Limiting Current Density

Chapter 11

11.1. CONCENTRATION POLARIZATION, WATER DISSOCIATION

AND LIMITING CURRENT DENSITY

When an electric current is passed through an ion exchange membrane, saltconcentration on a desalting surface of the membrane is decreased due to con-centration polarization, and reduced to zero at the limiting current density. In thiscircumstance, there are no more salt ions available to carry the electric current.Thus, the voltage drop across the boundary layer increases drastically resulting ina higher energy consumption and generation of water dissociation. As describedabove, the limiting current density is closely related with the concentrationpolarization and water dissociation, and these phenomena were already discussedin Chapter 7 (Concentration polarization) and Chapter 8 (Water dissociation).In this chapter we discuss the mechanism of the limiting current density.

11.2. DIFFUSION LAYER AND BOUNDARY LAYER

Tobias et al. (1952) discussed an electrode reaction between an electrodeand a solution surrounding it, and suggested that in absence of fluid turbulence,ions are transported from a solution to an electrode by three principal mech-anisms: (a) migration, (b) diffusion and (c) convection. They illustrated theconcentration profile in the vicinity of an electrode during electrolysis showingthe curved line as depicted in Fig. 11.1, in which the boundary layer (thickness d)is formed on the electrode. Diffusion layer (thickness d0) is defined as the layercreated from an intersection between the concentration line C0 in bulk and theconcentration gradient line (dC/dx)x ¼ 0 at an electrode/solution interface. Thistheory is perfectly applicable to the phenomenon occurring on the surface of anion exchange membrane. Referring to Fig. 11.1, the material balance of counter-ions in the membrane phase and at the membrane/solution interface (x ¼ 0) isrepresented by the following equation.

i

Ft ¼

i

FtþD

dC

dx

� �x¼0

(11.1)

dC

dx

� �x¼0

¼C0 � C0

0

d0(11.2)

t and t are transport numbers for counter-ions in the membrane and in thesolution, respectively. C0 and C0

0 are the ionic concentrations in bulk (desalting

DOI: 10.1016/S0927-5193(07)12011-8

Ele

ctro

lyte

con

cent

ratio

n

Diffusion layer

Boundary layer

C'0

C'C'

(dC / dx) x =0

Distance from membrane

Figure 11.1 Concentration profile in the boundary layer formed on the desalting surfaceof an ion exchange membrane (Tobias et al., 1952).

Ion Exchange Membranes: Fundamentals and Applications246

cell) and at x ¼ 0 (solution/membrane interface), respectively. D is the diffusionconstant of an electrolyte dissolving in this system.

In the absence of fluid turbulence, the transport of ions i, Ji, in the boundarylayer is generally described by the following extended Nernst–Planck equation.

Ji ¼ �Di

dCi

dx�

FDiziCi

RT

dcdx

þ Civ (11.3)

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, v the solution velocity dueto the natural convection, F the Faraday constant, R the gas constant and T is theabsolute temperature. In this situation, the greater part of the ionic transport isdue to the natural convection v, which is a horizontal component of ascendingflow produced by the decrease in the solution density in the vicinity of the mem-brane surface in the boundary layer (Tanaka, 2004).

Under the forced flowing circumstance in a desalting cell, the naturalconvection is suppressed and the solution velocity v in Eq. (11.3) is caused onlyby the electro-osmosis and concentration-osmosis passing through the mem-brane. In this situation, the convection term in Eq. (11.3) substantially disap-pears and the concentration distribution profile in the boundary layer isapproximated by a straight line (Tanaka, 2003). This event means that the

Limiting Current Density 247

diffusion layer thickness d0 in Fig. 11.1 becomes to be equivalent to boundarylayer thickness d; d ¼ d0, and the limiting current density equation of the mem-brane ilim is expressed by Eq. (11.4) introduced by putting C0

0 ¼ 0 in Eqs. (11.1)and (11.2). Equation (11.4) is already suggested by Peer (1956) and Rosenbergand Tirrell (1957).

ilim ¼FD

t� t

dC

dx

� �x¼0

¼FDC0

ðt� tÞd0(11.4)

11.3. LIMITING CURRENT DENSITY EQUATION INTRODUCED

FROM THE NERNST–PLANCK EQUATION

Spiegler (1971) introduced the limiting current density equation in anelectrolyte solution dissolving monovalent cations A and monovalent anions Xfrom the Nernst–Planck equation as follows.

Jþ ¼ �Dþ

dC

dx

� �þ

FC

RT

� �dcdx

� �� �(11.5)

J� ¼ �D�

dC

dx

� ��

FC

RT

� �dcdx

� �� �(11.6)

In the membrane, the numerical value of the flux ratio, J+/J�, is equal to thetransport number ratio tþ=t�:

Jþ

J�

¼ �tþ

t�¼ �

tþ

1� tþ(11.7)

Here, in the membrane, J+ and J� have opposite signs, while tþ and t� arealways taken positive.

From Eqs. (11.5)–(11.7):

Jþ ¼�2Dþ

1�Dþ t�

D� tþ

dC

dx(11.8)

J� ¼�D�

1�D� tþ

Dþ t�

dC

dx(11.9)

It is often more convenient to express the fluxes in terms of the diffusioncoefficient of the electrolyte, D, rather than the ionic diffusion coefficients, D+

and D�. Noting that in a free solution

Dþ

D�

¼tþ

t�¼

tþ

1� tþ(11.10)

Ion Exchange Membranes: Fundamentals and Applications248

Using the Nernst expression for the diffusion coefficient of 1-1 electrolyte in adilute solution

D ¼2DþD�

Dþ þD�

¼ 2Dþð1� tþÞ (11.11)

Taking account of Eqs. (11.10) and (11.11), we rewrite Eqs. (11.8) and (11.9) asfollows:

Jþ ¼ �Dtþ

ðtþ � tþÞ

dC

dx(11.12)

J� ¼ �Dt�

ðt� � t�Þ

dC

dx(11.13)

We can calculate the electric current density, i, from the Faraday law:

i ¼ F ðJþ � J�Þ (11.14)

Substituting Eqs. (11.12) and (11.13) into Eq. (11.4) we obtain

For a cation exchange membrane:

i ¼ �FD

ðtþ � tþÞ

dC

dx(11.15)

For an anion exchange membrane:

i ¼FD

ðt� � t�Þ

dC

dx(11.16)

Since i, D and the transport numbers are constants, the concentration gradientdC/dx in Eqs. (11.15) and (11.16) is linear. The limiting current density ilimpresented by Eq. (11.4) is introduced by substituting dC/dx ¼ C0/d0 into Eqs.(11.15) and (11.16).

Yamabe et al. measured the limiting current density ilim from an inflectionof an electric current vs. electric potential curve under a flowing circumstancein a desalting cell (Yamabe et al., 1967). The diffusion layer thickness d0

was calculated by substituting ilim into Eq. (11.4). Results are presented inTable 11.1.

11.4. DEPENDENCE OF LIMITING CURRENT DENSITY ON

ELECTROLYTE CONCENTRATION AND SOLUTION VELOCITY OF A

SOLUTION

A drawback of Eq. (11.4) is that definite diffusion layer thickness d0 is notdetermined before hand, and further that the phenomenological meaning of d0 isobscure to some extent as defined in Eq. (11.2). In order to determine the

Table 11.1 Diffusion layer thickness

NaClConcentration (M)

SolutionVelocity (cm s�1)

Diffusion Layer Thickness (10�2 cm)

Selemion CMG-10 Selemion AMG-10

0.1 0.026 2.60 2.460.226 2.15 2.290.857 1.05 1.27

0.3 0.025 1.98 2.500.260 1.64 2.650.720 1.25 1.85

0.5 0.024 1.86 1.840.250 1.44 1.58

0.75 0.027 2.010.226 1.680.785 0.98

Source: Yamabe et al., 1967.

Limiting Current Density 249

limiting current density of an ion exchange membrane, it is rather practicallyreasonable to measure directly the limiting current density and evaluate theeffects of electrodialysis conditions on the limiting current density. An exampleof such an experiment (Tanaka, 2005) is introduced below.

The experimental apparatus as shown in Fig. 11.2a was assembled incor-porated with cation exchange membranes (Aciplex K-172, Asahi Chemical Co.)and anion exchange membrane (Aciplex A-172). The thickness of cell D wasadjusted to 0.075 cm. Width and length of the flow pass in cell D were adjustedto 1 and 2 cm, respectively (Fig. 11.2b). A diagonal net spacer was put into cellD. A 251C NaCl solution was supplied to cell D. Passing an electric current andchanging the current density incrementally, the limiting current density ilim of acation exchange membrane (K� in the figure) was measured from the inflectionof V/I vs. 1/I plot. The experiment was repeated by changing NaCl concentra-tion C and solution velocity u in D step by step. ilim of an anion exchangemembrane was measured in the same manner. ilim is plotted against C with u as aparameter, and shown in Fig. 11.3 (cation exchange membrane, ilim,K) andFig. 11.4 (anion exchange membrane, ilim,A). The plots except for u ¼ 0 areexpressed by the following equation:

ilim ¼ mCn (11.17)

m and n are expressed by the following functions of u (Fig. 11.5).

For cation exchange membrane Aciplex K-172:

m ¼ 83:50þ 24:00u

n ¼ 0:7846þ 8:612� 10�3u(11.18)

(a) Cell arrangement

G

A K K* A K

E D' D C D' E

NaCl soln.

(b) D

Figure 11.2 Apparatus for measuring limiting current density of an ion exchangemembrane (Tanaka, 2005).

Ion Exchange Membranes: Fundamentals and Applications250

For anion exchange membrane Aciplex A-172:

m ¼ 66:36þ 14:72u

n ¼ 0:7404þ 3:585� 10�3u(11.19)

11.5. LIMITING CURRENT DENSITY ANALYSIS BASED ON THE

MASS TRANSPORT IN A DESALTING CELL

Concentration polarization is a phenomenon occurring in a boundarylayer formed on the desalting surface of an ion exchange membrane. Thelimiting current density is influenced by the solution flow in a desalting cell andionic transport in the boundary layer. This phenomenon is widely investigatedby means of chemical engineering techniques, which are exemplified in thissection.

-3

-2

-1

0

10-6 10-5 10-4 10-310-3

10-2

10-1

1

C (mol/cm3)

i lim

,K (

A/c

m2 )

Figure 11.3 C vs. ilim,K plot (Aciplex K-172) (Tanaka, 2005).

Limiting Current Density 251

11.5.1 Analysis Based on the Chilton–Coburn Transfer Factor

Rosenberg and Tirrell (1957) applied the following Chilton–Coburntransfer factor jD for analyzing the boundary layer thickness d:

jD ¼k

u

� �mD

� �2=3

(11.20)

where k is the mass transfer coefficient, u the linear velocity in the desalting cell,m the viscosity and D the diffusion coefficient of the salt.

d is expressed as follows:

d ¼D

k(11.21)

jD in a flat duct is presented as follows:

For streamline flow (Reynolds number Reo2100)

jD ¼ 1:85 Re�2=3 l

de

� ��1=3

(11.22)

where l is path length over which the flow is developed, de the equivalentdiameter of the path.

-3

-2

-1

0

10-6 10-5 10-4 10-310-3

10-2

10-1

1

C (mol/cm3)

i lim

,A (

A/c

m2 )

Figure 11.4 C vs. ilim,A plot (Aciplex A-172) (Tanaka, 2005).

Ion Exchange Membranes: Fundamentals and Applications252

Using Eqs. (11.20)–(11.22) and taking account of Re ¼ deur/m(r ¼ 1), boundarylayer thickness d ( ¼ diffusion layer thickness d0) is introduced as:

d ¼1

1:85

Ddel

u

� �1=3

(11.23)

Substituting Eq. (11.23) into Eq. (11.4), ilim/C is obtained as

ilim

C¼ 1:85

FD

t� t

� �u

Ddel

� �1=3

(11.24)

For turbulent flow (Re>2100):

jD ¼ 0:023 Re�0:2 (11.25)

d and ilim/C are computed in a similar way as for the streamline flow:

d ¼1

0:023

� �D1=3m2=3

u

� �Re0:2 (11.26)

ilim

C¼ 0:023

Fu

t� t

� �D

m

� �2=3

Re�0:2 (11.27)

0 2 4 6 8 10 12 14 16

00 2 4 6 8 10 12 14 160

200

400

600

800

1000

n = 0.7404 + 3.585 × 10-3u

n = 0.7846 + 8.612 × 10-3u

m = 66.36 + 14.72u

m = 83.50 + 24.00u

0

1

m

n

u (cm/s)

Figure 11.5 Coefficients m and n in the limiting current density equation (Tanaka,2005).

Limiting Current Density 253

11.5.2 Analysis Based on the Frank–Kamenetskii Equation

Cowan and Brown (1959) employed d introduced from the Frank–Kamenetskii equation:

d ¼ amRe1=8

0:395u

� �(11.28)

a is dimensionless constant. ilim/C is obtained by substituting Eq. (11.28) intoEq. (11.4):

ilim

C¼ 0:395

FD

t� t

� �Re7=8

ade

� �(11.29)

11.5.3 Analysis Based on the Stanton Number, Peclet Number and Potential

Difference Number

Kitamoto and Takashima (1967) expressed the mass transport and energyconsumption in an electrodialysis system by the following Stanton number St,

Ion Exchange Membranes: Fundamentals and Applications254

Peclet number Pe and potential difference number C. These parameters aredefined as follows:

St ¼i=F

Cu(11.30)

Pe ¼au

D(11.31)

C ¼FDcRT

(11.32)

Here, St stands for the dimensionless ratio of the material flux transportingacross the membrane i/F against that in the solution flowing into the desaltingcell Cu. Pe expresses the dimensionless solution velocity in the desalting cell u. ais the thickness of the desalting cell. D is the diffusion constant. C is thedimensionless potential difference between the potential at the cation exchangemembrane surface and at the anion exchange membrane in the concentratingcell. Dc is approximated by the difference between the potential differenceapplied to a membrane pair Dcapplied and the membrane potential as follows:

Dc ¼ Dcapplied �2RT

FlnCcon

Cde¼

ade

kdeþ

acon

kconþ rK þ rA

� �i (11.33)

where Cde and Ccon are electrolyte concentration in a desalting and a concen-trating cell, respectively. ade and acon are the thickness of a desalting and a con-centrating cell, respectively. kde and kcon are specific conductivity of electrolytesolution flowing in a desalting and a concentrating cell, respectively. rK and rA areelectric resistance of a cation and an anion exchange membrane, respectively.

Plotting St againstC/Pewas confirmed to be expressed by the straight line asshown in Fig. 11.6 by means of the electrodialysis of a NaCl solution using theapparatus (effective membrane area; 5� 50 or 8� 24 cm2, thickness of a desaltingcell; 0.71–5.6 cm) integrated with Selemion CMV-10/AMT-10 (Asahi Glass Co.) orAciplex CK-1/CA-1, CA-2 (Asahi Chemical Co.) membranes with spacers andfeedinga10�3–0.6MNaCl solution.Fig. 11.6 is presentedby the followingequation.

St ¼ 1:04CPe

� �¼ 1:04

FD

au

� �DcRT

� �(11.34)

Further, Kitamoto and Takashima (1968) measured the mass transport in a desalt-ing cell applying the limiting current density (measured from I–pH or I–V curves)and obtained Fig. 11.7, which is presented by the following equation.

clim ¼ 0:09Pe0:65

CPe

� �lim

¼ 0:09Pe�0:35 (11.35)

With spacer

10-2

10-3

10-4

10-4 10-3 10-2

Key

1.5×105

9×104

6.5×104

3×104

6×104

4×104

2×104

1.2×104

7×103

2×103

0.54

0.54-0.55

0.54-0.55

0.54-0.55

0.25

0.25

0.25, 0.075

0.25, 0.075

0.25, 0.075

0.075

(+) CMV-10

(−) AMT-10

made byAsahi Glass Co., Ltd.

(+) CK-1

(−) CA-1, CA-2Made byAsahi Chem. Ind. Co., Ltd.

Pe D [cm]Names of a pairof membranes

Key

6×104

1.5×105

9.6×104

9.6×104

6×104

0.58-0.60

0.58-0.60

0.8-0.9

0.8-0.9

0.8-0.9

(+) CMV-10

(−) AMT-10

made byAsahi Glass Co., Ltd.by Y. Oda20)

(+) CSG, (−) ASG

(+) CMG, (−) ASG

(+) CMG, (−) AMG

Pe D [cm]Names of a pairof membranes

Figure 11.6 Relationship between St and C/Pe. (Kitamoto and Takashima, 1967)

Limiting Current Density 255

Substituting Eq. (11.34) into Eq. (11.35), the limiting current density in a NaClsolution system is obtained as follows:

Stlim ¼ 0:094Pe�0:35 (11.36)

or

ilim ¼ 1:88� 102Cu0:65D�0:35 (11.37)

103

102

10104 105

Pe

106

NaCI

Na 2SO 4

lim

Figure 11.7 Relationship between Clim and Pe (Kitamoto and Takashima, 1968).

Ion Exchange Membranes: Fundamentals and Applications256

11.5.4 Analysis Based on the Sherwood Number

Miyoshi et al. expressed the ionic electrolyte flux Ji on a y-axis perpen-dicular to the membrane surface in a desalting cell incorporated with spacers bythe following equation (Miyoshi et al., 1988).

Ji ¼t

Fi þDe

@C

@y(11.38)

where t is the transport number of the ionic electrolytes and De the eddy diffu-sivity of the electrolytes in the solution. Based on Eq. (11.38) and the equation ofcontinuity, the limiting current density ilim was introduced as in the form of thefollowing Sherwood number Sh exhibiting the dimensionless ratio of currentdensity i against electrolyte concentration in a desalting cell C:

Sh ¼a

DF

� � ilim

C

� �¼

at� t

a

l

� �1=3M1=3 Reð1þ2bÞ=3Sc1=3 (11.39)

in which Sc is the Schmidt number ¼ n/D,M is a variable, a and b are the constants.Equation (11.39) was experimentally analyzed using the electrodialyzer

(membrane area: 20 cm length� 4 cm width, desalting cell thickness: 0.08–0.40 cm) in which cation exchange membranes (Neocepta CL-25T, TokuyamaInc.) and anion exchange membranes (Neocepta AV-4T) were incorporated withhoneycomb net or pointed twill net spacers. Supplying a 0.005–0.10 eq dm�3 NaClsolution or a 0.05 eq dm�3 KCl, MgCl2, CaCl2, NaHCO3 or Na2SO4 solution intothe electrodialyzer, ilim was measured from the inflection of V/I vs. 1/I plots. Based

Limiting Current Density 257

on the experiment mentioned above and Eq. (11.38), Sh was introduced as in thefollowing equation, presented in Fig. 11.8:

Sh ¼0:095

ðt� tÞfnðds � d f Þg1=2

ða=lÞ1=3M1=3

fð1� �Þ2=�3g1=5� Re1=2 Sc1=3 (11.40)

where ds is the thickness of a spacer, n and df are the number and fiber diameter ofthe spacer, e the void fraction of the spacer andM a spacer parameter representedby using the eddy viscosity parameter m originated by the spacer as follows:

M ¼m3

ðmþ 1Þ2 lnðmþ 1Þ � 1:5m2 �m(11.41)

m ¼ 2:1� 105fnðds � df Þg2:4 ð1� �Þ2

�3

� �0:8

(11.42)

11.5.5 Analysis Based on the Reynolds Number, Schmidt Number and Shape

Factor

Huang and Yu investigated the effects of Reynolds number, Schmidtnumber, and shape factor of the electrodialysis cell on the mass transfer rate inelectrodialysis for various systems at limiting current density (Huang and Yu,1988). In this study, the electrolyte concentration C in the electrodialysis cell wasexpressed by

@C

@tþ u � rC ¼ rðDrCÞ (11.43)

D is the molecular diffusion coefficient of the electrolytes in a solution. Consideringthe flow of an electrolyte solution through two flat parallel ion exchange membranesat steady state, for flow velocity in the x-direction nx and constant molecular diffu-sion coefficient D, Eq. (11.43) is written on the coordinate along the membranesurface from the upstream end, x, and that normal to the membrane surface, y, as

vx ¼@C

@x¼ D

@2C

@y2(11.44)

with the boundary conditions under limiting current density:

C ¼ 0; x40; y ¼ B

C ¼ C0; x ¼ 0; �BfyfB

C ¼ C0; x40; y ¼ �1

(11.45)

Here, B is the half-thickness of the channel and C0 is the electrolyte concentrationin bulk. If the Schmidt number is large, the thickness of the diffusion boundarylayer will be smaller than that of the viscous boundary layer, so the velocitycomponent nx can be expressed approximately by

103

102

101

102

12 3 4 5 6 7 8 9 10

11 12 13 14 15

16 17 18 19

Run Run Run

103

Figure 11.8 Relationship between Re, Sc and Sh (Miyoshi et al., 1988).

Ion Exchange Membranes: Fundamentals and Applications258

nx ¼ 3hui 1�y

B

� �(11.46)

Next, if the diffusivity in the solution of the cation is smaller than that of the anion,concentration polarization develops first on the surface of the cation exchangemembrane. In this situation, the dimensionless mass transfer rate, the Nusseltnumber Nu showing the dimensionless ionic flux across the membrane N againstthe electolyte concentration in the desalting cell C on the cation exchange mem-brane is defined as

Nu ¼Nþjy¼Bde

CþDþ

(11.47)

Nþ ¼ilim

zþF¼ �nþ 1�

zþ

z�

� �Dþ

@C

@y

����y¼B

(11.48)

N� ¼ 0 (11.49)

Here, N+ and N� are molar fluxes of cations and anions, respectively. de is theequivalent diameter of the channel. n+ is the number of cations produced by the

Limiting Current Density 259

dissociation of one molecule of electrolytes in the solution. z+ and z� are thevalence of cations and anions, respectively.

Solving Eqs. (11.44)–(11.48), Nu on the cation exchange membrane isintroduced as follows:

Nu ¼ 1�zþ

z�

� �ð4=3Þ1=3

Gð4=3ÞRe Scde

x

� �1=3

(11.50)

The average value of the Nu over the length L, (Nu)ave of the cation exchangemembrane is introduced as

ðNuÞave ¼1

L

Z L

0

Nudx ¼ 1:849 1�zþ

z�

� �Re Scde

L

� �1=3

(11.51)

which is expressed by the function of the Reynolds number Re, Schmidt numberSc and dimension factor de/L.

On the other hand, if the diffusivity in the solution of the anion is smallerthan that of the cation, concentration polarization develops first on the surfaceof the anion exchange membrane. In this case, (Nu)ave on the anion exchangemembrane is expressed by the equation equivalent to Eq. (11.51), however Nu

and N� are expressed as follows:

Nu ¼�N�jy¼�Bde

C0�D�

(11.52)

N� ¼ilim

z�F(11.53)

The theory presented above was examined through the limiting currentdensity measurement using the electrodialysis apparatus (effective membranearea; 20 cm2) incorporated with a desalting channel and an ion exchange mem-brane pair (Selemion CMV/AMV or Aciplex K-102/A-102). Supplying aCH3COONa, a NaCl, a CuSO4 or a NiSO4 solution or a mixed solution ofH2SO4+glucose or xylose, into the channel, and passing an electric current, thelimiting current density was determined from the first inflection point of thecurrent density vs. voltage curve. The relationship between the experimentalNusselt number (Nu)exp and Re, Sc and de/L was shown in Fig. 11.9, and waspresented by the following equation, which is similar to the theoretical Eq. (11.51).

ðNuÞexp

1�zþ

z�

� � ðfor cation exchange membraneÞ

¼ðNuÞexp

1�z�

zþ

� � ðfor anion exchange membraneÞ

¼ 1:793 Re0:341Sc0:329de

L

� �0:301

ð11:54Þ

CH3COONa

NaCI

CuSO4

NiSO4

H2SO4 + 6% glucose

130

120

110

100

90

80

70

60

50

(Nu)

exp

1793Re0.340 Sc0.329 (de/L)0.301

40

30

20

10

0200 40 60 80 100 120

H2SO4 + 6% xylose + 1.8% glucose

Figure 11.9 Relationship between Nu, Sc, de/L (Huang and Yu, 1988).

Ion Exchange Membranes: Fundamentals and Applications260

11.6. SOLUTION VELOCITY DISTRIBUTION BETWEEN DESALTING

CELLS IN A STACK (Tanaka, 2005)

Limiting current density of an electrodialyzer is strongly influenced by thedistribution of solution flow in desalting cells, which is discussed below. Sea-water was supplied to desalting cells in an unit-cell type electrodialyzer (Ta-ble 11.2) incorporated with Selemion CMV/AST membranes and cross piecespacers, and electrodialyzed at the linear velocity of 3 cm s–1 and the currentdensity of 2Adm–2. The solution velocity distribution in stacks was evaluated bymeasuring the solution volume Q flowing out of the stacks. After that, the stackswere disassembled and washed, and then the solution velocity distribution wasmeasured again in the same manner. Frequencies of stacks Nstack are plottedagainst solution volume ratio x ¼ ðQ� QÞ=Q; (Q: the average solution volume

Table 11.2 Specifications of the electrodialyzer

Membrane Selemion CMV/AMT

Numbers of stacks in the electrodialyzer 44Numbers of desalting cells in a stack 12Membrane area 0.96m2 (98 cm� 98 cm)Distance between membranes in a desalting cell 2mmSpacer Cross piece

Source: Tanaka, 2005.

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.40

2

4

6

8

10

12

14

16

Nst

ack

before disassembly Q = 23.7l/min = 0.079

after washing and assembly Q = 28.9l/min = 0.037

Figure 11.10 Velocity distribution of desalted solutions between stacks (Tanaka, 2005).

Limiting Current Density 261

in a stack), and shown in Fig. 11.10 which is equated with the normal distri-bution. The standard deviation of the normal distribution s are measured as0.037. Next, the solution velocity distribution in desalting cells in one of thestacks before assembly was evaluated by injecting coloring liquid into the inletsof each desalting cell and measuring elapsed time until the coloring liquid

Ion Exchange Membranes: Fundamentals and Applications262

appears in the outlets of the desalting cells. The average linear velocity u wasadjusted to 3 cm s–1. The frequencies of desalting cells Ncell are plotted againstthe solution velocity ratio x ¼ ðu� uÞ=u; and shown in Fig. 11.11 which is alsoequated with the normal distribution and s is measured as 0.078.

Seawater was supplied to desalting cells in a filter-press type elect-rodialyzer (Table 11.3) and electrodialyzed at the current density of 3A dm–2.The electrodialysis was repeated changing the linear velocity of solutions indesalting cells incrementally. The solution velocity in each desalting cell u wasevaluated by measuring the electrolyte concentration at the inlets Cin and theoutlets Cout of cells. Ncell vs. x ¼ ðu� uÞ=u was confirmed to be equated with thenormal distribution. Standard deviation s of solution velocity ratio x is deter-mined as presented in Table 11.4. s is distributed in the range of 0.017–0.222.

Here, we define the solution velocity ratio x in desalting cells integrated ina stack in an electrodialyzer by Eq. (11.55)

x ¼u� � u

u(11.55)

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.70

2

4

6

8

10

12

14

16

18

20

Nce

ll

=(u*- u ) /u

u = 3 cm/s = 0.078

Figure 11.11 Velocity distribution of desalted solutions between cells (Tanaka, 2005).

Table 11.3 Specifications of the electrodialyzer

Membrane Aciplex K-172/A-172

Numbers of stacks in the electrodialyzer 1Numbers of desalting cells in a stack 36Membrane area 0.15m2 (96.6 cm� 15 cm)Distance between membranes in a desalting cell 0.75mmSpacer Diagonal net

Source: Tanaka, 2005.

Table 11.4 Standard deviation of solution velocity ratio s in desalting cells

u (cm s�1) C0in (eq dm�3) C0

out (eq dm�3) s

1.12 0.590 0.276 0.1171.58 0.557 0.276 0.2221.73 0.606 0.392 0.1412.07 0.603 0.430 0.0172.24 0.606 0.426 0.1343.12 0.603 0.488 0.0203.54 0.606 0.491 0.1224.93 0.603 0.535 0.102

Source: Tanaka, 2005.

Limiting Current Density 263

where u� is the linear velocity in every desalting cell and u the average linearvelocity in a stack. The frequency distribution of x is equated by the normaldistribution, so that the minimum of x and u� may be equated with �3s and u,respectively, where s is the standard deviation of the normal distribution and u

is the minimum value of linear velocities within all desalting cells in a stack.Putting x ¼ �3s and u� ¼ u in Eq. (11.55) yields Eq. (11.56)

u ¼ uð1� 3sÞ (11.56)

11.7. LIMITING CURRENT DENSITY OF AN ELECTRODIALYZER

(Tanaka, 2005)

11.7.1 Limiting Current Density Equation

When current density reaches the limit of an ion exchange membrane ilimat the outlet of a desalting cell in which linear velocity and electrolyte concen-tration are the least, the average current density applied to an electrodialyzer isdefined as its limiting current density (I/S)lim, in which I is the total electriccurrent and S is the membrane area. (I/S)lim is strongly related to the circum-stances distributing in an apparatus. Namely, (I/S)lim is influenced by the dis-tribution of solution velocity (cf. Section 11.6). It is also influenced by the

Ion Exchange Membranes: Fundamentals and Applications264

distribution of electrolyte concentration in desalting cells and an electric currentin a stack (cf. Section 9.1). Further, it is naturally influenced by the limitingcurrent density of an ion exchange membrane ilim (cf. Section 11.4). We discuss(I/S)lim in this section based on the suggestions mentioned above.

Referring to Eqs. (11.17)–(11.19), ilim, is expressed by the function ofelectrolyte concentration C0

out and linear velocity uout at the outlet of thedesalting cell in which the linear velocity is the least as follows:

ilim ¼ ðm1 þm2uoutÞC0n1þn2uoutout (11.57)

where m1, m2, n1 and n2 are the constant. uout is expressed by the followingequation referring to Eq. (11.56):

uout ¼ uoutð1� 3sÞ (11.58)

The limiting current density of an electrodialyzer is defined as follows usingEq. (9.19)

I

S

� �lim

¼ilim

zout(11.59)

Substituting Eq. (11.57) into Eq. (11.59):

I

S

� �lim

¼ðm1 þm2uoutÞC

0n1þn2uoutout

zout(11.60)

Next, when the limiting current density (I/S)lim is applied to an electrodialyzer,the following material balance is realized in the desalting cell in which the linearvelocity is the least:

I

S

� �lim

¼aF

Zl

� �uavðC

0in � C0

outÞ (11.61)

a and l are, respectively, the thickness and flow pass length of a desalting cell. Fis the Faraday constant. uav is the average of linear velocity at the inlet andoutlet of a desalting cell in which the velocity is the least.

uav ¼uin þ uout

2(11.62)

Z is the current efficiency in the desalting cell and indicated by the overall masstransport equation (cf. Chapter 6) as follows:

Z ¼FJS

ðI=SÞlim¼

F lðI=SÞlim � mðC00 � C0avÞ

� ðI=SÞlim

(11.63)

JS is the flux of ions transported across a membrane pair (cf. Eq. (6.1)). C0av is

the average of electrolyte concentration at the inlet and outlet of a desalting cellin which the velocity is the least.

C0av ¼

C0in þ C0

out

2(11.64)

Limiting Current Density 265

C00 is the electrolyte concentration in concentrating cells and assumed to beinvariable at every point in the cells because the concentrated solution isextracted through an overflow extracting system.

On the other hand, the linear velocity uout at the outlet of desalting cell inwhich the velocity is the least is given using the overall mass transport equation(cf. Eq. (6.2)).

uout ¼ uin �S

ab

� �JV ¼ uin �

l

a

� �f

I

S

� �lim

þ rðC00 � C0avÞ

� �(11.65)

uin ¼ uinð1� 3sÞ (11.66)

b is the flow pass width of the desalting cell. C00 is introduced as follows using theoverall mass transport equation (cf. Eqs. (6.10)–(6.12)).

C00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ 4rB

p� A

2r(11.67)

l, m, j and r in Eqs. (11.67)–(11.69) are, respectively, the overall transportnumber, the overall solute permeability, the overall electro-osmotic permeabilityand the overall hydraulic conductivity, which are defined in the overall masstransport equation. l, m and j are expressed by the functions of r (cm4 eq�1 s�1)as below (cf. Eqs. (6.5)–(6.7)):

A ¼ jI

S

� �lim

þ m� rC0av (11.68)

B ¼ lI

S

� �lim

þ mC0av (11.69)

lðeq C�1Þ ¼ 9:208� 10�6 þ 1:914� 10�5r (11.70)

mðcm s�1Þ ¼ 2:005� 10�4r (11.71)

jðcm3 C�1Þ ¼ 3:768� 10�3r0:2 � 1:019� 10�2r (11.72)

11.7.2 Procedure for Calculating the Limiting Current Density of an

Electrodialyzer

The limiting current density of an electrodialyzer (I/S)lim is computedusing trial and error calculation as follows:

11.7.2.1 Process 1

The outlet current density uniformity coefficient zout is calculated usingthe procedures described in Chapter 9 (Section 9.1.1), assuming appropriate

Ion Exchange Membranes: Fundamentals and Applications266

s, uin, C0in; I/S, a, I and r. Here, l, m and j are computed using r and

Eqs. (11.70)–(11.72).

11.7.2.2 Process 2

We adopt an approximated equation (11.73) for convenience

u�in ¼ u�av ¼ u�out

u�in ¼ u�av ¼ u�out

uin ¼ uav ¼ uout

(11.73)

Equation (11.73) means that the volume fluxes transporting from desaltingcells to concentrating cells are zero, so that computed result includes errors.Due to this approximation, Eq. (11.60) is presented by Eq. (11.74) using uininstead of uout:

I

S

� �lim

¼ðm1 þm2uinÞC

0n1þn2uinout

zout(11.74)

The second term m(C00–C0) in Eq. (11.63) is sufficiently small compared to the firstterm l(I/S)lim, so that Z is expressed by canceling the second term as follows:

Z ¼ Fl (11.75)

Substituting Eq. (11.75) and uin ¼ uav in Eq. (11.73) into Eq. (11.61):

I

S

� �lim

¼a

ll

� �uinðC

0in � C0

outÞ (11.76)

Setting as Eq. (11.74) ¼ Eq. (11.76):

C0n1þn2uinout

C0in � C0

out

¼ Z (11.77)

Z ¼azoutll

� �uin

m1 þm2uin

� �(11.78)

Here, we calculate as follows:

(a)

Calculate uin by substituting uin and s into Eq. (11.66). (b) Calculate C0

out 111 by substituting uin into Eqs. (11.77) and (11.78) andequating Eqs. (11.77) and Eq. (11.78).

(c)

Calculate (I/S)lim 111 by substituting C0out 111 into Eq. (11.74).

(d)

Calculate C0av 111 by substituting C0

out 111 into Eq. (11.64).

(e) Calculate C00 111 by substituting (I/S)lim111 and C0

av 111 into Eqs. (11.67)–(11.69).

(f) Calculate uout 111 by substituting (I/S)lim 111 , C0

av and C00 111 into Eq. (11.65).

(g) Calculate uav 111 by substituting uin and uout 111 into Eq. (11.62). (h) Calculate Z 111 using Eq. (11.75).

Limiting Current Density 267

11.7.2.3 Process 3

We calculate as follows:

Figden

(a)

ure 1sity o

Calculate C0out 2 by substituting (I/S)lim 111 and uout 111 into Eq. (11.79)

introduced from Eq. (11.61).

(11.79)

Calculate C0 2 by substituting C0 2 into Eq. (11.64).

(b) av out

(c)

Calculate Z 2 by substituting C0av 2 , C00 111 and (I/S)lim 111 into Eq. (11.63).

(d)

Calculate (I/S)lim 2 by substituting C0out 2 and uout 111 into Eq. (11.60).

(e)

Calculate uav 2 by substituting uin and uout 111 into Eq. (11.62). (f) Calculate C00 2 by substituting (I/S)lim 2 and C0

av 2 into Eqs. (11.67)–(11.69).

(g) Calculate uout 2 by substituting (I/S)lim 2 , C0

av 2 and C00 2 into Eq. (11.65).

Process 3 is proceeded using correct equations, but the results are assumed to haveerrors due to the approximation (Eqs. (11.73) and (11.75)) adopted in Process 2.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350

2

4

6

8

10

12

14

16

18

20

(I/S

) lim

(A/d

m2 )

3

u in = 5 cm/s, C'in = 0.6 eq/dm3, I/S = 3 A/dm2

a = 0.075 cm, b = 100 cm, l = 100 cm, S = 1 m2

1.12 Effect of standard deviation of solution velocity ratio on limiting currentf an electrodialyzer.

Ion Exchange Membranes: Fundamentals and Applications268

11.7.2.4 Process 4

Repeat the calculation in process 3 until the errors becomes zero. Re-peating two times is necessary and sufficient.

11.7.3 Computation of the Limiting Current Density of an Electrodialyzer

We calculate here the limiting current density of an electrodialyzer ac-cording to the processes described above and assuming the specifications of theelectrodialyzer as S ¼ 1m2, l ¼ 1m, b ¼ 1m, a ¼ 0.075 cm, I/S ¼ 3Adm–2 andthe overall water permeability r ¼ 1� 10�2 cm4 eq�1 s�1 (cf. Chapter 6). In thiscalculation, we use the outlet current density uniformity coefficient zout obtainedby the process described in Section 9.1.1.

Fig. 11.12 shows the relationship between s and (I/S)lim computed bysetting uin ¼ 5 cm s–1 and C0

in ¼ 0:6 eq dm�3. Taking account of that this appa-ratus is operating at I/S ¼ 3Adm�2, we can evaluate the limiting standarddeviation of solution velocity slim ¼ 0.290, which corresponds to a permissiblelimit of s value. The relationship between uin and (I/S)lim is presented inFig. 11.13, obtained by setting s ¼ 0.1 and C0

in ¼ 0:6 eq dm�3. In this situation,

14

12

10

8

6

4

3

2

00 1 2 3

u in, lim = 0.912cm/s

u in (cm/s)

(I/S

) lim

= (

A/d

m2 )

4 5 6

Figure 11.13 Effect of solution velocity at the inlets of desalting cells on limiting currentdensity of an electrodialyzer.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

2

4

6

8

10

12

14

3

(I/S

) lim

(A

/dm

2 )

C'in,lim = 0.117eq/dm3

C'in(eq/dm3)

= 0.1, uin = 5 cm/s, I/S = 3 A/dm2

a = 0.075 cm, b = 100 cm, l = 100 cm, S = 1 m2

Figure 11.14 Effect of electrolyte concentration at the inlets of desalting cells on limitingcurrent density of an electrodialyzer.

Limiting Current Density 269

the permissible limiting solution velocity is evaluated as uin,lim ¼ 0.912 cm s�1.By setting s ¼ 0.1 and uin ¼ 5 cm s�1, the relationship between C0

in and (I/S)limis calculated as in Fig. 11.14. Here, the permissible limiting electrolyte concen-tration is recognized to be C0

in;lim ¼ 0:117 eq dm�3.

REFERENCES

Cowan, D. A., Brown, J. H., 1959, Effect of turbulence on limiting current in electro-dialysis cells, Ind. Eng. Chem., 51(12), 1445–1448.

Huang, T. C., Yu, I. Y., 1988, Correlation of ionic transfer rate in electrodialysis underlimiting current density conditions, J. Membr. Sci., 35, 193–206.

Kitamoto, A., Takashima, Y., 1967, Mass transfer by electrodialysis using ion-exchangemembranes–Studies on electrodialytic concentration of electrolytes, J. Chem. Eng.,Jpn., 31(12), 1201–1207.

Kitamoto, A., Takashima, Y., 1968, Studies on electrodialysis, maximum attainableconcentration, limiting current density and on energy efficiency in electrodialysis usingion-exchange membranes, J. Chem. Eng., Jpn., 32(1), 74–82.

Ion Exchange Membranes: Fundamentals and Applications270

Miyoshi, H., Fukumoto, T., Kataoka, T., 1988, A method for estimating the limitingcurrent density in electrodialysis, Sep. Sci. Technol., 23(6 & 7), 585–600.

Peer, A. M., 1956, Discuss. Faraday Sci., 21, 124 (Communication in the membranephenomena special issue).

Rosenberg, N. W., Tirrell, C. E., 1957, Limiting currents in membrane cells, Ind. Eng.Chem., 49(4), 780–784.

Spiegler, K. S., 1971, Polarization at ion exchange membrane-solution interfaces,Desalination, 9, 367–385.

Tanaka, Y., 2003, Concentration polarization in ion-exchange membrane electrodialysis–the events arising in a flowing solution in a desalting cell, J. Membr. Sci., 216, 149–164.

Tanaka, Y., 2004, Concentration polarization in ion-exchange membrane electrodialysis.The events arising in an unforced flowing solution in a desalting cell, J. Membr. Sci.,244, 1–16.

Tanaka, Y., 2005, Limiting current density of an ion-exchange membrane and of anelectrodialyzer, J. Membr. Sci., 266, 6–17.

Tobias, C. W., Eisenberg, M., Wilke, C. R., 1952, Diffusion and convection in electrolysis–A theoretical review, J. Electrochem. Soc., 99(12), 359C–365C.

Yamabe, T., Yamana, S., Yamagata, K., Takai, S., Seno, M., 1967, On the concentrationpolarization effect in the electrodialysis using ion-exchange membranes (IV) Concen-tration polarization effect multi-compartment electrodialysis cell, J. Electrochem. Soc.,Jpn., 35, 578–582.