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

The microwave heating of two-dimensional slabs in a long rectangularwaveguide propagating the TE₁₀ mode is examined. The temperaturedependency of the electrical conductivity and the thermal absorptivityis assumed to be governed by the Arrhenius law, while both theelectrical permittivity and the magnetic permeability are assumedconstant. The governing equations are the forced heat equationand the steady-state version of Maxwell's equation while theboundary conditions take into account both convective and radiativeheat loss. Approximate analytical solutions, valid for smallthermal absorptivity, are found for the temperature and theelectric-field amplitude using the Galerkin method. As the Arrheniuslaw is not amenable analytically, it is approximated by a rational-cubicfunction. At the steady state the temperature versus power relationshipis found to be multivalued; at the critical power level thermalrunaway occurs when the temperature jumps from the lower (cool)temperature branch to the upper (hot) temperature branch ofthe solution. In the steady-state limit the approximate analyticalsolutions are compared with the numerical solutions of the governingequations for various special cases. These are the limits ofsmall and large heat loss and an intermediate case involvingradiative heat loss. Results are also presented for a case wheredifferential cooling occurs on the different sides on the slab.An alternative heating scenario, where one end of the waveguideis blocked by a short, is also considered. The approximate solutionsare found for this geometry and compared in the small Biot-numberlimit to Kriegsmann (1997). Also, a control process is presented,which allows thermal runaway to be avoided and the desired finalsteady state to be reached. Various special cases of the feedbackparameters associated with the control process are examined.     

Modelling thermal front dynamics in microwave heating

The formation and propagation of thermal fronts in a cylindricalmedium that is undergoing microwave heating is studied in detail.The model consists of Maxwell's wave equation coupled to a temperaturediffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading-ordertemperature solution of a singular perturbation analysis isused to reduce the system to a free boundary problem. This approximationis then used to derive predictions for the steady-state penetrationand profiles of the temperature and electric fields. These solutionsare valid for arbitrary values of the electric conductivity,and thus extend the previous (small conductivity) results foundin the literature. A quasi-static approximation for the electric field is thenused to obtain an ordinary differential equation for the relaxationdynamics to the steady state. This equation appears to accuratelydescribe the time scale of the electric field's evolution bothwith and without the presence of a strongly coupled temperaturefront, and may be of wider interest than the model for microwaveheating studied here.