Formulation of model equations for heating by microwave radiation

Abstract

In general, heating by microwave radiation involves simultaneously solving Maxwell's equations of electromagnetism coupled with the heat equation and for which all electrical, magnetic, and thermal parameters are nonlinearly dependent upon the temperature T. The solution of this coupled nonlinear system, either analytically or numerically, is formidable even if sufficient data existed to properly specify the various electrical and magnetic parameters. Accordingly, the proposition of reducing the problem to studying the heat equation alone with a certain nonlinear body heating Q(T) is an extremely attractive one. In order to identify the body heating term Q(T) as accurately as possible, an exhaustive examination is made of existing experimental results of von Hippel, which have been available for almost 40 years. We show that for the higher frequencies the experimental data are very accurately approximated by both parabolic and linear fits for the body heating Q(T). At the lower frequencies one of two types of exponential curves provides an adequate fit of the experimental data. At intermediate frequencies the experimental data exhibit both parabolic and exponential behavior, and we propose a body heating term Q(T), consisting of a linear combination of the previously used exponential curves, which very accurately fits the data over the full range for which it is available. Finally, we demonstrate the changes in variation of Q(T) with changing frequency but for which insufficient data exist to properly quantify. The principal object of this investigation is to obtain simple analytical forms for Q(T) rather than an application of elaborate curve-fitting procedures, and the mathematical consequences of the specific forms established here will be the subject of future work.