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Concentration polarization in electrodialysis—III. Practical electrodialysis systems
Electrochimica
Acta,
1961. Vol.
5, pp. 216 to 228.
PergamonPressLtd. Printedin NorthernIreland
CONCENTRATION POLARIZATION IN ELECTRODIALYSIS-III. PRACTICAL ELECTRODIALYSIS SYSTEMS* B. A. COOKE? and S. J. VAN DER WALT National Chemical Research Laboratory, South African Council for Scientific and Industrial Research, Pretoria, Union of South Africa. Abstract-Concentration overpotential has been measured in systems having mass transfer by forced convection caused by flow through a compartment bounded by ion-exchange membranes. The dependence upon current density of the difference between the bulk solution concentration and the apparent interfacial concentration derived from the measured overpotential is found to be non-linear. This is ascribed to the changing nature of current distribution with increasing current, together with the presence of a diffusion layer of varying thickness. Interposition of a perforated corrugated spacer between the membranes, besides assisting depolarization, widens the range of layer thicknesses to an extent which increases with increasing flow velocity. Systems with mass transfer by stirring are briefly reported upon. R&urn&La surtension de concentration a tt6 mesurk dans des systtmes oti le transport de masse g convection for&e est dQ au passage au &avers d’un compartiment form& par des membranes &hangeurs d’ions. La corr&lation entre la densite du courant et la diffkrence entre la concentration de la solution principale et la concentration interfaciale apparente-indiqute par le surtension mesureeparait &tre non-lineaire. Ce pht?nom&ne est attribue aux variations de la distribution du courant qui accompagnent tout accroissement de ce courant, ainsi qu% la presence d’une couche de diffusion dont l’kpaisseur varie. L’interposition entre les membranes d’un distanceur pIis& et perfork-part de faciliter la ddpolarisation-fait accroEtre Y&endue des variations d’epaisseur de ces couches dans une mesure qui accroit avec la velocite du flux. Des systtmes B transport de masse par agitation ont dtC dCcrits d’une faGon succincte. Zusammenfassung-Die Konzentrationsiiberspannung wurde in Systemen mit Stofftransport gemessen, wobei eine Konvektion mittels Strtimung durch einen von Ionenaustauschermembranen abgeschlossenen Abteil erzwungen wurde. Die Differenz zwischen den Gesamtkonzentrationen, sowie die aus der gemessenen ijberspannung errechnete scheinbare Grenzfl$chenkonzentration, sind nicht linear von der Stromdichte abhlngig. Dies wird der Anderung der Stromverteilung, sowie der Die Einfiihrung einer gelochten und Gegenwart einer Diffusionsschicht Dicke zugeschrieben. gewellten Distanzscheibe bewirkt neben einer Begiinstigung der Depolarisation eine mit wachsender Flussgeschwindigkeit zunehmende Erweiterung des Variations-bereiches der Schichtdicken. Systeme mit stofftransport durch Riihren werden kurz behandelt. THE
need for effective depolarization in the operation of electrodialysis apparatus is one of the basic considerations in the design of, and choice of operating conditions for, such apparatus. Depolarization is invariably achieved by forced convection, the solution being passed through a narrow space bounded by the ion-exchange membranes. In many cases, eddy formation in the liquid stream is deliberately promoted * Manuscript received 16 January 1961 t Present address: Research and Development (Nobel Division), Stevenston, Ayrshire, Scotland. 216
Department,
Imperial Chemical Industries
Ltd.
Concentration
polarization
in electrodialysis-III
217
so as to improve depolarization, e.g. by the insertion of obstacles in the liquid path1 or by interposing between adjacent membranes a corrugated, perforated sheet material, or something similar, which aids in supporting the membranes while increasing the tendency towards eddy formation.2v3 No entirely satisfactory method of estimating concentration polarization in these systems seems to have been put forward. Apart from the suggestion1 of using pH change as a criterion for the limiting current in such systems, which can yield misleading conclusions,4 most estimates have consisted of limiting current determinations, e.g. from current/voltage relationships at constant flow, or by varying dialysate flow rate at constant current. Because the curves of voltage vs. current or flow rate do not display sharp inflexions under all conditions, a measure of ambiguity arises in assigning the limiting current. The difficulties are increased by the need to allow for the current distribution over the length of membrane through which electrodialysis takes place: in practical apparatus this path length is frequently large, in which case the diluting or concentrating effect resulting from the alternate arrangement of anionand cation-exchange membranes is considerable. In the present paper, the study of these systems by means of concentration overpotential measurements4 is considered. The approach is similar to that adopted towards systems with natural convection, 5 attention being paid to short-range current distribution in the boundary cases of very low and limiting current densities respectively. In the experimental work, short paths have been employed together with a membrane sequence which causes the least possible demineralization or concentration effect, so that the long-range current distribution remarked on in relation to practical apparatus does not arise, the intention being to isolate the polarization effect. Some results are also presented on stirred membrane systems. THEORETICAL
In this section,
the following
symbols
ANALYSIS
will be used:
Subscript Subscript b = b’ = c, = c’ =
obs denotes an experimentally observed value. app denotes a theoretically predicted value for an experimental constant (equation (1)). constant (equation (11)). concentration of bulk solution. interfacial concentration. D = diffusion coefficient of electrolyte.
e, = 2$(i
observation.
- t).
F = Faraday’s constant. i = current density (local value).
i
= mean current density.
i* = the value of i for which the extrapolated
vs ; curve cuts the line 0,,, = c,,. k = (f - i)b -Z--’ k, = (f - t)b --Z-’ p=-i
k Yli2 co
or
k’Y
-
CO
i.
line representing
the initial slope of the O,,,,
218
B. A. OXIKE and S. J. R = 7 = I = T = y = Y= 6 = 7 = 13 =
VAX DER WALT
gas constant. transport number of the counterion in the membrane. transport number of the same ion in free solution. absolute temperature. ordinate from leading edge of membrane. length of membrane in the direction of flow. thickness of Nernst diffusion layer. concentration overpotential. concentration difference between bulk and interface co - c’.
As is the case with natural convection, parallel laminar flow of a fluid across a plane surface sets up a boundary layer the thickness of which increases with distance from the leading edge of the plane. Levich6 has given a relation for the thickness of the Nernst diffusion layer in such a case, and it will suffice for the present purpose to reduce his equation to 6 = byi’2, (1) which holds for a given velocity of flow past the plane. solution system
ez(f--t)
From this, for the membrane/
.
6z = kQ1/2.
f;D
(2)
If, under all conditions, i varied with y so that 0 remained constant, the measured overpotential would provide a direct indication of 8 and a plot of 0 against i would be a straight line up to the limiting condition 0 = cO (the discussion is confined to the dialysate side). While the condition of a single-valued 8 clearly holds in the limiting current region, analogy with the case of natural convection517 suggests that the current distribution is much more uniform at low current densities, and, in what follows, it will be taken that 8 = kp1j2 for small i, (3) the validity of which may be expected to improve as i -+ 0. It will also be assumed that concentration overpotentials are measured with capillary tips placed at a distance from the polarized membrane, which is considered to be small in area. It is then not necessary to consider the exact situation of the tips in relation to the membrane as was required for the relatively large area of membrane exposed in the experiments on natural convection. Using (3) and ignoring differences in activity coefficients, the local concentration overpotential is given by
-q/e 0 = and the observed
value for a membrane VaPP -_=
_
co
1 y
E y1i2)
ln (1 -
co-
of length
_
s
Yin the direction
y o In (1 - zy1j2)
of flow is
dy,
from which %PP --_=
co
1 --
(P2
1 ln(1 1
-P)+I+I,
2
P
(4)
Concentration
polarization
where
in electrodialysis-III
219
k Y1f2i
p=-.
(5)
co On expansion
(4) becomes 2P P2 P3 = y- + 4 + 7 $- . . . . . .
%PP
-
e0
Since the apparent
interfacial
concentration c’aPP
c,exp
=
is
(
-F
)
,
(6)
%q, . the slope IS dP dCiPP
_
-C-$+$t
. . . .&exp(-$+
. . . .),
dP and its limiting
value asp --f 0 [in which case equation dc;pp _ o--3’ H dp
Then, since %!!Z = - 5 dP dP
(3) may be considered
exact] is
2~0
= ky1/2 , the initial slope of the eapp vs. i curve is
and g
CO
(7) By analogy with (1) and (2), this may be expressed in terms of an apparent layer thickness, a:,,, calculated from the initial slope, 6Zpp = $bY1’2;
@a)
or in terms of the current density value, i*, at which the extrapolated the initial slope cuts the line eapp = co. In this case, p=l
i -
at
and the mean limiting
current
=
cO
XY
-l/2
(8b) are carrying
1
s y
Yo
)
ili,dy =
2coY-1'2. k-
From (5), it is clear that i = ilimat p = 2, while the apparent for the limiting condition is K, = 4 b Y1’2. (8) and (9),
their limiting
is
z,im = -
From
line representing
2i* 3 *
At the other extreme, when all regions of the membrane current, the condition 8 = co everywhere yields from (2) zlim
Nernst
so apq = 413. %PP
Nernst
layer thickness (9) (10)
220
B. A. COOKEand S. J.
VAN DER WALT
Two features of the eapp vs. i curve predicted for the idealized system with laminar flow are now clear and are illustrated in Fig. l(a). The extrapolated line A, representing the initial slope according to (7), cuts the line ospl, = c0 at i = i* or p = 1.5, but the limiting condition occurs only atp = 2 (point L). Apparent Nernst layer thicknesses derived from these two boundary cases are therefore in the ratio 4/3 as required by (10). The nature of the dependence between the low and the limiting current regions must depend on the current distribution prevailing: if it were uniform right up top = 1 (the theoretical limit), the line B, calculated from (4) and (6), would result, with a continuation up to L such as the broken line C drawn. A more continuous change in current distribution might yield a smooth curve such as D. Proceeding in the same manner, the corresponding expressions for a hypothetical system in which 6 = b’_y (11) are found to be %pp -= e0
In this casep
= ‘zi;
1 --1
ln(l-p)$l.
!P
1
(12)
then
CO 6&,
p=l
=
at
$
b’Y,
i = i*/2,
&I,, = 0.
(134
(13b) (14)
These are mentioned not because of any supposed direct bearing on a real system, but as an indication of the direction in which the behaviour of a system might be expected to tend if a wider variation of diffusion layer thicknesses than is implied by (1) obtains over the membrane surface. Features of the predicted eapp vs. i curve for The produced line A, representing the initial this case are shown in Fig. l(b). of perfectly uniform current slope, cuts 13~~~= co at p = 2. Forp & 1, assumption C is subject to the requiredistribution gives the curve B. Forp > 1, the continuation ment of equation (14), namely that the system does not display a true limiting current. If current distribution is not perfectly uniform throughout the rangep G 1, a smooth curve such as D might be expected. In either event, whether a system approximates in behaviour to (1) or to (1 l), the condition p = 1 corresponds to the smallest current density at which any region of the membrane/solution interface can become polarized to the limiting extent. This will happen only if current distribution is uniform at the current corresponding to p = 1. In a study of polarized membrane systems having mass transfer by natural convection,5 the experimental evidence indicated substantial uniformity of the current distribution for most of the range p G I, and it proved possible, by observing pH change, to establish the existence of a “first limiting current”, iiim, corresponding providing a approximately to p = 1. In the present case, with forced convection wider variation per unit distance in the Nernst layer thickness than is the case with natural convection, highly uniform current distribution up to p = 1 is less probable because of the larger resulting potential gradients parallel to the membrane/solution interface generated by the concentration polarization. Qualitatively, therefore, the
Concentration
polarization
in electrodialysis-III
P
&,= c,
1.c )
I.0
c,-e
I.5 __---
4’ /’
/ /’
D
B
0” \
g 0.5
w
A
/
1
T
(4
P
P) FIG. 1. Features of the predicted curve of 0app/~, against i for systems obeying (a) equation (I), (b) equation (11). The straight lines A show the initial slopes, and curves B the predicted dependence if current distribution is perfectly uniform up top = l(i = &,), with their continuations C up to the point of limiting current density L [not possible if system obeys (1 l)]. The smooth curves D represent what might be expected if current distribution alters continuously from uniformity at low p to the ultimate limiting condition.
221
222
B. A. COOKEand S. J. VANDER WALT
actual value of $, in a system with forced corresponds to p = 1. EXPERIMENTAL
convection
is likely
to be higher
than
TECHNIQUE
All determinations were made at 25 + 0*2”C. Small apparatus for use with current interrupter. The same cell and electrical circuitry were used as in the experiments on natural convection.5 In experiments on the effect of stirring, anion-exchange membranes (A.M.F. from American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A., and Permaplex A20 from Permutit Co., Ltd., London) were used and both stirrers operated at the same speed; 8 was then obtained by assuming it to have numerically identical values on both sides. In experiments on flow past a membrane, the polarization on one side of a single membrane was isolated by arranging the membranes in the following sequence (for testing a cation-exchange membrane) :
Here a and k denote anode and cathode, +, - and 0 anion-exchange, cation-exchange and non-selective (bleached parchment paper) membranes respectively. The compartment G was 0.09 cm wide and the exposed area of membrane 3 (the membrane under test) was reduced to 0.64 cm2 by means of thin mica plates clamped to it; the small area had the double advantage of reducing the current requirement of the system and of minimizing bending of the membrane. Mica plates were used on both sides of 3 to prevent current refractions that could reduce the effective current density on the side facing G. In all tests the bulk solution used was 0.05 M NaCl at 25°C but compartment D contained O-2 M NaCl and the cathode side of 3 was depolarized by stirring this solution at 1,50Orev/min, allowance being made for the imperfect depolarization by means of data obtained in connexion with the natural convection work. The capillary tips connected to reference electrodes were fixed in compartments C and D on the axis of the exposed membrane area. It was assumed that negligible polarization arose at the bleached parchment membrane. When corrugated spacer was present in G, the mean pressure in this compartment was kept slightly below that in C and D, but pressures were accurately balanced when spacer was not used. In the latter case, two pieces of spacer material were placed in the liquid path-one reaching to 2 mm below the working membrane surface and the other commencing 2 mm after it. These aided in sealing compartment G against leakage into or from C or D. The spacer material had an amplitude of 0.09 cm and wave length of 0.19 cm, with 1,100 perforations of 0.24 cm diameter (0.045 cm2 area) each per 100 cm2 (Stanley Smith and Co., Isleworth, Middx.). It was arranged in the compartment so that the corrugations lay at 20 degrees to the flow path. Large apparatus. Although the results to be discussed were all obtained on the compartment, measurements have small apparatus with a “model” electrodialysis also been obtained on a larger apparatus, the current demand of which exceeded the The general findings on the larger range of a normal electronic current interrupter. apparatus were similar to those on the small one, but reproducibility was poorer. As,
Concentration
polarization
in electrodialysis-III
223
however, the method used is in principle applicable to any size of apparatus, it will be described briefly. In Fig. 2, T, and T, are capillary tips passing through the walls of the compartments between which it is desired to measure the overpotential. The use of Ag/AgCl wire electrodes in place of the tips (with calomel electrodes) was found to be unsatisfactory as large, irreproducible e.m.f.‘s were found on the depolarized system after a period of polarization. A very slow flow of saturated KC1 was maintained through the tips by applying a small pressure externally. The measuring
-
I
Rh
c
FIG. 2. Arrangement for overpotential measurements on electrodialysis apparatus of any size (manual current interruption). a, k are respectively the working anode and cathode of the apparatus; T,, Tz capillary tips connected to the reference electrodes; r the external balancing resistor; E external potentiometer; R,, Rz input resistors of the differential oscilloscope amplifier V,, Vz; S, a relay controlled either by S3 or by “flash” contacts on the camera viewing the oscilloscope screen. Point B was joined to the external triggering input connection of the oscilloscope. S, enabled the voltage between cathode and ground to be changed from zero to 1.W.
arrangement with balancing resistor r, potentiometer E, and differential oscilloscope amplifier V,V, was as described previously,4l5 x being adjusted until the vertical position of the trace (before polarizing current was passed) was not affected by operation of the switch S,. On interrupting the current to the electrodialysis apparatus by closing S,, the collapse of the voltage between B and ground caused the oscilloscope to make a single sweep commencing at the instant of interruption. On repeated manual interruption, the balanci point (trace passing through zero at the chosen time after interruption) could be obtained approximately by adjusting E. The difference between the voltage appearing across T, and T, and the value of E as obtained approximately by viewing the trace could then be obtained accurately by photographing it, the camera shutter being synchronized with the relay S,. A source of error in measurements on large apparatus is the internal, and possibly external, short-circuiting that occurs through interconnected dialysate, brine and electrode rinse streams. If the ohmic resistance between the compartments in which T, and T, are inserted is small compared with that in the short-circuit path, as is the case in a well-designed apparatus, the error is small.
224
B. A. COOKEand S. J. RESULTS
Stirred membrane
AND
VAN DER WALT
DISCUSSION
systems
Fig. 3 shows values of cobs found at various stirrer speeds (plotted as r213, r being in rev/mm) on two anion-exchange membranes. The stirrers were identical, of the “tulip” type, 2.6 cm in diameter, and were mounted 2.6 cm from the membrane face on either side. The dependence on the 213 powers of r is obeyed approximately, except that the extrapolated line for the A20 membrane does not appear to pass 0.02 /’ /
.‘*
i .s’ ; 0.0
I
r 1: 8”
/’ 0
5
FIG. 3. Effect of stirring with two different anion-exchangemembranes: A.M.F. (0)and Permaplex A20 (0). Y = stirrer speed in rev/min. 0.05M NaCl solutions used at current density 14.1 mA/cm”.
through the origin, suggesting the presence of a diffusion layer which is not entirely eradicable by stirring. This may be an inherent property of heterogeneous membranes, i.e. those consisting of particles of ion-exchange resin imbedded in an inert polymer matrix. The slopes of the lines do not differ very greatly, the majority of the difference observed resulting from the difference in intercept, while no significant difference was found when these two membranes were compared in natural convection systems for which, it may be supposed, mass transfer is governed entirely by processes in the liquid boundary layer. Flow through compartments
Fig. 4 shows the general form of dependence of Oobs upon i found in these forced convection systems. The initial rise is sensibly linear, but this region is followed by one of gradually decreasing slope, making it difficult to ascribe a definite limiting the relatively slow rise of total overpotential provides little help in this current; regard. It is clear that the short-range current distribution at all values of i could not be such that 8 is single-valued, otherwise an essentially linear rise would persist up to a well-defined limiting current. The curves found are similar in form to line D (Fig. l), indicating a gradual change in the nature of current distribution with increasing current. In order to make
Concentration
polarization
in electrodialysis-III
225
P 0
I
2
> E
la)
500 2
I
0.04
/
<
/
.-5 3 zf
0.03
2
-5 00
-w 82
2
0.02
o.ol
fYJi[ ..-•-
0
.-•
;.
;* 0
50 : ‘,
I00
mA/cm2
FIG. 4. Experimental values of &bs (C) and qt (0) as a function of current density obtained with A.M.F. cation-exchange membrane and 0.05M NaCl flowing through spacer material. Values ofp derived from the initial slopes are also shown as abscissae. (a) Apparent velocity = 0.93 cmisec Values of p based on equation Sb. (b) Apparent velocity = 12.3 cm/set Values of p based on equation 13b.
(b)
226
B. A. COOKE
and S.J.
VAN DERWALT
a more detailed comparison between predicted and observed 8 vs. i curves, an exact knowledge of the limiting current is required. This does not seem possible in practice because of the gradual approach to the limiting condition evident in the experimental curves. For purposes of discussion, it is desirable, therefore, to stipulate some value, 8’,,,,_ of 6,,bs which is less than c0 but close enough to it to be representative of the limiting current region. A value of SA,, can then be computed from
%,s s:,,= FD (t - t)i’ ’ 7
where i’ is the current density producing 19:~~. The values of a,&, listed in Table I have been obtained
in this manner
by choosing
f&,, = 0.980 c,,,
(15)
and must therefore be regarded as no more than representative measures of the limiting condition, the sole value of which is that the corresponding current can be measured with some accuracy. TABLE 1. VALUESOFNERNSTLAYERTHICKNESSESREPRESENTATIVEOFTHEINITIALSLOPE(~~~,)ANDOFTHE LIMiTINGCuRUENTREGION(~~bs)FORCATION-EXCHANGEMEMBRANESAND0~05MNACLSOLUTIONFLOWING AT VARIOUS VELOCITIES, u. MEMBRANE
I-A.M.F.
CATION.
MEMBRANE
2-PERMAPLEX
c20
(THE
PERMUTIT CO., LTD, LONDON)
Flow system
__----~ Without spacer
, /
1 With spacer
(cm)
-
:
1 1
1 1 1 1 1 2
) i
0.93 3.10 12.3 0.93 3.10 12.3 18.3 3.10
I
low o!Js
u (cm/set)
Membrane
7.10 4.94 3.25
I /
6.89 4+Kl 2.20 1.55 5.42
-
(cm)
i
643 4.31 2.94
I / )
G,
lOv&
~
5.08 2.81 1.21 0.77 3.24
&M
) ~ ~
1.10 1.14 1.11 1.36 1.42 1.82 2.02 1.67
The general shape of the 8 ObS vs. i curves found was the same whether or not Important differences between corrugated spacer was present in the compartment. the two cases are, however, evident from Table 1. Firstly, the depolarizing action of the spacer is clear, the effect being small at low velocity but of definite practical value at high velocity (both flow velocities and current densities used in computing the results obtained with spacer are apparent values, no allowance having been made in either case for its presence). Secondly, the ratio #,,/c?L, found is small and independent of velocity when spacer is absent. There are two reasons why the ratio found without spacer might be smaller than the value of 4/3 predicted for laminar flow in equation (IO) : (i) appreciably higher values-between 1.25 and 1.3-could be obtained by selecting t9&,,closer to c,, than in (15), but without proceeding beyond the apparent ultimate limiting current; (ii) the experimental arrangement could not be expected to meet the theoretical requirement upon which (10) is based, namely, that the uniform flow pattern and the electrodialysis should commence at exactly the same point.
Concentration
polarization
in electrodialysis-411
227
The most interesting point concerning the values of 6$,,/8~,, in Table I is that, while this ratio is independent of velocity when spacer is absent, it increases markedly with increasing velocity when spacer is used. It should not be inferred from the value of 1.36 found with spacer at the lowest velocity used that the system is close to the ideal, i.e. that it obeys (1) under those conditions, since, as Fig. 4(a) shows, the true limiting current was reached in this case at p > 2, sayp N 2.4, so that it must be concluded that the range of diffusion layer thicknesses effective over the membrane surface is greater than is implied by equation (1), with a slight tendency in the direction of the other case dealt with [equations (1 I) to (14)]. As the velocity is increased, even the superficial resemblance to equation (10) disappears, the curvature of the cobs vs. i curve becoming very pronounced [Fig. 4(b)]. This behaviour can be interpreted qualitatively by supposing that there exist relatively stagnant regions between the perforated corrugated spacer and the membrane which are not effectively swept even by the rapidly flowing liquid, but which nevertheless carry current and contribute to polarization. It may be supposed that such a flow pattern repeats itself regularly over the membrane surface in accordance with the geometry of the membrane/spacer boundary. As was found with the respective anion-exchange membranes in stirred systems, a Permaplex cation-exchange membrane gives rise to significantly greater polarization than the A.M.F. type in a system with flow. The S,“,,/S,&,, ratio was also greater, and may be an indication of the existence of regions on the membrane surface which conduct current but which are less accessible to the flowing liquid even than in the other case; the effect might have the same basic cause as the failure of the extrapolated line on Fig. 3 for the Permaplex membrane to pass through the origin. Criterion of depolarization
in electrodialysis
apparatus
A convenient method of characterizing the initial slope of a O,,bs vs. i curve is by means of the value of i*. While this is a simple experimental quantity, the relation between i and p can be deduced only by making an assumption concerning the nature of the variation of Nernst layer thickness over the membrane surface. If, as the experimental results suggest, the dependences (1) and (11) can be regarded as extreme possibilities for practical systems, it follows that the condition p = 1 occurs at a fraction of i* which is between 5 and +, i.e. a conservative estimate of lli,,, (based upon uniform current distribution at lower currents) would be _I
s>-),+,
llim 1*
the smaller value being more probable as behaviour tends towards equation (1 I), or, as the experimental results suggest, as the flow velocity is increased when spacer is used. In Fig. 4(a), the p scale shown has been deduced from equation (8b) for no better reason than the superficial resemblance in this case to the predictions for a system obeying (1). In the case of Fig. 4(b), equation (13b) has been used and, although the discussion which follows applies qualitatively to both cases, that of Fig. (4b) will be used as an illustration chiefly because the flow velocity employed (12.3 cm/set) is in the practical range.
228
B. A. COOKE and S. J. VAN DER WALT
For the case shown in Fig. 4(b), i* = 55 mA/cm2 so that illi,,, could be as low as 27.5 mA/cm2. The total overpotential at the latter current density is circa 30 mV, which is quite small compared with the voltage drop per compartment ( 2 1 V) in an electrodialysis apparatus run under typical conditions. The position can thus arise that there are regions of the membrane/solution interface conducting their limiting current in a system which is being operated at a current density remote from the limiting condition evident from the current/voltage dependence. The contribution of the associated overpotential to the power consumed in the process may be commercially insignificant, but a possible consequence of operating above $, is the localized occurrence of whatever consecutive process supplants normal conduction in the limiting current region. Of particular importance is the possibility of OHtransfer through anion-exchange membranes, which can cause precipitate formation if, for example, Mg2+ or Ca2+ and HCOa- are present. This argument provides an explanation of the frequently encountered experience in the electrodialysis of hard waters that scales containing Mg(OH), and CaCO, accumulate on the anode sides of the anion-exchange membrane even when the apparatus is being operated in an apparently depolarized condition. The localized occurrence of the limiting condition of concentration polarization is of less general importance in the case of cation-exchange membranes because of the small contribution of Hf transfer to conduction above the limiting current.*T5 Acknowledgements-The experimental work, and mission to publish.
writers thank Messrs. W. E. Brooks and J. G. Goodey
the South
African
Council
for Scientific
and
Industrial
for assistance with Research for per-
REFERENCES 1. N. W. ROSENBERGand C. E. TIRRELL, Industr. Engng. Chew. (Zndustr.) 49, 780 (1957). 2. Centraal Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek (Netherlands), Brit. par. 750 220, 736 888, 750 238 (1956). 3. J. R. WILSON, Trans. Znstr. Chem. Engnrs, Lond. 37, 198 (1959). 4. B. A. COOKE, Electrochim. Acta, 3, 307 (1961). 5. B. A. COOKE, Electrochim. Acta, 4, 179 (1961). 6. B. LEVICH, Disc. Faraday Sot. 1, 37 (1947). 7. K. ASADA, F. HINE, S. YOSHIZAWA and S. OKADA, J. Electrochem. Sot. 107,242 (1960). 8. F. L. TYE, Disc. Faraday Sot. 21, 200 (1956). 9. C. W. TOBIAS, M. EISENBERGand C. R. WILKE, J. Electrochem. Sot. 99, 359C (1952).
Acta,
1961. Vol.
5, pp. 216 to 228.
PergamonPressLtd. Printedin NorthernIreland
CONCENTRATION POLARIZATION IN ELECTRODIALYSIS-III. PRACTICAL ELECTRODIALYSIS SYSTEMS* B. A. COOKE? and S. J. VAN DER WALT National Chemical Research Laboratory, South African Council for Scientific and Industrial Research, Pretoria, Union of South Africa. Abstract-Concentration overpotential has been measured in systems having mass transfer by forced convection caused by flow through a compartment bounded by ion-exchange membranes. The dependence upon current density of the difference between the bulk solution concentration and the apparent interfacial concentration derived from the measured overpotential is found to be non-linear. This is ascribed to the changing nature of current distribution with increasing current, together with the presence of a diffusion layer of varying thickness. Interposition of a perforated corrugated spacer between the membranes, besides assisting depolarization, widens the range of layer thicknesses to an extent which increases with increasing flow velocity. Systems with mass transfer by stirring are briefly reported upon. R&urn&La surtension de concentration a tt6 mesurk dans des systtmes oti le transport de masse g convection for&e est dQ au passage au &avers d’un compartiment form& par des membranes &hangeurs d’ions. La corr&lation entre la densite du courant et la diffkrence entre la concentration de la solution principale et la concentration interfaciale apparente-indiqute par le surtension mesureeparait &tre non-lineaire. Ce pht?nom&ne est attribue aux variations de la distribution du courant qui accompagnent tout accroissement de ce courant, ainsi qu% la presence d’une couche de diffusion dont l’kpaisseur varie. L’interposition entre les membranes d’un distanceur pIis& et perfork-part de faciliter la ddpolarisation-fait accroEtre Y&endue des variations d’epaisseur de ces couches dans une mesure qui accroit avec la velocite du flux. Des systtmes B transport de masse par agitation ont dtC dCcrits d’une faGon succincte. Zusammenfassung-Die Konzentrationsiiberspannung wurde in Systemen mit Stofftransport gemessen, wobei eine Konvektion mittels Strtimung durch einen von Ionenaustauschermembranen abgeschlossenen Abteil erzwungen wurde. Die Differenz zwischen den Gesamtkonzentrationen, sowie die aus der gemessenen ijberspannung errechnete scheinbare Grenzfl$chenkonzentration, sind nicht linear von der Stromdichte abhlngig. Dies wird der Anderung der Stromverteilung, sowie der Die Einfiihrung einer gelochten und Gegenwart einer Diffusionsschicht Dicke zugeschrieben. gewellten Distanzscheibe bewirkt neben einer Begiinstigung der Depolarisation eine mit wachsender Flussgeschwindigkeit zunehmende Erweiterung des Variations-bereiches der Schichtdicken. Systeme mit stofftransport durch Riihren werden kurz behandelt. THE
need for effective depolarization in the operation of electrodialysis apparatus is one of the basic considerations in the design of, and choice of operating conditions for, such apparatus. Depolarization is invariably achieved by forced convection, the solution being passed through a narrow space bounded by the ion-exchange membranes. In many cases, eddy formation in the liquid stream is deliberately promoted * Manuscript received 16 January 1961 t Present address: Research and Development (Nobel Division), Stevenston, Ayrshire, Scotland. 216
Department,
Imperial Chemical Industries
Ltd.
Concentration
polarization
in electrodialysis-III
217
so as to improve depolarization, e.g. by the insertion of obstacles in the liquid path1 or by interposing between adjacent membranes a corrugated, perforated sheet material, or something similar, which aids in supporting the membranes while increasing the tendency towards eddy formation.2v3 No entirely satisfactory method of estimating concentration polarization in these systems seems to have been put forward. Apart from the suggestion1 of using pH change as a criterion for the limiting current in such systems, which can yield misleading conclusions,4 most estimates have consisted of limiting current determinations, e.g. from current/voltage relationships at constant flow, or by varying dialysate flow rate at constant current. Because the curves of voltage vs. current or flow rate do not display sharp inflexions under all conditions, a measure of ambiguity arises in assigning the limiting current. The difficulties are increased by the need to allow for the current distribution over the length of membrane through which electrodialysis takes place: in practical apparatus this path length is frequently large, in which case the diluting or concentrating effect resulting from the alternate arrangement of anionand cation-exchange membranes is considerable. In the present paper, the study of these systems by means of concentration overpotential measurements4 is considered. The approach is similar to that adopted towards systems with natural convection, 5 attention being paid to short-range current distribution in the boundary cases of very low and limiting current densities respectively. In the experimental work, short paths have been employed together with a membrane sequence which causes the least possible demineralization or concentration effect, so that the long-range current distribution remarked on in relation to practical apparatus does not arise, the intention being to isolate the polarization effect. Some results are also presented on stirred membrane systems. THEORETICAL
In this section,
the following
symbols
ANALYSIS
will be used:
Subscript Subscript b = b’ = c, = c’ =
obs denotes an experimentally observed value. app denotes a theoretically predicted value for an experimental constant (equation (1)). constant (equation (11)). concentration of bulk solution. interfacial concentration. D = diffusion coefficient of electrolyte.
e, = 2$(i
observation.
- t).
F = Faraday’s constant. i = current density (local value).
i
= mean current density.
i* = the value of i for which the extrapolated
vs ; curve cuts the line 0,,, = c,,. k = (f - i)b -Z--’ k, = (f - t)b --Z-’ p=-i
k Yli2 co
or
k’Y
-
CO
i.
line representing
the initial slope of the O,,,,
218
B. A. OXIKE and S. J. R = 7 = I = T = y = Y= 6 = 7 = 13 =
VAX DER WALT
gas constant. transport number of the counterion in the membrane. transport number of the same ion in free solution. absolute temperature. ordinate from leading edge of membrane. length of membrane in the direction of flow. thickness of Nernst diffusion layer. concentration overpotential. concentration difference between bulk and interface co - c’.
As is the case with natural convection, parallel laminar flow of a fluid across a plane surface sets up a boundary layer the thickness of which increases with distance from the leading edge of the plane. Levich6 has given a relation for the thickness of the Nernst diffusion layer in such a case, and it will suffice for the present purpose to reduce his equation to 6 = byi’2, (1) which holds for a given velocity of flow past the plane. solution system
ez(f--t)
From this, for the membrane/
.
6z = kQ1/2.
f;D
(2)
If, under all conditions, i varied with y so that 0 remained constant, the measured overpotential would provide a direct indication of 8 and a plot of 0 against i would be a straight line up to the limiting condition 0 = cO (the discussion is confined to the dialysate side). While the condition of a single-valued 8 clearly holds in the limiting current region, analogy with the case of natural convection517 suggests that the current distribution is much more uniform at low current densities, and, in what follows, it will be taken that 8 = kp1j2 for small i, (3) the validity of which may be expected to improve as i -+ 0. It will also be assumed that concentration overpotentials are measured with capillary tips placed at a distance from the polarized membrane, which is considered to be small in area. It is then not necessary to consider the exact situation of the tips in relation to the membrane as was required for the relatively large area of membrane exposed in the experiments on natural convection. Using (3) and ignoring differences in activity coefficients, the local concentration overpotential is given by
-q/e 0 = and the observed
value for a membrane VaPP -_=
_
co
1 y
E y1i2)
ln (1 -
co-
of length
_
s
Yin the direction
y o In (1 - zy1j2)
of flow is
dy,
from which %PP --_=
co
1 --
(P2
1 ln(1 1
-P)+I+I,
2
P
(4)
Concentration
polarization
where
in electrodialysis-III
219
k Y1f2i
p=-.
(5)
co On expansion
(4) becomes 2P P2 P3 = y- + 4 + 7 $- . . . . . .
%PP
-
e0
Since the apparent
interfacial
concentration c’aPP
c,exp
=
is
(
-F
)
,
(6)
%q, . the slope IS dP dCiPP
_
-C-$+$t
. . . .&exp(-$+
. . . .),
dP and its limiting
value asp --f 0 [in which case equation dc;pp _ o--3’ H dp
Then, since %!!Z = - 5 dP dP
(3) may be considered
exact] is
2~0
= ky1/2 , the initial slope of the eapp vs. i curve is
and g
CO
(7) By analogy with (1) and (2), this may be expressed in terms of an apparent layer thickness, a:,,, calculated from the initial slope, 6Zpp = $bY1’2;
@a)
or in terms of the current density value, i*, at which the extrapolated the initial slope cuts the line eapp = co. In this case, p=l
i -
at
and the mean limiting
current
=
cO
XY
-l/2
(8b) are carrying
1
s y
Yo
)
ili,dy =
2coY-1'2. k-
From (5), it is clear that i = ilimat p = 2, while the apparent for the limiting condition is K, = 4 b Y1’2. (8) and (9),
their limiting
is
z,im = -
From
line representing
2i* 3 *
At the other extreme, when all regions of the membrane current, the condition 8 = co everywhere yields from (2) zlim
Nernst
so apq = 413. %PP
Nernst
layer thickness (9) (10)
220
B. A. COOKEand S. J.
VAN DER WALT
Two features of the eapp vs. i curve predicted for the idealized system with laminar flow are now clear and are illustrated in Fig. l(a). The extrapolated line A, representing the initial slope according to (7), cuts the line ospl, = c0 at i = i* or p = 1.5, but the limiting condition occurs only atp = 2 (point L). Apparent Nernst layer thicknesses derived from these two boundary cases are therefore in the ratio 4/3 as required by (10). The nature of the dependence between the low and the limiting current regions must depend on the current distribution prevailing: if it were uniform right up top = 1 (the theoretical limit), the line B, calculated from (4) and (6), would result, with a continuation up to L such as the broken line C drawn. A more continuous change in current distribution might yield a smooth curve such as D. Proceeding in the same manner, the corresponding expressions for a hypothetical system in which 6 = b’_y (11) are found to be %pp -= e0
In this casep
= ‘zi;
1 --1
ln(l-p)$l.
!P
1
(12)
then
CO 6&,
p=l
=
at
$
b’Y,
i = i*/2,
&I,, = 0.
(134
(13b) (14)
These are mentioned not because of any supposed direct bearing on a real system, but as an indication of the direction in which the behaviour of a system might be expected to tend if a wider variation of diffusion layer thicknesses than is implied by (1) obtains over the membrane surface. Features of the predicted eapp vs. i curve for The produced line A, representing the initial this case are shown in Fig. l(b). of perfectly uniform current slope, cuts 13~~~= co at p = 2. Forp & 1, assumption C is subject to the requiredistribution gives the curve B. Forp > 1, the continuation ment of equation (14), namely that the system does not display a true limiting current. If current distribution is not perfectly uniform throughout the rangep G 1, a smooth curve such as D might be expected. In either event, whether a system approximates in behaviour to (1) or to (1 l), the condition p = 1 corresponds to the smallest current density at which any region of the membrane/solution interface can become polarized to the limiting extent. This will happen only if current distribution is uniform at the current corresponding to p = 1. In a study of polarized membrane systems having mass transfer by natural convection,5 the experimental evidence indicated substantial uniformity of the current distribution for most of the range p G I, and it proved possible, by observing pH change, to establish the existence of a “first limiting current”, iiim, corresponding providing a approximately to p = 1. In the present case, with forced convection wider variation per unit distance in the Nernst layer thickness than is the case with natural convection, highly uniform current distribution up to p = 1 is less probable because of the larger resulting potential gradients parallel to the membrane/solution interface generated by the concentration polarization. Qualitatively, therefore, the
Concentration
polarization
in electrodialysis-III
P
&,= c,
1.c )
I.0
c,-e
I.5 __---
4’ /’
/ /’
D
B
0” \
g 0.5
w
A
/
1
T
(4
P
P) FIG. 1. Features of the predicted curve of 0app/~, against i for systems obeying (a) equation (I), (b) equation (11). The straight lines A show the initial slopes, and curves B the predicted dependence if current distribution is perfectly uniform up top = l(i = &,), with their continuations C up to the point of limiting current density L [not possible if system obeys (1 l)]. The smooth curves D represent what might be expected if current distribution alters continuously from uniformity at low p to the ultimate limiting condition.
221
222
B. A. COOKEand S. J. VANDER WALT
actual value of $, in a system with forced corresponds to p = 1. EXPERIMENTAL
convection
is likely
to be higher
than
TECHNIQUE
All determinations were made at 25 + 0*2”C. Small apparatus for use with current interrupter. The same cell and electrical circuitry were used as in the experiments on natural convection.5 In experiments on the effect of stirring, anion-exchange membranes (A.M.F. from American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A., and Permaplex A20 from Permutit Co., Ltd., London) were used and both stirrers operated at the same speed; 8 was then obtained by assuming it to have numerically identical values on both sides. In experiments on flow past a membrane, the polarization on one side of a single membrane was isolated by arranging the membranes in the following sequence (for testing a cation-exchange membrane) :
Here a and k denote anode and cathode, +, - and 0 anion-exchange, cation-exchange and non-selective (bleached parchment paper) membranes respectively. The compartment G was 0.09 cm wide and the exposed area of membrane 3 (the membrane under test) was reduced to 0.64 cm2 by means of thin mica plates clamped to it; the small area had the double advantage of reducing the current requirement of the system and of minimizing bending of the membrane. Mica plates were used on both sides of 3 to prevent current refractions that could reduce the effective current density on the side facing G. In all tests the bulk solution used was 0.05 M NaCl at 25°C but compartment D contained O-2 M NaCl and the cathode side of 3 was depolarized by stirring this solution at 1,50Orev/min, allowance being made for the imperfect depolarization by means of data obtained in connexion with the natural convection work. The capillary tips connected to reference electrodes were fixed in compartments C and D on the axis of the exposed membrane area. It was assumed that negligible polarization arose at the bleached parchment membrane. When corrugated spacer was present in G, the mean pressure in this compartment was kept slightly below that in C and D, but pressures were accurately balanced when spacer was not used. In the latter case, two pieces of spacer material were placed in the liquid path-one reaching to 2 mm below the working membrane surface and the other commencing 2 mm after it. These aided in sealing compartment G against leakage into or from C or D. The spacer material had an amplitude of 0.09 cm and wave length of 0.19 cm, with 1,100 perforations of 0.24 cm diameter (0.045 cm2 area) each per 100 cm2 (Stanley Smith and Co., Isleworth, Middx.). It was arranged in the compartment so that the corrugations lay at 20 degrees to the flow path. Large apparatus. Although the results to be discussed were all obtained on the compartment, measurements have small apparatus with a “model” electrodialysis also been obtained on a larger apparatus, the current demand of which exceeded the The general findings on the larger range of a normal electronic current interrupter. apparatus were similar to those on the small one, but reproducibility was poorer. As,
Concentration
polarization
in electrodialysis-III
223
however, the method used is in principle applicable to any size of apparatus, it will be described briefly. In Fig. 2, T, and T, are capillary tips passing through the walls of the compartments between which it is desired to measure the overpotential. The use of Ag/AgCl wire electrodes in place of the tips (with calomel electrodes) was found to be unsatisfactory as large, irreproducible e.m.f.‘s were found on the depolarized system after a period of polarization. A very slow flow of saturated KC1 was maintained through the tips by applying a small pressure externally. The measuring
-
I
Rh
c
FIG. 2. Arrangement for overpotential measurements on electrodialysis apparatus of any size (manual current interruption). a, k are respectively the working anode and cathode of the apparatus; T,, Tz capillary tips connected to the reference electrodes; r the external balancing resistor; E external potentiometer; R,, Rz input resistors of the differential oscilloscope amplifier V,, Vz; S, a relay controlled either by S3 or by “flash” contacts on the camera viewing the oscilloscope screen. Point B was joined to the external triggering input connection of the oscilloscope. S, enabled the voltage between cathode and ground to be changed from zero to 1.W.
arrangement with balancing resistor r, potentiometer E, and differential oscilloscope amplifier V,V, was as described previously,4l5 x being adjusted until the vertical position of the trace (before polarizing current was passed) was not affected by operation of the switch S,. On interrupting the current to the electrodialysis apparatus by closing S,, the collapse of the voltage between B and ground caused the oscilloscope to make a single sweep commencing at the instant of interruption. On repeated manual interruption, the balanci point (trace passing through zero at the chosen time after interruption) could be obtained approximately by adjusting E. The difference between the voltage appearing across T, and T, and the value of E as obtained approximately by viewing the trace could then be obtained accurately by photographing it, the camera shutter being synchronized with the relay S,. A source of error in measurements on large apparatus is the internal, and possibly external, short-circuiting that occurs through interconnected dialysate, brine and electrode rinse streams. If the ohmic resistance between the compartments in which T, and T, are inserted is small compared with that in the short-circuit path, as is the case in a well-designed apparatus, the error is small.
224
B. A. COOKEand S. J. RESULTS
Stirred membrane
AND
VAN DER WALT
DISCUSSION
systems
Fig. 3 shows values of cobs found at various stirrer speeds (plotted as r213, r being in rev/mm) on two anion-exchange membranes. The stirrers were identical, of the “tulip” type, 2.6 cm in diameter, and were mounted 2.6 cm from the membrane face on either side. The dependence on the 213 powers of r is obeyed approximately, except that the extrapolated line for the A20 membrane does not appear to pass 0.02 /’ /
.‘*
i .s’ ; 0.0
I
r 1: 8”
/’ 0
5
FIG. 3. Effect of stirring with two different anion-exchangemembranes: A.M.F. (0)and Permaplex A20 (0). Y = stirrer speed in rev/min. 0.05M NaCl solutions used at current density 14.1 mA/cm”.
through the origin, suggesting the presence of a diffusion layer which is not entirely eradicable by stirring. This may be an inherent property of heterogeneous membranes, i.e. those consisting of particles of ion-exchange resin imbedded in an inert polymer matrix. The slopes of the lines do not differ very greatly, the majority of the difference observed resulting from the difference in intercept, while no significant difference was found when these two membranes were compared in natural convection systems for which, it may be supposed, mass transfer is governed entirely by processes in the liquid boundary layer. Flow through compartments
Fig. 4 shows the general form of dependence of Oobs upon i found in these forced convection systems. The initial rise is sensibly linear, but this region is followed by one of gradually decreasing slope, making it difficult to ascribe a definite limiting the relatively slow rise of total overpotential provides little help in this current; regard. It is clear that the short-range current distribution at all values of i could not be such that 8 is single-valued, otherwise an essentially linear rise would persist up to a well-defined limiting current. The curves found are similar in form to line D (Fig. l), indicating a gradual change in the nature of current distribution with increasing current. In order to make
Concentration
polarization
in electrodialysis-III
225
P 0
I
2
> E
la)
500 2
I
0.04
/
<
/
.-5 3 zf
0.03
2
-5 00
-w 82
2
0.02
o.ol
fYJi[ ..-•-
0
.-•
;.
;* 0
50 : ‘,
I00
mA/cm2
FIG. 4. Experimental values of &bs (C) and qt (0) as a function of current density obtained with A.M.F. cation-exchange membrane and 0.05M NaCl flowing through spacer material. Values ofp derived from the initial slopes are also shown as abscissae. (a) Apparent velocity = 0.93 cmisec Values of p based on equation Sb. (b) Apparent velocity = 12.3 cm/set Values of p based on equation 13b.
(b)
226
B. A. COOKE
and S.J.
VAN DERWALT
a more detailed comparison between predicted and observed 8 vs. i curves, an exact knowledge of the limiting current is required. This does not seem possible in practice because of the gradual approach to the limiting condition evident in the experimental curves. For purposes of discussion, it is desirable, therefore, to stipulate some value, 8’,,,,_ of 6,,bs which is less than c0 but close enough to it to be representative of the limiting current region. A value of SA,, can then be computed from
%,s s:,,= FD (t - t)i’ ’ 7
where i’ is the current density producing 19:~~. The values of a,&, listed in Table I have been obtained
in this manner
by choosing
f&,, = 0.980 c,,,
(15)
and must therefore be regarded as no more than representative measures of the limiting condition, the sole value of which is that the corresponding current can be measured with some accuracy. TABLE 1. VALUESOFNERNSTLAYERTHICKNESSESREPRESENTATIVEOFTHEINITIALSLOPE(~~~,)ANDOFTHE LIMiTINGCuRUENTREGION(~~bs)FORCATION-EXCHANGEMEMBRANESAND0~05MNACLSOLUTIONFLOWING AT VARIOUS VELOCITIES, u. MEMBRANE
I-A.M.F.
CATION.
MEMBRANE
2-PERMAPLEX
c20
(THE
PERMUTIT CO., LTD, LONDON)
Flow system
__----~ Without spacer
, /
1 With spacer
(cm)
-
:
1 1
1 1 1 1 1 2
) i
0.93 3.10 12.3 0.93 3.10 12.3 18.3 3.10
I
low o!Js
u (cm/set)
Membrane
7.10 4.94 3.25
I /
6.89 4+Kl 2.20 1.55 5.42
-
(cm)
i
643 4.31 2.94
I / )
G,
lOv&
~
5.08 2.81 1.21 0.77 3.24
&M
) ~ ~
1.10 1.14 1.11 1.36 1.42 1.82 2.02 1.67
The general shape of the 8 ObS vs. i curves found was the same whether or not Important differences between corrugated spacer was present in the compartment. the two cases are, however, evident from Table 1. Firstly, the depolarizing action of the spacer is clear, the effect being small at low velocity but of definite practical value at high velocity (both flow velocities and current densities used in computing the results obtained with spacer are apparent values, no allowance having been made in either case for its presence). Secondly, the ratio #,,/c?L, found is small and independent of velocity when spacer is absent. There are two reasons why the ratio found without spacer might be smaller than the value of 4/3 predicted for laminar flow in equation (IO) : (i) appreciably higher values-between 1.25 and 1.3-could be obtained by selecting t9&,,closer to c,, than in (15), but without proceeding beyond the apparent ultimate limiting current; (ii) the experimental arrangement could not be expected to meet the theoretical requirement upon which (10) is based, namely, that the uniform flow pattern and the electrodialysis should commence at exactly the same point.
Concentration
polarization
in electrodialysis-411
227
The most interesting point concerning the values of 6$,,/8~,, in Table I is that, while this ratio is independent of velocity when spacer is absent, it increases markedly with increasing velocity when spacer is used. It should not be inferred from the value of 1.36 found with spacer at the lowest velocity used that the system is close to the ideal, i.e. that it obeys (1) under those conditions, since, as Fig. 4(a) shows, the true limiting current was reached in this case at p > 2, sayp N 2.4, so that it must be concluded that the range of diffusion layer thicknesses effective over the membrane surface is greater than is implied by equation (1), with a slight tendency in the direction of the other case dealt with [equations (1 I) to (14)]. As the velocity is increased, even the superficial resemblance to equation (10) disappears, the curvature of the cobs vs. i curve becoming very pronounced [Fig. 4(b)]. This behaviour can be interpreted qualitatively by supposing that there exist relatively stagnant regions between the perforated corrugated spacer and the membrane which are not effectively swept even by the rapidly flowing liquid, but which nevertheless carry current and contribute to polarization. It may be supposed that such a flow pattern repeats itself regularly over the membrane surface in accordance with the geometry of the membrane/spacer boundary. As was found with the respective anion-exchange membranes in stirred systems, a Permaplex cation-exchange membrane gives rise to significantly greater polarization than the A.M.F. type in a system with flow. The S,“,,/S,&,, ratio was also greater, and may be an indication of the existence of regions on the membrane surface which conduct current but which are less accessible to the flowing liquid even than in the other case; the effect might have the same basic cause as the failure of the extrapolated line on Fig. 3 for the Permaplex membrane to pass through the origin. Criterion of depolarization
in electrodialysis
apparatus
A convenient method of characterizing the initial slope of a O,,bs vs. i curve is by means of the value of i*. While this is a simple experimental quantity, the relation between i and p can be deduced only by making an assumption concerning the nature of the variation of Nernst layer thickness over the membrane surface. If, as the experimental results suggest, the dependences (1) and (11) can be regarded as extreme possibilities for practical systems, it follows that the condition p = 1 occurs at a fraction of i* which is between 5 and +, i.e. a conservative estimate of lli,,, (based upon uniform current distribution at lower currents) would be _I
s>-),+,
llim 1*
the smaller value being more probable as behaviour tends towards equation (1 I), or, as the experimental results suggest, as the flow velocity is increased when spacer is used. In Fig. 4(a), the p scale shown has been deduced from equation (8b) for no better reason than the superficial resemblance in this case to the predictions for a system obeying (1). In the case of Fig. 4(b), equation (13b) has been used and, although the discussion which follows applies qualitatively to both cases, that of Fig. (4b) will be used as an illustration chiefly because the flow velocity employed (12.3 cm/set) is in the practical range.
228
B. A. COOKE and S. J. VAN DER WALT
For the case shown in Fig. 4(b), i* = 55 mA/cm2 so that illi,,, could be as low as 27.5 mA/cm2. The total overpotential at the latter current density is circa 30 mV, which is quite small compared with the voltage drop per compartment ( 2 1 V) in an electrodialysis apparatus run under typical conditions. The position can thus arise that there are regions of the membrane/solution interface conducting their limiting current in a system which is being operated at a current density remote from the limiting condition evident from the current/voltage dependence. The contribution of the associated overpotential to the power consumed in the process may be commercially insignificant, but a possible consequence of operating above $, is the localized occurrence of whatever consecutive process supplants normal conduction in the limiting current region. Of particular importance is the possibility of OHtransfer through anion-exchange membranes, which can cause precipitate formation if, for example, Mg2+ or Ca2+ and HCOa- are present. This argument provides an explanation of the frequently encountered experience in the electrodialysis of hard waters that scales containing Mg(OH), and CaCO, accumulate on the anode sides of the anion-exchange membrane even when the apparatus is being operated in an apparently depolarized condition. The localized occurrence of the limiting condition of concentration polarization is of less general importance in the case of cation-exchange membranes because of the small contribution of Hf transfer to conduction above the limiting current.*T5 Acknowledgements-The experimental work, and mission to publish.
writers thank Messrs. W. E. Brooks and J. G. Goodey
the South
African
Council
for Scientific
and
Industrial
for assistance with Research for per-
REFERENCES 1. N. W. ROSENBERGand C. E. TIRRELL, Industr. Engng. Chew. (Zndustr.) 49, 780 (1957). 2. Centraal Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek (Netherlands), Brit. par. 750 220, 736 888, 750 238 (1956). 3. J. R. WILSON, Trans. Znstr. Chem. Engnrs, Lond. 37, 198 (1959). 4. B. A. COOKE, Electrochim. Acta, 3, 307 (1961). 5. B. A. COOKE, Electrochim. Acta, 4, 179 (1961). 6. B. LEVICH, Disc. Faraday Sot. 1, 37 (1947). 7. K. ASADA, F. HINE, S. YOSHIZAWA and S. OKADA, J. Electrochem. Sot. 107,242 (1960). 8. F. L. TYE, Disc. Faraday Sot. 21, 200 (1956). 9. C. W. TOBIAS, M. EISENBERGand C. R. WILKE, J. Electrochem. Sot. 99, 359C (1952).