I
DONALD A. C O W A N and JERRY
H. BROWN’
Texas Electric Service Co., Fort Worth, Tex.
€ffect of Turbulence o n Limiting C u r r e n t in
Electrodialysis Cells Conversion of brackish and saline water i s important if industry is to have sufficient process water in the future, aside from the need for potable water in residential areas existing and still undeveloped. The results reported here offer a n explanation to some of the basic problems in a n electrodialysis system, and it i s from essentially basic studies such as this that a vastly improved method could develop
IN
DEMISERALIZIKG saline water by electrodialysis, polarization limits the amount of current which can pass through the cells. Although polarization in the bulk of the stream is destroyed by turbulent flow through the cell, above a critical current density polarization is again noticeable. If a laminar boundary layer exists wherein turbulent mixing does not occur, the current is diffusion-limited and therefore should be a function of seven variables: viscosity, flow velocity, and temperature of the fluid ; dimensions of the cell; and the diffusion coefficient, mobility, and concentration of the salt. Current density in an electrodialysis cell has three critical values: above one, a polarizing voltage opposes current flow adding a cost element; above another, pH changes rapidly causing membrane damage; above the third, a charge gathers on the membrane. “Limiting current” designates the first value in that current is diffusion limited. T h e second value is nearly identical with the first; but p H change is greatest at the membrane-fluid interface where damage is most likely to result. T h e p H measured in midstream does not indicate the magnitude and, because it responds to freed ions only, changes oppositely to the p H in the fluid a t the membrane, which polarizes first. A paradox of electrodialysis is thus accounted for, and a complete theory of polarization, p H change, and charged surfaces emerges. Its immediate practical importance will be in fixing the flow velocity conditions needed in a practical electrodialysis demineralizer,
Present address, Science Division, University of Dallas, Dallas, Tex.
Experimental
T h e electrodialysis cell consisted of several channels through which water could flow, alternate streams being conirolled separately for velocity and concentration, Electrodes were placed so that electric current was transverse to stream velocity. Cell channels were 2.54 cm. wide with a centerline run of 429 cm.; 10 U-turns reduced the total exposed area to 1036
sq. cm. Spacer thicknesses of 0.16, 0.32, and 0.64 cm. were used. S o obstructions were in the channels. With two exceptions, the salt used was sodium sulfate in treated municipal water. Runs were made with cation-selective, anion-selective, and alternate membranes as in a demineralizing system. Membranes tests were available commercially (Ionics, Inc.), except for experimental, low-resistance membranes supplied by another manufacturer.
Polarization of sodium sulfate with concurrent pH change. Two diluting channels 0.32 cm. thick with 71 8 pmho/cm. conductivity, flowing at 94 cm./second
I 2 6-
64
I
Figure 1. Small decrease in current efficiency at the highest current density shown indicates that ionic current i s still largely furnished b y the electrolyte VOL. 51, NO. 12
DECEMBER 1959
1445
--I
/
I Y
ZOr
/
-/
/
"r 1
I 04 05 Figure 2. Point of limiting current density, at which negative slope cuts positive slope, was chosen from polarization of potassium chloride in distilled water ,B
1
,
01 02 03 R E C I P R O C I L CURRENT AMPS"
A. 6.
0.64-cm. channels, Re = 4 8 4 0 0.32-cm. channels, Re = 5 4 0 0
Electrical measurements of voltage to current ratios for the cell were made directly with a Wheatsone bridge arrangement of which the cell was one leg and a 30-amp., 50-mv. shunt in the main line to the cell was another; the other branch was made u p of a 10-turn variable resistor and a set of fixed precision resistors. A potentiometer across the same shunt resistor effectively measured the current to the cell. T h e relative conductivity before and after dilution was measured with conductivity cells in the influent and effluent streams as part of an alternating current bridge employing 1000-c.p.s. signal, amplifier, and null detector. T h e p H was read on a standard calomel electrode p H meter. Electrical measurements were accurate to 2%, flow velocities to 570, and temperature to 1%. Data were occasionally erratic, probably because of entrapped charges in the membranes.
Results T h e voltage drop across an electrodialysis cell consists of electrode potentials V,, concentration potentials V E polariza, tion voltages V,, and ohmic voltages ZR. Accordingly, the relation
v/r = R
+ ( K + v,+ V ~ Y I
(1)
indicates that a plot of V/Z against I-' will have the resistance of the cell as an
intercept and the cell voltages plus their derivatives as the slope. Polarization voltage manifests itself as a rapid change of slope. T h e p H of a diluting stream begins to change at a current value very near the value at which the resistance slope changes and continues to decrease as current density increases. At low current densities, to the right of this point in Figure 1, readings of V/Z a t various Z values can be taken randomly with corresponding values repeated, but at higher current densities to the left of the slope-change point, values cannot be repeated upon a reduction of current density. T h a t V/Z remained above normal even after a current interruption of 20 minutes while the water continued to flow indicates a charge on or in the membrane opposite in sign to that which the membrane exchanges. T h e current efficiency for the cell (some current was shunted through the electrode streams) in the run shown in Figure 1 was 827, a t low values of current density and decreased only to 78% a t the highest current density shown, evidence that the fluid against the membrane was not exhausted of electrolyte; had the current above the p H change point been carried only by hydrogen and hydroxyl ions, current efficiency would be less than 507, a t the high current density shown.
Table 1.
Runs Were M a d e in the Dialysis Cell with Anion-Permeable Membranes Cation-Permeable Membranes, and Combinations of Both K O . of C",Q Fluid Membranes pmho: remp., Ri,C I1,B Tfr, R/,b A0 Cm. Ohms Red Amp. iw'/caef Ch Cm. Ohms a c. 5 5 5 5 5 5 5 5 5
5 5 5 5
5 3 3 3 3 3 3 3 3 4 4 8 I
17-
!
/
t/i,
16
I
,
,
,
I
,
,
02 03 04 RECIPROCAL CURRENT - A M P - ' 01
Figure 3. limiting current density increases as flow velocity increases Conductivity, 3 0 0 0 pmho/cm.; 0.64-cm. channels; velocities, 29, 43, 58, and 87 cm./second
1446
3 3 3 3 3
5 2 2 3 3 3 3 3 3 4 4 8 3i 3i 3i 3i 3i
0.64 0.64 0.64 0.64 0.32 0.32 0.32 0.32 0.32 0.16 0.16 0.16 0.16 0.64 0.32 0.64 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.64 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32
3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 3000 718 1300 1300 1300 1300 1300 3310i 3310i 3310 3310 1250 1900 2500 3425 3000
1.48 1.50 1.59 1.50 0.76 0.79 0.75 0.72 0.74 0.38 0.38 0.39 0.39 1.43 0.71 1.42 0.72 1.14 0.70 0.71 0.73 0.76 0.76 1.63 0.82 0.85 0.84 2.47 1.63 1.24 0.90 1.03
2.32 2.19 2.59 2.32 1.38 1.25 1.32 1.20 1.20 0.905 0.785 1.015 1.015 1.68 0.98 1.78 1.04 1.94 1.17 1.18 1.21 1.42 1.39 2.02 1.62 1.76 1.72 2.63 2.09 1.63 1.19 1.27
14.5 14 10 14 13 14.6 14.0 18 16.5 13 13.6 12 12 19.4 18.9 18.7 17.2 29 29.2 28.6 27.3 24.8 25.2 26.7 26.3 24.0 24.5 26.5 26.0 25.5 25.8 24.7
3300 4900 6610 9920 3250 4990 6570 8080 9730 3300 3300 6550 7930 3300 3270 3300 3240 6160 4490 4860 8275 9850 11830 4840 5450 4400 4400 4040 4040 4040 4040 4040
5.6 7.6 9.5 13.0 9.1 11.5 14.3 18.9 22.2 10.0 13.3 14.7 16.0 5.3 7.5 5.0 7.5 3.8 4.5 6.0 8.8 8.8 10.0 10.7 15.6 15.1 10.7 3.8 5.6 7.5 10.4 11.0
79 108 136 186 72 91 113 150 176 42 56 62 67 84 67 80 67 141 93 129 187 187 213 116 96 108 87 81 79 80 86 92
Calculated total rea Conductivity of streams in which boundary layer limits current. sistance of fluid in cell. Total resistance of cell, including membranes, as determined by Calculated as if temperature were 25' C. e Limiting current. f Limitintercepts of graphs. ing current number. 0 Anion permeable. Cation permeable. Potassium chloride. Low resistance membranes.
INDUSTRIAL AND ENGINEERING CHEMISTRY
J'
E L E C T R O D I A L Y S I S CELLS
2
r 7
OC
Ii
Analysis T h e deduction of an empirical relation for limiting current density is complicated by the fact that this critical value is a function of seven variables. The conditions of continuity are: Div j = 0 (2) Div K E = 4?r u d N / d t = Div (jtF-' D grad .V) ( 3 )
+
At an interface, these become :
- Kj/Cf)J
4Tu = (K,/C,
(4)
0.47 so that e- = 0.0234 volt and e+ = 0.0205 volt. The effect of V, becomes noticeable at some value u, small but greater than 0; polarizing limiting current can be designated by
Change of pH. T h e charge per unit volume q in the boundary layer can be obtained frcm Equations 4 and 8 :
C: is related to normality : C
=
-
ah'
T h e mass condition becomes 27-
T h e transfer number t now refers to the ion not being transferred through the interface; its value changes from near zero to t at the membrane, and the diffusion coefficient changes from 0 to D. For the steady state, d.Y/dt = 0, the current density at the boundary is J
IC,
1
01 02 R E C I P R O C A L CURRENT
04
03
-
AMP-'
Figure 4. Thickness of the stream has a small effect on limiting current density 0.32-cm. channels; velocities, 51.4, 79, 104, 126, and 154 cm./second
T h e point a t which the negative slope of Figure 1 cuts the positive slope when continued downward is designated as "limiting current density" because of its apparent relation to diffusion. T h e dependence of this point on fluid velocity, channel width, and concentration has been determined experimentally. This point was selected as the one corresponding to the point defined by such an intersection in a study made with a 3300pmho per cm. solution of potassium chloride in distilled water of about 12 pmhos per cm. (Figure 2). T h e shape of the curves in Figures 1, 3, 4, and i is affected by the presence of other ionic species in the supply water. A series of runs was made at a fixed concentration of 2120 p.p.m. of sodium sulfate (0.0303 S), as indicated by a 3000-pmho per cm. reading on a conductivity meter compensated to 25' C. With five anion-permeable membranes and no cation membranes, runs a t various velocities were made with 0.64-cm. channel spacers (Figure 3), with 0.32-cm. spacers (Figure 4), and with 0.16-cm. spacers. For these graphs the intercepts have been moved to coincide, the actual intercepts appearing as R, in Table I.
= ( F D / t ) (b.V/by)
+ constant
- AVj)/6
(7)
where is the normality in midstream. Similarlv, conductivity has the gradient bC/by = (C, - C/)/S and in the boundary layer is
c
=
c/
d
25-
24-
(6)
T h e constant is zero if t is zero in the membrane. For a boundary layer of thickness 6, in which the diffusion coefficient D is constant, if t changes only at the interface the concentration gradient, b S f b j , may be expressed as bN/by = (No
2 6-
(8)
m s =
l-
: 23I
~
z w
e u
-
2220
4
c A
-
/
> 21
c
-
I -
L
-
-
-
01 RECIPROCAL
i
-
02 CURRENT
03
-
04
AMPS-'
Figure 5. Transfer numbers can b e determined from the ratio of limiting current a t the two different membranes A.
+ ( C , - C,)2/6
(9)
Cation membranes. 6. 3 anion-2 cation membranes. C. Anion membranes. Velocity, 2 9 cm./second; 0.64-cm. channels
T h e conductivity a t the interface can be eliminated from the expression through the gradients in Equations 6 and 8 Co - Cy - . at bC/by = ___ - I FD
=
j/e
where a term is introduced: , g = - FD at
c
=
co - J / e ( 6 - V
)
(10)
Polarization. The voltage drop across a boundary layer is 60 ' O
50
For sodium sulfate a t 25O C., with D = 1.Z24 10-5 sq. cm. per second, a = 99 mho sq. cm. per equiv, t = 0.38 (4) at the anion-permeable membrane, 0- has the value 0.0285 volt, and at the cation permeable membrane 8+ is 0.0175 volt. However, Figure 5 shows that the effective transference numbers are 0.53 and
r *
t-
40
.
I
3000
4000
'AL
5000 6000 REYNOLDS
B O 0 0 10,000 NUMBER
Figure 6. Limiting current i s inversely proportional to the thickness of the unstirred boundary layer in a turbulently flowing cell l o w values are from narrow channels 0.16 cm. thick
VOL. 51, NO. 12
0
DECEMBER 1959
1447
At the polarizing current (Equation 12):
If 6 has a value of about 10-3 cm. and u of 0.05 volt, a t polarizing current, q
has a charge density a t the interface corresponding to the hydrogen ion concentration for p H 7, and the charge increases without limit as the limiting current number approaches unity. T h e previous definition for limiting current will therefore serve as the limit defined by charge concentration. T h e immobilization of charge in the boundary layer frees charge of the opposite sign so that a divergence of current exists. I n general, this will not be precisely offset by the corresponding source in the other boundary layer but must find its sink in the concentrate stream. Because cations usually have smaller transfer numbers than anions, the imbalance is such that a drop in p H of the diluting stream results. Charge on t h e Membrane. The charge on the membrane face is governed by Equations 4 and 10.
the value L = 4.4; from Nikuradze the value 6.8; from experimental data on kinetics of dissolution at high viscosities, values between 2.5 and 4.6, and from heat transfer data, the value 4.0. T h e dimensionless boundary layer thickness in units of w‘ is :
(The value off employed is the Blasius coefficient, which is four times that shown in Equation 24) Frank-Kamenetskii has derived from the work of Prandtl
1448
Nomenclature = ratio of conductivity to normality, mho sq. cm./equiv. C = conductivity, mhos/cm. D = diffusion coefficient, sq. cm./ sec. Div = operation divergence E = electric field, volts/cm. F = Faraday, 96,500 coulombs f = friction coefficient Grad = operation gradient I = eiectrical current, amperes = limiting current density, ma./ j sq. cm. (with N ) K = dielectric constant L = characteristic length, dimensionless = normality of fluid, equiv./liter = volume charge density, 1.1 X f? coulombs/liter (in Esuations 13 and 14, p H = i.lG X 10-14 a) R = electrical resistance, ohms Re = Reynolds number t = transfer number of ion V = electrical potential difference, volts U = arbitrary polarizing voltage W = channel thickness, cm. W’ = hydraulic thickness, cm. = distance normal to membrane Y 6 = thickness of laminar sublayer = surface charge density, coulU ombs/sq. cm. e = characteristic ion voltage, FD/ at, volts = kinematic viscosity, sq. cm.,/sec. 1‘ Subscripts m = membrane = fluid at membrane interface f 0 = midstream c = concentration e = electrode ,b = polarization - = anion-permeable membrane = cation-permeable membrane a
Discussion
This “limiting current number” on the left of Equation 19 is plotted against Reynolds number in Figure 6. If the 0.16-cm. channel thickness data are disregarded because of the possibility of intermittent blocking, the fit of the ’/a power is satisfactory, and L has a value close to that derived from Nikuradze’s work. T h e expression for limiting current density or for thickness of the laminar sublayer is ‘
I
coe= L.’s
= 0.058 (Re)7!8
(20)
FD or j = 0.058 I(Re)7’* w t
Membranes whose K J C , is about that of the water will collect only a small surface charge until the polarizing current is approached. Then a charge opposite in sign to that which the membrane conducts collects on the surface and apparently becomes entrapped so that it does not disappear when the excess current is halted but continues to effect an added, erratic voltage in subsequent tests for some time. Membranes of high specific conductivity exhibit the effect at currents below the polarizing limit so testing is difficult. Organization of Data. T h e limiting current defined by polarization (Equation 12) is the value observed in these experiments. T h e boundary layer thickness is expected to be a function of Reynolds number VUJ’/Vand w ’ = 4 (area/perimeter). Because Cola is a constant, ,TO,the values of Co and a a t 25’ C. are used. Alhough both the diffusion coefficient and the Reynolds number change with temperature, the changes are compensatory so that the data are not sensitive to temperature. -4ccording to Frank-Kamenetskii ( Z ) , the laminar sublayer must be proportional to a length related to viscosity, velocity, and friction coefficientf: 6 = Lr/V(f/2)1‘2 (16) where L is dimensionless. With the friction described by Blasius’ law: 6 = L(vRe”8/0.395 V) (17)
which limiting current is exceeded and is, probably, related to overvoltages in polarography. During these experiments a value of 1.9 volts was measured for sodium sulfate in distilled water.
T h e Chilton-Colburn relation (5) does not fit. The Colburn relationship, w ’ / 6 = (f/2) Re ( V / O ) ~ ’ (21) ~ when combined with the relation frequently used in analogies of momentum transfer and heat transfer, f = 0.46 (Re)o.8 (22) yields for water the relation w ’ / 6 = 0.22 (Re)o.8 (23) This relation, used elsewhere ( 7 , 3 ) , is shown in Figure 6. A value of the friction factor more widely used (2) is f = 0.079 Re-114 (24) which would adjust the Colburn relationship to w ‘ / 6 = 0.38 (Re)3/* (25) T h e value a t Reynolds number of 10,000 would still be twice that determined in this work. The von KArmAn-Sherwood relation ( 6 ) , which could be stated: w‘/6 =
0.0395 (Re)3’4 1
+
890 895 (Re)-‘,8 ( 2 6 )
yields values which lie between the bulk of the data shown here and the 0.16-cm. data-at Reynolds number 10,000 the value would be 123-but the slope does not fit. T h e Frank-Kamenetskii relationship is favored by the data. An additional point of interest is the resumption of a negative slope at high currents (left in Figures 1 to 6). T h e divergence in transfer numbers has moved into the turbulent zone, so the boundary layer is no longer diluting. T h e exchange process demands additional energy, as indicated by the higher voltage of the slope which is proportional to the number of boundary layers in
INDUSTRIAL AND ENGINEERINGCHEMISTRY
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Acknowledgment T h e authors are grateful for the contributions of H. R. Drew, Texas Electric Service Co., and H. E. McDowell, Ebasco Services. They have also benefited from discussions with John Allred, University of Houston. literature Cited (1) Chilton, T. H.: Colburn, A . P., IND. ENG.CHEM.26, 1183 (1934). (2) Frank-Kamenetskii, D. A,, “Diffusion and Heat Exchange in Chemical Kinetics,’’ transl. N. Thon, Princeton Univ., Press, Princeton, N. J., 1955. (3) Harned, H. S., Owen, B. D., “The Physical Chemistry of Electrolytic Solutions,” Reinhold, New York, 1958. (4) Knudsen, J. G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” Univ. of Michigan Press, Ann Arbor, Mich., 1954. (5) Rosenberg, N.W., Tirrell, C. E., Zbid., 49, 780 (1957). (6) Sherwood, T. K., Trans. A m . Znst. Chem. Engrs. 36, 817 (1940). RECEIVED for review September 8, 1958 ACCEPTED August 6, 1959
