The steady-state microwave heating of slabs

The steady-state microwave heating of finite one and two dimensional slabs is examined. The temperature dependency of the electrical conductivity and the thermal absorptivity is assumed to be governed by the Arrhenius law, while both the electrical permittivity and magnetic permeability are assumed constant. The governing equations are the steady-state versions of the forced heat equation and Maxwell's equations while the boundary conditions take into account both convective and radiative heat loss. Approximate analytical solutions, valid for small thermal absorptivity, are found for the steady-state temperature and the electric-field amplitude using the Galerkin method. As the Arrhenius law is not amenable analytically, it is approximated by a rational-cubic function. At the steady-state the temperature versus power relationship is found to be multivalued; at the critical power level thermal runaway occurs when the temperature jumps from the lower (cool) temperature branch to the upper (hot) temperature branch of the solution. The approximate analytical solutions are compared with the numerical solutions of the governing equations in the limits of small and large heat-loss and also for an intermediate case involving radiative heat-loss.