Page - 46 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating For instance, a rectangular dielectric foil is heated to 350 K, and the temperature of surrounding air is 300 K. For the dielectric materi- als, it is assumed the corresponding heat conductivities are below 10 W/(m ·K). For example, the conductivity of silicon rubber (the mate- rial that ismostlyusedinourexperiments) isbetween0.2W/(m·K) to 1.3 W/(m ·K) [Sil12], and the conductivity of CFRP using epoxy ma- trices is 5W/(m ·K) to 7W/(m ·K) in plane [Tia11]. The second spa- tial derivative of the temperature is difficult to be defined. Based on themoderatelyhomogeneousEMfielddistributionwithinHEPHAIS- TOS, it is feasible to use±0.1K/cm2 as the second spatial derivatives. Otherparametersareassumedas %= 0.9, h=20W/(m2 ·K), Using above values, for a simple cell on the surface (1 cm×1 cm×1 mm)theratesof thermalconduction, convectionandradiationcanbe calculatedas ∣∣Pcd∣∣<0.01W, Pcv= −0.1W, Prd= −0.035W. (3.14) Above results show that the dissipated power of either thermal con- vection or thermal radiation is much larger than that of thermal con- duction, therefore the thermal conduction term in equation 3.13 can be omitted. Furthermore, the dissipated power of both thermal con- vectionandthermalradiationareproportional tothetemperaturedif- ferenceT−Ta and the area of the unit cell l2. In this case, the overall dissipatedpower is rewrittenas Pdiss≈Pcv +Prd ≈ − h˜(T)(T−Ta) · l2, h˜(T) :=h+%σ′(T2+T2a) ·(T+Ta), (3.15) where h˜(T) isdefinedas theoverallpowerdissipationcoefficient that contains effects of both thermal convection and thermal radiation. Substituting the above equation 3.15 into equation 3.12, the resulted system models has a similar ARMAX form like the models used in [RCVI99]and[SBA00]. 46