The gas channel geometry in solid oxide fuel cells (SOFCs) influences the reacting thermo-fluid process and, thus, overall cell performance. This paper presents a dimensionless approach to the study of the transport phenomena in the gas channels of planar anode-supported proton-conducting SOFC. Out-of-scale modeling reduces the number of variables that should be investigated and offers generalized results, giving insight into similar fuel cells. A 2D numerical model for the multiphysics process in SOFC is developed. A dimensionless form of the governing equations is derived in order to identify the dimensionless quantities that characterize the transport phenomena in SOFC. Reynolds, Peclet, and Sherwood are the important parameter groupings of flow channels that influence mass and temperature distribution. The efficacy of the computational fluid dynamic model is confirmed by comparing simulated results with experimental data from the literature. The effect of fuel and air channels’ dimensionless parameters on cell performance is discussed. Similar changes in fuel and air channels exert various influences on SOFC electrical performance. It is found that reducing Pe in the fuel channel improves power generation. However, Sh and Re reduction effect neutralize the increase in power generation due to Pe reduction in the air channel.