Page 257 - High Temperature Solid Oxide Fuel Cells Fundamentals, Design and Applications

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

simplified equivalent circuit can be used to describe the process, using what is
known as the Warburg element, which consists of a number of resistors and
capacitors [5]. The presence of capacitors ensures that the response time or time
constant is non-zero. Since the relevant time dependences are not describable by
simple first order kinetics, it is not appropriate to describe response time as a time
constant. Nevertheless, a characteristic time can be defined, which depends on
electrode thickness, electrode microstructure and the representative diffusivity.
In terms of physically measurable parameters, analytical expressions for
anodic concentration polarisation have been derived which allow its explicit
determination as a function of a number of parameters. One of the important
parameters is the anode-limiting current density, which is the current density at
which the partial pressure of the fuel, e.g. H2, at the anode/electrolyte interface,
is near zero such that the cell is starved of fuel. If this condition is realised during
operation, the voltage precipitously drops to near zero. This anode-limiting
current density, i,,, has the following form [6]

where D,(,J) is the effective gaseous diffusivity through the anode, and I, is
the anode thickness. The effective anode diffusivity contains the binary
diffusivity of the relevant species, namely Hz and H20, DH*-H,o, the volume
fraction of porosity, VVca), and the tortuosity factor, t, [3,4]. If the fuel contains
hydrocarbons, multi-component nature of gaseous diffusion must be addressed.
The tortuosity factor is a measure of the tortuous nature of the anode through
which diffusion must occur. In very fine microstructures, the tortuosity as a
phenomenological parameter may include effects of Knudsen diffusion, surface
diffusion, and possible effects of adsorption/desorption. The anodic
concentration polarisation is then of the form [6]

(10)

Note that as the current density approaches the anode limiting current
-
density, that is when i --f ias, the first term approaches infinity. The maximum
value of q:onc is limited by the OCV. Thus, the maximum achievable current
density will always be less than ias. The dependence of the anodic concentration
polarisation given by equation (10) on various parameters can be qualitatively
described as follows: From the standpoint of physical dimensions, and
microstructural parameters, the lower the volume fraction porosity, the higher
the tortuosity factor, and the greater the anode thickness, the higher is q;onc.
From the standpoint of fuel gas composition, the lower the partial pressure of
hydrogen, p&, the higher is the q:onc. The temperature dependence is
complicated. It is seen that i,, cx T1l2, since D,(,J) cx T3/2, which would mean $one
increases as temperature decreases. At the same time, as seen from equation
(10). Y $ ~ ~ ~linearly dependent on temperature, which would mean q:onc
is