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Concentration polarization in electrodialysis—II. Systems with natural convection
Electrocl~imica Acta, 1961. Vol. 4, pp. 179 to 193. Perganmn Press Ltd. Printed in Northern Ireland
CONCENTRATION POLARIZATION IN ELECTRODIALYSIS-II. SYSTEMS WITH NATURAL CONVECTION* B. A. CHOKED National Chemical Research Laboratory, South African Council for Scientific and Industrial Research, Pretoria, Union of South Africa Abstract-The significance of apparent interfacial concentration values derived from overpotential measurements on a membrane system having mass-transfer by natural convection is considered in relation to current distribution over the height of the vertical membrane. Because current distribution is essentially uniform at low currents the system exhibits a first limiting current which is appreciably lower than the average for the system as a whole. With due regard paid to this point the experimental results are shown to be in agreement with previous theoretical and experimental results on natural convection. Events above the limiting current are discussed qualitatively, the suggestion being advanced that the lack of H+ conduction through cation-exchange membranes under these conditions is a consequence of cation/water interaction. R6sum&-La signification des valeurs apparentes des concentrations interfaciales, d&iv&es de mesures de surtensions dans un systkme A membrane oh le transport de masse s’effectue par convection naturelle, est CtudiQ en connexion avec la distribution du courant le long de la hauteur d’une membrane verticale. La distribution du courant &ant essentiellement uniforme aux courants faibles, le systbme pos&de un premier courant limite qui est considbablement infkrieur il la moyenne du systeme int@ral. En tenant compte de ce fait, il est montre que les Aultats obtenus sont en accord avec des Aultats anterieurs theoriques et exp&imentaux, se rapportant A la convection naturelle. Les phknom&es observks au dessus du courant limite sont discutes qualitativement; il est sugg6r.5 que, dans ces conditions, I’absence de transport d’ions H+ B travers des membranes 6changeurs decations est causQ par l’interaction cation/eau. Zusammenfassung-Die Bedeutung der Werte der scheinbaren GrenzlXchenkonzentration, welche durch Messung der uberspannung an einem Membransystem mit Stofftransport durch natiirliche Konvektion bestimmt wurden, wird im Zusammenhang mit der Stromverteilung iiber die H6he der vertikalen Membran untersucht. Weil die Stromverteilung bei niedrigen Striimen im Wesentlichen gleichm&sig ist, zeigt das System einen ersten Grenzstrom, der bedeutend niedriger liegt als der iiber das Gesamtsystem gemittelte Wert. Bei Beriicksichtigung dieser Tatsache kann gezeigt werden, dass die experimentellen Ergebnisse mit denjenigen aus friiheren theoretischen und experimentellen Untersuchungen in Einklang sind. Die Vorglnge oberhalb des Grenzstromes werden qualitativ diskutiert; das Fehlen der H+-Leitung durch Kationenaustauschermembranen wird unter diesen Bedingungen als Folge einer Kation-Wasser-Wechselwirkung gedeutet.
IN A previous paper,l an account was given of the applicability of overpotential measurements to the study of concentration polarization in electrodialysis, and the view was advanced that it provides a useful alternative to the measurement of current vs. voltage curves which has been the chief method of studying this phenomenon. The intention is now to consider the significance of interfacial concentration values derived from overpotentials measured on a membrane system of which the mass-transfer characteristics are known with some certainty. A substantial body of evidence, * Manuscript received 23 August 1960. t Present address: Research and Development Department, Imperial Chemical Industries Ltd. (Nobel Division), Stevenston, Ayrshire, Scotland. 179
B. A. COOKE
180
recently reviewed by Ib12, indicates a high order of agreement between theory and experiment for systems in which heat or mass transfer takes place to or from a vertical plane surface by natural convection. Rather than seeking to confirm this agreement for yet another system the approach adopted in the present paper is to make use of the previous findings, notably those of Wilke et al3 in interpreting results obtained on the membrane system. SIGNIFICANCE OF OVERPOTENTIAL VALUES OBTAINED ON A MEMBRANE SYSTEM WITH NATURAL CONVECTION*
Over a wide range of experimental conditions heat and mass-transfer by natural convection are described by the relation4 Nu = a(Sc.Gr)1/4.
(1)
For the present case the non-migrational electrolyte flux is, ignoring electra-osmosis and electrolyte diffusion through the membrane,6
from which it follows that 6=
ka44/6y1/5i4/5.
Two extreme cases of current distribution may now be considered. (a) 8 independent of y If this is the case, i must vary withy, and the mean current density over a membrane of height Y is obtained by integration from (3): i= $
iyidy
=
a
(!f)6’4y--li4,
(4)
where d = 4a/3. This yields for 0 0 =
k@fb
yWi4/5,
(5)
while the mean flux is generally expressed in the form %
= rt(Sc.?%$/4.
It is this relation, with d - 0.68 indicating good agreement with the theoretical result of Wilke et aL3 of a = O-508, that has been highly successful in correlating a variety of experimental findings on liquid systems with natural convection.2 This suggests that these systems fulfil the requirement of 8 being independent of y, a fact which is not surprising in the case of many electrode reactions, the experimental study of which has consisted of limiting current determinations. Whatever the current distribution may be at low currents it is clear in such cases that 8 must have the constant value of c0 in the limiting situation provided the consecutive process available to the system above the limiting current necessitates a marked increase in electrode potential. In the case of membrane systems there is no great certainty as regards the nature of the consecutive process (cf. the various views of Frilettes, Rosenberg and Tirrell’, and * A list of symbols is given at the end.
181
Concentration polarization in electrodialysis-II
Peer@), nor is there in general any reason to suppose that 8 should be independent ofy at currents well below the limiting value.g Clearly, if the membrane system were to obey the requirement of a single-valued 13 at each value of i, the concentration overpotential found would furnish an unambiguous estimate of 8 and would be unaffected by the position of the capillary tips (attached to reference electrodes) in relation to the membrane in the experimental arrangement for determining 9. (b) i independent ofy. In Fig. l(a), let the capillary tips T,, T,, (forming salt bridges with, e.g. saturated ‘I
5
(0) (b) FIG. 1. (a) Polarized membrane, M, of height Y with capillary tips T1, T,. (b) Representative network for polarized membrane shown in (a).
calomel reference electrodes) be placed as shown in relation to the polarized membrane 44, and let the membrane separate bulk solutions of the same concentration, that on side (1) being unstirred while (2) is efficiently stirred. Let the polarizing current cause
interfacial concentrations on the unstirred side (1) to be lower than the bulk solution value. If, as a simplification, all relevant activity coefficients are considered identical, the concentration overpotential at a height y is given by
where
e, = F(i
- t).
The voltage, rap*, registered between T, and T2 immediately after interrupting the polarizing current, will be a mean value analogous to eappfound with the representative network of Fig. l(b). The resistance associated with the overpotential 11at y is taken to be proportional to the length of path connecting Tl and T, through y, i.e. 2x cosec v; in so doing additional resistance contributed by the dilute solution in the layer is ignored and the membrane is assumed to be thin and narrow. Since, for the representative network, e
Ze/r)
app = yjqF) ’
B.A. COOKE
182
the overpotential registered is given by
2
(cos fpl.-
cos po) = [sinpl.ln[l
-$(/r+xcot~)1~5]+.
(7)
If the tips are remote from the membrane, the result is 1
rltpp _ _--
y
Y s0
e0
=
In 1 - s5 (
yl15) dy
1
( 11n(l-p)+f+~+$i2~+~(X~Y), ---1
(8)
P5
ki4/5y1/5
where
P=
coa415
It has already been indicated1 that experimentally observed overpotentials can be converted to interfacial concentration values without recourse to a simplified expression of the form , CaPP
= co exp
VaPP
(
e0
1
9
so that the most satisfactory way of comparing the theory and experiment would be by computing predicted ctappvalues with the aid of (9) from qapp/eoas yielded by (7) or (8). Experimental c’,,~*values (or Oobs= co - clobs) may then be compared with predicted ones rather than direct comparison of overpotential values. Quite apart from the inherent assumption of i being independent ofy, equations (7) and (8) must break down for p > 1. At p = 1 the uniformly distributed current just causes the attainment of the limiting condition at the top edge of the membrane, and, without specific information on overpotential in the limiting current region it is not possible to take the matter further. EXPERIMENTAL
The essential features of the cell used are shown in Fig. 2. Five annular Perspex sections of inner diameter 3.2 cm were clamped together to form compartments A to E,
k
FIG. 2. Electrodialysis
tration polarization
cell (diagrammatic sectional view) for determination of concenwith natural convection at membrane 3. a = anode, k = cathode; for other features see text.
Concentration
polarization
in electrodialysis-II
183
the end compartments holding the anode and cathode, respectively. A sub-assembly, H, holding the membrane under test, 3, was mounted between the compartments C and D : it consisted of a pair of Perspex plates having rectangular openings 2 cm high and 1 cm wide at their centres to limit the exposed area of membrane 3 to 2 cm2. Glass tubes, T, and T,, terminating in capillary tips, passed through compartments C and D so that the tips reached (in most experiments) to within 2 mm of the membrane face on either side, with both tips on the perpendicular axis of the exposed portion of membrane. T, and T, were filled with saturated KC1 and led to saturated calomel electrodes mounted externally. The tips were sealed with asbestos fibre so as just to furnish TI
Tz
--& I2
3
4
I2
3
4
+- -++II B
// 12
Cl b 33' t
Fix-
E
4
5
G
(a) (b) (cl FIG. 3. Sequence of membranes used for studying overpotential at (a) a cation-exchange membrane, (b) an anion-exchange membrane, (c) arrangement for studying pH changes or current efficiency at polarized cation (3) or anion (3’) exchange membrane. + denotes anion-exchangers, - cation-exchangers, a denotes anodes, k cathodes.
electrolytic contact when a small hydrostatic head was applied on the external KC1 solution; excessively permeable tips gave false results for the polarization. A small platinum wire P passed through one plate to provide the earthing point1 for the system. Identical “tulip” stirrers of 2.6 cm diameter, Si, S,, were mounted in C and D equidistant from the membrane under test. The sequence of membranes used in the arrangement of Fig. 2 for the testing of a cation-exchange membrane is shown in Fig. 3(a); it is such that the bulk concentration in C and D could not change significantly with passage of current during a period of several minutes. The solution in B was several times more concentrated than that in C or D, while the electrode compartments were filled with phosphate buffer solutions to permit extended operation without replenishment. The analogous arrangement for testing an anion-exchange membrane is shown in Fig. 3(b), the membrane under test being again numbered 3. The essentials of the electrical circuitry have been described.l The interrupted current source was of conventional design and was used with an interruption time of 350 ps at a frequency of 35 s-l (current broken during 1.2 per cent of the total time). Its output was not grounded. The oscilloscope used was a Tektronix model 502 (Tektronix Inc., Portland, Oregon, U.S.A.), use being made of the differential amplification facility of this instrument; the differential balance of the amplifiers, which must be highly accurate for this type of measurement, was adjusted with the aid of the square wave generator which is one of the auxiliary circuits of this oscilloscope. A fraction of the grid signal controlling the series valve in the interrupter was introduced into the time-base circuit of the oscilloscope to ensure accurate triggering of that instrument at all times, including those occasions when the signal was disconnected to examine the zero-signal position of the trace in the event of drift at high sensitivities. The results
184
B. A. COOKE
reported have all been derived from null-point potentiometric measurements on the trace 40 ps after interruption by which time the initial very rapid decay attributable to capacitive effects had completely disappeared. In order to study-polarization with natural convection on a single side of a membrane, e.g. the side on which the interfacial concentration is less than the bulk solution value (the “dialysate” side), it is desirable to depolarize the opposite (“brine”) side efficiently. It is essential then to verify that depolarizing the brine side does not influence polarization on the other side, e.g. by mechanically transmitted vibration from a stirrer. This point was tested in the following manner. For a given solution, membrane and current density below the limiting value, four overpotentials were measured: with neither side stirred (Q), the brine side only stirred (qZ), the dialysate side only (q3), and both sides (q&. Ideal depolarization would give q4 = 0. In cases in which this has been approached it has invariably been found that Q > Q + 1;/a indicating that Q and ~a are less than their true values because of spurious agitation on the dialysate and brine sides, respectively, caused by the stirring on the other side. It has been found necessary to accept 1;/4> 0 and seek the most efficient stirring conditions for which Q = 1;72+ Q - q4, indicating an absence of the unwanted agitation. This criterion was found to be satisfied when “tulip” stirrers were used at ~2000 r.p.m. provided the cell mounting did not permit direct transmission of vibration from the stirrer motors to the cell. With the cell independently suspended the stirrer speed used throughout work on systems with natural convection was 1500 r.p.m. (adjusted stroboscopically). Under these conditions small but significant values of q4 were found which, at a fixed current density and bulk solution concentration, varied markedly with the type of membrane used. To study polarization on a single unstirred side, qa and either Q or Q were measured, depending on whether dialysate or brine side polarization was desired. By assuming the concentration differences between bulk and interface to be numerically equal when both sides were stirred, the interfacial concentration on the single stirred side was estimated from q4, so that the apparent interfacial concentration on the unstirred side giving rise to Q or q3 was obtained. Clearly, the error possible in this procedure increases with the magnitude of q4 in relation to Q or Q; for this reason, the membranes used in most tests were of a type displaying small values of q4. These were A.M.F. membranes (American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A.). For the membranes in positions 1, 2 and 4 (Fig. 3), Permaplex A20 and C20 (The Permutit Co. Ltd., London) were used. A modification of the basic cell arrangements of Fig. 3(a) or (b) that was used to estimate the current efficiency displayed by a polarized membrane, as well as OH- or H+ transport at high currents, is shown in Fig. 3(c). The two membranes, 3 and 3’, separated by a thin (0.9 mm) gasket were clamped between the plates forming the sub-assembly H; a stream of solution was passed by means of suitable channels in the plates of the sub-assembly through the space formed between the membranes. Either membrane could be depolarized on its dialysate side by means of the stirrers in compartments C and D. Pressures on either side of membranes 3 and 3’ were balanced manometrically. All work was carried out in a constant temperature room at 25 -f 0*2”C. Analytical grade reagents were used, either in distilled water, or when H+ or OH- transport was to be detected, “conductivity” water prepared by mixed bed demineralization.
Concentration
polarization
RESULTS
Signl@ance
AND
185
in electrodialysis-II DISCUSSION
of overpotential measured
4 shows a variety of results in the form of a plot of log 6 against log (SC.~), the values of cobs used in computing these groups having been obtained by direct application of the empirical method described previouslyi for obtaining interfacial concentrations from observed overpotentials, while allowance was made in the manner Fig.
.
2.00.o
. 1.900 . /
c+ +
I.70 . I.65 /', 72
) 75
8.0
'85
90
log (SC z;) FIG. 4. Plot of log % vs. log (SC& for various combinations of membrane and electrolyte solution at 25”C, derived from overpotential measurements with capillary tips near the centre of the working area of the 2 cm high vertically mounted membrane. cc was Values of D and t used were from Conway, .I0 i was 0.99 for all combinations; taken as 41.6 or 47.6 cm* mole-l for KC1 and NaCl solutions respectively. The combinations of membrane and bulk solution concentration were, for polarization on the dialysate side : 0 A.M.F cation, @05 M NaCl A A.M.F cation, 0.02 M NaCl q A.M.F cation, 0.20 M NaCl 0 A.M.F cation, 0.05 M KC1 + A20 anion, 0.05 M KC1 x A.M.F anion, 0.05 M KC1 and on the brine side: l A.M.F cation, 0.05 M NaCl A A.M.F cation, 0.02 M NaCl n A.M.F cation, 0.20 M NaCl A.M.F = American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A. A20 = The Permutit Co. Ltd, London.
described above for imperfect stirring on the side of the membrane not being polarized., In these experiments the capillary tips were placed adjacent to the geometrical centre of the membrane, the arrangement being, referring to Fig. l(a), x = 0.2 cm, h = 1-Ocm (Y = 2-00 cm). The experiments covered variation of current density (always below the limiting value for dialysate side polarization), of the nature and concentration of bulk solution, the sign of exchange activity of the membrane, the type of membrane used and of the side on which concentration polarization was allowed to develop. The line drawn in Fig. 4 is that of equation (6) with d = O-68, and it describes the results reasonably well. Although results for only one type of membrane other than the A.M.F. type have been shown in Fig. 4, no significant differences have been
B. A. COOKE
186
found in the polarization exhibited by other types examined in systems with natural convection. As the homogeneous A.M.F. type and the heterogeneous Permaplex A20 type of membrane probably represent opposite extremes in modern commercial ionexchange membrane preparation, it seems fair to conclude that the membrane plays no significant part in determining the hydrodynamic characteristics of these systems; as mentioned previously, this was not found to be the case in stirred membrane systems. Fig. 4 also shows that it is immaterial, as far as the value of 8 observed at a given i is concerned, on which side of the membrane the polarization is allowed to take place.
1
3
2 : ‘,
4
5
6
7
fllA/ClT12
FIG. 5. Comparison of polarization observed at A.M.F anion-exchange membrane in 0.05 M NaCl at 25°C when capillary tips were mounted in various positions relative to the working surface of the membrane. Experimental points : 0 tips “remote” (x = 2.6, h = 1.0 cm), + tips close but low (x = 0.1, h = 0.1 cm), x tips close but central (x = 0.2, h = 1.0 cm). Theoretical lines : A equation (5), B and C equations (7) and (9) for the low and central positions respectively, D equations (8) and (9).
An important reason for concentrating attention upon the dialysate side of the membrane is simply that higher overpotentials are obtained at a given current when this side is polarized; any error made in correcting for imperfect stirring on the brine side then has relatively little effect. The successful correlation furnished by equation (6), with ti = 0.68, in covering experimental results over a range of bulk solution concentrations and current densities below the various limiting values, makes it tempting to conclude that the condition of 0 being independent ofy, upon which (6) is based, is obeyed. This point was tested by determining overpotentials with the capillary tips mounted close to the lower edge of the membrane, with x = 0.1 cm, h = 0.1 cm (Fig. la), and comparing the resulting cobsvalues with those obtained when the tips were mounted in a close central position (x = O-2, h = 1-Ocm) or at a distance from the membrane (x = 2.6, h = 1-Ocm). For a reason mentioned later an anion-exchange membrane was used in these experiments. Fig. 5 shows the results: there is a marked difference between the values observed with centrally and with low mounted tips. Line B, computed from equations (7) and
Concentration
polarization
in electrodialysis-II
187
(9) taking a = 051, accounts well up to a certain current density for the small polarization detected by the low mounted tips; the fit is probably fortuitous in view of the simplified nature of (7), integration having been carried out over the height rather than both dimensions of the membrane (quite apart from the other assumptions*). The result does suggest, however, that assuming uniform current distribution below the limiting current is nearer the truth than supposing the current distribution to be that necessary to produce a single-valued 6. Line B has been drawn up to the line p = 1 (equation 8) above which (7) fails. Deviations are in fact noticeable somewhat below this limit and probably indicate the onset of non-uniform current distribution as the top of the membrane approaches the limiting condition. Lines C and D, for the close and remote centrally mounted tips respectively, practically coincide (which justifies the use of (8) for D even though x N Y) and the corresponding experimental points could not with certainty be ascribed to one or the other. In fact, both these theoretical lines and the experimental values lie fairly close, especially at the lower current densities, to line A, which was computed from (5) and is therefore based on the supposition of a single-valued 8 for each i value. It follows that overpotential determinations made below the condition of limiting current, and obtained with the aid of centrally mounted or remote capillary tips, provide an apparent mean interfacial concentration of a similar order to that predicted on the grounds of the mean transfer rate given by (4), but the actual averaging process is much closer to that implied by (7) and (9). The transition, with increasing current density, from the incipient limiting condition at p = 1, when only the extreme top edge is conducting its limiting current, to the full limiting condition causes a sharp rise in the curve found with low mounted tips. Although the rise is smaller with the other tip positions, a “hump” is also apparent and the final limiting currents registered by the various arrangements agree approximately with one another and with that predicted by (5); this is to be expected as the singlevalued 0 condition must hold when the entire membrane is in the limiting condition. Confirmatory evidence for the existence of a fully polarized portion of the conducting membrane surface at a current below the full limiting value was obtained with the aid of the arrangement of Fig. 3(c). The same anion-exchange membrane as was used in obtaining the results of Fig. 5 was placed in position 3’ and the pH of the effluent from compartment G determined, compartment C being continuously stirred. C and D contained pure 0.05 M NaCl and the same solution was passed through compartment G. At a current density of 5 mA/cm2, the effect of ceasing stirring in compartment D was to raise the pH of the effluent from G from 6.70 to 6.74-a negligible effect. At lower current densities no pH change was detected. At i = 5.5 mA/cm2 (just abovep = l), the pH increase was 0.70 unit and at 6.0 mA/cm2, 2.7 unit was found. Such an extent of OH- conduction could result only from the local attainment of a very low interfacial concentration although, as Fig. 5 shows, the limiting current had not been reached at all points at these current densities. An anionexchange membrane was chosen for this study because of the much smaller pH change observed when cation-exchange membranes are operated above the limiting current. * The value given by WranglerP for the coefficient a in the case of a concentration-polarized electrode with uniformly distributed current is 0.558, indicating theoretical lines 7.0 per cent below B, C and D as shown in Fig. 5. A slightly poorer fit is evident if Wranglen’s value is used, but it remains clear that the observed effect of varying the position of the capillary tips is attributable to the uniformity of current distribution at low current densities.
B. A. COOKE
188
Summarizing, the polarized membrane system having mass-transfer by natural convection exhibits a first limiting current, i’hm, appreciably below the average for the system as a whole, Iiim. The value of i’lim, as well as other behaviour of the system, are at least approximately accounted for by supposing uniform current distribution to obtain at lower currents, while between i’Ilmand ilimthe current distribution alters to give a constant-virtually zero-interfacial concentration at all points. If overpotentials are measured with centrally placed capillary tips, near to the membrane or remote, the findings correspond approximately to expectations based on the average mass-transfer relation Nu = ri(S~.z)‘/~ with d = O-68. Total overpotential
For a simple Nernst film, with single-valued 8 and constant concentration gradient up to a distance 6 from the membrane, the ohmic overpotential while polarizing current is flowing is readily calculated to be
in which A is regarded as constant in the range q, to (c,, - 6). Since in the Nernst idealization 6 is defined phenomenologically by id=:.
FD0 t-t
(10) may be written
(11) For a natural convection system obeying (5) below the limiting current (11) becomes
wherep’ = i/&;
and, by adding the concentration 11 =
’ e,ln 1 - cp’)4/5
overpotential ’
the total overpotential while current is flowing is obtained,
vt =
11+ 771=
e, + C
-q
(f
_
t)n
ln [l _;J
- f;“i’;;.
(12)
This relation gives a good account of experimental findings with capillary tips in the central position (Fig. 6), but the results with the low tip position indicate that this agreement is as specious as is the case with that apparent in Fig. 4. Nevertheless, (12) is a fair working approximation for a normal arrangement of capillary probes. Events above the limiting current
Simple considerations suggestll that the consecutive process available to a polarized membrane system above the limiting current might be H+ and OH- conduction through
Concentration
polarization
in electrodialysis-II
189
and anion-exchange membranes respectively. While this might be largely true for anion-exchangers, there is a good deal of evidence1~s~7~8that the expected H+ conduction does not occur through cation-exchangers. For this reason, it is found that the dialysate becomes acid in electrodialysis apparatus handling NaCl solutions at high currents, rather than alkaline as might be expected on the grounds of the lower limiting current at the cation-exchange membrane. Kressman and Tye12 found that acidity developed in the dialysate compartment of their apparatus for determining transport numbers when the current was high and only natural convection possible; stirring eliminated the effect. The same observation has repeatedly been made in the electrodialysis of saline waters in apparatus with forced convection.
cation
* 0
0.2
0.4
0.6
0.8
I.0
P’
FIG. 6. Total overpotential found with A.M.F anion-exchange membrane in 0.05 M NaCl at 25°C. Line shown is equation (12) and experimental points were obtained with x = 0.2, h = 1.0 cm (0) and x = 0.1, h = 0.1 cm (+).
In the present study the arrangement of Fig. 3(c) has been used to determine the nature of ion transport above the limiting current, polarization being allowed to develop on the anode side of membrane 3 or the cathode side of 3’ by appropriate use of the stirrers in compartment C or D. With 0.05 N solutions and polarization at membrane 3, the H+ flux was found to be O-1 f 0.05 per cent of the current in excess of the limiting value for a variety of electrolytes with the membrane in the corresponding cationic form. The forms were Na+, Li+, K+, Cs+, NMe,+ and Mg2+. The OH- flux when 3’ was polarized, which could be evaluated more accurately by passing acidulated NaCl solution through compartment G, was found to be 57 i 3 per cent of the excess current when NaCl was used. Values of the same order were found with other anions: F-, Br-, I- and SOJ2-. Peers*, noting the small H+ conduction by cation-exchange membranes, reported that the excess current is carried by co-ions in the case of a Permaplex C-10 membrane in NaCl: this observation formed the basis of the treatment by Partridge and Peers6 of current efficiency above the limiting current. This has not been confirmed in the present work. The current efficiency found for the single membrane 3 was O-996when compartment C was stirred and O-990 when it was not, the current in both cases being three times the limiting value for the unstirred condition. It cannot, therefore, be the case that the membrane leaks co-ions to any appreciable extent above the.limiting current. The expected fall in current efficiency was observed with the polarized anionexchange membrane.
190
B.
A. COOKE
interfacial concentration might, for some reason, not reach a value low enough to be comparable with that of H+ and OH- in water, i.e. lo-’ N, whereas this low concentration is easily reached at the anion membrane; (ii) the concentrations dissociation of the water at the cation membrane interface, while it readily occurs at the anion membrane; further conduction through the cation membrane must then involve some very difficult process-either counter-ion transport by some further unknown mechanism across the fully polarized film (found by Frilette6 and in the present work). A knowledge of the concentration in the limiting current region should permit a decision between the two possibilities. successfully for total overpotential below the limiting current, is assumed to be applicable to the limiting situation, it is possible to estimate the concentration provided it is known that no ionic species other than those provided by the in solution are present in the layer (the speciousness polarized).
Noting that 0 N c,, for i > ilim,(11) may
be written
and the total overpotential is Tit = Q(i > flim) =
e, +
& (t -
t)A 1
-
(13)
From (13) it is estimated that, if NaCl is used at 25°C and c0 = 0.05 M, an interfacial concentration of c’ = lo-’ M causes a total overpotential of -690 mV at a cationexchange membrane. It has been found experimentally that qlt is a highly irreproducible quantity for currents much in excess of ilim,but all determinations with cation membranes have indicated a rapid rise in q’$from 700 to 450 mV for small increases in current above ilim, the irreproducibility setting in only above about 950 mV. It is thus reasonable to conclude that the interfacial concentration reached is, at most, small enough to permit H+ transport as the consecutive process if the water ionizes in the normal way. Possibility (ii) above is therefore indicated. The postulated suppression of the dissociation of water at the cation-exchange membrane, which evidently does not apply at the anion-exchanger, is very possibly a consequence of ion/solvent interaction. The interaction between a cation and the immediately neighbouring water molecules may be viewed, following Haggis et &.,I3 as causing the “irrotational binding” of a number of water molecules. The affected water molecules, because of the breakdown of the normal spatial relationship of dipoles, become relatively inactive as a dielectric medium, or, as Ubbelohde14 has expressed it, nearer in properties to a fluid like H,S, the strong ionizing power being lost. This effect, which may be considered the basic feature of primary hydration, is a good deal less marked with anions;13y16thus, it is generally agreed that anions suffer smaller primary hydration than cations-cf. the successful account by Bascombe and BellI of
Concentrationpolarization in electrodialysis-II
191
the acidity function of concentrated acids in which anionic hydration was ignored. It may therefore be suggested, in the present connexion, that the water at the membrane side of the diffusion layer at a cation membrane is relatively undissociated as a result of the action of the cations crossing the interface, whereas that at the anion membrane surface is normal water (or deviates structurally from normal water to a much lesser degree). To account for the observed large difference in behaviour between cation and anion membranes above the limiting current it is clear that a marked difference between the counter-ion/solvent interactions would have to exist, and it would have to be shown that the fixed cation in the anion membrane does not exert the same effect upon water as the mobile cations in the cation membrane. Some evidence on this point has been obtained from measurements of the partial molar volume of water in partially humidified resins. l7 The depression of partial molar volume, which persists in cation-exchangers up to eight or more sorbed water molecules per equivalent of ionexchange capacity, was found to disappear after the sorption of 2 moles water per equivalent of an anion-exchanger. It may be supposed from this that the fixed cation effect in the anion-exchanger extends to no more than 2 molecules per exchange site, and that the structure-modifying effect of the anionic counterions is much smaller than that normal for cations. If the above suggestion concerning the smallness of H+ conduction through cation membranes above the limiting current is correct, it is a striking manifestation of ion/solvent interaction. It remains to account for the observation that Na+ continues to reach the cation membrane surface above the limiting current, Frilette6 made the same observation and suggested that electro-osmotic how associated with the cation flux causes sufficient disturbance of the diffusion layer to increase the mass transfer in the necessary degree. Certainly, it does not seem possible to account for the Na+ flux as a primary electro-osmotic effect, since the water transport normally found cannot significantly affect the value of the limiting current except at very high bulk solution concentrations.5 In the present study water transport above the limiting current in a natural convection system was compared with that occurring at the same current but with mass transfer by stirring, and only small differences were found (a two-compartment cell with Ag/AgCl electrodes was used and accurate weight analysis of the solutions before and after electrolysis enabled the water transport to be determined). It does not, at present, seem possible to improve on Frilette’s suggestion, but certain observations made during the present work are at least consistent with it. These were: (i) when a high bulk solution concentration (0.5 M NaCl) was used (and therefore a large ion flux below the limiting current with consequently large electro-osmotic effect per cm2 of membrane) the OH- flux through an anion membrane above the limiting current was found to be equivalent to only ca. 0.15 per cent of the excess current; (ii) with a low bulk solution concentration (0.001 M NaCl) and a cation membrane the H+ flux increased to ca. 3 per cent of the excess current. It appears, therefore, that the interfacial concentration was an order of magnitude higher in case (i), and lower in case (ii), than was the case with the intermediate bulk solution concentration used in most work (0.05 M) for which the stoichiometric OH- and H+ fluxes through anion and cation-exchangers were 57 and O-1 per cent respectively. Acknowledgemenrs-Thewriter thanks Mr. S. J. van der Walt for assistance with experimental work, Mr. N. van der Vlist who constructed the current interrupter used and the South African Council for Scientific and Industrial Research for permission to publish.
B. A.
192
COOKE
REFERENCES 1. B. A. COOKE,Electrochim. Actu 3, 307 (1960). 2. N. IBL, Electrochim. Actu 1, 117 (1959). 3. C. R. WILKE, M. EISENBERG and C. W. TOBIAS,J. Electrochem. Sot. 100, 513 (1953); Chem. Engng. Progr. 49,663 (1953). 4. J. N. AGAR, Disc. Faraday Sot. 1, 26 (1947). 5. S. M. PAR-E and A. M. PEERS, J. Appl. Chem. 8,49 (1958). 6. V. J. FRILETTE, J. Phys. Chem. 61, 168 (1957). 7. N. W. ROSENBERG and C. E. TIRRELL,Industr. Engng. Chem. 49,780 (1957). 8. A. M. PEERS,Disc. Faraday Sot. 21, 124 (1956). 9. K. ASADA,F. HINE,S. YIXH~ZAWAand S. OKADA,J. Electrochem. Sot. 107,242 (1960). 10. B. E. CONWAY, Electrochemical Data. Elsevier, London (1952). 11. F. HELFFERICH, Zonenaustuuscher. Vol. I, p. 363. Verlag Chemie, Weinheim (1959). 12. T. R. E. KRE~SMAN and F. L. TYE, Disc. Faraday Sot. 21, 185 (1956). 13. G. H. HAGGIS, J. B. HASTEDand T. .I. BUCHANAN, J. Chem. Phys. 20, 1452 (1952). 14. A. R. UBBELOHDE, Disc. Faruduy Sot. 21,137 (1956). 15. J. B. HASTED,D. M. R~TXINand C. H. COLLIE,J. Chem. Phys. 16, 1 (1948). 16. K. N. BAXOMBEand R. P. BELL,Disc. Faraday Sot. 2.4, 158 (1957). 17. B. A. COOKE, unpublished results. 18. G. WRANGLBN, Acta Chem. Stand. 12,1143 (1958). -
NOTATION Subscript obs denotes an experimentally observed value. Subscript app denotes a theoretically predicted value for an experimental observation. a = numerical coefficient in equation (I), taken as 0.51.
ii = 4a/3. c,, = concentration of bulk solution. c’ = interfacial concentration.
D = diffusion coefficient of electrolyte.
2y(” -
e, = -
t).
F = Faraday’s constant. g = acceleration due to gravity. h = height of capillary tip above lower edge of membrane. i = current density (local value). i = mean current density. i’um = first limiting current. ilim = mean limiting current. j = non-migrational electrolyte flux (local value). j = mean non-migrational electrolyte flux. k =
(g)“”
(3”“.
k yl/si4/5
p=-.
coa415
p’ = i/&m. R = gas constant. i = transport number of the counterion in the membrane. t = transport number of the same ion in free solution. T = absolute temperature.
Concentration
polarization
in electrodialysis-II
193
x = normal distance of capillary tip from membrane face. y = ordinate from bottom edge of membrane. Y = total height of membrane. Gr = Grashof’s number (local value) = Y%qe y2 . Gr=
YsgUB y2’
Nu = Nusselt’s number
local value = g
8D I
’
SC = Schmidt’s number = v/D. CI= specific density coefficient of electrolyte in solvent used = -1dp - . 6 =
rj = qr = qt = 13= A = v= v = vO =
pdc
Nernst layer thickness. concentration overpotential. ohmic overpotential. total overpotential = 7 + rjT. concentration difference between bulk and interface =I co - c’. equivalent conductivity. kinematic viscosity. angle indicated in Fig. l(a). 77- tan-r (x/h).
CONCENTRATION POLARIZATION IN ELECTRODIALYSIS-II. SYSTEMS WITH NATURAL CONVECTION* B. A. CHOKED National Chemical Research Laboratory, South African Council for Scientific and Industrial Research, Pretoria, Union of South Africa Abstract-The significance of apparent interfacial concentration values derived from overpotential measurements on a membrane system having mass-transfer by natural convection is considered in relation to current distribution over the height of the vertical membrane. Because current distribution is essentially uniform at low currents the system exhibits a first limiting current which is appreciably lower than the average for the system as a whole. With due regard paid to this point the experimental results are shown to be in agreement with previous theoretical and experimental results on natural convection. Events above the limiting current are discussed qualitatively, the suggestion being advanced that the lack of H+ conduction through cation-exchange membranes under these conditions is a consequence of cation/water interaction. R6sum&-La signification des valeurs apparentes des concentrations interfaciales, d&iv&es de mesures de surtensions dans un systkme A membrane oh le transport de masse s’effectue par convection naturelle, est CtudiQ en connexion avec la distribution du courant le long de la hauteur d’une membrane verticale. La distribution du courant &ant essentiellement uniforme aux courants faibles, le systbme pos&de un premier courant limite qui est considbablement infkrieur il la moyenne du systeme int@ral. En tenant compte de ce fait, il est montre que les Aultats obtenus sont en accord avec des Aultats anterieurs theoriques et exp&imentaux, se rapportant A la convection naturelle. Les phknom&es observks au dessus du courant limite sont discutes qualitativement; il est sugg6r.5 que, dans ces conditions, I’absence de transport d’ions H+ B travers des membranes 6changeurs decations est causQ par l’interaction cation/eau. Zusammenfassung-Die Bedeutung der Werte der scheinbaren GrenzlXchenkonzentration, welche durch Messung der uberspannung an einem Membransystem mit Stofftransport durch natiirliche Konvektion bestimmt wurden, wird im Zusammenhang mit der Stromverteilung iiber die H6he der vertikalen Membran untersucht. Weil die Stromverteilung bei niedrigen Striimen im Wesentlichen gleichm&sig ist, zeigt das System einen ersten Grenzstrom, der bedeutend niedriger liegt als der iiber das Gesamtsystem gemittelte Wert. Bei Beriicksichtigung dieser Tatsache kann gezeigt werden, dass die experimentellen Ergebnisse mit denjenigen aus friiheren theoretischen und experimentellen Untersuchungen in Einklang sind. Die Vorglnge oberhalb des Grenzstromes werden qualitativ diskutiert; das Fehlen der H+-Leitung durch Kationenaustauschermembranen wird unter diesen Bedingungen als Folge einer Kation-Wasser-Wechselwirkung gedeutet.
IN A previous paper,l an account was given of the applicability of overpotential measurements to the study of concentration polarization in electrodialysis, and the view was advanced that it provides a useful alternative to the measurement of current vs. voltage curves which has been the chief method of studying this phenomenon. The intention is now to consider the significance of interfacial concentration values derived from overpotentials measured on a membrane system of which the mass-transfer characteristics are known with some certainty. A substantial body of evidence, * Manuscript received 23 August 1960. t Present address: Research and Development Department, Imperial Chemical Industries Ltd. (Nobel Division), Stevenston, Ayrshire, Scotland. 179
B. A. COOKE
180
recently reviewed by Ib12, indicates a high order of agreement between theory and experiment for systems in which heat or mass transfer takes place to or from a vertical plane surface by natural convection. Rather than seeking to confirm this agreement for yet another system the approach adopted in the present paper is to make use of the previous findings, notably those of Wilke et al3 in interpreting results obtained on the membrane system. SIGNIFICANCE OF OVERPOTENTIAL VALUES OBTAINED ON A MEMBRANE SYSTEM WITH NATURAL CONVECTION*
Over a wide range of experimental conditions heat and mass-transfer by natural convection are described by the relation4 Nu = a(Sc.Gr)1/4.
(1)
For the present case the non-migrational electrolyte flux is, ignoring electra-osmosis and electrolyte diffusion through the membrane,6
from which it follows that 6=
ka44/6y1/5i4/5.
Two extreme cases of current distribution may now be considered. (a) 8 independent of y If this is the case, i must vary withy, and the mean current density over a membrane of height Y is obtained by integration from (3): i= $
iyidy
=
a
(!f)6’4y--li4,
(4)
where d = 4a/3. This yields for 0 0 =
k@fb
yWi4/5,
(5)
while the mean flux is generally expressed in the form %
= rt(Sc.?%$/4.
It is this relation, with d - 0.68 indicating good agreement with the theoretical result of Wilke et aL3 of a = O-508, that has been highly successful in correlating a variety of experimental findings on liquid systems with natural convection.2 This suggests that these systems fulfil the requirement of 8 being independent of y, a fact which is not surprising in the case of many electrode reactions, the experimental study of which has consisted of limiting current determinations. Whatever the current distribution may be at low currents it is clear in such cases that 8 must have the constant value of c0 in the limiting situation provided the consecutive process available to the system above the limiting current necessitates a marked increase in electrode potential. In the case of membrane systems there is no great certainty as regards the nature of the consecutive process (cf. the various views of Frilettes, Rosenberg and Tirrell’, and * A list of symbols is given at the end.
181
Concentration polarization in electrodialysis-II
Peer@), nor is there in general any reason to suppose that 8 should be independent ofy at currents well below the limiting value.g Clearly, if the membrane system were to obey the requirement of a single-valued 13 at each value of i, the concentration overpotential found would furnish an unambiguous estimate of 8 and would be unaffected by the position of the capillary tips (attached to reference electrodes) in relation to the membrane in the experimental arrangement for determining 9. (b) i independent ofy. In Fig. l(a), let the capillary tips T,, T,, (forming salt bridges with, e.g. saturated ‘I
5
(0) (b) FIG. 1. (a) Polarized membrane, M, of height Y with capillary tips T1, T,. (b) Representative network for polarized membrane shown in (a).
calomel reference electrodes) be placed as shown in relation to the polarized membrane 44, and let the membrane separate bulk solutions of the same concentration, that on side (1) being unstirred while (2) is efficiently stirred. Let the polarizing current cause
interfacial concentrations on the unstirred side (1) to be lower than the bulk solution value. If, as a simplification, all relevant activity coefficients are considered identical, the concentration overpotential at a height y is given by
where
e, = F(i
- t).
The voltage, rap*, registered between T, and T2 immediately after interrupting the polarizing current, will be a mean value analogous to eappfound with the representative network of Fig. l(b). The resistance associated with the overpotential 11at y is taken to be proportional to the length of path connecting Tl and T, through y, i.e. 2x cosec v; in so doing additional resistance contributed by the dilute solution in the layer is ignored and the membrane is assumed to be thin and narrow. Since, for the representative network, e
Ze/r)
app = yjqF) ’
B.A. COOKE
182
the overpotential registered is given by
2
(cos fpl.-
cos po) = [sinpl.ln[l
-$(/r+xcot~)1~5]+.
(7)
If the tips are remote from the membrane, the result is 1
rltpp _ _--
y
Y s0
e0
=
In 1 - s5 (
yl15) dy
1
( 11n(l-p)+f+~+$i2~+~(X~Y), ---1
(8)
P5
ki4/5y1/5
where
P=
coa415
It has already been indicated1 that experimentally observed overpotentials can be converted to interfacial concentration values without recourse to a simplified expression of the form , CaPP
= co exp
VaPP
(
e0
1
9
so that the most satisfactory way of comparing the theory and experiment would be by computing predicted ctappvalues with the aid of (9) from qapp/eoas yielded by (7) or (8). Experimental c’,,~*values (or Oobs= co - clobs) may then be compared with predicted ones rather than direct comparison of overpotential values. Quite apart from the inherent assumption of i being independent ofy, equations (7) and (8) must break down for p > 1. At p = 1 the uniformly distributed current just causes the attainment of the limiting condition at the top edge of the membrane, and, without specific information on overpotential in the limiting current region it is not possible to take the matter further. EXPERIMENTAL
The essential features of the cell used are shown in Fig. 2. Five annular Perspex sections of inner diameter 3.2 cm were clamped together to form compartments A to E,
k
FIG. 2. Electrodialysis
tration polarization
cell (diagrammatic sectional view) for determination of concenwith natural convection at membrane 3. a = anode, k = cathode; for other features see text.
Concentration
polarization
in electrodialysis-II
183
the end compartments holding the anode and cathode, respectively. A sub-assembly, H, holding the membrane under test, 3, was mounted between the compartments C and D : it consisted of a pair of Perspex plates having rectangular openings 2 cm high and 1 cm wide at their centres to limit the exposed area of membrane 3 to 2 cm2. Glass tubes, T, and T,, terminating in capillary tips, passed through compartments C and D so that the tips reached (in most experiments) to within 2 mm of the membrane face on either side, with both tips on the perpendicular axis of the exposed portion of membrane. T, and T, were filled with saturated KC1 and led to saturated calomel electrodes mounted externally. The tips were sealed with asbestos fibre so as just to furnish TI
Tz
--& I2
3
4
I2
3
4
+- -++II B
// 12
Cl b 33' t
Fix-
E
4
5
G
(a) (b) (cl FIG. 3. Sequence of membranes used for studying overpotential at (a) a cation-exchange membrane, (b) an anion-exchange membrane, (c) arrangement for studying pH changes or current efficiency at polarized cation (3) or anion (3’) exchange membrane. + denotes anion-exchangers, - cation-exchangers, a denotes anodes, k cathodes.
electrolytic contact when a small hydrostatic head was applied on the external KC1 solution; excessively permeable tips gave false results for the polarization. A small platinum wire P passed through one plate to provide the earthing point1 for the system. Identical “tulip” stirrers of 2.6 cm diameter, Si, S,, were mounted in C and D equidistant from the membrane under test. The sequence of membranes used in the arrangement of Fig. 2 for the testing of a cation-exchange membrane is shown in Fig. 3(a); it is such that the bulk concentration in C and D could not change significantly with passage of current during a period of several minutes. The solution in B was several times more concentrated than that in C or D, while the electrode compartments were filled with phosphate buffer solutions to permit extended operation without replenishment. The analogous arrangement for testing an anion-exchange membrane is shown in Fig. 3(b), the membrane under test being again numbered 3. The essentials of the electrical circuitry have been described.l The interrupted current source was of conventional design and was used with an interruption time of 350 ps at a frequency of 35 s-l (current broken during 1.2 per cent of the total time). Its output was not grounded. The oscilloscope used was a Tektronix model 502 (Tektronix Inc., Portland, Oregon, U.S.A.), use being made of the differential amplification facility of this instrument; the differential balance of the amplifiers, which must be highly accurate for this type of measurement, was adjusted with the aid of the square wave generator which is one of the auxiliary circuits of this oscilloscope. A fraction of the grid signal controlling the series valve in the interrupter was introduced into the time-base circuit of the oscilloscope to ensure accurate triggering of that instrument at all times, including those occasions when the signal was disconnected to examine the zero-signal position of the trace in the event of drift at high sensitivities. The results
184
B. A. COOKE
reported have all been derived from null-point potentiometric measurements on the trace 40 ps after interruption by which time the initial very rapid decay attributable to capacitive effects had completely disappeared. In order to study-polarization with natural convection on a single side of a membrane, e.g. the side on which the interfacial concentration is less than the bulk solution value (the “dialysate” side), it is desirable to depolarize the opposite (“brine”) side efficiently. It is essential then to verify that depolarizing the brine side does not influence polarization on the other side, e.g. by mechanically transmitted vibration from a stirrer. This point was tested in the following manner. For a given solution, membrane and current density below the limiting value, four overpotentials were measured: with neither side stirred (Q), the brine side only stirred (qZ), the dialysate side only (q3), and both sides (q&. Ideal depolarization would give q4 = 0. In cases in which this has been approached it has invariably been found that Q > Q + 1;/a indicating that Q and ~a are less than their true values because of spurious agitation on the dialysate and brine sides, respectively, caused by the stirring on the other side. It has been found necessary to accept 1;/4> 0 and seek the most efficient stirring conditions for which Q = 1;72+ Q - q4, indicating an absence of the unwanted agitation. This criterion was found to be satisfied when “tulip” stirrers were used at ~2000 r.p.m. provided the cell mounting did not permit direct transmission of vibration from the stirrer motors to the cell. With the cell independently suspended the stirrer speed used throughout work on systems with natural convection was 1500 r.p.m. (adjusted stroboscopically). Under these conditions small but significant values of q4 were found which, at a fixed current density and bulk solution concentration, varied markedly with the type of membrane used. To study polarization on a single unstirred side, qa and either Q or Q were measured, depending on whether dialysate or brine side polarization was desired. By assuming the concentration differences between bulk and interface to be numerically equal when both sides were stirred, the interfacial concentration on the single stirred side was estimated from q4, so that the apparent interfacial concentration on the unstirred side giving rise to Q or q3 was obtained. Clearly, the error possible in this procedure increases with the magnitude of q4 in relation to Q or Q; for this reason, the membranes used in most tests were of a type displaying small values of q4. These were A.M.F. membranes (American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A.). For the membranes in positions 1, 2 and 4 (Fig. 3), Permaplex A20 and C20 (The Permutit Co. Ltd., London) were used. A modification of the basic cell arrangements of Fig. 3(a) or (b) that was used to estimate the current efficiency displayed by a polarized membrane, as well as OH- or H+ transport at high currents, is shown in Fig. 3(c). The two membranes, 3 and 3’, separated by a thin (0.9 mm) gasket were clamped between the plates forming the sub-assembly H; a stream of solution was passed by means of suitable channels in the plates of the sub-assembly through the space formed between the membranes. Either membrane could be depolarized on its dialysate side by means of the stirrers in compartments C and D. Pressures on either side of membranes 3 and 3’ were balanced manometrically. All work was carried out in a constant temperature room at 25 -f 0*2”C. Analytical grade reagents were used, either in distilled water, or when H+ or OH- transport was to be detected, “conductivity” water prepared by mixed bed demineralization.
Concentration
polarization
RESULTS
Signl@ance
AND
185
in electrodialysis-II DISCUSSION
of overpotential measured
4 shows a variety of results in the form of a plot of log 6 against log (SC.~), the values of cobs used in computing these groups having been obtained by direct application of the empirical method described previouslyi for obtaining interfacial concentrations from observed overpotentials, while allowance was made in the manner Fig.
.
2.00.o
. 1.900 . /
c+ +
I.70 . I.65 /', 72
) 75
8.0
'85
90
log (SC z;) FIG. 4. Plot of log % vs. log (SC& for various combinations of membrane and electrolyte solution at 25”C, derived from overpotential measurements with capillary tips near the centre of the working area of the 2 cm high vertically mounted membrane. cc was Values of D and t used were from Conway, .I0 i was 0.99 for all combinations; taken as 41.6 or 47.6 cm* mole-l for KC1 and NaCl solutions respectively. The combinations of membrane and bulk solution concentration were, for polarization on the dialysate side : 0 A.M.F cation, @05 M NaCl A A.M.F cation, 0.02 M NaCl q A.M.F cation, 0.20 M NaCl 0 A.M.F cation, 0.05 M KC1 + A20 anion, 0.05 M KC1 x A.M.F anion, 0.05 M KC1 and on the brine side: l A.M.F cation, 0.05 M NaCl A A.M.F cation, 0.02 M NaCl n A.M.F cation, 0.20 M NaCl A.M.F = American Machine and Foundry Co., Inc., Stamford, Conn., U.S.A. A20 = The Permutit Co. Ltd, London.
described above for imperfect stirring on the side of the membrane not being polarized., In these experiments the capillary tips were placed adjacent to the geometrical centre of the membrane, the arrangement being, referring to Fig. l(a), x = 0.2 cm, h = 1-Ocm (Y = 2-00 cm). The experiments covered variation of current density (always below the limiting value for dialysate side polarization), of the nature and concentration of bulk solution, the sign of exchange activity of the membrane, the type of membrane used and of the side on which concentration polarization was allowed to develop. The line drawn in Fig. 4 is that of equation (6) with d = O-68, and it describes the results reasonably well. Although results for only one type of membrane other than the A.M.F. type have been shown in Fig. 4, no significant differences have been
B. A. COOKE
186
found in the polarization exhibited by other types examined in systems with natural convection. As the homogeneous A.M.F. type and the heterogeneous Permaplex A20 type of membrane probably represent opposite extremes in modern commercial ionexchange membrane preparation, it seems fair to conclude that the membrane plays no significant part in determining the hydrodynamic characteristics of these systems; as mentioned previously, this was not found to be the case in stirred membrane systems. Fig. 4 also shows that it is immaterial, as far as the value of 8 observed at a given i is concerned, on which side of the membrane the polarization is allowed to take place.
1
3
2 : ‘,
4
5
6
7
fllA/ClT12
FIG. 5. Comparison of polarization observed at A.M.F anion-exchange membrane in 0.05 M NaCl at 25°C when capillary tips were mounted in various positions relative to the working surface of the membrane. Experimental points : 0 tips “remote” (x = 2.6, h = 1.0 cm), + tips close but low (x = 0.1, h = 0.1 cm), x tips close but central (x = 0.2, h = 1.0 cm). Theoretical lines : A equation (5), B and C equations (7) and (9) for the low and central positions respectively, D equations (8) and (9).
An important reason for concentrating attention upon the dialysate side of the membrane is simply that higher overpotentials are obtained at a given current when this side is polarized; any error made in correcting for imperfect stirring on the brine side then has relatively little effect. The successful correlation furnished by equation (6), with ti = 0.68, in covering experimental results over a range of bulk solution concentrations and current densities below the various limiting values, makes it tempting to conclude that the condition of 0 being independent ofy, upon which (6) is based, is obeyed. This point was tested by determining overpotentials with the capillary tips mounted close to the lower edge of the membrane, with x = 0.1 cm, h = 0.1 cm (Fig. la), and comparing the resulting cobsvalues with those obtained when the tips were mounted in a close central position (x = O-2, h = 1-Ocm) or at a distance from the membrane (x = 2.6, h = 1-Ocm). For a reason mentioned later an anion-exchange membrane was used in these experiments. Fig. 5 shows the results: there is a marked difference between the values observed with centrally and with low mounted tips. Line B, computed from equations (7) and
Concentration
polarization
in electrodialysis-II
187
(9) taking a = 051, accounts well up to a certain current density for the small polarization detected by the low mounted tips; the fit is probably fortuitous in view of the simplified nature of (7), integration having been carried out over the height rather than both dimensions of the membrane (quite apart from the other assumptions*). The result does suggest, however, that assuming uniform current distribution below the limiting current is nearer the truth than supposing the current distribution to be that necessary to produce a single-valued 6. Line B has been drawn up to the line p = 1 (equation 8) above which (7) fails. Deviations are in fact noticeable somewhat below this limit and probably indicate the onset of non-uniform current distribution as the top of the membrane approaches the limiting condition. Lines C and D, for the close and remote centrally mounted tips respectively, practically coincide (which justifies the use of (8) for D even though x N Y) and the corresponding experimental points could not with certainty be ascribed to one or the other. In fact, both these theoretical lines and the experimental values lie fairly close, especially at the lower current densities, to line A, which was computed from (5) and is therefore based on the supposition of a single-valued 8 for each i value. It follows that overpotential determinations made below the condition of limiting current, and obtained with the aid of centrally mounted or remote capillary tips, provide an apparent mean interfacial concentration of a similar order to that predicted on the grounds of the mean transfer rate given by (4), but the actual averaging process is much closer to that implied by (7) and (9). The transition, with increasing current density, from the incipient limiting condition at p = 1, when only the extreme top edge is conducting its limiting current, to the full limiting condition causes a sharp rise in the curve found with low mounted tips. Although the rise is smaller with the other tip positions, a “hump” is also apparent and the final limiting currents registered by the various arrangements agree approximately with one another and with that predicted by (5); this is to be expected as the singlevalued 0 condition must hold when the entire membrane is in the limiting condition. Confirmatory evidence for the existence of a fully polarized portion of the conducting membrane surface at a current below the full limiting value was obtained with the aid of the arrangement of Fig. 3(c). The same anion-exchange membrane as was used in obtaining the results of Fig. 5 was placed in position 3’ and the pH of the effluent from compartment G determined, compartment C being continuously stirred. C and D contained pure 0.05 M NaCl and the same solution was passed through compartment G. At a current density of 5 mA/cm2, the effect of ceasing stirring in compartment D was to raise the pH of the effluent from G from 6.70 to 6.74-a negligible effect. At lower current densities no pH change was detected. At i = 5.5 mA/cm2 (just abovep = l), the pH increase was 0.70 unit and at 6.0 mA/cm2, 2.7 unit was found. Such an extent of OH- conduction could result only from the local attainment of a very low interfacial concentration although, as Fig. 5 shows, the limiting current had not been reached at all points at these current densities. An anionexchange membrane was chosen for this study because of the much smaller pH change observed when cation-exchange membranes are operated above the limiting current. * The value given by WranglerP for the coefficient a in the case of a concentration-polarized electrode with uniformly distributed current is 0.558, indicating theoretical lines 7.0 per cent below B, C and D as shown in Fig. 5. A slightly poorer fit is evident if Wranglen’s value is used, but it remains clear that the observed effect of varying the position of the capillary tips is attributable to the uniformity of current distribution at low current densities.
B. A. COOKE
188
Summarizing, the polarized membrane system having mass-transfer by natural convection exhibits a first limiting current, i’hm, appreciably below the average for the system as a whole, Iiim. The value of i’lim, as well as other behaviour of the system, are at least approximately accounted for by supposing uniform current distribution to obtain at lower currents, while between i’Ilmand ilimthe current distribution alters to give a constant-virtually zero-interfacial concentration at all points. If overpotentials are measured with centrally placed capillary tips, near to the membrane or remote, the findings correspond approximately to expectations based on the average mass-transfer relation Nu = ri(S~.z)‘/~ with d = O-68. Total overpotential
For a simple Nernst film, with single-valued 8 and constant concentration gradient up to a distance 6 from the membrane, the ohmic overpotential while polarizing current is flowing is readily calculated to be
in which A is regarded as constant in the range q, to (c,, - 6). Since in the Nernst idealization 6 is defined phenomenologically by id=:.
FD0 t-t
(10) may be written
(11) For a natural convection system obeying (5) below the limiting current (11) becomes
wherep’ = i/&;
and, by adding the concentration 11 =
’ e,ln 1 - cp’)4/5
overpotential ’
the total overpotential while current is flowing is obtained,
vt =
11+ 771=
e, + C
-q
(f
_
t)n
ln [l _;J
- f;“i’;;.
(12)
This relation gives a good account of experimental findings with capillary tips in the central position (Fig. 6), but the results with the low tip position indicate that this agreement is as specious as is the case with that apparent in Fig. 4. Nevertheless, (12) is a fair working approximation for a normal arrangement of capillary probes. Events above the limiting current
Simple considerations suggestll that the consecutive process available to a polarized membrane system above the limiting current might be H+ and OH- conduction through
Concentration
polarization
in electrodialysis-II
189
and anion-exchange membranes respectively. While this might be largely true for anion-exchangers, there is a good deal of evidence1~s~7~8that the expected H+ conduction does not occur through cation-exchangers. For this reason, it is found that the dialysate becomes acid in electrodialysis apparatus handling NaCl solutions at high currents, rather than alkaline as might be expected on the grounds of the lower limiting current at the cation-exchange membrane. Kressman and Tye12 found that acidity developed in the dialysate compartment of their apparatus for determining transport numbers when the current was high and only natural convection possible; stirring eliminated the effect. The same observation has repeatedly been made in the electrodialysis of saline waters in apparatus with forced convection.
cation
* 0
0.2
0.4
0.6
0.8
I.0
P’
FIG. 6. Total overpotential found with A.M.F anion-exchange membrane in 0.05 M NaCl at 25°C. Line shown is equation (12) and experimental points were obtained with x = 0.2, h = 1.0 cm (0) and x = 0.1, h = 0.1 cm (+).
In the present study the arrangement of Fig. 3(c) has been used to determine the nature of ion transport above the limiting current, polarization being allowed to develop on the anode side of membrane 3 or the cathode side of 3’ by appropriate use of the stirrers in compartment C or D. With 0.05 N solutions and polarization at membrane 3, the H+ flux was found to be O-1 f 0.05 per cent of the current in excess of the limiting value for a variety of electrolytes with the membrane in the corresponding cationic form. The forms were Na+, Li+, K+, Cs+, NMe,+ and Mg2+. The OH- flux when 3’ was polarized, which could be evaluated more accurately by passing acidulated NaCl solution through compartment G, was found to be 57 i 3 per cent of the excess current when NaCl was used. Values of the same order were found with other anions: F-, Br-, I- and SOJ2-. Peers*, noting the small H+ conduction by cation-exchange membranes, reported that the excess current is carried by co-ions in the case of a Permaplex C-10 membrane in NaCl: this observation formed the basis of the treatment by Partridge and Peers6 of current efficiency above the limiting current. This has not been confirmed in the present work. The current efficiency found for the single membrane 3 was O-996when compartment C was stirred and O-990 when it was not, the current in both cases being three times the limiting value for the unstirred condition. It cannot, therefore, be the case that the membrane leaks co-ions to any appreciable extent above the.limiting current. The expected fall in current efficiency was observed with the polarized anionexchange membrane.
190
B.
A. COOKE
interfacial concentration might, for some reason, not reach a value low enough to be comparable with that of H+ and OH- in water, i.e. lo-’ N, whereas this low concentration is easily reached at the anion membrane; (ii) the concentrations dissociation of the water at the cation membrane interface, while it readily occurs at the anion membrane; further conduction through the cation membrane must then involve some very difficult process-either counter-ion transport by some further unknown mechanism across the fully polarized film (found by Frilette6 and in the present work). A knowledge of the concentration in the limiting current region should permit a decision between the two possibilities. successfully for total overpotential below the limiting current, is assumed to be applicable to the limiting situation, it is possible to estimate the concentration provided it is known that no ionic species other than those provided by the in solution are present in the layer (the speciousness polarized).
Noting that 0 N c,, for i > ilim,(11) may
be written
and the total overpotential is Tit = Q(i > flim) =
e, +
& (t -
t)A 1
-
(13)
From (13) it is estimated that, if NaCl is used at 25°C and c0 = 0.05 M, an interfacial concentration of c’ = lo-’ M causes a total overpotential of -690 mV at a cationexchange membrane. It has been found experimentally that qlt is a highly irreproducible quantity for currents much in excess of ilim,but all determinations with cation membranes have indicated a rapid rise in q’$from 700 to 450 mV for small increases in current above ilim, the irreproducibility setting in only above about 950 mV. It is thus reasonable to conclude that the interfacial concentration reached is, at most, small enough to permit H+ transport as the consecutive process if the water ionizes in the normal way. Possibility (ii) above is therefore indicated. The postulated suppression of the dissociation of water at the cation-exchange membrane, which evidently does not apply at the anion-exchanger, is very possibly a consequence of ion/solvent interaction. The interaction between a cation and the immediately neighbouring water molecules may be viewed, following Haggis et &.,I3 as causing the “irrotational binding” of a number of water molecules. The affected water molecules, because of the breakdown of the normal spatial relationship of dipoles, become relatively inactive as a dielectric medium, or, as Ubbelohde14 has expressed it, nearer in properties to a fluid like H,S, the strong ionizing power being lost. This effect, which may be considered the basic feature of primary hydration, is a good deal less marked with anions;13y16thus, it is generally agreed that anions suffer smaller primary hydration than cations-cf. the successful account by Bascombe and BellI of
Concentrationpolarization in electrodialysis-II
191
the acidity function of concentrated acids in which anionic hydration was ignored. It may therefore be suggested, in the present connexion, that the water at the membrane side of the diffusion layer at a cation membrane is relatively undissociated as a result of the action of the cations crossing the interface, whereas that at the anion membrane surface is normal water (or deviates structurally from normal water to a much lesser degree). To account for the observed large difference in behaviour between cation and anion membranes above the limiting current it is clear that a marked difference between the counter-ion/solvent interactions would have to exist, and it would have to be shown that the fixed cation in the anion membrane does not exert the same effect upon water as the mobile cations in the cation membrane. Some evidence on this point has been obtained from measurements of the partial molar volume of water in partially humidified resins. l7 The depression of partial molar volume, which persists in cation-exchangers up to eight or more sorbed water molecules per equivalent of ionexchange capacity, was found to disappear after the sorption of 2 moles water per equivalent of an anion-exchanger. It may be supposed from this that the fixed cation effect in the anion-exchanger extends to no more than 2 molecules per exchange site, and that the structure-modifying effect of the anionic counterions is much smaller than that normal for cations. If the above suggestion concerning the smallness of H+ conduction through cation membranes above the limiting current is correct, it is a striking manifestation of ion/solvent interaction. It remains to account for the observation that Na+ continues to reach the cation membrane surface above the limiting current, Frilette6 made the same observation and suggested that electro-osmotic how associated with the cation flux causes sufficient disturbance of the diffusion layer to increase the mass transfer in the necessary degree. Certainly, it does not seem possible to account for the Na+ flux as a primary electro-osmotic effect, since the water transport normally found cannot significantly affect the value of the limiting current except at very high bulk solution concentrations.5 In the present study water transport above the limiting current in a natural convection system was compared with that occurring at the same current but with mass transfer by stirring, and only small differences were found (a two-compartment cell with Ag/AgCl electrodes was used and accurate weight analysis of the solutions before and after electrolysis enabled the water transport to be determined). It does not, at present, seem possible to improve on Frilette’s suggestion, but certain observations made during the present work are at least consistent with it. These were: (i) when a high bulk solution concentration (0.5 M NaCl) was used (and therefore a large ion flux below the limiting current with consequently large electro-osmotic effect per cm2 of membrane) the OH- flux through an anion membrane above the limiting current was found to be equivalent to only ca. 0.15 per cent of the excess current; (ii) with a low bulk solution concentration (0.001 M NaCl) and a cation membrane the H+ flux increased to ca. 3 per cent of the excess current. It appears, therefore, that the interfacial concentration was an order of magnitude higher in case (i), and lower in case (ii), than was the case with the intermediate bulk solution concentration used in most work (0.05 M) for which the stoichiometric OH- and H+ fluxes through anion and cation-exchangers were 57 and O-1 per cent respectively. Acknowledgemenrs-Thewriter thanks Mr. S. J. van der Walt for assistance with experimental work, Mr. N. van der Vlist who constructed the current interrupter used and the South African Council for Scientific and Industrial Research for permission to publish.
B. A.
192
COOKE
REFERENCES 1. B. A. COOKE,Electrochim. Actu 3, 307 (1960). 2. N. IBL, Electrochim. Actu 1, 117 (1959). 3. C. R. WILKE, M. EISENBERG and C. W. TOBIAS,J. Electrochem. Sot. 100, 513 (1953); Chem. Engng. Progr. 49,663 (1953). 4. J. N. AGAR, Disc. Faraday Sot. 1, 26 (1947). 5. S. M. PAR-E and A. M. PEERS, J. Appl. Chem. 8,49 (1958). 6. V. J. FRILETTE, J. Phys. Chem. 61, 168 (1957). 7. N. W. ROSENBERG and C. E. TIRRELL,Industr. Engng. Chem. 49,780 (1957). 8. A. M. PEERS,Disc. Faraday Sot. 21, 124 (1956). 9. K. ASADA,F. HINE,S. YIXH~ZAWAand S. OKADA,J. Electrochem. Sot. 107,242 (1960). 10. B. E. CONWAY, Electrochemical Data. Elsevier, London (1952). 11. F. HELFFERICH, Zonenaustuuscher. Vol. I, p. 363. Verlag Chemie, Weinheim (1959). 12. T. R. E. KRE~SMAN and F. L. TYE, Disc. Faraday Sot. 21, 185 (1956). 13. G. H. HAGGIS, J. B. HASTEDand T. .I. BUCHANAN, J. Chem. Phys. 20, 1452 (1952). 14. A. R. UBBELOHDE, Disc. Faruduy Sot. 21,137 (1956). 15. J. B. HASTED,D. M. R~TXINand C. H. COLLIE,J. Chem. Phys. 16, 1 (1948). 16. K. N. BAXOMBEand R. P. BELL,Disc. Faraday Sot. 2.4, 158 (1957). 17. B. A. COOKE, unpublished results. 18. G. WRANGLBN, Acta Chem. Stand. 12,1143 (1958). -
NOTATION Subscript obs denotes an experimentally observed value. Subscript app denotes a theoretically predicted value for an experimental observation. a = numerical coefficient in equation (I), taken as 0.51.
ii = 4a/3. c,, = concentration of bulk solution. c’ = interfacial concentration.
D = diffusion coefficient of electrolyte.
2y(” -
e, = -
t).
F = Faraday’s constant. g = acceleration due to gravity. h = height of capillary tip above lower edge of membrane. i = current density (local value). i = mean current density. i’um = first limiting current. ilim = mean limiting current. j = non-migrational electrolyte flux (local value). j = mean non-migrational electrolyte flux. k =
(g)“”
(3”“.
k yl/si4/5
p=-.
coa415
p’ = i/&m. R = gas constant. i = transport number of the counterion in the membrane. t = transport number of the same ion in free solution. T = absolute temperature.
Concentration
polarization
in electrodialysis-II
193
x = normal distance of capillary tip from membrane face. y = ordinate from bottom edge of membrane. Y = total height of membrane. Gr = Grashof’s number (local value) = Y%qe y2 . Gr=
YsgUB y2’
Nu = Nusselt’s number
local value = g
8D I
’
SC = Schmidt’s number = v/D. CI= specific density coefficient of electrolyte in solvent used = -1dp - . 6 =
rj = qr = qt = 13= A = v= v = vO =
pdc
Nernst layer thickness. concentration overpotential. ohmic overpotential. total overpotential = 7 + rjT. concentration difference between bulk and interface =I co - c’. equivalent conductivity. kinematic viscosity. angle indicated in Fig. l(a). 77- tan-r (x/h).