The microwave heating of two-dimensional slabs with small Arrhenius absorptivity

The microwave heating of two-dimensional slabs in a long rectangular waveguide propagating the TE10 mode 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 the magnetic permeability are assumed constant. The governing equations are the forced heat equation and the steady-state version of Maxwell's equation 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 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. In the steady-state limit the approximate analytical solutions are compared with the numerical solutions of the governing equations for various special cases. These are the limits of small and large heat loss and an intermediate case involving radiative heat loss. Results are also presented for a case where differential cooling occurs on the different sides on the slab. An alternative heating scenario, where one end of the waveguide is blocked by a short, is also considered. The approximate solutions are found for this geometry and compared in the small Biot-number limit to Kriegsmann (1997). Also, a control process is presented, which allows thermal runaway to be avoided and the desired final steady state to be reached. Various special cases of the feedback parameters associated with the control process are examined. Keywords:microwave heating; thermal runway; Arrhenius law; control process; small thermal absorptivity