Fuel cells are promising for future energy systems, because they are energy efficient and able to use renewable fuels. A fully coupled computational fluid dynamics (CFD) approach based on the finite element method, in twodimensions, is developed to describe a solid oxide fuel cell (SOFC). Governing equations for, gas-phase species, heat momentum, ion and electron transport are implemented and coupled to kinetics describing electrochemical and internal reforming reactions. Both carbon monoxide and hydrogen are considered as electrochemical reactants within the anode. The predicted results show that the current density distribution along the main flow direction depends on the local concentrations and temperature. A higher (local) fraction of electrochemical reactants increases the Nernst potential as well as the current density. For fuel mixtures without methane, the cathode air flow rate needs to be increased significantly to avoid high temperature gradients within the cell as well as a high outlet temperature.