Page 243 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN005B-205 June 15, 2001 20:24

150 Electrochemical Engineering

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migration, the movement of charged species under the in- v ·∇c = D∇ c, (23)
fluence of an electric field. The second term is the flux
where
due to diffusion, and the third term is the flux due to con-
vection. This expression is strictly correct for extremely z + u + D − − z − u − D +
D = . (24)
dilute solutions; however, it is generally applied to more z + u + − z − u −
concentrated solutions and used as a reasonable engineer- The convective diffusion equation is analogous to equa-
ing approximation. tions commonly used in dealing with heat and mass trans-
The current is due to the motion of charged species: fer. Similarly, if migration can be neglected in a multicom-

i = F z i N i . (18) ponentsolution,thentheconvectivediffusionequationcan
be applied to each species,
i
At steady state the net input of a reacting species in the v ·∇c i = D i ∇ c i . (25)
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electrolyte is zero. If we assume that reactions occur only
at the electrode surface, then the material balance can be The hydrodynamic conditions influence the concen-
expressed as tration distribution explicity through the velocity term
present in the convective diffusion equation. For certain
∇· N i = 0. (19)
well-defined systems the fluid flow equations have been
Because the electrical forces between charged species solved, but for many systems, especially those with turbu-
are so large, the positive and negative particles have a lent flow, explicit solutions have not been obtained. Conse-
strong tendency to associate. On a macroscopic level, quently, approximate techniques must frequently be used
charge separation cannot be detected in the bulk elec- in treating mass transfer.
trolyte, and the solution is electrically neutral:

z i c i = 0. (20) B. Mass-Transfer Boundary Layer
i
Consider the process of plating copper on a plane elec-
These four equations form the basis for a description of
trode. Near the electrode, copper ions are being discharged
the mass transport in electrolytic solutions. To solve these
on the surface and their concentration decreases near the
equations, we must calculate the bulk solution velocity
surface. At some point away from the electrode, the cop-
from a knowledge of the fluid mechanics.
per ion concentration reaches its bulk level, and we ob-
Solutions to this system of equations depend on the
tain a picture of the copper ion concentration distribution,
cell geometry and on the boundary conditions; therefore,
shown in Fig. 6. The actual concentration profile resem-
generally valid solutions cannot be obtained. With certain
bles the curved line, but to simplify computations, we as-
simplifying assumptions, the equations reduce to familiar
sume that the concentration profile is linear, as indicated
forms, and solutions can be obtained for large classes of
by the dashed line. The distance from the electrode where
problems.
the extrapolated initial slope meets the bulk concentration
If temperature and concentration variations are ne-
line is called the Nernst diffusion-layer thickness δ.For
glected, then an expression for the potential distribution
order of magnitude estimates, δ is approximately 0.05 cm
in the bulk electrolyte is given by Laplace’s equation,
in unstirred aqueous solution and 0.01 cm in lightly stirred
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∇ φ = 0. (21) solution.
If we neglect the overpotential at the electrodes, then the
boundary conditions for solving this problem are the con-
stant electrode potentials. This type of problem has exact
analogs in electrostatics, and many generalized solutions
for symmetric configurations are available. In this type of
problem, the current density is proportional to the poten-
tial gradient, and the current distribution can be calculated
from Ohm’slaw:
i =−κ∇φ. (22)
For the solution of a salt composed of two ionizable
species (binary electrolyte), the four basic equations can
FIGURE 6 Nernst diffusion-layer model. The solid line represents
be combined to yield the convective diffusion equation for the actual concentration profile, and the dashed line for c 0 the
steady-state systems: Nernst model concentration profile.