Page - 43 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling indicates the heat is flowing from the load to the surrounding air. According to equation 3.3, the dissipated power due to the heat convectionPcv canbecalculatedas Pcv=pcv · l2= −h(T−Ta) · l2, (3.4) where l2denotestheareaofthecontactingsurfacebetweenthecell andthesurroundingair. • Dissipatedpowerduetoheat radiation The maximum radiation flux density prd from a black surface is definedbytheStefan-BoltzmannLaw[LNS04],which is prd= −σ′T4, (3.5) whereσ′= 5.669×10−8W/(m2 ·K4) is theStefan-Boltzmanncon- stant. But in practice for a setup as in figure 3.1 with a non-black surface,wheretheloadiscompletelysurroundedbyamuchlarger volume of air, the flux density prd can be also calculated by the equation[LNS04] prd= −%σ′(T4−T4a), (3.6) where% is the emissivity of the surface (0≤%≤ 1, 1 for black sur- face), indicating the radiation ability of the material [Sie01]. The dissipatedpowerduetotheheatradiationPrdcanbesimilarlyrep- resentedas Prd=prd · l2= −%σ′(T4−T4a) · l2. (3.7) • Dissipated/heatingpowerduetoheatconduction According to aforementioned assumptions 1 and 2, the provided or dissipated power by the heat conduction can be decomposed into thecomponents in thex- andy-directions [LNS04], suchas Pcd= ∂ ∂x ( κxl 2∂T ∂x ) · l+ ∂ ∂y ( κyl 2∂T ∂y ) · l, (3.8) where κx and κy indicate thermal conductivities in the x- or y- direction,respectively. Iftheheatedloadhasthesamethermalcon- ductivity in different directions likeκx = κy = κ, then the above 43