Thermal runaway in microwave heated ceramics: A one-dimensional model

Abstract

The heating of a ceramic slab under TM-illumination is modeled and analyzed in the small Biot number regime. The temperature distribution is almost spatially uniform in this limit and its evolution in time is governed by a first-order nonlinear amplitude equation. This equation admits a time independent solution which is a multivalued function of the microwave power. The graph of this steady temperature as a function of the microwave power gives an S-shaped response curve when the electrical conductivity is modeled either as an exponential function of temperature or an Arranius law. The dynamics of the heating process are deduced from the amplitude equation and the multivalued response, and are dependent upon the microwave power and initial conditions. For certain initial conditions and power levels the system evolves to the upper branch of the response curve which corresponds to thermal runaway. Other initial conditions and power levels force the system to evolve to the lower branch and a safe sintering temperature. This heating process can be controlled in some sense by varying the microwave power in time at a rate commensurate with the thermal changes. Specifically, the power is allowed to change in an exponential fashion from a higher to a lower power level. This relationship is turned into a differential equation, which is appended to the amplitude equation to form an autonomous system of the second order. This system is analyzed using a phase-plane method. The analysis shows the existence of a stable manifold which divides the phase plane into two parts: Trajectories in the region above this curve correspond to runaway heating while those below yield stable sintering. Various heating scenarios are presented and discussed.