The invention provides a fuel cell including an elongate substrate the length of which is the greatest dimension such that the elongate substrate exhibits thermal expansion along a dominant axis that is coextensive with the length. A reaction zone is provided along a first portion of the length for heating to an operating reaction temperature, and at least one cold zone is provided along a second portion of the length that remains at a low temperature below the operating reaction temperature when the reaction zone is heated. An electrolyte is disposed between an anode and a cathode in the reaction zone and the electrolyte is monolithic with an interior ceramic support structure of the elongate substrate. The anode and cathode each have an electrical pathway extending to an exterior surface of the at least one cold zone for electrical connection at low temperature.