I
N. W. ROSENBERG and C. E. TIRRELL lonics, Inc., Cambridge, Mass.
limiting Currents in Membrane Cells
Hydrodynamic control may solve the problem of capacity and investment cost in electrodialysis for demineralizing waters in arid regions
Durum THE LAST several years, demineralization of saline waters by electrodialysis has been investigated extensively and appears promising for large scale use in arid areas ( 7 7 , 75). Major costs are distributed between energy consumption and equipment amortization (74). I n many waters of interest (below 5000 p.p,m. of total dissolved solids), costs for equipment amortization are higher than for power consumption (7) because of current density limitations. The mechanism of ion transferral in an electrodialysis cell is such that about half of the ions transferred must arrive a t the membrane interface by a diffusional (nonelectrical) process and transfer rates are limited by diffusion rates to the membrane surfaces. Diffusional processes operating in this system are similar to those operating in electrode processes (7, 9 ) and polarographic analysis (4, 8 ) . Close relationship also exists with chemical engineering studies of heat transfer and mass transfer. Use of theoretical and empirical formulas developed in chemical engineering practice has allowed certain projections to be made in the similar problem of limiting currents at electrodes (70).
Theoretical Consider a region about the interface between a n anion-selective membrane
780
and a sodium chloride solution where electrical current is being carried by negative ions from solution to membrane (Figure 1). If the transfer number of chloride ions in solution is t i and in the membrane t,, and if volumes are so large that bulk concentrations are unaffected by the passage of current, then a t current density i (ma. per sq. cm.) an electrical flow of chloride ions will take place across boundary AB within the membrane equal to t i i / F meq./sec. sq. cm. and across boundary CD within the solution equal to ti i / F meq./sec. sq. cm. where F is Faraday's constant (96.5 amp. sec./ meq.). 'This electrical transfer is unbalanced, and a net deficit of [ ( t , t i ) i / F meq./sec. sq. cm.] occurs. Therefore, the solution concentration of sodium chloride in the region drops, and nonelectrical diffusion of sodium chloride into the region takes place. At steady state, concentration at the interface is determined by equating diffusion into the region with net electrical transfer out of the region:
where D L is the diffusion coefficient of the salt (sq. cm. per sec.); c, bulb solution concentration (meq. per cc.); c,, solution concentration at the membrane surface; 6, solution film thickness in centimeters, across which the concentra-
INDUSTRIAL AND ENGINEERING CHEMISTRY
tion gradient exists; and F, Faraday's constant. Chloride ion transfer can be increased by increasing current density until the concentration of such ions at the interface reaches substantially zero. Any additional current is passed by transfer of hydroxyl ion from water in the film. Operation at current densities in excess of limiting current densities presents three distinct operating difficulties in electrodialysis-Le., current carried by hydroxyl ions is ineffective in desalting; low film concentrations represent high electrical resistances; and pH changes at the interface are high and pH-sensitive solution components precipitate. Actual limiting current density is not a point function. because competition between chloride and hydroxyl ions becomes significant at increasing currents as the chloride concentration in the film approaches the hydroxyl concentration. Furrher current lowers the hydroxyl concentration (\vhich must mainrain its dissociation equilibrium with hydrogen ions lefr in the film) as well as the chloride concentration. Limiting current density can be obtained from Equation 1 by setting the surface concentration. c,, equal to zero.
at an anion membrane
B
D
I I
Effective film thickness, 6, is determined by the hydrodynamics of the system. I n systems where natural convection is the only source of agitation (5),6 is about 0.05 cm., and hence (i/t)llm is about 50 (ma./sq. cm.)/(meq./cc.) if t: = 0.5. However, if hydraulic flow parallel to the interface is established, film thickness is reduced and limiting current is increased.
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I BY CURRENT
There are two types of flow parallel to surfaces-Le., streamline flow, where radical convection is absent and diffusion provides the only mechanism for mass transfer from bulk to surface; and turbulent flow where radial convection is high and a physical slowmoving surface film exists through which rate-limiting diffusion occurs. I n both types, nonelectrical diffusion is governed by Fick’s law.
I I
f I I
I n chemical engineering practice, the hydrodynamic factors and their influence on transfer rates have been intensively studied. I n summary, an empirical correlation defined subsequently, has been found between Reynolds number and the Chilton-Coburn factor, j D . This correlation allows estimation of an effective film thickness, 6, applicable to Equation 2. Figure 2 shows the j, correlation and Figure 3 indicates effective values of 6 computed for aqueous solutions. The experimental range reported here is limited to streamline flow conditions, but extension to turbulent regions is now under investigation in this laboratory.
A
~~
Figure 1.
The Chilton-Coburn transfer factor, is defined by j D = KL/V ( p / D L ) 2 / 5 . T h e Reynolds number of a flowing
jD,
Mechanism of ion transfer at membrane interface
stream is defined as N R ~= D , V p / p where K -, is the mass transfer coefficient (meq./sq. cm. sec.)/(meq./cc.); V, average linear velocity (cm./sec.) ; p, viscosity (g./sec. cm.); p, density in grams per cc.; and De, equivalent hydraulic diameter in centimeters, defined as the product of four times the cross section of the duct divided by its ~ perimeter. For streamline flow ( N R< 2100), in flat ducts (IZ),
De
j D =
1.85 ( N R , ) - ” ~ ( L / D ~ ) - ” ~
where L is path length in cm. over which the flow is developed. For turbulent flow ( N R , > 2100) = 0.023 N ~ ~ - 0 . 2
(5)
Equation 5 represents a purely empirical correlation, but Equation 4 is based on a solution of Fick’s law as
Lfa
I
I 0-3 I o2
i I It1111 lo3
I
I I I I Ill1
NM I 0’ Figure 2. Chitton-Coburn mass transfer correlation for parallel surfaces. iD K L / V ( ~ / ~ D L and ) ~ / ~NRE
Figure 3. Effective film thickness 6 for diffusion predicted for various hydrodynamic conditions. p = 0.01, p = 1.O,
DBVP/P
DL = 10-5
=
=
VOL. 49, NO. 4
APRIL 1957
781
I'
IFigure 4.
Cell assembly for studying polarization
D~(d'/bx' f b'c/by') = V ( y ) b c / x (6)
where x is direction of hydraulic flow; y , distance from interface; and V O ) , linear velocity of the stream in the x direction at a point y from the wall. The solution to Equation 6 has been obtained by Norris and Streid (72) for flat ducts, and may be expressed as Equation (7). 6 = D L / K L = 0.54 ( D L D , L / V ) " ~ ( 7 )
Physically, film thickness, 6, has a reality even in streamline flow. At the entrance to a channel, before any salt has been removed from the stream, bulk concentration extends uniformly to the interface. A short distance, x , downstream, some salt will have been removed from the layers nearest the wall, and the gradient for salt diffusion will have decreased-Le., distance 6 over which the concentration gradient extends into the bulk has increased. Thus, as a given portion of liquid proceeds downstream, the region of lower concentration extends further into the bulk so the concentration gradient, and hence the diffusion rate, decreases. The value of 6 extends to the center of the spacer as a limiting case for long path length. Experimental
The effect of increasing current densities on the p H of solutions flowing past membrane surfaces was determined in cells constructed of alternating spacers and membranes, with electrodes located a t each end of the assembly. Seven-cell assemblies were used (Figure 4). Cell 1 was bounded by the cathode and a cation-selective membrane; cell 7 by the anode and an anion-selective membrane; all other cells were bounded by membranes as shown. Polarization was studied in the central cell (cell 4) which was fed with a 0.005
782
to 0.05N sodium chloride solution. Other cells were fed with a salt 3 to 10 times more concentrated. Thus, at the current densities used, polarization was restricted to the membrane faces in the central cell. This technique, suggested by Edgardo Parsi of this laboratory, allowed polarization at the two membranes to be studied independently. At a series of increasing applied voltages, current, influent and effluent concentrations of cell 4, and effluent pH values in cells 3, 4, and 5 were determined. Concentrations were determined by chloride or sulfate analysis, and p H values with a Beckman p H meter. The spacer used is shown in Figure 5. The holes in the margins register with holes punched in membranes to provide ducting. By suitable orientation and feeding from both ends of the assembly, up to 12 different solutions may be fed simultaneously and independently. The obstructions across the path in Figure 5 are half the thickness of the spacer, and thus form bridges which do not completely block flow. They provide some degree of eddying to mix the saltdepleted surface layers with the bulk stream-Le., to reduce the effective path length, L. This spacer is 0.14 to 0.16 cm. thick with outside dimensions of 22 X 22 cm. The spiral path for solution flow was 240 cm. long by 0.5 cm. wide, with an area of 120 sq. cm. for current passage. The bridges were spaced at 45-degree intervals around the spiral path, alternating in contact with the two membrane faces. As there are 60 bridges, the average distance between them is 4 cm. Since the eddy currents a t the bridges tend to mix the depletion surface layers with the bulk solution, the applicable length, L, in Equation 7 is taken as 4 cm. Both cation-selective and anion-selective membranes were used. The cationselective membranes were sulfonated
INDUSTRIAL AND ENGINEERING CHEMISTRY
divinylbenzene polystyrene types (2) formed as homogeneous sheets about 0.08 cm. thick on glass cloth backing, with 2.6 meq. exchange capacity per dry gram, 40Yo water content, and a resistance of 14 ohm (per sq. cm.). Anionselective membranes of quaternized divinylbenzene pyridine types ( 3 ) were formed in the same manner, except with 2.0 meq. exchange capacity per dry gram, 4470 water content, and a resistance of 20 ohm (per sq. cm.). Both membrane types were highly selective; their transport numbers were above 0.95 in the concentration range studied. The anode was constructed from platinum sheet and the cathode from stainless steel. The spacers were made by steel-rule die punching of plasticized poly(viny1 chloride) sheet. Two unplasticized poly(viny1 chloride) sheets "4 inch thick were used as distributors a t the ends of the assembly. Steel bolts bearing against two steel plates clamped the assembly so that the spacers formed leak-free gaskets. Results and Discussion
A typical history of pH and sodium chloride concentration in central cell 4 us. current is given in Figure 6. As current increases, the interface between the anion membrane and cell 4 solution becomes depleted of sodium chloride. Transfer of hydroxyl ions from cell 4 to cell 3 causes p H of the effluent to decrease in cell 4 and increase in cell 3. At this current, there is no indication of cation membrane polarization. As the current is increased, however, the cation membrane interface becomes depleted; hydrogen ion transfer from cell 4 into cell 5 occurs, and the pH of cell 5 falls. However, rhe transfer of hydroxyl ions through the anion membrane interface increases at a faster absolute rate, and the pH of cell 4 continues to fall. The bulk concentration of cell 4 de-
Figure 5. Design of the spacer used in this study
ELECTRODIALYSIS
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creases significantly on its passage through the cell. Figure 6 shows the effluent normality us. current, from which a n average current efficiency can be computed. As expected, in the polarized region average current efficiency falls significantly. A differential current efficiency, defined as d (meq. C1 transfer)/F dl a t a total current I, is more pertinent (Figure 6). The p H difference between feed and effluent streams is the most sensitive measure of polarization. I n this report, on an arbitrary basis, the anion membrane limiting current is taken as that at which the p H difference between cell 3 and cell 4 effluents equal 2 p H units. Cation membrane limiting current is taken as that at which the p H of cell 5 falls to 6.0. Because of the rapid change of p H with current, it is believed that limiting currents are close to those which would be obtained by any other definition of polarization. The path length of the cell was 2.4 meters in order to transfer sufficient hydrogen and hydroxyl ions so that p H changes could be readily measured. Therefore, significant concentration changes occurred along the path. Experimental data included total current I (amperes); total area, A (sq. cm.); and influent concentration ci, and effluent concentration c, (meq./cc.). For evaluation of 6 from Equation 2, point values of the ratio of current density to concentration ( i / c ) are required. A solution is possible if the approximation is made that i/c is constant along the path a t a given current in a given run. With this assumption, consider passage of a solution along a cell path w cm. wide, t, cm. thick, a t a linear velocity, I' cm. per second, and a t a current efficiency, E. The concentration reduction along the path is given by dc/dx
=
-(iE/F)
(l/Vte)
(8)
Substituting i = ck (our assumption) and integrating
k
In c J c e = ( k E / F ) ( x / % ) (9) But the total reduction in concentration is given by 6% - C, = ( Z E / F ) ( l / V w t , ) (10) Combining Equations 9 and 10 and rearranging, letting area A = wt,
T h e assumption that i / c is independent of position along the path, is based on four approximations : 1. The potential difference between membrane faces in the central cell is considered independent of position along the path. The solutions and membranes in other compartments in the system have much higher conductivity than the solution in the central compartment;
Table 1. Experimental Data of a Typical (Cell 4 feed, 0.0113N NaC1; flow rate, 1.8 ml./sec.; velocity;22.5 spacer thickness, 0.16 cm.) Cell No. Cell 4 Exit Current, 3 4 5 Concn., Amp. PH PH PH 10s X N 0.00
0.79 0.91 1.00 1.07 1.17 1.26 1.31 a
7.0 7.1 7.4 8.8 9.3 9.9
.. ..
Lt
7.0 7.0 6.9 6.4 5.0 3.9 3.7 3.7
7.0 7.0 7.1 7.3 7.2 7.1 5.9 4.5
Runa cm./sec.; temp., 25O C.;
Current Efficiency, % Av. Differential
11.3 6.9 6.4 6.1 5.9 5.7 5.6 5.6
..
..
96 92 89 87 82 78 75
96 72 57 49 34 19 0
Run 12. I
thus, redistribution of current necessary to maintain the membrane faces a t constant potentials is probably good. 2. Equivalent conductance of the bulk solution is approximately constant, and variation of that in dilute solutions used is less than 10% for a tenfold concentration reduction. 3. Current efficiency is approximately constant along the path. This assumption is valid if polarization occurs uniformly along the path, since transport numbers of membranes and of solution d o not vary significantly below 0.1N solution concentrations. 4. The ratio of film resistance to bulk solution resistance is approximately constant. If the gradient from the surface is linear, this approximation is adequate. With these approximations, it is believed that i/c values computed from Equation 11 have a probable accuracy within &IO%, which is entirely adequate to establish trends, if not for exact quantitative correlation. Tirrell is carrying out a separate study in which identical membranes bound the central cell. I n such a cell, concentration changes are negligible, while p H changes are measurable; thus, uncertainty introduced by Equation 11 is eliminated. Table I1 presents a summary of 13 experimental runs made with the spacer design (Figure 5 ) . The values of i/clirn (ma./sq. cm.)/(meq./cc.) were computed from Equation 11. Variables included velocity, temperature, concentration, membrane surface, and anion species. Table I11 presents effective film thicknesses predicted from Equation 7 , theoretical ( i / c ) ~values ,~ obtained for anion and cation membranes by use of Equation 2 and the experimental i / c values reported in Table 11. T h e ratio of experimental to theoretical (il c)llrn values are also tabulated. The diffusion constants were estimated from Harned and Owen (6). D Lfor sodium chloride and sulfate was estimated as 1.3 x 10-6 sq. cm. per second a t 25' C. and 1.7 X 10-5 sq. cm. per second, a t 43' C. respectively. Solution transport num-
bers were taken as t i = 0.61 in both sodium chloride and sulfate (6). De was taken as 0.22 cm. for a path 0.50 X 0.14 cm. and as 0.24 cm. for a path 0.50 X 0.16 cm. The most significant single observation is that in all experimental runs, anion polarization occurred first at i/c values close to those predicted from theory; cation polarization, however, was not evident until nearly twice its
PH
100
I-
0 06
08
10
1.2
Current, Amps.
Figure 6. Data from run 13, sodium chloride feed, ci = 0.01 13, flow 1.8 ml. per second, spacer thickness, 0.16 cm. VOL. 49,
NO. 4
APRIL 1957
783
Table II. Summary of Limiting Current Density Observations (Temp,, 2 j 0 C.; membranes sanded; solution, NaC1; spacer path, 240 X 0.6 cm.; spacer areas, 120 sq. em.)
Run No.
Spacer Thickness, Cm.
Average Velocity,
1 2 3
0.14 0.14 0.14
6.60 6.60 7.10
4
0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.16 0.16
7.35 7.35 7.35 7.35 7.60 7.85 1.01 1.16 22.5 21.5
5
6“ 7a 8h Qh 10 11 12 13c a
*
Cm./Sec.
Anion Polarization Ci, Ziim, - C. Meq./Cm. amp. ci (%)iim 0.0055 0.0060 0.0451 0.0435 0.0438 0.0435 0.0435 0.0435 0.0425 0.0447 0.0447 0.0113 0.0113
0.186
...
0.69
..
480
1.69 1.63 1.50 1.59 1.67 2.15 2.10 0.46 0.485 0.98
0.68 0.64 0.65 0.68 0.68 0.84 0.79 0.98 0.94 0.47
0.57
0.26
..
Cation Polarization 1 - ‘3 ci (%),im amp
liirn,
...
..
..
520 500 480 510 520 900 810 340 275 980
0.283 1.76 1.80 1.70 1.63 1.78 2.25 2.27 0.485 0.49 1.26
0.68 0.70 0.68 0.71 0.68 0.68 0.84 0.83 0.99 0.94 0.50
660 560 580 560 520 575 940 940 410 285 1300
490
1.05
0.31
930
Membranes unsanded. Temp., 43’ C. Solution, NagSOa.
Table 111. Comparison of Theoretical and Experimental Limiting Currents [Theoretical film thickness 6 = 0.54 ( D L D , L / V )1’3; theoretical limiting current (i/c) = DL/GF(trn - & ) I Anion Membrane Cation Membrane (i/c)I m (ilc) 1im Run V, Ci Theor., Exptl./ Exptl./ No. Cm./Sec. N X 103 Cm. X 103 Theor. Exptl. theor. Theor. Exptl. theor. !
1 2 3
4 5 6
7 8
9 10 11 12 13
6.60 6.60 7.10 7.35 7.35 7.35 7.35 7.60 7.85 1.01 1.16 22.5 22.5
5.5 6.0 45.1 43.5 43.8 43.5 43.5 43.5 42.5 44.7 44.7 11.3 11.3
6.5 6.5 6.3 6.2 6.2 6.2 6.2 6.6 6.6 12.1 11.5 4.4 4.7
530 530 550 560 560 560 560 680 680 290 300 780 780
predicted i / c values were reached. Also, theory predicts rhat, because of the low sodium transport number in sodium chloride solutions. cation polarization should occur at 6574 of the anion polarizing currents; the experiments showed, ho\rever. that cation polarization is negligible a t anion-polarizing currents. This effect cannot be explained by simple diffusional phenomena, and some membrane interaction may be involved. .4 second effect which shows that membrane interaction may be significant appeared in run 13, lvhere sodium sulfate was utilized instead of sodium chloride. The sulfate ion polarized a t a n z/c value half that of chloride ion in run 1 2 a t the same hydrodynamic conditions. If hydrodynamic conditions alone determined polarization, the chemical nature of the species would have had no effect on ( i / c ) l i m . Anion-polarizing z IC values were generally in good agreement with theo-
784
480
..
0.89
520 500 480 510 520 900 810 340 275
...
..
980
0.95 0.89 0.86 0.91 0.93 1.32 1.19 1.18 0.91 1.25
340 340 350 360 360 360 360 440 440 185 193 500
660 560 580 560 520 575 940 940 410 285 1300
1.94 1.60 1.62 1.55 1.45 1.60 2.12 2.12 2.20 1.48 2.60
490
0.63
500
930
1.85
. ..
retical predictions. As expected in streamline flow, the effect of surface roughness was negligible. Runs 6 and 7 were carried out with unsanded glossysurfaced membranes; all other runs used membranes sanded with 3/0 sandpaper. T h e variation in i / c with concentration, as expected, was insignificant (compare runs 1 and 2 with runs 3, 4, 5, and 12). Its increase with temperature was expected, but the actual increase was considerably above that expected from the increased diffusion coefficient (compare runs 8 and 9 a t 43’ C. with other runs). Increase in velocity from 1 to 20 cm. per second increased i / c values in the same direction and to approximately the same degree as expected. Conclusions
Ion transfer rates under electrical current across selective membranes are
INDUSTRIAL AND ENGINEERING CHEMISTRY
determined by diffusion rates to membrane surfaces. Diffusion rates are controlled by hydrodynamic flow conditions. Good predictions of limiting currents a t anion membranes are obtained by using the theoretical equations. Limiting currents a t cation membranes are significantly higher than predicted from present theoretical considerations. T o maximize permissible current densities in electrodialysis, high linear velocities, thin cell spacings, and in the streamline flow region, closely spaced eddy-promoting obstructions, are desirable. To achieve high degrees of demineralization in a given cell, a tortuous path is desirable to provide a reasonably long contact time while maintaining high velocities (73). In the streamline region, the ratio o f limiting current to normality can be increased with (V/DLD,L)1’3. I t is predicted that turbulent flow, to be effective in permitting increased i / c values, must be obtained by increasing velocity rather than increasing cell thickness. Acknowledgment
T h e authors wish to thank Edgardo Parsi and N. E. Saliba for their direct assistance, and the staff of Ionics, Inc., for many helpful suggestions. Literature Cited
(1) Agar, J. N., Discussions Faraday Soc. l 926 (1947). (2) Clarke, J. T., U. S. Patent 2,731,411 (1956). (3) Ibid., 2,732,351 (1956). (4) Glasstone, S., “Introduction to Electrochemistry,” pp. 445-55, Van Nostrand, New York, 1954. ( 3 ) Glasstone, S., Trans. Electrochem. SOC. 59, 277 (1931). (6) Harned,H. S., Owen,B. B., “Physical Chemistry of Electrolytic Solutions,” Reinhold, New York, 1943. ( 7 ) Kirkham, T. A., Chem. Eng. 63, 185 (1956). (8) Kolthoff, I . M., Lingane, J. J., “Polarography,” vol. I, Interscience New York, 1952. (9) Levich, B., Discusszons Faraday SOC.1, 37 (1947). (10) Lin, C. S., others, IXD. END. CHEM. 43,2136 (1951). 11) Nachod, F. C., Schubert, J., “Ion Exchange Technology,” pp. 11881, Academic Press, New York, 1956. 12) Norris, R. H., Streid, D. D., Trans. Am. SOC. Mech. Enprs. 62, 525 (1940). 13) Rosenberg, K. W., U. S. Patent 2,708,658 (1955). 14) Rosenberg, N. W., Kirkham, T. A., Tirrell, C. E., Saliba, N. E., Research and DeveIopment Rept. 1, Saline Water Program, Dept. of Interior, Washington, D. C., 1954. (15) U. S. Dept. of Interior Saline Water Program Annual Reports, Washington 1952-55.
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RECEIVED for review August 9, 1956 ACCEPTED Xovember 12, 1936
