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Influence of the relative electric conductivity of the two phases on the pontential distribution in flow-though porous electrodes under limiting current conditions
Elecrrmhbnica Acta, Vol. 23.pp 1197-1203. 0 PergamonPms Ltd. 1978.Pnnti m Great dritain.
@X9-WM/V8/1101-1197 SOZOO,CI
INFLUENCE OF THE RELATIVE ELECTRIC CONDUCTIVITY OF THE TWO PHASES ON THE POTENTIAL DISTRIBUTION IN FLOW-THROUGH POROUS ELECTRODES UNDER LIMITING CURRENT CONDITIONS A. GAUNANDand F. COEURET* Laboratoire des Sciences du G&e Chimique, C.N.R.S.-E.N.S.I.C. 1, rue Grandville, 54042Nancy, France (Received 18 October 1977; and infimlform 10 March 1978) Abshaet -The paper examines the influence of the relative electric conductivity of the solution and matrix phases on the potential distribution in flow-through porous electrodes working under limiting current conditions. The solution giving the potential distribution is deduced and is experimentallytested for packed bed electrodes of graphite microspheres using the reduction of ferricyanide ions as the electrochemical reaction. A satisfactory agreement between theory and experiment is observed.
INTRODUCTION
Potential distributions within flow-through porous electrodes operating under limiting current conditions have been examined for the particular case of a highly conductive dispersed phase (or matrix) the potential of which being considered as constant[l-51. This interesting case corresponds to fixed bed electrodes tiving a metallic matrix as in the works studying the influence of flow conditions on the limiting current[6-121. However the electrical conductivity of the matrix may not always be infinitely high and the influenceofitstinitevaluehas tobe takenintoaccount, as for example : (a) For fixed bed electrodes of graphite granules or carbon grains which industrially might be prefered to metallic grains - in order to restrict the investment costs - for electrochemical operations without phase change. In such systems indeed the contact resistances between the grains limit the electrical conductivity of the mat& Some experimental works of Sioda[l3,14] used such poorly conducting beds but without considering the matrix conductivity. (b) For fluidized bed electrodes of highly conductive particles and operated at very low expansion ratios ie in hydrodynamic conditions such that tbe interparticle conduction exists but not so highly as for the fixed bed state. As reviewed by Newman[lS] the influence of the electrical conductivities of the dispersed and continuous phase8 respectively has been studied theoretically for porous electrodes working either under Tafel kinetics or under the linear electrode kinetics. Tbeaim was to show how theseconductivities influence the uniformity of the reaction distribution. The present paper concerns Row-through porous
electrodes working under limiting current conditions. For the case where electrolyte and current flow in the same direction, it examines how the above two conductivities influence the potential distribution within the electrodes. THEORElK4L ELECTRODEPOTENTIAL DISI’RiBUTION
One considers the cathodic fixed bed electrode of Fig. 1 for which the electrolyte flows downwards at the uniform superficial velocity u. The bed is supposed one-dimensional; its porosity, considered as uniform, is E and its height L. It is made of a granular material the specific area of which is a&, = 6/d,, for spherical grains of diameter d,) and having the electrical conductivity u,,; the specific area of the bed itself is A, = a,. (1- E).The column cross-sectional area is R; the electrical conductivity of the electrolyte is y. and its concentration in reacting ions at the bed entrance is c,,. As in[S] the anode is situated upstream the bed.
* Present Address: Laboratoire Etudes Adrodynamiques et Thermiques, 40, avenue du Recteur Pineau46022, Poitiers, France. Correspondence concerning this paper should be addressed to F. Coeuret. 1197
Fig. 1. Schematic view of the fixed bed electrode.
A. GAUNAND AND F. Coamurr
1198
The local electrode potential R(X) at the reduced height X = x/L measured from the bottom of the bed is the difference :
between the local matrix potential 4,(X) and the local solution potential q%s(X).As proposed in the literature for porous electrodes[M], the potential distribution can be deduced assuming that both continuous and dispersed phases are continuous media with the electrical conductivities y and B respectively : y corresponds to the electrolyte within the pores of the bed ; it depends on y0 and on the tortuasity of the current lines in the electrolyte around the granular material. -u characterizes the electrical conductivity of the matrix and depends on u0 and on the intergranular contact resistances. By integrating the differential equation (A-7) deduced in the Appendix, one obtains for the electrode potential distribution :
t
L. I4
E(X) = E(O)+ -
Y
[(l-R)-X+Xln(l-R)-l]
(1) / where E(0) is the current feeder potential, R the overall fractional conversion of the reacting ions in the electrode and i, the overall cd relative to n. Expression (1) shows that the difference E(X)-S(0) not only depends on y/u and R, but also on I, y, Co and II owing to (A-4); on the other hand, as expressed below in (3), u and R arc interdependent through the mass transfer coefficient k, the value of which depends on the hydrodynamics ie on u. Thus one has to recognize that the theoretical analysis of (1) under its form would be difficult without choosing a priori values for I,, y, C,, u thus obviously restricting to such a choice the conclusions which could be deduced from the analysis. Considering the quantity :
Z(X) = & CW) - WI
(2)
one sees that it only depends on v/u and R. The multiplicative term y/Lli, 1in (2) represents physically the inverse of the overall potential drop which would be observed if the matrix of the bed was no conductive at all (electrochemical reaction on the current feeder alone). Through i, this term depends on the operating conditions. For the purpose of the following theoretical analysis we consider it as coustant, thus allowing the respective influence of v/u and R on Z(X) to be representative of their influence on the electrode potential distribution. In the discussion of the results, theoretical distributions calculated from (1) will be presented. 1. Influence of v/u Figure 2 shows this influence for R = 0.9. For a matrix much more conductive than the electrolyte (y/o + 0), Z is always negative and the more so as X is high;
-0.5
7
0
0.5
Fig. 2. Inlkace of y/o on 2 far R = 0.9.
this means through (2) that the electrode potential E(X) goes to more cathodic values as X increases from X = 0 to X = 1. When y/u deviates from zero, a maximum appears in the potential distribution and the potential at the top of tbe bed tends to be eq@ to the current feeder potential. For higher r/u values one sees that Z(X), ie the difference E(X)-E(O), remains positive for every X: the current feeder is the most cathodic part of the electrode. The existence of a maximum-more or less pronounced according to the value presented by y/a or R-merits some attention. Indeed it traduces the fact that parts of the bed would operate at cathodic potentials out of the range bounded by the current feeder potential E(0) and the potential E( 1) at the top of the bed; consequently this form of potential distribution for sufficiently high r/a values signifies that by only considering the range delimited by E(0) and E(1) it would not be possible to guarantee the operation of the whole electrode at the limiting diffusion current. The situation of the maximum potential appears as given by :
x ma= 2. lnqkence
In(1 -R)
ofR
Figure 3 puts in evidence the influence of R on the potential distribution for a value of y/a which corresponds to the presence of maxima in the distri-
The intlucnceof the relativeelectricconductivity butions. One sees clearly that for low R values, as X is increased E(X) is first less cathodic than E(0) and second morecathodic than E(0) as the top of the bed is approached. Furthermore with increasing R the potential at X = 1 becomes equal at E(0) and further increases of R are traduced by electrode potentials all less cathodic than the current feeder potential. Also it is evident that as R is increased from low values the position of the maximum deviation of E(X) to E(0) moves to the top of the bed. When R tends to 1, the distribution is simply given by : xli,l R(X) - E(0) = d
which signifies that the potential distribution obeys Ohm’s law applied to the dispersed phase for the overall electric current. Both examples of Figs 2 and 3 show that when the matrix is not highly conductive, three local potentials would have to be taken into account in the design of the electrode: E(O),E( 1) and E(X,,,& Three particular cases of (l), corresponding to y/a = 0; 1 and cc respectively could be examined but the most interesting seems to be the first one (y/u = 0) which has been considered separately in a previous paper[5] and the potential distribution of which appears here as a limiting case of (1). EXPERIMENTAL
1. Cell, electrolytes and electrodes The hydraulic circuit, the current feeders to the cell, the measuring probes, the electrical equipment and the operating mode are the same as in[S]. The electrolytes consist in solutions of about 0.001 M potassium ferricyanide in sodium hydroxide,
1199
the true ferricyanide concentration being known by amperometric titration with cobalt sulphate. Two sodium hydroxyde concentrations were used in order to have two values of the electrolyte electrical conductivity yO,as indicated below: 0.75 N NaOH
y0 = 0.13 Ohm-’ cm-’
0.4 N NaOH
y,, = 0.08 Ohm-’ cm-’
The porous electrodes were fixed beds of graphite microspheres of diameter d,, = 0.064cm and
0.0825cm respectively. The bed height was varied between 1 and 3 cm and the bed porosity was constant and equal to 0.42.The electrolyte superficial velocity u was varied between 0.01 and 0.08 cm/s. 2. Relative conductiuity v/a of the systems As in[4,5] i is calculable from y. and y by: Y 2.E Yo 3--E In what concerns u, the value of which highly depends on contact resistances between the graphite grains and consequently on the settling state of the bed, it was determined “in situ” by two methods after previously verifying in a separated dispositive the applicability of Ohm’s law to the electrical conduction in fixed beds of dry graphite grains. Method (a) applies Ohm’s law to electronic current flow in the fixed bed itself (Fig. 1).For that the solution filling the column is only sodium hydroxyde thus avoiding ionic conduction within the pores of the bed. This bed of graphite microspheres is surmounted of a few layers of gold plated microspheres in which an electrical conductor is immersed; they play the role of counter-electrode in such a realized conductivity cell. Method (b) is taken from the first steps of the theoretical calculation (Appendix, paragraph 3). From (A-5),near the grid supporting the bed, we have :
05
WV = -L!! C (-1dX X=0 which follows from local application of Ohm’s law. Thus the experimental potential distributions in the layers of the bed immediately neighbouring the grid may lead to crby means of the preceeding relation. The values such deduced for r/a are tabulated in Table 1. It is seen that both methods lead to values in satisfactory agreement although in one case method (b) had given values which appear as depending on L; probably these deviations are due to the difficulty to reproduce the settling state of the bed.
Table 1. Method (a) y. -
0.130hn-‘cm-’
I=4.4 0
Method (b) 1, 0
d, = 0.064cm -0 5
Fig, 3. Influence of R on 2 for y/a = 0.5.
yo=
O.oaOhm-‘cm-’
dp =
0.0825~111
4.6 for L = I cm
I
3.3 for L = 2cm
Y -=0.44
J!=os
0
#
1200
A. GAUNAND
AND
F. COBUREI
t 1,
IO
I
0.I
Rep Fig.4. Exp~&~~talvsriadensof%, withRet,forfixedbedsofgrsphitemicrospheres characterized by y/u = 0.5
and comparison with correlations for highly conductive microspheres.
electrodes of highly conductive grains that in the present work the graphite microspheres were acting as monopolar electrodes. The variations of R with u for various values of the bed thickness L are similar to thase p?~iQ~~ pubiislxd for electrodes of highly conductive grains[ll] and of graphite granules[l4]. More important in the R= 1 present paper is to examine if the mass transfer and, as inC11], the values of R calculated from coefficientsdeduced from (3) for fixed bed electrodes of experimental overah limiting currents I, (expression A- graphite microspheres are in agreement with those 5) allow the variations of the mass transfer coefficient presented in[li, 171 for gold plated microspherea. k, beben the~w~gl~~d and &hegrains of the fixed For that, Fig 4 reports the ex~rimental Sherwood bed to be known. The main interest of such an numbers Sh, = k,. d,/D us Reynolds numbers Re, exploitation of the overall data is to insure, by = u. $,/v for beds of a class of particles and a given comparison with results for flow-through fixed bed Schrmdt number (SC = 1950); also two empirical lU2%JLTS AND DISCUSSION
1. Fr~t~~~ c~~v~s~~s R Without axial dispersion in’ the electrode, R is expmssed as [see Ap~dix):
-exp{ _!?$}
(3)
Electrolyte
0.3
4
=6.40xlO%m Co-0,861xlO”M L=lcm y=i3xlO”otnn*! cm“ --fcbm(1) with % = 4.6
t
-I x h
f
Fig. 5. Experimentalpotential dist~bution~ showing the inttuenee of R and comparison with tbooq.
1201
The influence of the relative electric conductivity
-2
7
d, =6.40xlO~%m L*2cm %
s3.30 x
0.1
0.
0.2 E( X)-E(O)
thso.
[volt]
Fig. 6. Comparisonbetweentheoreticaland calculatedpotentialvalues for a givenfixedbed electrode. correlations established elsewhere[11,173 for highly conductive microspheres are reported in that figure. T’hegood agreement between the experimental points and both correlations allows to conclude that the flowthrough fmed bed electrodes of graphite microspheres were operated at the limiting diffusion current what prove the validity of the experiments reported in the pr=*t pap= 2. Potential distributions Figure 5, given as an example, presents experimental distributions of the differenceE(X)-E(O) within a fixed bed characterized by y/u = 4.6; the dotted curves are the corresponding theoretical distributions calculated from (1).It has to be noted that high values of R, ie low electrolyte supc&iai velocities, are considered. Such a choice is intentional because it allows well defined diff~@onplateaux over a potential range sufficiently extended to permit accurate potential determinations. The agreement between theory and experiment is good except near the top of the bed (X = 1); these de viations could be due to the fact that the beds were not surmounted by a calming section of nonconducting microspheres. Also the bad quality of the contact grain-probe in zones where the settling state of the bed may be altered by the displacement of the measuring probe could influence the potential distribution in that zones localized at the top of the bed. For other type of bed, Fig. 6 compares theory and experiment and confinns that (1) is a good representation of the potential distribution within electrodes wiih y higher than 0. Experimental potential distributions for an electrode characterized by y/u = 0.5, which was chose-nin the theoretical calculations of Fig. 3, are plotted in
Fig. 7. Maximum deviations of E(X) to E(0) appear within the electrod& their importance decreasing with increasing R The experimental evidence of the existence of such maxima confirms the qualitative validity of the theoretical approach. Then it proves that in the most general case it is important in design to take into account that the potential distribution may present an extremum in the core of the bad and to not restrict the overall potential considerations to B(1) and E(0) only. Furthermore, as seen in Fig. 7, the calculated distributions (dotted curves) agree satisfactorily with -the corresponding experimental distributions except, as previously, in the layers immediately next to the top of the bed. The deviation between theory and experiment at X = 1 increases with R ie when the superficial velocity is decreased; this seems to support the explanation given above relatively to the absence of a calming section at the bed entrance. CONCLUSION
The assimilation of both dispersed and continuous phases of a flow-through porous electrode to pseudocontinuous media with the electrical conductivities 0 and y respectively,allows to modelize the behaviour of such an electrode operating under limiting current couditions. This modelization could be useful in the design of practical flow-through porous electrodes the matrix of which would not be highly conductive, as for example for fixed he& of carbon or graphite grains. In spite of the difficulty encountered to known CT, the experience supports this mod&&on. Depending on the value of y/u, the potential distributions present an extremum more or less pronounced which one would have to consider when the application of a flowthrough porous electrode corresponds to strict con-
A. GAUNA~ANO F. QIIURET
1202
Ekctrolytt
0.4
d, *8.25x10+cm Ca=099xlO+M L=3cm Y 8xlO”ohm“ cm” ----Froni (f) wItoh’yb mO.5 -antal
Fig. 7. Experimental potential distributions for a low v/u value and comparison with theory,
ditions in what concerns the extent of the potential range.
1. Eleetrfcal charge balance
REFERENCES
i, = i,(X) + is(X)
1. 2. 3. 4. 5.
R. E Sioda, Efectrochim.Acta 16, 1569(1971). R. E. Sii 1. electromwf. Ckm 34,399 (1972). R. E. Sii J. electrounal. Chem. 56, 149 (1974). F. Cocuret, Electrochim.Acta 21, 203 (1976). A. Oaunand, D. Hutin and F. Coeurct, Electrochlm. Acta g 93 (1977). 6. R. E Sii Electrochim.Acta 13, 375 (1968). 7. R. E !Goda,J. ekceermaaal.Chem.34,411 (1972). 8. R. E. S&da, E&-o&m. Acta 15, 783 (1970). 9. W. I. Blacdcl and S. L Bayer, AMlyt. Chem.45, (2),258 (1973). 10. R. Alkire and B. Gracon, J. electrochem Sot. 122 (12), 1594(1975). 11. F. Coaurct, ixlectrochim. Acta 21, 185 (1976). 12. R. B. Siodp, Etectr&m. Act0 Z& 439 (1977). 13. R. E. Sioda. E&ctrocLlm. Acta 13, 1559 (1968). 14. R. JI. Sioda, J. appl. Electrochem 3,221 (1975). 15. J. Newman and W. Ticdcman, AJ.Ch.E. JI 21(l), 25 (1975). 16. J. Newman and C. W. Tobias, J. elecrrochem.Sot. 109, 1183(1962). 17. A. Oawumd, Tbe&, Nancy, Frame (1977). APPENDIX
From the overall call current f,, three cds are defined:
i, = -4
n
L(X)
i&X) = -
n
and Z&Y)are the algebraic valuesrelative to Ox axis of tbc currents flowing at the reduced height X in the matrix and in the solution respectively. Ttu deduction of E(X) requires the following step: whm I&‘)
i,(X)
I
F
or &(X) dX
dis(X) +--0 dX
(A-1)
with the boundary conditions:
is(l)= i, iM(l)=O is(O)= 0
b(l) = il
assuming that the contribution of the current feeder to the elcctrocbcmical process is negligible against that of tht bed. 2 Maw baalaneein the elecrrode at the d#iibn current If(hcelectrodcisCDll~~GdBSaplugEowclcdrochomieal rGector without any axial dispersion, a differential mass balance lads to : IL dC(X) ---=E,.A,.L~.C(X) e dX
(A-2)
in which C(X) is tbc concentration at X and k# the maas transfer codficiulf supposed as ill-t of x, bctwun the matrix and the sobltbn An evaatual axial dispersion would need to complete the above t&ncc cquation[ll]. Intqration of (A-2) gives the concentration profile : 44 C(X) = co exp - --L.(l II I
-X)
(A-3)
1203
The intluence of the relative electric conductivity lheconc-entration C,
at the electrode exit is C(X =0) and the fractional conversion R of the ions through the bed is M
R,,_c”, co
z.F.~.i-l.C~
ie by linear combination of thcsc two equalities:
64-4)
with z thenumber of electrons in the electrochemicalreaction.
3. ApplicationOfOhm’slaw to the e&crricalconducrionin borh
4. Appkarfon oj Faraday’s law For a diiTereatialclement of heightdX :
WX)
-=
dX
ph4ses
For the matrix : L -.idX)= 0
_-
&(X)
dX
for theelectrolyte witbin the pores : L -.js(X)= Y
-_
d&(X) dX
-z.F.k,.L.A,.C(X)
(A-6)
The derivation of (A-5) with respect to X and the combination of (A-6),(A-l) and (A-3)conduce to the clilk~atial equation :
@X9-WM/V8/1101-1197 SOZOO,CI
INFLUENCE OF THE RELATIVE ELECTRIC CONDUCTIVITY OF THE TWO PHASES ON THE POTENTIAL DISTRIBUTION IN FLOW-THROUGH POROUS ELECTRODES UNDER LIMITING CURRENT CONDITIONS A. GAUNANDand F. COEURET* Laboratoire des Sciences du G&e Chimique, C.N.R.S.-E.N.S.I.C. 1, rue Grandville, 54042Nancy, France (Received 18 October 1977; and infimlform 10 March 1978) Abshaet -The paper examines the influence of the relative electric conductivity of the solution and matrix phases on the potential distribution in flow-through porous electrodes working under limiting current conditions. The solution giving the potential distribution is deduced and is experimentallytested for packed bed electrodes of graphite microspheres using the reduction of ferricyanide ions as the electrochemical reaction. A satisfactory agreement between theory and experiment is observed.
INTRODUCTION
Potential distributions within flow-through porous electrodes operating under limiting current conditions have been examined for the particular case of a highly conductive dispersed phase (or matrix) the potential of which being considered as constant[l-51. This interesting case corresponds to fixed bed electrodes tiving a metallic matrix as in the works studying the influence of flow conditions on the limiting current[6-121. However the electrical conductivity of the matrix may not always be infinitely high and the influenceofitstinitevaluehas tobe takenintoaccount, as for example : (a) For fixed bed electrodes of graphite granules or carbon grains which industrially might be prefered to metallic grains - in order to restrict the investment costs - for electrochemical operations without phase change. In such systems indeed the contact resistances between the grains limit the electrical conductivity of the mat& Some experimental works of Sioda[l3,14] used such poorly conducting beds but without considering the matrix conductivity. (b) For fluidized bed electrodes of highly conductive particles and operated at very low expansion ratios ie in hydrodynamic conditions such that tbe interparticle conduction exists but not so highly as for the fixed bed state. As reviewed by Newman[lS] the influence of the electrical conductivities of the dispersed and continuous phase8 respectively has been studied theoretically for porous electrodes working either under Tafel kinetics or under the linear electrode kinetics. Tbeaim was to show how theseconductivities influence the uniformity of the reaction distribution. The present paper concerns Row-through porous
electrodes working under limiting current conditions. For the case where electrolyte and current flow in the same direction, it examines how the above two conductivities influence the potential distribution within the electrodes. THEORElK4L ELECTRODEPOTENTIAL DISI’RiBUTION
One considers the cathodic fixed bed electrode of Fig. 1 for which the electrolyte flows downwards at the uniform superficial velocity u. The bed is supposed one-dimensional; its porosity, considered as uniform, is E and its height L. It is made of a granular material the specific area of which is a&, = 6/d,, for spherical grains of diameter d,) and having the electrical conductivity u,,; the specific area of the bed itself is A, = a,. (1- E).The column cross-sectional area is R; the electrical conductivity of the electrolyte is y. and its concentration in reacting ions at the bed entrance is c,,. As in[S] the anode is situated upstream the bed.
* Present Address: Laboratoire Etudes Adrodynamiques et Thermiques, 40, avenue du Recteur Pineau46022, Poitiers, France. Correspondence concerning this paper should be addressed to F. Coeuret. 1197
Fig. 1. Schematic view of the fixed bed electrode.
A. GAUNAND AND F. Coamurr
1198
The local electrode potential R(X) at the reduced height X = x/L measured from the bottom of the bed is the difference :
between the local matrix potential 4,(X) and the local solution potential q%s(X).As proposed in the literature for porous electrodes[M], the potential distribution can be deduced assuming that both continuous and dispersed phases are continuous media with the electrical conductivities y and B respectively : y corresponds to the electrolyte within the pores of the bed ; it depends on y0 and on the tortuasity of the current lines in the electrolyte around the granular material. -u characterizes the electrical conductivity of the matrix and depends on u0 and on the intergranular contact resistances. By integrating the differential equation (A-7) deduced in the Appendix, one obtains for the electrode potential distribution :
t
L. I4
E(X) = E(O)+ -
Y
[(l-R)-X+Xln(l-R)-l]
(1) / where E(0) is the current feeder potential, R the overall fractional conversion of the reacting ions in the electrode and i, the overall cd relative to n. Expression (1) shows that the difference E(X)-S(0) not only depends on y/u and R, but also on I, y, Co and II owing to (A-4); on the other hand, as expressed below in (3), u and R arc interdependent through the mass transfer coefficient k, the value of which depends on the hydrodynamics ie on u. Thus one has to recognize that the theoretical analysis of (1) under its form would be difficult without choosing a priori values for I,, y, C,, u thus obviously restricting to such a choice the conclusions which could be deduced from the analysis. Considering the quantity :
Z(X) = & CW) - WI
(2)
one sees that it only depends on v/u and R. The multiplicative term y/Lli, 1in (2) represents physically the inverse of the overall potential drop which would be observed if the matrix of the bed was no conductive at all (electrochemical reaction on the current feeder alone). Through i, this term depends on the operating conditions. For the purpose of the following theoretical analysis we consider it as coustant, thus allowing the respective influence of v/u and R on Z(X) to be representative of their influence on the electrode potential distribution. In the discussion of the results, theoretical distributions calculated from (1) will be presented. 1. Influence of v/u Figure 2 shows this influence for R = 0.9. For a matrix much more conductive than the electrolyte (y/o + 0), Z is always negative and the more so as X is high;
-0.5
7
0
0.5
Fig. 2. Inlkace of y/o on 2 far R = 0.9.
this means through (2) that the electrode potential E(X) goes to more cathodic values as X increases from X = 0 to X = 1. When y/u deviates from zero, a maximum appears in the potential distribution and the potential at the top of tbe bed tends to be eq@ to the current feeder potential. For higher r/u values one sees that Z(X), ie the difference E(X)-E(O), remains positive for every X: the current feeder is the most cathodic part of the electrode. The existence of a maximum-more or less pronounced according to the value presented by y/a or R-merits some attention. Indeed it traduces the fact that parts of the bed would operate at cathodic potentials out of the range bounded by the current feeder potential E(0) and the potential E( 1) at the top of the bed; consequently this form of potential distribution for sufficiently high r/a values signifies that by only considering the range delimited by E(0) and E(1) it would not be possible to guarantee the operation of the whole electrode at the limiting diffusion current. The situation of the maximum potential appears as given by :
x ma= 2. lnqkence
In(1 -R)
ofR
Figure 3 puts in evidence the influence of R on the potential distribution for a value of y/a which corresponds to the presence of maxima in the distri-
The intlucnceof the relativeelectricconductivity butions. One sees clearly that for low R values, as X is increased E(X) is first less cathodic than E(0) and second morecathodic than E(0) as the top of the bed is approached. Furthermore with increasing R the potential at X = 1 becomes equal at E(0) and further increases of R are traduced by electrode potentials all less cathodic than the current feeder potential. Also it is evident that as R is increased from low values the position of the maximum deviation of E(X) to E(0) moves to the top of the bed. When R tends to 1, the distribution is simply given by : xli,l R(X) - E(0) = d
which signifies that the potential distribution obeys Ohm’s law applied to the dispersed phase for the overall electric current. Both examples of Figs 2 and 3 show that when the matrix is not highly conductive, three local potentials would have to be taken into account in the design of the electrode: E(O),E( 1) and E(X,,,& Three particular cases of (l), corresponding to y/a = 0; 1 and cc respectively could be examined but the most interesting seems to be the first one (y/u = 0) which has been considered separately in a previous paper[5] and the potential distribution of which appears here as a limiting case of (1). EXPERIMENTAL
1. Cell, electrolytes and electrodes The hydraulic circuit, the current feeders to the cell, the measuring probes, the electrical equipment and the operating mode are the same as in[S]. The electrolytes consist in solutions of about 0.001 M potassium ferricyanide in sodium hydroxide,
1199
the true ferricyanide concentration being known by amperometric titration with cobalt sulphate. Two sodium hydroxyde concentrations were used in order to have two values of the electrolyte electrical conductivity yO,as indicated below: 0.75 N NaOH
y0 = 0.13 Ohm-’ cm-’
0.4 N NaOH
y,, = 0.08 Ohm-’ cm-’
The porous electrodes were fixed beds of graphite microspheres of diameter d,, = 0.064cm and
0.0825cm respectively. The bed height was varied between 1 and 3 cm and the bed porosity was constant and equal to 0.42.The electrolyte superficial velocity u was varied between 0.01 and 0.08 cm/s. 2. Relative conductiuity v/a of the systems As in[4,5] i is calculable from y. and y by: Y 2.E Yo 3--E In what concerns u, the value of which highly depends on contact resistances between the graphite grains and consequently on the settling state of the bed, it was determined “in situ” by two methods after previously verifying in a separated dispositive the applicability of Ohm’s law to the electrical conduction in fixed beds of dry graphite grains. Method (a) applies Ohm’s law to electronic current flow in the fixed bed itself (Fig. 1).For that the solution filling the column is only sodium hydroxyde thus avoiding ionic conduction within the pores of the bed. This bed of graphite microspheres is surmounted of a few layers of gold plated microspheres in which an electrical conductor is immersed; they play the role of counter-electrode in such a realized conductivity cell. Method (b) is taken from the first steps of the theoretical calculation (Appendix, paragraph 3). From (A-5),near the grid supporting the bed, we have :
05
WV = -L!! C (-1dX X=0 which follows from local application of Ohm’s law. Thus the experimental potential distributions in the layers of the bed immediately neighbouring the grid may lead to crby means of the preceeding relation. The values such deduced for r/a are tabulated in Table 1. It is seen that both methods lead to values in satisfactory agreement although in one case method (b) had given values which appear as depending on L; probably these deviations are due to the difficulty to reproduce the settling state of the bed.
Table 1. Method (a) y. -
0.130hn-‘cm-’
I=4.4 0
Method (b) 1, 0
d, = 0.064cm -0 5
Fig, 3. Influence of R on 2 for y/a = 0.5.
yo=
O.oaOhm-‘cm-’
dp =
0.0825~111
4.6 for L = I cm
I
3.3 for L = 2cm
Y -=0.44
J!=os
0
#
1200
A. GAUNAND
AND
F. COBUREI
t 1,
IO
I
0.I
Rep Fig.4. Exp~&~~talvsriadensof%, withRet,forfixedbedsofgrsphitemicrospheres characterized by y/u = 0.5
and comparison with correlations for highly conductive microspheres.
electrodes of highly conductive grains that in the present work the graphite microspheres were acting as monopolar electrodes. The variations of R with u for various values of the bed thickness L are similar to thase p?~iQ~~ pubiislxd for electrodes of highly conductive grains[ll] and of graphite granules[l4]. More important in the R= 1 present paper is to examine if the mass transfer and, as inC11], the values of R calculated from coefficientsdeduced from (3) for fixed bed electrodes of experimental overah limiting currents I, (expression A- graphite microspheres are in agreement with those 5) allow the variations of the mass transfer coefficient presented in[li, 171 for gold plated microspherea. k, beben the~w~gl~~d and &hegrains of the fixed For that, Fig 4 reports the ex~rimental Sherwood bed to be known. The main interest of such an numbers Sh, = k,. d,/D us Reynolds numbers Re, exploitation of the overall data is to insure, by = u. $,/v for beds of a class of particles and a given comparison with results for flow-through fixed bed Schrmdt number (SC = 1950); also two empirical lU2%JLTS AND DISCUSSION
1. Fr~t~~~ c~~v~s~~s R Without axial dispersion in’ the electrode, R is expmssed as [see Ap~dix):
-exp{ _!?$}
(3)
Electrolyte
0.3
4
=6.40xlO%m Co-0,861xlO”M L=lcm y=i3xlO”otnn*! cm“ --fcbm(1) with % = 4.6
t
-I x h
f
Fig. 5. Experimentalpotential dist~bution~ showing the inttuenee of R and comparison with tbooq.
1201
The influence of the relative electric conductivity
-2
7
d, =6.40xlO~%m L*2cm %
s3.30 x
0.1
0.
0.2 E( X)-E(O)
thso.
[volt]
Fig. 6. Comparisonbetweentheoreticaland calculatedpotentialvalues for a givenfixedbed electrode. correlations established elsewhere[11,173 for highly conductive microspheres are reported in that figure. T’hegood agreement between the experimental points and both correlations allows to conclude that the flowthrough fmed bed electrodes of graphite microspheres were operated at the limiting diffusion current what prove the validity of the experiments reported in the pr=*t pap= 2. Potential distributions Figure 5, given as an example, presents experimental distributions of the differenceE(X)-E(O) within a fixed bed characterized by y/u = 4.6; the dotted curves are the corresponding theoretical distributions calculated from (1).It has to be noted that high values of R, ie low electrolyte supc&iai velocities, are considered. Such a choice is intentional because it allows well defined diff~@onplateaux over a potential range sufficiently extended to permit accurate potential determinations. The agreement between theory and experiment is good except near the top of the bed (X = 1); these de viations could be due to the fact that the beds were not surmounted by a calming section of nonconducting microspheres. Also the bad quality of the contact grain-probe in zones where the settling state of the bed may be altered by the displacement of the measuring probe could influence the potential distribution in that zones localized at the top of the bed. For other type of bed, Fig. 6 compares theory and experiment and confinns that (1) is a good representation of the potential distribution within electrodes wiih y higher than 0. Experimental potential distributions for an electrode characterized by y/u = 0.5, which was chose-nin the theoretical calculations of Fig. 3, are plotted in
Fig. 7. Maximum deviations of E(X) to E(0) appear within the electrod& their importance decreasing with increasing R The experimental evidence of the existence of such maxima confirms the qualitative validity of the theoretical approach. Then it proves that in the most general case it is important in design to take into account that the potential distribution may present an extremum in the core of the bad and to not restrict the overall potential considerations to B(1) and E(0) only. Furthermore, as seen in Fig. 7, the calculated distributions (dotted curves) agree satisfactorily with -the corresponding experimental distributions except, as previously, in the layers immediately next to the top of the bed. The deviation between theory and experiment at X = 1 increases with R ie when the superficial velocity is decreased; this seems to support the explanation given above relatively to the absence of a calming section at the bed entrance. CONCLUSION
The assimilation of both dispersed and continuous phases of a flow-through porous electrode to pseudocontinuous media with the electrical conductivities 0 and y respectively,allows to modelize the behaviour of such an electrode operating under limiting current couditions. This modelization could be useful in the design of practical flow-through porous electrodes the matrix of which would not be highly conductive, as for example for fixed he& of carbon or graphite grains. In spite of the difficulty encountered to known CT, the experience supports this mod&&on. Depending on the value of y/u, the potential distributions present an extremum more or less pronounced which one would have to consider when the application of a flowthrough porous electrode corresponds to strict con-
A. GAUNA~ANO F. QIIURET
1202
Ekctrolytt
0.4
d, *8.25x10+cm Ca=099xlO+M L=3cm Y 8xlO”ohm“ cm” ----Froni (f) wItoh’yb mO.5 -antal
Fig. 7. Experimental potential distributions for a low v/u value and comparison with theory,
ditions in what concerns the extent of the potential range.
1. Eleetrfcal charge balance
REFERENCES
i, = i,(X) + is(X)
1. 2. 3. 4. 5.
R. E Sioda, Efectrochim.Acta 16, 1569(1971). R. E. Sii 1. electromwf. Ckm 34,399 (1972). R. E. Sii J. electrounal. Chem. 56, 149 (1974). F. Cocuret, Electrochim.Acta 21, 203 (1976). A. Oaunand, D. Hutin and F. Coeurct, Electrochlm. Acta g 93 (1977). 6. R. E Sii Electrochim.Acta 13, 375 (1968). 7. R. E !Goda,J. ekceermaaal.Chem.34,411 (1972). 8. R. E. S&da, E&-o&m. Acta 15, 783 (1970). 9. W. I. Blacdcl and S. L Bayer, AMlyt. Chem.45, (2),258 (1973). 10. R. Alkire and B. Gracon, J. electrochem Sot. 122 (12), 1594(1975). 11. F. Coaurct, ixlectrochim. Acta 21, 185 (1976). 12. R. B. Siodp, Etectr&m. Act0 Z& 439 (1977). 13. R. E. Sioda. E&ctrocLlm. Acta 13, 1559 (1968). 14. R. JI. Sioda, J. appl. Electrochem 3,221 (1975). 15. J. Newman and W. Ticdcman, AJ.Ch.E. JI 21(l), 25 (1975). 16. J. Newman and C. W. Tobias, J. elecrrochem.Sot. 109, 1183(1962). 17. A. Oawumd, Tbe&, Nancy, Frame (1977). APPENDIX
From the overall call current f,, three cds are defined:
i, = -4
n
L(X)
i&X) = -
n
and Z&Y)are the algebraic valuesrelative to Ox axis of tbc currents flowing at the reduced height X in the matrix and in the solution respectively. Ttu deduction of E(X) requires the following step: whm I&‘)
i,(X)
I
F
or &(X) dX
dis(X) +--0 dX
(A-1)
with the boundary conditions:
is(l)= i, iM(l)=O is(O)= 0
b(l) = il
assuming that the contribution of the current feeder to the elcctrocbcmical process is negligible against that of tht bed. 2 Maw baalaneein the elecrrode at the d#iibn current If(hcelectrodcisCDll~~GdBSaplugEowclcdrochomieal rGector without any axial dispersion, a differential mass balance lads to : IL dC(X) ---=E,.A,.L~.C(X) e dX
(A-2)
in which C(X) is tbc concentration at X and k# the maas transfer codficiulf supposed as ill-t of x, bctwun the matrix and the sobltbn An evaatual axial dispersion would need to complete the above t&ncc cquation[ll]. Intqration of (A-2) gives the concentration profile : 44 C(X) = co exp - --L.(l II I
-X)
(A-3)
1203
The intluence of the relative electric conductivity lheconc-entration C,
at the electrode exit is C(X =0) and the fractional conversion R of the ions through the bed is M
R,,_c”, co
z.F.~.i-l.C~
ie by linear combination of thcsc two equalities:
64-4)
with z thenumber of electrons in the electrochemicalreaction.
3. ApplicationOfOhm’slaw to the e&crricalconducrionin borh
4. Appkarfon oj Faraday’s law For a diiTereatialclement of heightdX :
WX)
-=
dX
ph4ses
For the matrix : L -.idX)= 0
_-
&(X)
dX
for theelectrolyte witbin the pores : L -.js(X)= Y
-_
d&(X) dX
-z.F.k,.L.A,.C(X)
(A-6)
The derivation of (A-5) with respect to X and the combination of (A-6),(A-l) and (A-3)conduce to the clilk~atial equation :