Modeling hotspot dynamics in microwave heating

Abstract

The formation and propagation of hotspots in a cylindrical medium that is undergoing microwave heating is studied in detail. A mathematical model developed by Garcia-Reimbert, C., Minzoni, A. A. and Smyth, N. in Hotspot formation and propagation in Microwave Heating, IMA, Journal of Applied Mathematics (1996), 37, p. 165-179 is used. The model consists of Maxwell's wave equation coupled to a temperature diffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading order temperature solution of a singular perturbation analysis is used to reduce the system to a free boundary problem. This approximation accurately predicts the steady-state solutions for the temperature and electric fields in closed form. These solutions are valid for arbitrary values of the electric conductivity, and thus extend the previous (small conductivity) results of Garcia-Reimbert et.al. A time-dependent approximate profile for the electric field is used to obtain an ordinary differential equation for its relaxation to the steady-state. This equation appears to accurately describe the time scale of the electric field's evolution even in the absence of a temperature front (with zero coupling to the temperature), and can be of wider interest than the model for microwave heating studied here. With sufficiently small thermal diffusivity and strong coupling, the differential equation also accurately describes the time evolution of the temperature front's location. A closed form expression for the time scale of the formation of the hotspot is derived for the first time in the literature of hotspot modeling. Finally, a rigorous proof of the existence of steady-state solutions of the free boundary problem is given by a contraction mapping argument.