The microwave heating of three-dimensional blocks: semi-analytical solutions

The microwave heating of three-dimensional blocks, by the transverse magnetic waveguide mode TM11, is considered in a long rectangular waveguide. The governing equations are the forced heat equation and a steady-state version of Maxwell's equations, while the boundary conditions take into account both convective and radiative heat loss. Semi-analytical solutions, valid for small thermal absorptivity, are found using the Galerkin method. The electrical conductivity and the thermal absorptivity are assumed to be temperature dependent, while both the electrical permittivity and magnetic permeability are taken to be constant. Both a quadratic relation and an Arrhenius-type law are used for the temperature dependency. As the Arrhenius-type law is not amenable analytically, it is approximated by a rational-cubic function. A multivalued steady-state temperature versus power relationship is found to be possible for both types of temperature dependency. 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 semi-analytical solutions are compared with numerical solutions of the governing equations for various special cases such as the limits of small and large heat loss at the edges of the block. An excellent comparison is obtained between the semi-analytical and numerical solutions, on both temperature branches for the Arrhenius-type law. For the quadratic temperature dependency the comparison is excellent on the low branch but the semi-analytical theory significantly underpredicts the temperature on the upper solution branch.