Popov K.I., Djokic S.S., Grgur B.N. Fundamental Aspects of Electrometallurgy

Подождите немного. Документ загружается.

102

Chapter 4

where is the slope of the cathodic activation overpotential-current

density dependence,

is the conductivity of solution and l is a characteristic

length.

Before the parameter

was used, according to Kasper

, Hoar and Agar

The Wagner number represents the ratio of the polarisation resistance to

the solution resistance. The larger it is, the more even is the current

distribution in spite of non-uniform geometry. In general, the current

distribution is more uniform if

the smaller characteristic length of the system is,

the larger the conductivity of the solution is, and

the larger the slope of the activation overpotential-current density

curve is.

Obviously, the Wagner number can be used only to compare the current

distribution in the cell with non-uniform geometry, which contains different

electrolytes.

The same situation appears if the ability of an electrolyte to uniformly

distribute the current density is experimentally determined using the method

of Haring and Blum

The current distribution on a macroprofile is very important in technical

metal electrodeposition. In electroplating, the current distribution determines

the local variations in the thickness of the coating. In electrowinning and

electrorefining of metals, a non-homogenous current distribution can cause a

short circuit with the counter electrode, and the corner weakness effect in

electroforming. This is very important in the three-dimensional electrodes, as

well as in some storage batteries. In all the cited cases a uniform current

density distribution over the macroprofile is required.

The aim of this chapter is to present the procedure, based on simple

equations of electrode kinetics, by which the condition in which a desired

current density distribution can be obtained, or an undesired one avoided,

under the assumption that the limiting diffusion current density does not vary

over the whole electrode surface, including the edges of flat and the tips of

wire electrodes.

4. The Current Distribution in Electrochemical Cells

103

4.1

TWO EQUAL PLANE PARALLEL ELECTRODES

ARRANGEMENT

The cell with two equal plane parallel electrodes represents the

elementary cell of electrode arrangement in electrochemical refining and

winning processes.

It is a well-known fact that in a cell with parallel electrodes (if the

electrode edges do not touch the side walls of the cell), the current density is

higher at the edges than at the centre of the electrode

1,9

. This is because the

current flow passes partially around the rectangular space between the

electrodes. The increased current density at the edges of the electrodes can

be easily noticed by observing the quality of the metal electrodeposit at the

cathode. In some cases the deposit in the central part of the cathode may be

compact and flat whereas the occurrence of dendrites is observed at the

edges. The appearance of dendrites at the edges of the cathodes in such

situations is the most important problem of the current density distribution,

because the growing dendrites could cause short circuits followed by a

decrease in the current efficiency, or even damage the power supply.

The aim of this section is to show in which way dendritic growth at the

cathode edges can be avoided in electrowinning and refining processes.

4.1.1

Ohmic resistance of the cell

The current density distribution in a rectangular electrolytic cell in which

parallel electrodes cover only part of the wall is illustrated in Fig. 4.1.

104

Chapter 4

The linear approximation of the current distribution in the cell with plane

parallel electrodes shown in Fig. 4.1 is presented schematically in Fig. 4.2.

The analysis performed here for the current distribution between the

electrode edges and the cell side walls is obviously valid also for the

situation in which there is the distance between the upper edges of the

electrodes and free surface of ssolution and lower edges to the bottom of the

cell. In the case under consideration these two distances are zero.

The resistance dR of a section of the electrolyte of thickness dc is given

by:

where B is the height of the electrode and is the specific resistance of the

electrolyte. From the linear approximation:

is obtained. The parameters d and c are indicated in Fig. 4.2.

The resistance of the whole electrolyte is then given by

and for by:

4. The Current Distribution in Electrochemical Cells

105

where corresponds to the resistance of a system with a homogeneous

current density distribution (the side walls touch the edges of the electrodes).

For L can be related to A by a linear coefficient k as follows:

which transforms Eq. 4.3 to:

and

taking into account Eq. 4.4, where represents the interelectrode distance

in a cell with L=0, the resistance of which is equal to the resistance of a cell

in which the interelectrode distance is l and L > 0

The cell for the determination of the equivalent resistance is shown in

Fig. 4.3.

The side screens enabled the distance

between the edges of the

electrodes and the side walls to be varied for given values A and

C. The

resistance of the system for various adjusted values of L was measured by

the bridge method using platinum electrodes in a KC1

solution. The electrodes were 2 cm long and 1 cm wide. The interelectrode

distance was 2 cm. Hence, in this case A=C. The back sides of the electrodes

were insulated. The upper edges of the electrode touch the free surface of the

solution and the lower edge the bottom of the cell.

106

Chapter 4

The dependence of the total resistance of a system with plane parallel

electrodes on the distance between the electrode edges and the cell side walls

is shown in Fig. 4.4.

The open circles represent the experimental values, arid the curve is

obtained using Eq. 4.6. The good agreement between the experimental

results and the values predicted by Eq. 4.6. extends to It can be

concluded that for this system Eq. 4.6. is valid for k < 1. This means that the

maximum penetration of the current lines occurs when L = C = A in this

case, and that the maximum length of the current line, l’ is

4.1.2 The very edge ohmic resistance

This consideration of the very edge current density can be elaborated

mathematically in the following way

. Assuming total ohmic control, the

voltage drop in the solution between the electrodes inside the homogenous

field is given by:

and outside of the homogeneous field by:

4. The Current Distribution in Electrochemical Cells

107

where U is the cell voltage, E is the equilibrium potential difference, is the

specific resistivity of the electrolute, l the interelectrode distance, j is the

current density, is the length of the i-th current line and is the current

density corresponding to the i-th current line, as can be seen from Fig. 4.5.

The difference in the current lines outside of the homogeneous field is

given by:

or in the differential form

When Eq. 4.11 is integrated from the inter-electrode distance l to the

maximum length of the current line, l', j', the maximum contribution to the

edge current density due to current line propagation between the electrode

edges and the side walls of the cell, is obtained:

108

Chapter 4

Taking into accounts Eq. 4.8 one obtains

The edge current density can be written as

The maximum value of j' is obtained from Eq. 4.13 as:

Combining Eq. 4.15 and 4.14, the maximum edge current density can be

given as

for as follows from Figs. 4.2 and 4.4.

This means that the very edge resistance is lower than in the homogenous

field and that the minimum effective interelectrode distance, between

the edges of the anode and cathode will be:

because of

4.1.3

The edge effect

In a cell with parallel plate electrodes, if the electrode edges do not touch

the cell side walls, the potential difference between two points in the

homogenous field symmetrically positioned on the electrodes is given by:

Analogously, the cell voltage at the edges can be expressed as:

4. The Current Distribution in Electrochemical Cells

109

where and are, respectively, the anodic and cathodic overpotentials

corresponding to the homogenous field, and and are, respectively,

the anodic and cathodic overpotentials corresponding to the edges.

Elimination of U from Eqn. 4.19 and 4.20 gives

In this case because the increasing of the current density

leads to the increasing of the cathodic and anodic overpotentials also.

In this way, a part of the ohmic potential drop in a homogenous field

transforms into electrochemical overpotential for points at the plane

electrode edges, or in a similar position, meaning the edge current density is

larger than in the homogenous field. In this way it is possible to explain the

change in the quality of the metal deposit near the edge and at the very edge

of an electrode. It should be noted, however, that according to the proposed

model the entire edge current is located at the very edge of the electrode. In

other words, a homogeneous electric field and, consequently, a uniform

current distribution is assumed over the entire electrode surface up to the

very edge of the electrode, where the current density increases abruptly,

which is quite close to the real state described by other authors

2-5

. The

illustration of this effect will be given in Section 4.2.2.

4.1.4

The depth of the penetration of a current line between the

electrode edges and the cell side wall

4.1.4.1 Mathematical model

Equation 4.12 can be rewritten in the form

and j’ can be majorized by giving:

110

Chapter 4

as the maximum length of a current line. U – E in Eqs. 4.22 and 4.23 is the

ohmic potential drop, but it can be substituted by the cell potential due to the

following facts.

The current along each line should be very low, and because of this the

electrochemical overpotentials at the edges of electrodes due to one current

line can be neglected relative to the ohmic potential drop. Hence, the cell

potential transforms into the ohmic potential drop along each current line

and U – E in Eq. 4.23 can be substituted by the cell potential from Eq. 4.19.

Substitution of U – E in Eq. 4.23 by the cell voltage from Eq. 4.19 gives:

or after rearrangement:

Assuming a linear approximation of the propagation of a current line the

relation between L’, the maximum depth of the propagation of a current line

penetration in the space between the edges of the electrodes and the cell side

walls, l and l’ is given by:

Substituting l’ from Eq. 4.25 into Eq. 4.26 and rearranging gives:

It can be shown that if:

4. The Current Distribution in Electrochemical Cells

111

then and and in the opposite case and

This shows that the ability of an electrolyte to distribute the current density

uniformly increases with decreasing

product, i.e., with decreasing ohmic

polarization. Furthermore, is to be expected that the larger the spacing (the

distance between the polarization j – U curves for L = 0 and L > 0), the

worse current density distribution becomes.

4.1.4.2 Cell voltage-current density dependencies

Taking into account Eqn. 4.5 and 4.7 for L > 0 (assuming that the current

density in the homogenous field is equal to the overall one, i.e. that current

density distribution is uniform over all the electrode, except for the very

edge), Eq 4.19 can be rewritten in the form

L’

< L, L in Eq. 4.29 should be substituted by

L’.

Hence, calculations of the current density-cell voltage dependencies must

begin with the determination of the relation between L and L’.

It is obvious that if:

electrolysis is predominantly under electrochemical control, and if:

the electrolysis is predominantly under ohmic control.

In the first case an S-shaped polarization curve can be expected, and a

straight line in the second one. Hence, the shape of the polarization curve is a

good indicator of the nature of the electrolysis process. On the other hand, the

degree of current line penetration into the solution between the edges of the