A generalized, three-dimensional (3D) mathematical model of solid oxide fuel cells (SOFCs) for various geometries is constructed in this paper. A finite-volume method is applied to calculate the electric characteristics, which is based on the fundamental conservation law of mass, energy and electrical charge. The electrical potential distribution, the current density distribution, the concentrations distribution of the chemical species and the temperature profile are calculated by solving the governing equations of a single-unit model with double channels of co-flow and counter-flow pattern using the commercial computational fluid dynamic software Fluent. The internal steam reforming and the water shift reactions are taken into account in the mathematical model. The Knudsen diffusion is considered for computation of the gases diffusion in the porous electrodes and the concentration overpotential. The Butler-Volmer equation and the function of the reaction gases composition for the exchange density are used in the model to analyze the activation overpotential. Numerical simulations are performed for a planar geometry solid oxide fuel cell and the detailed features of the temperature, the electrical potential distribution and the gases composition are illustrated. The simulation results agree well with the Benchmark results for planar configuration. With the simulated temperature profile in the planar SOFC, the finite-element method is employed to calculate the thermal stress distribution in the planar solid oxide fuel cell. A 3D finite-element model consists of positive electrode-electrolyte-negative electrode (PEN) and interconnects assembly is constructed by using commercial finite-element code Abaqus. The effects of temperature profile, electrodes and electrolyte thickness, and coefficients of thermal expansion (CTE) mismatch between components are characterized. The calculated results indicate that the maximum stress appears on the electrode and electrolyte interface. The value and distribution of the thermal stress are the functions of the applied materials CTE, applied temperature profiles and the thicknesses of electrode and electrolyte. The calculated results can be applied as the guide for the SOFC materials selection and the SOFC structure design.

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