Thickness and concentration profile of the boundary layer in electrodialysis

ELSEVIER Desalination 109 (1997) 95-103

DESALINATION

Thickness and concentration profile of the boundary layer in electrodialysis

M. Law a, T. Wen b*, G.S. Solt c a Department of Civil Engineering (TORG), University of Newcastle, Newcastle upon Tyne, NE1 7R U, UK

bBritish Sugar Technical Centre, Norwich Research Park Colney, Norwich, NR4 7UB, UK Tel. +44 (0) 1603 250000; Fax +44 (0) 1603 455874

CSchool of Water Sciences, Cranfield University, Cranfield, Bedford, MK43 OAL, UK

Received 17 September 1996; accepted 3 November 1996

Abstract

Back electrical motive force (emf) measurements with spiral electrodialysis (SPED) modules showed that obtaining the profile of the back emf transient curves during depolarization is difficult from the Nernst model, and the assumption of a linear concentration profile in a stirred polarized boundary layer is oversimplified. A non-linear concentration distribution model derived from the error function is introduced.

Keywords: Spiral electrodialysis; Concentration profile; Boundary layer thickness

1. Introduct ion

A literature survey shows that, in many books about electromembrane processes, discussions of concentration polarization in electrodialysis are based on an idealized model (i.e., the so-called Nernst idealization) in which the following are assumed [1,2]: (a) there are boundary layers adjacent to the membranes in which the solutions are completely static, Co) the solution in the

*Corresponding author.

interior of a solution compartment (i.e., between the boundary layer) is thoroughly mixed so that the concentration of electrolyte at any point in this zone is the same as that at any other point, and (c) there is no change of either the thickness of the boundary layer or the concentration gradients along the flow channel.

The Nernst model, used extensively in this century, provides a simplified approach to mathematical developments, which results in expressions that are easy to use in the design of electromembrane processes.

0011-9164/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0011-9164(97)00055-6

96 M. Law et aL / Desalination 109 (1997) 95-103

Like any other theories, the Nemst model has its limitations which will be discussed in this paper. However, this research should not, of course, be taken to belittle the important role of the Nernst model in the development of electrochemical processes.

2. Polarization and membrane interface concentrations

2.1. Boundary layer and limiting current

Fig. 1 shows a typical description of concentration gradients in electrodialysis [2] in which the Nernst model of boundary layers has been depicted. Theoretically, in NaCI solution, the limiting current for desalination can be evaluated [3] by

FC°D " . : (1)

where 6 is the diffusion boundary layer thickness which is determined by the hydrodynamics of the system; but since it is a characteristic of an idealized system, its value in any practical apparatus cannot be predicted from hydraulic data alone. Values of 6 are usually obtained by determining experimentally the values Ofl l i m and calculating them from Eq. (1).

2.2. Back emf and membrane interface concentrations

When an ion exchange membrane separates two solutions of electrolytes of different concentration, a membrane potential difference arises, which opposes the applied emf and is termed as back emf. In electrodialysis, the resulting back emf of one cell pair, V/,, is calculated from the Nemst Equation [4].

V b : 2__ log, C;V:,

(2)

Enriching solution

(Conc)

Cation- exchange

membran~

Depleting solution

( D i l )

t

/,, I I I I I

6

V Boundary layers

C b

q-

Fig. I. Concentration gradients in electrodialysis (taken from [2]).

In NaCI solution, at 298K, with the value of R as 8.314 JK -1 mol -l and F as 96,500 Cmol -l, assuming activity coefficients 5,~ and y,~ of unity, the resulting back emf of one cell pair may be evaluated as

$ Cc

V b = 0 .1181og- - (3)

where C~ and C,~ are interface concentration of concentrate (Conc) and diluent (Dil) respectively.

Fig. 1 demonstrates that in the polarization, the concentration at the interface is accumulated in the Conc compartment but depleted in the Dil compartment, so that the differential C~/C,~ across the membrane is increased by an unknown amount, leading to an increased back emf, according to Eq. (3). It is therefore possible to study the membrane surface concentration changes by back emf measurements.

M. Law et al. / Desalination 109 (1997) 95-103 97

In the electrodialysis, the concentration depletion in Dil is more of a troublesome problem than the concentration build-up in Cone because the extremely low concentration at the interface results in water splitting, and hence the pH changes• In the following sections of this paper, emphasis will be laid mainly on the concentration distribution in the Dil compartment.

3. Back emf transient

3.1. The back emf transient predicted by the Nernst model

The following is the mathematical expression of Fick's law of diffusion [5],

0 s 0 0 dc Cd - Cd Ca - Ca

- 0 (7)

the maximum speed of surface concentration change should be at time 0 when the concentration difference is the maximum, the speed should then drop with time because of decreasing concentration difference.

Similarly, in the Cone compartment (see Fig. 1), at time 0, the concentration difference is also the maximum,

dc[ _ c : - c ° (8)

d r dw = - D = d t (4)

dx

and surface concentration change should be at the maximum speed.

From Eq. (4), the rate of diffusion can be presented as

dw dc - D - - - (5)

dt dx

where D is the diffusion coefficient and dc/dx is the concentration gradient within the boundary layer, which could be regarded as uniformly linear during polarization, according to the idealized Nemst model of boundary layer [6].

Fig. 2 demonstrates that in a fully polarized ED system, the membrane face concentration in the Dil compartment changes vs time after the interruption of external emf, which is due to diffusion. Setting the zero time at the point when the external emf is interrupted and assuming time ~2 is needed to restore the concentration in the depleted boundary, according to the linear concentration distribution within the boundary layer, one obtains

]d~] C ° - C ; C °

t2=0 6 6 (6)

C

.......... ~ . : . ~ .. . . . . . . . . . . . . . . . . . . . l . . . . . . . .

. . ' ' ' '•*] . . . . ' " .°.~

• I ,•. , ,""

°

0 ,"

0 I,." . . 1 0 5

_-- ×

Fig. 2. Concentration changes in depolarization.

98 M. Law et al. / Desalination 109 (1997) 95-103

According to Eq. (3), the idealized model would therefore lead to a rapid fall in ~ at first, followed by progressive slowing of the diffusion rate.

3.2. Experimental results

Two single-start, parallel flow, spirally-wound electrodialysis (SPED) modules [7,8] were used for the back emf tests. Structure and dimensional data of these modules could be found in [9,10], and the experimental system and the module voltage transient measurement procedure could be found in [ 11,12].

Fig. 3 demonstrates the module voltage measurements vs time of a similar set of experiments all with the same hydraulic conditions but at varying applied voltages. Zero time has been set at the points at which the last applied emf voltages were recorded. The results show a "plateau" on the voltage transient curve. The length of the "plateaus" represents the relaxation time needed for ions in the bulk solution to diffuse across the boundary layer.

Increasing the applied voltage increases the back emf V~, represented by the height of the plateau on curves 4 to 6, proportionally. Curves 5 and 6 show above an applied voltage of about 27 volts; the relaxation time is near constant.

It should be pointed out that in these experiments, the back emf measurement was the sum of 14 turns (i.e., cell pairs, each containing two membranes); as there was only one long spiral cell pair in the SpED module tested, hydrodynamics in each cell would be the same in all these tums.

3.3 Comparing the experimental results with the Nernst model

According to the Nernst model, it was expected that between the applied voltage and back emf caused by bulk concentrations, there would be an initial value representing the back emf due to polarization, followed a die-away

35

30

o 25

2O

5

0

G

2

i . . . . . . . . . .

0 5 10 Time (ms)

Fig 3. Voltage transient at different applied voltages.

25

2 2 0

~ 1 5

o

10

5

0

Measured

Predicted

0 2 4 6 8 10 12 Time (ms}

Fig. 4. Comparing the Nernst model with the experimental results.

voltage transient curve during relaxation, with the maximum decreasing rates at the time just when the external emf was interrupted, due to the maximum concentration difference.

However, experimental results (Fig. 3) show that in the depolarization process, the back emf decay is uneven: the rate of this fall was steady first, with the maximum decreasing rates at the time just before the back emf dropped to the values due to bulk concentration.

Fig. 4 compares the differences, which hints that perhaps the assumption of a linear concentration profile in a stirred polarized layer is wrong.


4. The non-linear concentration distribution model

I11 contrast with the Nernst model, the non- linear concentration distribution model represents a more realistic presentation: within the boundary layer, there is a series of smaller liquid layers undergoing laminar flow. The velocity of these liquid layer increases with increasing distance from the membrane surface. The velocity would be zero only at the membrane surface and increases to bulk flow velocity gradually at about 6 position [13]. Ion transport by convection would be promoted at the higher liquid velocities away from the membrane surface. Fig. 5 depicts a concentration distribution derived from the error function.

erf (X) =

k

2 fe_Y2dy (9)

and

f e _y2 dy = 0

1 1 ! 3

1 ~5 1 ~7 - - _ _ _ _ + d - " ' "

2l 5 3! 7

(10)

Fig. 6 shows the basic characteristics of the error function (eft) and its complement (erfc). The most important properties of the error function are:

4.1. The error equation

For the model of non-linear concentration distribution in the boundary layer, the mathematical description is based on the error function (erf), which is defined as

a. ~, = 0, erf (~,) = 0

b. X = _>2, erf(X) = 1

c. The slope of the curve at X=0 is

(11)

(12)

1.o

0

..~ 0.5

0.0

I flIt ! / I / /I

lit ll l l ~

/t

Fig 5. Concentration distribution in boundary layer (taken from [13]).


1.0

0.5

0.0 0

I i l

t ¢ ~

l II

llll

2

f

4 7 i 2 2

h

Fig 6. Characteristics of the error function.

d e f t ( t ) ] _ 2

d~, ]xoo V/~ (13)

d. From Eq. (13), we get

2~, i < 0.2, ef t (1) = --_

4.2. Concentration step and potential step

In electrochemistry, one of the transient techniques is a "concentration step", which is usually obtained by a jump in applied potential, so the technique is also named as "potential step". The concentration distribution near the electrode surface can be described by a function

where x is the distance from the electrode surface, and t I is the elapsed polarization time after the potential step. There are two ways to realize the concentration step.

First, if the electrode reaction deals with only one kind of ion and the electrochemical equili- brium on the electrode surface can be maintained approximately, the electrode surface concentration, C s, can be kept constant by maintaining a certain applied potential, i.e.

C(O, tl) = C s = constant (14)

Second, if the applied potential is high enough, the electrode surface concentration is always so small that it can be ignored compared with bulk concentration C O , then, keeping the applied potential at certain value (even not exactly) leads to "complete polarization", i.e.

C(x, tl) C(O, tl) = C s = 0 (15)


4.3. Mathematical model of the concentration distribution

If

C(x, o) = c o

layer, 6, is given by

C o 6 - - ~fC-~ll

(t~C) (18)

"~X x=O

C(oo, t l ) -- c o

C(0, q) = C s = constant

for a normal potential step experiment [13,14], then the transient concentration distribution could be described by

C~, tl) = Cs +(C°-CS) e r f l - s - ~ ) (16) [ 2~/Dtl J

If C(0, t0=C~=0, Eq. (16) can be simplified as

(17)

Fig. 5 shows the concentration distribution near electrode surface at time *r It is clear that the form of this curve is completely the same as the error fungtion curve in Fig. 6, and ~. corre- sponds to x/ (2D~l ) . It can be seen that

x at - - - O, o r x = O : C(O,~I ) = 0

at x ~ 4 D ~ 1 C(x,.cl)~C o - >_ 2 , o r x •

It is recognized [13] that at time zl, the "total thickness" of the diffusion layer is about 4 D ~ l , and the "effective thickness" of the diffusion

Fig. 7 shows the schematic variation of C(x,q), Eq. (17), with time t I and distance x from the surface of the electrode [ 14].

Ill at~

/

= C o derf(~.) ( x ~ 0 d ~ [ 4DO"St, '5

(19)

Eq. (19) gives the partial differentiation result of Eq. (17), the derivative of C(x, tl) with respect to time tl, which is less than zero. The physical meaning of Eq. (19) is that at any given distance

m

C t~

C o

×

Fig. 7. Schematic variation of C(x,t) (taken from [14]).


from the electrode (membrane) surface, x, the slope of the concentration curve will decrease with the time t 1 increase, as shown in Fig 7.

If it can be assumed that transient concentration profiles for depolarization follow the same function as that given in Eq. (17) but in reverse order, then, in this case,

x

C0:, t2 ) : C0 erf 21/D0:1 _ t2) (20)

According to Eq. (20), depolarization in the semi-stagnant liquid layer can take place without any change in the surface concentration until t2=zl, according to Eq. (19), during the depolarization, the slope of the concentration profiles will be more and more steep with the elapsed time t 2 (Fig. 7), these would describe the observed experimental trend more nearly than the idealized Nernst model.

5. Evaluation of boundary layer thickness fi

From Eq. (17) and Fig. 7, it can be seen that at any point the value of C(x,q) decreases with increase of time tl; if t l -% at any point, there would be

C ( x , ~ ) ~ C°erf(0) = 0 (21)

This means that on the plate electrode surface, the stable mass transfer process cannot be reached by pure diffusion. However, if there is any convection (natural or disturbed) when lhe value of 4. Dv/-D-~I approaches to that of diffusion layer 4 DX/~I, the stable mass transfer gradually sets up on the electrode surface.

In the back emf measurements, if the applied voltage is high enough, the membrane surface concentration reaches to almost zero, and when the operation is stable, the value of ~ D x , would be equal to that of hydraulic boundary layer g.

Assuming the concentration changes in polarization and depolarization are reversible, i.e. the time needed to set up depletion polarization, zl, is approximately equal to the time needed to restore the concentration in the depleted boundary, z2, and one may then use 1; 2 to calcu- late the effective boundary layer thickness 6.

For z2=12.5 ms (see Fig. 3, curves 5 and 6), D=0.002 mm 2 s -1 [15]. According to Eq. (18), a 6 value of about 8.9 #m is obtained, which is in good correlation with the values mentioned by [6]: 10 #m for a planar electrode in contact with a rapid-stirred solution, and that quoted by [15]: 9 /~m for a similar system. Besides, the "total thickness" of the diffusion layer, 4 DV/-~I, is about 20 #m (see Fig 5).

6. Conclusion

The non-linear distribution profiles derived from the error function can describe the actual boundary concentration distribution in electrodialysis compartments more nearly than the discontinuous linear profiles of the Nernst model.

7. Acknowledgement

This research was funded by the Process Engineering Committee of the British Science and Engineering Research Council. The experimental work was carried out at the School of Water Sciences, Cranfield University.

8. Symbols

C~ - - membrane surface concentration of the concentrated, mg eqv/l

C O - - bulk concentration of the concentrated, mg eqv/l

C~ - - membrane surface concentration of the diluted, mg eqv/l

C O - - bulk concentration of the diluted, mg eqv/l


C •

O . _ _

D m

F - -

J i m m

t 1 - -

t 2 - - -

T s - - -

V ~ - - -

V~b ---

X " - -

Greek

"~ | . m

-C2 m

electrode surface concentration, mg eqv/l bulk concentration, mg eqv/l diffusion coefficient, m 2 s-1 Faraday constant, C mol-1 limiting current density, A m -2 elapsed polarization time after the potential step, s elapsed depolarization time after external emf interruption, s ion transport number in the membrane ion transport number in the solution back emf due to membrane surface concentrations, volts back emf due to bulk concentrations, volts distance from the electrode (membrane) surface, m

transient time of full polarization, s transient time of full depolarization, s variable in the error function effective thickness of boundary layer, m

References

[1] W. Nernst, Z. Physik Chem. (Leipzig), 47 (1904) 52.

[2] T.A. Davis and G.F. Brockman, in: Industrial Processing with Membranes, R.E. Lacey and S. Loeb, eds., Wiley-Interscience, New York, 1972, p. 28.

[3] G.S. Solt, Electrodialysis, in: Membrane Separation Process, P. Meares, ed., Elsevier, Amsterdam, 1976, p. 245.

[4] K.S. Spiegler, Electrochemical Operations, in: Ion Exchange Technology, F.C. Nachod and J. Schubert, eds., Academic Press, New York, 1956, p. 137.

[5] S. Glasstone, Textbook of Physical Chemistry, 2nd ed., Macmillan, New York, 1946, p. 1257.

[6] D.R. Crow, ed., Principles and Applications of Electrochemistry, 3rd ed., Chapman & Hall, London, 1991, p. 178.

[7] G.S. Solt and T. Wen, IChemE Sym. Ser. 127, (1992) 11.

[8] T. Wen, G.S. Solt and Y.F. Sun, Desalination, 103 (1995) 165.

[9] T. Wen, G.S. Solt and Y.F. Sun, Desalination, 101 (1995) 79.

[10] T. Wen, Spirally Wound Electrodialysis (SPED) Modules, PhD Thesis, Cranfield University, UK, 1993.

[11] G.S. Solt, T. Wen and S.J. Judd, J. Applied Electro- chemistry, 23 (1993) 1117.

[12] T. Wen, G.S. Solt and D.W. Gao, Electrical resistance and Coulomb efficiency of ED apparatus in polarization, J. Membr. Sci., 114/2 (1996) 255.

[13] Q.X. Zha, Kinetics of Electrode Processes, 2nd ed., Science Press, Beijing, 1987, pp. 89-121.

[14] D.D. MacDonald, Transient Techniques in Electro- chemistry, Plenum, London, 1977, 72.

[15] B.A. Cooke, Electrochimica Acta, 3 (1961) 307.

Documents

Thickness and concentration profile of the boundary layer in electrodialysis