Page 319 - High Temperature Solid Oxide Fuel Cells Fundamentals, Design and Applications

Cell, Stack and System Modelling 295

distribution and gas flow rate. Undesirable or even dangerous operating
conditions may arise from the flow distribution [I]. Due to differences in
coefficients of thermal expansion, temperature gradients during transient or
stationary operation cause stresses that may lead to failure. Interdiffusion of
materials used for the anode, the electrolyte, and the cathode may lead to
gradual performance degradation. In order to calculate flow and temperature,
the conservation laws in fluid mechanics are used [2].

77.2.7 Mass Balance
A species' mass in a reacting mixture of gases is determined by solving the species
continuity equations:

api/dt + O.[pi(v + Ui)] = mi (1)

where pi is the species density, v is the fluid velocity, Ui is the species diffusion
velocity, t is time, and mi is the rate of production of species i due to chemical
(or electrochemical) reactions. The mass flux of species i (pi Ui) due to diffusion
can be approximated for most applications using Fick's law:

piU = -PD~VG (2)

where ci is the species mass fraction (pi/p), and Dim is the multicomponent
diffusion coefficient of species i in the mixture. Dim is a weighted average of
binary diffusion coefficients Dij, that is, of the diffusion coefficients of species i
with respect to each of the other species, j. Depending on the composition of the
gas mixture, Dim can often be assumed to remain fairly constant. If there is one
dominant species, k, in the mixture, the multicomponent diffusion coefficient Dim
may often be approximated by the binary diffusion coefficient Dik.

77.2.2 Conservation of Momentum

Mass balances must be used with the flow pattern (known apriori from theory
or experimental measurements) to establish species concentrations and fluxes
at any point in the fuel cell. When the flow pattern is a priori unknown,
conservation-of-momentum equations (also called equations of motion) must be
used with mass balance equations to establish the velocity and concentration
profiles. Conservation of momentum for gases leads to the following equations
(Navier-Stokes equations), in which k represents one of the three orthogonal
directions in the coordinate system (x, y, and 2):
d(pVk)/at + v'(pvkv) = pgk - ap/axk + V'(/-kvvk) + Qk f (3)
rk
Here P is the pressure, g is the acceleration due to gravity, and pe is the effective
viscosity. The term zk represents other than Newtonian viscous losses and may be