Page 59 - Lindens Handbook of Batteries

2.16 PRINCIPLES OF OPERATION

Electrode geometry design can increase migration slightly by altering electrode curvature. The fields
at convex surfaces are greater than those at flat or concave surfaces, and thus migration is enhanced
at convex curved surfaces.
The third process, diffusion in a concentration gradient, is the most important of the three
processes and is the one which typically is dominant in mass transport in batteries. The analysis of
diffusion uses the basic equation due to Fick which defines the flux of material crossing a plane
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at distance x and time t. The flux is proportional to the concentration gradient and is represented
by the expression:
δ C
=
qD (2.34)
x δ
where q = flux, D = diffusion coefficient, and C = concentration. The rate of change of concentration
with time is defined by

2
δC = D δ C (2.35)
δt δx 2
This expression is referred to as Fick’s second law of diffusion. Solution of Eqs. (2.34) and (2.35)
requires that boundary conditions be imposed. These are chosen according to the electrode’s expected
“discharge” regime dictated by battery performance or boundary conditions imposed by relevant
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electrochemical techniques. Several of the electrochemical techniques are discussed in Sec. 2.6.
For application directly to battery technology, the three modes of mass transport have meaning-
ful significance. Convective and stirring processes can be employed to provide a flow of electroactive
species to reaction sites. Examples of the utilization of stirring and flow processes in batteries are
the circulating zinc/air system, the vibrating zinc electrode, and the zinc-chlorine hydrate battery. In
some types of advanced lead-acid batteries, circulation of acid is provided to improve utilization of
the active materials in the battery plates. Migration effects are in some cases detrimental to battery
performance, in particular those caused by enhanced electric fields (potential gradients) around sites
of convex curvature. Increased migration at these sites tends to produce dendrite formations, which
eventually lead to a short-circuit and battery failure.

2.5.1 Concentration Polarization

Diffusion processes are typically the mass-transfer processes operative in the majority of battery
systems where the transport of species to and from reaction sites is required for maintenance of
current flow. Enhancement and improvement of diffusion processes are an appropriate direction of
research to follow to improve battery performance parameters. Equation (2.34) may be written in
an approximate, yet more practical form, remembering that i = nFq, where q is the flux through a
plane of unit area. Thus,
DA C - C )
(
=
inF B E (2.36)
δ
where symbols are defined as before, and C = bulk concentration of electroactive species, C =
E
B
concentration at electrode, A = electrode area, δ = boundary-layer thickness, that is, the layer at the
electrode surface in which the majority of the concentration gradient is established (see Fig. 2.15).
When C = 0, this expression defines the maximum diffusion current, i , that can be sustained in
E
L
solution under a given set of conditions,
DAC
i = nF δ L B (2.37)
L