Page - 44 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating termcanbefurtherwrittenas Pcd= ∂ ∂x ( κxl 2∂T ∂x ) · l+ ∂ ∂y ( κyl 2∂T ∂y ) · l = ∂ ∂x ( κ ∂T ∂x ) · l3+ ∂ ∂y ( κ ∂T ∂y ) · l3 =∇·(κ∇T) · l3, (3.9) where∇ is the two-dimension differential operator. If the sign of ∇·(κ∇T) is positive, the energy goes from other parts of the load to thiscell, andviceversa. •Microwaveheatingpower According to equation 2.6, the microwave heating power density iswrittenas pmw= 1 2 σe(T) ∥∥∥−→E∥∥∥2 , (3.10) whereσe(T) is theelectricalconductivityoftheheatedloadattem- perature T (equation 2.6). The electric field strength as well as the entire EM field distribution is determined by the well-known Maxwell’s equations [Mer98] [Y+66]. Unlike numerical simula- tionapproaches, theexactsolutionsoftheMaxwell’sequationsare not required for the temperature control system. Instead the mi- crowave heating power densities are approximated and estimated using different system identification methods that will be intro- duced later this chapter. For a small cell as defined in figure 3.1, it is assumed that the electric field is constant within the cell, and correspondingly themicrowaveheatingpower isgivenby Pmw=pmw · l3. (3.11) Replacing the corresponding terms in equation 3.1 by above expres- sions, thereis thecompleteheat transferequationforanylocation(cu- 44 biccell) onthesurfaceof theheated load