Microwave Heating and Joining of Ceramic Cylinders: A Mathematical Model

A thin cylindrical ceramic sample is placed in a single mode microwave applicator in such a way that the electric eld strength is allowed to vary along its axis. The sample can either be a single rod or two rods butted together. We present a simple mathematical model which describes the microwave heating process. It is built on the assumption that the Biot number of the material is small, and that the electric eld is known and uniform throughout the cylinder's cross-section. The model takes the form of a nonlinear parabolic equation of reaction-di usion type, with a spatially varying reaction term that corresponds to the spatial variation of the electromagnetic eld strength in the waveguide. The equation is analyzed and a solution is found which develops a hot spot near the center of the cylindrical sample and which then propagates outwards until it stabilizes. The propagation and stabilization phenomenon concentrates the microwave energy in a localized region about the center where elevated temperatures may be desirable. The work of the rst authorwas partially supportedby the NationalAeronautics and Space Administration under NASA Contract No. NAS1-19480 while in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665, and by the National Science Foundation under grant No. DMS-9403798. The work of the second author was supported by the Air Force O ce of Scienti c Research under grant No. AFOSR 91-0252, the Department of Energy under grant No. DE-FG02-94ER25196, and the National Science Foundation under grant No. DMS-9305828.