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The variation of current density and electrode potential with electrode resistance and its role in cell design
THE VARIATION OF CURRENT DENSITY AND ELECTRODE POTENTIAL WITH ELECTRODE RESISTANCE AND ITS ROLE IN CELL DESIGN Technisch-Chemisches
PmmM. ROBWTSON Labor, ETHZ, Universitiitstrasse 6, 8006 Ziirich, Switzerland
(Received 1 June 1976; and In&vised fom
11 August 1976)
potential drop along parallel electrodes of:finite resistance has been calculated (a) assuming a pure resistive representation of the cell and (b) by considering the polarization characteristics of an irreversible electrode reaction. It was found that a much more uniform potential (and also
Abstract-The
current) distribution was obtained with the current feeders at opposite ends of the electrodes rather than at the same ends. With this configuration it is advantageous to have electrodes of equal resistance. Equations are given that enable the potential and current density distribution to be easily calculated. was derived which is a measure of the e.lCctive utilization of the electrode area. A factor I&,,, It may be used for calculating the required electrode area with non-uniform cd distribution.
NOMENCLATURE electrode
area (m’) effective electrode area = A (I,.,/!, ,) (mr) cost of electrode/unit volume (g/m’) concentration of electro-active species at electrode interface (mole/m3) thickness of electrode (m) thickness of anode (m) thickness of cathode (m) diffusion coefficient of electroactive species (m’/s) local potential of anode in the electrode phase (V) local potential of cathode in the electrode phase
(v) local anode potential (measured with reference electrode) (V) local cathode potential (measured with reference electrode) standard potential potential
(V) potential for anodic reaction (V) across current feeders (v) on anode at opposite end of electrode to current feeder (V)
characteristic potential for linearized treatment (V) electrode separation (m) local cd at anode (A/m*) local cd at cathode (A/m’) limiting cd (A/m*) minimum observed cd (A/m’) maximum observed cd (A/m’) local current flowing along anode (A) local current flowing along cathode (A) theoretical cell current = electrode area x min cd encountered (A) theoretical cell current = electrode area x max cd encountered (A) total ceil current (A) 2 pldh P< Cm- ‘1 (Rc - Ri) (diJdE),, (A/m’) (d(,/d& (Aim’) rate of electrode reaction (m/s) cost of electrodes/unit area (S/m’) cost of electrodes/unit effective area (S/m2) breadth of electrode (m)
L L,
n x Y P
P. PC P.
length of electrode (in direction of current flow) (m) length of electrode wasted because of current feeder connections (m)
number of electrons involved in electrode reao tion distance along electrode in direction of current flow (m) L/4 + pJ4)/p,h (m-*) resistivity resistivity resistivity resistivity
of of of of
electrode material (Szm) anode (&I) cathode (am) electrolyte (Qm)
1. INTRODUCTION
A major contribution to the cost of electrolytic alls is the electrode material. Sometimes inexpensive metals eg iron may be employed, but this occurs seldom and usually only for the cathode. In the case of the anode, corrosion resistance often dictates the use of noble metals eg platinum or platinised titanium. There are of course the less expensive graphite and magnetite that may be acceptable but their electrical conductivities are lower than for metals and could lead to significant potential drops in the electrodes themselves. This may even occur with metal electrodes if the cd is very high, or the thickness has been reduced on account of cost or because of design requirements. The problem of potential drop in the electrodes is a well known one, which in addition to wasting power may also give non-uniform current distribution resulting in lower production capacity per unit electrode area. When designing an electrolytic cell the resistance and therefore dimensions of the electrodes must be chosen carefully so as to minimize the cost. The effect of electrode resistance on the potential and current distribution has been discussed for several two dimensional electodes [l-9]. Tobias and Wijsman calculated the distributions at parallel plate electrodes with the current feeders to the electrodes at one end (see Figure la) [l]. This geometry is
4f2
PETERM. ROBERTSON E’
Figs l(a), l(b). Current feeder configurations. encountered in tank electrolysers and also in accumulators, which have been studied by Shepherd [2,3]. Other electrode shapes that have been treated are wires or cylindrical forms [4-Y] and rectangular ducts [S]. Arrangements in which the power connections were made to opposing ends of the electrodes (Figure lb), or cells having unequal anode and cathode resistances have not* been discussed in great detail. A particular problem encountered in this lahoratory was the use of very thin metal foils or sheets as electrodes. In the design of cells with thin electrodes the cd distribution had to be known so as to estimate the required electrode surface area. Thin metal foils are used in the “Swiss-roll” cell that we are investigating [ll]. The main feature of this cell is a sandwich consisting of thin anode and cathode sheets with separator cloths in between to prevent electrical short-circuits. The total thickness of this sandwich can be well under 1 mm. Typical dimensions are 300 x 15 x 0.04cm. A three dimensional structure is obtained by rolling the sandwich up around an axis. The power connections to the electrodes in this small laboratory sized cell were made by utilizing the axis of the roll and container as current feeders. It is easy to appreciate that with the large electrode area obtained (9000cm2), large currents may be encountered. This fact together with the electrode length and thickness may easily result in substantial potential drops in the electrodes if care is not taken. In this publication the optimum current feeder arrangement and relationship between the anode and cathode resistances will be derived. The potential and current distributions for the optimum arrangement will then be examined as a design aid for the determination of the optimum electrode dimensions. 2.OPTIMUM
CURRENT
Fig. 2. Simplified model of cell. cathode current feeder (Volt), p,, and pc are the resistivities of the anode and cathode (Qn), pe is the resistivity of the electrolyte (Qm) and I, and I, are the local currents flowing in the anode and cathode (A). The electrolysis current for a small length dx of electrode is dl and the corresponding cd i = (dl/dx)/l. We apply first Ohm’s law to the electrode phases.
Differentiating we obtain the change of local current density. d2En-L._! dl dx=
* An exception are three dimensional electrodes eg porous, that can be represented by similar equivalent electrical circuits [IO].
d2E C=
a,
PC
dx’i;i;
dx2
(2&b)
- dl, dl, dx=dx
(3)
This is also the current flowing between the electrodes, which may be written: dl 2 = 6% - E,)Uhihpe
(4)
Combining (2)-(4) and differentiating we obtain:
The solution of which is: E, = a exp(y’%) $ b exp(-y”%)
+ EX + e
(6)
+ cx +P
(7)
with
(a) Cell model The electrochemical cell consists of two parallel plate electrodes of breadth I (m), length L (m) and separation h (m), and the anode and cathode have thicknesses d, and d,, respectively. It is assumed in the derivations that follow that the current flows in a straight path through the electrolyte perpendicular to the electrodes and with negligible end effects. We consider first that the cell may be represented by pure resistance, as is the case when the electrolyte resistance is very much greater than the Faradaic impedance or a linear approximation of the polarisation characteristic is taken. E, and E, are the local potentials in the metal phase of the anode and cathode us the
md
Now an increase of current in the anode is equivalent to a decrease of current in the cathode, thus:
FEEDER
CONFtCURATlON
P
dx Id,
y = ~/&+pJ&) and from (2a), (4) and (6): E, = - ‘.d.aexp(g*‘*x) pad, - $hexp(-y’%) Dc
The boundary conditions must now be introduced in order to determine the coefficients ~1,b, c and e. We treat both cases as depicted in Figs l(a) and l(b). Case (i). Current feeders at the same ends of the electrodes. With both current feeders at x = -L/2 we obtain the following boundary conditions: at _
x=-L/2
J&=0
and
E,=E*
(8)
Electrode resistaace and its role in cell design n(eY”‘L _ e- Y”‘L)Rly”2
(E* is the cell voltage) X = G/2
and at
C=
I, = I, = 0
(9) since no current flows out of the electrodes at x = L/2. Introducing these boundary conditions into ‘(6) and (7) the coefficients are found to be:
0, * = d,p, + d,p,
(12)
e=*_.g* t&p, + d,p,
(13)
Case (if). Current feeders at opposite ends of annde and cathode. In this case the cathode feeder is at x = -L/2 and the anode feeder at x = + L/2. The boundary conditions are therefore:
x = L/2
E, = 0 E, = E*
(14) (15)
Again because no current flows out of or into the ends of the electrodes without the current feeder we have : at
x=
-L/2
I, = 0
(16)
and at
x = L/2
I, = 0
(17)
Introduction of these conditions into (6) and (7) gives the following expressions for the coefficients a, b, c and e: a =
E*RI(R, ey”* + R2e-Y”‘LIZ) (R: + R;)(&“L + e-Y’ilL)
+
RIR2y1’2L(eY”‘L- e-Y”zL) + 4R,R2
e-$J’L)z)’
R2
(20)
e-
y”‘L
1
+ R,RZy’/2L(er”‘L - e_y”2L)/2 f 2R1Rz
d=
with
c=o
x = -L/2
+ +
1 RL(Rl&'rL/Z +&e-V"'L/Z) (21)
pL
(10) (11)
and at
(Rley”‘L/2
E* exp(y”2L/2) 1 + exp(2y”‘L)
b = (I exp(y”2L)
at
413
Rt = p,ld,
and
Rz = p,ld,
(b) Results and discussion The above equations for the potential distribution may be written in terms of the dimensionless parameters: WE*), (WE*), (d,&W, (2nald,W1’2L and (x/L). The pd’s (En - l&)/E* for both cases are shown in Figs 3 and 4 as a function of (x/L) and (d,pJd,p.) at a constant (2pJd,hp,)‘?L value of 1. As ( clearly demonstrated in Fig. 3 the maximum potential difference for Case (i) is obtained at the end of the electrodes connected to the current feeder ie at x/L = -0.5 and decreases with increasing x/L. In contrast when the current feeders are connected to opposite ends of the electrodes, ie Case (ii), there are regions of high potential difference at both ends of the electrodes with a minimum in between. The variation of potential difference is much less than for Case (i) which also implies more uniform current distribution. In Fig. 3 it may also be observed that in the case of the current feed at the same ends of the electrodes a more uniform potential (and therefore current distribution) is obtained at low values of d, p,/ d,p, for a fixed value of (2pJd,hp,)‘@L ie the electrode resistance pJd, should be low. This also applies of course to the other electrode resistance p,jd,. The lower the electrode resistance, the more uniform the potential and current distributions are. For the case in which the current feeders are at opposite ends of the electrodes (Fig. 4) the most uniform distribution is obtained with log(d.p,/d,p,) = 0 ie with pJ d, = PC/d,.The uniformity of distribution under these circumstances is very much better than that achieved for Case (i) and makes possible the use at very much higher cds. This configuration appears lo be the optimal one and will therefore be discussed exclusively in the development that follows. For Case (ii) with p,/d, = pdd, = p/d the solution simplifies to:
(18) 1
E, = 2a cash (k1’2x)+ cx + e
(22)
Ez = -2acosh(k%)
(23)
and + cx + r
with
Fig. 3. Potential differeace for current feeders at same ends.
E* ’ = 2Lk”* smh(k”2L/2) + 4 cosh(k”2L/2)
(24)
e = E*/2
(26)
k = 2p/dhp,
(27)
and
,414
PEm M. ROBERTSON
Fig. 4. Potential difference for current feeders at opposite ends.
I
I
2
I
I
3
k’=L/Z
3. EFFECTIVE UTILIZATION OF ELECTRODE AREA
Fig. 5. The electrode utilization 1,,,/1,,, and cd variation imiJi_ factors.
Electrochemical processes usually have an optimum current density. This current density may simply be a limiting cd, it may arise from a consideration of electrode investment and power costs [12] or it may be a maximum allowable cd at which no undesirable side reaction takes place. For whatever reason the optimum cd arises the electrode area will be lCO% effectively used only when the cd over the whole area is constant and equal to the optimum value. Because of the finite resistance of the electrodes the cd is not uniform and the electrodes will not be utilized to the full. It is useful to define the factor I,a/l,,,~Xwhich gives a measure of the effective utilization of the electrodes (electrode utilization factor). I,,, is the total current fed to the electrodes and I,, is the total current that would flow if the cd were uniform and equal to the maximum value observed, ie that at the ends of the electrodes. Now the current density i. is given by:
The function (33) is plotted in Fig. 5. In a similar fashion the cd variation may be expressed by the factor : bJim.x
=
(cU,/dx),,o/(dr,/dx),=,,,
= sech (k’/‘L/2)
(34)
This function is also plotted in Fig. 5. For large values of the argument k”‘L/2, I,,/l,., = 2/k”‘L. 4 EXTENSIONS OF THE SIMPLE MODEL FOR ELEmOCHEMlCAL CHARACTERISTICS
(a) Reaction nt the limiting current density We now assume that the local cd is always equal to the limiting cd. The current flowing along the electrodes is therefore linearly dependent on the distance from the current feeder. Thus 1, = i&L/2 + xbl
(35)
I, = i&L/2 - x).1
(36)
From (4), (22) and (23) we obtain: i, = 4n cash (k”‘x)/hp,.
(29)
Substituting in (laj and (lb), taking
Therefore LIZ I,,, = 1
I -L/Z
P. PC -c-zda d,
i,dx = 8al sinh (k’i2L/2)/hp,k1i2 (30)
Now the maximum cd is observed at x = L/2 or -L/2, thus i,,,., = 4a cash (k”‘L/Z)/hp,
d
and integrating we obtain: E, = pi&Lx/2 + x2/2)/d + c,
(37)
E, = pi,(Lx/2 - x2/2)/d + c2
(38)
and
(31)
and
P
from which I InaX= 4uIL cash (k”‘L/2)/hp,
(32)
Therefore I**LX
= ,$& t anh (!&‘L/2)
(33)
E. - E, = p&x2/d + (cl - c2) (39) where i, is the limiting cd and c, and c2 are integration constants that are dependent on the standard potentials of the system. The validity of (39) is restricted to systems that have a limiting current plateau
Electrode resistance and its role in cell design
that is a least as broad as the potential variation pi&‘/4d. The factor &,/I,,, is of course equal to 1 since the cd is always i,.
this section the complete model for the system,
including reactions below and/or at the limiting cd will be discussed. Since most proeessa of industrial interest are operated irreversibly eg deposition of a metal ion or oxidation of an alcohol to an acid, we shall consider only the irreversible case and in particular an anodic oxidation*. The description of the electrode reaction is that used by Delahay [13] for irreversible electrode reactions in stirred solutions. The current density i, is given by: i, = nFk,Cexp((l-a)(E:
- EO)nF/RTJ
C = 6 (i, - iJttFD
(41)
This model considers therefore a concentration change from the solution bulk up to the electrode surface through the diffusion layer, ie a change of concentration perpendicular to the direction of electrolyte flow through the cell. Substituting for C in (40) we obtain: (D/k,d)exp{
-(l
- ol)(Eb- E”)nF/RT}
>
Substituting (49) into (2a) and solving as before gives: E, = 2a cash (2/1ka/d)“~n + cx + e
Now the potential difference E, - E, may be expressed in terms of the electrode potentials EL and E’<(measured with a reference electrode in the solution) and the iR drop term for the current flow through the electrolyte. E, - E, = E:, - E’=+ iR
(43)
For the sake of simplicity and to minimize the number of independent variables it will be assumed that the concentration of the depolarizer for. the cathodic reaction is so high that the cathode electrode potential EL is virtually constant. It will also be assumed that the iR drop is negligible. Substituting for Eb in (42) gives: _~
‘9
1 + (D/kJ)exp{ -(l
EC = - 2acosh(2pk#)%
+ cx + e f k,/kl
E* + kl/k, a =
2_Q2pk,/d)liz sinh [(pkJ2d)“‘L]
E* + kJk2 ’ = L + (2d/pk#2
coth [(pkz/2d)“2L]
(50)
and e = (E* - k,/k,)/2
(51)
The electrode utilization factor for this case is similar to that obtained in Section 3 :
LtiLax= CP4& )“*/~ItanhCW&4”*~1W) In a more empirical way the range of applicability of this linearization may be extended. If only a small variation of cd is tolerated then a short section of the polarization curve may be regarded as a straight line of slope (di/dE),,,. This line ab is shown in Fig 6. It is the tangent to the polarization curve and crosses the abscissa at some potential &,. The cd may therefore be written as (53)
Again we may express as (46) with k, = (EC - E,J (diJd&, kl = (d&/da,. The solutions (47), (48) and (52) are therefore valid with these substitutions. The calculation of I,&,,, is now relatively simple: we must only measure the slope of the polarization curve at the average cd to be encountered. The accuracy of the approximate solutions discussed up to this point may be determined if the exact solutions are also known. (ii) Exact numerical solution. The potential and current density distribution is now solved for the full unapproximated polarization behavior with equation (44). This incorporates concentration polarization, which has not been considered in previous treatments [lo] but assumes constant concentration of the eledroactive species in the solution bulk.
(4)
- E, + E: - E”)nF/RT}. (45)
* Although the derivation here considers an ancdic process. the results are equally applicable to cathodic ones.
(49)
+ 4 cash [(pk&f)“*L]
- a)@, -EC + El -E”)nF/RT}
+ (1 - G%
(48)
with
(i) Approximate solution by Rnearization. Assume that D/k, S s 1 and that the exponent is small. We may therefore write: i, = +{I
(47)
and
i, = (EL - E,&(di/dE), (42)
i. = ~
.
i. = kr + k,(E, - E,) = f ‘2
(4)
where a is the charge transfer coefficient, k, the rate constant of the electrode reaction at the standard potential ED and C the interfacial concentration of the electro-active species. The interfacial concentration is then expressed in terms of the limiting cd for a particular Nemst diffusion layer thickness 6, and diffusion coefficient D:
ia=1 +
This equation is of the form:
(
(b) Reuction below the limiting current density In
415
Fig. 6. Linearization
of the p&rimtion curve.
416
Pmm M. Roam-
Table 1. Dimensionless groups used in calculations
computer
x, = E,nF/RT x1 = E,nF/RT off 3 E*nF/RT n, = i/i, lij = x/L
16 = II, = cg = ng = x,~ =
D/k,b a i,p L’nF/RT d (2p/dhp,)“‘L (E: - EO)nF/RT
In order to minimize the number of variables it is preferable to introduce dimensionless parameters. Those chosen are listed in Table 1. Equations (2a), (2b), (3) and (44) may be written as a system of 4 first-order differential equations with the dimensionless parameters of Table 1:
P ._
o-
(54)
-025
0
025
0.5
X/L
dn,z
As = 1+
__ dn,
-0.5
Fig. 8. Computer derived normaiized cd distribution (ilie) for integer values of ztg from 1 to 10.
x,exp[(~, - 1)(x, - E2 + nro)]
-dnz= ni3 dx, h, dntz -=-dn, dr,
(57)
These equations have been solved in terms of an initial value problem wing the standard Runge, Kutta-Gill method. The initial values are given by (14) and (16). The other two conditions (15) and (17) are of course boundary values. Boundary condition (15) presents no problems since it is an independent variable and our interest is in tabulating the solution for several values of E*. This tabulation is more conveniently performed by replacing (15) by an initial condition (E,), = _L,2 = E**. Tbe other boundary condition is more difficult to treat. An iterative technique must be used to find a suitable initial value of i, (at x = - L/Z) such that the required boundary condition (17) is fulfilled. Some cd distributions
obtained numerically are plotted in Figs 7-10. The numerical solutions for I,,JI,,X are tabulated in Table 2 for various values of na, x6 and ns. Also tabulated are the ltO,/i_ values obtained by applying the linearization treatment of Section 4b (i) with the slope of the polarization curve (diJdE)& at the average cell cd I,,&. As can be seen the agreement between the linearized and exact treatments is good at values of LJLX close to 1. At I,,,/I_ i 0.8 the agreement is less than ideal but this should not detract from the value of the approximation since smaller I,,,/&,,,, values have in any case less interest. (iii) Numerical examples. As an illustration I,JIIILy will be calculated for a typical ‘Swiss-roll” cell and also a diaphragm chlorine cell. We consider first the “Swiss-roll” cell. The electrodes are of nickel of resistivity 6.84 x lo-*am. The dimensions of each electrode are 3.0 x 0.15 x O.C001m. The slope of the polarization curve is now required. We shall take a fairly typical value of 5CQA/mzV (=(diJdE),) as in for example the oxidation of ahphatic alcohols to
lb0
IO
LOO,
075
0.75 I
t P
0.50
050
1 f
5:’
t 0.25
-x
Fig. 7. Computer derived nmmak&d,@ f&rihion for integer valued of Itj froLn -10.
,i.C” . ,.
(i/i.)
Fig 9. Computer derived normalized cd distribution (i/i,) for integer values of 2rj from l-10.
Electrode resistance and its role in cell design
417
Table 2. Comparison of approximate and exact numerical solution of cd distribution
X/L
Fig. 10. Computer derivet$ normalized cd distribution for integer values of zj from t-10. acids at nickel anodes
in alkaline
solutions
(i/i,,)
[ll].
Thus [p(di,JdE),,J2dj1’ZL = 1.24
1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 IO.00 lO.ctl 10.00 10.00 10.00 10.00 10.00 10.00
1.00 l.ctl 1.00 1.00 10.00 10.00 10.00 10.00 1.00 1.00 1.00 1.00 10.00 lO.cNl 10.00 lO.Otl
0.980
0.962 0.966 0.991 0.836 0.680 0.628 0.614 0.991 0.95 1 0.908 0.934 0.914 0.699 0.549 0.480
0.981 0.960 0.962 0.987 0.833 0.644 0.539 0.404 0.990 0.95 1 0.907 0.918 0.916 0.719 0.570 0.396
We now make the assumption that the electrode cost is proportional lo the volume of metal. Thus K = a&J = a,Ad
and from (52) I,, &_=
0.68.
The second example is a diaphragm cell with current feeders also as depicted in Fig. l(b). We take electrodes of RuOI covered titanium (1 m x 1 m) of thickness 2 x 10e3 m and assume that the RuOa coating has negligible conductivity in comparison to the substrate. Titanium has a resistivity 4.2 x 10-l Qm. Now in typical chlorine cells the electrolyte resistance is much greater than the Faradic impedance and therefore the pure resistive model may be used. For this example we take an interelectrode resistance (hpJ of 6.8 x 10m4 (Qm2)[14]. Equation (33) value for the pure resistive case: gives the LlallLmBx ULll,, = 0.834. As can be seen from the calculated L,0&,X values the chlorine cell has a very even cd. This arises from the high electrolyte resistance in comparison to the electrode resistance. The cd distribution is worse for cells with small inter-electrode gaps and low electrolyte resistances (as for the “Swissroll” cell). 5. ROLE OF F, IN ELECI’ROCHEMICAL CELL DESIGN The results obtained may be readily used for calculating the minimum electrode thickness for essentially uniform potential or current density. Another aspect is the calculation of the electrode area for a particular production rate and Iror/lm~x~ 1. In this case a larger surface area is required than for the ideal case with L,,lLln,X= 1. The effective surface area A, may be
written:
where A is the actual surface area and 1the electrode breadth perpendicular to the direction of current flow. * Similar conclusions are also reached from a consideration of the hydrodynamic properties of the ceil.
(59)
where a, is the cost of electrode material per unit volume (g/m”) and K is the corresponding electrode cost. We now define an electrode unit cost to be the cost of the electrode per unit effective surface area: KU = KIA, = a&L,&,,,) Substituting expression (52) for IJ,,, K
=
”
(60) we obtain:
a,WwW’* tanh (L(&/2d)“*)
(61)
This function has the value zero at zero L and d and shows that the electrochemical and electrical properties of the system on their own give little help in designing the cell. A minimum cost is obtained with the unrealistic value 0 for &&rode thickness d and electrode length *L and a very non-uniform cd distribution. The optimum electrode dimensions that come from the cd distribution calculations or indeed from hydrodynamics are not practical propositions. The calculation leads to these conclusions because certain important factors have been omitted. These are the constructional costs which oppose the tend to zero values of d and L. One such construction cost concerns the working down of the electrode sheets to the required thickness. The cost of thick metal sheets is more or less proportional to the weight, however, as the thickness of the sheet is reduced then the amount of metal working and the associated cost increase considerably per unit weight.
The costs of some nickel
sheets are
given in Table 3. Another rest&ion on the design is the availability of materials. In the case of nickel sheet the minimum thickness normally available is 0.1 mm. A second assumption that was made was that the whole metal area could be used as electrode. In practise this is hard to achieve. There is usually ‘a certain area that cannot be used as active electrode, ey that
k
F’ETERM. ROFX~R-~SON
418
Table 3. Costs of nickel sheets for use as electrodes
Thickness (mm)
cost (SwFrjkg)
0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2.0
44.75 41.60 40.80 40.00 39.55 38.50 38.50 37.00 36.25 35.40
area to which the current feeders make contact. We shall illustrate these features by the development of a design for the “Swiss-roll” cell. A laboratory sized “Swiss-roll” cell has previously been described [ll]. A feature of the design was the use of the cell axis and container as current feeders to the ends of the electrodes. From a consideration of the cd distributionas presented here it is preferable to minimize the path length over which the current flows. This becomes particularly important when scale-up is considered. With the “Swiss-roll” cell it is convenient to scale-up the radius. Rather than using the axis and container as current feeder a very much shorter path length for the flaw of electricity is provided when the current feeders are attached to the ends of the roll. To enable the connection of an electrode to a current feeder there must be a certain area of each electrode unobstructed by the other. This is best achieved by displacement of the electrodes with respect to each other as shown in Fig. 11. The unit cost must therefore be rewritten:
dfmm
I
0
20
40 L,
60 cm
Fig. 12. Unit electrode costs for different electrode dimcnsions (d&/d& = 500R 1 m 2.
U”d(1 + L,/L) (62) Ku = K,,/&l.X 1 where 15, is the length of metal not used as electrode because of the connection to the current kxders. The unit cost has been calculated as before for the descrete sheet thicknesses given in Table 3 with their corresponding costs and L, = 0.1 m. The results are shown in Fig. 12 as a function of electrode length L. The optimum electrode thickness is once again the thinnest available (0.1 mm). The effect of the unused electrode strip of breadth L, is to give an optimum electrode breadth dependent on the value of kZ. For low current densities the cost curve about the optimum
is very flat, however, at large cd’s when I,~,/l_x deviates considerably from unity the cost is more sensitive to L and the optimum electrode breadth decreases with increasing cd. This effect is shown in Fig. 13 for several c&s. Contour lines for I,,&_.= 0.9, 0.8 and 0.7 are also plotted in the figure. They show that at the optimum I,,,,/Z,,, is close to unity. The effect of increasing L, is to increase the optimum electrode length L and to decrease 1,,,/Z,, at the optimum. The derivations presented here represent only a small part of the complete cell design problem. The cost represented by (62) is of course not complete, missing are terms such as pumping cost electrolysis current cost and even independent variables eg current density. The simplified equation was justified, however, in that it was intended to show only the effect of cd distribution. Inclusion of the extra cost factors may well shifk considerably the position of the optimum for the variables L and d. In concluding this section attention is drawn to some restrictions in applying the simplified treatment presented here. The concentration change of the eleo troactive species has been assumed to be negligible. This may not always be the case. A second problem concerns the reaction at the counter electrode. Often
Fig. 11. Cross section of the “Swiss-roll” cell with current feeders at the ends of the roll.
Fig. 13. Unit electrode costs at different (di,jdE), values. 1-500R~1m~Z; 2-5OOO~~‘m-‘: 3-1000051-‘m~Z.
419
Electrode resistance and its role in cell de-sign
a gas eg hydrogen is generated since this reaction can proceed at high cd and with catalytic electrodes at low overpotentials. Gases however change the apparent electrolyte resistance [15]. In systems with the electrolyte resistance dominating over the Faradic impedance the dependence of electrolyte resistance on gas content and its variation with x/L and i would have to be incorporated into the simple treatment presented here.
AcknowledgementsIt is a pleasure to acknowledge helpful discussions with Professor N. Ibl.
REFERENCES
1. C. W. Tobias and R. Wijsman, .I. electrochem. Sot. 100. 459 (1953). 2. C. M. Shepheid, J. electrochem. Sot. 120, 851 (1973). 3. C. M. Shepherd, J. electrocham. Sot. 112, 252 (1965). 4. W. W. H&y, J. electrochem. Sot. 109, 638 (1962). 5. J. Wojtowicz, L. Lalibertt and B. E. Conway, Electro6. CONCLUSIONS AND SUMMARY chim. Acta, 13, 361 (1968). 6. B. E. Conway, E. Gileadi and H. G. Oswin, Can. 1. The most uniform cd distribution in an electrolytic Chem. 41, 2447 (1963). cell is obtained when the current feeders are con7. S. K. Ranearaian. M. J. Dienam and B. E. Conwav. nected at opposite ends of the electrodes as shown Can. J. C&n,bs; 422 (19675 in Fig. l(b). In this configuration the electrode resist8. R. Brown, J. electrochea. Sot. 112, 1228 (1965). ances should ideally be equal. The actual cell current 9. F. H. Edeiman and R. W. Klopfenstein, J. electrochem. Sot. 112, 109 (1965). may be related to the ideal cell current with uniform 10. J. S. Newman and C. W. Tobias, J. electrochsm. Sac. that has been called the eleccd by a factor I,,/I,,, 109, 1183 (1962). trode utilisation factor. For the linear model it is a Il. P. M. Robertson, F. Schwager and N. Ibl, J. elecfunction of the ratio of electrode to electrolyte resisttroanal. Chem. 65, 883 (1975). ance. It has been derived for some simple cases, in12. N. Ibl and E. Adam. Chemie-lng-Tech. 37, 573 (1965). cluding an irreversible electrode reaction. For I,o,/lmar 13. P. Deiabay, New Instrumental Methods in Electraclose to the ideal ( = I) the problem may be simplichemistry, p. 222. Inlerscience, New York (1965). fied considerably by taking a linearized form of the 14. P. Gallone, Trattatodi lngegneria Efettrachimica, p. polarization characteristic. The importance of ItoJlrnax 358. Tamburini Editore, Milan (1970). 15. F. Hine, M. Yasuda, R. Nakamura and T. Noda, J. in cell design is to predict the cd distribution variation electrochem. Sot. 122, 1185 (1975) and correct the required electrode area.
PmmM. ROBWTSON Labor, ETHZ, Universitiitstrasse 6, 8006 Ziirich, Switzerland
(Received 1 June 1976; and In&vised fom
11 August 1976)
potential drop along parallel electrodes of:finite resistance has been calculated (a) assuming a pure resistive representation of the cell and (b) by considering the polarization characteristics of an irreversible electrode reaction. It was found that a much more uniform potential (and also
Abstract-The
current) distribution was obtained with the current feeders at opposite ends of the electrodes rather than at the same ends. With this configuration it is advantageous to have electrodes of equal resistance. Equations are given that enable the potential and current density distribution to be easily calculated. was derived which is a measure of the e.lCctive utilization of the electrode area. A factor I&,,, It may be used for calculating the required electrode area with non-uniform cd distribution.
NOMENCLATURE electrode
area (m’) effective electrode area = A (I,.,/!, ,) (mr) cost of electrode/unit volume (g/m’) concentration of electro-active species at electrode interface (mole/m3) thickness of electrode (m) thickness of anode (m) thickness of cathode (m) diffusion coefficient of electroactive species (m’/s) local potential of anode in the electrode phase (V) local potential of cathode in the electrode phase
(v) local anode potential (measured with reference electrode) (V) local cathode potential (measured with reference electrode) standard potential potential
(V) potential for anodic reaction (V) across current feeders (v) on anode at opposite end of electrode to current feeder (V)
characteristic potential for linearized treatment (V) electrode separation (m) local cd at anode (A/m*) local cd at cathode (A/m’) limiting cd (A/m*) minimum observed cd (A/m’) maximum observed cd (A/m’) local current flowing along anode (A) local current flowing along cathode (A) theoretical cell current = electrode area x min cd encountered (A) theoretical cell current = electrode area x max cd encountered (A) total ceil current (A) 2 pldh P< Cm- ‘1 (Rc - Ri) (diJdE),, (A/m’) (d(,/d& (Aim’) rate of electrode reaction (m/s) cost of electrodes/unit area (S/m’) cost of electrodes/unit effective area (S/m2) breadth of electrode (m)
L L,
n x Y P
P. PC P.
length of electrode (in direction of current flow) (m) length of electrode wasted because of current feeder connections (m)
number of electrons involved in electrode reao tion distance along electrode in direction of current flow (m) L/4 + pJ4)/p,h (m-*) resistivity resistivity resistivity resistivity
of of of of
electrode material (Szm) anode (&I) cathode (am) electrolyte (Qm)
1. INTRODUCTION
A major contribution to the cost of electrolytic alls is the electrode material. Sometimes inexpensive metals eg iron may be employed, but this occurs seldom and usually only for the cathode. In the case of the anode, corrosion resistance often dictates the use of noble metals eg platinum or platinised titanium. There are of course the less expensive graphite and magnetite that may be acceptable but their electrical conductivities are lower than for metals and could lead to significant potential drops in the electrodes themselves. This may even occur with metal electrodes if the cd is very high, or the thickness has been reduced on account of cost or because of design requirements. The problem of potential drop in the electrodes is a well known one, which in addition to wasting power may also give non-uniform current distribution resulting in lower production capacity per unit electrode area. When designing an electrolytic cell the resistance and therefore dimensions of the electrodes must be chosen carefully so as to minimize the cost. The effect of electrode resistance on the potential and current distribution has been discussed for several two dimensional electodes [l-9]. Tobias and Wijsman calculated the distributions at parallel plate electrodes with the current feeders to the electrodes at one end (see Figure la) [l]. This geometry is
4f2
PETERM. ROBERTSON E’
Figs l(a), l(b). Current feeder configurations. encountered in tank electrolysers and also in accumulators, which have been studied by Shepherd [2,3]. Other electrode shapes that have been treated are wires or cylindrical forms [4-Y] and rectangular ducts [S]. Arrangements in which the power connections were made to opposing ends of the electrodes (Figure lb), or cells having unequal anode and cathode resistances have not* been discussed in great detail. A particular problem encountered in this lahoratory was the use of very thin metal foils or sheets as electrodes. In the design of cells with thin electrodes the cd distribution had to be known so as to estimate the required electrode surface area. Thin metal foils are used in the “Swiss-roll” cell that we are investigating [ll]. The main feature of this cell is a sandwich consisting of thin anode and cathode sheets with separator cloths in between to prevent electrical short-circuits. The total thickness of this sandwich can be well under 1 mm. Typical dimensions are 300 x 15 x 0.04cm. A three dimensional structure is obtained by rolling the sandwich up around an axis. The power connections to the electrodes in this small laboratory sized cell were made by utilizing the axis of the roll and container as current feeders. It is easy to appreciate that with the large electrode area obtained (9000cm2), large currents may be encountered. This fact together with the electrode length and thickness may easily result in substantial potential drops in the electrodes if care is not taken. In this publication the optimum current feeder arrangement and relationship between the anode and cathode resistances will be derived. The potential and current distributions for the optimum arrangement will then be examined as a design aid for the determination of the optimum electrode dimensions. 2.OPTIMUM
CURRENT
Fig. 2. Simplified model of cell. cathode current feeder (Volt), p,, and pc are the resistivities of the anode and cathode (Qn), pe is the resistivity of the electrolyte (Qm) and I, and I, are the local currents flowing in the anode and cathode (A). The electrolysis current for a small length dx of electrode is dl and the corresponding cd i = (dl/dx)/l. We apply first Ohm’s law to the electrode phases.
Differentiating we obtain the change of local current density. d2En-L._! dl dx=
* An exception are three dimensional electrodes eg porous, that can be represented by similar equivalent electrical circuits [IO].
d2E C=
a,
PC
dx’i;i;
dx2
(2&b)
- dl, dl, dx=dx
(3)
This is also the current flowing between the electrodes, which may be written: dl 2 = 6% - E,)Uhihpe
(4)
Combining (2)-(4) and differentiating we obtain:
The solution of which is: E, = a exp(y’%) $ b exp(-y”%)
+ EX + e
(6)
+ cx +P
(7)
with
(a) Cell model The electrochemical cell consists of two parallel plate electrodes of breadth I (m), length L (m) and separation h (m), and the anode and cathode have thicknesses d, and d,, respectively. It is assumed in the derivations that follow that the current flows in a straight path through the electrolyte perpendicular to the electrodes and with negligible end effects. We consider first that the cell may be represented by pure resistance, as is the case when the electrolyte resistance is very much greater than the Faradaic impedance or a linear approximation of the polarisation characteristic is taken. E, and E, are the local potentials in the metal phase of the anode and cathode us the
md
Now an increase of current in the anode is equivalent to a decrease of current in the cathode, thus:
FEEDER
CONFtCURATlON
P
dx Id,
y = ~/&+pJ&) and from (2a), (4) and (6): E, = - ‘.d.aexp(g*‘*x) pad, - $hexp(-y’%) Dc
The boundary conditions must now be introduced in order to determine the coefficients ~1,b, c and e. We treat both cases as depicted in Figs l(a) and l(b). Case (i). Current feeders at the same ends of the electrodes. With both current feeders at x = -L/2 we obtain the following boundary conditions: at _
x=-L/2
J&=0
and
E,=E*
(8)
Electrode resistaace and its role in cell design n(eY”‘L _ e- Y”‘L)Rly”2
(E* is the cell voltage) X = G/2
and at
C=
I, = I, = 0
(9) since no current flows out of the electrodes at x = L/2. Introducing these boundary conditions into ‘(6) and (7) the coefficients are found to be:
0, * = d,p, + d,p,
(12)
e=*_.g* t&p, + d,p,
(13)
Case (if). Current feeders at opposite ends of annde and cathode. In this case the cathode feeder is at x = -L/2 and the anode feeder at x = + L/2. The boundary conditions are therefore:
x = L/2
E, = 0 E, = E*
(14) (15)
Again because no current flows out of or into the ends of the electrodes without the current feeder we have : at
x=
-L/2
I, = 0
(16)
and at
x = L/2
I, = 0
(17)
Introduction of these conditions into (6) and (7) gives the following expressions for the coefficients a, b, c and e: a =
E*RI(R, ey”* + R2e-Y”‘LIZ) (R: + R;)(&“L + e-Y’ilL)
+
RIR2y1’2L(eY”‘L- e-Y”zL) + 4R,R2
e-$J’L)z)’
R2
(20)
e-
y”‘L
1
+ R,RZy’/2L(er”‘L - e_y”2L)/2 f 2R1Rz
d=
with
c=o
x = -L/2
+ +
1 RL(Rl&'rL/Z +&e-V"'L/Z) (21)
pL
(10) (11)
and at
(Rley”‘L/2
E* exp(y”2L/2) 1 + exp(2y”‘L)
b = (I exp(y”2L)
at
413
Rt = p,ld,
and
Rz = p,ld,
(b) Results and discussion The above equations for the potential distribution may be written in terms of the dimensionless parameters: WE*), (WE*), (d,&W, (2nald,W1’2L and (x/L). The pd’s (En - l&)/E* for both cases are shown in Figs 3 and 4 as a function of (x/L) and (d,pJd,p.) at a constant (2pJd,hp,)‘?L value of 1. As ( clearly demonstrated in Fig. 3 the maximum potential difference for Case (i) is obtained at the end of the electrodes connected to the current feeder ie at x/L = -0.5 and decreases with increasing x/L. In contrast when the current feeders are connected to opposite ends of the electrodes, ie Case (ii), there are regions of high potential difference at both ends of the electrodes with a minimum in between. The variation of potential difference is much less than for Case (i) which also implies more uniform current distribution. In Fig. 3 it may also be observed that in the case of the current feed at the same ends of the electrodes a more uniform potential (and therefore current distribution) is obtained at low values of d, p,/ d,p, for a fixed value of (2pJd,hp,)‘@L ie the electrode resistance pJd, should be low. This also applies of course to the other electrode resistance p,jd,. The lower the electrode resistance, the more uniform the potential and current distributions are. For the case in which the current feeders are at opposite ends of the electrodes (Fig. 4) the most uniform distribution is obtained with log(d.p,/d,p,) = 0 ie with pJ d, = PC/d,.The uniformity of distribution under these circumstances is very much better than that achieved for Case (i) and makes possible the use at very much higher cds. This configuration appears lo be the optimal one and will therefore be discussed exclusively in the development that follows. For Case (ii) with p,/d, = pdd, = p/d the solution simplifies to:
(18) 1
E, = 2a cash (k1’2x)+ cx + e
(22)
Ez = -2acosh(k%)
(23)
and + cx + r
with
Fig. 3. Potential differeace for current feeders at same ends.
E* ’ = 2Lk”* smh(k”2L/2) + 4 cosh(k”2L/2)
(24)
e = E*/2
(26)
k = 2p/dhp,
(27)
and
,414
PEm M. ROBERTSON
Fig. 4. Potential difference for current feeders at opposite ends.
I
I
2
I
I
3
k’=L/Z
3. EFFECTIVE UTILIZATION OF ELECTRODE AREA
Fig. 5. The electrode utilization 1,,,/1,,, and cd variation imiJi_ factors.
Electrochemical processes usually have an optimum current density. This current density may simply be a limiting cd, it may arise from a consideration of electrode investment and power costs [12] or it may be a maximum allowable cd at which no undesirable side reaction takes place. For whatever reason the optimum cd arises the electrode area will be lCO% effectively used only when the cd over the whole area is constant and equal to the optimum value. Because of the finite resistance of the electrodes the cd is not uniform and the electrodes will not be utilized to the full. It is useful to define the factor I,a/l,,,~Xwhich gives a measure of the effective utilization of the electrodes (electrode utilization factor). I,,, is the total current fed to the electrodes and I,, is the total current that would flow if the cd were uniform and equal to the maximum value observed, ie that at the ends of the electrodes. Now the current density i. is given by:
The function (33) is plotted in Fig. 5. In a similar fashion the cd variation may be expressed by the factor : bJim.x
=
(cU,/dx),,o/(dr,/dx),=,,,
= sech (k’/‘L/2)
(34)
This function is also plotted in Fig. 5. For large values of the argument k”‘L/2, I,,/l,., = 2/k”‘L. 4 EXTENSIONS OF THE SIMPLE MODEL FOR ELEmOCHEMlCAL CHARACTERISTICS
(a) Reaction nt the limiting current density We now assume that the local cd is always equal to the limiting cd. The current flowing along the electrodes is therefore linearly dependent on the distance from the current feeder. Thus 1, = i&L/2 + xbl
(35)
I, = i&L/2 - x).1
(36)
From (4), (22) and (23) we obtain: i, = 4n cash (k”‘x)/hp,.
(29)
Substituting in (laj and (lb), taking
Therefore LIZ I,,, = 1
I -L/Z
P. PC -c-zda d,
i,dx = 8al sinh (k’i2L/2)/hp,k1i2 (30)
Now the maximum cd is observed at x = L/2 or -L/2, thus i,,,., = 4a cash (k”‘L/Z)/hp,
d
and integrating we obtain: E, = pi&Lx/2 + x2/2)/d + c,
(37)
E, = pi,(Lx/2 - x2/2)/d + c2
(38)
and
(31)
and
P
from which I InaX= 4uIL cash (k”‘L/2)/hp,
(32)
Therefore I**LX
= ,$& t anh (!&‘L/2)
(33)
E. - E, = p&x2/d + (cl - c2) (39) where i, is the limiting cd and c, and c2 are integration constants that are dependent on the standard potentials of the system. The validity of (39) is restricted to systems that have a limiting current plateau
Electrode resistance and its role in cell design
that is a least as broad as the potential variation pi&‘/4d. The factor &,/I,,, is of course equal to 1 since the cd is always i,.
this section the complete model for the system,
including reactions below and/or at the limiting cd will be discussed. Since most proeessa of industrial interest are operated irreversibly eg deposition of a metal ion or oxidation of an alcohol to an acid, we shall consider only the irreversible case and in particular an anodic oxidation*. The description of the electrode reaction is that used by Delahay [13] for irreversible electrode reactions in stirred solutions. The current density i, is given by: i, = nFk,Cexp((l-a)(E:
- EO)nF/RTJ
C = 6 (i, - iJttFD
(41)
This model considers therefore a concentration change from the solution bulk up to the electrode surface through the diffusion layer, ie a change of concentration perpendicular to the direction of electrolyte flow through the cell. Substituting for C in (40) we obtain: (D/k,d)exp{
-(l
- ol)(Eb- E”)nF/RT}
>
Substituting (49) into (2a) and solving as before gives: E, = 2a cash (2/1ka/d)“~n + cx + e
Now the potential difference E, - E, may be expressed in terms of the electrode potentials EL and E’<(measured with a reference electrode in the solution) and the iR drop term for the current flow through the electrolyte. E, - E, = E:, - E’=+ iR
(43)
For the sake of simplicity and to minimize the number of independent variables it will be assumed that the concentration of the depolarizer for. the cathodic reaction is so high that the cathode electrode potential EL is virtually constant. It will also be assumed that the iR drop is negligible. Substituting for Eb in (42) gives: _~
‘9
1 + (D/kJ)exp{ -(l
EC = - 2acosh(2pk#)%
+ cx + e f k,/kl
E* + kl/k, a =
2_Q2pk,/d)liz sinh [(pkJ2d)“‘L]
E* + kJk2 ’ = L + (2d/pk#2
coth [(pkz/2d)“2L]
(50)
and e = (E* - k,/k,)/2
(51)
The electrode utilization factor for this case is similar to that obtained in Section 3 :
LtiLax= CP4& )“*/~ItanhCW&4”*~1W) In a more empirical way the range of applicability of this linearization may be extended. If only a small variation of cd is tolerated then a short section of the polarization curve may be regarded as a straight line of slope (di/dE),,,. This line ab is shown in Fig 6. It is the tangent to the polarization curve and crosses the abscissa at some potential &,. The cd may therefore be written as (53)
Again we may express as (46) with k, = (EC - E,J (diJd&, kl = (d&/da,. The solutions (47), (48) and (52) are therefore valid with these substitutions. The calculation of I,&,,, is now relatively simple: we must only measure the slope of the polarization curve at the average cd to be encountered. The accuracy of the approximate solutions discussed up to this point may be determined if the exact solutions are also known. (ii) Exact numerical solution. The potential and current density distribution is now solved for the full unapproximated polarization behavior with equation (44). This incorporates concentration polarization, which has not been considered in previous treatments [lo] but assumes constant concentration of the eledroactive species in the solution bulk.
(4)
- E, + E: - E”)nF/RT}. (45)
* Although the derivation here considers an ancdic process. the results are equally applicable to cathodic ones.
(49)
+ 4 cash [(pk&f)“*L]
- a)@, -EC + El -E”)nF/RT}
+ (1 - G%
(48)
with
(i) Approximate solution by Rnearization. Assume that D/k, S s 1 and that the exponent is small. We may therefore write: i, = +{I
(47)
and
i, = (EL - E,&(di/dE), (42)
i. = ~
.
i. = kr + k,(E, - E,) = f ‘2
(4)
where a is the charge transfer coefficient, k, the rate constant of the electrode reaction at the standard potential ED and C the interfacial concentration of the electro-active species. The interfacial concentration is then expressed in terms of the limiting cd for a particular Nemst diffusion layer thickness 6, and diffusion coefficient D:
ia=1 +
This equation is of the form:
(
(b) Reuction below the limiting current density In
415
Fig. 6. Linearization
of the p&rimtion curve.
416
Pmm M. Roam-
Table 1. Dimensionless groups used in calculations
computer
x, = E,nF/RT x1 = E,nF/RT off 3 E*nF/RT n, = i/i, lij = x/L
16 = II, = cg = ng = x,~ =
D/k,b a i,p L’nF/RT d (2p/dhp,)“‘L (E: - EO)nF/RT
In order to minimize the number of variables it is preferable to introduce dimensionless parameters. Those chosen are listed in Table 1. Equations (2a), (2b), (3) and (44) may be written as a system of 4 first-order differential equations with the dimensionless parameters of Table 1:
P ._
o-
(54)
-025
0
025
0.5
X/L
dn,z
As = 1+
__ dn,
-0.5
Fig. 8. Computer derived normaiized cd distribution (ilie) for integer values of ztg from 1 to 10.
x,exp[(~, - 1)(x, - E2 + nro)]
-dnz= ni3 dx, h, dntz -=-dn, dr,
(57)
These equations have been solved in terms of an initial value problem wing the standard Runge, Kutta-Gill method. The initial values are given by (14) and (16). The other two conditions (15) and (17) are of course boundary values. Boundary condition (15) presents no problems since it is an independent variable and our interest is in tabulating the solution for several values of E*. This tabulation is more conveniently performed by replacing (15) by an initial condition (E,), = _L,2 = E**. Tbe other boundary condition is more difficult to treat. An iterative technique must be used to find a suitable initial value of i, (at x = - L/Z) such that the required boundary condition (17) is fulfilled. Some cd distributions
obtained numerically are plotted in Figs 7-10. The numerical solutions for I,,JI,,X are tabulated in Table 2 for various values of na, x6 and ns. Also tabulated are the ltO,/i_ values obtained by applying the linearization treatment of Section 4b (i) with the slope of the polarization curve (diJdE)& at the average cell cd I,,&. As can be seen the agreement between the linearized and exact treatments is good at values of LJLX close to 1. At I,,,/I_ i 0.8 the agreement is less than ideal but this should not detract from the value of the approximation since smaller I,,,/&,,,, values have in any case less interest. (iii) Numerical examples. As an illustration I,JIIILy will be calculated for a typical ‘Swiss-roll” cell and also a diaphragm chlorine cell. We consider first the “Swiss-roll” cell. The electrodes are of nickel of resistivity 6.84 x lo-*am. The dimensions of each electrode are 3.0 x 0.15 x O.C001m. The slope of the polarization curve is now required. We shall take a fairly typical value of 5CQA/mzV (=(diJdE),) as in for example the oxidation of ahphatic alcohols to
lb0
IO
LOO,
075
0.75 I
t P
0.50
050
1 f
5:’
t 0.25
-x
Fig. 7. Computer derived nmmak&d,@ f&rihion for integer valued of Itj froLn -10.
,i.C” . ,.
(i/i.)
Fig 9. Computer derived normalized cd distribution (i/i,) for integer values of 2rj from l-10.
Electrode resistance and its role in cell design
417
Table 2. Comparison of approximate and exact numerical solution of cd distribution
X/L
Fig. 10. Computer derivet$ normalized cd distribution for integer values of zj from t-10. acids at nickel anodes
in alkaline
solutions
(i/i,,)
[ll].
Thus [p(di,JdE),,J2dj1’ZL = 1.24
1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00 1.00 3.00 5.00 10.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 IO.00 lO.ctl 10.00 10.00 10.00 10.00 10.00 10.00
1.00 l.ctl 1.00 1.00 10.00 10.00 10.00 10.00 1.00 1.00 1.00 1.00 10.00 lO.cNl 10.00 lO.Otl
0.980
0.962 0.966 0.991 0.836 0.680 0.628 0.614 0.991 0.95 1 0.908 0.934 0.914 0.699 0.549 0.480
0.981 0.960 0.962 0.987 0.833 0.644 0.539 0.404 0.990 0.95 1 0.907 0.918 0.916 0.719 0.570 0.396
We now make the assumption that the electrode cost is proportional lo the volume of metal. Thus K = a&J = a,Ad
and from (52) I,, &_=
0.68.
The second example is a diaphragm cell with current feeders also as depicted in Fig. l(b). We take electrodes of RuOI covered titanium (1 m x 1 m) of thickness 2 x 10e3 m and assume that the RuOa coating has negligible conductivity in comparison to the substrate. Titanium has a resistivity 4.2 x 10-l Qm. Now in typical chlorine cells the electrolyte resistance is much greater than the Faradic impedance and therefore the pure resistive model may be used. For this example we take an interelectrode resistance (hpJ of 6.8 x 10m4 (Qm2)[14]. Equation (33) value for the pure resistive case: gives the LlallLmBx ULll,, = 0.834. As can be seen from the calculated L,0&,X values the chlorine cell has a very even cd. This arises from the high electrolyte resistance in comparison to the electrode resistance. The cd distribution is worse for cells with small inter-electrode gaps and low electrolyte resistances (as for the “Swissroll” cell). 5. ROLE OF F, IN ELECI’ROCHEMICAL CELL DESIGN The results obtained may be readily used for calculating the minimum electrode thickness for essentially uniform potential or current density. Another aspect is the calculation of the electrode area for a particular production rate and Iror/lm~x~ 1. In this case a larger surface area is required than for the ideal case with L,,lLln,X= 1. The effective surface area A, may be
written:
where A is the actual surface area and 1the electrode breadth perpendicular to the direction of current flow. * Similar conclusions are also reached from a consideration of the hydrodynamic properties of the ceil.
(59)
where a, is the cost of electrode material per unit volume (g/m”) and K is the corresponding electrode cost. We now define an electrode unit cost to be the cost of the electrode per unit effective surface area: KU = KIA, = a&L,&,,,) Substituting expression (52) for IJ,,, K
=
”
(60) we obtain:
a,WwW’* tanh (L(&/2d)“*)
(61)
This function has the value zero at zero L and d and shows that the electrochemical and electrical properties of the system on their own give little help in designing the cell. A minimum cost is obtained with the unrealistic value 0 for &&rode thickness d and electrode length *L and a very non-uniform cd distribution. The optimum electrode dimensions that come from the cd distribution calculations or indeed from hydrodynamics are not practical propositions. The calculation leads to these conclusions because certain important factors have been omitted. These are the constructional costs which oppose the tend to zero values of d and L. One such construction cost concerns the working down of the electrode sheets to the required thickness. The cost of thick metal sheets is more or less proportional to the weight, however, as the thickness of the sheet is reduced then the amount of metal working and the associated cost increase considerably per unit weight.
The costs of some nickel
sheets are
given in Table 3. Another rest&ion on the design is the availability of materials. In the case of nickel sheet the minimum thickness normally available is 0.1 mm. A second assumption that was made was that the whole metal area could be used as electrode. In practise this is hard to achieve. There is usually ‘a certain area that cannot be used as active electrode, ey that
k
F’ETERM. ROFX~R-~SON
418
Table 3. Costs of nickel sheets for use as electrodes
Thickness (mm)
cost (SwFrjkg)
0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2.0
44.75 41.60 40.80 40.00 39.55 38.50 38.50 37.00 36.25 35.40
area to which the current feeders make contact. We shall illustrate these features by the development of a design for the “Swiss-roll” cell. A laboratory sized “Swiss-roll” cell has previously been described [ll]. A feature of the design was the use of the cell axis and container as current feeders to the ends of the electrodes. From a consideration of the cd distributionas presented here it is preferable to minimize the path length over which the current flows. This becomes particularly important when scale-up is considered. With the “Swiss-roll” cell it is convenient to scale-up the radius. Rather than using the axis and container as current feeder a very much shorter path length for the flaw of electricity is provided when the current feeders are attached to the ends of the roll. To enable the connection of an electrode to a current feeder there must be a certain area of each electrode unobstructed by the other. This is best achieved by displacement of the electrodes with respect to each other as shown in Fig. 11. The unit cost must therefore be rewritten:
dfmm
I
0
20
40 L,
60 cm
Fig. 12. Unit electrode costs for different electrode dimcnsions (d&/d& = 500R 1 m 2.
U”d(1 + L,/L) (62) Ku = K,,/&l.X 1 where 15, is the length of metal not used as electrode because of the connection to the current kxders. The unit cost has been calculated as before for the descrete sheet thicknesses given in Table 3 with their corresponding costs and L, = 0.1 m. The results are shown in Fig. 12 as a function of electrode length L. The optimum electrode thickness is once again the thinnest available (0.1 mm). The effect of the unused electrode strip of breadth L, is to give an optimum electrode breadth dependent on the value of kZ. For low current densities the cost curve about the optimum
is very flat, however, at large cd’s when I,~,/l_x deviates considerably from unity the cost is more sensitive to L and the optimum electrode breadth decreases with increasing cd. This effect is shown in Fig. 13 for several c&s. Contour lines for I,,&_.= 0.9, 0.8 and 0.7 are also plotted in the figure. They show that at the optimum I,,,,/Z,,, is close to unity. The effect of increasing L, is to increase the optimum electrode length L and to decrease 1,,,/Z,, at the optimum. The derivations presented here represent only a small part of the complete cell design problem. The cost represented by (62) is of course not complete, missing are terms such as pumping cost electrolysis current cost and even independent variables eg current density. The simplified equation was justified, however, in that it was intended to show only the effect of cd distribution. Inclusion of the extra cost factors may well shifk considerably the position of the optimum for the variables L and d. In concluding this section attention is drawn to some restrictions in applying the simplified treatment presented here. The concentration change of the eleo troactive species has been assumed to be negligible. This may not always be the case. A second problem concerns the reaction at the counter electrode. Often
Fig. 11. Cross section of the “Swiss-roll” cell with current feeders at the ends of the roll.
Fig. 13. Unit electrode costs at different (di,jdE), values. 1-500R~1m~Z; 2-5OOO~~‘m-‘: 3-1000051-‘m~Z.
419
Electrode resistance and its role in cell de-sign
a gas eg hydrogen is generated since this reaction can proceed at high cd and with catalytic electrodes at low overpotentials. Gases however change the apparent electrolyte resistance [15]. In systems with the electrolyte resistance dominating over the Faradic impedance the dependence of electrolyte resistance on gas content and its variation with x/L and i would have to be incorporated into the simple treatment presented here.
AcknowledgementsIt is a pleasure to acknowledge helpful discussions with Professor N. Ibl.
REFERENCES
1. C. W. Tobias and R. Wijsman, .I. electrochem. Sot. 100. 459 (1953). 2. C. M. Shepheid, J. electrochem. Sot. 120, 851 (1973). 3. C. M. Shepherd, J. electrocham. Sot. 112, 252 (1965). 4. W. W. H&y, J. electrochem. Sot. 109, 638 (1962). 5. J. Wojtowicz, L. Lalibertt and B. E. Conway, Electro6. CONCLUSIONS AND SUMMARY chim. Acta, 13, 361 (1968). 6. B. E. Conway, E. Gileadi and H. G. Oswin, Can. 1. The most uniform cd distribution in an electrolytic Chem. 41, 2447 (1963). cell is obtained when the current feeders are con7. S. K. Ranearaian. M. J. Dienam and B. E. Conwav. nected at opposite ends of the electrodes as shown Can. J. C&n,bs; 422 (19675 in Fig. l(b). In this configuration the electrode resist8. R. Brown, J. electrochea. Sot. 112, 1228 (1965). ances should ideally be equal. The actual cell current 9. F. H. Edeiman and R. W. Klopfenstein, J. electrochem. Sot. 112, 109 (1965). may be related to the ideal cell current with uniform 10. J. S. Newman and C. W. Tobias, J. electrochsm. Sac. that has been called the eleccd by a factor I,,/I,,, 109, 1183 (1962). trode utilisation factor. For the linear model it is a Il. P. M. Robertson, F. Schwager and N. Ibl, J. elecfunction of the ratio of electrode to electrolyte resisttroanal. Chem. 65, 883 (1975). ance. It has been derived for some simple cases, in12. N. Ibl and E. Adam. Chemie-lng-Tech. 37, 573 (1965). cluding an irreversible electrode reaction. For I,o,/lmar 13. P. Deiabay, New Instrumental Methods in Electraclose to the ideal ( = I) the problem may be simplichemistry, p. 222. Inlerscience, New York (1965). fied considerably by taking a linearized form of the 14. P. Gallone, Trattatodi lngegneria Efettrachimica, p. polarization characteristic. The importance of ItoJlrnax 358. Tamburini Editore, Milan (1970). 15. F. Hine, M. Yasuda, R. Nakamura and T. Noda, J. in cell design is to predict the cd distribution variation electrochem. Sot. 122, 1185 (1975) and correct the required electrode area.