This paper focuses on the thermal management of a hydrogen-selective low temperature water-gas-shift (WGS) membrane reactor for simultaneous high-purity hydrogen production and carbon capture. A mathematical model of the reactor is developed consisting of a set of first-order hyperbolic PDEs. Open-loop simulations under a step change in the syngas inlet composition reveal the existence of large temperature gradients along the reactor. A control strategy is proposed whereby multiple distributed cooling zones are placed across the reaction zone in order to regulate the temperature profile. A nonlinear distributed controller is derived, and its performance is evaluated for disturbance rejection and set-point tracking case studies.