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Electrochemical Engineering 151

It is clear from Fig. 6 that the concentration of a reacting two variables control the reaction rate, there is a possibil-
species decreases at the electrode surface as the current ity that the current distribution has been calculated over a
is increased. The minimum concentration is zero at the range of operating variables, and we can use these solu-
surface, which corresponds to the maximum rate at which tions directly. If this is not the case, then it is likely that a
the electrodeposition reaction can proceed. The current model must be constructed, and computer techniques are
density corresponding to this maximum rate is called the required in the solution.
limiting current density i l , which can be approximated by Current distribution problems are usually classified ac-
cording to the rate-limiting process:
nFD i c b
i l = , (26)
δ
1. Primary current distribution. The current distribution
where c b is the bulk concentration of copper ions. Pro- is governed solely by the electric field. No other effects
cesses occurring at the limiting current represent the case are considered.
in which only the mass-transfer limitations must be con- 2. Secondary current distribution. Both field effects and
sidered, and the kinetic limitations and ohmic effects can the effects of sluggish reaction kinetics are considered.
be neglected. Because there are numerous correlations for 3. Tertiary current distribution. Field effects, ki-
the limiting current density, many cases of engineering netic limitations, and mass-transfer limitations are all
interest can be treated in an approximate manner. considered.

The complexity of a model increases as we proceed
C. Concentration Overpotential
from the primary to the tertiary distribution and as the
The concentration of reacting species can vary signifi- number of spatial dimensions that are considered in-
cantly across the relatively thin mass-transfer boundary creases. Essentially all published solutions have been re-
layer. When the reacting species are ions, a potential dif- duced to one or two dimensions, and most include only
ference, called the concentration overpotential, arises be- simulations of the primary and secondary current distri-
cause of these gradients. When operation is occurring at butions. For the special case in which only mass transport
less than 90% of the limiting current density, the magni- is limiting, a large number of correlations for the current
tude of the concentration overpotential is relatively small distribution are available.
(of the order of 10 mV). Approximate expressions for
estimating concentration overpotential are available for
B. Primary Current Distribution
binary electrolyte and for the case in which supporting
electrolyte is present. In the latter case the expression for The primary current distribution represents the distribu-
concentration overpotential is tion resulting solely from resistance to current flow in
the electrolyte. Since temperature and concentration vari-

RT i
η c = ln 1 − . (27) ations as well as overpotential are neglected, this type of
nF i l
current distribution is usually easy to calculate.
Laplace’s equation governs the potential distribution
[Eq. (21)]. Since overpotential is ignored, the potential
IV. CURRENT DISTRIBUTION
immediatelyadjacenttotheelectrodesisconstant.Atinsu-
lated surfaces the normal potential gradient must be zero.
A. Classification
These two requirements dictate the boundary conditions
Overall power requirements for an electrolytic process are for the differential equation.
determined from a knowledge of the total current and the Models for phenomena such as heat conduction, fluid
applied potential; however, more detailed knowledge of flow, and diffusional mass transfer are also based on
the distribution of reaction rates (current distribution) is Laplace’s equation. Consequently, many solutions to the
required in an optimization of system performance. Al- potential distribution problems or the analogous problems
though local current densities can usually be measured, it in other fields are available. The current distribution can
is always desirable to develop a mathematical model of the be obtained from the potential distribution through Ohm’s
process and to simulate the effects of changes in operating law [Eq. (22)].
conditions. If the assumptions inherent in the primary current distri-
In making a calculation of the current distribution, we bution model are reasonable for the system being consid-
need to select only the important variables for use in the ered, then a simulation of the system behavior is relatively
simulation. If the geometry is symmetric and only one or straightforward.