Three-dimensional (3D) multi-physics models of co-, counter- and cross-flow planar solid oxide fuel cell (SOFC) stack units are described. The models consider electronic conduction in the electrodes, ionic conduction in the electrolyte, mass transport in the porous electrodes and electrochemical reactions on the three phase boundaries. Based on the analysis of the ionic conducting equation for the thin electrolyte layer, a mathematically equivalent method is proposed to scale the electrolyte thickness with the corresponding change in the ionic conductivity to moderate the thin film effect in the meshing step and decrease the total number of degrees of freedom in the 3D numerical models. Examples of applications are given with typical physical fields illustrated and the characteristic features discussed for co-, counter and cross-flow designs. The 3D models are also used to optimize the rib widths in SOFC stacks as a function of interconnect-electrode contact resistance. (C) 2009 Elsevier B.V. All rights reserved.