The Rolls-Royce integrated-planar solid oxide fuel cell (IP-SOFC) consists of ceramic modules with electrochemical cells printed on the outer surfaces. The cathodes are supplied with oxygen from air flowing over the outside of the module and the anodes are supplied with fuel diffusing from the internal gas channels. Natural gas is reformed into hydrogen in a separate reformer module of similar design except that the fuel cells are replaced by a reforming catalyst layer. The performance of the modules is intrinsically linked to the behavior of the gas flows within their porous structures. The multi-component convective-diffusive flows are simulated using a new theory of flow in porous material, the cylindrical pore interpolation model. The effects of the catalyzed methane reforming and water-gas shift chemical reactions are also considered using appropriate kinetic models. It is found that the shift reaction, which is catalyzed by the anode material, has certain beneficial effects on the fuel cell module performance. The shift reaction enables the fuel cells to make effective use of carbon monoxide as a fuel when the supplied fuel has become depleted of hydrogen. In the reformer module the kinetics of the reaction make it difficult to sustain a high methane conversion rate. Although the analysis is based on IP-SOFC geometry, the modeling approach and general conclusions are applicable to other types of SOFCs.