A hydrodynamic theory of desalination by electrodialysis

Desulinarion - Ekvicr

A

Publishing Company.

HYDRODYNAMIC

THEORY

Amsterdam - Printed in The Ncthcrlanus

OF

DESALINATION

BY

ELECTRODIALYSIS

has been developed fcx a multichannel system with steady laminar flow between plane, paraftef membranes. The modeling of the system is found to be governed by four basic similarity parameters: (i) a dimensionless appIied potential. (ii) the product of the channel aspect ratio and the inverse Pfclet number, (iii) the ratio of brine and (iv) a parameter measuring membrane and dialysate inlet concentrations. resistance_ For sutficientlg long channels it is shown that there are two distinct regions: a “developing” region where the concentration diffusion layers are growing. ‘and a “developed” region where the ditrusion layers fill the channel. Parabolic and uniform velocity profiles arc considered and self-consistent solutions are derived for the distributions of salt concentration. electric field and current density in the A hydrodynamic

theory

of demineralization

by efectrodialysis

system. as well as for the total current. An intcgrat method In the limits of low and high polarization analytic solutions

of solution

when matched at their point of equatity cioseiy approximate cal solutions. It is found that under a \iide range of operating solution

for the total current

i = [I

-

exp(-

is represented

by the empirical

is used. -

are obtained which the complete numericonditions,

the

formula

W)J1!3,

where i and q are, respectively, a dimensionless current and potential embodying the four similarity parameters mentioned. Comparison is made of the calculated limiting total current with experiment. SYhlBOLs

(MKS units are given for illustration) c c#n

-

cOB

-

C

-

dimensionless dimensionless

ion concentration, C/Co, ion concentration at membranes in dialysate dimensionless ion concentration at inlet of brine channel, COB/Co, reduced ion concentration, Z+C+ = Z-C,, kg equiv/m’ DesaJinatJon. 5 (1968) 293-329

A. A. SOSIN

293

c,

-

<-OS

-

c on J D D,

-

%i F

-

i 1. f r’

-

i

-

J

-

k lit i

-

P@

-

R

-

t

-

T I’ Y r;,

-

s

-

x 1 J’* IT ; 2 zr

-

Fr s A -= A* z#

-

=z. $L

-

Q,

-

concentration of positive/negative reduced ion con&ntration at inlet reduced ion con&ntration at infet haff width of channefs, m effeciive drffusion coefficient (Eq.

AND

R. F. PRO&STEIN

ions, kg mole/m’ of brine channef, kg equiv/m” of diafysate channel, kg equiv/& 2.8). m’fstc

diffusion cucfkient of positive/negative ions. m’/sec dtmctrsiontess ekctric field at membranes in &a&sate

(Eq. 3.40) Faradajr’s constant. 9.65 x IO’ coul/kg equiv dimensionless current density (Eq. 3.6) value of i where diffusion, lagers merge dimensionless total current (Eq. 4.32) correlating dimensionless total current (Eq. 7.2 or 7.7) current density, ampim’ total current per unit channel breadth, amp/m parameter defined in Eq. 5.3a parameter defined in Eq_ 5.3b channel length, m Pdclet number. I’;,dlD universal gas constant, S.ji x f03 joutei“K - kg mole time, set temperature. ‘K dimensionless flow velocity, V/V,, fiuid flow velocity, m;sec average flow vefocity in channel, mfsec dimensiontess transverse coordinate, _Y.ulr$ transverse coordinate (Fig. I), m dimensionfess lengthwise coordinate (Eq. 3.6) vafue of _r where diffusion layers merge fengthwise coordinate (Fig:. t). m I-s charge number charge number of positive/negative ions fiux density of positive/negative ions, kg mole/m’ - set dimensionless diffusion iaycr thickness in developing region (Fig. 4) half thickness of cation!anion selective membrane. m dimensionless membrane thickness, A*fd dimensionfess membrane resistance parameter (Eq. 3.5 or 4.123) dimensionkss parameter (Eq. 2.9) conductivity of cation/anion selective membrane, mholm dimensionless membrane conductivity (Eq. 3.6) eiectrical potential, I/ applied potential per channet pair, V Desalinatian, 5 (I !Y68)293-329

HYDRODYNAMIC

+

-

v @

-

THEORY

t?Y ELECTRODIALYSIS

potential, ZFd/RT applied potential per channel

dimensionless dimensionless correlatmg

OF DESALINATIOS

dimensionless

applied

potential

pair,

295

ZF@/RT

per channt-4

pair

(Eq.

7.3 or

saline

has

been under

7.8) 1.

lSTRODUCTtOS

The

electrodialttic

method

of demineralizing

water

investigation and development for over twenty-five >mrs (1. 2), and now finds application in the desalination of brackish as dell as sea waler (3). The cc;mmon form of the electrodialysis system is shown schematically in Fig. I. Alternating cation and nnion selcctivc membrrtnes form narrow channels into which the saline water is passed. An clcctric field is applied transverse to the membranes, and under its action cations such as Na+ pass through the cation selective membranes and anions such as CI- pass through the anion selective membranes. In the setup shown in Fig. I rhis reduces the salt content in the channel formed by the left pair of membranes (the din&ate channel) and increases it in the channel formed by the right pair of membranes (the brine channel). In practice stacks made up of a large number of alternating dialvsatc and brine channels between one pair of electrodes xc employed. and a number of these stacks may be used in series to achieve the desired

Iekel of desalination.

2A_ '+-+d:

25

Z&+

28

2a_

-i--1 7

No*

,Cf-

-7

-

J

Cathode

4,node *

Y i

I.2

f

1

J Flow

I

Fig. 1. System and coordinates. Dcsulinuriurt.

5 (1968)

293-329

A. A. SOSIN

296

ASV

R. F. I’RQIW-EIN

The fundamental problems in electrodialysis fall into two major areas. The first of these includes the !argely chemical problems associated with the membranes: their pcrmselectivrty. their rcsistivity. their susceptibility to deterioration and fouling. etc. At the time the method was conceived suitable membranes did not exist and it is this arca that has 10 date received most attention. At the present time the technology of the ion selective membranes is fairly wvell developed. The second class of probiems arc of hydrodynamic origin and depend on the derailed flow phenomena in rhc system. In this category are the several effects which arise from concentration polarization near the membrane-fluid boundaries and which arc recognized to be critical importance in system design. Polarization sets a limit to the usable current. It also gives rise to a low salt concentration and high el.ectric field near the membranes. which in turn may lead to membrane fouling.

of

precipitate

formation. and other effects deleterious to membrane life (4. 5). This hydrodynamic area has’reccived less attention and to date a self-consistent. rational theory for the hydrodynamic aspects of the electrodialvsis process is lacking. The present work is an attempt to develop such ~1 thcoq for steady lnminar flow. We take the membrane properties as given and derive a self-consistent theory for the distributions of concenrration, current density, and electric field strength in a multicompartment electrodialysis system. It is expected that the results of the analysis can provide conditions.

2.

MODEL

AS0

a rational

basis for the optimization

of design

and operating

EQUATlOWi

In this section we outline the basic physical. geometrical. and operational assumptions defining the model. and the governing equations which result. Apart from the basic assumption that the Reynolds numbers are low enough that the Row is a laminar one with the concentration diffusion governed by laminardiffusion equations there are three basic physical assumptions: (I) The fluid consists of an unionized solvent and an infinitely dilute, fully ionized sak (2) The fluid is incompressible

and at constant temperature. (3) The membranes are perfectly selective, haveconstant electrical conductivity with respect to the permeable ion, and are impermeable to water molecules. WC would emphasize that any one or all of these assumptions could be r&axed at the expense of additionally complicating the analysis. It is felt, however, that the restrictions imposed still provide for a realistic model of an actual electrodiaIysis process. A consequence of the first assumption is that it separates the problems of determining the velocity and concentration distributions in the channels (so long as free convection effects are negligible). Furthermore, the diffusion equations for the ions are simplified in that concentration replaces activity, and the diffusion Desaknariott,

5 ( 1968) 293-329

HYDRODYF,\MlC

THEORY

OF DESALINXTIOS

l3Y ELECTRODIALYSIS

297

cotllicient and mobility arc related by Einstein‘s relation, And finrrlly, there is only one positive and one negative charged species and these do not enter into any reactions in the bulk of the fluid. The second assumption is consistent with the fact that the flow velocities in eIectrodial1 sis systems arc low enough that the fluid may be treated as incompressible. Further. if the power inputs mto the system are not too Iarge the fluid will be essentiality at constant temperatureThe third assumption is essentially that the concentrations of fixed charges in the membranes arc uni!brm and very larse compared with the ion concentrations in the fluid. The potcntinl drop within the membrane can therefore be calculated from a simple Ohm’s type law and the boundary conditions on the ion fluxes at the membranes become relnti\cly simple. AS a conscquencc of the first two assumptions, the equations which _povcrn the flow of charged species in the fiuid are their mass conservation equations t3T

f +

I3

v* r, =

and the condition

(2.1)

0.

of locnl elcctricai

IZ+C, - z_c_.i

&

neutrality

z_c+.

(3.2)

Here. C is the ion conccntmtion. I- the ion flux density. Z the number of electronic charges. while the subscripts + and - refer to the positive and negative ions, respectively. The flux densities of the ions (in moles per unir area per unit time) are given by

r,

=

C+V

-

D,VC, -

-

I(Z+FD_ff?TKaV& -

-

(2.3)

where V is the bulk flow velocity. D the ion diffusion coefficient, # the electrical potential. T the temperature, F Faraday’sconstant, and R the universal gas constant.

The first term on the right hand side represents convective flux. the second diffusive tlur. and the third migration in the electric field. The unknown variables in Eqs. 2.1 to 2.3 arc C,. C_, and (p. The bulk flow eelocity V is assumed to be known separately from the equation of mass conservation for the solution as a whole

v-v=o.

(2.4)

and the equation of motion, which for brevity we shall not write down here since we shall not use it direct@. We now introduce the reduced ion concentration

c = z,c,

= z_c_,

(2.5) Dcsulinurion,5 (1968) 293-329

*

A.

295

A. SOSIN

AND

R. F.

PROl3STElS

and on making use of Eq. 2.4 reduce Eqs, 2.1 to 2.3 to _2C -- -+-V+VC = DV”C,

(2.6)

i;t

Here, D is an efiective diffusion coetiicient (2.8)

and

Q-9) is a

whose absolute

vague is less than unity (usually small compared witb unity]. Eq. 2.7 is simply an expression of current continuity. as can be seen from the equation for current density obtained by taking the difference of the positive number

and negari% ion ffux densities given by Eqs. 2.3 j=

-

F(z,D,

+ Z-D-)

(OVC

-i-

(2fO)

where A j = 0. l

From the third assumption the potential drop within the membrane can be catculated from the simple Ohm’s type relation j=l

-@nbb

(2.11)

where cr is the electrical conductivity of the membrane with the permeable ions as char@ carriers and is constant throughout the membrane. From charge continuity. V *j = 0. so that the potential satisfies Laplace’s equation within the membrane. In addition to the resistive potential drop across the membranes it is necessary to take into account the Dannan potential drop (6), which results from the discontinuitics in charge concentration at the membrane-~uid boundaries. Let us now introduce several seometricat and operational assumptions which in genera! are characteristic of elcctrodiaiysis systems and which at the same time simplify the analysis. We shaif confine our attention to the twodimensional geometry with plane paratfet membranes shown in Fig. I, and make the following additionai assumptions and approximations: ( I) Tfz flow is steady. (2) Free convection efEcts may be negtected. (3) Derivatives in the Y direction along the channel may be neglected in Desi&ration, 5 (1%8) 293429

HYDRODYNAMIC

THEORY

OF DESALtNATIOS

299

RY ELECTRCiLXALYS1.S

comparison with the corresponding derivatives in the Xdircction across the channel. This results in there being on& a component of the current in the X direction, transverse to the ffow. When the ratio of channel length to channel width is large. as is usuatly the case in practice. this approximation is cIoseiy sritisfied. (4) There we a large number of dialysate and brine channel pairs in the system, so that the potentinf drop across the fwo channets adjacent to the electrodes is smaft compared with the drop across the rest of the system. ft foliotss that the

potential drop across ;a chanpd pair (one dialysate channel rend one brine channel. including the two associated membranes) is constant and equal !o the total applied voltage divided by the number of channel pairs. (5) The dialysatc and brine channels have the same separation. 2rL between the membranes whose thicknesses arc uniform and equal to 2A+ and 2A_ for the onion and cation selective membranes, respectively. (6) The fluid enters the region where current is allowed to ftow with a fully dcvekped velocity profile. kc*-. t/ = c/(X). and the velocity profiles and average flow speeds arc the %me in both channek (7) The diffusion coetiicients nnd charge numbers of the positive and negative ions are taken equaf. ix., 2,

= z_

= 2.

D,

= D- = 0.

where D is the ctTectivc diffusion

(2.12)

coeflicient defined in Eq. 2.8. From

Eq. 2.9 it

follows that 0 = 0.

(2.13)

This approximation is not essential to the analysrs but its introduction simpIifzes the equations and the form of the boundary conditions considerably. So long 3s the absolute \aiue of 0 is small compnred with unity. as is the case in m~lnyconlrnon electrolytes, the basic features of the solution wifl remain unchanged_ tntroducing the above approximations into Eqs. X5.2.7 and 2-f 1 they reduce, with the aid of Eq. XJO, to c2c .--

‘1

t;X’

I’(X)

cc

----.

----1

24 II

3-T =

.

RTAY) -.

(2.14)

c/“Y

D

.

..-..I_

in the fluid,

(2.15)

in the membranes,

(2.M)

ZZF’?DG

where CT+ and a_ are, respectively, sefective membranes_

the conductivity

of the cation

Desalitmtiott.

and anion

5 ( 1968) 293-239

300

A. A_ SOSIN

ASD

The corresponding boundary conditions (see Fig. 1) are: At the inlets of the channels the salt concentrations specified, i.e., at 1‘ = 0 (i)

C(xc.0)

R. F. PROUSTEIN

are unifcrm

and

= co,

in the dialysate,

(2.17a)

= CM?

in the brine.

(2.17b)

(ii) In the dinlysate channel at the anion selective membrane, X = - (I. the ion flux is zero. 1.9 the brine channel at the cation selective membrane. - (1. 1he negative iou fluu is zero. Using Eq . 2.3 with V set equal to zero at x= the membrane surface and Eqs. 2.5, 2.10. 2.12. and 2.13 these conditions can bc written positive

SC -.-

=: +_-KY.)

Here, the upper channeIs. (iii)

(2.18)

ZFD‘

dX

Similarly,

and lower signs apply we obtain

respectively

to the dialysate

and brine

for X = 0. (2.19)

(iv) The total potential drop across a channel pair is constant value. @_ This condition couples the solution for the two channels. 3.

SlMtLARIN

PARAMFTERS

AND

DIMENSIONLESS

at 3. given

VARIARLES

The system geometry. governing equations, and boundary conditions show the present problem to be specified b_ttthe folIowin_g independent parameters: I_ Geometrical* (i) Width of channels, 2d (ii) Length of channels (i.e., length of portion which carries current), 1. (iii) Thickness of cation selective membranes, 2A+. (iv) Thickness of anion selective membranes, 2A_. 2. Fluid and flow properties (i) Characteristic ffow speed, e.g., the average speed V,, (taken to be the same in both channels). (ii) Temperature, T_ 3, Salt properties (i) Ion diffusion coefficient, D. (ii) Ion charge number, Z. l

The breadth of the

(two-dimensional

channel does not enter since it is taken to be large compared approximation).

with the width

Desalinarion, 5 (1968) 293-239

HSDRODYNAXIIC

(iii) (iv) J.

THEORY OF DtSALlSAflOE

Concentrntion Concentration

at inlet at inlet

BY ELECTROf>lhLYSIS

301

of dialysate channel, Co,,. of brine channel. Co,.

Membrane properties (i) Conductivity of cation (ii) Conductivity of anion

selective selective

membranes. membranes.

G+_ G_,

5. Applied voltage per channel pair. +. In addition. rhe equations also contain Faraday’s constant F and the universal gas constant R. On the bnsls of formal dimensional antiiysis, tire problem is seen to contain nine dimensionless similarity parameters. I

ri

A+

A_

COH

7 11

d

z

I o,tl

RT

D

co,,

o,RT a-RT - ‘-.., ‘i --‘--F’DC,, F-DC,,,

t-CD

-

(3-U

We have here discsrdcd V,,j(RI)‘.’ (rsscntially the Mach number) which is neelieibly small in the present problem. and the ion concentration itself since the + soiurion is tahcn to be infinitely dilute. i\s we shall :xc below some of the above nine pxameters may be grouped ~ch that in fact ONI~_~~VWGmiiitude parameters sufftce to specify the problem. These quantities sre the :tsptxt ratio of the channel, i/c/, times the inverse of the Pticict number (3.2)

( /,rl) PC - ’ = iD;d;”I’;, ; tht dimensionless

potential

drop per channel

pair (3.3)

V = ZF@jRT; the ratio of brine fo diaiysate %a

concentration

at the inlet (3.4)

= Coil!C*lG

a quantity which measures the ratio of the resist3ince with a channel pair to the fluid resistance

of the membranes

associated

(3.5)

rt = . -- ---.--. Consistent and parameters X _y=-_.

with these parameters

lf JI

=.?5!3,i =: RT

Y

I

y=-._ *...___c=_--, li

Pe

jcf ___--.-.., ZFDC,,

In terms of these dimensionless

we define

c

v vat*

c=____

c

reduced

variables

(3.6)

OD

a: RT 0_+ = jxC-0; variables

the following

A-+ &2 = - tl -__

Eqs. 2.14-2.16

become Desalination, 5 (1968) 293-239

A. A. SOSW

AND

ft. F. PROBSTEIX

13.7) a+4 = -_.t’s z-

The boundary (i)

iW c

in the fluid.

(3.8)

i(_r)):~,

in the

(3.9)

conditions

(ii)

Eqs. 2.17-Z. 19 tz ke the form: in the dialysate,

At .I‘ = 0. c = 1 -_ At s = -

mcmbraneG.

(3. IO)

in the brine.

‘OS

1, -?- = + i in the dialysatr. t:s = - i in the brine.

?C (;ii) At .Y = I, -;-I = - i

C’S

==+i

(3.11)

in the dialysate, (3.fZ)

in the brine.

(iv) Total drop in JI across a chmnel

pair is constant

and equal to Y’.

Before proceeding to the method of solution we first outline the quafitatlve behavior of the concentration and potential distributions in the dialysate and brine channels. A schematic representation of these distributions is shown in Fiss_ 2 and 3 for the case tihere the infct concentrzxtions are the same in both charm&. In the present model at the point the fluid enters the region where current is allowed to flow the velocity profile is already fully developed and the ion concentration is uniform (see Fig. 2). As a result of the nearly uniform conccntration close to the iniet, the fluid responds to the appiied electric field simply like a medium with constant electricaf conductivity, i.e., there is a linear potential drop throtqh the fluid as we11 as the membranes and the Donnan potential drops are negligible (see Fig. 3). Further downstream. concentration gradients devetop adjacent to the membranes as a resuft of the ion loss or gain {depending on the channel) and these concentration boundary layers grow in thickness as illustrated in Fig. 2. The ion concentration decreases at the membrane surfaces in the diaiysate and increases in the brine. In the dialysate channel as a result of the large concentration gradients the potential drop is larger than that associated with the average conductivity

tIYDROI>YSAXllC

THEORY

OF DFS4LISATKPl

303

RY ELECCRODIALYSIS

v

.. \. -

‘---

i -i-’ !

0;

.

: t

L

J* :

~

:

.. -.

-I_-

;.*

*. L ~. --!

.

t O~OI,POh3 channel

F,g. 2. Skctsh

of salt concmwation

Fig. 3. Shctch

of’ porenti~1

distribution

dldr8buriun

t

Br.ne Chonn*l

in cbanriels.

in chmrds.

in the channel. with the drop bein=oxsharpest _ at the membrane bound~~ries. In the brine channel the opposite situation prevails. In addition. the Donnan potential

drop at the fluid-membrane boundnrles becomes significant because of the increased concentration discontinuitic(;. The electric field is. of course, a maximum at the membranes in the dialysa;e channel and is continually increasing as the concentration boundary layer> develop, with the decrease in conductivity leadins to a decrease in current density. Eventually. the diffusion layers fill the channels and thereafter the ion concentration begins to decrease in the center of the dialysate channel and to increase

in the brine channel. If the lengths of the channels were to increase indefinitely the concentrations in the dialysate and brine channels would tend to limiting values corresponding to the total applied potential drop being taken up by the Donnan potential resulting from the concentration discontinuities at the membrane interfaces. It should be noted that the concentration in the dialysate channel will not become zero. At these large lengths the electric field and the current density tend to zero. Desalination. 5 (1968) 293-329

A.

304

A. SOSIX

ASD

R. F. PRORSlEIN

By analogy with velocity profile development in the entrance region of a pipe or a channel, WC call the re_gion where the diffusion layer is growing the “developin_g” region, and the region where the diffusion layer has filled the channel the ‘Sdeveloped”

region. The use of this latter term is not quite analogous to its use in connection with a fully developed velocity profile. which ceases to change once the viscous boundary layers fill thechannel, since theconcentration profilecontinues to deform as long LIScurrent flows.

To sohe Eqc;. 3.7-3.9 wz adopt an integral method similar to the one familiar in bounda? layer theor?. In this method we assume r, functional form of the concentration profile containing unknown functions of J which are subsequently determined by solving an integrated form of the ion conservation equation under the constraint, already noted. that the total potential drop across a dialysnte-brine channel

pair. including the associated membranes. be constant. In the region of where the ion diffusion layer is growing in thickness, the unknown functions are expressed in terms of the current density and the diffusion layer thickness, while in the region further downstream. where the diffusion layers 611 the entire chrtnnel, the role of the diffusion layer thickness is taken over by the maxithe flow

mum ion concentration in the channel. The regions in which the concentration profile is developing and debTloped are considered separately below. 3.1 The developing cmcetUraIiotl profikc

In the region work with a

where the diffusion

coordinate

measured

layer is growing

it is more convenient to

out from the membrane

from the center of the channel. Let us consider the diffusion the right hand membrane in either channel (Fig. 4) and set -_=

I -

surface

rather

layer developing

s.

than on

(4.1)

Eq. 3.10 remain unWith x replaced by z, Eq. 3.7 and the boundary condition changed. The boundary condition Eq. 3.12, appropriate to the right hand membrane. is applied in terms of; at z = 0. With S defined to be the z coordinate (see Fig. 4) we also impose the conditions

of the outer edge of the diffusion

layer

for _- 2 S in the dialysate,

C=l

= cos

(4.2)

for f 2 d in the brine,

and dc

-

s-

= 0

for z B 6 in both channels.

These latter constraints that the concentration diffusion layer.

(4-3)

must be added in an integral method of solution to ensure and its first derivative

are continuous

at the “edge” of the

Desalinarion,

5 (1968) 293-329

tIYDRODYSAXlIC

-I

.I

0 (b)

Fully

THEORY

CiCveloDed

OF Dt3ALlNATlOS

-I

+I

0

ConcenIrot~on

proflIes

Fig. 4. Parrtmctcrs defining dimensionless

conccntr.don

distribution.

Integating Eq. 3.7 from z = 0 to S and applying Eqs. 3. f 2, 4.2. and 4.3 we obtain -

ti

NS) - f r(z)

d_s . 0

[c(z.y)

-

305

BY ELECTRODfALYSIS

t] dz

=

-

i(y)

the boundary

conditions

in the dialysate, (4.4)

a03 d ---- [ r(z) [C(ZJ) d_v L

- COB-Jdz = i(y)

in the brine.

0

We now assume a quadratic

distribution

c = a&)

G,

of concentration

in the diffusion-

layer + n*(y) z + n&)

(4.5)

and evakite the three functions a,(~-) by again applying Eqs. 3.I2, 4.2, and 4.3. This yields, for z < s, c=(1-

G/2) + iz - (i/26) 2’

in the dialysate,

= (COB+ is/21 - iz -I- (i/Z&) zz in the brine.

the boundary

conditions (4.6) (4.7)

5 (1968) 293-329

A. A.

306

velocity

The solution

profile in the diffusion

of the equations

used in the

channels

and

governing

the

flow

SOSIN

layer is assumed

the solute flow. When

is laminar

ASD

R. F. PRORSTEIS

to be known

from a

no spacer material is

and debeloped,

then the velocity

profile is parabolic r(z) = 3/2 ;(2 - z).

t4.8)

The determination

of the velocity profile for the case when there is a spacer material in the channe!s is a separate problem and will not be dealt with here. For the present purpose. we note that when the Reynolds number is low enouph (so that the Row is laminar) and the spacer material is uniformly distributed. its primary effect is merely to make the velocity profile more uniform on the average. Thus. in many practical cases the velocity profile should fall between the parabolic shape of unimpeded laminar flow and the completely uniform profile V(Z) = 1. The

however, governed

(4.9)

presen:

theory

is applied

to these

two

limiting

casts.

We emphasize,

that with a spacer material present the concentration protile may not be by the laminar diffusion equations which are used here. Rather. the

concentration

diffusion may be more akin to a turbulent process. despite the fact

that the flow itself

IS a laminar

one.

For the parabolic wlocir~projik, we insert Eqs. 4.6 or 4.7 and 4.8 into Eq. 4.3 and on integrating obtain

f..$

[i6’(1

- g)]

= i.

(4. IO)

For the uniform ve~ocif_t-pmjiieEq. 4.9 we obtain in a similar 1 d _ -_ (id”) 6 d_r

manner

= j.

(4.11)

Eqs. 4.10 and 4.11 apply to both the dialysate and brine channels, indicating that the diffusion layers in both channels have a common rate of growth. This is a consequence of our assumption that the flow velocities in the two channels arc equal. Furthermore, it is evident from the symmetry of the boundary conditions on the concentration that Eqs. 3.10 and 4.1 I apply to the diffusion layers on both sides of each channel. This simplification arises from our assumption that the positive and negative ions are equally mobile (Eq. 2. ! 3). The equations for the diffusion layer thickness must be solved subject to the initial condition d = 0 at which

_V= 0,

is an alternative

(4.12) statement

of Eq. 3.10 within

the integral

formulation.

In

Desalination, 5 ( 1968) 293-329

HYDRODYSAMIC

THEORY OF DFSALISAIIDS

BY ELECTRODIALYSIS

307

addition, the ttlrcady noted constraint that the total potential drop across a channel pair is constant must be imposed. This constraint provides the necessary relation bctv\ecn the difTusion layer thickness 5 and the current density i. The potential

distribution

is given

by Eqs. 3.8 and

3.9 with s replaced

by =

on the right hand sides (because of the coordinate transformation). The potential drop across the dialpsate channel is obtained by integrating Eq. 3.S across that channel. with c given by Eq 4.6 for z G 5 and c = I outside the diffusion la>er. and

with positive

instead

A$ = ~(iS,‘2)‘tarln Similarly.

of negative

- ’ (iS/Z)i

signs

+ 2i( 1 -

(5).

with c given by Eq. J-7 we find the potential

A@ = j(iii;2L’Or,l*tan-’

(ib/2c,,)’

The combined resistive potential with a channel pair is obtained from

+ Zi(l

-

drop across Eq. 3.9 as

(4.13) drop

across

the brinechannel

3)/c,,.

(4.1-t)

the two membranes

w = 2i(A,i5_ + A_;,_>.

associated

(4.15)

B= are the dimensionless thicknesses of the membranes defined in Eq. 3.6. Finall?;. we have the potential drop which arises from the discontinuities in concentration at the boundaries of the two membranes (the Donnan equilibrium potential) where

(4.16) where the argument of the log term is the ratio of the ion concentration at the membrane in the brine channel to the ton concentration at the membrane in the dialysate channel (see Eqs. 4.6 and 4.7). The factor of 2 accounts for the two membranes, which contribute equally to the potential drop. The total potential drop Y across one channel pair is therefore obtained by summing

Eqs. 4:134.16

1/2‘u

= In cOR -+- (1 + L-G: + 11)i -

+ 2(iS/2cos)ftan-

’ (ifS,/2c,,)*

+ In

(I f ?&a) ici + 2(i6/2)ftanh-’ 1 +__~ id/k,, _[ 1 - ib/Z

1-

(is/Z)* (4.17)

Here (4.18) is the dimensionless membrane resistance basic similarity parameters of the problem.

defined earlier by Eq. 3.5 as one of the For t7 << I, TV-. I, 9 >> I the membrane Desahafion.

5 (I 968) 293-329

A. A.

308 resistance is. respectively. fluid resistance. Eq. 4-17 npphes

to either

aping

concentration

ously

with this equation

parameters 42

region

Here

to the

velocity profile. so that the solution for the dcvelby solving either Eq. 4.10 or 4.11 simuitane-

is given

for i and c5in terms of _r and the dimensionIess

similarity

pro,FIc to the original coordinatr

collccvlrrariorr

WC: ;evcrt

Eq. 3.7 across the channel Eq. 3.1 I and 3.12, the ion conser\ation d -.-

R. F. PROBSTEIN

of the same order. or large compared

Integrating

ditions

AXD

Y. coB, and P,?-

The ifcrciq7ed

center.

negligible.

SOSIN

Al ’

s measured

and applying equation

from

the channel

the boundary

con-

becomes

r(s) C(S._I) ds = I lit_v).

(4. IP)

dx . I --I

where the upper sign applies to the dialysate snd the lower sign to the brine. Again. we assume a quadratic concentration profile of the form c =

b,(y)

+

b,(y)

-I- i

b,(y)

(420)

2.

In this case we evaluate the functions h,(_r) in terms of the current densit) I either the local maximum or minimum ion concentration at _I*(see Fig. -I) c = c&r) =

c,(,‘)

at

s = 0

in the dialysate.

at

s = 0

in the brine.

From the above relations c = co = cg +

(4.21)

and Eqs. 3.1 I and 3.12

l/2 is’

in the diafysate.

(4.221

1/2is’

in the brine.

(423)

For the case of the parabolic

velocity

profile in terms of x

L(_x) = 3/2 (1 - x2), while

and

(4.24)

for the uniform flow V(X) = 1. Substituting the expressions for c and

into

Eq. 4.19 we obtain

for the

parabolic velocity profile

= - i for the dialysate,

2

fcr, - iij

&

(cB + i_%i = i

(4.25)

forthebrine.

(4.26) Desdinatiott, 5 t 1968)

293-329

~iYDRODY~~~~liC

THEORY OF DESALINATIDS

the corresponding

With a tmif;wwl whei/y

equations

309

are

i for the diatysate.

(4.27)

for the brine.

(4.B)

The initial conditions

for cn and C~ arc

cD = 1 and cB ==<08

where _I; reprzxnts side of the channel

RY ELECTRDCtfALYSfS

at

(4.29)

_r = ?;,

the stntion where the diffclsion iaycrs which develop on each rncrgc, th& is. the point at which S = 1 and the developed

region be&s. The value of _r. is obtained from c region. Note tkwt Eq. C?9 applies to the particular

the solution of the developing case considered. where the flow

~clocrties in the brine and dial>sate channcts ;wc equal. If this were not the CXX, the diffusion layers in the two chrtnnels would not merge at the same point. From

Eqs. -I_25 and 4.26 or Eqs. 4.27 and &28 we have that cD + c5 = CUIL~~, so that from the above c‘,,

i-

c)j

condition =

1 f

(4.30)

eg,y

The potential drop across a channel pkr is obtained. a5 in the cws of the developins concentration profile, by integrating Eys. 3.5 nnd 3.9 across thechanncls and mcmbrancs and nddinp the Donnan potential drop which arises from the concentration discontinuities at the fluid-membrane boundaries:

(4.31) As with Eq. 4.17, this equation applies for either velocity profile. The terms on the right hand side of this equation represent, respectively, the potential drop across the dialysatc channel, the potential drop across the brine channel, the resistive potential drop across the two membranes, and the Donnan potential drop. Hew, the dimensionless brine concentration cB has been eliminated by moans of Eq. 4.30. The problem for the developed region is thus reduced to solving Eq. 4.25 (or Eq. 427) and Eq_ 4.31 simuitaneous~y for i and c, in terms of _r and the dimensionless 4.3

simitarity

Total ~w8wzf a&

parameters

efpcr fit

There are in addition

Y, coBI and $1.

JE+i~ ifisfrihiifiotl

to the basic variabtes

S, i, and co certain

other quan-

Dwziinur~~rr, 5 ( 196s) 293-329

A. A. SOKIN

3 10

ASD

R. F. PROIISTEIS

tities of practical interest. One of these is the total current. or the total rate of dcsslination. The ion concentration and the clcctric field at the membranes in the &al> sate are also of importance ns indicators of the onset of significant pH ehnn_gc. which in turn may lead to scale formation and other undesirable side effects (4.5). We note that although the present theory ceases to apply when the salt concentration is no longer htrge compared with the H’ or OHconcentration, it can be used to predict the point at which the pH begins to change rtppreciably. The H * AXI OH- fluxes become of the order of the salt ion flu\ when the air icn concentration nt the membranes approaches the Hater ion conccntrxtion. As a measure of the total current WC delinc the dimensionless qu.tnttt)

(4.32)

The physical significance of I becomes

dimensional

quantities

for i and _r and

clear obtain

when

we substitute

from

Eq.

3.6

(4.33s) \v here r J=

r

(J.33b)

j(,‘)dY

-0

is the real total of the incoming

current per unit channei breadth. Clearly, I represents the fraction salt removed from a dialysate channel between the channel inlet

and the station I_ If Y is the coordinate of the end of the channel, then f represents the fraction of the incoming salt removed in one pass through the dialysate. Note that I - I gives the dimensionless salt concentration the efflus from a dir&ate channel of dimensionless length rese~oir. Integrating Eqs. 4.10 and 4.25 (over the appropriate the parabolic velocit_r pro~?h. in the developing

in the developed

Similarly

from

one would obtain if y were dumped into a limits)

we obtain

for

(4.34)

region,

(4.35)

region.

Eqs. 4.1 I and 4.37 in the case of the uniform vcfocit_r profilE Desahation.

5 ( 1968) 293-329

Ht-DROI~YS;ASllC

THEORY

OF lXSSALISATlOS

1W ELECTRODIALYSIS

in the dcvcfoping

= I -

cD

Eqs. 4.6 and 4 .22_ For either

from

in tfle dc\efoping

= c,,( I -

in the dclcfopcd

ij2cD)

(4.37)

in the diafysatc.(sec

Fig.

4)

profile

iefocity

C, = 1 - i&/2

(4.36)

regron.

c,,, at the membranes

The salt concentration is obtained

region.

in the developed

ii

+

311

(4.38)

region.

(4.39)

region.

The dimcnsionkss

electric field distribution in the fluid is given by Eq. 3.8. of J the field is a maximum at the membrane surfaces in the diafysate

At any \afue where

c has a minimum E,(y)

=

v;1fuc. The \afuc

(‘9 . = t c’s ) .c= z1

-

i

1 -

I -

5.

ASYMPTOTIC

in the de\clopmg

These

SOLU TIOSS FOR P~RAROLIC

equations

(4.4)

(4.43)

region.

must

the diffusion

region

in pncraf

can be obtained

\ EL0ClT-Y

PROFILE

for flow with a parabolic where

in the developed

equations

solutions

protile

region.

in the developed

and 4-f 7 in the region

4.25 and 43f

vetocity

ii%,

The governing

4.10

(J.-w

i&,1

i/co ___ _._.

=

t * c-m

for cm WC havs fur an arbitrary

so that from the exprcssious &,=_

of this field is

layers

velocity

where

the diffusion

be solved

numerically.

in analytic

form

profile

are developing. layers

Eqs.

fiff the channel.

However.

for the two limiting

are Eqs. and

asymptotic

cases of very high

and very low polarization. As it turns out, these two asymptotic solutions yield a _eood approximation to the compkte numerical solution. It is necessary to define what we mean by polarization in this context. Vetter (6) in his monograph gives a rigorous definition of polarization as it is used in electrochemistry. However, the word has been variously interpreted in the literature on etectrodiafysis. In the present paper we use the word in a simple qualitative sense. lf when a current is passed through a solution the potential drop differs

by only a small

fraction

from

the simple

ohmic

drop

based on the average

then we shall say the polarization is Iott= On the other hand, if as a result of sharp concentration gradients the potential drop is substantinliy higher than the ohmic drop based on the average conductivity, then the polarization will

conductivity.

Dmctlinuriorr.

5 ( 1968) 293-329

A. A. SOSIS

312

Ah.n

R. F. PROBSTElS

be termed /fjs~l. This definition is useful f0r di~Terenti~~tin~ between the inescapabIe ohmic potential drop and the addition:11 potential drop caused by the presence of the diffusirrn laers which, in principle. may be controlled by .qqmrprinte design and choicr of oper&ing conditions. The asymptotic solutions for low and high polarization are presented betow. The h~j~od~nami~at~y triviai sotution for the final conditions in a system of infinite length is also giw.x~. Complete numerkd solutions for pZi?icuhr given in Section 7. where they-are compared with the analytic as> mptotic of this and the following srxtion.

(0)

f3aditpitfg

Whether

cases are

solutions

pmfik (_r -c .t; f is high or low depends.

nmxvm-ffriotl

in the region where the i&j2 When iti;2 --+ 0. it is clear from Eq. 4.17 that the potential drop across a channel pair is simply the sum of the potential drops based on uniform conductivity and the Donnnn porentinf: in other words. there is no polarization as WC have defined it. Thisisthr situation at the inlet of the system, where _Y= 0. For iii,9 $ I, the potential drop differs by only a small fraction (of the order i&Q) from the v&e without polarization, LE., polarization is IOH. Polarization is always tow sufficiently near the channel entrance_ where i5fZ is small because S is small. or it is low throughout the system at sufficient& few potentials. in which ixse A513is imall be&use i is everywhere small (6 C I). We note here that the limit of high polarization occurs when i&2 -+ 1, Eq. 4.10 for the growth of the diffusion layer thickness in a parabolic vefocity profile can be integrated by parts to yield diffusion

polarization

layer is growing.

on the magitude

of the quantity

where i and 6 are related by Eq. 4.17. For &IS polari=affort we expand powers of icSi2 and, retaining only terms to first order in i&2, obtain

Eq. 4.17 in

(5.2)

ffere k = (I K = k(l

are parameters

-I-

co; -II-

q)-‘.

c$),

6 I for all values of c oB and tg. With this low polarization

(5.3a)

(53b) approxi-

II\ DRODYSALlIC

TIIEORY OF IXSALINATIOX

BY ELFCTROi~lALYSIS

313

mation for i(6). Eq. 5.1 con be readily integrated exactly. A good approximation simple analytic form may be obtained. however, by expanding the solution powers of S with the result

in in

(5.4) where we have retained terms only to the frst To this order. Eq. 5.4 can bc in:;erted to read

order

in S within the brackts. .

(5.5) This rel:ition is :tCtWily ;I good appro.uimatinn for S throughout the developing region. It is of mtcrest to note that the diffusion layer thickness is independent of the applied potential. which may be expected in the present limit. but in addition it is also independent of the brine concentration and membrane resistance to zero order. We would also note here that the point at which the diffusion layers merge and the developed region besins (_r = _r.) is given approximately by Eq. 5.4 with 6 = I. Substituting Eq. 5.5 into Eqs. 5 2, 4.34. 4.38. and 4.41 and keeping terms to the same order as in Eq. 3.5. we gc’ the folkwing asymptotic low polnrization solutions for the current densit). total current. and concentration and electric field itt the ntembrancs in the dialysate: j = k(‘Yi7I-

cm =

1 -

-

In toe) (I - K_v”~).

kc Y!:!

-

h-ac&r

1;3

(5.6)

.

E,, = k(‘Yj2 -- In cDH) (t + [k(V/1

(5.S)

-

In coEI) -t- K]_r”3)_

(5.9)

Although the ahoke equations are derived for reasonably small values of y’j3. they are in fact very good approximations to the exact solution for low polarization for itI1 J < 0.1. that is. throughout the developing rc_rion. The criterion for low polarization is

equation shows explicitly that poinrimtion can always be maintained low either by using suffkiently low Y (low potential) or sutkiently low _V(short channel length. high flow velocity, large channel spacing). This

Dcsalirratiotr. 5 (1965) 293-329

A. A. SOXIS ASD R. F. PHOnSrt3N

314

At low polarization. then. the diffusion layer thickness grows essentially profile is parabolic. The current density starts at the as J 1!3 when the velocity simple polarization-free value af the inlet and then decreases monotonically as _I increases. The salt concentration at the membrane in the dialysnte decreases with _r_

The electric field. however. may either increase or decrease from its pofariznrionfree \aIuc at the inlet. At low potentials [V/Z -C (1 + c;,’ + in co,)]. E,,, decreases, white at higher applied potentials E,,, increases wirh _v. We also note that the total fraction of salt removed in one pass I is always small compared with unity when polarization is low throughout the channel. . (b)

Drw/~ptvi

cowtvrrrutiml

prty?lc (;(’ >

_v.)

of developed flow is characterized by the quantity ii2cLI- This is evident from Eq. 4.22. which shows that the salt concentration is virtually uniform across the channel when i/2c, G I (low polnriThe degree of polarizarion

in the region

zation). whife when i/2c, - 1 the concentration membrane in Ihe dialysate (high polarization).

The distribution by Eq. 4.35, which

of the centerline

can be integrated

Jde, l

?’ -Y*=;

.-.-i

cn

’

concentration

to D \ery

(

low vnlue at the

r, in thedialysate

isgoverned

to yield

&_

In

10

drops

(5.1 I)

i’ )

Here i. is the value

developed

of i alt the point where the diffusion layer% merge and the region begins, i.e., where S = cD = I and _r = _I-.-In this case _r. is given

approximately by Eq. 5.4 with 5 = 1. For low polarization we expand Eq. 4.3 t in powers only

of i/Z,

and on retaining

first order terms obtain _

* -;

v/2

-_-_ -

lnt(l + coB --. --_._

cD)/cD] ._____.__.

(5.12)

3_ -._. _._..--_____.__ (1 + COB) ._ + d 2 c,(l + COB - CD) With the above expression for the current density. Eq. 5.1 I can be integrated although a term in q appears as an integral over L’~ since it is not expressible in

terms of simple functions_ However, we observe that by the iow polarization limit we mean the limit where polarization is low over the whole length of the channel, that is, for all values of_r smaller than the one considered. We shall see below that this can be the case only when cD remains close to unity. i.e., when CD = 1 - AC,, where AC, is small. We therefore expand the functions in Eq. 5. t 2 (and 5. I I) for small AC, and retain only terms to first order in AcD. After intcgation and some rearrangement,

we obtain

the sohttion Desalitxarion. 5 (1968) 293-329

{I==--+

7

0 “I

5

_$’

(513b)

t + cwi

~hc cwrent density and total current the sitme order in AC, we find

fuflaw from Eqs. 5_tZ and 4.35. Tu

Some cctncfusions may nuw be drawn ahoul: operation at few ~~~r~~t~on in it tong channel where the concentration profiic reaches the fuify developed state. First. the desrce of pofmizotion is highest at the point y = y. where the diffusion ky.xs merge, for is:‘, incwascs with _r in the developing rcgton and ii2ca decreases with _x-in the developed re$on_ Secondty. the assumptiun that c, cannot drop substantially &low unity at low polrttization is cltarfy justified. The criterion for law polarization is most critical at the point where the diffusion layers merge and hence can be ckpressed a~ (5.f6) it folhxvs from Eq. 5-I fa that c-~ must cvcrywherc be close to unity, and therefore from Eq_ 5.15 that I 4 g. Thus, we reach the important conclusion that even in a long chrtnnef. where the flow reaches the fuity developed state, a high fraction& suit removal per pass through the system cannot be :tchicied without high pokkzation in some part of the channel. At Iotv poiwization the electric field f$,, at the membrane in the diatysatc is not of practical importance. We note, however. that E, q i decreases with _I*in the developed region. Thus, ~5~ is either ;I maximum at the channel entrance or, if Y is sufficiently lnrg (see Eq. 5.9) it first increases with J’ in the deve!oping region. reacher a maaimum ut J’ = y.. and then decreases asin.

316

A.

applied potentials.

A.

SOMS

AND

R. F. PROflSTElN

In the limit Y -+ ;Y, we can set

Aid = 1

9 (5.17)

over the entire len_eth of the developing

region. and integrate

Eq. 4. IO to obtain (5.18)

We can also write the obvious relation lltm = “ii.

(5.19)

.

and from Ey. 4.3-t (5.20) tt here it,,,, represents

the limiting current density distribution in the channel and I f,m the limiting value of- the total current. The limiting current is here defined to be the value reached as~niptotically at high applied potentials. The limiting value of the total current is aIwa>s its maximum value. However. as will be discussed in Section 7. at a given _r the limiting current density may be somewhat lower than the maximum current density. Note that ilim and Olin,. as weli as 6. arc functions of_ralone and do not depend on the brine concentration or the membrane resistance. The point at which the concentration profile becomes fully developed is found from Eq. 5.1s to be r’. = 0.0646.

(5.21)

This value of F. is always less than the value obtained in the fow polarization case (cf: Eq. 5.4 with S = 1) as a resuh of the faster growth of the diffusion Iayer (c$ Eq. 5.5 and Eq, 5.22 below). For small S, Eqs. 5.18 and 5.20 may be rewritten more convenientiy as explicit functions of _r in the forms (5.22)

(5.23) These last equations are actual& developing region. The concentration c,,, = i 4.38) is very small compared with totic solution for c,,, we expand dominant terms

within 2% of the exact sokttion

over the entire

- i&f2 at the membranes in the dialysate (Eq. unity at high polarization. To obtain an asympEq. 4.17 in powers of c,, on retaining the

Desalina?ion,

5 (1968) 293-329

HYDRODYXAMIC

THEORY

i/k = Y/Z

f

OF DFSALISATIOS

2(1 + cod)

-

Zc,l”

RY ELECTRODIALYSIS

tan-’

-I/?.

COB

where k is given by Eq. 5.3a. In the limit c,,, -+ 0, i -

-

In [*I

317 + co,)/c~].(5.24)

2/h and the above equation

yields c, = 2( 1 + COR)‘8z exp [-

‘Fj4 - (I + C,,:,

+ L-o;.) tan-

I c&y

+ r;iwJ, (5.25)

where 5t r) is given by Eq. 5.22. We note that 5 increases wi:h _a-,so that c,,, decreases as it must. More important. however. is that at anv, -riven _Y.c,,, decreases exponentially wth applied potentutl. The electric field at the membrane is given from Eqs. 1.40 and 5.19 by E,,, = 2!c,,,S. or from the above calue of c,,,

the developing region, E,,, mcredses monotonically with J+. At a given _r it is important to note that &,,,increases exponentially with the applied potential once high polarization is approached. so that very high fields can be set up near the membranes even at moderate applied potentials. Since ii d I throughout

Der~~lopr~ri concwrrmiorr

ih)

profiles (_v > y. = 0.0616)

In the developed region the polarization becomes high when ii2c, + I (see Eq. 4.31). Analogous to the developing region. in the limit ‘Y 4 CGwe set i -= ZC,

(5.27)

1

over the whole length of the developed resion boundary conditions that co = 1 at _t*= 0.0646.

co =

exp[-

2.5(_r -

and integrate

Eq. 5.1 I with

This gives the timitin~

0.0646)],

the

solution

(5.25)

from which -

= 2 exp[-

'lim

3.5(y - 0.0646)-J,

(5.29)

and using Eq. 4.35 Ilim = 1 -

0.80 cup[ - 2.5(_r - 0.0646)-j.

(5.30)

The concentration at the membranes c,.,, is obtained in the same manner as for the devetoping region. We expand Eq. 4.31 in powers of (I - i/2c,J, retain only

the dominant c,

=

2c&

terms.

and

set i = 2c,.

with the result

+ co,)@

exp[:-- VI4 -t- {Ml

(5.3 1)

-t co6 - co)) * tan- * {c,/( 1 + co8 - ce))$

+ rjcD].

Desalinariott. 5 (I 968) 293-329

318

A.

The electric field at the membranes E-#$= &-&,,

SQ that from EC+ 5.3X

A.

SQNIN

AND

R. F_ PRQBS”CEfS

is given from Eqs. 4,&l and 5.X

by

HYDRODYNAMIC

6.

THEORY

sowno%

ASYMPTOTIC

OF DESALISATIOS

FOR UXIFORM

BY ELECTRODIALYSIS

VELOCITY

319

PROFILE

The case with uniform flow velocity is solved precisely as in the previous section. the only difference being that Eqs. 4.11 and 4.27 replace Eqs. 4.10 and 4.25. respectively. We therefore only briefly indicate the steps and simply list the resuhs analogous to those obtained for the parabotic velocity profile.

With a uniform velocity profile the growth of the diffusion layers ckr&piny re@l is governed by Eq. J-1 I. which WC write in the form

in the

(6.1) At low polarization

Eq. 5.2 applies and the above equation

integrates

to

where the constant K is defined by Eq. 5.3b. For small 6 the soiution can be written explicitly in terms of y and we find for J

<

J-.:

5 =

\!6 j-‘/z

;_t!, -\

1 +

s”r

,

(6.3)

which leads to

(6.5) c,

=

I

(6.6)

E,,, = k(Y/2

In coB) ,1 + 2J6

-

Ck(Vl2

In the region p > J.. where the diffusion to be solved is Eq. 4.27, which may be written

Ln coa) + K]_?\

I

-

F-7)

layers are ck\*eloped, the equation

1

dco _.___-:In C__

Y = _s*if CD

At low polarization

i

i(c,)

()

(6-g)

i

is approximated

by Eq. 5.12 and the deviation

of cD

Desalination.5 (1968) 293-329

A. A. SONIN AND R. F. PROBSTEIX

320

from unity is smail. Proceeding as in Section 5.2, we integrate Eq. 6.S and obtain solutions identical to Eqs. 5.13a, 5.14 and 5.15, except that the constant a is in this case giveen by Q = 4._ .+ ...__.I!___ I + c,,“ 3 and ;

is &en

(6.9)

by Eq. 6.2 with S = 1.

In the limit V + cn we set i&/Z =I I throughout integrate Eq. 4.11 to obtain the limiting solution t ?’

<

?‘*

LT

3 =

-.:

the

&-~~~5pit~g

(12Jp2,

region

and

(6. lOI

12

2 .f/2 * 1,,m = “‘-,- - j \‘3

(6.12)

The asymptotic soWions Eqs. 5.25 and 5.26 for r, and &, apply, eaccpt that in this case S(J) is given bp Eq. 6-10. In the dmeluped rqiotr the limit Y -+ 30 implies that i,/3eg - 1, and Eq. 4.27 integratrs to yield

~firn= 2 cxp

i

lrrn

=

Here c,,, and E, are @en

1 -

[-

3(_r- j;)l_

(6_$4)

;exp

(6.15)

[ - 3(_r - /J]*

by Eqs. 5.31 and 5.32. with cI) obtained

from Eq. 6.f 3

above. 7. IXSCUSStOXOF REStiLlS AKD fO%fPARiSOXWfTH E;YP&RI~lEKT Complete numerical solutions have been obtained for the developing region by solving Eq. 5.1 (for a parabolic velocity profile) or Eq. 6.1 (for a uniform velocity profile) simultaneously with Eq_ 4.17. For the developed region Eq_ 5.1 i (for a parabolic velocity profile) or Eq. 6.8 (for a uniform vetocity profile) was solved Desalirroriun, 5 ( 1968) 293-329

HYDHODYNAMIC

THEORY

OF DESALINATIOS

321

RS ELEC-TRODIALYSIS

simultaneously with Eq. 4.31. in the developed region the value of _r. was obtained from the solution of the developing region. The special case of negligible membrane resistance (rl = 0) and equal brine and dialysatc inlet concentrations (coB = I) was chosen for illustration in most cases. However. the total current as a fun&on of applied potential was also calculated for both velocity profiles with other values of cos and tf_ As will be seen below. from the totai. current plots. the nature of the solusolution for arbitrary values of cotc and ~1is easily inferred from the analytic tions for high and low polarization. Figs_ 5-9 are for a parabolic velocity profile with co8 = 1 and q = 0. The current density distribution i(_r) is shown in Fig. 5. At an applied potential ‘I’ large compared with unity. i starts at the channel entrance with a value based on a simple resistive potential drop and then decreases gradually with _vas the diffusion layers develop. In this initial re_eion the asymptotic low polarization solution (Eq. 5.6) is a good approximation. Polarization becomes si_gnificnnt approximately at the point where the low and high polarization solutions (Eqs. 5.6 and 5.19) are equal. i.e.. approximately when k(‘Yj2

-

In c,,)_v’!~

-

1.

(7-l)

r\t larger values of _r. the solution follows the limiting current density closely. The point _a’.at which the diffusion layers merge is marked with a circle. Although J* decreases somewhat with increasing potential. its value lies in the neighborhood

1

,j’ IO"

-0

-4

:0-2

10-J

iO_’

I

YD )’ =&&far Fk

5_ Current den&v

distribution for cmrabolic velocitv

I.

t)

=

Oh

of something less than 0.1 (note that ; - 0.09 from Eq. 5.4 and 0.065 from Eq. 5.21). It should be noted that Fig. 5, as well as most of the succeeding figures, are logarithmic plots which appear to stretch the coordinates at the low values. For example, in a system with an exit y coordinate of 0.1 and an applied potential Resalinari’n.

5 (I 968)

293-329

A. A. SONIS

323.

AND

R. F. PROBSTEIN

Y = 60, polarization becomes severe already very near the channel entrance, in less than a percent of the lengrh, and the current density drops by a factor of four in the initial 10% of the channel length. A curious result is that i tends to “overshoot” its limiting value. in other words. at :t given _Ythe current density increases with ‘P to a maximum value and

Fig. 6. Ion concentration at the mcmbrann profile. Cf)U I= 1. l] :- 0). Circles: ;I .-.=I.

: -...c ’

in the dialysatr

channel (pxnboIic

wlocizy

__... _.._. -__ __--__. __. .___ .._--.z zc-: ,yf aa-” I’=‘ 1 ST +-5.

Fig. 7. Elwtric field at rhe membranes in the dialysate channel (paabolic COB = 1. q = 0). Circles: J =- I.

velocity profile.

then decreases somewhat before it reaches i,,,,,. The reason why this occurs in the developing region can be traced to the fact that the diffusion layer tends to thicken somewhat as the potentia1 is increased (cf- Eqs. 5.5 and 5.22). At low potentiais. i increases with Y. but as pofarization becomes severe and i becomes more sensitive to the diffusion layer thickness, the increase in 6 actuatly results in a decrease in i with further increase in Y. At a point downstream of .,; the current density can reach values considerably higher than ilim. In this case the maximum occurs because c, decreases with potential. Desalination, 5 (1968) 293-329

HYDRODYKAMIC

THEORY

OF DFSALINATION

RY ELECTRODIALYSIS

323

The salt concentration (;n and the electric field E,,, at the membranes in the dialysate are plotted in Figs. 6 and 7. Again. at a point given approximately by Eq. 7.1, polarization sets in and c, becomes small compared with unity and E,,, large compared with the field based on resistive drop at the inlet conditions. The low polarization solution indicates the behavior at smalkr values of y. while at lar_gcr .r_the high potarizntion sotutim pives a good approximation. Note that E, reaches a maximum somewhat downstream of _I-. and then decreases. Thus. in flows where the diffusio;l layers do not develop, E,,, reaches a maximum at the exit of the channel, whereas in long channels with the larger values of _r required to achieve a large fractional salt removal in one pass through the system (see Figs. 8 and 9) the makrmunr electric field occurs within the channel. Note also from the asymptotic solution (Eqs. 5.25 and 5.26) that at a given _I’, c,,, decreases and E,,, increases clcponentiall> with Y/-t once polarization becomes high.

Y

Fig. 8. Dimensionless total curr~‘nt &Spotential for parabolic velocity profile, _V 2 0.1

‘, ;= 0).

Fig. 9. Dimcnsionk%

totA current IX potential for parabolic velocity profile (COD = I,

Desalination.

5 (1965) 293-329

A. A. SOWS

321 Figs. S potenrinl.

We

and 9 show the total dimensionless recall

current

ASD

R. F. PROBSTEIS

I as a function

of applied

in one pass through a system of dimensionless length J’. Each curve in Figs. S and 9 applies to a +en sslue ofy. .4t low patential the poiltrization is low and I increases linearly with Y (Eqs. 5.7 and 5.15). At a potential given approximately by Eq. 7.1, polarization sets in and I begins to level off and soon reaches its limiting vdue (Eqs. 5.23 or 5.30). approaching it monotonic~~iy. The similarity of the current-voltage cur’ccs suggests that-it might be possible to find a single empirical function which expresses the current-\oltagc relation for the general

that I is the fraction

case. This

is in fact possible

Guided by the asymptotic solutions dimensionless current and potentia!

of the incoming

for ail _r <

salt remoled

0.1. i.e.. for undeveloped

for the developing

region,

Rows.

we deiine a new

(7.2)

(7.3) >-

I

L

I *of

A0

,

,

_-

.___-

r.‘_.t’..

-7

.__.

COB

*-.

o’lL%7.l0“’ ‘2 A--

. .._

I-

_

0

IO

-_..

y:*rro“

Lx.___-.._

I

0

I

-_-

2

___._.ro

’ t looa%o-’ .___-

’

0

-.

‘I

‘I

0

-

.,-x-1

I

- _-.. 0

I I

0

zj 2:o

1.0 ,p

Fig. 10. Empirical corrclatinp function for total current for all _V< 0.1 (parabolic velocity defined in Eqs. 7.2 and 7.3. Points represent cxtactnumerical solutions.

profile). fand + are

These are defined such that Fig. 10 is a plot of 1 1’s q velocity profile. Clearly, the collapse onto a single curve / = Cl - exp(-

for small y. 1 % \F at small 9 and I + 1 at large q_ obtained from numericaf solutions for the parabolic solutions for the various values of _v, cos. and PI ail given to a good approximation by

*‘)Jt/‘.

(7.4)

Eqs. 7.2 to 7.4, therefore, express the total current I for a parabolic in terms of the parameters _r, ‘I”, cos. and q for all p < O-l_

velocity profile

Desahatiun. 5 ( 1968)293-329

HYDRODYNAhlIC

The similar

solutions

to those

distribution The major

THEORY OF DESALINATION

for with

density

polnrizatton

dmsify

a uniform profile.

velocity

is not

much

dtstribution

different

for

untform

in the

profile

I I shows

Fig.

cake

are

are

the

qualitatively

current

2 -

the limiting current

density

with ~1 = 0 and cos = 1. two:

First,

although

the

velocity

region

where

polarization

sets in at ;). higher value of _I*.given in this case approximately

ktV;Z - In C,,))” Secondly.

with

parabolic

for the uniform velocity, again for the &xts of the change in \clocity profile

FIQ. It _ Current C1rclcs: 8) 1: 1. current

flow

the

325

RY ELECTRODIALYSIS

1.

is !ow,

by C7.5)

density is higher with a uniform

veIocity. Note also

that the ‘*overshoot” of i over its limitrng value is more pronounced because the thickening of the diffusion layer with potential (cf. Eqs. 6.3 and 6.10) is greater in this case. Fig. 12 compares the total current-potential relations for uniform and parabolic velocity distributions in a system of given dimensionless length _I+.The velocity profile does not influence the initial slope of the I-Y relation significantly. but with a uniform profile polarization is delayed to a higher current and the limiting current is considerably higher (Eq. 6.12 w Eq. 5.23). For the two velocity profiles the limiting currents, which arc Independent of coB and 9. are compared in Fig. 13. An interesting point to note is the dependence of the current on the ffow velocity_ The real total current per unit channel breadth J (say, in ampjm) is given by (see Eq_ 3.333) .I = F Co,> 2dV,,. l(_r, ‘f‘. cos, qh where Co, is the real reduced salt concentration for I from the expressions for flimr we find

(7.6) at the dialysate inlet. for a uniform velocity

Substituting profile, Jttm

Desulimzrion, 5 (1968)

293-329

A. A. SONIN

326 varies as V,,cr’2 at low J*and approaches

a linear

variation

with

AND

R. F, PROBSTEIN

V,, as J’ approaches

unity

(in which case practica& all the incoming salt is removed). For a parabolic at low p but also approaches a linear variation with profile. Jlim varies aS VULXG3 V#, as ,r nears unity.

r 0 I5 -

01IO i

oc 15

0 0

5’:

25

Y \&city

Fig. 12. Comparison of total current-voltage rclntion for flows with mformand parabolic profilcs in channel of dtmensionlcss length y = 2.17 Y IO -1 (co11 = I. ~1 =-- 0).

IO+-

IO-'

10-2 ‘=c

zo

IO-

Fig. 13. Limiting total current and comparison

I

IO

with euperimcnt.

The current-voitage relation fo r a uniform velocity profile with co8 = 1 shown for J 2 O-1 in Fig. 14. For _V,( 0.1, we can again find a single empirical function expressing I in terms of J’, Y, coB. and q. For the uniform profile the asymptotic solutions suggest the definitions

and q = 0 is

Desaiinatfmi,

5 (1968) 293-329

HYDRODYNAMIC

THEORY

OF DESALINATION

I

0

(con

Itm ._

IO

Fig. 14. Dimensionless =- 1. I} = 0).

327

RY ELECTRODtALYSlS

I

20

Y total current

for uniform

w potential

velocity

profile,

p >, 0.1

(7-7)

(7.8) Numerical

solutions

for

different

values

profile).

and 11 are plotted in Fiz. IS in with the parabolic profile, all the described by Eq. 7.4. For y d 0.1, / in terms of the four similarity in Eqs. 72-7.4 for the parabolic the case of uniform velocity.

of _I’, cos,

terms of the above coordinates and again. as results are seen to collapse onto the single curve therefore. we hahe an analytic expression for parameters ; Y, coI1, and rl_ This is embodied v&city profile. and in Eqs. 7.7. 7.8 and 7.4 for

Fig. 15. Empirical correlating function for total current for all y c 0.1 (uniform velocity iand q arc defined in Eqs. 7.7 and 7.8. Poil:ts represent exact numerical solutions. Desalination,

5 (I 968) 293-329

A. A. SON19 AXI> R. F. PR013STEIS

328

!k\eral clectrodialysis a form which ever.

the total

have carried out evperimcntcll studies of polarization in systems. e.g.. (3. 7-10) but unfortunately their published data is in does nof allow a detailed comparison with the present theory. Howworkers

limiting

current

has been

carefully

documented

in most cases.

and

earf\r studies of Rosenberg and Tirrell (7) and Cowan and Brown (8) are not suitable for comparison_ Roscn-

a comparison

can be made_ We note that the oft-quoted

berg and Ttrrelt used an apparatus in which regularly spaced. obstructions were employed to deliberately break up the steady

eddy-promoting flow pattern for

which the theory is developed. and Cowan and Brown performed their experiments at high Reynolds numbers \vhere the fiow was turbulent. The limiting currents measured theor)-.

by both these groups which

were higher

than those

predicted

b_v our

lnminar

is in the espected

Experiments expected to apply,

direction. at lower Reynolds numbers. where the present theory may be are reported in (3. 9-11). In all these experimenrq some reason-

ably dense spacer material was used in the channels, so that the elective velocity profite was presumably somewhere intermediate to the parabolic and the uniform.

The data of Mandersfoot

and Hicks (9) reflect the VOFi’ vari;~tion of JItm predicted

by the theory for a uniform veIocity profile for low _r. but fall about 35 7; betow that theory (Fig. !3) and agree better in magnitude with the solution for a pnraboiic velocity profile. The data of Rapier er ul. (/I)* represent measurements at large _\ It is quite likely that some of this and also fall about 35% below the theory. discrepancy reflects ambiguities in the method of determining or defining limiting current in the experiments rather than any basic disagreement with theory.Tsunoda and Kato (3) performed their tests in an apparatus designed for use in fishing vessels. Their

results**

agree

reasonably

well

in magnitude

with

our

theory.

the slope appears to differ from the theory and from the results of (9) in the range of _Vcovered by their studies. It should be pointed out that none of the workers whose data we have used has employed the scaling parameters derived in the present paper to correlate their results. In particular. the dependence of the axerage limiting current density J,,,// on the path length (see Eq. 7.6 and Eqs. 5.23 or 6.12) was not recognized (W also (I), pp. 15-l 7 and pp. 220-Z I ). Athough

l Refs. (IO) and (II) report me;lsurements in tuo systems. One of them was set up so that inlet conditions were not the same in all channels within the stack. as we hake assumed. The other appararus also differed from the geometry assumed in the prevent model in that it employed channeis with tapering breadth to provide a varying fiow velocity. We refer hcrc to the latter data, intsrpreted in terms of the average ffow befucity. l * It is not clear from {3) which of the two systems described was used for the polarization study. WC hate assumed it was their Model SV-3. and ascribe it an effcctix path length of 17 cm to reduce their data to the form gixcn hem.

Desdinarion, 5 ( 1968) 293-329

tlYDRODYNAM1C

THEORY

OF DESALISATIOS

BY ELECTRODIALYSIS

329

ACRXOWLEDGEMEKT

The authors wish to thank Mr. Gershon Grossman of M.I.T. for introducing them to the hydrodynamic problems in rIcctrodi:tlysis and for suggesting the uniform velocity profile as a limiting supported by the Office of Sstine Grant No, 14-Ot-tXHi-172t _

fiow C;W.KThe research reported herz has been Water, U.S. Department of the Interior under

REFEREXCf5

5.

6. 7. 8. 9.

t)edinufioff.

f (1%8) 293-329