Page 344 - High Temperature Solid Oxide Fuel Cells Fundamentals, Design and Applications

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

subject to the electrical conservation equation (under assumption of
electroneutrality):

(V-i) = 0 (35)
where i is the current density vector. This yields for the potential distribution
Laplace’s equation:

(02Q) = 0 (36)

with the appropriate boundary conditions for conducting and nonconducting
boundaries. Equations (35) and (36) may be applied separately to the ionic
current and its associated electric potential, respectively, and to the
electronic current and its associated electric potential. This is very helpful in
formulating the electrochemical rate at each point of the electrode/electrolyte
interface by equating the local potential difference @electronic - @ionic with the
total overpotential q = V - E,, at that point of the interface. This coupling of
the potential distribution, which obeys Eq. (36), with the electrochemistry and
thermodynamics of the electrode reaction leads to a generalised potential
balance equation, of which Eq. (7) is a specific form valid for thin pIanar cells.
In a similar manner, the species mass balance equations, Eqs. (l), (2), may be
coupled with the electrochemical rate at each point of the reaction zone (at or
near the TPB). In the continuum-level modelling discussed in Section 11.2, the
concentration polarisation of the electrode, qconc, was related to a limiting
current of the reactant, e.g., Eq. (9). A more fundamental and general expression
for the concentration overpotential (the term ‘overpotential’ denotes exclusively
the local polarisation) at any point of the electrode reaction zone is the so-called
Nernst equation; for example

?k = RT/nF 1n[(C,.eIectrode/Gr,bulk)/(Cp,eIectrode/,~~~~)] (3 7)

This is valid for a simple electron transfer reaction r + ne- = p but may easily be
generalised. Because the fundamental mass balance equations, Eqs. (l), (2), in
the absence of convection become diffusion equations, the solution of the
diffusion equations for species r and p yields the concentration overpotential, qc,
at any point of the reaction zone.
Once the local concentration overpotential is known, the activation
overpotential, qa, is obtained by subtracting qc from total q. The local activation
overpotential is the actual driving force of the electrochemical reaction. It is
related to the local current density at any point of the reaction zone by an
electrochemical rate equation such as the Butler-Volmer equation (Eq. (loa)).
Therefore, the rate equation, the Nernst equation (Eq. (37)), and the potential
balance in combination couple the electric field with the species diffusion field. In
addition, the energy balance applies also at the electrode level. Although this
introduces another complication, a model including a temperature profile in the
electrode is very useful because heat generation occurs mainly by
electrochemical reaction and is localised at the reaction zone. while the