Microwave-induced combustion: A one-dimensional model

A model for the heating and ignition of a combustible solid by microwave energy is formulated and analysed in the limit of small inverse activation energy and small Biot number B. The high activation energy limit implies that the heating process is effectively inert until the temperature within the material reaches a critical ignition value, while the small Biot number limit implies that during this stage spatial variations in temperature throughout the material are always small. Analysis of the inert stage includes determination of the dynamics of inert hot-spots. As the ignition temperature is approached chemical energy is released rapidly in the form of heat, and the evolution then enters an ignition stage which develops on a fast time-scale. A reduced system is derived governing small-amplitude departures of the temperature from the inert value during the ignition stage under the significant scaling relation between the expansion parameters, which is shown to be ϵ ∼ B. This reduced system recovers both of the familiar canonical systems describing (i) localized ignition by in-depth absorption of radiation and (ii) spatially homogeneous blow-up, in the limits of small and large values of µ = ϵ/B, respectively. Numerical integration of the reduced system in parameter regimes relevant to production of materials by combustion synthesis shows that ignition can occur either on the boundary or in the interior of a solid sample, and that there are regimes where the ignition site changes abruptly with variation of system parameters.