Page - 45 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling Pab︷ ︸︸ ︷ ρcp dT dt · l3= Pcd︷ ︸︸ ︷ ∇·(κ∇T) · l3 Pcv︷ ︸︸ ︷ −h(T−Ta) · l2 Prd︷ ︸︸ ︷ −%σ′(T4−T4a) · l2 + 1 2 σe(T) ∥∥∥−→E∥∥∥2 · l3︸ ︷︷ ︸ Pmw . (3.12) It shouldbenotedthat in theaboveequationthepowerabsorptionor dissipation isalreadyreflectedbythesignof individual terms. 3.2.2. ApproximationandSimplification Theaboveequation 3.12isacomplicatedparabolicpartialdifferential equation (PDE). This equation can be only solved in special cases like the one-dimensional scenario. For a microwave heating system like HEPHAISTOS, where the electromagnetic field distribution is com- plex and varying all the time during the heating, a general solution is impossible tobeobtained. Inorder tomakeequation 3.12suitable for a control system, approximation and simplification have to be done here, regardingdifferent termsontheright-handside. Approximationsof thethermodynamic terms From equation 3.12, the overall dissipated power can be deduced, suchas Pdiss =Pcd +Pcv +Prd =∇·(κ∇T) · l3−h(T−Ta) · l2−%σ′(T4−T4a) · l2, (3.13) where thethree termsontheright-handsidecorrespondtothepower ofconduction,convectionandradiation, respectively. Averyeffective scheme to approximate this heat loss part is to ignore the rate of con- duction and only keep the rate of convection and radiation, because the effect of conduction is quantitatively much smaller than the effect of theconvectionor theradiation. 45