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Electrochemical Engineering 153

distributions.Forafewsystems,importantinelectrochem-
ical applications, the detailed fluid behavior is known.
Even with this knowledge, finding a solution to the current
distribution problem for all but the simplest geometries is
a formidable task. The hydrodynamic conditions for lam-
inar flow at a rotating disk and between plane parallel
electrodes have been quantitatively described. These are
among the few systems for which fairly rigorous tertiary
current distributions have been obtained.
When a system is operating at the limiting current,
rather than at an appreciable fraction of the limiting cur-
rent, the problem is very much simplified. Such problems
can be classified as mass-transport limited. Usually, the
limiting current density is correlated with dimensionless
numbers. Most forced-convection correlations take the
form
Sh = f (Re, Sc), (29)

where Sh (Sherwood number) is related to the limiting
current density, Re (Reynolds number) characterizes the
hydrodynamics, and Sc (Schmidt number) is related to FIGURE 10 Current distribution on a disk electrode. The primary
transport properties of the fluid. Both laminar and turbu- current distribution approaches infinity at the junction of the elec-
lent flow problems are treated over a wide range of oper- trode and the coplanar insulator. The secondary current distri-
bution is more uniform. Average current density is i avg and the
ating and physical parameters in this manner.
electrode radius r 0 .
E. Current Distribution Characteristics As the disk spins, it pumps fluid to the surface. For laminar
Several cell configurations are common in electrochemi- flow, analytical solutions describing the fluid motions have
cal research and in industrial practice. The rotating disk been obtained. In modeling the system the disk is assumed
electrode is frequently used in electrode kinetics and in to be immersed in a large volume of electrolyte with the
mass-transport studies. A cell with plane parallel elec- counterelectrode far away.
trodes imbedded in insulating walls is a configuration used The primary potential distribution is, by definition, uni-
in research as well as in chemical synthesis. These are form adjacent to the electrode surface, but the current dis-
two examples of cells for which the current and poten- tribution is highly nonuniform (Fig. 10). It is a general
tial distributions have been calculated over a wide range characteristic of the primary current distribution that the
of operating parameters. Many of the principles governing current density is infinite at the intersection of an elec-
current distribution are illustrated by these model systems. trode and a coplanar insulator. This condition obtains at
The rotating disk electrode appears in Fig. 9. It consists the periphery of the disk electrode, and the current density
of a cylindrical electrode imbedded in an insulating disk. becomes infinite at that point. Additional resistance due
to kinetic limitations invariably reduces the nonuniformity
of the current distribution. In this system the current dis-
tribution becomes more uniform as the Wagner number
increases. Theoretically, the current distribution is totally
uniform as the Wagner number approaches infinity.
In general, the effects of mass-transport limitations are
not as easy to characterize. The direction of fluid flow, the
flow regime, and the local fluid velocity all influence the
current distribution. Fluid flow to the rotating disk is un-
usual in that fluid velocity normal to the disk is dependent
only on the normal distance from the disk surface, and not
on radial distance. Because the disk surface is uniformly
FIGURE 9 Rotating disk electrode. Fluid is drawn uniformly to
the electrode surface, and the reactant concentration depends accessible to incoming reactants, mass-transport limita-
only on the normal distance from the electrode. tions tend to reduce the current density in regions of high