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Show 5 The system of N integral equations (7) relates the spectral black body emitted fluxes E . (T) to the spectral heat fluxes q Integration of the spectral fluxes over the entire spectrum (from 0 to 1, K <») gives the total radiation flux, which together with the convective flux [see equation (1)] is supplied to a given surface i. The integration over the spectrum is facilitated by dividing the entire wavelength region into a finite number of finite-spectral width bands n such that AX, Defining the mean emissivity for band k, e. (8) i.AA. ' K •X, +AX, _ * » SAi^** 'i.AX, X, +AX r EbX(T.)dA (9) the net radiative flux for band k given by equation (7) can be expressed as N I j = l 5.. u e.J. AX, (T:) 1 L ej.AXt<Tj> "I Hqj,AXk I^-M^^^-^^JK' ij = 1,2... N (10) j=i where the fractional black body function f ( L T.) is defined as f(Xk^ = JokEbx(T^/rEbxmdX (11) In writing equation (10), the differential configuration factor dFdA _dA (r.,r.) has been replaced by a configuration factor F|_j because the areas of the two zones (Aj and Aj) are small size. The system of N equations (10) must be solved for each zone N of the furnace enclosure and every band n to calculate the band fluxes qiJc. Since the temperatures of the surfaces are not known a priori and must be obtained as part of the problem solution, an iterative procedure must be used. This is discussed in a later subsection. Convective Heat Transfer. In the absence of forced circulation of furnace gases in indirectly-fired furnaces, convective heat transfer to the load is negligible compared to the radiative transport. Since the modeling of local gas composition and temperature within the furnace enclosure is a very complex undertaking, the entire enclosure is treated as a single gas zone. The well stirred gas model approach greatly simplifies the gas energy balance and yet provides a reasonable estimate of the net energy transfer in the furnace. An instantaneous overall energy balance on the gas can be expressed as |