1346 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 5, MAY 2002 Modeling Microwave and Hybrid Heating Processes Including Heat Radiation Effects Jens Haala, Member, IEEE, and Werner Wiesbeck, Fellow, IEEE Abstract—This paper presents an efficient simulation tool for conventional, microwave, and combined heating. Two heat-transfer mechanisms are included: conductive and radiant heat transfer. The conductive heat transfer is modeled by a finite-difference algorithm. A modeling technique for radiant heat transfer in nonuniform grids has been developed and is here presented for the first time. For the radiant heat transfer a finite-difference scheme is not applicable, as radiation from a material surface is not bounded to the immediate vicinity, as is conductive heat transfer. Therefore, ray optical methods are used. Rays connecting mutually visible surfaces are obtained by a new fast method. Necessary, but acceptable simplifications allow fast computations. The algorithms are integrated conveniently together with an electromagnetic finite-difference time-domain program to one simulation tool. Representative simulations are presented for an oven heated conventionally, by microwaves, and by a combination of both. Index Terms—FDTD, heat radiation, microwave heating. I. INTRODUCTION T HERMAL modeling is mandatory for the optimization of heating processes in ovens. Especially in combination with microwaves, the heating process has to be carefully designed to achieve a fast and uniform heating. Various re- ports [1]–[10] reported electromagnetic-field computations of microwave ovens, but only few authors [11], [12] include a thermal model in their procedures. To the authors’ knowledge, all models consider only heat transfer by conduction, while radiation is usually neglected. This simplification becomes questionable at higher temperatures. In fact, with increasing temperature, radiant heat exchange becomes more and more important since the energy emitted from a material surface increases proportional to . In contrast, energy transported by heat conduction is only proportional to . Radiant heat transfer eventually prevails over conductive heat transfer. This paper specifically includes radiant heat transfer and, therefore, closes the gap of neglected radiant heat exchange. Pure thermal problems are mostly simulated by finite-el- ement modeling, some of which include radiation. Heating by microwaves is sometimes considered, but the variation of the electromagnetic field with increasing temperature due to changing material properties is not. Additionally, these models normally perform only steady-state calculations. Manuscript received October 28, 2000. J. Haala is with Marconi Communications Software Systems, D-71522 Back- nang, Germany (e-mail: J.Haala@ieee.org). W. Wiesbeck is with the Insitut für Höchstfrequenztechnik und Elek- tronik, Universität Karlsruhe (TH), D-76128 Karlsruhe, Germany (e-mail: Werner.Wiesbeck@etec.uni-karlsruhe.de). Publisher Item Identifier S 0018-9480(02)04056-5. For the optimization of ovens, one needs to determine the dy- namic heat process. Hence, a combination of thermal and elec- tromagnetic simulations must be used. Both radiation and the influence of increasing temperature on the electromagnetic field must be considered. The finite-difference time-domain (FDTD) method has proven to be an excellent algorithm for the calculation of electromagnetic fields, especially in closed structures like ovens. This method needs fewer computational resources as methods in the frequency domain. Broad-band calculations are easily performed. The FDTD modeling of thermal processes is also economical in memory usage. A combination with an electromagnetic FDTD leads to a very efficient and powerful simulation tool. Self-consistent modeling is possible, as well as analyzing a dynamic heating process. Both the conductive thermal and electromagnetic algorithm are of local character as the temperature and fields in one dis- cretization cell are only related to their neighboring cells. This local scheme applies only to conductive heat transfer. When considering radiant heat transfer, energy may be transported through the whole computational space. Mutually visible sur- faces, view factors, and material parameters have to be deter- mined and used for the calculation. This usually means high computational effort. However, the proposed method reduces this effort considerably. The main task when including radiant heat transfer is to de- termine surface pairs that are mutually visible and exchanging radiant energy. A very fast algorithm has been developed that is optimized for detecting mutually visible surfaces in a rectan- gular nonuniform grid. This algorithm determines surface pairs very quickly. They need to be allocated and stored only once before the calculation starts. Based on this initial step, the cal- culation of the resulting temperature at each time step is very fast and easy. After discussing the mechanism and modeling of the conduc- tive and radiant heat transfer, an efficient algorithm for the com- putation of view factors and transfer rays is presented. The in- clusion of the electromagnetic simulation is then shown. A hy- brid oven simulation verifies the applicability of the method. II. THERMAL MODELING The first sentence of thermodynamics states that if tempera- ture varies within a closed environment, energy has to be trans- ported over the borders of that environment. This is expressed by (1) 0018-9480/02$17.00 © 2002 IEEE