- Email: [email protected]
An electrical analogue for electrodialysis
lle.wfina:kwz - Elr~icr
AN ELECTRICAL
Publishing Company.
ANALOGUE
Amsterdam
- Printed in The Netherlands
FOR ELECTRODIALYSIS
This paper is 3 first attempt to perform an analysis of clectrodialysis by considering the process as an electrical network composed of resistive elements representative of various electrochemical subprocesses. The total etrect of all subprocesses is unified into the single mathematical equation for the network. The treatment gises a breakdoitn of the various factors that contribute to efectricaI resistance and pinpoints those factors that must be improved to make technological improvements in the process. Application of the analysis to the Webster. S. D. and Buckeye. Ariz.. plants enables the resistance of the sepdrnte stages to be calculated. The importance of ohmic scale polarization in these plants is emphasized and predicted by the use of an empirical equation (16). The average calculated values for the siu stages of these plants agree to within 947; of the aberage measured values. SYMBOLS
a A
A, b t3 C
Cd=)
-
-
actual: membrane cross-sectional area, cm’ (= ISI x n) temperature dependent constant in the Onsagcr equation
active membrane area. cm* subscript denoting brine bulk stream temperature dependent constant in the Onsager equation average concentration of salt, in the water. g equivjliter average concentration of a cell pair at a distance = along the path iength. g equivjliter brine or dialysate concentration at some point s away from the membrane surface and along the fluid path length (see Fig. I ), g equivfliter dialysate inlet and exit (product) bulk concentration, 3 equivfliter concentration at the membrane surface, along the flow path z, of the brine and dialysate streams, respectively. g equiv/liter subscript denoting diaiysate bulk stream Desahation,
5 (1968) 267-291
268
G.
-
DL &
E
-
7’ == (_C& (II)x
-
F
-
F* r;b i
-
4 ‘lrm LPcr
-
IL
-
k k’
-
K,, K,. K,
-
m N N’ N,, N,
-
n
-
0
-
Ci
-
r
-
R CaPR,
-
R CP
-
R CO"C R rhrm
-
R,, 4, 4 Rl -
-
c”dW)~O,.,
-
R “P R dsm
-
dR/dt
-
ROS
-
RP
-
-
DELFORT
AND
G. A. CUTER
diffusion coefficient. cm”jsec stack voltage, volts anion and cation membrane potential respectively, volts fraction desalted fraction in product and feed streams in e.p.m. of ionic species .X Faraday ISo., coulombs/g equiv concentration stream exit flow rate. llsec, channel dilute stream exit flow rate. I/set. channel stack current density. amps/cm” current leakage due to duct losses, amps/cm’ limiting current density, mA/cm’ operating density, mA/cm” leakage current due to co-ion transport. amps ratio of the operating to limiting current density = ioppr/i,im ratio of brine to diaiysate compartment thickness = ybjyd constants in the generalized R, VS. C, Eq. 22 overall membrane length. cm number of cell pairs/stack N/2 number of anion and cation membranes, respectively overall membrane width, cm initial = (dialysate inlet), cm fraction of membrane area available for desalination power index, defined by Eq. 53 normalized leakage resistance = (RJR,N) cationic or anionic exchange membrane resistances respectively, ZL-cm” concentration polarization resistance. Q-cm2~ceIl pair concentration overpotential resistance (electrodes), R/stack chemical overpotential resistance (electrodes), R/stack dialysate and brine stream resistances respectively, R/stack parasitic duct loss resistance, R-cmZ/cell pair equivalent leakage resistances, Q/stack equivalent channet resistance, Q/stack total membrane potential, Q/stack total membrane potential defined by Eq. 40, Q-cmZ/cell pair ohmic o~~erpotential (electrodes), R/stack rate of change in ohmic polarization resistance of stack, ohms hr-’ resistance due to ohmic scale polarization, Gcm2@Al pair equivalent cell pair resistance, R/cell pair
Desalination,
5 (I 968)
267-291
AN ELECTRICAL ANALOGUE
-
-
-
-
cell pair resistance at some specific distance z up from the bottom (z = 0) of the stack, Q-cm’/cell pair hypothetical membrane selectivity resistance, Q-cm2~cell pair electrode solution resistance, Q/stack electrode diffusion layer resistance. Q/stack time of operation. hours transport number of counter ion in buik sofution transport number of counter ion in anion and cation exchange membranes, respectively transport numbers of the co-ions in the cation and anion exchange membranes, respectively where r‘l = I - f; and 1% = 1 - z‘!, linear fluid velocity within the membrane compartment. cm/src primary hydration number for both sodium and chloride ions
diaIysate or brine compartment -_ coordinate system away from -
-
269
FOR ELECTRODIALYSIS
thickness,
and along
cm
a membrane
surface
(refer to Fig. 1) (Ifnz) In [CC& O)fC(S, nt)] diffusion layer thickness. cm true coulombic cffrciency coulombic
efliciency
-
chemical
overpotential
-
concentration overpotential (electrode), volts ohmic overpotential (electrode), volts potential drop due to electrode solution resistance, vohs potential drop due to the electrode diffusion layers, volts equivaIent conductance of dialysate and brine streams equivalent conductance of solution at average concentration
-
-
-
excluding
water transfer
(electrode),
losses
volts
of salts and 2°C equivalent conductive at infinite dilution and t’C resistivity of the cationic or anionic membranes respectively, Q-cm resistivity of the solution at coordinate points s and z, R-cm dimensionless channel resistance. = &/R,N
INTRODUCTIOX
The paper is divided into three sections. In Section 1, Su~omponent Analysis, expressions are developed based on electrical analogies for each of the subcomponent factors influencing the operation of an electrodialysis system. The factors considered are concenttation polarization, ohmic scale polarization, composite ceil pair resistance, parasitic electrical losses, water transfer processe6, membrane potential, electrode polarization and membrane selectivity and temperamathematiczl
Desufinurion. 5 (1968) 267-29 1
G.
270
BELFORT
ASD
C.
A. CUTER
ture effects.
The effects of hydrodynamic factors and temperature are also included. In Section 11. lntegrntion of Subcomponent Mathematical Elements into the interrelationships between the resistive a General Analytical Expression. elements are studied and combined intoageneral expression for a resistance network. In expression for total stack resistance is then written in terms of the separate resistive elements and their combination into R resistance network_ in Section III. Applications of Generalized Mathematical Solution to Specific Situations. the expression for the derived resistance network is applied to specific plant situations at Webster, South Dakota and Buckeye. Arizona. The conclusion includes recommendations for future research and development work in Improving electrodialysis technology. These recommendations are based on the analysis of the Webster and Buckeye plants. This paper represents n starting point for a realistic optimization of the operation and design of electrodialysis plants. SUBCOMPO?iEZ;F
AXALYSIS
A discussion and reciew of a number of subcomponent factors of the electrodialysis process will be undertaken. For each subcomponent factor, the resistanceanologue (ohm-cm”~cell pair). wit1 be preceded by a summary of the present state of the art. Thereafter. all the individual subcomponent resistances will be combined together into a general analytical expression. This general analytical expression will represent a model or electrical rtnalogue of the electrodialysis process. M etttbrutie polurizalion Extensive experimental work has been (I), and is being (_‘I. done to quantitatively explain the phenomena of membrane polarization. The dialyzing current faces a two-ford polarization effect close to the membrane surface. A concentration gradient across the diffusion layer and scale formation are the respective causes of such polarization, The former is termed concentration polarization, while the latter is called ohmic scale polarization. Each is separately discussed and evaluated belOW.
Cottcenlrutiotl polarixrion It is possible to estimate the approximate resistance due to concentration polarization that the dialyzing current faces. provided two important system pafameters can be calculated. These are the thickness (6) of the concerttration gradient and profile across the diffusion layer. Several empirical approaches (31, such as use of the Chifton-Colburn transfer factors and the flux equation of Fick, are able to predict the diffusion layer thickness for nonspacer-filled compartments. H. P. Cregor, et al. (4) have studied and measured experimentally, using various size spacers at different compartment Desalinarimt.
5 ( 1968) 267-29 I
AN ELECTRICAL
FOR ELECTRODlALYSIS
ANALOGUE
271
flow rates and Reynolds numbers, the relation of the diffusion layer-thickness with flow rates. Refer to Fig. I to calculate the diffusion layer resistance that the dialyzing
Colcentratron Prof1lr
_____
--
-
-
I-le:
x !L
~(6.0)
Fig. 1. Diagram of the concentration protile and the diffusion layer on the dinlysatc side of an elecrroddysis IOII exchange membrane.
current must pass through. a double integration is performed layer (x coordinate) and atong the flow path (Z coordinate).
From the Onsager equation (A&c
= fA,),*c
and the basic definition
across the diffusion
of resistivity we get
(2)
- CA + B CA.,x=cl GA*
1000 p,,, = __L!!__ = _.._____.._-_....___ (A,),ec C,., - C,‘!: CA + B (A ,X=J (A,)PC Cx.2
(3)
After pl.= is substituted in Eq.1, C,., as a function of x and z has to be evaluated from the boundary conditions to relate the concentration at the membrane surface C(0, Z) with the diffusion layer-bulk interface concentration, C(is, z). The Nernst idealized equation will be used. as most empiricai (I) and theoretical (5) equations available to do this are also not applicable to spacer filled configurations. Desalination,
5 (1968) 267-29 I
G. RELFORT ASD C. A. CUTER
272
Together with several simplifying assumptions presupposes a linear concentration gradient.
(I). the Nernst
equation
z) - _..._ CfO, z) f_) = C(3, dc = i(C_ ____._-. __..-._ _____- -..-ds
w
d
F4.
From the Nenst equation and realizing that at the limiting current tlirn, ~.~rface concentration c’(O, :) npproachec; zero,
densit)
C-(0. 1) = (1 - k)C(S, z) The following general equation assuming a linear concentration nential bulk stream concentration
(5) for C(x., z) can bc written from linear anrllysis. gradient across the diffusion layer and an eupodeea? from the inlet to the stream outlet.
The diffusion layer thickness (6). the limiting current (i&. and the operriting current density (i). must be estimated before the above equation can bc used. Limiting current density stream C(0, W) shown
diffusion
fili,) wilt occur when the exit concentration of the diiutc in Fig. I reaches zero. providing the fluid velocitg and
layer thiclcness are constant.
From the Nernst equation.
s = ___ C(& ~)FDL ._.. _.. i,im(i!_ -
(7)
I-)
Severat empirical formulae (6.7) have been developed to obtain the Iimitingcurrent density as a function of concentration and finc?aarfluid velocity. The general form of these equations is _.--.JJirn-
C(&
=
Al(c)
at a given
temperature.
(8)
nr)
Some question arises as to exactly when the limiting current occurs. defined from the “Cohan” plot (8) or from pH changes. Once iJi,
It could be is obtained
experimentally, S can be calculated from Eq. 7 and iopCrfrom Eq. 10, defined Hence. k can be evaluated. Using Faraday’s Law and a material balance, the capacity is detined iqrQ,tz = zip = --=_
FddYdlf
g equiv
below. as,
transferred/see
and,
iowr = -f FdCJtf-
amps/cm
’
9.4,
We obtain
A,, the equivalent
solution
conductance,
from
IksalinaJton,
the Onsager 5 f 1968) 267-29 1
AN ELECTRICAL
equation
ANALOGUE
at various
FOR ELECTROI~IAL%SIS
temperatures
and concentrations
273 with a second term correcting
for non-ideality.
(W
The concentration polarization resistance component R,, is evaluated for the dialysate stream and :tnion exchange membrane using Eq.1. The remaining three ditfusion layer resistances ore calculated in a similar manner.
Ohmic scale polarization invoives a number of complex and interrelated fixtors. It is associated with scale formation and other effects that contribute to resistance rise with operating time in the order of hours and days. Scales differ in their chemical and physical character as welt as the processes by which they are formed. Scales. such as calcmm suiphate and carbonate. magnesium hydroxide, ferric hydroxide, can precipitate out of solution and deposit on the m~rnbr~nc surfaces at points of poor hydrodynamic activity. Ohmic scale polarization can thus be expected to vary with electrolyte composition_ current density, limiting current densit), concentration polarization, local hydrodynamic conditions as influenced by spacer design and flow velocity, and pretreatment procedures. The task of writing a resistance analogue expression for the ohmic polarization factor is further encumbered by a lack of plant and experimental data on this phenomenon. Some data have been collected usin, a ‘at small test stack at the Webster plant. Fig. 2 shows the rate of resistance change with continuous operating current. The influence of current is obvious. This data was obtained usinS Webster water which had not been treated with sulfuric acid. Resistance buildup is due to dcposition of calcium carbonate and calcium sulfate scales as we11 as ferric hydroxide floe and inorganic materials which evade the pretreatment filters. The orientation polarization due to the reorientation of internal polar groups, may also be operative. of the general type The data of Fig. 2 were found to satisfy an equation Desahatiorr,
5 (1968)
267-291
G. RELFORT
Fig. 2. Relationship
dR -
between rate of resistance change and reciprocal
AND
G. A. GUTER
of stack ctwrcnt.
a)p
(15.W
dt
where n’ and h’ are constants. The ohmic scale polarization following equation and definition.
analogue
resistance
will be evaluated
using the
= 0.150 + 0.92k
$$ =
1
(15B)
41+‘2k
dt
and for a 20 hour period between current restored to the stack.
reversal with the virgin resistance
being
The constants in Eq. 16 were chosen to give the best fit to the avaifabfe data. A comparison ofcalcutated values and observed values for ohmic scale poktrization is given in Table I. The observed values for the Webster and Buckeye plants were calculated by assuming that the major differences between observed cell pair Desalination,
5 ( 1968) 267-29 I
AS
ELECTRICAL
ANALOGUE
FOR ELECTRODIALYSIS
275
TABLE I
l~*chsrcr plurlr 0.64 0.63 0.52 0.56
7.6 7.3 58 -l.i
5.5 5.4 .a.3 A.6
Bttckc*y e pianr 0.20 0.16
2.1 2.1
22 2.1
. ... _---_..---__
.
.- _
Ohmscmzjh per ccl1 pair. ** Fi\c pow test srnck. 260 cm2 membnncs. Darn suppled b\ Dr. John Nordin. Wcbsrcr,
l
So.
Dakot:;.
resistance and the \Aue calculated using ail factors other than ohmic scale poiaritation were due to the latter factor. The equation gives surprisin@y good results based on the theory and data limitations. It should be pointed out. howvcr, that Eq.16 is only an approGmation at best. It should also he noted that R,, is sery scnsttive to i/iti,_ This ratio can vary from time to time during stack operation_ It also may not be constant along the flow path. iIlm was evaluated using Eq. 8 with the constants M and n supplied by the membrane manufacturer at varioub concentrations.
The cell pair resistance is defined as the sum of the anion exchange membran;t resistance R,,. the cation exchange membrane resistance R,,,, and the dialysate Rd. and brine R,. stream resistances, all cakulated at some position along the flow path length. WA
= R,
+ L
f R, f R,
the membrane resistances can be obtained where H,,, L,, evaiuated at some average concentration, C,
($7) from literature
or
276
(i.
6ELFORT
AND
G. A. Gl.‘TER
Integration over the mtmbrane area (pu) available for dewlination of the pair resrstance (R,), at chosen path length z, from the bottom of the stack, result in the total resiswnce per cell pair. R, tihere
(19)
ff ue assume that (f?,,), is constan! z direction), then the area equ&. pu
= pnz
flu
= pnd: cm’
transposing
in the iateral direction
(perpendicular
to the
cm’
variables,
According to Mason and Kirkham (6) cell pair resistance expressed in terms of borne average conccntratiun. 1
= 3
CJ:)
at coordinate
z con be
k’ *. .- + -.I.CJ=, C,(z) I
(21)
and (22) Variation
of CJz) and L‘,Iz) with Z, wili make C=(z) a function
C,(z) = j-C=)
of Z. (231
(24) Substituting I
R,
Eq. 24 into Ey. 20, .
=
+
Pff 0
K,
-
K,f(z)
-I
I
(25)
dz
where Desdinarbn,
5 (1968) 267-291
AN ELECI-RICAL
ASALOGlJE
277
FOR ELECTRODIALYSIS
K. K2 and K, are constants for a given space geometry, ionic mixture, membrane type. and temperature as defined in Eq. 22. Based on the traditional treatment 49) of electrodiatysis optimization. an exponential decay relationship for the dialysate bulk stream concentration and the path distance. =_ wit1 be assumed. It has also tacitly been assumed up untii now, that equipctential surfaces exist parallel to the membranes and electrodes. The exponential assumption can be used to prove the existence of the equipotentiai surfacf5. Evaluating each bulk solution as a function of path distance, 1, and combining each to give c’, tie get for the Dialysate c-&J-_)= C&)e-”
(26)
and from a material balance across the unit at a distance r: along the path length and the top of the unit (at z -= ~PZ).Refer to Fig. 3.
“,
‘,
c
c
b
fcJf
Fig. 3. The concentration
and brim
profile
kqxxentla!)
stremns for co- and uountcr-current
and material h.dances of both the dialysatc
operation. Dtwdinatiun. 5 ( 1968) 267-.29 I
G.
27s Brine
(Counter-current C,(z)
from
XSD
G. A. CUTER
flow)
= [C,(m)
By definitron
BELFORT
-
C,(m)]
(F,/FJ
Eq. 21. evaluate
+ [(FJF,)
C~o)]e-”
(25)
C,, (29)
substituting from Eqs. 26 and Co-current case
and
Thus,
substituting into Counter-current
substituting
Eq_ 29 from ase
into
R,. can be evaluated.
27 for the
Eqs. 26 and
2s for the
Eq. 25 for f(c) or C,(z). the composite cell pair resistance. (AZ) can be chosen to evaluate the integral
Small intekals
as a summation.
--
I
=
RF
-7 -.-__,.____ ___ _.____.___ p’r-L_1 c-J=) C,(z) Ar
_
Parasiric ekctricai Because
K, i- fcl C,(z)
-
(32)
KJ
0
dicer Iosses
of the
inherent
engineering
design
of electrodialysis
units,
each
concentration stream within the stack is connect. via conduits, to all other concentration streams. The same applies for the dilution streams. Since the concentration stream is more conductive to electricity than the dilute stream. it would be expected that a high fraction of the nonproductive-leakage current would flow through the concentration stream. Leakage current calculations have been developed by Wilson (ft9) and Mandersloot and Hicks (If). Wilson uses an approximation of Manderdoot and Hicks’ generalized leakage equation by truncating the second term onward for both the numerator and denominator_ This approximation gives a slightly lower result. Dcsulinmiotr. 5 ( 1968) 267-291
Wilson’s truncated
equations
for
the
leakage
fraction
are given &tow.
[Z(tV’ f #)(I\;” + l)f3!-j if = _ --_._ -__. .._._ - .-...-___ -. _ i
t- i- ft
+
Y)
~!V’(N’
+
f33)
1)/q
Refer to Fig. 4.
If IV’ is very high compared
fi ._=. i
to r in Eq.33. the right hand side reduces to
2.-.-3fi + ‘Y)
(341
Van der Held (I.?) who considered a two-dimcnsionnl continuum membrane pack, derived the above equation. but without the 2!3 factor. Hence, we need only calculate 7 ratio in order to cstimatc the current Ieakage as a function of the totnf stack current. To calculate v we must cvatuate ‘iii,, channel rcsisrancc, and R. ceil pair resistance. R, is obtained by summing in par&A the brine and dialysatc chrtnnet resistances. R is obtaitwd in J. similar n~nncr to that discussed in the composite &I pair section. The anatogue duct leakage resistance can now be evaluated. knowing i, E, the stack current and potential difference. respectively_ &r = ;
[---:--_a
ohms-cm”@1
pair
(35)
Water can muvc from the dirifysate to the brine con~~~~rtrng~t and vice versa via several mechanisms including “primary h_vdration” of ions--either by counreror co-ion transport. Mandersloot ( !_Sj nlathenlaticall~ describes the detrimental effect water losses have on the coulomb etticicncy with the following equation. Desdittatiurt, 5 ( 1968) 267291
G. BELFORT
280
AND
Ci. A.
GUTCR
- Cl 2 rK(cd)d*l Cl - 2 I<( C*)i] _____ _ ._--.-_-..- _..-__ --._-I - 2 l~‘C),nme,n 2 W(C ) ] ln[l 9 = _____ ...__ ’ “:(‘d),l![’ --. . ---- --- __1- .J o 2= _ ,...._.._.. _L_-__._.. ___ rlD
1 -
rt;jc&
1 -
wc,),
Mandersfoot ( Z_3)has plotted tf/rlo rerms It’, for various desalination ranges. This enables us to get ~1immediately from Fig. 5 or Eq. 36. The true coulombic efiiciency, ‘I. is used to correct the coulombic efficiency (*lo) obtained from the membrane manufacturer.
Fig. 5. Ratio r~/~)n for various values of water transferred (it:) and diaiysnte tration at start (CJ)C when Cl,Atlct = 0.01 (drinking water) (13) (Eq. 36).
concen-
Hence, no actual analogue resistance to represent rhe water transfer tosses is fnvisaged. The correction and replacement of ‘lo by rt will represent the effect water transfer has on the system as a whole.
Membrane poteatid Membrane potentials arise from the concentration cell potentials which cdn exist across both types of membranes. In the derivation of the equations for membrane potentials a concentration celi with reversible electrodes and with transference of ions through the membrane is considered. The membrane potential is then obtained by subtracting the electrode potentials from the potential of such a cell. The homogeneity and refated phenomena of membranes used in efectrodialysis is being studied by Spiegler, et al. (14). it is hoped that a better understanding of this probiem will result and increase our knowledge of membrane potential_ Subtracting the electrode potential of a concentration cell with reversible Derolination, 5 (1968) 267-291
AS
ELECTRICAL
electrodes
ASALOCUE
FOR ELECTROI~IALYSIS
from the concentration
cell potential
E,,,,. and the cation
brane potential, Substituting
elcctrolytc
membrane
concentrations
281 itself. we can get the anion memE,,,,.
potential,
for activities, (37)
and
(C,,,,), and (C,,,,): are the concentrations of electrolyte at the membrane boundary layer on the brine side and dialysate side respectively and at position z along the ffow path of the membrane. the flow of ions is from In a concentration cell operatin, m spontaneously, higher to lower concentrations and a potential is set up accordingly_ The direction of flow of ions and the potential set up in each case opposes the direction of ion tlow and the potential applied when the electrodialysis process operates. Consequentiy. Eqs. 37 and 3S represent potcntkls which oppose the applied clectrodialysis potenti&. The total resistance of a membrane stack due to membrane potcnrial then where
becomes (
Rmp),o,~,
=
%!Y?+ i
In resistance-area R =v=
NcE“C i
R-cm’jstack
units per CC11pair.
(Rn&otat IL-cm’/ceII pair -- --G--
(40)
An electrode analogue resistance is developed (Ronode nnrl crrbodc).that will describe the various resistive components within the electrode compartment due to the bulk solution, the diffusion layer and kinetic phenomena at the electrode surface. The passage of current through each of the electrode compartments involves three steps: I_ The transfer of ions from the bulk of the solution to the surface of the electrode. 2. The ekctrochemicaI reaction at the electrode, 3. The formation of the final products of the reaction
from the electrode
and their removal
surface.
Coricentration overpotwtial
When the current
is flowing,
the ions that discharge
migrate
towards
the
Desalination,5 (I %8) 267-29 I
G. EELFORT AND G. A. GUTER
282
electrode and cause a concentration gradient across the thin diffusion layer at the electrode surface. This phenomenon is exictiy analogous to the concentration gradient that occurs at the ion exchange membranes_ The ~~ncentratjon gradient leads to a change in eiectrode potentiai of
Thus, w.: get
The chemical overpotentiat. &hem? is defined to be that potentiaf in excess of the discharge potentini for the given reaction which must be applied to the ceI1 in order to maintain a finite rate of discharge. Chemical averpotential occws PS a resuft of steps (2) and (3) shove. The value of t],-&*,,,, for the etectrude reaction is given by Ttifel”s Formula: ffchem
=
Cl
f
.-
where 2’ =s +
and
G = constant
(depends
on nature of cathode)
and &hem R ckeln = ---Ii
The Ohmic overpotential (Aq.& consists of two parts, namely the voltage drop which occurs in the bulk solution of constant concentration plus the voltage drop across the diffitsion layer where the concentration gradient varies linearly with the current density. Aqoam = &or
(49
+ Arls
which results in two series resistances, R ohm
=
R,t
-i-
Ra
(46-I
The first term, Rsolr is evaluated by the same procedure as used in the above discussion on composite cell pair resistance. providing the bulk solution chemical analysis is known. Desalination.
5 (1968)
267-291
AN ELECTRICAL
ANALOGUE
FOR ELECI RODlALYSlS
253
R, may be evaluated by specifying the rcsistivity at any point within the dtfksion Iaver as a function ~;(s, Z) and by computing the double integral of this function (as in the section for the case of membrane polarization). Thus. the anodic and cathodic resistance andope can be evaluated by summing ihe separate resistive components. respectively.
Basic definitions defining selectivity are available (17) in order to compare various membranes on a relative basis. The theoretical explanation of ionic transport in permselrctive membranes is in the embryonic stage. Umxplaincd anomalies occur. For example, at current densities above the critical value, it is expected that
hydrogen ions (H-‘), resulting from “water-splitting”. would start to carry the current through the cntionic membranes. but this is not always the case (18). The fact that the selectivity. F, in practice is not equal to one suggests that some hypothetical resistance R, could be placed in parallel with thecurrent etfecting separation so as to lower the coulomb efficiency. This allows the establishment of a resistance analogue for membrane selectivity. The following relationship between leakage current (jL). due to co-ion transport can be used to calculate R,. i rt
=
_.--.-
i
IL
=
1 -
(C. + r”,,
(48)
where tf is the coulomb efkicncy, considering losses due only to co-ion transport. Solving Eq. 48 for iL and substituting in the following equation we obtain, R, = __F._= $. ___ = .___A_- ._ 1, - 11) i(t”_ + f”,)
(49)
Studies (6, 19) of temperature effects on electrodialysis have produced overall temperature coefficients. Now that each subcomponent of the electrodialysis process can be quantitatively estimated, the effect of temperature on each individual subcomponent is possible. For esample, the equivalent conductance, (A,),-c. and hence the resistivity, px._, can be evaluated from Eq. 3 as a function of temperature for various solution concentrations. This will be used as a temperature correction for the equivatent resistances of concentration polarization, composite cetl pair and efectrode bulk streams. GENERAUZED
Table
ANALYTICAL
IL summarizes
EXPRESSION
the important
mathematical
relationships
derived
in
Desalinaiiun, 5 (1968) 267-291
II
II
1
II
p_-___
, Electrode polarization
Membrane potential
-_-
Rrol
R,
1
Rrhcm
R,,,,
_
E,,,
._-.-
-
i
_-__-___-
Ill
1
I
‘/chrm ‘-;-
A,
P 4’ = _!?!!-
=-
=
= -!‘f~!!!~
__---
(34)
_---
_---~.
__
__---P
(47)
(1)
(43
_______I-___
..-.
__..__-_____.-
_._1--
--
.
__“_--
-_-we-e--v-
---_
Rratoran = (Rror t Rd)n,mt-I- Rconc+ Rchrm
_ - _____-------.---
A, _- - -~~---.. Rmp = (RrnJOlJl N
__
_ _____..____------.-
(37)
_.._.-_ __-___.-- _
!’ _ y,,) - (1 2yG~ _ _ _-^ -_ -._.In( I - 2w,C,)/(I - 20,CT) (coulomb cfficicncy) ‘I
________ .______ ____ __ __ ---
E ” _ ..__ &if = _“._ iN
__.- _.- _ ______ _ __ ___ __ _
= !!.j! (I_ - I+) In .!:!!!:! dm:
__ _________ _-__._
-..
= R/R,N
-_ _.-_-_-
Y
i
= -I- 2-_-_ 3(1 t ‘I9
-ii
__...____-___--
l
Water transfer processes
_.______
Parasiticelectr&l duct losses
_
_
---
--.--
_---
(40)
W
286
G_ BELFORT
AND
G_ A. GUTER
the preceeding section. To develop a general mathematical equation. the individual resistance analogues can be combined as parallel or series resistances as expressed in the following equations: I
LI
p.l,r
=
--.. 1
1 -+ &crier
(50)
-ii- prd!d --
cquir
=q”‘.
1 ___-. =-_1 _+ -J’ R pW;lllCl
Rs
(51)
RL
cqu*r
where each component, R,. RAP, etc., is independently evaluated. For a list of the nomenclature and the equivalent circuit in above equations, refer to Figs. 6 and 7.
Fig. 6. Equhalent 1s
=
RAP
==-
RE
=
RAE
=
circuit for electrodialysis
process. Each R is an equivalent
Effective current for salt transfer EIectrode polarization effect, anode Electrode bulk stream Diffusion layer (DL) of anion membrane (AiW), electrode Bulk AN DL of AM, dilute stream side Dilute bulk stream DL of cation membrane (CM), dilute stream side Bulk C&f DL of C&f, concentrate stream side Concentrate bulk stream DL of Anf, concentrate stream side DL of Cbf, electrode stream side Electrode polarization effect, cathode Electrical leakage through dilute stream ducts Electrical leakage through concentrate stream ducts Membrane selectivity Resistance due to ohmic scale polarization
resistance.
stream side
= 2: == RD = RCD = RCJ~ = Rcc = Rc = RAC = RCE = Rcr RL
= =
Rs
= =
Ros
Desul
AN ELECTRICAL
ASALOCXJE
f*cd;
*
I B+_
.
3r.s..
Fig. 7. Equklenr
Also:
287
FOR ELECTRODIALYSIS
%Tki
rcsistnnce circuits for ZI sin&
cell pair. Symbols same ZLSin Fig. 6.
Rsu -= Rack dtffusion of cko2trol~te. (&-I.)T =:: Resistance for smglc ccl1 pair.
A power index. P,, for a given stack design is then defined as (53) where I is the stack current. R, the stack resistance, and tl is the current efficiency and AC is the change in dialysate concentration. This index is proportional to the power cost required for electrodialysis processing. A comparison of power indexes is possible for stacks of various designs. when product water rzte. feed water concentration. and amount of total dissolved solids removed are held constant. APPLICATIOSS
OF TIIE ELECTRICAL
The mathematical
ASALOGUE
derived above wili be applied to and compared with the actual operating data for two electrodialysis plants at Webster, South Dakota and Buckeye. Arizona.* Most of the plant data for Webster was obtained from Nordin (7) and Seko (20) and for Buckeye from Parsi (21) and Katz (22) of lonics. Inc. (23). A tabulation af results and comparison of actual values are given in Tables 111 and IV. The separate resistive component values are listed as well as their combined values. The composite cell pair resistance listed in these tables consists of membrane resistance and electrolyte resistance each of which were computed separately and listed in Table V. The composite cell pair resistance in generai increases with inlet feed concentration, as can be noticed in the Tables III and IV in progressing from the first stage to the latter stages. Webster water contains the least electrolyte and gives the highest value for composite cell pair resistance. A comparison of membrane l
equations
A computer program in Fortranws usedlo obtain the results depicted in this section. Desalination.
5 (I 968) 267-291
VI
$” _
**
l
13
9
---
zzz
--.
---
109.59 106.17 84.21 93.98
PLANT
3.07 5.35
44 42 .-.__--_
9,097 9,x95
per cell pilir for LL20 hour period.
--.--
225,363 247,869 273,hXX 284,472
All rCsiSlilfKX MlllCS in olimcm~ per Cdl pilit. Rise on ohm-en+ per cdl pair for ;I 20 hour pwotl,
-.......--_-___--I
68.68 107.67
** Rise in ohm-d!
! II
-
_-
__. -
7.16 10.06 13.20 19.1 I
wi.Im~
+ All rcsiwrncc vnlucs in ohm-cn$ per cdl pair,
102.25 I36,42 IXS,IB 249,7Y
1 II III IV
CFLL PAIR RF”wrANCI * 01
_.--
c-_--
-.
._--
__
X,486 11),3lY
--
4,4Y6 5.674 7,230 9,166
.-
-..--
I
--__
_.__
“--_
7.39 43.0x
_
56,3 94.6 173.1 260. I _._.__
_“.^
..__
______
0.114 0.216
.
0.234 0.309 O,hlO 0,617’) _-__.
.._
__..__-___
I IX.0 192.6
.
254.3 127.1 42x.x 581%
--.-*....--“_
I IX.0 193.2
303,‘) x+,4 4SY.O 567.3 ______
.
--
32.98 8.85
6,YO 4.07 1.75 1.34
S! io
s
t
= .” 3r
z
AS ELECTRICAL ASALOGUE
I’OR ELECTROIXALYSIS
289
resistance to electrolyte resistance at various positions along the flow path is shown in Table V. The membrane resistance in the Webster plant is low and insignificant, as the feed water changes in composition to that in the Buckeye plant, the membrane rcsistancc becomes greater, and is eighteen percent of the total cell pair resistance. In the Webster plant. membrane resistance is only two pcrccnt of the total stack resistance. TABLE
V
Chirs
Buckry
sru<=r
_.-
_.__
_.._ _-_
If&,) Composite cell patr wsistancc H Mcmbranr p.irh l’ocrIi@n* 0.25 In 0.50 MI 0.75 !,I
c
.__ . ._. I _, __._. ,.-----..--
ffSm’j~~:cll pair 6Y.6P
i!cm’
~klll
planr -_.._. II .---107.67
11.6W4
I I.5974
1 I .M3Y
i I S969
I I .fw9
I I.5961
52.3493 55.4473 61 s979
83.0379 94.7128 I l2.zOxo
1vt4Jsrrr plcmr
-
sru@2 --_.--.I --.--
-
-111 - -.--.
II
-.---
102.25
136.42
.-_.._-IV
l85.16
-.
249.79
Not calculntcd
4.806 4.805 4.805
4.804 4.504 4.804
4.804 4.803 4.803
Not calculated
I 16.765 126.738 138.957
163.862 176.541 191.770
223.638 241.246 262.507
The concentration polarization values represent the resistance of the diffusion layers adjacent to the membrane surface. Compared with other factors, this resistance is minor and represents at most five percent of total cell pair resistance. The resistance due to ohmic scale polarization is a major factor in total cell pair rcsistancc. It represents over forty percent of the total resistance in Stage I at Webster. Duct
leakage
and selectivity resistances are parallel to the desaltiq
resistances
in the network equations. The high resistance values found for these factors desirable and make only minor contributions to total stack resistance. Membrane
polarization
is a major factor contributing
are
as much as 45 percent
of the total resistance in the fourth stage of Webster plant. Electrode polarization is a minor factor, making up about one-tenth of one percent of the total resistance. The power index was calculated for each stase using Eq. 53. These values arc reported in the last column of Tables 111 and IV. The index is proportional to the electrical power required to treat 1009 gallons of water. This index decreases Desaharion.
5 ( 1968) 267-29
I
G. BELFORT AND C. A. CXJTER
290
from first to last stage for all plants. and increases from plant to plant as the salinity of the feed water increases. as is expected. The index can be used to compare stack or plant design oni) when the same feed water is used and when the same change in salinity is obtained. The power index can be used to indicate cffcctiveness or stack and pfent design.
This study represents dialysis
plant
design
a starting
and operation.
poi It for a realistic optimization Novel
engineering
equations
of clectro-
are derived
by
combining all kno*rn factors. including polarization phenomena. These equations etpress the complex nonohmic (non-linear) character of the electrodialysis process. Previous analyses were based on simpiified models which did not take into consideration the many subprocesses of electrodinlysis and their nonohmic behavior. This could, therefore. lead to substantiat errors. especially ii? the high current density region.
Several recommendations can be made for future research and development work to advance electrodialysis technology. The recommendations follow. I . The rtitttittation or subsrantiul reduction ofohmic scale poiarizntiott would si~nt$cant[v intprow the eronotttj- of the process. This factor is the least understood of all those subprocesses encountered in this paper. Although ohmic scale polarization is complex in nature. there appears to be no theoretical reason for not expecting improvement in this area. Ohmic scale polarization is considered to be due to build up of hard and soft scale, floes. and opposing potentials that can build up within the membrane. 7 iUurlio& of redttciitg nteftthranc potentials ttttm he considereci 10 guin -. ~igatjicant redtwrion ift ekrriral power costs. Membrane potentials arise due to concentration differences across the membrane coupled with sefective transport properties and are augmented by concentration gradients in the diffusion layers adjacent
to the membrane.
Membrane
potential
is a major resistive element and
could possibly be eliminated by improved hydrodynamics. 3. rlnu&tical studies to opritnizc pIam desigtt slroukd be tttade. The breakdown of total stack rcsistnnce and capability to calculate separate resistive elements can readily
be adopted to optimizing plant design and operating procedures. 4. Data .frunt operatitrg plartrs mm be ubraimxi ro further rqfinc mtd ~&~~fop
mow csrensiw md ntratriqqfui eiecrricaf anulogue expressiorts_ As more plant operating data becomes available, continued refinements to the analogue expressions will be possibte. 5. Membrane re:ear&t mm be pursued from rite slandpoim of redttcitt~ horh shot-r atrd long term polarizariott effecrs. Membrane resistance is a minor considemtion in seeking these improvements. Polarization effects determine cost factors exclusive of the electrical power costs. Scale and floe formation require special Desahation.
5(
1968) 267-29 I
AN ELECTRlCAL
ANALOGUE
operating procedures and replacement.
291
FOR ELECTRODIALYSIS
pretreatment.
pulsizq. acid backwashing,
membrane breakage
ACRSOWLEfXESfEN-f-
Thrs paper is the result of work completed 0001-676, “Study of Electrical Analogue Study for are indebted to the Ofke of Saline Water. U.S. ‘Nashington, D.C., for its support of this work and
under OSW Contract 14-OlElectrodiafysis.” Yhe’authors Department of the Interior, for its permission to present
and publish this paper. We arc also grateful to Drs. K. S. Spiegler and Carl Berger for their advice during the course of this study. REFkRESCES
R. A. Cooa. Some phcnamtn.x assoaated with concentration polnrizttion in clectrodralysts, Pmt. Fit cr Intern. S>ntp. art tf brer Dt~suImarion. Wushingfm. D.C.. Ocfoher 3-9, fW(55, Wpkx SWDll8. l_&wawr_I. Kep;trf I%. 65-6, Unixrsity of California, 2. S. A. \VLIXER. Sm gj’urer Cmwrtim Bcrk&y. 3. N. W. ROSEW~~RGASU C. E. TIRRFLL, Limiting currents in mcmbranc cells. in;/. Enp_ Ch~nr., -19 (1957) 7SI. Boundary layer dlmcnsions in daalysis, 4. H. P. GREGOR. PWL F. BRUIW ZIXD M. Rormswm;. Id. &kg. Char. Process Des&n Drrelop.. 4 ( 1965) 3. Saline Water Comcrsion Research in Israel, %linc Water Conversion 11, 5. IL S. SptttitxR, rldwnce.r in 0wmisfr.v Series. So. 335. Am. Chcm. Sot., Washington. D.C.. 1963, p. 179. 6. E. A. MMOU fun T. A. Knw~tz\\~. C.E.P. Spnp. St’ries So. 24, 55 (1959) 175. with Dr. John Nordin from the Demonstration Unit at Webster, 7. Prixatc Communication South Dakot.~ 8, D. A. COWAY AW J. H. BRO\VN, In& Enx. Cherrr.. 5 1 (1959) 1445. q/* Desaiinu~ictm, K. S. SPIEGLER(Edrtor), 9. L. ti. SH-\FFER a%~ M. S. MISTZ, in: Prinripk Academtc Prrs?;, N-Y., l966. London, 1960, p_ 268. 10. J. R. Wtt.sos, D;*minera?ix?tLn hy Efcrtrodialysis, Buttetworths, I (1966) 178-193. 11. W. G. 3. MA-.DERSLOOT AND R. E. HICI~S. Dcdmufion, rcsolts. discussed in Ref. (II). 12. E. F. h-I. VA’: DER H~t.t>. unpublished SM., 1I I (1964) 838. 13. W. G. B. MASDE~L~O’~, J. Elecmd~em. 14. K. S. SP~E(~LER,Private communication. En&wood. N.J., 1962, Chnptcr 6. 15. V. LEvIat. P/~~siuc/ternicaf ffyrlrodynun~icx, Prentice-Hall, New York, 1966, pp. 53 f. 16. C. L. MavrEt.t_. f%TIroci~emica~ Engineering, McGraw-Hill, 17. F. BERCSXW,Dechemu .\fom~gr.. 17 ( 1962) 449. 1% G. S. Sot_r, Proc. FirAt Intern. Symp. on Watu Desalina!ion, Wadrin~tarr, D.C., October 3-9, 1965, Paper SWD/3. I?. K. S. SPIEGLER, in: Ekctrorftenrical Operarions in IWI Ekcitan~e Techmo!og_s’.F. C. NICHOU AXD F. SCHUBERT (Editors), Acixdcmic Press. NY., 1956, p. 16% 20. M. SELO. Decltcntu, rtfomxerv 47 (1962) 575. 21. E. J. PXXSI, Private communication. 22_ W. E. KATZ. Private communication* 13. T. A. KIRKHAM, Papw presented April 4, 1963, Division of Water and W-s!e Chemistry, A.C.C.. Los Angeles, California. I.
Desalination,
5 ( l96R) 267-29 I
AN ELECTRICAL
Publishing Company.
ANALOGUE
Amsterdam
- Printed in The Netherlands
FOR ELECTRODIALYSIS
This paper is 3 first attempt to perform an analysis of clectrodialysis by considering the process as an electrical network composed of resistive elements representative of various electrochemical subprocesses. The total etrect of all subprocesses is unified into the single mathematical equation for the network. The treatment gises a breakdoitn of the various factors that contribute to efectricaI resistance and pinpoints those factors that must be improved to make technological improvements in the process. Application of the analysis to the Webster. S. D. and Buckeye. Ariz.. plants enables the resistance of the sepdrnte stages to be calculated. The importance of ohmic scale polarization in these plants is emphasized and predicted by the use of an empirical equation (16). The average calculated values for the siu stages of these plants agree to within 947; of the aberage measured values. SYMBOLS
a A
A, b t3 C
Cd=)
-
-
actual: membrane cross-sectional area, cm’ (= ISI x n) temperature dependent constant in the Onsagcr equation
active membrane area. cm* subscript denoting brine bulk stream temperature dependent constant in the Onsager equation average concentration of salt, in the water. g equivjliter average concentration of a cell pair at a distance = along the path iength. g equivjliter brine or dialysate concentration at some point s away from the membrane surface and along the fluid path length (see Fig. I ), g equivfliter dialysate inlet and exit (product) bulk concentration, 3 equivfliter concentration at the membrane surface, along the flow path z, of the brine and dialysate streams, respectively. g equiv/liter subscript denoting diaiysate bulk stream Desahation,
5 (1968) 267-291
268
G.
-
DL &
E
-
7’ == (_C& (II)x
-
F
-
F* r;b i
-
4 ‘lrm LPcr
-
IL
-
k k’
-
K,, K,. K,
-
m N N’ N,, N,
-
n
-
0
-
Ci
-
r
-
R CaPR,
-
R CP
-
R CO"C R rhrm
-
R,, 4, 4 Rl -
-
c”dW)~O,.,
-
R “P R dsm
-
dR/dt
-
ROS
-
RP
-
-
DELFORT
AND
G. A. CUTER
diffusion coefficient. cm”jsec stack voltage, volts anion and cation membrane potential respectively, volts fraction desalted fraction in product and feed streams in e.p.m. of ionic species .X Faraday ISo., coulombs/g equiv concentration stream exit flow rate. llsec, channel dilute stream exit flow rate. I/set. channel stack current density. amps/cm” current leakage due to duct losses, amps/cm’ limiting current density, mA/cm’ operating density, mA/cm” leakage current due to co-ion transport. amps ratio of the operating to limiting current density = ioppr/i,im ratio of brine to diaiysate compartment thickness = ybjyd constants in the generalized R, VS. C, Eq. 22 overall membrane length. cm number of cell pairs/stack N/2 number of anion and cation membranes, respectively overall membrane width, cm initial = (dialysate inlet), cm fraction of membrane area available for desalination power index, defined by Eq. 53 normalized leakage resistance = (RJR,N) cationic or anionic exchange membrane resistances respectively, ZL-cm” concentration polarization resistance. Q-cm2~ceIl pair concentration overpotential resistance (electrodes), R/stack chemical overpotential resistance (electrodes), R/stack dialysate and brine stream resistances respectively, R/stack parasitic duct loss resistance, R-cmZ/cell pair equivalent leakage resistances, Q/stack equivalent channet resistance, Q/stack total membrane potential, Q/stack total membrane potential defined by Eq. 40, Q-cmZ/cell pair ohmic o~~erpotential (electrodes), R/stack rate of change in ohmic polarization resistance of stack, ohms hr-’ resistance due to ohmic scale polarization, Gcm2@Al pair equivalent cell pair resistance, R/cell pair
Desalination,
5 (I 968)
267-291
AN ELECTRICAL ANALOGUE
-
-
-
-
cell pair resistance at some specific distance z up from the bottom (z = 0) of the stack, Q-cm’/cell pair hypothetical membrane selectivity resistance, Q-cm2~cell pair electrode solution resistance, Q/stack electrode diffusion layer resistance. Q/stack time of operation. hours transport number of counter ion in buik sofution transport number of counter ion in anion and cation exchange membranes, respectively transport numbers of the co-ions in the cation and anion exchange membranes, respectively where r‘l = I - f; and 1% = 1 - z‘!, linear fluid velocity within the membrane compartment. cm/src primary hydration number for both sodium and chloride ions
diaIysate or brine compartment -_ coordinate system away from -
-
269
FOR ELECTRODIALYSIS
thickness,
and along
cm
a membrane
surface
(refer to Fig. 1) (Ifnz) In [CC& O)fC(S, nt)] diffusion layer thickness. cm true coulombic cffrciency coulombic
efliciency
-
chemical
overpotential
-
concentration overpotential (electrode), volts ohmic overpotential (electrode), volts potential drop due to electrode solution resistance, vohs potential drop due to the electrode diffusion layers, volts equivaIent conductance of dialysate and brine streams equivalent conductance of solution at average concentration
-
-
-
excluding
water transfer
(electrode),
losses
volts
of salts and 2°C equivalent conductive at infinite dilution and t’C resistivity of the cationic or anionic membranes respectively, Q-cm resistivity of the solution at coordinate points s and z, R-cm dimensionless channel resistance. = &/R,N
INTRODUCTIOX
The paper is divided into three sections. In Section 1, Su~omponent Analysis, expressions are developed based on electrical analogies for each of the subcomponent factors influencing the operation of an electrodialysis system. The factors considered are concenttation polarization, ohmic scale polarization, composite ceil pair resistance, parasitic electrical losses, water transfer processe6, membrane potential, electrode polarization and membrane selectivity and temperamathematiczl
Desufinurion. 5 (1968) 267-29 1
G.
270
BELFORT
ASD
C.
A. CUTER
ture effects.
The effects of hydrodynamic factors and temperature are also included. In Section 11. lntegrntion of Subcomponent Mathematical Elements into the interrelationships between the resistive a General Analytical Expression. elements are studied and combined intoageneral expression for a resistance network. In expression for total stack resistance is then written in terms of the separate resistive elements and their combination into R resistance network_ in Section III. Applications of Generalized Mathematical Solution to Specific Situations. the expression for the derived resistance network is applied to specific plant situations at Webster, South Dakota and Buckeye. Arizona. The conclusion includes recommendations for future research and development work in Improving electrodialysis technology. These recommendations are based on the analysis of the Webster and Buckeye plants. This paper represents n starting point for a realistic optimization of the operation and design of electrodialysis plants. SUBCOMPO?iEZ;F
AXALYSIS
A discussion and reciew of a number of subcomponent factors of the electrodialysis process will be undertaken. For each subcomponent factor, the resistanceanologue (ohm-cm”~cell pair). wit1 be preceded by a summary of the present state of the art. Thereafter. all the individual subcomponent resistances will be combined together into a general analytical expression. This general analytical expression will represent a model or electrical rtnalogue of the electrodialysis process. M etttbrutie polurizalion Extensive experimental work has been (I), and is being (_‘I. done to quantitatively explain the phenomena of membrane polarization. The dialyzing current faces a two-ford polarization effect close to the membrane surface. A concentration gradient across the diffusion layer and scale formation are the respective causes of such polarization, The former is termed concentration polarization, while the latter is called ohmic scale polarization. Each is separately discussed and evaluated belOW.
Cottcenlrutiotl polarixrion It is possible to estimate the approximate resistance due to concentration polarization that the dialyzing current faces. provided two important system pafameters can be calculated. These are the thickness (6) of the concerttration gradient and profile across the diffusion layer. Several empirical approaches (31, such as use of the Chifton-Colburn transfer factors and the flux equation of Fick, are able to predict the diffusion layer thickness for nonspacer-filled compartments. H. P. Cregor, et al. (4) have studied and measured experimentally, using various size spacers at different compartment Desalinarimt.
5 ( 1968) 267-29 I
AN ELECTRICAL
FOR ELECTRODlALYSIS
ANALOGUE
271
flow rates and Reynolds numbers, the relation of the diffusion layer-thickness with flow rates. Refer to Fig. I to calculate the diffusion layer resistance that the dialyzing
Colcentratron Prof1lr
_____
--
-
-
I-le:
x !L
~(6.0)
Fig. 1. Diagram of the concentration protile and the diffusion layer on the dinlysatc side of an elecrroddysis IOII exchange membrane.
current must pass through. a double integration is performed layer (x coordinate) and atong the flow path (Z coordinate).
From the Onsager equation (A&c
= fA,),*c
and the basic definition
across the diffusion
of resistivity we get
(2)
- CA + B CA.,x=cl GA*
1000 p,,, = __L!!__ = _.._____.._-_....___ (A,),ec C,., - C,‘!: CA + B (A ,X=J (A,)PC Cx.2
(3)
After pl.= is substituted in Eq.1, C,., as a function of x and z has to be evaluated from the boundary conditions to relate the concentration at the membrane surface C(0, Z) with the diffusion layer-bulk interface concentration, C(is, z). The Nernst idealized equation will be used. as most empiricai (I) and theoretical (5) equations available to do this are also not applicable to spacer filled configurations. Desalination,
5 (1968) 267-29 I
G. RELFORT ASD C. A. CUTER
272
Together with several simplifying assumptions presupposes a linear concentration gradient.
(I). the Nernst
equation
z) - _..._ CfO, z) f_) = C(3, dc = i(C_ ____._-. __..-._ _____- -..-ds
w
d
F4.
From the Nenst equation and realizing that at the limiting current tlirn, ~.~rface concentration c’(O, :) npproachec; zero,
densit)
C-(0. 1) = (1 - k)C(S, z) The following general equation assuming a linear concentration nential bulk stream concentration
(5) for C(x., z) can bc written from linear anrllysis. gradient across the diffusion layer and an eupodeea? from the inlet to the stream outlet.
The diffusion layer thickness (6). the limiting current (i&. and the operriting current density (i). must be estimated before the above equation can bc used. Limiting current density stream C(0, W) shown
diffusion
fili,) wilt occur when the exit concentration of the diiutc in Fig. I reaches zero. providing the fluid velocitg and
layer thiclcness are constant.
From the Nernst equation.
s = ___ C(& ~)FDL ._.. _.. i,im(i!_ -
(7)
I-)
Severat empirical formulae (6.7) have been developed to obtain the Iimitingcurrent density as a function of concentration and finc?aarfluid velocity. The general form of these equations is _.--.JJirn-
C(&
=
Al(c)
at a given
temperature.
(8)
nr)
Some question arises as to exactly when the limiting current occurs. defined from the “Cohan” plot (8) or from pH changes. Once iJi,
It could be is obtained
experimentally, S can be calculated from Eq. 7 and iopCrfrom Eq. 10, defined Hence. k can be evaluated. Using Faraday’s Law and a material balance, the capacity is detined iqrQ,tz = zip = --=_
FddYdlf
g equiv
below. as,
transferred/see
and,
iowr = -f FdCJtf-
amps/cm
’
9.4,
We obtain
A,, the equivalent
solution
conductance,
from
IksalinaJton,
the Onsager 5 f 1968) 267-29 1
AN ELECTRICAL
equation
ANALOGUE
at various
FOR ELECTROI~IAL%SIS
temperatures
and concentrations
273 with a second term correcting
for non-ideality.
(W
The concentration polarization resistance component R,, is evaluated for the dialysate stream and :tnion exchange membrane using Eq.1. The remaining three ditfusion layer resistances ore calculated in a similar manner.
Ohmic scale polarization invoives a number of complex and interrelated fixtors. It is associated with scale formation and other effects that contribute to resistance rise with operating time in the order of hours and days. Scales differ in their chemical and physical character as welt as the processes by which they are formed. Scales. such as calcmm suiphate and carbonate. magnesium hydroxide, ferric hydroxide, can precipitate out of solution and deposit on the m~rnbr~nc surfaces at points of poor hydrodynamic activity. Ohmic scale polarization can thus be expected to vary with electrolyte composition_ current density, limiting current densit), concentration polarization, local hydrodynamic conditions as influenced by spacer design and flow velocity, and pretreatment procedures. The task of writing a resistance analogue expression for the ohmic polarization factor is further encumbered by a lack of plant and experimental data on this phenomenon. Some data have been collected usin, a ‘at small test stack at the Webster plant. Fig. 2 shows the rate of resistance change with continuous operating current. The influence of current is obvious. This data was obtained usinS Webster water which had not been treated with sulfuric acid. Resistance buildup is due to dcposition of calcium carbonate and calcium sulfate scales as we11 as ferric hydroxide floe and inorganic materials which evade the pretreatment filters. The orientation polarization due to the reorientation of internal polar groups, may also be operative. of the general type The data of Fig. 2 were found to satisfy an equation Desahatiorr,
5 (1968)
267-291
G. RELFORT
Fig. 2. Relationship
dR -
between rate of resistance change and reciprocal
AND
G. A. GUTER
of stack ctwrcnt.
a)p
(15.W
dt
where n’ and h’ are constants. The ohmic scale polarization following equation and definition.
analogue
resistance
will be evaluated
using the
= 0.150 + 0.92k
$$ =
1
(15B)
41+‘2k
dt
and for a 20 hour period between current restored to the stack.
reversal with the virgin resistance
being
The constants in Eq. 16 were chosen to give the best fit to the avaifabfe data. A comparison ofcalcutated values and observed values for ohmic scale poktrization is given in Table I. The observed values for the Webster and Buckeye plants were calculated by assuming that the major differences between observed cell pair Desalination,
5 ( 1968) 267-29 I
AS
ELECTRICAL
ANALOGUE
FOR ELECTRODIALYSIS
275
TABLE I
l~*chsrcr plurlr 0.64 0.63 0.52 0.56
7.6 7.3 58 -l.i
5.5 5.4 .a.3 A.6
Bttckc*y e pianr 0.20 0.16
2.1 2.1
22 2.1
. ... _---_..---__
.
.- _
Ohmscmzjh per ccl1 pair. ** Fi\c pow test srnck. 260 cm2 membnncs. Darn suppled b\ Dr. John Nordin. Wcbsrcr,
l
So.
Dakot:;.
resistance and the \Aue calculated using ail factors other than ohmic scale poiaritation were due to the latter factor. The equation gives surprisin@y good results based on the theory and data limitations. It should be pointed out. howvcr, that Eq.16 is only an approGmation at best. It should also he noted that R,, is sery scnsttive to i/iti,_ This ratio can vary from time to time during stack operation_ It also may not be constant along the flow path. iIlm was evaluated using Eq. 8 with the constants M and n supplied by the membrane manufacturer at varioub concentrations.
The cell pair resistance is defined as the sum of the anion exchange membran;t resistance R,,. the cation exchange membrane resistance R,,,, and the dialysate Rd. and brine R,. stream resistances, all cakulated at some position along the flow path length. WA
= R,
+ L
f R, f R,
the membrane resistances can be obtained where H,,, L,, evaiuated at some average concentration, C,
($7) from literature
or
276
(i.
6ELFORT
AND
G. A. Gl.‘TER
Integration over the mtmbrane area (pu) available for dewlination of the pair resrstance (R,), at chosen path length z, from the bottom of the stack, result in the total resiswnce per cell pair. R, tihere
(19)
ff ue assume that (f?,,), is constan! z direction), then the area equ&. pu
= pnz
flu
= pnd: cm’
transposing
in the iateral direction
(perpendicular
to the
cm’
variables,
According to Mason and Kirkham (6) cell pair resistance expressed in terms of borne average conccntratiun. 1
= 3
CJ:)
at coordinate
z con be
k’ *. .- + -.I.CJ=, C,(z) I
(21)
and (22) Variation
of CJz) and L‘,Iz) with Z, wili make C=(z) a function
C,(z) = j-C=)
of Z. (231
(24) Substituting I
R,
Eq. 24 into Ey. 20, .
=
+
Pff 0
K,
-
K,f(z)
-I
I
(25)
dz
where Desdinarbn,
5 (1968) 267-291
AN ELECI-RICAL
ASALOGlJE
277
FOR ELECTRODIALYSIS
K. K2 and K, are constants for a given space geometry, ionic mixture, membrane type. and temperature as defined in Eq. 22. Based on the traditional treatment 49) of electrodiatysis optimization. an exponential decay relationship for the dialysate bulk stream concentration and the path distance. =_ wit1 be assumed. It has also tacitly been assumed up untii now, that equipctential surfaces exist parallel to the membranes and electrodes. The exponential assumption can be used to prove the existence of the equipotentiai surfacf5. Evaluating each bulk solution as a function of path distance, 1, and combining each to give c’, tie get for the Dialysate c-&J-_)= C&)e-”
(26)
and from a material balance across the unit at a distance r: along the path length and the top of the unit (at z -= ~PZ).Refer to Fig. 3.
“,
‘,
c
c
b
fcJf
Fig. 3. The concentration
and brim
profile
kqxxentla!)
stremns for co- and uountcr-current
and material h.dances of both the dialysatc
operation. Dtwdinatiun. 5 ( 1968) 267-.29 I
G.
27s Brine
(Counter-current C,(z)
from
XSD
G. A. CUTER
flow)
= [C,(m)
By definitron
BELFORT
-
C,(m)]
(F,/FJ
Eq. 21. evaluate
+ [(FJF,)
C~o)]e-”
(25)
C,, (29)
substituting from Eqs. 26 and Co-current case
and
Thus,
substituting into Counter-current
substituting
Eq_ 29 from ase
into
R,. can be evaluated.
27 for the
Eqs. 26 and
2s for the
Eq. 25 for f(c) or C,(z). the composite cell pair resistance. (AZ) can be chosen to evaluate the integral
Small intekals
as a summation.
--
I
=
RF
-7 -.-__,.____ ___ _.____.___ p’r-L_1 c-J=) C,(z) Ar
_
Parasiric ekctricai Because
K, i- fcl C,(z)
-
(32)
KJ
0
dicer Iosses
of the
inherent
engineering
design
of electrodialysis
units,
each
concentration stream within the stack is connect. via conduits, to all other concentration streams. The same applies for the dilution streams. Since the concentration stream is more conductive to electricity than the dilute stream. it would be expected that a high fraction of the nonproductive-leakage current would flow through the concentration stream. Leakage current calculations have been developed by Wilson (ft9) and Mandersloot and Hicks (If). Wilson uses an approximation of Manderdoot and Hicks’ generalized leakage equation by truncating the second term onward for both the numerator and denominator_ This approximation gives a slightly lower result. Dcsulinmiotr. 5 ( 1968) 267-291
Wilson’s truncated
equations
for
the
leakage
fraction
are given &tow.
[Z(tV’ f #)(I\;” + l)f3!-j if = _ --_._ -__. .._._ - .-...-___ -. _ i
t- i- ft
+
Y)
~!V’(N’
+
f33)
1)/q
Refer to Fig. 4.
If IV’ is very high compared
fi ._=. i
to r in Eq.33. the right hand side reduces to
2.-.-3fi + ‘Y)
(341
Van der Held (I.?) who considered a two-dimcnsionnl continuum membrane pack, derived the above equation. but without the 2!3 factor. Hence, we need only calculate 7 ratio in order to cstimatc the current Ieakage as a function of the totnf stack current. To calculate v we must cvatuate ‘iii,, channel rcsisrancc, and R. ceil pair resistance. R, is obtained by summing in par&A the brine and dialysatc chrtnnet resistances. R is obtaitwd in J. similar n~nncr to that discussed in the composite &I pair section. The anatogue duct leakage resistance can now be evaluated. knowing i, E, the stack current and potential difference. respectively_ &r = ;
[---:--_a
ohms-cm”@1
pair
(35)
Water can muvc from the dirifysate to the brine con~~~~rtrng~t and vice versa via several mechanisms including “primary h_vdration” of ions--either by counreror co-ion transport. Mandersloot ( !_Sj nlathenlaticall~ describes the detrimental effect water losses have on the coulomb etticicncy with the following equation. Desdittatiurt, 5 ( 1968) 267291
G. BELFORT
280
AND
Ci. A.
GUTCR
- Cl 2 rK(cd)d*l Cl - 2 I<( C*)i] _____ _ ._--.-_-..- _..-__ --._-I - 2 l~‘C),nme,n 2 W(C ) ] ln[l 9 = _____ ...__ ’ “:(‘d),l![’ --. . ---- --- __1- .J o 2= _ ,...._.._.. _L_-__._.. ___ rlD
1 -
rt;jc&
1 -
wc,),
Mandersfoot ( Z_3)has plotted tf/rlo rerms It’, for various desalination ranges. This enables us to get ~1immediately from Fig. 5 or Eq. 36. The true coulombic efiiciency, ‘I. is used to correct the coulombic efficiency (*lo) obtained from the membrane manufacturer.
Fig. 5. Ratio r~/~)n for various values of water transferred (it:) and diaiysnte tration at start (CJ)C when Cl,Atlct = 0.01 (drinking water) (13) (Eq. 36).
concen-
Hence, no actual analogue resistance to represent rhe water transfer tosses is fnvisaged. The correction and replacement of ‘lo by rt will represent the effect water transfer has on the system as a whole.
Membrane poteatid Membrane potentials arise from the concentration cell potentials which cdn exist across both types of membranes. In the derivation of the equations for membrane potentials a concentration celi with reversible electrodes and with transference of ions through the membrane is considered. The membrane potential is then obtained by subtracting the electrode potentials from the potential of such a cell. The homogeneity and refated phenomena of membranes used in efectrodialysis is being studied by Spiegler, et al. (14). it is hoped that a better understanding of this probiem will result and increase our knowledge of membrane potential_ Subtracting the electrode potential of a concentration cell with reversible Derolination, 5 (1968) 267-291
AS
ELECTRICAL
electrodes
ASALOCUE
FOR ELECTROI~IALYSIS
from the concentration
cell potential
E,,,,. and the cation
brane potential, Substituting
elcctrolytc
membrane
concentrations
281 itself. we can get the anion memE,,,,.
potential,
for activities, (37)
and
(C,,,,), and (C,,,,): are the concentrations of electrolyte at the membrane boundary layer on the brine side and dialysate side respectively and at position z along the ffow path of the membrane. the flow of ions is from In a concentration cell operatin, m spontaneously, higher to lower concentrations and a potential is set up accordingly_ The direction of flow of ions and the potential set up in each case opposes the direction of ion tlow and the potential applied when the electrodialysis process operates. Consequentiy. Eqs. 37 and 3S represent potcntkls which oppose the applied clectrodialysis potenti&. The total resistance of a membrane stack due to membrane potcnrial then where
becomes (
Rmp),o,~,
=
%!Y?+ i
In resistance-area R =v=
NcE“C i
R-cm’jstack
units per CC11pair.
(Rn&otat IL-cm’/ceII pair -- --G--
(40)
An electrode analogue resistance is developed (Ronode nnrl crrbodc).that will describe the various resistive components within the electrode compartment due to the bulk solution, the diffusion layer and kinetic phenomena at the electrode surface. The passage of current through each of the electrode compartments involves three steps: I_ The transfer of ions from the bulk of the solution to the surface of the electrode. 2. The ekctrochemicaI reaction at the electrode, 3. The formation of the final products of the reaction
from the electrode
and their removal
surface.
Coricentration overpotwtial
When the current
is flowing,
the ions that discharge
migrate
towards
the
Desalination,5 (I %8) 267-29 I
G. EELFORT AND G. A. GUTER
282
electrode and cause a concentration gradient across the thin diffusion layer at the electrode surface. This phenomenon is exictiy analogous to the concentration gradient that occurs at the ion exchange membranes_ The ~~ncentratjon gradient leads to a change in eiectrode potentiai of
Thus, w.: get
The chemical overpotentiat. &hem? is defined to be that potentiaf in excess of the discharge potentini for the given reaction which must be applied to the ceI1 in order to maintain a finite rate of discharge. Chemical averpotential occws PS a resuft of steps (2) and (3) shove. The value of t],-&*,,,, for the etectrude reaction is given by Ttifel”s Formula: ffchem
=
Cl
f
.-
where 2’ =s +
and
G = constant
(depends
on nature of cathode)
and &hem R ckeln = ---Ii
The Ohmic overpotential (Aq.& consists of two parts, namely the voltage drop which occurs in the bulk solution of constant concentration plus the voltage drop across the diffitsion layer where the concentration gradient varies linearly with the current density. Aqoam = &or
(49
+ Arls
which results in two series resistances, R ohm
=
R,t
-i-
Ra
(46-I
The first term, Rsolr is evaluated by the same procedure as used in the above discussion on composite cell pair resistance. providing the bulk solution chemical analysis is known. Desalination.
5 (1968)
267-291
AN ELECTRICAL
ANALOGUE
FOR ELECI RODlALYSlS
253
R, may be evaluated by specifying the rcsistivity at any point within the dtfksion Iaver as a function ~;(s, Z) and by computing the double integral of this function (as in the section for the case of membrane polarization). Thus. the anodic and cathodic resistance andope can be evaluated by summing ihe separate resistive components. respectively.
Basic definitions defining selectivity are available (17) in order to compare various membranes on a relative basis. The theoretical explanation of ionic transport in permselrctive membranes is in the embryonic stage. Umxplaincd anomalies occur. For example, at current densities above the critical value, it is expected that
hydrogen ions (H-‘), resulting from “water-splitting”. would start to carry the current through the cntionic membranes. but this is not always the case (18). The fact that the selectivity. F, in practice is not equal to one suggests that some hypothetical resistance R, could be placed in parallel with thecurrent etfecting separation so as to lower the coulomb efficiency. This allows the establishment of a resistance analogue for membrane selectivity. The following relationship between leakage current (jL). due to co-ion transport can be used to calculate R,. i rt
=
_.--.-
i
IL
=
1 -
(C. + r”,,
(48)
where tf is the coulomb efkicncy, considering losses due only to co-ion transport. Solving Eq. 48 for iL and substituting in the following equation we obtain, R, = __F._= $. ___ = .___A_- ._ 1, - 11) i(t”_ + f”,)
(49)
Studies (6, 19) of temperature effects on electrodialysis have produced overall temperature coefficients. Now that each subcomponent of the electrodialysis process can be quantitatively estimated, the effect of temperature on each individual subcomponent is possible. For esample, the equivalent conductance, (A,),-c. and hence the resistivity, px._, can be evaluated from Eq. 3 as a function of temperature for various solution concentrations. This will be used as a temperature correction for the equivatent resistances of concentration polarization, composite cetl pair and efectrode bulk streams. GENERAUZED
Table
ANALYTICAL
IL summarizes
EXPRESSION
the important
mathematical
relationships
derived
in
Desalinaiiun, 5 (1968) 267-291
II
II
1
II
p_-___
, Electrode polarization
Membrane potential
-_-
Rrol
R,
1
Rrhcm
R,,,,
_
E,,,
._-.-
-
i
_-__-___-
Ill
1
I
‘/chrm ‘-;-
A,
P 4’ = _!?!!-
=-
=
= -!‘f~!!!~
__---
(34)
_---
_---~.
__
__---P
(47)
(1)
(43
_______I-___
..-.
__..__-_____.-
_._1--
--
.
__“_--
-_-we-e--v-
---_
Rratoran = (Rror t Rd)n,mt-I- Rconc+ Rchrm
_ - _____-------.---
A, _- - -~~---.. Rmp = (RrnJOlJl N
__
_ _____..____------.-
(37)
_.._.-_ __-___.-- _
!’ _ y,,) - (1 2yG~ _ _ _-^ -_ -._.In( I - 2w,C,)/(I - 20,CT) (coulomb cfficicncy) ‘I
________ .______ ____ __ __ ---
E ” _ ..__ &if = _“._ iN
__.- _.- _ ______ _ __ ___ __ _
= !!.j! (I_ - I+) In .!:!!!:! dm:
__ _________ _-__._
-..
= R/R,N
-_ _.-_-_-
Y
i
= -I- 2-_-_ 3(1 t ‘I9
-ii
__...____-___--
l
Water transfer processes
_.______
Parasiticelectr&l duct losses
_
_
---
--.--
_---
(40)
W
286
G_ BELFORT
AND
G_ A. GUTER
the preceeding section. To develop a general mathematical equation. the individual resistance analogues can be combined as parallel or series resistances as expressed in the following equations: I
LI
p.l,r
=
--.. 1
1 -+ &crier
(50)
-ii- prd!d --
cquir
=q”‘.
1 ___-. =-_1 _+ -J’ R pW;lllCl
Rs
(51)
RL
cqu*r
where each component, R,. RAP, etc., is independently evaluated. For a list of the nomenclature and the equivalent circuit in above equations, refer to Figs. 6 and 7.
Fig. 6. Equhalent 1s
=
RAP
==-
RE
=
RAE
=
circuit for electrodialysis
process. Each R is an equivalent
Effective current for salt transfer EIectrode polarization effect, anode Electrode bulk stream Diffusion layer (DL) of anion membrane (AiW), electrode Bulk AN DL of AM, dilute stream side Dilute bulk stream DL of cation membrane (CM), dilute stream side Bulk C&f DL of C&f, concentrate stream side Concentrate bulk stream DL of Anf, concentrate stream side DL of Cbf, electrode stream side Electrode polarization effect, cathode Electrical leakage through dilute stream ducts Electrical leakage through concentrate stream ducts Membrane selectivity Resistance due to ohmic scale polarization
resistance.
stream side
= 2: == RD = RCD = RCJ~ = Rcc = Rc = RAC = RCE = Rcr RL
= =
Rs
= =
Ros
Desul
AN ELECTRICAL
ASALOCXJE
f*cd;
*
I B+_
.
3r.s..
Fig. 7. Equklenr
Also:
287
FOR ELECTRODIALYSIS
%Tki
rcsistnnce circuits for ZI sin&
cell pair. Symbols same ZLSin Fig. 6.
Rsu -= Rack dtffusion of cko2trol~te. (&-I.)T =:: Resistance for smglc ccl1 pair.
A power index. P,, for a given stack design is then defined as (53) where I is the stack current. R, the stack resistance, and tl is the current efficiency and AC is the change in dialysate concentration. This index is proportional to the power cost required for electrodialysis processing. A comparison of power indexes is possible for stacks of various designs. when product water rzte. feed water concentration. and amount of total dissolved solids removed are held constant. APPLICATIOSS
OF TIIE ELECTRICAL
The mathematical
ASALOGUE
derived above wili be applied to and compared with the actual operating data for two electrodialysis plants at Webster, South Dakota and Buckeye. Arizona.* Most of the plant data for Webster was obtained from Nordin (7) and Seko (20) and for Buckeye from Parsi (21) and Katz (22) of lonics. Inc. (23). A tabulation af results and comparison of actual values are given in Tables 111 and IV. The separate resistive component values are listed as well as their combined values. The composite cell pair resistance listed in these tables consists of membrane resistance and electrolyte resistance each of which were computed separately and listed in Table V. The composite cell pair resistance in generai increases with inlet feed concentration, as can be noticed in the Tables III and IV in progressing from the first stage to the latter stages. Webster water contains the least electrolyte and gives the highest value for composite cell pair resistance. A comparison of membrane l
equations
A computer program in Fortranws usedlo obtain the results depicted in this section. Desalination.
5 (I 968) 267-291
VI
$” _
**
l
13
9
---
zzz
--.
---
109.59 106.17 84.21 93.98
PLANT
3.07 5.35
44 42 .-.__--_
9,097 9,x95
per cell pilir for LL20 hour period.
--.--
225,363 247,869 273,hXX 284,472
All rCsiSlilfKX MlllCS in olimcm~ per Cdl pilit. Rise on ohm-en+ per cdl pair for ;I 20 hour pwotl,
-.......--_-___--I
68.68 107.67
** Rise in ohm-d!
! II
-
_-
__. -
7.16 10.06 13.20 19.1 I
wi.Im~
+ All rcsiwrncc vnlucs in ohm-cn$ per cdl pair,
102.25 I36,42 IXS,IB 249,7Y
1 II III IV
CFLL PAIR RF”wrANCI * 01
_.--
c-_--
-.
._--
__
X,486 11),3lY
--
4,4Y6 5.674 7,230 9,166
.-
-..--
I
--__
_.__
“--_
7.39 43.0x
_
56,3 94.6 173.1 260. I _._.__
_“.^
..__
______
0.114 0.216
.
0.234 0.309 O,hlO 0,617’) _-__.
.._
__..__-___
I IX.0 192.6
.
254.3 127.1 42x.x 581%
--.-*....--“_
I IX.0 193.2
303,‘) x+,4 4SY.O 567.3 ______
.
--
32.98 8.85
6,YO 4.07 1.75 1.34
S! io
s
t
= .” 3r
z
AS ELECTRICAL ASALOGUE
I’OR ELECTROIXALYSIS
289
resistance to electrolyte resistance at various positions along the flow path is shown in Table V. The membrane resistance in the Webster plant is low and insignificant, as the feed water changes in composition to that in the Buckeye plant, the membrane rcsistancc becomes greater, and is eighteen percent of the total cell pair resistance. In the Webster plant. membrane resistance is only two pcrccnt of the total stack resistance. TABLE
V
Chirs
Buckry
sru<=r
_.-
_.__
_.._ _-_
If&,) Composite cell patr wsistancc H Mcmbranr p.irh l’ocrIi@n* 0.25 In 0.50 MI 0.75 !,I
c
.__ . ._. I _, __._. ,.-----..--
ffSm’j~~:cll pair 6Y.6P
i!cm’
~klll
planr -_.._. II .---107.67
11.6W4
I I.5974
1 I .M3Y
i I S969
I I .fw9
I I.5961
52.3493 55.4473 61 s979
83.0379 94.7128 I l2.zOxo
1vt4Jsrrr plcmr
-
sru@2 --_.--.I --.--
-
-111 - -.--.
II
-.---
102.25
136.42
.-_.._-IV
l85.16
-.
249.79
Not calculntcd
4.806 4.805 4.805
4.804 4.504 4.804
4.804 4.803 4.803
Not calculated
I 16.765 126.738 138.957
163.862 176.541 191.770
223.638 241.246 262.507
The concentration polarization values represent the resistance of the diffusion layers adjacent to the membrane surface. Compared with other factors, this resistance is minor and represents at most five percent of total cell pair resistance. The resistance due to ohmic scale polarization is a major factor in total cell pair rcsistancc. It represents over forty percent of the total resistance in Stage I at Webster. Duct
leakage
and selectivity resistances are parallel to the desaltiq
resistances
in the network equations. The high resistance values found for these factors desirable and make only minor contributions to total stack resistance. Membrane
polarization
is a major factor contributing
are
as much as 45 percent
of the total resistance in the fourth stage of Webster plant. Electrode polarization is a minor factor, making up about one-tenth of one percent of the total resistance. The power index was calculated for each stase using Eq. 53. These values arc reported in the last column of Tables 111 and IV. The index is proportional to the electrical power required to treat 1009 gallons of water. This index decreases Desaharion.
5 ( 1968) 267-29
I
G. BELFORT AND C. A. CXJTER
290
from first to last stage for all plants. and increases from plant to plant as the salinity of the feed water increases. as is expected. The index can be used to compare stack or plant design oni) when the same feed water is used and when the same change in salinity is obtained. The power index can be used to indicate cffcctiveness or stack and pfent design.
This study represents dialysis
plant
design
a starting
and operation.
poi It for a realistic optimization Novel
engineering
equations
of clectro-
are derived
by
combining all kno*rn factors. including polarization phenomena. These equations etpress the complex nonohmic (non-linear) character of the electrodialysis process. Previous analyses were based on simpiified models which did not take into consideration the many subprocesses of electrodinlysis and their nonohmic behavior. This could, therefore. lead to substantiat errors. especially ii? the high current density region.
Several recommendations can be made for future research and development work to advance electrodialysis technology. The recommendations follow. I . The rtitttittation or subsrantiul reduction ofohmic scale poiarizntiott would si~nt$cant[v intprow the eronotttj- of the process. This factor is the least understood of all those subprocesses encountered in this paper. Although ohmic scale polarization is complex in nature. there appears to be no theoretical reason for not expecting improvement in this area. Ohmic scale polarization is considered to be due to build up of hard and soft scale, floes. and opposing potentials that can build up within the membrane. 7 iUurlio& of redttciitg nteftthranc potentials ttttm he considereci 10 guin -. ~igatjicant redtwrion ift ekrriral power costs. Membrane potentials arise due to concentration differences across the membrane coupled with sefective transport properties and are augmented by concentration gradients in the diffusion layers adjacent
to the membrane.
Membrane
potential
is a major resistive element and
could possibly be eliminated by improved hydrodynamics. 3. rlnu&tical studies to opritnizc pIam desigtt slroukd be tttade. The breakdown of total stack rcsistnnce and capability to calculate separate resistive elements can readily
be adopted to optimizing plant design and operating procedures. 4. Data .frunt operatitrg plartrs mm be ubraimxi ro further rqfinc mtd ~&~~fop
mow csrensiw md ntratriqqfui eiecrricaf anulogue expressiorts_ As more plant operating data becomes available, continued refinements to the analogue expressions will be possibte. 5. Membrane re:ear&t mm be pursued from rite slandpoim of redttcitt~ horh shot-r atrd long term polarizariott effecrs. Membrane resistance is a minor considemtion in seeking these improvements. Polarization effects determine cost factors exclusive of the electrical power costs. Scale and floe formation require special Desahation.
5(
1968) 267-29 I
AN ELECTRlCAL
ANALOGUE
operating procedures and replacement.
291
FOR ELECTRODIALYSIS
pretreatment.
pulsizq. acid backwashing,
membrane breakage
ACRSOWLEfXESfEN-f-
Thrs paper is the result of work completed 0001-676, “Study of Electrical Analogue Study for are indebted to the Ofke of Saline Water. U.S. ‘Nashington, D.C., for its support of this work and
under OSW Contract 14-OlElectrodiafysis.” Yhe’authors Department of the Interior, for its permission to present
and publish this paper. We arc also grateful to Drs. K. S. Spiegler and Carl Berger for their advice during the course of this study. REFkRESCES
R. A. Cooa. Some phcnamtn.x assoaated with concentration polnrizttion in clectrodralysts, Pmt. Fit cr Intern. S>ntp. art tf brer Dt~suImarion. Wushingfm. D.C.. Ocfoher 3-9, fW(55, Wpkx SWDll8. l_&wawr_I. Kep;trf I%. 65-6, Unixrsity of California, 2. S. A. \VLIXER. Sm gj’urer Cmwrtim Bcrk&y. 3. N. W. ROSEW~~RGASU C. E. TIRRFLL, Limiting currents in mcmbranc cells. in;/. Enp_ Ch~nr., -19 (1957) 7SI. Boundary layer dlmcnsions in daalysis, 4. H. P. GREGOR. PWL F. BRUIW ZIXD M. Rormswm;. Id. &kg. Char. Process Des&n Drrelop.. 4 ( 1965) 3. Saline Water Comcrsion Research in Israel, %linc Water Conversion 11, 5. IL S. SptttitxR, rldwnce.r in 0wmisfr.v Series. So. 335. Am. Chcm. Sot., Washington. D.C.. 1963, p. 179. 6. E. A. MMOU fun T. A. Knw~tz\\~. C.E.P. Spnp. St’ries So. 24, 55 (1959) 175. with Dr. John Nordin from the Demonstration Unit at Webster, 7. Prixatc Communication South Dakot.~ 8, D. A. COWAY AW J. H. BRO\VN, In& Enx. Cherrr.. 5 1 (1959) 1445. q/* Desaiinu~ictm, K. S. SPIEGLER(Edrtor), 9. L. ti. SH-\FFER a%~ M. S. MISTZ, in: Prinripk Academtc Prrs?;, N-Y., l966. London, 1960, p_ 268. 10. J. R. Wtt.sos, D;*minera?ix?tLn hy Efcrtrodialysis, Buttetworths, I (1966) 178-193. 11. W. G. 3. MA-.DERSLOOT AND R. E. HICI~S. Dcdmufion, rcsolts. discussed in Ref. (II). 12. E. F. h-I. VA’: DER H~t.t>. unpublished SM., 1I I (1964) 838. 13. W. G. B. MASDE~L~O’~, J. Elecmd~em. 14. K. S. SP~E(~LER,Private communication. En&wood. N.J., 1962, Chnptcr 6. 15. V. LEvIat. P/~~siuc/ternicaf ffyrlrodynun~icx, Prentice-Hall, New York, 1966, pp. 53 f. 16. C. L. MavrEt.t_. f%TIroci~emica~ Engineering, McGraw-Hill, 17. F. BERCSXW,Dechemu .\fom~gr.. 17 ( 1962) 449. 1% G. S. Sot_r, Proc. FirAt Intern. Symp. on Watu Desalina!ion, Wadrin~tarr, D.C., October 3-9, 1965, Paper SWD/3. I?. K. S. SPIEGLER, in: Ekctrorftenrical Operarions in IWI Ekcitan~e Techmo!og_s’.F. C. NICHOU AXD F. SCHUBERT (Editors), Acixdcmic Press. NY., 1956, p. 16% 20. M. SELO. Decltcntu, rtfomxerv 47 (1962) 575. 21. E. J. PXXSI, Private communication. 22_ W. E. KATZ. Private communication* 13. T. A. KIRKHAM, Papw presented April 4, 1963, Division of Water and W-s!e Chemistry, A.C.C.. Los Angeles, California. I.
Desalination,
5 ( l96R) 267-29 I