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The analysis of an isothermal batch electrochemical reactor with a secondary cathodic reaction
Electrochimico
THE
Acta.
1974, Vol.
19, pp. 239-244.
ANALYSIS REACTOR
Pergamon
Press. Printed
in Great Britain.
OF AN ISOTHERMAL WITH A SECONDARY D. J.
BATCH ELECTROCHEMICAL CATHODIC REACTION
PICKE’IT
Department of Chemical Engineering, The University of Manchester, Institute of Science and Technology, Manchester M60 lQD, England (Received 18 June 1973; in revised form 20 September 1973)
Abstract-An
analysis has been made to examine the effect of a constant applied electrolysing voltage on the performance of an isothermal batch electrochemical reactor in which the rate of the cathodic reaction is limited by mass transfer of a reactant to the electrode surface, a simultaneous secondary reaction occurring at higher cathode potentials. Numerical solutions are presented for a hypothetical process study to show how average production rates for various percentage conversions of reactant are influenced by the magnitude of the electrolysing voltage and the corresponding electrical energy requirement. The most noticeable effect is that the electrical energy requirements for a given production rate are very strongly influenced by both the degree of conversion of reactant and by the extent to which the secondary reaction occurs.
Superscripts standard (298”K, 1 at., unit activity) interfacial above the limiting current average.
NOTATION
0 * + -
constants in Tafel equations (V) electrode area (m’) constants in Tafel equations (V) constant in Tafel equation (V) molar concentration (kmol/m3) Diffusion coefficient (mz/s) reversible emfof overall cell reaction (V) Faraday constant (96487 C/mol) current density (A/m’) total current (A) electrical energy requirement (J) stoichiometric numbers gas constant (J/mol *K) electrolysis time (s) absolute temperature (“K) Applied cell voltage (V) net cell volume (m’) effective inter-electrode spacing (m) electrochemical valencies dimensionless current density or concentration gradient * diffusion layer thickness (m) current efficiency effective electrolyte conductivity (75- l/m) overpotential (V) average production rate of%pecies P (kmol/s)
INTRODUCTION
In view of the size, number
and importance of industrial electrochemical processes it is perhaps surprising that not too much work of a general nature has been done on the analysis of electrochemical systems. A very simple conception of an electrochemical cell is to regard it as a type of chemical reactor in which the free energy necessary for the desired reactions is supplied by electrical energy. In such a reactor an increase in reaction rate (current) is achieved in most cases by raising the external applied voltage and providing that only one irreversible reaction occurs at each electrode it is possible to set up design equations analogous to those for chemical reactors with the current substituted for the specific reaction rate. Obviously desirable design equations for an electrochemical cell would go further and ideally relate the rate of any given electrode reaction to the applied voltage. Unfortunately, even in the simplest situations an increase in applied voltage ultimately shifts the electrode potentials to values such that secondary reactions occur. To then be able to predict the rate of any of the reactions occurring would require knowledge of all current/electrode potential relationships applicable. In most cases these relationships have such complex interdependence that useful expressions for the variation
Subscripts
A C M,iV,P,Q 0 IR L min
t
anode cathode chemical species at zero time ohmic drop limiting minimum at time t 239
D. J.PICKETT
240
of the rate of a reaction with appIied voltage are unobtainable. A good deal of progress has been made on somewhat less ambitious fronts and a number of workers have developed models for a single reaction occurring at electrodes of various types. These have resulted in expressions to describe the spatial potential, current and concentration distributions under various flow conditions. Recent and noteworthy among such investigations are those of Newman and co-workers[l-31 for plane, porous and packed bed electrodes and Fleischmann et aZ[4-61 for fluidised beds. The behaviour of various types of flow reactors under constant current density operation has been investigated by Fahidy[7-91 for steady state and transient conditions. Rousar et al[lG 131 have discussed the current density distributions arising in flow cells in which gas is produced and compared with analyses with data from sodium chlorate cells. An analysis considerably less rigorous than those above but capable of approximately describing the behaviour of a flow cell operating in the turbulent region has been given by Pickett and Stanmore[l4]. In this model an approximate voltage balance is set up for the cell and used to determine the concentration distribution along a parallel plate cell numerically. This permits numerical integration of a simple electrochemical rate equation to give the required electrode area for a given conversion. Future electrochemical processes are not likely to be large and may well involve the use of general purpose electrolysers operating batchwise. A simple analysis along the lines of [14] for a batch system and extended to cover possible secondary reactions may be of potential use in providing information about the operation of such units. This paper attempts to make such an analysis for a system in which the primary cathodic reaction is mass transfer controlled with a secondary reaction occurring at elevated cathode potentials. This procedure enables the effect of applied voltage on current efficiency and production rates for various reactant conversions to be calculated. ANALYSIS
Consider an electrolytic reaction scheme in which species A4 and N are consumed and species P and Q produced. The component electrode processes are Cathodic
;
mM + z,e--+pP.
Anodic:
MN--t 9Q + z2e,
I z,F’
For a mass transfer controlled cathodic reaction with species M dissolved in a large excess of supporting electrolyte such that the migration flux is insignificant compared to the diffusion flux. then
_
i
z,F
=
gee,-
C&).
The current density expressed by equation (3) is equal to the mean current density in equation (2) if there is perfect mixing in the bulk of the electrolyte, uniform electrode potentials and the diffusion layer has a constant thickness. Additionally if the diffusion coefficient is independent of concentration then equations (2) and (3) give
The minimum theoretical time of electrolysis will occur if the cell always operates at the limiting current density (i = ir. Ct = 0) or above it.
If the minimum electrolysis time is achieved by operation with i greater than iL then such operation will be with reduced current and energy efficiencies due to secondary cathodic reactions. Operation with i = iL throughout the electrolysis if a constant voltage is maintained is impossible since CM decreases with time. To enable satisfactory constant voltage operation without secondary reactions occurring it is necessary to choose a voltage such that at the end of the electrolysis the limiting current is not exceeded. To be able to do this a voltage balance on the system is required for CM = CMt together with further voltage balances throughout the range of concentration of M in order to determine the actual time of electrolysis, A first approximation to the voltage balance ignoring metal and contact potential drops is u = E + VA + lqci +
VIR.
(6)
The voltage components on the right-hand side equation (6) can be given in the following forms:
and the overall reverse reaction is eliminated by use of a diaphragm. The rate ofelectroreduction of M at any time assuming 100 per cent current efficiency is described by -VdC, m dl
Expressing the total current in terms of the mean current density over the whole cathode area and rearranging equation (1) enables the electrolysis time to be written as
(‘I
(i) The theoretical voltage of the cell, (assumed to the er??fof the reversible overall reaction)
of
equal
.E=,!P+
(7a)
The analysis of an isothermal batch electrochemical (ii) The anodic
reactor with a secondary cathodic reaction
241
overpotential qa = a, + b1 log i.
(iii) The cathodic l’lcl =a2 (iv) The resistance
t-1
m
overpotential 1L
+b~logi+c,log-
iL -
(7c)
i’
overpotential
(74 It must be emphasised that the Tafel-like forms of equations (7b) and (7~) are not rigorously applicable for very low overpotentials and when the concentrations of the species present change. However, for computations made on systems for which experimental polarisation data are available it will often be satisfactory to use these equations with average coefficients over the concentration and potential ranges considered. The restrictions in the use of Tafel relationships at very low overpotentials are unlikely to be important for operation of practical systems. Both equations (7b) and.(7c) are referenced to the respective reversible half cell potential. The form qIR for the resistive part of the overpotential is used to emphasise that this property could vary over the cell due to local current density differences even though this case is excluded here. The values of x and K in equation (7a) include the resistive characteristics of the diaphragm as well as the electrolyte with being constant. It will be note.d,that the equality of anode and cathode area is imphclt in equations (7b) to (7d). i and iL in equations (7b) to (7~) by use Eliminating of equation (3) and substituting into equation (6).
Examination of equation (10) reveals that for a fixed value of U, y may approach unity as C, decreases. This implies that the cathodic overpotential will become infinitely large. In reality a value of y will be reached for a given value of CM such that the potential necessary for the secondary cathodic reaction (usually hydrogen evolution) is obtained. Above this value of y equation (7~) can not be used and the cathodic overpotential must then be described by another equation applicable to the secondary reaction, ie, [VA =a3
+b,logi.
(11)
In conformity with equations (7b) and (7~) a, and b3 are average values with a3 chosen with respect to the reversible half-cell potential of mM + zre = pp. The choice of reference is arbitrary. Referencing a3 to the reversible potential of the secondary reaction involves considering the current density formally in two parts, one part for the reduction of M and the other for the secondary process. This alternative reference is possibly preferable in general although the form adopted above can be more readily obtained from cathodic polarisation experiments on real systems. For conditions under which the secondary reaction occurs the voltage balance can be given the form
-I- as + b, log i+ + i;,
+c, log 2
M
(8) A dimensionless current driving force) defined by
density
(or
mass
y=pM;C~
iL and substituted tionally superior
2.303RT + ;‘,_log(&
M
into equation form.
(8) yields
+ a, + a,
transfer
(9) a computa-
(12)
i* being the total current density flowing when the secondary reaction occurs. The minimum value of i+ can be obtainable by equating (7~) and (1 I) so that (b3-bz)logi+
=az-a,+c,log.
7.
1L-l
IL
(13)
It will be noticed that equation (13) can only give a very approximate idea of the onset of the secondary reaction since the Tafel form of equation (11) will not be valid at such low partical current densities applicable to that reaction. To determine the electrical energy requirement, current efficiency and electrolysis time for a given value of U for a reactant conversion from CM0 to CM, y values have to be computed at intervals over the entire concentration range using either equation (10) or (12) as applicable. The electrolysis time can then be obtained by numerical integration of equation (5). The total quantity of electricity per unit electrode area is
242
D. J. Prcxrrr
obtained by summation for each computational interval. The derived quantities of importance are the average production rate II given by Z-Z= V(C MO- C,) the current efficiency for the reduction ~ _ lmv
zlF(cM, m&
-
(14) of species M,
cd
(1%
idt
and the electrical energy required f .j = U i dt. I0 The procedure will be illustrated by an example.
(16)
EXAMPLE
(b) Computational details For given values of y and CM,,, the applied voltage U is calculated from equation (10). With U kept constant y is calculated for various values of CM between CM0and the final concentration CM,. For each interval a check is made with equation (13) to see if the cathodic potential has reached the magnitude necessary for the side reaction. When this occurs equation (12) is used for calculating the current density. The integral in equation (15) is then evaluated and the quantity of electricity determined by summation of the current densities for each interval over the operation. The procedure was followed for 25, SO, 75 and 90 per cent conversions of M for y values between 0.10 and 0.95 in intervals of O-05.Table 1 presents part of the computed data for a 50 per cent conversion of M at U = 2.756 V. For all results IZ, E and j were calcuIated from equations (14), (15) and (16).
(a) Data DISCUSSION
(i) Cell .speciJication. V = 1O-6 m3.
A = 10m4m2,
Anode/cathode
effective separation
x = 10 mm.
(ii) Cell reaction and electrochemical data. The overall reaction is the formation of metanilic acid and oxygen from m-nitrobenzenesulphonic acid in supporting acid electrolyte. Cathodic reaction: C,H,. SO,H.NO, + 6H+ + 6e-+C,H,.S03H.NH2
+ 2H,O.
Anodic reaction: 3H,O--, 6H+ + 140, + 6e. CM, = 0.2 kmol/m3, c p,,’ = lo- 3 kmol/m3, 21 = 6. With constant hydrogen ion concentration and oxygen pressure the terms C, and C, become redundant, as well as the necessity for including additional concentration terms in the cathodic process. E” = 0.43 V (estimated
from standard
Figure 1 presents current-voltage plots for the cell to show the voltage drop contributions due to the electrolyte resistance and overpotentials but excluding the theoretical electrolysing voltage which varies with concentration during electrolysis. Line I presents the total overpotential at the start of electrolysis and Line II (broken) after 50 per cent conversion approximately, which is reflected in the ratio 2: 1 between limiting currents. The apparent linearity and constancy of the anode overpotential is a reflection of the small current density range plotted. The actual computation range for the current density extended from about 300A/m2 to 3300A/m2 for which limits the Tafel equations (7b, 7c and 11) were applicable. Figure 2 shows the variation of current efficiency with applied voltage for the four percentage conversions. For a given voltage current efficiency increases as the percentage conversion is decreased. For the low Table 1. Computed values of Y as a function of concentration CM for one case involving 50 per cent conversion of reactant A4
free energy
data[lSj).
WV
Tafel coefficients,
2.756
0.20 0.19 @18 @l? 0.16 015 0.14 013 0.12 0.11 0.10
t = 2857s
E = 96.1% I7 = 3.50 x lo- 5kmol/s
= 027 V, b, = o-22 V, b, a3
=
@32 V,
=0.12 V per current decade = O-07 V per current decade = 0.07 v 0.12 V per current decade.
Fh;=
The Tafel data represent average values obtained at a copper cathode and lead anode for m-nitrobenzenesulphonic acid systems. The values for a3 and b3 are for hydrogen evolution in this system[16]. D = 259 x iO_ lo m’/s, 6 = to-sm, ic=20Umm, T = 298°K.
CM kmole/m3
* Secondary reaction occurring.
Y 0.75 0.798 0.840 0.884 0.929 0,970 0.994 0.999 1OoO* 1QOO* 1Ooof j = 1661J
The analysis of an isothermal batch electrochemical
U,
reactor with a secondary cathodic reaction
v
Fig. 1. Current density us voltage to show the overpotential conversion.
conversions throughout
the current density remains more uniform electrolysis due to smaller changes in reac-
tant concentration with correspondingly high current efficiencies. At the high voltages the current density is so close to the limiting current density initially that the secondary reaction occurs soon after the start of electrolysis. Figure 3 demonstrates the variation of average production rate with cell voltage. At very low voltages very slightly higher production rates are obtained for greater reactant conversions although individual differences are too small to be adequately represented graphically. These arise due to propressively longer electrolysis times required for equal changes of reac-
243
magnitudes
initially and after 50 per cent
tant concentration as electrolysis proceeds and are approximately demonstrated by equation (5). As the cell voltage is increased and departure from 100 per cent current efficiency occurs, the higher production rates are obtained for the smaller degrees of conversion. At very high electrolysing voltages the production rate becomes constant since the current density is always greater than the limiting current density, the reduction of A4 then occurring in the minimum time predicted by equation (5).
25
56
90%
Fig. 2. Current efficiency as a function of applied voltage for various percentage conversion.
Fig. 3. Average production rate as a function of applied voltage for various percentage conversions.
D. J. PICKEIT
244
would require more equations for certain overpotentials ranges, and experiments are in hand to cover these problems. Possible extensions of the model to cover the important cases of ionic migration are also being considered.
REFERENCES
Sot. 1. J. S. Newman and C. W. Tobias, J. electrochem. 109, 1183 (1962). Sot. 2. J. S. Newman and W. R. Parrish, J. electrochem. 117, 43 (1970). 3. D. N. Bennion and J. S. Newman, J. appl. Ekctrochem. 2, 116 (1969). and J. W. Oldfield, J. electroand. 4. M. Fleischmann
Chrm. 29, 211 (1971). J. W, Old&Id, 5. M. Fleischmannand Fig.
4. Electrical energy requirement as a function applied voltage for various percentage conversions.
of
29, 231 (1971). 6. M. Fleischmann.
I. W. Oldfield
J. rlectr-oad. and
Chem.
D. F. Porter.
J.
rlectroanal. Chek. 29, 241 (1971). Figure 4 which presents the electrical energy requirements shows similar features to Fig. 3. Initially steep rises in production rate with energy input are obtained until at high voltages the rate of the main reaction starts to become constant. Obviously the use of even such an elementary analysis in predicting the behaviour of real systems is restricted especially due to the many assumptions made concerning the overpotential contributions. Particularly vulnerable arc the use of an estimated theoretical cell voltage and averaged cathodic overpotential data. It is possible-that good agreement would be obtained for systems in which more accurate experimental voltage component data were available. Undoubtedly, this
I. T. Z. Fahidy and A. B. Babijide, Catr. .I. them. Engng 46, 253 (1968). 8. T. Z: Fahidy, Chem. Engng Sci. 23,469 (1968). 9. T. Z. Fahidv, Chem. Enan~ Sci. 24, 141 (1969). 10. A. Regner, c. Cezner and’i. Rous&, Colln Czech. them. Commun. 31 (1 l), 4193 (1966). 11. I. Rousar and V. Cezner, Colln Czech. them. Commun. 32 (3), 1137 (1967). 12. I. Rousar and V. Cezner, Colln Czech. them. Commun. 33 (3), 808 (1968). Sot. 116, 48 (1969). 13. I. Rousar, J. electrochem. Paper presented at 14. D. J. Pickett and B. R. Stanmore, Inst. Chem. Engrs Symp. on Eiectrochem. Engng, Newcastle upon Tyne (1971). Circ. No 500 15. F. D. Rossini, ed. Nat. Bur. Standards. U.S. Gov. Print Off., Washington DC. (1952).
THE
Acta.
1974, Vol.
19, pp. 239-244.
ANALYSIS REACTOR
Pergamon
Press. Printed
in Great Britain.
OF AN ISOTHERMAL WITH A SECONDARY D. J.
BATCH ELECTROCHEMICAL CATHODIC REACTION
PICKE’IT
Department of Chemical Engineering, The University of Manchester, Institute of Science and Technology, Manchester M60 lQD, England (Received 18 June 1973; in revised form 20 September 1973)
Abstract-An
analysis has been made to examine the effect of a constant applied electrolysing voltage on the performance of an isothermal batch electrochemical reactor in which the rate of the cathodic reaction is limited by mass transfer of a reactant to the electrode surface, a simultaneous secondary reaction occurring at higher cathode potentials. Numerical solutions are presented for a hypothetical process study to show how average production rates for various percentage conversions of reactant are influenced by the magnitude of the electrolysing voltage and the corresponding electrical energy requirement. The most noticeable effect is that the electrical energy requirements for a given production rate are very strongly influenced by both the degree of conversion of reactant and by the extent to which the secondary reaction occurs.
Superscripts standard (298”K, 1 at., unit activity) interfacial above the limiting current average.
NOTATION
0 * + -
constants in Tafel equations (V) electrode area (m’) constants in Tafel equations (V) constant in Tafel equation (V) molar concentration (kmol/m3) Diffusion coefficient (mz/s) reversible emfof overall cell reaction (V) Faraday constant (96487 C/mol) current density (A/m’) total current (A) electrical energy requirement (J) stoichiometric numbers gas constant (J/mol *K) electrolysis time (s) absolute temperature (“K) Applied cell voltage (V) net cell volume (m’) effective inter-electrode spacing (m) electrochemical valencies dimensionless current density or concentration gradient * diffusion layer thickness (m) current efficiency effective electrolyte conductivity (75- l/m) overpotential (V) average production rate of%pecies P (kmol/s)
INTRODUCTION
In view of the size, number
and importance of industrial electrochemical processes it is perhaps surprising that not too much work of a general nature has been done on the analysis of electrochemical systems. A very simple conception of an electrochemical cell is to regard it as a type of chemical reactor in which the free energy necessary for the desired reactions is supplied by electrical energy. In such a reactor an increase in reaction rate (current) is achieved in most cases by raising the external applied voltage and providing that only one irreversible reaction occurs at each electrode it is possible to set up design equations analogous to those for chemical reactors with the current substituted for the specific reaction rate. Obviously desirable design equations for an electrochemical cell would go further and ideally relate the rate of any given electrode reaction to the applied voltage. Unfortunately, even in the simplest situations an increase in applied voltage ultimately shifts the electrode potentials to values such that secondary reactions occur. To then be able to predict the rate of any of the reactions occurring would require knowledge of all current/electrode potential relationships applicable. In most cases these relationships have such complex interdependence that useful expressions for the variation
Subscripts
A C M,iV,P,Q 0 IR L min
t
anode cathode chemical species at zero time ohmic drop limiting minimum at time t 239
D. J.PICKETT
240
of the rate of a reaction with appIied voltage are unobtainable. A good deal of progress has been made on somewhat less ambitious fronts and a number of workers have developed models for a single reaction occurring at electrodes of various types. These have resulted in expressions to describe the spatial potential, current and concentration distributions under various flow conditions. Recent and noteworthy among such investigations are those of Newman and co-workers[l-31 for plane, porous and packed bed electrodes and Fleischmann et aZ[4-61 for fluidised beds. The behaviour of various types of flow reactors under constant current density operation has been investigated by Fahidy[7-91 for steady state and transient conditions. Rousar et al[lG 131 have discussed the current density distributions arising in flow cells in which gas is produced and compared with analyses with data from sodium chlorate cells. An analysis considerably less rigorous than those above but capable of approximately describing the behaviour of a flow cell operating in the turbulent region has been given by Pickett and Stanmore[l4]. In this model an approximate voltage balance is set up for the cell and used to determine the concentration distribution along a parallel plate cell numerically. This permits numerical integration of a simple electrochemical rate equation to give the required electrode area for a given conversion. Future electrochemical processes are not likely to be large and may well involve the use of general purpose electrolysers operating batchwise. A simple analysis along the lines of [14] for a batch system and extended to cover possible secondary reactions may be of potential use in providing information about the operation of such units. This paper attempts to make such an analysis for a system in which the primary cathodic reaction is mass transfer controlled with a secondary reaction occurring at elevated cathode potentials. This procedure enables the effect of applied voltage on current efficiency and production rates for various reactant conversions to be calculated. ANALYSIS
Consider an electrolytic reaction scheme in which species A4 and N are consumed and species P and Q produced. The component electrode processes are Cathodic
;
mM + z,e--+pP.
Anodic:
MN--t 9Q + z2e,
I z,F’
For a mass transfer controlled cathodic reaction with species M dissolved in a large excess of supporting electrolyte such that the migration flux is insignificant compared to the diffusion flux. then
_
i
z,F
=
gee,-
C&).
The current density expressed by equation (3) is equal to the mean current density in equation (2) if there is perfect mixing in the bulk of the electrolyte, uniform electrode potentials and the diffusion layer has a constant thickness. Additionally if the diffusion coefficient is independent of concentration then equations (2) and (3) give
The minimum theoretical time of electrolysis will occur if the cell always operates at the limiting current density (i = ir. Ct = 0) or above it.
If the minimum electrolysis time is achieved by operation with i greater than iL then such operation will be with reduced current and energy efficiencies due to secondary cathodic reactions. Operation with i = iL throughout the electrolysis if a constant voltage is maintained is impossible since CM decreases with time. To enable satisfactory constant voltage operation without secondary reactions occurring it is necessary to choose a voltage such that at the end of the electrolysis the limiting current is not exceeded. To be able to do this a voltage balance on the system is required for CM = CMt together with further voltage balances throughout the range of concentration of M in order to determine the actual time of electrolysis, A first approximation to the voltage balance ignoring metal and contact potential drops is u = E + VA + lqci +
VIR.
(6)
The voltage components on the right-hand side equation (6) can be given in the following forms:
and the overall reverse reaction is eliminated by use of a diaphragm. The rate ofelectroreduction of M at any time assuming 100 per cent current efficiency is described by -VdC, m dl
Expressing the total current in terms of the mean current density over the whole cathode area and rearranging equation (1) enables the electrolysis time to be written as
(‘I
(i) The theoretical voltage of the cell, (assumed to the er??fof the reversible overall reaction)
of
equal
.E=,!P+
(7a)
The analysis of an isothermal batch electrochemical (ii) The anodic
reactor with a secondary cathodic reaction
241
overpotential qa = a, + b1 log i.
(iii) The cathodic l’lcl =a2 (iv) The resistance
t-1
m
overpotential 1L
+b~logi+c,log-
iL -
(7c)
i’
overpotential
(74 It must be emphasised that the Tafel-like forms of equations (7b) and (7~) are not rigorously applicable for very low overpotentials and when the concentrations of the species present change. However, for computations made on systems for which experimental polarisation data are available it will often be satisfactory to use these equations with average coefficients over the concentration and potential ranges considered. The restrictions in the use of Tafel relationships at very low overpotentials are unlikely to be important for operation of practical systems. Both equations (7b) and.(7c) are referenced to the respective reversible half cell potential. The form qIR for the resistive part of the overpotential is used to emphasise that this property could vary over the cell due to local current density differences even though this case is excluded here. The values of x and K in equation (7a) include the resistive characteristics of the diaphragm as well as the electrolyte with being constant. It will be note.d,that the equality of anode and cathode area is imphclt in equations (7b) to (7d). i and iL in equations (7b) to (7~) by use Eliminating of equation (3) and substituting into equation (6).
Examination of equation (10) reveals that for a fixed value of U, y may approach unity as C, decreases. This implies that the cathodic overpotential will become infinitely large. In reality a value of y will be reached for a given value of CM such that the potential necessary for the secondary cathodic reaction (usually hydrogen evolution) is obtained. Above this value of y equation (7~) can not be used and the cathodic overpotential must then be described by another equation applicable to the secondary reaction, ie, [VA =a3
+b,logi.
(11)
In conformity with equations (7b) and (7~) a, and b3 are average values with a3 chosen with respect to the reversible half-cell potential of mM + zre = pp. The choice of reference is arbitrary. Referencing a3 to the reversible potential of the secondary reaction involves considering the current density formally in two parts, one part for the reduction of M and the other for the secondary process. This alternative reference is possibly preferable in general although the form adopted above can be more readily obtained from cathodic polarisation experiments on real systems. For conditions under which the secondary reaction occurs the voltage balance can be given the form
-I- as + b, log i+ + i;,
+c, log 2
M
(8) A dimensionless current driving force) defined by
density
(or
mass
y=pM;C~
iL and substituted tionally superior
2.303RT + ;‘,_log(&
M
into equation form.
(8) yields
+ a, + a,
transfer
(9) a computa-
(12)
i* being the total current density flowing when the secondary reaction occurs. The minimum value of i+ can be obtainable by equating (7~) and (1 I) so that (b3-bz)logi+
=az-a,+c,log.
7.
1L-l
IL
(13)
It will be noticed that equation (13) can only give a very approximate idea of the onset of the secondary reaction since the Tafel form of equation (11) will not be valid at such low partical current densities applicable to that reaction. To determine the electrical energy requirement, current efficiency and electrolysis time for a given value of U for a reactant conversion from CM0 to CM, y values have to be computed at intervals over the entire concentration range using either equation (10) or (12) as applicable. The electrolysis time can then be obtained by numerical integration of equation (5). The total quantity of electricity per unit electrode area is
242
D. J. Prcxrrr
obtained by summation for each computational interval. The derived quantities of importance are the average production rate II given by Z-Z= V(C MO- C,) the current efficiency for the reduction ~ _ lmv
zlF(cM, m&
-
(14) of species M,
cd
(1%
idt
and the electrical energy required f .j = U i dt. I0 The procedure will be illustrated by an example.
(16)
EXAMPLE
(b) Computational details For given values of y and CM,,, the applied voltage U is calculated from equation (10). With U kept constant y is calculated for various values of CM between CM0and the final concentration CM,. For each interval a check is made with equation (13) to see if the cathodic potential has reached the magnitude necessary for the side reaction. When this occurs equation (12) is used for calculating the current density. The integral in equation (15) is then evaluated and the quantity of electricity determined by summation of the current densities for each interval over the operation. The procedure was followed for 25, SO, 75 and 90 per cent conversions of M for y values between 0.10 and 0.95 in intervals of O-05.Table 1 presents part of the computed data for a 50 per cent conversion of M at U = 2.756 V. For all results IZ, E and j were calcuIated from equations (14), (15) and (16).
(a) Data DISCUSSION
(i) Cell .speciJication. V = 1O-6 m3.
A = 10m4m2,
Anode/cathode
effective separation
x = 10 mm.
(ii) Cell reaction and electrochemical data. The overall reaction is the formation of metanilic acid and oxygen from m-nitrobenzenesulphonic acid in supporting acid electrolyte. Cathodic reaction: C,H,. SO,H.NO, + 6H+ + 6e-+C,H,.S03H.NH2
+ 2H,O.
Anodic reaction: 3H,O--, 6H+ + 140, + 6e. CM, = 0.2 kmol/m3, c p,,’ = lo- 3 kmol/m3, 21 = 6. With constant hydrogen ion concentration and oxygen pressure the terms C, and C, become redundant, as well as the necessity for including additional concentration terms in the cathodic process. E” = 0.43 V (estimated
from standard
Figure 1 presents current-voltage plots for the cell to show the voltage drop contributions due to the electrolyte resistance and overpotentials but excluding the theoretical electrolysing voltage which varies with concentration during electrolysis. Line I presents the total overpotential at the start of electrolysis and Line II (broken) after 50 per cent conversion approximately, which is reflected in the ratio 2: 1 between limiting currents. The apparent linearity and constancy of the anode overpotential is a reflection of the small current density range plotted. The actual computation range for the current density extended from about 300A/m2 to 3300A/m2 for which limits the Tafel equations (7b, 7c and 11) were applicable. Figure 2 shows the variation of current efficiency with applied voltage for the four percentage conversions. For a given voltage current efficiency increases as the percentage conversion is decreased. For the low Table 1. Computed values of Y as a function of concentration CM for one case involving 50 per cent conversion of reactant A4
free energy
data[lSj).
WV
Tafel coefficients,
2.756
0.20 0.19 @18 @l? 0.16 015 0.14 013 0.12 0.11 0.10
t = 2857s
E = 96.1% I7 = 3.50 x lo- 5kmol/s
= 027 V, b, = o-22 V, b, a3
=
@32 V,
=0.12 V per current decade = O-07 V per current decade = 0.07 v 0.12 V per current decade.
Fh;=
The Tafel data represent average values obtained at a copper cathode and lead anode for m-nitrobenzenesulphonic acid systems. The values for a3 and b3 are for hydrogen evolution in this system[16]. D = 259 x iO_ lo m’/s, 6 = to-sm, ic=20Umm, T = 298°K.
CM kmole/m3
* Secondary reaction occurring.
Y 0.75 0.798 0.840 0.884 0.929 0,970 0.994 0.999 1OoO* 1QOO* 1Ooof j = 1661J
The analysis of an isothermal batch electrochemical
U,
reactor with a secondary cathodic reaction
v
Fig. 1. Current density us voltage to show the overpotential conversion.
conversions throughout
the current density remains more uniform electrolysis due to smaller changes in reac-
tant concentration with correspondingly high current efficiencies. At the high voltages the current density is so close to the limiting current density initially that the secondary reaction occurs soon after the start of electrolysis. Figure 3 demonstrates the variation of average production rate with cell voltage. At very low voltages very slightly higher production rates are obtained for greater reactant conversions although individual differences are too small to be adequately represented graphically. These arise due to propressively longer electrolysis times required for equal changes of reac-
243
magnitudes
initially and after 50 per cent
tant concentration as electrolysis proceeds and are approximately demonstrated by equation (5). As the cell voltage is increased and departure from 100 per cent current efficiency occurs, the higher production rates are obtained for the smaller degrees of conversion. At very high electrolysing voltages the production rate becomes constant since the current density is always greater than the limiting current density, the reduction of A4 then occurring in the minimum time predicted by equation (5).
25
56
90%
Fig. 2. Current efficiency as a function of applied voltage for various percentage conversion.
Fig. 3. Average production rate as a function of applied voltage for various percentage conversions.
D. J. PICKEIT
244
would require more equations for certain overpotentials ranges, and experiments are in hand to cover these problems. Possible extensions of the model to cover the important cases of ionic migration are also being considered.
REFERENCES
Sot. 1. J. S. Newman and C. W. Tobias, J. electrochem. 109, 1183 (1962). Sot. 2. J. S. Newman and W. R. Parrish, J. electrochem. 117, 43 (1970). 3. D. N. Bennion and J. S. Newman, J. appl. Ekctrochem. 2, 116 (1969). and J. W. Oldfield, J. electroand. 4. M. Fleischmann
Chrm. 29, 211 (1971). J. W, Old&Id, 5. M. Fleischmannand Fig.
4. Electrical energy requirement as a function applied voltage for various percentage conversions.
of
29, 231 (1971). 6. M. Fleischmann.
I. W. Oldfield
J. rlectr-oad. and
Chem.
D. F. Porter.
J.
rlectroanal. Chek. 29, 241 (1971). Figure 4 which presents the electrical energy requirements shows similar features to Fig. 3. Initially steep rises in production rate with energy input are obtained until at high voltages the rate of the main reaction starts to become constant. Obviously the use of even such an elementary analysis in predicting the behaviour of real systems is restricted especially due to the many assumptions made concerning the overpotential contributions. Particularly vulnerable arc the use of an estimated theoretical cell voltage and averaged cathodic overpotential data. It is possible-that good agreement would be obtained for systems in which more accurate experimental voltage component data were available. Undoubtedly, this
I. T. Z. Fahidy and A. B. Babijide, Catr. .I. them. Engng 46, 253 (1968). 8. T. Z: Fahidy, Chem. Engng Sci. 23,469 (1968). 9. T. Z. Fahidv, Chem. Enan~ Sci. 24, 141 (1969). 10. A. Regner, c. Cezner and’i. Rous&, Colln Czech. them. Commun. 31 (1 l), 4193 (1966). 11. I. Rousar and V. Cezner, Colln Czech. them. Commun. 32 (3), 1137 (1967). 12. I. Rousar and V. Cezner, Colln Czech. them. Commun. 33 (3), 808 (1968). Sot. 116, 48 (1969). 13. I. Rousar, J. electrochem. Paper presented at 14. D. J. Pickett and B. R. Stanmore, Inst. Chem. Engrs Symp. on Eiectrochem. Engng, Newcastle upon Tyne (1971). Circ. No 500 15. F. D. Rossini, ed. Nat. Bur. Standards. U.S. Gov. Print Off., Washington DC. (1952).