Page 263 - High Temperature Solid Oxide Fuel Cells Fundamentals, Design and Applications

240 High Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications

A phenomenological theory, which gives a quantitative relation between
current density and r&t, is known as the Butler-Volmer equation, and is of the
form[2,26]

where j3 is a dimensionless, positive number, less than one (for a one-step charge
transfer process), which is known as the transfer coefficient, and i: is known as
the exchange current density. Note that the relationship between and i is
nonlinear and implicit, that is, it does not allow an explicit determination of qLt
as a function of current density. Rather, the equation gives net current density
for a given vict. However, limiting forms of the Butler-Volmer equation allow one
to express qiCt as a function of current density, i. The low current density and
high current density regimes are described in what follows.
In the low current density limit, it is possible that
I (l-B)zFd << 1. In such a case, the Butler-Volmer equation
RT

or
RT
I&tl (19)

The term $has the units of area specific resistance, Qcm2, and is referred to
as the charge transfer resistance, denoted by R:t, and is given by Rgt = g, An
important point to note here is that a linear relationship between vict ana the
current density, i, in the low current density Iimit does not imply ohmic
relationship, since the response time for the process is long, and is determined by
whatever is the underlying physical process. In the simplest case, the charge
transfer process is describable by a parallel R - C circuit, in which case the time
constant is given as RC. Thus, in DC measurements, the capacitive part is not
reflected. At the same time, in the current interruption experiment, the voltage
drop across the interface is usually not separable from the other time-dependent
parts of the impedance. Measurement of frequency response, however, allows
one to estimate both R and C. More about this is discussed later.
An experimental measurement of q&t as a function of current density
(particularly in the low current density regime) allows one to estimate the R& or
i:. The i: is a measure of the rate of charge transfer process, and depends upon a
number of material properties, microstructure, temperature and also on the
In the high current density regime, 1-1 >> 1 and the Butler-Volmer
atmosphere.

equation can be approximated by