A limiting case of solid-fluid heat transfer is examined, in which a gas passes through a porous wall of high specific surface with heat generation within the solid material. Dimensionless temperature profiles in the wall are presented in terms of the rate of heat generation, rate of flow, and thermal properties of the gas and solid. The pressure drop across the wall is approximated by using an average wall temperature and assuming isothermal conditions. Temperature profiles, pressure drops, and pumping-power/power-output ratios are calculated for the hypothetical case of a heated graphite wall cooled by helium. It is found that the thermal dependence of the gas viscosity produces a minimum in the pressure-drop versus flow-rate curve, and it appears that favorable pumping-power/power-output ratios can be obtained by the use of high pressures. The problem of temperature stability in a gas-cooled porous solid is pointed out and the need for experimental work emphasized. Use of the sweat-cooling technique for high-pressure, high-temperature ducts is suggested.