In a catalytic reactor the solid heat capacity is relatively high compared to that of the gas so that even relatively slow periodic flow reversal approximates infinite switching. An internally consistent model is developed which assumes plug-flow of the gas, neglects conduction in the direction of flow and, by having different gas and solid temperatures, allows heat transfer between the gas and solid. It is shown that this reactor can in certain situations achieve higher conversions in a given volume of catalyst than an adiabatic plug-flow reactor for exothermic reversible reactions. It is further shown that these equations also model the limit of a more industrially desirable counter-current steady-state reactor.