figure 7.30 Convection velocity and NaCl concentration fluctuation in a boundarylayer formed on a cation exchange membrane at limiting current density (illustration).


(3)
NaCl concentration near the membrane C is decreased. At the sametime, dC/dx is increased. These events generate the concentration noiseand voltage noise.
(4)
Jconv is decreased further. (5) C is decreased further. (6) Jmigr is decreased and Jdiff ( ¼ JNa,diff+JCl,diff) is increased. (7) imigr is decreased and idiff ( ¼ iNa,diff+iCl,diff) is decreased. Accordingly,
i ( ¼ imigr+idiff) is decreased.1

(8)
v is increased for maintaining the material balance and electric currentbalance. This event generates the optical noise.
(9)
Jconv is increased. (10) C is increased and dC/dx is decreased. These events generate the
concentration fluctuation and voltage fluctuation.
(11) Jconv is increased further. (12) C is increased further. (13) i is increased. (14) v is decreased for maintaining the material balance and electric current
balance.

vdecrease

i Jconv =Cvincrease decrease

Optical noise

Jdiff, Jmigr Cincrease decrease

Concentration noiseVoltage noise

C Jdiff, Jmigrdecreaseincrease

Optical noise

Jconv=Cv iincrease decrease

vincrease

Figure 7.31 Mechanism of NaCl concentration fluctuation in a boundary layer formedon a cation exchange membrane at limiting current density.


The process mentioned above is repeated 10 times per second on a cationexchange membrane placed in a NaCl solution, and do not get to steady state(cf. Section 7.4).

Experimental and theoretical studies on fluctuation presented so far (Lifsonet al., 1978; Li et al., 1983; Krol et al., 1999; Tanaka, 2004) are understandablebased on Fig. 7.31.

7.8.6 Potential Gradient in a Boundary Layer

Potential gradient dc/dx is divided into the terms of Ohmic potential (dc/dx)ohm and diffusion potential (dc/dx)diff:

dcdx

¼dcdx

� �ohm

þdcdx

� �diff

(7.56)

dcdx

� �ohm

¼1

k(7.57)

dcdx

� �diff

¼ �RT

F

� �ð2tNa � 1Þ

1

C

� �dC

dx

� �(7.58)

0.00 0.01 0.02 0.03 0.04 0.050

5

10

15

20

25

30

35(d

�/d

x)di

ff, (

d�/d

x)oh

m(V

/cm

)

x (cm)

Cation exchange membrane 0.1 M NaCl

1.13 A/dm2; 2.10 A/dm2; 2.93 A/dm2 (LCD); 4.30

A/dm2 (over LCD), open: ohmic gradient; filled: diffusion potential gradient.

Figure 7.32 Ohmic potential gradient and diffusion potential gradient in a boundarylayer (cation exchange membrane).


where k is the specific electric resistance of a NaCl solution in a boundary layer.Equation (7.58) is the Henderson equation. In this electrodialysis system, theeffect of the Donnan potential term is negligible. Substituting the Schlieren-diagonal observation (Figs. 7.8 and 7.9) into Eqs. (7.56)–(7.58), (dc/dx)ohm and(dc/dx)diff are computed and shown in Fig. 7.32 (cation exchange membrane)and Fig. 7.33 (anion exchange membrane). (dc/dx)ohm on the cation exchangemembrane is positive, and that on the anion exchange membrane is negative.(dc/dx)diff is positive both on the cation exchange membrane and on theanion exchange membrane. Integrating dc/dx vs. x plots, current density i vs.voltage drop DV in the boundary layer is calculated and shown in Fig. 7.34, inwhich the overlimiting current phenomenon generated in Region III in Fig. 7.1is found. Note here that the overlimiting current is generated more easily (atlower voltage) on the cation exchange membrane than on the anion exchangemembrane.

0.00 0.01 0.02 0.03 0.04 0.05-50

-40

-30

-20

-10

0

10

(d�

/dx)

ohm

, (d�

/dx)

diff (

V/c

m)

x (cm)

Anion exchange membrane 0.1 M NaCl

1.19 A/dm2; 2.27 A/dm2; 3.75 A/dm2; 3.99 A/dm2

(LCD), open: ohmic potential gradient; filled: diffusion potential gradient.

Figure 7.33 Ohmic potential gradient and diffusion potential gradient in a boundarylayer (anion exchange membrane).


7.9. CONCENTRATION POLARIZATION ON A CONCENTRATING

SURFACE OF AN ION EXCHANGE MEMBRANE

Concentration polarization is usually discussed on the desalting surface ofan ion exchange membrane as described in the preceding sections. In this section,however, we treat the phenomenon on the concentrating surface of the membrane.

In order to observe the concentration distribution in a boundary layerformed on the concentrating surface of an ion exchange membrane, a three-compartment optical glass cell was set up with a central concentrating cell,desalting (electrode) cells put on both outsides of the central cell and ionexchange membranes (Aciplex CK-2/CA-2) (Takemoto, 1969). Effective mem-brane area was adjusted to 1.5 cm2 (1.5 cm height, 1 cm width). Distance betweenthe membranes in the central concentrating cell was maintained at 2.6–2.8mm.At 25 1C, 0.5M NaCl solutions were put in the desalting cells and concentratingcell, and an electric current density of 2A dm–2 was applied through Ag–AgClelectrodes. The changes of NaCl concentration distribution in the boundary

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

i (A

/dm

2 )

�V (V)

LCD

: Cation exchange membrane : Anion exchange membrane0.1 M NaCl

Figure 7.34 Current density vs. voltage drop in a boundary layer.


layer formed on the concentrating surface of the membrane were observed bythe Schlieren-diagonal method (cf. Section 7.4).

The concentration changes in the boundary layer observed in this exper-iment are understandable on the basis of Fig. 7.35. In this experiment, ionicconcentration in the cation exchange membrane is assumed to be larger thanthat in the anion exchange membrane. NaCl concentration in the concentratingcell is 0.5M before electric current passing as shown by 0 in the figure. Applyingan electric current, the concentrated counter-ions in the membrane flows outinto the concentrating cell and flows down with concentrated co-ions along themembrane surface due to the electro-gravitational movement (Frilette, 1957).Successively, the concentration distribution is changed as 0-1-2-3, and fi-nally attains to 4 in a steady state. Fig. 7.36 shows the concentration profilecorresponding to 4 in the steady state observed by Takemoto (1969), indicatingthat the NaCl concentration on the cation exchange membrane becomes largerthan that on the anion exchange membrane. From the experiment describedabove, the mechanism of the concentration polarization occurring on the con-centrating surface of the membrane is assumed to be not the same in principle tothat occurring on the desalting surface of the membrane.

C (

M)

x (mm)

Figure 7.36 NaCl concentration distribution on a concentrating surface of a cationand an anion exchange membrane in a steady state due to concentration polarization(Takemoto, 1969).

C" 0K

C" 0A

4 C"

3

2

electro-gravitation

10

Cation Concentrating cell Anionexchangemembrane

exchangemembrane

0: before current passing, 1: just after current passing,2,3: at a transitional state, 4: at a steady state

Figure 7.35 Electrolyte concentration changes in boundary layers formed on the con-centrating surface of ion exchange membranes.



NOTE

1. On the cation exchange membrane i (o0) ¼ idiff (>0)+imigr (o0) ¼ F(JNa,diff(o0)– JCl,diff(o0))+F(JNa,migr (o0) – JCl,migr (>0)) holds, and we have idiff > 0 because of9JNa,diff9o9JCl,diff9. So,9i9is decreased owing to the increase of 9Jdiff9. On the other hand,on an anion exchange membrane, i (>0) ¼ idiff (>0)+imigr (>0) ¼ F(JNa,diff (o0) –JCl,diff (o0))+F(JNa,migr (>0) – JCl,migr (o0)) holds, and we have idiff > 0 because of9JCl,diff 9>9JNa,diff9. So 9i9 is increased owing to the increase of 9Jdiff9. Accordingly, on theanion exchange membrane, the process represented in Fig. 7.31 is interrupted at step (4)and (5) and is never possible to continue (cf. Eqs. (7.48)–(7.50), Fig. (7.16) and Figs.(7.20)–(7.23)).

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