A mathematical model for the concentration field in an electrodialysis cell

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A MATHEMATICAL

MODEL

ELECTRODIALYSlS

CELL

FOR THE CONCENTRATIDN

FIELD

IN AN

A mathematical model is proposed for the concentration fietd between two membranes in an electrodiaitysis dc~~i~~tio~ cell. A spacer tilts this space and the, flow and the concentration field are considered in the presence of this spacer and without the assumption of infinite mixing. An integral method is used to obtain a soiution, which is Jater shown to be a very good approximation near the membrane and a rather good approximation everywhere else. The form of this solution is relatively simpte, which makes it usefuf in further investigations. SYMBOLS

R,S

-

c

-

G

-

CD CL -

co D

-

d

-

F

-

ft

-

N

-

N._

-

---__-.

coefficients irr integral sotution concentration mean brine concentration mean dialysate concentration mean diaiysate concentration concentration at the entrance diffusion coefficient of sodium half cell width Faraday number current density cell Iength negative ion transport number negative ion transport number

* On ubbaticaf

at the exit (X = tf (x = 0) chloride in water

leave from the Tazhnion-Israel

through

negative

membrane

Institute of Technology,

Hnifa.

D.

298

N+ P

p*

-

P-

-

Q t

-

11 ~~, I2

s

-

-

PNUELI

AND

G. GROSSMAN

negative ion transport number through positive membrane positive ion transport number positive ion transport number through positive membrane positive ion trenspori number through negative membrane mass flux time velocity component in the sdirection mean veiocity velocity component in the y-direction Cartesian coordinate in the flow direction; changes from s = 0 at the celI entrance to x = L at the exit from the cell Cartesian coordinate perpendicular to the membranes; changes from = 0 at the cell center to _V = d at the membrane )’ non-dimensionaI characteristic number of the spacer exponent of the rate of change of the current density a positive number, 0 < E < 1. a new inde~ndent variabie. C = fi (1 --I& y-dependent part of the concentration q at the membrane (y = d, 5 = 0) a mean value of dr dummy variable eigenvalue. cp2 = Y2Q4WV3W111

fNTRODUCTlON

Consider the process of efectrodialysis desalination of water, shown schematicalty in Fig. 1. The imposed voltage causes the Na+ ions to move toward the negative electrode, and the Cl- ions to move towards the positive one. The membranes through which the ions must pass are selective. They are ahernately permeable to Na+ ions only and to Cl- ions only. This results in increased NaCl concentration in the brine cells and in decrease in the NaCI concentration in the desalinate cells. The detailed description of the concentration field is of some interest for many reasons. it contributes to the general understanding of electrodialysis. It may also be immediately useful in the* following examples for applications. It is known (I) that an increase in the applied voltage causes an increase in current and higher desalination rates onfy up to a certain value_ Above this value a boundary layer type behavior of the. concentration near the membranes (Le., very low concentration at the entrance and very high concentration at the exit from the membrane) makes higher desalination rates impossible. Further voltage increase causes water dissociation. This “polarization” phenomenon is undesirable because of power waste and because of scale formation at the high concentration side. Detailed description of the flow (5) and the concentration field before “polarization” Desahation, 7 (1970) %i-%

MATHEhlATlCAL

-t-

MODEL FOR CONCENTRATION

I’

I

I

DIALYSATE I

I

NO+

f

t _

I--

I

I

No*

-

2

2d

w

I

I

vo* _ t

299

FIELD IN ED CELL

No*

r

I

t+1

w P

Cl

*

Cl-

-l

Cl

x

0

5

4 I

&f--J--WATER

ER&CKISH Fig.

-)Y F

IN

I. Ekctrodiafysis

&s&nation

scheme.

took ptace may help in the investigation and the possible elimination of the phenomenon. Another important example is that although it is believed that there are cases in which electrodialysis is the best desalination process, the design of the system is highly experimental; it is very expensive, and once the system works, modifications are hardly made, because of the large expenses. Thus a successful experiment on a large scale contributes relatively little to the understanding of the phenomenon. Detailed description of the flow and concentration fietd may tcad to better designs and to more organized experiments. Several investigations in this field suggested analytic solutions to the CORcentntion field (24). However, the simplifying assumptions which they made were either no spacer between the membranes, or completely mixed flow (infinitely fine spacer). Their results apply therefore to these extreme cases. This inves:igation also contains some simplifying assumptions. Still it proposes a mathematical model for the concentration field in an electrodialysis cell, in the presence of a spacer which is not infinitely fine. Thus it covers the gap between the two extreme cases mentioned before. It describes the present industrial operation and contributes to the understanding of desalination by electrodialysis

in this practical range. THE EQUATION

OF MASS TRANSPORT

The flow field between iwo membranes in the presence of a spacer has been solved in Ref. (5). Thiisolution is an approximation in the sense that it assumes a certain length scale below which no details may be observed, but which still Desahbarion, 7 (1970) 297-308

300 (-

D.

( UC+

a&-(UC)da) dy

&dY Fig. 2,

Oiffermtiaf

D&;* $$dx)dy

PNUELI

AND

G.

GROSShtAN

(+I

s&e

control volume.

permits the spacer to be considered isotropic. Essentiatly it is an integral-solution approach with some assumptions on what happens inside the integration regions. Since the velocity field is borrowed from Ref. (5) (for a lack of a better one) the length scale for the considerations here is assumed to be the same as in Ref. (5). Consider the control volume in. Fig. 2. Mass can be transported into the controt voiume by three mechanisms: by ion exchange, by diffusion, and by convection.

The electric current is carried through the solution by the ions. Consider an imaginary sotution in which only one species of ion exists, say of positive charge. These ions move towards the cathode. where they gain electrons and become electrically neutral: then they diffuse towards the anode, give up someeiectrons, become charged again, and so on. Also there is an exchange of electrons between ions and neutral atoms, reversing their roles. The electric conductivity of a solution is determined by the mobility of the ions and atoms in it, and by the strength of the bonds between ions and electrons. When two or more species of ions are present, the mechanism of electron transport is still the same, except that electrons may also be transferred from one species to another and that molecules may also be formed. The whofe phenomenon is of course governed, by the thermodynamics of the system. An ion transport number is defined as the percentage of the total current carried by that particular ion. When the ion transport numbers of two .species differ considerably the concentration field is influenced. Consider, for example, Desahation.

7 (1970) 297-308

MATHEMATICAL

MODEL FOR CONCENTRATION

FIELD IN ED CELL

301

a case where one set of ions is much more mobile than another. The mobile one will run back and forth between the electrodes, while the slow ions concentrate near one electrode, forming a sort of a boundary layer.* The ion transport numbers of Na+ and of Cf- are approximately the same, and to simplify matters they are assumed equal. Because the solution is electricMy neutral the numbers of these ions are the same and therefore so are their nobilities. Therefore they do not contribute to the mass balance in the J-direction.

dn

Fig 3. Second order differential control volume.

The diffusion contribution (see Fig. 2):

to the mass balance comes out directly to be

Conwction As shown in Fig. 2 there is a convection contribution of - u(dC/%r)dxdy. There is no contribution in the y-direction. This point needs some elaboration. On the length scale considered in Ref. (5), u = 0, and this is the length .xale used here too. However, this is not enough to deduce dy = 0. For this particular argument a smaller length scale must be used. Consider the second order differentiai control volume in Fig. 3. On the length scale used in * This boundary layer may act as an extension of the ekctrodc, forcing the mobile ions toerchange &a&x only with it. In such a case the definition of the ion transport number becomes more

dit?icult. Dcsa/inatiott. 7 ( 1970) 297-308

302

D. PNUELI

AND

C. GRCBSBIAN

Ref. (5) u = 0 which on the second order differential scale must be interpreted as Rx . J 0

fxq = 0.

The mass carried into the first order differential control volume by convection in the _r-direction is

. J dr

i:Cdc

0

but because o and C are assumed continuous

The first term on the right-hand side is identically zero; the mean value theorem is applied twice to the second term to yield

Taking into account the convection mass flux through the opposite side, the net con?ribution in the y-direction becomes

which is of a higher order than the terms obtained for diffusion and convection in the x-direction. Collection of all contributions yields

= D a% ~ a2c dc-tu ax’ ay2 at ax

I

-1

(1)

In all practical cases the diffusion in the x-direction is small (compared with convection in the x-direction), and can be neglected. If steady state is also assumed, Eq. (1) becomes dC

x=

D

a%

v

(2) Dexdinarion,

7 (1970)

297408

MATHEMATICAL

MODEL FOR CONCENfRAftON

303

FIELD tN ED CELL

Where rdis copied from Ref. (5s):

and /I is a characteristic of the spacer. Let an average concentration

in the didysate

and in the brine be defined as d

c,

=

.!_ Q

.

J

in diaIysate

uCd_s

-d

d

s

in brine

trCd_s

-If

Since matter is exchanged only between brine and dialysate, fur a sufficiently large number ofceils c,

+ cs = CO

(5)

where C, is the inlet concentration. Let P denote the positive ions transport number in the solution, N denote that of the negative ions, and P+, N,, P_. IV.+, denote transport numbers of the corresponding

ions through

the positive

and the negative membranes (for ideal mem-

branes the fast two are zero). Because the solution is e1ectricaIty neutral P - N = P, - N,

= P_ - N_ = I

and because current is carried by ions only

which is a boundary condition for Eq. (2)_ In general i is dependent on the concentration. Data on the ohmic resistance of electrolytes (6,7) indicates a slight decrease in the current along the cell, which in practical cases may amount to a few percent. Since the whole drop in the current density amounts to a few percent only, it is approximated by a monotonously decreasing function which exhibit a generally similar behavior. This function is chosen to make the mathematics simpler. L(in Eq. (3) depends on 4’ only. Therefore separation of variables in Eq. (2) yields C = &) Substitution

exp (-

u’.u)

(8)

in Eq. (4) now makes C, have the form

CD = CO exp (-

y*X)

(9) Desahztion,

7 (1970)

297408

D. PNUELI

AND G. GROSSMAN

where C, is the inlet concentr3tion. The boundary conditions of Eq. (7) now beeome

ac i$

ix = --__

- y2c, *Q ---

cxp (-

y2x) Tit _r =

d

(membrane) (IO)

0

at_r=O

(eel1 middle)

& where the assumption of symmetry with respect to the s-axis is made to symplify computations. The separation of variables in Eq. (8) indicates that the function which approximates i in Eq. (7) is just i = i,exp(-

p’r)

(1ll

and

where CL is the average diaiysate conce:entratioa at the exit. Indeed one may consider the argument that ican be represented approximately by such a fun&ion (because i changes a few percent only) as a justification for the separation of variables with only one term retained, instead of a series solution representation. Substitution of Eq. (8) in Eq. (2) and in Eq. (IO) yields

P’Q @ u -d2rl + _-....-_-._2Dd p-1 d_s2

Let a new i~d~~ndent

variable

- ev(-

PU - r/d&i rl= 0

be defined

<=/Xl -Yld) and a constant #pz =r

Y’Qd - 1)

2DflUl

In these notations

Eq. (13) become

(c&i middle)

(membrane)

MATHEMATICAL

MODEL FOR CONCENTRATION

FIELD

IN ED CELL

305

THE SOLUTION

Eq. (16) constitutes an eigenvatue problem, with the eigenvalue being cp_ (Physically the more “natural” eigenvalue is y* which determines the rate at which the current density changes atong the cell; however, mathematically the form of Eq_ (16) is more convenient). This eigenvalue probferr is not the Strum-tiouvilte classical problem because the eigenvalue appears in the boundary condition too. An additional complication is the form of the differential equation and the iarge values that /? may attain (3 < /I < 30, (8)). Numerical methods can be utilized, that through the use of computers will yield **exact” tabulated solutions to this problem. It is felt, however, that this is not the most important object of this investigation. Approximate methods are suggested, which yield solutions having the foltowing properties: I) They satisfy the boundary conditions. 2) They satisfy the differential equation in the mean. 3) They are very good approximations near the membrane and in the cell middle. 4) They have simple analytic description and can be relatively easily used in further investigations of electrodiaiysis (e.g., in the reduction of membrane polarization, which is not done in this paper). Let q in Eq. (16) be expanded in power series 00

Let exp (-c) be expanded into series too. Substitution of both series into Eq. ($6) and the use of the second boundary condition yield:

+

- i;4/12 %u - (~~(1~16

+

p/i20

-

. ..)I

(18)

The terms shown in Eq. (18) are sufkient to describe q with an error of less than I %, for c G I. This series becomes useless once c attains values of the order of fi, which it must attain because 0 S C < /I is the region considered. Therefore, Eq. (18) is used to “accurately” describe q near the origin. Because 0.1 < cp2 < 0.6 (Ref. (7)). ‘1 = C&

- 1) V2C*

0 < i < OS

which is a straight line, is also a good approximation Now consider Eq. (16) rewritten as:

-d*q = 4’

(p2[1

- ew(-

for q.

jnlrt Desalinaritm,7 ( 1970) 297-308

D. PNUELl

AND G. GROSSMAN

Near C z 0, q behaves as a straight line. Near i ;2: j?, q behaves-like cos rp 4. Because both (I- exp ( -QT] and q are non-negative and because both increase with i, (dzq/d
dtl =

Co@ -

rp2 at c = 0

1)

-iif to

at

-z-=0 *

c=/I

Such a behavior of q can roughly be described by g z A cos cp(B -

c) i- B.

(21)

rp G Nw)

Let Eq. (20) be integrated once between the Iimits c = 0 and t: = 8. Because of the boundary condition at 5 = fl P

~o(p_~)rp22!Li

dF Ir=o

=cp2

[l

- exp (-

01 ttQ<

0

which may be considered as an application of the Karman-Polhausen integral method. The form of q from Eq. (21) is chosen to be used in Eq. (22) because it has the general desired behavior and because it becomes more and more accurate as 5 + b. It can become more accurate (locally) at C -+ 0 too if p = n/(2/J) is chosen. Substitutjon in Eq. (22) yield: Acp

sin cpj? = Co@ - 1) cpz

or

(23) A 2c,@-l

2

B

I

and b

Co@ -

1) =

s

Cl - expC- 01

0 -f-

B-jd5 =

0

=

+

Co# - l)+

BLIP-- 1

[

nc.~][exp1_8)-~]1[l+nZ]

+exp(-_

4F

+ (24)

DesalinaGon,7 (1970)297408

MATHfMATlCAL

B

=- f

hiODEL

co-B-1 B

FOR CONCENTRATION

Ex -

28 exp (-

FlELD

IN ED CELL

8111((4B2 f a2)LY - 1 f exp (-

Substitution of A and B in Eq. (21) yield:

307

S)J>

.

The value of qa in Eq. (18) may now be set to equal B in Eq. (I 8). However, computation of B/A for 3 d p < 30 yield values of about + 1% or less. Hence, in accord with the approximation used here both 4, and S may be approximated by zero.

e’

I

I

1.0

0.6

,

I

0.6

I

I

0.4

I

I

0.2

1

’

0

_L d

Fig. 4. The concentration field.

One universal curve, for alI j? and all Co, can describe the solution. Such a curve is given in Fig. 4. CLOSING REMARKS

The results obtained in Eq. (25) and shown in Fig. 4 may require some brief discussion. One may note that if exp (-0 is neglected with respect to 1 in Eq. (16), these results are obtained as an exact solution. However, this is a good approxim-

308

D. PNUELI

AND G. GROSSMAN

ation at < near p, and not so at < near zero. The reason for the integral solution yielding these results is that the vetocity profiles are rather “flat”, as can be seen in Fig. 3, Ref. (5), or by drawtng.the graph of Eq. (3). In other words, the velocity is nearly a constant over the mqjor region of integration, and since the right-hand side of Eq. (221 is just J$trqdC,it makes the solution resemble such an approximation everywhere. indeed, over that part of the region in which the vetocity is “almost” constant Eq. (25) describes an “almost” analytic solution. At the membrane the solution must be justified in a different fashion. In particular it must be shown that the integration over the large region of almost constant velocity does not “average out” the details near the membrane, by the shear weight of the large integration region. This does not happen here. Just at the wall the differential equation is satisfied (Le., both (d2g/diz) and I - exp (-JJ are zero). and the boundary condition is satisfied too. Moreover, inspection of Eq. (25) and Eq. (IS) (the exact solution near the wall) yields that there is a region near the membrane, 0 6 < f 0.5, in which the difference between the exact solutiort and the integral one is Iess than iYO_ Thus Eq. (25) is a solution which is good at both ends of the region, over the major part of the region (where the velocity profile is almost flat) and also integrally over the whole region. REFERENCES

I. L. ?I.SHAFFER AND &I_S. MINTZ. E!ectrodialysis. in: Principkof Desalinafion, K. S. SPIEGLER. Ed., Academic Press. New York, N-Y.. 1966. pp. 199-289. Z A. !!&AN AND Y. W~NUGRAD. Boundary Layer Analysis of Polarization in Ekctrodia!ysis in a Twc+DimensionaI Larninar Flow, TME-101. Fluid Mcch. Group. Department of Mechanica Engineering, Tcchnion. I.I.T., Haifa. Israel, 1968. 3. L. J. MAS, P. M. PERRARD, P. A. PRAX AND J. C. SOHM,Behaviorof an ElectrodialysisUnit Cell,Thomson Houston Co. Report. Chatou, France, 1948. 4. A. A. So%x~h‘ ASD R. F. PROBSTEIH. A Hydrodynamic Theory of Desalination by Ekctrodialysis. Pd. No. 684, Fluid Mech. Lab.. Dcpart;ment of Mechanical Enginwing. M.I.T., Cambridge, Mass., 1968. 5. D. Phvnr AND G. GIC~SSIAS. A Mathematical Model for the Flow in an Elcctrodialysis Cell. Dcsahtarion, 6 (1965) 303.

R. W. GURSEY, IonicProcessesin Sohions, McGraw Hill, New York, N-Y.. 1953. 7. B. A. ROBIPSSON .\ND R. H. STOKES,EIrcrrolyre Solutions, Butterworth, London, 1959. 8. J. R. WI-N, Demherafization by Uectrodialysis, Buttcrworth. London, !960.

4.

Desalitt~tion, 7 (1970) 297-308