Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics


Semi-analytical models are presented describing the microwave heating of materials in waveguides in a number of different scenarios. Coupling occurs between the heating of the slab and the microwave radiation via temperature dependency the electrical conductivity and the thermal absorptivity (which are related to dielectric constant and dielectric loss). Two forms of temperature dependency considered, an Arrhenius-type law, which is bounded for large temperatures, and a quadratic relationship. The governing equations are the forced heat equation and a steady-state version of Maxwell's equations while the boundary conditions on the material take into account both convective and radiative heat-loss. Semianalytical solutions, valid for small thermal absorptivity, are developed for temperature and the electric-field amplitude using the Galerkin method. For the heating scenarios considered in this thesis an excellent comparison is obtained between the semi-analytical and numerical solutions.

In chapter 2 the heating of a three-dimensional block in an infinitely long waveguide is considered, using both the TE10 and TM11 waveguide modes. 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 semi-analytical solutions are compared with the numerical solutions of the governing equations in the limits of small and large heat-loss, an excellent comparison obtained. Also, a single mode assumption is used here, in which it is assumed that the heating does not generate any other waveguide modes in the slab or waveguide. The usefulness of this assumption is examined by comparison with multi-mode numerical solutions.

In the third chapter, linear feedback control is examined, as a method to prevent thermal runaway. Semi-analytical solutions are developed for the heating of one and two dimensional slabs. A local stability analysis is performed on semi-analytical solution and the region of parameter space is found, in which limit-cycles can occur. This semi-analytical parameter region is found to closely correspond to that found numerically. The amplitude and period of the limitcycles are also accurately predicted by the semi-analytical model.

In the fourth chapter, the steady-state heating of a two-dimensional slab by the TE10 mode in a microwave cavity, which has an iris with an variable aperture is considered. A semi-analytical model is developed to examine the effect of varying the aperture width. It is found that an optimal aperture setting and short position exists which minimises the input power needed to obtain a given processing temperature.